Vehicle Dynamics

Vehicle Dynamics
Vehicle Dynamics
Compendium for Course MMF062
steady state cornering, driver tries to keep constant path radius
16
0.1*vehicleYawRate
[0.1*deg/s]
vehicleSpeed [m/s]
14
vehicleLateralAcceleration
[m/(s*s)]
vehicleLongitudinalAcceleration [m/(s*s)]
12
road wheel steering angle [deg]
10
8
5.25
6.75
6
4
0
2
0
0
10
20
30
40
50
time [s]
60
70
80
90
2015
Bengt Jacobson et al, Vehicle Dynamics Group, Division of Vehicle and Autonomous Systems,
Department of Applied Mechanics, Chalmers University of Technology, www.chalmers.se
Preface 2015
This edition has various changes and additions. Some of these are: brush model with parabolic
pressure distribution, typical numerical data for heavy vehicle, added “2.3.1 Tyre design and
function”, “4.5.3.2 (Example of) Explicit form model”, more about tyre relaxation, introduction of
neutral steering point, introduction of steady state roll-over wheel lift diagram. Thanks to Anton
Albinsson, Edo Drenth (VCC), Gunnar Olsson (LeanNova), Manjurul Islam, Mathias Lidberg, Mats
Jonasson (VCC), Niklas Fröjd (Volvo GTT), Ola Benderius, Pär Pettersson, and Zuzana Nedelkova
among other.
Bengt Jacobson, Göteborg, 2015
Preface 2014
This edition has various small changes and additions. The largest changes are: Function definitions added and major update
of section 1.2, 3.1.1, 4.1.1, 5.1.1.
Thanks to Lars Almefelt from Chalmers, Jan Andersson from VCC, Kristoffer Tagesson from Volvo GTT and Gunnar Olsson
from Leannova and Karthik Venkataraman.
Bengt Jacobson, Göteborg, 2014
Preface 2013
This edition has various small changes and additions. The largest additions were in: Functional architecture, Smaller
vehicles, Roll-over, Pendulum effect in lateral load transfer and Step steer.
Thanks to Gunnar Olsson from LeanNova, Mathias Lidberg, Marco Dozza, Andrew Dawkes from Chalmers, Erik Coelingh
from Volvo Cars, Fredrik Bruzelius from VTI, Edo Drenth from Modelon, Mats Sabelström, Martin Petersson and Leo Laine
from Volvo GTT.
Bengt Jacobson, Göteborg, 2013
Preface 2012
A major revision is done. The material is renamed from “Lecture notes” to “Compendium”. Among the changes it is worth
mentioning: 1) the chapters about longitudinal, lateral and vertical are more organised around design for vehicle functions,
2) a common notation list is added, 3) brush tyre model added, 4) more organised and detailed about different load
transfer models, and 5) road spectral density roughness model is added.
Thanks to Adithya Arikere, John Aurell, Andrew Dawkes, Edo Drenth, Mathias Lidberg, Peter Nilsson, Gunnar Olsson, Mats
Sabelström, Ulrich Sander, Simone Sebben, Kristoffer Tagesson, Alexey Vdovin and Derong Yang for review reading.
Bengt Jacobson, Göteborg, 2012
Preface 2011
Material on heavy vehicles is added with help from John Aurell. Coordinate system is changed from SAE to ISO. Minor
additions and changes are also done.
Bengt Jacobson, Göteborg, 2011
Preface 2007
This document was developed as a result of the reorganization of the Automotive Engineering Master’s Programme at
Chalmers in 2007. The course content has been modified in response to the redistribution of vehicle dynamics and power
train education.
These lecture notes are based on the original documents developed by Dr Bengt Jacobson. The text and examples have
been reformatted and edited but the author is indebted to the contribution of Dr Jacobson.
Rob Thomson, Gothenburg, 2007
Cover picture, left column, from top: Test of lane change support for combination-vehicle A-double in VTI’s simulator,
Chalmers’ Saab93 test vehicle in steady state cornering at AstaZero, and Tilt table test of timber trailer at Volvo GTT.
This compendium is also available as pdf file at http://pingpong.chalmers.se/public/courseId/5512/lang-en/publicPage.do .
© Copyright: Chalmers University of Technology, Bengt Jacobson
3
Contents
1
INTRODUCTION
1.1
11
About this compendium
11
1.2
Attributes and Functions
1.2.1
Attributes
1.2.2
Functions
12
12
14
1.2.2.1
1.2.2.2
1.2.2.3
1.2.3
1.2.4
Seeing versus Blind functions
Dynamic versus Driver-informing functions
Other categorisations
14
15
15
Requirements
Models, methods and tools
16
16
1.3
Technical References
1.3.1
Engineering
1.3.2
General modelling, drawing and mathematics
1.3.3
Notations
1.3.4
From general “Mechanical engineering”
1.3.4.1
1.3.4.2
1.3.5
1.3.6
1.3.7
1.3.8
1.3.9
1.3.10
1.3.10.1
2
17
17
17
20
24
Free-body diagrams
Operating conditions
24
25
Verification methods with real vehicles
Verification methods involving theoretical simulation
Tools & Methods
Coordinate Systems
Terms with special meaning
Architectures
26
26
27
29
32
34
Reference architecture of vehicle functionality
34
1.4
Heavy truck versus passenger cars
1.4.1
General differences
1.4.2
Vehicle dynamics differences
35
36
36
1.5
Smaller vehicles
36
1.6
Automotive engineering
38
1.7
Typical numerical data
1.7.1
For passenger vehicle
1.7.2
For heavy vehicle
39
39
40
VEHICLE INTERACTIONS
43
2.1
Introduction
43
2.2
Introduction to Tyre Terminology
2.2.1
Wheel angles
2.2.1.1
2.2.1.2
2.2.1.3
2.2.1.4
2.2.1.5
43
44
Steering angle
Camber (Angle)
Caster Angle
Toe Angle
Steering axis inclination
44
44
44
45
45
2.3
Tyre Construction
2.3.1
Tyre design and function
45
47
