The Automotive Chassis: Engineering Principles

The Automotive Chassis: Engineering Principles
The Automotive Chassis:
Engineering Principles
SECOND EDITION
Chassis and vehicle overall
Wheel suspensions and types of drive
Axle kinematics and elastokinematics
Steering - Springing - Tyres
Construction and calculations advice
Prof. Dipl.-Ing. Jornsen Reimpell
Dipl.-Ing. Helmut Stoll
Prof. Dr.-Ing. Jurgen W. Betzler
Translated from the German by AGET Limited
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INTERNATIONAL
Published on behalf:
Society of Automotive Engineers, Inc.
400 Commonwealth Drive
Warrendale, PA 15096-0001
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Butterworth-Heinemann
An imprint of Elsevier Science
Linacre House, Jordan Hill, Oxford OX2 8DP
225 Wildwood Avenue, 'Nobum, MA 01801-2041
Original copyright 1986 Vogel-Buchverlag, Wiirzburg
Fourth German edition published by Vogel-Buchverlag, Wiirzburg 1999
First English edition published by Arnold 1996
Second edition published by Butterworth-Heinemann 2001
Reprinted 2002
Copyright © 2001, Elsevier Science. All rights reserved.
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Library of Congress Cataloguing in Publication Data
A catalogue record for this book is available from the Library of Congress
ISBN 0 7680 06570
Order number R-300
Society of Automotive Engineers, Inc.
400 Commonweatlth Drive
Warrendale, PA 15096-0001 USA
Phone: (724) 776-4841
Fax: (724) 776-5760
E-mail: [email protected]
http://www.sae.org
Composition by Cambrian Typesetters, FrimIey, Surrey
Printed and bound in Great Britain by BiddIes Ltd, Guildford & Kings Lynn
Contents
Preface
1
Tyres of suspension and drive
1.1 General characteristics of wheel suspensions
1.2 Independent wheel suspensions - general
1.2.1 Requirements
1.2.2 Double wishbone suspensions
1.2.3 McPherson struts and strut dampers
1.2.4 Rear axle trailing-arm suspension
1.2.5 Semi-trailing-arm rear axles
1.2.6 Multi-link suspension
1.3 Rigid and semi-rigid crank axles
1.3.1 Rigid axles
1.3.2 Semi rigid crank axles
1.4 Front-mounted engine, rear-mounted drive
1.4.1 Advantages and disadvantages of the front-mounted
engine, rear-mounted drive design
1.4.2 Non-driven front axles
1.4.3 Driven rear axles
1.5 Rear and mid engine drive
1.6 Front-wheel drive
1.6.1 Types of design
1.6.2 Advantages and disadvantages of front-wheel drive
1.6.3 Driven front axles
1.6.4 Non-driven rear axles
1.7 Four-wheel drive
1.7.1 Advantages and disadvantages
1.7.2 Four-wheel drive vehicles with overdrive
1.7.3 Manual selection four-wheel drive on commercial and
all-terrain vehicles
1.7.4 Permanent four-wheel drive; basic passenger car with
front-wheel drive
1.7.5 Permanent four-wheel drive, basic standard design
passenger car
1.7.6 Summary of different kinds of four-wheel drive
--------------------
Xl
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VI
2
Contents
Tyres and wheels
2.1 Tyre requirements
2.1.1 Interchangeability
2.1.2 Passenger car requirements
2.1.3 -Commercial vehicle requirements
2.2 Tyre designs
2.2.1 Diagonal ply tyres
2.2.2 Radial ply tyres
2.2.3 Tubeless or tubed
2.2.4 Height-to-width ratio
2.2.5 Tyre dimensions and markings
2.2.6 Tyre load capacities and inflation pressures
2.2.7 Tyre sidewall markings
2.2.8 Rolling circumference and driving speed
2.2.9 Influence of the tyre on the speedometer
2.3 Wheels
2.3.1 Concepts
2.3.2 Rims for passenger cars, light commercial vehicles
and trailers
Wheels
for passenger cars, light commercial vehicles
2.3.3
and trailers
Wheel
mountings
2.3.4
2.4 Springing behaviour
2.5 Non-uniformity
2.6 Rolling resistance
2.6.1 Rolling resistance in straight-line driving
2.6.2 Rolling resistance during cornering
2.6.3 Other influencing variables
2.7 Rolling force coefficients and sliding friction
2.7.1 Slip
2.7.2 Friction coefficients and factors
2.7.3 Road influences
2.8 Lateral force and friction coefficients
2.8.1 Lateral forces, slip angle and coefficient of friction
2.8.2 Self-steering properties of vehicles
2.8.3 Coefficients of friction and slip
2.8.4 Lateral cornering force properties on dry road
2.8.5 Influencing variables
2.9 Resulting force coefficient
2.10 Tyre self-aligning torque and caster offset
2.10.1 Tyre self-aligning torque in general
2.10.2 Caster offset
2.10.3 Influences on the front wheels
2.11 Tyre overturning moment and displacement of point of
application of force
2.12 Torque steer effects
2.12.1 Torque steer effects as a result of changes in normal
force
-T
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Contents
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2.12.2 Torque steer effects resulting from tyre aligning torque
2.12.3 Effect of kinematics and elastokinematics
146
146
Wheel travel and elastokinematics
3.1 Purpose of the axle settings
3.2 Wheelbase
3.3 Track
3.4 Roll centre and roll axis
3.4.1 Definitions
3.4.2 Body roll axis
3.4.3 Body roll centre on independent wheel suspensions
3.4.4 Body roll centre on twist-beam suspensions
3.4.5 Body roll centre on rigid axles
3.5 Camber
3.5.1 Camber values and data
3.5.2 Kinematic camber alteration
3.5.3 Camber alteration calculation by drawing
3.5.4 Roll camber during cornering
3.5.5 Elasticity camber
3.6 Toe-in and self-steering
3.6.1 Toe-in and crab angle, data and tolerances
3.6.2 Toe-in and steering angle alteration owing to wheel
bump-travel kinematics
3.6.3 Toe-in and steering angle alteration due to roll
3.6.4 Toe-in and steering angle alteration due to lateral forces
3.6.5 Toe-in and steering angle alteration due to
longitudinal forces
3.7 Steer angle and steering ratio
3.7.1 Steerangle
3.7.2 Track and turning circles
3.7.3 Kinematic steering ratio
3.7.4 Dynamic steering ratio
3.8 Steering self-centring - general
3.9 Kingpin inclination and kingpin offset at ground
3.9.1 Relationship between kingpin inclination and
kingpin offset at ground (scrub radius)
3.9.2 Braking moment-arm
3.9.3 Longitudinal force moment-arm
3.9.4 Alteration to the kingpin offset
3.10 Caster
3.10.1 Caster trail and angle
3.10.2 Caster and straight running
3.10.3 Righting moments during cornering
3.10.4 Kingpin inclination, camber and caster alteration
as a consequence of steering
3.10.5 Kinematic caster alteration on front-wheel travel
3.10.6 Wheel travel-dependent rotation of the rear steering
knuckle
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Vll1
4
5
Contents
3.10.7 Resolution of the vertical wheel force on caster
3.10.8 Settings and tolerances
3.11 Anti-dive and anti-squat mechanisms
3.11.1 Concept description
3.11.2 -Vehicle pitch axis front
3.11.3 Pitch axes rear
3.12 Chassis alignment
3.12.1 Devices for measuring and checking chassis
alignment
3.12.2 Measuring the caster, kingpin inclination, camber
and toe-in alteration
251
254
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258
260
Steering
4.1 Steering system
4.1.1 Requirements
4.1.2 Steering system on independent wheel suspensions
4.1.3 Steering system on rigid axles
4.2 Rack and pinion steering
4.2.1 Advantages and disadvantages
4.2.2 Configurations
4.2.3 Steering gear, manual with side tie rod take-off
4.2.4 Steering gear, manual with centre tie rod take-off
4.3 Recirculating ball steering
4.3.1 Advantages and disadvantages
4.3.2 Steering gear
4.4 Power steering systems
4.4.1 Hydraulic power steering systems
4.4.2 Electro-hydraulic power steering systems
4.4.3 Electrical power steering systems
4.5 Steering column
4.6 Steering damper
4.7 Steering kinematics
4.7.1 Influence of type and position of the steering gear
4.7.2 Steering linkage configuration
4.7.3 Tie rod length and position
266
Springing
5.1 Comfort requirements
5.1.1 Springing comfort
5.1.2 Running wheel comfort
5.1.3 Preventing 'front-end shake'
5.2 Masses, vibration and spring rates
5.3 Weights and axle loads
5.3.1 Curb weight and vehicle mass
5.3.2 Permissible gross vehicle weight and mass
5.3.3 Permissible payload
5.3.4 Design weight
j
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3.
Contents
5.3.5 Permissible axle loads
5.3.6 Load distribution according to ISO 2416
Springing curves
5.4.1 Front axle
5.4.2 Rear axle
5.4.3 Springing and cornering behaviour
5.4.4 Diagonal springing
Spring types
5.5.1 Air- and gas-filled spring devices
5.5.2 Steel springs
5.5.3 Stops and supplementary springs
5.5.4 Anti-roll bars
Shock absorbers (suspension dampers)
5.6.1 Types of fitting
5.6.2 Twin-tube shock absorbers, non-pressurized
5.6.3 Twin-tube shock absorbers, pressurized
5.6.4 Monotube dampers, pressurized
5.6.5 Monotube dampers, non-pressurized
5.6.6 Damping diagrams and characteristics
5.6.7 Damper attachments
5.6.8 Stops and supplementary springs
Spring/damper units
McPherson struts and strut dampers
5.8.1 McPherson strut designs
5.8.2 Twin-tube McPherson struts, non-pressurized
5.8.3 Twin-tube McPherson struts, pressurized
5.8.4 Damper struts
Variable damping
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381
Chassis and vehicle overall
6.1 Vehicle and body centre of gravity
6.1.1 Centre of gravity and handling properties
6.1.2 Calculating the vehicle centre of gravity
6.1.3 Axle weights and axle centres of gravity
6.1.4 Body weight and body centre of gravity
6.2 Mass moments of inertia
6.3 Braking behaviour
6.3.1 Braking
6.3.2 Braking stability
6.3.3 Calculating the pitch angle
6.3.4 Influence of radius-arm axes
6.3.5 Anti-dive control and brake reaction support angle
6.4 Traction behaviour
6.4.1 Drive-off from rest
6.4.2 Climbing ability
6.4.3 Skid points
6.5 Platform, unit assembly and common part systems
386
5.4
5.5
5.6
5.7
5.8
5.9
6
IX
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386
387
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392
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397
399
402
407
410
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410
414
416
419
x
Contents
Bibliography
422
Glossary of symbols
424
Index of car manufacturers
433
435
437
Index of car suppliers
Subject index
Preface
This translation of the fourth German edition is published by ButterworthHeinemann as the second English edition of The Automotive Chassis.
We are fortunate to have Prof. Dr.-Ing. Jiirgen W. Betzler as co-author; he has
been an expert in the field of chassis/simulation technology and design studies
at the University of Cologne since 1994. Jointly, we revised The Automotive
Chassis: Engineering Principles to include a large number of technical innovations.
The clear and easy descriptions, many example designs and calculations and
the inclusion of 434 illustrations and tables are easily understood and have, over
the years, proven to be the best way of imparting information.
The authors' many years of experience in chassis engineering support the
practical bias and will help engineers, inspectors, students and technicians in
companies operating in the automotive industry and its suppliers to understand
the context. The comprehensive index of key words and numerous cross-references make this book an invaluable reference work.
We should like to thank Dipl.-Ing. Achim Clasen for collating the test results
in the Automotive Engineering Laboratory at the Technical University in
Cologne and Sabine Jansen M.A. for her hard work in converting the symbols.
Jomsen Reimpell
Helmut Stoll
Jiirgen W. Betzler
Cologne/Rosrath
r
--------------r
1
Types of suspension and
drive
This chapter deals with the principles relating to drives and suspensions.
1. 1
General characteristics of wheel
suspensions
The suspension of modem vehicles need to satisfy a number of requirements
whose aims partly conflict because of different operating conditions
(loaded/unloaded, acceleration/braking, level/uneven road, straight running/
cornering).
The forces and moments that operate in the wheel contact area must be
directed into the body. The kingpin offset and disturbing force lever arm in the
case of the longitudinal forces, the castor offset in the case of the lateral forces,
and the radial load moment arm in the case of the vertical forces are important
elements
.
. whose effects interact as a result of, for example, the angle of the steermg aXIS.
Sufficient vertical spring travel, possibly combined with the horizontal movement of the wheel away from an uneven area of the road (kinematic wheel) is
required for reasons of ride comfort. The recession suspension should also be
compliant for the purpose of reducing the rolling stiffness of the tyres and shortstroke movements in a longitudinal direction resulting from the road surface
(longitudinal compliance, Fig. 1.1), but without affecting the development of
lateral wheel forces and hence steering precision, for which the most rigid wheel
suspension is required. This requirement is undermined as a result of the necessary flexibility that results from disturbing wheel movements generated by
longitudinal forces arising from driving and braking operations.
For the purpose of ensuring the optimum handling characteristics of the vehicle in a steady state as well as a transient state, the wheels must be in a defined
position with respect to the road surface for the purpose of generating the necessary lateral forces. The build-up and size of the lateral wheel forces are determined
:2
The Automotive Chassis
Or
2
1 4
Fig. 1.1
A multi-link rear axle - a type of suspension system which is progressively
replacing the semi-trailing arm axle, and consists of at least one trailing arm on each
side. This arm is guided by two (or even three) transverse control arms (Figs 1.62 and
1.77). The trailing arm simultaneously serves as a wheel hub carrier and (on four-wheel
steering) allows the minor angle movements required to steer the rear wheels. The
main advantages are, however, its good kinematic and elastokinematic characteristics.
BMW calls the design shown in the illustration and fitted in the 3-series(1997) a
'central arm axle'. The trailing arms 1 are made from GGG40 cast iron; they absorb
all longitudinal forces and braking moments as well as transfering them via the points
2 - the centres of which also form the radius arm axes (Figs 3.158 and 3.159) - on
the body. The lateral forces generated at the centre of tyre contact are absorbed at
the subframe 5, which is fastened to the body with four rubber bushes (items 6 and
7) via the transverse control arms 3 and 4. The upper arms 3 carry the minibloc
springs 11 and the joints of the anti-roll bar 8. Consequently, this is the place where
the majority of the vertical forces are transferred between the axle and the body.
The shock absorbers, which carry the additional polyurethane springs 9 at the top
(Fig. 5.50), are fastened in a good position behind the axle centre at the ends of the trailing arms. For reasons of noise, the differential 10 is attached elastically to the subframe
5 at three points (with two rubber bearings at the front and one hydro bear1ng at the
back). When viewed from the top and the back, the transverse control arms are positioned at an angle so that, together with the differing rubber hardness of the bearings at
points 2, they achieve the desired elastokinematic characteristics. These are:
•
•
•
•
toe-in under braking forces (Figs 3.64 and 3.82);
lateral force compliance understeer during cornering (Figs 3.79 and 3.80);
prevention of torque steer effects (see Section 2.10.4);
lane change and straight running stability.
For reasons of space, the front eyes 2 are pressed into parts 1 and bolted to the
attachment bracket. Elongated holes are also provided in this part so toe-in can be
set. In the case of the E46 model series (from 1998 onwards), the upper transverse
arm is made of aluminium for reasons of weight (reduction of unsprung masses).
T'----------------L-
Types of suspension and drive
3
by specific toe-in and camber changes of the wheels depending on the jounce
and movement of the body as a result of the axle kinematics (roll steer) and operative forces (compliance steer). This makes it possible for specific operating
conditions such as load and traction to be taken into consideration. By establishing the relevant geometry and kinematics of the axle, it is also possible to
prevent the undesirable diving or lifting of the body during braking or accelerating and to ensure that the vehicle does not exhibit any tendency to oversteer
and displays predictable transition behaviour for the driver.
Other requirements are:
• independent movement of each of the wheels on an axle (not guaranteed in the
case of rigid axles);
• small, unsprung masses of the suspension in order to keep wheel load fluctuation as low as possible (important for driving safety);
• the introduction of wheel forces into the body in a manner favourable to the
flow of forces;
• the necessary room and expenditure for construction purposes, bearing in
mind the necessary tolerances with regard to geometry and stability;
• ease of use;
• behaviour with regard to the passive safety of passengers and other road users;
• costs.
The requirements with regard to the steerability of an axle and the possible
transmission of driving torque essentially determine the design of the axis.
Vehicle suspensions can be divided into rigid axles (with a rigid connection of
the wheels to an axle), independent wheel suspensions in which the wheels are
suspended independently of each other, and semi-rigid axles, a form of axle that
combines the characteristics of rigid axles and independent wheel suspensions.
On all rigid axles (Fig. 1.23), the axle beam casing also moves over the entire
spring travel. Consequently, the space that has to be provided above this reduces
the boot at the rear and makes it more difficult to house the spare wheel. At the
front, the axle casing would be located under the engine, and to achieve sufficient jounce travel the engine would have to be raised or moved further back. For
this reason, rigid front axles are found only on commercial vehicles and fourwheel drive, general-purpose passenger cars (Figs 1.3 and 1.4).
With regard to independent wheel suspensions, it should be noted that the
design possibilities with regard to the satisfaction of the above requirements
and the need to find a design which is suitable for the load paths, increase with
the number of wheel control elements (links) with a corresponding increase in
their planes of articulation. In particular, independent wheel suspensions
include:
• Longitudinal link and semi-trailing arm axles (Figs 1.13 and 1.15), which
require hardly any overhead room and consequently permit a wide luggage
space with a level floor, but which can have considerable diagonal springing.
• Wheel controlling suspension and shock-absorber struts (Figs 1.8 and 1.57),
which certainly occupy much space in terms of height, but which require little
space at the side and in the middle of the vehicle (can be used for the engine
;I
4
The Automotive Chassis
Fig. 1.2 An extremely compact four-bar twist beam axle by Renault, with two
torsion bar springs both for the left and right axle sides (items 4 and 8). The V-shape
profile of the cross-member 10 has arms of different lengths, is resistant to bending
but less torsionally stiff and absorbs all moments generated by vertical, lateral and
braking forces. It also partially replaces the anti-roll bar.
At 23.4 mm, the rear bars 8 are thicker than the front ones (0 20.8 mm, item 4). On
the outside, part 8 grips into the trailing links 1 with the serrated profile 13 and on the
inside they grip into the connector 12. When the wheels reach full bump, a pure torque
is generated in part 12, which transmits it to the front bars 4, subjecting them to
torsion. On the outside (as shown in Fig. 1.63) the bars with the serrated profile 11 grip
into the mounting brackets 7 to which the rotating trailing links are attached. The pivots
also represent a favourably positioned pitch centre Or (Fig. 3.159). The mounting
brackets (and therefore the whole axle) are fixed to the floor pan with only four screws.
On parallel springing, all four bars work, whereas on reciprocal springing, the
connector 12 remains inactive and only the thick rear bars 8 and the cross-member
10 are subject to torsion.
The layout of the bars means soft body springing and high roll stability can be
achieved, leading to a reduction of the body roll pitch during cornering.
To create a wide boot without side encroachments, the pressurized monotube
shock absorbers 9 are inclined to the front and therefore are able to transmit forces
upwards to the side members of the floor pan.
or axle drive) and determine the steering angle (then also called McPherson
suspension struts).
.
• Double wishbone suspensions (Fig. 1.7).
• Multi-link suspensions (Figs 1.1, 1.18 and 1.19), which can have up to five
guide links per wheel and which offer the greatest design scope with regard to
------------.------.....,------.--------I
...:::...L
Types of suspension and drive
4
5
6
7
1
3
5
3
Fiig. 1.3 Driven, rigid steering axle with dual joint made by the company GKN Birfield AG for four-wheel drive special-purpose vehicles, tractors and construction
machinery.
The dual joint is centred over the bearings 1 and 2 in the region of the fork carriers; these are protected against fouling by the radial sealing rings 3. Bearing 1 serves
as a fixed bearing and bearing 2 as a movable bearing. The drive shaft 4 is also a sun
gear for the planetary gear with the internal-geared wheel 6. Vertical, lateral and
longitudinal forces are transmitted by both tapered-roller bearings 6 and 7. Steering
takes place about the steering axis EG.
the geometric definition of the kingpin offset, pneumatic trail, kinematic
behaviour with regard to toe-in, camber and track changes, braking/starting
torque behaviour and elastokinematic properties.
In the case of twist-beam axles (Figs 1.2, 1.31 and 1.58), both sides of the
wheels are connected by means of a flexurally rigid, but torsionally flexible
beam. On the whole, these axles save a great deal of space and are cheap, but
o1fer limited potential for the achievement of kinematic and elastokinematic
balance because of the functional duality of the function in the components and
require the existence of adequate clearance in the region of the connecting beam.
They are mainly used as a form of rear wheel suspension in front-wheel drive
6
The Automotive Chassis
70
=
=
i
-----r
I
T--'-'
'.
Fig. 1.4 Top view of the dual joint (Fig. 1.3). The wheel end of the axle is turned
about point P in the middle of the steering pivot during steering. The individual joints
are constrained at points A and B so that point A is displaced to position A', P is
displaced to P' and B is displaced along the drive axle by the distance Xto B In order
to assimilate the variable bending angle {3 resulting from the longitudinal displacement
of point B, the mid-point of the joint P is displaced by the distance Y. The adjustment
value Y depends on the distance between the joints and the steering angle at which
constant velocity is to exist. Where large steering angles can be reached (up to 60°),
there should be constant velocity at the maximum steering angle.
The adjustment value Yand the longitudinal displacement X should be taken into
consideration in the design of the axle.
f
•
_.,-,------------------'---
Types of suspension and drive
7
vehicles up to the middle class and, occasionally, the upper middle class, for
example, the Audi A6, and some high-capacity cars.
11.2
1.2.1
Independent wheel suspensions - general
Requirements
The chassis of a passenger car must be able to handle the engihe power installed.
Ever-improving acceleration, higher peak and cornering speeds, and deceleration lead to significantly increased requirements for safer chassis. Independent
wheel suspensions follow this trend. Their main advantages are:
• little space requirement;
• a kinematic and/or elastokinematic toe-in change, tending towards understeering is possible (see Section 3.6);
• easier steerability with existing drive;
• low weight;
• no mutual wheel influence.
The last two characteristics are important for good road-holding, especially on
bends with an uneven road surface.
Transverse arms and trailing arms ensure the desired kinematic behaviour of
the rebounding and jouncing wheels and also transfer the wheel loadings to the
body (Fig. 1.5). Lateral forces also generate a moment which, with
unfavourable link arrangement, has the disadvantage of reinforcing the roll of
the body during cornering. The suspension control arms require bushes that
yield under load and can also influence the springing. This effect is either reinforced by twisting the rubber parts in the bearing elements, or the friction
Fig. 1.5
On front independent wheel suspensions, the lateral cornering force Fy,w,f
causes the reaction forces Fv.E and Fv.G in the links joining the axle with the body.
Moments are generated on both the outside and the inside of the bend and these
adversely affect the roll pitch of the body. The effective distance c between points E
and G on a double wishbone suspension should be as large as possible to achieve
small forces in the body and link bearings and to limit the deformation of the rubber
elements fitted.
8
The Automotive Chassis
I
F..Y,W,f,o
F..Z,W,f,i
F'Z,W,f,o
..
Fig. 1.6 If the body inclines by the angle cp during cornering, the outer independently suspended wheel takes on a positive camber BW,o and the inner wheel takes
on a negative camber BW,i. The ability of the tyres to transfer the lateral forces Fy,w,t,o
or FY,W,t,i decreases causing a greater required slip angle (Fig. 3.53 and Equation 2.16),
mBo,t is the proportion of the weight of the body over the front axle and Fe,Bo,t the
centrifugal force acting at the level of the centre of gravity Bo. One wheel rebounds
and the other bumps, i.e, this vehicle has 'reciprocal springing', that is:
FZ,w,t,o = Fz.w,t + fi.Fz,w,f
Fz.w,f.i = Fz,w,f - fi.Fz,w,t
increases due to the parts rubbing together (Fig. 1.11), and the driving comfort
decreases.
The wheels incline with the body (Fig. 1.6). The wheel on the outside of the
bend, which has to absorb most of the lateral force, goes into a positive camber
and the inner wheel into a negative camber, which reduces the lateral grip of the
tyres. To avoid this, the kinematic change of camber needs to be adjusted to take
account of this behaviour (see Section 3.5.4) and the body roll in the bend should
be kept as small as possible. This can be achieved with harder springs, additional
anti-roll bars or a body roll centre located high up in the vehicle (Sections 3.4.3
and 5.4.3).
1.2.2
Double wishbone suspensions
The last two characteristics above are most easily achieved using a double wishbone suspension (Fig. 1.7). This consists of two transverse links (control arms)
either side of the vehicle, which are mounted to rotate on the frame, suspension
subframe or body and, in the case of the front axle, are connected on the outside
to the steering knuckle or swivel heads via ball joints. The greater the effective
distance c between the transverse links (Fig. 1.5), the smaller the forces in the
suspension control arms and their mountings become, i.e. component deformation is smaller and wheel control more precise.
The main advantages of the double wishbone suspension are its kinematic
Types of suspension and drive
9
Fig. 1.7 Front axle on the VW light commercial vehicle Lt 28 to 35 with an
opposed steering square. A cross-member serves as a subframe and is screwed to
the frame from below. Springs, bump/rebound-travel stops, shock absorbers and
both pairs of control arms are supported at this force centre. Only the anti-roll bar,
steering gear, idler arm and the tie-rods of the lower control arms are fastened to the
longitudinal members of the frame. The rods have longitudinally elastic rubber bushin~Js at the front that absorb the dynamic rolling hardness of the radial tyresand
reduce lift on uneven road surfaces.
possibilities. The positions of the suspension control arms relative to one another
- in other words the size of the angles a and f3 (Fig. 3.24) - can determine both
the height of the body roll centre and the pitch pole (angles ex' and ~', Fig.
3.155). Moreover, the different wishbone lengths can influence the angle movements of the compressing and rebounding wheels, i.e. the change of camber and,
irrespective of this, to a certain extent also the track width change (Figs 3.50 and
3.7). With shorter upper suspension control arms the compressing wheels go into
negative camber and the rebounding wheels into positive. This counteracts the
change of camber caused by the roll pitch of the body (Fig. 1.6). The vehicle
pitch pole 0 indicated in Fig. 6.16 is located behind the wheels on the front axle
10
The Automotive Chassis
and in front of the wheels on the rear axle. If Or can be located over the wheel
centre (Fig. 3.161), it produces not only a better anti-dive mechanism, but also
reduces the squat on the driven rear axles (or lift on the front axles). These are
also the reasons why the double wishbone suspension is used as the rear axle on
more and more passenger cars, irrespective of the type of drive, and why it is
progressively replacing the semi-trailing link axle (Figs 1.1, 1.62 and 1.77).
1.2.3 McPherson struts and strut dampers
The McPherson strut is a further development of double wishbone suspension.
The upper transverse link is replaced by a pivot point on the wheel house panel,
which takes the end of the piston rod and the coil spring. Forces from all directions are concentrated at this point and these cause bending stress in the piston
rod. To avoid detrimental elastic camber and caster changes, the normal rod
diameter of 11 mm (in the shock absorber) must be increased to at least 18 mm.
With a piston diameter of usually 30 mm or 32 mm the damper works on the
twin-tube system and can be non-pressurized or pressurized (see Section 5.8).
The main advantage of the McPherson strut is that all the parts providing the
suspension and wheel control can be combined into one assembly. As can be
seen in Fig. 1.8, this includes:
•
•
•
•
•
the spring seat 3 to take the underside of the coil spring;
the auxiliary spring 11 or a bump stop (see Fig. 5.49);
the rebound-travel stop (Fig. 5.54);
the underslung anti-roll bar (7) via rod 5;
the steering knuckle.
The steering knuckle can be welded, brazed or bolted (Fig. 5.53) firmly to the
outer tube (Fig. 1.56). Further advantages are:
• lower forces in the body-side mounting points E and D due to a large effective
distance c (Fig. 1.5);
• short distance b between points G and N (Fig. 3.30);
• long spring travel;
• three bearing positions no longer needed;
• better design options on the front crumple zone;
• space at the side permitting a wide engine compartment; which
• makes it easy to fit transverse engines (Fig. 1.50).
Nowadays, design measures have ensured that the advantages are not outweighed
by the inevitable disadvantages on the front axle. These disadvantages are:
• Less favourable kinematic characteristics (Sections 3.3 and 3.5.2).
• Introduction of forces and vibrations into the inner wheel house panel and
therefore into a relatively elastic area of the front end of the vehicle.
• It is more difficult to insulate against road noise - an upper strut mount is
necessary (Fig. 1.9), which should be as decoupled as possible (Fig. 1.10, item
lOin Fig. 1.8 and item 6 in Fig. 1.56).
________-----
...
~----.------------:.::..L.
I
Types of suspension and drive
10
/
11
E
9
2
5
13
8
6
G
7
o
w
Fi!e. 1.8
Rear view of the left-hand side of the McPherson front axle on the Opel
Omega (1999) with negative kingpin offset at ground (scrub radius) (IT and pendulumlinked anti-roll bar. The coil spring is offset from the McPherson strut to decrease
friction between piston rod 2 and the rod guide. Part 2 and the upper spring seat 9
are fixed to the inner wheel house panel via the decoupled strut mount 10.
The additional elastomer spring 11 is joined to seat 9 from the inside, and on the
underside it carries the dust boot 12, which contacts the spring seat 3 and protects
thEl chrome-plated piston rod 2. When the wheel bottoms out, the elastomer spring
rests on the cap of the supporting tube 1. Brackets 4 and 13 are welded to part 1,
on which the upper ball joint of the anti-roll bar rod 5 is fastened from inside. Bracket
13 takes the steering knuckle in between the U-shaped side arms.
The upper hole of bracket 13 has been designed as an elongated hole so that the
camber can be set precisely at the factory (see Fig. 3.102). A second-generation
double-row angular (contact) ball bearing (item 14) controls the wheel.
The ball pivot of the guiding joint G is joined to the steering knuckle by means of
clamping forces. The transverse screw 15 grips into a ring groove of the joint bolt
and prevents it from slipping out in the event of the screw loosening.
The subframe 6 is fixed to the body. In addition to the transverse control arms,
details of which are given in Ref. 5, Section 10.4, it also takes the engine mounts 8
and the back of the anti-roll bar 7. The drop centre rim is asymmetrical to allow negative wheel offset (not shown) at ground (scrub radius) (Figs 2.10, 2.11 and 2.23).
Ii
tl
il
---_._---,-----------------------------------"-
i
i
12
The Automotive Chassis
4
F
7t--.,----.---r---,--.-~
kN
6 t---r----+---+---+-_+::_~
st--+--+---+----H~~
3t---r----t----:;~~¥I--__I
2t---r-----:;~-v'_l_--+--_I
~-_r_-+---:-l:---+--~s
4
8
12
16mm20
Fig. 1.9
McPherson strut mount on the VW Golf III with a thrust ball bearing,
which permits the rotary movement of the McPherson strut whereas the rubber
anchorage improves noise insulation. Initially the deflection curve remains linear and
then becomes highly progressive in the main work area, which is between 3 kN and
4 kN. The graph shows the scatter. Springing and damping forces are absorbed
together so the support bearing is not decoupled (as in Fig. 1.10).
In the car final assembly line the complete strut mount is pressed into a conical
sheet metal insert on the wheel house inside panel 1. The rubber layer 2 on the
outside of the bearing ensures a firm seat and the edge 3 gives the necessary hold
in the vertical direction. The rubber ring 5 clamped on plate 4 operates when the
wheel rebounds fully and so provides the necessary security (figure: Lemforder
Fahrwerktechnik AG).
• The friction between piston rod and guide impairs the springing effect; it can
be reduced by shortening distance b (Figs 1.11 and 3.30).
• In the case of high-mounted rack and pinion steering, long tie rods and, consequently, more expensive steering systems are required (Figs 1.57 and 4.1); in
addition, there is the unfavourable introduction of tie-rod forces in the middle
of the shock-absorbing strut (see Section 4.2.4) plus additional steering elasticity.
• Greater sensitivity of the front axle to tyre imbalance and radial runout (see
Section 2.5 and Refs 1 and 4).
• Greater clearance height requirement.
• Sometimes the space between the tyres and the damping element (Fig. 1.41)
is very limited.
This final constraint, however, is only important on front-wheel drive vehicles as
it may cause problems with fitting snow chains. On non-driven wheels, at most
Types of suspension and drive
Fig. 1.10 The dual
path top mount support
of the Ford Focus
(19198) manufactured by
ContiTech Formteile
GmbH. The body spring
and shock-absorber
forces are introduced
into the body along two
paths with variable
rigidity. In this way, it is
possible to design the
shock-absorber bearing
(inner element) in the
region of small amplitudes with little rigidity
and thus achieve good
insulation from vibration and noise as well
as iimprove the roll
behaviour of the body.
With larger forces of
approximately 700 N
anel above, progression
cams, which increase
the rigidity of the bearing,. come into play. A
continuous transition
between the two levels
of rigidity is important
for reasons of comfort.
The bearing must have
a high level of rigidity in
a transverse direction
in order to ensure that
unwanted displacements and hence
changes in wheel position do not occur. The
forces of the body
springs are directed
along the outer path,
which has a considerably higher level of
rigidity.
13
inner path
F (inner path)
2500
J
N
J
1500
J.
1000
b'l
J
500
-~ V'
l."'"7
-2.00
-1.00
~
I
I
/'
-0.00
1.00
3.00 S
mm
500
1000
1500
2000
I
2500
F (outer path)
6000
N
V
4000
3000
2000
1000
a
/
0.00
/
/
/
v
i/
S
0.50
1.00
1.50
2.00
mm
3.00
14
The Automotive Chassis
(J
/
b
F'z,w
Fv.e = Fz,w . ---c+ 0
Fig. 1.11 If lateral force Fz w moves
lever arm b round guiding joint G, the
lateral force Fsp continually acts in the
body-side fixing point E of the
McPherson strut as a result of the force
Fv.E. This generates the reaction forces
Fv.c and Fv.K on the piston rod guide and
piston. This is Fv.c + Fv.E = Fv.K and the
greater this force becomes, the further
the frictional force Gr increases in the
piston rod guide and the greater the
change in vertical force needed for it to
np away.
As the piston has a large diameter
and also slides in shock-absorber fluid,
lateral force Fv.K plays only a subordinate
role (see Fig. 5.54). Fv.K can be reduced
by offsetting the springs at an angle and
shortening the distance b (see Figs 1.56
and 3.30, and Equation 3.4a).
_ _ _ 20
Direction
19
m-------l0
1IWlv.;---20
19
16
17
Fig. 1.12 The McPherson strut rear axle on the Lancia Delta with equal length
transverse links of profiled steel trunnion-mounted close to the centre the crossmembers 7 and 8. As large a distance as possible is needed between points 6 and 14
on the wheel hub carrier to ensure unimpaired straight running. The fixing points 13
of the longitudinal links 16 are behind the wheel centre, exactly like mounting points
17 of the anti-roll bar 18. The back of the anti-roll bar is flexibly joined to the body via
tabs 19. The additional springs 10 attached to the top of the McPherson struts are
covered by the dust tube 20. The cross-member 15 helps to fix the assembly to the
body. An important criterion for dimensioning the control arm 16 is reverse drive
against an obstruction.
Types of suspension and drive
15
the: lack of space prevents wider tyres being fitted. If such tyres are absolutely
necessary, disc-type wheels with a smaller wheel offset e are needed and these
lead to a detrimentally larger positive or smaller negative kingpin offset at
ground r o (Figs 2.8 and 3.102).
McPherson struts have become widely used as front axles, but they are also
fitted as the rear suspension on front-wheel drive vehicles (e.g. Ford Mondeo
sedan). The vehicle tail, which has been raised for aerodynamic reasons, allows
a larger bearing span between the piston rod guide and piston. On the rear axle
(Fig. 1.12):
• The upper strut mount is no longer necessary, as no steering movements
occur.
• Longer cross-members, which reach almost to the vehicle centre, can be used,
producing better camber and track width change (Figs 3.15 and 3.48) and a
body roll centre that sinks less under load (Fig. 3.30).
• The outer points of the braces can be drawn a long way into the wheel to
a.chieve a shorter distance b.
• The boot can be dropped and, in the case of damper struts, also widened.
• However, rubber stiffness and the corresponding distance of the braces on the
hub carriers (points 6 and 14 in Fig. 1.12) are needed to ensure that there is no
unintentional elastic self-steer (Figs 3.79 and 3.80).
1.2.4 Rear axle trailing-arm suspension
This suspension .- also known as a crank axle - consists of a control arm lying
longitudinally in the driving direction and mounted to rotate on a suspension
subframe or on the body on both sides of the vehicle (Figs 1.13 and 1.63). The
control arm has to withstand forces in all directions, and is therefore highly
subdect to bending and torsional stress (Fig. 1.14). Moreover, no camber and toein changes are caused by vertical and lateral forces.
The trailing-arm axle is relatively simple and is popular on front-wheel drive
vehicles. It offers the advantage that the car body floor pan can be flat and the fuel
tank and/or spare wheel can be positioned between the suspension control arms.
If the pivot axes lie parallel to the floor, the bump and rebound-travel wheels
undergo no track width, camber or toe-in change, and the wheel base simply
shortens slightly. If torsion springs are applied, the length of the control arm can
be used to influence the progressivity of the springing to achieve better vibration
behaviour under load. The control arm pivots also provide the radius-arm axis 0;
i.e. during braking the tail end is drawn down at this point (Fig. 3.159).
The tendency to oversteer as a result of the deformation of the link (arm)
when subject to a lateral force, the roll centre at floor level (Fig. 3.33), the
extremely small possibility of a kinematic and elastokinematic effect on the
position of the wheels and the inclination of the wheels during cornering
consistent with the inclination of the body outwards (unwanted positive
camber) are disadvantages.
16
The Automotive Chassis
Fig. 1.13 Trailing-arm rear suspension of the Mercedes-Benz A class (1997). In order
to minimize the amount of room required, the coil spring and monotube gas-pressure
shock absorber are directly supported by the chassis subframe. The connecting tube is
stress optimized oval shaped in order to withstand the high bending moments from
longitudinal and lateral wheel forces which occur in the course of driving. The torsionbar stabilizer proceeds directly from the shock-absorber attachment for reasons of
weight and ease of assembly. When establishing the spring/shock-absorber properties,
the line along which the forces act and which is altered by the lift of the wheel is to be
taken into consideration, as a disadvantageous load-path can occur with jounce. The two
front subframes are hydraulically damped in order to achieve a good level of comfort
(hydromounts). The chassis subframe can make minor elastokinematic control movements. When designing subframe mounts, it is necessary to ensure that they retain
their defined properties with regard to strength and geometry even with unfavourable
conditions of use (e.g. low temperatures) and for a sufficiently long period of time,
because variations in the configuration have a direct effect on vehicle performance. The
longitudinal arms which run on tapered-roller bearings and which are subject to both
flexural as well as torsional stress are designed in the form of a parallelogram linkage.
In this way, the inherent disadvantage of a trailing arm axle - unwanted toe-in as a result
of the deformation of the link when subject to a lateral force - is reduced by 75%,
according to works specifications.
a
I
.~.
I
I
I
I
+~Z77777/
.
Fig. 1.14 On rear axle trailing-link
suspensions, the vertical force Fz.w together
with the lateral forces Fy,w cause bending
and torsional stress, making a corresponding (hollow) profile, e.g. a closed box profile
necessary. A force from inside causes the
largest torsional moment (see Chapter 4 in
Ref. [3]):
T = Fz,w x a + Fy,w x
(dyn
-----------_._------..,.----_._--------------'"I
Types of suspension and drive
1.2.5
17
Semi-trailing-arm rear axles
This is a special type of trailing-arm axle, which is fitted mainly in rear-wheel
and four-wheel drive passenger cars, but which is also found on front-wheel
drive vehicles (Fig., 1.15). Seen from the top (Fig. 1.16), the control arm axis of
rotation EO is diagonally positioned at an angle a = 10° to 25°, and from the rear
an angle f3 : : :; 5° can still be achieved (Fig. 3.36). When the wheels bump and
Fig. 1.15
Tilted-(Multiple) Staft Steering Rear Axle of the Opel Omega (1999), a
further development of the tilted shaft steering axle. The differential casing of the
rear-axle drive is above three elastic bearings, noise-isolated, connected with
subframe (1), and this subframe is again, with four specially developed elastomer
bearings on the installation (pos. 2 to 5). On top of part seated are the bearings (6) for
the back of the stabilizer. Both of the extension arms (8) take up the inner bearings of
the tilted shafts, which carry the barrel-shaped helical springs (9). In order to get a flat
bottom of the luggage trunk, they were transferred to the front of the axle drive
shafts. The transmission isp (wheel to spring, see equation 5.14 and paragraph 5.3.2
in (3)), becomes thereby with 1.5 comparatively large. The shock absorbers (10) are
seated behind the centre of the axle, the transmission is with iD = 0.86 favourable.
The angle of sweep of the tilted shafts amounts to alpha = 10° (Fig.3.35) and the
Dachwinkel, assume roof or top angle beta = 1°35'. Both of these angles change
dynamically under the influence of the additional tilted shaft (11). These support the
sidE~forces, comin~~ from the wheel carriers directly against the subframe (1). They
raise the lateral stability of the vehicle, and provide an absolute neutral elastic steering under side-forces and also, that in driving mode, favourable toe-in alterations
appear during spring deflection, and also under load (Fig. 3.20). The described reaction of load alteration in paragraph 2.12 disappears - in connection with the arrangement and adaptation of bearings 2 to 5 - almost entirely.
----,-------------
18
The Automotive Chassis
Fig. 1.16 Flat, non-driven air-suspended semi-trailing-arm rear axle of the
Mercedes-Benz V class, whose driven front axle with spring-and-shock absorber
strut has conventional coil springs. The air-spring bellows are supplied by an electrically powered compressor. The individual wheel adjustment permits the lowering or
lifting of the vehicle as well as a constant vehicle height, regardless of - even onesided - loading. It is also possible to counteract body tilt during cornering. The damping properties of the shock absorbers are affected by spring bellow pressure
depending on the load. The short rolling lobe air-spring elements make a low load
floor possible; its rolling movement during compression and rebound results in selfcleaning. In the case of semi-trailing arm axles, roll understeer of the rear axle can
be achieved (Fig. 3.73) by means of a negative verticle angle of pivot-axis inclination
(Fig. 3.36); the kinematic toe-in alteration is also reduced (Fig. 3.49).
rebound-travel they cause spatial movement, so the drive shafts need two joints
per side with angular mobility and length compensation (Fig. 1.17). The horizontal and vertical angles determine the roll steer properties.
When the control arm is a certain length, the following kinematic characteristics can be positively affected by angles a and {3 (Fig. 3.20):
•
•
•
•
height of the roll centre;
position of the radius-arm axis;
change of camber;
toe-in change;
Camber and toe-in changes increase the bigger the angles a and
axles have an elastokinematic tendency to oversteering.
f3:
semi-trailing
------------.------r-------.---------'
I
Types of suspension and drive
19
Fig" 1.17 Constant velocity sliding joints by GKN Automotive. In front-drive vehicles, considerable articulation angles of the drive axles occur, sometimes even
during straight running, as a result of the installation situation, short propshafts and
lifting movements of the body due to torque steer effects. These result in force and
moment non-conformities and losses which lead to unwanted vibration. The full-load
sliding ball joint (top, also see Fig. 1.53) permits bending angles of up to 22° and
displacements of up to 45 mm. Forces are transmitted by means of six balls that run
on intersecting tracks. In the rubber-metal tripod sliding joint (bottom), three rollers
on needle bearings run in cylindrically machined tracks. With bending angles of up to
25° and displacements of up to 55 mm, these joints run particularly smoothly and
hence quietly.
1.2.. 6 Multi-link suspension
A form of multi-link suspension was first developed by Mercedes-Benz in 1982
for the 190 series. Driven and non-driven multi-link front and rear suspensions
have since been used (Figs 1.1, 1.18, 1.19 and 1.44).
Up to five links are used to control wheel forces and torque depending on the
geometry, kinematics, elastokinematics and force application of the axle. As the
T
20
The Automotive Chassis
Fig. 1.18 Multi-link suspension of Ford Werke AG. Derived from the Mondeo
Turnier model series, multi-link suspension is used by Ford for the first time in the
Focus models (1998) in the segment of C class vehicles. This is called the 'control
sword axle' after the shape of the longitudinal link. As there are five load paths available here instead of the two that exist in twist-beam axles and trailing arm axles,
there is great potential for improvement with regard to the adjustment of riding
comfort, driving safety and noise and vibration insulation. As a result of a very elastic front arm bush, the high level of longitudinal flexibility necessary for riding comfort
is achieved. At the same time, very rigid and accurate wheel control for increased
driving safety is ensured by the transverse link, even at the stability limit. The longitudinallink is subject to torsional stress during wheel lift and to buckling stress when
reversing. By using moulded parts, it was possible to reduce the unsprung masses
by 3.5 kg per wheel.
arrangement of links is almost a matter of choice depending on the amount of
available space, there is extraordinarily wide scope for design. In addition to the
known benefits of independent wheel suspensions, with the relevant configuration the front and rear systems also offer the following advantages:
• Free and independent establishment of the kingpin offset, disturbing force and
torque developed by the radial load.
• Considerable opportunities for balancing the pitching movements of vehicles
during braking and acceleration (up to more than 100% anti-dive, anti-lift and
anti-squat possible).
• Advantageous wheel control with regard to toe-in, camber and track width
behaviour from the point of view of tyre force build-up, and tyre wear as a
function of jounce with almost free definition of the roll centre and hence a
very good possibility of balancing the self-steering properties.
• Wide scope for design with regard to elastokinematic compensation from the
_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _...".....
I
.
L-..'
Types of suspension and drive
5
8
21
9
Fig. 1.19 Multi-link rear suspension of the BMW 5 series (E39, 1996). For the first
time in large-scale car production, mainly aluminium is used for the suspension
system derived from the geometry of the BMW 7 series.
The subframe (rear-axle support) (1), produced from welded aluminium tubes, is
attached to the bodywork by means of four large rubber mounts (2). These are soft
in a longitudinal direction for the purposes of riding comfort and noise insulation and
rigid in a transverse direction to achieve accurate wheel control. The differential gear
also has compliant mounts (3). The wheel carrier is mounted on a U-shaped arm (5)
at the bottom and on the transverse link (7) and inclined guide link (8) at the top. As
a result of this inclined position, an instantaneous centre is produced between the
transverse link and guide link outside the vehicle which leads to the desired brake
undHrsteer during cornering and the elastokinematic compensation of deformation
of the rubber bearings and components. The driving and braking torque of the wheel
carrier (11) is borne by the 'integral' link (9) on the swinging arm (5), which is subject
to additional torsional stress as a result. This design makes it possible to ensure
longitudinally elastic control of the swinging arm on the guide bearing (10) for
reasons of comfort, without braking or driving torque twisting the guide bearings as
would be the case with torque borne by pairs of longitudinal links. The stabilizer
behind presses on the swinging arm (5) by means of the stabilizer link (6), whereas
the twin-tube gas-pressure shock absorber, whose outer tube is also made of
aluminium, and the suspension springs provide a favourably large spring base
attached directly to the wheel carrier (11). For reasons of weight, the wheel discs are
, also made of aluminium plate. The wheel carrier is made of shell cast aluminium. The
rear axle of the station wagon BMW Tourer is largely similar in design. However, the
shock absorber extends from the U-shaped swinging arm in order to allow for a wide
and low loading area.
I
:!
I
I
II'
:1
:1
I
, 'i
-
II
--'-~-------r
I
I
22
The Automotive Chassis
point of view of (a) specific elastokinematic toe-in changes under lateral and
longitudinal forces and (b) longitudinal elasticity with a view to riding comfort
(high running wheel comfort) with accurate wheel control.
As a result of the more open design, the wheel forces can be optimally
controlled, i.e: without superposition, and introduced into the bodywork in an
advantageous way with wide distances between the supports.
The disadvantages are:
• increased expenditure as a result of the high number of links and bearings;
• higher production and assembly costs;
• the possibility of kinematic overcorrection of the axle resulting in necessary
deformation of the bearings during vertical or longitudinal movements;
• greater sensitivity to wear of the link bearings;
• high requirements with regard to the observation of tolerances relating to
geometry and rigidity.
1.3
1.3.1
Rigid and semi-rigid crank axles
Rigid axles
Rigid axles (Fig. 1.20) can have a whole series of disadvantages that are a
consideration in passenger cars, but which can be accepted in commercial
vehicles:
• Mutual wheel influence (Fig. 1.21).
• The space requirement above the beam corresponding to the spring bump
travel.
• Limited potential for kinematic and elastokinematic fine-tuning.
• Weight - if the differential is located in the axle casing (Fig. 1.20), it produces
a tendency for wheel hop to occur on bumpy roads.
• The wheel load changes during traction (Fig. 1.22) and (particularly on twin
tyres) there isa poor support base bsp for the body, which can only be
improved following costly design work (Fig. 1.42).
The effective distance bsp of the springs is generally less than the tracking width
br , so the projected spring rate C'P is lower (Fig. 1.23). As can be seen in Fig. 1.61,
the springs, and/or suspension dampers, for this reason should be mounted as far
apart as possible (see also Section 5.3 and Chapter 6 in Ref. [3]).
The centrifugal force (Fe,Bo, Fig. 1.6) acting on the body's centre of gravity
during cornering increases the roll pitch where there is a rigid axle (see Section
5.4.3.5).
Thanks to highly developed suspension parts and the appropriate design of
the springing and damping, it has been possible to improve the behaviour of
rigid drive axles. Nevertheless, they are no longer found in standard-design
passenger cars, but only on four-wheel drive and special all-terrain vehicles
(Figs 1.43 and 1.68).
_______..._-----_----_----.--------------L..
i
Types of suspension and drive
23
Fig. 1.20 Rear axle on the VW LT light commercial vehicle. The long, parabolashaped rolled-out, dual leaf springs cushion the frame well and are progressive. The
rubber buffers of the support springs come into play when the vehicle is laden.
Spring travel is limited by the compression stops located over the spring centres,
which are supported on the side-members. The spring leaves are prevented from
shifting against one another by the spring clips located behind them, which open
downwards (see also Fig. 1.68).
The anti-roll bar is fixed outside the axle casing. The benefits of this can be seen
in Fig. 1.23. The shock absorbers, however, are unfortunately located a long way to
the inside and are also angled forwards so that they can be fixed to the frame sidemembers (Fig. 5.23).
Fig. 1.21 Mutual influence of the two wheels of a
rigid axle when travelling
along a road with pot-holes,
shown as 'mutually-opposed
springing'. One wheel
extends along thl3 path $2
and the other compresses
along the path $1.
; -,t."","
}I..,..,.--+51
Because of its weight, the driven rigid axle is outperfonned on uneven roads
(and especially on bends) by independent wheel suspension, although the deficiency in road-holding can be partly overcome with pressurized mono-tube
dampers. These are more expensive, but on the compressive stroke, the valve
characteristic can be set to be harder without a perceptible loss of comfort. With
this, a responsive damping force is already opposing the compressing wheels.
24
The Automotive Chassis
Rear view
/
F:Y,W,r
F:Y,W,r
!:1Fy,,W, r
Fig. 1.22
If the differential is located in the body of the rigid axle, the driving
torque M A coming from the engine is absorbed at the centres of tyre contact, resulting in changes to vertical force ±1i.Fr,w,r.
In the example, M A would place an additional load on the left rear wheel
(Fr,w,r + 1i.Fr,w,r) and reduce the vertical force (Fr,w,r -1i.Fv.w,r) on the right one,
On a right-hand bend the right wheel could spin prematurely, leading to a loss in
lateral force in the entire axle and the car tail suddenly breaking away (Fig. 2,37; see
also Section 6.5 in Ref. [3]).
Sp __
I
bs
w
w
Fig. 1.23 When considering the roll pitch of the body with the rigid axle the
distances bsp (of the springs F) and bs (of the anti-roll bar linkage points) are included
in the calculation of the transfer with mutually opposed springing. i<p is squared to
give the rate c<p:
i<p = brlbsp and c<p = cri~
The greater the ratio, the less the roll reaction applied by the body, i.e. the springs
and anti-roll bar arms should be fixed as far out as possible on the rigid axle casing
(see Section 5.4.3.5 and Equations 5.20 and 5.21).
T
Types of suspension and drive
25
This is the simplest and perhaps also the most economic way of overcoming the
main disadvantage of rigid axles. Section 5.6.4 contains further details.
In contrast to standard-design vehicles, the use of the rigid rear axle in frontwheel drive vehicles has advantages rather than disadvantages (Fig. 1.24). As
Section 6.1.3 explains, the rigid rear axle weighs no more than a comparable
independent wheel suspension and also gives the option of raising the body roll
centre (which is better for this type of drive, see Fig. 3.42). Further advantages,
including those for driven axles, are:
• they are simple and economical to manufacture;
• there are no changes to track width, toe-in and camber on full bump/reboundtravel, thus giving
• low tyre wear and sure-footed road holding;
• there is no change to wheel camber when the body rolls during cornering (Figs
1.6 and 3.54), therefore there is constant lateral force transmission of tyres;
• the absorption of lateral force moment My = FT,x hRo,r by a transverse link,
which can be placed at almost any height (e.g. Panhard rod, Fig. 1.25);
• optimal force transfer due to large spring track width bsp
• the lateral force compliance steering can be tuned towards under- or oversteering (Figs 3.81 and 1.29).
Fi~l. 1.24 The rear axle on a Ford Escort Express de~ivery vehicle. Single leaf
springs carry the axle and support the body well at four pOln~s. The s~ock absorber~
(fitted vertically) are located close to the wheel, made possible by slim wheel-earners/hub units. The additional elastomer springs sit over the axle tube and act on the
side members of the body when at full bump.
26
The Automotive Chassis
Fy, W,r,o
tAF..
/
Z,W,r
Fig. 1.25
On rigid axles the axle body absorbs the bending moments which arise
as a result of lateral forces. Only the force Fr occurs between the suspension and
the body, and its size corresponds to the lateral forces Fv.w,,;o and Fv.W,r,i. On a horizontal Panhard rod, the distance hRo,r is also the height of the body roll centre. The
higher this is above ground, the greater the wheel force change ±L'1Fz,wr.
There are many options for attaching a rigid axle rear suspension beneath the
body or chassis frame. Longitudinal leaf springs are often used as a single suspension control arm, which is both supporting and springing at the same time, as
these can absorb forces in all three directions as well as drive-off and braking
moments (Figs 1.26 and 5.20). This economical type of rear suspension also has
the advantage that the load area on lorries and the body of passenger cars can be
supported in two places at the back: at the level of the rear seat and under the boot
(Fig. 1.27). This reduces the stress on the rear end of the car body when the boot
is heavily laden, and also the stress on the lorry frame under full load (Fig. 1.20).
The longitudinal leaf springs can be fitted inclined, with the advantage that
during cornering the rigid rear axle (viewed from above) is at a small angle to
the vehicle longitudinal axis (Fig. 1.28). To be precise, the side of the wheel base
on the outside of the bend shortens somewhat, while the side on the inside of the
bend lengthens by the same amount. The rear axle steers into the bend and, in
other words, it is forced to self-steer towards 'roll-understeering' (Fig. 1.29).
Fig. 1.26
Longitudinal force""'---'ro.,
Vertical force
Longitudinal leaf springs can
absorb both forces in all directions and
the drive-off, braking and lateral force
moment. (See Section 6.2 in Ref. [3]).
Types of suspension and drive
27
l
Fig. 1.27
Longitudinal rear le'af springs support the body of a car in two places under the back seats and under the boot - with the advantage of reduced bodywork
stress.
This measure can, of course, have an adverse effect when the vehicle is travelling on bad roads, but it does prevent the standard passenger car's tendency to
oversteer when cornering. Even driven rigid axles exhibit - more or less irrespective of the type of suspension - a tendency towards the load alteration
(torque steering) effect, but not to the same extent as semi-trailing link suspensions. Details can be found in Section 2.12.2 and in Ref. [2] and Ref. [9].
On front-wheel drive vehicles, the wheels of the trailing axle can take on a
negative camber. This improves the lateral grip somewhat, but does not promote
perfect tyre wear. This is also possible on the compound crank suspension (a
Fiig. 1.28 Angled longitudinal leaf
springs fixed lower to the body at the front
than at the back cause the rigid rear axle to
self-steer towards understeering (so-called
roll pitch understeering). Where there is
body roll, the wheel on the outside of the
bEmd, which is compressing along the path
S1, is forced to accommodate a shortening
of the wheel base D.-/1, whilst the wheel on
the inside of the bend, which is extending
by 52, is forced to accommodate a lengthening of the wheel base by D.-/2 • The axle is
displaced at the steering angle fir (see also
Fig. 3.75).
~DirectiO:
I
I
'-,
II
-,---
28
The Automotive Chassis
Fig. 1.29
If a rigid rear axle steers
with the angle lir towards understeer,
the tail moves out less in the bend and
the driver has the impression of more
neutral behaviour. Moreover, there is
increased safety when changing lanes
quickly at speed.
The same occurs if the outside
wheel of an independent wheel
suspension goes into toe-in and the
inside wheel goes into toe-out (see
Fig. 3.79).
suspension-type halfway between a rigid axle and independent wheel suspension) which, up to now, has been fitted only on front-wheel drive vehicles.
Details are given in Fig. 1.2 and Section 1.6.4.1.
1.3.2
Semi rigid crank axles
The compound crank suspension could be described as the new rear axle design
of the 1970s (Figs 1.30 and 1.2) and it is still used in today's small and mediumsized front-wheel drive vehicles. It consists of two trailing arms that are welded
to a twistable cross-member and fixed to the body via trailing links. This member
absorbs all vertical and lateral force moments and, because of its offset to the
wheel centre, must be less torsionally stiff and function simultaneously as an antiroll bar. The axle has numerous advantages and is therefore found on a number
of passenger cars which have come onto the market.
From an installation point of view:
•
•
•
•
•
the whole axle is easy to assemble and dismantle;
it needs little space;
a spring damper unit or the shock absorber and springs are easy to fit;
no need for any control arms and rods; and thus
only few components to handle.
From a suspension point of view:
• there is a favourable wheel to spring damper ratio (See Section 5.3.5 in Ref. [3]);
• there are only two bearing points 0\ and Ors, which hardly affect the springing
(Fig. 1.31);
• low weight of the unsprung masses (see Section 6.1.3); and
• the cross-member can also function as an anti-roll bar.
T
Types of suspension and drive
29
Fig. 1.30 Twist-beam suspension of the VW Golf IV (1997), VW Bora (1999) and
Audi A3 (1996). The rubber-metal bearings of the axle body are set at 25° to the
transverse suspension of the vehicle in order to improve the self-steering properties
of the suspension together with the rigidity of the bearings which varies in three
directions in space. Compared with the previous model, it was possible to reduce
unwanted lateral-force toe-out steer resulting from link deformation by 30% to
approximately 1 mm per 500 N of lateral force. Figure 1.72 shows the four-wheel
drive version of the VW Golf IV.
/Direction
b
r
F'v,w,r,o
r
F'v,w,r,l
Fig. 1.31 The lateral forces Fy,w,r,o and Fy,W,r,i occurring at the centres of tyre
contact during cornering are absorbed at the bearing points OJ and Ors. This results
in a moment My = (Fy,w,r,o + FY,W,r,i) Xr = Fx,o be which (depending on the elasticity
of the rubber bearing) can cause 'lateral force oversteering'. The longer the control
arms (distance r) and the closer the points 0 1 and Ors (distance be), the greater the
longitudinal forces ±FxQ.
=
=
30
The Automotive Chassis
From a kinematic point of view:
• the~e i.s negligible toe-in and track width change on reciprocal and parallel
spnngmg;
• there ~s a loW change of camber under lateral forces (Figs 3.54 and 3.57);
• there IS low load-dependent body roll understeering of the whole axle (Fig.
3.38 and 3.78); and
• good radius-arm axis locations 0 1 and Ors (Fig. 1.31), which reduce tail-lift
during braking.
The disadvantages are:
• a tendency to lateral force oversteer due to control arm deformation (Fig.
3.72);
• torsion and shear stress in the cross-member;
• high stress in the weld seams; which means
• the permissible rear axle load is limited in terms of strength;
• the limited kinematic and elastokinematic opportunities for determining the
wheel position;
• the establishment of the position of the instantaneous centre by means of the
axle kinematics and rigidity of the twist-beam axle;
• the mutual effect on the wheel;
• the difficult decoupling of the vibration and noise caused by the road surface;
and
• the considerable need for stability of the bodywork in the region of those
points on the front bearings at which complex, superposed forces have to be
transmitted.
1.4
Front-mounted engine, rear-mounted
drive
In passenger cars and estate cars, the engine is approximately in the centre of
the front axle and the rear wheels are driven (Fig. 1.32). To put more weight on
the rear axle and obtain a more balanced weight distribution, Alfa Romeo,
Porsche (928, 968 models) and Volvo integrated the manual transmission with
the differential. This is also the case with the Chevrolet Corvette sports car
(1998; Figs 1.33 and 1.34). With the exception of light commercial vehicles, all
lorries have the engine at the front or centrally between the front and rear axles·
together with rear-wheel drive vehicles. The long load area gives hardly any
other option. Articulated lorries, where a major part of the trailer weight - the
trailer hitch load - is carried over the rear wheels, have the same configuration.
On buses, however, the passengers are spread evenly throughout the whole interior of the vehicle, which is why there are models with front, central and rear
engines.
~
Fig. 1.32 Front-mounted engine, rear-mounted drive (BMW 3 series E46, 1998). The manual transmission is flange-mounted on
the engine, which is longitudinally positioned over the front axle. The rear-axle differential is driven by means of a propshaft. The fuel
tank is situated in tront of the rear axle for safety in case of an accident. The battery was placed in the boot in order to achieve a
balanced 50:50 axle-load distribution. Figure 1.1 shows the rear axle in detail.
· 32
The Automotive Chassis
Fig. 1.33 Chevrolet Corvette (1998). In order to achieve balanced axle-load distribution, a more rigid overall system (necessary on account of the greater flexibility of
the plastic bodywork) and more leg room, the gearbox is integrated with the rear-axle
differential. Compared with standard drives, the cardan shaft turns higher (with
engine speed) but is subject to correspondingly less torque. The front and rear axles
have plastic (fibreglass) transverse leaf springs.
Compared with the previous model, unwanted vibration, particularly on an uneven
road surface, is reduced as a result of the shorter length of the wheel spindles of 63
mm and the small steering-axle angle of 8.8 degrees. Owing to the combination of
a castor angle of 6.5 degrees with a castor trail of 36 mm (previous model: 5.9
degrees, 45 mm), a good compromise is achieved between high lateral rigidity of the
axle and good feedback properties.
1.4.1
Advantages and disadvantages of the front-mounted
engine, rear-mounted drive design
The standard design has a series of advantages on passenger cars and estate cars:
• There is hardly any restriction on engine length, making it particularly suitable
for more powerful vehicles (in other words for engines with 8-12 cylinders).
• There is low load on the engine mounting, as only the maximum engine torque
times the conversion of the lowest gear without differential transmission has
to be absorbed.
• Insulation of engine noise is relatively easy.
• Under full load most of the vehicle mass is on the driven rear axle (important
for estate cars and trailers (Figs 1.36 and 6.22)).
• A long exhaust system with good silencing and catalytic converter configuration.
• Good front crumple zone, together with the 'submarining' power plant unit,
i.e. one that goes underneath the floor panel during frontal collision.
Types of suspension and drive
33
1
2
Fig. 1.34
Rear axle (left side of wheel) of the Chevrolet Corvette (1998). Links 1,
2 and wheel carrier 3 of the multi-link suspension are made from aluminium in order
to reduce the unsprung masses. The plastic leaf spring 4 is mounted at two places
on the right and left sides of the body (5) so that it also helps to make the body more
resistant to roll. Roll spring stiffness is further increased by stabilizer 6. This is
attached to subframe 7, which is also made of aluminium. The design of the wheel
carrier 3 on the front and rear axles is the same, but not the wheel links 1 and 2. The
toe-in control of the rear axle is exercised very stiffly and precisely, via tie rod 8.
•
•
•
•
Simple and varied front axle designs are possible irrespective of drive forces.
More even tyre wear thanks to function distribution of steering/drive.
Uncomplicated gear shift mechanism.
Optimum gearbox efficiency in direct gear because no force-trans~itting
bevel gear is in action (Fig. 6.19).
• Sufficient space for housing the steering system in the case of a recirculating
ball steering gear.
• Good cooling because the engine and radiator are at the front; a power-saving
fan can be fitted.
• Effective heating due to short hot-air and water paths.
The following disadvantages mean that, in recent years, only a few saloon
cars under 2 I engine displacement have been launched internationally
using this design, and performance cars also featured the front-mounted
design:
34
rt
The Automotive Chassis
D'Irec Ion
• Fx,w,a
Fig. 1.35 On a front-wheel drive
(left) the vehicle is pulled. The result
is a more stable relationship
between the driving forces Fx.w.a and
the inertia force Fc.v Conversely, in
the case of driven rear wheels an
unstable condition is theoretically
evident; front axle settings ensure
the necessary stabilization.
U
Ur
rf-
!
Direction
I
• I~
Fx,w.a
I
~,v
~,v
n-
I.
I
I
-..,...
I
4
Fx,w.a
Fx,w,a
• Unstable straight-running ability (Fig. 1.35), which can be fully corrected by
special front suspension geometry settings, appropriate rear axle design and
suitable tyres.
• The driven rear axle is slightly loaded when there are only two persons in the
vehicle, leading to poor traction behaviour in wet and wintry road conditions linked to the risk of the rear wheels spinning, particularly when tight bends are
being negotiated at speed. This can be improved by setting the unladen axle
load distribution at 50%/50% which, however, is not always possible (Figs 1.36
and 6.22). It can be prevented by means of drive-slip control (see Ref. [7]).
• A tendency towards the torque steer effect (Fig. 2.53) and, therefore,
• complex rear independent wheel suspension with chassis subframe, differen.
tial gear case and axle drive causing
• restrictions in boot size
• The need for a propshaft between the manual gearbox and differential (Fig.
1.32) and, therefore,
• a tunnel in the floor pan is inevitable, plus an unfavourable interior to vehicle
-length ratio.
Fig. 1.36
Average proportional axle load distribution based on drive type and loading condition. With the standard design saloon, when the vehicle is fully laden, the
driven rear wheels have to carry the largest load. With the front-wheel drive,
however, with only two persons in the vehicle, the front wheels bear the greater load.
Empty
2 passengers at the front
4 passengers
5 passengers and luggage
Front-wheel drive
Rear wheel drive
Rear engine
front
rear
front
rear
front
rear
61
60
55
49
39
40
45
51
50
50
47
44
50
50
53
56
40
42
40
41
60
58
60
59
¥
A
Types of suspension and drive
35
1.4.2 Non-driven front axles
The standard design for passenger cars that have come onto the market in
recent years have McPherson struts on the front axle, as well as double wishbone or multi-link suspensions. The latter type of suspension is becoming
more and more popular because of its low friction levels and kinematic
advantages. Even some light commercial vehicles have McPherson struts or
double wishbone axles (Fig. 1.7). However, like almost all medium-sized and
heavy commercial vehicles, most have rigid front axles. In order to be able to
slltuate the engine lower, the axle subframe has to be offset downwards (Fig.
1.37).
The front wheels are steerable; to control the steering knuckle 5 (Fig. 1.38)
on double wishbone suspensions, there are two ball joints that allow mobility
in all directions, defined by full bump/rebound-travel of the wheels and the
steering angle. The wishbone, which accepts the spring, must be carried on a
supporting joint (item 7) in order to be able to transmit the vertical forces. A
regular ball joint transferring longitudinal and lateral forces (item 8) is generaJlly sufficient for the second suspension control arm. The greater the distance
between the two joint points, the lower the forces in the components. Figure
1.39 shows a front axle with ball joints a long way apart.
Fig. 1.37 The front rigid
a)(le on the Mercedes-Benz
li~lht commercial vehicle of
the 207 D/308 series with
recirculating ball steering gear
and steering rod 1 parallel to
the two-layer parabolic spring.
This rod has to be slightly
shorter than the front side of
the spring, so that both parts
take on the same motion
curve when the axle bottoms
out (see also Fig. 4.6). The
brace 3, running from the
steering column jacket 2 to
the body, bends on impact.
The T-shaped axle casing 4,
which is cranked downwards
and to which the springs are
fastened, can be seen in the
section. The elastomer spring
5 sits on the longitudinal
member of the frame and the
two front wheels are joined
by the tie rod 6. The safety
steering wheel has additional
pcldding.
36
The Automotive Chassis
Fig. 1.38
c
Front hub carrier
(steering knuckle) on the
Mercedes-Benz S class
(W40, 1997) with a large
effective distance c (see also
Fig. 1.4). The upper transverse control arm 6 forms
the casing for the ball pivot of
the guiding joint, whereas
the lower supporting joint 7
is pressed into the hub
carrier 5. The ventilated brake
disc 34 (dished inwards), the
wheel hub 9, the doublehump rim 43 with asymmetrical drop centre and the space
for the brake caliper (not
included in the picture) are
clearly shown.
The base on McPherson struts is better because it is even longer. Figure 1.40
shows a standard design and Fig. 1.8 the details.
The coil spring is offset at an angle to reduce the friction between piston rod
2 and the rod guide. The lower guiding joint (point G) performs the same function as on double wishbones, whereas point E is fixed in the shock tower, which
is welded to the wheel house panel. As the wheels reach full bump, piston rod 2
moves in the cylinder tube (which sits in the carrier or outer tube, see Fig. 5.53)
and when there is a steering angle the rod and spring tum in an upper strut
mount, which insulates noise and is located at point E (Fig. 1.9).
Wheel controlling damper struts do not require such a complex mount. The
piston rod turns easily in the damping cylinder (Fig. 1.41). Only the rod needs
noise insulation. The coil spring sits separately on the lower control atm, which
must be joined to the steering knuckle via a supporting joint. The damper is
1,1
Types of suspension and drive
37
Fi!~. 1.39
Multi-link front suspension of the Mercedes-Benz model W220 ($
class, 1998). Based on a double wishbone axle, two individual links (tension strut
and spring link) are used instead of the lower transverse link in order to control the
steering axle nearer to the middle of the wheel. As a result, the kingpin offset and
disturbing force lever arm are reduced and vibrations are caused by tyre imbalances and brake-force fluctuations is consequently minimized. Crash performance
is also improved by the more open design. The air-spring struts with integrated
shock absorber (see Fig. 5.19) proceed directly from the spring link. The laterally
rigid rack and pinion steering in front of the middle of the wheel leads to the
desired elastokinematic understeer effect during cornering owing to the laterally
elastic spring link bearings. The manufacturing tolerances are kept so small by
means of punched holes that the adjustment of camber and camber angles in
production is not necessary.
lighter than a shock-absorbing strut and allows a greater bearing span across the
damping cylinder, permits a wider, flatter engine compartment (which is more
streamlined) and is easier to repair. However, it is likely to be more costly and
offsetting the spring from the damper (Figs 1.8 and 1.11) may cause slip-stick
problems with a loss of ride comfort.
In the case of front-wheel drive vehicles, there may be a problem in the lack
of space between the spring and the drive axle.
38
The Automotive Chassis
Fig. 1.40
Spring strut front axle of the BMW Roadster Z3, which Lemforder
Fahrwerktechnik produce in the USA and supply directly to the assembly line there.
The additional springs 2 are positioned in the coil springs (Fig. 1.11) which are offset
at an angle in order to reduce friction. The stabilizer 6 is connected to the lower links
by the struts 3.
The cross-member 7 which serves as the subframe takes the hydraulically
supported rack and pinion steering 1 at the front and the transverse link 4 on its
lower side. The L-shape of the transverse link makes good decoupling of the lateral
rigidity and longitudinal elasticity possible: lateral forces are introduced directly into
the rigid front bearing, while longitudinal forces produce a rotational movement
about the front bearing as a result of the laterally elastic rear bearing 5. As shown in
Fig. 3.84, these rubber elements ensure a defined lateral springing. The large-diameter internally ventilated brake discs (15" rim) and the third-generation, two-row
angular ball bearings, whose outside ring also acts as a wheel hub, are clearly shown.
The kingpin offset at ground (scrub radius) depends on the tyre width and thus the
wheel offset (Fig. 3.1 au; it is (a = + 10 mm on 185/65 R 15 tyres and (2 = +5 mm on
205/60 R 15 tyres.
Types of suspension and drive
39
IFig. 1.41 Front axle of the Mercedes-Benz Sprinter series (1995). The wheelcontrolling strut is screwed on to the wheel carrier, which is, in turn, connected to
the lower cross-member by means of a ball joint. Both the vehicle suspension and
roll stabilization are ensured by means of a transverse plastic leaf springmounted on
rubber elements. Large rubber buffers with progressive rigidity act as additional
springs and bump stops.
1.4.3
Driven rear axles
Because of their cost advantages, robustness and ease of repair rigid axles are
fitted in practically all commercial and off-road vehicles (Fig. 1.43) in combination with leaf springs, coil springs or air springing (Figs 1.20 and 1.42).
They are no longer found in saloons and coupes. In spite of the advantages
described in Section 1.3, the weight of the axle is noticeable on this type of
vehicle.
For independent suspension, the semi-trailing arm axle, shown in Figs 1.15
and 1.45, is used as independent wheel. suspension in passenger and light
commercial vehicles. This suspension has a chassis subframe to which the
differential is either fixed or, to a limited degree, elastically joined to give additional noise and vibration insulation. The springs sit on the suspension control
arms. This gives a flat, more spacious boot, but with the disadvantage that the
forces in all components become higher.
40
The Automotive Chassis
2
5
5
..
Direction
2 ----t--t-+-~~...,,;
Fig. 1.42
Driven rear axle with air springs of the Mercedes-Benz lorry 1017 L to
2219 L 6 x 2. The axle is carried in the longitudinal and lateral directions by the two
struts 1 and the upper wishbone type control arm 2. The four spring bellows sit
under the longitudinal frame members and, because of the twin tyres, they have a
relatively low effective bs p • The tracking width br divided by bs p yields approximately
the ratio i~ = 2.2. As shown in Equation 5.19, with reciprocal springing the rate is C~,r
which amounts to only 21 % of the rate Cr with parallel springing.
To reduce body roll pitch the anti-roll bar 3 was placed behind the axle and is
supported on the frame via the rop 4. The four shock absorbers 5 are almost vertical
and are positioned close to the wheels to enable roll movements of the body to fade
more quickly.
Iii
Types of suspension and drive
41
Fi!l. 1.43
The rear axle on the all-terrain, general-purpose passenger car,
Mitsubishi Pajero. The rigid axle casing 1 is taken through the longitudinal control
arms 2. These absorb the drive-off and braking forces (and the moments which arise)
and transmit them to the frame. The rubber mountings 3 in the front fixing points 1,
which also represent the vehicle pitch pole Or (Fig. 3.160), are designed to be longitudinally elastic to keep the road harshness due to the dynamic rolling hardness of
the radial tyre away from the body. The Panhard rod 4 absorbs lateral forces. The
anti-roll bar 5 is (advantageously) fastened a long way out on the frame (Fig. 1.23).
The disc brakes, coil springs and almost vertical shock absorbers can be clearly seen.
Further details are contained in Section 3.5 in Ref. [2].
Because of its ride and handling advantages, more and more passenger cars
have double wishbone suspension rear axles or so-called multi-link axles (Figs
1.1, 1.19, 1.34 and 1.72).
Most independent wheel suspensions have an easy-to-assemble chassis
subframe for better wheel control and noise insulation. However, all configurations (regardless of the design) require drive shafts with length compensation.
This is carried out by the sliding CV (constant velocity) joints fitted both at the
wheel and the differential. Figure 1.17 shows a section through a joint of this
type, and Fig. 1.44 shows a typical modern bearing of a driven rear wheel.
1.,5
Rear and mid engine drive
The rear-mounted power plant consists of the engine and the differential and
manual gearbox in one assembly unit, and it drives the rear wheels. The power
plant can sit behind the axle (Fig. 1.45, rear-mounted engine) or in front of it
(Fig. 1.46, central engine). This configuration makes it impossible to have a rear
seat as the engine occupies this space. The resulting two-seater is only suitable
as a sports or rally car.
42
The Automotive Chassis
11 4
8
14----+.
24---;
5 9
7
6
1-r-----2
1r--'''"'r---1
3---
Fig. 1.44 Rear axle wheel hub carrier with wheel and brake. The drive shaft 7 is
butt-welded to the CV slip joint 6. The drive shaft transmits the driving torque to the
wheel hub 15 via a serrated profile. Part 15 is carried by the maintenance··free, tworow angular (contact) ball bearing 5. The one-part outer ring is held in the hub carrier
4 by the snap ring 16.
The seal rings on both sides sit in the permanently lubricated bearing unit. The
covering panel 11 (that surrounds the brake disc 12) acts as additional dirt-protection
outside, as does collar 9 of the CV joint on the inside. This grips into a cut-out in the
wheel hub carrier 4 and creates a cavity. The centrifugal effect of the bell-shaped
joint housing prevents ingress of dirt and water. The brake disc 12 is pulled from
outside against the flange 15 and fixed by dowel 14 until the wheel is mounted. The
jaws 20 of the drum brake acting as a handbrake act on the inside of part 12. At the
lower end, the illustration shows the fixed calliper 1 of the disc brake. Two hexagonal bolts (item 2) fix it to the wheel hub carrier 4. Piston 3 and the outer brake pad
are shown cut away (illustration: Mercedes-Benz).
The disadvantages of rear and central engine drive on passenger cars are:
• moderate straight running abilities (caster offset at ground angles of up to T =
8° are factory set);
• sensitivity to side winds;
• indifferent cornering behaviour at the stability limit (central engine);
• oversteering behaviour on bends (rear-mounted engine, see Fig. 2.42);
• difficult to steer on ice because of low weight on the front wheels;
• uneven tyre wear front to rear (high rear axle load, see Fig. 1.36);
• the engine mounting must absorb the engine moment times the total gear ratio;
=1
I
Fig. 1.45 VW Transporter, a light truck which could be used either as an eight-seater bus or for transporting goods, and which has
the optimal axle load distribution of 50%/50% in almost all loading conditions. The double wishbone suspension at the front, the
semi-trailing link rear axle and the rack and pinion steering, which is operated via an additional gear set in front, can be seen clearly.
To achieve a flat load floor throughout, VW changed the Transporter to front-wheel drive in 1990.
44
The Automotive Chassis
Fig. 1.46 The Porsche Boxster (1996) has a water-cooled engine which is longitudinally installed in front of the rear axle. The front axle is designed as a spring strut-type
axle. The transverse link is arranged almost in extension of the wheel axle; it is
connected to the longitudinal link by a strut bush which is soft for reasons of comfort.
This open design and link geometry make it possible to combine a high level of driving
precision, a result of rigid wheel control, with riding comfort, owing to the longitudinal
elasticity of the axle. At a camber angle of 8 degrees, good straight running results
from the large castor displacement of 41 mm. The kingpin offset is -7 mm and the
disturbing force lever arm is 83 mm. The pitch centre of the front axle was located near
to the road to achieve kinematic wheel recession of the axle, which is important for
riding comfort, with the result that braking-torque compensation is only 10%.
The rear axle is also a spring strut-type axle in an open link design; the wheel
carrier, hub and bushes as well as the transverse link are the same as those found
on the front axle. The open design makes it possible to have an inwardly inclined
elastokinematic axis of rotation, so that a stabilizing toe-in position of the rear wheels
is produced during braking. The axle can also be designed to understeer when
subject to lateral forces.
The main disadvantages of the mid-engine design are apparent from the boot
space: only 130 I are available at both the front and back.
•
•
•
•
•
the exhaust system is difficult to design because of short paths;
the engine noise suppression is problematic;
complex gear shift mechanism;
long water paths with front radiators (Fig. 1.46);
high radiator performance requirement because of forced air cooling, the electric fan can only be used on the front radiator;
Types of suspension and drive
45
• the heating system has long paths for hot water or warm air;
• the fuel tank is difficult to house in safe zones;
• the boot size is very limited.
In the case of vehicles with a short wheel base and high centre of gravity with
the engine on or behind the rear axle, there is a danger that the vehicle will overturn if it is rolling backwards down a steep slope and the parking brake, which
acts upon the rear axle, is suddenly applied.
As a result of the logical further development of the kinematics and elastokinematics of the axles, Porsche have succeeded in improving straight running
as well as cornering in the steady state (vehicles now understeer slightly up to
high lateral accelerations) and transient state as well as when subject to torque
steer effects. Even in the case of the Boxsters (with mid-engine, see Fig. 1.46,
since 1996) and 911 (water-cooled since 1997), Porsche are adhering to rearwheel drive (whereas the VW Transporter, Fig. 1.45, has not been built since
1991) and, in so doing, obtain the following benefits:
• very agile handling properties as a result of the small yawing moment;
• very good drive-off and climbing capacity, almost irrespective of load (Fig.
6.22);
• a short power flow because the engine, gearbox and differential form one
compact unit;,
• light steering due to low front axle load;
• good braking force distribution;
• simple front axle design;
• easy engine d.ismantling (only on rear engine);
• no tunnel or only a small tunnel in the floor pan;
• a small overhang to the front is possible.
1.6
Front-wheeldrive
The engine, differential and gearbox form one unit, which can sit in front of,
over, or behind the front axle. The design is very compact and, unlike the standard design, means that the vehicle can either be around 10G-300 mm shorter,
or the space for passengers and luggage can be larger. These are probably the
main reasons why, worldwide, more and more car manufacturers have gone over
to this design. In recent years only a few saloons of up to 2 I capacity without
front-wheel drive have come onto the market. Nowadays, front-wheel drive vehicles are manufactured with V6 and V8 engines and performances in excess of
150 kW.
However, this type of drive is not suitable for commercial vehicles as the rear
wheels are highly loaded and the front wheels only slightly. Nevertheless, some
light commercial vehicle manufacturers accept this disadvantage so they can
lower the load area and offer more space or better loading conditions (Fig.
1..47). The propshafts necessary on standard passenger cars would not allow
this.
46
The Automotive Chassis
Fig. 1.47
The low cargo
area on the Peugeot light
commercial vehicle J 5/J 7
is achieved due to frontwheel drive and a semitrailing link axle to the rear
(similar to the one in Fig.
1.63).
1.6.1
Types of design
1.6.1.1
Engine mounted longitudinally 'North-South' in front of the axle
In-line or V engines mounted in front of the axle - regardless of the wheelbase
- give a high front axle load, whereby the vehicle centre of gravity is pushed a
long way forwards (Fig. 1.48). Good handling in side winds and good traction,
especially in the winter, confirm the merits of a high front axle load, whereas the
heavy steering from standing (which can be rectified by power-assisted steering), distinct understeering during cornering and poor braking force distribution
would be evidence against it.
This type of design, as opposed to transverse mounting, is preferred in the
larger saloons as it allows for relatively large in-line engines. The first vehicles
of this type were the Audi 80 and 100. Inclining the in-line engine and placing
the radiator beside it means the front overhang length can be reduced. Automatic
gearboxes need more space because of the torque converter. This space is readily available with a longitudinally mounted engine.
A disadvantage of longitudinal engines is the unfavourable position of the
steering gear: this should be situated over the gearbox. Depending on the axle
design, this results in long tie rods with spring strut (McPherson) front axles
(Fig. 1.57).
1.6.1.2 Transverse engine mounted in front of the axle
In spite of the advantage of the short front overhang, only limited space is available between the front wheel housings (Figs 1.49 and 1.50). This restriction
means that engines larger than an in-line four cylinder or V6 cannot be fitted in
a medium-sized passenger car. Transverse, asymmetric mounting of the engine
and gearbox may also cause some performance problems. The unequal length of
the drive shafts affects the steering. During acceleration the vehicle rises and the
drive shafts take on different angular positions, causing uneven moments around
the steering axes. The difference between these moments to the left and to the
right causes unintentional steering movements resulting in a noticeable pull to
one side (Fig. 3.88); drive shafts of equal length are therefore desirable. This also
prevents different drilling angles in the drive shaft causing timing differences in
drive torque build-up.
The large articulation angle of the short axle shaft can also limit the spring
Types of suspension and drive
47
Fig. 1.48 In front-wheel drive vehicles the engine can be mounted longitudinally·
in front of the front axle with the manual gearbox behind. The shaft goes over the
transverse differential (illustration: Renault).
Fig. 1.49 Compact power train unit on the Vauxhall Corsa (1997). The engine is
transverse mounted with the gearbox on the left. The McPherson front axle and
safety steering column can be seen dearly.
48
The Automotive Chassis
Direction
Fig. 1.50
Layout of transverse engine, manual gearbox and differential on the VW
Polo. Because the arrangement is offset, the axle shaft leading to the left front wheel
is shorter than that leading to the right one. The shifter shaft between the two can
be seen clearly. The total mechanical efficiency should be around n:::: 0.9.
travel of the wheel. To eliminate the adverse effect of unequal length shafts,
passenger cars with more powerful engines have an additional bearing next to
the engine and an intermediate shaft, the ends of which take one of the two sliding CV joints with angular mobility (Figs 1.51 and 1.17). Moreover, 'flexing
vibration' of the long drive shaft can occur in the main driving range. Its natural
frequency can be shifted by clamping on a suppression weight (Fig. 4.1).
1.6.2
Advantages and disadvantages of front-wheel drive
Regardless of the engine position (see Fig. 1.52), front-wheel drive has numerous advantages:
• there is load on the steered and driven wheels;
• good road-holding, especially on wet roads and in wintry conditions - the car
is pulled and not pushed (Fig. 1.35);
Types of suspension and drive
49
II
Fig., 1.51 Gearbox unit on the Lancia Thema, located beside the transverse
engine and betwel3n the front axle McPherson struts. Owing to the high engine
performance, the design features two equal-length axle shafts joined by an intermediate shaft. There are also internally ventilated disc brakes.
• good drive-off and sufficient climbing capacity with only few people in the
vehicle (Fig. 6.22);
• tendency to understeer in cornering;
• insensitive to side wind (Fig. 3.125);
• although the front axle is loaded due to the weight of the drive unit, the steering
is not necessarily heavier (in comparison with standard cars) during driving;
• axle adjustment values are required only to a limited degree for steering alignment (see Section 3.8);
• simple rear axle design - e.g. compound crank or rigid axles - possible;
• long wheelbase making high ride comfort possible;
• short power flow because the engine, gearbox and differential form a compact
unit;
• good engine cooling (radiator in front), and an electric fan can be fitted;
• effective heating due to short paths;
• smooth car floor pan;
• exhaust system with long path (important on cars with catalytic converters);
• a large boot with a favourable crumple zone for rear end crash.
The disadvantages are:
• under full load, poorer drive-off capacity on wet and icy roads and on inclines
(Figs 1.36 and 6.22);
i
i
50
The Automotive Chassis
Fig. 1.52
Arrangement of the gearbox beneath the motor, which is inclined
towards the rear, and the differential gear placed behind it. A single oil-economy
undertakes the supply, in this case, of the driving unit, narrow in its design.
(Works Illustr. Fa Peugeot)
•
•
•
•
•
•
•
•
with powerful engines, increasing influence on steering;
engine length limited by available space;
with high front axle load, high steering ratio or power steering is necessary;
with high located, dash-panel mounted rack and pinion steering, centre takeoff tie rods become necessary (Figs 1.57 and 4.39) or significant kinematic
toe-in change practically inevitable (Fig. 3.67)
geometridll difficult project definition of a favourable interference force lever
arm and a favourable steering roll radius (scrub radius);
engine gearbox unit renders more ditlicult the arrangement of the steering
b
,
Packaae·
the power plant mounting has to absorb the engine moment times the total gear
ratio (Figs 3.110, 6.20 and Equation 6.36)
it is difficult to design the power plant mounting (see Ref. [5]) - booming
Types of suspension and drive
51
Fi!J. 1.53 Front~wheel output shaft of GKN Automotive. A constant-velocity sliding joint is used on the gearbox side and a constant-velocity fixed joint is used on the
wheel side (Fig. 1.17). The maximum bending angles are 22° for the sliding joint and
47" for the fixed joint. For reasons of weight, the sliding joint is placed directly into
thE! differential and fixed axially by a circlip. A central nut secures attachment on the
wheel side. The intermediate shaft is designed as a carburized, shaped hollow shaft.
•
•
•
•
•
•
•
•
noises, resonant frequencies in conjunction with the suspension, tip in and let
off torque effects etc., need to be suppressed;
with soft mountings, wavy road surfaces excite the power plant to natural
frequency oscillation (so-called 'front end shake', see Section 5.1.3);
there is bending stress on the exhaust system from the power plant movements
dluring drive-off and braking (with the engine);
there is a complex front axle, so inner drive shafts need a sliding CV joint (Fig.
1.53);
the turning and track circle is restricted due to the limited bending angle (up
to 50°) of the drive joints (see Section 3.7.2);
high sensitivity in the case of tyre imbalance and non-uniformity on the front
wheels;
higher tyre wear in front, because the highly loaded front wheels are both
steered and driven;
poor braking force distribution (about 75% to the front and 25% to the rear);
complex gear shift mechanism which can also be influenced by power plant
movements.
The disadvantage of the decreased climbing performance on wet roads and
those with packed snow can be compensated with a drive slip control (ASR, see
Chapter 6 in Ref. 7) or by shifting the weight to the front axle. On the XM models,
Citroen moved the rear axle a long way to the rear resulting in an axle load distribution of about 65% to the front and 35% to the back. The greater the load on the
front wheels, the more the car tends to understeer, causing adverse steering angles
and heavy steering, which makes power steering mandatory (see Section 4.2.5).
1.6.3
Driven front axles
The: following are fitted as front axles on passenger cars, estate cars and light
commercial vehicles:
•
•
•
•
double wishbone suspensions;
multi-link axles;
NlcPherson struts, and (only in very few cases);
damper struts.
On double wishbone suspensions the drive shafts require free passage in those
places where the coil springs are normally located on the lower suspension
52
The Automotive Chassis
c?ntrol arms. This means that the springs must be placed higher up with the
dIsadvantage that (as on McPherson struts) vertical forces are introduced a long
way up on the wheel house panel. It is better to leave the springs on the lower
suspension control arms and to attach these to the stiffer body area where the
upper control arms are fixed. Shock absorbers and springs can be positioned
behind the drive shafts (see Fig. 1.54) or sit on split braces, which grip round
the shafts and are jointed to the lower suspension control arms (Fig. 1.55). The
axle is flatter and the front end (bonnet contour) can be positioned further
down. The upper suspension control arms are relatively short and have mountings that are wide apart. This increases the width of the engine compartment
and the spring shock absorber unit can also be taken through the suspension
control arms; however, sufficient clearance to the axle shaft is a prerequisite.
Fig. 1.54 Double wishbone front axle assembly of the Audi A4. The Audi A6 of 1997,
the Audi A8 (1996) and the VW Passat of 1996 are similar. Four individual transverse
'arms' on each side form what is effectively a double wishbone arrangement which
provides lateral and longitudinal wheel location. The two upper members (1 and 2) are
attached to the spheroidal graphite iron hollow-section stub-axle post (18) by low-friction
ball and socket joints. The track rod (3) provides the steering input through a horizontal
extension of this stub-axle post which forms a steering arm. The two lower suspension
members consist of the radius arm (4) and the transverse arm (5). This latter must be
capable of reacting high loads from the anti-roll bar (6) and spring/damper (7) attachment
points. The co-axial spring/damper assembly incorporates a polyurethane rubber bumpstop, as well as the hydro-mechanical tension rod stop (Fig. 5.51). The spring/damper unit
(7) and the inner bearings of the upper members (1) and (2) are mounted on the upper
suspension bracket.
The inner ends of suspension members (4 and 5) are located by substantial rubber
mountings on the inside of the sub-frame (10). The rear mounting (11) is hydraulically
damped to absorb any harshness associated with radial tyres. The vehicle body is
mounted on four rubber mountings (12 to 15) of specified elasticity to ensure a high standard of ride comfort.
The inner drive shafts are located to the rear of the spring dampers (7) and are
connected to the drive-line by 'tripot' flexible couplings (16). The outer ends of the drive
shafts transmit the drive to the wheels through double-row angular contact bearings. The
inner races of these bearings are integral with the wheel hubs.
The hydraulically assisted steering rack is mounted ~n the ve.hicle's scuttle, with ~he
steering damper (17) located on one side of the steering hOUSing, and the other Side
attached to the steering rack.
The high location of the wheel-joint facilitates space saving and a consequent reduction of the lever-arm forces, and allows the inner valences of the mudguard to be located
further outboard.
The advantages of this type of four-link suspension .include the lo.cation of t~e points
E and G of the paired arms 1 and 2, likewise 4 and 5 (Fig. 3. ~ 45), which ~re subjected to
outward thrusts resulting from steering input to the steering-arm, which are thereby
compressed through r = 10 mm (see para 3.9.3). More~ver the high loc~tion of the point
E (Fig. 1.5) - together with the negative steering roll radius: = -7 mm (Fig. 3.106) - helps
to reduce the loads in all components of the front suspension system.
Other design parameters of the suspension arrangement are:
King pin inclination
Caster angle
Camber angle
Caster linear trail
e = 30'
r = +3°50'
a = +3°45'
1] = +5.5mm
I
{
t
11;1
,..
C\I
L
,..o
54
The Automotive Chassis
Fig. 1.55 Double wishbone front suspension on the Honda models Prelude and
Accord with short upper wishbones with widely spaced bearings, lower transverse
control arms and longitudinal rods whose front mounts absorb the dynamic rolling stiffness of the radial tyres. The spring shock absorbers are supported via fork-shaped struts
on the transverse control arms and are fixed within the upper link mounts. This point is
a good force input node. Despite the fact that the upper wheel carrier joint is located
high, which gives favourable wheel kinematics, the suspension is compact and the
bonnet can be low to give aerodynamic advantages. The large effective distance c
between the upper and lower wheel hub carrier joints seen in Fig. 1.5 results in low
forces in all mounts and therefore less elastic deflection and better wheel control.
Fig. 1.56
Lancia front axle. The McPherson strut consists of the wheel hub carrier 1
and the damping part 2; the two are connected by three screws. The lower spring seat
3 sits firmly on the outer tube and also acts as a buffer for the supplementary spring 4.
This surrounds the outer tube 2 giving a longer bearing span (path 1-0, Fig. 1.11). The
supporting bearing 5 is arranged diagonally and thus matches the position of the coil
spring which is offset to reduce damping friction. The rubber bearing 6 absorbs the
spring forces, and the rubber bearing 7 absorbs the forces generated by the damping.
Disc 8 acts as a compression buffer and plate 9 acts as a rebound buffer for this elastic
bearing. Both parts come into play if the damping forces exceed certain values.
The centre of the CV joint 10 lies in the steering axis and the wheel hub 11 fits
onto a two-row angular (contact) ball bearing. Guiding joint 12 sits in a cone of the
wheel hub carrier 1 and is bolted to the lower transverse control arm 13. Inelastic ball
joints provide the connection to the anti-roll bar 14. The steering axis inclination (J'
between the centre point of the upper strut mount and guiding joint 12 and the (here
slightly positive) kingpin offset at ground (scrub radius) r~ are included.
r-~fI___=;::==_---_. 5
8
-\+-------3
--1--------2
1__
10
.
I
13
12
14
56
The Automotive Chassis
Due to the slight track width change, the change of camber becomes favourable.
Furthermore, the inclination of the control arms provides an advantageous
radius arm axis position and anti-dive when braking (see also Fig. 1.75).
Most front-wheel drives coming on to the market. today have McPherson
struts. It was a long time after their use in standard design cars that McPherson
struts were used at the front axle on front-wheel drive vehicles. The drive shaft
requires passage under the damping part (Fig. 1.56). This can lead to a shortening of the effective distance 1-0, which is important for the axle (Fig. 1.11), with
the result that larger transverse forces Fy,c and FY,K occur on the piston and rod
guide and therefore increase friction.
On front-wheel drive vehicles there is little space available to fit rack and
pinion steering. If the vehicle has spring dampers or damper struts, and if the
steering gear is housed with short outer take-off tie rods, a toe-in change is
almost inevitable (Figs 4.4 and 3.67). A high steering system can readily be
attached to the dash panel (Fig. 1.57), but a centre take-off is then necessary and
the steering system becomes more expensive (Figs 4.1, 4.11 and 4.39).
Moreover, the steering force applied to the strut is approximately halfway
between mountings E and G (Fig. 1.11). The inevitable, greater yield in the
transverse direction increases the steering loss angle and makes the steering less
responsive and imprecise.
1.6.4 Non-driven rear axles
If rear axles are not driven, use can frequently be made of more simple designs
of suspension such as twist-beam or rigid.
1.6.4.1 Twist-beam suspension
There are only two load paths available on each side of the wheel in the case of
twist-beam axles. As a result of their design (superposed forces in the links, only
two load paths), they suffer as a result of the conflicting aims of longitudinal
springing - which is necessary for reasons of comfort - and high axle rigidity which is required for reasons of driving precision and stability. This is particularly noticeable with the loss of comfort resulting from bumpy road surfaces. If
the guide bearings of the axle are pivoted, the superposition of longitudinal and
lateral forces should particularly be taken into consideration. As a result of the
design, twist-beam suspensions exhibit unwanted oversteer when subject to
lateral forces as a result of deformation of the swinging arms. In order to reduce
the tendency to oversteer, large guide bearings which, as 'toe-in correcting' bearings, permit lateral movements of the whole axle body towards understeer when
subject to a lateral force are provided. As the introduction of longitudinal and
lateral forces into the body solely occurs by means of the guide bearings, it must
be ensured that the structure of the bodywork is very rigid in these places (see
Figs. 1.30 and 1.58, also Sections 3.6.3 and 3.6.4).
1.6.4.2 Rigid axle
Non-driven rigid axles can be lighter than comparable independent wheel
suspensions. Their advantages outweigh the disadvantages because of the almost
Types of suspension and drive
57
Fi!~.
1.57 Driven McPherson front axle on the Audi 6 (Audi 100, 1991). The
dynamic rolling hardness of the radial tyre is absorbed by the rubber bearings shown
in Fig. 3.85, which sits in the lower transverse control arms. The inner sleeves of
these bearings take the arms of the anti-roll bar, which also act as a trailing link (classic McPherson construction). To avoid greater toe-in changes when the wheels are
at 'full bump/rebound-travel, centre take-off tie rods are used on the rack and pinion
steering higher up and in the centre (Fig. 4.39). Together with these rods, the steering damper located on the right is fastened to the end of the steering rack. The
en!~ine is mounted longitudinally, which means the drive shafts are of equal length
(see Section 3.6.5.3).
The development of the axle since 1997 is shown in Fig. 1.54.
non-variable track and camber values during drive. Figure 1.24 illustrates an
inexpensive yet effective design:
• axle casing in steel tubing;
• suspension on single leaf springs.
The lateral and longitudinal wheel control characteristics are sufficient for
passenger cars in the medium to small vehicle range and delivery vehicles. The
---------,
\
Types of suspension and drive
59
Fiig. 1.58 Twist-beam suspension of the Audi A6 (1997). An advantageously large
support width of the guide on the links - important for force application - was chosen
bHcause of the overhung arrangement. The flexurally resistant, but torsionally soft V
sE~ction profile of the axle is in an upright position in order to ensure that the suspension has roll understeer properties through the high position of the centre of thrust
of the profile. The instantaneous centre height is 3.7 mm and the toe-in alteration is
0.21 min/mm. Braking-torque compensation of 73% is reached. The stabilizer situated in front of the axis of rotation increases the lateral rigidity of the axle design,
bE~cause it accepts tension forces upon the occurrence of lateral wheel forces. The
linear coil springs mounted on noise-insulating moulded rubber elements on both
sides are separated from the shock absorbers to allow the maximum loading width
of the boot as a result of their location under the side rails. The gas-pressure shock
absorbers support additional springs made of cellular polyurethane which act softly,
through specific rigidity balancing, to avoid uncomfortable changes in stiffness when
reaching the limits of spring travel. Owing to the rigid attachment of the shock
absorbers to the bodywork, these also work at low amplitudes; so-called 'parasitic'
springing resulting from the unwanted flexibility of wheel suspension or bodywork
components is thereby reduced.
Fig. 1.59 Rear wheel bearing on the Fiat Panda
wilth a third-generation, two-row angular (contact)
ball bearing. The wheel hub and inner ball bearinQ
ring are made of one part, and the square outer ring
is fixed to the rigid axle casing with four bolts
(picture: SKF).
------,----------------'
60
The Automotive Chassis
resultant hard springing is acceptable and may even be necessary because of the
load to be moved. The wheel bearing can be simple on such axles (Fig. 1.59).
Faster, f?-ore comfortable vehicles, on the other hand, require coil springs and,
for preCIse axle control, trailing links and a good central guide (Fig. 1.60) or
Panhard rod. This is generally positioned behind the axle (Fig. 1.61).
1~6.4.3
Independent wheel suspension
An independent wheel suspension is not necessarily better than a rigid axle in
terms of handling properties. The wheels may incline with the body and the
lateral grip characteristics of the tyres decrease (Figs 3.53 and 3.54), and there
are hardly any advantages in terms of weight (see Section 6.1.3). This suspension usually needs just as much space as a compound crank axle.
Among the various types, McPherson struts (Fig. 1.12), semi-trailing or trailing
Fig. 1.60
'Omega' rear wheel suspension on the Lancia Y 10 and Fiat Panda, a
trailing axle with a U-shaped tube, drum brakes, inclined shock absorbers and additional elastomer springs seated inside the low positioned coil springs. The rubber
element in the shaft axle bearing point, shown separately, has cut-outs to achieve
the longitudinal elasticity necessary for comfort reasons; the same is true for the
front bearings of the two longitudinal trailing links. The middle bearing point is also
the body roll axis.
The body roll centre is located in the centre of the axle but is determin~d by the
level of the three mounting points on the body. The lateral forces are absorbed here.
The angled position of the longitudinal trailing links is chosen to reduce the lateral
force oversteering that would otherwise occur (shown in Figs 1.31 and 3.79). The
coil springs are located in front of the axle centre and so have to be harder, with the
advantage that the body is better supported on bends (for details, see Section 3.4 in
Ref. [2]).
Types of suspension and drive
61
Fi!g. 1.61 Torsion crank axle on the Audi A6 (Audi 100, 1991) with spring
dampers fixed a long way out at points 6 and which largely suppress body roll vibrations.
The longitudinal control arms therefore had to be welded further in to the U-profile
acting as a cross--member and reinforced by shoe 5. The U-profile is also raised at
the side to achieve higher torsional resistance. The anti-roll bar is located inside the
U-profile.
Brace 2 distributes the lateral forces coming from the Panhard rod 1 to the two
body-side fixing points 3 and 4. Bar 1 is located behind the axle, and the lateral force
understeering thus caused and shown in Fig. 3.81 could be largely suppressed by
the length of the longitudinal control arms. Furthermore, it was possible to increase
the comfort and to house an 80 I fuel tank as well as the main muffler in front of the
axle.
The only disadvantage is that the link fixing points, and therefore the body roll axis
Or, moves further forward and this reduces the 'anti-dive' described in Fig. 3.159,
and that the suspension requires much space when assembled.
link axles (Figs 1.2, 1.13 and 1.63) and - having grown in popularity for some
years now - double wishbone suspensions, mostly as so-called multi-link axles
(Figs 1.1, 1.8 and 1.62; Section 5.3 in Ref. [2]) are all used. The latter are
currently the best solution, due to:
•
•
•
•
•
kinematic characteristics;
elastokinematic behaviour;
space requirements;
axle weight;
the possibility of being able to retrofit the differential on four-wheel drive
(Figs 1.77 and 1.1).
(See also Section 1.4.3.).
62
The Automotive Chassis
_-12
11
10
---~-2
6
Fig. 1.62
Top view of the double wishbone rear axle on the Honda Civic. The trailing arm 2, which is stiff under flexure and torsion, and the wheel hub carrier 1 form
a unit and, along with the two widely spaced lower transverse control arms 7 and 11,
ensure precise wheel control and prevent unintentional toe-in changes. The rubber
bearing in point 3, which represents the so-called 'vehicle roll axis' Or, provides the
real longitudinal wheel control of the axle (Fig. 3.159). The lateral control of wheel
carrier 1 is performed by the short upper transverse control arm 6 and the longer
lower one 7, which accepts the spring shock absorber 8 in point 9. The length difference in the control arms gives favourable camber and track width change (Figs 3.19
and 3.49).
During braking, bearing 3 yields in the longitudinal direction and, due to the angled
position of the links 11 when viewed from the top, the front point 4 moves inwards
(Fig. 3.82) and the wheel goes into toe-in. Behaviour during cornering is similar: the
axle understeers due to lateral force and body roll (see Sections 3.6.3 and 3.6.4 and
Figs 1.1 and 1.77). The wheel is carried by 'third-generation' angular (contact) ball
bearings on which the outside ring is also designed as a wheel hub. In models with
smaller engines, brake drums (item 10) are used, which are fixed to the wheel hub.
Types of suspension and drive
63
6
3
Fig. 1.63 Compact trailing arm rear axle, fitted by Renault to less powerful
medium-sized vehicles. The short torsion bar springs grip into the guide tubes 2 and
3 in the centre of the vehicle. Parts 2, 3 and 4 are jointly subjected to torsional
stresses and so the torsional stiffness of the. transverse tubes contributes to the
spring rate. On the outside, the cast trailing arms 1 are welded to the transverse
tubes, which (pushed into each other) support each other on the torsionally elastic
bE3arings 5 and 6. This creates a sufficiently long bearing basis, which largely
prevents camber and toe-in changes when forces are generated.
The entire assembly is fixed by the brackets 7 which permits better force transfer
on the body side sill. Guide tubes 2 and 3 are mounted in the brackets and can rotate,
as well as the outer sides of the two torsion bars 4. The two arms thus transfer all
vertical forces plus the entire springing moment to the body. The anti-roll bar 8 is
connected to the two trailing arms via two U-shaped tabs. The two rubber bearings
5 and 6 located between the tubes 2 and 3 also contribute to the stabilizing effect.
The bump and rebound travel stops are fitted into the shock absorber 9 (see
Sl3ction 5.6.8). As shown in Fig. 1.2, on the newer models the dampers would be
inclined so that they can be fixed to the side members of the floor pan which also
leads to more space between the wheel housings.
- - - ' - - , - - - - - - - - - - - - - - - - - -....- - - - '
64
The Automotive Chassis
7.2
6.4
5.6
/
4.8
r
z
4.0
,:,(.
(J,)
()
l-
.E
3.2
()
co
~
il
I
I >.::: -- --.....
I, ~
If'
V/ ---
Front-wheel d rive with winter tyres
....
c
0
.;::
.J ~:::rive with winter:;:""
2.4
,~
/'
--- ---
Fig. 1.64 With a loaded
Vauxhall Cavalier on
compacted snow (/-lX,W = 0.2)
driving forces are measured
on the flat as a function of
the slip (Fig. 2.33). The illustration shows the advantage
of four-wheel drive, and the
necessity, even with this
type of drive, of fitting
correct tyres. Regardless of
the type of drive, winter tyres
also give shorter braking
(stopping) distances on these
road surface conditions .
/
4-wheel dnve with summer
tyres
1.6
____
-\I-_-I---I--~
0.8
Front-wheel drive wit 1 summer tyres
,0
10
20
30
40
Slippage %
1.7
50
60
70
80
..
Four-wheel drive
In four-wheel drives, either all the wheels of a passenger car or commercial vehicle are continuously - in other words permanently - driven, or one of the two
axles is always linked to the engine and the other can be selected manually or
automatically. This is made possible by what is known as the 'centre differential
lock' . If a middle differential is used to distribute the driving torque between the
front and rear axles, the torque distribution can be established on the basis of the
axle-load ratios, the design philosophy of the vehicle and the desired handling
characteristics. That is why Audi choose a 50%:50% distribution for the V8
Quattro and Mercedes-Benz choose a 50%:50% distribution for M class off-road
vehicles, whereas Mercedes-Benz transmits only 35% of the torque to the front
axle and as much as 65% to the rear axle in vehicles belonging to the E class.
This section deals with the most current four-wheel drive designs. In spite of
the advantages of four-wheel drive, suitable tyres - as shown in Fig. 1.64 should be fitted in winter.
1.7.1
Advantages and disadvantages
In summary, the advantages of passenger cars with permanent four-wheel drive
over those with only one driven axle are:
Types of suspension and drive
65
• better traction on surfaces in all road conditions, especially in wet and wintry
weather (Figs 1.64, 1.65 and 1.66);
• an increase in the drive-off and climbing capacity regardless of load;
• better acceleration in low gear, especially with high engine performance;
• reduced sensitivity to side wind;
• stability reserves when driving on slush and compacted snow tracks;
• better aquaplaning behaviour;
• particularly suitable for towing trailers;
• balanced axle load distribution;
• reduced torque steer effect;
• even tyre wear.
40 r----.-,------,----r-----.r----:;.,.----,-------r------,~--"?I
%
Fully locked
4-wheel drive
4-wheel drive
50%:50%
30 I-----l-----+------.hol---+----+-~~,_L_f~--+-I - - - - ;
Front-wheel drive
t...
c:
Q)
-g....
201-----+----i-J1----1----F-f----t---.-t---t------j
<:l
10 ~--~-~~---+---+--______1r---+---+_--_t
0.1
0.2
0.3
0.4
Friction
fJ.X,W
0.5
0.6
0.7
0.8
..
Fig. 1.65 Hill-climbing capacity on a homogeneous surface with front, rear-wheel
and four-wh'eeI drive, and with locked centre differential and a driving force distribution of 50%/50% on four-wheel drive. Of the cars studied, the front/rear axle load
distribution was (Fig. 1.36):
front-wheel drive 57%/43%
rear-wheel drive 51 %/49%
four-wheel drive 52%/48%
(see also Fig. 6.22),
---r--
66
The Automotive Chassis
Four-wheel drive (4WD) fully locked
100,-------------------------.
%
t
751-----------------------------------1
Fig. 1.66 Influence of the type of drive and differential lock on the propulsion
force with 'IL split', in other words a slippery road surface with ILx,w = 0.1/0.8 on one
side only. 100% locking of the rear axle differential gives most benefits.
Some car manufacturers offer this option as ASR (or EDS) or using a hydraulic
manual selection clutch (see details in Chapter 6 of Ref. [7]). However, only 25% to
40% locking is provided on the multi-disc limited-slip differentials that have usually
been fitted on vehicles to date (see Section 5.3.2 in Ref. 9 and Section 6.4 in Ref. 8).
According to ED Directive 701156IEWG, a 'towed trailer load' of 1.5 times
the permissible total weight has been possible for multi-purpose passenger vehicles (four-wheel passenger vehicles) since 1994.
However, the system-dependent, obvious disadvantages given below should
not be ignored:
•
•
•
•
•
acquisition costs;
around 6% to 10% higher kerb weight of the vehicle;
generally somewhat lower maximum speed;
5% to 10% increased fuel consumption;
in some systems, limited or no opportunity for using controlled brake gearing,
for instance for anti-locking or ESP systems;
• not always clear cornering behaviour;
• smaller boot compared with front-wheel drive vehicles.
Predictability of self-steering properties even in variable driving situations, traction, toe-in stability and deceleration behaviour when braking, manoeuvrability,
behaviour when reversing and interaction with wheel control systems are the
principal characteristics of the vehicle movement dynamics which are taken into
consideration for an assessment of four-wheel drive systems.
To transmit the available engine torque to all four wheels, interaxle differentials (such as cone, planet or Torsen differentials), which are manually or automatically lockable, or clutches (such as sprag, multi-disc or visco clutches) must
be installed on the propshaft between the front and rear axles. Differentials must
be present on both drive axles. However, on roads with different coefficients of
friction on the left and right wheels, known as 'p,-split', and with traditional
____________- - - - -
------------,---,---..1.--'
Types of suspension and drive
67
differentials, each driven axle can, at most, transmit double the propulsion force
of the wheels running on the side with the lower coefficient of friction (JL-Iow).
Higher driving forces can be achieved with an 'axle differential lock' or
controlled wheel brake gearing which creates the need for 'artificial' torque on
the spinning wheel. Differential-locking can only be 100% effective on the rear
axle as, at the front, there would no longer be problem-free steering control. The
lock partially or completely stops equalization of the number of revolutions
between the left and right wheel of the respective axle and prevents wheelspin
on the JL-Iow-side.
In passenger cars, automatic locking differentials are used between front and
rear axles. These can operate mechanically (multi-disc limited slip differentials
(see Section 5.3.2 in Ref. [9], Torsen differential, Fig. 1.71) or based on fluid
friction (visco lock, Fig. 1.74) and produce a locking degree of usually 25% to
40%. Higher values severely impede cornering due to the tensions in the power
train (Fig. 1.69) Nevertheless, up to 80% locking action can be found in motor
sJP0rt.
The locking action of uncontrolled or slip-dependent differential locks necessitates increased expenditure with the use of brake-power control systems (ABS,
ESP). Thus in the case of the visco lock, a free-wheel clutch is required that is
engaged during reversing. Here the advantage of controlled differentials (Haldex
clutch, automatically controlled locking differential, see Sections 1.7.4 and
1.7.5) becomes apparent: They can be used to maximum effect in any operating
conditions and with any brake-power control system, because the locking action
is: produced by an electronically controlled, hydraulically activated multi-disc
clutch (Fig. 1.67).
Traditional clifferentiallocks are increasingly being replaced because of the
Fig. 1.67
The Mercedes G all-terrain vehicle, according to DIN 70, a so-called 'allpurpose passenger car', has high ground clearance and short overhangs both front
and rear. This, together with the large ramp angles (af, ar) and the overhang angle f3,
makes it particularly suitable for off-road driving.
-------------------
68
The Automotive Chassis
use of wheel control systems in both front- and rear-wheel drive as well as fourwheel drive vehicles. In these systems, the wheel speed is measured, usually
with the use of ABS sensors. If the speed of a wheel is established, this wheel is
retarded by means of the wheel braking device. In the case of the differential,
this corresponds to the build-up of torque on the side of the spinning wheel and
it can now transmit torque at the higher coefficient of friction up to the adhesion
limit of the wheel. Volkswagen AG calls this system electronic differential lock,
as front-wheel drive forces which correspond to those of a driven axle with
differential lock and 100% locking action, and which can even be exceeded in
intelligent (slip-controlled) systems, such wheel gearing systems produce. The
system can ensure that the driving torque that is to be applied to the side with the
retarded wheel is equal to the torque on the side with the higher coefficient of
friction. This 'lost torque' must be generated by the transmission, on the one
hand, and retarded by the wheel brake, on the other, so that loss of engine power
and heating of the wheel brakes are produced. The braking temperatures are
calculated on the basis of the braking torque and period of application of the
brakes. If the temperatures calculated exceed the permissible limits, application
of the brakes is discontinued during the front-wheel drive phase until a calculated cooling of the system has taken place; the transmission then corresponds
to that found in a conventional vehicle.
Another possibility for maximum utilization of grip is afforded by traction
control systems in which engine power is reduced by means of the throttle, injection and ignition point so that the spinning wheels work in the region of lower
slip and consequently higher adhesion. Both systems are used together, even
without four-wheel drive. In models of the E and M class with an electronic traction system (ETS), Mercedes-Benz uses electronic locks instead of mechanical
differential locks.
1.7.2
Four-wheel drive vehicles with overdrive
In four-wheel drive vehicles with overdrive the middle differential is not used.
The engine torque is distributed to all four wheels by means of a clutch on the
propshaft, as required. The clutch can be engaged manually, or automatically in
response to slip. With the use of sprag clutches, which are usually engaged
manually, the torque is transmitted in a fixed ratio between the front and rear
axles; multi-disc or visco clutches permit variable torque distribution. As these
systems have essential similarities with permanent four-wheel drive varieties,
they are discussed in Section 1.704.
With sprag-clutch engaged transmissions, the design complexity, and therefore the costs, are lower than on permanent drive. Usually there is no rear axle
differential lock, which is important on extremely slippery roads; while this
results in price and weight advantages, it does lead to disadvantages in the traction.
Front-wheel drive is suitable as a basic version and the longitudinal engine
has advantages here (Fig. 1048). With the transverse engine, the force from the
manual gearbox is transmitted via a bevel gear and a divided propshaft, to the
rear axle with a differential (Fig. 1.68). There is relatively little additional
I
'
•
",'...",_._-
Fig. 1.68
The Fiat Panda Treking 4 x 4, a passenger car based on front-wheel drive with transverse engine. The vehicle has
McPherson struts at the front and a rigid axle with longitudinal leaf springs at the back. The propshaft leading to it is divided into three
to be able to take the rotational movements of the rigid axle around the transverse (y) axis during drive-off and braking and to absorb
movements of the drive unit. The Fiat Panda is an estate car with the ratio
" =
2159 mm
3689 mm = 0.59 (see Equation 3.1)
70
The Automotive Chassis
complexity compared with the front-wheel drive design, even if, on the Fiat
Panda (Trecking 4 X 4), there is a weight increase of about 11 % (90 kg), not
least because of the heavy, driven, rigid axle. It is possible to select rear-wheel
drive during a journey using a shift lever that is attached to the prop-shaft tunnel.
Manual selection on the Subaru lusty operates pneumatically at the touch of
a button (even while travelling). This vehicle has independent rear-wheel
suspension and weighs only 6% more than the basic vehicle with front-wheel
drive. Traction is always improved considerably if the driver recognizes the need
in time and switches the engine force onto all four wheels. In critical situations,
this usually happens too late, and the abrupt change in drive behaviour becomes
an additional disadvantage.
Conversely, if the driver forgets to switch to single axle drive on a dry road,
tensions occur in the power train during cornering, as the front wheels travel
larger arcs than the back ones (Figs 1.69 and 3.91). The tighter the bend, the
greater the stress on the power train and the greater the tendency to unwanted
tyre slip.
A further problem is the braking stability of these vehicles. If the front axle
locks on a wet or wintry road during braking, the rear one is taken with it due to
the rigid power train. All four wheels lock simultaneously and the car goes into
an uncontrollable skid.
\
\
\
Os
Fig. 1.69 The front wheel on the outside of the bend draws the largest arc duri~g
slow cornering, the track circle diameter Os, whil~ the in~er. wheel d,raws the consIderably smaller arc or,i. This is the reason for the differential In the dnven front axle of
the front-wheel drive. The bend diameters oS,r and or,i to the rear are even smaller,
so the rolling distance of the two wheels of this axle decreases further .and ~here can
be tensions in the drive train if both axles are rigidly connected, a bend is being negotiated and when a dry road surface makes wheel slip more difficult because of high
coefficients of friction.
Fig. 1.70
Complex power distribution on the Fiat Campagnolo, a four-wheel drive, all-purpose passenger car. The drive moment
is transferred from the manual gearbox via a centrally located two-gear power take-off gear to the differentials of the front and rear
axles. Efficiency is not likely to be especially good.
~
-~~_¥~¥-#-=-
72
ThE~ Automotive Chassis
1.7.3
Manual selection four-wheel drive on commercial and
all-terrain vehicles
The basis for this type of vehicle is the standard design which, because of the
larger ground clearance necessary in off-road vehicles (Fig. 1.67), has more
space available between the engine and front axle differential and between the
cargo area and the rear axle. Figure 1.70 shows the design details:
• a central power take-off gear with manual selection for the front axle, plus a
larger ratio off-road gear, which can be engaged if desired;
• three propshafts;
• complex accommodation of the drive joints if there is a rigid front axle (Fig. 1.3).
1.7.4
Permanent four-wheel drive; basic passenger car with
front-wheel drive
All four wheels are constantly driven; this can be achieved between the front and
rear axle with different design principles:
• a bevel centre differential with or without manual lock selection;
• a Torsen centre differential with moment distribution, based on the traction
requirement (Fig. 1.71);
• a planet gear central differential with fixed moment distribution and additional
visco clutch, which automatically takes over the locking function when a
difference in the number of revolutions occurs or a magnetic clutch (which is
electronically controlled, Fig. 1.79);
• electronically controlled multi-disc clutches (Haldex clutch, Fig. 1.73);
5
8
6
6
7
4
4
Fig. 1.71 Torsen central differential fitted in Quattro models (apart from the TI)
by Audi. It consists of two worm gears, which are joined by spur gears and, depending on the traction requirement, can distribute the driving torque up to 75% to the
front or rear axle. Under normal driving conditions 50% goes to each axle.
Fig. 1.72 Four-wheel drive Golf 4motion (1998). In the four-wheel drive vehicle, Volkswagen uses a multi-link suspension consisting of one longitudinal and two transverse links mounted on a subframe. The driving torque is transmitted to the rear axle via a wet
multi-disc clutch by the Swedish company Haldex which is flange-mounted on the rear axle drive and runs in oil. This electronically
controlled clutch can build up a coupling torque of up to 3200 Nm even at small cardan-shaft rotation angles of 45° and can be
combined to good effect with brake-power control systems. The drive train of the Audi n Quattro (1998) built on the same platform
is built to almost the same design.
~
__
5~-<"~~_
74
The Automotive Chassis
• a visco clutch in the propshaft power train, which selects the initially undriven
axle depending on the tyre slip (Figs 1.72, 1.74 and 1.75).
Here too, ~he front-wheel drive passenger car is suitable as a basic vehicle. In
1~79, Audl was the first company to bring out a car with permanent four-wheel
dnve, the Quattro., and t?day vehicles with this type of drive are available
through?ut the ~ntlr~ A.udl range. On a longitudinally mounted engine, a Torsen
ce?tre dIfferential dIstnbutes the moment according to the traction requirement
(FIg. 1.71). The four-wheel drive increases the weight by around 100 kg.
6
7
8
Fig. 1.73 Multi-disc clutch of the Swedish company Haldex, used in the Golf
4motion (1998) and Audi
Quattro (1998). When there is a difference in speed
between the front and rear axles, the disc cam 6 on the output shaft activates the
working elements of the axial-piston pump 12 by means of the rollers 7. Via the
control valve 14, the pressure produced activates the working piston which moves
the discs. The torque transmitted is adjusted continuously by the control unit up to
the maximum value, depending on the driving situation described by the wheel
sensors, the signals from the slip and brake-power control systems, the position of
the accelerator pedal, the engine speed etc. The clutch is disengaged when the ABS
function is used. 1 electronic control unit, 2 connector vehicle (voltage, CAN, K
leads), 3 oil filter, 4 shaft bevel wheel exit (rear axle gearing), 5 lamella, 6 cam plate,
7 coil, 8 relief valve, 9 pressure regulating valve, 10 accumulator, 11 input shaft, 12
axial piston pump, 13 pre-load pump, 14 control valve, 15 intermittent or step motor.
n
Types of suspension and drive
Fi!l. 1.74
75
Visco clutch with slip-dependent drive moment distribution. Two different packages sit in the closed drum-shaped housing: radially slit steel discs, which
are moved by the serrated profile of the hollow shaft, and perforated discs which grip
(as can be seen below) into housing keys. The shaft is joined with the differential and
thE! casing with the propshaft going to the rear axle.
The discs are arranged in the casing so that a perforated disc alternates with a slit
one. The individual parts have no definite spacing but can be slid against one another
axially. The whole assembly is filled with viscous silicone fluid and the torque behaviour (therefore the locking effect) can be adjusted via the filling level.
If slip occurs between the front and rear axle, the sets of discs in the clutch rotate
relative to one another and shearing forces are transferred via the silicone fluid.
ThE~se increase with increasing slip and ensure a torque increase in the rear axle. The
power consumed in the visco clutch leads to warming and thus to growing inner
pre!ssure. This causes an increase in the transferable torque which, under conditions
of 13xtreme torque requirement, ultimately leads to. an almost slip-free torque transfer (rigid drive). With ABS braking, a free-wheeling device disengages the clutch; the
latter must be engaged again when reversing.
r
76
The Automotive Chassis
Fig. 1.75
Driven front axle of the Porsche 911 Carrera 4 (1996, 1998). The visco
clutch is flange mounted directly on the front axle to achieve a better distribution of
axle load. With corresponding slip of the rear wheels, up to 40% of the driving torque
is transmitted to the front axles. Particular attention was paid during the adjustment
of the four-wheel drive to predictable self-steering properties independent of drive
distribution and to controllability of the handling characteristics even at the stability
limit. Instead of differential locks, specific wheel brake engagements are made in
order to retard spinning wheels. Four-wheel drive is integrated into the Porsche
Stability Management (PSM), a system for controlling the dynamics of vehicle
movement with brake actuation.
Fig. 1.76
Double wishbone rear axle on the Audi A4 Quattro. The suspension
subframe 1 is fixed to the body with four widely spaced rubber mountings (items 2
and 3) and houses the differential casing 8 and transverse control arms (items 4 and
5). The sprin£Js and shock absorbers are mounted next to the fixings for the upper
control arms 7. The location 6 of the wheel hub carrier 5 was raised (long base c, Fig.
1.4) and drawn outwards. The lower transverse control arm 4 is fixed to part 1 with
widely spaced mountings. These measures ensure a wide boot and low forces,
making it easier to attain the desired kinematic characteristics.
----------------,-----
---------r----r--
78
The Automotive Chassis
VW used a visco clutch in the power train (without centre differential) for the
first time on the Transporter (Fig. 1.74) and then subsequently used it in the Golf
syncro. The clutch has the advantage of the engine moment distribution being
dependent on the tyre slip. If the slip on the front wheels, which are otherwise
driven at the higher moment, increases on a wet or frozen surface or off-road,
more drive is applied to the rear wheels. No action on the part of the driver is
either necessary or possible. The transverse engine makes a bevel gear in front
of the split propshaft nece~sary. The visco clutch sits in the rear differential
casing and there is also an overrunning clutch, which ensures that the rear
wheels are automatically disengaged from the drive, on overrun, to guarantee
proper braking behaviour. This type of drive is fully ABS compatible. When
reverse is engaged, a sliding sleeve is moved, which bridges the overrunning
clutch to make it possible to drive backwards.
When selecting their rear axle design, manufacturers choose different paths.
Audi fits a double wishbone suspension in the A4 and A6 Quattro (Fig. 1.76),
Honda uses the requisite centre differential on the double wishbone standard
suspension itn the Civic Shuttle 4WD (Figs 1.77 and 1.62).
Visco clutch
Rear differential
Fig. 1.77
Double wishbone rear axle of the Honda Civic Shuttle 4 WD. The visco
clutch sits (held by two shaft bearings) in the centre of the divided propshaft. The
rear axle diffE~rential has been moved forwards and is mounted to the rear on the
body via a cross~member. Apart from the different type of wheel bearings and the
lower transverse control arm positioned somewhat further back (to make it possible
to bring the drive shafts through in front of the spring dampers), the axle corresponds
to Fig. 1.62 and resembles the suspension shown in Fig. 1.1.
,
,,
,,,
;11
I
:
:i
I
'II
I
:
:i
i
I
Fig. 1.78 Drive train of the four-wheel drive Mercedes-Benz E class 4MATIC (from 1997). In order to be able to control the drive
shafts to the front wheels, an integrated spring-and-shock absorber strut in the shape of a fork on the lower transverse link is used.
In the almost identical suspension design of other than off-road varieties, the springs and shock absorbers are separate.
III
:1
I I '
l'
J
1,'
i
J
}:
}:
I'
I'
i I
~..::.
. ._';::::;F,"""'~"",
80
1.7.5
The Automotive Chassis
Permanent four-wheel drive, basic standard design
passenger car
Giving a standard design car four-wheel drive requires larger modifications, greater
design complexity and makes the drive less efficient (Fig. 1.78). A power take-off
gear is required, from which a short propshaft transmits the engine moment to the
front differential. The lateral offset must be bridged, for example, with a toothed
chain (Fig. 1.79). The ground clearance must not be affected and so changes in the
engine oil pan are indispensable if the axle drive is to be accommodated (Fig. 1.80).
The power take-off gear (Fig. 1.79) contains a planet gear centre differential
which facilitates a variable force distribution (based on the internal ratio); 36%
of the drive moment normally goes to the front and 64% to the rear axle. A multidisc clutch can also be installed that can lock the differential electromagnetically
up to 100%, depending on the torque requirement (front to rear axle). Moreover,
there is a further electrohydraulically controlled lock differential in the rear axle
which is also up to 100% effective.
1
5
6
2
Fig. 1.79 The torque coming from the engine is apportioned by the Planet WheelCentric Differential 1 in such one, to the rear cardan shaft 2 (64%) and to the front
one 3 (36%). The offset to this shaft is bridged-over by the inserted tooth type chain
4. The adaptation of the distribution of driving power is taken over the the multipledisk clutch 5, which is driven (controlled) by the electromagnet.
Power Divider A110 of the Fa. ZF. (Zahnradfabrik Friedrichshafen)
'----------r--
Types of suspension and drive
6
4
81
3
Fig. 1.80
Front cross-section view of the engine; and drive axle of a standard fourwheel drive vehicle (BMW assembly diagram). The basic vehicle has rear-wheel drive
and, in order to also be able to drive the front wheels, the front axle power take-off
4 had to be moved into the space of the oil pan. The intermediate shaft 1 bridges the
distance to the right inner CV joint and thus ensures drive shafts of equal length to
both wheels (items 2 and 3 and Fig. 1.51). Part 1 is mounted on one side in the nonlockable differential 4 and on the other side in the outrigger 5. This, and the casing
6, are screwed to the oil pan.
The two differentials with variable degrees of lock offer decisive advantages:
• to reach optimal driving stability, they distribute the engine moments during
overrun and traction according to the wheel slip on the drive axles;
• they allow maximum traction without loss of driving stability (Fig. 1.66).
The locks are open during normal driving. By including the front axle differential, they make it possible to equalize the number of revolutions between all
wheels, so tight bends can be negotiated without stress in the power train and
parking presents no problems. If the car is moved with locked differentials and
the driver is forced to apply the brakes, the locks are released in a fraction of a
second. The system is therefore fully ABS compatible.
In its four-wheel drive vehicles of the E class (Fig. 1.78), Mercedes-Benz uses
82
The Automotive Chassis
9
7
8
5
1/
2
3
Fig. 1.81
Front suspension and drive axle of the Mercedes-Benz off-road vehicle
of the M series. In off-road vehicles, rigid axles are mostly used. Instead of these,
Mercedes-Benz installs double wishbone suspensions at the front and rear. In this
way, the proportion of unsprung masses can be reduced by approximately 66%;
driving safety and riding comfort are increased. For space reasons, torsion-bar
springs are used for the suspension of the front axle.
1 lower transverse link in the form of a forged steel component because of the
introduction of torque by the torsion bars (2) and notch insensitivity off road conditions; 2 torsion bars (spring rate of 50 Nm/degree); 3 vertically adjustable torque
support which can be placed in any position in a transverse direction; 4 integral bearers (subframe) attached to the box-type frame by 4 bolts; 5 upper transverse link in
the form of a forged aluminium component; 6 rack and pinion power steering, 7 twintube shock absorber with integrated rubber bump stop, 8 transverse link mounting
points; 9 stabilizer application of force to lower transverse link.
a transfer gear with central differential situated on the gearbox outlet and a front
axle gear integrated into the engine-oil pan. The (fixed) driving torque distribution
is 35%:65%. Instead of traditional differential locks, the wheel brakes are activated
on the spinning wheels as in off-road vehicles of the M class. This system permits
maximum flexibility, its effect not only corresponds to differential locks on front
and rear axles as well as on the central differential, but also makes it possible for
other functions such as ABS and electronic yaw control (ESP) to be integrated
without any problem. Design complexity - and thus cost - is considerable.
1.7.6
Summary of different kinds of four-wheel drive
The list in Fig. 1.83 shows the increasing use of slip-controlled clutches (visco
and Haldex clutches) for the transmission of torque instead of an interaxle
---ri-
Types of suspension and drive
83
2
/
Fig. 1.82
Rear axle of the Mercedes-Benz off-road vehicle of the M series.
Suspension and damping are ensured by the spring strut (1) whose spring is tapered
for reasons of construction space (spring rate gradually increasing from 70 to 140
N/mm), 2 brake disc with integrated drum parking brake, 3 upper transverse link
(forged aluminium component), 4 lower transverse link (forged aluminium component), 5 tie rod (forged steel component), 6 integral bearer (subframe), 7 stabilizer, 8
transverse link mounting points.
Common characteristics of front and rear axles: camber and castor are adjusted
by positioning the transverse link mounting points (8) in long holes during assembly.
Technical data: spring travel ± 100 mm, kingpin offset -5 mm, disturbing force
moment arm 56.7 mm, kingpin inclination 10.5°, camber angle -0.5°, castor for front
axle/rear axle 7/-8.5°, castor trail for front axle/rear axle 37/-55 mm, wheel castor trail
for front axle/rear axle 5/-4.5 mm, instantaneous centre height for front axle/rear axle
80/119 mm, braking-torque compensation for front axle/rear axle 38/21 %, startingtorque compensation for front axle/rear axle -7/-3%. The axle concept was designed
and developed by Mercedes-Benz. Mass production and assembly is undertaken by
Zahnradfabrik Friedrichshafen AG who, via LemfOrder Fahrwerktechnik AG, supply
the complete subassemblies to the assembly line as required.
differential and the importance of electronic brake application systems which are
used instead of lockable differential gears. Modern four-wheel varieties operate
without functional restrictions with antilocking, slip and driving stabilization
systems.
I
I
!
,I
Fig. 1.83
Motor
position
Different kinds of four-wheel drive.
Reduction
drive
Drive
on
Four-wheel drive
switched
by
-I
I
slip
dependent
perm.
by
Middle
differential
locks via
Front axle
differential
Rear axle
differential
Example
locking
brake
locking
brake
longit.
2.05:1
rear
sprag-clutch man.
N/A
n
n
n
n Opel Frontera
longit.
1.425:1
rear
sprag-clutch man.
N/A
n
n
n Mitsubishi Pajero
longit.
2.15:1
rear
sprag-clutch man.
N/A
n
n
multi-disc
clutch
n
longit.
2.43:1
rear
sprag-clutch man.
N/A
n
n
longit.
2.48:1
rear
multi-disc.
electron.
N/A
n
n
transv.
longit.
front
rear
multi-disc
2.72:1
electron.
visco
N/A
N/A
n
n
n
n
multi-disc
clutch
multi-disc
clutch
n Suzuki Jimny Cross
Country
n Chevrolet Blazer
n Ford Explorer
transv.
transv.
front
front
visco
visco
N/A
N/A
n
n
n
y
n
multi-disc
clutch
n
n
longit.
rear
visco
N/A
n
y
n
transv.
front
visco
N/A
n
y
transv.
front
visco
NiA
n
y
transv.
front
Haldex
multi-disc
N/A
n
y
multi-disc
clutch
multi-disc
clutch
n
longit.
f +r
claw clutch
n
n
n
n Audi n Ouattro,
Golf 4motion
n Daihatsu Terios
N/A
n
y
n
y Mercedes-Benz M series
I
n Honda CR-V
n Jeep Grand Cherokee
n Chrysler Voyager 4WD
y Land Rover
Freelander Discovery
y Porsche Carrera 4
--
longit.
2.64:1
I~~~~
f +r
longit.
diff.
long it.
diff.
,
,.t.tl1~
n Volvo V70 R AWD
n VW Multivan Synchro
H_
longit.
y
f +r
longit.
f +r
longit.
f +r
1.21 :1
f +r
y
f +r
longit.
1.93:1
f +r
longi!.
1.20: 1
f +r
longit.
1.45:1
f +r
longit.
transv.
f +r
transv.
f +r
longit.
ditto
longit.
ditto
longit.
diff.
longit.
ditto
longit.
ditto
longit.
diff.
longit.
ditto
longit.
ditto
Torsen
ditto
Torsen
ditto
y
y
n
y
n
Mercedes-Benz G series
N/A
n
y
n
y
Mercedes-Benz E series
N/A
n
y
n
y
N/A
n
y
n
y
BMW Sports Activity
Vehicle (SAV, E53)
Land Rover Discovery
visco + man.
Toyoto LandCruiser
visco + man.
n
n
sprag-clutch n
Mitsubishi Pajero
visco lock
n
n
n
n
Subaru Legacy Outback
visco lock
n
n
n
n
Subaru Forester
self-locking
n
y
n
y
self-locking
n
y
n
y
Audi A4/A6 Quattro.
VW Passat Synchro
Audi A4/A6 Quattro.
VW Passat Synchro
.
2
Tyres and wheels
2.1
Tyre requirements
The tyres are crucial functional elements for the transmission of longitudinal,
lateral and vertical forces between the vehicle and road. The tyre properties
should be as constant as possible and hence predictable by the driver. As well as
their static and dynamic force transmission properties, the requirements
described below - depending on the intended use of the vehicle - are also to be
satisfied.
As tyres significantly affect the handling properties of vehicles, the properties
of original tyres - the tyres with which the vehicle is supplied to the customer are specified by the vehicle manufacturers in conjunction with the tyre manufacturers. However, spare tyres usually differ from the original tyres, despite
their similar designation; hence handling characteristics can change. Individual
vehicle manufacturers have therefore decided to identify tyres produced in
accordance with their specifications by means of a symbol on the sidewall of the
tyre or to sell tyres which meet the specifications of original tyres at their manufacturing branches.
2.1.1
Interchangeability
All tyres and rims are standardized to guarantee interchangeability, i.e. to guarantee the possibility of using tyres from different manufacturers but with the
same designation on one vehicle and to restrict the variety of tyre types worldwide.
Within Europe, standardization is carried out by the European Tyre and Rim
Technical Organization or ETRTO, which specifies the following:
• tyre and rim dimensions;
• the code for tyre type and size;
• the load index and speed symbol.
r
I .
Tyres and wheels
87
Passenger car tyres are governed by UNO regulation ECE-R 30, commercial
vehicles by R 54, spare wheels by R 64, and type approval of tyres on the vehicle by EC directive 92/23/EC.
In the USA the Department of Transportation (or DOT, see item 9 in Fig.
2.18) is responsibl~ for the safety standards. The standards relevant here are:
Standard 109
Standard 119
Passenger cars
Motor vehicles other than passenger cars.
The Tire and Rim Association, or TRA for short, is responsible for standardization.
In Australia, binding information is published by the Federal Office of Road
Safety, Australian Motor Vehicle Certification Board.
ARD23
Australian Design Rule 23/01:
Passenger car tyres
is the applicable standard.
In Germany the DIN Standards (Deutsches Institut ftir Normung) and the
WdK Guidelines (Wirtschaftsverband der Deutschen Kautschukindustrie
Postfach 900360, D-60443, Frankfurt am Main) are responsible for specifying
tyre data. All bodies recognize the publications of these two organizations.
At the international level, the ISO (International Organization for
Standardization) also works in the field of tyre standardization and ISO
Standards are translated into many languages.
2.1.2
Passenger car requirements
The requirements for tyres on passenger cars and light commercial vehicles can
be subdivided into the following six groups:
•
•
•
•
•
•
driving safety
handling
comfort
service life
economy
environmental compatibility.
To ensure driving safety it is essential that the tyre sits firmly on the rim. This is
achieved by a special tyre bead design (tyre foot) and the safety rim, which is the
only type of rim in use today (Figs 2.5 and 2.21). Not only is as great a degree
of tyre-on-rim retention as possible required, but the tyre must also be hermetically sealed; on the tubeless tyre this is the function of the inner lining. Its job is
to prevent air escaping from the tyre, i.e~ it stops the tyre from losing pressure.
However, this pressure reduces by around 25-30% per year, which shows how
important it is to check the tyre pressure regularly.
88
The Automotive Chassis
In order to guarantee driving safety, the aim is also to ensure that tyres are as
insensitive to overloading and as puncture-proof as possible and that they have
emergency running properties which make it possible for the driver to bring the
vehicle safely to a halt in case of tyre failure.
Handling characteristics include the properties:
•
•
•
•
•
•
high coefficients of friction in all operating conditions;
steady build-up of lateral forces without sudden changes;
good cornering stability;
direct and immediate response to steering movements;
guarantee requirement of sustained maximum speed;
small fluctuations in wheel load.
Riding comfort includes the characteristics:
•
•
•
•
good suspension and damping properties (little rolling hardness);
high smoothness as a result of low radial tyre run-out and imbalances;
little steering effort required during parking and driving;
low running noise.
Durability refers to:
• long-term durability
• high-speed stability.
Both are tested on drum test stands and on the road.
Economic efficiency is essentially determined by the following:
• purchase cost;
• mileage (including the possibility of profile regrooving in the case of lorry
tyres);
• wear (Fig. 3.46);
• rolling resistance;
• the necessary volume, which determines
• the amount of room required in the wheel houses and spare-wheel well;
• load rating.
Of increasing importance is environmental compatibility, which includes:
• tyre noise;
• raw material and energy consumption during manufacture and disposal;
• possibility of complete remoulding inherent in the construction.
The importance of
• tyre design, profile design and the 'radius-width appearance' must not be
neglected either.
Further details are available in Refs [4], [6], [7] and [9].
-r--'
Tyres and wheels
2.1.3
89
Commercial vehicle requirements
In principle, the same requirements apply for commercial vehicles as for passenger cars, although the priority of the individual groups changes. After safety,
economy is the main consideration for commercial vehicle tyres. The following
properties are desirable:
•
•
•
•
•
•
high mileage and even wear pattern
low rolling resistance
good traction
low tyre weight
ability to take chains
remoulding/retreading possibilities.
Compared with passenger car tyres, the rolling resistance of commercial vehicle
tyres has a greater influence on fuel consumption (20-30%) and is therefore an
important point (Fig. 2.32).
2.2
2.2.1
Tyre designs
Diagonal ply tyres
In industrialized countries, cross-ply tyres are no longer used on passenger cars,
either as original tyres or as replacement tyres, unlike areas with very poor roads
where the less vulnerable sidewall has certain advantages. The same is true of
commercial vehicles and vehicles that tow trailers, and here too radial tyres have
swept the board because of their many advantages. Nowadays, cross-ply tyres
are used only for:
• temporary use (emergency) spare tyres for passenger cars (due to the low durability requirements at speeds up to 80 or 100 km h- I );
• motor cycles (due to the inclination of the wheels against the lateral force);
• racing cars (due to the lower moment of inertia);
• agricultural vehicles (which do not reach high speeds).
Cross-ply tyres consist of the substructure (also known as the tyre carcass, Fig.
2.1) which, as the 'supporting framework' has at least two layers of rubberized
cord fibres, which have a zenith or bias angle ~ of between 20° and 40° to the
centre plane of the tyre (Fig. 2.2). Rayon (an artificial silk cord), nylon or even
steel cord may be used, depending on the strength requirements. At the tyre feet
the ends of the layers are wrapped around the core of the tyre bead on both sides;
two wire rings, together with the folded ends of the plies, form the bead. This
represents the frictional connection to the rim. The bead must thus provide the
permanent seat and transfer drive-off and braking moments to the tyre. On tubeless tyres it must also provide the airtight seal.
The running tread, which is applied to the outer diameter of the substructure,
90
The Automotive Chassis
Cap of the tyre (protector)
Shoulder
Wall
rubber
Installation
curve
Drop rim
Fig. 2.1 Design of a diagonal ply tubeless car tyre with a normal drop rim and
pressed-in inflating valve (see also Fig. 2.6).
Fig. 2.2
The diagonal ply tyre has crossed-bias
layers; the zenith angle ~ was 30° to 40° for passenger
cars. The 4 PR design should have two layers in each
direction. Smaller angles ~ can be found in racing cars.
Rolling resistance, lateral and suspension stiffness are
significantly determined by the zenith angle.
provides the contact to the road and is profiled. Some tyres also have an intermediate structure over the carcass as reinforcement.
At the side, the running tread blends into the shoulder, which connects to the
sidewall (also known as the side rubber), and is a layer that protects the substructure. This layer and the shoulders consist of different rubber blends from the
running tread because they are barely subjected to wear; they are simply
deformed when the tyre rolls. This is known as flexing. Protective mouldings on
the sides are designed to prevent the tyre from being damaged through contact
with kerbstones. There are also GG grooves, which make it possible to see that
the tyre is seated properly on the rim flange.
Cross-ply design and maximum authorized speed are indicated in the tyre
marking by a dash (or a letter, Fig. 2.12) between the letters for width and rim
----,-------'-------------ri-------------·-----JL-'
Tyres and wheels
91
diameter (both in inches) and a 'PR' (ply rating) suffix. This ply rating refers to
the carcass strength and simply indicates the possible number of plies (Fig. 2.5).
The marking convention is:
5.60-15/4 PR
7.00-14/8 PR
9.00-20/14 PR
(VW rear-engine passenger car, tyres authorized up to
150 km h- 1)
(VW Transporter, tyres authorized up to 150 km h- I )
(reinforced design for a commercial vehicle)
and on the temporary use spare wheel of the VW Golf, which requires a tyre
pressure of PT = 4.2 bar and may only be driven at speeds up to 80 km h- 1
(F symbol)
T 105170 D 14 38 F
2.2.2
Radial ply tyres
The radial ply tyre consists of two bead cores joined together radially via the
carcass (Fig. 2.3) - hence the name radial tyres. A belt of cords provides the
necessary stiffness (Fig. 2.4), whereas the external part of the tyre consists of
the tread and sidewall and the interior of the inner lining, which ensures the tyre
is hermetically sealed (Figs 2.5 and 2.1). In passenger car tyres, the carcass is
made of rayon or nylon, the belt of steel cord or a combination of steel, rayon
or nylon cord, and the core exclusively of steel. Due to the predominance of
steel as the material for the belt, these tyres are also known as 'steel radial
tyres'. The materials used are indicated on the sidewall (Fig. 2.18, points 7 and
Fig. 2.3 Substructure of a radial tyre.
The threads have a bias angle between
88° and 90°.
Fig. 2.4 The belt of the radial tyre
sits on the substructure. The threads
are at angles of between 15° and 25° to
the plane of the tyre centre.
92
The Automotive Chassis
1
2
>----
5
6
7
Fig. 2.5
Radial design passenger car tyres in speed category T (Fig. 2.12); the
number of layers and the materials are indicated on the sidewall (see Fig. 2.18). The
components are: 1 running tread; 2 steel belt; 3 edge protection for the belt, made
of rayon or nylon; 4 sidewall; 5 substructure with two layers; 6 cap; 7 inner lining; 8
flipper; 9 bead profile; 10 core profile; 11 bead core.
8). In commercial vehicle designs this is particularly important and the carcass
may also consist of steel.
The stiff belt causes longitudinal oscillation, which has to be kept away from
the body by wheel suspensions with a defined longitudinal compliance, otherwise this would cause an unpleasant droning noise in the body, when on cobbles
and poor road surfaces at speeds of less than 80 km h- 1 (see Sections 3.6.5.2 and
5.1.2). The only other disadvantage is the greater susceptibility of the thinner
sidewalls of the tyres to damage compared with diagonal ply tyres. The advantages over cross-ply tyres, which are especially important for today's passenger
cars and commercial vehicles, are:
• significantly higher mileage
• greater load capacity at lower component weight
Tyres and wheels
•
•
•
•
•
93
lower rolling resistance
better aquaplaning properties
better wet-braking behaviour
transferable, greater lateral forces at the same tyre pressure
greater ride comfort when travelling at high speeds on motorways and trunk
roads.
2.2.3
Thbeless or tubed·
In passenger cars, the tubeless tyre has almost completely ousted the tubed tyre.
The main reasons are that the tubeless tyre is
• easier and faster to fit
• the inner lining is able to self-seal small incisions in the tyre.
In tubeless tyres the inner lining performs the function of the tube, i.e. it prevents
air escaping from the tyre. As it forms a unit with the carcass and (unlike the
tube) is not under tensional stress, if the tyre is damaged the incision does not
increase in size, rapidly causing loss of pressure and failure of the tyre. The use
of tubeless tyres is linked to two conditions:
• safety contour on the rim (Fig. 2.21)
• its air-tightness.
Because this is not yet guaranteed worldwide, tubed tyres continue to be fitted
in some countries. When choosing the tube, attention should be paid to ensuring
the correct type for the tyre. If the tube is too big it will crease, and if it is too
small it will be overstretched, both of which reduce durability. In order to avoid
confusion, the tyres carry the following marking on the sidewall:
tubeless (Fig. 2.18, point 3)
tubed or tube type.
Valves are needed for inflating the tyre and maintaining the required pressure.
Various designs are available for tubeless and tubed tyres (Figs 2.6 and 2.7). The
most widely used valve is the so-called 'snap-in valve'. It comprises a metal foot
valve body vulcanized into a rubber sheath, which provides the seal in the rim
hole (Fig. 2.20). The functionality is achieved by a valve insert, while a cap
closes the valve and protects it against ingress of dirt.
At high speeds, the valve can be subjected to bending stress and loss of air
can occur. Hub caps and support areas on alloy wheels can help to alleviate this
(see Fig. 2.24 and Section 7.2 in Ref. [4]).
2.2.4 Height-to-width ratio
The height-to-width ratio H/W - also known as the 'profile' (high or low) influences the tyre properties and affects how much space the wheel requires
94
The Automotive Chassis
Vg 8
I
¢d
d
<1>57
DIN
I
Diameter d
43 GS 11.5
43
15.2
43 GS 16
43
19.5
Fig. 2.6 Snap-in rubber valve for
tubeless tyres, can be used on rims
with the standard valve holes of
11.5 mm and 16 mm diameter. The
numerical value 43 gives the total
length in mm (dimension I). There is
also the longer 49 GS 11.5 design.
Fig. 2.8 Tyre sizes and associated rims used on the VW Golf
III. All tyres fit flush up to the
outer edge of the wing (wheel
house outer panel) K. To achieve
this, differing wheel offsets
(depth of dishing) e are used on
disc-type wheels (Fig. 2.23) with
the advantage of a more negative rolling radius [a on wider
tyres (Fi!l 3.102). A disadvantage then is that snow chains
can no longer be fitted and
steering sensitivity changes very
slightly.
Valve specification
d
38/11.5
11.7
38/16
16.5
Fig. 2.7
Rubber valve vulcanized
onto tubes. Designations are 38/11.5
or 38/16.
K
\
1'15170 R13 5Y2Jx13
1135/60 R14 6Jx14
6Jx15
6Y2Jx15
Tyres and wheels
95
(Fig. 2.8). As shown in Fig. 2.9, the narrower tyres with a H/W ratio = 0.70
have a reduced tread and therefore good aquaplaning behaviour (Fig. 2.35).
Wide designs make it possible to have a larger diameter rim and bigger
brake discs (Fig. 2.10) and can also transmit higher lateral and longitudinal
forces.
W is the cross-sectional width of the new tyre (Fig. 2.11); the height H can
easily be calculated from the rim diameter given in inches and the outside diameter of the tyre ODT • The values ODT and Ware to be taken from the new tyre
I
175/80 R 14 88T
ContiEcoContact EP
195/65 R 15 91 V
ContiEcoContact CP
205/55 R 16 91W
ContiSportContact
225/45 ZR 17
ContiSportContact
Fig. 2.9 If they have the same outside diameter and load capacity the four tyre
sizes used on medium-sized passenger cars are interchangeable. The series 65, 55
and 45 wide tyres each allow a 1" larger rim (and therefore larger brake discs). The
different widths and lengths of the tyre contact patch, known as 'tyre print', are
clearly shown (Fig. 3.119), as are the different designs of the standard road profile
and the asymmetric design of the sports profile (see also Section 2.2.10). The 65
series is intended for commercial vehicles, and the 60, 55 and 45 series for sports
cars. (Illustration: Continental; see also Fig. 2.19.)
,
96
The Automotive Chassis
Fig. 2.10 The flatter the tyre, i.e. the larger the rim diameter d (Fig. 2.11) in
comparison with the outside diameter oar, the larger the brake discs or drums that
can be accommodated, with the advantage of a better braking capacity and less
tendency to fade. An asymmetric well-base rim is favourable (Figs 1.8 and 2.11).
12
221
200
Wheel rim diameter in inches
Brake disc outer diameter in mm
Brake drum inner diameter in mm
13
256
230
14
278
250
15
308
280
16
330
300
17
360
325
w
H
b
D
r
d
Fig. 2.11
Tyre dimensions specified in standards and directives. B is the crosssection width of the new tyre; the tread moulding (as can be seen in Fig. 2.1) is not
included in the dimension. For clearances, the maximum running width with the
respective rim must be taken into consideration, as should the snow chain contour
for driven axles. The tyre radius, dependent on the speed, is designated r (see
Section 2.2.8). Pictured on the left is an asymmetrical well-base rim, which creates
more space for the brake caliper and allows a larger brake disc (Fig. 2.10).
mounted onto a measuring rim at a measuring tyre pressure of 1.8 bar or 2.3 bar
on V-, W- or ZR tyres, Fig. 2.15):
H = 0.5 (ODT
1" = 1 in
-
d)
=25.4 mm
(2.1)
(2.1a)
The 175/65 R 1482 H tyre mounted on the measuring rim 5J X 14 can be taken
as an example:
The cross-section ratio is rounded to two digits and given as a percentage. We
talk of 'series', and here the ratio profile is 65% as shown in the tyre markingin other words it is a 65 series tyre. A wider rim, e.g. 6J X 14 would give a
smaller percentage.
2.2.5 Tyre dimensions and markings
2.2.5.1
Designations for passenger cars up to 270 km h-1
The ETRTO standards manual of the European Tire and Rim Technical
Organization
includes all tyres for passenger cars and delivery vehicles up to 270
1
km h- and specifies the following data:
•
•
•
•
•
tyre width in mm
height-to-width ratio as a percentage
code for tyre design
rim diameter in inches or mm
operational identification, comprising load index; LI (carrying capacity index)
and speed symbol GSY.
The following applies to the type shown in Fig. 2.15:
175
/
65
R
14
82
H
L
speed symbol (authorized up to
210 km h- 1, Fig. 2.12).
load index (maximum load capac' - - - - - ity 475 kg at 2.5 bar and 160
km h- 1, Figs 2.13 and 2.14).
rim diameter in inches (Fig. 2.20).
code for tyre design (R = radial,
diagonal tyres have a dash '-' here
(see Section 2.2.1 and Chapter 6
in Ref. 4).
L..-
L-
Gross-section ratio profile as a
% (can be omitted on 82 series or
replaced by 80; see Section
2.2.5.2).
width of the new tyre on the
measuring rim and at measuring
pressure of 1.8 bar.
98
The Automotive Chassis
Fig. 2.12 Standardized speed categories for radial tyres, expressed by means of
a speed symbol and - in the case of discontinued sizes - by means of the former
speed marking. Sizes marked VR or ZR may be used up to maximum speeds specified by the tyre manufacturer. The symbols F and M are intended for emergency
(temporary use) spare wheels (see Chapter 6 in Ref. [5]).
V max
in km/h- 1
Speed symbol
80
130
150
160
170
180
190
210
240
270
300
Identification
F
M·
p
Q
R
S
T
H
V
W
y
over 210
over 240
VR
ZR
(old system)
The old markings can still be found on individual tyres:
155
S
L
R
13
rim diameter in inches
radial tyre
' - - - - - - - - speed symbol (authorized up to 180 km h- 1)
~--------
2.2.5.2
width of the new tyre and 82 series, when details of
the cross-section ratio missing
Designations of US tyres and discontinued sizes for passenger cars
Tyres manufactured in the USA and other non-European countries may also bear
a 'P' for passenger car (see Fig. 2.17) and a reference to the cross-section ratio:
P 155/80 R 13 79 S
The old system applied up until 1992 for tyres which were authorized for speeds
of over V = 210 km h- 1 (or 240 km h- 1, Fig. 2.12); the size used by Porsche on
the 928 S can be used as an example:
225/50
VR
, - - I-
16
-
-
-
radial tyre
speed symbol V
' - - - - - - - (authorized over 210 km h- 1)
--rr--
Tyres and wheels
99
Fig. 2.13
Load capacity/air pressure category specified in the directives. The
load capacity on the left - also known as 'load index' (L1) - applies for all passenger
cars up to the speed symbol W; they relate to the minimum load capacity values up
1
to 160 km h- at tyre pressure 2.5 bar (see Section 2.2.6). Further criteria, such as
maximum speed, handling etc., are important for the tyre pressures to be used on
the vehicle. For LI values above 100, further load increases are in 25 kg increments:
LI
LI
LI
= 101 corresponds to 825 kg,
= 102 corresponds to 850 kg etc. to
= 108 corresponds to 1000 kg.
Wheel load capacity in kg
with tyre pressure measured in bars
Load
index
1.5
1.6
1.7
1.8
1.9
2.0
2.1
2.2
2.3
2.4
2.5
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
215
225
230
235
245
250
255
265
275
280
290
300
305
315
325
330
340
350
360
370
385
400
410
420
430
445
460
470
485
500
515
530
225
235
240
250
255
260
270
280
290
295
305
315
325
330
340
350
360
370
380
390
405
420
430
440
455
470
485
495
510
525
540
560
240
245
255
260
270
275
285
295
305
310
320
330
340
350
360
365
380
390
400
410
425
440
450
465
475
490
505
520
535
550
570
590
250
260
265
275
280
290
300
310
315
325
335
345
355
365
375
385
395
4'10
420
430
445
460
475
485
500
515
530
545
560
575
595
615
260
270
275
285
295
300
310
320
330
340
350
360
370
380
390
400
415
425
440
450
465
480
495
505
520
540
555
570
585
600
620
640
270
280
290
295
305
315
325
335
345
355
365
375
385
395
405
420
430
445
455
470
485
500
515
525
545
560
575
595
610
625
650
670
285
290
300
310
315
325
335
350
360
370
380
390
400
415
425
435
450
460
475
485
505
520
535
550
565
585
600
620
635
650
675
695
295
300
310
320
330
340
350
360
370
385
395
405
415
430
440
450
465
480
490
505
525
540
555
570
585
605
625
640
660
675
700
720
305
315
325
330
340
350
360
375
385
400
410
420
430
445
455
470
480
495
510
525
545
560
575
590
610
625
645
665
685
700
725
750
315
325
335
345
355
365
375
385
400
410
425
435
445
460
470
485
500
515
525
540
560
580
595
610
630
650
670
685
705
725
750
775
325
335
345
355
365
375
387
400
412
425
437
450
462
475
487
500
515
530
545
560
580
600
615
630
650
670
690
710
730
750
775
800
100
The Automotive Chassis
Fig. 2.14 The tyre load capacity shown in the ETRTO standards manual in the
form of the load index LI is valid for V tyres up to vehicle speeds of 210 km h-1 ; for
W tyres up to 240 km h- 1 and for Y tyres up to 270 km h- 1 • At higher speeds, lower
percentages of the load capacity must be incurred; for VR and ZR tyres, which are
no longer made, these values were determined by vehicle and tyre manufacturers.
Tyre load capacity (%)
Top speed of car
(km h- 1)
210
220
230
240
250
260
270
280
290
300
Speed symbol
V
W
Y Tyres
100
97
94
91
100
100
100
100
95
90
85
100
100
100
100
100
100
100
95
90
85
The following should be noted for VR tyres:
• over 210 kIn h- 1 and up to 220 kIn h inclusive, the load may only be 90% of
the otherwise authorized value;
• over 220 kIn h- 1 the carrying capacity reduces by at least 5% per 10 km h- 1
speed increment.
2.2.5.3 Designation of light commercial vehicle tyres
Tyres for light commercial vehicles have a reinforced substructure compared
with those for passenger cars (Fig. 2.5), so they can take higher pressures, which
means they have a higher load capacity. The suffix 'C' followed by information
on the carcass strength (6, 8 or 10 PR) used to indicate suitability for use on light
commercial vehicles, or the word 'reinforced' simply appeared at the end of the
marking. The current marking (as for passenger cars) retains the speed symbol
as well as the load index which, behind the slash, gives the reduced load capacity on twin tyres (Fig. 3.4). Compared with the previous marking, the new
system is as follows:
Former
Current
185SR14
185 SR 14 reinforced
185 R 14 C 6 PR
185 R 14 C 8 PR
205/65 R 15 98 S (Fig. 2.15)
185 R 1490 S
185 R 1494 R
185 R 14 99/97 M
185 R 14 102/1 00 M
The 185 R 14 tyre is a passenger car size which is also fitted to light commercial vehicles.
1'_1.---'-----------~I r-
Tyres and wheels
101
2.2.5.4 Tyre dimensions
Figure 2.15 shows the important data for determining tyre size:
• size marking;
• authorized rims and measuring rim;
• tyre dimensions: width and outside diameter new and maXImum during
runnmg;
• static rolling radius (Fig. 2.11);
• rolling circumference (at 60 km h- I , Fig 2.16, see also Section 2.2.8);
• load capacity coefficient (load index LI, Fig. 2.13);
• tyre load capacity at 2.5 bar and up to 160 km h- I (see Section 2.2.6).
2.2.6
Tyre load capacities and inflation pressures
The authorized axle loads mV, f,max and mv,r.max (see Section 5.3.5), and the maximum speed V max of the vehicle, determine the minimum tyre pressure. However,
the required tyre pressure may be higher to achieve optimum vehicle handling
(see also Section 2.10.3.5 and Fig. 2.44).
2.2.6.1 Tyre load capacity designation
The load capacities indicated in the load index (item 6, Fig. 2.18) are the maximum loads per tyre permitted for all tyres up to the speed symbol 'H'. They are
valid up to speeds of 210 km h- I for tyres marked 'V' and up to 240 km h- 1 for
those marked 'R' 'W' or 'ZR'. For vehicles with a higher top speed, the load
capacity has to be reduced accordingly.
Consequently, for tyres with speed symbol 'V', at a maximum speed of 240
1
km h- the load capacity is only 91 % of the limit value (Fig. 2.14). Tyres designated 'W' on the sidewall are only authorized up to 85% at 270 km h- I . In both
cases the load capacity values between 210 km h- 1 ('V' tyre) and 240 km h- 1
('W' tyre) and the maximum speed must be determined by linear interpolation.
For higher speeds (ZR tyres), the interpolation applies to the 240-270 km h- 1
speed range. At higher speeds, the load capacity as well as the inflating pressure
will be agreed between the car and tyre manufacturers. However, this approval
does not necessarily apply to tyres which are specially produced for the US
market and which bear the additional marking 'P' (Fig. 2.17 and Section
2.2.5.2).
2.2.6.2 Tyre pressure determination
For tyres with speed symbols 'R' to 'V' and standard road tyres the minimum
pressures set out in the tables and corresponding with load capacities are valid
up to 160 km h- 1 (see Fig. 2.15 and Section 2.1.1).
Special operating conditions, the design of the vehicle or wheel suspension
and expected handling properties can all be reasons for higher pressure specification by the vehicle manufacturer.
Further, for speeds up to 210 km h- 1 the linear increase of basic pressure has
to be by 0.3 bar (i.e. by 0.1 bar per Av = 17 km h- 1; see also end of Section
2.84) and at speeds above 210 km h- 1 the tyre load capacity has to be reduced
~
Radial 65 series tyres, sizes, new and running dimensions, authorized rims and load capacity values (related to maximum
160 km h-1 and 2.5 bar); the necessary increase in pressures at higher speeds can be taken from Section 2.2.6. The tyre dimensions apply
to tyres of a normal and increased load capacity design (see Section 2.2.5.3) and to all speed symbols and the speed marking ZR.
Fig. 2.15
Manufacturer's measurements
Dimensions of new tyre
Width of
crosssection
Outer
diameter
157
532
157
558
170
544
170
570
177
558
Tyre size
Measuring rim
155/65 R 13
4.50 B
155/65 R 14
4lf2 J
165/65 R 13
5.00 B
165/65 R 14
5J
175/65 R 13
5.00 B
175/65 R 14
5J
x
13
177
584
175/65 R 15
5J
x
15
177
609
185/65 R 13
5.50 B
189
570
185/65 R 14
5lf2 J
189
596
x
x
x
13
14
x
13
14
x
x
x
13
14
14
----
Permissible
rims
according to
DIN 7817
and DIN 7824
Max.
width
4.00 B X 131
4.50 B x 13 1
5.00 B x 13 1
5.50 B x 131
4 J x 142
4lf2J x 142
5 J x 142
51/2J x 142
4.50 B x 131
5.00 B X 131
5.50 B x 131
6.00 B x 131.3
4lf2J x 142
5 J x 142
51hJ x 142
6 J x 14
5.00 B x 13 1
5.50 B x 131
6.00 B x 13 1.3
5 J X 142
51,12 J x 142
6 J x 14
5 J x 152
5lf2 J x 152
6 J x 15
5.50 B x 131
5.50 B x 131
6.00 B x 131.3
6% J x 13
5 J x 14
5lf2 J x 14
6 J x 14
6V2 J x 14
158
164
169
174
158
164
169
174
171
176
182
187
171
176
182
187
184
189
194
184
189
194
184
189
194
191
197
202
207
191
197
202
207
Max.
outer
diametet
Static
radius
:±2.0%
Circumference
+1.5%
-2.5%
Load
index (LI)
Wheel
load
capacity
540
244
1625
73
365
566
257
1700
74
375
533
248
1660
76
400
579
261
1740
78
425
567
254
1700
80
450
593
267
1780
82
475
618
279
1855
83
487
580
259
1740
84
500
606
272
1820
86
530
- ~ - - ~ · - - " - - d
• $iii
• t.
!f"~
185/65 R 15
5lJ2 J x 15
189
621
195/65 R 14
6 J x 14
201
610
5 J x 15
5Vz J x 15
6 J x 15
6Vz J x 15
5lJ2 J X 14
6 J X 14
61JzJx14
195/65 R 15
6 J X 15
201
635
205/65 R 14
6 J X 14
209
622
205/65 R 15
6 J X 15
209
647
215/65 R 15
61Jz J
X 15
221
661
215/65 R 16
6lJ2 J X 16
221
686
225/65 R 15
6% J X 15
228
673
7 J X 14
5lJ2 J X 15
6 J X 15
6lJ2 J X 15
7 J X 15
5lJ2 J X 14
6 J X 14
6lJ2 J X 14
7 J X 14
7 1h J X 14
51Jz J X 15
6 J X 15
6lJ2 J X 15
7 J X 15
71Jz J X 15
6 J X 15
61Jz J X 15
7 J X 15
7lJ2 J X 15
6 J X 16
6lJ2 J X 16
7 Jx 16
7lfz J X 16
6 J X 15
61Jz J X 15
7 J X 15
7lJ2 J X 15
8 J X 15
191
197
202
207
204
209
215
220
204
209
215
220
212
217
222
227
233
212
217
222
227
233
225
230
235
240
225
230
235
240
232
237
242
248
253
631
284
1895
88
560
620
277
1860
89
580
645
290
1935
91
615
633
282
1895
91
615
658
294
1975
946
670
672
300
2015
967
710
697
312
2090
98
750
685
304
2055
99
775
~nstead of wheel rims with the identification letter B, same-sized rims with the identification letter J may be used. For example 5¥z J X 13 instead of 5.50 B X 13. (See Section
2.3.2.)
2 Instead of wheel rims with the identification letter J, same-sized rims with the identification letter B may be used. For example 4.50 B X 14 instead of 4% J X 14.
3 The wheel rims without identification letters mentioned in the table are expected to be identified with DIN 7824 Part 1.
4 The outer diameter of wheels with M & S - tread can be up to 1 % bigger than the standard tread.
5 Maximum in kg at 2.5 bar.
6 Reinforced model, 750 kg at 3.0 bar (L1 98).
7 Reinforced model, 800 kg at 3.0 bar (L1 100).
1
104
The Automotive Chassis
Fig. 2.16 Factor kv , which expresses the speed dependence of the rolling circumference of passenger vehicle radial tyres above 60 km h- 1 as a percentage. The
permissible tolerances fi.kv have to be added (see Section 2.2.8), all taken from the
German WOK Guideline 107, page 1.
v (km h-')
Factor kv (%)
Deviation !:J.k v (%)
60
90
+0.1
±0.1
150
+0.4
±0.4
120
+0.2
±0.2
180
+0.7
±0.7
210
+ 1.1
±1.1
240
+1.6
±1.6
Fig. 2.17 ZR tyres manufactured
specially for the American market and
marked with a 'p' do not meet the
European standard and are therefore
not authorized here (photograph:
Dunlop factory).
in accordance with item 2.2.6.1. If the tyre load is lower than the maximum
load capacity, a lower additional safety pressure can be used in consultation
with the tyre manufacturer.
For tyres with the speed symbol 'W', the pressures in Fig. 2.13 apply up to
190 km- I . After this it has to be increased by 0.1 bar for every 10 km h- I up to
240 km h- I . For higher speeds, the load capacity must be reduced (see Section
2.2.6.1).
On vehicles, pressure should be tested on cold tyres, i.e. these must be
adjusted to the ambient temperature. If the tyre pressure is set in a warm area
in winter there will be an excessive pressure drop when the vehicle is taken
outside.
On M & S winter tyres it has long been recommended that inflation pressures
be increased by 0.2 bar compared with standard tyres. Newer brands of tyre no
longer require this adjustment.
2.2.6.3 Influence of wheel camber
Wheel camber angles Ew considerably influence tyre performance and service
life. The camber angle should therefore not exceed 4° even in full wheel jounce
condition. For angles above +2° (see Section 3.5.1), the loadability of the tyres
reduces at
EW
> 2° to 3° to 95%
Ew> 3° to 4° to 95%
Intermediate values have to be interpolated. Compensation can be achieved by
increasing the inflation pressure. The values are as follows:
¥
I
!
Tyres and wheels
Camber angle
Pressure increase
105
3°40' 4°
11.5% 14.1 %
Taking all the influences into account, such as top speed, wheel camber and axle
load, the minimum tyre pressure required can be calculated for each tyre category (size and speed symbol). Formulas are shown in the 'WdK 99' guidelines
from the Wirtschaftsverband der Deutschen Kautschukindustrie.
2.2.6.4
Tyre pressure limit values
Tyre pressure limit values should be adhered to. These values are
Q and T tyres
H to Wand ZR tyres
M & S tyres (Q and T tyres)
2.2.7
3.2 bar
3.5 bar
3.5 bar
Tyre sidewall markings
All tyres used in Europe should be marked in accordance with the ETRTO standards (see Section 2.1.1).
In the USA, Japan and Australia, additional markings are required to indicate
the design of the tyre and its characteristics. The characters must also bear the
import sizes - the reason why these can be found on all tyres manufactured in
Europe (Fig. 2.18).
2.2.8
Rolling circumference and driving speed
The driving speed is:
v ~ 0.006( I -
CR,dyn x
SX,w,a)
.
nM
.
(km/h)
(2.1b)
to x ta
This includes:
SX,W,a
CR,dyn
nM
to
ta
the
the
.the
the
the
absolute traction slip (Equation 2.4f)
dynamic rolling circumference in m (Equation 2.1d)
engine speed in rpm
ratio in the axle drive (differential)
ratio of the gear engaged (Equation 6.36)
The following can be assumed for slip
1st gear
2nd gear
3rd gear
0.08
0.065
0.05
SX,W,a:
4th gear
5th gear
0.035
0.02
106
The Automotive Chassis
la
Fig. 2.18 Explanation of the marking on the sidewall of a tyre manufactured by
Pneumatiques Kleber SA:
Legal and industry
standard markings on
the sidewalls of tyres
according to:
FMVSS and CIR 104
UTQG (USA)
CSA Standard (Canada)
ADR 23B (Australia)
ECE-R30 (Europe)
1 Manufacturer (brand)
1a Product name
2 Size marking
195 = nominal tyre
wideth in mm
60 = height-width
ratio (60%)
radial type
construction
14 rim diameter in
inches
3 Tubeless
_ _ _ .-
4 Trade code
5 Country of
manufacture
6 Load capacity index
(L1)
7 Maximum load
capacity for the USA
8 Tread: under the tread
are 6 plies carcass
rayon, 2 plies steel
belt, 2 plies nylon)
Sidewall: the substructure consists of 2 plies
rayon
9 Maximum tyre
pressure for the USA
10, 11, 12 USA:
manufacturer's
guarantee of
compliance with the
Uniform Tire Quality
,_,
Grade (UTOG) which
specifies: 10 tread
wear: relative life
expectancy compared
with US-specific
standard test values;
11 traction: A, B, C=
braking performance
on wet su rfaces 12
temperature
resistance: A, Bor C
=temperature
resistance at higher
test stand speeds; C
fulfills the legal
requirement in the
USA
13 E4 = tyre fulfils the
ECE R30 value
requirements
4 = country in which
--
approval was carried
out
(4 =The Netherlands)
14 identity number
according to ECE
R-30
15 DOT =tyre fulfils the
requirements
according to FMVSS
109 (DOT =
Department of
Transportation)
16 Manufacturer's code:
CU =factory
(Continental)
L2 = tyre size
AXCT = model
127 =date of
manufacture:
production week 12,
1987
-..1._
Tyres and wheels
107
According to DIN 75020 Part 5, the rolling circumference CR given in the tyre
tables relates to 60 km/h and operating pressure of 1.8 bar. At lower speeds it
goes down to CR,stat:
CR,stat
= rstat 27r
(2.1c)
The values for rstat are also given in the tables. At higher speeds, C R increases due
to the increasing centrifugal force. The dynamic rolling circumference CR,dyn at
speeds over 60 km h- l can be detennined using the speed factor k v • Figure 2.16
shows the details for kv as a percentage, increasing by increments of 30 km h- l .
Intennediate values must be interpolated. The circumference would then be:
CR,dyn
=
CR
(l + 0.01 X kv ) (mm)
The dynamic rolling radius can be calculated from
(2.1d)
CR,dyn
as
or, at speeds of more than 60 km h- l ,
(2.2)
Taking as an example the tyre 175/65 R 14 82 H at v = 200 km h- l (Fig. 2.15)
gIves
kvl80
=0.7% and k
v210
= 1.1 %
and interpolation gives:
kv200 = 0.007 + 0.0027 = 0.0097
k v200 = 0.97%
The rolling circumference CR taken from Fig. 2.15, according to Equation 2.1d,
gIves
CR,dyn200 = 1780 X (l + 0.0097)
= 1797 mm
and thus the dynamic radius in accordance with Equation 2.2 is:
rdyn60
=283 mm and rdyn200 =286 mm
The outside diameter (construction measure) is
ODT = 584 mm and thus ODT /2 = 292 mm
a value which shows the extent to which the tyre becomes upright when the
vehicle is being driven: rdyn is only 9 mm or 6 mm less than ODT/2. Chapter 3 of
Ref. [3] gives further details.
108
The Automotive Chassis
2.2.9
Influence of the tyre on the speedometer
The speedometer is designed to show slightly more than, and under no circumstances less than, the actual speed. Tyres influence the degree of advance,
whereby the following playa role:
•
•
•
•
the degree of wear
the tolerances of the rolling circumference
the profile design
associated slip.
The EC Council directive 75/443, in force since 1991, specifies an almost linear
advance ~v,
+ ~ v ::::; 0.1 X v + 4 (km h -I)
(2.2a)
On vehicles registered from 1991 onwards the values displayed may only be as
follows:
Actual speed (km h- I )
Max displayed value (km h- 1)
30
37
60
70
120
136
180
202
240
268
As Fig. 2.15 indicates, at 60 km h- I the rolling circumference CR has a tolerance
range of ~CR =+1.5% to -2.5%, and according to Fig. 2.16 with a speed factor
of k y, deviations of up to ~ky = + 1.6% are possible. When related to the dynamic
rolling circumference CR,dyn (Equation 2.1d), the following tolerance limits
(rounded to the nearest figure) may prevail and result in the displayed values
when only the minus tolerances are considered, and if the speedometer has the
maximum authorized advance:
Actual speed (km h -1)
Possible overall tolerance (%)
Max display value at minus
tolerance (km h -I)
60
+1.5
-2.5
72
120
+1.7
-2.7
140
180
+2.2
-3.2
208
240
+3.1
-4.1
279
The slip should be added directly to this, which in direct gear amounts to around
2% (see equations 2.1 band 2At), in other words
SX,W,a
= 0.02
If the manufacturer fully utilizes the advance specified in Equation 2.2a, it is
possible t?at although the speed?meter ind~cates ~40 km h- I , the vehicle is
only movIng at 120 km h- . ThIS occurs, In partIcular, when the tyres are
worn:
3 mm wear gives an advance of around 1%
Tyres and wheels
109
Fig. 2.19 Designs of Continental tyre. (Top) Summer tyre (tyre foot prints see Fig.
2.9) EcoContact EP (size 185/65 R14T) and Sport Contact (size 205/55 R16W).
(Below) Winter tyre WinterContact TS760 (size 185/65 R14T) and WinterContact
TS770 (size 235/60 R16H).
Tyres with an M & S winter profile can, however, have a 1% larger outside diameter so that the profile can be deeper (Fig. 2.15, note 5 and Fig. 2.19). They
would therefore reduce the degree by which the speedometer is advanced if the
tyres are not yet worn. The same applies where the positive tolerances given in
the above table are used. In this instance it is also possible that even a very
precise speedometer could display too low a speed.
The Automotive Chassis
110
2.2.10 Tyre profiles
The design of tyre profiles (Fig. 2.19) depends on the intended use, taking into
account the parameters of height-to-width ratio, construction and mixture and
design. The aquaplaning properties are improved by increasing the negative
proportion (light places in the tyre impression, Fig. 2.9). The shoulder region
with its transverse water-drainage grooves is particularly important for its properties in a lateral direction and the middle region with straight longitudinal
grooves is important for its properties in a longitudinal direction. An asymmetrical profile design ('sports' profile) is chosen for wide tyres, tread lugs in the
outside shoulder, which are subject to greater stress during cornering, can be
designed to be more rigid. By adjusting the correct balance between profile
rigidity and belt rigidity, it must be ensured that no conical forces are produced.
Profiled bands around the middle region increase noise reduction and improve
the steering response properties and, via the increase in circular rigidity, the
brake response properties.
Winter tyre profiles are improved, in terms of their force transmission properties in the wet, snow and ice, by a higher negative profile component, trans..;
verse grooves and a large number of sipes. Directional profiles (TS770) can be
used to increase water dispersal, the longitudinal force coefficient and selfcleaning by means of transverse grooves which run diagonally outwards. Noise
control is improved by variation in block length, sipes cut up to under the groove
base or ventilation grooves running around the tyre.
2.3
2.3.1
Wheels
Concepts
Tyres are differentiated according to the loads to be carried, the possible maximum speed of the vehicle, and whether a tubed or tubeless tyre is driven. In the
case of a tubeless tyre, the air-tightness of the rim is extremely important. The
wheel also plays a role as a 'styling element'. It must permit good brake ventilation and a secure connection to the hub flange (see Chapter 9, in Ref. [6]).
Figure 2.20 shows a passenger car rim fitted with a tubeless tyre.
2.3.2
Rims for passenger cars, light commercial vehicles and
trailers
For these types of vehicle only well-base rims are provided. The dimensions of
the smallest size, at 12" and 13" diameter and rim width up to 5.0", are contained
in the standard DIN 7824. The designation for a standard rim, suitable for the
145 R 13 tyre (Fig. 2.1) for example is:
DIN 7824 - drop base rim 4.00 B X 13
Tyres and wheels
111
Wid~h .of cross-section
Height of horn
r---t~~
Base deptht--_t-ei-f-''''--'
I
I
I
Base of rim
I
I
I
....
<D
.....
<D
E
co
"'0
E
cr:
Fig. 2.20
Series 66 wide tyre designs, mounted on a double hump rim with the
inflating valve shown in Fig. 2.6. The actual rim consists of the following:
• rim horns, which form the lateral seat for the tyre bead (the distance between the
two rims is the jaw width a);
• rim shoulders, the seat of the beads, generally inclined at 60 ± 10 to the centre
where the force transfer occurs around the circumference (Fig. 2.6);
• well base (also known as the inner base), designed as a drop tim to allow tyre
fitting, and mostly shifted to the outside (diagram: Hayes Lemmerz).
This type of rim used on passenger cars up to around 66 kW (90 PS) has only a
14 mm high rim flange and is identified with the letter B. The DIN standard can
generally be dropped.
In order to make it possible to fit bigger brakes (Fig. 2.10), more powerful
vehicles have larger diameter rims as follows:
• series production passenger cars: 14" to 17" rims
• sports cars: 16" to 18" rims.
The J rim flange applied here is used on rims from 13" upwards and is 17.3 mm
high. The rim base can (as shown in Fig. 2.1) be arranged symmetrically or
shifted outwards. The rim diameter, which is larger on the inside, creates more
space for the brake (Figs 1.8, 1.56, 2.10, 2.11 and 2.20). DIN 7817 specifies the
rim widths from 3i" to 8¥'. The definition of a normal asymmetrical rim with a
5" width, J rim flange and 14" diameter is:
DIN 7817 drop base rim - 5 J X 14
The symmetrical design is identified by the suffix'S' . The standards also contain
precise details on the design and position of the valve hole (see also Figs 2.20
and 2.24).
C tyres for light commercial vehicles require a broader shoulder (22 mm
112
The Automotive Chassis
Hump
Flat hump_______.
.
Special ledge
, __
-----~
._/~:=j::.:......._//
i -----/ I
!
I I
~/
//
N
I·
orma rim
Fig. 2.21
Standard rim and contours of the safety shoulders which can be used
on passenger cars and light commercial vehicles.
instead of 19.8 mm), which can be referred to by adding the letters LT (light
truck) at the end of the marking:
DIN 7817 drop base rim - 5~ J X 15 - LT
There is a preference worldwide for using tubeless radial tyres on passenger cars
and light commercial vehicles. Where these tyres are used, it is essential to have
a 'safety contour' at least on the outer rim shoulder. This stops air suddenly
escaping if the vehicle is cornering at reduced tyre pressure.
The three different contours mainly used are (Fig. 2.21):
Hump
(H, previously HI)
Flat-hump
(FH, previously FHA)
Contre Pente (CP)
Sheets 2 and 3 of DIN 7817 specify the dimensions of the first two designs. The
'hump' runs around the rim, which is rounded in H designs, whereas a flat hump
rim is simply given a small radius towards the tyre foot. The fact that the bead
sits firmly between the hump and rim flange is advantageous on both contours.
An arrangement on both the outside and inside also prevents the tyre feet sliding
into the drop bases in the event of all the air escaping from the tyre when travelling at low speeds, which could otherwise cause the vehicle to swerve. The
disadvantage of hump rims is that changing the tyre is difficult and requires
special tools.
A French design, intended only for passenger car rims, is the 'Contre Pente'
rim, known as the CP for short. This has an inclined shoulder towards the rim
base, which for rim widths between 4" and 6" is provided on one or both sides.
For years, the rims of most passenger cars have had safety shoulders on
both sides, either a double hump (Figs 2.20 and 2.24) or the sharp-edged flathump on the outside and the rounder design on the inside (Fig. 2.23). The
desired contour must be specified in the rim designation. Figure 2.22 gives the
possible combinations and abbreviations which must appear after the rim
diameter data. A complete designation for an asymmetrical rim would then be
as follows:
Tyres and wheels
Drop base rim DIN 7817 - 5
J
x
L
13
113
H2
double hump
rim diameter in inches
reference to drop base
for tyre-fitting
L..-
' - - - - - - - - - rim flange design
1....----------- rim width in inches
1....------------
number of standard
(only in Germany; can
be dropped)
Fig. 2.22
Marking of the various safety shoulders when used only on the outside
of the rim or on both the inside and outside. Normal means there is no safety contour
(Fig. 2.1). Further details are contained in standard DIN 7817.
I:
,.
Nature of safety shoulder
Denomination
Outside of rim
Inside of rim
One-sided hump
Double hump
One-sided flat hump
Double-sided flat hump
Combination hump
Hump
Hump
Flat hump
Flat hump
Flat hump
Normal
Hump
Normal
Flat hump
Hump
1
2
Identification letters
In place of the identification letters FH the identification letters FHA were also permitted.
In place of the identification letters CH the identification letters FH 1-H were also permitted.
Jaw width
Vent
hole
...
I
Q)
..c
E
I
:::l
c:
Q)
0
Q)
.:!:
.c
.c
0
..c
I
~
t.l
.::
u
Q)
0
::c
lSl
Fig. 2.23
Q)
I
0
Q)
"'0
~
~
lSl
I
I Depth of
I
eimpression
01
c:
ctl
::;::
..c
:::l
::c
lSl
E
i:i:
lSl
The sheet metal disc-type
wheel used in series production vehicles
consists of a rim and disc. To avoid fatigue
fractures, the wheel hub flange diameter
should be greater than the dish contact
surface. Wheel offset e (depth of impression) and kingpin offset at ground r are
directly correlated. A change in e can lead to
an increase or a reduction in r
The dome-shaped dish leading to the
negative kingpin offset at ground is clearly
shown (diagram: Hayes Lemmerz).
IT
IT •
\
\
\
114
The Automotive Chassis
I
I
1~~/'l-.1~57E9 (:8:~)
-
Fig. 2.24 Hayes Lemmerz alloy wheel for the Audi
80, made of the aluminium alloy GK-AI Si 7 Mg wa.
The wheel has a double-hump rim (H2) and middle
centring and is fixed with four spherical collar bolts.
The different wall thicknesses, which are important
for the strength, the shape of the bolt hole, the
different shape of the drop-rim and the position of
the valve hole are clearly shown. At high speeds the
. snap-fit valve (Fig. 2.6) is pressed outwards by the
centrifugal force and supported below the rim base.
I
1""-,
I
I
2.3.3
Wheels for passenger cars, light commercial vehicles and
trailers
Most passenger cars and light commercial vehicles are fitted with sheet metal
disc wheels, because these are economic, have high stress limits and can be readily serviced. They consist of a rim and a welded-on wheel disc (also known as
an attachment face, Fig. 2.23). Cold-formable sheet metal, or band steel with a
high elongation, can be used (e.g. RSt37-2 to European standard 20) depending
on the wheel load, in thicknesses from 1.8 to 4.0 mm for the rim and 3.0 to 6.5
mm for the attachment faces.
There is a direct correlation between wheel offset e and 'kingpin offset at
ground' ra; the more positive ra, the smaller can be the depth dimension e.
However, a negative kingpin offset -ra, especially on front-wheel drive, results
in a significant depth e and severe bowing of the attachment faces (as can be seen
in Figs 2.8, 2.23, 2.25 and 3.102 and Section 7.3 in Ref. [6]).
The wheel disc can be perforated to save weight and achieve better brake
cooling. Despite the fact that they cost almost four times as much as sheet metal
designs, alloy wheels are becoming increasingly popular (Figs 1.56 and 2.24).
Their advantages are:
•
•
•
•
lower masses;
extensive styling options; and therefore
better appearance;
processing allows precise centring and limitation of the radial and lateral
ronout (see Section 2.5);
• good heat transfer for brake-cooling (see Chapter 9 in Ref. [6]).
Tyres and wheels
r--r--~~'--~~~~---..fi!!!"
r-:----~~':"-:I"~~~-~150907
,,"--~t"""-\:~~-\-~-6Jx15H2
.l'---'l-:~~+-tt--ET37
TJ-----t-->O";~-f---H--
am
""'----"-------:~~---,f-,f-,H--Germany
\:::::=F==---::;:::~....-'=;'L....:.,L--#----1 0.98
115
"Yheel manufacturer's
sign
Wheel manufacturer's
part number
Wheel size and hump
type
Depth of impression
Car manufacturer's
sign
Car manufacturer's
part number
Country of
manufacture
Date of manufa cture
Fig. 2.25 Double-hump sheet metal disc-type wheel with openings for cooling
the brakes. Also pictured is the stamp in accordance with the German standard DIN
7829, indicating manufacturer code, rim type and date of manufacture (week or
month and year).
Also specified is the wheel offset (ET37) and, in the case of special wheels with
their own ABE (General operating approval), the allocation number of the KBA, the
German Federal Vehicle Licensing Office. If there is not much space the stamp may
be found on the inside of the dish. The date of manufacture also points to when the
vehicle was manufactured (diagram: Hayes Lemmerz).
Often incorrectly called aluminium rims, alloy wheels are mainly manufactured
using low-pressure chill casting, occasionally forging or aluminium plate, and
generally consist of aluminium alloys with a silicon content (which are sometimes heat hardenable), such as GK-Al Si 11 Mg, GK-AI Si 7 Mg T (T =
tempered after casting) etc.
Regardless of the material, the wheels must be stamped with a marking
containing the most important data (Fig. 2.25).
2.3.4 Wheel mountings
Many strength requirements are placed on the wheel disc sitting in the rim (or the
wheel spider on alloy wheels); it has to absorb vertical, lateral and longitudinal
forces coming from the road and transfer them to the wheel hub via the fixing bolts.
116
The Automotive Chassis
Fig. 2.26
Depression design with special springing
characteristics on a passenger car sheet metal disc-type wheel.
The wheel can be centred using the fixing bolts or by fitting
into the toleranced hole (Fig. 2.24).
The important thing here is that the contact area of the attachment faces, known
as the 'mirror', should sit evenly and, for passenger cars, that the hub flange should
have a slightly larger diameter (Fig. 2.23), otherwise it is possible that the outer
edge of the hub will dig into the contact area, with a loss of torque on the bolts.
The notch effect can also cause a fatigue fracture leading to an accident.
The number of holes and their circle diameter are important in this context.
This should be as large as possible to introduce less force into the flange and
fixing bolts. If the brake discs are placed onto the wheel hub from the outside which is easier from a fitting point of view - it is difficult to create a hole larger
than 100 mm on 13" wheels, and using a 14" or 15" wheel should make for the
best compromise (Figs 1.8, 1.41, 1.44 and 2.10). German standard DIN 74361
contains further details.
The brake disc can also be fixed to the wheel hub from the inside (Fig. 1.38).
However, the disadvantage of this is that the hub has to be removed before the
disc can be changed. This is easy on the non-driven axle, but time-consuming on
the driven axle (see Section 2.5 in Ref. 2 and Chapter 9 in Ref. 6). This brief look
shows that even the brakes playa role in the problems of fixing wheels.
Nowadays, wheels are almost always fixed with four or five metric M12 X 1.5
or M14 X 1.5 DIN 74361 spherical collar bolts. The high friction between the
spherical collar and the stud hole prevents the bolts from coming loose while the
vehicle is in motion. For this reason, some car manufacturers keep the contact
surface free of paint. On sheet metal disc wheels with attachment faces up to 6.5
mm thick, the spring action of the hole surround (Fig.. 2.26) is an additional safety
feature, which also reduces the stress on the wheel bolts as a result of its design
elasticity. Sheet metal rings are often inserted in the alloy wheels to withstand high
stresses underneath the bolt head.
Generally, the spherical collar nuts also do the job of centring the wheels on the
hub. Hub centring has become increasingly popular because of a possible hub or
radial run-out and the associated steering vibrations. A toleranced collar placed on
the hub fits into the dimensioned hole which can be seen in Fig. 2.24.