4
2.4
Longitudinal Properties of Tyres
2.4.1
Tyre Rolling and Radii
2.4.2
The Rolling Resistance of Tyres
2.4.2.1
2.4.3
2.4.3.1
2.4.3.2
2.4.3.3
2.4.3.4
49
49
51
Variation of rolling resistance
52
Longitudinal forces
55
Tyre brush model for longitudinal slip
Empirical tyre models
Tyre with both rolling resistance and slip
Relaxation
56
60
62
63
2.5
Lateral Properties of Tyres
2.5.1
Tyre brush model for lateral slip
2.5.2
Empirical tyre models
2.5.3
Influence of vertical load
2.5.4
Relaxation
2.5.5
Other effects than lateral force due to lateral slip
2.5.5.1
2.5.5.2
2.5.5.3
3
Overturning moment
Tyre self-aligning moment
Camber force
64
64
66
67
69
69
69
70
70
2.6
Combined Longitudinal and Lateral Slip
71
2.7
Vertical Properties of Tyres
76
2.8
Tyre wear
77
2.9
Vehicle Aerodynamics
2.9.1
Longitudinal wind velocity
2.9.2
Lateral wind velocity
77
77
78
2.10 Driver interactions with vehicle dynamics
2.10.1
Open-loop and Closed-loop Manoeuvres
2.10.2
Pedals
2.10.3
Steering
78
79
80
81
2.11
81
Traffic interactions with vehicle dynamics
LONGITUDINAL DYNAMICS
83
3.1
Introduction
3.1.1
References for this chapter
83
83
3.2
Steady State Functions
3.2.1
Propulsion System
83
83
3.2.1.1 Prime movers
3.2.1.2 Transmissions
3.2.1.2.1
Main transmissions
3.2.1.2.2
Distribution transmissions
3.2.1.3 Energy buffers
3.2.2
3.2.3
3.2.4
3.2.5
3.2.6
3.2.7
3.2.8
3.2.8.1
3.2.8.2
3.2.8.3
84
85
86
86
87
Traction diagram
Driving Resistance
Top speed *
Starting with slipping clutch
Steady state load distribution
Friction limit
Uphill performance
87
88
89
90
90
91
92
Start-ability *
Grade-ability *
Towing capacity *
92
93
93
5
3.3
Functions over longer events
3.3.1
Driving cycles
3.3.2
Other ways of defining the driving event
3.3.2.1
3.3.2.2
3.3.3
3.3.4
3.3.5
3.3.6
3.3.7
3.3.8
3.3.9
3.3.10
3.3.11
3.3.11.1
3.3.11.2
3.3.11.3
Driving pattern
Transport task/Operating cycle
96
96
Rotating inertia effects
Traction diagram with deceleration
Fuel or energy consumption *
Emissions *
Tyre wear *
Range *
Acceleration reserve *
Load Transfer, without delays from suspension
Acceleration – simple analysis
Acceleration performance *
Solution using integration over time
Solution using integration over speed
3.4.3
3.4.4
3.4.5
3.4.6
3.4.7
3.4.8
108
109
109
Braking efficiency *
Braking Distance *
Stopping Distance *
109
109
110
(Friction) Brake system
Pedal Response *
Pedal Feel *
Brake proportioning
Body heave and pitch due to longitudinal wheel forces
Load Transfer including suspension linkage effects
3.4.8.1
3.4.8.2
3.4.8.3
3.4.8.4
3.4.8.5
3.4.9
3.4.10
3.4.11
3.4.12
3.4.13
Load Transfer model with Pitch Centre
Load Transfer model with Axle Pivot Points
Steady state heave and pitch due to longitudinal wheel forces
Examples of real suspension design
Additional phenomena
Dive at braking *
Squat at propulsion *
Anti-dive and Anti-squat designs
Deceleration performance *
Acceleration performance *
3.5.3
4
117
117
118
119
119
123
123
123
Pedal driving *
Cruise Control and Adaptive Cruise Control (CC and ACC) *
Anti-lock Braking System, ABS *
Electronic Brake Distribution, EBD *
Traction Control, TC *
Engine Drag Torque Control, EDC *
Automatic Emergency Brake, AEB *
Longitudinal Motion Functionality shown in a reference architecture
LATERAL DYNAMICS
4.1
110
111
111
111
113
116
120
120
120
121
123
3.5
Control functions
3.5.1
Longitudinal Control
3.5.2
Longitudinal Control Functions
3.5.2.1
3.5.2.2
3.5.2.3
3.5.2.4
3.5.2.5
3.5.2.6
3.5.2.7
97
99
100
103
103
103
103
104
106
106
106
108
3.4
Functions in shorter events
3.4.1
Typical test manoeuvres
3.4.2
Deceleration performance
3.4.2.1
3.4.2.2
3.4.2.3
94
94
96
123
124
124
125
126
126
126
128
131
Introduction
131
6
4.1.1
References for this chapter
131
4.2
Low speed manoeuvrability
4.2.1
Path with orientation
4.2.2
Vehicle and wheel orientations
4.2.3
Steering System
4.2.3.1
4.2.3.2
4.2.4
4.2.5
4.2.6
4.2.7
4.2.8
4.2.9
4.2.10
4.2.11
4.2.12
132
132
133
133
Chassis steering geometry
Steering system forces
133
134
One-track models
Ideally tracking wheels and axles
One-track model for low speeds, with Ackerman geometry
Turning circle *
Swept path width and Swept Area *
Off-tracking *
Steering effort *
One-track model for low speeds, with non-Ackerman geometry
One-track model for low speeds, with trailer
137
139
140
142
143
144
144
145
148
4.3
Steady state cornering at high speed
4.3.1
Steady state driving manoeuvres
4.3.2
Steady state one-track model
148
148
150
Relation 𝜹𝒇 𝒗𝒙 and 𝑹
Relation 𝜹𝒇 𝒗𝒙 and 𝜷
Relation 𝒗𝒙 𝑹 and 𝜷
Simpler model
153
153
154
154
4.3.2.1
4.3.2.2
4.3.2.3
4.3.2.4
4.3.3
4.3.3.1
4.3.3.2
4.3.3.3
4.3.3.4
4.3.4
4.3.4.1
4.3.4.2
4.3.4.3
4.3.4.4
4.3.4.5
4.3.4.6
4.3.4.7
4.3.4.8
4.3.4.9
4.3.4.10
4.3.5
4.3.5.1
4.3.6
4.3.6.1
4.3.6.2
4.3.6.3
4.3.6.4
4.3.7
4.3.8
4.3.9
Under-, Neutral- and Over-steering *
Understeering as built-in measure
Understeering as varying with steady state lateral acceleration
Understeering as variable during a transient manoeuvre
Neutral steering point
How to design for a desired understeer gradient
Tyre design, inflation pressure and number of tyres
Roll stiffness distribution between axles
Steering system compliance
Side force steer gradient
Roll steer gradient
Camber steer
Toe angle
Wheel Torque effects