2.4
Springing behaviour
The static tyre spring rate CT - frequently also known as spring stiffness or (in the
case of a linear curve) spring constant - is the quotient of the change in vertical
Tyres and wheels
117
force Ii.Fz,w in Newtons and the resultant change Ii.sT - the compression in mm
within a load capacity range corresponding to the tyre pressure PT (Fig. 2.27; see
also Section 2.2.5.4):
(2.3)
The parameter CT forms part of the vibration and damping calculation and has a
critical influence on the wheel load impact factor (see Section 5.2 in Ref. [3],
Section 4.1). The stiffer the tyre, the higher the damping must be set and the
greater the stress experienced by the chassis components. The following parameters influence the spring rate:
•
•
•
•
•
•
vertical force
tyre pressure
driving speed
slip angle
camber angle
tim width
6
kN
5
z
--
oX
i
Q.)
...0u
-
,
II
LA.,
<I
3
co
u
''2 2
~
L-_ _---l
O ~
o
10
20
Static compression Sf
~s=6mm
---L.
---l
30
mm
'0
..
Fig. 2.27 The static tyre spring rate Cr is the quotient of the force and the deflection travel shown on the radial tyre 175/70 R 13 80 S at Pr = 1.8 bar, 2.1 bar and 2.4
bar; the example shown gives:
CT
=
t:,.Fz.w
t:,.ST
1000 N
=-6mm
= 167N/mm
----'------------------------,--------
118
The Automotive Chassis
I---t---t---t--+--+--+--+---i--t-----l N/mm
200
I
t
'-
Q)
c.
E
Q)
....
.;:
c.
00
-10° _8° _6° _4° _2°
0°
+2° +4° +6° +8°
Slip angle
•
Fig. 2.28 Tyre springing rate as a function of slip angle and road speed, measured
on a radial tyre 185/70 R 13 86 S at Pr = 2.1 bar. Speed increases the springing rate
as the belt stands up due to the centrifugal force. However, the slip angle makes it
softer because the belt is pushed away to the side and the shoulders take over part
of the springing effect.
•
•
•
•
height-to-width ratio
construction of tyre (bias angle, material)
tyre wear and tear
wheel load frequency.
As can be seen in Fig. 2.27, apart from in the low load range, the spring rate is
independent of the load. A linear increase can be seen as the speed increases
(Figs 2.16 and 2.28; see also Equation 5.5a), which persists even when the tyre
pressure changes.
During cornering, the force Fy,w (Fig. 3.119) shifts the belt in a lateral direction, and so it tips relative to the wheel plane. This leads to a highly asymmetrical distribution of pressure and (as can be seen from Fig. 2.28) to a reduction in
the spring rate as the slip angles increase.
2.5
Non-uniformity
The tyre consists of a number of individual parts, e.g. carcass layers, belt layers,
nmning tread, sidewall stock and inner lining, which - put together on a tyre
Tyres and wheels
119
rolling mach~n~ - g~ve t~e tyre blank ~Fig. 2.5). In the area where it is put
together, varIatIOns III thIckness and stIffness occur, which can lead to nonuniformity.
Owing to the irregularities caused during manufacture, the following occur
around the circumference and width of the tyre:
• thickness variations
• mass variations
• stiffness Variations.
These cause various effects when the tyre rolls:
•
•
•
•
•
•
•
•
imbalance
radial tyre runout
lateral tyre runout
variation in vertical and/or radial force
lateral force variations
longitudinal force variation
ply steer (angle) force
conicity force.
Imbalance U occurs when an uneven distribution of mass and the resulting
centrifugal forces are not equalized. Because the uneven distribution occurs not
only around the circumference, but also laterally, we have to differentiate
between static and dynamic imbalance (Fig. 2.29). This is calculated in size and
direction on balancing machines and eliminated with balancing weights on the
rim bead outside and inside the wheel.
Radial and lateral runout are the geometrical variations in the running tread
and the sidewalls. They are measured with distance sensors on a tyre-uniformity
machine. The German WdK Guideline 109 contains full details.
The most important of the three force variations is the radial force variation. For
greater clarity, it is shown on the model in Fig. 2.30, where the tyre consists of
different springs whose rates fluctuate between CI and Cg. The resulting phenom-
u
I
I
I
I
- + -+
Fig. 2.29 Different forms of
imbalance U: (a) static, (b) dynamic.
The imbalance is equalized in (c).
----
(a) Static
imbalance
U2
U2
(b) Dynamic
imbalance
(c) Moment of
imbalance
U1 '# U2
U1 =U2
-----------------------r------
120
The Automotive Chassis
Fig. 2.30 The tyre spring rate can
fluctuate depending on the manufacturing
process, shown as Cl to Ca.
ena should be indicated on the 175 R 14 88 S steel radial tyre, loaded at Fz,w =
4.5 kN and pressurized to PT = 1.9 bar. Assuming this had a mean spring rate CT
= 186 N m- 1, which fluctuates by +5%, the upper limit would be CT,max = 195 N
mm- I and the lower limit would be CT,min = 177 N mm- I • Under vertical force Fz,w
=4.5 kN =4500 N the tyre would, according to Equation 2.3a, have as its smallest jounce travel
Fz,w
ST,min
=--
=
4500
CT,max
195 '
ST,min
= 23.1 mm
(2.3a)
and
ST,max
= 25.4 mm
as the greatest travel. The difference is
LlsT =
ST,max -
ST,min
= 2.3 mm
This difference in the dynamic rolling radius of LlST = 2.3 mm would cause variations in vertical force LlFz,w, which nevertheless is still smaller than the friction
in the wheel suspension bearings. At a speed of perhaps 120 km/h and travelling
on a completely smooth road surface, this would nevertheless lead to vibration
that would be particularly noticeable on the front axle.
The vehicle used as an example should have a body spring rate of Cr = 15
N/mm per front axle side. The travel LlsT would then give a vertical force difference, in accordance with Equation 5.0a, of:
LlFz,w,r
= Cr LlST = 15
X 2.3;
LlFz,w,f = 34.5 N
The friction per front axle side is, however, not generally below
Frr = ±100 N (Fig. 5.6)
lyres and wheels
121
so it can only be overcome if greater variations in vertical force occur as a result
of non-unifomlity in the road surface. The more softly sprung the vehicle the
more the variations in radial force in the tyre make themselves felt (see Se~tion
5.1.2).
The ~ateral forc~ variations of the tyre influence the straight-running ability of
the vehIcle. Even with a tyre that is running straight, i.e. where the slip angle is
zero, lateral forces occur, which also depend on the direction of travel (see
.
Chapter 11 in Ref. [4]).
The variations in longitudinal force that occur must be absorbed on the chassis side by the lubber bearings described in Section 3.6.5.2.
The ply steer force dependent on the rolling angle results from the belt design
~ecause of the lateral drift of the tyre contact area as a consequence of flat spottmg. In contrast, the conicity force, resulting from a change in diameter across
the width of the tyre, is not dependent on the rolling angle. Both forces disturb
the straight running of the vehicle (see Chapter 11 in Ref. [4]).
2.6
2.6.1
Rolling resistance
Rolling resistance in straight-line driving
Rolling resistance is a result of energy loss in the tyre, which can be traced back
to the deformation of the area of tyre contact and the damping properties of the
rubber. These lead to the transformation of mechanical into thermal energy,
contributing to warming of the tyre.
Sixty to 70% of the rolling resistance is generated in the running tread (Fig.
2.5) and its level is mainly dependent on the rubber mixture. Low damping
running tread mixtures improve the rolling resistance, but at the same time
reduce the coefficient of friction on a wet road surface. It can be said that the
ratio is approximately 1: 1, which means a 10% reduction in the rolling resistance leads to a 10% longer braking distance on a wet road surface. The use of
new combinations of materials in the running tread (use of silica) has led to
partial reduction of the conflict between these aims.
Rolling resistance is either expressed as a rolling resistance force F R or as the
rolling resistance factor kR - also known as the coefficient of rolling resistance:
(2.4)
The factor k R is important for calculating the driving performance diagram and
depends on the vertical force Fz,w and the tyre pressure PT. Figure 2.31 shows the
theoretical kR curve of tyres of different speed classes as a function of the speed.
Although the coefficient of rolling friction of the T tyre increases disproportionally from around 120 km h- I , this increase does not occur in H and V tyres until
160 to 170 km h- I . The reason for this behaviour is the shape of the rolling hump
that occurs at different speeds depending on the speed class, and is dependent on
the stiffness of the belt, in other words on its design. The lower kR values for the
T tyres result from the usually poorer wet skidding behaviour of this speed class.
The Automotive Chassis
122
Rolling resistance
1.6 --,----..,...--...,---r-----,----r------,---,---..,...---,.---rl----,
I
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-- - ,-. - - -- ---
_____ J,__ n __ :_ -----Ji-n--H
en
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--- - - i - - - - -i --- - - - i -- -- - i - - -- -:- - -- -i - - - - - ~ - - - --- i- - -- -:- ._-/-; - --- -
t) 1.2
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-----j-------f-----i------:---- -;----- -1175/80 R 1488 H L-~--l+
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:
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+----l...:.:::::=.:.:....:.:...:::.~J_---i---+---+---f----+----+---+-----l
o
20
40
60
80
100
120
140
160
180
200
220
Speed in km/h
Fig. 2.31
Rolling resistance coefficients kR,a, average values of radial tyres as a
function of the speed, measured on a drum test rig. Tyres authorized up to 210 km h-1
have a lower rolling resistance below 160 km h-1 (than the V and W designs) whilst
the value rises sharply above this speed (measurements: Continental),
Asphalted roads cause kR,a to increase by around 20% as kR and rough concrete to
at least 30%. The ratios iR are then 1.2 or 1,3 to 1.4 and the actual value of kR is:
(2.4a)
The difference is due to the different design emphases during development of
the tyres. The design priorities for H, V and W tyres are high-speed road holding and good wet skidding and aquaplaning behaviour, whereas T tyres are
designed more for economy, i.e. lower rolling resistance (which plays an important role at lower speeds and influences urban driving fuel consumption, Fig.
2.32) and long service life.
2.6.2
Rolling resistance during cornering
Rolling resistance can change dramatically during cornering; its value depends
on the speed and the rolling radius R, in other words on J.LY,W (see Equations 2.9
and 2.11 and Fig. 2.43) and (Xforr. The rolling resistance kR,co, which is included
in some calculations (see Equation 3.35), comprises the coefficient kR for
straight running and the increase f1k R :
(2.4b)
The following data can provide an example:
Tyres and wheels
123
Resistances (Golf)
100,--%
Ac~eleration
_
resistance
80
I
I Rolling resistance
o
60..--
Air resistance
40
20
o
L--~::--_....l-..-...L..::-:-:----}L---l-:-::-::-:--+
90 km- 1
constant
City
traffic
..J
120 km- 1
constant
Fig. 2.32
In town and when the vehicle is travelling at low speeds on rural roads,
f~el consumption is determined up to 40% by the rolling resistance, whereas at
higher spe~ds the air drag is the determining factor see Section 2.1 and Section 2.2
in Ref. [3]). The figure shows a study carried out by VW on the Golf.
Front axle force FZ,V,f
Tyres 155 R 13 78 S
= 7kN;
J.LY,W
= 0.7 (asphalted road)
PT = 1.8 bar, v ~ 120 Ian h
In accordance with Equation 2.11 related to one wheel:
Fy,w,f
FY,W,f
= p.,Y,W FZ,W,f
= 2.45 kN
=
J.LY,W F Z
,v,r/2
= 0.7
X 3.5 kN
The slip angle read off at Fy,w,f in Fig. 2.44 is 4° and corresponds to the values
in Fig. 2.43.
However, the dynamic wheel load transfer seen in Fig. 1.5 plays a role during
cornering, leading to a greater slip angle on the wheel on the outside of the curve
(and thus also on the inner wheel), than resulted from test rig measurements. On
'82' series tyres, ex is about 5°, in accordance with Fig. 2.38:
ex : : : 7
J.LY,W
(2.4c)
With sin 5° in accordance with Equation 2.4b there is an increase of
flk R
::::::
0.7 X 0.087 = 0.061
Assuming a value of kR,o = 0.012, In accordance with Equation 2.4a, on
asphalted road
,----,-------------
The Automotive Chassis
124
kR
= i R kR,o = 1.2
x 0.010
= 0.012
and therefore the rolling resistance during cornering is
kR,co
= 0..012 + 0.061 = 0.073
In the case of the understeering vehicles (Fig. 2.41) kR,co increases as a result of
the additional steering input and - if the wheels are driven - f.Lrsl should be
inserted for J.LY,W (see Equation 2.18); the slip angle increases further. '65 Series'
tyres, on the other hand, require a smaller steering input and thus make the vehicle easier to handle:
ex. = 3 X
2.6.3
J.LY,W
(2.4d)
Other influencing variables
The rolling resistance increases in certain situations:
• in the case of a large negative or positive camber (the influence can be ignored
up to +2°);
• due to a change to track width (Fig. 3.6);
• in the case of deviations in zero toe-in around 1% per 3 = 10' or v = 1 mm;
• on uneven ground.
In general it can be said that the ratio i R (see Fig. 2.31) will take the following
values:
•
•
•
•
around 1.5 on cobbles
around 3 on potholed roads
around 4 on compacted sand
up to 20 on loose sand.
2.7
2.7.1
Rolling force coefficients and sliding
friction
Slip
If a tyre transfers drive or braking forces, a relative movement occurs between
the road and tyre, i.e. the rolling speed of the wheel is greater or less than the
vehicle speed (see Equation 2.1b). The ratio of the two speeds goes almost to 00
when the wheel is spinning, and is 0 when it locks. Slip is usually given as a
percentage. The following equation applies during braking:
SX,W,b
vehicle speed - circumferential speed of wheel
= ----------------vehicle speed
...
Tyres and wheels
=
SX.W.b
v - Vw
v
X 100 (%)
125
(2.4e)
Drive slip is governed by:
=
SX.W,a
Vw -
V
X
100 (%)
(2.41)
Vw
The different expressions have the advantage that, in both cases where the wheel
is spinning or locked, the value is 100% and is positive.
Further details can be found in Section 2.2.8, in Ref. 6 (Section 1.2), Ref. 7
(Chapter 1) and in Ref. 9 (Section 2.2).
2.7.2
Friction coefficients and factors
The higher the braking force or traction to be transmitted, the greater the slip
becomes. Depending on the road condition, the transferable longitudinal force
reaches its highest value between 10% and 30% slip and then reduces until the
wheel locks (100% slip). The quotient from longitudinal force Fx and vertical force
Fz.w is the coefficient of friction, also known as the circumferential force coefficient
j..Lx,w
(2.5)
= FX,wlF z,W
when it relates to the maximum value, and the coefficient of sliding friction,
also called sliding friction factor
j..LX,W,lo
(2.5a)
= F X,wiF z,W
when it is the minimal value (100% slip) (Fig. 2.33). F x is designated
during braking and FX,W,a during traction.
In all cases /.J..x,W is greater than j..LX,W.lo; in general it can be said that
,
1
Fig. 2.33
Coefficient of
friction p,x.w of a summer tyre
with 80 to 90% deep profile,
measured at around 60 km/h
and shown in
relation to the slip on road
surfaces in differEmt
conditions (see also Fig.
1.64). Wide tyres in the '65
series' and below have the
greatest friction at around
10% slip, which is important
for the ABS function (see
Chapter 1 in Ref. ,[7]).
r'8
c 0.6
o
f
/'
II/
u
C 0.2
Q)
'(:5
~
Q)
o
u
0
~ I'--
:--
II
oJ'
/
0.4
o
-
r- ""'-
-- --
J
',p
.~
I.-
FX,W,b
Dry asphalt
Wet asphalt
Loose gravel
~
If
il
Loose snow
'I ""
'"
Ice
20
40
60
Slip
%
100
."
The Automotive Chassis
126
on a dry road J-Lx.w ~ 1.2
on a wet road J-Lx.w
2.7.3
~
1.3
J-LX.W.lo
(2.6)
J-LX.W.lo
(2.6a)
Road influences
2.7.3.1
Dry and wet roads
On a dry road, the coefficient of friction is relatively independent of the speed
(Fig. 2.34), but a slight increase can be determined below 20 km/h. The reason
lies in the transition from dynamic to static rolling radius (see the example in
Section 2.2.5.4) and is therefore linked to an increasing area of tyre contact. At
speeds a little over zero, on a rough surface, a toothing cogging effect can occur,
which causes a further increase in the coefficient of friction, then:
J-Lx,w ;::: 1.3
(2.6b)
When the road is wet, the coefficient of friction reduces, but is still independent
of the speed. This situation changes as the amount of water increases and also
with shallower profile depth. The water can no longer be moved out of the
profile grooves and the JL value falls as speed increases.
2.7.3.2 A<luaplaning
The higher the water level, the greater the risk of aquaplaning. Three principal
factors influence when this occurs:
• road
• tyres
• speed.
1.0
Dry
t--...
0.8
Damp
r 1'::
c
0.6
f"........
0
'';:;
(,)
.;::
'+-
CJ)
c
"'-....
I
~wet
I'--........
0.4
r--...-
"0
lJl
-
'+-
0
0.2
c
OJ
'(3
~
OJ
0
U
0
Fig. 2.34
20
40
Speed
60
km h- 1
...
100
Dependency of the
coefficient of sliding friction P-X,W,lo
on speed on different road
conditions.
Tyres and wheels
127
Fig. 2.35
Coefficients of
friction J.l-x.w of a summer tyre
with an 8 mm deep profile
dependent on speed at different water levels. Hardly any
influence can be detected
under 60 km h-1 ; at higher
speeds and 3 mm water
depth, the curve shows a
lowering of fJ-X.W which
indicates the aquaplaning
effect.
I
0.8 r---t---t----1--Water level (mm)
t
~.
~~=---+--+-_~I 0.2
0.7
0.5
0.6 t---+--i"r--~--+-~ 1.0
::1.
c:
o
'E
-
0.5 t---t-----j--\-+--~---l
'i:
'0
....
c:
0.4 r--+-----ir---f\---+--~
2.0
Q)
'0
lE
Q)
o
0.3 -r---t----Ir----t-4--4----j
U
0.2 t---t-----j'----+---P~--l
0.1 t - - - + - - - - j - - + - - - + - - - l
60
80
100
km h-1
Speed
140
~
With regard to the road, the water level is the critical factor (Fig. 2.35). As the
level rises, there is a disproportionate increase in the tendency towards aquaplaning. When the level is low, the road surface continues to playa role because
the coarseness of the surface absorbs a large part of the volume of water and
carries it to the edge of the road. Following rainfall, the water levels on roads are
generally up to 2 mm; greater depths can also be found where it has been raining for a long time, during storms or in puddles.
On the tyre, the tread depth has the greatest influence (Fig. 2.47). There can
be up to a 25 km h- 1 difference in speed between a full tread and the legal minimum tread depth of 1.4 mm. High tyre pressure and low running surface radius
r (Fig. 2.5) lead to the area of contact becoming narrower, giving the advantage
of improved aquaplaning behaviour as the distribution of ground pressure
becomes more even (Fig. 2.9). Lower tyre pressure and contours with larger radii
make aquaplaning more likely; this also applies to wider tyres (Fig. 2.19) particularly when tread depths are low. However, the greatest influence by far is the
speed, especially when the water level increases and tread depths are low. This
is why reducing speed is the best way to lessen the risk of aquaplaning, and is a
decision drivers can make for themselves.
2.7.3.3 Snow and ice
Similar to aquaplaning, low coefficients of friction occur on icy roads, although
these are highly dependent on the temperature of the ice. At close to QOC, special
128
The Automotive Chassis
0.7,-------,.-----r------,.------,...----------.
Speed
---+-----1 10
1---1--1-~::;t;~==__=:=t==:::::=::::::120
1
~---4----140
3:
km h-1
~
c
o
'~
';::
o
.....
c
Q)
u
~
Q)
o
U
0.1 p------t-----+------t-----I----~
o'---
-l-
-5
I-.-
-10
L.--_ _- - - l
-l-
-20°C -25
-15
Ice temperature
~
Fig. 2.36
Influence of ice temperature and car speed on the coefficient of friction
/Lx.w of an 82 series winter tyre; the extremely low values at oDe can be seen clearly,
conditions occur; compression of the surface can lead to the formation of water
which has a lubricating effect and reduces the coefficient of friction to }.Lx,w ~
0.08 (Fig. 2.36). At -25°C, a temperature that is by no means rare in the Nordic
countries, values of around }.Lx,w = 0.6 can be reached. At low temperatures,
coefficients of friction and sliding friction are further apart:
}.Lx,w ~ 2
2.8
2.8.1
(2.7)
}.LX,W,lo
Lateral force and friction coefficients
Lateral forces, slip angle and coefficient of friction
Lateral forces on a rolling tyre can be caused by the tyre rolling diagonal to the
direction of travel (so-called slip), the tendency of a tyre to move from its position vertical to the road, camber or conical effects. The build-up of lateral forces
as a result of slip will be discussed next.
If a disturbing force Fc,v acts at the centre of gravity of the vehicle (e.g. a wind
or side negative lift force), lateral wheel forces Fy,w,f,o; FY,W,f,i, Fy,w,r,o and FY,W,r,i are
needed to balance the forces (Fig. 2.37). To build up these forces, the vehicle
must alter its direction of travel about the angle a, the slip angle. The size of the
slip angle depends on the force transmission properties of the tyre and the
disturbing force (Fig. 2.38).
,
-------------------.2--
Tyres and wheels
129
When cOlmering, the interference force should be equal to the centrifugal
force Fe,v, which results from the speed v in rnIs and the radius of the bend R in
m, on which the vehicle centre of gravity V (Fig. 2.29a) moves. With the total
weight mV,1 of the vehicle the equation is:
(2.8)
The centrifugal or disturbance force is just as large as the lateral forces on the
wheels (Fig. 2.37):
F y,v
= FY,W,f,o
+ FY,W,f,i + FY,W,r,o + F Y,W,r,i
= '2..Fy,w
(2.8a)
and
'2..Fy,w =:
JLY,W
X '2..Fz,w =
JLY,W
X FZ,V,1
Together the two equations give
JLY,W
FZ,v'1
=
JLY,W
X
mV,1
g =mV,1 X Q y
(2.9)
,Direction
Original direction
New direction
t
F..V,W,f,1
Fig. 2.37
Tyn3s are only able to
transfer a lateral force Fr. vacting on the
vehicle if they are rolling at an angle to
the vehicle. Regardless of whether
these are Fv. v or the centrifugal force
Fc,v during cornering, the lateral forces
Fr. w should be rElgarded as being
perpendicular to the wheel centre
plane.
Fig. 2.38 The higher the lateral force
Fy, w, the greater the tyre slip angle a.
,---------------------'--------------
130
The Automotive Chassis
Fig. 2.39 Increasing lateral forces Fv.w during cornering
caused by the centrifugal force Fc,v leads to increasing slip
angles a.
and
jLY,W
= g/ay
The coefficient of friction jLY,W is not dependent on the radius of the curve and
driving speed and is therefore more suitable for calculating cornering behaviour
(see also Equation 6.13a).
The faster the vehicle negotiates a bend, the higher the coefficient of friction
used and the greater the slip angles (Fig. 2.39).
2.8.2
Self-steering properties of vehicles
The self-steering properties of a vehicle describe the lateral force and hence
slip angle ratios produced during steady-state cornering (radius and driving
speed constant; no external disturbances). In the case of an understeering vehicle, a larger slip angle is required on the front axle than at the rear axle (ar >
a r , Fig. 2.41). During cornering with an increase in lateral acceleration, the
driver must force the vehicle into the bend by increasing the steering angle (see
Fig. 5.2). If the necessary slip angles on the front and rear axles are the same
(ar = a r , Fig. 2.40), one speaks of neutral handling characteristics. Over-steering behaviour is present if the tail of the vehicle moves outwards during
cornering and the slip angle on the rear axle is greater than on the front axle
(ar < a r , Fig. 2.42). The driver must respond to this by reducing the steering
angle.
As understeering behaviour is consistent with the expectations and experience
of the driver, it is this which needs to be aimed for. In normal driving conditions
Tyres and wheels
M
Fig. 2.40 ~f, during cornering, af - ar,
the handling of a vehicle can be
described as neutral.
131
M
Fig. 2.41
If there is a greater slip
angle at on the front wheels than ar on
the rear, the vehicle understeers.
Fig. 2.42
If there is a greater slip angle
on the rear wheels than on the front (at),
the vehicle oversteers. The positive angle
describes the angle between the vehicle.
longitudinal axis and its speed at the
centre of gravity.
ar
M
(anti-skid roadway, lateral acceleration of less than 6 m!s), all vehicles, therefore,
are now designed to understeer. With increasing lateral acceleration, the understeering behaviour should be as linear as possible and then, also as a warning to
the driver that the stability limit is about to be reached, increase progressively. If
the handling characteristics change to oversteer at the stability limit, for instance
with very high acceleration, this is an unpredictable driving situation which the
untrained driver can only control with difficulty. For active riding safety, the
predictability of self-steering properties in all kinds of conditions (vehicle loading, the distribution of driving torque in four-wheel drive vehicles, different
coefficients of friction, acceleration or braking procedures, changes in tyre pressure, etc.) is of paramount importance.
For a simplified representation of the relationships described, the so-called
single-track model is used, in which the wheels of the vehicle are drawn together
in the middle of the vehicle, without taking into account the height of the centre
of gravity (flat model).
Since in greater bend radii the average steering angle Om is less than 50, it can
be assumed that the sine and radius values of the angle are equal, and the angles
80 and Oi correspond to this (Fig. 3.91 and Equation 3.17):
-
132
The Automotive Chassis
Using Equation 3.12 it is now possible to determine the relationship between
steering angle, turning circle diameter Ds (Figs 1.69 and 3.89) and slip angles at
a constant cornering speed:
2 X I
(2.10)
Ds
The kingpin offset at ground r a is so negligable in comparison to D s that it can
be ignored.
2.8.3
Coefficients of friction and slip
To determine the cornering behaviour, the chassis engineer needs the lateral
forces (or the coefficient of friction) based on the slip angle and the parameters:
•
•
•
•
vertical force (or wheel load) in the centre of tyre contact
tyre pressure
wheel camber
tyre type.
The measurements are generally taken on test rigs, up to slip angles of a = 10°.
The drum surface with its friction values of j.Lo = 0.8-0.9 sets limits here, and
larger angles hardly give increasing lateral coefficients of friction:
(2.11)
Conditions on the road are very different from those on the test rig; the type of
road surface and its condition playa role here. As can be seen in Fig. 2.43, the
coefficient of friction on rough, dry concrete increases to ex = 20° and then falls.
In precisely the same way as with the longitudinal force the slip Sy,w (in the
lateral direction) is also taken into consideration; this is as a percentage of the
sine of the slip angle times 100:
SY,W
= sin a X 100 (%)
(2.12)
In conjunction with the drum value a = 10°, this would give a slip of Sy,w = 17%,
and on the street at a = 20° slip values of up to Sy,w = 34%. If the tyre is further
twisted to a = 90°, it slides at an angle of 90° to the direction of travel; sin a
would then be equal to one and Sy,w = 100%. The coefficient of friction then
becomes the coefficient of lateral sliding friction J.LY,W.lo, which on average is
around 30% lower:
J.LY,W,lo ::::::
0.7 X
J.LY,W
(2.13)
In contrast to dry concrete (as also shown in Fig. 2.43) on asphalt and, in particular on wet and icy road surfaces, no further increase in the lateral cornering
forces can be determined above a = 10° (i.e. Sy,w:::::: 17%).
1".-- - - - - - - - - - - - - - - - , - - - - - . - - - -
Tyres and wheels
133
1.2
t
1.0
~
~
~ 0.8
c
o
'u
:E
..o
...c
.~
0.6
0.4
/
~
Q)
o
(.)
......co
co
Q)
...J
0.2
/
/
/
V
o
-
~
---- CD
~
---..~0
l.--"""
5°
CD Dry, rough concrete
--.....
---
10°
o
~8
15°
20°
Slip angle a --..
25°
30°
Dry, smooth concrete G)Snow cover CDRough ice cover
Fig. .~~43
Latt3ral coefficients of friction !-Lv.w as a function of slip angle and road
for an '82 series' summer tyre with around 90% deep profile. The
Ice temperature IS around -4°e. The vertical force Fz,w was kept constant during the
measurements to obtain the dimensionless values of !-Lv.w. The maximum at a = 20°
on a very skid-resistant road can be seen clearly. The further !-Ly,w sinks, the further it
moves towards smaller angles.
~ondltlon, show~
2.8.4 Lateral cornering force properties on dry road
Figure 2.44 shows the usual way in which a measurement is carried out for a
series 82 tyre. The lateral force appears as a function of the vertical force in kilonewtons and the slip angle a serves as a parameter. A second possibility can be
seen in Fig. 2.45; here, for the corresponding series 70 tyre, J.LY,W = Fy,wIFz,w is
plotted against a and Fz,w serves as a parameter. The degree of curvature of the
graphs in both figures shows that slope at any point changes as a function of Fz.w
or ~y,w. The maximum occurs with large angles and small vertical forces. A less
stressed tyre in relation to its load capacity there~re pennits greater coefficients
of friction and higher cornering speeds than one whose capacity is fully used.
This result, which has been used for a long time in racing and sports cars, has
also become popular in modern cars, A mid-range standard car can be taken as
an example. The ear manufacturer specifies PT = 2.2 bar/2.5 bar under full load
for the front and rear wheels 185/65 R 15 88H. At these pressures, the load
capacity, in accordance with Figs 2.13 and 2.15, is:
front 505 kg and rear 560 kg
Figure 5.10 contains the authorized axle loads from which the wheel load
(divided by two) results:
134
The Automotive Chassis
4 -------t-------t---
t------+-------::7'.oC.....J
10°, PT = 2.0 bar
lOo,PT = 1.8 bar
kN
6°
3 t------+--------j-------hl,;r--~~__!~==~ 10°, PT = 1.4 bar
1
~
Lt
Q)
o~
2 t-----t----~~",r----~"-I-----__+---_____i
'+-
...
<tl
--+------12 °
....<tl
Q)
....I
O~----l-----..J-----I--------'-----
o
2
Vertical force Fz,w
3
4
kN
5
...
Fig. 2.44 Lateral cornering forces of the 155 R 13 78 S '82 series' steel radial tyre,
measured on a dry drum at Pr = 1.8 bar. The load capacity at this pressure is around
360 kg, corresponding to a vertical force Fz,w = 3.53 kN. Also shown are the forces
at a = 10° and PT = 1.4 bar and 2.0 bar to indicate the influence of the tyre pressure
on the lateral cornering properties.
front 375 kg and rear 425 kg
As described in Section 2.2.6, at speeds up to 210 Ian h- 1 (H tyres), an increase
in tyre pressure of 0.3 bar is necessary or there is only a correspondingly lower
load capacity. This then is, with PT = 1.9 bar at the front or 2.2 bar at the back,
450 kg and 505 kg
Thus, the actual load factor k m at 210 km/h becomes:
front km,f = (375/450) X 100 = 83%
back km,r = (425/505) X 100 = 84%
2.8.5
2.8.5.1
(2.14)
Influencing variables
Cross-section ratio H/W
The 185/65 R 15 88H size used as an example in the previous section is a 65 series
wide tyre; the 15" diameter also allows a good sized brake disc diameter (Fig. 2.10).
Tyres and wheels
1.2
135
r-----r---r------r----r-----,------.
Fz,w =1.0 kN
2.0
1.0
1-·---r---+------:b~;...-~~--~:::+~3.:-0-~
_J----r 4.O
__-1----15.0 kN, 17sn
r
:;: 0.8
~
R13
1------t--~~r:::::::.--;;;:;;~::::::::.--t-----=4-=-:::-;~:-=-J
I
c:
o
'fl
:E
'0
0.6
1------
'E
,/
CJ]
/
./'
/'
/'
:g'0 0.4 l-ffII7t--r--+---+----+---~------J
o
(.)
co
...
....
CJ]
j
0.2
H'f/TT·-t----+---+----+----!----J
o ~-__- : - - - - - L . . . - - - . . . L . - . - - - - l - - - - L - - - . J
o
2
6
Slip angle
8
a---..
10
0
12
Fig. 2.45 Lateral coefficients of friction f.LY,W as a function of the slip angle a and
the vertical force Fz,w, measured on a dry drum on a 175/70 R 1382 S tyre at Pr = 2.0
bar. The tyre, which has been inflated in such a manner, carries 395 kg or
Fz,w = 3.87 kN. In order to indicate the influence of the cross-section on the transferable lateral forces the 82 series 155 R 13 78 S tyre was also included.
In contrast to the 82 series standard tyre, the sizes of the 70 series and wide
tyres (H/W = 0.65 and below) generate higher lateral cornering forces at the
same slip angles (Figs 2.9, 2.45 and 2.46). As can be seen in Fig. 1.6, these, as
Fy,w,o = /-LY,W (Fz,w + /),.Fz,w), are all the greater, the faster the vehicle takes a bend.
2.8.5.2 Road condition
The force transmission ratios between the tyres and road are determined by the
state of the road (see construction, surface roughness and condition; Figs 2.43
and 2.47).
2.8.5.3 Track width change
The track width change that exists, in particular on independent wheel suspensions described in Section 3.3, causes undesirable lateral forces at the centres of
tyre contact on both wheels when the vehicle is moving unimpeded in a straight
line. Figures 3.5 and 3.6 show this, and also what lateral forces can occur if a
series 82 radial tyre rolling in a straight line is brought out of its direction by an
136
The Automotive Chassis
7000
N
5 degree slip angle
6000
......- ...
5000
/;'"
v'-, / [......./ .....
1
/~.
L(
./
~
4000
//
Q)
....u
....
./
3000
0
co
....
Q)
.....
co
~
2000
-I
1000
/
V
1
195/60 R14
I
;;185170 R13
~
r--.... 165 R1 3
~
~/
~
1/
1000
2000
3000. 4000
5000
6000
Fz,w
...
Vertical force
7000 8000 N 9000
Fig. 2.46 Lateral force FY,w dependent on vertical force Fz,w and tyre sizes of
different H/Wratios: 165 R 1382 H, 185/70 R 1385 Hand 195/60 R 1485 H.
Up to Fz,w = 4000 N the CUNes are more or less the same, but at higher loads the
more favourable lateral cornering properties of the wide tyre are evident.
suspension-kinematic dependent change. This effect is magnified by an increase
in slip rigidity, as, for example, in wide tyres.
2.8.5.4 Variations in vertical force
During cornering, vertical force variations + AFz.w in the centre of tyre contact
cause a reduction in the transferable lateral forces F y,w as the tyre requires a
certain amount of time and distance for the build-up of lateral forces. The loss
of lateral force AFy,w,4 depends on the effectiveness of the shock absorbers, the
tyre pressure PT (which can enhance the 'springing' of the wheels, see Equation
5.6) and the type of wheel suspension link mountings. Further influences are
wheel load and driving speed. To calculate cornering behaviour, an average loss
of lateral force AFy,w,4 due to variations in vertical force and dependent only on
tyre design and slip angle a, should be considered:
AFY,W,4
2.8.5.5
~
40 N per degree a
(2.15)
Camber change
Wheels that incline with the body during cornering have a similar, detrimental
influence on the transferability of lateral forces. As can be seen from Fig. 1.6, positive angle (+sw) camber changes occur on the outside of the bend and negative
Tyres and wheels
1.0 ~
t
-r-----
137
10....
-
D.B
Dry
~
»
:::l.
0.2mm
§ 0.6
.~
:E
o
..... 0.4
c
Q)
"(3
~
Q)
8
0.2
...
l~
~
~
~
1.0mm
2.0mm
10---
V-- ~
~
V
Q)
>
Q)
E
t;::
...
Q)
.....
Ctl
~
Q)
.....
Ctl
...J
o
20
40
Depth of profile
60
80
%
100
..
Fig. 2.~7 Possible lateral friction coefficients }LY,W of a steel radial tyre 155 R 13 78 S
depending on the depth of the tyre profile as a percentage (starting from 8 mm == 100%)
at Pr == 1.8 bar, (¥ == 10°, v 60 km/h and varying water film levels in mm.
, T~e i.n:proved grip of the tread less tyre on a dry road can be seen clearly as can
It~ s.,gnl:,cantly poorer grip in the wet; a fact which also applies to the coefficient of
frictIon In the longitudinal direction (see Section 2.7.2),
=
angles (-Ew)on the inside of the bend as a consequence of the body roll. The
lateral forces are directed to the centre point of the bend (Fig. 3.13). If a wheel
is 'cambered' against this, in other words inclined at the top towards the outside
of the bend, the possibility of transferring lateral forces reduces; on a dry road
surface, depending on the tyre size, the change is
~FY,W,3
= 40 N to 70 N per degree of camber
(2.16)
To counteract this, a greater slip angle must occur and greater steering input
becomes necessary for the front wheels. This makes the vehicle understeer more
(Fig. 2.41) and appear less easy to handle. Furthermore, the steering aligning
moment (see Section 3.10.3) also increases. If this effect occurs on the rear axles
- as is the case with longitudinal link axles (Fig. 1.14) - the vehicle has a
tendency to oversteer. Negative camber -Sw on the outside of the bend and positive +sw on the inside would have exactly the opposite effect. Wheels set in this
manner would increase the lateral forces that can be absorbed by the amount
stated previously for ~FY,W,3 and cause a reduction in the tyre slip angle.
2.8.5.6 Lateral force due to camber
Wheels according to the body roll inclined towards the outside edge of the bend
(Fig. 1.6) try to roll outwards against the steering direction, so that additional
138
The Automotive Chassis
camber forces are required in the tyre contact patches to force the wheels in the
desired steering direction. As these camber forces act in the same direction as the
centrifugal force Fe,Bo orY in the case described, greater lateral slip forces Fy,w,f,o,
F Y,\V,f,i, F Y,W,r,o and F Y,W,r,i and hence greater slip angles must be applied to maintain
the balance of forces on the part of the tyres.
The average force Few with the standard camber values for individual wheel
suspensions on a dry road are (see Section 2.2.3 in Ref. 9):
Few = Fz,w X sin Bw
2.9
(2.17)
Resulting force coefficient
Rolling resistance increases when negotiating a bend (see Equation 2.4a), and
the vehicle would decelerate if an increased traction force FX,W,A did not create
the equilibrium needed to retain the cornering speed selected. In accordance
with Equation 6.36, FX,\v'A is dependent on a series of factors and the type of
drive system (front- or rear-wheel drive); on single-axle drive (see Sections 1.4
to 1.6), the traction force on the ground stresses the force coefficient of friction
(the coefficient of)
j.Lx,w = FX,W,A,forrlFZ,y,forr
(2.15)
and thus greater slip angles at the driven wheels. With given values for cornering speed and radius (see Equation 2.8) the resulting force coefficient j.Lrsl can be
determined:
(2.18)
/Lrsl
cannot be exceeded because the level depends on the road's surface and the
condition.
When braking on a bend, additional longitudinal forces F X,\V,b occur on all
wheels (see Section 6.3.1), and act against the direction of travel. In this case
Equation 2.18 also applies.
On standard vehicles and front-wheel drives, the front wheels take 70-80% of
the braking force and the rear wheels only 20-30%. This means that the slip
angles increase on both axles, but more at the front than the rear and the vehicle
tends to understeer (Fig. 2.41 and Equation 6.20). If the wheels of an axle lock,
the friction becomes sliding friction and the vehicle pushes with this pair of
wheels towards the outside of the bend (Figs 6.8 to 6.10).
Taking into consideration the maximum possible values in the longitudinal
and lateral direction of the road - known respectively as j.LX,W,max and j.LX,W,min the increasing force coefficient can be calculated:
j.Lx,w =
j.LX,W,max
[1 _
(_j.L_Y._,W_)2]!
j.LY,W,max
(2.19)
Tyres and wheels
3000
N
f
~
2000
.t- /
I-
0
1000 10 %
co
I-
Q)
~ ~()
K1/
~
I
i
.
I
"
/
I
..... -8°
--- ------~
~
-6°_
-5°_
-4°_
,f
I
/'
/
/ 1/
ill'
~
I
"
I
...,;:
2°_
1-4-
~~
~~
,
I
..
~
r--:
I
/
SXWb
"""""
~3°-
I
I
-Slip angle a
~~ ---Brake slip
J
1/
-7
dJ P":/ L.--1"'"
Q)
()
..J
e;
/
~ ~/
J
....co
~1100
.~ ~ ,..7
139
1°
-""
~
J
'
I
7%"- IIto- 15
% '3 %'2 % . /1 %
LJ ; 1 ' I '
I
1
0
3000 N 2000
1000
o
0°
1000
2000
Traction force Fx.w.a
.....'4l---Brake force FX•W•b
N
3000
....
Fig. 2.48 Tyre-tangential lateral force performance characteristics with slip angles
and brake slip as parameters. The study was carried out on a 18565 R 14 86 S radial
tyre loaded at 300 kg at Pr :::: 1.5 bar. The shape of the curves indicates that, with
increasing longitudinal forces, those which can be absorbed laterally reduce. At 1.5
bar, the tyre carries a weight of 350 kg, i.e. it is only operating at 86% capacity.
Consider as an example a braking process on a dry road at 100 km/h on a bend
with R = 156 m. Using Equation 2.9 the calculation gives ~Y,W = 0.5.
Figure 2.48 shows a measurement on the tyre in question where the greatest
coefficient of friction in the lateral direction at F z.w = 2490 N, Bw = 10% and a
=: 4 0 (see Equation 2.11) amounts to
J.LY,W.max
= Fy,wIFz.w = 2850/2940 (NIN)
J.LY,W.max
= 0.97
In the longitudinal direction the possible braking force FX.W.b = 3130 N is at a =
0 0 and therefore (see Equation 2.5),
J.LX.W.max
= Fx.w.JFz.w = 3130/2940 (NIN)
= 1.06
and
05
/Lx.w = 1.06 [ 1 - ( 0.97
)2]~
= 0.91
The lateral forces that the tyre can absorb during braking can also be calculated:
J.Ly,w =
J.LY,W.max
[1 _
(_J.L_X_'w_\
J.LX.W,max
)
]~
(2.19a)
The Automotive Chassis
140
!J.x,W = 0.7 should be given. The lateral force coefficient (which can be used) is:
At SX,W,b = 10% and a = 4° the transferable lateral force is
Fy,w
= JLY,W X
Fz,w
= 0.73
X
2940
= 2146 N
and the available braking force is
FX,W,b = JLx,w X Fz,w = 0.7 X 2940
= 2058 N
2.10
2.10.1
Tyre self-aligning torque .and caster
offset
Tyre self-aligning torque in general
The focal point of the force of the tyre contact patch lies behind the middle of the
wheel because of its load- and lateral-force-related deformation. As a result, the
point of application of the lateral force alters by the amount rT,T, known as the caster
offset, and comes to lie behind the centre of the wheel (Fig. 3.119). On the front
wheels, the lateral cornering force FY,W,f together with rT,T (as the force lever) gives
the self-aligning moment MZ,T,Y which superimposes the kinematic alignment
torque and seeks to bring the input wheels back to a straight position (Section 3.8).
The self-aligning torque, lateral force and slip angle are measured in one
process on the test rig. MZ,T,Y is plotted as a function of the slip angle (Fig. 2.49),
the vertical force Fz,w serves as a parameter. The higher Fz,w, the greater the selfalignment and, just like the lateral force, the moment increases to a maximum
and then falls again. MZ,T,Y,max is, however, already at a ~ 4° (as can be seen in
Fig. 2.43) and not, on a dry road, at a ;:::: 10°.
2.10.2
Caster offset
Caster offset, rT,T, is included in practically all calculations of the self-aligning
moment during cornering (see Section 3.10.3). The length of this can easily be
calculated from the lateral force and moment:
(2.20)
rT,T = MZ,T,y/Fy,w (m)
This requires two images, one which represents Fy,w = f(Fz,w and a) or JLY,W =
f(Fz,w and a), and another with MZ,T,Y =f(Fz,w and a). The values of the 175170R
1-.=
Tyres and wheels
140
141
r---.------.------r-----r----...,------,
Nm
120,----t--------t-----t-----+----+-----l
100
1
...
80
c:
IV
E
0
E 60
Ol
c:
c:
Ol
m
I
40
~
IV
en
5kN
20
10
Slip angle a
0
..
I=ig. 2.49 Self-aligning torques of a 175/70 R 13 82 S steel radial tyre measured
on a dry drum as a function of the slip angle at Pr = 2.0 bar. The vertical force Fzw in
kilonewtons is used as a parameter. The torques increase sharply at low angles,
reach a maximum at a = 3° to 4° and then reduce slowly. As the cornering speed
increases, the tyre self-aligning torque decreases, while the kinematically determined torque increases (see Section 3.8).
13 82 S steel radial tyre shown in Figs 2.45 and 2.49 and measured at PT = 2.0
bar serve as an example. At a = 2° and Fz,w = 5.0 kN the coefficient of friction
P'Y,W = 0.44 and therefore:
Fy,w
= f.LY,W
X Fz,w
= 0.44
X 5.0
= 2.2 kN
= 2200 N
At the same angle and with the same wheel force, the self-aligning torque is
MZ,T,Y = 95 Nm and therefore
rT,T
= MZ,T,yIFy,w = 95/2200 = 0.043 m
= 43mm
Figure 2.50 shows the caster (caster offset trail) calculated in this manner.
Higher lateral forces necessitate greater slip angles, and the latter result in
smaller self-aligning moments and a reduced caster offset. The explanation for
this fact is that, at low slip angles, only the tyre profile is deformed at the area
The Automotive Chassis
142
50
mm
40 t----,---+-~---+----t_---_+_---_+---__J
t.::
30
I-
Q)
en
~
0
~
20
Q)
en
co
U
10
5kN
4kN
o ""'-
..l--
--J-
" - -_ _~==__=;.;.:.:..:,_ ___I..._..;..:3~k~N"____I_
Slip angle a - - .
Fig. 2.50
Caster offset of tyre (T,T calculated from Figs 2.45 and 2.49 for 175/70 R
13 82 S steel radial tyres at Pr = 2.0 bar. The higher the vertical force Fz,W (in kN) and
the smaller the angle a, the longer is (T,T.
of contact. The point of application of the lateral force can therefore move further
back, unlike large angles where, principally, the carcass is deformed. High vertical wheel forces cause the tyre to be severely compressed and therefore an
increase both in the area of tyre contact and also in the caster offset occur.
2.10.3
Influences on the front wheels
The tyre self-aligning torque is one of the causes for the steering forces during
cornering; its level depends on various factors.
2.10.3.1 Dry roads
The self-aligning torque is usually measured on a roller test bench with the drum
allowing a coefficient of friction of fLo = 0.8 to 0.9 between its surface and the
tyre. If the resultant self-aligning torque on the open road is required, it is possible to approximate the value MZ,T,Y,fJ. using a correction factor:
(2.21)
A cement block with fLy,w ~ 1.05 (Fig. 2.43) and the 175/70 R 13 82 S radial tyre
can be used as an example. In accordance with Fig. 2.49,
l
Tyres and wheels
MZ,T,Y
143
= 40 N m with Fz,w = 3 kN and a = 4°
As a correction factor this gives
kJ.l = II . _ro_a_d_ =
roller
- 1.31
JLY,W
= 1.05
JLo
0.80
and thus
Mz,T,y,1J.
= klJ. X Mz,T,Y = 1.31 X 40
= 52.4 N m
2.10.3.2
Wet roads
Provided that klJ. is independent of tyre construction and profile, the approximate
value for a wet road can also be determined. In accordance with Fig. 2.47, with
1 mm of water on the surface and full profile depth the JLY,W value reduces from
0.86 to 0.55.· Owing to the reduced coefficient of friction, only a smaller value
MZ,T,Y,IJ.' can be assumed; in other words,
kJ.l
wet
= JLY,W -roller
--
MZ,T,Y, IJ.
0.55
-
--
0.86
= 0.64, and
= 0.64 X 40 Nm
= 25.6 Nm
A greater water film thickness may cause the coefficient of friction to reduce but
the self-aligning moment increases and the water turns the wheel back into the
straight position. Furthermore, the self-aligning maximum shifts towards smaller
slip angles when the road is wet.
2.10.3.3 Icy roads
Only with greater vertical forces and small slip angles is the smoothness of the
ice able to deform the area of tyre contact and generate an extremely small
moment, which is nevertheless sufficient to align the tyre. Low front axle loads
or greater angles a arising as a result of steering corrections would result in a
negative moment - MZ,T,Y (in other words in a 'further steering input' of the
tyres). The wheel loads at the front, which were only low, were already a problem on rear-engine passenger vehicles.
2.10.3.4 Longitudinal forces
As shown in Fig. 3.119, traction forces increase the self-aligning torque; the
equation for one wheel is
MZ,w.a = Fy,w .
rT,T
+ FX,w.a . rT = Fz,w (ILy,w .
rT,T
+ ILx,w . h) (2.22)
The Automotive Chassis
144
During braking the moment fades and reduces to such an extent that it even
becomes negative and seeks to input the wheels further. The formula for one
wheel is
MZ,W,b
= Fy,w . rT,T = Fz,w (J-Ly,w •
The length of the paths
2.10.3.5
rT,T
FX,W,b • rT
rT,T
-
and
rT
JLx,w .
rT)
(2.23)
can be found in the details of Fig. 3.117.
Tyre pressure
When the tyre pressure is increased the self-aligning torque reduces by 6-8% per
0.1 bar, and increases accordingly when the pressure reduces, by 9-12% per 0.1 bar.
A reduction in pressure of, for example, 0.5 bar could thus result in over a 50%
increase in the moment, a value which the driver would actually be able to feel.
2.10.3.6
Further influences
The following have only a slight influence:
• positive camber values increase the torque slightly, whereas negative ones
reduce it;
• MZ,T,Y falls as speeds increase because the centrifugal force tensions the steel
belt which becomes more difficult to deform (Fig. 2.16);
• widening the wheel rim width slightly reduces self-alignment.
2.11
Tyre overturning moment and
displacement of point of application
of force
A tyre which runs subject to lateral forces on the tyre contact patch is subject to
deformation; there is a lateral displacement between the point of application of
the normal force (wheel load; Fig. 3.119) and the centre plane of the wheel.
Figure 2.51 shows the lateral drift of the normal (wheel load) point of application which is dependent on the size of the tyre, the lateral force and the camber
angle and to a large extent on the construction of the tyre. Low section tyres with
a small height-to-width ratio and a high level of sidewall rigidity exhibit greater
lateral displacement. The rollover resistance of the vehicle is considerably
reduced, as there is a decrease in the distance between the point of contact of the
wheel and the centre of gravity of the vehicle.
This displacement results in the emergence of tyre overturning moments
MX,T,u about the longitudinal axis of the tyre (Fig. 2.52).
Both the lateral displacement of the point of application of the normal force
and the tyre overturning moments must be taken into account when considering
the overturning behaviour of vehicles, as they can considerably reduce rollover
resistance, if, for example, a vehicle has a high centre of gravity and a small
track dimension.
Tyres and wheels
145
mm
r
f---
-
Wheel load 8000 N
- - Wheel load 6700 N
- - - Wheelload'5300 N
.....
c:
25
E
eI)
20
~V'
15
// VJ...-- . .
eI)
(,)
co
c.
ell
:.s
10
co
//~...-
L-
ei)
.....
co
5
.....I
-15
10
5
...- ...-
........
-
-- ... -- -- ~:4'
.-------
-
~
______
-~--
~-'-------
~~/
-,.."","
0
-5
5
10
Degree 15
Slip angle
-10
~/
.... ~
-15
-20
25
~ig. 2.5~
Lateral displacement of normal (wheel load) point of application dependIng on slip angle and wheel load; measurements by Continental on a tyre of type
205/65 R 15 94 V ContiEcoContact CPo
Nm
200
'--
Wheel load 8000 N
- - Wheel load 6700 N
........... Wheel load 5300 N
Cl
c:
'c 150
::;,
L-
t:
Q)
::0
/
100
Q)
L-
~
-10
-15
. . . . .... .. . .. . .... .... .. . . . . ... .. ,
.....::---
Fig. 2.52
r,::..:.;....-""
~7
/
//
~...... '
~~o
........C;.
-5
50
-50
V
~
V
...... " ....
---
.... .... _............ ...... " ..............................
10
5
Degree
15
Slip angle
-100
-150
200
Tyre overturning moments MX,T,Ct on the wheel as a result of the buildup of lateral forces at different slip angles and wheel loads Fz,w; measurements by
Continental on a tyre of type 205/65 R 15 94 V ContiEcoContact CPo
146
2. 12
The Automotive Chassis
Torque steer effects
Torque steer effects, i.e. changes in longitudinal forces during cornering, are an
important criterion for the definition of transient handling characteristics. The
torque steer effects depend on the size of the change in the longitudinal force
.
'
the adherence potentIal between the tyres and the road, the tyres and the kinematic and elastokinematic chassis design.
2.12.1
Torque steer effects as a result of changes in normal
force
Torque steer effects usually occur during cornering when a driver has to slow
down on a wrongly assessed bend by reducing the amount of acceleration or
applying the brake.
The reaction force acting at the centre of gravity of the vehicle causes an
increase in front axle load with a simultaneous reduction in the load on the rear
axle. At an initially unchanged slip angle, the distribution of lateral forces
changes as a result. If the force coefficient relating to the simultaneous transfer
of longitudinal and transverse forces is sufficient, e.g. in the case of torque steer
effects owing to reduction in acceleration or gentle braking (cf. Fig. 2.48), the
increased lateral force corresponding to the increase in normal force on the front
axle results in a yawing moment which allows the vehicle to tum into the bend.
If the adhesion potential is exceeded as a result of fierce braking or a low
force coefficient, the tyres are no longer able to build up the necessary lateral
forces. This results in an over- or understeering vehicle response depending on
the specific case, be it a loss of lateral force on the front axle or rear axle or both.
2.12.2
Torque steer effects resulting from tyre aligning torque
The lateral displacement of the tyre contact area as a result of lateral forces leads
to longitudinal forces being applied outside the centre plane of the wheel (Fig.
2.53).
This effect causes an increase in tyre aligning torque in driven wheels. In rearwheel drive vehicles, this torque has an understeering effect with tractive forces,
whereas it has an oversteering effect where there is a change in braking power.
In front-wheel drive vehicles, the resultant tractive force vector applies about
lever arm lr X sin Of offset from the centre of gravity of the vehicle (Fig. 2.54),
so that an oversteering yawing moment is produced during driving which alters
with application of a braking force to a (small) understeering yawing moment.
2.12.3
Effect of kinematics and elastokinematics
An attempt is made to keep the torque steer effects of a vehicle low by means
of specific chassis design. The above-mentioned changes in forces produce
Tyres and wheels
FX,W,A'
r't
FX,W,B'
147
rr
F X,W,A =2FX,W,a
rr
FX,W,B
= 2F x,W,b
Fig. 2.53 The deformation of the tyre contact area during cornering results in
aligning torque of the lateral forces which is further intensified by tractive forces and
produces an understeer;ng yawing moment. If there is a change in load, the braking
forces produce an oversteering yawing moment.
F X,W,A --2F X,W,a
FX,w,A'
(If . sin bf
-
rr)
Fig. 2.54
With front-wheel drive,
an oversteering yawing moment is
produced, because the resultant
tractive force vector is applied about
lever arm If X sin 8f displaced to the
centre of gravity of the vehicle.
·t·~
~
F.X,W,r,o
--.
FX,W,r,1
148
The Automotive Chassis
bump and rebound travel movements on the axles. The results, depending on
the design of the chassis, in kinematic and elastokinematic toe-in and camber
changes which can be used to compensate for unwanted changes in lateral
forces, particularly in the case of multi-link suspensions. With unfavourable
axle design. and construction, there is, however, also the possibility of an
increase in the torque steer effects.
3
_
Wheel travel and
elastokinematics
'Kinematics' - wheel travel, according to DIN often also called wheel (or steering/suspension) geometry - describes the movement caused in the wheels during
vertical suspension travel and steering, whereas 'elastokinematics' defines the
alterations in the position of the wheels caused by forces and moments between
the tyres and the road (Fig. 3.1. and Section 3.6.5), or the longitudinal movement
of the wheel, against suspension anchorage required to prevent compliance,
kinematic changes (Fig. 3.2). The changes are the result of the elasticity in the
suspension parts. The coordinate directions (within which everything is to be
considered) and the kinematic formulas are laid down in the German Standards
DIN 70 000 and DIN 74 250 (Figs 3.3 and 3.101), as well as in the International
Standards ISO 4130 and ISO 8855.
Fig. 3.1
Spring strut type front
axle of the VW Passat (1995). As
well as the vertical springing, the
lon9itudinal springing shown is
required in order to reduce the
rolling hardness of the tyres and
short-stroke movements caused by
the road surface. This longitudinal
springing is achieved by the lateral
flexibility of the rear bearing 4;
unwanted steering effects are
corrected by the appropriate arrangement of steering tie rod points U and
T (also see Fig. 3.83). The suspension arm is L-shaped in order to
enable introduction of lateral wheel
forc(~s directly into the rigid bearing
D to achieve a high level of lateral
rigidity without a force component
acting on the bearing 4.
4
~"'~'\\li,~
I
',.<::_._, . '
- ------------'--------
150
The Automotive Chassis
,~
I!
I ~--"'"-~
:
I
~
k~
.
I
' __ I
6
~9::""----_--I.._-=r-o::
,
X,W,b,r
.
I
fDireclion
5
I.
__ -
I
Fig. 3.2
If the front transverse link 5 on the bottom pair of suspension control
arms of a rear McPherson axle is shorter than the rear one 6, and if the longitudinal
forces are absorbed by a trailing link (not illustrated), its front bearing, which is fixed
to the underbody, can comply in a defined manner when braking forces FX•W •b•r occur.
The outer point 1 of the link 5 then moves in an arc around D1 to 3 and point 2 of the
link 6 around D2 to 4. Due to the different radii of the two arcs, a toe-in angle ~8k.r
occurs which opposes the returning moment M b = FX.W.b.r (b (Fig. 3.109) and produces
braking force understeering effects in the handling.
Fig. 3.3 Axis of coordinates in
accordance with ISO 4130 and DIN
70000. The positive Z direction
points upwards and, when viewed
into the direction of travel (X direction), the Yarrow points left (see Fig.
3.101).
3. 1
Purpose of the axle settings
To ensure the required road holding and directional stability and to prevent
excessive tyre wear, automobile manufacturers specify certain settings, including the permissible tolerances for the front axles of all models and for the rear
axles, provided these are not driven rigid axles. Toe-in can be set via the tie rods
or eccentric discs (Fig. 3.62) and camber and caster angles can also be adjusted
on some vehicles. The remaining manufacturers' data for kingpin inclination,
kingpin offset at ground (scrub radius), caster offset and differential toe angle are
Wheel travel and elastokinematics
151
design data and not easy to measure and are actually only used for checking the
roadworthiness of a vehicle which has been damaged in an accident or has
reached a given age.
As shown in the figures in the following sections, the axle settings depend on
load and load distribution. In order to make the measurements easier for garages
to carry out, only the curb weight, in accordance with recommendation DIN
70 020 (see Section 5.3.1.1) should be used as the basis for measurements.
3.2
Wheelbase
The wheelbase 1, measured from the centre of the front to the centre of the rear
axle (Fig. 6.1), is an important variable in the vehicle's ride and handling properties. A long wheelbase relative to the overall length of the vehicle makes it
possible to accommodate the passengers easily between the axles and reduces
the influence of the load on the axle load distribution (see Section 5.3.6). The
short body overhangs to the front and rear reduce the tendency to pitch oscillations and make it possible to fit soft springing, normally associated with a high
level of ride comfort. A short wheelbase, on the other hand, makes cornering
easier, i.e. gives a smaller swept turning circle for the same steering input (see
Section 3.7.2).
Vehicle designers seek to achieve a long wheelbase on both front-wheel drive
passenger cars and on conventional designs. However, this depends on the body
shape. (See Section 1.1 in Ref. [8] and Ref. [20]). A hatchback estate saloon
(Figs 1.68 and 1.72) can be of a more compact design, giving a longer wheelbase relative to the vehicle length than notchback saloons and the estate cars
developed from them. The ratio
wheel base
z=----vehicle length
(3.1 )
can be used as a reference and should be as large as possible:
it = 0.57-0.67 on estate saloons, and
it = 0.56-0.61 on notchback saloons
In coupes the value can be below 0.56 and on small cars it is up to 0.72.
The wheelbase is quoted in the manufacturers' brochures and the trade press
and lies between:
1= 2160 and 3040 mm
The Automotive Chassis
152
Fig. 3.4
On twin tyres the track specification br relates to the mean distance; the
lower load capacity of the tyres should be noted here (see Section 2.2.5.3).
roll (see Section 5.4.3.1). It should be as large as possible but cannot exceed a
certain value relative to the vehicle width. On the front axle the compressing,
fully turned wheel may not come into contact with the wheel house (arch) (Fig.
2.8) and on the driven axle (regardless of whether front, rear or both) there has
to be enough space for snow chains to be fitted. When the wheels compress or
rebound, they must not come into contact with any part of the chassis or the
bodywork.
The tread width on passenger cars is normally:
b rorr = 1210 to 1602 mm
and ib can be used as a ratio for the width utilization and should be as large as
possible:
lb
=
tread width
vehicle width
= 0.84 to 0.87
(3.1a)
When the wheels travel in bump and rebound-travel direction, the track changes
on almost all independent wheel suspensions, which may be the result of functional factors or, as the following section shows, unavoidable if a higher body
roll centre is necessary. However, the track size alteration causes the rolling tyre
to slip (Figs 3.5 and 3.6) and, on flat cross-sections in particular, causes lateral
forces, higher rolling resistance and a deterioration in the directional stability of
the vehicle, and may even influence the steering.
Track variation on the front and rear axle must be checked on the drawing
when the vehicle is at an early design stage. On a double wishbone suspension,
arcs with suspension control arm lengths c and f must be drawn around points C
and D (i.e. the suspension control arm axes of rotation), and the centres of the
outer ball joints marked as points 1 and 2 (Fig. 3.7). A template can be prepared
to show the steering knuckle and wheel (Fig. 3.8) and, in addition to points 1 and
2, must also have holes indicating the centre of tyre contact Wand, if necessary,
the central point U of the outer tie rod joint (see Section 4.6.3).
As shown in Fig. 3.7, points I and 2 of this template must be drawn upwards
Wheel travel and elastokinematics
153
Fig. 3.5
On independent wheel
suspensions, the bump and reboundtravel of the wheels as they go over a
bump can lead to a track alteration and
this, in turn, to the tyres running at the
slip angle ex. This causes disturbing lateral
forces, particularly if bump travel occurs
on one side; directional stability and
rolling resistance deteriorate.
Direction
f
200
'I
N
t
150
II
/
V
OJ
~
.E
J
.... 100
(1J
V
OJ
+-'
(1J
-l
Lateral forces Fv.w from
the tyre to the road resulting from
an alteration in track - shown on a
radial 175/65 R 114 82 H tyre
inflated to 1.9 bar under a load of
380 kg and at a :speed of
80kmh-1 •
/
50
Fig. 3.6
V
7
1
V
2
3
it
5
6 mm 7
Tread width alteration-'"
154
The Automotive Chassis
u
w
Fig. 3.7
Calculation by drawing of
the alteration in track of a wheel (in the
centre of tyre contact W) and the path
of the outer tie rod joint U on the
double wishbone suspension, using the
template shown in Fig. 3.8.
Fig. 3.8
Template for easy
calculation of alteration in track,
can be used on double wishbone
suspensions (Fig. 3.7) and longitudinal
link axles (Fig. 3.9).
along the arcs around C and D until point W of the template has reached the end
of the bump travel SI, previously indicated by a parallel to the ground, and
downwards over the rebound travel sz. The motions of Wand U are then filled
in step by step with a pencil. The line linking the points, which have been found
in this way, gives the alteration of the track and the travel of the tie rod joint,
but takes no account of any elasticity in the suspension control arm bearings
(see Fig. 3.18).
In the case of the longitudinal control arm axle an arc must be drawn around
D at the bottom, whilst a vertical line must be drawn on the suspension control
arm axis of rotation (Fig. 3.9) and must go through point 1. At the same time a
template as per Fig. 3.8 is moved along the arc and the vertical line to determine
the tread width alteration.
McPherson struts have a mounting point E (Fig. 1.7) in the wheel house.
When the wheel is in bump travel, the distance of the lower ball joint 2 to point
C shortens and then lengthens again when the wheel rebounds (Fig. 3.11). The
template has to take this length alteration into account (Fig. 3.10) and it has a
slot in the direction of the strut damper centre line EE (only in the direction of
the steering axis E2 if point 2 lies in its extension, see Figs 3.29, 3.30 and 4.46).
Using point 2, which also has to appear on the template, a movement is made
along the arc around D, whilst the slit is shifted over point C. A needle should
mark this point on the drawing board.
If an arc is drawn around poles P, the track alteration of the dual joint swing
axle can easily be drawn. Figure 3.12 shows both this and the advantages of
lowering the tail end of the vehicle, i.e. achieving a smaller and thus more
favourable camber angle and a higher lateral camber force on bends.
On all-independent wheel suspensions the position of the pole P determines the
momentary alteration + db (present in a small springing range, Fig. 3.14). Tread
width alteration is avoided completely if P is at ground level and the lengths of the
Wheel travel and elastokinematics
155
u
w
Fig. 3.9 Determination of the track
alteration and the track of the outer tie
rod joint U using the template shown in
Fig. 3.8 on the longitudinal link axle. The
description of this wheel suspension can
be seen in Figs 3.32 and 3.157 and
Section 9.4 in Ref. [2].
Fig. 3.10 The template needed
to calculate, by drawing, the tread width
alteration on the McPherson
strut and strut damper must have a
slot in the direction of the damper
centre line E.
suspension control arms on a double wishbone suspension have been determined
so that the pole moves horizontally from side to side on it when the wheels
compress and rebound (Fig. 3.13). This can be demonstrated up to wheel travel
s = +70 mm using a drawing, calculation or models whereby any elasticity has
been ignored (Fig. 3.18).
The tread width alteration can be measured as a function of the bump and
rebound travel (SI and S2) on the finished vehicle by determining the lateral shift
of two parallel plates on which the two wheels of an axle are standing. It is
necessary to run them parallel because a kinematic toe-in alteration when the
wheels reach full bump/rebound travel (see Section 3.6.2) could turn the plates
slightly and distort the measurement results.
Represented as a graph, the wheel travel should be plotted on the y-axis (Fig.
Fig. 3.11
Calculation by drawing of the
alteration in track of one wheel and the
path of the outer tie rod joint U on the
McPherson strut and strut damper using
the template shown in Fig. 3.10. C is the
centre of the upper strut mount; this point
is marked as E in Figs 1.8 and 3.139.
156
The Automotive Chassis
Fig. 3.12
Lowering the suspension control arm pivots P reduces the alteration in
track on the dual swing axle, causes the body roll centre to drop from R0 1 to Ro z and
a wider track. With two people in the vehicle, there is already negative camber on
the wheels - giving the advantage of accepting more of the lateral forces by the
tyres, but the disadvantage of reduced bump travel (see description of swing axle in
Ref. 2, Section 9.1).
P
/7
Fig. 3.13 An almost zero alteration in track requires a body roll
centre at ground level (or at infinity,
Fig. 3.25). Better kinematic
properties are also obtained if the
roll-centre axis is on the ground.
3.14) and - in accordance with the direction in which the axle is moving - bump
travel can be shown as positive and upward (s!), and rebound travel as downward
(S2). The zero position should correspond to the design weight (see Section
5.3.4), in other words the weight when three (or even two) people~ each weighing 68 kg, are in the vehicle. An empty vehicle would be unrealistic.
The track alteration lib of the two wheels (orlib/2 of one wheel) appears on
the x-axis, with the increase (as a positive value) entered to the right and the
reduction (as a negative value) to the left. The existing track dimension bf or r in
the zero position is an important dimension that should be stated. The tread
width difference lib to fully laden (or empty) can be determined using the
spring-rate characteristic. The spring travel lis! from the zero position to the
permissible axle load (or the bump travellis2 to the 'empty status') can be read
off this to obtain the track alteration curve lib as a function of lis.
Figure 5.9 shows the front wheel springing of a front-wheel drive vehicle,
where the dimension 80 mm must be deducted from 115 mm to get the rebound
travel, Iis 2 = 35 mm starting from the zero position (here, two people each
weighing 68 kg). The vehicle moves in bump-travel (from the zero position) by
lis! = 92 - 50 = 42 mm at the permissible wheel load (half the axle load). The
paths are marked in Fig. 3.14; lis! gives lib! = +4 mm and Iis 2 gives Iib 2 =
- 8 mm. The track should be specified for the kerb weight: b f = 1286 mm.
Wheel travel and elastokinematics
157
80 Travel
mm
51
60
5 passengers and luggage
Zero position
2 fJassengers
mm -30
Narrowing
of track
-~b
-20
-10
+10
b= 1286 mm
+30 mm
Widening
of track
II)
('t)
+~b
Curb
weight
-L--4
-----i----,L-I-
Track
+ 20
40
8
60
mm
Fig. 3.14 The track (b) between the two wheels of an independent wheel
suspension depends on the loading.
Figures 3.7, 3.15 and 3.18 show the track alteration of double wishbone
suspensions and McPherson struts and the lower alteration values in bump travel
can be clearly seen. As described in more detail in Section 3.4.1, the shape of the
curve determines the level of the body roll centre. On all three passenger vehicle body configuration RO r is above the ground and falls perceptively (with the
exception of the Honda, Fig. 3.15) when the vehicle is laden.
If the vehicle manufacturer has designed it at ground level as standard and the
vehicle is subsequently lowered (Fig. 3.16) the body roll centre then moves into
an adverse position; ROr drops below ground level and directional stability is
likely to be impaired, particularly with wide tyres.
In double wishbone suspensions, the springs sit on the upper or lower suspension control arms and, in both cases, a moment arises (Figs 3.17 and 1.6) which,
as a result of the elasticity in the suspension control arm bearings, causes the
tread width alteration curve to take on a slightly different shape, thereby slightly
altering the position of the roll centre (Fig. 3.18). The alteration curve, determined by measurements on the vehicle (with springs), gives the correct height in
any case.
The tread width alteration curves of typical rear wheel suspensions are shown
158
The Automotive Chassis
80
Bump
mm
,
I Audi
I
I
I
I
I
20
...
Narrowing of track
mm -15
I
Widening of track--..
-10
-5
+5
+15 mm
I
Curb weight: Audi
Curb weight: Honda
Cu rb weight: OpeINauxhall---?"I~~--r-#-----t-20 - - - - 1 - - - - - - + - - -
40 - - - - 1 - - - - - - + - : - - -
---t------+-~~__..~4A----_+
60 - - - - t - - - - - - t - - -
--+-~=--_'*--_f_---::l~--_+_----_1_
mm
- - - t - - - - - - + - - - - - - t - - - - - - + 80 -
Rebound
Fig. 3.15
Alteration in track of one wheel measured on the front axles of frontwheel drive Audi A6 (1996), Opel Astra (1996) and Honda Accord (1996) (Figs 1.57,
5.52 and 1.55). The Honda is the only passenger car to have double wishbone
suspension; the kinematic advantages can be seen clearly.
The 'body roll centre height' hRo •f in mm is:
Vehicle
OpelNauxhal1
Audi
Honda
Design position
40
77
138
Permissible axle load
15
30
111
Wheel travel and elastokinematics
mm
159
Q)
(,)
+----+-----120
Design position,
lowered vehicle
mm -15
Narrowing of track
...
o
§
...,o
~
+5 ~ +15 mm
.::: Widening of track
-+--\-T+-~ I
·
Normal position
Production vehicle
"0 --'-+e
;:,
o
.Q
-+---:If---+--I100
£-t--
mm
- - 1 - - - - + - - 1 120~f----+--
Fig. 3.16
Track alteration of both wheels measured on the front axle of a lowered
VW Golf" Gli. In the normal position, specified by the manufacturer, the body roll
centre is around road level. Lowering the vehicle by 30 mm means the body roll
centre moves 115 mm below ground, resulting in a longer body roll lever and a theoretically increased roll. However, due to the early acting jounce bumper and virtually
non-existent bump travel, the cornering inclination is greatly reduced (see Fig. 5.16
and Section 5.5.3).
a
c
The force h.w at the centre of
the tyre contact and FG.z on the lower
supporting ball joint form a moment, which
is absorbed laterally on the suspension
control arms causing the force pair +FE•y and
-h. y here. For reasons of simplification,
upper and lower suspension control arms
are assumed to be horizontal.
Fig. 3.17
d
160
The Automotive Chassis
51
mm
I
I
--40Hf-With springs
- - - Without springs
mm
-10
+10 mm
Fig.3.18
Alteration in track of both wheels,
measured with and without springs as a function of
the spring travel on a double wishbone suspension.
The curvature differs, being equivalent to a higher
body roll centre on the drivable vehicle than the
theoretical value (without opposing spring force)
calculated or drawn on the drawing board (see
Fig. 3.7).
in Figs 3.12, 3.19, 3.20 and 3.74. Non-driven rigid and twist-axle suspensions
experience an increase or decrease in track as a result of the elastic camber alteration (Fig. 3.55).
3.4
Roll centre and roll axis
In all independent wheel suspensions, there is a direct correlation between the
alteration in track and the height of the roll centre, so the two should be examined together. See Refs [2] and [9] for details.
3.4.1
Definitions
According to the German standard DIN 70 000, the body roll centre Ro is the
point in the vertical plane which passes through the wheel centre points (Fig.
3.21), and in which transverse forces (y-direction) can be exerted on the sprung
mass, in other words the body, without kinematic roll angles occurring.
The body roll centre is therefore the point in the centre of the vehicle (from
the front), and in the centre of the axle (when viewed from the side), around
which the body begins to roll when a lateral force acts, and at which reaction
forces are absorbed between axle and body. Based on the existing track alteration curve of a wheel, the body roll centre is the point Ro in the centre of the
vehicle (Fig. 3.22), which is intersected by a vertical, drawn on the tangent AB
laid on the alteration curve in the centre of tyre contact. The height hRo,f of point
Ro at the front (or hRo,r at the rear) can be determined in this way using the paths
Wheel travel and elastokinematics
161
I
~
Bump
80
mm
,I
6n
\
Honda}
1/,
40
. ,/
~BMW
" J
V(
I
/
Mercedes
1/
20
I/!//
-ft.b
15mm
-10
-5
/"
..,.-/ L
..,...,
,..... ,.....
-L/
/
/
"
.
/
1
.
/'
V
/
/
/
/
V
+10 mm
I
I
I
I
Curb weight: Honda, BMW
20
V '/",// /
.,
+ft.b
+5
/':
I
/
Curb weight: Mercedes
40
6n
mm
80- Rebound
I
r
Fig. 3.19
Track alteration of one wheel, measured on the driven rear axle of a
Mercedes (see Section 5.3.4 in Ref. 2), a BMW 3-series (Fig. 1.1) and the non-driven
axle of a Honda Accord (Fig. 1.55). The shape of the curve indicates that, with the
multi-link axle of the Mercedes, the body roll centre falls under load (Fig. 3.22). The
levels hRo,r (in mm) are as follows:
Vehicle
Design position
Permissible axle load
BMW
Honda
Mercedes
122
92
74
58
65
162
The Automotive Chassis
120.-.----r----,-T"T"\'-,..----,-----,.---;---,--,....---.---....------.
"'. ~
\\"
\1\
mm
\\\.
I
100 f----+;-\·,+-\-+---t\,..:,~,v-Wheel.base
t
'\t'"
\\ \
JI.//" Toe in/out
®
I
Camber/
~:~ \ 'v
::J
/
60 f---1----ll~__\_-fr.?4--+----t_i_-I_+_--+--J._____l
~'. \ It' ~
co
I/
\~. \ \ \
I
40 kID)·-+--I---\=~-\-_t+-_Hf-I-_l_-_+_--+----I
~. ~
Anti-squat
angle
" \
\\ \ ,
; /
1:/
20 1---+---t----.l~\.I--;~r-\-;\---1,:+---t'lJ.l--l---+---1---1
Q.l
>
....ro
.....
/ \ ~~ \ \\'
®
Q.l
Q.l
of---+T~ack change "
.c
:s:
(one wheel)
(j)
-20
-401-----+
C
-r
I
\ \
\\
Normal ride height
®f
/
I..-/" Brake reaction
I '\ i'J I
~ {\
support angle
-
;1 \,
Roll centre
height
,
'\ \I
::J
Q.l
CD
I
',", alteration
\
E
o
..c
-+---!-I--=-1+----1---1
80 f---'---1-----'!\-+-.\--\-I-'+'--+---+-.J--I-J:---+---+------1
0..
"0
r
I
@
-60 1----1---+--+---.f---,i-1':.\I\------I\'.---t---+---+-----.J~____l
/
a:
; \\ f,
/
I
-80 f---+---+----J.1+--I-' -H\r--I\\- +----+---!----+----l
V.-,-'
{~.\
.'
I·\\~.\
-1 00f---'---1---+-/--+--J~_H'-\-_¥t.\..---+_--+--~____l
'
mm
"
/
I.
\~
V \ \'\
-1201---_--L-_--L._....-JL...-_...L-_-i-_---l_ _. . L - _ - L _ - - . J
-40
-30 -20 -10
10
o
40 min 50
20
30
I
-8
I
660
I
-40
I
-32
I
18
I
-20
Fig. 3.20
I
-6
I
680
I
-'-30
I
-24
I
22
I
20
I
-4
I
700
I
-20
I
-16
I
26
I
60
I
-2
I
720
I
-10
I
-2
I
30
I
100
I
0
I
740
I
0
I
0
I
34
I
140
I
2
I
760
I
10
I
8
I
38
I
180
I
4
I
780
I
20
I
16
I
42
I
220
·1
6
I
800
I
30
I
24
I
46
I
260
CD
®
®
I
I
I
I
I
I
@
I
I
I
I
®
®
I
I
8 deg 10
820 mm 840
40 mm 50
32 deg 40
50 deg 54
3QO mm 340
(j)
Kinematics of the semi-trailing rear axle of an Opel Omega (1996). This
measurement shows the change in track of one wheel only. The variation of toe or
steer with suspension vertical deflection curve indicates a roll-steer effect on the rear
axle tending towards understeering. This was achieved by the addition of a 'toe
control link' on each side. The lowering of the rear body roll centre under load
favourably reduces the dynamic wheel load transfer on the bend at permissible axle
load (relative to that on the front): it allows the vehicle to understeer more.
Brake reaction support angle e and anti-squat (diagonal springing) angle X are
shown in Fig. 3.160. The axle is shown in Fig. 1.15.
Wheel travel and elastokinematics
163
Fig. 3.21 The body roll centre is in the
centre of the vehicle (viewed from the front)
and in the centre of the axle (viewed from
the side).
. bt or
Ro I----~--~
V~hiCle witti 2
. hRo , for r
As
passengers
,-Ab
+Ab
hRo,f
Lib
bf
= -- X -Lis
2
hRo,r
=-- X --
Lib
b~
Lis
2
Fig. 3.22
The height hRo,f or r of the body roll centre can be determined using a
tangent from the measured track alteration curve in the respective load condition.
D..s and D..b drawn at the tangents, considering all elasticities in the suspension
control arm bearings (Fig. 3.18). It behaves as follows:
hRo,ror r
-= tan
As O.Sb rorr
(3.2)
ct
and therefore the height of the body roll centre related to one wheel is
front
D..b b
D..b br
r
=-and rear hRo,r =-- ,
D..s 2
As 2
h Ro r
Where br = 1400 mm, Ab
- ....
=6 mm per wheel and As =40 mm,
(3.2a)
The Automotive Chassis
164
hRo f
,
6 1400
2
=-40
= 105 mm
The greater the tread width alteration in the point corresponding to the respective load (Fig. 3.14), the steeper the vertical on the tangent becomes and the
higher the body roll centre lies above ground. However, in the case of small
track alterations, Ro is only slightly above, or on, the ground if the tangent AB
is parallel to the y-axis (Fig. 3.13). If (as partly shown in some figures in
Section 3.3) the track alteration due to both wheels is entered, the height of the
body roll centre can be determined in the same way but only half the alteration
travel, i.e. Abl2, has to be considered. The equation is therefore related to both
wheels:
hRoforr
,
Ab brorr
= --As 4
(3.3)
In Fig. 3.15, in the Audi and Opel, tangents drawn on the upper curve are always
parallel to the y-axis when the wheels compress, this being the equivalent of a
drop in the body roll centre under load, a characteristic of McPherson struts.
However, on the double wishbone suspension the tangent angle, and therefore the
height of point Ro, alters less under load (Honda and Fig. 3.18). The same applies
to this type of rear axle (Figs 3.19 and 3.20). With varying deflections to left and
right, the body roll centre is generally no longer located at the vehicle centre.
3.4.2
Body roll axis
The position of the roll centres at the front and back and the course of the direction line joining these - the roll axis C (Fig. 3.23) - is of decisive importance for
the handling properties: the height of the roll centres determines both the wheel
load differences of an axle and hence the self-steering properties of the vehicle
through the tyre properties, as well as the necessary roll suspension, which is
again crucial to comfort in the case of unilateral deflection where a high level of
roll rigidity is required and a stabilizer is used. The position of the roll centre
also depends on the instantaneous position of the wheel links, i.e. the roll centre
usually only lies in the centre plane of the vehicle if there is symmetrical wheel
displacement and alters its position both horizontally and as vertically with
unilateral displacement (cornering), resulting in the unwanted support effects of
the wheel link forces on the body. A roll centre which decreases with symmetrical displacement helps to remedy this.
The height of the roll centre and the change in the roll centre with wheel
travel is consequently a compromise between the following requirements:
• defined changes in wheel load during cornering to achieve the required (understeering) self-steering properties;
• track changes with wheel travel which are not critical for the dynamics of
vehicle movement;
Wheel travel and elastokinematics
165
c
m771177 A-:-~~
c-- .--- ROt r---~.!.-.-~I---~~...J
/
Fi~. 3.23 .Line C joining the front and rear body roll centre represents the theoretIcal roll aXIs (here at an angle). The path dhso is the body roll lever pointing vertical to the ground b~t:",een this Iin~ and the body centre of gravity 80. If the
pa.ssenger c~r has ~ n~ld rear axle, thIs angled disposition is beneficial. The body roll
aXIs o~ a vehIcle wIth Independent wheel suspensions front and rear should only be
at a slight angle (hso see Equations 6.7 and 6.24).
•
•
•
•
roll spring stiffness which is not crucial to comfort
desired - or permissable - camber change;
,
as small as possible reaction forces acting on the body;
the position of the roll axis.
The roll axis should rise slightly towards the rear in order to make use of fractions of the body damping to damp the yawing movements of the vehicle. Roll
centre heights in the design of independent wheel suspensions are
h = 30 to 100 mm at the front
h = 60 to 130 mm at the rear.
Particular attention has to be paid to the superposition of high wheel loads with
traction forces and hence a reduction in lateral force potential.
Depending on the curvature of the track alteration curve, the body roll
centres fall under load to a greater or lesser degree (Figs 3.15, 3.19, 3.20 and
3.22).
The design of a chassis firstly requires the determination of the height hRo,f of
the front body roll centre (dependent on the track alteration) so that, in a second
step, an appropriate rear axle can be provided; in the case of independent wheel
suspensions with a slightly higher hRo,r •
If the vehicle is fitted with a rigid axle, the body enjoys less anti-roll
support on bends (i'P = bsrfb r, Fig. 1.23) as a result of the shorter effective
distance bsp of the springs relative to the track br. To balance this out, it is
recommended that the body roll centre be designed slightly higher at the rear
(as shown in Fig. 3.23). The possibilities for this can be taken from Ref. [2].
The additional lines A and B drawn in Fig. 3.23, are the actual body roll axes,
which are mostly parallel to the ground. The precise location depends on the
166
The Automotive Chassis
The following dimensions have to be known:
c, d, bf , r." a, (3, (T
p
Fig. 3.24 Determination by
drawing and calculation of the
paths hRo and p on double wishbone suspensions and a multi-link
as well as longitudinal transverse
axes (Figs 1.1 and 3.32).
d
~
p
k cos {3 + d tan
t
if
p = k sin {3 + d
+ r"
/
0
= c sin (90 + (J"- o.l= k
sin (a + (3)
angular position of the steering control arms. The body inclines around A and B
under the influence of a lateral force.
3.4.3
Body roll centre on independent wheel suspensions
The height of the (instantaneous centre of rotation) P determines the position of
the body roll centre Ro (Fig. 3.24). If P is above ground level, Ro will also be
above ground. As can be seen in Fig. 3.22, the tangent drawn at the zero point
on the track alteration curve varies by the angle ex from the vertical. However,
the shape of the curve at this point depends on the distance between virtual
centre of rotation P and the centre of tyre contact W. The further the two are apart
(i.e. the longer the path q, Fig. 3.30), the less pronounced the curvature and the
lower the camber alteration (see Section 3.5.2). The following figures show the
determination of height h Ra of the body roll centre and path p by drawing. The
virtual centre of rotation distance q from virtual centre of rotation to tyre
contact-patch centre can be measured or calculated simply:
p brar r
(3.4)
As can be seen in Figs 3.24 and 3.7, on the double wishbone suspension only the
position of the steering control arms is important (i.e. the sizes of the angles a
and f3). The lines connecting the inner and outer steering control arm pivots need
to be extended to fix virtual centre of rotation P and, at the same time, its height
p. P linked with the centre of tyre contact W gives the body roll centre Ro in the
intersection with the vehicle centre plane. In the case of parallel control arms, P
is at co, and a line parallel to them needs to be drawn through W (Fig. 3.25).
Where the virtual centre of rotation is a long way from the wheel centre of
contact, it is recommended that the distances p and hRa be calculated using the
formulae in Fig. 3.24. Steering control arm axes of rotation, which are sloped
Wheel travel and elastokinematics
167
Fig. 3.25
Determination of the
body roll centre on parallel double
wishbones; the virtual centre of
rotation is at infinity.
~Pis
at
--
00
\
\--
--
p
Fig. 3.26
If the suspension control arm axes of rotation are atan angle to one
another when viewed from the side, a vertical should first be drawn to the ground
through the points E, and G,; the intersections with the axes of rotation C,C 2 and
0, O2 yield the points E2 and G2, needed for determining the virtual centre of rotation
when viewed from the rear.
when viewed from the side (designed this way to obtain a vehicle pitch axis - Fig.
3.155), need E( and G 1 to be moved perpendicularly up or down (Fig. 3.26). The
points E2 and G2 obtained in this way - linked with E 1 and G 1 when viewed from
the rear - give the virtual centre of rotation P, and the line from this axis to the
centre of tyre contact (as shown in Fig. 3.24) gives the body roll centre. If the axle
is controlled by transverse leaf springs, where these are held in the middle (Fig.
3.27), the kinematic lever L 3 is important for calculating the body roll centre and,
p
Fig. 3.27
Determination of Ro and
P on a high, centrally
anchored transverse
leaf spring.
168
The Automotive Chassis
Fig. 3.28 Determination of Ro
and P on a low transverse leaf spring
supported in two places.
~
Fig. 3.29 The greater
the tread width bt , the
higher the body roll centre
Ro, shown using the
example of a McPherson
strut (Fig. 1.56).
b f ,2
-2-
bt,l < bt,2 therefore h Ro , 1 < hRO , 2
if the springs are attached at two points, the distance ~ to the spring attachment
point is important (Fig. 3.28). Further details are given in Section 4.7.3.1.
On McPherson struts, or strut dampers, a vertical must be created in the body
side fixing point E to the centre line of the shock absorber piston rod, and the
lower steering control arm must be extended. The intersection of the two lines
will then give P (Fig. 3.29). The illustration also shows how increasing the track
from br,1 to b r,2 results in the body roll centre being raised from RO l to R0 2 • A
negative kingpin offset at ground makes it necessary to shift the lower swivel
joint in to the wheel (Fig. 3.102) which separates the kingpin axis from the shock
absorber centre line. Figure 3.30 shows the path EP, which is then vertical to the
shock absorber centre line and also that hRo is not dependent on the steering
control arm length, which is the decisive factor for the kinematic properties.
Where the suspension control arm lies flat, it is recommended that the heights h Ro
and p be calculated because, if drawn, the virtual centre of rotation would be too
far outside the drawing board (Fig. 3.31). Section 4.7.3.2 contains further details.
On the longitudinal link axle (Fig. 3.32), the direction of movement of the
upper point E (vertical to the suspension control arm axis of rotation) plays a
role. A parallel to CF must be drawn through E to obtain P and Ro. The calculation can be seen in Fig. 3.24. On the McPherson strut, the height of the body
roll centre can only be influenced by placing the lower suspension control arm
at an angle and only marginally by changing the angle between steering axis EG
and the McPherson strut centre line (Fig. 3.30), which is a disadvantage of this
type of suspension. On the longitudinal control arm axle it is possible to increase
Wheel travel and elastokinematics
169
p
p
q
Fi~. 3.30 The more vertical the McPherson struts and dampers and the more
hOrizontal the lower control arm GD 1 , the closer the body roll centre Ro is to the
ground. This res~lts in an adverse ca~ber alteration when the wheels are in bump
travel. Lengthening the lower suspension control arm (point 0 1 to O2) improves the
kinematic properties.
To achieve a small or negative kingpin offset at ground rcr , point G must be drawn
outwards into the wheel, giving the benefit of a shorter lever b for the vertical force
Fz.w . The shorter can be path b, the less friction occurs between the piston rod and
rod guide, as well as at the piston, and ~he smaller the forces in bearing points 0, E
and G (see also Fig. 1.11). A long path q means tread width alteration can be
restricted. Fig. 1.8 shows the precise position of points E and G.
The lever b is easy to calculate:
b
= rcr + d tan
(J'
(3.4a)
Depending on the design, either +rcr or -rcr has to be included in the equation (see
Section 7.2 in Ref. [3]).
the angle of the axis of rotation CF further and therefore to raise Ro. At the same
time, the virtual centre of rotation moves closer to the wheel, giving the additional
advantage that the compressing wheels move more strongly into negative camber.
The heights hRo,f of the front body roll centres determined in accordance with
Figs 3.24 to 3.32 only agree in the case of bearings which, although they can be
rotated, are otherwise not flexible, and only at body roll angles up to c.p = 2°. The
elasticity of the rubber elements used slightly alters the height available on the
vehicle (Fig. 3.18). Furthermore, calculations and studies have both shown that,
in the case of larger body roll angles, the left and right pivot axes take on a
different position, but that the body roll centre in the centre of the vehicle experiences an alteration of only tlh Ro = ::!: 10 mm. Parallel measurements carried out
on passenger cars showed a deviation of up to tlh Ro = 20 mm.
170
The Automotive Chassis
Fig. 3.31 Calculation of the paths
hRo and p in the standard configuration of a McPherson strut and strut
damper.
p
d
bf
T
P
.cos~+ dIan
k=c
~ ksin
f3+
d
ff+ /
c+o =k
sin (0: + {3)
(¥Z
>
(¥1
therefore h Ro , Z > hRo,1
Fig. 3.32
With the longitudinal transverse suspension, a parallel to CF should be
drawn through E and this made to intersect with the extension of the path GD to determine the roll centre Ro. Pole P is then connected to W to give Ro in the vehicle centre
plane. The greater the angle of the upper suspension control arms, when viewed from
the rear (0:2 right), the closer P moves to the vehicle centre; tread width and camber
alteration increase and R0 1 becomes R0 2 at a higher level (see also Fig. 4.49).
In contrast to the front independent wheel suspensions, rear ones sometimes
have only one control ann on each side; here, too the position of the virtual centre
of rotation determines the height of the body roll centre, with the direction of
movement of the wheel providing additional information. If the axis of rotation
lies horizontal (Fig. 3.33) on the link axle, the wheel moves vertically and the roll
centre Ro is at ground level. If the axis of rotation is inclined (Fig. 3.34), Ro
moves above ground or, if the angle is in the other direction, below ground.
The single joint swing axle (Fig. 3.35) has its point of rotation in the centre
of the vehicle. The pole is, at the same time, the body roll centre, unlike the dual
joint swing axle on which point P is to the side next to the differential and Ro is
Wheel travel and elastokinematics
171
Fig. 3.33 If, with longitudinal links, the axis of rotation is horizontal, the body roll
centre is at ground level and P is at 00; the magnitude of torsional springing ± f depends
on the suspension control arm length (diagonal springing angle X' see Fig. 3.58).
therefore disproportionally high. Figure 3.12 shows how Ro is calculated, with
the fall in the body roll centre in the case of negative camber -ew (left) clearly
indicated.
In the case of the semi-trailing link axle, the movement of the wheel vertical
to the three-dimensional axis of rotation EG plays a role (Fig. 3.36). The point
at which the extension of the axis of rotation intersects a vertical plane in the
centre of the axle gives the virtual centre of rotation PI (= P2), from which the
height h Ro of the body roll centre in the middle of the vehicle can be determined.
To find this, first draw the top view, taking into account the angle ex, and in it the
extension of the suspension control arm axis of rotation made to intersect with
the axle centre. The pole PI obtained in this way is moved perpendicularly down
in to the rear view and made to intercept with the extension of the axis of rotation - this time using the angle r3. Finally, the pole P 2 found in the rear view must
be linked with W. With small angles ex and r3, it may be sensible to calculate h Ro
and p as a function of the dimensions specified by the designer. Figure 3.36 also
contains the formulas for these relationships.
w
Fig. 3.34
If, with longitudinal links,
the axis of rotation is at an angle, the
body roll centre will lie above ground (or
below it, if the angle is reversed); P is
at 00 in both cases (see Fig. 3.158).
w
Fig.3.35
On the single-joint swing
axle, the suspension control arm pivot,
which is approximately at the centre of
the vehicle, is both the rear pivot axis and
roll centre (see Section 9.2 in Ref. [2]).
l
172
The Automotive Chassis
The following dimensions have to be known:
e, f, k, b, a, {3
p = k- tan
hRo
b!
=-2
X --
f+ d
Fig. 3.36
On the semi-trailing
suspension, the positions of the
virtual centre of rotation P and roll
centre Ro are determined by the
length r of the suspension control
arms and the top view angle ex and
rear view angle 13. The equations
are used for calculating the height
hRo in the vehicle centre. When the
vehicle is laden, points E and G
(and therefore also P and Ro)
move down. The momentary tread
width alteration results from an arc
around P2 (see also Figs 3.20 and
3.160).
f3 d
t
d
= e cot IX
3.4.4 Body roll centre on twist-beam suspensions
The kinematic or static body roll centres of this suspension are the bearing points
o (Figs 3.37 and 1.31) at which - as specified in DIN 70 000 and described in
Section 3.4.1 - the lateral forces are absorbed. The elastokinematic body roll
centre, on the other hand, determines the alteration to toe-in and camber on reciprocal springing. Owing to the low torsion resistance of the transverse members
the wheels swing (precess) during cornering, as on the semi-trailing link suspension, around the line connecting the points 0 1 and Ors with the thrust centre point
SM (Fig. 3.38). Toe-in and camber alteration are shown in Figs 3.54 to 3.57.
3.4.5
Body roll centre on rigid axles
As shown in Figs 1.25 and 1.26, on rigid axle suspension the lateral forces are
absorbed in only one or two places. The body roll centre can therefore only occasionally be determined using the theory of transmission kinematics. It is the laws
of statics which mainly apply, and the spring axle mounting point - at which the
forces are transferred between body and axle - which should be observed.