Transient vehicle motion effects on yaw balance
Some other design aspects
Normalized required steering angle
Critical and Characteristic speed *
155
155
156
157
158
159
159
160
160
160
161
161
161
162
162
162
163
163
Steady state cornering gains *
164
Yaw Rate gain
Curvature gain
Lateral Acceleration gain
Side slip gain as function of speed
165
165
166
166
Manoeuvrability and Stability
Handling diagram
Steady state cornering at high speed, with Lateral Load Transfer
4.3.9.1 Load transfer between vehicle sides
4.3.9.2 Body heave and roll due to lateral wheel forces
4.3.9.2.1
Steady-state roll-gradient *
4.3.9.3 Lateral load transfer
4.3.9.3.1
Load Transfer model with Wheel Pivot Points
4.3.9.3.2
Load Transfer model with Axle Roll Centres
4.3.9.4 Influence of anti-roll bars
4.3.9.5 Axle suspension system
7
167
167
169
169
170
172
172
172
173
177
178
4.3.9.6
4.3.10
4.3.11
Lateral load transfer influence on steady state handling
Steering feel *
Roll-over in steady state cornering
4.3.11.1
4.3.11.2
4.3.11.3
4.3.11.3.1
4.3.11.3.2
4.3.11.3.3
4.3.11.4
Roll-over threshold definitions
Static Stability Factor, SSF
Steady-state cornering roll-over
Model with fore/aft symmetry
Model without fore/aft symmetry
Using a transient model for steady-state roll-over
Roll-over and understeering/propulsion
4.4
Stationary oscillating steering
4.4.1
Stationary oscillating steering tests
4.4.2
Transient one-track model
4.4.2.1
4.4.2.3
4.4.3
Simpler model
Validity of solution
Single frequency response
Lateral Velocity and Yaw Rate response
Lateral Acceleration response
Other responses to oscillating steering
Random frequency response
4.5.4
4.5.4.1
4.5.4.2
4.5.5
(Example of) Mathematical model
(Example of) Explicit form model
5
193
199
200
201
203
Step steering response *
213
Mild step steering response
Violent step steering response
213
214
Rearward Amplification *
Off-tracking *
Yaw Damping *
214
215
216
216
4.6
Lateral Control Functions
4.6.1
Lateral Control Design
4.6.2
Lateral Control Functions
4.6.2.1
4.6.2.2
4.6.2.3
4.6.2.4
186
186
187
203
207
Long heavy vehicle combinations manoeuvrability measures
4.5.5.1
4.5.5.2
4.5.5.3
182
183
184
184
185
185
186
193
196
198
199
199
4.5
Transient handling
4.5.1
Transient driving manoeuvres
4.5.2
Transient one-track model, without Lateral load transfer
4.5.3
Transient model, with Lateral load transfer
4.5.3.1
4.5.3.2
181
181
191
192
Steering frequency response gains *
4.4.3.1
4.4.3.2
4.4.3.3
4.4.3.4
4.4.3.5
180
217
217
217
Lane Keeping Aid, LKA *
Electronic Stability Control, ESC *
Roll Stability Control, RSC *
Lateral Collision Avoidance, LCA *
217
218
220
220
VERTICAL DYNAMICS
221
5.1
Introduction
5.1.1
References for this chapter
221
221
5.2
222
Suspension System
5.3
Stationary oscillations theory
5.3.1
Time domain and time frequency domain
5.3.1.1 Mean Square (MS) and Root Mean Square (RMS) of variable
5.3.1.2 Power Spectral Density and Frequency bands
5.3.1.2.1
Differentiation of PSD
5.3.1.3 Transfer function
5.3.2
5.3.2.1
Space domain and Space frequency domain
Spatial Mean Square (MS) and spatial Root Mean Square (RMS) of variable
8
224
225
225
226
227
227
228
229
5.3.2.2
Spatial Power Spectral Density and Frequency bands
5.4
Road models
5.4.1
One frequency road model
5.4.2
Multiple frequency road models
5.4.2.1
230
231
231
Transfer function from road spectrum in spatial domain to system response in time domain234
5.5
One-dimensional vehicle models
5.5.1
One-dimensional model without dynamic degree of freedom
5.5.1.1
5.5.1.2
5.5.2
Response to a single frequency excitation
Response to a multiple frequency excitation
One dimensional model with one dynamic degree of freedom
5.5.2.1
5.5.3
Response to a single frequency excitation
235
235
236
237
238
238
One dimensional model with two degrees of freedom
5.5.3.1
5.5.3.2
230
Response to a single frequency excitation
Simplified model
242
244
247
5.6
Ride comfort *
5.6.1
Single frequency
5.6.2
Multiple frequencies
248
248
249
5.7
Fatigue life *
5.7.1
Single frequency
5.7.2
Multiple frequencies
253
253
253
5.8
Road grip *
5.8.1
Multiple frequencies
254
254
5.9
Variation of stiffness and damping
5.9.1
Varying suspension stiffness
5.9.2
Varying suspension damping
5.9.3
Varying unsprung mass
5.9.4
Varying tyre stiffness
255
255
256
257
257
5.10 Two dimensional oscillations
5.10.1
Bounce (heave) and pitch
5.10.2
Bounce (heave) and roll
258
259
261
5.11
Transient vertical dynamics
261
5.12
Other excitation sources and functions
262
BIBLIOGRAPHY
221
INDEX
266
9
10
INTRODUCTION
1 INTRODUCTION
1.1 About this compendium
This compendium is intended for first-time-studies studies of vehicle dynamics. At Chalmers
University of Technology this is in the course “MMF062 Vehicle Dynamics”. The compendium covers
more than actually included in the course; both in terms of subsystem designs, which not necessarily
has to be fully understood in order to learn the vehicle level; and in terms of some teasers for more
advanced studies of vehicle dynamics. Therefore, the compendium can also be useful in general
vehicle engineering courses, e.g. in the Chalmers course “TME121 Engineering of Automotive
Systems”; and as an introduction to more advance courses, which at Chalmers is the course “TME102
Vehicle Dynamics Advanced”.
The overall objective of the compendium is to contribute to education of automotive engineers with
good skills to design for, and verify, requirements on complete vehicle level functions, regarding
vehicle dynamics or motion.
The compendium focus on road vehicles, both passenger cars and commercial vehicles. Smaller road
vehicles, such as bicycles and single-person cars, are only very briefly mentioned. It should be
mentioned that a lot of ground-vehicle types are not covered at all, such as: off-road/construction
vehicles, tracked vehicles, horse wagons, hovercrafts or railway vehicles.
The vehicle can be seen as a component (or sub-system) in a superior transport system, consisting of
vehicles with drivers, other road users and roads with wind and weather. A vehicle is also, in itself, a
system, within which many components (or sub-systems) interact. In order to approach the vehicle
dynamics in a structured way, the course is divided into the following parts:



Longitudinal dynamics (e.g. propulsion and braking sub-systems)
Lateral dynamics (e.g. steering sub-systems)
Vertical dynamics (e.g. suspension sub-systems)
These 3 parts are found as chapters 3-5, and the goal is that the compendium should lead all the way
to “functions”. Functions are needed for requirement setting, design and verification. The overall
order within the compendium is that models/methods/tools needed to understand the functions are
placed before the functions.
It is important to qualitatively understand the characteristics of the vehicle’s sub-systems and, from
this, learn how to quantitatively analyse the vehicle’s behaviour. These skills support the overall goal
of engineering design for desired attributes and functions, see Section 1.2.
The reader of this compendium is assumed to have knowledge of mathematics and mechanics, to the
level of a Bachelor of Engineering degree. Previous knowledge in dynamic systems, e.g. from Control
Engineering courses, is often useful. There are no prerequisites on previous knowledge in automotive
engineering, because basic vehicle components (or sub-systems) are presented to the necessary level
of detail. This could be considered as unnecessary repetition for some readers. References and
suggestions for further reading are provided and readers are directed to these sources of information
for a deeper understanding of a specific topic.
11
INTRODUCTION
1.2 Attributes and Functions
Development of a vehicle is driven by requirements which come from:



Use case based needs of the customer/users
Legislation and rating from the authorities and consumer organisation, and
Engineering constraints from the manufacturer’s platform/architecture on which is it to
certain vehicle should be built.
One way of organising most of those, in particular the first ones, is to define Attributes and
Functions. The terms are not strictly defined and may vary between vehicle manufacturers and over
time. With this said, it is assumed that the reader understand that the following is an approximate
and simplified description.
In this compendium, both attributes and functions implicitly concern the complete vehicle; not the
sub-systems within the vehicles and not the superior level of the transportation system with several
vehicles in a road infrastructure.
Attributes and Functions are used to establish processes and structures for requirement setting and
verification within a vehicle engineering organisation. Such processes and structures are important to
enable a good overall design of such a complex product as a vehicle intended for mass production at
affordable cost for the customers/users. Figure 1-1 tries to give an overview, with reference to the
well-known V-model, how a vehicle program generally is organised. The figure is idealized in many
ways. First, one have more levels of functions and subsystems. Secondly, it is difficult to keep such a
clean hierarchical order between functions and subsystems. Thirdly and most conceptually difficult,
is the fact that each subsystem gets requirements from many functions as showed by the dashed
lines; this makes the complexity explode.
Traditional V-model
for product development
Design
An extended V-model for a vehicle programme
Product
planning
Describe Complete
Vehicle Attributes
Customer/User
perception
Design loops on
Vehicle program level
(pre-series, virtual and real)
Set Requirements on Complete Vehicle
[from/per (Complete Vehicle) Function]
Validation of
Complete Vehicle
Verify Requirements
on Complete Vehicle
Design loops per
Subsystem
Verify Requirements
on Subsystem
Break down to Requirements
on Subsystem
Figure 1-1: V-model applied to a vehicle program.
1.2.1
Attributes
An attribute is a high-level aspect of how the user of the vehicle perceives the vehicle. Attributes
which are especially relevant for Vehicle Dynamics are listed in Table 1.1. The table is much
generalised and the attributes in it would typically need to be decomposed into more attributes
when used in the engineering organisation of an OEM. Also, not mentioned in the table, are
12
INTRODUCTION
attributes which are less specific for vehicle dynamics, such as Affordability (low cost for user),
Quality (functions sustained over vehicle lifetime), Styling (appearance, mainly visual), etc.
Attributes can be seen as one way to categorise or group functions, especially useful for an OEM
organisation and vehicle development programs. Functions can, in turn, be seen as one way to group
requirements. Legal requirements are often, but not always, possible to trace back to primarily one
specific attribute. Requirements arising from company-internal platform and architecture
constraints are often more difficult to trace in that way. Hence, legal and platform/architecture can
be seen as two “attribute-like requirement container”, beside the other user-derived attributes.
Table 1.1: Attributes
Attribute
Transport
Efficiency
Abbreviation
Description
This attribute means to maximize transport output and minimize costs for
transportation. Transport output can be measured in 𝑝𝑒𝑟𝑠𝑜𝑛 ⋅ 𝑘 or 𝑡𝑜𝑛 ⋅ 𝑘 .
The costs are mainly energy costs and time, but also wear/life time of vehicle
parts influence. The attribute is most important for commercial vehicles, but
becomes increasingly important also for passenger vehicles. The attribute is
mainly addressing long-term vehicle usage pattern, typically 10 min to 10 hours.
There are different ways to define such usages, e.g. (Urban / Highway / Mixed)
driving cycles. So far, the attribute is mainly required and assessed by the
vehicle customers/users.
E
The attribute can also be seen to include “Environmental Efficiency”, which
means low usage of natural resources (mainly energy) and low pollution, per
performed transport task. This is to a large extent required and assessed by
society/legislation.
Safety
Minimizing risk of property damages, personal injuries and fatalities both in
vehicle and outside, while performing the transportation. This attribute is to a
large extent required and assessed by society/legislation. In some markets,
mainly developed countries, it is also important for vehicle customers/users.
S
Driving
Dynamics
How the driver experiences driving, from relaxed transport driving (comfort) to
active driving (sensation). Driving Dynamics contains sub-attributes as:
 Driveability, Handling and Road-holding describes how the vehicle
responds to inputs from driver and disturbances, and how driver gets
feedback from vehicle motion e.g. through steering feel. (It can be
noted that same response aspects are similarly important for a “virtual
driver”, i.e. a control algorithm doing some automatic/autonomous
control.) Driveability often refers to longitudinal (acceleration, braking
gear shifting). Handling and Road-holding often refer to lateral
manoeuvres.