If longitudinal leaf springs are used as the suspension, the lateral force is
concentrated on the main leaves, and Ro is at their centre within the clamp (Fig.
---_._----------, ---------,----- ,----,
Wheel travel and elastokinematics
173
Fig. 3.37
On what is sometimes called
the 'compound crank axle' (also called the
torsion or twist beam axle) lateral loads are
reacted by the two trailing links, which are
stiff in torsion and bending. The height
above ground of pivot points 0 determines
the roll centre location. The position of 0 is
dependent on the arm length r and its
angle ±x to the horizontal. On the linked
trailing arm or torsion beam (sometimes
called 'twist beam') rear suspension the
later~1 forces are reacted by the two trailing arms, which are stiff in torsion and
b~ndlng: The roll ~e~tr~ position is ?etermined by the height above ground of the
PIV?t POints O. ThIs IS Itself determined by the arm length r and its angle to the
hOrizontal plane ±x.
Rear view
Wheel centre
.~
~.
Top view
'Direction
0 ..
I
~I
C·
~I
:EI
i
Fig. 3.38
~
Determination of the height
hRo,r
of the elastokinematic roll centre RO r
a~ound which the body inclines under the influence of the centrifugal force acting on
tile body centre of gravity for the twist-beam suspension. The thrust centre point SM
of the cross-member, which must be linked in the top view with the bearing points
o and intersected with a straight line through the wheel centres, must be known.
The resulting centres of rotation must be moved vertically upwards to the wheel
centre axis in the rear view and linked with the centres of tyre contact W to obtain
point RO r in the vehicle centre.
The position of the 'thrust centre point' also determines camber and caster alteration on counteracting bump/rebound-travel springing (Figs 3.54 and 3.55) as well as
the lever arm ratio between the spring and shock absorber. (For more details see
Ref. [2], Section 4.3.)
174
ThE~
R0 1 (lorry)
Automotive Chassis
Ro (passenger vehicle)
2
I
Fig. 3.39
If the rigid axle is carried by
longitudinal leaf springs, the lateral forces
are concentrated in its main bearings. The
body roll centre is on the axle mounting
in the middle of the main leaf, regardless
of whether the spring is fixed above (left
side and Figs 1.24 and 1.37) or below the
axle (right and Fig. 1.26),
Fig. 3.40
If a panhard rod provides
lateral force reaction support, the body
roll centre is at the intersection of the rod
with the vehicle centre line.
3.39). To keep it flat for a low underbody-ground clearance on a passenger car,
the spring is underslung below the axle (right-hand side of the picture), whereas
commercial vehicles need a high body roll centre to reduce the body inclination.
The spring is then above the axle (left-hand side, see also Fig. 1.37) with the
advantage that the fixing bolts are not subject to further tensile forces.
If the lateral force is supported by a panhard rod (Fig. 3.40), the body roll
centre will be at the intersection of the panhard rod with the vehicle centre line
(and not, as sometimes thought, in the centre of the bar). During cornering, the
position of the bar changes and therefore so does the height of Ro. However, if
a watt linkage supports the forces in a lateral direction, the point at which it is
fixed to the axle housing is the decisive point of reference (Fig. 3.41).
The upper pair of longitudinal control arms and the panhard rod can be
replaced by an A-arm (Fig. 3.42), which transfers lateral and longitudinal forces
Fig. 3.41
Watt linkage on a
passenger car rear axle. This
allows the axle to be carried
without any lateral deviation.
When the springs deflect in
bump and rebound-travel, the
linkage turns around the mounting point on the axle housing,
which is also the roll centre.
Wheel travel and eJastokinematics
Fig. 3.42 If a longitudinal A-arm
supports the rigid axle, its fixing position
on the axle housing is also point Ro.
175
Top view
Rear view
~ the body. The body roll centre Ro is then the fixing point on the axle. In
~ontrast to the panhard rod, point Ro maintains its height hRo when subjected to
l?ad.
Instead of the upper A-arm, two suspension control arms at an angle to one
atother can be used (Fig. 3.43). In this case, the intersection of the extension of
t e suspension control arm from the top view gives the virtual centre of rotation
PI which must be brought down perpendicularly in the side view. In the case of
p~allellowersuspension control arms, a line drawn in the same direction as the
a~s intersects with the axle centre in the body roll centre Ro.
Unlike the rigid axle suspensions discussed so far, on the drawbar (longitudin11-pivot) axle (also known as the A-bracket axle) lateral forces can be absorbed
jqintly on the front bearing point Or and two lateral struts (Fig. 1.60). The body
rqll centre is then at the height at which these three parts are attached to the body.
If) instead of the two struts, there is a panhard rod, the forces are supported on
th~s and point Or. The side view shown in Fig. 3.44 next to the top view clearly
sh\ows both reaction forces F O,y and FT,y. The body roll centre is therefore on the
li~ linking the two points, which can be seen in the side view. If (as shown in
Fi . 3.40) the panhard rod is at an angle, the mean height of the rod in the rear
vi w must be determined and then transferred to the side view.
I
i,
I
3.~.1
In
Camber values and data
~ccordance with the standards DIN 70 000, camber is the a~gle b~t~e~n the
w~eel
centre plane and a vertical to the plane of the road. It IS pOSItIve If the
The Automotive Chassis
176
Top view
~ ..
Fig. 3.43
If the two upper suspension
links, which lie at an angle to one another in
the top view, absorb the lateral forces, their
extensions give virtual centre of reflection P1.
To determine Ro in the side view, a parallel
must be drawn to the lower suspension
control arms through P1 . As these two
suspension links point in the same direction,
as can be seen in the top view, their virtual
centre is at 00.
I
~.
Sideview
Top view
FyW,o
Fo,y
~==F.~V,§iW~,i *+1 Fv,w,o +
u;;
•
FT,y
Fy, W,i
F T,y
Side view
Rear view
Panhard rod T
Fig. 3.44 The lateral forces Fv.w,o and Fv.W,i are transferred from the axle to the
body at the foreward differential-housing exension mounting and the rear panhard
rod. The reaction forces Fo,vand Fr,v occur. The body roll centre RO r must therefore
lie on the line connecting points T and Or from the side view. (The" drawbar" mounting is described in Ref. [2], Section 3.4.)
t
Wheel travel and elastokinematics
177
Fig. 3.45
Positive camber +ew is the inclination of
the wheel plane outwards from the vertical. The
wheel shown would roll to the left because of the
AT,e 'lateral camber force', if a right-hand counterweight did not restore the balance (i.e the direction
straight ahead).
.
-
Fy,W
wheel is inclined outwards (Fig. 3.45) and negative, as -ew, when inclined
inwards.
When a vehicle is loaded with two or three persons (design weight, see
Section 5.3.4), a slightly positive camber would be useful on passenger cars to
make the tyres roll as upright as possible on the slightly transverse-curved road
surface and give more even wear and lower rolling resistance. As Fig. 3.46
shows, the optimum value for this purpose would be
ew = 5' to 10', i.e. around 0.1 °
To give better lateral tyre grip on bends and improve handling, nowadays this
rule is generally no longer adhered to and, on passenger cars, the setting is negative even when the vehicle is empty. Front axle values are as follows on newer
production vehicles:
eW,f,u!
= 0° to - 1° 20'
100 v
V
. / 0/0
/
/'
v
V
....
i'--.
>(,)
c:
....
l'Cl
V
0.
x
Q)
Q)
Inner shoulder tyre wear :.J
:;;.
"'-
~ r--...
~
(,)
Q)
1/
..........
1-50
_10
"""
,Outer shoulder tyre wear
l-
00
Camber ew
Studies have shown that a camber of ew = +5' to 10' leads to the most
even tyre wear; more positive camber would lead to more pronounced ,w~ar on the
outer shoulder and negative camber to more pronounced wear on the inSide of the
tyre tread.
'Fig. 3.46
ZCX:__ CLS-.: __
U&~-"'"~-
_ -
"'"~-_-_-_-_
--2">-:...--
-_-_-..c_v~~:~~-::o'-~- ~~_~-~UB~~~~~~~~~~~;S~K~~~~":J&""~~..,-~-W0-'-~~~y~~..i--~~%-~~~~~
- - - ~--A-~~~-=---'-"'"--""----e--~
-'~ ,~~...;c -
-..::.: ~ ~ c~ C:~~€~::-5:" -::..-__ ~ _ -":-~__._~
_
The Automotive Chassis
178
In addition to the absolute camber, the tolerance values are important, i.e. both the
deviation from the permitted value and also the difference between the left and right
wheel. A + 30' deviation is usual to enable the components of the front axle to be
manufactured economically. This is why it is not always possible to adjust the
camber on front wheel suspensions. The various designs are described in Ref. [2].
To avoid the steering pulling to one side when the vehicle is moving in a
straight line, the difference in the kingpin inclination angle between left and
right wheels should not exceed £lcr = 30'. As can be seen in Fig. 3.103, camber
and kingpin inclination are directly related, i.e. if the camber deviation is too
great, so is the kingpin inclination angle. This is why no camber difference
greater than 30' should be allowed as a factory setting. The information in the
subassembly drawing of the front axle would then be as follows, for example:
Camber - 40' + 30';
maximum difference between left and right 30'.
(3.4b)
The measurement condition, which must relate to the kerb weight (i.e. the
unoccupied vehicle, see DIN 70 020), must also be added. In the case of rear
independent wheel suspensions and compound crank axles, designers prefer to
use negative camber to increase lateral tyre grip; the mean value for the kerb
weight can then be:
Camber - 10 30' + 20';
maximum difference between left and right 20'.
(3.4c)
The existing setting options allow tighter tolerances here. On semi-trailing
link axles there is a danger of too negative a value in the fully laden condition
(Fig. 3.49); this could lead to the risk of the tyres becoming excessively warm
and the protective cover coming free. This is the reason why passenger car
manufacturers have reduced the kinematic camber alteration on this type of
suspension by means of the angles ex and r3 of the control arm axis of rotation
(see Fig. 3.36 and Section 2.2.6.5).
3.5.2
Kinematic camber alteration
As described in Section 1.2.1, one disadvantage of independent wheel suspension is that the wheels incline with the body on a bend, i.e. the wheel on the
outside of the bend goes into positive camber relative to the ground, and the
lateral grip of the tyre under the greatest load (unlike the one on the inside of the
bend) reduces (Figs 3.54 and 3.55). To balance this out, manufacturers tend to
design the suspension on passenger cars such that the wheels go into negative
camber as they travel in bump and into positive camber as they rebound (Figs
3.47 and 3.48).
On the x-axis, negative camber is given in degrees on the left and positive
camber on the right, whereas wheel travel is plotted on the y-axis; wheel bump
travel s\ is plotted in mm upwards and rebound travel S2 downwards. The curve
for the double wishbone suspension, which bends sharply into the negative
Wheel travel and elastokinematics
Fig. 3.47
In independent
wheel suspensions, the wheels
incline with the body when the
vehicle ;s cornering (Fig. 1.6). To
even this out, the wheels, in
bump travel, should go into
negative camber and the
rebounding ones into positive
camber.
r-----------I
"-
~
1--\.
~\
"'\
I
80
~
\
' - - Compressed
------ Normal
,
Mercedes
,
179
I
Bump travel of wheel
I
-
mm
60
\
'
\
40
BMW~
.
\~~~
\
\
_4°
_3°
_2°
_1°\ \
Negative camber -Ew
I.
Curb weight: BM
~,Honda
.\\ ~.
\\
\.
\
Curb weight: Mercedes
1°
2°
3°
Positive camber +Ew
\.
20
\
1\\
\
1\
\\
40"\
~
60
~\
mm
"
80-Rebound-travel of wheel
I
I
Fig. 3.48
I
Camber alteration on the front double wishbone suspension of a Honda
Accord (Fig. 1.55) as a function of the wheel jounce travel 5, and rebound travel 52 in
comparison with the McPherson suspension of a 3-series BMW (Fig. 1.40) and the
strut damper axle of a Mercedes.
,I
180
The Automotive Chassis
I
I
80 ~ Bump travel of wheel
,
"'," .~
mm
~
6n.
\
""- ~
-.. .
~
""~.~
\
\
- - 4°
_3°
'"
\\
'.
_2°
\"'
negative camber
\,
I
I
curb weight BMW
I
I
I
I
curb weight Honda
4:
---./
I
I
curb weight Mercedes
"
\.
\".
~
\
.
\.
20
'\.
'\
\
\
2°
1°
\
\
~
~\-
positive camber
3°
-
_Honda
40 \ . .
\
~ ~BMW
Mercedes----\
\, 6:
mm
\
'"
-...
80-, Rebound-travel of wheel
I
I
1
Fig. 3.49
Camber alteration on the rear wheels of a Mercedes, a 3-·series BMW
and a Honda Accord. The multi-link independent suspension of the Mercedes has a
fairly precise camber setting. In the empty condition this was eW,O.l = -55' and
eW,O,rs = -35' and increased to around -1 °30' when there were three people in the
vehicle. When the springs compress, the curve shape is slightly progressive. The
manufacturer's specification for the empty condition is ew = -50' ± 30' (see Ref. [2],
Section 5.3.4).
The multi-link axle of the BMW (Fig. 1.1) exhibits a straight-·line curve; when the
springs deflect in bump travel, the negative camber is less than on the Mercedes.
The double wishbone suspension of the Honda (Fig. 1.55) has zero camber in the
design position, but the wheels take on higher alterations (negative values) when the
springs deflect in bump travel than on the two other suspensions.
Wheel travel and elastokinematics
181
during the compression, shows the advantage of this axle. For the McPherson
strUt or strut damper the curve bends (unfavourably) in the other direction.
However, the wheel on the strut dampers takes on more positive camber during
rebound, this being the equivalent of better lateral force absorption on the (less
loaded) wheel on the inside of the bend.
The camber alteration curves for rear independent wheel suspensions are
shown in Figs 3.20, 3.49 and 3.74, where improved properties can be seen than
on the front ones. As there is no steering input to be considered, the semi-trailing links or transverse links can adopt an improved position. From the zero position shown, as can be seen in Fig. 5.14, the Mercedes compresses by 53 mm
under full load" The camber is then SW.t = -2°50' and remains above the critical
value sWmax = -4°, which should not be exceeded.
3.5.3
Camber alteration calculation by drawing
From a construction point of view, the camber alteration on the front wheels can
easily be determined as a function of the wheel travel over the angle of alteration
~a of the kingpin inclination if elasticities are ignored. On double wishbone
suspensions, arcs with the suspension control arm lengths e and f must be drawn
around the points C and D (in other words the suspension control arm axes of
rotation) and, in the normal position, the centres of the outer ball joints marked
as points 1 and 2 (Fig. 3.50). A point 3 is determined on the upper arc and an arc
with the path 1,2 drawn around it to give point 4. The line connecting them, 3,4,
then has the alteration angle ~a to the path 1,2, if the wheel compresses by the
path S!. If it goes into negative camber (as in the example), ~a must be
subtracted from the camber angle Sw,O in the normal position i.e.
BW
= Bw,O -- ~a (e.g. -40' -
2°
= -2°40')
(3.4d)
In the case of positive camber, Lla would have to be added:
Normal
- - - - Bump
e
......................
Fig. 3.50
Construction determination of the kingpin inclination alteration Llcr on double
wishbones which is equal to the
camber alteration.
------
-,----_. - ---_._-------
(
"'~
....
182
The Automotive Chassis
Normal
- - - Normal
-==~'it-..-=-c
Bump
-- -- .........
----
-.
Bu m p
........ -.
2
Fig. 3.51
Construction for determining the camber and kingpin inclination
alteration on the McPherson strut and
strut damper.
BW
=
Bw,o
Fig. 3.52
Construction for determining the camber and kingpin inclination
alteration on the longitudinal and transverse axes.
+ Llcr
On McPherson struts and strut dampers, the distance 1,2 is shortened when the
wheel is in bump travel, the upper mounting point is in the wheel house and only
the lower point 2 moves to 3. Llcr is again the angle between the two connecting
lines (Fig. 3.51).
The upper suspension control arm of the longitudinal link suspension (Fig.
3.52) requires a vertical to be created on the axes of rotation CC through the
point I so that point 4 can be obtained using an arc around 3 and the length 1,2.
If the axes CC were to deviate more from the horizontal, Llcr (and therefore the
camber alteration, Fig. 3.32) would improve.
An arc around vertical axis P must be drawn on the swing axle (Fig. 3.12).
The tangents drawn to this one after the other give the camber alteration
which must be subtracted from or added to Bw,O. The same applies to the semitrailing link axle where the arc needs only to be drawn around Pz (rear view,
Fig. 3.36).
3.5.4 Roll camber during cornering
When the body rolls, the camber of individually suspended wheels also changes,
on the outside of a bend by the angle LlBw,k,o and on the inside by ABw,k,i (Fig.
1.5). The mean value of the two LlBw,lf> = 0.5 (LlBw,k,o + LlBw,k,i) together with the
kinematic body roll angle <Pk gives the
roll camber coefficient ke,w,lf> = dBw/d<p
(3.5)
A wheel that is cambered positively to the ground on the outside of a bend by
the angle Bw,o = Bw,O + LlBw,k,o and one that is inclined on the inside of the bend
Wheel travel and elastokinematics
183
by the angle BW,i = Bw,O - .1B\v'k,i can experience an additional camber due to the
vertical force elements (Fig. 3.53):
Fe,\V,o
=Fz,w,o sin Bw,o
and
Fe,\v'i
=FZ,W,i sin BW,i
(3.5a)
The softer the suspension control arm bearings have to be, and the shorter the
path c on double wishbones (Fig. 1.5) or the distance i-o between piston and rod
guide on McPherson struts and strut dampers (Fig. 1.11), the worse the roll
camber becomes. The diameter of the piston rod (see Section 5.8.1) and the basic
kinematics of the suspension also have an influence.
The body roll camber factor can be determined by tilting the body over to
both sides and measuring the body roll angle and the camber angle. The wheel
travel in compression and rebound can be plotted on the y-axis instead of the
body roll angle (Figs 3.54 and 3.55), and the body roll angle can be easily calculated from this using the tread width br or 1':
SI + S2
d<p = - - (rad) and d<p = 57.3 d<p (degree)
(3.6)
brol' I'
The compound crank axle of the VW Golf has a track br = 1444 mm and where
the path is SI + S2 = 80 mm, the body roll angle is
d<p = 80/1444 = 0.00554 rad
= 3.17° = 3°10'
The progressive spring characteristic of this passenger car means the wheel on
the outside of the bend only moves in bump a little relative to the amount by
Inside of bend
Outside of bend
I
I
Fig. 3.53
When the
body (and therefore
also the wheels)
incline, the vertical
force element Fz,w,o sin
ew,O on a left-hand bend
I
--- --
I
I
-.l
pushesthewheE~lon
the outside of the bend
(here the right-hand
one) further into positive camber and the
force Fz,W,i sin eW,i
pushes the one on the
inside into an (equally
unfavourable) ne\Jative
camber (see also Fig.
1.6).
F-.Z,W,o
F:.Z,W,o =£Z,W,f + /j, F.:.Z,W, f
---'---'------------.----------------
The Automotive Chassis
184
f:
Q)
60 --+---f---+---f---::::::>"""""=t'--r-t-r--r-+
1
~
....
....
Twist- earn suspension
Rigid suspension _ _- - 1
~ 40-Longitudinallink suspension -+--~+-'7"-+--+--~
Semi-trailing suspension --t-----J
McPherson strut
20--+---+--;---t---:f-T-JrCI---t--+--
O--+---+--+--~:"'-_+---t--+
-20--+---+--;..'-:H--t---+"t:l
§ -40 --..-----+-.L--:L-1f---t--o
.c
Q)
::
1
0::
-80-r-H~+--.--+_-_t_-__ir_-+
00
Camber
20
Fig. 3.54
Camber alteration relative to the ground of various rear-wheel suspensions in the case of reciprocal springing; with the exception of the rigid axle, the
wheel on the outside of the bend goes into positive camber and the one on the
inside into negative camber on all configurations. Wheel travel when the wheels
compress and rebound is entered on the axis of the ordinate. The body roll angle 'P
is easy to calculate using the path differences ~Sl and ~S2 (see Equation 3.6).
which the opposite wheel rebounds (see Section 504.2). Given the permissible
axle load, the following paths are assumed:
SI
= 27 mm
and
S2
= 53 mm
The following values arise:
Camber
Camber alteration
SW,o = -0.1 0; SW,i = -3.55°
dSw,k = (sw,o - sW,i)/2
dSw,k
[-0.1 - (-3.5)]12 1.7°
=
=
and (referring to Equation 3.5) as a body roll camber factor
ke,w,'fl = dsw,k/d<p = 1.7/3.44 = 0.49
The average roll camber factors for the following axles are:
(3.7)
Wheel travel and elastokinematics
185
120
Q)
It
mm
v/
Fermissible ax e load
Starting point .
~-80 I/Ior measuring
Z
Ai
Camber
(in re lation to the road)
8°
_6 0
/
-r;
/
/
f' 40
~
Toe-in
~o
0
_2 0
/
7
•
20
0
,-
40
60
8°
Angie
-20
-4~
-0
c
-60
g
.0
Q)
a:
mm
~ It
-100
Fig. 3.55
Values for toe-in and camber angle measured by VW on the compound
crank axle of a Golf with reciprocal springing, entered as a function of the wheel
travel relative to the body. The bump-travel wheel on the outside of the bend goes
into positive camber and the rebound-travel one on the inside of the bend goes into
negative camber relative to the ground. The vehicle was measured with permissible
rear axle load. Toe-in does not alter favourably. Figure 1.6 shows the body roll angle
<p and Fig. 3.38 the relevant thrust centre point SM.
longitudinal link axles
McPherson struts
double wishbone suspensions
compound crank axles
rigid axles
1.05
0.85
0.80
0.55
0.0
3.. 5.5 Elasticity camber
In addition to the body roll camber, the camber alteration caused by the lateral
forces must also be taken into consideration. In accordance with DIN 70 000,
LleW,k,e is the proportion of the camber of a wheel that can be ascribed to the elasticity in the suspension and the steering, and is caused by forces acting between
the tyre and road or by their moments.
---~-----,---------------------------_._--'--
The Automotive Chassis
186
Figure 3.56 shows the values calculated on the McPherson strut front axles of
two passenger cars and Fig. 3.57 those measured on various rear axles. If there
are no test results available, the following can be taken as the elasticity camber
coefficient (per kilonewton):
dew,k/dF::::: 22'/1 kN
(3.7a)
• For further details, see Refs [2] and [9] .
Fig.3.56
Toyota
5+---+-\+--I-I-',-+----t---+--
t
~
1/
kN
4
,Vfvw
3
i _ ;1
<tl
Camber alteration
measured on the driven McPherson
front suspension of a lower mid-size
passenger car with lateral forces
directed inwards and applied statically at
the centre of tyre contact. Wheel disc
elasticity was eliminated on the
measurements, and caster (which has
no influence here), was ignored.
,/
~ ~
V
o
1°
2°
Camber angle
5
f-
kN
oyota
Fig. 3.57
Elastic camber change
measured on various non-driven rear
axles of mid-size passenger cars with
lateral forces introduced statically in the
middle of the centres of tyre contact.
The type of axles were:
t4
Q)
....u
0
'+-
3
<tl
....
Q)
.....
<tl
....J
2
1+---J'.../-T:.....-.J----.J----f------ir--
o
1°
2°
Camber angle
Opel:
Fiat:
Lancia:
Toyota:
Renault:
twist beam suspension
twist beam suspension
McPherson strut
McPherson strut
trailing link suspension
The low elasticity of the compound
crank axles is clearly visible.
Considering the caster would also give
the same results.
Wheel travel and elastokinematics
Fig.3.58 The toe-in
of both wheels in
accordance with the German standard DIN
70 020 is the difference in dimension b - c
in mm, measured on the rim flanges at the
level of the wheel centre.
187
t
fA.t
Direction
+OV,O
~-'\
c
"""'---=--1-"
,,\
)
r-o------o-"\ \
I
J----..=.b-_-.-4 \
r/l. t
=b -
C
k--- ""
c-fC)l
"'-----/
,3.6
3.6.1
Toe-in and self-steering
Toe-in and crab angle, data and tolerances
In accordance with standard DIN 70 000, the static toe-in angle 8'1,0,1 or rs is the
angle that results in a standing vehicle (reference status), between the vehicle
centre plane in the longitudinal direction and the line intersecting the centre
plane of one left or right wheel with the road plane. It is positive, when the front
part of the wheel is turned towards the vehicle longitudinal centre plane and
negative ('toe-out') when it is turned away.
The total toe-in angle 8V,o,t is obtained by adding the toe-in angle of the right
and left wheels. The total value is sometimes still given in millimetres (as stated
in DIN 70 020, part 1). The toe-in is then the dimensional difference rA,t = b c (Fig. 3.58), by which the rim flanges at the back are further apart than at the
front. The toe-in should be measured at the height of the wheel centre, when the
vehicle is empty, with the wheels pointing straight forward; rA,t therefore relates
to both wheels of one axle. Expressed in degrees, the toe-in angle 8'1,0 of a wheel
corresponds to the tyre slip angle Ctf (see Section 2.8.1); i.e. where there is toein, the front wheels of a vehicle are set to slip (drift), with the disadvantage of
an increase in rolling resistance (Equation 2.4) of
AFR ::::: 0.01 F R per
The toe-in dimension
angle 8'1,0 (i.e. rA,t/2):
in radians
in minutes
8'1,0
rA
8'1,0
= 10'
(3.7b)
of just one wheel is included in determining the toe-in
(3.8)
= rAID
8''1,0
= rAID
X
57.3
X
60
(3.8a)
rA. should be taken at the rim flanges, which is why its distance D must be
considered. With a given toe-in dimension, e.g. rA = 2 mm there is a larger angle
._-----,--------,--------_-_---0_----------
188
The Automotive Chassis
Rim diameter: 12" 13" 14" 15"
60
min
SO
t
0
;>
40
30
l()
20
10
2
3
5
4
't::.
6
7
8
9 mm 10
...
Fig. 3.59
Toe-in angle Ov.o as a function of rim size and toe-in rli in mm, measured
on one front wheel.
on small 12" rims than on ones with a 15" diameter. Figure 3.59 shows the influence of the rim diameter and Fig. 2.11 the individual dimensions: D = d + 2 b.
A tyre moving in a straight line has the lowest tyre wear and rolling resistance. When it rolls, a rolling resistance force F R, directed from front to back,
arises at the centre of tyre contact, which generates a moment with the lever arm
r a, which is absorbed via the tie rod to the steering (Figs 3.60 and 3.111, and
Equation 2.4).
As a result of existing compliance, particularly in the suspension control arm
bearings, this moment pushes the wheel backwards slightly and, in order to make
it run straight when the vehicle is moving, 'slip' is set as toe-in when it is stationary. In front-wheel drive vehicles, the traction forces directed from back to front
Fig. 3.60
The rolling resistance causes a longitudinal force FR in the wheel centre, which pushes
the wheel backwards into toe-out via the lever ra;
for reasons of simplification, the steering axis EG
(Fig. 3.103) is assumed to be vertical in this and
the next illustration. The moment M R = FRra causes
the force Fr to arise in the tie rod. Braking force
Fx.W,b operates in the same direction as FR but has
a different lever (Figs 3.108 and 3.109, and Section
7.1.7 in Ref. 3).
I
Direction
fFx,w,b)
Fj
110-_
<.----->
Wheel travel and elastokinematics
Fig. 3.61 On front-wheel drive vehicles, tractive
forces Fx.w.a attempt to push the wheels into toe-in.
The tie-force Fr arises on both sides; the same applies
to driven rear axles (Fig. 3.64).
189
t
Direction
.I
Fx,w••
n
attempt to push the wheels together at the front edge (Fig. 3.61), so toe-out (i.e.
negative toe-in) alignment can be beneficial. As a result of the built-in elastokinematics (Figs 3.83 and 3.86) and in order not to cause a deterioration in the driving
stability in the overrun (coasting) condition (i.e. when the driver removes his foot
from the accelerator), front-wheel drive vehicles may also be set with toe-in.
In addition to the absolute value of the total toe-in, tolerances must be specified for both front wheels which, because they can be adjusted by changing the
tie rod length (Fig. 4.13), only need to be aBv,o,t = 5' per wheel. Average values
in factory information for toe-in are
on rear-wheel drive vehicles aBv,o,t = +15' ± 10'
(3.8b)
on front-wheel drive vehicles aBv,o,t = 0° + 10'
(3.8c)
With semi-trailing links it is possible to alter the toe-in on the rear axle by swivelling the axis of rotation of the suspension control arms (Figs 1.15, 1.16 and
3.62) and, on 'double wishbone suspensions', by a lateral length alteration on
one suspension control arm (Figs 1.1 and 1.62!. Tolerances of aBv,o,~ = +?' can
be maintained where there is a setting. If thIS has not been prOVIded In the
design, values of aBv,o,t = +25' are almost inevitable, if tight component tolerances are not to render manufacturing uneconomical.
'Fig. 3.62
Hexagonal bolts with eccentric discs,
which come into contact with lateral collars, can be
provided for setting camber and toe-in on both
semi-trailing links (illustration: Ford).
- - - - - - - - - - - - - - ---_._-----------_._-------
190
The Automotive Chassis
Fig. 3.63 The difference between the
toe-in angle OV,O,r,1 on the left and Ol/,O,r,rs on
the right rear wheel determines the size of
the axle drive (heading) angle ±f3'. It is
positive if the median points forward and
left (see also Fig. 3.75).
X
X'
~
DV,O,r,1
\
\.
\
8v,o,r,rs
,.
1\
x x'
Regardless of whether the rear axle is steered or not, toe-in angles of the same
size, both left and right, are required to ensure that the direction of movement
x'-x' of the vehicle corresponds to its longitudinal axis X-X (Figs 3.63 and
3.75). The German standard DIN 70 027 therefore specifies that the so-called
crab angle W, must be quoted, i.e. half the total toe-in angle of the rear axle:
W=(oV,O,r,rs -
0v,0,r,I)/2
(3.8d)
Where it is possible to set the toe-in, W= + 10' can be maintained; if there is no
facility for setting toe-in on independent wheel suspensions or the vehicle is
fitted with a twist beam suspension or a rigid axle (Fig. 3.75), up to ~' = +25'
must be allowed to enable economical production.
Taking as an example a passenger car with OV,O,r,1 = -10' and OV,O,r,rs = +5' in
accordance with Fig. 3.63:
W= [+5' -
(-10')]12 = +7.5'
This means the angle is positive.
The toe-in on the rear axle of passenger cars is Ov,o,t = 10-20'; the drawing
information for a vehicle with independent wheel suspension would then, for
example, be:
toe-in 15' + 10', crab angle maximum + 15'
BMW already specifies this condition for all models:
geometrical crab angle 0° + 15'
and, on vehicles with compound crank axles, VW specifies
maximum permissible deviation from the direction of travel 25'.
Wheel travel and elastokinematics
191
3.6.2 Toe-in and steering angle alteration owing to wheel
bump-travel kinematics
Even more important than a toe-in which has been correctly set on the stationary
vehicle, is whether- this is maintained when the vehicle is moving or whether it
changes as a consequence of the wheels travelling in bump and rebound. This can
be the fault of inadequate steering kinematics (see Section 4.6) or deliberately
introduced to achieve certain handling properties. A change in the gradient of the
toe-in characteristic as a function of wheel travel should be avoided, as handling
properties then change unpredictably for the driver with variations in the load.
To avoid increased tyre wear and rolling resistance or impeding directional
stability (as shown in Fig. 3.64 and curve 1 in Fig. 3.65) no toe-in change should
Wheel travel
b ump (mm )
100
,
\
80
\,
Accelerating
Fx.w.a•r = 3 kN
r- ~
\
60
40
1---\\
. I
Free motion
\
\
I
,---
Braking
I
FX•W•b•r = 1.89 kN
~
\
Fiig. 3.64 Kinematic toe-in
alteration of one wheel on the
multi-link independent rear suspension of the Mercedes Benz S class
with barely any deviation from the
static value 3vo r = 12'. The illustration also shows the behaviour of the
wheel when subjected to a constant
drive-off force Fx.w.a = 3 kN (Fig.
3.113) introduced in the wheel
centre and an opposed braking force
FxWb = 1.89 kN acting at the centre
ot'tyre contact (Fig. 3.108), all beginning in the design position (see .
Section 5.3.4). As the tyre and spring
compresses when the vehicle
moves off, it goes +Ll3 e .r = 3' further.
into toe-in and, for elastokinematic
reasons , further into toe-in by +Ll3
. e •r
= 10' when the brakes are applied.
The rear axle stabilizes the braking
process (see Section 3.6.5.1).
20
1
r
\ \
Wheel position
when accelerating
10 ~o
-10
.... \.
\.\
-20
-40
-60
-80
•
I
positIon
\\- Wheel
when braking
\\
\~,.
~
l\
\\
\
-100
I
Toe-in (mm)
:Wheel position in
design position
\1\
Rebound (mm)
1
I
_-.,-------o-------r----------.--r
\
192
The Automotive Chassis
Fig. 3.65 Possible alteration of
toe-in of one wheel (in minutes) as
it bump and rebound travels, due
to an incorrect tie rod length or
position.
60'
80'
Toe-in
Fig. 3.66
Too short a tie rod (point 2) causes
rebound both the bump and rebound travelling
wheel to go into toe-out. However, too long a
tie rod (point 3) causes toe-in in both directions
(see Fig. 3.65).
/ 3
1
2
1--....
3
,
occur when the wheels compress or rebound. The wheel travel upwards (St) and
downwards (sz) is plotted on the y-axis of the figures, whereas on the x-axis positive toe-in is plotted to the right for one wheel each time, and negative toe-in (i.e.
toe-out) plotted to the left. The ideal curve 1 would be difficult to achieve at the
design stage and certain deviations from the ideal shape have to be accepted.
A toe-in alteration can be the result of incorrect tie rod length or position.
Provided that the steering arms are behind the front axle (Figs 3.60, 4.3 and 4.4),
the example of a double wishbone suspension can be used to explain how different length tie rods act (Fig. 3.66). If they are too short (point 2), they pull the
wheels together at the back both during bump and rebound travel, and go into
toe-out as shown in curve 2 of Figs 3.65 and 3.67. Tie rods, which are too long,
push the wheels apart in the direction of toe-in, curve 3; in both cases the graph
displays a high curvature.
If, when the tie rods are the correct length, the inner joint 4 is too high (or the
outer one too low, Fig. 3.68), when the wheel rebounds, the back of the wheel is
drawn inwards and toe-out occurs; whereas, when it compresses, the wheel goes
into toe-in. This results in approximate straight line running but at an angle
Wheel travel and elastokinematics
Fig. 3.67 As can be seen in Figs
3.66 (point 2) and 4.46, too short tie
rods on McPherson struts cause the
toe-in curve to be bent. If the steering arms are behind the axle, both
bump and rebound travelling wheels
go into toe-out. The figure shows
values measured on the left front
wheel on three front~wheel drive
vehicles of the lower mid-size
range. Curve 3 shows a 'roll-steer
E~ffect' on the front axle. This
measure, which tends towards
understeering, is achieved due to
the difference in height between
the inner and outer tie rod joint
(which can be seen in Fig. 3.68
under point 5).
3
193
2
1
80
mm
-t----..t+-4---~
c.
E
::J
co
Q)
>
-r------1f-----\t'Ol\-- ~
a;
-----f-
Q)
'..r:::
3:
-2 0
+1°
Toe-in
-.
.
aIteratfon
"C
§---~
o
.0
Q)
c:::
Fiig. 3.68 An inner tie rod joint that is too high
(point 4) produces curve 4 (Fig. 3.65), and one that
is too low generates curve 5.
.-0
.-'-
'-'-'-'0
4
1
5
(curve 4 in Fig. 3.65). A tie rod joint 5, which sits too low on the inside or too
high on the outside has, as the corresponding curve shows, the opposite effect as. do steering arms, which point forwards in all observed cases (Figs 4.37 and
4.39).
3.6.3
Toe-in and steering angle alteration due to roll
As a further example, Figs 3.69 and 3.70 show a deliberate steering alteration.
W"hen cornering, the outer bump-travelling wheel goes into toe-out and the inner
rebounding one into toe-in. The steering angle is thus reduced slightly under the
r
I
194
The Automotive Chassis
"100t-------;------j------t------l-
1
Cl.
E
ci5
80+---;--;:----;--;+------+-----+-----1Both
Left wheel
wheels
Right wheel
20
Design position
Q)
>
co
........
Q)
Q)
..c
0
-30'
+30'
+1°
+1° 30'
Toe-in
20
S
40
60+------t-----\-\:--t--~-__+-----l-
80+------+-----~l\_----*------l"'0
C
:::l
o
.0
Q)
c::
~
100 + - - - - - - ! - - - - - - t - - - J . . - - - - - + - - - - - - - I -
120+------!------t-------+-------I-
Fig. 3.69
Toe-in alteration recorded on an Opel Omega (1999) indicating body roll
understeering on the front axle. The individual wheels were measured to obtain the
total toe-in. The design position relates to the vehicle with three passengers each
weighing 68 kg; the height of the unladen vehicle is also marked.
influence of the body inclination in order to achieve body roll understeering on
the front axle or to improve handling when changing lanes (Fig. 3.71, see also
curve 3 in Fig. 3.67).
As described in Section 3.6.4, rear axles can tend to lateral force oversteer which can lead to an overswing of the vehicle's rear end (Fig. 3.72). To compensate for this and make the overall handling of the vehicle neutral, designers like
to make the rear axle body roll understeer (Fig. 3.73). On individual wheel
Wheel travel and elastokinematics
195
Co
Left wheel
Right wheel
Overall toe-in
t
0--0
lC-·-lC
E
til
~-I
~
-2
mm
I
C]
-1
~
I
Design position, lowered vehicle
+1
I
+2
mm
L
+3
Toe-in
Normal
position
40
60
1· is equivalent to 6 mm
80
"'0
C
~ a:
::::l
0
.0
Q)
100
mm
120
\
0
.
\
Fig.3.70
Toe-in alteration measured on a VW Golf Gli that has been lowered by
As = 30 mm. In the normal position (also marked as specified by the manufacturer),
ciS the wheels bump and rebound, the alteration values (which have a negative influE~nce on directional stability and tyre wear) are less than in the lowered condition. The
(now minimal) residual small compression spring travel can be seen clearly.
Direction
t
Fig. 3.71
If the bump-travelling wheel on
the outside of the bend goes into toe-out
and the rebounding one on the inside of the
bend into toe-in under the influence of the
body roll inclination (or due to lateral forces),
the steering input is slightly reduced by the
angle ~O'l',f. The axle understeers.
r
The Automotive Chassis
196
--.-- --
-
---'--- ---.
--'-'-
/
~._~-
~e.r
._.-.- ---'--' ---'--
Fig. 3.72
Under the influence of a lateral force, the rear axle can take on the
proportion of the steering angle ilSe,r - or the suspension links may be deformed
accordingly -- so that the vehicle oversteers to the inside edge of the bend (left and
Fig. 2.42). To correct this, VW install track-correcting bearings which largely prevent
oversteering (see Section 2.3.5 in Ref. [2)). Another possibility is to allow body roll
understeering of the axle, (see Figs 1.30, 1.31, 3.77 and 3.78).
Fig. 3.73
To reduce the tendency to oversteer, the rear wheel suspension can be
designed so that body roll or lateral force understeering of the axle is possible, i.e. under the
influence of the body roll (or lateral forces) the
compressing wheel on the outside of the bend
goes into toe-in and the rebounding one on the
inside of the bend into toe-out in proportion to
the steering angle ilSe,r.
Direction
,
I
I
I
,
I
I
,
I
I
suspensions the bump-motion wheel on the outside of the bend in this case must
go into toe--in and the rebounding inner one into toe-out; Figs 3.20 and 3.74 show
this type of alteration curve (see also Section 2.12).
As they are directly linked to one another, the. wheels of rigid-axle and twistbeam suspension have no toe-in alteration where the springing is parallel.
However, due to design tolerances or incorrect installation, the axle can sit at an
angle in the vehicle, i.e. one wheel has toe-in and the other toe-out in respect of
the longitudinal axis of the vehicle. In this case, the direction of movement x'-x'
•
Wheel travel and elastokinematics
1
100
Q)
--+-~~---+---
197
Track
alteration
mm
>
...E -80tT-----t--I-----+Q)
Q)
..c:
~
60
--+------+--4,--- 40t-t----/--f--------+20
--+------+------..!~
Or--f----f--------+-
-60
i
..
_3·
i
i
I
_I,.
_2·
:Camber
i
i
i
,
i
I
b
I
I
_1·
-80
mm
-~o
-~o
.-Narrowing a tread wi th
..
i
-0.8 0
i
-0.'6 0
I
Toe-out
-0.'4
0
-0.'2 0
0
b
.i·
.1·
I
i
.3·I
I
Camber
,
I
i
...,.
(
I
. '0
m.m
80
arrowtng of tread wldth--.
O.~o
0.4 • 0.6
0:8
~o
i
0
0
i
0
Toe-in--'
Fiig. 3.74 Kinematic properties of an Audi A6 Quattro (1996) as the rear wheels
compress and rebound. The relatively small tread width alteration of the two wheels,
the favourable negative camber as the springs compress and the toe-in alteration (of
one whee!), which points to roll understeering of the rear axle, are clearly visible.
of the vehicle and its longitudinal axis X-X deviate from one another by the crab
angle (Figs 3.63 and 3.75).
On compound crank axles, the bearing points a shown in Fig. 3.37 move under
the centre of the wheel when the vehicle is loaded, resulting in negative angles K.
This results in increasing body roll understeering, respectively, decreasing roll
oversteering with load and therefore an improved roll-steer factor (Fig. 3.77).
r
198
The Automotive Chassis
x
Fig. 3.75
\
Front wheel is
parallel to rear
wheel
I1<5v,O,f -<5
- V,O,r
I
I
I
I
I
If the rigid rear axle is
not fitted at a right angle to the vehicle's longitudinal axis X-X, i.e. if the
vertical on it deviates from the direction of movement x'-x' by the crab
angle l3', a slight steering input is
necessary to make the vehicle move
in a straight line. The figure also
shows how the self-steering of the
rear axle makes it necessary to turn
the front wheels if the vehicle is to
move in a straight line on an uneven
road surface under reciprocal springing (Fig. 1.21). The axle can displace
by the angle 8v.o,r = f3' (Figs 1.28 and
3.63).
'\
I\
x' X
Direction
....
c
SI~G
r---
_---
o
---
...........
--- -
Fig.3.76
If the body of a rigid rear axle on the outside of the bend led by two trailing link pairs has bump-travel in path 51 - caused by trailing links at an angle to one
another and of different lengths, as can be seen in Fig. 3.161 - the axle centre is
drawn forwards slightly by the path il/1 (left) and pushed backwards by il/2 on the
inside by 52 rebounding. As a result of this, the rigid axle moves by an angle and roll
understeers. This reduces the tendency to oversteer of standard vehicles.
Even with rigid axles, body roll understeering can be achieved by - as shown
in Figs 1.28, 1.29 and 3.76 - the axle being drawn forwards on the outside of
the bend when the body inclines and backwards on the inside of the bend. The
alteration Llo<p,r of the steer angle in the axle as a whole, divided by the alteration Llcp of the kinematic roll inclinations, is termed the 'roll steering factor'
(Fig. 3.77).
A rigid rear axle which self-steers when the body inclines also self-steers
when going in a straight line on an uneven road. The steering effect this
causes occurs not only on reciprocal bend springing (Fig. 1.21) but also on
unilateral springing. This is the reason why 'self-steering', which can only be
compensated for by spontaneously turning the front wheels (see Fig. 3.74), is
limited.
-_.' -,_.. -----,-------------------------------
Wheel travel and efastokinematics
199
-r---f--4---1--+30'-r---t_ _+-_-+-_---+_
-q;
•
•
o
-4 0
4 passengers
-r--+--+---I--30't---t----J----I-_ _+_
-6mT,r
-q;
~'
......
'?J
Bendto
~ the right
..Gl.c
Bend to \
the left
Angle? po~iti.on by the angle d8<p,r ~s a function of the roll angle cp
measured on the driven rigid rear axle of a conventIonal passenger car occupied with
two and four people. When there are two people in the vehicle and <p = 4°, d8<pr = 6'.
'
The body roll-steering factor would then be d8<p,rld<p = 0.1 °/4° = 0.025.
~hen t~ere are four pe~ple in the vehicle, this increases to 0.075; the tendency
of thIS vehicle to oversteer IS therefore reduced, depending on the load.
':ig. 3.77
3.6.4
Toe-in and steering angle alteration due to lateral forces
Increasing lateral forces try to push the turned-in front wheels with the lever of
the kinematic caster r'T,k and the caster offset r'T,T (Fig. 3.120) into the straight
running position. As a result of the elastic compliance in the system this reduces
the steering angle and lateral force understeering takes place.
To achieve this on the rear wheels (as shown in Fig. 3.73), the wheelan the
outside of the bend has to go into toe-in and the one on the inside of the bend in
the direction of toe-out.
To some extent, exactly the opposite of this can be seen in Fig. 3.79. The rear
wheels of the twist-beam crank axle (Opel and Fiat) on the outside of the bend
are pushed into toe-out by the lateral force Fy,w,o and the ones on the inside of the
bend into toe-in by FY,W,i. The result is lateral force oversteering (Fig. 3.72),
which is also noticeable on the longitudinal link axle of the Renault (Fig. 1.63)
and is also slightly evident on the McPherson struts of the Lancia (Fig. 1.12).
Toyota moves the two transverse links 1 and 2 (Figs 3.2 and 3.80) backwards in
parallel, therefore achieving the elastokinematic steering angle alteration .6.8e ,r on
the outside of the bend and (as shown in Fig. 3.79) toe-out on the inside.
r
I
200
The Automotive Chassis
Degree
~,
11
Radiant
0.01
b-'--r'+---,-£--f7'--7"l--r-t+--;;i''----7'
... 6lJl, ~ 0.5 t---t--t--i-;:-Pe-r-lm-:-is-:si7""":bI+-e-ax"7,e-+I-oa-d-J
4 passenger~
2 passengers
+0.1
Radi~nt
,Understeering
Fig.3.78
Roll-steering measured on a VW Polo; increasing the load increases the
understeering of the twist-beam suspension. At <p = 4°, the roll-steer factor is 0.025,
0.07 and 0.1 depending on the load.
For the measurement, the lateral force was applied statically in the centre of
the tyre contact; shifting it backwards by the caster offset rT,T = 10 to 40 mm
would cause all toe-in curves to turn counter-clockwise. The Toyota Corolla
would then have a slight tendency to lateral force understeer, whereas there is an
increased tendency on all other passenger cars to oversteer.
j
Another way of reducing this would be to give the rear wheels negative c a s t e r l
-hr;,kh(~igSlf3.1d17and 3· 144)1;' howelver, t~is must b(pe.gre ater thadn3th1at90fTthh~ tyre,
1.
w lC Itse re uces as the s Ip aug es ex. Increase IgS.
i
2 5O an . 1). IS can
be achieved on double wishbone suspensions (Pig. 3.145 and Section 5.3.4 i n '
Ref. 2); the negative caster increases on the bump-travelling wheel on the
outside of the bend and under load.
Even with rigid axles lateral force understeering is possible. If the panhard rod
is behind the axle casing (Pigs 1.61 and 3.81), the effective distance a between
the lateral forces Fy,w,r,o and FY,W,r,i on the two rear wheels and the rod force FT,y
results in a pair of forces that generate the forces + F x in the trailing links and due to the elasticity in the rubber bearings - causes the desired self-steering.
3.6.5
Toe-in and steering angle alteration due to longitudinal
forces
3.6.5.1 During braking
Toe-in leads to stabilization of the vehicle braking. This means better straightrunning behaviour and it can be achieved both by negative kingpin offset (Fig.
6.12) and by an elastokinematic toe-in alteration.
The front end of the vehicle moves towards bump when the brake is activated.
If (as shown in Fig. 3.69) the body roll has been kinematically designed to
understeer, both front wheels go into toe-out, i.e. with a positive kingpin offset,
Wheel travel and elastokinematics
201
1:'
o C
~Q)
.0
. Q)
..c'
Q).c
Opel
~~
o
0
..... Q)
(ij1:'
~.-
Q)
(I)
-+----+--\-+10 '5
Renault
,
"",
,
-oJ
0
Toyota/
\.
\.
4--+---t--+-+-_
/
\.
", "
Fiat'
-45'
-30'
I
Toe-out
+30'
/
/
._ 1:'
~c
..c'~
Q)
Q)
....
U.c
~
,E .....
,
+45'
I •
Toe-m
,
\
\
_0
co
Q)
~1:'
Q).-
....
(1)
-oJ
co _C
Fiig. 3.79 Lateral forces introduced statically in the centre of the tyre contact of
different rear axles produce, in the Toyota, a steering angle change ~6e.r in the direction of the toe-in on the outside of the bend, but toe-out on the other vehicles tested;
these exhibit a lateral force steering tending towards oversteering. The vehicles are
fitted with twist-beam suspension (OpelNauxhall and Fiat), McPherson strut (Lancia
and Toyota) and trailing link suspension (Renault, Fig. 1.63). If the lateral force operates in the other direction (i.e. from the inside out), there is toe-in instead of toe-out
on the vehicles. The toe-in alteration in minutes appears on the x-axis and the force
in kilonewtons on the y-axis.
Fig. 3.80 Under the influence of the lateral force
Fy,w.o acting on the outside of the bend behind the
wheel centre by the tyre caster (T.T, the mountings
of the transverse link 1 flex more than the brace 2,
which is offset backwards; point 6 moves to 7 and
thH toe-in angle A3 e.r occurs elastokinematically (see
also Fig. 3.2).
r
f
I
I
'---
I
I
202
The Automotive Chassis
i
Direction
~
Y,W,r,o
Fv,W,r,i
a
Fig. 3.81 The effective distance a between the lateral forces FY,w,r,o and i on the
wheels of the rigid axle and the force Fr. v on the panhard rod at the back leads to a
force pair which generates the forces ±Fx in the longitudinal links and can cause
lateral force understeering due to the compliance of the rubber mountings. If the rod
is in front of the axle, oversteering is possible.
and they continue to travel in the same direction in which they were already
being pushed by braking forces FX,W,b (Figs 3.60 and 6.11). To limit this effect,
the necessary counter-steering in the direction of toe-in can be achieved, r = 0
or there is a small positive kingpin offset at ground. The only prerequisite for this
is a top view angle ~ between transverse link 1 and tie rod 7 (Fig. 3.82).
Using a Mercedes model as an example: the front of the longitudinal rod 4 is
anchored at point 6 on the suspension control arm, and the back carries the
supporting bearing 5. Under the influence of the braking force FX,W,b the defined
longitudinal elasticity of part 5 yields, the lower guiding joint G moves out to 4
and the outer tie rod joint U moves to 9. As points G and U move in different
arcs and the tie rod joints are also less laterally compliant than the bearing D of
the transverse link 1, both front wheels are pushed into toe-in in spite of the
opposing moment Mb = FX,W,b rb seen in Fig. 3.109.
In the same way, individually suspended rear wheels can experience an elastokinematic toe-in alteration during braking (Figs 3.2 and 3.64). For further
details, see Refs [2] and [6].
(J"
tDireClion D
Fig. 3.82 A positive top view angle
i; between the tie rods 7 and the
transverse links 1 close to them (mostly
the lower ones) can cause an
elastokinematic toe-in alteration during
braking.
II
Wheel travel and elastokinematics
203
:3.6.5.2 Longitudinal suspension without toe-in alteration
Nowadays, manufacturers fit only steel radials to series production vehicles.
However, unlike the cross-ply tyres used in the past, these have the disadvantage
of dynamic rolling stiffness (see Section 2.2.2). The very stiff belt causes longitudinal oscillations ~hich, on independent wheel suspension, are transferred to the
body via the steering knuckle and the tie rod and - particularly on cobbles, rough
concrete and at speeds below 80 km h -I - can cause an unpleasant droning noise
inside the vehicle. The vibrations can be stopped if the steering knuckle is given a
precisely defined longitudinal mobility (compliance). This is a task that is not easy
to fulfil at the design stage because neither a toe-in alteration may occur, nor a
lateral force at the centre of tyre contact (Fig. 3.6) under the influence of the paths
of s ::> + 10 mm (Fig. 3.1), as straight rolling ability and rolling resistance would
deteriorate.
On the front axle it can be solved using a transverse link, which has an outrigger pointing backwards (or forwards; Figs 3.83 and 3.84), and which is
supported at the side in a rubber bearing with a highly progressive, precisely
defined spring rate. The important thing is that stiff bearing elements, which
only yield a little under cornering lateral forces and braking forces, sit in the
pivots D and G.
If a transverse link anchored at point D controls the wheel, it can have a hole
,---
Direction
t
o
\...- _ ---'---+
Fig. 3.83
To achieve the necessary longitudinal spring.ing BMW .fits the
boomerang-shaped (bell-crank) control arm 1 (shown separa~ely .In the next f.lgure) on
thl3 front axle of the Z3 Roadster. Under .the effec~ of longltudlna~ forces, It ~otates
around the (only slightly compliant) ball jOint 0 and IS supported wl~h th.e out~lgger 4
by means of a large rubber mounting on the bo.dy. In t~e .lateral direction this bearinC) has an initially soft, but then highly progressive, ~pnngrng curve.
.,
~ 1ie-rod 7 lies at the height of the control arm and IS almost parallel to the Irne IrnkinC) the bearing points GO; points U and G therefore move on an arc o! around .the
sa~me radius and longitudinal wheel movements do not. cause .any.toe~rn alteration.
As shown in Fig. 3.111, the rolling resistance FR, which vanes In Size, must be
observed in the wheel centre as F'R.
r
i il
I
,I
The Automotive Chassis
204
I
G
6
5
A
C
B
10r--r----r---Ir--rr-----.
kN
8r---r---H---f..-----t
t
6
IJ..>' 41---1-+-----.(---+----1
4r--,.----,.----.,...-----,
kN
3 r---+----.-+--- - - I
t
4
8~
2 r----t--+-f-----++---.l
1 1--~<t_--f:1L-_I_--l F.'
y
--~~I-*-
o~___=_'"=__-:!-----:-.L:_--.-J
o
0.5
1
s
1.5 mm
....
2
4
6 mm8
S - - I....
~
Fig. 3.84
Front boomerang-shaped suspension control arm of the BMW Z3
Roadster. The guiding joint 5 links the suspension control arm 1 with the suspension
strut and is press-fitted from below into hole G; the inner joint 6 sits in hole D. The
suspension control arm rotates around this part under the influence of longitudinal
forces and is supported by outrigger 4 on the transversely elastic bearing 8. Its
progressive compliance in the y direction is shown by the illustration on the right. In
the vertical direction (z) the bearing is stiffer.
In part 5 the rubber ring C is vulcanized in between the joint housing A and the
outer ring B, and is - as can be seen from the illustration on the left - laterally more
compliant (Fy) than in the longitudinal (x) direction (illustration: Lemf6rder
Fahrwerktechnik).
Ii
II
Wheel travel and elastokinematics
205
23± 0.2
4
3
21.5
2
4
LO
o
+1
to
r----t.-........,;~
0
+t
C\l
M cO
(0
1
LO
,...
,...
lSI
__........-I
"O"---+~
12 r - - - - - - - r - - - - - - .
12r----,-----..-....
kN
kN
Fz
8 t------+----r:--1
81-----+---1----1
4 t-------t7'-----1
41-----+.'------1
o
2
mm
5 - - - 1.......
Fig. 3.85
4
o
2
mm
4
5
Mounting of the anti-roll bar fitted at the front in the transverse links on
the Audi 6 (built until 1996) (Fig. 1.57). The two rubber parts in the suspension control
arms are vulcanized to the inner tube 1 and ring 2. Under the influence of longitudinal forces Fx one part comes into contact at the dome-shaped washer 3 and the
other part relaxes. As can be seen on the left, the rubber part 4 projects beyond the
sleeve 1; when fitted this achieves the necessary pre-tensioning. Ring 2 ensures that
it sits firmly in the suspension control arm, so that the mounting can transmit vertical forces Fz without complying too much. The diagrams show the longitudinally
progressive characteristic curve and the almost vertical linear characteristic curve of
both bearings when fitted (illustration: Lemf6rder Fahrwerktechnik).
206
The Automotive Chassis
Fig.3.86
u
1
Direction
o '
An A arm can be
replaced by two individual suspension arms: one is transverse (position 1) and carries lateral forces',
the other is longitudinal (position 5)
and transfers forces in this direction. A longitudinally compliant
bearing (position 4) in a hole in part
1 absorbs the dynamic rolling hardness of the radial tyres. As in the
Audi A6 (Fig. 1.57), and in the classic McPherson strut, part 5 can
also be the arm of the anti-roll bar.
in which a longitudinally elastic rubber bearing sits (Figs 3.85, 3.86 and 1.57).
The inner tube of this part is supported on the anti-roll bar 5 or a tension or
compression rod strut 4, pointing either backwards or forwards.
On driven independent rear wheel suspensions it is especially important that
the trailing or semi-trailing arms be controlled as well as possible to avoid elastic camber and toe-in alterations. The three or four rubber bearings, which link
the suspension subframe and the differential with the body, then have to be
designed so that the dynamic rolling hardness of the radial tyres is absorbed
(points 2, 3 and 4 in Fig. 1.15). This task is carried out by the bearings in the
longitudinal struts on rigid axles and by the rubber elements sitting in the pivot
points 0 on compound crank axles (as shown in Figs 1.30, 1.61 and 3.87).
3.6.5.3 Toe-in alterations due to front-wheel tractive forces
As can be seen in Fig. 1.50, on a transverse engine the differential is relocated .
from the middle of the vehicle to the manual transmission that is sited at the side,
resulting in drive shafts of different lengths. When the vehicle moves off in the
lower gears, the front end rebounds and the shorter (left-hand) shaft takes on a
steeper working angle <Xl to the wheel axis than the longer one (right-hand, Fig.
3.88). The clockwise/anti-clockwise moments about the steering axes EG which
combine to bring about toe-in result from the bending deflections due to rotation
of the drive-shafts:
MZ,W,a,1 or rs
= 112 F X,W,A
rstat
tan
(3.8e)
<Xl or rs
(For FX,W,A see Equation 6.36 and rstat in Section 2.2.5.)
As the angle <X is larger on the left, a slightly higher moment can arise there
than on the other side, with the risk of the vehicle pulling to the right. If the
driver takes his foot off the accelerator quickly, a braking moment is generated
by the engine, the front end dips and a steering effect in the other direction is
inevitable. This is the main reason why (as shown in Figs 1.51 and 1.57) frontwheel drive vehicles with powerful engines necessarily have an intermediate
shaft, and drive shafts of equal lengths.
Ii
Wheel travel and e/astokinematics
207
1
Z
..x::
-
1
2
3
s
...
mm
5
Fig. 3.87
Elastic bearing in the front eyes of the twist-beam suspension of the
Audi A6 (1996). The indents in the rubber part give the necessary elasticity, The bearing must be soft enough in the longitudinal direction to absorb the dynamic rolling
hardness of the tyres (when the axle shown in Fig, 1.61 is controlled precisely) and
not very compliant in the vertical direction to be able to absorb safely the forces Fz,o
which occur during braking (Fig, 3.160) (illustration: Lemforder Fahrwerktechnik).
VMiddle of vehicle
Steering
axis EG
¥
l
Direction of turning
r
~\
'
'ddl
f d'ff
'I
Lesser steenng moment,
MI ~
e 0 I erentra '\ MZ,W,a,rs
u""
/u
Driving torque
Greater steering moment,
Fig. 3.88
MZ,W,a,1
When the engine is transverse, the differential is no longer in the centre
of the vehicle and an intermediate shaft is necessary (Fig. 1.51), otherwise the drive
shafts are not the same length. If they are at different angles 0., different moments
can occur around the steering axes, causing the steering to pull to one side. a' = o.rs
can be achieved by tipping the differential by up to 2°.
The Automotive Chassis
208
-
Fig. 3.89
Kinematic relationships in accordance with Ackermann between the
steering angle 8A.o on the wheel on the outside of the bend and 8i on the inside of
the bend. The illustration also shows the ~8A and the track circle diameter Os (see
Fig. 1.69).
3.7
Steer angle and steering ratio
~:~~~~~ces 1 and 9 deal with this area in detail. Section 4.7 covers steering kine-
3.7.1
Steer angle
When the vehicle is moving very slowly and 'free of lateral forces', it will only
comer precisely when the verticals drawn in the middle of all four wheels meet
at one point - the centre of the bend M. If the rear wheels are not steered, the
verticals on the two front wheels must intersect with the extension of the rear
axle centre line at M (Figs 3.89 and 1.69) whereby different steer angles OJ and
OA,o occur on the front wheels on the inside and outside of the bend. The nominal value OA.o of the outer angle - also known as the Ackerman angle - can be
calculated from the larger inner angle OJ:
cot OA,o = cot OJ + jll
(3.9)
where l is the wheelbase and j the distance between the two steer axis extensions
(Figs 3.90 and 3.103), measured at the ground, i.e.
j = be - 2
(3.10)
fIT
here the kingpin offset fIT is negative, the integer is positive (Fig. 3.113).
The differential steer angle ilo A included in Fig. 3.89 (also known as the toe
.....
11
I
Wheel travel and elastokinematics
209
Fig. 3.90 Path designations
on the front axle; bt is the tread
width on the front and fa the (in
this case) positive kingpin offset
on the ground (scrub radius),
differ~nce angl~) ~ust always be positive for the nominal values calculated
(nommal curve m FIg. 3.92).
(3.11)
T~e theoret~cal trac~ circle diameter Ds can be calculated using the angle 8A ,o
(FIg. 3.89), I.e. the dIameter of the circle which the outer front wheel traces with
the largest steering angle (see also Equation 2.10). The turning circle of a vehicle should be as small as possible to make it easy to tum and park. The fonnula
Ds
=2 (
1
sin 8A ,O',max
+ TO' )
.
(3.12)
derived using the illustration shows that this requirement necessitates a short
wheelbase and a large steer angle 8A ,o on the outer wheel of the bend. This in tum
requires an even greater steering angle applied to the wheel at the inside of the
turning circle, though this is limited by the fact that the tyre must not come into
contact with the wheel arch or any of the front-axle components. The wheel
house cannot be brought too far into the sides of the front foot well as the pedals
(on both left and right-hand drive vehicles) would then be at an angle to the
direction in which the driver faces and foot-space would be restricted. In frontwheel drive vehicles, room also needs to be allowed for snow chains (Figs 2.8
and 3.102) and the largest working angle of the drive joints (Figs 1.3 and 1.53).
3.7.2
Track: and turning circles
The inner angle 8i is therefore limited, whereas the wheel angle on the outside
(for functional reasons a smaller ,angle) is not. This may be the same size as the
inner one. The disadvantage is that it impairs the cornering behaviour of the
vehicle (Fig. 3.91), but with the advantage that the track circle becomes smaller
and the lateral tyre force capacity on the outside of the bend increases. For this
reason, the outer steering angle is larger on most passenger cars, i.e. the actual
value 80 (without index A) is greater than the nominal angle 8A ,o calculated
according to Ackennan (Fig. 3.92) by the steering flaw Ll8 F• In other words, the
required steering deviation is as follows:
The Automotive Chassis
210
Fig. 3.91
To use the space available in the wheel house, it is an obvious idea to
turn the wheel on the outside of the bend inwards by as much as the wheel on the
inside of the bend; the wheels are then turned parallel and ~8A is zero. It is possible to increase the cornering force by turning the outer wheel more (compared
with the wheel on the inside of the bend, Fig. 3.92).
(3.13)
where ~o = OJ - 00 expresses the so-called differential steer angle.
The turning circle diameter D s shown in Fig. 3.89 can be reduced by deliberately accepting a steering deviation. In addition to ~OF, the angle OA,o,max, in other
words the largest outer nominal angle according to Ackermann calculated using
Equation 3.9, must also be known. A series of test measurements has shown that
a reduction by ~Ds : : : 0.1 m per 1° steering deviation can be achieved; the
formula which should include all dimensions in metres would then be
D s = 2 (.
sm
I
+ ra )
OA,o,max
-
O.l~oF(m)
(3.14)
A front-wheel drive vehicle with conventional steering geometry can be used as
an example.
The data when the wheels are turned to the right are:
I
= 2.677 m; bf = 1.47 m; r = -0.015 m; OJ.max = 42°; Oo,max = 35°40'
j = 1.47 - [2 (-0.015)] = 1.5 m
cot OA,o = cot 42° + 1.5/2.677 = 1.671; OA,o = 30°55'
a
~OF
= 35°40' - 30°55' = 4°45'
D s = 2 [2.677/sin 30°55' + (-0.015)] - 0.1 X 4.75°;
D s = 9.91 m
The turning circle diameter measured on the passenger car was DS,t = 9.92 m.
The turning circle radius is basically only a theoretical value which can be
calculated at the design stage; for the driver it is the swept turning circle kerb to
kerb that is important, in other words the distance between two normal height
kerbs (Fig. 3.93) standing parallel to one another, between which the driver can
just turn the vehicle. This circle diameter Dte,kb can be measured but can also be
II
,
10°
,
8°
1
o
c-o
<l
...
6°
v5
<]
a)
CJ)
c
,
4°
CO
....
Q)
Q)
.....
(J)
CO
'E
2°
~
....
Q)
~
o
o
_1°•
~
Nominal steering curve
according to Ackerman
-~t:-- ---
------
----10°
_-
V
/
----
--
/'
.v----
.~
_I-- - 20°
~
--
V
.........
/
[)/~
/
,...... ~ teering deviation BMW
I--
.J
V
~-
1/
I-' • Steering deviaVtion Mercedes
V-
V -"
/'
,/
Actual
curve BMW~r--Actual curve
Mercedes
./"'/
V
30°
Steer angle, inside wheel, OJ
Required, nominal steeri~g curve !or two standard passenger cars with the same wheelbase and approximately the
same track calculated in accordance with Equation 3.9. The mean value of the actual curve measured when the wheels are turned
to the left and right is included, and the st.eering deviation AOF (~Iso known a~ the s~eering error) is also marked. The steering angle
OJ of the wheel on the inside of the bend IS entered on the x-aXIS, and the differential steer angle Ao = OJ - 00 (which relates to the
actual curve) and AOA = OJ - OA,o (which is valid for the nominal curve according to Ackermann) are marked on the y-axis.
In the workshop manuals Ao must be indicated with a tolerance at OJ = 20°; on the BMW 3-series it would be do = 3° and on the
Mercedes Ao = 10', The differential steer angle of the Mercedes, which is negative up to OJ =:: 20° indicates that the wheel on the
outside of the bend is turned more than the one on the inside and so the lateral force absorbed by the front axle when it corners _
and with it the steering response - is increased.
Fig. 3.92
212
The Automotive Chassis
nl------~~
I
I
I
Fig.3.93 Turning
circle kerb to kerb
Dte,kb; an important
dimension for the
driver when turning
the vehicle,
calculated easily using the turning circle diameter Ds and the actual width of the
tyre (Figs 2.11 and 2.15):
Dte,kb
= D s + B (m)
(3.15)
However, the swept turning circle, the diameter of which D te is greater than that
of the circle by the front overhang length Lex,f (see the caption to Fig. 1.67), is
probably a more important dimension.
According to DIN 70 020, Dte is the diameter of the smallest cylindrical envelope in which the vehicle can turn a circle with the largest steering input angle
(Fig. 3.94). The smallest turning circle can be calculated at the design stage, but
is easier to measure and appears as manufacturer's information in the specifications or as a measurement value in test reports.
The radius Rr,o of the turning circle, which the rear wheel on the outside of the
bend traces, or Rr,i - that traced by the wheel on the inside of the bend - can be
calculated from the known turning circle diameter Ds (see also Fig. 1.69). These
are:
I
~'
---
----
---------------
Fig. 3.94 The swept turning circle Dte is the arc described by the parts of the vehicle protruding furthest outwards when the wheels are turned in at the largest steering angle.
Ii
Wheel travel and elastokinematics
Rr.i = Rr.o -- br .
213
(3.16a)
The formulae indicate that the longer the wheelbase I, with the radius of the track
circle D unchanged, the vehicle requires more width (R r•o and Rr•i become smaller).
3.7.3 Kinematic steering ratio
The kinematic steeri~g. ratio is is ~e ratio of the alteration BH of the steering
wheel angle to the mInImal alteratIOn Bm of the mean steer angle, of a pair of
steered wheels, where steering is operated free of moments and begins from the
on-centre (straight ahead) position. Initially, the steering compliance and the
alteration of the ratio during steering are ignored:
mean steering angle Bm = (B o + Bi)/2
(3.17)
kinematic ratio is = BH/Bm
(3.18)
The equations are only valid when there is a greater input range (e.g. Bm = 20°)
or a ratio which remains approximately constant over the whole steering range
(Fig. 3.95). However, if this changes (Fig. 3.96), the steering wheel angle
19
----~
f
f
18
BMW ._(1) 17
16
15
14
-
.......
-......... ,
I
"\..Mercedes
"'\
13
Opel' ,
12
Left turning
......
Right turning
:~
o
50°
Mean steering angle, Om
Fig. 3.95 Overall steering ratio is (see Section 4.3), measured on three conventional
passenger cars with power-assisted recirculating ball steer.ing. Altho~gh the BMW has
a ratio which remains almost constant throughout the turning range It reduces on both
sides from around 8m = 20° on the Vauxhall/Opel and the Mercedes, so the driver
needs fewer turns of the steering wheel to park. These two model groups have an
opposed steering square positioned behind the axle (Figs 1.41, 4.12 and 4.30),
whereas the BMW uses a synchronous one which also sits behind the axle (Fig. 4.3).
iii
I
"l"
214
The Automotive Chassis
l
2:
.--24
en 25
./
/
/
/
/
/
~
7
~
~ ~
~/
/'
//
/ f/
'"
~
--
~
10°
23
22
21
2019
18
17
16
15
14
~ Audi 80 OpelNauxhal I Vectra
---
"""'" ~
~
~'
...........
~
~
/
!/
l' ~
'- "Z.
"
~
".J'"
/~
/
"'"
i'...
'"
"
~
OpelNaux~all Corsa VW PolO'
0°
10°
Mean steering angle, 8m
Right turning
....
~It---Left turning
_ - I.....
Fig.3.96
Total steering ratio is (Equation 3.19) measured on four front-wheel drive
passenger cars with manual (non-power-assisted) rack and pinion steering, superimposed on the mean steering angle 8m of the wheels (Equation 3.17). It is important
to note the relatively severe drop in ratio as the wheels are turned more (due to the
steering kinematics, see Section 4.2). To limit the forces on the steering wheel when
the vehicle is being parked, heavy vehicles, such as the Audi 80 and the
OpelNauxhall Vectra, have the larger ratio is,o = 24.2 or 22.2 in the straight running
position. All vehicles have a constant steering gear ratio i's, i.e. not the varying split
ratio seen in Fig. 3.97.
proportions .18 H must be assumed, as well as the resulting minimal mean steering wheel proportion .18 m,min relating to both wheels:
(3.19)
If the overall steering ratio relates to the on-centre position a zero should be
given as the index is,Q.
As shown in Figs 4.3 and 4.36 to 4.38, steering gears with a rotational movement need a steering square arrangement of the linkage, in which the length and
position of the tie rods and steering arms allow almost every type of steering
ratio as a function of the input angle. However, the entire steering system has
more component parts and is more expensive (see Section 4.3).
The more economical design is rack and pinion steering, although this has the
disadvantage that - as can be seen in Fig. 3.96 - for kinematic reasons the ratio
reduces as steering angles increase. On power-assisted steering systems, the
reduction in ratio has a favourable effect on the handling properties. In the
straight running position, a more generous ratio is desirable on passenger cars at
Ii
Wheel travel and elastokinematics
215
hi~h speeds in order not to mak.e the steerin¥ too sensitive, whereas reducing tfie
ratIO could be better for comenng and makIng parking and manoeuvring possible with fewer turns of the steering wheel.
The hydraulics (or electrics) (see Ref. [1]) support the increasing activation
forces at greater steering angles, however this is not the case on vehicles without power-assis~ed s~eering. Here, forces can b~come disproportionately high
because the fall In ratIO cannot be reduced, espeCIally on front-wheel drive vehicles. The reasons for this are:
• the steering gear is located in the narrow space available between the dashpanel and engine;
• the fixing points have to be laterally stiff;
• toe-in alteration (Fig. 3.67) must be avoided;
• the need to produce the actual steering curve (Fig. 3.92).
The design position of the tie rods in the top view is also a consideration. It
makes a difference whether these - as shown in Figs 4.4 and 4.39 to 4.41 - are
situated in front of or behind the centre of the axle (or intersect with it) and
whether the inner joints are screwed into the sides of the steering rack (outer
take-oft) or must be fixed in the centre (centre take-oft). The influence of the
kingpin inclination angle and caster offset angle and the size of the steering arm
angle, (Fig. 4.32) also have to be taken into aCCOunt.
Series measurements have shown ~ that~ oJ:! front-wheel drive vehicles, the
reduction in ratio from the 'on-centre position' to full lock is 17-30%.
Standard passenger cars have space under the engine-gearbox-block; this is
the reason for the significantly lowerreduction of only 5-15%.
Rear-engine vehicles offer even more space under the front-end boot. Of
these, passenger cars were found with rack and pinion steering systems in which
the ratio does not change throughout the entire input range.
The curve shown in Fig. 3.96 of the steering ratio of the Vauxhall Cavalier
exhibits is,o ::::: 22.2, with the wheels in the straight-ahead position and at a mean
steering angle of 8m ::::: 35° the value is,min::::: 17.7; is.min/is,O::::: 0.80, i.e. the reduction is 20%.
The steering gear manufacturer ZF has developed a system to counteract
the disadvantage of the reduction in ratio on non-assisted steering systems.
For this purpose the steering rack varies its pitch from t, to t2 (Fig. 3.97).
This causes the rolling circle diameter of the pinion gear to reduce on both
sides from d, to d z when the wheels are turned to the off-centre range position. The path Sz shortens as the wheels are turned more and therefore the
ratio is in the steering gear itself increases. The consequence is more turns of
the steering wheel from stop to stop, but a reducing steering wheel moment
(Fig. 3.98).
3.7.4
Dynamic steering ratio
The true steering ratio, as experienced by ~he driver, wo~ld be the dynamic rat~o
i dyn ; this comprises the proportions resultIng from steenng 88 H and the elastIc
216
The Automotive Chassis
I... 51 .. I
Fig. 3.97 If the steering rack is designed in such a way that the pinion gear is
given a larger pitch circle (d1, left) in the middle than on the outside (d2 , right), the
rack travel reduces from S1 to S2 as the wheels are turned more: the ratio becomes
larger, the rack moment smaller (illustration: ZF).
Steering to the left
I
Middle steering
wheel angle
Steering to the right
Ratio is, generated in the steering gear itself when (as shown in Fig.
3.97) the steering rack has a varying split (illustration: ZF).
Fig. 3.98
LlOH,e (Fig. 3.99) portion. To calculate the group of curves a given steering angle
range LloH must be assumed on both wheels (e.g. 0° to 5°, 0° to 10°, 0° to 15°,
etc.) and the respective mean value determined in each case (here Om = 2.5°, 5°,
7.5° etc.) to be able to take the kinematic steering ratio is at these points on the
respective curves. The dynamic ratio depends on the height of the steering wheel
moment M H, so that only one point of a given curve can be considered in each
instance. The equation is
(3.20)
Figure 3.100 shows the dynamic steering ratio measured on a standard passenger vehicle. As an example i dyn at M H = 5 N m in the range Llo H = 0°_5° can be
calculated. Taken from the lower curve (for is) the overall steering ratio is
Ir
I
.
Wheel travel and elastokinematics
Fig. 3.99
Characteristic result of a
steering compliance measurement on a
passenger car with rack and pinion
steering that records the steering wheel
angle as a result of elasticities. It shows
the compliance ~8H:e when the wheels
are turned to the left and right and the
steering wheel moments MH increase;
the wheels were locked during the
measuring process. If the curve is steep,
there is a high CH = MH/~8H.e value, i.e.
low steering elasticity, The greatest
moment M H = ± 70 N m corresponds to
a force of FH = 184 N per hand with a
steering wheel diameter of 380 mm.
This should be enough to permit conclusions about the elasticity behaviour
during driving. The hysteresis also
shows the residual angles ~8H,Re that
remain when the wheels are turned and
the vehicle is stationary.
217
70
Nm
t
....c:
60
50
Q)
E
0
E 40
Q)
Q)
.r:
~
0)
30
c:
';:
Q)
Q)
....
CJ)
... -AO H,e
20
10J---.-hf---+-
Left turning
80°
40°
40°
80°
Right turning
+AO H•e
..
U)
....
CD
CD
'
--+-+--+-+---120 ~'
to
~
:::1'"
30
CD
CD
3
o
----fl---+--+------l 40 ~
:::l
.....
--H---r--+---150
~
~++-+----l---l60
Nm
-++---~--l70
is = 21. In accordance with Fig. 3.99, the mean value of the steering wheel
proportions as a result of elasticities is dOH,e = 17°. This gives:
i dyn
=21 + 1715 =24.4
This value should then be entered at Om = 2.5°. The smaller the steering angle
range, and the greater M H becomes, the more the dynamic ratio increases; if, for
example, M H is 15 N m, idyn already has a value of 31.
218
The Automotive Chassis
idyn •
I
I
34
I 1I32
I
I
~
~~
:::::::
--: ~
\
I i~ \
/~
/
\
I---"'"
is
24 \
r--...././
--....
22
---
1\
V; 'j 2\\ \
I~
~~
e:::::~
,\
"
~
5°
0°
5°
5N m
r--
I--
=--
~
t:::::
"'"---~
10°
Mean steering wheel angle 8m
Steering to the left
10N m
k::r----- r--:: E:::::::
20
10°
15 N m
Steering to the right
Fig. 3.100 Typical curve of the dynamic steering ratio idyn of a vehicle with rack
and pinion steering entered as a function of the mean steering angle Om and the
steering wheel moments M H = 5, 10 and 15 N m. The kinematic total ratio is
measured on the same vehicle was entered for comparison; this falls from is,o = 21
(in the centre position) to iS,min = 19.7 (where Om = ±35°), in other words by only 6°.
3.8
Steering self-centring - general
If there were no self-centring torque on the steering axle (Z) on the front wheels
of a vehicle, straight-ahead driving would be impaired and only a small force
would be needed to turn it into the bend; no feedback on steering torque - the
most important source of information about lateral force conditions - would be
forthcoming. When the bend had been negotiated, the steering wheel would have
to be turned back and would not go back of its own accord to the straight-ahead
position. The driver would. have no feel for cornering speed and handling with
the risk that they would not be able to return the steering to the normal position
fast enough when coming out of a bend. Sections 1.4.1, 1.5 and 1.6.2 refer to the
correlations with the various types of drive and Fig. 1.35 shows the differences.
There are several ways of self-centring the steering at the end of a bend with,
in each instance, one of the three forces acting on the centre of tyre contact
(vertical force Fz,W, lateral force Fy,w or longitudinal force Fx,w) having a lever
to generate moments. They have been given indices to differentiate them and
n
r
:1
Wheel travel and elastokinematics
Fig. 3.101 The forces occurring between
tyres and road tyre contact point Ware transferred from the suspension W to the body.
This is shown on the left front wheel for the
vertical force +Fz.w , the rolling resistance or
braking force -Fx.W.b and the lateral force +Fv.w
(see also Fig. 3.3) acting from the inside out
which increases the moment.
219
z
y
Fzw /l
' [ / Direction
which indicate their point of action the direction of the righting force (Fig.
3.101) or other associated aspects whose length is contingent on the type of tyre
(see index T):
Mz,w,z moment from vertical force Fz,w, kingpin offset ra and kingpin
inclination IT (Figs 3.105 and 3.107).
MZ,T,Y moment from lateral force Fy,w and lateral force lever nT,k (Figs
3.119 and 2.49).
MZ,T,X moment from rolling resistance force F R , and lateral force lever nT,k
(Fig. 3.123).
Mz,w,Y moment from lateral force Fy,w and caster nT,k (Figs 3.121 and
3.127).
In addition, there can be self-centring torques on front-wheel drive vehicles
caused by the tractive forces (MZ,W,a,1 and rs; Fig. 3.129), by the body roll when
drive shafts lie diagonally (Figs 1.6 and 3.88) and drive joints, whose centres lie
outside the steering axis (MZ,W,A,f; Fig. 3.102). Braking forces would also right
the wheel on the outside of the bend while turning the (lesser loaded) wheel on
the inside of the bend further (see MZ,W,b; Equation 3.26a).
In accordance with the German standard DIN 70 000, the steering moment
Ms is the sum of all moments around the steering axis of the steered wheels.
These 'self-induced' moments are introduced by the driver, whereas selfcentring after the vehicle has negotiated a bend is a question of the driving
condition and the coefficients of friction. This difference is merely being
pointed out. The vertical force Fz,w, which influences all righting moments and
is also called wheel load, is half the weighed front axle force FZ,Y,f determined
( I
,.
\
The Automotive Chassis
220
Fig. 3.102
E
W'
Left front axle of an Audi with
negative kingpin offset TcT =-18 mm and an
almost vertical damper unit; the spring was
angled to reduce the friction between the
piston rod and rod guide. For reasons of
space, the CV-joint centre Q had to be shifted
inwards; the space allowed for snow chains
can be seen here (see Fig. 2.8).
in the design position (see Section 5.3.4), i.e. when there are three people each
weighing 68 kg in the vehicle:
Fz,w
=F Z,v,r/2
and FZ,V,f =mV,fg (kN)
(3.21)
As can be seen, the level of the front axle load mV,f is also a consideration here
and so we sometimes speak of 'weight self-centring'.
Using Fz,w we can obtain:
the lateral force
Fy,w = /LY,W Fz,w
the rolling resistance force FR = kR Fz,w, sometimes also
the tractive force
FX,w,a = /Lx,w Fz,w (see Equations 6.36 and
6.37a)
the braking force
FX,W,b = /Lx.w Fz.w
The values for lJ.y,w, k R and IJ.x,w are given in Sections 2.8.3, 2.6.1 and 2.7, and
Section 3.10.3 contains a summary of all righting moments.
The opinion still sometimes expressed that the steering is centred by the vehicle front end lifting when the wheels are turned would only apply at zero caster.
As Fig. 3.165 shows, at l' = 0° the body lifts on both wheels (-fill), but if there
is caster, the wheel on the outside of the bend moves upwards, the most highly
r
II
Wheel travel and elastokinematics
221
loaded side of the body sinks and instead of self-centring it, the weight would turn
the steering further. However, the less loaded side on the inside of the bend lifts.
3.9
Kingpin inclination and kingpin offset
at ground
3.9.1
Relationship between kingpin inclination and kingpin
offset at ground (scrub radius)
According to ISO 8855, the kingpin inclination is the angle (J' which arises
between the steering axis EG and a vertical to the road (Figs 3.103 and 3.107).
The kingpin offset is the horizontal distance rrr from the steering axis to the intersecting point of line N'N in the wheel centre plane with the road. Values on
present passenger cars are:
(J'
= 11 ° to 15°30' and
r rr = -18 mm to +20 mm
As shown in Fig. 2.8, rrr can also depend on the tyre width.
Fig. 3.103 The precise position of the
steer axis - also known as kingpin inclination axis - can only be determined if the
centre points E and G of the two ball
joints are known. The total angle of kingpin inclination and camber (a + ew) must
also be included when dimensioning the
steering knuckle as an individual part.
!
I
222
The Automotive Chassis
Larger kingpin inclination angles are necessary to give the vehicle a small or
negative kingpin offset. In commercial vehicles, tractors and building-site
lorries, the inclination of the kingpin is often equivalent to the angle (J" (Fig. 1.3),
whereas the wheels are controlled by ball joints on the front axles of passenger
cars. On double wishbone suspensions, the steering axis therefore goes through
the centres of the ball sockets E and G indicated (Figs 1.38, 3.120 and 3.103);
the engineering detail drawing must show the total angle of camber and kingpin
inclination.
The McPherson strut and strut damper have a greater effective distance
between the lower ball joint G and the upper mounting point· E in the wheel
house (Figs 1.8 and 3.102); however, the upper axle parts are next to the wheel,
so attention should be paid to creating enough clearance for the rotating tyre
(possibly for snow chains). As a result, a higher inclination of the steering axis
and a higher angle (J" have to be accepted. In addition, as can be seen in the illustrations, point G has been shifted to the wheel to obtain a negative kingpin offset.
The steering axis then no longer matches the centre line of the suspension strut
(Figs 1.8, 3.30 and 3.104).
Due to the relationship between camber and kingpin inclination shown in Fig.
3.103, the angle (J" does not need to be toleranced on double wishbone suspensions. The permissible deviations on the overall angle 8w + (J" are given in the
detailed drawing of the steering knuckle. If the camber has been set correctly on
this type of suspension, the kingpin inclination angle will also be correct.
However, the important thing is (as specified in the camber tolerance) that the
deviation between left and right does not exceed 30', otherwise the steering
could pull to one side if the caster angle 'T on the left- and right-hand sides differ
(see Section 3.10.7).
On McPherson struts and strut dampers, the steering knuckle is usually bolted
to the damping unit (Figs 1.56 and 5.54). In this case there may be play between
the bolts and holes or the position may even be used for setting the camber (Fig.
3.104). In this case it is sensible to tolerance the kingpin inclination angle
Detail C
Fig. 3.104 Camber can be set at the
bracket between steering knuckle and
suspension strut using an excenter on the
upper bolt C; the lower screw is then used
as a pivot. The kingpin inclination, which is
important for driving behaviour, cannot be
corrected in such cases. The steering axis
entered here does not lie in the damper
centre line.
r
I
i
Wheel travel and elastokinematics
Fz,w
Fig. 3.105 For static observation, the
vertical force Fz.w must be shifted to the
wheel axis and resolved into its components. The distance to the steering axis
is equivalent to the vertical force lever q,
the size of which depends on the kingpin offset (a and the angle 0'.
223
cos (J
Fig.3.106 The negative kingpin
offset reduces the vertical force lever q.
However, its length is one of the
determining factors in the self-aligning
torque M z.w.z. To maintain its level, the
kingpin inclination angle 0' would have
to be increased.
because, provided the camber is correct, the kingpin inclination does not have to
be.
There is also a direct correlation between the alteration to camber and
kingpin inclination when the wheels bump and rebound. As described in
Section 3.5.2, the aim is to make the compressing wheel go into negative
camber, as this leads to small changes in camber at body roll, but an increase
in kingpin inclination by the same angle. Strictly speaking, the calculation by
drawing of the camber alteration, shown and described in Figs 3.50 to 3.52,
relates to the kingpin inclination, and for this reason the angle alteration Llcr
is also entered.
To obtain the self-aligning torque Mz,w.z which is important for righting, the
vertical force Fz.w, which is always present on the centre of tyre contact, must,
for static consideration, be shifted up to the wheel axis and resolved there in the
direction of the steering axis:
F z,W cos cr and, vertical to it, F z,w sin cr (Figs 3.105 and 3.106)
The steering lever or vertical force lever q at the resolution point is:
q = (ra +
rdyn
(3.21a)
tan cr) cos cr
The equation will apply provided that cos Bw = 1, a condition that applies to
normal camber angles. If the vehicle has caster, the force components Fz,w sin cr
must be further resolved by the angle 'T (see Equation 3.3). The parameter rdyn
can be calculated using Equation 2.2.
I,
,i
224
The Automotive Chassis
Fig. 3.107
Direction
I
Fz,w
Fz,w sin a
Fz,w
sin a cos 6
When the wheels are turned by
the angle 5, the vertical force component Fz.w
sin IT gives the self-aligning torque M z.w.z; the
extent of this weight-related self-alignment
depends on the kingpin inclination angle IT,
the lever q, the front axle load ffiv.f and the
caster (Fig. 3.147).
sin a sin 6
When the wheels are turned, the force Fz,w sin a is at the angle 0 to the wheel
axis (Fig. 3.107) and the component Fz,w sin a sin 0 will, with smaller steering
angles, give the approximate righting moment based on the whole axle:
Mz,w,z
= FZ,Y,f
sin a sin omq
(3.22)
(For FZ,Y,f see Equation 3.21 and Om in Equation 3.17.)
The exact solution has to take the changing kingpin inclination angle (due
to lateral forces when the wheels are turned and due to the body roll) into
account as well as the positive and negative caster that occurs (Figs 3.48,
3.53, 3.56 and 3.132). The influence of the paths rT.T and rT in the tyre contact
area (Figs 3.119 and 3.120) also has to be considered. Both can have a significant influence on the size of M z.w.z during cornering. On the outside of the
bend, the lateral force F y,w,O reduces the kingpin offset by -rT (or causes it to
become more negative) whereas on the inside of the bend, it increases or
becomes less negative (Fig. 3.127).
There is also a load alteration during cornering, whereby Fz,w,f,o > FZ,W,f.i and
also OJ and 00 are not always of the same size, so that different moments always
occur on individual wheels. The kingpin offset r a, which appears in Equation
3.21 a, influences the level of the self-aligning torque Mz,w,z; if this offset is large,
the righting increases, if r a decreases or even becomes negative (owing to the
shorter lever q), the moment reduces (Fig. 3.106).
The more M z .w.z increases, the more the front axle becomes longitudinally
sensitive. There is, therefore, a clear tendency towards a small positive or negative kingpin offset.
If Mz.w.z is to remain at the same level, the kingpin inclination angle has to be
enlarged with the disadvantage that, when the wheels are turned, the wheel on
the outside of the bend goes in the more positive camber direction, which makes
more space necessary because the brake disc has to be shifted into the disc wheel
(Figs 3.102 and 2.23). With a given path ra,I, the necessary angle al can be calculated from the existing values ra (in mm) and ao:
r
Wheel travel and elastokinematics
225
(3.23)
where
A=
B
(ra
+
= rdyn -
rdyn
tan
0'0)
sin
0'0
cos
0'0
A
The dynamic tyre radius rdyn can be determined from the rolling circumference
C R (or CR,dyn, see Section 2.2.8):
(3.24)
Taking as an example a standard passenger car with the tyre size 185 R 1490 S,
which has a rolling circumference of 1965 mm, the axle settings were 0'0 = 5°54'
and ra = 73 mm.
The aim is to find the kingpin inclination angle 0'1 with a negative kingpin
offset ral = -18 mm:
= 1965/21T = 313 mm
A = (+73 + 313 tan 5°54') sin 5°54' cos 5°54'
A = 11 mm: B = 311 - 11 = 302 mm
rdyn
tan
0'1
=
-18 ] 2 11
+= 0.0298 + 0.1912
[ 2 X 302
302
-18 ) +
( 2 X 302
The following would then appear on the drawing and in the workshop manual:
kingpin inclination 12°30'
a normal value for a negative kingpin offset. The parameter ra,l can be
more easily calculated as a function of the amended kingpin inclination angle
0'1 :
ra,l
= .
sm
A
0' 1 cos 0' 1
rdyn
tan O't
(3.25)
3.9.2 Braking moment-arm
During a braking process carried out with the brake mounted on the steering
knuckle or wheel carrier, the braking force FX,W,b tries to tum the wheel with the
brake acting on the moment-arm (Fig. 3.108):
The Automotive Chassis
226
Direction
/
F..'
X,W,b
Fig.3.108
The braking force FX,W,b has the lever rb = [" cos (J" to the steering axis
EG; shifted vertically on this axis, Fx,W,b acts by the amount a below ground and
causes the greatest force in point G: FG,x = FX,W,b + FE,x (see also Fig. 3.155). When
there is caster, FX,W,b must be resolved at the centre of tyre contact around the angle
T (Fig. 3.115).
rb
= r a cos
(3.26)
(J
around the steering axis, i.e. the moment
MZ,W,b
= F X,W,b cos
T
(3.26a)
rb
is generated, which, as Fig. 3.109 shows, results in the tie rod force F T and,
where r a is positive, pushes the wheel into toe-out (for caster angle T see Fig.
3.115).
The longer the path r a, the more the moment MZ,W,b increases and the larger
the influence of uneven front brake forces on the steering - which is the reason
for keeping r a as low as possible or even making it negative (Figs 3.102 and
3.106). Thus, as shown in Fig. 6.12, brakes that do not respond equally cause a
counter-steering effect, which can reduce or eliminate the yaw response of the
I
Direction
Top view
Fig. 3.109
If the brake is in the wheel, the
causes the moment
MZ,W,b = Fx,W,b rb, which tries to push the wheel
into toe-out and causes the tie rod force Fr. The
steering axis is assumed to be vertical to simplify
the calculation.
braking force
r
r
Fx,W,b
Wheel travel and elastokinematics
Engine
227
0~
Fig.3.110
If a front-wheel drive vehicle has an inside brake, the engine mounting must absorb not only the drive-off moment, but also the braking moment; the
reaction forces ±D.Fz, the size of which depends on the effective distance C, occur
in the rubber buffers.
body, which is also true for an elastokinematic toe-in alteration (Figs 3.2 and
3.82). The longitudinal force FX,W,b arising on the ground produces the reaction
forces FE,xand Fa,x in the pivot points of the steering knuckle. In order to be able
to determine their size, FX,W,b must be shifted towards the braking force lever on
the extension of the steering axis EG. Therefore, with positive kingpin offset
F X,W,b lies below the ground by the amount a and is shown in the side view of
Fig. 3.108 as Fx.,W,b:
a = rb sin a = +r(] cos a sin a
(3.27)
If ra is negative, FX,W,b moves above ground (Fig. 3.156) and F a.x becomes
smaller.
If the brake is on the inside on the differential, the braking moment is transmitted via the universal joints to the engine and causes the bearing reaction
forces I1Fz in the engine mounting (Fig. 3.110):
The smaller the wheel radius (rdyn) and the larger the effective distance c, the
lower the forces and therefore also the compliance in the rubber mountings. The
braking force FX,W,b which occurs at the centre of tyre contact must, in such
cases, be shifted to the centre of the wheel (like the rolling resistance force F~ in
Fig. 3.111), because a shaft bearing can only transfer forces, and not moments,
in its effective direction. Just like F R, F'X,W,b acts on the longitudinal force lever
r a, also known as the disturbance and traction force moment-arm:
r a = r(] cos a + rdyn sin (a + Bw)
With it,
FX,W,b
MZ.W,b
(3.28)
causes the moment:
= FX,W,b cos 'T r a
(3.28a)
The Automotive Chassis
228
Fig. 3.111 When the wheel is rolling in a straight line, the rolling resistance force
FR must be observed as FAin the wheel centre; its distance to the steering axis is
ra• This so-called longitudinal force lever depends on the kingpin offset rcr and the
smaller this can be, the further up FR acts as FRon the steering axis and the more
evenly points E and G are stressed in the longitudinal direction. The same static
conditions apply to the braking force if the brake is located on the inside on the differential (see also Figs 3.113 and 3.154).
which also occurs when r cr = O. In the equations, note must be taken of the plus
or minus signs; in the case of a negative kingpin offset, the first element (- r cr cos
IT) must be subtracted from the second. Figure 3.115 contains cos IT.
Because ra > rb, there is a higher moment when the brake is on the inside at
the differential, which has a more pronounced influence on the steering. The
reaction force FG.X in the lower ball joint, however, becomes much smaller. To
determine the forces FE,x and FG,X F~,W,b has to be shifted vertical to the steering
axis and, in the side view, comes to lie below the wheel centre as F~,W,b (Fig.