 Performance describes how the vehicle can perform at the limits of its
capabilities; acceleration, deceleration or cornering. Most often, it
refers to longitudinal limitations due to propulsion and brake systems
limitations.
 Ride comfort. Ride comfort often refers to vibrations and harshness of
the occupants’ motion, primarily vertical but secondly longitudinal. So,
V and H in NVH (=Noise, Vibration and Harshness) is included. If
expanding to “comfort” it would include also N (noise) and
compartment air conditioning, but these sub-attributes are less related
to vehicle dynamics.
This attribute is to a large extent required and assessed by the vehicle
customers/users, both through own experience but to a large extent indirectly
via assessments by experts, e.g. in motor journals.
D
13
INTRODUCTION
1.2.2
Functions
A function supplies certain functionality to the customer/user of the vehicle. A function is more
specific than an attribute, which is needed to set (quantitative) requirements, see Section 1.2.3. The
(complete vehicle) function does not primarily stipulate any specific subsystem. However, the
realisation of a function, in one particular vehicle programme, normally only engages a limited subset
of all subsystems. So, the function will there pose requirements on that subsystem. Hence, it is easy
to mix up whether a function origins from an attribute or a subsystem. One way to categorise
functions is to let each function belong to the subsystem which it mainly implies requirements on
rather than the source attribute. Categorizing functions by subsystems tends to lead to “carry-over”
function realisations from previous vehicle program, which can be good enough in many cases.
Categorizing functions by source attribute facilitates more novel function realisations, which can be
motivated in other cases.
The word “function” has appeared very frequently lately along with development of electrically
controlled systems. The function “Accelerator pedal driving” in Section 3.5.2.1 has always been
there, but when the design of it changed from mechanical cable and cam to electronic
communication and computational algorithms (during 1990’s) it became much more visible as a
function, sometimes referred to as “electronic throttle”. The point is that the main function was
there all the time, but the design was changed. The change of design enabled, or was motivated by,
improvement of some sub-functions, e.g. idle speed control which works better in a wider range of
engine and ambient temperature.
At some places, the compendium emphasizes the functions by adding an asterisk “*” in section
heading and a “Function definition” in the following typographic form:
Function definition: {The Function} is the {Measure} … for {Fixed Conditions} and certain
{Parameterized Conditions}.
The {Measure} should be one unambiguously defined measure, such as time, velocity or force. The
{Fixed Conditions} should be unambiguously defined and quantified conditions for the vehicle and its
surroundings. The keyword “certain” identifies the {Parameterized Conditions}, which are not fixed in
this definition, but need to be fixed before using the Function definition for requirement setting. The
{Measure} is ideally a continuous, objective and scalar quantity, subjected for setting a requirement
on. A requirement can then be: “{Measure} shall be =, , > or < {value*unit}”, see more in Section
1.2.3.
Since the term “Function” is defined very broadly in the compendium, these definitions become very
different. One type of Function definition can be seen in beginning of Section “3.2.4 Top speed”,
which includes a well-defined measure. Another type of Function definition is found in Section
“3.5.2.3 Anti-lock Braking System, ABS” and “4.3.3 Under-, Neutral- and Over-steering”. Here, the
definitions are more on free-text format, and an exact measure is not so well defined.
The compendium defines functions on complete vehicle level, i.e. not cascaded to requirements on
specific sub-systems.
There are different ways of categorising functions, where of two are shown below.
1.2.2.1
Seeing versus Blind functions
This categorisation is relevant for control functions and environment sensors.
1.2.2.1.1 Seeing functions
These functions use information about the (vehicle’s) environment, sensed through camera, radar,
etc. Normally, these functions do something instead of the driver, e.g. brake, accelerate or steer the
vehicle. The driver is important mainly in two ways: the driver’s actions can be used to identify when
14
INTRODUCTION
the driver needs help (e.g. when he has not identified a threat in the environment), then an
automatic intervention is triggered; the driver’s action can also be used to identify when the driver
wants to take back control, then the automatic intervention is cancelled. Examples of these functions
are Automatic Emergency Brake, Lane Departure Warning and Forward Collision Warning.
1.2.2.1.2 Blind functions
These are functions which do not rely on information about the environment, but only about the
vehicle itself. ABS and ESC are examples of blind functions.
1.2.2.2
Dynamic versus Driver-informing functions
This categorisation is relevant for control functions and driver interface.
1.2.2.2.1 Dynamic functions
These functions change the dynamics of the vehicle, using actuators such as brake system,
propulsion system and steering system. Normally, the functions help the driver to control the vehicle
motion and path/trajectory in the desired way. The driver’s actions are then interpreted as requests
to the control functions. Also, sensing of vehicle’s state and environment can be used to further
improve the interpretation of driver’s intention, and/or to correct a control error. The control is then
performed by vehicle motion actuation, e.g. changing engine torque, brake torque or steering angle.
In today’s vehicles, the vehicle motion is mainly measured by the vehicle’s speeds and accelerations,
but in future one could expect that also environment sensors will give inputs to actual
motion/position versus object in the environment. Examples of these functions are ABS, ESC and
AEB, see Section 3.5.2 and 4.6.2.
1.2.2.2.2 Driver-informing functions
These are functions which do not change the dynamics of the vehicle. Instead, they might actuate
audio, visual or haptic (touch) warnings to driver. Examples of these functions can be Forward
Collision Warning (FCW), Lane Departure Warning (LDW) and Blind Spot Detection (BSD). FCW is
mentioned in Section 3.5.2.7. LDW is a function that warns the driver via visual, audio and/or haptic
signals when the vehicle is tending to drift out of the lane. BSD is a function which warns when there
is a vehicle beside and behind the subject vehicle (in the “blind spot” for the driver).
1.2.2.2.3 Un-categorized
There are interventions which are both of information type and dynamic, or difficult to categorize.
Such as a superimposed steering-wheel torque in a traditional steering system. It will actually steer
the vehicle if driver does not hold on to steering wheel. Alternatively, if driver holds steering wheel
fixed, it would just result in a haptic force in driver’s hand.
Similar difficulties to categorize could appear with “active pedals”, but such solutions are hardly on
market yet.