3.154) by
a = ra sm
3.9.3
(3.28b)
IT
Longitudinal force moment-arm
Figure 3.111 shows the rolling resistance force F R, which always occurs when
the vehicle is moving. It generates the same moment left· and right:
MZ,T,X
= F R cos
'T
ra
(3.28c)
which is absorbed at the tie rods (Fig. 3.112); any caster angle 'T must be considered. If the moments are of the same size, the vehicle moves in a straight line,
but if they are different it can pull to one side. Tyres that have a different rolling
circumference (Fig. 2.15) or front axles where the angles IT + ew differ to the left
and the right can be the reason for this. The factor rdyn sin (IT + ew) primarily
determines the length of the moment-arm r a (see Equation 3.28). On a bend, the
outer wheel experiences a force increase (Fz,w + ~Fz.w, Fig. 1.6) and the inner
one a reduction equivalent to FR,o > FR,i. Where there is no caster, the wheel on
r'---_---------------..JLI
Wheel travel and elastokinematics
MZ,T,X,I
I
MZ,T,X,rs
Lr==
-
229
t
Dir e ction
......
...::L
I
I
~,I
.
1
((
ll
1
I
I
Fig. 3.112
The rolling resistance force FR pushes the wheels backwards via the
longitudinal force moment-arm (a, i.e. into toe-out -(j..t. A moment arises on both
sides, which is absorbed and cancelled out at the tie rods. In the case of caster the
angle'T must also be observed (see Fig. 3.115).
the outside of the bend rights itself more than the one on the inside is trying to
tum into the bend.
The tractive forces F X,W,a, which occur at the contact points of the front wheels
on front-wheel drive vehicles, cause moments acting in the opposite direction,
but also have to be taken into consideration in the centre of the wheel (Fig.
3.113), i.e. in vehicles of this design, a smaller longitudinal force moment-arm
r a will be particularly important. Citroen has achieved this by shifting the ball
joints E and G in to the wheel centre plane (Fig. 3.114). This means that
BW
+
0"
= 0,
r(J
=
°
and, therefore, r a =
°
The longitudinal force moment-arm should be as short as possible.
Comparison with the formula for the vertical force lever q (Equation 3.21a)
shows the difficulties:
.
... Direction
Fig. 3.113
The negative
kingpin offset on the ground
favourably shortens the longitudinal force moment-arm (a. The
tractive force Fx.w.a relates to
one wheel and, in case of
caster, must be resolved around
the angle 'T in the wheel centre
FX.w.a.
The Automotive Chassis
230
Fig. 3.114
Section through the centre axle steering of the
model GSA which Citroen no longer builds; swivel and
supporting joints are in the wheel centre plane and re, is zero.
q = r a cos a + rdyn sin a and
r a = r a cos a + rdyn sin (a + ew)
If, for example, the camber is ew = 0°, q = ra if there is only a small or no caster
angle 'T and the vehicle moves unimpaired in a straight line.
However, during cornering, q changes significantly while r a remains virtually
unchanged (Fig. 1.6).
3.9.4
Alteration to the kingpin offset
To improve cornering behaviour, disc wheels with lower wheel offsets e are
sometimes used (Fig. 2.23) or (in the past) spacer rings were laid between the
wheel and brake disc to give the advantage of a slightly wider track (around
2-4%), but with the disadvantage of up to 100% greater kingpin offset at ground.
The result is a more noticeable disturbance effect on the steering when the road
is uneven and particularly when the front brakes do not pull evenly.
If, as on almost all passenger cars with negative kingpin offset, the two brake
circuits are designed to be diagonal, these measures cannot be implemented. The
negative kingpin offset would either change from negative to positive or become
too positive when there is an elastokinematic toe-in alteration on the front axle
(Fig. 3.82), and toe-out would occur during braking instead of toe-in.
Figure 2.8 shows the alteration of r a depending on the tyre width, using the
example of the VW Golf III.
3.10
3.10.1
Caster
Caster trail and angle
We differentiate between the kinematic caster trail r,.,k of the wheel, the caster
angle 'T, the caster offset n,." the tyre caster r,.,T, the lateral force lever n,.,k and the
elastokinematic caster r,.,e. Dynamic measurements are contained in Section
5.2.3 of Ref. [9].
r
Wheel travel and elastokinematics
231
rlbCOST
~W,b
J5<,W,b
sin 1"
Fig. 3.115
If the extension of the steering axis goes through the ground at point
K in front of the wheel centre, the distance arising is the kinematic caster trail (T,k
(Case 1). A vertical to EG, drawn through the centre of tyre contact W, when
projected onto the xz plane, gives the lateral force lever nT,k (Equation 3.30).
Longitudinal forces which arise, such as the braking force FX•W.b (or the rolling
resistance FR), must be resolved at the centre of tyre contact (or as FR in the wheel
centre, Fig. 3.111) by the angle 'T.
In accordance with DIN 70 020 (and also DIN 70 000), 7 is the angle
between the steering axis EG projected onto an xz plane and a vertical, drawn
through the wheel centre (Case 1, Fig. 3.115), and rT,k the distance between the
points K and W on the ground. The castering of the wheel centre of contact W
behind the intersection K can also be achieved by shifting the axes of rotation
in front of the wheel centre: +nT (Case 2, Fig. 3.116). On some front-wheel
drive vehicles, owing to the increased self-centring moment caused by tractive
forces, negative caster was designed, which can be achieved with a reversed
angled steering axis (Case 3, Fig. 3.117) or by positioning an axis EG behind
the wheel centre and inclining it by the angle 7, which leads to negative caster
offset -nT (as can be seen in Figs 3.118 and 3.139). For the following reasons,
linking a positive caster angle and - n is popular from the point of view of
construction:
T
Direction
Fig. 3.116 Caster can also be achieved by
shifting the wheel centre behind the steering
axis (Case 2); if this is vertical, as shown, the
(here) positive caster offset is equal to the
moment arm: n T = (T,k = +nT,k. Rolling resistance
forces FR acting at the centre of tyre contact
must be observed as R. in the wheel centre.
i
i
The Automotive Chassis
232
..
Direction
Fig. 3.117
Caster (Case 3): a steering axis,
which is inclined opposite by the angle -'1",
results in negative caster -(T,k, and the associated disadvantage of a more positive camber
on the outside of the bend when the wheels
are turned. However, where the angles -'1" are
small, the tyre caster (T.T balances out the negative caster trail (Fig. 3.121). On the independent
rear wheel suspensions the steering knuckle
(here not the steering axis) can be given negative caster to achieve lateral force understeering (see Figs 3.144 and 3.145).
Fig. 3.118
Front axle properties can be
improved by a negative caster offset nT ; the
caster trail (T.k on the ground shortens by this
amount and the camber alteration when the
wheels are turned becomes more favourable.
• the kinematic caster trail rT,k is smaller, i.e. the influence on the steering resulting from uneven road surfaces reduces;
• the camber alteration is increased when the wheels are turned (Fig. 3.132).
The trail rT,k and the moment-arm n T of the lateral force (i.e. the path projected
onto the vertical plane xz) both with and without negative offset - nT can be
easily determined using the dynamic rolling radius rdyn (see also Section 3.10.7):
=
rT,k
= rdyn
tan
tan
nT,k
nT,k
rT,k
rdyn
'T
-nT
(3.29)
= r dyn
SIn 'T
=
SIll 'T - n T cos 'T
(3.30)
rdyn
'T
During a bend, the area of tyre contact deforms due to the slip angle Ci (Fig.
3.119). The lateral force Fy,w therefore is offset by the amount rT,T - known as
tyre caster - behind the wheel centre (Figs 3.120 and 2.50). The tyre caster of
practically rT,T = 10 mm to 40 mm must therefore be included in all static and
elastokinematic observations. Without and with caster offset the overall path rT,t
is then as follows (Fig. 3.121):
(3.31)
r
r
f
I
I
Wheel travel and elastokinematics
Fig. 3.119
The tyre contact patch (also known
as the 'tyre print', Fig. 2.9) of a tyre rolling at an
angle under the influence of lateral forces deforms
in the shape of a kidney; this means the point of
application of the vertical force Fz,w and the lateral
force Fxw moves by the trail (T,T - the tyre casterbehind the wheel centre and the tyre self-aligning
torque Mz,T.Y = Fxw x (T,T occurs. If the vehicle has
front-wheel drive, Fx,w,a acts at a point in the tyre
contact area, offset by (T from the wheel centre
plane, as does the rolling resistance force FR,co on a
bend. Tyre caster is between (T,T = 10 mm and
40 mm; lateral offset is (T :::: 3 mm per f.LV.W = 0.1
(see Section 2.10.2 and Figs 3.127 and 3.128).
If the slip angle 0'. is specified instead of the
coefficient of friction f.Lv.w, Equation 2.4c will apply.
Camber has a similar effect; negative camber
diminishes (T and positive increases it:
±(T
233
Direction
Wheel
centre
= 6 mm per t1ew,k = ±1°
Direction
Bottom breakthrough point
1
",.
~.-+.-f..----+--I--~~
I~I--""'*Q
Fig. 3.120
The extension of the steering axis EG, which is three-dimensionally at
an angle due to kingpin inclination and caster, penetrates the ground in front of the
wheel centre and gives (in the example) the positive kingpin offset (cr and the kinematic caster trail (T,k. On a bend, the lateral force acts offset to the tyre caster (T,T in
the tyre contact area. The total trail (index 7, t) is therefore (T,t = (T,k + (T,T and - in
accordance with Fig. 3.119 - the kingpin offset (overall on the outside of the bend)
(cr,t
= (cr -
(T.
(3.32)
(3.32a)
If precise calculations are required, the elastokinematic caster rT,e must be used
instead of rT,k, although this can only be determined by experiment on the vehicle (see Section 5.2.3 in Ref. [9]).
The Automotive Chassis
234
Direction
Fig. 3.121
Due to the tyre caster (T,T,
which is always present during cornering, the
lateral force lever is extended and becomes
(Fig. 2.50)
•
n.T,t
= nT,k + (T,T cos T
Lateral
force
In order to make them clearer, the path rT,T is not shown in some of the following figures.
3.10.2
Caster and straight running
Caster can be compared with the tea trolley effect, where the pulled wheel takes
on the direction of pull and the wheel centre adopts a position behind the axis of
rotation 1 (Fig. 3.122). The tensile force and the opposed force FR generated by
the rolling resistance are on an effective line, in other words in a stable ratio to
one another because the guiding and wheel axis lie behind one another. The
same effect also exists (despite kingpin offset and kingpin inclination) on the
wheels of a vehicle if these can be rotated around axes. The wheels are set to
caster on both sides and are linked by tie rods.
If unevenness in the road surface or a steering input pushes the wheels out
from the straight-ahead direction, by the angle 0, the rolling resistance components F R sin 0 (as shown in Fig. 3.123) move both wheels back via the force lever
nT,k (or the overall lateral force lever nT,t) until they roll in a straight line again.
The components F R cos 0 (left and right) compensate, and only subject the tie
rods to pressure. Negative caster on the wheels could have the opposite effect
and the vehicle would become unstable.
On a vehicle moving in a straight line, caster would not only have advantages
but also disadvantages. Uneven road surfaces cause alternating lateral forces on
the centres of tyre contact (see Section 4.2 in Ref. [3]) and these, together with
1
Tractive
power
Fig. 3.122 If the rolling resistance force
FR acts behind the steering axis 1, the
wheel follows in a stable manner in the
direction in which it is pulled.
r
Wheel travel and elastokinematics
i
Fig. 3.123 When the
vehicle is travelling in a
straight line, caster has a
stabilizing effect. Fig. 3.147
shows the necessary
further resolution of the
force components by the
angle T.
235
Direction
Fig. 3.124
Lateral forces fY,W.f caused
by uneven ground, in conjunction with
the caster moment-arm nT k cause the
forces Fr in the tie rods.
Direction
~
F:Y,W,f
the lever in nT,k (or nT,t, Equations 3.30 to 3.32a) cause moments around the steering axis (Fig. 3.124), which are supported on the tie rods and can cause steering
disturbances and vibrations. Furthermore, there is increased wind sensitivity due
to the fact that a wind force acting on the body (Fig. 3.125) causes lateral forces
Fy,w in the opposite direction, on the centres of tyre contact. In addition, the front
forces Fy,w,f together with the caster moment-arm n..,k (or nT,t) result in moments
that turn the vehicle in the direction of the wind, i.e. further in the direction in
which the body is already being pushed by the wind. This also applies to driving
on (diagonally) inclined roads and leads to increased steering moment.
3.10.3
Righting moments during cornering
The alteration to the caster and kingpin inclination (or camber) angle, which is
influenced by the body roll inclination (Figs 3.53 and 3.143) and is caused by
the steering angle of the wheels (Figs 3.132 and 3.135), results in an alteration
to all levers on which vertical, lateral and longitudinal forces are acting.
Observation of each individual wheel would mean looking at these very complicated kinematic relationships, and errors would almost be inevitable because of
the additional elastokinematic movements.
Determination of the righting moments based on the whole axle and on the
position of the vehicle parallel to the ground is - particularly in the case of small
steering angles and low cornering speeds - sufficiently precise. The caster 'T,T =
10 mm to 40 mm (Fig. 3.120) (not present when the vehicle is moving in a
The Automotive Chassis
236
--0-Fv,w
u
Wind force
Original
direction
~
\
v
Deviation
I I c::
.
U rV.w
Fig. 3.125 Caster can increase the wind sensitivity of a vehicle. The point at
which the wind acts is usually in front of the centre of gravity V; a moment arises
which seeks to turn the vehicle and causes the wheels to steer in the same direction.
straight line) must, nevertheless, be included in the equation. Righting moments
should be indicated in newtons, whereby 1 kN mm = 1 N m. The front axle
caster angle will affect the Equation 3.22 righting moment, due to the vertical
forces generated:
Mz,w,z
= FZ.V,f sin
(J"
cos
T
sin Om q
(3.33)
The caster angle in the lever nT,k is considered in the case of the righting moment
due to lateral force. Here, the kingpin inclination must also be included in the
calculation (Fig. 3.126):
Fy,w,f,o
+
FY,W,f,i
=
J..LY,W
Fz,v
= Fy,w,f,t
(3.33a)
acts around the lateral force lever nT,t offset behind the wheel centres (Figs
3.120 and 3.127, and Equations 3.32 and 3.32a):
Fy,W,f,t
Rear view
FY,W,f,O
cos a
Fig. 3.126 The lateral forces acting on the
centres of tyre contact of the front wheels must
be resolved in the direction of the steering axis
and vertical to it, shown here for the wheelan
the outside of the bend. Fv.Wf 0 cos (J' then has a
righting effect and Fr,w,f,o sin (J' strengthens the
vertical force Fz,W,f,o.
Wheel travel and elastokinematics
237
Direction
Fz,W, f,i --t-Tr.V.
Fig. 3.127
On wheels that roll at an angle of af, the lateral cornering forces Fv.W,f
act behind the wheel centres, offset by the tyre caster nT,T and push the centres of
tyre contact (and therefore also the vertical forces Fz,w.f, Fig. 3.119) to the bend
centre by the trail (T. The marked forces and paths are of different sizes on the
outside (0) and inside (i) of the bend:
= Fz,w + Ii Fz,W,f,i and Fz,W,f,i = Fz,w - IiFz,w
Fv.W,f,o = f.LY,W FZ,w,f,o and Fv.W,f,i = f.LY,W Fz,W.f,i
Fz,W,f,o
The steering axis is shown - simplified - standing vertically.
Mz,W,Y =
Fy,W,f,t cos
(J
nT,t
(3.34)
Section 2.8.4 describes the coefficient of friction J.1Y,W.
If the axis of rotation is vertical in the side view, Case 2 (7 = 0, Fig. 3.116), the
fonnula remain unchanged; only n r + rT,T appears for nr,t, i.e. the path around which
the wheel centre is located behind the steering axis, together with the tyre caster.
If the vehicle has negative caster (Case 3, Fig. 3.117), the lateral force could
cause the wheels to steer into the bend if this were not counterbalanced by caster
rT,T and by the tractive force FX,w,a which also has a righting effect on front-wheel
drive vehicles (Figs 3.119 and 3.129). In the case of negative caster, only rT,T rT,k needs to be inserted into the equation.
The increased rolling resistance force FR,co,f during cornering on the outside
and the inside is
FR,co,o
=
kR,co FZ,W,f,o
and
FR,CO,I. -- kR,co FZ,W"If'
and seen together this must be resolved into two components:
F R,co cos etf and
FR,co sin (Xf
In the case of positive caster, Case 1, the last component has a righting effect on
both wheels (Fig. 3.128 and Equations 3.32 and 3.32a):
;
I
238
The Automotive Chassis
Direction
FA,co, i sin at
Fig.3.128
The rolling resistance forces FR•co .o and FR•co .i, which have increased on
a bend due to the tyre slip, must be resolved by the angle af; the component FR•co
cos af then appears in the wheel centre with the lever (a. The greater af, and the
longer the caster trail (T.k, the stronger the self-righting due to FR•co sin aft
For reasons of clarity, the tyre caster (T.T and lateral offset (T (Fig. 3.119) have been
ignored here and the st~ering axis has been shown vertically.
MZ,T,X,1
= kR,co
FZ,Y,f
sin
(J'
sin
af
n,.,t [or (n,. + r,.,T)]
(3.35)
The rolling resistance force is (in accordance with the transfer of wheel forces
dFz,w,f during cornering, Fig. 1.6) greater on the outside of the bend than on the
inside, so that the difference in vertical force dFz,w,f, together with cos af, can be
a factor:
Fz,w,o - FZ,W,i
= 2 dFz,W,f
(3.35a)
MZ,T,X,2 = kR,co 2 dFz,w,f cos
af
cos
T
(ra - 2 rT)
(3.35b)
This deals with the longitudinal forces that act at the wheel centres - shifted to
the middle of the bend (Figs 3.111 and 3.119); it is possible to calculate the coefficient of rolling resistance kR,co required using Equations 2.4a to 2.4c.
The previous figures also refer to the tractive forces FX,w,a on front-wheel
drive vehicles (related to one wheel). These must be resolved first in the wheel
centre by the angle T (Fig. 3.113) and considered offset by rT, in the rolling direction of the wheel. Provided that the differential divider the moment equally to
each front wheel when the wheels are turned as the wheel load changes + dFz,w,
the tractive force component FX,W,A = 2Fx,w,a (relating to the entire axle) would
cause the following moments (Fig. 3.129):
MZ,W,A
= -Fx,w,a cos T(ra - rT) + FX,w,a cos T (ra - rT)
= F X,W,a cos T 2rT
=F X,W,A cos,. rT
(3.36)
The size of the force FX,W,A depends either on the coefficient of friction J.Lx,w
(FX,W,A = J.Lx,w FZ,Y,f, see Equation 2.5) or on the drive torque (see Equation 6.36);
--r
I
Wheel travel and elastokinematics
239
Direction
At (a + (T, the tractive force Fx.w.a.i on the inside of the bend has a larger
moment-arm than that on the outside of the bend Fx.w.a.o at (a - (T; the steering axis
is shown vertically, for simplification.
f:ig. 3.129
the lateral offset length rT is contained in the caption to Fig. 3.119 (see also
Section 2.10.3.4).
3.10.4
Kingpin inclination, camber and caster alteration as a
consequence of steering
Due to the spatial movement of the steering axis (onto which the vertical force
Fz,w must be shifted, see Figs 3.105 and 3.107) the righting moment for one
wheel can only be calculated precisely if the kingpin inclination, when the
wheels are turned, is taken into account. If, in the zero position, the steering axis
is inclined exclusively by the angle 0"0, i.e. there is either no caster or this has
been achieved by shifting the wheel centre (Fig. 3.116), then it is easy to deternline the kingpin inclination angle 0"0 or j which becomes smaller in both input
directions:
outside of the bend: tan
inside of the bend: tan
0"0
O"j
= tan
= tan
0"0
0"0
(3.37)
cos 00
cos OJ
(3.37a)
As shown in Fig. 3.103, kingpin inclination and camber are directly related, i.e.
if either one changes the other one must change too. This means that the camber
values SW.o or j, adopted by the wheel on the outside and the one on the inside of
the bend when the wheels are turned, can be determined simultaneously:
Sw,O = (0"0
+ sw,o) -
0"0
and
SW,i = (0"0
+ sw,o) -
O"j
(3.38)
and Sw,O are the angles prevailing when the wheels point straight ahead in the
design load or the particular load position (this applies equally to To). If the steering axis is also inclined by the positive caster angle 'To, the two auxiliary angles
0,1 and 0' must first be calculated using 0"0 and To:
0'0
240
The Automotive Chassis
tan 0' =
tan 'To
and tan a' = - - tan ao
sin 0'
tan
'To
(3.39)
They can then be used to determine directly the angles 00 or i on the wheels on the
outside and inside of the bend:
outside of the bend: tan a 0 = tan a' cos (0' - 00 )
(3.39a)
inside of the bend: tan ai = tan a' cos (0' + Oi)
(3.39b)
Equation 3.38 again applies to the camber SW;oori. Using a passenger car with the
following axle settings as an example:
SW,o = 15',
ao = 9°53'
and
'To
= 10°4'
This gives 0' = 45.54° and a' = 13.97° and, where 00 and Oi = 20°, the values are
as follows:
The wheel on the outside of the bend therefore goes into negative camber on this
vehicle, while the inner one goes into positive camber. This is not the same as
the case of a front-wheel drive vehicle with the following axle settings (Fig.
3.130):
SW,o = +40',
ao = 12°25'
and
'To
= +36'
+'wt 4°
3°
~Measured
\""---..
...... h:.---~
Calculated
curve
30°
"'-0; Inside
20°
--
2°
r--.
1°
~---
10°
--
~
1-------
0°
10°
~../
~
......
20°
30°
00 Outside - - .
Fig. 3.130 Camber alteration measured and calculated as a function of the steering angle on a front-wheel drive vehicle. Due to the large kingpin inclination
(To = 12°25', the wheels on both the inside and outside of the bend go into positive
camber.
The values measured are higher than those calculated because the camber of the
vehicle tested was in the plus tolerance, The calculation was made on the basis of
the manufacturer's information (ew = 20' and T = 0°) and this accounts for the slightly
different inclination of the curves.
,
\
Wheel travel and elastokinematics
~\
+7°
'"
+6°
'\
40°
241
30 c,
Ins ide steering angle
f... +5°
I\.
'"'"
20°
Q)
.0
+'"
E
co +3°
u
+2°
~
' " 1°
lbo
0
'"
-1°
04
-2-3°
-4°
10°
"'~
20°
30°
..
Outside steering angle
'"
40°
-
~
-5°
Fig. 3.131
Camber alteration measured on a Mercedes as a function of the steering angle. The axle settings in the design position were ew = 0°, (T = 14°40',
1" = 10°10', (a = --14 mm and the negative caster offset nT = -28 mm.
Owing to the front-wheel drive and the manual non-power-assisted steering, the
vehicle has a minor· caster and therefore positive camber on the turned front
wheel on the outside of the bend.
Mercedes Benz designed their passenger car with a non-driven strut damper
front suspension to have negative caster offset - n-r (Fig. 3.118) and a large angle
T. Figure 3.131 shows the success of this design: the wheel on the outside of the
bend goes into severe negative camber and the one on the inside of the bend goes
favourably into positive camber.
For demonstration purposes, the camber alteration based on ew,O =0°, 0"0 =6°
and various caster angles were calculated (Fig. 3.132); a larger kingpin inclination would only have resulted in a higher curvature for all curves. It can be
clearly seen how an increase in the angle To improves the lateral grip properties
of the entire front axle as the wheel on the outside of the bend goes into more
negative camber and the one on the inside into positive camber.
Trail and caster angle alter in exactly the same way as kingpin inclination and
camber alteration when the wheels are turned. In a passenger vehicle with rearwheel drive for example, rT,k is 6.5 mm when the wheels are pointing straight
ahead. The trail increases on the inside of the bend when the wheel is turned,
with a decrease on the outer wheel (Fig. 3.133). Negative caster occurs as oa ~
8°, and at 00 = 30° it amounts to rT,k :::::: - 30 mm, which would lead to the outer
wheel turning into the bend under lateral force if there were no caster.
242
The Automotive Chassis
+£w
8° r---+---+---+---+----I
40°
30°
20°
8°+-_-+_ _+-_-+-_ _+-__-1
r----l------II----l-----If---
-EW
Fig. 3.132
Camber angles SW,o and SW,i, as a function of the steering angle 00
(outside of bend) and Oi (inside of bend). The influence of the various caster angles T
can be clearly seen. Values given: a = 6° and Sw = 0°.
The caster angle alteration can be calculated just as simply as that of the kingpin inclination:
outside of the bend: tan
inside of the bend: tan
'To
'Tj
= tan
= tan
(j'
(j'
sin (0' +
oD
If the vehicle has 'To ~ 0°, only the kingpin inclination angle
thereby simplifying the formulas as follows:
outside of the bend: tan
inside of the bend: tan
'To
'Tj
= -tan
= +tan
(jo
(3.40)
sin (0' - 00)
(jo
sin 00
sin OJ
(3.40a)
(jo
plays a role
(3.41)
(3.41a)
The equation shows that negative caster can occur on the wheel on the outside
of the bend even with small steering input; this is clearly demonstrated in Figs
3.134 to 3.136, which show a comparison of curves calculated with various 'To
and (jo angles and also one measured curve.
a
Wheel travel and elastokinematics
t-+50
Lock angle,
inner
--•
OJ
'al
III "0
:l::c:
o
~
30·
...............
243
Lock angle.
outer
mm _
+JD
:::l
(j) e"X)
t)Cl
.......
20·
co
'lti ..2O
~"'Q
o~
10·
0
....
-10
Q)
III
Q)
>
-+:i
:l::
0
-,
'10·
20·
"'"
-20
...
30· 0
0
ItO·
........
~
~!
-X)
Q) III
z~
fltO
""-
Fig. 3.133
The length of a caster trail (T,k at the ground alters depending on the
steering input, shown using the example of a standard passenger car and the axle
settings:
= +20', = 11 °5', 'T = 8°20'
n = -32.5 mm and re, = +56 mm
€w
(f
T
The large kingpin inclination angle results in tl1edeviation of the curve from the horizontal.
..
l'0-go
1'0=6°
----
t
..........
~
1'0=00
sao OJ
I~ '-
Q)
~ "'-
Cl
c:
(0)
co
-10
..........
Q)
....
III
"'" ~
""'"
-12
(~
~ '\
I~ "
4
......
2
~
100
300 20 0
400
Steer angle, inner wheel
~
'"" "
'\
"
"""
"""
I'-...
"\
~Oo
200 30 0 ~o 00 500
........ Steer angle. outer whee
-2
-4
-6
~~
Fig.3.134 Caster angles calculated as a function of (f = 9° and 'To = 0°,6° and 9°.
The smaller the 'To value in the normal position. the faster negative caster occurs on
the wheelan the outside of the bend.
,.1
244
The Automotive Chassis
12
0'0=150~
f
C70=12°
100=9°
100=6°
10
(0)
~
~
......
~~ u
~
~ ~ 14
~
Q)
0)
c: 8
co
Q)
II)
co 6
~
"'" ~
~
50i 15; IIJo 300 20° 10°
S teering angle, outer wheel
K>
~ r--....
100'
1-2
Steering angle, i nner wheel
400 60 50°
KK
~~
"4
1"0
1-6
Fig.3.135 Caster angles calculated as a function of the steering input with T = 3°
and ao = 6°, go, 12° and 15°, The larger the kingpin inclination, the sooner the wheel
on the outside of the bend goes into negative caster (-'j"),
"'-
~
t-16 0
Q)
0)
c
co
Q.; 14°
~
...
""
40°
30°
Inner steer angle
20°
II)
co
u
~\
100
10
•
8°
6'"
4°
Fig. 3.136
1'\
20°
..
30°
10· Outer steer angle
\
1\
\
1\
Caster alteration measured as a function of the steering angle on the
wheels of a Mercedes. The kingpin inclination angle of a = 14°40' is the determining factor for the severe curvature of the curve and the caster angle T := 10°10' for
the angle position.
Wheel travel and elastokinematics
3.10.5
245
Kinematic caster alteration on front-wheel travel
If there are two people seated in the front of a vehicle, the body moves into bump
travel almost parallel and the caster hardly changes. However, if two or three people
are seated in the baGk, or the boot at the back of the vehicle is loaded, it is a very
different story. The rear axle springing complies more strongly than that of the front
axle and the body's position, which was almost parallel to the ground, alters by
A8 Bo•t = 1° to 2!O (Fig. 3.137). The caster angle increases by the same amount AT something which designers should bear in mind when specifying axle settings.
The increase in the caster angle under load is likely to be the main reason
why the steering is heavier on a fully laden vehicle even though this sometimes
causes the front axle load to be reduced. An alteration in caster has its disadvantages, as this in tum causes the self-righting torque to alter, but it is unavoidable if the brake dive on the front axle is to be kept within limits by means of
vehicle pitch poles (see Section 6.3.2).
On double wishbone suspensions, the axes of rotation 1 and 2 of the two
suspension control arms are usually parallel to one another (Fig. 3.138); in the
standard configuration of the McPherson strut and strut damper there is a right
angle between the centre line of the damping part and suspension control arm
(Fig. 3.139). In such cases - regardless of the position of the compressed or
rebounded wheel - the caster is retained. This is not the case where there are
different angles between the suspension control arm axes of rotation (Fig.
3.140), or the damper centre and suspension control arm (Fig. 3.141).
Direction
<==
l::1,
= ~eBo,t
Fig. 3.137
When
loaded, the body tail
sinks further than the
front; the caster angle
increases by its angle
alteration 6.0 so ,t (see
also Fig. 6.15).
'T
Direction
...
Fig. 3.138
On most double wishbone suspensions, the axes of rotation 1 and 2 are parallel to one
another; in such cases, caster does
not change when the wheels
compress and rebound.
-r-
Direction of
travel during
jounce
I I
The Automotive Chassis
246
..
Direction
2
i_,
\
\~~G
)
,
I
"".
Fig.3.139 If the line EG and
control arm axis form a right angle
on the McPherson strut and strut
damper there is no caster alteration. Point G moves vertical to the
suspension control arm axis when
the wheels bottom out, i.e. parallel
to the line EG. The axle shown has
a negative caster offset -nT and
the lower link G shifted forwards.
The line EG gives a small caster
angle and the trail {T,k.
The steering arm 1 is positioned
high up and inclined backwards;
the disc brake calliper 2 is at the
front, giving the disadvantage of a
higher wheel bearing load during
braking, (See Section 7,4 in Ref.
[6] ,)
/
Direction
....
Direction of travel during jounce
Fig.3.140 To create a virtual centre
of rotation pole on the front axle (see
Fig. 3.155) on double wishbone suspensions, the axes of rotation C and 0 must
be inclined against one another. The
disadvantage of this is that when the
wheels compress, point 1 moves to
point 3 and point 2 to 4, increasing the
caster angle by Lh, equivalent to twisting
the steering knuckle by this angle.
When the front wheel compresses, the upper ball joint 1 of the steering
knuckle moves backwards and the lower one forwards, resulting in an increase
in caster. Rebounding has the opposite effect - the caster (if in the normal position) decreases and may even become negative. In the case of McPherson struts
and strut dampers, point 2 moves to 4, parallel to the axes of rotation, and
compresses the damping element, which is fixed in point 1. This shortens and
there is rotation by the angle ~'T.
As can be seen in Figs 3.155 and 3.156, a virtual centre of rotation Or, which lies
behind the front axle, would be more readily achieved with a double wishbone
Wheel travel and elastokinematics
247
Direction
Direction of travel
during jounce
Fig. 3.141
When the McPherson strut or strut damper compresses, point 2
moves to 4 and the caster angle increases by dT. The intersection of a parallel to the
suspension control arm axis of rotation (drawn through point 2) and a vertical on the
damper centre line in point 1 gives the vehicle the pitch pole Ot. The steering knuckle
fixed to the damping part also rotates by this angle.
Of or r
Direction
~
/
Fig. 3.142
So as not to reduce the ground clearance of the front axle and the
front overhang (Fig. 1.67), the back 1 of the anti-roll bar must be raised. The arms 2,
which support the lower transverse links in the longitudinal direction, therefore drop
backwards. The result is a vehicle virtual centre of rotation Ot in front of the axle,
which causes the front end to be pulled further down during braking and adverse lift
of the front end when a front-wheel drive vehicle moves off (Figs 4.1 and 3.143).
In the case of a rear wheel suspension, this position of the vehicle pitch pole Or
would be favourable.
suspension than with McPherson struts and strut dampers (Fig. 3.141). If the antiroll bar is located in front of the axle and used for the longitudinal wheel control,
its rear-end must be raised to provide enough ground clearance (Fig. 3.142). This
will cause the centre of rotation to lie in front of the axle and will also draw the front
end down when the brakes are applied. Figure 3.143 shows the kinematic caster
-I
248
The Automotive Chassis
100
1
a.
mm
/
80
/
E
::J
\
OJ
60
/Uno
V
Polo
40
\
20
Q)
>
co
....
o
Q)
Q)
.c
~
20
10
20
\
\
/V
30
\
\
"'0
C
::J
o
.0
Q)
0::
!
80
mm
I
I-- Mercedes
go
10" /
j
~1---,
I
II"
..
Caster angle
/
( (
40
60
/
\T
I
12"
140
I
Design position _
I
Curb weight
/
/
I
100
Fig. 3.143
Typical caster alteration curves for McPherson struts and strut
dampers measured on three front axles. The strut damper of the Mercedes has a
large caster angle that increases even further when the springs compress, i.e. a positive anti-dive mechanism. There is no such mechanism on the suspension strut of
the Fiat Uno (the almost vertical curve shape indicates this) and the McPherson
suspension on the VW Polo (1995) has a pro-dive mechanism.
The front end is further drawn down when the vehicle brakes; this phenomenon
becomes more pronounced the more it dips. The reasons for this anti-dive are the
vertical position of the suspension strut and the high location of the anti-roll bar back;
the virtual centre of rotation is therefore far in front of the axle. Figures 3.139, 3.143
and 4.1 give details. On the Mercedes, the lateral force lever nT.k on the compressing wheel on the outside of the bend increases; this means, therefore, that speeddependent lateral force understeering occurs.
Wheel travel and elastokinematics
249
alteration measured on three passenger cars with spring dampers or McPherson
axles. The curve shape clearly shows whether there is an 'anti-dive' or a 'pro-dive'
mechanism.
From a design point of view, the alteration angle AT = f (s) can easily be determined by drawing.verticals to the suspension control arm axes of rotation C and
D through the centres 1 and 2 of the wheel joints, as shown in Fig. 3.140. Fixed
paths must be marked off on one of the two verticals and, using a compass with
the path 1-2 the corresponding point on the other determined. The angle AT of
the connecting line 3-4 to the initial position 1-2 is the caster alteration. In the
case of McPherson struts and strut dampers (Fig. 3.141) the upper point 1 is
fixed in the wheel house, so that the distance 1-2 shortens when the spring
compresses (path 1-4) and lengthens when it rebounds.
Figure 3.140 shows that when the vehicle is designed with a virtual centre of
rotation, the steering knuckle rotates by the angle AT - clockwise at the front
\
1\
\
20
\
o
Q.l
_4°
-ATr
>
...
co
~
Design position (0 W)
/
~20
0
.... 20
Q.l
Q.l
..c
40
3';:
'"
6 \T
-g
80
::J
o
.0
Q.l
100
0::
!
mm
12 0
I
+ 2°
+tT'i
I
'"
I
I
Empty height (C W)
'\.
'\
~
"
"" ""
~
Fig. 3.144 Alteration Ll'Tr in the theoretical negative caster angle, measured as a
function of the compression and rebound travel on the rear axle of a Mercedes. The
company specifies the trail as (T,k = -15 mm; in the design position this would correspond to an angle of 'Tr - _3°. This increases as the springs compress, and it
decreases or goes into positive caster as they rebound. The inclined position of the
curve indicates high virtual centre of rotation that move further upwards when the
wheels rebound and therefore progressively reduce the brake dive. Furthermore, the
negative caster trail increases on the compressing outer wheel during cornering,
resulting in favourable lateral force understeering, increasing with speed.
-- _ _ 5
=
250
The Automotive Chassis
(Figs 3.141 and 3.143) and anticlockwise at the rear axle (Fig. 3.144); this is
demonstrated by the shape of the curves in the above figures. This wheel traveldependent rotation causes a changing relative speed between stator and rotor in
the wheel sensors of all wheel slip control systems, which adversely affects the
response speed of the ABS and ASR traction control. Comprehensive information is given in Chapter 3 of Ref. [7].
3.10.6
Wheel travel-dependent rotation of the rear steering
knuckle
On the multi-link independent rear suspensions fitted in Mercedes models, five
rods are used to control the steering knuckle with four of them providing lateral
force reaction support (see Section 5.3.4 in Ref. [2]). The extensions of the two
upper rods intersect in pole E and those of the lower rods in G. The lines
connecting the two poles give the theoretical steering axis EG. A negative caster
was set (Figs 3.117 and 3.145) to obtain lateral force understeering (Fig. 3.73).
The tyre caster rT,T, which reduces this, must also be considered. The lateral
force lever is then (Fig. 3.120, see also Equation 3.32):
(3.41b)
If anti-dive behaviour is desired, the centre of rotation Or must lie in front of the
rear axle, as shown in Figs 3.142 and 3.153.
•
,,~
E
Fr
1 F.
.~.
!\G
Fig. 3.145
If, on a multi-link rear suspension, there are four bars supporting the
lateral forces, when viewed from the rear, their extensions meet in the points E and
G. When they are connected in the side view, the result can be the theoretically
negative caster angle -'Tr and the caster trail -(T.k on the ground.
Where the brake is on the outside, the braking force Fx.W,b should be regarded as
acting at the centre of tyre contact. The rolling resistance force FR and the tractive
force Fx.w.a have to be shifted into the wheel centre.
____,
.------~----------_,_----------,--..1-
Wheel travel and elastokinematics
251
A parallel development is the multi-link rear axle, which is becoming more
and more popular. This contains a trailing link (that forms one piece with the
steering knuckle), with a pivot in front of the axle centre which, simultaneously,
represents the centre of rotation Or (Figs 1.1, 1.18, 1.62 and 1.77; see also
Section 5.3 in Ref. [2]). The kinematic movement of the wheel carrier corresponds to that of the longitudinal link suspension (Figs 3.159 and 6.17).
3.10.7
Resolution of the vertical wheel force on caster
If the steering axis EO on a double wishbone suspension is angled by the caster
angle 'T, the lower ball joint lies in front of the wheel centre and the upper one
behind it. If the spring is supported on the lower suspension control arm, its force
Fa,Z may be the same size as the vertical wheel force less the weight of the axle
side (Fig. 3.146, Equation 5.3), but the moment M z = F a,Z (f - e) occurs, causing the forces FE•x and Fa,x. The compliance present causes the caster angle to
reduce. If the spring were on the top, it would increase.
Where there is caster (Case 1), the vertical force component F z.w cos (J", shown
in Fig. 3.105, would be further resolved by the angle 'T, i.e. in F z.w cos (J" cos 'T and
Fz,w cos (J" sin 'T (Fig. 3.147). The last component tensions the wheels via the lever
q at the front (Fig. 3.148). If the caster angles 'T on the left and right wheels are
different, the same will apply to the tensioning forces, i.e. the vehicle could deviate from the direction of travel if the steering wheel were let go, and would pull
to one side when held (Fig. 3.149 and Equation 3.41c). A 2° ditIerence means that
there is a 30-40 N higher tie rod force on the side with the greater angle 'T.
If the caster is achieved by relocating the wheel centre to the back (Case 2),
the component Fz,w sin (J" pushes the wheels together at the front via the force
lever n, (Figs 3.150 and 3.151), i.e. even here the parallel position of the left and
right steering axes to one another plays a role.
Side view
Direction
~
fr
Fig. 3.146
If the spring is supported
on the lower suspension control arm
and if the front axle has caster, the
supporting ball joint will be in front of
the wheel centre. Forces Fz,w and FG,z
form a moment, which generates the
reaction forces -FE,x and +FG,x in the
direction of the suspension control arm
axes of rotation. In the example these
are assumed to be parallel to the
ground.
'dyn
Fz,w
252
The Automotive Chassis
Fig. 3.147 If the steering axis
is at the caster angle T in the side
view, the vertical force component
FVN cos (J calculated in the rear
view in Fig. 3.105 must be further
resolved.
Direction
<;:==
sinT
Fz,w cos C1 COST
Fz,w
Fz,w cos (1. sin r
i--t<::::=:=
q~,~
...
Direction
q+--I<:;::::::J
Fz w cos (1 sin T
Fig. 3.148 The forces Fz,w cos (J
sin T push the front wheels together
at the front via the levers q (i.e. into
toe-in) both when the vehicle is in a
stationary position and moving in a
straight line, and generate the forces
Fr in the tie rods. The caster angles T
left and right may therefore only
deviate slightly from one another
(see Equation 3.42a).
_+-.a-_.....: '
MZ,w,'t,rs
r
q
Direction
-Fz,w
+Fz,W
cos q sin T
cos q sin T
q
Fig. 3.149 Caster on the left and negative caster on the right front wheel (or
caster angles T of different sizes) cause the vehicle to pull to the right when travelling in a straight line. This is caused by opposed moments:
MZ,W,T
____,
= ±Fz,w cos
,---
(J
sin Tq
(3.41 c)
--------------::J'-
Wheel travel and elastokinematics
Fig. 3.150 With
caster (Case 2, Fig.
3.116) achieved by
setting the wheel centre
back, the vertical force
component Fz.w sin <r
comes to be located
behind the steering axis.
253
Direction
Fig. 3.151 Left and right vertical force
components Fz.w sin (J' push the front
wheels into toe-in when the vehicle is
stationary and when it is moving in a
straight line, and put the tie rods under
stress (forces Frl.
The camber angle ew (and therefore also
the kingpin inclination angle (J') should be
largely the same left and right (see
Equations 3.4b and Fig. 3.103).
Direction
In addition, equal kingpin inclination angles are required on both wheels, and
because these are generally directly related to the camber (see Section 3.9), only
a small camber deviation between left and right front wheels is permissible (see
Equation 3.4c).
Where the (J angles are different, the length of the vertical force lever q =
(ra + rdyn tan (J) cos (J (Equations 3.46 and 3.21a), which appears in all formular,
changes and, with caster as in Case 2 above, the vertical force Fz,w sin (J is no
longer the same on the right and the left. Both instances cause the steering wheel
to pull to one side.
The negative offset nT shown in Fig. 3.118 - together with the angle T requires a more in-depth look at the correlations (Fig. 3.152). The vertical force
Fz,w on the wheel axes resolved in the direction of the kingpin inclination, gives
Fz,w cos (J and Fz,w sin (J. The first component must be further divided up in the
side view into Fz,w cos (J sin T and Fz,w cos (J cos T. As can be seen in the top
view, when the vehicle is moving in a straight line, there are two opposing
moments on each wheel (which can cancel each other out):
MZ,W,T,t
= Fz,w (cos
(J
sin
T
q- sin
(J
cos
T
nr )
Further details are given in Sect,ion 7.2 of Ref. [3].
r
(3.42)
The Automotive Chassis
254
E
17'
Ff..w cos a sin r '.
Ff.,w cos a
Ff.,w cos a cos r
Ff..w
/
Fig. 3.152 Force ratios on front axles with negative caster offset -n The
opposed moments Fz.w sin an, cos ,- and Fz,W cos a sin ,-q an cancel one another out.
T
3.10.8
•
Settings and tolerances
The caster value of the empty vehicle should appear on the drawing and in workshop manuals. Optical measurement is also carried out in this load condition, as
specified in DIN 70 020.
Where there is no caster offset, in order to ensure favourable steering selfcentring, passenger cars of a standard design have caster angles of around 4° to
8°. However, in the case of a designed offset -nT' the values can rise to 'T = 8°
to 11 0. The type of steering system is also a factor here. If it is power assisted
(see Sections 4.2.5 and 4.3.3), the steering moment must also right the parts in
the hydraulics. In such cases a greater caster angle is preferable. If no power
steering is available, lower angles have generally to be designed to limit the
steering effort, especially during parking manoeuvres.
Front-wheel drive vehicles are set to 'T = 1° to 4°. The righting moment, which
is strengthened by the tractive forces MZ.T,Y (Fig. 3.119), means that caster values
are not absolutely necessary.
In addition to the absolute value, a tolerance is required, which is usually
around +30' but can be as much as + 1°30' to make manufacturing more costeffective. The additional requirement (as in the case of camber, see Equation
3.4c) that there should be no greater difference than 30' between left and right
wheels is necessary to prevent the vehicle pulling to one side (Fig. 3.149). The
details given on the drawing would then be:
'T
=4°
-
+ 30'
maximum difference between left and right 30'
(3.42a)
Wheel travel and elastokinematics
3. 11
3.11.1
255
Anti-dive and anti-squat mechanisms
Concept description
The anti-dive mechanism reduces the amount by which the front end of the vehicle dips or the tail rises when the brakes are applied. It can - in the case of brakes
which are outside in the wheels - only be achieved if there are pitch poles Of and
Or between the axles at the front, at the rear, or on both axles (Fig. 3.153).
The anti-squat mechanism reduces the amount by which the tail drops on
rear-wheel drive vehicles or the front end lifts (on front-wheel drive vehicles). It
acts during acceleration and only on the driven axle. On independent wheel
suspensions it is important for the pole to be higher than the wheel centre of the
driven axle (as can be seen in Figs 3.156 and 3.160) or, on a rigid axle, the differential is located in the axle housing (Figs 1.22 and 1.43). For further details, see
Sections 6.3.2 and 6.4.1 in Ref. 2.
The anti-dive and anti-squat angle is also a consideration here; e and K are
entered in Fig. 3.160; the greater these can be the better is the pitch equalization.
3.11.2 Vehicle pitch axis front
Left and right suspensions are generally identical so the pivot axes determined
by the momentary position of the suspension control arms are in the same position on both sides, which leads to the so-called pitch axes. If these are at infinity (i.e. for practical purposes they do not exist, Fig. 3.138) the longitudinal
forces are concentrated in the wheel centre, which applies if the brake is located
on the inside (on the differential). Here the brake dive can be countered by the
two double wishbones being set at an angle in the same direction (Fig. 3.154).
As can be seen from the illustration, the brake force operating as F~,W,b, shifted
from the wheel centre vertical to the steering axis (shown in Fig. 3.111 for the
rolling resistance), causes the reaction forces F E,x and F a,X in the suspension
Direction
Fig. 3.153
The pitch axis is obtained by linking virtual centres of rotation front and
rear. If these are available with Ot (at the front) and Or (at the rear), the body is
supported at this point in the longitudinal direction when the brakes are applied,
assuming the brakes are on the outside of the wheels.
r
256
The Automotive Chassis
..
Car body
Direction
?+F.
I e,z +Ft ,z
G
~
Fig.3.154
If the front brake is on the inside on the differential, brake dive can be
compensated by disposing the suspension control arms in the same direction, but at
an angle. The braking force must be regarded as being under the wheel centre by
a = fa sin (J' (see Equation 3.28b). When it compresses, the wheel moves forward.
The diagonall springing angle is K = (ex + ~)/2.
control arms, which (due to the angled position) cause the vertical component
- FE,z = FE,x tan ex and - Fo,z = Fo,x tan ~. The sum of forces in one effective
direction must be zero, i.e. +FE,z and +Fo,z work against the vehicle front-end
bump travel. Two suspension control arms, which are placed at an angle in this
manner, have the advantage of no caster change but the disadvantage that they
move forward during jounce (in other words in the direction of the obstacle
force), The Citroen GSA had this type of suspension control arm configuration
and therefore an almost 100% anti-dive system (see Section 5.2.4 in Ref. [2]).
Where the brake is on the outside (as shown in Figs 3.153 and 6.16), it is also
necessary for the suspension control arms to be at an angle to achieve a pitch
axis and, therefore, reaction forces in the vertical direction. However, the two
suspension control arms must be inclined against one another. The right-hand
side of Fig. 3.155 shows the statics with the (compared with Fig. 3.154) significantly increased components Fo,z, caused by the higher force Fo,x = FX,W,b + FE,x
in the case of outside brakes (in the case of inside brakes Fo,x = FX,W,b -- FE,x). All
front-wheel drive vehicles built in Germany have a negative kingpin offset. The
prerequisite for the counter-steering effect, which can be achieved in this way
(Fig. 6.12), is a brake inside the wheel. By angling the lower suspension control
arm, on a double wishbone suspension it is possible to reduce both the brake dive
and the squat. The brake force F'x.,W,b acting now on the steering axis by the
amount a above ground causes the component F G,z supporting the body (Fig.
3.156) and -- as shown on the right - the drive-off force F'x.,W,a acting below the
wheel centre causing the force - Fo,Z pulling downwards. The upper suspension
control arm is horizontal. Its job can also be done by a vertically positioned
McPherson strut or strut damper. In this type of suspension there is an anti-dive
and anti-squat mechanism.
The pitch axis on double wishbones can be shown on a drawing using parallels to the suspension control arm axes of rotation C and D, drawn through the
......
.
'
YVheel travel and elastokinematics
257
r
Car body
Direction
____
j +FE
,z +F(;,z
.
F'X,W,b F6 ~
l
Fig.3.155 To reduce brake d~' e when the brakes are on the outside, the suspension control arms must be inclin d against one another. The forces FE,x and FG,x must
be calculated ~ssu~in~ the braki~' g force FX,W,b below ground by a = rb sin ( j (or in the
case of negative kingpin offset, bove ground by the same amount, Equation 3.27).
The components acting against ront-end dip are then +FE,z and +FG,z; in the case of
-r all forces are smaller. In the rase of caster FX,W,b = Fx,W,b cos or (Fig. 3.115).
CT ,
Car body
..
Direction
Car body
~
~
E
=+--
-F,G,l
F"
X,W,b
FG
~-Fr.=-G,x
Starting
Fig.3.156
In front-wheel driv
off and front-end brake dive can
only at an angle, if (as is usually
In the case of a negative kin
amount a above ground (see Eq
vehicles, both the lifting of the vehicle as it moves
e reduced by disposing the lower suspension links
he case) the brake is in the wheel.
pin offset, FX,W,b acts on the steering axis by the
ation 3.27).
centres of the ball joints E an G (Fig. 3.155). The McPherson strut and strut
damper require a vertical to e set up on the direction of movement of the
damper in point E, the interse tion of which with the suspension control arm
parallel passing through G, giv s the point Or (Fig. 3.141), or an extension of the
tension rods or anti-roll bars a sorbing the longitudinal forces (Fig. 3.142).
258
The Automotive Chassis
Direction
Fig. 3.15'1 To determine the vehicle virtual centres of rotation Of on the longitudinal transverse axle the upper suspension control arm must be lengthened and a
parallel must be drawn to the suspension control arm axis of rotation through the ball
joint centre. When the front end moves towards bump, Of moves towards the
wheel, this being the equivalent of a progressive anti-dive mechanism.
On the trailing link axle, in order to get the axis Of, the upper control arm
must be extended and again a parallel to the axis of rotation must be drawn
through the lower wheel joint (Fig. 3.157). When the front end of the vehicle
jounces, the upper suspension control arm moves to a greater angle of inclination and the pole 0 moves closer to the wheel. This means a progressively
increasing anti-dive system, which also applies if, in the case of a double wishbone suspension, the braking forces are absorbed upwards through a trailing or
semi-trailing link (Figs 1.39 and 3.32) or by the legs of the anti-roll bar.
3.11.3
Flitch axes rear
The requirement for a reduction in the brake dive demands a pitch axis that is close
to the wheel and as high as possible; however, both of these result in a severe caster
change on the front axle. Here - particularly where the vehicle is fitted with ABS
(see the end of Section 3.10.5) - a compromise must be struck between both criteria. On the rear axle, the picture is different. The virtual centres of rotation Or can
be positioned close in front of the axle whereby the length of the suspension
control arms and the ABS performance objectives represent the limits.
Too short a suspension control arm gives unfavourably large rotation angles
+ K (Fig. 3..158) to achieve the desired spring travel s\ and Sz. The wheel base
change associated with the pitch axis and ill should not affect the handling properties. As proof, we can look at the earlier Renault models on which there were
different wheel bases left and right.
The trailing link suspension and the compound crank axle (Figs 1.2, 1.13 and
1.31) have the best position of the pitch pole among the wheel suspensions
commonly used as rear axles. These lie in the centre of the suspension control
arm axis of rotation and the force - FO,z, which draws the rear end down during
braking, is :in accordance with Fig. 3.159:
Fo,z
= FX,W,b g/d
(3.44)
I
~heel
...
travel and elastokinematics
259
Directiorl
r
I'
al
51
/
Fig. 3.158 Longitudinal links
virtual centre of rotation Or. The s
however, with the required sprin
may arise must not be too large.
the driver will hardly be aware of
n a rear axle have the advantage of the favourable
spension control arm should be as short as possible,
travel 51 and 52, the angular deflections ±K, which
These would lead to significant diagonal springing f,
he change in wheel base associated with this.
Fig. 3.159 On trailing link an multi-link
suspensions with axes of rotatio parallel to
the ground, the mounting point n the body
is also the pitch axis. The higher the axis
lies (path g) and the closer it is t the
wheel (path d), the more the for e -Fo,z
pulls the tail end down during br king.
%~
-F:.o
,I
PO,x
'dyn
d
i.e. the higher the height g and he shorter the distance d can be, the stronger the
effect.
Figure 3.159 also applies to 11 'multi-link axles' as well as rigid axles carried
on two trailing links:
•
•
•
•
the
the
the
the
twist-beam axle (Fig. 1.6
drawbar or A-bracket axl
off-road vehicle axle sho
multi-link axles in Figs 1
)
(Fig. 1.60)
n in Fig. 1.43.
1, 1.62 and 1.77.
For the semi-trailing link axle, he top view must be drawn first to ascertain the
virtual centre of rotation (Figs l160 and 3.36). Using the angle (x, the distance d
(pitch pole 0 to wheel centre) i determined and then, in the rear view, the height
g of point 0 is also determined. The side view then shows the actual position.
If a watt linkage or a pair of ontrol arms per side is used to link the rigid rear
axle (Fig. 3.41), the centre lin s of the suspension arms must be extended and
made to intersect to obtain Or ( ig. 3.161 and Section 3.4.5). The shorter upper
suspension control arms ensure that the pitch pole moves favourably towards the
axle when the vehicle is loaded (in other words the tail sinks at points E and G)
and therefore the anti-dive mec anism is reinforced.
The Automotive Chassis
260
Side view
7777777/
Fig.3.160
On the semi-trailing link suspension the point at which the extension
of the axis of rotation goes through the plane of the wheel centre gives thl~ pitch axis
O. The brake reaction support angle B can be calculated from the existin~J paths:
tan
B
:= g/d
(3.45)
The same applies to the anti-squat (or diagonal springing) angle
and Fig. 3.30):
tan
K
= (g -
K
(see Section 5.4.4
(3.46)
{dyn)/d
except that here the sign of the integer is important. In the case of +K (as shown)
when the vehicle accelerates, the squatting tail is pushed upwards and in the case
of -K, it is pulled further down.
r=O
Fig. 3.161
If a rigid rear axle is
controlled by two trailing link pairs, its
extensions give the pitch axis Or. When
the vehicle is laden, points E and G on the
body side move down, i.e. Or moves
towards the wheel in a favourable manner.
If, as can be seen in Fig. 1.43, the differential is contained in the axle housing, the moments coming from the engine and driving the wheels are vertical to
one another (Fig. 1.22); because the forces on the axle housing are jointly
supported, the anti-squat mechanism is, at the same time, supported at the pitch
aXIS.
3. 12
3.12.1
Chassis alignment
Devices for measuring and checking chassis alignment
The handling properties of a vehicle, both in the steady-state as well as the transient region, are determined by the kinematics - the change in position of the
-,,-----_._--_._--------,----
--...-,
heel travel and elastokinematics
261
Fig. 3.162
Computer-aided heel kinematics measuring device of the Chassis/Simulation Technology Laborato of the University of Applied Science, Cologne.
Changes in the position of the Fheels in the course of body lifting and lowering
movements can also be measur d by means of actuators which are integrated into
the test stand. .
wheel during travel or roll mo ements of the body - and the elastokinematics the movements of the wheel wih longitudinal or lateral forces in the tyre contact
area - of the wheel suspensio . As even slight changes in the position of the
wheels can have a big effect n handling properties and the wear and tear of
tyres, particular importance is attached to the accuracy of the measuring technology used and a precise kno ledge of the way in which control of the wheels
depends on forces and movem nts.
The wheel kinematics are m asured by mechanical, optical or computer-aided
measuring devices (Fig. 3.16 ). The vehicle, loaded in accordance with the
construction or other specificati ns of the manufacturer, is placed on four sliding
plates which are made level wi h each other. These permit the forceless horizontal movement of the wheels. Th castor, toe-in, camber, kingpin and crab angle as
well as the track change are measured by sensors attached to the spin axes.
In order to measure the elas okinematic properties of wheel suspensions, test
stands are used that introduce oth longitudinal and lateral forces into the tyre
contact area. Different wheel tr vel positions and contact area angles can also be
represented to simulate the roll movement of the vehicle body during cornering
(Fig. 3.163). Forces are introduc d into the vehicle either by fixing the vehicle body
on the test stand and applying forces and movements to the four wheels, or by
applying forces and movement to the vehicle body, with the tyre contact areas
representing a fixed plane. Parti ular attention is paid to the attachment of the body
to the test stand regardless of th kind of forces which are introduced, as unwanted
flexibility leads to false results. Where possible, the body should be held in the
'------'------,_._-----,........-----------'---!
262
The Automotive Chassis
Fig. 3.163
Elastokinematics test stand of the Chassis/Simulation Technology
Lab-oratory of the University of Applied Science, Cologne. The movements of the
wheels are measured by a combined system of filament potentiometers and inclinometers constructed on the basis of a system of error and fault minimization;
I
forces are measured by means of force transducers. A computer system is used for
measurement, stipulation of the required values and evaluation. Forces can be
1<
applied with the normal tyres left in place or via wheel replacement carriers; a i r - ,..
sprung sliding plates are used to minimize f r i c t i o n . ,
I
immediate vicinity of the wheel suspensions, for example on or in the longitudinal
frame side rails, the slispension strut domes or the auxiliary frame connection
points. In order to prevent unwanted elastic deformation of the tyres, wheel replacement carriers are used which reproduce the exact force application conditions (such
as tyre and diameter, offset, effective lever arm with traction or braking forces).
The standards governing the accuracy of measurement of the forces, displacements and angles to be ascertained are very high; Fig. 3.164 gives reference
values for these.
3.12.2
Measuring the caster, kingpin inclination, camber and
toe-in alteration
3.12.2.1
Measurement conditions
In repair workshop manuals, the axle settings (with a few exceptions) relate to
the vehicle when empty, and when checking only the manufacturer's specified
values, this condition should be assumed. To eliminate the influencing friction
.----
•
• _ _'--r-
._ _
~
II
I
I
Wheel travel and elastokinematics
Wheel load
Wheel vertical travel
Longitudinal force
Longitudinal (fore-and-aft) travel
Lateral force
Lateral displacement
Camber angle
Steering wheel angle
Steering angle
Caster angle
Steering wheel moment
Steering wheel angle
Fig. 3.164
263
Measurement
area
Measurement
precision
Unit of
measurement
20000
±150
±10 000
±75
±10 000
±75
±10
±5
±45
±10
±20
±900
40
0.5
20
0.2
20
0.2
0.01
0.025
0.2
0.02
0.2
N
mm
N
mm
N
mm
1
degrees
degrees
degrees
degrees
Nm
degrees
Necessary measur ng ranges and accuracy of elastokinematic charac-
teristics
in the suspension parts, the ve icle should be briefly settled by hand on both
axles before measurement begi s.
The initial basis for all altera ions resulting from the wheels compressing and
rebounding is the design posit on. The vehicle carries a load based on three
people with a weight of 68 kg each. Preferably, if this is done, use dummies
which can be filled with wate , as these exactly reflect the masses and mass
distributions of the occupants 0 be represented. The static settings are determined in this condition. Even oad distribution (apart from the 'swinging' or
dynamic load transfer) is imp rtant because otherwise the body can tilt and
therefore take on different cam er on the left and the right. It is therefore essential that the third person should it in the middle of the rear seat. The bump travel
between the empty condition a d the design position should be taken from the
wheel house arches so that the ehicle can later be drawn down as far as possible against a fixed resistance fo the static measurements.
3.12.2.2 Measuring the cam er angle
A spirit level or electronic mea uring device can be used to measure the static
camber angle precisely if the ero position of the device corresponds to the
wheel centre plane. If the wh el is turned slowly, the device holder can be
aligned.
3.12.2.3 Measuring the cast r angle
Determining the static angle In (regardless of the measurement condition)
require a steering angle input 011 3 = + 20°. The greater the angle 'T the more the
body sinks down over the whe I on the outside of the bend and is accordingly
pushed up over the wheel on th inside of the bend (Fig. 3.165). The body tilts
slightly, and the positive cambe on the side of the vehicle on the outside of the
264
The Automotive Chassis
1--_-+-_ _i--_-+_--t+/iH 8t----t---+----+--~1F---~
mm
6+---+--+---,~--+---l
4+---+-~+---t-~~+---l
2t--""'~~+---+-T
=40 1----1
400
30 0
300
40 0
2+---+--+----+
4+----+---1----+--+---1
6+---+---1----+---1-----\
1----+---I----+---+_
6H
8
Fig. 3.165 Lift heights flH calculated for the wheelan the outside and the one
on the inside of the bend as a function of the steering angle with the settings (Jo =
6 0 , ra = +25 mm and various caster angles. Where 'T = AD, the centre of tyre contact
of both wheels moves below ground (flH becomes negative), which is equivalent to
lifting the body. The larger is 'T, the more the body is raised on the inside of the bend
(-flH at Oi), but drops on the outside of the bend. When kingpin inclination and caster
are measured, these relationships must be taken into account. In the case of ra = 0,
straight lines rather than sets of curves are produced and when the kingpin offset on
the ground is negative, the curves bend in the other direction.
bend consequently increases, and that on the inside of the bend reduces. The
associated decrease in the kingpin inclination on the outside (and increase on the
inside) of the bend can lead to a measurement error, if the body is not braced
against a fixed resistance to obtain the necessary horizontal position during the
measurement process.
3.12.2.4 Measuring the caster alteration
To avoid a pitch angle distorting the measurement (Fig. 3.137), the body should
be drawn down parallel (or pushed up parallel). Only the alteration values + AT,
are ascertained and these must be deducted from or added to the initial data in
the design position. The simplest way of doing this is to determine the rotation
of the wheel with a measuring device. It is important that the brakes be locked.
The floating plates (on which the wheels stand) flex longitudinally and laterally.
if
I
Wheel travel and elastokinematics
meth~
265
No other measurement
can be used on the rear axle; Figs 3.143 and
3.144 show curves recorded inl this way.
I
3.12.2.5 Measuring the kin pin angle
Once the static caster angle ha been determined in the empty condition, a mean
value between left and right sh uld be calculated to eliminate the angle To calculated in this manner, by raisin the tail end (or lowering it in the case of negative caster) and thereby obtai ing steering axes that are vertical from the side
view. To measure the angle 0", teering inputs (where possible up to B + 20°) are
necessary, and the kingpin inc ination angle is determined via the three-dimensional movement of the wheel centre plane. The modification values should be
the same for left and right inpu s. Figure 3.132 indicates the correlations clearly.
If the vehicle has the wheelba e l, the necessary lift height AhT in the middle of
the rear axle would be:
I
AhT = l sin
I ·
To
(3.43)
In the raised position, the castfr must then be zero, but it is always worthwhile
checking.
i
~nclination
3.12.2.6 Checking kingpin
and camber
As shown in Fig. 3.103 the su of camber and kingpin inclination (sw + 0") left
and right should be the sam. If the deviation exceeds 30', this may be a
measurement error, the result of an accident, or an assembly inaccuracy. on
McPherson struts and strut da pers (Fig. 3.104).
3.12.2.7 Measuring kingpi inclination and camber alteration
The two are identical and p re alteration values can easily be determined.
(+ ASW,k = + AO", see Figs 3.50 nd 3.51). Only spirit levels or electronic measuring devices need to be fixed tb the wheels and corrected by the caster angle,
which changes as the vehicl~is drawn up or down parallel with the brakes
locked. The values should the be added to or subtracted from the data determined in the design position. igures 3.48 and 3.49 indicate curves measured in
this way.
I
I
3.12.2.8 Measuring the toeJn alteration and drive axle angle
The static toe-in angle Av.o,f
(at the front or back, see Equation 3.8) can be
determined nowadays with 0 to-electronic measuring devices. The alteration
values for the front and rear a Ie should then be recorded as a function of the
wheel travel Sl and S2 - separa ely from the basic values - for the left and right
wheel and added to the basic alues. The wheel position is measured relative to
the body, so it is sensible to wOfk with optical devices and to fix the scale (or the
mirror) to the body itself. Lateral movements of the vehicle, when it is raised or
pulled down, could otherwise 1ead to errors when reading off the figures. Drive
oJ
axle angle 13' indicated in Fig. .63 can be detennined with the aid of the stationary toe-in angle.
I
I
i
i
4
Steering
This chapter gives only the essential aspects of the subject: details are given in
Refs [1] and [2] and the connections relating to four-wheel drive passenger cars
are described in Ref. [9], Section 5.2.
The steering system is type-approved on all new passenger cars and vans
coming on to the market; it is governed by the following EC directives.
70/311IEWG
74/297IEWG
911662IEWG
92/62/EWG
Figures 4.1, 1.46, 1.57 and 1.72 show the complete steering system of a frontwheel drive passenger vehicle with left-hand steering.
4.1
4.1.1
Steering system
Requirements
On passenger cars, the driver must select the steering wheel angle to keep deviation from the desired course low. However, there is no definite functional relationship between the turning angle of the steering wheel made by the driver and
the change in driving direction, because the correlation of the following is not
linear (Fig. 4.2):
•
•
•
•
turns of the steering wheel;
alteration of steer angle at the front wheels;
development of lateral tyre forces;
alteration of driving direction.
This results from elastic compliance in the components of the chassis. To move
a vehicle, the driver must continually adjust the relationship between turning the
steering wheel and the alteration in the direction of travel. To do so, the driver
f
Steering
267
-Fig. 4.1
Damper strut front xle of a VW Polo (up to 1994) with 'steering gear',
long tie rods and a 'sliding c1utc ' on the steering tube; the end of the tube is stuck
onto the pinion gear and fixed wi h a clamp. The steering arms, which consist of two
half shells and point backwards, re welded to the damper strut outer tube. An 'additional weight' (harmonic dampe ) sits on the longer right drive shaft to damp vibrations. The anti-roll bar carries !he lower control arm. To give acceptable ground
clearance, the back of it was dtSigned to be higher than the fixing points on the
control arms. The virtual pitch a is is therefore in front of the axle and the vehicle's
front end is drawn downwards i hen the brakes are applied (Figs 3.142 and 3.143).
i
i
will monitor a wealth of infotmation, going far beyond the visual perceptive
faculty (visible deviation fro~ desired direction). These factors would include
for example, the roll inclinatio~ of the body, the feeling of being held steady in
the seat (transverse acceleratio ) and the self-centring torque the driver will feel
through the steering· wheel. T e most important information the driver receives
comes via the steering moment or torque which provides him with feedback on
the forces acting on the wheel~.
i
i
I
268
The Automotive Chassis
1600
8°
Slip angle, front right
t
6°
120·
Q,)
0)
c:
co
Steering wheel
angle to left
I
'0
c.
80° ~
0)
c:
co
Slip angle, rear right
(J)
1
Q,)
Q,)
2°
40°
.c:
~
0)
V=
0
c:
.;::
100km h-1
0
1.0
0.5
1.5
Time (5)
..
2.0
2.5
5
Q,)
Q,)
+-'
(J)
3.0
Fig. 4.2
Delayed, easily manageable response of the right front wheel when the
steering wheel is turned by 100° in 0.2 s, known as step steering input. A slip angle
of at :::: 7° on both front tyres is generated in this test. The smaller angle ar on the rear
axle, which later increases, is also entered. Throughout the measurement period it
is smaller than at (x-axis), i.e. the model studied by Mercedes Benz understeers and
is therefore easy to handle.
•
I
t
Direction
I
Fig. 4.3
Synchronous steering A-bar on the front suspension of a left-hand drive
passenger car or light van; on the right-hand drive vehicle, the steering gear is on the
other side. The steering arm (3) and the pitman arm (4) rotate in the same direction.
The tie rods (2) are fixed to these arms.
____,_,
,
-----r------------..:l..-
Steering
269
t
Direction
I
121
7
1
/ ~
8
\
7
3
I
Rack .and pinion ~teelrin~ with. t~e steering linkage 'triangle' behind the
front axle. The spigots of the mner tIe rod jomts 7 are fixed to the ends of the steering rack 8 and the outside ones
the steering arms 3 (see also Figs 1.40 and 1.54).
Fig. 4.4
tq
I
II
It is therefore the job of thel steering system to convert the steering wheel
angle into as clear a relationshi~ as possible to the steering angle of the wheels
and to convey feedback about thie vehicle's state of movement back to the steering wheel. This passes on the J'ctuating moment applied by the driver, via the
steering column to the steering gear 1 (Fig. 4.3) which converts it into pulling
forces on one side and pushing rces on the other, these being transferred to the
steering arms 3 via the tie rods . These are fixed on both sides to the steering
knuckles and cause a turning ovement until the required steering angle has
been reached. Rotation is aroun the steering axis EO (Fig. 3.103), also called
kingpin inclination, pivot or ste ring rotation axis (Fig. 1.3).
I
i
4.1.2
Steering system on
~ndependent wheel
suspensions
I
i
If the steering gear is of a type~1 mploying a rotational movement, i.e. the axes
of the meshing parts (screw sha t 4 and nut 5, Fig. 4.15) are at an angle of 90 0
to one another, on independent~heel suspensions, the insides of the tie rods are
connected on one side to the pit an arm 4 of the gear and the other to the idler
arm 5 (Fig. 4.3). As shown in igs 4.12 and 4.36 to 4.38, parts 4 and 5 are
connected by the intermediate rpd 6. In the case of steering gears, which operate using a shift movement (racJ and pinion steering), it is most economical to
fix the inner tie rod joints 7 to t~e ends of the steering rack 8 (Fig. 4.4).
i
4.1.3
Steering system on
~gid axles
I
Rack and pinion steering system~ are not suitable for steering the wheels on rigid
front axles, as the axles move i a longitudinal direction during wheel travel as
a result of the sliding-block gui ,e. The resulting undesirable relative movement
between wheels and steering I gear cause unintended steering movements.
Therefore only steering gears ,ith a rotational movement are used. The intermediate lever 5 sits on the stee~ing knuckle (Fig. 4.5). The intermediate rod 6
I I
270
The Automotive Chassis
f
Direction
6
5
3
2
Fig. 4.5
On rigid axles, apart from
the two steering arms 3, only the tie
rod 2, the idler arm 5 and the drag
link 6 are needed to steer the
wheels. If leaf springs are used to
carry the axle, they must be aligned
precisely in the longitudinal direction,
and lie vertical to the lever 5 when
the vehicle is moving in a straight
line. Steering arm angle A is an
essential factor in the relationship
between the outer and the inner
curve steering angles.
links the steering knuckle and the pitman arm 4. When the wheels are turned to
the left, the rod is subject to tension and turns both wheels simultaneously,
whereas when they are turned to the right, part 6 is subject to compression. A
single tie rod connects the wheels via the steering arm.
However, on front axles with leaf springs, the pitman arm joint 4, which sits
on the steering gear 1, must be disposed in such a manner that when the axle is
at full suspension travel, the lower joint 8 describes the same arc 9 as the centre
of the front axle housing (Figs 4.6 and 1.37). The arc 9 must be similar to the
curved path 7, otherwise there is a danger of the wheels experiencing a parallel
Direction
...
Fig. 4.6
Side view of a
rigid front axle showing the
movement directions 9 and
7 of the drag link and axle
housing during bump and
rebound-travel. The path of
point 7 is determined by the
front half of the leaf spring
and can be calculated on a
spring-balance by
measuring the change in
length when a load is added
to and removed from the
spnng.
;;~'h"
.\ij,~,,:;
0~¥~t-',\
J~J~'
;':'~:~J!
Steering
--
271
-
Direction
~
-I
Fig. 4.7 If the movement cut:e 7 of the axle housing and curve 9 of the rear
steering rod joint do not match w en the body bottoms out, the wheels can turn and
therefore an unwanted self-steeri, g effect can occur.
I
"
toe-in alteration when the suspdnsion reaches full travel, i.e. both being turned
in the same direction (Fig. 4.7). f a rigid axle is laterally controlled by a panhard
rod, the steering rod must be pa allel to it.
Its construction is similar t that of the intermediate rod of the steering
linkage shown in Fig. 4.13; len th adjustment and ball joints on both sides are
necessary.
I
I
i
!
4.2
Rack and
pi~ion steering
II
4.2.1
Advantages and diJadvantages
m~vement
This steering gear with a shift
is used not only on small and mediumsized passenger cars, but also onJheavier and faster vehicles, such as the Audi A8
and Mercedes E and S Class, PtUS almost all new light van designs with independent front wheel suspension The advantages over manual recirculating ball
1
steering systems are (see also Slfction
4.3.1):
!
•
•
•
•
simple construction;
economical and uncomPlicate to manufacture;
easy to operate due to good d gree of efficiency;
contact between steering rac and pinion is free of play and even internal
damping is maintained (Fig. 4!.10);
• tie rods can be joined directl;I't~ the steering rack;
• minimal steering elasticity cofupliance (Fig. 3.99);
• compact (the reason why thi~ type of steering is fitted in all European and
Japanese front-wheel drive vepicles);
!
t
i
i
272
The Automotive Chassis
• the idler arm (including bearing) and the intermediate rod are no longer
needed;
• easy to limit steering rack travel and therefore the steering angle.
The main disadvantages are:
• greater sensitivity to impacts;
• greater stress in the case of tie rod angular forces;
• disturbance of the steering wheel is easier to feel (particularly in front-wheel
drivers);
• tie rod length sometimes too short where it is connected at the ends of the rack
(side take-off design, Fig. 3.67);
• size of the steering angle dependent on steering rack travel;
• this sometimes requires short steering arms 3 (Fig. 4.4) resulting in higher
forces in the entire steering system;
• decrease in steering ratio over the steer angle (Fig. 3.96) associated with heavy
steering during parking if the vehicle does not have power-assisted steering;
• cannot be used on rigid axles.
4.2.2
Configurations
There are four different configurations of this type of steering gear (Fig. 4.8):
Type 1 Pinion gear located outside the vehicle centre (on the left on left-hand
drive and on the right on right-hand drive) and tie rod joints screwed into the
sides of the steering rack (side take-off).
Type 2
Pinion gear in vehicle centre and tie rods taken off at the sides.
-_._.-++-+
Fig.4.8
The three most common
types of rack and pinion steering on
left-hand drive passenger cars; righthand drive vehicles have the pinion
gear on the other side on the top and
bottom configurations (shown in Fig.
4.39). The pinion gear can also be positioned in the centre to obtain longer
steering rod travel.
Steering
273
Type 3 Pinion gear to the side and centre take-off, i.e. the tie rods are fixed in
the vehicle centre to the steeringl rack.
Type 4 'Short steering' with off-centre pinion gear and both tie rods fixed to
one side of the steering rack (Fi~. 4.1).
Types 1 and 3 are the solutions Igenerally used, whereas Type 2 was found in
some Porsche vehicles, and Typ~ 4 used to be preferred by Audi and VW. For
design details, see Section 3.2.4 ~n Ref. 1.
I
4.2.3
Steering gear, manual with side tie rod take-off
Type 1 (Fig. 4.8) is the simplest solution, requiring least space; the tie rod joints
are fixed to the sides of the steering rack (Fig. 4.9), and neither when the wheels
are turned, nor when they bottom out does a moment occur that seeks to tum the
steering rack around its centre line. It is also possible to align the pinion shaft
pointing to the steering tube (Figs 1.57, 4.24 and 4.29) making it easy to connect
the two parts together. Using an intermediate shaft with two joints (Figs 1.49 and
4.26) enables the steering column to bend at this point in an accident. In this
event the entire steering gear is turned when viewed from the side (i.e. around
the y-axis).
Figure 4.10 is a section showing how, on all rack and pinion steering systems,
not only can the play between the steering rack and the pinion gear be easily
eliminated, but it also adjusts automatically to give the desired damping. The
pinion gear 21 is carried by the grooved ball bearing 20; this also absorbs any
axial forces. Ingress of dirt and dust are prevented by the seal 31 in a threaded
ring 43 and the rubber cap 45. The lower end of the pinion gear is supported in
the needle bearing 23.
In a left-hand drive passenger car or light van, the steering rack 3 is carried
on the right by a plastic bearing shell and on the right by guide 15, which presses
the steering rack against the pihion gear. On a right-hand drive vehicle this
arrangement is reversed. The hal~-round outline of the guide 15 does not allow
radial movement of the steering r~ck. To stop it from moving off from the pinion
gear, when subject to high stehing wheel moments (which would lead to
reduced tooth contact), the underside of the guide-bearing 15 is designed as a
buffer; when it has moved a distance of s ~ 0.12 mm it comes into contact with
the screw plug 16.
Depending on the size of the'! steering system, coil spring 14 has an initial
tension force of 0.6 kN to 1.0 kN, which is necessary to ensure continuous
contact between steering rack and pinion gear and to compensate for any machining imprecision, which might occur when the toothing is being manufactured or
the steering rack broached or the pinion gear milled or rolled. The surface of the
two parts should have a Rockwell hardness of at least 55 HRC; the parts are not
generally post-ground due to the existence of a balance for the play. Inductionhardenable and annealed steels such as Cf 53, 41 Cr 4 and others are suitable
materials for the steering rack, case-hardened steels such as 20 MnCr 5, 20 MoCr
4, for example, are suitable for the pinion gear. In order to ensure a good
I
co
c:l:
c
c
o
".j:;
u
Q)
.::
o
o
'.j:;
u
Q)
(J)
Steering
31
43
275
45
20
10-+---::'"'\--------15
r-...--\-_----19
W--7-:;~..:::::.Ir------14
-18
..... ""...........~~...-\-----16
Fig. 4.10 Rack- and- pinion steering by ZF; section through pinion gear, bearing
and rod guide. The distance ring 18 is used for setting the plays, and the closing
screw 16 is tightened against it. The O-ring 19 provides the damping function and
prevents rattling noises.
response and feedback of the steering, the frictional forces between guide-bearing 15 and gear rack 3 must be kept as small as possible.
Sealing the steering rack by means of gaiters to the side (Fig. 4.9) makes it
possible to lubricate them with grease permanently, and lubrication must be
provided through a temperature range of -40°C to +80°C. It is important to note
that if one of the gaiters is damaged, the lubricant can escape, leading to the
steering becoming heavier and, in the worst case, even locking. Gaiters should
Fig. 4.9 Rack and pinion steeririg on the Vauxhall Corsa (1997). The tie rod axial
joints 4 bolted to the side of the st$ering rack and the sealing gaiters 5 can be seen
clearly. To stop them from being c,rried along when t.he toe-in is set (which is done
by rotating the middle part of the r d) it is necessary to loosen the clamps 6.
The pinion 1 has been given a 'helical cut', due to the high ratio, and is carried
from below by the needle bearing 2. The bearing housing has been given a cover
plate to facilitate assembly and prevent dirt ingress.
276
The Automotive Chassis
therefore ~~ checked at every service inspection. They are also checked at the
German TUV (Technischer Uberwachungs Verein) annual vehicle inspection.
4.2.4 Steering gear, manual with centre tie rod take-off
As shown in Figs 1.57 and 4.1, and as described in Section 4.7.3.2, with
McPherson struts and strut dampers the tie rods must be taken off from the
centre" if the steering gear has to be located fairly high up. This is because the
steering tie rods must thus be very long in order to prevent unwanted steering
movements during wheel travel (Fig. 4.46).
In such cases the inner joints are fixed in the centre of the vehicle to the steering rack itself, or to an isolator that is connected to it. The designer must ensure
that the steering rack cannot twist when subject to the moments that arise. When
the wheels rebound and compress, the tie rods are moved to be at an angle,
something which also happens when the wheels are steered. The effective
distance a between the eye-type joints of the tie rods and the steering rack centre
line, shown in Fig. 4.11, gives a lever, via which the steering could be twisted.
Two guide pieces which slide in a groove in the casing stop this from happening. However, the need to match the fit for the bearing of the steering rack and
the guide groove can lead to other problems. If they are too tight, the steering
will be heavy, whereas if they are too loose, there is a risk of rattling noises when
the vehicle is in motion.
As the steering forces are introduced at a relatively large distance from the
bearing points of the steering axle (suspension strut support bearings at the top,
ball-and-socket joint on the transverse link at the bottom), elastic (flexural)
deformations occur on the suspension strut and shock-absorber strut. As a result,
steering precision and response characteristics worsen.
Fig. 4.11
Top view of the rack and pinion steering of the front-wheel drive Opel
(Vauxhall) Astra (up to 1997) and Vectra (up to 1996); the steering arms on the
McPherson strut point backwards and the steering gear is located relatively high. For
this reason the tie rods have to be jointed in the middle and (in order not to come
into contact with the gear housing when the wheels are turned) have to be bent. The
guide-bearing in the groove of the housing prevents the steering rack from twisting.
On the inside, both tie rods have the eye-type joint shown in Fig. 5.45; the distance
a to the steering rack centre, which causes a bending moment, and a torque (when
the wheels bump and rebound) is also shown. The two bolts 6 gripping into the
steering rack are secured.
Once the screws 3 and 4 have been loosened, toe-in to the left and right can be
set by turning the connecting part 5.
The steering gear has two fixing points on the dashpanel, which are a long way
apart and which absorb lateral force moments with minimal flexing.
As also shown in Fig. 4.10, the pinion is carried by a ball and a needle bearing
(positions 20 and 23) and is also pressed onto the steering rack by a helical spring.
The illustration shows the possible path s of the rack guide. Figures 4.46 to 4.48
show the reason for the length of the tie rods on McPherson struts and strut
dampers.
LO __
.~
...
M
N
Q)
...---
-
C5
(0
o
N
~.
LO
C)
J:
c
o
.~
Q)
CJ)
278
4.3
4.3.1
The Automotive Chassis
Recirculating ball steering
Advantages and disadvantages
Steering gears with a rotating movement are difficult to house in front-wheel
drive passenger cars and, in a standard design vehicle with independent wheel
suspension, also require the idler arm 5 (see Fig. 4.3) and a further intermediate
rod, position 6, to connect them to the pitman arm 4; the tie rods are adjustable
and have pre-lubricated ball joints on both sides (Figs 4.13 and 4.14).
This type of steering system is more complicated on the whole in passenger
cars with independently suspended front wheels and is therefore more expensive
than rack and pinion steering systems; however, it sometimes has greater steering elasticity, which reduces the responsiveness and steering feel in the on-centre
range (see Section 3.7.4).
Comparing the two types of configuration (without power-assisted steering)
indicates a series of advantages:
• Can be used on rigid axles (Figs 4.5 and 1.37).
• Ability to transfer high forces.
• A large wheel input angle possible - the steering gear shaft has a rotation
range up to ±45°, which can be further increased by the steering ratio.
~
Direction
Fig. 4.12
Top view of the strut damper front axle on a Mercedes vehicle. The
intermediate rod and the tie rods are fixed side by side on the pitman and idler arms
and one grips from the top and the other from the bottom into the two levers; the
steering square is opposed. The steering damper is supported on the one side at the
intermediate rod and on the other side on the suspension subframe.
The anti-roll bar is linked to the lower wishbone type control arms whose inner bearings take large rubber bushings. The defined springing stiffness of these bearings,
together with the inclined position of the tie rods (when viewed from the top) means
that when the vehicle corners, there is a reduction in the steering input, i.e. elastic
compliance in the steering, tending towards understeering (Fig. 3.82). The strut dampers
are screwed to the steering knuckles; the negative kingpin offset is fa = -14 mm.
Steering
279
L± 10
_~.-or..,.
/
_
\
'ffi\
p
Fig. 4.13
Configuration of an adjustable tie rod with pre-lubricated joints and
buckling-resistant central tube, the interior of which has a right-hand thread on one
side and a left-hand thread on the other. It can usually be continuously adjusted by
± 10 mm. When toe-in has been set, the length on the right and left tie-rod may differ,
resulting in unequal steering inputs and different size turning circles; for this reason,
the central tube should be turned the same amount on the left and right wheel.
The configuration shown in the illustration is used on rigid front axles and as a
drag link (illustration: Lemfbrder Fahwerktechnik).
4 --;;;~~""'"
3--I4iw
1-~,-1ll
- - i
t
t----
6----:;;~~iiiiiii~--L..!...l::====::d'
5---7
2
8
Fig. 4.14
Lemfbrder Fahrwerktechnik pre-lubricated tie rod joint, used on
passenger cars and light vans. The joint housing 1 has a fine thread on the shaft
(M14 x 1.5 to M22 x 1.5) and is made of annealed steel C35V; surface-hardenable
steel 41 Cr4V is used for the ball pivot 2.
The actual bearing element - the one-part snap-on shell 3 made from polyacetal
(e.g. DELRIN, made by Dupont) - surrounds the ball; the rolled-in panel cover 4
ensures a dirt- and waterproof seal. The polyurethane or rubber sealing gaiter 5 is
held against the housing by the tension ring 6. The gaiter has a bead at the bottom
(which the second tension ring 7 presses against the spigot) and a sealing lip, which
comes into contact with the steering arm.
The ball pivot 2 has the normal 1:10 taper and a split pin hole (position 8). If there
is a slit or a hexagonal socket (with which the spigot can be held to stop it twisting),
a self-locking nut can be used instead of a slotted castle nut and split pin.
280
The Automotive Chassis
• It is therefore possible to use long steering arms.
• This results in only low load to the pitman and intermediate arms in the event
of tie rod diagonal forces occurring.
• It is also possible to design tie rods of any length desired, and to have steering
kinematics (hat allow an increase in the overall steering ratio is with increasing steering angles. The operating forces necessary to park the vehicle are
reduced in such cases (see Section 3.7.3).
4.3.2
Steering gear
The input screw shaft 4 (Fig. 4.15) has a round thread in which ball bearings run,
which carry the steering nut 5 with them when the steering wheel is rotated. The
balls which come out of the thread at the top or the bottom (depending on the
8
Fig. 4.15 Mercedes Benz recirculating ball st~ering suitable. f~r passe0ger cars
and light vans; today, apart from in a few exceptIonal cases, thIs IS only fitted as a
hydraulic power-assisted version. Pit~an arm 9 IS mounted onto the tapered toothed
profile with a slotted castle nut 11 (Fig. 4.24).
Steering
281
direction of rotation) are returned through the tube 6. The nut has teeth on one
side which mesh with the toothed segment 7 and therefore with the steering
output shaft 8. When viewed from the side, the slightly angular arrangement of
the gearing can be seen top right. This is necessary for alignment bolt 1 to overcome the play of the wheels when pointing straight ahead, by axial adjustment.
If play occurs in the angular ball bearings 2 and 3, the lock-nut must be loosened
and the sealing housing cover re-tightened.
Only a few standard design larger saloons can be found on the road with
manual recirculating ball steering. For reasons of comfort, newer passenger cars
of this type have hydraulic power-assisted steering. The same applies to
commercial vehicles; only a few light vans are still fitted with manual configurations as standard and even these are available with power-assisted steering as
an option.
4.4 Power steering systems
Power steering systems have become more and more widely used in the last few
years, due to the increasing front axle loads of vehicles on the one hand and the
trend towards vehicles with more agile steering properties and hence direct
transmission steering systems on the other. With the exception of some members
of the 'sub-compact' class, power steering systems are optionally or automatically included as one of the standard features.
Manual steering systems are used as a basis for power steering systems, with
the advantage that the mechanical connection between the steering wheel and the
wheel and all the components continues to be maintained with or without the help
of the auxiliary power. The steering-wheel torque applied by the driver is detected
by a measurement system located in the region of the input shaft of the steering
gear or in the steering tube, and additional forces or moments are introduced into
the system. This follows a characteristic curve (valve characteristic) or group of
curves depending on the height of the steering-wheel torque, if another quantity,
e.g. driving speed, is entered as a signal. The steering boost is thereby reduced,
with the aim of achieving better road contact at higher speeds. An exact functional
description of such systems can be found in Chapter 10 in Ref. [1].
4.4.1
Hydraulic power steering systems
Hydraulic power steering systems are still the most widely used. The method of
using oil under pressure to boost the servo is sophisticated and advantageous in
terms of cost, space and weight. Sensitivity to movements caused by the road
surface and hence the effect of torsional impacts and torsional vibrations passing
into the steering wheel is also noticeably reduced, particularly with rack and
pinion steering. This can be attributed to the hydraulic self-damping. It might also
be the reason why it is possible to dispense with an additional steering shock
absorber in most vehicles with hydraulical rack and pinion steering, whereas it is
required for the same vehicles with manual steering (see Section 4.6).
282
The Automotive Chassis
The oil pump is directly driven by the engine and constantly generates
hydraulic power. As hydraulic power steering systems have to be designed in
such a way that a sufficient supply volume is available for fast steering movements even at a low engine speed, supply flow limiting valves are required.
These limit the supply flow to about 8 I per minute in order to prevent the
hydraulic losses which would otherwise occur at higher engine speeds.
Depending on the driving assembly and pump design, the additional consumption of fuel can lie between 0.2 and 0.7 I per 100 km.
Assemblies which are added to provide auxiliary power are shown in Fig.
4.16, taking the example of the rack and pinion steering used by Opel in the
Vectra (1997). The pressure oil required for steering boost is supplied direct to
the steering valve 6 located in the pinion housing from vane pump 1 via the high-
Fig. 4.16 Hydraulic power steering system of the Opel Vectra (1997). The individual components are:
1
2
3
4
5
6
7/8
9
10
vane pump, driven by V-belts
high-pressure line
cooling circuit
return line from the steering valve to the pump
steering g~ar with external drive, attached to the auxiliary frame
steering valve
pressure lines to the working cylinder
steering column with intermediate shaft
steering wheel with integrated airbag.
Steeri ng
283
pressure line 2 and the cooling circuit 3. From here, depending on the direction
of rotation of the steering wheel and the corresponding counterforce on the
wheels, distribution to the right or left cylinder line takes place (items 7 and 8).
Both lead to the working cylinder which is integrated in the steering-gear housing 5. A disc locat~d on the gear rack divides the pressure chamber. Differences
in pressure generate the required additional axial force in the gear rack F pi via
the active areas of the disc:
(4.1)
where APi is the effective piston surface, here the difference between the disc and
gear rack surfaces, and phyd.1 or 2 are the pressures acting on the working piston.
In a situation where there is no torque, for example during straight running, the
oil flows direct from the steering valve 6 back to the pump 1 via the return line
4.
The method of operation of the steering valve is shown in Fig. 4.17, using the
example of recirculating-ball steering. In a similar way to rack and pinion steering, it is integrated into the input shaft of the steering gear. As is the case with
most hydraulic power steering systems, the measurement of the steering-wheel
torque is undertaken with the use of a torsion bar 18. The torsion bar connects
the valve housing 5 (part of the steering screw) to the valve pistons 9/10 in a
torsionally elastic way. Steering-wheel torque generates torsion of the torsion
bar. These valve pistons then move and open radial groove 13 or 14, depending
on the direction of rotation. This leads to a difference in pressure between pressure chambers D1 and D2. The resultant axial force on the working piston 2 is
calculated using Equation 4.2. Because phyd,2 also operates in the interior space
of the piston behind the steering screw 5, the surface areas are the same on both
sides:
F pi = phyd,l or2APi = phyd,l or2
1tD~i
4
(4.2)
The exact description is contained in Section 5.2 in Ref. [1].
4.4.2
Electro-hydraulic power steering systems
With electro-hydraulic power steering systems, the power-steering pump driven
by the engine of the vehicle via V-belts is replaced by an electrically operated
pump.
Figure 4.18 shows the arrangement of the system in an Opel Astra (1997).
The electrically operated power pack supplies the hydraulic, torsion-bar
controlled steering valve with oil. The pump is electronically controlled - when
servo boost is not required, the oil supply is reduced.
The supply of energy by electricity cable allows greater flexibility with regard
to the position of the power pack. In the example shown, it is located in the
immediate vicinity of the steering gear. Compared with the purely hydraulic
284
The Automotive Chassis
Steering valve
Vane pump
15
17
i"===¥l--20
1It--H--
8
3
Fig. 4.17
Illustration of the principles of the ZF recirculating ball steering in the
n~utral position (vehicle travelling in a straight line). The steering valve, the working
piston and the mechanical gear sit in a common housing. The two valve pistons of
the steering valve have been turned out of their operating plane to make the diagram
easier to see. The individual parts are:
1
2
3
4
5
6
7
8
gear housing
piston with steering nut
steering spindle connection
steering shaft with toothed segment
steering worm roller with valve body
balls
recirculation tube
fluid flow limitation valve
9/10
11/12
13/14
15/1 6
17
18
19
20
valve piston
inlet groove
radial groove
return groove
fluid reservoir
torsion bar
hydraulic pump
pressure-limiting valve
system, the lines can be made considerably shorter and there is no cooling
circuit. The steering gear, power pack and lines are installed as a ready-assembled and tested unit.
To sum up, electro-hydraulic power steering systems offer the following
advantages:
• The pressure supply unit (Fig. 4.19) can be accommodated in an appropriate
location (in relation to space and crash safety considerations).
• Servo boost is also guaranteed by the electrical pressure supply even when the
engine is not running.
• Pressure-controlled systems generate only the amount of oil required for a
Steering
285
Fig. 4.18
Electro-hydraulic power steering system of the Opel Astra (1997). The
individual components are:
1 electrically operated power-steering pump with integrated reserve tank ('power
pack')
2 pump-steering valve hydraulic lines
3 rack and pinion steering gear with external drive, attached to auxiliary frame
4 steering valve.
Pressure hose
To the
power
supply
+
Sensor cable
Control system,
e.g. steering
speed signal
Return
hose
Pressure supply unit with
integrated control device
Standard rack and pinion hydraulic steering (by ZF)
Fig.4.19 Open-centre control system from ZF. The pressure supply unit designed
as a modular unit can befitted with different electric motors (DC motor with or without brushes) and pump fuel feed volumes (1.25-1.75 cm 3 per rpm) depending on its
particular function. Oil tanks for horizontal or vertical installation are also available.
Operating pressure is up to 120 bar, with a maximum power consumption of 80 A.
The Automotive Chassis
286
particular driving situation. Compared with standard power steering systems,
energy consumption is reduced to as little as 20%.
• The steering characteristics (nature and amount of steering boost, sensitivity,
speed dependency) can be adjusted by the control electronics individually for
the particular vehicle.
4.4.3
Electrical power steering systems
The bypass of the hydraulic circuit and direct steering boost with the aid of an
electric motor has additional advantages in terms of weight and engine bay space
compared with electro-hydraulic steering, because of the omission of all the
hydraulic components. Other advantages are obtained through more variations
of the steering boost because of the purely electrical signal processing.
The electrical servo unit can be installed on the steering column (Fig. 4.20),
pinion (Fig. 4.21) or gear rack (Fig. 4.22). The steering axle loads and maximum
gear rack forces are, depending on the particular arrangement, about 650 kg and
6000 N, 850 kg and 8000 N or 1300 kg and 10000 N.
Fig. 4.20
Steering column with power-steering assembly of the Opel Corsa
(1997). The individual components are:
1
2
3
4
5
6
7
column tube
steering tube
sliding sleeve with groove
rotary potentiometer with tap
servomotor
drive worm
worm gear.
Steeri ng
287
, - - - - Safety
steering column
I Servo unit - - -......
with electronics
Intermediate shaft
with universal joints
I and 2 are
assembled
Manual steering gear 2
Ratio
• constant
• variable
Fig. 4.21
Electrical power steering system by ZF. The servo unit acts directly
upon the pinion of the rack and pinion steering. Consequently, the amount of stress
to which the pinion is subjected increases by the amount of steering boost,
compared with a mechanical or hydraulic power steering system.
The systems only have limited power because the current is limited by an operating voltage of 12 V. They are of interest though for smaller vehicles. In this class
of vehicle in particular, electric power steering systems show their advantages,
not least because of the small amount of energy required. The introduction of the
increased voltage of 42 V will make the use of electrical power steering systems
and wheel brakes much easier.
Figure 4.23 shows the steering system of the Opel Corsa (1997) with electric
power steering. It is a system with steering-tube transmission, i.e. the intermediate spindle transmits the whole of the torque resulting from the steering wheel
force and servo boost. Due to the more direct steering transmission, this torque
is clearly higher than in a comparable manual steering system, something which
must be taken into consideration when deciding on the size of the components
which control performance.
In Fig. 4.20, the method of operation of the servo assembly (EPAS system by
NSK) becomes clear: a plastic worm gear 7 is applied to the steering tube 2. This
is engaged by the worm 6, which in its tum is connected to the shaft of the servomotor 5. Steering-wheel torque generates a torsional movement of the torsion bar
(concealed by the sliding sleeve 3). The steering tube area is axially grooved
above the torsion bar and spindle-shaped below. As the spindle rises, the sliding
288
The Automotive Chassis
r----
Safety
steering column
Intermediate shaft
with universal joints
Sensors and - - -.......
torsion bar
Manual steering gear - - - - - - - . . .
Ratio
• constant
• variable
" ' - - - - Electronics
' - - - - Servo unit with
• recirculating ball gearing
• electric motor
• angle transducer
Fig.4.22 Electrical power steering system by ZF. The servo unit acts on the gear
rack itself. This system is suitable for high axle loads and steering forces. The maximum current strength is 105 A with a 12 V electric system; with a 42 V system, it is
only 35 A.
sleeve makes an axial movement on the steering tube proportional to the torsion
of the torsion bar. This axial movement is transmitted to the rotary potentiometer
4 via a tap. Corresponding to a group of curves, the servo boost is detennined
from the steering-wheel torque and driving speed signals and the servomotor 5
controlled accordingly.
More detailed functional descriptions, also of other systems, are contained in
Chapter 8 in Ref. [1].
4.5
Steering column
In accordance with the German standard DIN 70 023 'nomenclature of vehicle
components', the steering column consists of the jacket tube (also known as the
outer tube or protective sleeve), which is fixed to the body, and the steering shaft,
also called the steering tube. This is only mounted in bearings at the top (or top
and bottom, positions 9 and 10 in Fig. 4.26) and transfers the steering-wheel
moment M H to the steering gear.
A compliant cardan joint (part 10 in Fig. 4.24) can be used to compensate for
small angular deviations. This also keeps impacts away from the steering wheel
I
Steering
"
.
.......
..----tr---....
.-", '" "
( ---"= :~..
-/'
---~
.-.-
~
'"'- '.., "- --',
'-...-
\"
,
289
If',
"'--;'
-1: i.=:I
.'
-.f
____ 2
;~...
-. -
I
-=
..- " ' - -
\'\
"
3
.,
....
_-
1
.......
\
I
,
Fig.4.23
Electric power steering system of the Opel Corsa (1997), The individual
components are:
1 steering-column assembly
2 steering column with intermediate spindle
3 rack and pinion steering with external drive.
and, at the same time, performs a noise insulation function on hydraulic powerassisted steering. If the steering column does not align with the extension of the
pinion gear axis (or the steering screw), an intermediate shaft with two universal
joints is necessary (part 6 in Fig. 4.26). When universal joints are used, attention
should be paid to their transmission properties, which are dependent on their
angle of inflexion for steering wheel angle and moment, because a non-linear
steering moment above the steering angle, noticeable for the driver, can occur.
The steering tube should be torsionally stiff to keep the steering elasticity
low. On the other hand, it should show, together with the jacket tube, a deformation behaviour which is defined in a longitudinal direction, as steering wheel
intrusion in case of a head-on crash is to be avoided while the absorption of
force necessary for the functionality of the airbag (Fig. 4.25) must be safeguarded. As there is a requirement in some US states that the airbag should cushion a driver who is not wearing his seatbelt in a crash, despite the fact that seat
belts are mandatory, the steering column must be designed for this borderline
contingency.
Three types of steering tube configuration meet these requirements with vehicle-specific deformation paths on passenger cars:
• steering tubes with flexible corrugated tube portion (Fig. 4.24);
• collapsible (telescopic) steering tubes (Figs 4.27 and 4.28);
• detachable steering tubes (Figs 4.1 and 4.29).
r
290
The Automotive Chassis
Fig. 4.24 Mercedes Benz safety steering tube and dished steering wheel; it is
fixed to the recirculating ball steering gear with a compliant 'joint'. The bottom illustration shows the corrugated tube bending out in a head-on crash. The illustration
also shows the energy-absorbing deformation of the steering wheel and the flexibility of the steering gear mounting.
To increase ride and seating comfort, most automobile manufacturers offer an
adjustable steering column, either as standard or as an option. The position of the
steering wheel can then be altered backwards and forwards as well as up and
down (positions 1 and 2 in Fig. 4.30). As can be seen in the illustrations, electrical adjustment is also possible.
On light vans, which have a steering gear in front of the front axle, the steering
-
Steering
Fig. 4.25
291
BMW passenger car with air bags for the front, sides and head (front
and back).
5
Fig. 4.26
Steering column of the VW Golf III and Vento (1996). The collapsible
steering tube (Fig. 4.27) is carried from the bottom by the needle bearing 9 and
through the top by the ball bearing 10 in the jacket tube; the spigot of the steering
lock grips into part 5. The almost vertical pinion gear of the rack and pinion steering
is linked to the inclined steering tube via the intermediate shaft 6 with the universal
joints 7 and 8. The dashpanel is sealed by the gaiter 11 between this and the steering gear (illustration: Lemf6rder Fahrwerktechnik).
292
The Automotive Chassis
-======61;•.;3ii;;••
II
Fig.4.27 Telescopic collapsible steering tubes consist of a lower part 1, which is
flattened on the outside, and a hollow part 2, which is flattened on the inside. The
two will be fitted together; the two plastic bushes 3 ensure that the assembly does
not rattle and that the required shear-off force in the longitudinal direction is met. The
tab 4 fixed to part 1 ensures the passage of electric current when the horn is operated. The spigot of the steering wheel lock engages with the welded-on half shells
5 (illustration: Lemforder Fahrwerktechnik).
1
-----.
\
/
2
Fig. 4.28 Volvo steering column. Both the corrugated tube 1 i~ the intermediate
shaft and the collapsible steering tube 2 meet the safety req~lrement~. To s~ve
weight, the universal joints are made of aluminium alloy AI Mg SI 1 F31 (Illustration:
Lemforder Fahrwerktechnik).
Steering
293
Fig.4.29
'Release clutch'
used by VW on steering
columns. A half-round plate sits
on the short shaft that is linked
to the steering pinion gear, and
carries the two pins 1 which
point downwards. They grip into
the two holes of the clutch 2
sitting on the steering tube from
the top. The jacket tube is
connected to the dashboard via
a deformable bracket. As shown
in a head-on crash, this part 3
flexes and the pins 1 slide out of
part 2.
- - Release clutch
2
1
Fig. 4.30
Electrically adjustable steering column manufactured by Lemf6rder
Fahrwerktechnik. The electric motor 3 turns a ball nut via the gears 4 and this
engages with the grooves 5 of the steering tube and shifts it (position 6) in the longitudinal direction (position 1). To change the height of the steering wheel (position 2),
the same unit tips around the pivot 8 by means of the rod 7.
294
The Automotive Chassis
I
,
Fig. 4.31
The VW Bus Type /I has an almost vertical steering column. In a headon crash, first the steering wheel rim gives and then the retaining strut 1, which is
designed so that a given force is needed to make it bend inwards.
column is almost vertical (Figs 1.7 and 1.37). In a head-on crash the outer tube
bracket 1 and the steering wheel skeleton must flex (Fig. 4.31).
4.6
Steering damper
Steering dampers absorb shocks and torsional vibrations from the steering wheel
and prevent the steering wheel over-shooting (also known as free control) on
front-wheel drive vehicles - something which can happen when the driver pulls
the steering wheel abruptly. The dampers therefore increase ride comfort and
driving safety, mainly on manual steering gears. The setting, which generally
operates evenly across the whole stroke range, allows sufficiently light steerability but stops uncontrollable wheel vibrations where the front wheels are
subjected to uneven lateral and longitudinal vibrational disturbances; in this
event the damper generates appropriate forces according to the high piston
speeds involved (see Section 11.4 in Ref. [5]).
The dampers are fitted horizontally. As shown in Fig. 1.57, on rack and pinion
steering, one side of the damper is fixed to the steering rack via an eye or pintype joint and the other to the steering housing. On recirculating ball steering
systems, the pitman arm on independent wheel suspensions or the intermediate
rod can be used as a pivot point (Figs 1.39 and 4.12) or the tie rod on rigid axles.
As shown in Fig. 4.5, this is parallel to the axle housing. Section 5.6.5 describes
how the non-pressurized monotube damper works.
4.7
4.7.1
Steering kinematics
Influence of type and position of the steering gear
Calculating the true tie rod length Uo (Fig. 4.32) and the steering arm angle A
(Fig. 4.3) creates some difficulties in the case of independent wheel suspensions.
-----------------..---'--1""-'--------I
Steering
295
z
x
y--_.-
u·
Fig.4.32 On independent wheel suspensions. the tie rod UT is spatially inclined.
The path ti (i.e. the lateral distance of points U and T from one another) or the angle
l( must be determined when viewed from the rear. From the top view, the distance
d or the angle COo is more important; the projected lengths which appear in both
views are u, and U2. The true tie rod length is then:
Uo
= (ti 2 + c 2 + d 2)1/2
The position of the steering column influences the position of the steering gear
by the type of rotational movement. If this deviates from the horizontal by the
angle 0) (Fig. 4.33), a steering gear shaft, which is also inclined by the angle 0),
becomes necessary. The inner tie rod joint T which sits on the pitman arm, is
carried through a three-dimensional arc, influenced by this angle 0) when the
wheels are turned. However, the outer joint U on the steering knuckle whose
steering axis is inclined inwards (Fig. 4.34) by the kingpin inclination angle cr
and is often also inclined backwards by the caster angle t (shown in Fig. 4.33).
This joint therefore moves on a completely different three-dimensional arc (Figs
3.7,3.9 and 3.11).
The construction designer's job is to calculate the steering arm angle A (and
possibly also the angle 0 of the pitman arm, Fig. 4.37) in such a manner that
when the wheels are turned, the specified desired curve produced comes as close
as possible. The achievement of the necessary balance is made more difficult
still by the movements of the wheel carrier during driving: for example, wheel
travel, longitudinal flexibility and vertical springing.
Figure 3.92 shows two curves that are desirable on passenger cars with an
initially almost horizontal shape (~o ::::: +30') and a subsequent rise in the curve
to nearly half the nominal value when the wheels are fully turned. The more
highly loaded wheel on the outside of the bend can even be turned further in than
the inner wheel (and not just parallel to it, ~o::::: -30'); due to the higher slip angle
that then has been forced upon it, the tyre is able to transfer higher lateral forces.
"Then the wheels are fully turned, the actual curve should, nevertheless, remain
below the nominal curve to achieve a smaller turning circle (see Equation 3.14).
The steering angle 00 of the wheel on the outside of the bend depends on the
angle of the one on the inside of the bend OJ via the steering difference angle ~o:
~O
= OJ - 00 (axis of the ordinate, Fig. 3.92)
296
The Automotive Chassis
Direction
~
Fig. 4.33
The central points of the tie rod joints (T on the inside and U on the
outside) change their position relative to one another, based on the wheel travel
(vertical and horizontal) on independent wheel suspensions. The reasons for this are
the different directions of movement of pitman arm and steering arm. The former
depends on the inclined position of the steering gear (angle ro) and that of the point
U from the inclination of the steering axis EG, i.e. the kingpin inclination cr and the
caster angle 't.
Fig.4.34
T
- .......
4.7.2
When
viewed from the rear, the
inner tie rod joint T on
rack and pinion steering
moves parallel to the
ground, whereas the outer
tie rod joint U moves on
an arc running vertical to
the steering axis EG. Any
caster angle 't must also
be considered.
Steering linkage configuration
The main influences on Ao are the steering arm angle A, the inclined position of
the tie rod when viewed from the top (angle <po, Fig. 4.32) and the angle 0 of the
pitman and idler arms on steering gears with a rotational movement. The tie rod
position is determined by where the steering gear can be packaged. The amount
of space available is prescribed and limited and the designer is unlikely to be
1--'-
Steering
297
able to change it by more than a little. The task consists of determining the
angles Aand 0 by drawing or calculation. Both also depend on the bearing elasticities, which are not always known precisely.
The configuration of the steering kinematics on rack and pinion steering is
comparatively simple; here, it is only necessary to transfer a straight-line lateral shift
movement into the three-dimensional movement of the steering knuckle (Fig. 4.34).
However, the extension of the tie rod UT must point to virtual centre of rotation P
(Fig. 4.35); this is necessary on all individual wheel suspensions for determining the
body roll centre Ro and is therefore known (see Sections 3.4.3 and 4.6.3).
On steering gears with a rotational movement, the 4-bar linkage can be either
in front or behind the axle and can be opposed or synchronous; Figs 4.3 and 4.36
to 4.38 show four different configurations.
From a kinematic point of view, rack and pinion steering systems have
a triangular linkage that can either be in front of or behind the axle or even
across it. Figures 4.4 and 4.39 to 4.41 show the individual options for left- and
Fig. 4.35
Path and movement points necessary for determining the tie rod length
and position. The position of the tie rods is given by the connecting line UP (to the
pole). The illustration also shows the roll centre Ro.
tDireClion
Fig. 4.36
'Synchronous' 4-bar linkage with steering arms pointing forwards. The
inner joints are fixed to the sides of the intermediate rod.
---_._------------.--------,--------
298
The Automotive Chassis
Fig. 4.37. 'Opposed' 4-ba.r linkage.loca~ed i.n front of the wheel centre. Steering
arm a.nd pitman arm rotate In opposite directIons towards one another similar to
meshing gears. The tie rods are fixed directly to pitman and idler arms. Fo~ kinematic
reasons, these can have the pre-angle 0 (see also Fig. 1.7).
t
Direction
Fig. 4.38
'Opposed' 4-bar linkage located behind the wheel centre. The inner tie
rod joints can be fixed to the middle part of the intermediate rod or directly to the
pitman and idler arm (see Fig. 4.12).
right-hand drive vehicles and also where the pinion gear must be located above or below the steering rack - to make the wheels turn in the direction in
which the steering wheel is turned. The steering arms (negative angles A)
which point outwards, shown in Fig. 4.41, allow longer tie rods; something
which is useful when the inner joints are pivoted on the ends of the steering
rack (Fig. 3.67).
The significantly simpler steering kinematics on rigid axles are shown in Figs
4.5 to 4.7 and are described in Chapter 2 of Ref. [1] and Chapter 5 of Ref. [10].
f
Steering
299
fDirection
Fig. 4.39
The rack-and-pinion steering is behind and above the wheel centre and
the steering arms point forward (shown for a right-hand drive vehicle). For kinematic
reasons, the inner tie rod joints are fixed to a central outrigger - known as a central
take-off. This type of solution (also shown in Fig. 1.57) is necessary on McPherson
and strut damper front axles with a high-location steering system as the tie rods have
to be very long to avoid unwanted steering angles during jounce.
t
Direction
bf
j'
Fig. 4.40
The steering is in front of the wheel centre and the triangular linkage
behind it, with the inner joints fixed to the ends of the steering rack.
4.7.3
Tie rod length and position
When the wheels compress and rebound as well as in longitudinal movement,
there should not be any, or only a very specific, toe-in alteration; both depend
primarily on the tie rods being the correct length and on their position. Various
illustrations in Section 3.6 show the results of incorrect toe-in and the possibility of achieving a roll-steer effect on the front wheels and steer-fight during
---_._-----------_._-----,---------
300
The Automotive Chassis
t
Direction
bf
j'
L
Fig. 4.41 Where rack and pinion steering and the steering triangle are shifted in
front of the wheel centre, for kinematic reasons the steering arms must point
outwards, making longer tie rods possible (see also Fig. 1.40).
braking. The elasticity in the steering system (Figs 3.99 and 3.100) or that in the
bearings of the steering control arms, is also a contributory factor. Chapter 7 of
Ref. [3], gives the calculation of the forces required for such elasticity.
4.7.3.1 Double wishbone and multi-link suspensions
There are two ways of determining the central point T of the inner tie rod joint
as a function of the assumed position U of the outer joint, the template and
'virtual centre' procedure. Both methods consider one side of the front axle
when viewed from the rear (here the left side, Fig. 4.42). The projected length u'
of the tie rod shown in Fig. 4.32 and the angle K, which determines its position,
must be calculated. This must match the line connecting the outer joint U with
pole P, which is also needed for calculating the roll centre (see Section 3.4.3).
Initially, the position of the outer tie rod joint U is unknown when viewed
from the rear; to obtain an approximation of this point, the height of the steering
gear must be specified (Fig. 4.35). The angle Ais assumed so that, together with
the known steering arm length r, the path required for configuring it
k = r sin A
(4.3)
can be calculated (for r and A see Fig. 4.40).
The templates that are used for finding point T by drawing have already been
described in Section 3.3 and can be seen in Figs 3.7 to 3.11. All figures contain
point U and the curve of its movement. It only remains to find point T on the
connecting line UP. T would be the centre point of the arc which best covers the
path of point U.
Steering
301
o
Fig. 4.42
Double wishbone suspension with steering arm pointing inwards. The
tie rod is above the lower control arm.
It is likely to be simpler and more precise to determine the point T graphically, using virtual centres. First, as shown in Figs 4.42 and 3.24 to 3.28, the
virtual-centre at P (marked here as PI) must be calculated so that it can be
connected to U. The extension of the paths EG and DC gives Pz, which is also
required and from which a line to PI must be drawn. If the path UP I is above GD,
the angle a enclosed by the two must be moved up to PIPz; if UP I were to lie
below it, the line would have to be moved down. A line drawn from PI at the
angle a must be made to intersect with the extension of the connecting path UE
to give the tie rod virtual-centre P3 • To calculate the desired point T - i.e. the
centre of the inner joint - P3 is connected to C and extended.
The path k (i.e. the distance of point U from the steering axis EG, Fig. 4.35
and Equation 4.3) is the determining factor for the position of virtual-centre P3
in the lateral direction. Figure 4.43 shows the case of point U, which lies left of
the path EG. This is something that is only possible where the steering gear is
located in front of the axle (Fig. 4.41). P3 moves to the right, resulting in an inner
link T moving further away from the centre of the vehicle. This is beneficial if it
is to be fixed to the end of the steering rod.
A tie rod that is located above the upper suspension control arm (Fig. 4.44)
causes a large angle a and P3 that is shifted a long way to the right. Where the
control arms are parallel to one another (Figs 4.45 and 3.25), PI is at 00. In such
cases, a line parallel to the path GD must be drawn through U and, at the same
302
The Automotive Chassis
t
Fig. 4.43
In the case of a steering gear located in front of the wheel centre, the
centre of the tie rod joint U lies outside the steering axis EG.
II
j
M
,!
!
I
I
1
Fig. 4.44
A
high~location
steering gear can involve a tie rod above the upper
control arm. The steering arm points backwards and towards the inside in the exam~
pie.
1
II
"
1
')I
r
Steering
303
Fig. 4.45 Suspension control arms,
which are parallel to one another in the
design position of the vehicle, have to
have a tie rod in the same position.
distance, a further one drawn through the virtual centre P2 • The intersection of
this second parallel with the extension of the path UE gives P3, which must be
linked to C to obtain T.
4.7.3.2 McPherson struts and strut dampers
When the vehicle is fitted with McPherson struts or strut dampers - due to the
alteration in distance between E and G when the wheels compress and rebound
- point T is determined by a different method. To obtain pole PI, a vertical to
the centre line of the shock absorber is drawn in the upper mounting point E and
made to intersect with the extension of the suspension control arm GD (Figs
3.29 and 4.46); PI linked with U gives the position of the tie rod. A line parallel to EP, must be drawn through G; the intersection with the extension of ED
then gives the second virtual-centre P2 • The angle Cl, included by the paths EP,
and UP" must be entered downwards to the connection P IP2 to obtain P3 as the
intersection of this line with the extension of the path UG. The extension of the
connecting line P3D then gives the central point T of the inner tie rod joint on
UP,.
If, in the case of A = 0°, point U is on the steering axis EG which dominates
the rotation movement (Figs 3.30 and 4.47), P3 is on the extension of this path.
The determining factor for the position of PI is the direction of the shift in the
damping part of the McPherson strut; for this reason, the vertical in point E must
be created on its centre-line (not on the steering axis EG). The important thing
in this calculation is the position of point U, i.e. the extension of the connecting
line UG downwards. U is shown on the steering axis EG simply for reasons of
presentation.
A low mounted tie rod causes the virtual-centre P3 to move to the right (Fig.
4.48) and this then causes a shorter rod. This situation is favourable if the inner
joint needs to sit on the ends of the steering rack. The figures clearly show that
the higher U, which constit.utes the connection between steering arm and tie-rod,
---_._------------,------.,._._-------------:1
304
The Automotive Chassis
P,
Fig. 4.46 On the McPherson strut or strut damper, the tie rod is above the lower
control arm; the steering arms point inwards with the result that the outer joint U lies
more to the vehicle centre.
1
I\
1
is situated, the longer the tie rods must be, i.e. a centre take-off becomes necessary on a high-mounted rack and pinion steering (Figs 1.57, 4.11 and 4.39).
4.7.3.3 Longitudinal transverse axles
On longitudinal wishbone axles the upper point E moves in a straight line
vertical to the steering axis CF and the lower point G on an arc around D (Figs
3.32 and 4.49). To obtain PI, a parallel to CF must therefore be drawn through
E and made to intersect with the control arm extension GD. A parallel to EP,
laid through point D gives the virtual centre Pz on the connecting line EG. The
angle a enclosed by the paths EP, and UP, must be drawn downwards to the
connecting line P,P z to obtain the virtual centre P3 as the intersection with the
extension of the path UG. P3 linked with D then gives the centre T of the inner
tie rod joint.
4.7.3.4 Reaction on the steering arm angle A,
Figures 4.40 to 4.49 indicate that shifting the outer joint U to the side results in
a slight alteration in the distance UT. However, this shift is necessary if the angle
A has to be reduced or increased with a given steering arm length r. The
projected length u' of the tie rod, and therefore also. its overall length Uo (Fig.
r
T
I
I
Steering
305
I
Fig. 4.47 On a McPherson strut with the joint G shifted to the wheel, the outer
one, U of the tie rod, can lie in the plane of the steering axis (i.e. on the connecting
line EG) when viewed from the rear. Extending the path UG is crucial for determining the virtual centre P3 , whereas the direction of movement of the damper, i.e. a
vertical on the piston rod in point E, must be the starting point for calculating P1 •
E
Fig. 4.48 The tie rod can also lie under the control arm when the steering arm
points inwards.
--------------------------.,.-'--------------_. - - !
306
The Automotive Chassis
Fig. 4.49
Longitudinal transverse axle with the tie rod located above the lower
control arm and the steering arm pointing inwards.
4.32), changes when viewed from the rear. However, the latter is one of the
determining factors for the aspects relating to the steering angles 8i (inside) and
80 (outside), i.e. for the actual steering curve (Fig. 3.92). It is, therefore, likely to
be essential to check the desired position of point T with the tie rod, which has
become longer or shorter.
5
Springing
5. 1
Comfort requirements
Springing and damping on a vehicle are mainly responsible for:
• ride comfort and dynamic wheel load.
They also play an important part in:
• handling (Fig. 5.2)
• the tendency of the body to roll and pitch.
Other important influences on the handling are the kinematic changes, and the
elastokinematics, of the wheels as they bump and jounce. Details are given in
Chapter 3, and in Refs 2 and 9.
The ride comfort experienced by the vehicle occupants depends on their
(Stainz) fitting position (Fig. 5.1) - also in relation to the controls such as steering wheels and pedals - as well as on the acceleration and mechanical vibration
acting upon them. The critical frequency range is 1-80 Hz. It is sensible to
subdivide this into two ranges to which different comfort terms are allocated:
• springing or ride comfort, which lies below n = 240 min- 1, i.e. 4 Hz;
• wheel comfort or road harshness (/> 4 Hz).
The split is sensible because the two frequency ranges are experienced differently by the human body and become important if individual parts of the
suspension, such as springs, shock absorbers, suspension link bearings etc., are
to be evaluated for their influence on comfort. The springing balance (which
expresses how well the front and rear axles are matched to one another) also
needs to be taken into consideration. If a vehicle does not pitch when it goes
over bumps in the ground, but instead moves up and down in parallel translation, it has a good springing balance. To provide an objective evaluation of
----,-----,_.-------------_._--------,---Ii
308
The Automotive Chassis
The bottom should be placed as close
as possible to the seat-back. The gap
between the seat and the pedals should
be so located that the leg remains
slightly bent when the pedals are
completely depressed.
Adjust the front of the seat
(forward seat extension) so that
the thighs are supported almost to
the knees. Rule of thumb: There
should be space for two or three
fingers between the front of the
seat pad and the leg behind the
knee.
The shoulders should be as close as
possible to the seat-back. The inclination
of the seat-back should be such that the
• steering wheel can be easily controlled
with the arms remaining slightly bent.
The shoulders should be able to remain
in contact with the seat back when the
==:!!!===::.._-:::::::= steering wheel is being turned.
The RECARO airmatik is correctly
adjusted when the spinal column
adopts its natural shape.
The seat height should be set as high as
possible. This will give the driver vision
to all sides of the vehicle as well as
sight of all instruments that display
information.
The seat platform should be arranged
so that the pedals can be easily
depressed. The thighs should not exert
too much pressure on the seat cushion.
Ensure that the angle of inclination of
the back is satisfactory before starting.
"f
,
..
\,
"..
.
,
"
.. ,
.
The seat should be located laterally so that the upper torso is
comfortably located laterally without the need to draw in the arms
The upper edge of the head
restraint should be adjusted so
that it is level with the top of the
head. Note: The distance from the
head should be about 2 em.
Fig. 5.1
In addition to the body suspension and damping, the seat is of crucial
importance for driver comfort. The position of the seat must ensure safe, comfortable and tireless operation of the vehicle. Apart from the static properties of the seat
(general seat position, possibility of adjustment), the quasi-static characteristics
(possibility of slight body movements to achieve muscle relaxation), temperature and
air-conditioning properties and the transmission of vibration are also important. The
latter depends on the seat design (suspension and damping behaviour of the seat)
and the mass of the driver; the latter in particular determines the excitation of the
driver and hence the impression of comfort, specifically in the region of vertical vibration above 5 Hz.
To increase the safety of the driving environment, active ventilation of the seats
by electric fans in the seat cushion and backrest cushion can be used, as well as
dynamically operating pneumatic adjustment of the backrest in the shoulder and
lumbar region for muscle, pelvic and spinal column relaxation can be used. By means
of two hydraulic chambers attached to the surface of the seat, BMW produces a
movement of the spinAl column specifically intended for the prevention of scoliosis.
comfort, measurement devices are used which evaluate, based on the VDI
(Verein Deutscher Ingenieure) directive 2057, the vibration that occurs (vibration stroke, speed and acceleration) dependent on frequency, in accordance with
existing knowledge of how the human body experiences it. The measurement
result is then available as a numerical value, the so-called K-value. Low values
Springing
309
indicate good ride comfort, whereas high values indicate poor ride comfort (see
also Chapter 7 in Ref. 9).
5.1.1
Springing comfort
This comfort range is mainly influenced by the acceleration acting on the upper part
of the human body and lies in a frequency range of 1 to approximately 4 Hz. With
a given vehicle body mass mBo (see Equation 6.5), the critical variables are the
configuration of the rate of the body springing and rising from the body's resonant
frequency. In accordance with the simplified model of the sprung-mounted car body
shown in Fig. 5.7 (single mass vibrator) the mass portion mBo, for r exhibits free
undamped vibration at the natural frequency in accordance with Equation 5.4, and
the corresponding vibration rate in accordance with Equation 5.4a.
The softer the springing, i.e. the lower the springing rate Cf or r of the body
(front or rear), the lower the natural frequency for a specified body mass and,
accordingly, the greater the ride comfort. Unfortunately, at the same time the roll
increases on bends (it must be reduced by anti-roll bars, see Section 5.5.4 and
Fig. 5.2), as does the tendency to pitch when the brakes are applied or when
starting out (see Sections 3.11 and 6.3). Vibration values of nf or r = 60 min- 1
(where f = 1 Hz) are desirable, but cannot necessarily be easily achieved (see
Section 5.2).
Fig.5.2
Influence of
the anti-roll bar rate on the
steering angle, measured
while the vehicle is steadystate driving on a circular
path (R = 42 m) on a standard design passenger car
with rr1v.t = 1544 kg. The
understeering can be reinforced, or an incipient
tendency to oversteer
reduced by increasing the
rate of the front anti-roll
bar and/or reducing the
rate of the rear one. On
front-wheel drive vehicles
a more highly stabilized
rear suspension is usually
necessary.
In the case of low
lateral acceleration, and
therefore also on wet or
slippery roads, the antiroll bar rate has no effect
on the self-steering
behaviour.
140
o
A~ti-roll bar rate I
I
I
I
1
_Front c<p,f = 16.5 N mm- Rear c<p,r = 3.0 N mm- 1
--·Front C<p,f = 9.75 N mm- 1 Rear C<p,r = 9.75 N mm- 1
120
:I:
R =42m
CoO
05"
co
.c
~
0)
c
60
'i:
Q)
Q)
+-'
--
Ilx,w=O·V . / , /
~ 80
Q)
Q)
)
1/
--------
~
~ .....-
;;;.-
en
40
20
o
1
3
2
5
4
Lateral acceleration,
6
By
---_._-----------_._-----,--------:i
m 7
82
The Automotive Chassis
310
Fig. 5.3 When a wheel rebounds by the
path 52, the wheel load reduces by the
amount AFz.w. The level of the residual force
which still ensures wheel grip
F Re = Fz.w - AFz.w depends mainly on the
springing stiffness, defined by the rate Ct or r.
Direction
,.
Another advantage of soft springing would be the improvement in the
absorbency of bumps and the wheel grip. If, for example, a front wheel loaded
at Fz,w = 3000 N drops into a 60 mm deep pothole (Fig. 5.3), with soft linear
springing at the rate Cf = 15 N mm- 1 the residual force at the bottom of the
pothole is
FRe,f = Fz,w,f -
Cf 82
= 3000 - 15 X 60 = 2100 N
(5.0)
With 'sporty' hard springing at Cf = 30 N mm- 1, it is only F'Re,f = 1200 N. The
greater residual force equates with better road holding. The same can be said of
a vehicle travelling over a 40 mm high bump (Fig. 5.4). With hard springing, the
force transferred from the axle to the body as an impact, ignoring the damping and time influence, would be ~Fz,w = 1200 N; soft springing only transfers
600 N and therefore generates lower wheel load fluctuation.
The disadvantage (as already mentioned) is the greater body roll on bends and
the concomitant lower ability of the wheels to transfer lateral forces (see Section
5.4.3 and Equation 2.16). As shown in Fig. 1.6, the wheels incline with the body
on independent wheel suspensions. The wheel on the outside of the bend, which
absorbs most of the lateral forces, loses negative camber (or goes into positive
camber), resulting in the need for a larger tyre slip angle (see Section 2.8.5.5).
The springing comfort, and associated with it also the handling, depends not
only on the weight of the vehicle and the body springing rate, but also on other
variables and the interaction of the individual components:
• the load distribution (see Section 5.3.6)
• the design of the wheel suspension
• the type of mounting and design of the springs (see Section 5.3)
Direction
..
1--
Fig. 5.4
When the wheel displaces by
distance 51 the wheel load increases by AFz.w.
The size of the increase in force in the body
depends mainly on the springing hardness, i.e.
the rate Ct or r.
Springing
•
•
•
•
•
•
•
•
the
the
the
the
the
the
the
the
5.1.2
311
anti-roll bars (Fig. 5.2 and Section 5.5.4)
torsional rate of the rubber bushings (Figs 3.18, 3.84 to 3.87, and 5.5)
shock absorbers and their mountings (see Section 5.6.7)
weight of the axles (see Section 6.1.3)
type of engine and gearbox mounting (see Chapter 10 in Ref. [5])
wheelbase (see Section 3.2)
tread width (see Section 3.3) and
tyres in general (see Section 2.4).
Running wheel comfort
Even smooth-looking road surfaces have almost invisible slight irregularities
and bumps which are transferred to the body as high frequency acceleration and
jolts (4-80 Hz). The vehicle occupants feel them in the underbody of the vehicle, in the seat cushion, and the driver also feels them in the steering wheel and
the pedals. They determine the wheel comfort and the concomitant road harshness.
The cause of this is the often limited vibration insulation between the suspension parts and the body, i.e. the suspension links, suspension subframe and
McPherson strut mount, plus the friction in the suspension control arm bearings
(Fig. 5.5), the wheel joints (Fig. 1.38) and in the shock absorbers or spring
dampers (Fig. 5.51; see also section 4.2 in Ref. [2] and Ref. [5]).
On McPherson struts and strut dampers the friction in the piston rod guide
generated by transverse forces can be the cause (see Figs 1.8, 1.11 and 3.30; also
Section 6.43 in Ref. 5). The springing does not respond as well and today's everwider (and therefore harder) tyres no longer absorb the bump loads - these are
transferred directly to the body.
These relationships can easily be explained using the hysteresis of a springing curve (Fig. 5.6). The friction force is 200 N per wheel in the central range,
i.e. starting from the centre line:
Frr = + 100 N
The rate of body springing at the front should be
of a bump Sl = 6 mm results in a spring force of
/iFr =
Cr Sl
= 15 X 6 = 90 N
Cr
= 15 N mm- 1 and the height
(5.0a)
As Frr > /iFr, in this instance the soft springing would not absorb the bump and
the suspension would pass on the force to the body (see also Section 2.5).
However, if the spring rate is Cr = 30 N mm- 1, the force would be absorbed by
the spring. The problem here is reversed, as shown in Figs 5.3 and 5.4.
Soft springing creates greater difficulties in achieving the desired running
wheel comfort in terms of road harshness than harder springing, particularly on
front-wheel drive vehicles.
There is also the longitudinal vibration caused by the steel belts of the radial
tyre, particularly on rough cobbles. Section 2.2.2 contains details and Section
------~-------..;-'-------------'
!I
312
The Automotive Chassis
y
859
7 1
2
3
10
4
9
9
6
s
Y 2:1
5
8
10
X 2:1
kN
t
IJ...
6
4
2
0
9.4
Fig. 5.5
105.1
2
4
6
mm
10
s--.....
The mounting of the upper control arm of the double wishbone front axle
on the Mercedes C class, manufactured by Lemforder Fahrwerktechnik. The inner
tubes 1 within the two brackets 8 on the wheel UK usage panel are fixed using the
hexagonal bolt 11. Rubber parts (position 9) are vulcanized onto the intermediate
tubes 6, which are pressed into the suspension control arms. Flanges 5 on both
sides absorb the axial forces Fa,x. The compliance in this direction and the low compliance in the radial direction (Frad ) are indicated in the diagram.
To keep the friction moment M fr = 1 N m there are PTFE coated guide bushes 3
between the tubes 1 and 6 and the discs 2 between the lateral flanges 5. The lips 7
provide the seal to the maintenance-free mountings. The smaller the moment Mfr
can be, the more favourable the ride comfort and the absorbency of the springs
become. The outer tubes 6 are slightly shorter than the inner ones (position 1),
between them is the clearance 5, which evens out installation tolerances and
provides the longitudinal mobility to take the radial tyre rolling hardness (see Section
3.6.5.2). In the case of high (axial) braking forces, depending on the compliance of
the rubber flange 5, the outer tubes 6 butt up against one another and ensure the
necessary longitudinal stiffness.
For further details, see Section 2.3 in Ref. [2].
Springing
313
550
kg
Total wheel travel ~ 07 mm
450
t
J~
400
"0
co
o
/
350
/:~
300
~
250
200
150
100
;:V
V
~
V
V
~v
~
•
~
V
~
200 N friction force
IIII
50
o
o
20
40
60
80
100
Wheel travel
120
140
160
180
mm
220
..
Fig. 5.6
Hysteresis of the curve of front wheel springing shown in Fig. 5.9; the
line distance indicates the friction force in the suspension parts, i.e. the self-damping. This is 200 N in total, i.e. Fir = ± 100 N (taking the mean value as nominal).
3.6.5.2 explains how this vibration can be kept away from the body. The design
complexity is likely to be greater on driven wheels than on non-driven ones.
5.1.3 Preventing 'front-end shake'
'Front-end shake' is a term used to describe short, hard jolts (in the vertical
direction) in th~ body floor and the front end of the vehicle which, particularly
--------_._-----------~-'---------'
il
314
The Automotive Chassis
on front-wheel drive vehicles, are triggered by movement of the engine on the
rubber parts of the engine mountings and are in a frequency range of around
8-12 Hz. This vibration does not occur continuously, but whenever the engine
mounting, the frequency of which is often very close to that of the suspension,
begins to resonate. The softer these bearings can be, the less engine noise and
vibration will be transmitted to the vehicle interior, but the higher will be the
tendency to front-end shake. Conversely, hard mountings reduce front-end
shake, but more engine noise is transferred to the vehicle interior. To solve this
conflict of aims, hydraulically dampened engine mountings, so-called hydromounts, are used and these have a lower static spring rate and, in the event of
resonance, generate far higher damping than is possible with normal elastomer
mountings. See Section 10.4 in Ref. [5] for details.
5.2
Masses, vibration and spring rates
For determining the vibration rates nforr (front or rear) of the body and the spring
rate Cf or r the front axle load mY, f, pi (or mY, f, max) and the rear axle load mY, r, pi (or
mY, r, max) must be known in the design (normal ride height) position (see Section
5.3.4, index pI = partly loaded) and for a permissible gross vehicle weight (index
max). With maximum payload the permissible rear axle load mY, r, max is mostly
fully utilized; in this instance, the resulting front axle load mY, f,lo (index 10 =
loaded) needs to be calculated from the maximum gross vehicle weight mY, t, max
(see Equation 5.9):
mY, f, 10
=
mY,t,max -
mY, r, max
(kg)
(5.1)
The mass proportions ml,Bo,f and ml,Bo,r of the body (which at front and rear load
one axle side respectively) can be calculated using the axle load and the masses
mU,f and mu,r of the front and rear axles (unsprung masses, based on both wheel
stations of the suspension system; see Section 6.1.3).
mY,f-mU,f
ml, Bo, f
= ---2
ml, Bo, r
= ---2
mY,r-mu,r
(5.2)
(5.3)
The suspension masses comprise the mass of the wheels and wheel calTiers. The
latter can be the steering knuckles or, in the case of rigid axles, the axle housing
including the differential. There is also the proportional (sometimes half) mass
of the suspension parts, which flexibly connect the actual axle with the body or
frame. This includes:
• suspension control arms
• tie rods
Springing
315
- f - - - m'.Bo,f or r
' 5 0 - - - Ct or r
On the simple vibration system, the level of the body frequency nt or r
(front or rear) depends only on the weight or mass proportion m',Bo,t or r of the body
over a front or rear axle side and the spring rate Ct or r, which on linear springing is a
quotient of force and travel: Ctorr = F/s. On progressive springing, the change in force
flF over a minimum travel range fls plays a part Ct or r = t1Flfls (see Fig. 5.12).
Fig. 5.7
•
•
•
•
•
axle shafts
leaf or coil springs
shock absorbers
anti-roll bar arms
panhard rod etc.
The other half of the mass is accounted for by the body. Torsion bars are in the
underbody, so their mass forms part of the sprung mass.
Section 6.1.3 contains all details and Equation 6.4c contained therein makes
it possible to determine the approximate weight of an axle based on its design.
(see also Section 5.2 in Ref. [3]).
.
The spring rate Cforr (Fig. 5.9) is required for calculating the spring itself and
the configuration of the suspension. This should appear in N mm- I on drawings
and as a measurement value, where(is in all calculations the unit is N m- I. If this
stipulation is not complied with, there is a risk of calculation errors, unless these
are recognized when a dimension equation is done. With the international units
the equation for the angular frequency w is as follows (Fig. 5.7):
The conversion 1 N = 1
kgm
-2
s
results in:
To obtain the vibration rates nf or r (per minute) used in the springing layout, the
angular frequency needs to be multiplied by
60/211" = 9.55 (s min-I)
Related to the body, if the damping, the influence of the mountings and the tyre
were ignored, the equation (with indexes) would then be:
:1
The Automotive Chassis
316
Ct or r
kO,f or r
m',U,f or r
nforr=
9.55 ( Cforr
ml, Bo, for r
)i (.
mm _I)
The calculation of the vibration rate
includes half the axle mass
ml, u, forr
Fig. 5.8 The level of the wheel
vibration rate nU,f or r is a function of the
axle mass ml,U,t or r, of the body springing rate Cf or r, the tyre springing rate
Cr.t or r and the damping ko. tor r. The
driving speed also has an influence
(shown in Fig. 2.28).
(5.4)
nU,f
or r of one axle side (front or rear)
= mu, forr/2
(5.5)
in kilograms and the tyre spring rate CT, forr in N m- I. Figures 2.27 and 2.28 show
statically measured values which increase during driving (see also Section
2.2.8). The factor kT includes the springing hardening of around 1% per 30 km h- I
(see also Section 2.2.8):
kT
~
1.04 at 120 km h- I
(5.5a)
The equation for the axle vibration rate is then (Fig. 5.8):
)2 (.mIn-I)
1
kT . CT' f" or r + Cf or r
_ 9•55
nU,for r (
ml, u, forr
(5.6)
For passenger cars with steel springs the body vibration rate should be
Front: nf = 60-80 min- I
Rear: n r = 70-90 min-I
The natural frequency (vibration frequency) of the body over the rear axle is
chosen to be 10-20% higher than that of the body over the front axle. Thus the
vibrating motion results from vibration of the front axle caused by unevenness of the
road surface is 'overtaken' by the more quickly vibrating rear axle. Thus causes the
bouncing motion, which is desirable for comfort reasons, instead of the pitching
motion which is uncomfortable for occupants of the car. Particular importance is
attached to this design in vehicles with a short wheelbase and a high seat position.
For reasons of comfort, nr or r should be approximately 60 min-I, which is
rarely achieved on the front axles of small to medium-sized passenger cars and
can only be achieved at the back if the vehicle is fitted with an automatic level
control. The load difference between the loading conditions 'one person' and
Springing
317
'full load' (Figs 5.14 and 5.15) makes it difficult to design the springing on the
rear axle to be soft, as would be required for comfort.
There are further limitations on the front axle. The fact that the engine
bonnetlhood is low, both for aesthetic reasons and because of the requirement for
low air drag values, limits the space available for the springs, particularly in the
case of McPherson struts.' So as not to exceed the material stresses, soft springs
are longer and their block length is therefore larger; harder springing does not
have this disadvantage (Fig. 5.13). However, this reduces the comfort; on the
other hand, stnlt dampers do allow longer spring travel (Figs 1.41 and 5.12).
The spring rate Cr or r can be calculated on the basis of a specified vibration
level nrorr using the transformed equation 5.4a:
Crorr =
0.011 nf,orr ml,Bo,rorr (N m- I )
(5.7)
The frequency figure in min-I and the mass in kg must be inserted.
The front wheel springing of a front-wheel drive vehicle can be used as an
example; the specified loads correspond to the lower limit, i.e. with one person
in the vehicle:
Front axle load
Axle mass
Specified vibration level
mr
= 455 kg
55 kg
= 60 min- t
mU,r=
nr
In accordance with Equations 5.2 and 5.7:
= (475 - 55)/2 = 210 kg
0.011 X 602 X 210 = 8316 N m- I
= 8.3 N mm- 1
ml, Bo, r
Cr=
. Figure 5.9 shows the springing curve with the calculated rate (and associated
long paths). The design or zero position (i.e. when there are three people each
weighing 68 kg in the vehicle, see Section 5.3.4), is entered as a further point of
reference and the weighed wheel load as a function of the wheel travel is shown.
This load is observed in the centre of tyre contact.
In the reverse situation, the springing rate can be calculated from an existing
springing curve as a function of the various loading conditions. If a curve is
linear in the middle range (as shown in Fig. 5.9), it only needs to be extended
over the whole spring travel in order to read the load difference at the end points
(here 3.32 kN and 1.61 kN). This, divided by the total travel (St = 207 mm), gives
the spring rate.
In the case of a progressive curve, a tangent must be drawn to the curve for
the loading condition to be observed and for it to be possible to read the difference values of loads and paths from it. Figure 5.12 shows an example relating to
the design position.
The vibration rate can then be calculated from spring rate, axle load and estimated axle weight. This is usually more precise than settling because most vehicles have McPherson struts, strut dampers or spring dampers and the inherent
friction in these parts means a correct result is unlikely.
318
The Automotive Chassis
Empty position
Permitted wheel load
0)
C
.;::
5
St
= 207 mm
0-
w
>"co
.....
kN
z
Q)~
E ....
C
Q)
•
"Q.(\')
0- II
"'C
co
~w:
80mm
o
~
=115 mm
Force into the
compression stop
Q)
()
"-
z
F2 = 1.61 kN
~
Zero position
2.56 kN
C"l
.E
0)
C
.;::
~ ~
(\')
.....
II
~
..x .....
co
E Q)
ci. 'L{}c.9
2.61 kN
20
193mm
40
60
80 100 120 140 160 mm 200
iSo = 308 mm
Spring travel s
....
Fig. 5.9 Curve of the front wheel springing of a Renault model, the wheel load (in
kg) is entered as a function of the wheel travel (in mm). The soft springing shown
requires stops; if the bump stops were missing (Fig. 5.48). the front wheel could
jounce from the zero position (the vehicle occupied with three people each weighing
68 kg) by 50 = 308 mm. Where there is no supplementary spring (Figs 5.21 and 5.50).
at FSP.max = 3.32 kN the axle would make a hard contact. The residual forces to be
absorbed by the spring travel limiters are entered in kN. The progressivity achieved
by the supplementary spring can be seen clearly. If the stops are in the shock
absorber (Fig. 5.29). the compliance of the suspension parts also appears in the
curve. The rate of the body springing is:
~F
5.3
CI.pl
3.32 - 1.61
= ~5 = 0.0207
CI,pl
= 8.26
kN m-1
= 8.3
N mm-1
Weights and axle loads
Without knowledge of the weight in the empty or loaded condition and distribution of the load to the two axles, springing on a passenger car can neither be
configured nor evaluated. The variables of weight and load laid down in German
standard DIN 70 020, page 2, relate to the mass (in kilograms or tons) of the
Springing
319
vehicle occupants, the transportable items or goods and the vehicle itself. For
details, see Section 1.1 in Ref. 3 and Ref. 8.
The following information and details relate only to vehicles of class M I in
accordance with the directive 711320lEEC of the European Union. These vehicles must be used for carrying passengers and may not, apart from the driver's
seat, have more than eight seats. They must have at least four (or three) wheels
and a total mass of mV, t, max which does not exceed 1 t (ton force) when fully
loaded.
.
5.3.1
Curb weight and vehicle mass
The actual weighed curb weight mV,ul of the vehicle is essentially determined by:
• the weight of the body with interior trim and the fuel tank;
• the engine and gearbox weight with all necessary accessories, such as starter
motor, generator, exhaust system, etc.;
• the weight of the chassis;
• the optional equipment such as automatic gearbox, air-conditioning system,
sun roof, etc. (see Equation 5.8a).
According to German standard DIN 70 020 the curb weight also includes:
•
•
•
•
the charged battery
lubricant, coolant and brake fluid
the standard tool set
a fuel tank at least 90% full.
However, Section 42 of the German StraBen verkehrs-Zulassungsordnung
(StVZO, the regulations for vehicle approval) requires a full tank.
There are also various loose pieces of equipment, such as jack, spare wheel,
etc. which must be carried in the vehicle and, in most countries, the triangular
safety reflector and first aid kit. The international recommendation ISO/R 1170
contains further details.
5.3.1.1
Curb weight according to manufacturer's data
As the curb weight information required by law allows a tolerance of + 5% which for a vehicle means a weight range of f1mv = 110 kg mV, m = 1100 - vehicle manufacturers try to set the curb weight mV, ul, 0 shown in the vehicle identification card such that it is as low as possible (this governs the balance weight
class, which itself is important for the vehicle's fuel consumption and emission
rating) and yet still covers as many model versions as possible to keep the technical expenditure low (for example, for type approval). This leads to the optional
and supplementary equipment sometimes being ignored. In such a case it is not
easy for the vehicle's registered keeper to calculate the actual permissible
luggage, roof and trailer load and he or she will be held responsible if the maximum permissible gross vehicle weight is exceeded.
'11_
__I
The Automotive Chassis
320
5.3.1.2 Mass of driveable vehicle
Since 11111996, all new models of class M\, and from 11111998, all newly registered
vehicles must be tested in accordance with EU Directive 92/21/EEC and 95/48/EC.
These specify that all vehicle manufacturers must quote the mass mY, dr in
driveable condition, i.e. the weight of the vehicle driver at mp = 68 kg and the
baggage mass at mb = 7 kg must be included. Until now, in Germany, this
approval condition was only specified for vans and commercial vehicles (class
N in EU Directive 711320/EEC; see Section 5.3.6.3).
5.3.1.3
Mass of the driveable vehicle when towing a trailer
If the vehicle is intended for towing a trailer, the weight mTh of the towing device
and the permissible tongue load dMTr under static conditions must be added to
the mass mY, dr (see Section 5.3.3.4). The permissible rear axle load must in this
case usually be increased.
5.3.2
Permissible gross vehicle weight and mass
This is specified by the vehicle manufacturer taking into consideration the minimum load - which corresponds to the nominal payload mt (see Equation 5.7c) in
accordance with ISO 2416 - required by law, based on the number of seats
provided.
5.3.3
Permissible payload
The permissible payload mt,max of a passenger car is the load that the driveable
vehicle can carry without exceeding the permissible gross vehicle weight. It
therefore results from the difference between the permissible gross vehicle
weight mY, t, max and the actual curb weight mY, ul:
mt,max
=
mY, t, max -
mY, ul
(5.7a)
Vehicle manufacturers generally specify the payload higher than the regulations
demand. This is reflected in a larger permissible gross vehicle weight. The calculation takes into account the component and material stress to be guaranteed, the
tyre and wheel bearing load capacity and the loss of braking capacity and
handling usually associated with a higher load. The distribution of the goods
being transported and the spring travel limitation also playa part in this loss of
handling (see Sections 5.3.6 and 5.5.3 and Ref. 9).
There is also the risk of the permissible rear axle load mY, r, max being exceeded
with a full boot and it is possible that the front axle then might lift. This is bound
to lead to reduced steerability. On a front-axle drive vehicle, traction and climbing capacity are reduced (see Sections 1.1.7 and 6.4). The EU Directive 92/21
EEC therefore specifies that the front axle load mY, f may not be less than 30% of
the vehicle total weight mY,f, i.e.
mY,f
~
0.3
mY,t
----,,--------------------,------
(5.7b)
Springing
S.3.3.1
321
According to ISO 2416
This standard specifies the minimum payload for passenger cars, i.e. the nominal payload mi. This depends on the number n of seats provided by the vehicle
manufacturer and the passengers' luggage or on the number no of the actually
occupied seats and the luggage mass mlr of the goods then transportable.
To determine the number n, a weight of mp = 68 kg for each person - including clothing - must be assumed, plus a luggage mass of mb = 7 kg per person.
The nominal payload mt must then be
(5.7c)
The greatest value - Le. the luggage mass actually transportable mtr - is then
(5.7d)
mlr =
mv,t,max - mV,uI -
m p X no
(5.8)
Experience has shown that the actual or weighed curb weight mV,ul exceeds the
rnanufacturer's stated curb weight mo by the weight of the optional equipment
L~mv found in the vehicle
mV,ul = mxo
+ dmv
(5.8a)
A five-seater passenger car with a permissible payload of
20 kg optional equipment can be used as an example:
mlr ·= ml,max - dmv - m p X n
mlr = 400 - 20 - 68 X 5 = 40 kg
mt,max
= 400 kg and
(5.8b)
The transportable luggage mass mlr is therefore above the minimum value:
mb
5.3.3.2
=7
X 5
= 35 kg
Nominal payload
It is the manufacturer who specifies the payload - and therefore also the permissible gross vehicle weight - taking into consideration the expected use of the
vehicle (saloon, estate car, sports coupe, etc.) while complying with the legally
required nominal payload mt, i.e. based on the number n of seats provided. In
accordance with Equation 5.7c, ml will be
2 people:
3 people:
4 people:
5 people:
etc.
136 kg
204 kg
272 kg
340 kg
+ 14 kg luggage =
+ 21 kg luggage =
+ 28 kg luggage =
+ 35 kg luggage =
150 kg
225 kg
300 kg
375 kg
This means that for a nominal payload of mt = 375 kg a saloon will still be
legally approved as a five-seater. The precondition is that the other requirements
are met, e.g. in respect of seat belt anchoring.
322
The Automotive Chassis
If five people, each weighing 75 kg, occupy a five-seater passenger car, the
permissible payload of which, at 375 kg, is at the lower limit, this already gives
a figure of 375 kg. If the vehicle has retrofitted accessories not included in the
weight calculation or optional equipment t:,.mv beyond the normal amount (see
Equation 5.8a),.the vehicle is already overloaded and it would not be possible to
carry any luggage. If, without being aware of the situation, the driver nevertheless puts items of luggage into the boot, the vehicle will exceed the permissible
total weight and probably also the permissible rear axle load. If the resulting
deterioration in handling or the now insufficient tyre pressure leads to an accident, the driver would be regarded under law in Germany as responsible for the
overload. Legal decisions back this up.
5.3.3.3 According to ED directives 92/21/EEC and 95/48/EC
Contrary to Section 5.3.1.2 the curb (empty) weight (i.e. without occupants) is
assumed rather than the ready-to-drive weight.
5.3.3.4 When towing a trailer
When the vehicle is towing a trailer, EU Directives 92/21/EEC and 95/48/EC
specify that the weight mTh of the towing device and the maximum drawbarimposed load t:,.mTr allowed by the manufacturer must be included in the calculation of the necessary nominal payload (Section 5.3.1.3). A five-seater
passenger car would then be permissible for the following nominal payload
(Section 1.1.6 in Ref. [3]):
Minimum value for five people
Weight of optional equipment
including towing device (assumed)
Drawbar-imposed load when towing a trailer
m p = 375 kg
t:,.mv + mTh =
t:,.mTr -
Required nominal payload
70 kg
75 kg
mt = 520 kg
If the payload is 420 kg, the relationships are different:
Nominal payload
Optional equipment
Towing device
Drawbar-imposed load
420 kg
-30 kg
-15 kg
-75 kg
Minimum value
300 kg
According to Equation 5.7c, the vehicle would just about count as a four-seater
and the number of permissible seats would therefore have to be altered in the
vehicle identification papers.
The maximum static torque load is generally t:,.mTr = 50--75 kg; however,
according to Directive 92/21/EEC the maximum permissible load must not be
less than 25 kg.
--------------------
Springing
5.3.4
323
Design weight
The design weight mV, t, pi determines the axle weights mV, f. pi and mV, r, pi, as well
as the design position of the vehicle, also known as the normal or zero position.
Under the specified payload, starting from the empty condition, the body
compresses front and rear and the result is a particular position vis-a-vis the
ground. ISO/IS 2958 'Road vehicles: Exterior protection for passenger cars'
therefore internationally specifies the design position in relation to the number
of seats (specified/allowable number of passengers) as follows:
Number of
seats
Distribution
2 and 3
4 and 5
6 and 7
two people each weighing 68 kg on the front seats
two people on the front seats and one person on the rear seat
two people on the front and rear seats
Luggage is ignored. The vehicle should be shown with this number of passengers on the drawing board.
When vehicle manufacturers are exchanging vehicle dimensions, the design
weight is always specified for determining the design position. The German
Directive VDA 239-01 (Verband der Automobilindustrie - Automobile Industry
Federation) and Ref. [11] cover all aspects relating to this field.
:5.3.5
Permissible axle loads
5.3.5.1
According to Section 34 of the German Stra6enverkehrsZulassungsordnung (StVZO)
The permissible axle loads front and rear are specified by the vehicle manufacturer. Several points on which the axle loads have a direct effect must be considered:
n
41
41
41
component strength of the body and wheel suspension or axles;
load capacity and therefore minimum size of the tyres;
configuration of the brake and brake force distribution (Ref. 6);
springing and damping.
The permissible axle loads are included in the ABE (Allgemeine Betriebserlaubnis or General Operating Approval) in type-testing in Germany or, in the
case of the approval of an individual vehicle in accordance wi Section 21 of the
StVZO, are included in the report of an officially approved expert. The values
are indicated on the type plate.
To date, for passenger cars, this specification has not bee governed by any
particular legal regulations, with the result that only the n minal payload mt
(Equation 5.7c) in accordance with the number of seats a proved had to be
considered and that the sum of permissible axle loads fron mV, f. max and rear
The Automotive Chassis
324
mY, r. max has to be greater than, or at least equal to, the permissible gross vehicle
weight (see also Equation 5.1):
mY,f.max
+ mY,r,max
~
mY,t.max
(5.9)
To be able to match the payload to the load compartment in the vehicle better,
the gross vehicle weight is usually kept larger than the permissible total value
mY,t.max (see Fig. 5.11).
On drive tests and in vehicle behaviour simulations (see Sections 6.3 and 6.4),
the least favourable loading condition, i.e. the permissible rear axle load mY,r.max
must be assumed. The front axle load mY,f.lo which arises, is then usually below
the permissible mY,f.max (Equation 5.1).
The vehicle manufacturer is given the option of the residual hub paths, i.e.
there are no regulations on how far a fully laden axle may compress the springs.
If this is less than SRe = 50 mm, the desired springing effect would be compromised. Furthermore, the body can barely go any further down on the outside of
the bend when cornering, so its centre of gravity rises and the cornering behaviour changes and tends to oversteer, as a result of which situations can arise
which are beyond the competence of the driver (Figs 2.42, 5.15 and 5.16).
5.3.5.2 According to ED Directive 92/21IEEC
Directive 92/91IEEC (see Section 5.3.1.2) made the loading of the vehicle and
therefore the axle loads subject to stricter regulations. The permissible payload
mt.max (see Equations 5.7a and 5.8a) to be calculated from the difference between
the permissible gross vehicle weightmy,t,max and the actual curb weight mY,ul must
be divided up as a percentage into flat rate'mass:
91 % (90.7%, to be precise) were then allocated to the seats and 9% (or 9.3%)
evenly distributed throughout the boot (Section 5.3.6).
The manufacturer had to certify the resulting axle loads as permissible values.
Directive 65/481EEC, which was subsequently issued, withdrew this measure
and again requires the values according to ISO 2316 (see Section 5.3.3.1).
5.3.5.3
When towing a trailer
If the vehicle has a towing device, a reduced loading by its component weight must
be assumed and, furthermore, the maximum static drawbar-imposed load ~mT of
the trailer must also be included (see Section 5.3.1.3 and Section 1.1.7 in Ref. [3]).
The remaining payload is then set at 100% and distributed to the seats and boot.
The permissible rear axle load is then greater. Two options can be derived:
• The manufacturer specifies the higher axle load for all vehicles. This means
that the vehicle components listed above must be designed with this in mind,
with the disadvantage that stiffer springs reduce the comfort and, under certain
circumstances, tyres, axle parts and wheel bearings with a higher load capacity may become necessary.
• The manufacturer specifies two separate axle loads with and without a trailer
towing device; the manufacturer must then ensure that the requirements listed
under Section 5.3.5.1 are met.
Springing
325
Shock absorbers with variable damping (see Section 5.9) or an automatic level
control system (see Chapter 9 in Ref. 5) or supplementary springs (Figs 5.20 and
5.49) can balance the springing.
5.3.6 Load distribution according to ISO 2416
The springing of a vehicle, irrespective of whether it is a passenger car, commercial vehicle or trailer, can only be designed if the axle load distribution has previously been calculated or determined by weighing. The important thing is how
many kilograms of payload (and not what percentage) will be on the respective
axle, and whether the permissible axle load is fully utilized or exceeded.
The permissible roof load is between 50 kg and 100 kg (see Ref. [3]); it can
be taken from the service manual of the respective vehicle.
5.3.6.1 On passenger cars with a non-variable boot volume
Figure 1.36 shows the axle load distribution as a percentage. Where the axle load
weight is known, once the weight of the people has been added, the axle loads
in the various loading conditions can be calculated.
Section 5.3.3 describes calculation of the permissible axle load which gives
the axle load distribution. In industry, and at the TOV, this is determined with
weights placed on the seats at the hip centre H, i.e. the centre of gravity of a
person. The position of this point is laid down internationally in the standards
SAE-J 826a, ISO 6549 and in DIN 33408. See Sections 1.1.3 in Ref. [3] and 7.2
in Ref. [20] for details.
The adjustable front (and, where 'applicable,also rear) seats must be moved
into the end position for calculating the load distribution and, in accordance with
ISO 2416, the weight of the occupants arranged in such a manner that their Hpoints act 100 mm in front of the respective H-point of the seats. Where the rear
seats are not adjustable, the distance is only 50 mm. However, ED Directive
92/21/EEC specifies exclusively the furthest back steering or sitting position and
no shifting forwards of the H-points (see Section 5.3.5.2). Both cases are therefore a purely theoretical determination of the load distribution, which ignores
whether the vehicle can be steered and operated at all with the sitting position
set.
The permissible payload mt,max calculated using Equations 5.7a and 5.8a has
to be distributed in accordance with Section 5.3.3.1, and the luggage mass must
be put into the centre of the boot. The standard design passenger car shown in
Fig. 5.10 would, at mt,max at = 427 kg, m p = 68 kg and mb = 87 kg, would have
the following loads and axle loads.
For practical reasons, and because it would calculate the difference values
afterwards, less tiring than lifting many individual weights into the boot and the
passenger compartment, it would be easier to do the weighting with people of
any weight. In order to work as precisely as possible, the driver (who should
weigh around 68 kg and be approximately 1.70 m tall) should adjust the seat into
a comfortable position. Because of the centre of gravity of the passengers, the
weight of all the people should not deviate too greatly from this standard mass
m p • (See Sections 1.1.3 and 1.1.4 in Ref. [3] for details.)
,
=
=
326
The Automotive Chassis
Fig.5.10 Axle load distribution determined on a standard medium-size passenger
car by means of weighing. The vehicle was fitted with an electric sun roof. This and
further special features meant it weighed 1173 kg empty (instead of 1100, as specified by the manufacturer).
Manufacturer's
details
State of loading
Empty
2 passengers
2 passengers in front
and 1 in rear
4 passengers
5 passengers
Maximum load
Number of seats
Curb weight
Payload
Permissible
gross weight
5
1100 kg
500 kg
1600 kg
Load
Weight of
vehicle
(kg)
Permissible axle load
Front
750 kg
Rear
850 kg
Total
1600 kg
Axle load
Axle load distribution
Front
(kg)
Rear
(kg)
Front
Rear
(kg)
(%)
(%)
0
136
1173
1309
623
692
550
617
53.1
52.8
46.9
47.2
204
272
340
427
1377
1445
1513
1600
705
718
731
721
672
727
782
879
51.2
49.6
48.4
45.1
48.8
f50.4
f51.6
f54.9
The table (Fig. 5.10) shows the load distribution of a mid-range passenger car
which, because it is carrying additional equipment, has an unladen (empty)
weight 73 kg heavier than in the specified 'as-delivered' condition. In consequence the permitted luggage capacity is reduced from 500 kg to 427 kg.
Although the weight of luggage that can be carried now is still 87 kg, with 5
passengers on board, each with an average weight of 68 kg (= 5 X 68 kg, the
allowable rear axle load is exceeded by 29 kg. However, the 185/65 R 15 88 H
tyre fitted carries 490 kg at v :::; 190 kID h- 1 with the specified air pressure for full
load of PT = 2.5 bar (Fig. 2.15 and Equation 2.14), so the overload would affect
neither the tyres nor, as shown in Fig. 5.14, the springs.
The axle load distribution at 45%/55% (front to rear) in the fully laden condition is likely to cause a slight deterioration in the driving properties of this standard vehicle, while significantly improving the traction.
The situation on a front-wheel drive vehicle also studied in the Laboratory for
Chassis Engineering of the University of Cologne shows a different picture (Fig.
5.11). The axle load distribution of 46%/54% calculated under full passenger load,
indicates such a severe load alleviation on the driven front wheels that difficulties
will be encountered in wet weather conditions, during uphill driving and when the
vehicle is towing a trailer (Fig. 6.22). Passenger weights of 70 kg were used to
compensate somewhat for the manufacturer's specified excessively high additional
load of 500 kg. When empty, the vehicle weighs 6 kg more than shown on the
logbook; nevertheless, 144 kg of luggage weight had to be taken into consideration.
If this luggage is in the boot, the handling, braking and cornering properties deteriorate (see Figs 5.13, 5.15, 5.16 and 6.15). The ideal load distribution in accordance
with EU Directive 92/21/EEC would certainly give significantly better results.
Springing
327
Fig. 5.11
Axle load distribution determined on a front-wheel drive compact family
car by means of weighing. Empty, the vehicle weighed only 6 kg more than quoted.
The manufacturer's approved high payload of 500 kg (or here 494 kg) would be
extremely difficult to achieve. If it is fully utilized, serious effects on the driving safety
cannot be ruled out (Fig. 5.16). The rear axle load can be up to 780 kg which, at the
total weight of maximum 1400 kg, would mean a load of 620 kg on the driven front
wheels and an unreasonable axle load distribution of 44.2%/55.8% on a front-wheel
drive vehicle (Fig. 1.36 and Equation 5.7bl.
Manufacturer's
details
State of loading
Empty
2 passengers
2 passengers in front
and 1 in rear
4 passengers
5 passengers
Maximum load
Number of seats
Curb weight
Payload
Permissible
gross weight
5
Permissible axle load
Front
770 kg
Rear
750 kg
Total
1550 kg
893 kg
500 kg
1393 kg
Load
Weight of
vehicle
Rear
(kg)
Rear
(kg)
Front
(kg)
Front
(kg)
(%)
(%)
0
140
899
1039
548
623
351
416
60.9
60.0
39.1
40.0
210
280
350
494
1109
1179
1249
1393
635
647
659
643
474
532
590
750
57.2
54.8
52.7
46.1
42.8
45.2
47.3
53.9
Axle load
Distribution of axle
load
The 155 R 13 78 S tyres fitted have a load capacity of 410 kg at speeds of up
to 160 Ian h- 1 with a tyre pressure PT = 2.1 bar. The total of the two wheels (820
kg) is above the permissible rear axle load of 780 kg.
5.3.6.2 On passenger cars with a variable boot volume
On all estate cars, hatchback and fastback saloons (and some notchbacks) the
boot volume can be increased by folding the rear seats forwards. In this type of
passenger car design, the load distribution must be calculated in accordance with
ISO 2416, both for when the vehicle is carrying passengers only and when it has
been converted to carry goods. As specified by the vehicle manufacturer, to do
this the rear seat cushion must be folded forwards and the seat backs folded
down (or seat backs alone folded forwards) or the entire row of seats taken out.
One disadvantage can be that, on some· vehicles the front seats cannot then be
pushed back far enough; the driver seat travel is limited by the seat cushion
which has been folded forwards.
The axle loads must be calculated with two people, each weighing 68 kg, on
the front seats and the mass of luggage (or goods) determined in accordance with
Equation 5.7d. The numerical values of Equation 5.8b (and no for the number of
seats occupied) with two people in the vehicle give:
I'
The Automotive Chassis
328
/!>n'
.".,
I
mtr = mt,max - limv - limp X no
mtr =400 - 20 - 68 X 2 =244 kg
This large luggage mass can lead to the rear axle load mv'r,max being exceeded. To
avoid this, ISO _2416 allows the weight to be distributed in accordance with the
manufacturer's instructions.
Folding the rear seats forward can result in slight axle load changes of the
empty and driveable condition (including the driver), or if the rear row of seats
is removed, to a lower curb weight and a higher payload.
5.3.6.3
On vans and lorries
Where they have three or more wheels and a total weight exceeding 1 ton, these
types of commercial vehicle meet the conditions of class N in the ED Directive
71/320IEEC; the weight of 75 kg of the driver here, is therefore included in the
curb weight (see Section 5.3.1.2). Only the load distribution with any mass in the
centre of gravity of the cargo area and in the fully laden state needs to be determined, to calculate from this the axle loads at the design weight - calculated on
these vehicle types at 85% of the payload and in the fully laden condition.
Springing curves
5.4
5.4.1
Front axle
The springing on the front axle of a passenger or estate car should be soft, to give
a high level of comfort to the occupants, making it possible to transport goods
without them being shaken around and to give good wheel grip (see Section
5.1.1). At extremely low vibration frequency (n 30 min-I) people notice the
vibration paths and speeds 80% less than they do on hard springing with
frequencies around 100 min-I. However, the softness of the springing is limited
by the overall spring travel available:
::=;
St,f
=
Sl,f
+
SZ,f
(5.9a)
which comprises the compression and rebound travel of the wheels and should
be at least:
St,f ;:::
160 mm
It is almost as important that, on the front and rear axles, a residual spring travel
of SRe ;::: 50 mm is specified to keep the body centre" of gravity from rising too
much when the vehicle is cornering (see Fig. 5.11). Measurements on a variety
of passenger car models have shown that on comfortable vehicles (fitted with
steel springs), frequencies on the front axle are between nf =
60 min- I and 70 min-I, with a total travel path (from stop to stop) of 200 mm; Fig.
5.9 shows a springing curve of this type.
In automotive engineering, presentation of the paths on the x-axis and the
I
Springing
329
wheel loads on the y-axis has become the norm. To make it possible to read path
differences and the associated load changes on each wheel easily, it is necessary
for them to be entered in a sufficiently large scale, at least 1: 1 for the x-axis and
100 kg = 40 mm for the y-axis.
In Fig. 5.9, the spring rate in the linear range is Cf = 8.3 N mm- 1, and the wheel
would travel a path of So = 308 mm as it rebounds - starting from the zero position (Fz,w,pl = 2.56 kN). The travel can be calculated easily using the units of N
and mm.
So
=
F z, W,pl
Cf
=
2560
8.3
= 308 mm
(5.10)
From a ride and handling point of view, such a long travel is unnecessary and
cannot be designed in. For this reason, a rebound stop limits the rebound travel
Sz on all vehicles; on passenger cars and light lorries, this component is inside
the shock absorber (Figs 5.31, 5.51 and 5.54) or in the McPherson strut or strut
damper. In Fig. 5.9 Sz is relatively large at 115 mm. The kink in the curve at
around S = 30 mm indicates the point where the stop comes into operation. Soft
springing also demands that the compression travel be limited. If there were no
buffers the axle wo.uld make a hard contact. The buffer force (or load) in Fig. 5.9
IS
+ Cf X Sl = 2560 + 8.3 X 92 = 3324 N
= 3.32 kN (or 338.5 kg)
FSp,max= FZ,W,pl
FSp,max
(5.10a)
On roads with potholes, an impact factor of 2.5 is easily possible, i.e. based on
the normal force FZ,w,pl in the zero position, the maximum value F Z,W,max could
be:
FZ,W,max
=2.5
X
Fz,w,pl
=2.5
X 2.56
=6.4 kN
(5.10b)
The main spring, designed with a spring rate of 8.3 N mm- 1 absorbs FSp,max = 3.32
kN, whilst the additional rubber or polyurethane spring absorbs the residual
force F 1 :::;: 3.1 kN. Figures 5.21 and 5.50 show various configurations and characteristic curves; Fig. 5.9 shows where it comes into play after 140 mm spring
travel. If the vehicle compresses over a path of 67 mm from the zero position,
the spring begins to act in a way that is not noticed by the occupants and then
becomes highly progressive.
Figure 5.12 shows the curve of a soft-sprung standard passenger car (and
Fig. 5.10 the associated load distribution); the frequency nf,pl = 63 min- 1 is in
the soft range desired and, at St = 196 mm, there is a large total spring travel.
In contrast, the front-wheel drive vehicle shown in Fig. 5.13 has a high
frequency (i.e. stiff springing) at nf,pl = 84 min- 1 and, at St = 156 mm, still
reasonable total spring travel. The residual spring travel (54 mm) when there
are five people in the vehicle is sufficient but, if the very high, permissible
front axle load of 770 kg is fully utilized (Fig. 5.11), SRe returns to the too low
a value of 36 mm.
330
The Automotive Chassis
1o
kN
9
8
I
7
I
j
I
/
I
Overall whee travel 196
/
/
5
121
109
/
/
/
~
4
'/
..,,/"""'"
80
v
~
114
o
3
~
.... r>-O"'"
"tJ
III
/ /travel68
~
f
A~
/ Residual wheel
128
1
6
a> 0
OlCtl,a>
en
Q;
.8l~
en':l '
~
a>
I.:
I
~
Ctl
2
Ol Ol,"O 'a>
c c Ie: "-
.i?;
a> a> ,Ctl ' •
c.
en
en ~:~I~
E
w
Ctl Ctl:a>,E
0. o.,Ol
"0
N ~;c:
a> a>
en N
C/)'':
Ctl 0
o..c
LO'5
o
40
80
120
0.5
160
mm
200
o
Wheel travel - + -
Fig. 5.12
Soft front wheel springing with long travel and linear coil springs,
measured on a medium-size standard passenger car. The progressive characteristic
curve is achieved with supplementary spring (see Fig. 5.21); Fig. 5.10 contains the
wheel loads. To be able to determine the spring rate on the design weight (three
people each weighing 68 kg), a tangent must be drawn to the progressive curve
(path AS) which is then Llsed to read off two points:
wheel load 4.5 kN, wheel travel 183 mm
wheel load 3.0 kN, wheel travel 78 mm
The spring rate in the partly laden condition (index pi) is then:
Ctpl
,
=
150 x 9.81
-1
= 14.0 N mm
105
The axle weight needed to calculate the frequency figure is 59 kg and, in accordance
with Equation 5.4, it becomes nt, pi = 63 min-1 •
Springing
331
/---/----+-Ir-----t kN
r---/---Hr-----; 9
/---/---tHr-----t 8
/---/--rHr-----t7
:==~~~---o"'Tt-e-ra-lI.,l-p-ri-ng....,lr-ra-v-e-,~r-5-6==:===:+/~/f--;-+--1-+--_-_:6
1/
r----+---'-+---t--r---i----t---lt--H---I5
j
~/
t---+---I---1120 --+---1---!~-36,--+-+---14
/
102
Residual travel
/
94
f
54
/.
62
t--"f-'---t---j---+-:7"/''1-ti--+---t----t-t---J3
74
/v
V
v
82
]
t---+-"7'--t---t---IH--t-+t-Q) r-----jr---+i---I2
~
/
/
t-/-r--t---+--t--~
f!?
en
-
a.
~
~
~
E
Q)
-
H--+---+---/-...... ++-'M -
~...
~ -~
~ x
C'll
--1---+-+---1
C'll
a.. E r-----jr---+i---I
LO
"'0
Q)
N
'C
t - - - t - - - - t - - - I - - - I - t - - - I H + - , g r-----jr---+i----t 0.5
....:::l
«
~-.l---....L--...L....---I~----lL....L.L---L.---I..-...I...L.----'O
o
40
80
Wheel travel
120
mm
160
•
Fig. 5.13 Progressive front wheel springing measured on a compact front-wheel
drive passenger car. The residual spring travel is high, and at 156 mm the total path
is sufficient, which also applies to the residual bump travel of 54 mm when there are
five people in the vehicle. Luggage load in the boot would result in the front end
rebounding, i.e. it would increase the bump travel. As can be seen in Fig. 5.11, the
manufacturer allows a front axle load of 770 kg, which will be impossible to utilize
fully. On the wheel load of 385 kg then possible, the residual bump travel, at 36 mm,
is clearly too low. Frequency and rate indicate relatively hard springing; on the design
weight it is:
Cl.pl
= 21.8 N mm-1 and
nl.pl
= 84 min-1
332
The Automotive Chassis
5.4.2 Rear axle
The springing configuration on the rear axle is more difficult because of the
greater loading difference. Furthermore, the residual rebound travel S2,Re is also
included in the observation. The fuel tank is located in front of, behind or over,
the axle. If it is only part-full and there is only one person in the vehicle, the axle
load corresponds to the empty condition. The road-holding can be compromised
if the wheel cannot rebound far enough; ideally,
S2,Re
~
50 mm
At the front, the permissible axle load can be taken up, at most, by the roof
luggage. The amount of the difference between occupancy with one and five
people actually utilized is only
AmV,f = 73 kg
in accordance with Figs 5.10 and 5.11.
The weight of the people sitting on the front seats is distributed approximately
equally between the front and rear axle. However, if passengers sit on the rear
seat, on average 75% of their weight is carried on the rear axle springing.
Both standard design and front-wheel drive vehicles have the boot at the back.
When they are loaded, around 100% of the luggage weight is carried on the rear
axle. This is the reason for the significantly higher load difference between the
empty and permissible axle weight of
Amv,r = 300 daN or almost 400 daN
on the rear axles of the two vehicles studied. The result would be the value
Amv,r = 400 daN for each axle side. This would correspond to a wheel force
difference of AFz,w,r = 2000 N. If we assume linear springing at a rate Cr =
20 N mm- 1, due to AFZ,w,r a path of
As = AFz,wicr = 1962/20 = 98.1 mm
would be needed. There is also a residual jounce and bump-travel path of 50 mm
in each case so that total travel can barely be less than Sr,t = 200 mm.
Figure 5.14 shows the linear rear-wheel springing of a standard passenger car.
In spite of the soft springing at a rate of Cr,pl = 18.9 N mm- 1, there is residual
travel of 86 mm or 50 mm. The frequency on a partly laden vehicle (with three
people) is nr,pl = 77 min- 1 and, with additional loading, it reduces, increasing the
comfort (the spring rate remains constant but the mass increases, see Equation
5.4). This type of favourable configuration is achieved by:
•
•
•
•
a large total spring travel (Sr,t = 220 mm)
a payload level which only corresponds to 45% of the curb weight
a long wheelbase (l = 2665 mm)
a boot that does not protrude too far at the back.
Springing
333
8
- Overall wheel travel 22( mm
kN
7
J
/
170
./
162
V
J
V
~-
Qi
(1)
./
/
"
- ...
"0
/'
131
6
.c
3:-
V
4
"7
117
1.00'"
./
103
Residual wheel trave '
~
(rebound) 86 mm
~
/'
-
./
V
I
/
L7
/
V
"0
ctl
./
..Q
ctl
01
01
...
::J
ctl
-
.. "0 '
...
...
(J)
(J)
0)
...>
0.
E
40
80
01 I--~
c:
c:
0)
0)
(J)
(J)
01
c:
0t:
N
o
0)'
(J)
(J)
ctl
ctl
f--.o.
1
~
0)
"0
+--M
0.
E
.
,-~,
(J)
(J)
ctl
0.
. c:
~ ctl
(J)
0)
c:
(J)
ctl
<{
U'l
'5
1
~
.c
0.
0.5
V
o
160
120
Wheel travel
Fig. 5.14
01
x
0)
,"
0)
0)
ctl
W
o
Residual
3
wheel travel (compression) - 50 mm
mm
200
..
Almost linear and soft rear wheel springing with large travel, measured
on a standard passenger car; rebound stop and supplementary springs are located in
the shock absorber. Figure 5.10 contains the associated wheel loads. With a loading
of five people plus luggage (427 kg), the rear wheels still have a residual compression travel of 50 mm. The springing rate (with a design weight, fT/v.r,pl = 672 kg) is
1
Cr,pl = 18.9 N mm- 1 and the frequency nr,pl = 77 min- • The manufacturer specifies
mU,r = 91 kg as the weight of the unsprung mass.
334
The Automotive Chassis
The disadvantages are the dropping of the tail when the vehicle is laden and the
associated pitch angle e (Fig. 3.137), although the danger of dazzling other roadusers, which would otherwise be a problem, can be overcome by the headlight
height adjustment, which is fitted as standard.
Shortening of the spring travel and a less pronounced squat on the tail can
be overcome with progressive springing for bump travel beyond normal ride
height. Figure 5.15 shows this type of curve, measured on a front-wheel drive
vehicle. The frequency nt,pl = 93 min-I (where there are three people in the
vehicle) points to harder springing. In spite of the very high load difference
of 399 kg, which can be seen in Fig. 5.11, the axle bump travel of only ~Sr =
76mm.
The possible loading of 500 kg represents 56% of the manufacturer's stated
curb weight (893 kg). This unfavourable ratio leads to the severe load alleviation
on the driven front wheels, described in Fig. 5.11 and to the highly loaded rear
axle with a residual spring travel of only Sl,Re =28 mm, whereas at S2,Re =89 mm,
the rebound travel is large.
The springing curves (of front and rear axles seen together) lead to the
assumption that the vehicle initially stood higher by ~s ~ 20 mm.
The vehicle having been lowered by the owner (this is assumed due to the
unusually large jounce travel) and the high load are likely to be the reasons for
the too low residual compression paths front and back.
5.4.3
Springing and cornering behaviour
5.4.3.1 Wheel load change on independent wheel suspensions
As can be seen in Fig. 1.6, the centrifugal force related to the front axle
Fc,V,f
=:
(5.11)
mm,f ay = J..LY,W FZ,V,f
acts at the llevel of the vehicle centre of gravity. The wheel force change that
arises during cornering (outside of the bend +~Fz,w and on the inside of the bend
-~Fz.w) can be approximated separately for the two axles. For the rear axle the
equation is
(5.12)
+ ~Fz,w,r = J..Ly,w FZ,v,r hv/br
The values of the front-wheel drive vehicle inserted with a permissible axle load
and with the centre of gravity height hv ~ 530 mm, the tread width br = 1425 mm
and the lateral coefficient of friction J..Ly,w = 0.7, give a force of
+~Fzwr
, ,
=0.7
X
780 X 9.81 X 530/1425
= 1993 N
The wider the tread width and the lower the centre of gravity, the smaller
+ ~Fz,w will be.
The equation for the front axle is:
+ ~Fz,w,f = J..Ly,w FZ,V,f hv/bf
(5.12a)
T
Springing
335
8
kN
7
Overall wheel travel 19 mm
J
6
11
1/
"'0
III
J
..2- I-s
/
Authorized max.
rear axle load
165
>-r
V
[7
134
7
/
108
Residual wheel travel
(rebound) 50 mm
V
o
V
./
/
Q)
:E
.... - I -
Residual
wheel travel
(compression)
28 mm
/
4
A
-~
/
59
V
85
2
./
/
Qj
104
V
Q)
Cl
C'O
Cl
ClX_
:::lC'O-o
V
Q;
E
W
C:-oC'O Q) Q)
Q)
X
Cl
Cl
(I).!::!
Q)
(I)
(I)
Q)
(I)
(I)
Q) " ...
ro
Cl..c:
c:
c:
>.....
a.
-oE~
~
(I)
C'O
a.
C'O
a.
N
'<t
1
c: ..... C'O
Q)
(I)
:::l Q)
C'O ...
(1)-
C'O
a.
0,5
rl
160
120
80
Wheel travel
0
mm
200
..
Fig. 5.15 Progressive rear wheel springing, measured on a front-wheel drive vehicle; a poor example in respect of springing design with permissible axle load. Only
28 mm residual spring travel, in association with the very high load of 494 kg, jeopardizes the driving safety (see Fig. 5.16). The associated wheel loads are shown in
1
Fig. 5.11. With a design weight of rT7v,r,pl = 474 kg, the spring rate Cr,pl is 20.2 N mmand the frequency nr,pl = 93 min- 1 •
336
The Automotive Chassis
5.4.3.2 Spring travel on independent wheel suspensions
The calculated value of 1993 N corresponds to a load change of 203 daN and
therefore a wheel load
on the outside of the bend
on the insIde of the bend
593 daN
187 daN
Assuming a permissible value of 390 daN in Fig. 5.15, this leads to a compression travel of Lls1,f = 20 mm and a jounce travel of LlSZ'f =69 mm.
5.4.3.3 Change in the height of the centre of gravity of the body
The values inserted into the formula that is valid for the rear axle,
(5.13)
give the amount by which the body is pushed upwards above the rear axle (Fig.
6.15):
LlhBo,f:= (69 - 20)/2 = 24.5 mm
On the outside of the bend, the body only tilts downwards a little, but on the
inside it moves upwards. For this reason, Lls 1 must be deducted from Llsz (Fig.
5.16). The higher the centre of gravity R rises, the greater the wheel force change
(see Equation 5.12), particularly at the axle with the greater travel difference
LlhBo,f Of f· This is usually the rear axle. A tendency to oversteer and the torque
steer effect, increase, particularly when the tyre on the outside of the bend is
highly compressed, i.e. when its stress goes far beyond the possible load capacity (Figs 2.42 and 2.52).
The difference travel
(5.13a)
--,
-r-'-\
~,Bo,r \
F,.c,U,r
Fig.5.16 When the vehicle
corners, the centrifugal forces
Fe,Bo,r relating to the body act at
its centre of gravity. If the vehicle
has too Iowa residual bump
travel, it is not able to move in
bump as much on the outside of
the bend as it jounces on the
inside. This means that the body
centre of gravity moves up from
Bo to Bo' by the path flh Bo , and
critical oversteering, which is
difficult to control, can be the
result.
Sections 3.4.5 and 5.4.3.5 and
Fig. 1.25 contain further details.
80'
\
\
80
- --
o
CXl
..c::
----------------_._-----,--------,
I
Springing
337
by which the body moves upwards in the centre of the front axle, when (as shown
in Fig. 5.13) the spring curve is also progressive, can be calculated in the same way.
The distances IBo,f and IBo,r from the centre of the front or rear axles should also
be taken into consideration when calculating the path Jih Bo , around which the
body centre of gravity R changes position (Fig. 6.1, see also Equation 6.2.4):
l:1h
= l:1h Bo ,fl Bo,r + l:1h Bo,rIBo,f
I
Bo
(5.14)
If the axle loads or the weights are known, they can be inserted into the equation:
IBo,rll = mv'rlmv't = FZ,v,rlFz,v,t and
(5.14a)
IBo,rll = mv,r/mv't = FZ,v,JFz,v,t
(5.14b)
5.4.3.4
Body roll angle on independent wheel suspensions
The body roll angle of a torsionally stiff body is the same over the front and rear
axles. It can therefore only be determined for the vehicle as a whole, taking into
consideration any anti-roll bars fitted and the body roll centres front and rear (see
Section 3.4). A two-wheel trailer, which has the springing shown in Fig. 5.15,
can therefore be used as an example.
The body roll angle <p (Fig. 1.6) can easily be calculated in such cases:
lis l r + lis z r
,
, (radian) and, for 360/27l',
br
<p=
<p =
57.3
lis l r + Jis zr
'
, (degrees)
br
(5.15)
If the values of the examples are inserted on the existing progressive springing
(see also equation 6.23), the result is
<p =
57.3
20 + 69
1425
=
3.58° = 3°35'
In the case of linear springing over the whole range, compression and rebound
travel are equal and the level of the centre of gravity does not change. The larger
travel can be easily calculated using Equation 5.10:
liSl,r
=lisz,r = liFz,w,r!Cr,pl
If the values of Fig. 5.15 are inserted the result is:
Jis r = 1993/20.2 = 99 mm
The Automotive Chassis
338
This corresponds to a body roll angle of <p = 8°. The example in the calculation
should indicate the advantage of the progressive springing.
5.4.3.5 Body roll angle on rigid axles
The springs sit on the axle housing (Fig. 1.23) and the basis of support of the
body is the now smaller effective distance bsp • Furthermore, unlike on all independent wheel suspensions, the rigid axle does not support the tendency of the
body to roll.. Therefore the shortened body roll lever arm (h Bo - hRo,r) is included
in the equation, and this comprises the level hBo of the body centre of gravity and
the body roll centre height hRo,r (rear, Fig. 1.25) (see Section 3.4.5 and Fig. 5.16)
plus the proportion of weight FZ,Bo,r of the body (FZ,Bo,r = mBo,rg, see Equation
6.6b) and the weight Fz,u,r of the axle (see Section 6.1.3). In accordance with the
laws of sta1tic, parts that are flexibly connected must be separated. Almost all
previous equations are altered by this. The wheel force change (Equation 5.12)
becomes smaller,
±/).Fz,w,r = J-1Y,W FZ,Bo,r
h Bo - hRo,r
b
+
F
rdyn
Z,U,r
Sp
br
(5.16)
and the travels ~Sl,Sp and ~S2,Sp calculated from it relate to the springs that are
further to the inside. The ratio i<p is needed to obtain the values related to the
centre of tyre contact
i<p = bJb sp
~SI,r == ~SI,Sp i<p
(5.17)
and ~S2,r = ~S2,Sp i<p
(5.18)
In line with Equation 5.15, the equation for a single-axle vehicle with a rigid
axle (for the increased angle <p' in degrees) would then be:
I
_
<p-
57 3 (~SI,r
.
+ ~S2,r)
·2
(5.18a)
l<p
br
The further out the springs can be positioned, the smaller (i.e. more favourable)
i<p becomes:; this applies in particular to the drawbar axle (see Fig. 1.60 and
Section 3.4 in Ref. [2]).
5.4.3.6 Rates on reciprocal springing
Apart from slight deviations, the spring rates with parallel and reciprocal springing are equal on all independent wheel suspensions, if we ignore the (symmetric and asymmetric bumps) influence of the anti-roll bar:
Cf or r =: C<p,f or r
The picture is different for rear (or front) rigid axles:
----,--------------'
I
Springing
339
• If the springing is parallel, the rate on the centre of tyre contact Cr is equal to
that on the spring mounting CSp,r.
• However, if the springing is reciprocal (first wheel rises as second falls), the
rigid axle takes on an inclined position (Fig. 1.21).
As Equation 5.18 shows, the differences in travel dSI,r and dS2,r are greater than
those at the springs (dss p); however, the changes in force dFZ,w,r are smaller:
With a spring rate C~,r related to the centre of tyre contact, this yields
,
dFz,w,r
C<pr=
,
dS 1,2
,
dFsp,r
=---i<p,rSSp,ri<p,r
/'2
C <p,r = CSp,r l <p,r
(5.19)
In the case of reciprocal springing, the elastic parts in the guide joints and struts
are tensioned; the actual reciprocal springing rate C<p,r is around 7% higher than
the values
C<p,r =
1.07 C~,r
(5.19a)
calculated with Equation 5.19. The equations for a rigid front axle are similar;
except the index! appears.
5.4.4
Diagonal springing
Front and rear axle trailing links and longitudinal pairs of links mostly have a
vehicle pitch pole Oforr (per side, see Section 3.11). The wheels then no longer
move vertical to the ground when they rebound and compress, but on arcs + f
around the existing pole (Fig. 3.158). The driving safety is not impeded by the
wheels moving out, to the back or the front. When the wheels compress, they
move dl in the direction of the radius-arm axes (if these are at the height of the
wheel centre or below it) and away from them as long as they are above the
wheel centres. When the body moves downwards, the axes go down with it.
The diagonal springing angle X on trailing and semi-trailing links is entered
in Figs 3.158 and 3.160. This angle .also applies to compound crank suspensions
and multi-link suspensions and to appropriately sprung rigid suspension (Figs
1.1, 1.2, 1.43 and 1.61). However, on double wishbones disposed at an angle
(Fig. 3.155) this becomes:
X = (ex
+ ~)/2
(5.l9b)
On McPherson struts and strut dampers, the change in the caster angle, Le.
is a consideration (Fig. 3.137).
d'Tk
340
5.5
The Automotive Chassis
Spring types
Two springs, four stops, two shock absorbers and one anti-roll bar usually
control the springing of a pair of road wheels, the limitation of spring travel and
the reduction of body roll inclination for each wheel-station on passenger cars
and light commercial vehicles.
An exact description of the individual components and calculations is given
in Refs [2] and [5]. Here; the intention is merely to show the arrangement of
components and what they do.
The sprilngs can be distinguished by media and materials as follows:
•
•
•
•
•
steel springs
air and gas springs
composite (leaf) springs
rubber springs
springs of polyurethane elastomer.
These last two types are mainly used on passenger car two-wheel trailers or as
additional springs parallel to the steel spring. The polyurethane is stressed in
compression and the rubber in tension.
5.5.1
AJir- and gas-filled spring devices
For reasons of comfort, the natural frequency of the body above the rear axle
should be between 10% and 20% higher than on the front axle and should not be
dependent on the load. Unlike steel springs, air-spring systems allow the natural
frequencies of the body to remain unchanged even when there are changes in
vehicle load, and this is associated with other advantages:
•
•
•
•
vibration and suspension properties not dependent on load;
simple level adjustment;
a guarantee of complete wheel travel even with a load;
compact design with large spring and shock-absorber track widths.
Depending on the design of the system, it is also possible to ensure the followmg:
• the balance of unsymmetrical loading conditions on the left/right;
• a specific effect on vehicle roll movement, and thus dynamic wheel loads, by
support of the rolling moment;
• damping, which depends on the driving conditions (speed range, longitudinal,
vertical and lateral acceleration).
Because of these advantages, air-spring systems have long found almost universal application in buses and commercial vehicles used for long-distance travel
(in which the adjustment of the height of the loading surfaces is important as
Springing
341
Fig. 5.17 Air-sprung double-wishbone axle of the Audi A6 Quattro (1997). With
axle components that are as similar in design as possible (wheel carriers made of
nodular graphite cast iron, upper transverse link and auxiliary frame made of hydroformed tubes, lower transverse link and rod-shaped link in a sheet-steel shell
design), Audi supply the driven rear axle of the A6 Quattro with an air-spring and
shock absorber 1 instead of the single-tube gas-pressure absorber with coil spring.
The air is supplied by a vibration- and noise-damped air supply unit in the rear end
2 consisting of a 280 watt compressor, air dryer and control valves. A contactless
Hall rotational angle sensor 3 is attached to the levelling-value 4 in the middle of the
vehicle for the purpose of establishing the height of the vehicle. The time to adjust
from empty running to maximum load is about 60 s and the average current
consumption with a 1% on~period is 5 W.