1.2.2.3
Other categorisations
There are also other ways of categorizing functions:




Grouping in (sub-)systems on which the function imply main part of its requirements, such as
brake system and steering system.
Grouping in those who are realized solely mechanically and those who involve electronics.
Grouping in “customer functions” and “support functions”. Support functions are typically
used by many customer functions, see Section “1.3.10.1 Reference architecture of vehicle
functionality”.
Grouping in standard and option in the vehicle program or platform.
15
INTRODUCTION
One thing to remember about the concept of functions is that there are always several ways of
breaking down the total package of functions in a vehicle into a finite list of functions. Also, one can
define sub-functions as being parts of superior functions. A function can also be relying on another
function by needing input, or demanding actuation from the other. To summarise, defining separate
functions as parts in the whole vehicle is not very precise, although it is helpful when trying to
organise a combined hardware and software development process.
1.2.3
Requirements
A requirement shall be measureable. A requirement on complete vehicle level is typically formulated
as: “The vehicle shall … {< or > or =} {number} {unit}”. Examples:






The vehicle shall accelerate from 50 to 100 km/h on level road in <5 s when acceleration
pedal is fully applied.
The vehicle shall decelerate from 100 to 0 km/h on level road in <35 m when brake pedal is
fully applied, without locking any rear wheel.
The vehicle shall turn with outermost wheel edge on a diameter <11m when turning with
full-lock steering at low speed.
The vehicle shall have a characteristic speed of 70 km/h (10 km/h) on level ground and
high-friction road conditions and any recommended tyres.
The vehicle shall give a weighted RMS-value of vertical seat accelerations <1.5 m/s^2 when
driving on road with class B according to ISO 8608 in 100 km/h.
The vehicle shall keep its body above a 0.1 m high peaky two-sided bump when passing the
bump in 50 km/h.
In order to limit the amount of text and diagrams in the requirements it is useful to refer to ISO and
OEM specific standards. Also, it is often good to include the purpose, and/or a use case, when
documenting the requirement.
The above listed requirements stipulate the function of the vehicle, which is the main approach in
this compendium. Alternatively, a requirement can stipulate the design of the vehicle, such as “The
vehicle shall weigh <1600 kg” or “The vehicle shall have a wheel base of 2.5 m“. The first type (above
listed) can be called Performance based requirement. The latter type can be called Design based
requirement or Prescriptive requirement and such are rather “means” than “functions”, when seen in
a function vs mean hierarchy. It is typically desired that legal requirements are Performance based,
else they would limit the technology development in society.
1.2.4
Models, methods and tools
The attributes, functions and requirements are top level entities in vehicle development process. But
in order to define/formulate them, design for them and verify them, the engineers need knowledge
in terms of models, methods and tools.
As mentioned above, some sections in the compendium have an asterisk “*” in the section heading,
to mark that they explains a function, which can be subject for complete vehicle requirement setting.
The remaining section, without an asterisk “*”, are there to give the necessary knowledge (models,
methods, tools, etc.) to understand the function. It is the intention that the necessary knowledge for
a certain function appears before that function section. One example is that the Sections “3.2.1
Propulsion System”, “3.2.2 Traction diagram” and “3.2.3 Driving Resistance” are placed before
Section “3.2.4 Top speed *”.
The functions only appear in Chapter 3, Chapter 4 and Chapter 5.
16
INTRODUCTION
1.3 Technical References
This section introduces notations. The section also covers some basic disciplines, methods and tools.
Parts of this section probably repeat some of the reader’s previous education.
1.3.1
Engineering
Engineering Design or Engineering (in Swedish “Ingenjörsvetenskap”, in German
“Ingenieurwissenschaften”), is not the same as Analysis, Inverse analysis and (Nature) Science, see
Figure 1-2. In education, it is often easiest to do Analysis (or Inverse Analysis). However, in a product
based subject, such as Vehicle Dynamics, it is important to keep in mind that the ultimate use of the
knowledge is Engineering, which is to propose and motivate the design and actual values of design
parameters of a product. The distinction between Analysis and Inverse Analysis can only be made if
there is a natural direction (input or output) of interface signals.
Laws of Nature
Input
System
Given
Find
Output
Process
Input, Laws of Nature, System
Output
Analysis
Output, Laws of Nature, System
Input
Inverse Analysis
Input, Output, System
Laws of Nature
Science (induction)
Input, Output, Laws of Nature
System
Engineering Design
Dixon, J.R., (1966) Design Engineering, Inventiveness, analysis and Decision Making
Figure 1-2: Distinction between Engineering and its neighbouring activities.
Picture from Stefan Edlund, Volvo Trucks.
1.3.2 General modelling, drawing and
mathematics
A model is a representation of something during a certain event or course of events, such as the
model of the motion of a car during acceleration from stand-still to 100 km/h. Models are always
based on assumptions, approximations and simplifications. However, when using models as a tool for
solving a particular problem, the models at least have to be able to reproduce the problem one is
trying to solve. Also, models for engineering have to reflect design changes (e.g. changed parameters
or input signals) in a representative way, so that new designs can be proposed which solve/reduce
the problem. Too advanced models tend to be a disadvantage, since they produce a lot of data.
The modelling focused in vehicle dynamics (and many related areas) has no well-established name,
but some alternative names are: functional models, one-dimensional models, system models, daemodel, ode-models, etc. Functional models are typically multi-domain type, involving mechanics,
hydraulics, pneumatics, electric, control algorithms, etc. Functional models typically lumps or
discretises possible continuous parts of the system. Examples of modelling methods which are very
17
INTRODUCTION
seldom directly used for functional models are: Finite Elements, Computational Fluid Dynamics, CAD
geometry models, etc.
One can identify the modelling in different stages in the overall process of engineering:






Engineering design task, which describes which design decisions for a certain system (an
existing product or a concept/drawing) that is needed. Also, requirements on how the
system should work have to be present.
Analysis task is formulated, and a plan of how the result of this analysis task can support the
actual design decisions.
Physical model, in sketches (or thoughts & words, but sketches are really recommended!). It
clarifies assumptions for different parts of the system, e.g. rigid/elastic and inertial/massless.
How one model here should be based on what phenomena is needed to be captured, so it
might depend on both which system you model and what the analysis task is.