The system was developed by Continental AG and is supplied as a ready-assembled complete system.
well as protection of the load and road). Air-springs are also increasingly being
used in the van sector (Mercedes-Benz Vito). In the case of passenger cars, airsprings have up unt.il now only occasionally been used in comfort-orientated
vehicles (e.g. Mercedes 600) or off-road vehicles (Range Rover) for reasons of
cost; air suspension has recently been offered as an optional extra in vehicles of
the upper middle class (Audi A6 with front-wheel or all-wheel drive, BMW 5
series tourer). The traditional springs are replaced with air-springs and sometimes air-spring-and-shock absorbers (Figs 5.17 and 5.18). The front and rear
axles of the Mercedes Benz S class W220 are for the first time being air-sprung
as standard (Figs 1.39 and 5.19).
Citroen has installed gas springing - where the forces are transmitted
hydraulically via oil pressure, in its larger models since 1953 - as its so-·called
hydro-pneumatic springing.
342
The Automotive Chassis
1
4
2
Bump
I
3
_ 8 . 0 bar
-._·6.5 bar
---5.0 bar
......._ ......_ . . . L _ _......_....l
O...-.-'"'----..:.-._~_
o
0.13
0.26
0.39
0.52
0.65
0.78
'
(m
S-1)
1.04
Piston speed
Fig. 5.18
Air-spring-and-shock-absorber assembly and damping characteristic for
Audi A6 Quattro (manufacturing diagram of Continental AG). Air-spring 1 and shock
absorber 2 are coaxially arranged and form a spring-and-shock-absorber strut. By
means of a valve 3 connected to the air-springs, the air-spring pressure is used as a
load-proportional correcting variable for the continuously variable, load-dependent
adjustment of the damping force in rebound and compression, shown in the diagram
for the regions of 5.0, 6.5 and 8.0 bar.
With small excitation amplitudes (less than 3-5 mm), the dynamic rigidity of the
rubber bellows 4 reduces comfort. This hardness of response depends on the
strength of the material- durability and comfort have conflicting aims here - and the
nature of the material. The response was able to be improved by replacing polyamide
with aramide.
Fig. 5.19
Air-spring-and-shock-absorber assembly of the front axle of the
Mercedes Benz W220 series (1998). In order to reduce the inner friction which is
disadvantageous in air-sprung vehicles at small amplitudes and the dynamic hardening of the hose reel bellows with two X-shaped layers of textile fibres, Daimler
Chrysler uses single-layer rubber bellows with not very thick walls (1.6 mm) in the
Mercedes Benz W220 series. As a result, the hardening related to the total spring
rate of the system is reduced from 80% to 25% and the quality of the response
considerably improved at small excitation amplitudes. The limited capability for
expansion between the fibres of the single-layer rubber bellows requires a small gap
Springing
between the rolling piston and outer guide
and protection against fouling because of the
lack of self-cleaning of the original bellows;
the latter is ensured by the polyurethane
folding sleeve with labyrinth ventilation. The
radial deflection of the lower strut point
which occurs during wheel travel and particularly steering movements must be taken
into consideration when establishing the
position of the rolling piston and outer guide.
On the front axle, this occurs by means of
the deepset guided connection of the rolling
piston to the shock absorber by way of a
cardan soft sliding bearing. The pressure for
the system is supplied by a compressor with
an output of 400 W. In order to ensure short
filling times, a 4 e central pressure tank
whose tank pressure of 16 bar clearly lies
above the system pressure of 10 bar is used.
The weight-dependent spring rates are 15 N
mm- 1 on the front axle and 17 N mm-1 on the
rear axle; this results in natural body frequencies of 0.8 Hz and 0.9 Hz. The air-spring
system controls the height of the vehicle
regardless of the load, while taking into
account the driving speed (reduction of 15
mm above 140 km h-1 ), lifting of 25 mm
caused by a bad road surface and maintenance and wheel change positions. The
adjustable damping system operates automatically, taking into account the driving
conditions which are established by means
of the driving speed, ABS signal, bodyacceleration, steering angle signal and braking
pedal signal. The following shock absorber
characteristics can be produced by switching
on the valves which are flange-mounted
on the single-tube gas-pressure shock
absorbers:
Step 1: rebound and compression
soft - maximum driving
comfort.
Step 2: rebound soft compression
hard - increased damping.
Step 3: rebound hard, compression
soft - increased damping.
Step 4: rebound and compression
hard - maximum damping for
minimization of wheel load
fluctuations.
343
344
The Automotive Chassis
5.5.2 Steel springs
The following are manufactured in steel:
•
•
•
•
leaf springs
coil springs
torsion bars
anti-roll bars.
5.5.2.1 Leaf springs
Leaf springs are subdivided into longitudinal and transverse leaf springs.
Longitudinal leaf springs are used only on rigid axles, more commonly on
commercial vehicles and trailers. Figure 5.20 contains a weight comparison
between the previously exclusively used multi-layer leaf springs and modem
parabolic springs; Figs 1.20, 1.26 and 1.37 show various designs and also the
advantages. For reasons of cost and weight, springs with only a single layer, socalled single-leaf springs, are fitted to an increasing number of passenger cars
and light commercial vehicles; Fig. 1.24 shows these on the non-driven rear axle
of a van.
Transverse leaf springs, by contrast, can provide the springing on both sides
of the axle; they were previously used in independent wheel suspensions of
a. Conventionally multi-layered leaf spring with
smoothly cut layer-ends. 14 layers; height of bundle:
140 mm; weight: 122 kg
~
-lJBilfj
b. Improved multi-layered leaf spring with pressed layerends and plastic layers in between. 9 layers; height of
bundle: 127 mm; weight: 94 kg
c. Parabolic spring with pressed layer-ends (length approx. 1200 mm) and
plastic layers in between. 3 layers; height of bundle: 64 mm; weight: 61 kg
Fig. 5.20
Weight comparison between three different commercial vehicle rear
springs with the same data, carried out by Krupp-BrOninghaus; eye distance
L = 1650 mm, spring rate Cr = 200 N mm-1 and loadability Fsp = 33 kN; however, the
designs are different.
Springing
345
passenger cars (see Section 5.2.3 in Ref. [2] and are also built into (the 1955)
Daimler-Benz-Transporter Sprinter.
5.5.2.2 Coil springs
Coil springs with a linear curve over the entire wheel travel are used on the front
and rear axles of passenger cars (Figs 5.9 and 5.14). If necessary, a certain
progression can even be achieved by using various formings of conical spring
wire. Figures 1.7, 1.15, 1.39,1.41, 1.60 and 1.81 show springs in their fitted
condition, (see also Section 2.1.4 in Ref. [2]).
5.5.2.3 Torsion bars
Cylindrical torsion bars made of round steel are used to spring the body and
as anti-roll bars (see Section 5.5.4). To transfer the springing movement, both
ends have warm-upset heads, which carry a toothed profile or a square. Ushaped brackets can also be butt-welded and can be very easily fitted to the
suspension links. Figures 1.2 and 1.63 show torsion bars in the installed
condition.
5.5.3
Stops and supplementary springs
The following are differentiated:
• jounce stops
• bump stops
• supplementary springs.
As shown in Fig. 5.9, the jounce stops limit the jounce travel of the wheels on
soft and medium-hard springing. Apart from a few exceptions, jounce stops are
found in shock absorbers or McPherson struts and strut dampers (Figs 5.26,
5.47, 5.51 and 5.55). In this case, under tensile forces, the elastic attachment
parts of the damper and the elastomer, polyurethane or hydraulic jounce stop, all
flex (see Fig. 5.31 and Section 5.6.8.1; also Ref. [5]).
Bump stops limit the bump travel; they absorb high forces over a short path.
The elastic stops can also be accommodated in the shock absorber (Fig. 5.47).
They can sit within the coil springs (Figs 1.7 and 1.13) or be fixed on the axle
housing (Fig. 1.20) or come into contact with it when the springs reach full
bump.
In comparison with the relatively flat, hard bump stop, the supplementary or
additional springs are much longer, but act softly and, as shown in Figs 5.21,
5.50 and 5.51, have a favourable springing curve and absorb high forces when
fully compressed. The parts are made of rubber or polyurethane elastomer. The
air bubbles in the elastomer enable the bumpers to be compressed by 77% where
the diameter is increased by only 35%. Like compression stops, they then absorb
a force of F 1 = 7 kN (Fig. 5.49). Figures 1.24, 1.40, 1.41, 1.60 and 5.29 show
supplementary springs in the installed condition.
Almost any springing curve can be achieved by combining a linear steel
spring with a highly progressive supplementary spring (Figs 5.9 and 5.14).
346
The Automotive Chassis
8000
N
7000
I
0
C1J'
~
0
/
4000
VI
I/
V
~ J~
.E
g'
3000
'i:
0..
en
2000
V
1000
o
o
10
20
---~
30
c::::..- v
40
50
Spring travel,s
V
60
V
70
80
9Omml00
070
•
Fig. 5.21
Supplementary spring manufactured by the company Elastogran and
fitted by Ford. The part is made of a polyurethane elastomer and remains flexible
when cold down to an ambient temperature of -40°C. The 'buckling lip', which can
be seen at the lower end, ensures a soft contact and a low initial spring rate. The
upper end is kept tight against the body within the coil spring.
5.5.4 Anti-roll bars
The function of the anti-roll bars is to reduce the body roll inclination during
cornering (Figs 1.6 and 5.16) and to influence the cornering behaviour in terms
of under- or oversteering (Fig. 5.2), i.e. increasing the driving safety. In the case
of parallel springing, the back 1 turns (Fig. 5.22) in the bearings L; the anti-roll
bar remains inactive. The anti-roll bar rate Cs,<!' on reciprocal springing, which is
important for reducing roll inclination, related to both wheels of an axle,
depends, for independent wheel suspensions, on the ratio of the wheel joint G to
the attachment point T 2 on the suspension link or on rigid axles of distances br
Fig. 5.22
The anti-roll bar is mounted
to pivot with its centre-part 1 in the
points L. The connection between leg
ends T1 and wishbones (points T2 ) is
made by an intermediate rod. The ratio
is = b/a is much greater than one and
therefore increases the forces in the
suspension links and their mountings.
Springing
347
and bs (Fig. 1.23). With the rate Cs on the leg ends T 1 of the anti-roll bar, the rate,
related to the centre of tyre contact, becomes
CS,q>
= Cs/ I'2'P
and is = b/a or bibs
(5.20)
(5.21)
The closer to the wheel the anti-roll bar operates, the lighter and less expensive
it becomes and the lower the forces that occur in all the components.
An underslung-type anti-roll bar, shown in Fig. 1.8 and used only on
McPherson struts to date, provides a solution in this direction. The connecting
rod 5, whose path is around the same size as that of the wheels, is fixed to the
outer tube 1. The disadvantage of this arrangement is the effect of the anti-roll
bar on the McPherson struts during steering.
Figures 1.12, 1.42, 1.43, 1.54, 1.56, 1.57 and 1.63 show the configurations of
normal anti-roll bars and the various ways in which they are mounted. Apart
from the body roll inclination, the cornering behaviour can also be influenced by
anti-roll bars. The following rules will apply (see Fig. 5.2 and Section 5.2.1 in
Ref. [9]):
• A front-axle-mounted harder anti-roll bar promotes the tendency to understeer
and improves the behaviour when changing lanes.
• Higher rear axle stabilization means the front-wheel drive vehicle can become
more neutral, whereas the rear-wheel drive vehicle oversteers more.
However, the anti-roll bar also has disadvantages. The more the rate CS,'P related
to the wheels increases, and the more highly the elastic parts are pre-tensioned
in the various mountings (positions L, T 1 and T2 in Fig. 5.22 and positions 17
and 19 in Fig. 1.12), the less the total springing responds wheq the vehicle is
moving over a bumpy road; the vehicle 'copies' the road. Furthermore, the
engine begins to vibrate on its mountings (especially on front-wheel drive vehicles) and front-end shake occurs. The ride comfort also deteriorates (see
Sections 5.1.3 and 5.1.2). There is also an unfavourably harder reciprocal springing when the vehicle is moving along a pot-holed road (Fig. 1.21).
5.6
Shock absorbers (suspension dampers)
Running vehicles are exposed to almost constant vibration excitation; shock (i.e.
vibration) absorbers are consequently required for reasons of driving safety and
riding comfort. These aims partly conflict, because a taut suspension prevents
wheel hopping and thus a loss of road contact, whereas a soft suspension is
supposed to reduce body vibration and thus the annoying effects of acceleration
on the occupants. The establishment of the damping force is also made more
difficult by its dependence on the driving and load conditions, so that vehicle
manufacturers usually work on the assumption of an average load (two people
and 75 kg of luggage) as well as road surface excitation which is typical for the
----------------------,---------
348
The Automotive Chassis
use of the vehicle. Electronic components such as antilock, slip and stability
control systems must have operative shock absorbers, as wheel hopping with a
transmission of longitudinal forces as a result of a brief lack of normal power,
results in wheel lock (or spin) and thus gives a false input signal to the control
system.
Together with tyres and disc wheels, shock absorbers are one of the parts of
the chassis that are most frequently exchanged for models of the owner's choice.
The owner believes that the handling characteristics of the vehicle can be
improved. This can apply, although associated with the risk of premature wear
to the stops, if the dampers also have to limit spring travel (see Section 5.6.8). If
exchanging this part causes a change to the driving, steering or braking characteristics of the vehicle, and therefore represents a danger to other road-users, in
Germany the vehicle type approval and therefore also the insurance protection
would automatically lapse.
The correct tyre can be recognized from the size marking and the ECE
index (Fig. 2.18), just as a worn profile, the depth of which is no longer
permissible, is clearly visible. The shock absorber, in contrast, is located inside
the chassis, the type marking is embossed on to it, but mostly covered by dirt
and barely legible. Furthermore, with the variety of dampers available on the
market, it is likely only to be possible to find out whether the type fitted has
been approved by the manufacturer or is serviceable for the vehicle by referring to manuals.
The fact that a visual inspection only indicates failure where dampers are
leaky, and that inspections are rarely carried out when they are in the installed
condition is likely to be one of the reasons why there are more cars on our roads
with defective shock absorbers than ones with inadequate tyres.
For more details on the various systems and their practical applications, see
Ref. [5].
5.6.1
Types of fitting
The top of the shock absorber is fixed to the body or the frame and the bottom
to a suspension link or the axle itself. Both fixing points should be as rigid as
possible, so that the shock absorber also functions at more sensitive levels.
When the wheels rebound and compress, the rebound stage and the compression stage usually come into play; in both cases vibration is dampened (see
Section 5.2).
The shock absorber should be arranged vertically; if it is at the angle ~D to the
rigid axle (Fig. 5.23), the ratio iD is included in the calculation of the damping
related to the wheel on parallel springing:
iD = l/cos
~D
(5.22)
The larger ~D becomes, the smaller the force at the wheel and the lower the
path in the damper; the ratio iD is therefore squared in the damping calculation.
In the case of reciprocal springing, the distance b D also plays a role on rigid
axles:
Springing
349
Fig. 5.23
If the
dampers are fixed to a
rigid axle at an angle,
the angle ~iD increases
with compression with
the advantage of a more
unfavourable damping in
the loaded condition.
Moreover, the further in
the dampers are positioned, the less they
prevent the body roll
movement.
.
br
10l!'=---,
b o cos
£0
(5.23)
The further inside are the dampers, the less their effective distance bo in comparison with the tread width br. The ratio io,l!' for reciprocal springing increases,
leading to reduced body roll damping, the effect of which is unfavourable,
particularly on high bodies.
The deviation of the damping position from the vertical is a disadvantage - in
the rear and side view - even on individual wheel suspension and compound
crank axles (Fig. 1.2), except that here Equation 5.22 is valid both for parallel
and for reciprocal springing. All i o and io,l!' data can be found in Section 5.3 of
Ref. [3].
When establishing the damping forces, changes in the position of the shock
absorber with wheel travel are to be taken into consideration (Fig. 1.13).
Changes in the angle of the shock absorber can result in an unwanted decrease
in the damping force with an increase in jounce. The shock absorber connection points (eye and pin bearings) must also be designed for such changes in
angle.
5.6.2
Twin-tube shock absorbers, non-pressurized
5.6.2.1 Design of the damper
Figure 5.24 shows the design principle. The damper consists of the working
chamber A, the piston 1 fixed to the inner end of the piston rod 6, the bottom
valve 4 and the rod guide 8 (Figs 5.25 to 5.28); this also takes the sealS and,
together with the piston 1, transmits any bending moments that occur through
lateral forces to the eye-type joint of the damper. The reservoir C, also known as
350
The Automotive Chassis
Fig. 5.24
Diagram of the twin-tube principle to
explain the function.
1 piston
2 cylinder tube
3 outer tube
4 bottom valve
5 piston rod seal
6 piston rod
7 protective sleeve
8 piston rod guide
9 return holes
A working chamber
C equalization chamber
5
9
8
7
-- . --
:A-
6
-
-1(t-i~~U-- t
c
ft-Jr:~~U- 2
"101-"1---3
I:LJ:!'M~"'lh-1'tFrl:1.J1
~~~-4
the equalization chamber, which is around half filled with oil, is located between
cylinder 2 and outer tube 3. The remaining volume is used for taking both the oil
volume, which expands when it warms (temperatures up to +120°C are possible
and briefly up to +200°C where viton seals are used), and the oil volume which
is evacuated by the entry of the piston rod.
The level of the oil column in the equalization chamber must be at half full to
avoid air being sucked into the working chamber through the bottom valve in the
....- - - - 6
r-----9
,---u
o
$2
R2
5
..\---.-E
·~----G
::::'='-"1---$,
:.=:-.-.--A
1---3
.....--C
Fig. 5.25
Guide and seal set used by
Sachs Boge in series production of twin
tube dampers. The finished damper is
closed by rolling the outer tube 3 around
the edge U of the piston rod guide 8.
Springing
351
2
4
12
11
10
B,
K2
K3
9
K,
Z,
5
82
1
6
3
7
8
Fig. 5.26
Valve combination used by Sachs 80ge on twin-tube dampers (item 1
in Fig. 5.24).
1
2
3
4
5
6
7
8
9
10
11
12
K,
K2
K3
81
82
Z1
piston
piston rod
nut
cylinder tube
piston ring
valve disc
coil spring
nut
}
{ (cover plate)
rebound valve Y-spring
washer
top out (stop)
sealing edge 1
sealing edge 2
sealing edge 3
drill hole
channel
spigot
case of extreme driving conditions. This could occur if the piston rod extends
fully at extremely cold temperatures (-40°C).
The inclined position of the shock absorber in the vehicle, which leads to the
oil level in the equalization chamber C falling on one side, must also be considered. There is therefore a limit to the amount by which the angle ~D can deviate
352
The Automotive Chassis
F
F
Fig. 5.27 The damping curve can be progressive (top), linear (centre) or degressive (bottom). The curve shape and diagram shape are directly related. The smallest
area and therefore the lowest mean damping is in the diagram of the progressive
curve, while the largest area is that of the degressive damping. The shape of the
damping curve can be expressed in an equation by the exponent n:
Fo = ko v'b
c
A
1~-2
~~V"""""""~/~I~~H- 3
B2
5
---J-.I,:~~..>..I--+-''''''''''''
---t-:1.-~~rl-~"""'"
"-+-~I-f-- 1
1i\1--_
B1
84
Fig. 5.28
Bottom valve of the Sachs S27, S30 and S32 twin-tube dampers.
Springing
353
from the vertical (Fig. 5.23); a maximum of 45° may be reached in the fully
compressed condition.
5.6.2.2 Function
When the wheels, are subject to bump motion, the damper shortens, piston 1
moves down and part of the oil flows out of the lower working chamber through
the valve II into the upper one A (Fig. 5.24). The volume corresponding to the
immersed piston rod volume is thereby pushed into the equalization chamber C
through the valve IV in the valve body 4. This produces the main forces necessary for the compression damping and only if this does not suffice can the valve
II on the piston valve become effective.
As Fig. 5.26 shows, the valve II consists only of the Y-spring 10 loaded covering plate 9.
When the axle rebounds, there is overpressure between the piston 1 and
piston guide 8. As this happens, the main oil volume is pushed to the adjustable
valve I, which causes the jounce damping. The minor fluid volume is squeezed
through the gap between the guide and piston rod, indicated as SI in Fig. 5.25,
and the comer channels E and G. If the rod extends, this leads to a lack of oil in
the working chamber A. The missing volume is sucked from the chamber C (Fig.
5.24) and flows through the valve III, which is also only a simple return valve.
The oil pulsing back and forth between the working and equalization chamber is
cooled on the outer tube 3.
5.6.2.3 Air vent and volume equalization
Twin-tube shock absorbers have to be air vented, because air bubbles can form
in the working chamber - unavoidable in this type of damper. This happens
when:
• the damper is stored or transported horizontal prior to installation;
• the oil column in the working chamber falls when the vehicle has been standing for a long time;
• the shock absorber cools at the end of a journey, the oil in the working chamber contracts and air is sucked through the piston rod and rod guide.
Without special measures, an air pocket would arise and, particularly during cold
weather, unpleasant knocking, known as 'morning sickness', could occur.
Designers must ensure that the oil reaching to the top of the working chamber
cannot flow back into the equalization chamber when the vehicle is standing
and, furthermore, that fluid fills the space that has been freed as the oil has
contracted. Sachs Boge solves this problem with the angular ring 5, shown in
Fig. 5.25, and several channels E and G, disposed at a right angles and pressed
to the outside of the rod guide. Ring 5 creates the reservoir R2 from which the
oil can flow back via the two channels as it cools.
Another advantage is that the air that has been captured inside the working
chamber can escape better. Channels E and G are used for evacuation in such
cases; the air cushion quickly dissipates through the channels as a result of wheel
movements. The angular ring also prevents the oil jets, which shoot from the channel E as the piston rises, from hitting the outer tube 3 directly and foaming up.
354
The Automotive Chassis
As the piston lifts, over-pressure arises above the piston, which also pushes
some oil out upwards through the gap S \ (between piston rod and guide) and the'
comer channels E and G. This small amount lubricates, amongst other things,
the rod, collects in the reservoir R 2 and flows through the ring gap S2 (formed by
the angle ring 5 and the outer tube 3) back into the equalization chamber C. It is
then cooled in the tube 3 by the wind blast of the moving vehicle. Ring gap S \
as well as the size and number of the transverse channels G nevertheless represent a constant by-pass and their cross-sections must be considered when designing the orifices in the piston.
When subjected to compressive forces, the piston rod moves in, displacing a
certain volume and thereby creating over-pressure in the working chamber A, i.e.
in the compression stage oil is also pushed through the gap S\ and the channels
E and G and cools down on the outer tube 3 as it flows back.
5.6.2.4 Rebound valve
The rebound valve in twin-tube shock absorbers is generally a combination of
constant orifices and bores closed by spring-loaded valve discs (Fig. 5.26).
Piston 1 is fixed by the nut 3 to the lower end of the piston rod 2. Sealing to the
side of the cylinder tube 4 is provided by the piston ring 5, the mid-centring of
the piston by the spigot Z\. The valve consists of the valve disc 6, which is
pushed by the coil spring 7 against the sealing edge K\. The valve spring force
is regulated with the nut 8. Bypass or advance opening sections whose areas of
passage ensure a constant flow are impressed between the sealing edge K3 and
the top cover disc. As the piston rises, oil flows through hole B \ in order to
bypass the constant flow as well as the actual valve after the rising of the valve
head.
The height of the jounce damping is determined by several factors:
• the number and cross-sections of the impressed advance openings and (see
Fig. 5.25), the gap S\ between piston rod 6 and the hole in the guide A, as well
as the ventilation ducts E and G at a low piston speed;
• at medium speeds by the aperture of the valve disc 6, i.e. by the stiffness and
initial tension of the spring 7;
• at high piston speeds and with the valve wide open, by the number and crosssection of the holes B \.
By combining these options, any valve curve, from degressive via linear through
to progressive curves (Fig. 5.27) can be set.
The jounce movement of the shock absorber and hence the jounce travel of
the wheel suspension are limited by the jounce stop 12 which sits on the supporting plate 11; see Section 5.6.8.1.
The oil first flows through outer duct B2 in the direction of the pressure and
then lifts valve plate 9. This thin plate which only serves as a non-return valve
is movable in an axial direction and normally provides a seal along edges
K 2 and K 3 • The pressure force is applied by the soft star-shaped spring which
is attached between valve plate 9 and supporting plate 11; the latter also
serves as a stop and prevents too wide opening of the valve at high piston
speeds.
I
I*
i
I
iI
Springing
355
5.6.2.5 Compression stage valve
Parts 9 to 11 shown in Fig. 5.26, which sit on top of the piston, are simply a
check valve, as described at the start of Section 5.6.2.2; the bump damping
forces are primarily produced by the compression valve in the bottom of the
damper (part 4 in Fig. 5.24). Figure 5.28 shows a section through the configuration fitted by Sachs in shock absorber types S27, S30 and S32. The actual valve
body 1 has the holes B I, through which oil is sucked when the piston moves
upwards as the wheel jounces and the volume of the extending piston rod must
be replaced. The covering disc 3 loaded by the conical spring 2 lifts.
The piston rod has a diameter of 11 mm on passenger car and light van
dampers; the small cross-section area of only 95 mm2 must provide the fluid
displacement which then produces the bump damping (in comparison 478 mm2
is available for the jounce stage, corresponding to a 27 mm piston diameter
minus the rod cross-section surface).
When the piston rod enters, the bump stage valve is charged by the displacing oil. This valve consists of the set of spring washers 4, the upper washer of
which has the grooves S4 as a constant orifice. The required setting can be
achieved by nleans of the diameter of the hole B2 , the number and thickness of
the spring washers and the size of the by-pass grooves S4.
However, the constant by-pass has the disadvantage that when the vehicle is
standing, the oil in the working chamber A, which is at a higher level, can flow
back into the equalization chamber C. If the vehicle moves off again, after it has
travelled a certain distance this equalizes out, although it may be linked to a
certain unpleasant knocking noise, known as 'morning sickness'. Until the air
bubble at the top of the working chamber escapes, when the wheels rebound, the
oil is drawn suddenly against the piston guide. To avoid the noises this causes,
Sachs has added the anti-communication valve 5. This is upstream of the spring
washers 4, covers the holes B 2 and therefore prevents the oil from flowing back.
The compression damping curve arises through the interplay of the bottom
valve with the opening at the front of the piston S4 shown in Fig. 5.26 and the
check valve 9 on the piston. There are also the air vent channels E and G shown
in Fig. 5.25 and the nozzle clearance SI between piston rod and guide.
In order to prevent the quantity of oil being pushed out through the bottom
valve during compression and hence the introduction of the damping force, the
bottom valve must oppose the oil to be displaced with a fairly high level of resistance as the non-return valve 9 located on the piston opposes the quantity of oil
which flows through the piston.
5.6.3 Twin-tube shock absorbers, pressurized
The most economical form of damper design is the one that operates on the nonpressurized twin-tube system. Where certain vehicles or chassis conditions make
it appear sensible or necessary to use a gas-pressure damper, the low-pressure
twin-tube shock absorber is a good solution. The additional costs remain reasonable. Because compression damping continues to be provided via the bottom
valve, gas pressure of around 4 bar is sufficient. This means that the piston rod
extension force F pi , described in Section 5.6.4.1, remains low. This makes it
356
The Automotive Chassis
- - - Mounting bush/chassis eye
. _ - - Piston seal/gland packing
- - - Piston rod guide
---Gas
- - - Piston rod
- - - Oil reservoir
- - - Protective sleeve
- - - Reservoir tube
---Working cylinder
Fig. 5.29 Low-pressure
twin-tube shock absorber of
Sachs. In the shock absorber
preloaded with a gas pressure of 6-8 bar, particular
importance is attached to the
function of the piston rod
seal, because this must
provide a secure seal in all
operating conditions. Guided
by the piston rod, the jounce
stop which comes into play
on the piston-rod guide 8
during rebound sits above
the piston valve and thus
limits the jounce travel. The
rigidity properties of the
jounce stop are particularly
important for reasons of
comfort, as the jounce movement may not suddenly be
stopped. Figure 5.56 shows a
sectional view of the pistonrod guide.
Piston valve
Fixed Istatic) valve
'----- Mounting bush/axle eye
possible to use these absorbers without problems on McPherson struts, with
correspondingly thicker piston rods; see Fig. 5.55.
The basic design, the length and dimensions of the non-pressurized and pressurized designs are the same, so it does not matter which variety is used on a
vehicle (e.g. for sports models), as no change to the vehicle is necessary.
The advantages of the low-pressure twin-tube design are:
• more sensitive valve response at small amplitudes;
• ride comfort increases;
• damping properties under extreme conditions (e.g. on pot-holed roads) are
better;
Springing
357
• hydraulic hissing noises are reduced;
• shorter lengths and less friction than monotube gas pressure dampers - as the
required gas reservoir is not accommodated in the cylinder tube, but between
cylinder tube and tank reservoir tube;
• the shock absorbers remain functional even after loss of gas.
Unlike the unpressurized design described in the previous section, the oil reserve
(or equalization) chamber is also preloaded to 1/3 with gas of 6-8 bar with a
pressure-loaded twin-tube shock absorber (Fig. 5.29). As the gas chamber is not
located in the cylinder tube, as is the case with the single-tube gas-pressure
shock absorber, twin-tube shock absorbers have the advantage of a particularly
short length.
5.6.4 Monotube dampers, pressurized
5.6.4.1 Design and function
The design, used almost exclusively today, with a separator piston (position 1)
can easily be explained on the basis of the schematic diagram in Fig. 5.30. At the
top is the evacuation chamber 3, which (as in the twin-tube system) must absorb
the volume equalization by the oil warming and the volume displaced by the
piston rod. Gas and oil are separated by the piston 1, which seals off the actual
working chamber 2.
The damper piston 5 usually has a diameter of 30, 36, 45 or 46 mm and is
fixed to the piston rod 8. It carries the valves 6 and 7. As shown, the piston rod
can extend upwards or downwards (Fig. 5.31); the separator piston 1 makes it
possible to install the shock absorber in any position. If the damper cylinder is
fixed to the body or frame, the cylinder weight forms part of the sprung mass and
9
11
6
5
7
Sf="'--12
+--8
10
Fig. 5.30
Diagram of the pressurized monotube principle with separator piston (position 1),
--
--
-
-
--=-~-=~-=:-=:-~~~~
- "=----c""'-~~;;~~t~ii~~_i_~-~~iJ0-:""-ro?-~i£~m'i~~~'-;;~~~?~~-~~;~:5:-~~~~-~-=:::~~---,,-~~
--- ------------=-=-~~~~~~-~-"-~-~~~~-~----<-~-~-~~~~=~~~T~~?~~~~~~~
~~~~~~~~~~~;~~-~--- - - - -- ~-~-=-~.~~~~
- -",",
528 ± 2.5 Extended length of auxiliary spring
unten
,,,79
Compressed length 387 ± 2.5 Limit buffer 16 mm
37010.1
34 ....
35 .l~
38.5 ......
I
I>.
,...._ ..- .._ .._ .. _ .._-._..--.--._-._..- .._. -------------..-------------..-.-- ,["-"1
(I'" - - ..'\
I'
;
q
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M
El
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e 10.4 .0.'
I
,
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.
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.
__ . _ - -
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7'
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$1
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N
painted black
_ e r I e VOl>. Masc;t;ne 8151,,",
v... mls
Friction
100N max.
Zug
N
Druck N
Fig. 5.31
0.065
0.13
0.26
0.52
215.35
470.80
910.110
1730.140
130.25
210.35
345.50
660.65
\I-vi
unpainted
?/.~
After an original drawing by Bilstein of the front axle damper of the Mercedes Benz C (1997) class with a stroke of
mm, the fixed length Lfix = 246 mm, pin-type joint at the top (with a crimped spacer) and eye-type joint at the bottom; the
piston rod comes out at the top. The supplementary spring shown on the right is surrounded by a short, stable tube and comes into
contact with this and the support disc located above the piston rod guide when the spring compresses. The tube also carries the
actual plastic sleeve, which reaches up to the damper centre.
The mechanical compression stop sited above the piston is only fitted in this form on the performance models. The coil spring,
which forms part of this, helps to reduce pitch and roll movement on the body and its top is carried by the piston rod by means of a
washer. When the wheel rebounds, the washer comes into contact with the piston rod guide (Fig. 5.32). Further details are given in
Ref. [5], Section 8.3. Settings and tolerance values are shown at the bottom on the left of the figure. These are between 7.5% and
18%.
SD
= 141
Springing
359
only the light piston rod contributes to the unsprung mass. This is a reason for
preferring the installation position shown in Fig. 5.30.
When the wheel jounces, the oil flows through the jounce stage valve 6,
shown in Fig. 5.33 from the bottom to the top part of the operating chamber. The
gas pressure in th~ gas chamber 3 forces the separator piston to follow, equalizing out the reduction in volume (caused by the piston rod extending). If the
wheel goes into bump travel, the compression valve 7 is charged (Fig. 5.34) as
the dividing piston 1 moves upwards through the oncoming piston rod volume.
The entire piston surface is available for bump damping; this is then significantly
more effective than on the twin-tube system, and the valve 7 produces high
forces at lower fluid pressure - without loss of comfort - an advantage on vehicles with heavy rigid axles. The road-holding can be improved here by means of
responsive and correspondingly high compression damping.
The gas pressure at ambient temperature (20°C) is at least 25 bar. This value
is required to counteract the compression damping forces. If these exceed the
opposed force exercised by the gas pressure on the separator piston, the oil
column will rip off at the compression valve. Therefore, for a 36 mm piston
diameter, 2.8 kN are needed, and for a 46 rom diameter piston, 4.6 kN.
A disadvantage of the high gas pressure is the piston rod extensive force,
which amounts to
F Pi = 190 N to 250 N
If a vehicle has soft springing (e.g. Cf = 15 N mm- I ), where gas pressure dampers
are retrofitted, this can raise the body by
Sz
=Fp/Cf =250/15 = 17 mm
if the dampers are positioned close to the wheels. When the vehicle is running,
they warm up and, at an oil temperature of 100°C, extension force and body lift
increase by up to
F pi
:::=
450 Nand
Sz:::=
30 mm
If gas pressure dampers are fitted as standard, this influence has already been
taken into consideration by the vehicle manufacturer. If the owner subsequently
changes over from twin-tube to pressurized monotube dampers it is recommended that appropriately shortened springs be fitted.
5.6.4.2
Piston rod and rod guide
Figure 5.32 shows a section through the seal package with a piston rod guide
above the seal and therefore only slightly lubricated.
Unlike twin-tube dampers, a detachable piston rod guide (position 1), held by
the wire snap ring 2, is used to plug the damper. The g~id.e can be pushed do~n
to the second snap ring 3 and the ring 2 can then be laid mto the free groove m
tube 4. When the load is removed, the oil pushes the guide back against ring 2.
The O-ring 5 seals the rod guide to the outside and the mono-lip seal 6 to the
piston rod. The flange of this seal sits inside th~ guide 1 with its neck i.n the
'perbunane' disc 7. Intefnal pressure and clampmg load of the closure dISC 8,
--"--,---
360
The Automotive Chassis
9
2
6
7
5
8
3
4
Fig. 5.32 Seal package developed by Bilstein, which keeps the temperature
range of -40°C to +200°C demanded by the automobile industry. The outer piston
rod guide 1 has a hard-coated hole and is made of an aluminium wrought alloy (e.g.
AIMgSi 1 F 28). The piston rod 9 has the diameter d = 11- 0 .02 and the hole has the
tolerance range
d - 11
-
+0.07
+0.05
which corresponds approximately to the ISO fit D7/h7 with a play between 0.05 mm
and 0.09 mm.
which is secured to the guide, ensure that the sealing neck is also pressed against
the piston rod 9. The more the oil warms up when the vehicle is moving, the
more the inner pressure increases and the more tightly the seal is pressed on. If
a compression stop is fitted into the damper, it comes into contact with disc 8
when the wheel jounces.
The fluid seal on the pressurized monotube damper is more dependent on the
surface condition of the piston rod than on the gasket 6. The rod is therefore
manufactured with particular precision. In passenger cars and light commercial
vehicles monotube dampers made by Bilstein Ltd, the rods have 11 mm diameter
and are made of the heat-treatable Ck 45 QT steel. The strength properties are:
R m = 750 N mm-2 to 900 N mm-2 , R e ~ 530 N mm-2 and As ~ 6%
The surface is raised by induction hardening to a Rockwell hardness of 58+2
HRC and is then ground to achieve a roughness depth of R t = 0.8 f.1m to 1 f.1m.
A hard chrome layer over 20 f.1m thick, subsequently applied, raises the surface
hardness to 70 + 2 HRC and the subsequent super-finish treatment reduces the
roughness depth to the value R t = 0.2 f.1m needed for the seal.
5.6.4.3 Pistons and valves
Due to the equalization chamber being above the working chamber, the monotube
damper is longer than the one operating in the twin-tube system. To minimize this
Springing
361
Fig. 5.33 Space-saving compression stage
valve with spring plates and a supporting
washer found on almost all monotube
dampers. If, as shown, the piston rod moves
upwards, the lower valve is achieved. The
piston ring shown j'n the illustration is used to
prevent any unwanted flow in the gap
between this and the cylinder walls.
Fig. 5.34
If the piston rod moves
upwards, the spring plate valve for the
compression stage under the piston moves.
disadvantage, the separator piston 1 (Fig. 5.30) is hollowed out in the centre and
a flatter working piston fitted (flatter than the one in the twin-tube system). Flat
plate valves are also used.
When the piston rod extends, the oil flows past the compression valve at the
top through diagonal holes to the rebound stage valve (Fig. 5.33). Both thickness
and number of valve plates, as well as the support disc diameter do and the
amount of the constant orifices K d , are critical for the level of the damping
forces. The constant by-pass is created by a bottom valve plate on the compression valve (Fig. 5.34) which is smaller in diameter and does not completely
cover the inclined holes. Unlike in the twin-tube system, when the piston enters,
362
The Automotive Chassis
Fig. 5.35
Unshielded holes in the piston correspond to
a constant flow, also known as the advanced opening
cross-section or by-pass. On the monotube system they
give the highly progressive damping curve shown in Fig.
5.36. The compression and rebound forces are the same
size and have very high terminal values.
its larger diameter valve plates are charged by the entire oil column; this causes
much more intensive damping and prevents the wheels from oscillating - without reducing the ride comfort.
In all monotube dampers, the characteristic of the damping curve is determined exclusively by the valves on the piston and the holes.
If these just have constant orifices (Fig. 5.35) there is a highly progressive
curve shape with high forces (Figs 5.36 and 5.27 top) on both the compression
and the extension side; this also applies when there is a by-pass between the
piston and cylinder tube, i.e. if the piston ring were missing, or as is the case in
several variable dampers, if a by-pass nut is fitted to the cylinder tube (see Fig.
5.57).
Fig. 5.36
Highly progressive damping curve achieved by holes in the
piston or a gap between the piston and
cylinder wall.
Springing
Fig. 5.37 Spring-loaded valves over large
holes give a degressive damping curve. The
forces in the compression and rebound side
can be set to different levels. The piston ring 3
prevents an additional by-pass.
363
1
3
2 ~-.."",
Pre-tensioned valve plates over large holes (Fig. 5.37) cause the curve to take
on a degressive shape with the additional advantage of being able to set different forces on extension and compression sides (Fig. 5.38). At higher piston
speeds these only increase a little. The linear curve shown in Fig. 5.27 is
achieved either through low pre-tensioned valve plates or by using a combination of constant orifices and spring-loaded valve discs (Fig. 5.26).
5.6.4.4 Advantages and disadvantages
The pressurized monotube damper has a series of advantages over the non-pressurized twin-tube damper:
• good cooling due to the cylinder tube 11 (Fig. 5.30) with direct driving air
contact;
• a larger piston diameter is possible with the same tube diameter (e.g. 36 mm
instead of 27 mm), reducing the operating pressures;
• the compression stage valve 7 sits on the piston 5 and is charged by the
entire oil column;
• the oil level in the oil column does not fall as it cools, so no 'morning sickness' occurs (see Section 5.6.2.3);
Fig. 5.38
Degressive curve with
different force levels on the compression and rebound side, achieved by
spring-loaded valves (see also Fig.
5.27).
F,
364
The Automotive Chassis
• due to the pressurized oil column, the oil cannot foam, resulting in good damping of even small high-frequency vibrations;
• where there is a separator piston, the installation position is not restricted.
The disadvantages are that the high degree of manufacturing precision and the
essential gas seal lead to higher costs. Furthermore, the greater space requirement can amount to over 100 mm in the stroke length.
As a result of the sometimes considerable pressure preloading (25-30 bar),
the forces acting on the seals are greater; this results in unwanted friction which
reduces the response properties of the shock absorbers.
5.6.5
Monotube dampers, non-pressurized
Non-pressurized monotube dampers generally have a piston of only 20 or 22
mm diameter, an 8 to 9 mm thick piston rod and therefore absorb correspondingly lower forces. They are used as:
• engine vibration dampers (see Chapter 10 in Ref. [5])
• driver seat dampers
• steering dampers (see Section 4.5).
The first two designs are installed vertically and it is only necessary to fit a
compression valve (Fig. 5.28) instead of the separator piston (Fig. 5.30). As in
the twin-tube system, this ensures the necessary back-pressure when the piston
rod enters. The equalization chamber is above the working chamber and is
around half-filled with oil and air; the two media could mix if there were no
separator part, which is common on engine dampers.
Steering dampers must not have any extension force at the piston rod, otherwise the steering would be assisted in the compression direction and pulled to
one side. The dampers are fitted in a lying position, so only non-pressurized
monotube dampers (where the oil and air are separated) can be used.
Figure 5.39 shows a standard design, on which the flexible hose 1 performs
this function; it is fixed by rolling the outer tube 3. Part 3 is bevelled off on both
sides and presses the hose into pointed grooves of the cylinder tube to provide a
good seal. At the same time this measure prevents displacement when the vehicle is moving. When the piston rod 17 moves in, the oil flows through the two
apertures 4 of the valve in the valve body 5 and lifts the valve plate, which is
loaded by the spring 7; this produces part of the compression damping.
The area between the protective tube 3 and the hose 1 acts as an equalization
chamber. The hose 1 flexes when oil flows through the hole 9. As in the case of
all monotube dampers, the damping valve unit (consisting of the rebound stage
and the actual compression valve) is situated on the piston 10 (Figs 5.33 and
5.34). The piston ring 11 seals this off to the cylinder tube 2. The piston rod
guide 12, seal 13 and support disc 14 sit between the two rolled-in grooves; the
longitudinal hole in the guide acts as a pressure equalization. The eye-type joints
15 and 16 provide the installation.
The advantage of this design is the short length; increasing the stroke only
I
I
4
6
5
1
3
9
16
11
10
2
17
12 13 14
15
Fig. 5.39
Section through the Stabilus steering damper used on passenger cars and light vans, with its equalization chamber
consisting of the elastomer tube 1 and the upper part 8 above the working chamber. The piston lOis 20 mm and the rod is 8 mm in
diameter.
22
18
19
20
21
17
8
!;t:::::'~ittB
_ ·1
23
vrMHI!
I
-_.
r:H~
~
Fig. 5.40 Stabilus compact steering damper with pin-type joints on both sides (position 22 and 23), butt-welded equalization chamber 8 and spring-loaded cup seal 20.
366
The Automotive Chassis
makes it necessary to extend the tube 2 and the equalization hose 1 with the
protective tube 3. The longer tube 3 can then be a disadvantage. If it should not
prove possible to house this, an alternative design with a separator sleeve could
be used (Fig. 5.40), which has the same functional parts but also has an in-line,
welded-on, equalization chamber 8 with its inside diameter increased to 26 mm.
The coil spring 19 in flat rolled steel supported on the top 18 flexes under the
pressure of the oil displaced when the piston rod 17 enters. The opposed force
of the spring 19 is measured such that a light pressure is applied to the oil
column, but no extension force occurs. The seal between air and oil is provided
by the cup seal 21, which is inserted into the guiding part 20.
5.6.6 Damping diagrams and characteristics
The spring force is a function of the wheel travel, whereas the damping force
depends on the speed at which the two fixing points are pulled apart or pushed
together. A damper, which is subject to a constant force F o , flexes at a constant
speed over the whole stroke, whereas a spring flexes immediately, but only up to
a certain travel s" the length of which depends on the quotients of force and
spring rate CfOf f (see Fig. 5.27):
Fsp =
CfO f f
SI and Fo = ko
v~
The spring therefore stores work and usually releases it at a moment that is not
conducive to driving safety, whereas the damper annuls mechanical energy by
converting it into heat. The more energy that the damper absorbs, the hotter it
gets. In diagrams, the damping force F o appears as a function of the piston speed
•
-1
Vo III m s .
Figure 5.41 shows diagrams recorded on a standard test rig. At a constant rev
speed (no = 100 min-I), the stroke is changed step by step, but it is also possible
to keep the stroke fixed and to vary the engine and therefore the test rig speed
(Fig. 5.42). To record the damping curve, in both cases the maximum forces of
each stroke are taken and, as shown in Fig. 5.42, entered upwards and downwards on the y-axis as a function of a maximum piston speed. The equation for
calculating the individual values is
VO max
.
=
71" So
60
no
~I
(5.24)
(m s )
Damping force
Rebound
Fig. 5.41
Compression
The damping
forces on the production test
stand can be measured at
n = 100 min-1 with increasing
strokes to determine the curve.
Springing
Force - travel diagram
_
100 per min
Force - velocity curve
__.-
~~==...:..:::::..!::.:::.:...:.:.~.~=-.
25 per min
CD
. _ ._ _ .
g> Z
Compression
. - ti -.. velocity
=:::l
~
...
a. .E
[ ........1 1 - - - - (m S-1)
-t--------------~~g>.0.52 0.13
'0. -'i=:::j::::::+=4~
-
10-----
Stroke
= 100 mm
367
CD
... CD
- - - - - (/)
~ Q)
co
0.13
0.52
Rebound velocity
E
co
..
Q
eti
a.
Fig. 5.42
The maximum compression and rebound forces are taken from the individual diagrams to create the damping curve formerly known as the force speed
curve.
The value no = 100 min- I and So
VO,max
=
11'
X 0.1 X 100
60
= 100 mm gives the following speed:
-I
= 0.524 m s
Figure 5.43 shows the curve of the rear axle damping of a front-wheel drive
vehicle. Damping curve and diagram shape are closely related. A progressive
curve (Fig: 5.27, top, and 5.36) has a cornered diagram with a relatively small
surface, i.e. the actual mean damping, which is important for the springing
behaviour, is low. The degressive curves shown in Figs 5.27 (bottom), 5.38 and
5.44 have a rounded shape and so a high mean damping.
It would be correct, but too time-consuming with conventional methods, to
determine the size of the diagram's area in order to plot the resulting mean
damping over the corresponding mean piston velocity, or to oppose the mean
damping force to the mean piston speed VO,med by calculations:
VO,med
= vO, max/1.62
(5.25)
5.6.7 Damper attachments
5.6.7.1 Requirements
The damper attachments are used for fixing the damper to the frame, suspension
subframe or body at the top, and to the axle housing itself or a suspension control
arm at the bottom. Certain requirements must be fulfilled:
• maintenance-free and inexpensive to manufacture;
• angular flexibility (to absorb the movements in fixing points) with only low
reaction torque, so as not to subject the piston rod to bending stress;
• noise insulation (to prevent the transfer of road noise);
---'--r---
-----.....,..----------,-1-,_.- ---
The Automotive Chassis
368
16
fN
1.4
-z
~
Q)
()
"-
~/
1.2
/'
1.0
c:
0.8
0
.0
0.6
l/
::::l
Q)
a:
0.4
~
o. 52
0.2
-yo, max
(m 5-') 0.26 0.13 0.05
CD_______ l-/
V
t.--""'"
~/
L--V
v· . . . . .
1/''''''''''
/ /
1/
V
.....
/
V
I'
0
"C
.... '
Fig. 5.43 Rear axle
damping curve; 1 is the
standard setting and 2
that for the heavy-duty
version.
BY
V
0.05 0.13 0.26 (m 5-') 0.5 2
0.2 ~
+vo, max
~
-Ci) -
CIlZ
0.4e~
c.E~
0.6 0 0
U
kN 0.8
Mean damping force
-t--------------1I--..1.--¥-
Stroke
a-line
5
Fig. 5.44
The maximum piston speed VO. max and the greatest force F2 in the
rebound and F, in the compression direction are included in simplified form in the
determination of the wheel and body damping; both are easily measurable. The
actual form of the diagram, in this instance that of degressive damping (Fig. 5.27,
bottom) is ignored.
• precisely defined flexibility towards the damping forces - any unwanted loss
of travel in the rubber components reduces damping precision and road harshness.
On the vehicle side it must be ensured that the upper and lower fixing points
align with one another in the design (normal ride height) position (i.e. when
there are three people each weighing 68 kg in the vehicle); only in this way can
Springing
369
Fig. 5.45 The eye-type joint has 35 mm to 36 mm
outside diameter, a hole of 10+0.15 mm or 12+0 .15 mm
and is 32 mm wide. The maximum approved distortion
angles are a/2 = ± 15° and the cardan (conical) angles
~/2
= ±4°.
distortion, when the vehicle is running, and premature shock absorber wear be
avoided.
5.6.7.2 Eye-type joints
The requirements are best met by rubber joints. Figure 5.47 shows, on the top
and bottom of the damper, the type of suspension most used: the eye-type joint,
sometimes also known as a ring joint. The most common size in passenger cars
is 32 mm wide, 35 mm to 36 mm diameter and has a 10 mm or 12 mm fixing
hole with a +0.15 mm tolerance (Fig. 5.45). If compression stops are housed in
the shock absorber or if spring forces are also concentrated in the mountings,
40-60 mm wide joints may be necessary (Fig. 5.29).
The joint itself consists of a rubber bush that is in high radial pre-tension
between the outermost ring and the pressed-in inner tube. The rubber part has
beads at both sides as a measure to stop it sliding out when the vehicle is
moving. The size mostly used and shown in the illustration allows twisting
angles up to a/2 = + 15° and cardan (conical) deviations of up to ~/2 = +4°.
Greater twist angles would increase the bending moment in the piston rod and
therefore need different configurations (Fig. 5.31 and Section 5.2 in Ref. [5]).
5.6.7.3 Pin-type joints
If the same angle movement occurs in all planes at the upper or lower suspension when the vehicle moves, the design solution is to use a pin-type joint (Figs
5.46 and 5.40). This allows deviations up to +6° in all directions and consists of
two rubber snubbers, one above and one below the fixing point; the snubbers can
be separated or manufactured in one piece as a 'knob snubber'. The guide pin
usually has a cold-formed 10 mm diameter and an M 10 X 1 thread at the end.
The rubber parts are pre-tensioned via a dished washer and (as shown in the
figures) using a self-locking nut or two lock nuts. The distance between the
lower edge of washer and the damper, which is important for the function, can
be achieved using a loose spacer tube (usually of 2 mm wall thickness, i.e. 14
mm outside diameter) or by means of a rolled-in tube, as shown in Fig. 5.31.
--r--- -------r----------T'-,-----------
370
The Automotive Chassis
Ml0xl
On a pin~type joint, the preload on
the rubber parts should be ensured by a spacer
tube. Usually this has a wall thickness of 2 mm
and 14 mm outside diameter. To avoid contact in
the location hole, the upper snubber can be
centred by a washer. A self-locking nut is
frequently used for clamping the parts together
(illustration: Sachs).
Fig. 5.46
From a design perspective, it must be ensured that even at its greatest
compression and twist, the side of the pin or the spacer does not come into
contact with the bodywork or axle; this would lead to unpleasant noises and
increased bending stress. As shown in Fig. 5.46 on the upper snubber, contact
can be avoided by the use of a washer, the outer collar of which surrounds the
rubber part and grips into the hole with an edge that is turned downwards. In the
case of the lower snubber, the same effect is achieved by a vulcanized collar. The
fixing point itself can also be designed as a 'shim'.
5.6.8
Stops and supplementary springs
Installation of any end-stops means both the damper and the suspension strut
increase in length and there must be enough space in the vehicle to allow this.
5.6.8.1 Jounce stop
Figure 5.43 shows the maximum.jounce force 1.45 kN at Vo, max = 0.52 m S-I.
However, piston speeds of 3 m S-1 can occur, which lead to higher forces. If these
forces can no longer be absorbed hydraulically in the shock absorber valves,
jounce stops come into action (Fig. 5.9). On passenger cars and light commercial vehicles, the most economic solution is to locate the elastic limitation of the
jounce travel or the 'hydraulic stop' in the damper (see also Sections 5.3 and
8.3.1 in Ref. [5] ).
The other advantage is that the slight springing effect of the top and bottom
damper mountings can be additionally used to damp the jouncing wheel, and so
a relatively flat, more easily manufactured bumper 5 made of rubber,
polyurethane or Viton, polyamide or a similar plastic is completely adequate
(Figs 5.47 and 5.26). All that is needed to fit this is a groove turned into the
piston rod in which the collar on the stop disc 4 is rolled or a lock washer
inserted.
In the twin-tube system, when the piston rod is extended, the snubber 5 comes
into contact with the piston rod guide 6 which is smooth at the bottom (Fig.
5.47), or into contact with a disc 8 protecting the set of gaskets on monotube
dampers (Fig. 5.32). Figure 5.48 shows the shapes and the progressive springing
curve of the 4-12 mm high snubbers.
Springing
Fig. 5.47 Sachs S27 twin-tube damper with a
bump stop 2 carried by the piston rod 1. The
rebound stop 5 is supported on the disc 4 rolled
into a groove. The upper eye-type joint and the
outer and protective tube are also dimensioned
and toleranced.
371
32..0.3
24
2
~-3
6
~-5
!'I---4
I
~8.3
The durability of the elastic compression stop is determined by the shape and
material used. It must be able to withstand oil temperatures between -40°C and
+140°C without detrimental changes of elasticity and, in the case of sudden
loads, neither scuffing nor fissures may occur. Particle abraded off would get
into the valves and cause the damping to fail or lock.
Endurance tests carried out jointly by the respective vehicle and shock
absorber manufacturers, ensure that this type of damage does not occur. For this
reason, and to ensure wheel rebound travel is maintained, where dampers with
snubbers are used, only those authorized by the manufacturer should be fitted.
The same applies to spring dampers which, as an assembly unit, contain the
compression stop and the supplementary spring as shown in Fig. 5.51.
372
The Automotive Chassis
Rebound stop
2
3
l'
6
kN
I
II
2
o
/
/
II
J
~ y"
1
/
J
2
/
II
J
V
.Bt-~
~.B
It-'"~.---+
3
4
5
~--.
6 mm 8
2
Fig. 5.48 Sachs rebound stop in a twin-tube damper with 27 mm and 30 mm
piston diameter (types S26 and S30); shown here are body shapes and bump travel
$2 as a function of the tensile force F2 up to 6 kN. The heights 1
20 are at: position 1,
4 mm; position 2, 9 mm; and position 3, 12 mm. Snubbers up to 18 mm high are
used.
5.6.8.2 Bump stops
Bump stops act close to the end of the wheel travel and are designed to limit
bump travel without generating noise. The stop parts are housed in the top of the
protective tube (Fig. 5.47), which represents a low-cost solution and today
creates no difficulties, either from a technical point of view or in respect of the
service life. As explained in Section 5.6.8.1, the damper mountings are designed
in such a way that they can transfer relatively large forces and usually only a
slight reinforcement is necessary if additional forces occur through compression
stops or supplementary springs.
The bump stop 2 shown in Fig. 5.47 is carried by the piston rod 1; when the
wheels bottom out, it comes into contact with a cap surrounding the outer tube
and is supported - at full bump - on the steel protective tube 3. In the case of an
incorrect shape or non-wearproof rubber or plastic mixture, dust can get into the
piston rod seal and render it ineffective (Fig. 5.24). The consequence would be
escaping oil, a reduction in the damping effect and destruction of the (not always
oil-proof) bumper.
Figure 5.49 shows the progressive springing curves of three compression
stops of different length and the shape of those shown in part 2 of Fig. 5.47.
5.6.8.3 Supplementary springs
Flat compression stops barely allow any reasonably shaped springing curve.
Reduced impacts or the desire for a soft cushioning necessitate installation of
a supplementary spring made of polyurethane elastomer or a hollow bumper
(Figs 5.9 and 5.14). Figure 5.49 contains at position 4 a springing curve of a
44 mm high supplementary spring suitable for twin-tube dampers and Fig.
5.50 shows a design used for strut dampers. As shown in Fig. 5.51, this is
Springing
Fig. 5.49 Bump travel 51 on the Sachs
bump stops for the S27, S30 and S32
twin-tube dampers at forces up to
F1 = 7 kN. Configurations 1, 2 and 3 are
/12 ~ 8 mm, 15 mm or 23 mm high in
their unladen conditron and are the same
shape as part 2 in Fig. 5.47. The supplementary spring (position 4) is 44 mm
high.
7 1
373
3
2
4
kN
I
2
1
/
/
J
VL ~
o
10
./
/
20
/
/
mm
30
5 1 - - - 1.....
;56
7
kN
6
5
1._
3
9'>48
2
/
j
o
---
l.--"
20
~
40
Fig. 5.50
/'
60
5,
/
80
mm
120
~
Supplementary spring manufactured by Elastogram in Cellasto
polyurethane elastomer on the rear spring dampers of the VW Golf (III, 1996).
Material properties and shape make the highly progressive springing curve possible.
At 146 mm overall height, it can be compressed by 110 mm and accept an impact
load of over 700 kg or a force of F1 :;:,: 7 kN.
I -
374
The Automotive Chassis
17
8
6
9
'------16
_----7
10
12
18
1
4~_---...:~OO
19
r - - - - - 13
n-r~711_------ 2
_----3
_----20
40
Fig. 5.51 Sachs rear spring·
damper on the VW Golf (III, 1996)
and Vento with coil spring 1 and
jounce stop 2 visible in the crosssection. This is carried by the
11 mm thick piston rod and is
located 107 mm above the
27 mm diameter piston so that it
has an adequate minimum bearing
span in the fully extended condition; the stop ring 5 is rolled into a
groove of the piston rod.
The upper fixing is a pin-type
joint that transfers the springing
and impact forces to the body via
the large noise-insulating rubber
snubbers 6 and 7. The two parts
are drawn together by the hexagonal nuts 8 and 9; the tube 10 and
the bushes 16 and 17 ensure that
a precise preload is achieved. The
lower washer 11 comes into
contact with a wire snap ring
(which sits in a half-round groove)
and both the spacer tube 10 and
the upper spring seat 12 come
into contact with the washer. The
spring seat supports the coil
spring 1 via the elastic ring 18 and
also the polyurethane supplementary spring 4, which has a circular
bead at the bottom to take the
plastic protective tube 13.
If the suspension is in bump
travel, part 4 comes into contact
with the cap 14. This ensures the
piston rod seal is not damaged.
The cap has a groove (position 19)
through which the air in the
supplementary spring can escape
when it is compressed. The lower
spring seat is supported at three
points (position 15), which
protrude from the outer tube and
the outside diameters of which
must have a tolerance of
±0.5 mm.
To ensure the rubber part only
flexes a little under vertical forces,
the eye-type joint 16 was made
40 mm wide.
Springing
375
carried by the piston rod and comes into contact with a cap or disc when it
compresses.
5.7
Spring/damper units
The. spring~damper unit, which is described in detail in Ref. [5] Section 6.2, is a
devIce carned over from the m~tor cycle. It is used by more and more passenger
car manufacturers, not only on I.ndependent wheel suspensions, but also on rigid
a~d compound crank axles. ThIS forc~ centre, formerly described as a suspenSIOn strut, does not c~ the wheel-lIke McPherson struts, but comprises all
parts of a ~hee~ suspe~sIOn that are necessary for springing and damping. These
are the coIl spnng 1, Jounce stop 2, supplementary spring 4 (Fig. 5.51) and, as
the supporting element, the shock absorber.
The coil spring can be retrofitted and supported with rubber insulators on the
body or pre-assembled into the unit, in which case two bolts are used to fix the
entire assembly.
Installed spring/dampers can be seen in Figs 1.54, 1.55, 1.61, 1.62 and
1.77.
5.8
5.8.1
McPherson struts and strut dampers
McPherson strut designs
The McPherson strut also carries and controls the wheel. The piston rod, which
is strengthened from 11 mm to 18 mm up to 25 mm diameter on passenger cars
(and up to 28 mm on light commercial vehicles), can absorb longitudinal and
lateral forces and replaces the upper suspension link, including its three mountings. The designs, which are known today as McPherson struts, are divided into
two groups:
• those with the steering knuckle solidly fixed to the outer tube (Fig. 5.52):
• those with a bolted-on steering knuckle (Figs 1.8, 1.56, 5.54 and 5.55).
And, in terms of the damper part, into:
• those with wet suspension struts on which the damper part is directly mounted
into the carrier tube (Figs 5.54 and 5.55);
• cartridge designs in which the damper part is inserted into the carrier tube and
screwed together (Fig. 5.53).
A decision in favour of one of the solutions is mainly a question of the manufacturer's preferences, although whether the outer t~be needs ~o be i.ncl.uded for
transferring steering forces, i.e. whether the steenng arms SIt on It, IS also a
consideration (Figs 1.57,3.102,4.1,4.47 and 5.52).
376
The Automotive Chassis
Fig. 5.52 McPherson front drive axle
and suspension of an OpelNauxhal1 model.
The outer tube is press-fitted to the steering knuckle, with the steering arm 1 relatively high up.
2
3
1
Fig. 5.53
If the damping on the
OpelNauxhal1 suspension strut fails, the bolted
closure cap 2 must be undone and the shock
absorber cartridge 3 changed. The elastic ring
4, located above the coil spring, the supplementary spring 5 and the dust bellow 6 can be seen
clearly.
Springing
377
Wet suspension struts are bet~er at conducting. heat away from the damper
and, where they are detachably hnked to the steenng knuckle, offer the advanta¥e that they do not ?eed to be able to ?e dismantled and that, if the damping
~atls, t?e actual dampIng part can be easIly exchanged. This design also makes
It possible.to close, the ~trut.by means .of i~dentations in the outer tube (against
cover 5, FIg. 5.56), rollIng It (edge 6 In Fig. 5.54) or welding it to the sealing
cap.
If, as shown in Fig. 5.53, the steering knuckle is press-fitted to the suspension
strut, a screwed closure cap is necessary for exchanging the damper cartridge.
5.8.2
Twin-tube McPherson struts, non-pressurized
The suspension strut shown in Fig. 5.54 operates on the twin-tube principle; it
operates in the same way as the non-pressurized twin-tube damper (see Section
5.6.2). To have a sufficient minimum bearing span 1-0 (Fig. 1.11) in the fully
jounced condition, the jounce stop 13 has been set high. This measure, together
with the PTFE-coated guide bush 11, reduces friction.
5.8.3
Twin-tube McPherson struts, pressurized
The development of the pressurized McPherson strut has met with significant
difficulties for many years. Direct transfer of the monotube principle, as used in
the shock absorber, is not possible because of the high extension force. There are
solutions that keep the rod small and transfer the wheel control to the cylinder
tube, but these are expensive and involve high levels of damper friction (see
Section 6.4.6 in Ref. [5]).
The pressurized twin-tube system is a good compromise. Here, the oil is only
under a pressure of 6-10 bar (depending on the manufacturer) and the extension
force of the 18-28 mm thick piston rod is therefore limited.
Figure 5.55 shows a section through a McPherson strut. The spring seat 22
and the lower bracket for fixing to the steering knuckle are welded to the outer
tube 2. The piston rod 1 is solid but can be hollow to reduce the weight; the
piston has valve plates on both sides, depending on the desired damping curve,
or a twin-tube damper valve operating only in the extension direction (see
Sections 5.6.2.4 and 5.6.4.3). This can be an advantage where degressive valve
curves are requested.
The studs of the hollow piston rod are made in a special cold-forming
process; essentially, the upper one is given a hexagonal socket (?r two flat
surfaces) for holding during assembly and the lower tappet must be 011- and gastight. The rebound stop is made of plastic, is tight to ~he rod a~d transmits. the
vertical forces via the tube 14 to a zone of the rod that IS not subject to bendmg.
To keep friction low, the seal between the piston and cylinder wall is the broad
PTFE ring 15. The extension stage valve 16 is similar to that shown in Fig. 5.26.
The forces in the pressure stage are applied jointly by the valves 18 and 20 (see
Section 5.6.2.5).
378
The Automotive Chassis
Fig. 5.54
12
8
7 --=:::::;;:Gi:iiI:!
9----
10----
8
---6
11
2
13
3
L a - _ - - - 14
15 - - - - - i i W..
5
4
McPherson strut of the Fiat
Panda (1995) manufactured by Monroe.
The spring seat 2 for taking the coil
spring, the tab 3 (for fixing the steering
arm) and the bracket parts 4 and 5 to
which the steering knuckles are bolted
to the outer tube 1. The stop disc 7 is
supported on the rolled edge 6 of the
outer tube, and its two transverse
grooves 8 ensure that the supplementary spring cannot create overpressure in
the interior; this would press dirt and
deposits into the seal 9. The bush 11 is
pressed into the sintering iron rod guide
10 from the bottom and its surface
conditioned to reduce friction (to the
piston rod 12). The rod is 20 mm diameter and, in the mid-range, carries the
jounce stop 13; when the wheel is fully
extended, the minimum bearing span
(centre bush 11 to centre piston) is
120 mm.
The rod 12 is drawn in at the bottom
to provide space for the rebound stage
and check valve (see Fig. 5.26). The lowfriction ring 15 provides the seal
between the piston, which is 27 mm
diameter, and the cylinder tube 14.
Springing
379
Fig. 5.55
Low-pressure twin-tube
McPherson strut by Sachs, drawn with
the piston rod 1 fully in. The lower end 23
is drawn in and threaded to mount the
compression stage valve; the upper
tappet 24 gripping into the upper strut
mount on the wheel house has two
surfaces for retention.
24
5
6
2
9
22
14
------
18
$-
15
0
21
16
0
23
20
2
380
The Automotive Chassis
5
22
8
13--~~~N
-ro
I
I
I
3
12
11
-y---10
I
9
6
-t-I
16
17
7
1
Fig. 5.56
Rod guide and seal unit of the Sachs low-pressure twin-tube
McPherson strut.
The constant orifice on the piston, also known as a by-pass or advanced opening cross-section, is created by punched holes in the lower valve plate 21 and a
similar by··pass plate for the compressive stroke is used on the compression
valve 20. In order not to influence the efficiency of this constant opening on the
damping curve too much, the clearance area between guide bush 7 and piston
rod 1 is sealed in a controlled manner using the PTFE ring 13 (Fig. 5.56). In the
non-operative condition (as shown) it is at the bottom, but during operation, i.e.
when there is pressure in the working chamber 16, it comes into contact with the
spacer 8. This has transverse grooves of a precisely fixed cross-section which
provide the necessary ventilation.
As described in Section 5.6.2.3, when the oil cools after a journey, an air
bubble can form in the top of the pressurized twin-tube damper. On the strut
damper, the pressure in the oil column in the equalization chamber 9, together
with the inner tube valve 10, should significantly delay this. However, if at very
low temperatures pressure is reduced and the oil concentrated, the ventilation
facility becomes important again.
The internal pressure in the upper part 16 of the working chamber
increases on both jounce and bump damping. The residual oil volume flowing through the clearance between rod 1 and bush 7 collects in the high ring
channel 12 and is passed through the inclined holes 11 into the lower channel, which is formed by an angle ring and the tube valve 10. This latter part
lifts and allows the oil to flow back into the equalization chamber 9. The
chamber is around half full of oil and is pressurized by gas. The tube 10 acts
--------, ---..,.---'---'----------------------
Springing
381
as a lock and prevents ingress of gas in the reverse direction on the rod seal
3.
of.
the oil and forming of air bubbles in the valve ' known as cavitaFoaming
.
'
tlOn, IS prevented by the lOner pressure of p = 6-10 bar. If, for some reason, the
gas should escape, the damping function remains largely intact due to the existent bottom valve 20 (a designed-in safety feature). The suspension strut can be
closed by welding, or as shown in Fig. 5.56, by several beads, which press the
closure plat~ against the guide. unit 6, and press this in turn against the cylinder
tube 17, WhICh then presses the valve body 20 shown in Fig. 5.55 against the
bottom of the cold-sunk outer tube 2. The gasket 3, with the dust lip that
protrudes upwards, forms a unit with the closure plate 5, which is covered from
the top by the cap 22. The supplementary spring comes into contact with this cap
at full jounce.
5.8.4
Damper struts
Damper struts only carry the wheel without transferring vertical spnngmg
forces; there is no spring seat. However, rebound stops and supplementary
springs are arranged as in suspension struts (Fig. 1.41).
5.9
Variable damping
The dampers and McPherson struts described in the previous sections have
a fixed curve over the entire operating range that depends only on piston
velocity. It is determined by the vehicle manufacturer for a given vehicle
type and the loading condition, usually two people and 75 kg of luggage
in the case of passenger vehicles. This characteristic represents a compromise
between driving comfort and riding safety, i.e. soft and hard shock absorption.
Different loading and driving situations ideally require a damping characteristic specifically geared to these.
Figure 5.57 shows a shock absorber with impressed longitudinal grooves in
the cylinder tube. These grooves produce a by-pass flow around the shockabsorber piston in the normal state of the vehicle which corresponds to driving
conditions with a low load and a small roll angle and thus result in reduced
damping forces, that is to say, greater comfort. Outside the normal situation,
with a strongly jouncing or rebounding wheel, this bypass opening is not available; the damping force increases. Apart from the by-pass cross-section and the
length and position ot the grooves, the shock absorber can be made to suit the
individual vehicle with regard to comfort and riding safety requirements (Fig.
5.58).
Almost any adjustment of the damping characteristic is possible with the
continuously adjustable twin-tube shock absorber shown in Fig. 5.59 depending
on driving and loading conditions. As an input quantity for the control of the
382
The Automotive Chassis
Nut
Fig. 5.57
(Sachs).
Pressure-loaded single-tube shock absorber with bypass technology
Springing
383
Rebound
(tension levels/steps)
Standard
shock absorber
Sachs Vario
outside the
rim
Sachs Vario
inside the
rim
-50
100
Gain in comfort
Gain in safety
Spring
compressed
Fig. 5.58 Damping characteristic of a shock absorber with by-pass technology.
Compared with a traditional shock absorber, the damping force can be reduced for
the purposes of increasing comfort within the normal range of the vehicle, whereas
a higher damping force is made available for the purposes of riding safety outside the
normal range.
5
ulH+-----6
-iHHi----4
Fig. 5.59
Continuously adjustable
shock absorber of Sachs. The piston
valve 2 acts as a non-return valve during
rebound, so that the oil in the ring chamber 4 is displaced and directed through
the openings 5 and the intermediate
tube 6 by way of the proportional solenoid valve 1 into the gas-filled equalization chamber 7. Since the floor valve 3
closes during compression, the oil
displaced by the volume of the piston
rod must also flow over the solenoid
valve during compression. Compression
and rebound damping is largely ensured
by this solenoid valve.
1
7
2
3---H+-N
The Automotive Chassis
384
350
daN
250
200
- ---~
....-~....
150
...........: ~ ~
~V /
100
Q)
...
<.l
,'/) /
V7
50
//V/
0
'+-
Ol
C
.0..
E
0
~~ -----\'~ ~
~
ctl
0
·50
~
~
~
L---
I.---- ~
-
~
~
-------- ---~
l--
l---- '--"
L----
J.---
~
-
~
~
--'
- l----
~
-----0....;::: _ _
~
~
"
~
~"'"
~
~
~
~..,
~
-100
~
.- ~
~
~
........
~
.........
.... F=--:
------ - - ---- ----- -------,- ----~
~ ~~
~
·150
_I
r---
~ :::::::0-
-..;;;
~-
-.
r-- I---t--~
·200
~
--
-
--r---
r---,~
.......
~
I
-250
-300
1.048
0.524
0
m
5- 1
1.572
Damping velocity
I
I
--O.OA
Fig. 5.60
-+-0.3A
-.-0.6A
---0.9A
-1.2A
-1.5A
-'-1.8A
I
Damping characteristics of a continuously adjustable shock absorber of
Sachs. In the region of low shock absorber speeds, a slightly rising characteristic can
be set if low damping is required for reasons of comfort. If a high damping force is
required for reasons of riding safety, a very much steeper characteristic can be
chosen. The characteristics and height can be varied within a wide range for the
purposes of comfort or riding safety.
' I
!
Springing
385
electrically operated proportional valve, driving speed, lateral acceleration,
acceleration of the body at the front/rear, deceleration, accelerator and brake
actuation as well as steering angle are used in the same way as the bump movements on the wheels themselves, so that characteristic-controlled, adaptive
adjustment of the ~amping force takes place (Fig. 5.60).
6
Chassis and vehicle overall
6. 1
6.1.1
Vehicle and body centre of gravity
Centre of gravity and handling properties
Depending on the problem posed and the topic, the following are important variables in vehicle engineering:
• vehicle centre of gravity V
• body (sprung-mass) centre of gravity Bo
• axle (unsprung mass) centres of gravity Ur or Ur .
The distance of the centres of gravity V and Bo from the front or rear axle and
their height above ground are crucial for
•
•
•
•
•
•
braking and acceleration capacity
calculating the climbing ability
designing brake systems and four-wheel drives
designing body centre of gravity and aspects of vibration stability
driving stability investigations
determining mass moment of inertia.
Low centres of gravity are always desirable, as they are associated with fewer
driving dynamic problems and increased vehicle performance during cornering
and braking, but in practice the design options are relatively restricted.
The position of a vehicle centre of gravity V and the body centre of gravity
Bo is highly dependent on the load; when people get into the vehicle or luggage
is loaded in the boot or onto the roof, the centre of gravity changes vis-a-vis the
empty condition, both in the longitudinal direction (x-axis) and upwards (zdirection). The body lowers when it is loaded, i.e. its centre of gravity Bo drops.
The centre of gravity of the people and, in particular, that of the luggage carried
on the roof, is higher than that of the body so the end result is usually a higher
overall centre of gravity V (distance hy , Fig. 6.1).
'-.....,.---'_.,------
Chassis and vehicle overall
387
ISo r
Iso f
mV,f
I
Fig. 6.1
Designation of the paths for determining the centres of gravity V of the
overall vehicle and 80 of the body. The centres of gravity Uf and Ur of the front and
rear axles can be regarded as being in the centres of the wheels.
Details on these problems are dealt with exhaustively and further simple ways
of ascertaining centres of gravity are given in Chapter 1 of Ref. [3].
6.1.2
Calculating the vehicle centre of gravity
Calculating the position of the centre of gravity is likely to be possible only with
great difficulty and considerable effort. If the vehicle and all its individual
components are shown on a computer in the form of a digital model including
body surfaces and properties (digital surfaced model), modern CAE tools make
it possible to calculate the position of the centres of gravity of the components
and the whole vehicle.
It is much simpler to determine the position experimentally by weighing.
For this, both the empty vehicle should be observed and when it is occupied
by two or four people, approximately 170 cm tall and weighing around 68 kg.
6.1.2.1
Centre of gravity distance to front and rear axle
Figure 6.1 contains the paths and angles necessary for calculating the centres of
gravity and Fig. 3.3 the position of the coordinate system. When the vehicle is
weighed, it must be standing on a completely horizontal plane and with each
axle on a weighbridge. So as not to distort the weighbridge, it must be possible
388
The Automotive Chassis
to turn the wheels freely. The weighed front axle load mV,f and the rear axle load
mV,r give the total weight mv,t of the vehicle:
mV,t
=
mV,f
+ mV,r (kg)
(6.1)
The balance of moments around mV,f or mv'r, in conjunction with the wheelbase i
in the longitudinal direction, gives the centre of gravity distances lr to the front
and ir to the rear axle:
mV,f l - l
· -- mV,r
lt
- - l', l r- - mv,t
-
lr
(6.2)
mv,t
If the lateral distance of the centre of gravity (y-direction) from the vehicle
centre-line is required the wheel loads must be weighed to be able to calculate
first of all the lateral offset of the centres of the front and rear axles from the
centre-line via similar equations made up from the rear view, and then similarly for the vehicle centre of gravity from the top view (see Equations 5.14
and 6.24).
6.1.2.2
Centre of gravity height
To calculate hv , first the front and then the rear axle must be lifted as high as
possible (by the amount h) with an elevating mechanism (autohoist, jack, crane),
with the other axle standing in the centre of a weighbridge (Fig. 6.2). The
following would need to be ensured:
• The vehicle must be prevented from falling off by inserting wedges from the
outside on the axle to be raised. The brake must be released and the gearbox
must be in neutral. It must be possible to turn the wheels on the platform
easily; the platform would otherwise distort and the result be imprecise.
• The wheels are held still on the centre of the platform, the vehicle forward
movement must be even when the vehicle is raised, in order to prevent wrong
measured values as a result of different force application positions on the horizontal surface.
• If the change in axle load during lifting is measured by means of a crane over
a load cell, it is possible to ensure that the direction of lifting is completely
vertical.
• The vehicle should be in the on-road condition, i.e. full tank, tools, spare
wheel, etc. (as per the curb weight, see Section 5.3.1).
• Both axles must be prevented from compressing or rebounding before the
vehicle is raised. The locking device must be of an adjustable variety so that
the amount by which the body sinks when there are two or four people and
luggage in the vehicle can be taken into consideration.
• To eliminate tyre springing during the measurement, it is recommended
that the tyre pressure on both axles be increased to 3.0 to 3.5 bar.
Mathematical observation of the measurement is as follows (Fig. 6.2):
h/l = sina
_ _- -
...L'
Chassis and vehicle overall
389
Display
in kg
Fig. 6.2
Vehicle on a weighbridge with forces and paths for deriving the equation
for vehicle centre of gravity height hv included.
v
The angle a is known; but hv = h + rdyn is sought, whereby
hv= lilrltan
Ct
To be able to determine lilr, the equation of moments produced around the centre
of the front axle is set up:
mv'1
(lr + lilr) cos
Ct
= (mv,r + lim) l cos
a
Eliminating cos a
lil r =
(mv.r + lim) l
'
mV,r
- lr, whereas If = - - l
mV,1
mV,1
therefore,
lim
lilr =-- l, hence h
mv'1
l lim
v= mV,t tan
and
a
The Automotive Chassis
390
1
dm
hy = - mv,t tan
+ rdyn
Cl
(6.3)
In Equation 6.3the angle Clcan be expressed through the easily measurable vehicle stroke height h and so the equation can be simplified:
hy --
1 dm
-- --
mv,t
h
(1 2 - h2 )· 1/2 + ,...dyn
(6.4)
With dm/h or dm/tan Cl there is a constant in the equation. When it is weighed,
in each instance, only the changes caused by the vehicle lifting on one side,
namely dm and the raised dimension h, need to be determined. The other values
such as wheelbase I, vehicle weight mv,t and the dynamic rolling radius rdyn
remain the same. The centre of gravity height is required for calculating various vehicle conditions, i.e. for the travelling vehicle, so the dynamic rolling
radius rdyn of the tyre must be added to h~ and not the somewhat lower static
rolling radius that only applies to the standing vehicle. In accordance with
Equation 2.2, rdyn must be calculated from the rolling circumference CR (or CR,
dyn). The values of CR can be found in Fig. 2.15 and in Ref. [4].
As Equation 6.3 shows, the sensitivity of error is very large for small lifting heights (small values of Cl). Extensive tests have shown that exact, reproducible results can only be obtained with large lifting heights. The vehicle
should consequently be raised to the maximum possible lifting height several
times. Measurements of intermediate values at smaller heights can be
dispensed with.
The representation of axle load differences as a function of the lifting height
shown in Fig. 6.3 makes it possible to identify possible outliers which are not
taken into consideration in the evaluation. The assessment in the form of a linear
regression is computer-assisted, so that information about the accuracy of the
established result can also be provided.
6.1.2.3 Ratio i ut for the empty condition
The known centre of gravity height hv,o of the empty vehicle can be compared
with the empty height hut of the unladen vehicle. For the passenger car, this
would be hUI = 1380 mm, so that the ratio would be
i U1
=hv,o/h ut =0.377
If the centre of gravity height hv,o of a four- or five-seater passenger car is not
known, it can be judged using i u1 :
hv,o
= (0.38
+ 0.02) h u1
(6.4a)
6.1.2.4 Influence of loading
The value hv,o applies to the curb weight; when the vehicle is laden, the centre of
gravity generally moves upwards, i.e. the path hy increases, unlike the vehicle
Chassis and vehicle overall
391
x = Raised front axle
o = Raised rear axle
70 61'~==~-I----T----I---I--.==:::r-~
kg
60 59
~eadings
5052~==~_~~
te
~~
--i~
0.2
0.3
40
<J
30
20
10
0.1
tan a
0.4
0.5
...
Fig. 6.3
The measured axle load differences ilm are entered separately for the
front and rear axles as a function of tan a, depending on which axle was on the
weighbridge. A straight line, determined by linear regression and which must go
through the origin, can be used for the most precise possible determination of the
quotient ilm/tan a.
height, which reduces. The amount by which the centre of gravity of the vehicle
as a whole rises when there are two, four or five people in it, is a question of the
spring rate on the front and rear axles, the seat heights and the weights and sizes
of the occupants (Figs 5.12 to 5.15). The following'can be an approximate figure
for the centre of gravity height hv,pl (index pI = partial loaded or partly laden):
hv,o + flh v
two people
flhv'2 :::: + 12 mm
flhv,4 = -8 mm to +29 mm
four people
hv,pl =
(6.4b)
A fifth person on the rear seat or load in the boot causes the body to go down,
so the overall centre of gravity sinks (Fig. 6.4).
6.1.2.5 Roof load
Roof load will raise the vehicle and body centre of gravity. Section 1.3 in Ref.
[3] contains details.
392
The Automotive Chassis
6.1.3
Axle weights and axle centres of gravity
If, instead of the height of the centre of gravity hv of the vehicle as a whole, the
height hao of the body centre of gravity is required, it can be determined by
assuming that the centre of gravity of the unsprung mass mU,f (front) mU,r (rear)
is approximately at the centre of the wheel, Le. at the distance of the dynamic
rolling radius rdyn to the ground (Figs 6.1 and 6.5). Furthermore, their weight
should be known, determined by weighing or calculated by approximation.
lm,f mV,1
mur=--, 1 + im,f
(6.4c)
lm,r mV,r
mUr=---
,
1+
(6.4d)
im,r
The following approximate values can be included in the equation:
front axle
non-driven rear axle
driven rear independent wheel suspension
driven rear rigid axle
im,f ~
im,r ~
im,r ~
im,r ~
0.12
0.13
0.14
0.22
A passenger car which has a front axle load mv,r = 609 kg in the unladen condition can be used as an example:
mU,f
=
0.12 X 609
- - - - - = 65.3 kg
1 + 0.12
1 + im,r
lm,r mv,r
Section 5.2 and Ref. 3 (also Section 5.2) contain further details.
6.1.4
Body weight and body centre of gravity
Taking into consideration both axles, the body weight is:
mao
= mV,1 -
(mU,f
(6.5)
+ mU,r)
and the distances of the centres of gravity to the axle centres shown in Fig. 6.1
become:
lao,r
=
mao,r
l; lao,r
mso
=
mao,f
l
=
l - lso,f
(6.6)
mso
where mso,f and mSo,r are the proportions of the body weight over the front or rear
axle:
Chassis and vehicle overall
393
Fig. 6.4
Measuring sheet with values of a passenger car entered with additional
information on size and weight of the people in the vehicle during the measurement.
Source: Technical laboratory, Polytechnic of Cologne.
Vehicle
Year of manufacture
Passenger car
1994
2570 (mm)
296 (mml
195/65R14
M
Passengers in front
I
(dvn
Tyres
State when measured
o
Empty'
2 passengers
4 passengers
Size (em)
183
180
Weight (kg)
72.5
68,2
2
4
Passengers in rear
Weight (kg)
60
71
L140,7
M
h(mm)
L131
h
sin a =-
I
1200
6fT1v., (kg)
fTIv., (kg)
6fT1v., (kg)
0.53
683.2
678.0
577.4
573,6
54.2
672.6
54.5
49.3
43.9
0.39
0.35
22.96
0.48
0,42
20.56
0.38
667.0
38.3
562.8
800
700
0.31
0.33
662.3
657.6
33.6
558.8
0.27
18.06
15.66
600
0.23
13.30
28.9
24.5
553.5
549.3
544.1
30.3
26.1
20,9
523.2
o
654.9
648.5
57.9
900
0.28
0.24
653.2
500
0.20
11.54
0.20
648.9
20.2
o
o
o
o
628.7
0
27.92
1100
0.47
0,43
0.53
0,48
1000
900
0.39
0.35
800
700
0.31
0.27
600
500
o
1200
4
fTIv., (kg)
27.92
25,42
1000
2
tan a
0.47
0,43
1100
o
Size (em)
170
178
25.42
22.96
20.56
18.06
760.9
755.1
748.1
0,42
0.38
0,33
57.3
51,5
742.7
742,9
0.23
0.20
0.28
0,24
0.20
731.6
727.7
39.1
39.3
28.0
24.1
703.6
0
0,38
785.3
772,9
58.2
45.8
0,33
772.9
45.8
0.28
0.24
0,20
0,16
765.7
759.3
o
727.1
o
o
o
1100
0,43
1000
0,48
0,42
900
0.39
0,35
25.42
22.96
20,56
800
0.31
700
0.27
600
0.23
500
400
0.20
o
o
18.06
15,66
13.30
11,54
8,98
0.16
o
64.3
51.8
45.8
40.2
35.2
630.8
625.8
620.0
615.3
590.6
759,6
29.4
24,7
o
63.8
56,4
752,2
739.3
43.5
43.5
739.3
734.6
728,8
38.6
32.3
27.5
754.6
749,0
39.6
35,6
642.4
636,4
44.5
15.66
13.30
11.54
50.4
44.2
567.4
38.8
33.0
724.0
28.2
695.8
o
21.9
o
fTlv.t (kg)
0
1151.9
fTIv.,
1,=- I
fTIv.'
2
1294.2
4
1422.9
(mm)
0
52
hv =-
0
6m
I
X --
fTIv.,
tan a
mm
6mby
tan a = 0,5
2
59
4
67
(mm
=-
X kg
kg
+
1164.1
I,
=1- ~
0
1398.9
-
I
0
2.23
2
1.98
4
1.80
fTIv.,
2
1169.6
4
1253,3
rdyn
0
52
2.23 X + 296
0.5
hv,o
=528 mm
2
59
1.98 X + 296
0.5
hv,2
= 530 mm
4
67
1,8 X + 296
0.5
hv,4
= 537 mm
+ mm)
(mm)
2
1393.4
4
1309.7
(mm kg>l)
The Automotive Chassis
394
Fig. 6.5
hv
._~o
Vehicle shown tipped to derive the
equations of moments for the height hso of the
body centre of gravity.
.vjfrr
mBo
mv,t
mU,f
+
mU,r
mBo,f
=
mV,f -
mU,f
(6.6a)
mBo,r
=
mV,r -
mU,r
(6.6b)
The height hBo of the body centre of gravity B is easy to calculate by observing
the vehicle when it is tipped forwards (Fig. 6.5) using an equation of moments,
assuming that the individual weights act as forces at their respective distance on
the ground:
mv,th v - (mU,f + mU,r) rdyn
hBo = - - - - - - - - -
(6.7)
mBo
Depending on the loading condition and the weight of the unsprung mass, the body
centre of gravity hBo is 20-40 mm higher than that of the vehicle as a whole hy •
6.2
Mass moments of inertia
From the theory of mechanics it is known that when a body is accelerated in a
straight line the inertia Fe is given by
Fe = m ax = mass X acceleration (N)
In comparison to this, in the case of accelerated rotational movement, the acceleration moment is influenced by the rotation mass J.
Chassis and vehicle overall
395
The rotation mass - equivalent to the mass moment of inertia 1 (kg m2 ) and
also known as second degree mass moment - is a measure of inertia on rotating
bodies. In vehicles, three important rotational movements occur in the various
vehicle conditions, to which the variables of the mass moments of inertia 1 are
related.
• The vehicle moment of inertia lz,v around the vertical axis (z-axis, Fig. 3.3) is
required for driving stability studies or even for reconstructing road traffic
accidents.
• The body moment of inertia lX.Bo around the vehicle's longitudinal axis (xaxis) is essential for generally studying body movement (roll behaviour)
during fast lane changes in the driving direction.
• The body moment of inertia lY,Bo around the transverse axis (y-axis) is the
determining variable for calculating pitch vibration behaviour.
In addition to this, in general, the inertia moments of power units (engine-gearbox unit) and individual rotationally symmetrical elements, such as steering
wheels, tyred wheels, etc. are of importance. (See also Section 1.5 in Ref. [3].)
The position of its centre of gravity and the variables of the moment of inertia are usually determined with the basic design of a vehicle (drive, wheelbase,
dimensions and weight).
In addition to the type of drive, the vehicle's moment of inertia lz,v around the
vertical axis is the determining factor for its cornering performance.
Manoeuvrability increases as the inertia moment decreases, whereas driving
stability when the vehicle is moving in a straight line and on S bends decreases
by the same amount.
lz.v comprises the mass mV,T of the vehicle as a whole and the radius of gyration i z.v squared:
(6.8)
The magnitude of the radius of gyration i z.v depends on the length, width and
weight distribution of the body, the length and weight of aggregate units
(engine, gear box, differential) and the position and weight of the occupants
and the luggage. Series tests with saloons have shown that the radius of gyration is mainly a function of the load status and only varies within narrow
limits from vehicle to vehicle. Figure 6.6 shows the average values. Only the
vehicle weight mV,t in the occupancy or load condition to be investigated is
necessary for determining the approximate moment of inertia lz,v (see
Section 5.3.6). The values shown in Fig. 6.6 relate to medium-sized saloons.
If the vehicle has a five-, six- or eight-cylinder engine, a difference value must
be added:
i1i::::: 0.05 m to 0.1 m
If vehicle length Lt and wheelbase 1 are included in the following equation, an
accuracy of at least 98% can be achieved; only a correction factor needs to be
added:
396
The Automotive Chassis
Fig. 6.6 The approximate radius of gyration iX,Bo or iv.Bo (valid for medium-sized
saloon cars) for the inertia moment JX,Bo or JY,Bo of the body or Jz,v of the vehicle as a
whole, shown as a function of the loading condition and the pivot axis (Fig. 3.3).
Inertia radius in metres
Load
Car body only
x-axis
Empty
2 passengers in front
4 passengers
4 passengers and luggage
Formula sign
lz,v = 0.1269
mV,t
y-axis
Whole vehicle
z-axis
0.65
0.64
0.60
0.56
1.21
1.13
1.10
1.13
1.20
1.15
1.14
1.18
IX,Bo
!Y,So
Iz,V
Lt l(kg m2)
4,
'.~
l
1
'.~
"t'
(6.9)
Nevertheless, this equation only applies to the usual vehicle loading. Higher
loads in the boot (or a roof load) must be considered separately:
l1,v = 0.1269
mV,t
1 Lt + dm l~ (kg m2)
(6.10)
where lx is the distance of the loading mass dm to the vehicle centre of gravity.
The moment of inertia lX,Bo of the body is not so easy to calculate. In this
instance, the weights mu,f and mU,r of the unsprung masses must be known and
their distances to the respective coordinate axis drawn through the vehicle centre
of gravity (see Equations 6.4c, 6.4d and 6.6, and Fig. 3.3); it is easier, in this
case, to use Fig. 6.6:
'
(6.11)
A front-wheel drive passenger car with two occupants can be used as an example for the pitch vibration calculation (around the y-axis):
axle load front, partly laden (index pI) mv'f,p! = 609 kg
axle load rear, partly laden (index pI) mV,r,pl = 393 kg
The weight of the axle mass is:
front
mU,f
= 67 kg
and rear mU,r = 59 kg
The radius of gyration is
mBo,p!
mBo,pl
iY,Bo
= mv'f,pl + mV,r,p!
= 876 kg
-
= 1.13 m. Equation 6.5 gives:
(mU,f
+ mU,r) = 609 + 393 - (67 + 59)
Chassis and vehicle overall
397
In accordance with Equation 6.11 the mass moment of inertia of the body is
then:
JY,Bo
= mBo,pl i~Bo = 876 X 1.13 2 ,
JY,Bo
= 1118.6 kg m2
The same applies to body roll movements around the x-axis. The values in the
table should also be used here:
JX,Bo = mBo ikBo (kg m2)
(6.12)
For further details, see Ref. [3], Section 1.5.
6.3
Braking behaviour
Braking path SB in metres, starting speed v in metres per second (see Equation
2.1c) and delay ax are related as follows:
(6.12a)
6.3.1
Braking
When the driver brakes, the equivalent braking force acts as a reaction force at
the centre of gravity V of the vehicle as a whole (Fig. 6.1):
F X,Y,B --
II.
rX,W
F Z,Y,t
(6.13)
i.e. the coefficient of friction J-lx,w times the weight force Fz,y,t of the vehicle as
a whole, whereas J-lx,w can be equated to the deceleration ax in m S-2, divided by
gravity:
J-lx,w =
ax/g
(6.13a)
At an international level (DIN 74250), the formula
function:
z = J-lx,w and as a percentage:
Z = J-lx,w
100(%)
z is used for the braking
(6.13b)
i.e. during braking z =80% (corresponding to ax =7.85 m s -2) the coefficient of
friction J-lx,w = 0.8 is necessary (Fig. 2.33 and Section 1.3 of Ref. [6]).
The braking force FX,Y,B acting at the vehicle's centre of gravity causes longitudinal forces F X,W,B,f and F X,W,B,r at the centres of wheel contact of the front and
rear axles, and an increase in axle load +~Fz,v,o at the front and a reduction
- ~Fz,y,o at the back when the vehicle is observed as a rigid body. In accordance
with Fig. 6.7 the equations would then be
x = hv/l
(6.l3c)
398
The Automotive Chassis
Fig. 6.7
A braking force
acting at the centre of
gravity V of the vehicle
causes the axle load trans. fer ±tJ.Fz.v.o and the braking
forces FX.W.B,r on front and
FX,W.B.r on rear axle. If the
aerodynamic and rolling
resistances are ignored, the
forces can be easily calculated.
FX,V,B
I
I1Fz,v,o
FZ,Y,f,dyn
= J.Lx,w
=
FZ,v,t
X (kN)
(6.14)
+ I1Fz,v,o and FZ,V,r,dyn = FZ,v,r
FZ,Y,f
-
z,v,o
I1F
(6.15)
The lower the centre of gravity and the longer the wheelbase, the less is the
(undesirable) load transfer I1Fz,y,o. The braking force related to one axle is
then
front
rear
FX,W,B,f
= J.Lx,w
F X,W,B,r --
FZ,V,f,dyn
F X,V,B -
and
F X,W,B,r --
IL
F X,V,r,dyn
rX,W
(6.16)
(6.17)
Half the braking forces per axle multiplied by the dynamic rolling radius
rdyn, gives the braking moments M b at the wheels (see Equation 6.25a), which
are:
front:
Mb,f
= 0.5
rear Mb,r = 0.5
FX,W,B,f rdyn
FX,W,B,r rdyn
(6.18)
(6.19)
The larger is rdyn, the higher is the moment to be generated by the brake. This is
one reason for using tyres with an rdyn ::::; 300 mm on a medium size passenger
car (rdyn = CR/2'IT, see Equation 2.2).
The sizes of F X,W,B,f and F X,W,B,r depend both on the vehicle and its loading
condition and on the road, i.e. the coefficient of friction J.Lx,w possible on it (see
Section 2.7). A front-wheel drive vehicle and the calculation of the braking
forces for two possible cases with an unfavourable loading can be used as an
example to indicate the range of the braking force distribution:
CPf = FX,W,B,rIFx,V,B (times 100 as a percentage)
(6.20)
(6.20a)
with an unchanged centre of gravity height hv . As described in Sections 6.1.2.4
and 6.3.3.5, however, hv alters based on the load and the pitch angle.