Mathematical model (or Declarative model), in equations. Basically, it is about finding
exactly the right number of equations, not too many and not too few. Here, the assumptions
are transformed into equations. For dynamic systems, the equations form a “DAE”
(Differential-Algebraic system of Equations). It is seldom necessary to introduce derivatives
with respect to other independent variables, such as positions, i.e. one does seldom need
PDE (Partial Differential Equations). The general form of a DAE:
𝒇𝒛 (𝒛 𝒛 𝒖 𝒕)
;𝒚
𝒖(𝒕);
𝒛 (𝒛 𝒖 𝒕); 𝒖
where z are the dependent variables, u are the input variables, y the output variables and t is
time (independent variable).
Note that the DAE is generally on implicit form with respect to calculating the state
derivatives, as opposed to explicit form below. Note also that it is not only a question of
finding suitable equations, but also to decide parameterisation, which is how parameters
relate to each other. The parameterisation has to reflect a “fair” comparison between
different design parameters, which often requires a lot of experience of the product and the
full set of requirements on the vehicle.
Also, selection of output variables is important so that output variables are enough to
evaluate the requirements on the system. Selecting more might drive unnecessary complex
models.
Explicit form model (or Imperative model) is the equations rearranged to an (explicit) form
or an algorithm to calculate the state derivatives. You probably recognize this from solving
“ODE” (Ordinary Differential Equations) or “IVP” (Initial Value Problems). The general form is:
𝒙 𝒇𝒙 (𝒙 𝒖 𝒕); 𝒚
𝒖(𝒕);
𝒙 (𝒙 𝒖 𝒕); 𝒖
where x is the state variables and y the output variables.
Analysis or calculation. Several methods can be used depending of the analysis task, e.g.:
o
Simulation (e.g. Initial value problem, IVP or End value problem, EVP). There are
many advanced pre-programmed integration methods and tools which one can rely
on without knowing the details. It is often enough to know that the concept is similar
to the simplest of them, “Euler forward”, in which the state variables are updated in
each time step as follows:
𝒙𝒏𝒐𝒘 𝒇(𝒙(𝒕𝒏𝒐𝒘 ) 𝒖(𝒕𝒏𝒐𝒘 ) 𝒕𝒏𝒐𝒘 );
𝒙(𝒕𝒏𝒐𝒘 𝚫𝒕) 𝒙(𝒕𝒏𝒐𝒘 ) 𝚫𝒕 ∙ 𝒙𝒏𝒐𝒘 ;
Note: Simulation corresponds to running experiments in real world, while the 3
previous steps corresponds to building a real test object and instrument it.
o
Methods for linear systems, e.g. for studying stationary oscillation, eigen-oscillations,
step response, etc. With a linearized model, it is also possible to find explicit
solutions over time.
18
INTRODUCTION
o


Optimization. Either optimizing a finite number of defined design parameters or time
trajectories. Optimization is not further addressed in this compendium.
Synthesis of the numerical result. Understand and interpret analysis results, including judge
validity of model assumptions and parameterizations.
Solution, in terms of a concrete proposal of design, such as numerical parameter value,
drawing/control algorithm, …
In this process description, we can identify 3 modelling steps with 3 different kinds of models:
Physical model, Mathematical model and Explicit form model. The compendium spends most effort
on the first 2 of those, because the last step is more about mathematical science than vehicle
engineering.
A variable is something that varies in time. Constants and parameters are not varying, with the
distinction that a parameter does not change during the manoeuvre/experiment studied, but can be
changed between subsequent manoeuvres/experiments with the same model. Typical constant is pi
or gravity. Typical parameter is vehicle mass or road friction co-efficient.
A variable can be used as state variable, which means that it is given initial values and then updated
through integration along the time interval studied. Which variables to use as state variables is not
strictly defined by the physical (or mathematical) model. For mechanical systems, one often uses
positions and velocities of bodies as state variables, but it is quite possible and sometimes
preferable, to use forces in compliances (springs) and velocities of bodies as state variables. With
⁄ ;
mass/spring system as simple example, the two alternatives becomes [
𝑐 ∙ 𝑥; 𝑥
⁄ ;
⁄𝑐 ;], respectively.
;] and [
𝑐∙ ; 𝑥
Systems can be modelled with Natural causality. For mechanical systems, this is when forces on the
masses (or motion of the compliance’s ends) are prescribed as functions of time. Then the velocities
of the masses (or forces of the compliances) become state variables and have to be solved through
time integration. The opposite is called Inverse dynamics and means that velocities of masses (or
forces of compliances) are prescribed. For instance, the velocity of a mass can be prescribed and then
the required forces on the mass can be calculated through time differentiation of the prescribed
velocity. Cf. Analysis and Inverse Analysis in Figure 1-2.
Drawing is a very important tool for understanding and explaining. Beyond normal drawing rules for
engineering drawings, it is also important to draw motion and forces. The notation for this is
proposed in Figure 1-3.
Any vector or Translation (e.g. translational speed or force)
Along axis in plane of paper:
Quantity flow (e.g. Energy
flow = Power):
Perpendicular to paper, pointing out of paper:
Signal flow or Data flow
Perpendicular to paper, pointing in to paper:
Physical Connection
(no prescribed causal direction):
Rotation (e.g. rotational speed or moment/torque)
Around axis in plane of paper:
Parametrization
Around axis perpendicular to paper:
(at initialization of model):
Figure 1-3: Arrow like notation. Left: For motion and forces in drawings. Right: Other
It is often necessary to include more than just speeds and forces in the drawings. In vehicle dynamics
these could be: power flow and signal flows. These are often natural to draw as arrows
19
INTRODUCTION
When connecting components with signal flow, the resulting diagram is a data flow diagram. Physical
components and physical connections can be included in such diagram. It should be noted that a flow
charts and (discrete) state diagram represents a quite different type of diagram, even if they may
look similar; here an arrow between two blocks does not represent data flow from one component
to another, but stepping from one state or operation to another.
1.3.3
Notations
Generally, a variable is denoted x or x(t), where t is the independent variable time. In contexts where
one means that variable’s value at a certain time instant, t0, it can be denoted x(t0). In contexts
where one means the variable’s time history over a whole time interval (infinitely many values), it
can be denoted x(.).
𝑑
Differentiation (of x) with respect to time (t): 𝑑𝑡
𝑥
𝑑𝑒𝑟(𝑥).