Chassis and vehicle overall
6.3.1.1
Braking on dry concrete with only two people in the vehicle
FZ,V,f
J,Lx,w
= 6.9 kN; FZ,v,r = 4.2 kN; l = 2.49 m
= 0.9; h y = 0.58 m
~Fzv.o=0.9 X
"
FX,W,B,f
FX,W,B,r
6.3.1.2
399
11.15 X
0.58
2.49
=2.34kN
= 0.9 (6.95 + 2.34) = 8.36 kN
= 0.9 (4.2 - 2.34) = 1.67 kN
Braking on ice with a fully laden vehicle
= 7.1 kN; FZ,v,r = 7.0 kN; fLx,w = 0.15; land h y as previously
~Fz,v,o = 0.49 kN, FX,W,B,f = 1.14 kN and FX,W,B,r = 0.98 kN
FZ,V,f
In the first case, the front axle must accept as a percentage share:
<Pf X 100 =
8.36
11.15
X 100 = 75%
and accordingly the rear axle 25%. In the second example, the braking force
distribution is 54% and 46%. In the usual distribution of 75-80% on the front
and 20-25% on the rear in the case of non-ABS fitted cars, the axle could lock
in the first case (because its brake applies too high a torque); on ice it would be
the front axle. For details, see Chapter 3 of Ref. [6].
6.3.2
Braking stability
If both wheels of an axle lock (if ABS is not fitted), i.e. if they slide on the road,
there is not just reduced friction in the longitudinal direction (Fig. 2.33), but also
lower friction in the lateral direction. If the rear axle locks, as shown in Fig. 6.8,
lateral forces Fy,w,f will occur at the rolling wheels of the front axle, which will
intensify the problem, even in the case of a minor yawing effect, i.e. the condition is unstable.
Lateral forces or irregularities in the road acting on the body can cause the
vehicle which, to this point, has been travelling in a straight line to leave its direction of travel. A reinforcing yawing moment M~ occurs (Fig. 6.9), which seeks to
turn the vehicle sideways to its previous direction. There is a danger of lateral roll
over. However, if the front axle locks, the rear wheels, which will still be rolling,
will produce stabilizing lateral force Fy,w,r. The condition is stable (Fig. 6.10).
The position is different if the braking moments on the wheels of one axle are
of different sizes. The brakes pull to one side due to different lining coefficients
of friction or unequal coefficients of friction on the left and right wheels (fL split,
see Section 1.7.1 and Ref. [6] Section 2.4.4).
400
The Automotive Chassis
Direction
F.X,W,b,f
Fig. 6.8 Locking rear wheels
lead to an unstable driving
condition.
Direction
F-,
X,W,b,f
F.X,W,b,t
1
F.X,W,b,t
F:.X,V,B
F-,
X,W,b,f
F-,
X,W,b,f
IM*O
Fig. 6.9
When the rear wheels lock,
a reinforcing yawing moment occurs
even when the vehicle only slightly
leaves the direction of travel.
Fig. 6.10
When the front wheels
lock, the vehicle condition remains
stable although the vehicle can no
longer be steered.
'--~------------'--'-------------'
Chassis and vehicle overall
Fig. 6.11 As the static calculation below indicates, unequal
braking forces Fx,W,b,f,1 and Fx,W,b,f,rs
at the centres of tyre contact of
the front wheels cause the vehicle to rotate around the vertical
axis. In the case of a positive
wheel offset at ground (positive
scrub radius), there is also a
steering input in the same direction of rotation.
401
Direction
'i<,W;b",1
If,
X,W,b,f,rs
,
frurning of vehicle
F:.X,V,B
v
If.X,W,b,r
If.X,W,b,r
Vehicle
moment
~
FX,W,b.f,1
F..x,v,s
If.X,W,b,r
FX,w,b,r,rs
If.X,W,b,r
Figure 6.11 shows a higher braking force FX,W,b,f,1 on the left front wheel (than
on the right one). The difference force of the two wheels IiFx,w,b,r = FX,w,b,r,1 FX,W,b,f,rs, with the lever of half the tread width, gives the yawing moment MIjJ =
IiFx,w,b,r 0.5 b r which introduces rotation to the left into the vehicle. In addition,
there is also the steering moment MZ,W,b, which causes the steering to tum in the
same direction.
Where the brake is on the outside (at the wheel), the size of this moment
depends on the length of the wheel offset +rO', and is:
+MZ,W,b = IiFx,W,b,f rO' cos
(J'
(6.21)
In the case of negative - rO', there is counter steering (Fig. 6.12), and if rO' = 0,
only the yawing moment (-MIjJ, Fig. 6.13) occurs. This also applies to centre
axle steering (Fig. 3.114).
A differential braking torque is less noticeable on the rear axle. First, the
braking forces F X,W,b,r are smaller and second there is a stable condition. The
The Automotive Chassis
402
Direction
x,r ~¥rriing
£-X,W,b,f
£-
L-J--.
fr I~ehicle
I
FT.t
LJ
Fig.6.12
F:,.
X,V,B
v
In the case of a
negative wheel offset (or
elastokinematic toe-in alteration (Figs 3.82 and 3.102)
the steering is turned by the
front wheel, which must
transfer the greater braking
force FX,W,b,f (the left wheel
in the illustration) opposite
to the direction in which the
vehicle is turned by the
outer yawing moment. The
static calculation shown
indicates this. This leads to
an equalization which, even
in the case of different braking forces at the front,
largely prevents deviation
from the direction of travel.
I
Ff.X,W,b,r
FvX,W,b,r
Vehicle
moment
~
F..X,W,b,f
F..X,V,B
F:.X,W,b,r
F:.X,W,b,f
FvX,W,b,r
rM#-O
different sized forces FX,W,b,r and FX,W,b,f are behind the centre of gravity V (Fig.
6.14).
For further details, see Ref. [6] Section 2.4 and Ref. [9] Chapter 3.
6.3.3
Calculating the pitch angle
The pitch angle, i.e. the angle 8 s by which the body turns around the lateral axis
when the brakes are applied, can be calculated as a function of the braking force
Fx,v,s (Fig. 6.15, see also Section 5.4.3).
6.3.3.1 Data for the example calculation
A passenger car with the following data can be used to clarify the relationships:
Chassis and vehicle overall
Fig. 6.13
If, where the brake
is on the inside, the longitudinal
force lever ra = 0 or, where the.
brake is on the outside, the wheel
offset at ground (positive scrub
radius) is 0, then unequal braking
forces Fx,W,b,f on the front wheels
have practically no effect on the
steering. The steering rod forces
FT are almost zero.
,
403
Direction
If.
X,W,b,f
l
£X,W,b,f
F;=O
F;=O
F.X,V,s
v
£-X,W,b,r
If.
X,W,b,r
£-X,W,b,f
£-X,W,b,f
~
X,V,s
£-X,W,b,r
£-X,W,b,r
fDireClion
15<, w, b, f
~ "--o----.----e--t ~
F.X,V,s
v
Fig.6.14
A rear wheel brake, which is
unevenly pulled, hardly has any effect on the
steerability of a vehicle.
15<, w, b, f
404
The Automotive Chassis
80
Fig. 6.15 If the body goes down more at the front than it rebounds at the rear,
the body centre of gravity Bo moves down by ilh so . The braking force Fx,so,s would
then act at the height (hso - ilh so ) at point Bo. The pitch angle 6s is also shown (see
also Fig. 3.137).
Overall weight force (vehicle)
axle load front
axle load rear
axle weight force front
axle weight force rear
spring rate based on only one axle
dynamic rolling radius of tyre
braking
wheelbase
centre of gravity height
Fz,y,t = 11.15 kN
= 6.95 kN
FZ,y,r = 4.20 kN
FZ,U,f = 0.80 kN
FZ,u,r = 0.70 kN
front Cf = 11.5 N m- I ,
rear Cr = 14 N rn- I
rdyn = 0.288 m
z = 0.8, i.e. /-Lx,w = 0.8
(see Equation 6.13b)
l = 2.50 rn
hy = 0.58 m
FZ,Y,f
Details relating to the numerical values can be found in Sections 2.2.5.4, 5.3.6.1,
6.1.3 and 6.12.
6.3.3.2 Opposed springing forces
When the body is observed as a rigid mass, the spring opposed forces (related to
one axle's wheel track considered as one, front and rear, Fig. 6.7) correspond to
half the axle load transfer + !:J.Fz,y,o and, irrespective of whether the brakes are
outside of the wheel or inside on the differential, the forces can be calculated
easily on the basis of Equation 6.14.
+ Fz,y,o = 0.8 X 11.15 X (0.58/2.50) = 2.07 kN
The vehicle goes down at the front and rebounds at the back. The spring rates
are quoted in newtons per millimetre and also relate to one wheel. That is, 1 N
mm- I = 1 kN m- I , so we can assume !:J.Fz.y,o is multiplied by two. In accordance
with Equation 5.10, with linear springing the following theoretical values would
result:
Chassis and vehicle overall
Bump travel front
Jounce travel rear
SI,f
SZ,r
405
= tiFz,v,o/(2 cr) = 2.07/23 = 0.09 m
= tiFz,v'oI(2 Cr) = 2.07/28 = 0.074 m
6.3.3.3 Pitch angle with linear springing
The pitch angle 8B is. (see also Equation 5.15):
SI
r + SZr
8B = - ' - - '
l
(6.22)
and (times 360°/2'11")
8B = 57.3 X
Sl,f
+
l
SZ,r
(6.23)
In this example the result is then:
8B = 57.3 X
0.09 + 0.074
2.50
= 3.76° = 3°46'
'6.3.3.4 Pitch angle with progressive springing
In order to determine the travel on the front and rear axles, the spring characteristics must be known. Travel is entered here in mm and wheel load in kg. The
required values should therefore be calculated from the axle load:
FZ,v,r
6950
2g
2 X 9.81
wheel load, front, normal
mtY.f=--=
wheel load,
front, maximum
FZ,v,r + tiFz,v,o
"
6950 + 2070
= -2-X-9.-8-1-. = 460 kg
2g
ml,V,f,max =
FZ,V,f
4200
2g
2 X 9.81
wheel load, rear, normal
m I y.r = - - =
wheel load,
rear, mInImUm
FZ.V,f - tiFz,v,o
meV,min
"
=
2g
= 354 kg
= 214 kg
4200 - 2070
=
2 X 9.81 = 108 kg
In spite of the harder springing, the highly progressive curve shown in Figs 5.13
and 5.15 can be used as an example. The spring travel is
front at 354 kg = 112 mm, and at 460 kg = 134 mm
rear at 214 kg = 110 mm, and at 108 kg = 44 mm
406
II
The Automotive Chassis
,
The vehicle response would therefore be
front, goes down by
SI,f
= 134 -
rear, rebo,,!nds by S2,r = 110 - 44
112 =22 mm and
=66 mm
This would then give a pitch angle of only
6.3.3.5 Change of the centre of gravity height
Point B rises or falls when the brakes are applied based on how far the vehicle
rebounds front and rear and how far the body centre of gravity is away from the
axle centres. With the path entered in Fig. 6.15 and the weight forces, plus
Equations 5.14 to 5.14b, the result is then the change of height
Ah
U
Bo
F Z,Bo,f
= -Sl,f
F Z,Bo,r
+ S2,r--
FZ,Bo
(6.24)
FZ,Bo
When the springs compress, the body goes down and so
The values in accordance with Equations 6.5 to 6.6b are:
FZ,Bo,f
= FZ,V,f -
SI,f
becomes negative.
and
Fz,u,f
(6.24a)
F Z,Bo,r -F
Z,V,r -FZ,U,r
(6.24b)
+ F Z,Bo,r
(6.24c)
F Z,Bo := F Z,Bo,f
With the numerical values of the calculation example (see Section 6.3.3.1) and
with linear springing, the result is then:
FZ,Bo,f
= 6.95
- 0.80
= 6.15 kN; FZ,Bo,r = 4.20 -
FZ,Bo
= 6.15 + 3.5 = 9.65 kN, SI,f =-0.09 m
~hBo
= -0.09
6.15
and
0.70
S2,r
= 3.5 kN
= +0.074 m
3.5
+ 0.074 X - - =-0.03 m; ~hBo
9.65
9.65
X --
=-0.03m
The static centre of gravity height of the body, calculated using Equations 6.7
and 6.24a to c, is hBo = 0.625 m and that which occurs when the brakes are
applied is:
h Bo -
~hBo
=h~o =0.595 m
(6.24d)
hv
of the
The centre of gravity therefore goes down 4.8%. The resulting height
vehicle centre of gravity can be calculated from the value h~o = 0.595 musing
Chassis and vehicle overall
407
the transformed Equation 6.7. This can be more easily done if the axle weight
forces Fz,u,r and FZ,u,r are ignored. The error involved is less than 0.5%:
tihv
= -SI r
FZ,V,f
FZ,v,r
+ SZr-, F Z,V,t
' FZ,v,t
(6.25)
The axle weight forces (unsprung masses, see Section 6.1.3) must be known if
the pitch poles are to be included in the equation.
6.3.4 Influence of radius-arm axes
6.3.4.1 Pre-conditions for calculations
Radius-arm axes poles are only effective when the brakes are outside the wheel.
The entire calculation has to be done differently because not only do the braking force portions of the body FX,Bo,B,f and FX,Bo,B,r act at these axes Or (front) and
Or (rear), but so do the vertical forces tiFz,Bo,B,r and tiFz,Bo,B,r acting against brake
dive (B = axis-related).
Figures 3.108 and 3.113 show the static situation and, with Equations 6.13 to
6.16, the braking force related to one wheel can be calculated:
FX,W,b,f = Fx,w,B,r/2
(6.25a)
F X,W,b,r = F x,w,B,r/2
(6.25b)
6.3.4.2 Forces on the radius-arm axes of both axles
Figure 3.155 shows how the forces are calculated on one wheel station with
double wishbones and Fig. 6.16 shows the calculation based on the entire axlesuspension and using the pitch poles required in this instance.
To be able to calculate the forces FZ,Bo,B,f supporting the body vertically when
the brakes are applied, the equation of moments must be formed with the pivot
at the centre of tyre contact. Paths e and c define the point Or (present on both
sides) in the illustration:
~FZ,Bo,f
=F X,B,Bo,r e c+ F X,B,U,frdyn = AFz,v,r,z
U
(6.26)
The axle load difference tiFz,v,r,z is the same size as tiFZ,Bo,B,f and opposes
compression when the brakes are applied. The forces that also appear in
Equations 6.26 and 6.29 can be determined using Equations 6.6a and 6.6b:
FX,Bo,B,f =
FX,Bo,B,r
!J,X,W
=!J,X,W
F X,U,B,f
F X,U,B,r
FZ,Bo,f
FZ,Bo,r
= j.1x,w mBo,f g
=j.1x,w mBo,r g
= llx,w F Z,U,f = j.1x,w mU,f g
= llx,w F Z,U,r = Jlx,w mU,r g
(6.27 a and b)
(6.28 a and b)
•......
:'.1,
•...
> ~'
408
The Automotive Chassis
Car body ..--...-'-f.............
/
+F.,
Z,Bo,B,t
---- --- -
-F.,
Z,Bo,B,t
~
BoBr
e
/
c
!iF-.Z,V,t,2
Fig. 6.16
Paths and forces when the body is supported around the front virtual
pitch axis. The higher Of (path e) and the closer to the wheels (path c), the larger the
difference in force ilFz.Bo.B.f supporting the body and the smaller the pitch angle aB.
To calculate FZ,Bo,B,f (and also FZ,Bo,B,r) only the braking forces FX,W,B,f and FX,W,B,r,
which occur in the centres of tyre contact and relate to the axle as a whole, need
to be divided up into the proportion affecting the wheel suspension F X,U,B,f and
FX,Bo,B,f, which is critical to the body. The same applies to the rear axle. The index
r appears in this instance.
6.3.4.3 Numerical values
With the values of Section 6.3.3.1 the result for the front axle is then:
FX,U,B,f
= J-lx,w FZ,Y,f
= J-lx,w Fz,u,f
= 0.8 x 6.95 = 5.56 kN
= 0.8 x 0.8 = 0.64 kN
FX,Bo,B,f
= FX,W,B,f -
= 4.92 kN
FX,W,B,f
FX,U,B,f
and for the rear axle:
=J-lx,w FZ,y,r
::::: 0.8 x 4.20 = 3.36 kN
FX,U,B,r = J-lx,w FZ,u,r
= 0.8 x 0.70 = 0.56 kN
FX,Bo,B,r = FX,W,B,r - FX,U,B,r = 2.8 kN
FX,W,B,r
The following dimensions should apply to the pitch poles (Figs 6.16 and 6.17):
front
rear
front
c = 1.0 m, e = 0.15 m
d = 0.5 m, g = 0.25 m which gives the result
FX,Bo,B,f
FZ,Bo,B,f
e + FX,U,B,f rdyn
= -------C
(6.26)
Chassis and vehicle overall
409
Fig. 6.17 Paths and
forces when the body is
supported at the rear
pitch axis Or which are
relatively close to the
wheels. Figure 3.154
shows the forces,
based on only one axle
side, and Fig. 3.159
shows further kinematic
aspects.
rear
(4.92 x 0.15) + (0.644 x 0.288)
1.0
I1Fz ,Bo,B,f
=
I1F
_ FX,Bo,B,r
g + FX,U,B,r rdyn
Z,Bo,B,r -
d
(2.8 x 0.25) + (0.56 x 0.288)
=0.5
"
I1FzBoBr
,
=0.922 kN =I1Fz ,y,f,2
(6.29)
= 1.723 kN =M Z'"Y.r2
The pitch poles should be as close as possible to the wheel and as high as possible. Because of the more favourable position of the rear poles, f1F z ,Bo,B,r >
f1FZ ,Bo,B,f, i.e. also f1F Z ,Y,r,2 > f1FZ ,Y,f,2.
6.3.4.4 Spring-opposing forces
The spring-opposing forces f1FZ ,Y,l that result from the axle load transfer f1Fz ,y,o
(see Equation 6.14) and the weight force differences I1Fz ,y,2 with existing radiusarm axes are critical to the pitch angle desired OB:
front
f1F Z,Y,f,1 -- I1Fz,y,o - I1FZ,Y, f ,2
(6.30)
rear
f1F Z,Y,r,1 -- f1FZ,Y,O -
(6.31)
f1FZ,Y,r,2
Where I1Fz ,y,o = 2.07 kN (see Section 6.3.3.2) related to the entire axle, the
values are
11 F z,Y,f,1 = 1.148 kN
f1FZ,Y,r,l = 0.347 kN
6.3.4.5 Pitch angles
At the spring rates 2 X Cf = 23 N mm- 1 or 2 X Cr = 28 N mm- 1 based on one
axle side and in accordance with Equation 5.10, the paths needed for determining 8a (see also Section 5.4.3) are:
front
Sl,f
=0.05 m and Tear S2,r =0.012 m
410
W
i
The Automotive Chassis
'l~'f\
With the linear springing assumed, the angle decreases. Without considering the
axis it was 3°46' and now in accordance with Equation 6.23 it is
Os
= 57.3
0.05 + 0.012
.
.
2.50
X
= 1.42 = 1 25'
0
0
Section 6.3.3.4 contains the calculation of Os with progressive springing.
6.3.4.6
Radius-arm axes on one axle only
If, for example, only the rear axle suspension has radius-arm axes SZ,r must be
determined using /iFZ.V,r,l, whereas /iz,V,f,O alone is critical on the front axle. At the
value /iFz,v,o = 2.07 kN in Section 6.3.3.2 the bump travel was Sl,f = 0.09 m.
6.3.5
Anti-dive control and brake reaction support angle
Automobile manufacturers frequently quote the anti-dive control ke as a percentage in publications. It can easily be calculated on the basis of Figs 6.15 and 6.16.
front
rear
= /iFz, Y,f,Z//iFz,v,o
ke,f =0.922/2.07 = 0.45
ke,f = 45%
(6.32)
= /iFZ,y,r,z/ /iFz,v,o
= 1.723/2.07 = 0.83
= 83%
(6.32a)
ke,f
ke,r
ke,r
ke,r
The brake reaction support angle
6.16 and 6.17) is
front
tan Sf = e/c
tan Sf = 0.15/1.0
rear
tan Sr = g/d
tan Sr = 0.25/0.5
6.4.1
entered in Fig. 3.160 in the examples (Figs
(6.33)
= 0.15; Sf = 8°30'
(6.33a)
= 0.5; Sr = 25°32'
On production passenger cars,
40°.
6.4
S
Sf
is usually below 10° and
Sr
between 30° and
Traction behaviour
Drive-off from rest
The relationships when the vehicle moves off and accelerates are somewhat different to those when the brakes are applied. As shown in Fig. 3.113, the tractive force
FX,W,A must be shifted to the centre of the rolling wheel if the differential is fixed
------------,---,--------------
Chassis and vehicle overall
411
to the body or the engine (i.e. separately from the wheel suspension) and the driveoff moment is concentrated in its suspension (Fig. 3.110). This applies on all front
independent wheel suspensions and is equivalent to one virtual radius-arm axis Of
coming into effect. As shown in Fig. 3.154, in such cases, the squat can be reduced
by angling the two, double wishbones in the same direction. The same applies to
the rear axle in terms of the take-off dive (Fig. 3.160). The diagonal springing
angle X is then positive.
The picture is different when the differential is in the axle housing on a driven
rigid axle (Fig. 1.43). The drive pinion connected to the prop shaft is vertical to
the axle shaft connected to the wheels (Fig. 1.22), i.e. torque in and output form
a 90° angle. The result is that the tractive force F X,W,A, which occurs at the centres
of wheel contact, is supported exclusively in the suspension system of the axle.
Where there are pitch poles, the body is pushed upwards into these points Or and
the tail only dives a little, as shown in Fig. 3.159 using the example of the
opposed braking forces. The same effect is achieved with trailing link pairs at an
angle to one another (Fig. 3.161); there are pitch poles here too.
Figure 6.18 shows the forces generated during acceleration. Those acting on
the body are:
• the aerodynamic force F L , which can be ignored at speeds below 25
kIn h- I , and the
• excess force F X,ex, which is equal to the inertia in the x-direction.
The rolling resistance forces are
(6.34)
and the opposed tractive forces FX,W,A act at the wheels. As described in Ref. [9],
Section 2.1.4, the additional forces necessary to accelerate the turning masses,
can be determined using the rotating mass factor.
The equations for calculating the drive-off are:
FX,ex
(6.35)
= FX,W,A - (FL + FR,t) (N)
FL
..
..
..
..
rdyn
I
Fig. 6.18 Forces occurring in the vehicle centre of gravity V and at the centres of
tyre contact when a front-wheel drive vehicle accelerates.
The Automotive Chassis
412
F
-
X.W,A -
M
M.max
11
..
IOIG
r
(
N)
(6.36)
The following terms are used in the equation:
MM,max
11
'0
IG
r
the maximum engine torque in N m
the total efficiency (Figs 6.19 and 6.20)
the ratio of the final drive (differentials)
the ratio of the gear engaged
the static rolling radius rstat must be inserted in m at speeds below
25 kIn h- I , and above 60 kIn h- I ; the dynamic rolling radius rdyn
= CR,dynl2Tr (CR,dyn = rolling circumference in m, see Equations
2.2 and 2.2e).
Chapters 2 and 3 in Ref. [3] give details on resistances.
A compact front-wheel drive passenger car with a 1.3 I transverse engine can
be used as an example. When the acceleration in first gear from around 5 kIn h- I
is observed, the necessary data are
MM,max
=94 Nm,
io
=3.94,
tyres 155 R 13 78 S,
iG,1
= 3.55, 11 =0.90 (Fig. 1.50)
= 0.263 m
rstat
Fig. 6.19 When standard
vehicles are in the direct (mostly
fifth) gear, no pair of gears of the
manual gearbox is engaged.
However, the lower gears
require two pairs of gears to
transfer the engine moment.
c
1/
-T.
r
I
c
= 0.91 to 0.93
orI
1/
= 0.85 to 0.90
:c
t.
l
•
t
l
TL __L;
T-r__
I
arI
I
l·l.
'j'T
I- - -' _1-
Fig. 6.20
I
I
_L.
.1..
1.I
1/
= 0.90 to 0.95
--
If, on a front-wheel drive or rear-engine vehicle, the engine is longitudinal, on a manual gearbox one pair of gears is always engaged to transfer the drive
moment, regardless of what gear has been selected and whether the vehicle has a
four-, five- or six-speed box. On transverse engines (Fig. 1.50), the degree of efficiency can be better than "fl = 0.9.
Chassis and vehicle overall
F X,W,A
=
94 x 0.90 x 3.94 x 3.55
0.263
413
=4499 N =4.5 kN
Because the weight is taken off the front axle as the vehicle moves off (see
Equation 6.37), r becomes 10-15 mm greater than rstat and the driving force
around 5% smaller.
The vehicle has a curb weight of mv'ul = 875 kg. With two people each weighing 68 kg in the vehicle the actual weight would be
mV,t
= 1011 kg
and
FZ,v,t
=mV,t X g =9918 N =9.92 kN
The forces in the longitudinal direction at kR from Fig. 2.31 are
FR,t =kR FZ,v,t = 0.012 X 9.92 =0.12 kN, F L = 0
and FX,ex = FX,W,A - FR,t =4.5 - 0.12 =4.38 kN
The axle load transfer IiFz,v,o is determined using Equation 6.14; the vehicle data
are
l
=2.52 m,
hU1
= 1.4 m
and J.,Lx,w
= 1.05 (see Fig. 2.33)
The height of the centre of gravity hv,o can be obtained, using Equation 6.5, from
the unladen height h u1 of the vehicle:
hv,o ~ 0.38 X 1.4
~
0.532 m
When there are two people in the vehicle the centre of gravity rises by 10 to 15
mm (see Section 6.1.2.4); therefore hV,2 = 0.546 m is assumed:
IiFzy.o
, ,
hV,2
= rII.XW
Fzy.t - - = 1.05 X
,
"l
0.546
9.92 X 2.52
IiFz,v,o = 2.26 kN
As shown in Fig. 1.36, when there are two people in the vehicle, approximately
60% of the weight is carried on the front axle:
FZ,V,f = 0.6 FZ,v,t = 5.95 kN
Unlike when the brakes are applied, when the vehicle accelerates the weight is
taken off the front axle by IiFz,v,o:
FZ,v'f,dyn
= FZ,V,f - IiFz,v,o = 5.95 - 2.26 = 3.69 kN
(6.37)
The coefficient of friction required is then
II.
rX,W
--
F x, exIFZ,V,f,dyn
J.,Lx,w = 4.38/3.69 = 1.19
(6.37a)
414
The Automotive Chassis
When the vehicle accelerates fast from slow speeds, the driven front wheels
would spin due to the load alleviation. This disadvantage is particularly evident
in the range of maximum engine torque. The coefficient of friction needed f.Lx,w
= 1.19 is too high. Taking into consideration the load alleviated and therefore
larger tyre radius r, f.LH would drop to around 1.13 but not solve the problem.
With both values, i1Fz,v,o > 2.26 kN, which would be an even greater load alleviation of the front axle than the assumed jJ.x,W = 1.05.
On rear-wheel drive vehicles, Equation 6.35 is exactly the same. It is simply
a matter of shifting force F X,W,A shown in Fig. 6.18 to this axle and adding the
axle load shift to FZ,v,r. The result would be:
FZ,V,r,dyn = FZ,v,r + IiFz,v,o
(6.38)
IiF Z,v,r,l = IiFz,v,o - I1FZ,V,r,2
(6.31)
If the driven rigid axle of the vehicle under investigation has pitch poles (Figs
1.43 and 3.161), IiFz,v,o and I1FZ,V,r,2 must first be calculated to obtain I1FZ,v,r, 1
(Equations 6.14 and 6.31).
Equation 6.23 is again used for calculating the pitch angle SA and Equations
6.32a and 6.33a can be used for determining the take-off drive control ke,r and
the drive-off reaction support angle Xr as these values here are of the same size
as ke,r and c, i.e. are produced when the brakes are applied (Fig. 3.160). Only in
the case of independent wheel suspensions and rigid axles with a separate differential (De Dion axles) does the actual angle X need to be taken into consideration (Fig. 3.154 and see also Ref. [2] Section 3.6).
6.4.2
Climbing ability
The climbing ability q is quoted as a percentage and relates to the vertical height
hz reached at the end of a path Sx measured on the horizontal:
q = h/sx 100(%)
(6.39)
tan ex = h/sx
(6.40)
The inclination the vehicle can theoretically climb in first gear (e.g. in the range of
the greatest engine torque) can be calculated using the excess force FX,ex and the
total weight F X,V,l of the vehicle. In the previous example, in Equation 6.35 only the
rolling resistance would be somewhat smaller. The force F R,t must be multiplied by
cos ex ~ 0.9. FX,ex would increase from 4.38 kN to 4.39 kN, a negligibly small
difference of only 0.2%. The climbing ability of the example vehicle is:
sin ex = FX,cxlFz,v't = 4.38/9.92 = 0.44
sin ex = 26.1°, tan ex= 0.49 and q = 49%
On inclines, an axle load transfer of + F Z,V,3 occurs, i.e. a reduction of F Z,V,f on
the driven front axle. In accordance with Fig. 6.21 and Equation 6.35, it is
Chassis and vehicle overall
415
Ir
hV
F.:,
Z,V,t
I
Fig. 6.21 Paths and forces necessary for calculating the skid point, shown on a
front-wheel drive vehicle that is travelling up an incline at a constant speed.
~FZ,Y,3
=Fz,y,x hv/l =Fz,y,t sin ex (hv/l) (kN)
(6.41)
and
!-Lx,w = Fx,ex/(FZ,Y,f - ~FZ,Y,3)
(6.42)
Insertion of the example values give
~FZV.3
"
= 0.92 X 0.44 X
0.546
, ~FZV.3 = 0.946 kN
2.52"
and
4.38
!-Lx,w = 5.952 _ 0.946 = 0.87
To travel a 49% incline evenly the example vehicle only needs a coefficient of
friction of !-Lx,w' = 0.87 in the range of the greatest engine torque, a value to be
found on dry concrete.
416
The Automotive Chassis
For further details on climbing ability and resistance, see Section 2.3 and 3.3,
and Section 3.11 in Ref. [3].
6.4.3
Skid points
Theoretically, more powerful engines would be able to climb steeper inclines
with either front- or rear-wheel drive, if the grip of the road surface were to
permit it. To have realistic values the skid points should therefore be determined,
i.e. the inclination (as a percentage) on the road surface of which the driven
wheels do not yet quite slip; J,1x,w = 0.8 would be the correct coefficient of friction as an initial value. Using Fig. 6.21 the equations can be derived that are
necessary for calculating Cl = !(J,1x,w). In this the x' -direction is in the climbing
plane (slope of the incline) and the z'-direction is vertical to it. Breaking down
the total weight force FZ,v,t at the centre of gravity V gives
F!z,v,z
=FZ,v,t cos Cl
and F!z,v,x
=FZ,v,t sin Cl
and F!z,v,z causes the axle loads
F!z,v'f,z = FZ,V,f Cl and F!z,v,r,z = FZ,v,r cos
Cl
As can be seen in Fig. 6.21, FZ,V,f or FZ,v,r are the axle loads applied to the vehicle
standing on the flat. The component F!z,v.x, known as the vehicle load downhill, is
the same as the excess force F X,ex previously calculated. This causes a load reduction on the front axle by -!1FZ,V,3 (Equation 6.41) and an increase in axle load on
the rear axle of +!1FZ,V,3. The value hv/l, which appears in the equation, shows that
the longer the wheelbase I and the lower the centre of gravity V, the smaller is the
axle load transfer (which is unfavourable on front-wheel drive).
The condition that the sum of all forces in the x' -direction equals 0, would be
met if:
FX,W,A = F!z,v,x + FR,f + FR,r + FL
F R,f and F R,r together give
The solution, based on a driven front axle, in accordance with Equation 6.42 is:
F!z,v,x + k R F!z,v,z + FL
J,1x,w =
F!
_ !1F
Z,V,f,z
F Z,V,t sin
J,1x,w = FZ,V,f cos
Z,V,3
Cl
+ k R FZ,V,L cos
Cl -
F Z,V,t sm
.
Cl
+ FL
(h vII)
Cl
Numerators and denominators divided by FZ,v,t give
Chassis and vehicle overall
417
sin ex + k R cos ex + FJFz,v,t
~x,w=
.
(Fz,v,rIFz,v,t) cos ex - sm ex (hv/l)
The speeds achievable on steep inclines do not exceed 25 km h-( so F L can be
ignored. However, on flatter inclines this counter force must be included in the
equation. To simplify matters, numerators and denominators are divided by cos
ex:
front wheel drive
~x w
,
=
tan ex + k R + FL(Fz Vt cos ex)
',
FZ,v,fIFz,v,t - tan ex (hv/l)
(6.43)
The coefficient of friction needed to travel a given incline can be determined
using this equation. To obtain the gradient-ability, i.e. tan ex, as the result, it is
necessary to transform the equation, as follows:
front wheel drive
~x,w (FZ,v,fIFz,v,t) - kR - FL(Fz,v,t cos a)
tan ex = - - - - - - - - - - - - - - 1 + ~x,w (hv/l)
(6.44)
The value tan ex X 100 = gradient-ability q as a percentage (see Equations 6.39
and 6.40).
If F L needs to be considered, ex needs to be estimated provisionally for it to
be possible to insert cos ex. It may be necessary to correct this later in such cases.
However, the numerical value is relatively small,
The formula clearly indicates that the higher is the front axle load FZ,v,r and
the smaller the value hv/l, the greater the angle ex becomes. The picture is
completely reversed on a rear-wheel drive vehicle (Fig. 1.36); in this instance the
equation is:
,
~x,w (Fz,v,rlFz,v,t) - k R - FL(Fz,v,t cos a)
rear wheel dnve tan a = - - - - - - - - - - - - - 1 - ~x,w (hv/l)
(6.45)
On this type of drive hv/l and rear axle load should be large. If the coefficient of
friction necessary for a given incline is required, the following formula applies:
rear wheel drive
~x w =
,
tan ex + k R + FL(FzVt cos a)
',
Fz,v,rlFz,v,t + tan a (hv/l)
(6.46)
To produce the diagram of the driving and climbing performance, half the
payload in the total weight force should be considered, whereas to determine the
skid point the different loading conditions must be assumed. These do not just
lead to a change in FZ,v,t but also in the axle load distribution, which is included
in the equation as FZ,v,tlFz,v,t or Fz,v,rlFz,v,t. The three most important loading
conditions are (see Section 5.3.6 and Fig. 1.36):
418
The Automotive Chassis
• two people each weighing 68 kg in the front
• four people each weighing 68 kg
• full payload.
The payload FZ;t,max (see Section 5.3.3) must be distributed so that, in accordance
with Equation 5.1, the permissible rear axle load FZ,v,r,max is achieved. Therefore,
the front axle is usually not fully loaded. The wheelbase l and the changing
centre of gravity heights:
hV,2, hV,4
and
hv,max
must also be known (see Section 6.1.2.4). Figure 6.22 shows a diagram of the
(calculated) skid points on three different coefficients of friction:
IJ-x,w = 0.8 (dry), IJ-x,W= 0.5 (wet) and IJ-x,w = 0.15 (ice)
-
Standard model
- - - Rear engine
_._ .• Front-wheel drive
60
0'0
50
j.Lx,w
=0.8
j.Lx.w
=0.5
itO
t
Q)
c::
u
c::
30
"--
Fig. 6.22
Skid points as a
function of three different coefficients of friction /-Lx,w = 0.8, 0.5
and 0.15 and the loading condition and type of drive.
Fully loaded rear-wheel drive
vehicles can negotiate the
largest inclines, whereas frontwheel drive vehicles have the
best climbing capacity at low
loads, i.e. with only two passengers in the front and a relatively
empty fuel tank, particularly on
ice.
20
-......... -.......
10 t-----+-----+----~
j.Lx.W
=0.15
--'--
o~--:2--.J...------J4:------:JFull
passengers
in front
of vehicle
passengers
load
Chassis and vehicle overall
419
as a function of loading condition and configuration drive. The mean percentage
axle load distribution from Fig. 1.36 and hv/l = 0.23 is considered in this calculation, whereas F L is ignored. As a series of investigations showed, on standard
passenger cars the gradient-ability at IJ.x.w = 0.8 with two people in the vehicle
averages 45%, increasing to around 52% when the vehicle is fully laden.
Gradient-abilities over 60% quoted by manufacturers are not realistic. The
wheels would spin because of the lack of friction (see the calculation in Section
6.4.1).
Publications should therefore not base their values on the (purely theoretical)
engine performance, but rather on the climbing capacity as a function of the
coefficient of friction IJ.x,w = 0.8 produced by the road. This applies even more
so to a front-wheel drive vehicle.
The picture for four-wheel drive vehicles is rather different. Here, the engine
torque and the ratio in the manual gearbox and differential are the deciding
factors along with the higher rolling resistance on uneven roads kR (see Fig. 2.31
and Equation 6.36). These are:
tan a := IJ.x,w - k R
(6.47)
In the case of IJ.x,w = 0.8 and kR = 1.5 X 0.012 an incline of 38% could be
climbed.
With added roof load, the change in axle load distribution should first be
calculated, then the hv,t of the common centre of gravity, so that these values can
be used in Equations 6.41 to 6.46 (instead of h v ). Section 1.3.5 of Ref. [3] gives
details. Further information, including far trailers, is also contained in Ref. [3],
Sections 3.13 and 3.14.
6.5
Platform, unit assembly and common
part systems
The high cost pressure placed on vehicles makes it necessary to have systems
able to provide the individuality of products required by clients at a low cost.
As in other areas of mechanical engineering, unit assembly and common part
systems are more anti more frequently being used in vehicle technology within
vehicle series or even by different manufacturers within a concern for the
purpose of meeting these requirements.
These approaches offer the following benefits:
• acquisition of experience of the system and component characteristics of
complex functions;
.
• creation of a basis for structural and crash calculations which can be used
repeatedly;
.
.
• shorter development periods (e.g. from 30 to 19 months accordmg to vehIcle
manufacturers Nissan);
• reduced developments costs;
• lower and more easily calculable development risks;
- -~~~_~:~~-,;;..~~~~~~~~T~~WM~~~~~~=::-~~~---
~-~=~~~~-~~~~i"~l-~~}~~~~~~,~\.,-~~~"'-,~~~¥~~j~ii~$~~~~~~~~Yfi~~~ii~~i~~~~~~~~~~~~~~~~~,-----
-~~~~_~:~~~~~~~~~~})T~~~M-~~~~~~---=-=::-~~~--
~-~="'=~~-~~~~i"~l-~~}~~2~::?~,,\:\.~"\~~"'-,~?~¥~~j~ii~$~~~~
~~~~~~~~~~%~~~~~~~~~~--.~~""".,...---
420
The Automotive Chassis
• enormous cost savings as a result of the considerable decrease in the numbers
of units used for different series.
Apart from the cost of the drive train, the highest development and machine
costs relate to the platform, Le. the basic structure of the vehicle which consists
of the floor of the vehicle and the support structure for the assemblies. The realization of the platform concept results in the same basic platform being adapted,
for instance, for the suspensions or the drive train in different models of vehicle
with different wheelbases and different track widths by extension of the support
panels with unchanged connection conditions. Figure 6.23 shows this in the
example of the B/C platform for the Audi A4, Audi A6 and VW Passat models.
Using only four instead of the 17 original platforms, the Volkswagen group with
its Audi, Seat, Skoda and VW makes is able to manufacture over 40 different
types of vehicle. Even the Fiat group has been able to reduce its number of platforms from the original 20 to four. Nissan is seeking to achieve a reduction from
25 to five by the year 2005, with corresponding cost benefits.
The same part concept is not only confined to the actual platform with floor
panel and side rail, but includes the chassis with the front and rear axle, the
complete propulsion system including the engine and gearbox, the tank, the
steering system, the seat frame and even the central electrics and hence a total of
60% of all the development costs (Fig. 6.24). The front axle developed for the
!.
IN
Identical parts
I . '>1 Adaptable parts
• • Width/length modifications
Same and matching parts of the B/C ~Iatf~rn: of the Volkswagen A~d
With unchanged connection conditions, the track Width I.S Increased by 42 ~m a
the wheel base by 143 mm from the Audi A4 to the Audl A6. The re~ultant Incre.ase
in the overall resilience of the bodywork is completely offset by the Increased wl~th
of the sills with the use of identical sheet-metal mo~lded p~rts. The lowest bendln.g
t I frequency is 46 Hz in both vehicles and the first torslo~~l ~atural frequency IS
~~ ~: in th(3 A4 and 46 Hz in the A6. These data. ~efine the ngldlty of the bodywork
and are thus essential for safety, comfort and driVing accuracy.
Fig. 6.23
Chassis and vehicle overall
421
. , . Identical parts
I
I Adaptable
parts
. , . Width/length modifications
Fig. 6.24 Common and matching parts (chassis, drive train, steering system,
tank) for the SIC platform of Volkswagen AG. The Audi A4 and A6 as well as the VW
Passat are, for example, built on this platform.
Audi A8 is thus used with the necessary modifications in the Audi A4, Audi A6
and Volkswagen Passat models (Fig. 1.54); BMW uses the front axle of the 7
series (1994) in the 535i and 540i vehicles (1996); and Porsche uses similar
wheel carriers and hubs as well as transverse links on the front and rear axles of
the Boxster (Fig. 1.46) and the front axle of the Boxster is also used in the 911
model (type 996, from 1997). By standardizing the cylinder-centre distance and
confining themselves to two sizes of hole, Fiat have succeeded in obtaining 67
variants from eight basic engines. With the consistent application of the same
part philosophy, companies say that up to 30% of components can be used on
different types.
-----_.,--------,
Bibliography
Chassis Reference Books
[lJ STOLL, HELMUT: Lenkanlagen und Hilfskraftlenkungen. Wiirzburg: Vogel
Buchverlag, 1992.
[2J REIMPELL, JORNSEN: Radaufhangungen. Wiirzburg: Vogel Buchverlag, 2. Aufl. 1988.
[3J REIMPELL, JORNSEN/HoSEUS, KARLHEINZ: Fahrzeugmechanik. Wiirzburg: Vogel
Buc:hverlag, 2. Aufl. 1992.
[4J REIMPELL, JORNSEN/SpONAGEL, PETER: Reifen und Rader. Wiirzburg: Vogel
Buchverlag, 2. Aufl. 1988.
[5J REIMPELL, JORNSEN/STOLL, HELMUT: Stof3- und Schwingungsdampfer. Wiirzburg:
Vogel Buchverlag, 2. Aufl. 1989.
[6J BURCKHARDT, MANFRED: Bremsdynamik und Pkw-Bremsanlagen. Wiirzburg: Vogel
Buchverlag, 1991.
[7J BURCKHARDT, MANFRED: Radschlupf-Regelsysteme. Wiirzburg: Vogel Buchverlag,
1992.
[8J PREUKSCHAT, ALFRED: Antriebsarten. Wiirzburg: Vogel Buchverlag, 2. Aufl. 1988.
[9J ZOMOTOR, ADAM: Fahrverhalten. Wiirzburg: Vogel Buchverlag, 2. Aufl. 1991.
Reference Books
[10J PIPPERT, HORST: Karosserietechnik. Wiirzburg: Vogel Buchverlag, 2. Aufl. 1992.
[llJ LEHMANN, WOLFGANG: Reparatur- und Einstelltabellen 1999/2000. Wiirzburg:
Vogel Buchverlag, 1999.
[12J BOSCH: Kraftfahrtechnisches Taschenbuch. Berlin, Heidelberg: Springer-Verlag,
22. Aufl. 1998.
[13J FAKRA-Handbuch. Band 1 bis 4. Berlin: Beuth Verlag, 10. Aufl. 1987.
[14J TOV Bayern: Anderungen an Auto und Motorrad. Miinchen: 2. Aufl. 1978.
[15J WALLENTOWITZ, HENNING: Aktive Fahrwerkstechnik. Braunschweig: ViewegVerlag, 1991.
[16J REIMPELL, JORNSEN; STOLL, HELMUT: The Automotive Chassis: Engineering
Principles. London: Arnold Group, pI Edition 1996.
[17J REIMPELL, JORNSEN; STOLL, HELMUT; BETZLER, JURGEN: The Automotive Chassis:
Engineering Principles. Oxford: Butterworth-Heinemann, 2nd Edition 2001.
[18J SCHULE, ROLAND: Fahrwerktechnik. Wiirzburg: Vogel Buchverlag, 2000.
(Lernprogramm).
Bibliography
423
[19] MATSCHINSKY, WOLFGANG: Radfiihrungen der StrafJenfahrzeuge. Berlin;
Heidelberg; New York; London; Paris; Tokyo; Hong Kong, Barcelona; Budapest:
Springer, 2. Autl. 1998.
General books
DIXON, 1. C.: Tires, Suspension and Handling. London: Arnold.
MILLIKEN, W.: Race Car Vehicle Dynamics, Society of Automotive Engineers,
Warrendale, PA, 1995.
GILLESPIE, T. D.: Fundamentals ofVehicle Dynamics, SAE R-114, Society of Automotive
Engineers, Warrendale, PA, 1992.
MATSCHINSKY, W.: Road Vehicle Suspensions, Professional Engineering Publishing,
1999.
STOLL, H.: Lenkanlagen und Hilfskraftlenkungen. Wtirzburg: Vogel Buchverlag, 1992.
REIMPELL, J., HOSEUS, K.: Fahrzeugmechanik. Wtirzburg: Vogel Buchverlag, 1992.
BURCKHARDT, M.: Bremsdynamik und Pkw-Bremsanlagen. Wtirzburg: Vogel Buchverlag,
1991.
BURCKHARDT, M.: Radschlupf-Regelsysteme. Wtirzburg: Vogel Buchverlag, 1992.
ZOMOTOR, A.: Fahrverhalten. Wtirzburg: Vogel Buchverlag, 1991.
BOSCH: Automotive Handbook, Society of Automotive Engineers, Warrendale, PA,
1993.
Journals
1.
2.
3.
4.
5.
6.
7.
8.
9.
Auto-zeitung. KOln: Heinrich Bauer Verlag.
auto, motor und sport. Stuttgart: Vereinigte Motorverlage.
Automobil·.[ndustrie. Wtirzburg: Vogel-Verlag.
Automobiltechnische Zeitschrift (ATZ). Stuttgart: Franckh - Kosmos Verlag.
k/z-betrieb. Wtirzburg: Vogel-Verlag.
konstruieren und giessen. DW;seldorf: VDI-Verlag.
Kugellager-Zeitschrijt. Schweinfurt: SKF GmbH.
mot. Stuttgart: Vereinigte Motorve~lage:.
.
Verkehrsunfall und Fahrzeugtechmk: Kippenhelm: Verlag InformatlOn Ambs.
--------,----.....----- ,--------'
Glossary of symbols
The index of the symbols and dimensions follows the international standards:
ISO 31
Quantities
ISO 2416
Passenger cars - Mass distribution
ISO 8855
Road vehicles - Vehicle dynamics and road holding ability _
Vocabulary
ISO 1000
SI - Units and recommendations
SAE J670e Vehicle dynamics terminology
and the German standards:
DIN 1301
Einheiten
DIN 1304
Formelzeichen
DIN 70 000 StraBenfahrzeuge, Begriffe der Fahrdynamik
In a few cases, the connection between the various standards could not be
achieved. In these cases priority was given to the ISO standards or specific
suffixes or symbols have been selected.
1 Reference points in figures
In den Bildem werden die einzelnen Bezugspunkte mit gro~en, nicht kursiven Buchstaben, die gleichzeitig als «Index» bei Formelzeichen dienen, bezeichnet:
Bo
C to G
M
o
p
Q
Ro
Tand U
body centre of gravity
reference points, in general
centre point
pitchpole
rollpole
centre of driving joint
roll centre
tie rod or linkage point
Glossary
Uforr
V
W
425
wheel centre point, front or rear
vehicle centre of gravity
center of tyre contact
2 Suffixes
The majority of symbols require the usage of suffixes for clear identification. In cases
where more than one is needed, a comma is set between them. A few cases, where small
letters follow capital ones, deviate from this rule. The various suffixes have the following meanings.
a
ax
A
A
b
b
B
Bo
c
co
dr
dyn
D
D or
e
ex
E
f
fix
fr
F
G
H
1
k
kb
lor L
10
10
L
m
m or med
max
mm
M
o
o
P
pI
driven, accelerating (one wheel only)
axial
drive-off condition, accelerating in general
Ackermann steering angle
braking (one wheel only)
baggage
braking (overall vehicle)
body
inertia
comenng
drivable, incl. driver
dynamic
damping
.
axle drive
due to the elasticity, (compliances) .
excess
earth fixed
front
fixed, idle
friction
fault, flaw
gearbox
steering-wheel
inside of curve, inner wheel
kinematic
kerb
left, left side
slipping, sliding or lock
loaded condition
aerodynamic
mass
middle, mean, medium
maximum permissible
minimum
motor
outside of curve, outer wheel
orifice closing plate
person, passenger
...
partial loaded (or partly laden) or deSIgn pOSItIOn
-----------~------'
--------
1
;\~
426
Pi
Pr
r
rad
rs
rsl
R
Re
Ro
S
S
Sp
t
tc
tr
T
T
Th
Tr
ul
U
V
X or x
Yor y
Z or z
o
1
2
X
8
Li
£
cp
(J'
T
Glossary
piston
piston rod
rear
radial
right, right side
'resulting
rolling (wheel)
residual, remaining
body roll center
steering
anti-roll bar, stabilizer
spnng
total or nominal value
turning circle
transportable
tyre
rod, tie rod or linkage
trailer hitch
trailer, single-axle
unloaded, empty condition
unsprung weight or axle weight
overall vehicle
longitudinal direction (see also suffixes a and b)
lateral direction
vertical direction
zero-point position or starting point
to the top, in jounce, in compression, in or one
to the bottom, in rebound, out or two
acceleration reaction support, angle or diagonal springing angle
steer angle
static toe-in angle
camber angle
body roll angle
kingpin inclination angle
caster
3 LE!ngths and distances in mm, cm or m
a and b
b
bD
be or r
bs
bsp
Lib
CR
C R•dyn
dorD
Ds
D s.r
distances and length in general
distance between vertical force Fz.w and G
distance of shock absorber (damper) attachment points (at rigid axles)
track width, front or rear
distance of anti-roll bar (stabilizer) attachment points at rigid axles
effective spring distance at rigid axles
track-change or track offset at rigid axle
dynamic rolling circumference at 60 kmh'l
dynamic rolling circumference at top speed
diameter, in general
track circle diameter, front
track circle diameter, rear
--------------------------------------
Glossary
D te
Dte,kb
e
f
h orR
h Bo
hRo,f or r
h U1
hv
iX,Bo orY,Bo
lZ,V
J
l
lBo,f or r
If or r
Ltix
Lt
nr
nr.k
n r •t
ODT
q
r
ra
rb
rdyn
r stat
rT
r,d
r,d,
t
ra
ra,t
rr.e
rr.k
rr.t
rr,T
R
S
SRe
St
ST
SI
Sz
turning circle diameter, wall to wall
turning circle diameter, kerb to kerb
wheel offset
diagonal spring travel
height, in general
height of body centre of gravity
height of roll centre at front or rear axle
height of the unloaded vehicle
height of the vehicle center of gravity
radius of inertia of the body center of gravity in X or Y_
direction
radius of inertia of the vehicle center of gravity in Z-direction
distance between the two steering axis at the ground
wheelbase
distance of body centre of gravity of the middle of the front or rear axle
distance of vehicle centre of gravity to middle of front or rear axle
idle (fixed) length of the shock absorber
total length of the vehicle
caster offset at wheel centre
kinematic lateral force lever arm due to caster
lateral force arm, in total
outer diameter of the tyre
force lever of vertical force
effective control arm length or force lever in general
force lever of longitudinal or tractive force
force lever of brake force
dynamic rolling radius of the tyre at 60 kmh- I
static loaded radius of the tyre
force offset in the centre of tyre contact (+) inside or (-) outside of curve
static toe-in (one wheel only)
total static toe-in (both wheels of one axis)
transverse offset at ground, static
total transverse offset at ground
elastokinematic caster offset at ground
kinematic caster offset at ground
total caster offset at ground
caster offset tyre
path radius
travel or stroke, in genral
residual wheel travel
total wheel travel
static tyre deflection
wheel travel in jounce
wheel travel in rebound
4 Masses, loads and weights in kg
m
mb
mBo
427
mass, load or weight in general
mass of luggage (baggage) related to one passenger
vehicle body weight
--------_._-------,--------
428
Glossary
mBo, forI'
mp
mt
ml,max
mTh
mIl'
mTr
LimTr
mU,forr
mV,dr
mV,for I'
mV,f, 10 or 1',10
mV,f,max or r,max
mV,f,pl or r,pl
mv'1
mV,l,max
mV,ul
mv'ul,O
Limv
mW
ml,Bo,for I'
ml,U,for I'
ml,Y,f or I'
part of body mass on front or rear
mass of one passenger
nominal design pay mass (minimum required)
permissible payload
weight of the trailer hitch
nominal mass of transportable goods
trailer load
tongue load, trailer
unsprung axle' mass, front or rear
weight of drivable vehicle (with driver)
axle load, front or rear
axle load under full loaded condition, front or rear
maximum permissible axle load, front or rear
partial axle load (design load), front or rear
gross vehicle weight (aVW)
maximum gross vehicle weight
kerb weight (actual weight of unloaded vehicle without driver)
kerb weight as published by the car manufacturer (with driver)
weight of vehicle options
weight of one wheel
part of body mass on one side of the front or rear axle
weight of one side of front or rear axle
axle load front or rear
5 F()rces in Nand kN
A lower-·case subscript letter after the symbol F means that the force refers only to one
side of the axle; an upper-case letter refers to the whole axle. An exception is F R , the
rolling resistance of the tyre. However, this can also refer to a wheel, an axle or the whole
vehicle; the subsequent further subscript enables the difference to be recognized. The
forces at the reference points, or at the links C to U of the wheel suspension, are denoted
by the letter of that particular point and the direction.
dF or LiF
Fo
Fe,BoorV
Ffr
FH
FL
Fa
F pi
Frsl
FR
Fsp
FT
F X,Bo,B,f or r
Fx,cx
F X,U,B,f or I'
FX,Y,B
FX.W,aorA
change of force
damping force
centrifugal force at the body centre or vehicle
friction force in general or related to one side of the axle
steering-wheel force
aerodynamic drag
force at pitch center
piston rod extensive or aid force
resulting friction
rolling resistance of the tyre
spring force, one side of the axle
tie rod or push rod force
brake reaction force to the body, front or rear
excess force
brake reaction force to the front or rear axle
brake force at the centre of gravity of the vehicle
accelerating force in the centre of tyre contact of one wheel (a) or both
wheels (A)
--------'-----------------~----------
Glossary
FX,W,b
F X,W,B,f or r
FY,T.e
Fy,v
Fy,w
FZ,Bo
FZ,Bo,B,f or r
FZ,l,max
FZ.U,forr
LJFz,v
FZ,Y,for r
F Z,Y,f.dyn or r.dyn
FZ,Y,1
F'z,w
Fz,w
LJFz.w
F(
F2
429
brake force in the centre of tyre contact of one wheel
brake reaction force to the front or rear axle
lateral force dur to camber
lateral force at vehicle
lateral force at wheel
static body weight (force)
body lift or dive differential force during braking, front or rear
force of maximum payload
weight (force) of front or rear axle
axle load transfer
axle load front or rear
dynamic axle load, front or rear
cross vehicle weight
verticle force at the centre of tyre contact
verticle force without the axle weight of one axle side
change of verticle force at one wheel
compressive force
rebound force
6 Moments in NM
M ZTX orW denote the wheel torque around the steering axle (z), followed by either X, Y or
Z for the direction of the (wheel) aligning force or the causal force, and left (1) or right
(rt) may also be indicated.
If a lower case t appears, this signals that both axle wheels are meant, whereby r relates
to the rear axle only. All other moments are indicated in section 7 below.
M Z•T•X
M Z•T•Y
MZ.T,y,r,t
MZ.W,aorA
MZ,W,borB
Mz,w,y
M z.w.z
Mz.w,r
(tyre) self aligning torque due to longitudinal force
(tyre) self aligning torque due to sideforces at the front wheels
(tyre) self aligning torque due to sideforces at the rear wheels
(wheel) aligning torque due to the accelerating force at one wheel (a) or
both wheels (A)
(wheel) aligning torque due to the brake force at one wheel (b) or both
wheels (B)
(wheel) aligning torque due to the lateral force
(wheel) aligning torque due to the vertical force
steering torque due to differences in caster
7 Other moments in NM
M aorA
M borB
M fr
MH
MM
MR
M X•Bo
M X •T•a
MY,Bo
Mz,v
T
driving torque related to one wheel (a) or axle (A)
braking torque related to one wheel (b) or axle (B)
friction torque
steering-wheel torque
engine (motor) torque
rolling resistance torque
rolling torque, body
overturning torque
pitching moment, body
yawing moment, vehicle
torsional moment, torque
,-----,-----------'
1
430
Glossary
8 Spring rates in Nmm-1 or kNm- 1
Cr or r
Cs
cS,rp
CSp
CT
Crp,forr
rate of the body supporting spring at parallel springing, related to the
centre of tyre contact of one axle side, front or rear
. rate of the anti-roll bar (stabilizer) at receptrocal springing
rate of the anti-roll bar related to the centre of tyre contact
static rate of the spring
spring rate of the tyre
front or rear rate of the body supporting spring at reciprocal springing
related to the centre of tyre contact
9 Angles in degrees or radians
a
a
a
a or a'
torsional angle of a joint or bushing
top view angle of the semi-trailing arm twist axis
angle of gradient of the road
inclination (rear view) angle of upper control arm (double wishbone
axle)
aror r
slip angle of the front or rear wheel
rear view angle of the semi-trailing arm twist axis
sideslip angle (dynamic)
driving angle of the axis (static)
inclination (rear view) angle of lower control arm (double wishbone or
McPherson axles)
x
acceleration reaction support angle or diagonal springing angle
steer angle
OA,o
Ackermann steer angle, nominal value to outside of curve
OH
steering-wheel angle
Om
mean steer angle
00 or i
actual steer angle, outside or inside of curve
Or
steering or toe-in angle at rear wheels
Ov,o
static toe-in angle of one wheel
Ov,o,t
total static toe angle
Ll
static toe-in angle
Lloe
part of steer angle due to suspension pitch
do
change of steer angle of both wheels
Llo
differential steer angle (actual value)
LloA
differential steer angle according to Ackermann (nominal value)
Lloe
part of steer angle due to compliances
LloF
steering flaw
LloH
part of steering-wheel angle due to manu~ steer
part of steering-wheel angle due to complIances
LlOH,e
.
residual angle at the steering-wheel.
LlOH,Re
Llok
change of toe-in or steer angle due to kmematIcs
e
brake reaction support angle
camber angle
e orew
LleW,k or dew,k part of camber angle due to kinematics
part of camber angle due to steer
Ll e W,8
part of camber angle due to suspension roll
Llew,rp
o
-------'-----------r-----------------'-
Glossary
qJ
.1qJ or d({Jk
A,
e
AorB
.1e
(j
1'or 1'c
1'r
.11'k
.11'0
'If
~
~o
431
body roll angle
kinematic change of body roll angle
steering arm angle
body pitch angle under accelerating or braking
pitch angle change due to load changes
kingpin inclination angle
caster angle of the (steered) front wheels
caster angle at rear wheels (not steered)
part of caster angle due to kinematics
part of caster angle due to steer
yaw angle
top view angle between two control arms or roads
inclination of the shock absorber
10 Characteristics and data with no
dimensions
ldyn
lo
lo
la
II
lm
lR
ls
./
ls
lsp
luI
lq>
kb
ko
km
kR
kR•co
kR,o
kT
kv
kx
ko,v,rp
ke
ke,w,rp
k/1
n
no
SX,W,aorb
SY,W
Z
<Pc or r
dnamic steering ratio
axle differential ratio
ratio of shock absorber (damper) to the wheel
gearbox ratio
ratio of wheelbase to vehicle length
mass ratio
roll resistance ratio coefficient
overall kinematic steering ratio
steering gear ratio
ratio of spring to the wheel
ratio of vehicle centre of gravity to height of the unloaded vehicle
ratio of the wheel to the spring, shock absorber or anti-roll bar at reciprocal springing of a rigid axle
ratio of tread to width (breadth)
damping coefficient
load factor
rolling resistance coefficient
rolling resistance coefficient when concerning
rolling resistance coefficient measured on a tyre test rig
factor of the increase in tyre spring rate
velocity factor
anti-dive coefficient, accelerating
suspension roll steering coefficient
anti-dive coefficient, braking
suspension roll camber coefficient
friction coefficient correction factor, tyre
number of specified seats
number of seats engaged
longitudinal slip under accelerating or braking lateral slip
lateral slip
braking factor
brake force fraction front or rear
---------'--"'!""""------ ._--------
432
1]
/lrsl
/lx,w
/lX.W,lo
/lY,W
/lY,W,lo
Glossary
total efficiency
resulting coefficient of friction
coefficient of longitudinal force
coefficient of sliding braking force
~oefficient of lateral force
coefficient of sliding lateral force
11 Other symbols with dimensions
ax
ay
A
A5
Cs
E
f
g
HRC
I
JX,Bo
h,v
ko
n
Phyd
PT
q
Re
Rm
R pO.2
v or Vx
Vo
Vw
OJ
longitudinal acceleration or deceleration
lateral acceleration
area, cross-section area
ductile yield, elongation at rupture (La = 5 X do)
stiffness of the steering system
modulus of elasticity
frequency
acceleration due to gravity
Rockwell hardness
area moment of inertia
dynamic moment of inertia of body around the
longitudinal axis
dynamic moment of inertia of body around the
transverse axis
dynamic moment of inertia of vehicle around the
vertical axis
damping value
revolutions per minute or vibration frequency
hydraulic pressure
tyre pressure
climbing capability factor
yield strength
tensile strength
0.2% yield strength
longitudinal velocity
piston velocity in shock absorber
circumferential tyre velocity
circular frequency
m S-2
m
S-2
2
m
%
Nmrad- I
Nmm-2
Hz
m
S-2
cm4
kg m2
kgm 2
kgm2
Nsm- I
min-I
N cm-2
bar
%
Nmm-2
Nmm-2
Nmm-2
m S-I or km h- I
m S-I
m S-I
Hz
------,--,-------------------------
Index of manufacturers
(As mentioned in the text)
Audi
air sprung double wishbone axle 5.17, 5.18
anti-roll bar 3.85, 3.86
driven front axle
on A4 1.54
on A6 1.57
overall steering ratio 3.96
platform assembly 6.23, 420
rear axle on Quattro 1.76
steering gear 273
Torsen differential on Quattro 1.71
torsion crank axle on A6 1.61
track alteration on A6 3.15
twist beam suspension on A6 1.58
BMW
air bags 4.25
air springs 341
camber alteration on 3 series 3.48, 3.49
drive layout on 3 series 1.32
four wheel drive assembly 1.80
front axle on Roadster 1.40
multi-link rear axle 1.1
overall steering ratio 3.95
steering deviation on 3 series 1.32
suspension control arm on 23 3.83, 3.84
track alteration on 3 series 3.19
use of common parts 421
Chevrolet
drive layout of Corvette 1.33
rear axle of Corvette 1.34
Citroen
centre axle steering on GSA 3.114
hydro-pneumatic springing 341
Daimler-Benz-Transporter
leaf springs on Sprinter 345
Fiat
caster alteration on Uno 3.143
elastic camber change 3.57
four wheel drive on Compagnol0 1.70
layout of Panda Treking 1.68
McPherson strut on Panda 5.54
platform assembly 420
rear wheel bearing of Panda 1.59
toe-in alteration 3.79
Ford
McPherson strut 1.10, 15
rigid axle on Escort Express 1.24
Honda
camber alteration on Accord 3.48. 3.49
front suspension on Prelude 1.55
rear axle on Civic 1.62, 177
track alteration on Accord 3.15,3.19
Lancia
elastic camber change 3.57
front axle 1.56
front wheel drive on Thema 1.51
McPherson strut 1.12
rear wheel suspension 1.60
toe-in alteration 3.79
Mercedes Benz
air springing 341
all terrain vehicle 1.67
camber alteration 3.48,3.49,3.131, 241
--------_._--------- , - - - - - - - - -
434
Index of car manufacturers
caster alteration 3.143
double wishbone front axle on C class 5.5
driven rear axle on lorry 1.42
front axle
of Sprinter 1.41
on vans I.3Z
shock absorber 5.31
with four wheel drive 1.81, 82
front suspension on S class 1.39
multi-link suspension 19
overall steering ratio 3.95
rear engine drive 1.44
recirculating ball steering 4.15
steering
assembly 4.24
deviation 3.92
on S class 1.38
step steering input 4.2
strut damper front axle 4.12
track alteration 3.19
trailing arm rear suspension 1.13, 1.16
Mitsubishi
rear axle on Pajero 1.43
Nissan
vehicle assembly 419, 420
Opel
elastic camber change 3.57
electric power steering on Corsa 4.20, 4.23,
287
hydraulic steering
on Astra 4.18
on Vectra 4.16
kinematics of Omega rear axle 3.20
McPherson strut 1.8
overall steering ratio 3.95, 3.96
toe-in alteration 3.79
. toe-in angle on Omega 3.69
track alteration on Astra 3.15
Peugeot
front-wheel drive 1.47, 1.52
Porsche
front axle of Carrera 1.75
mid engine Boxster 1.46
stability management 1.75
type of steering gear 273
use of common parts 421
Renault.
elastic camber change 3.57
front wheel drive 1.48
springing curve 5.9
toe-in alteration 3.79
trailing arm rear axle 1.63
twist beam axle 1.2
Toyota
elastic camber change 3.56, 3.57
toe-in alteration 3.79
Vauxhall
driving forces on Cavalier 1.64
front wheel drive on Corsa 1.49
rack and pinion steering
on Astra 4.11
on Corsa 4.9
Volkswagen
camber angle on Golf 3.55
caster alteration on Polo 3.143
dampers on Golf 5.50, 5.51
double wishbone suspension 1.7
effect of loading on Polo 3.78
elastic camber change 3.56
four wheel drive 1.72
front axle of Passat 3.1
front wheel drive on Polo 1.50
McPherson strut 1.9
overall steering ratio on Polo 3.95
platform assembly 6.23, 6.24, 420
rear drive Transporter 1.45
steering column
for bus 4.31
on Golf 4.26
release clutch 4.29
steering gear on Polo 4.1
toe-in angle on Golf 3.70
track alteration on Golf 3.16
type of steering gear 273
-----,--,--------------._---------~'
Index 'of suppliers
(Suppliers to the car industry as mentioned in the text)
Bilstein Ltd
front axle damper 5.31, 360
monotube shock absorber 5.30
shock absorber seal 5.32
Continental
Tyres 2.9,2.19,2.51,2.52
Continental AG
air-sprung axle 5.17,5.18
ContiTech Formtelle GmbH
McPherson strut 1.10
Dunlop
ZR tyres 2.17
Dupont
bearing element 4.14
Elastogram
supplementary springs 5.21,5.50
GKN AutomotivE~
front wheel output shaft 1.53
sliding joints 1.17
GKN-Birtield AG
dual joint 1.3
HaIdex
multi disc clutch 1.72, 1.73, 72, 83
Hayes Lemmerz
sheet metal disc type wheel 225
Krupp-Briininghaus
steel springs 5.20
Lemrorder Fahrwerktechnik
adjustable tie rod 4.13
anti-roll bar 3.85
axle sub-assemblies 1.82
collapsible steering column 4.27
control arm of front axle 5.5
elastic bearing 3.87
electrically adjustable steering column 4.30
pre-lubricated tie rod joint 4.14
steering column on
Golf 4.26
Volvo 4.28
Monroe
McPherson strut 5.54
NSK
servo assemby 287
Pneumatiques Kleber SA
tyre markings 2.18
Sachs Boge
dampers with stops 5.47, 5.48, 5.49
McPherson strut 5.55, 5.56
non-pressurised shock absorbers 5.25, 5.26,
5.28
rear spring dampers 5.51
shock absorbers with variable damping 5.57,
5.59,5.60
Stabilus
steering dampers 5.39, 5.40
436
Index of suppliers
Zahnradfabrik Friedrichshafen
axle sub-assemlies 1.82
electric power steering 4.21
power divider 1.79
rack and pinion steering
pinion gear 4.10
with hydraulic power 4.19
recirculated ball power steering
4.17
variable ratio rack 3.97, 3.98
---------'--.,.....--------------
Subject index
Please note that:Figure and Table numbers are given in italics (e.g. 2.6) and come before page numbers
Equation numbers have the prefix e (e.g. e3.23b)
*
*
A-bracket axle see drawbar axle
Ackerman angle 3.89, 3.92, 208-9
Air bags 4.25
Allgemeine Betriebserlaubnis (ABE) see German
regulations
Alloy wheels
advantages 1.56,2.24, 114-15
Hayes Lemmerz type 2.24
Anti-dive and anti··squat mechanisms 255-65,
410
concepts 255
pitch axis
front 3.153, 3.154, 255-58
front wheel drive 3.156
rear wheel drive 3.154,3.155,256
rear 256-60, e3.44
longitudinal links 3.158
rigid 3.161
semi-trailing links 3.160
trailing links 3.159, 259
Anti-lock braking system (ABS) 2.33, 81, 82,
250,258
Anti-roll bars 5.2, 5.54, 309,346-47, e5.20,
e5.21
on Audi 3.85, 3.86, 3.87
Aquaplaning 2.35, 126-27
Assembly of vehicles using common parts
419-21
Axle drive angle 3.63, 190
Axle settings 150-51
Body roll centre 160-75
body roll axis 164-66
calculation 3.24
theory 3.23
calculations 3.2/, 3.22, 161-64, e3.2, e3.3
definition 3.19, 3.20, 160-63
DIN standards 160, 172
independent suspensions 166-72
calculation 3.25, 3.26, 3.27, 3.28, 166-67,
e3.4
McPherson struts 3.29,3.30, 3.31, 168-70,
e3.4a
rear axles 3.32,3.33,3.34,3.35,3.36, 170-72
kinematics of rear axle 3.20
on rigid axles 172-75
determination of height with 3.37, 3.38
drawbar 3.42, 3.44, 175
leaf springs 3.39
panhard rod 3.40, 174
Watt linkage 3.41, 3.43
on twist beam suspensions 172
Braking behaviour see vehicle braking behaviour
Bump stop 5.47, 5.49, 372
Camber 175-87 see also kingpin inclination
alteration 3.54,3.56, 3.57, 3.130, 3.131, 3.132,
239-41
angle 3.55, 183, e3.6
calculation 3.50,3.51,3.52, 181-82
during comering 182-85
coefficient 182, e3.5
factor 184, e3.7
forces 3.53, 183
definition and data 3.45, 3.46, 175-78, e3.4
elasticity 3.56, 3.57, 185-87
kinematic alteration 3.47, 3.48, 3.49, 178-81
Cardan joint 288-89
Caster 230-54
alteration 3.136, 3.144, 239-44
calculation 242, e3.40
on front wheels 245-50
with McPherson struts 3.139, 3.141,
3.142,3.143, 246-48
438
Subject index
Caster, on front wheels - cont.
vehicle loading 3.137,3.138,3.140,
245-47
and kingpin inclination 239-40
angle 230-34
calculation 3.134, 3.135
definition 3.115, 231
during cornering 3.119,3.120,3.121,232
kinematic caster trail 3.133, 230-34
calculation 232, e329, e330
definition 3.115, 231
offset 2.49,3.116, 140-42, e2.20
positive and negative 3.117,3.118,231
and the rear steering knuckle 3.145,250-51,
e3.4lb
resolution of the vertical wheel force 3.146,
3.147,3.148,3.149,3.150,3.151,3.152,
251-54, e3.42
righting moments 235-39
calculation 3.125,3.126, 235-36, e3.33
when cornering 3.127, 3.128, 3.129,
237-39
settings and tolerances 254
in straight running 3.122,3.123,3.124,
234-35
Centre of gravity of vehicle see vehicle centre of
gravity
Chassis alignment 260-65
caster angle and lift height 3.165, 264-65
measurements 3.164, 262-65
test equipment 260-62
Chassis/Simulation Technology Laboratory,
Cologne 3.162, 3.163
Coefficients of friction
with lateral forces on wheels 2.39, 2.45, 130,
e613a
with rolling forces on wheels 2.33, 125, e2.5a,
e2.6a
and skid points 6.22, 418
for wheels 2.33, 2.39, 2.45, 125, 130, 132-33
Compound crank axle see twist beam suspension
Contre Pente wheel rim 112
Cornering
effect of camber 182-85
effects on wheels 2.39,2.44, 122-24, 129-30,
133-34, e2.14
lateral forces on wheels 2.39, 129-30
rolling resistance of wheels 2.32, 122-24,
e2.4b-d
Crab angle see toe-in angle
Crank axle see rear axle trailing arm suspension;
rigid crank angles
Dampers see shock absorbers
Deutsche Institut filr Normung (DIN) standards
see German standards
Differential on front wheel drive car 1.71, 67
Double wishbone suspension see also multi-link
axles; rigid crank axles
VW design 8-10
on VW van 1.7
Drawbar axle 3.42, 175
Driven front axles 51-56
Audi A4 1.54
Audi A6 1.57
design 51-52
Honda Prelude 1.55
by Lancia 1.56
and McPherson struts 56
Driven rear axles see rigid crank axles
Dual joints
GKN design 1.3
top view 1.4
Dynamic steering ratio 3.99, 3.100, 215-18, e320
Elastokinematics 149-266 see also roll centre;
track; wheels
definition 3. I, 3.2, 3.3, 149
measurements 3. 164, 263-65
test equipment 260-62
European Tyre and Rim Technical Organisation
(ETRTO) 2.14, 86,97, 105
European Union directives
axle loads 324
curb weights 322
mass of vehicle 320
passenger cars 319
steering 266
towed trailer load 66, 322
tyres 87
vans and lorries 328
Four wheel drive 64-80
advantages and disadvantages 64-70
different kinds tabulated 1.83
on Fiat Campagnole 1.70
on front wheel drive car 72-79
differential on Audi Quattro 1.71
front axle of Porsche Carrera 1.75
layout by Mercedes Benz 1.78
layout by Volkswagen 1.72
rear axle of Audi Quattro 1.76
rear axle of Honda Civic 1.77
hill climbing capacity 1.65
layout of Fiat Panda Treking 1.68
Mercedes G all terrain vehicle 1.67
with overdrive 68
on rear wheel drive car 80-81
BMW assembly 1.80
front suspension on Mercedes Benz 1.81, 82
planet gear differential 1.79, 80-81
rear suspension on Mercedes Benz 1.82
Friction coefficient see coefficient of friction
---------'--~-------------
Subject index
Front engine, rear drive 30-41 see also
Non-driven front axles
advantages 32-33
axle load distribution 1.36
disadvantages 34
layout
of BMW 3 series 1.32
of Chevrolet Corvette 1.33, 1.34
stability 1.35
Front hub carrier see steering knuckle
Front-wheel drive 45-64
advantages and disadvantages 48-51
engine mountings 46
GKN output shaft 1.53
on Lancia Thema 1.51
by Peugeot 1.47, 1.52
by Renault 1.48
on Vauxhall Corsa 1.49
on VW Polo 1.50
German regulations
Allgemeine Betriebserlaubnis (ABE)
axle load 323
Strassen Verkehrs-Zulassungsordnung (StVZO)
axle load 323
curb weight 319
German standards see also European Union
directives; international standards
DIN (Deutsche Institut fUr Normung)
all purpose passenger car 67
axes of coordinates 3.3
body roll centre 160, 172
camber 175,185
caster angle 231, 254
curb weight 319
kinematics 149
nomenclature 288
steering moment 219
toe-in angle 3.58, 187
turning circle 212
tyres 2.22, 87, 110-12
weight and load 318-19
wheels 2.25
VDA (Verband der Automobilindustrie)
design weight 323
VDI (Verein Deutscher Ingenieure)
vibration 308
WdK (Wirtschaftsverband der Deutschen
Kautschukindustrie)
tyres 2.16, 87, 119
Haldex multi-disc clutch 1.72, 1.73, 72
Handling characteristics of vehicles I
Independent wheel suspensions see also rigid
crank axles; semi-rigid crank axles
439
general characteristics 1-7
non-driven rear axles 60-64
on Audi A6 1.6/
on Honda Civic 1.62
by Renault /.63
reaction forces /.5
reciprocal springing 1.6
requirements 3, 7-8
International standards (ISO) see also European
Union directives; German regulations;
German standards
axes of coordinates 3.3
axle loads 324
curb weight 319
exterior protection for passenger cars 323
kinematics 149
kingpin inclination 221
load dstribution 325-28
payload for passenger cars 320, 321
tyres 87
Joints
GKN design 1.3
top view 1.4
sliding 1.17
Jounce stop 370-71
Kinematic alteration
due to bumps 3.20, 3.74
due to camber 3.47, 3.48, 3.49, 178-81
Kinematic caster trail 3.133, 230-34
calculation 232, e329, e330
definition 3.115, 231
Kinematic steering ratio 213-15
definition 213, e317, e318
overall steering ratio 3.95,3.96, 214, e3.9
rack and pinion steering 3.97,3.98, 214-15
Kingpins 221-30
braking moment-arm 225-28
with brake on the inside 3.110,3.111
calculation 3.108, 3.109, 226-28, e3.26,
e3.27, e3.28
with front wheel drive 3.110
kingpin inclination angle 3.103, 221-25 (see
also camber)
calculation of forces 3.24, 3.25, 3.107,
223-24, e3.21a, e3.22, e3.23
and camber 3.104, 223
definition 221
and McPherson struts 222
kingpin offset 3.102, 3.105, 3.106, 222,
230
longitudinal force moment-arm 228-30
causes 3.112, 228-29
and kingpin offset 3.113.3.114, 229-30
- - - - - - - - - , - -.....------'---------
440
Subject index
Laboratory for Chassis/Simulation Technology,
Cologne 3.162,3.163
Lateral forces on wheels 2.37, 2.43, 2.46, 2.47,
128-30, 132-34, e2.8, e2.9, e2.l1, e2.12,
e2.l3, e2.16, e2.l7
coefficient of friction 2.39, 2.45, 130, e613a
during cornering 2.39, 129-30
and slip angle 2.38, 2.39, 128
variable factors 134-38
Leaf springs 5.20, 344-45
on rigid axles 3.39
on dgid crank axles 1.26, 1.27, 1.28, 26, 27
Mass moments of inertia see vehicle mass
moments of inertia
McPherson struts see also ddven front axles' nonddven front axles; shock absorbers '
advantages 10
and caster alteration 3.139,3.141,3.142,
3.143, 246-48
design 10-15,375-77
on driven front axles 5.52, 5.53, 56
forces on 1.11
on Ford Focus 1.10
on Ford Mondeo 15
on independent suspensions 3.29, 3.30, 3.31,
168-70, e3.4a
on Lancia Delta 1.12
on non-driven front axles 35, 36
on Opel Omega 1.8
and rack and pinion steering 4.1, 276
and running wheel comfort 5.5,311-12
and stl~edng kinematics 4.46, 4.47, 303-4
and toe-in angle 3.67, 192-93
twin tube struts
non-·pressudsed 5.54, 377
pressudsed 5.55,5.56, 377-81
on VW Golf 1.9
Multi-disc clutch 1.72, 1.73, 72
Multi-link suspension see also double wishbone
suspension; dgid crank axles; semi-trailing arm rear suspension; trailing arm rear
suspension
on BMW 1.1,1.19
descdption 1-2, 19-22
disadvantages 22
by Ford Werke AG 1.18
on Mercedes Benz 19
Non-driven front axles 35-39
on BMW Roadster 1.40
front suspension on Mercedes Benz S class
1.39
with McPherson struts 35, 36
on Mercedes Benz Sprinter 1.41
on Mercedes Benz vans 1.37
steedng on Mercedes Benz S class 1.38
Non-drive~ rear axles 56-64 see also dgid axles;
tWIst beam suspension
designs 56-60
independent suspension 60-64
on Audi A6 1.61
on Honda Civic 1.62
on Lancia YIO 1.60
by Renault 1.63
rear wheel bearing on Fiat Panda 1.59
se~i-trailing arm on Mercedes-Benz 1.16
tWIst beam suspension on Audi A6 1.58
Overdrive 68
Oversteer 2.40, 2.41, 2.42, 130-32, e2.10
and toe-in angle 3.72, 3.73
Panhard rod 3.40, 174
Pitch angle 402-7
Pitch axis
front 3.153, 3.154, 255-58
front wheel drive 3.156
rear wheel drive 3.154,3.155,256
rear 258-60, e3.44
longitudinal links 3.158
dgid 3.161
semi-trailing links 3.160
trailing links 3.159, 259
Pitman arm joint 4.15,270-71
Planet Wheel-Centric Differential 1.79, 80-81
Porsche Stability Management (PSM) 1.75
Power steering 281-88
electrical 286-88
advantages 286
assembly on Opel Corsa 4.20, 4.23, 287
layout on pinion 4.21
electro-hydraulic 283-86
control system 4.19
layout on Opel Astra 4.18, 283-84
hydraulic 281-83
design 283, e4.1, e4.2
layout on Opel Vectral 4.16, 282-83
pdnciples of operation 4.17, 283
Rack and pinion steering 3.97, 3.98, 214-15,
271-77 see also recirculating ball steering; steedng system
advantages and disadvantages 271-72
with centre tie rod 4.11, 276
configurations 4.8, 272-73
with McPherson struts 4.1, 276
section through pinion gear 4.10
with side tie rod 4.8,4.9, 272-75
Rear and mid engine drive 41-45
disadvantages 42, 44-45
by Mercedes Benz 1.44
Subject index
Porsche Boxster 1.46
VW Transporter 1.45
Rear axle trailing arm suspension see also semitrailing arm rear axles
design 15
forces on 1.14
on Mercedes Benz 1.13
Recirculating ball steering 278-81
adjustable tie-rod 4.13
advantages and disadvantages 278
layout and principles 4.17
pre-lubricated tie rod joint 4.14
steering gear 4.15, 280-81
strut damper front axle 4.12
Ride comfort 1
Rigid crank axles 22-30
advantages 25
design 3
disadvantages 22
forces on 1.22. 1.23, 1.25
on Ford Escort Express 1.24
on Mercedes Benz lorry 1.42
on Mitsubishi Pajero 1.43
mutually opposed springing 1.21
with non-driven rear axles 56, 60
and steering 1.29
on Volkswagen LT 1.20
Roll centre see body roll centre
Rolling forces on wheels
coefficients of friction 2.33, 125, e2.5a, e2.6a
road influences
aquaplaning 2.35, 126-27
snow and ice 2.36, 127-28, e2.7
wet and dry 2.34, 126, e2.6b
slip 124-25, e2.4e-f
Rolling resistance of wheels
variables 2.31, 124
when cornering 2.32, 122-24, e2.4b-d
when driving straight 2.31, 2.32, 121-22, e2.4
Safety shoulders on wheels 2.21, 112
Scrub radius see kingpin offset
Self-aligning torque of wheels 140, 142-44,
e2.21, e2.22, e2.23
Self-centring steering 130-32,218-21
forces involved 3.101,218-20, e3.21
Semi rigid crank: axles 28-30 see also twist beam
axle
disdavantages 30
forces on 1.3 J
on Volkswagen 1.30
Semi-trailing arm rear axles 17-19,39
on Mercedes Benz V class 1.16
on Opel Omega 1.15
and toe-in angle 3.62, 189-90
Sheet metal disc type wheels 2.23, 2.25, 2.26,
114-15
441
Shock absorbers 347-74 see also McPherson
struts
damper attachments 367-70
eye-type joints 5.45, 369
pin-type joints 5.46, 369-70
requirements 367-69
damping characteristics 5.41, 5.42, 5.44,
366-67, e5.24, e5.25
rear axle damping curve 5.43, 367
fitting 5.23, 348-49
monotube
non-pressurised 364-66
as steering dampers 5.39, 5.40, 364
pressurised 357-64
advantages and disadvantages 359,
363-64
damping curve 5.35, 5.36, 5.37, 362
design 5.30, 357-59
on front axle of Mercedes 5.31
piston rod and guide 5.32, 359-60
pistons and valves 5.33, 5.34, 360-63
spring/damper units 375
stops 370-75
bump stop 5.47, 5.49, 372
jounce stop 370-71
supplementary springs 5.50, 372-75
twin tube
non-pressurised 349-57
air venting 353-54
compression stage valve 5.26, 5.28, 355
damping curve 5.27
design 5.24, 349-53
function 353
guide and seal 5.25
rebound valve 5.26, 5.28, 354
pressurised 5.29, 355-57
variable damping 381-85
by-pass flow 5.57, 381
characteristics 5.58, 5.60
Sliding joints 1.17
Slip angle 2.38, 2.39, 128
Snow and ice 2.36, 127-28, e2.7
Springing 309-80
behaviour of wheels 2.27,2.28, 116-18, e2.3
comfort requirements 307-14
preventing front end shake 313-14
running wheel comfort 311-13
hysteresis of springing curve 5.6, 311
and McPherson struts 5.5, 311-12
vibration insulation 311
seating 5.1,307-9
springing comfort 309-11
effect of potholes 5.3,5.4, 310, e5.0
use of anti-roll bars 5.2, 309
variables involved 310-11
shock absorbers (see shock absorbers)
springing curves 328-39
cornering 5.16, 334-39
442
Subject index
Springing, springing curves - con!.
body roll angle 337-38, e5.15, e5.16
height change 336, e5.13
reciprocal springing 338-39
spring travel 336-37
wheel load change 5.15,334, e5.12, e511
diagonal springing 339
front axle 328-31
for front wheel drive car 5.13
spring design 328-29, e5.10
for standard passenger car 5.12
rear axle 332-34
spring design 332
for standard passenger car 3.14
spring types 340-47
air and gas filled 340-43
advantages 340-41
on Audi A6 5.17,5.18
combined shock absorber 5.18, 5.19
anti-roll bars 5.22, 346-47
steel leaf 5.20, 344-45
stops and supplementary springs 5.21,
345-46
vibration 314-18
calculation of rates 314-16, e5.1, e5.2, e5.3,
e5.4, e5.5, e5.6
forces on simple system 5.7
front wheel springing curve 5.9,317-18
standards 308
wheel vibration rate 5.8
weights and axle loads (see vehicle weights
and axle loads)
Steer angle see also steering ratio
calculations 3.11,3.12,208-9, e3.9, e3.10
kinematic relationships 3.89
with transverse engine 3.88
Steering column 288-94
adjustable 4.30, 290-91
assembly 4.23, 288-89
collapsible 4.26, 4.27, 4.28
configuration 4.24, 289-90
Steering damper 294
Steering kinematics 294-306
linkage configuration 296-99
opposed 4-bar linkage 4.37, 4.38, 297-98
rack and pinion steering 4.39, 4.40, 4.41
synchronous 4-bar linkage 4.36, 297-98
tie rod length and position 4.48, 299-306
double wishbone suspension
calculations 300-303, e4.3
control arms 4.45
geometry 4.42, 4.43,4.44, 300-303
longitudinal transverse axles 4.49, 304
McPherson struts 4.46, 4.47, 303-4
type and position of gear 294-96
tie rod
joints 4.33, 4.34
length 4.32, 4.35, 295
Steering knuckle 3./45,250-51, e3.41b
on non-driven front axles 1.38, 35
Steering ratio see also steer angle
dynamic 3.99,3./00,215-18, e3.20
kinematic 213-15
definition 213, e317, e318
overall steering ratio 3.95, 3.96, 214, e3.9
rack and pinion steering 3.97, 3.98, 214-15
Steering self-centring 130-32,218-21
forces involved 3.101, 218-20, e3.21
Steering system 266-71 see also power steering;
rack and pinion steering; recirculating ball
steering
on independent wheel suspensions 269
requirements 266-69
on rigid axles 269-71
with leaf springs 4.5, 4.6, 270
self-steering effect 4.7
steering gear on VW Polo 4.1
step steering input 4.2
synchronous steering A-bar 4.3
Strassen Verkehrs-Zulassungsordnung (StVZO)
see German regulations
Technical Laboratory, Polytechnic of Cologne
6.4
Technischer Uberwachungs Verein (TUV) 276
Til ted shaft steering rear axle 1.15
Toe-in angle 187-208
alteration in motion due to
bumps 3.64,3.65,3.66,3.74, 191-92
lateral forces 3.78, 3.79, 3.80, 3.81,
199-200
longitudinal forces 200-208
during braking 3.82, 200-202
front wheel tractive forces 206-8, e3.8e
radial tyres
Audi anti-roll bar 3.85, 3.86, 3.87
BMW control arm 3.83, 3.84
roll 3.69, 193-94
and axle drive angle 3.63, 190
definition 187, e3.8
forces 3.60, 188
and front wheel drive 3.61, 189, e3.8
oversteer effect 3.72, 3.73
and passenger loading 3.77
and rim size 1.88, 3.59
with semi-trailing arms 3.62, 189-90, e3.8
and steering 3.75
and tie rods 3.67, 3.68, 192-93
understeer effect 3.71,3.76, 198
Torque steer effects of wheels 2.53, 2.54,
146-48
Torsen differential 1.71, 67
Torsion beam suspension see twist beam
suspension
Torsion crank axle on Audi A6 1.61
Subject index
Track 151-60
alterations caused by bumps 3.5, 3.6, 3.7, 3.8,
3./5, 152-54
calculations involving McPherson struts 3.9,
3./0,3.11, 154-56
design methods 3.12, 3./3, 3.14, 3.15, 3.16,
3./7,3.18, 156-60
effect of tread width 3.4, 152, e3.la
Traction control 1.66, 68
Turning circle 209-13
calculations 209-10,212-13, e313, e314, e316
cornering force 3.91
kerb to kerb 3.93,210-12, e315
nominal steering curve 3.92
swept 3.94, 212
Twist beam suspension
advantanges 5-6
on Audi A6 1.58
with non-driven rear axles 56
Renault design 1.2
Tyres 86-110 see also Wheels
diagonal ply 2.1, 2.2, 89-91
dimensions and markings 2.15,97-101, 105
for light commercial vehicles 100
for passenger cars 2.12, 2.13, 2.14, 2.16,
2.17,2.18,97-100
DIN standards 2.22, 87, 110-12
European Tyre and Rim Technical Organisation
(ETRTO) 2.14, 86,97, 105
European Union directives 87
height-to-width ratio 2.8, 2.10, 2.11, 93-97
influence on speedometer accuracy 108-10,
e2.2a
interchangeability 2.9,86-87
International standards (ISO) 87
load capacity 2.14, 101
pressures 2.13, 101-4
profiles 110
radial ply 2.3,2.4, 2.5, 91-93
requirements 86-89
on commercial vehicles 89
on passenger cars 87-88
rolling circumference and speed 105-7, e2.l,
e2.2
standards 86--87
tubeless 93
typical designs 2.19
tyre print 2.9
U.S. designations 98
valves 2.6, 2.7
and wheel camber 104-5
Understeer 2.40, 2.41, 2.42, 130-32, e2.10
and toe-in angle 3.71,3.76, 198
University of Applied Science, Cologne 3.162,
3.163
University of Cologne 326
443
Vehicle
assembly with common parts 419-21
braking behaviour 397-410
braking forces 6.7, 397-99, e6.l3-e6.20
pitch angle 402-7
and springing 6.15,404-7,
e6.22-e6.25
radius-arm axes 407-10
anti-dive control 410, e6.32, e6.33
forces involved 6.16, 6.17,407-9,
e6.25-e6.31
pitch angles 409-10
stability 6.8-6.12, 399-402, e6.21
position of brake 6.13,6.14
centre of gravity 386-95
and axle weights 392, e6.4
and body weight 6.4, 392-94
calculation 6.1, 387-88, e6.1, e6.2
height 6.2, 6.5, 388-90, e6.3, e6.4
importance 386-87
influence of loading 6.3, 390-91
mass moments of inertia 394-97
calculation 395, e6.8
radius of gyration 6.6, 396, e6.9-e6.12
traction behaviour 410-19
acceleration 6.18,410-14, e6.34-e6.36
calculation for front wheel drive 6.19,
6.20, 412-14, e6.37, e6.38
climbing ability 6.21,414-16,
e6.39-e6.42
skid points 416-19
calculations 416-17, e6.43-e6.46
and coefficients of friction 6.22, 418
four wheel drive 419, e6.47
weights and axle loads 319-28
curb weight 319-20
international standards 321-28
load distribution 325-28
with front wheel drive 5.11, 326-27
hatchback and estate cars 327-28
standard passenger car 5.10, 325-26
vans and lorries 328
permissable
axle load 323-25
payload 320-23
when towing a trailer 322, 324
Verband der Automobilindustrie (VDA) see
German standards
Verein Deutscher Ingenieure (VOl) see German
standards
Vibration 314-18
calculation of rates 314-16, e5.1, e5.2, e5.3,
e5.4, e5.5, e5.6
forces on simple system 5.7
front wheel springing curve 5.9,317-18
standards 308
wheel vibration rate 5.8
Visco clutch 1.74, 78
444
Subject index
Watt linkage 3.41
Weights and axle loads see vehicle weights and
axle loads
Wheelbase 151, e3.1
Wheels see also tyres
advantages of alloy type 1.56, 2.24, 114-15
caster offset 2.49, 140-42, e2.20
coefficients of friction 1.25, 1.30, 1.32-1.33,
2.33-2.35, 2.39, 2.45
Contre Pent rim 112
cornering 2.39, 2.44, 122-24, 129-30, 133-34,
e2.14
Hayes Lemmerz alloy wheel 2.23, 2.24
lateral forces 2.37, 2.43, 2.46, 2.47, 128-30,
132-34, e2.8, e2.9, e2.11, e2.l2, e2.l3,
e2.l6, e2.17
coefficient of friction 2.39, 2.45, 130, e613a
during cornering 2.39, 129-30
and slip angle 2.38, 2.39, 128
variable factors 134-38
mountings 2.23, 2.24, 115-16
non·,uniformity, effects of 2.29, 2.30, 118-21,
e2.3a
overturning moments 2.51,2.52, 144-45
resulting force coefficient 2.48, 138-40, e2.l5,
e2.l8, e2.l9
rim
design 2.20, 110-14
markings 2.22, 2.25, 115
rolling forces
friction coefficients 2.33, 125, e2.5a,
e2.6a
road influences
aquaplaning 2.35, 126-27
snow and ice 2.36, 127-28, e2.7
wet and dry 2.34, 126, e2.6b
slip 124-25, e2.4e-f
rolling resistance
variables 2.31, 124
when cornering 2.32, 122-24, e2.4b-d
when driving straight 2.31, 2.32, 121-22,
e2.4
safety shoulders 2.21, 112
self-aligning torque 140, 142-44, e2.21, e2.22,
e2.23
self-steering properties 130-32
sheet metal disc type 2.23,2.25, 2.26, 114-15
slip angle 2.38,2.39, 124-25, 128
springing behaviour 2.27,2.28, 116-18, e2.3
torque steer effects 2.53, 2.54, 146-48
understeer and oversteer 2.40, 2.41, 2.42,
130-32, e2.l
°
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