Vectors exist of two kinds: Geometrical vector (or physical vector, or spatial vector), denoted by ⃗,
and mathematical vector, denoted [vx;vy]. The geometrical vector is the true vector, while the
mathematical is a representation of it, in a certain coordinate system. Matrices are denoted with
brackets, e.g. [A]. Row vectors are denoted with parentheses, e.g. (a).
Parentheses shall be used to avoid ambiguity, e.g. ( ⁄𝑏) ∙ 𝑐 or
⁄(𝑏 ∙ 𝑐), not
⁄𝑏 ∙ 𝑐.
Multiplication symbol (* or ∙) shall be used to avoid ambiguity, e.g. ∙ 𝑏𝑥, not 𝑏𝑥 (assuming 𝑏𝑥 is
one variable, not two). Stringent use of multiplication symbol enables use of variables with more
than one token, cf. programming. Stringent use of multiplication symbol reduces the risk of
ambiguity when using operators, e.g. it shows the difference between 𝑓(𝑥) and 𝑓 ⋅ (𝑥).
An interval has a notation with double dots. Example: Interval between
and 𝑏 is denoted . . 𝑏.
An explanation, between two consecutively following steps in a derivation of equations is written
} 𝑦 𝑥 or 𝑥 𝑦 7 ⇒ {
within {} brackets. Example: 𝑥 𝑦 {
𝑏 𝑏
𝑏 𝑏
} ⇒ 𝑦 𝑥 7.
An inverse function is denoted with 1 as superscript, e.g. 𝑦
𝑓(𝑥); ⇔ 𝑥
𝑓 − (𝑦).
Fourier and Laplace transforms are denoted ℱ(𝑓(𝑡)) and ℒ(𝑓(𝑡)), respectively.
1.3.3.1
Notation list
Table 1.2 shows notation of parameters, variables and subscripts used in this compendium. The
intention of this compendium is to follow International Standards (ISO8855). Some alternative
notations are also shown, to prepare the reader for other frequently used notation in other
literature.
The list does not show the order of subscripting, e.g. whether longitudinal (subscript x) force
(notation F) on rear axle (subscript r) should be denoted
or . The intention in this compendium
is to order subscript with the physical vehicle part (here rear axle) as first subscript and the
specification (longitudinal, x) as second:
. If there is also further detailed specifications, such as
coordinate system, e.g. wheel coordinate (subscript w), it would be the third subscript: 𝑤 .
20
INTRODUCTION
Table 1.2: Notation
21
INTRODUCTION
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INTRODUCTION
23
INTRODUCTION
1.3.4
From general “Mechanical engineering”
Vehicle dynamics originates from mechanical engineering. Therefore, it is important to be familiar
with the following basic relationships:






Energy = time integral of Power
[Torque or Moment] = Force ∙ Lever
Power = Torque ∙ Rotational speed or Power = Force ∙ Translational speed
(Torque) Ratio = Output torque / Input torque
(Speed) Ratio = Input speed / Output speed
Efficiency = Output power / Input power or, in a wider meaning, Efficiency = Useful / Used
1.3.4.1
Free-body diagrams
An example of a free-body diagram is given in Figure 1-4.
Division into sub-systems is often needed, e.g. to implement moment less connection points or other
models of the vehicle internal behaviour. The sub-system split typically goes through:



Connection point between towing unit and towed unit (interface quantities: 2D position, 2D
speed and 2D force)
Driveshaft close to each wheel (interface quantities: 1D angle and 1D torque)
Surface between driver’s hand and steering wheel (interface quantities: 1D angle and 1D
torque)
The free-body diagram is an important step between system model and equilibrium equations.
Relevant notes for this are:


A short-cut that can be taken when drawing the diagram is to assume forces are already in
equilibrium, which makes one equilibrium equation unnecessary. An example of this could
be to use
instead of
in Figure 1-4, which would make it pointless to set up the
vertical force equilibrium for the front axle (
0).
In the free-body diagram, one draws arrows and names them. These are typically to mark
forces and speeds. You can either use a standard sign rule and let mathematics decide the
sign ( ‘+’ or ‘-‘ ) of the variable, or, try to “feel” which direction the force or speed will have
and define your arrows and names according to this, to get positive numerical values. It is a
matter of taste which is best, but it is recommended that care is taken – it is very easy to
make sign mistakes.
You will notice that most mechanical models set up in Vehicle Dynamics are basically combinations of
(Static or Dynamic) Equilibrium, Compatibility and Constitutive relationships:


(Static) Equilibrium (using Newtonian mechanics):
o Sum of forces in one direction = 0
o Sum of moments around an axis in one direction = 0
Dynamic equilibrium (or “Equation of motion” or “Newton’s 2nd law”):
o Sum of forces (including “fictive forces” = “d’Alambert forces”) in one direction = 0,
or sum of forces in one direction = Mass * Acceleration in the forces’ direction
o Sum of moments (including “fictive moments”) in one direction = 0,
or sum of moments in one direction = Moment of inertia * Rotational acceleration in
the moment’s direction
o If fictive forces or moments are included on equation left-hand side, it is
recommended that they are introduced in the free-body diagram (see dashed force
arrow in Figure 1-4). These forces shall be opposite to the positive direction of
acceleration and act through the centre of gravity.
24
INTRODUCTION



(An alternative to Newtonian mechanics is Lagrange mechanics, which is sometimes an
easier way to find the same equations.)
Compatibility: Relation between positions or velocities, e.g. “Speed=Radius*Rotational
speed” for a rotating wheel
Constitutive relations: Relation between forces and positions or between forces and
velocities, e.g.
o “Force = constant * deformation” for a linear spring
o “Force = constant * deformation speed” for a linear damper
o “Force = constant * sign(sliding speed)” for a dry friction contact
acceleration
air resistance
m*acceleration
Fxf0
Draw free body diagrams
I.e. selecting the subsystems (=the
free bodies).
Here, these are:
• vehicle body together with rear
axle
• front axle alone
• remaining (ground & air)
m*g
Fzr
Tf
Fzf0
Fzf0
Tf
Fxf0
Fxf
Fzf
air resistance
Fzr
Fzf
Fxf
Figure 1-4: Free body diagram. The dashed arrow is a “fictive force”.
1.3.4.2
Operating conditions
The operating conditions of a vehicle lend expressions from general dynamics, as listed below.
•
(Static conditions, meaning that vehicle is standing still, are seldom relevant to analyse.
Static means that the all velocities are zero, i.e. that all positions are constant.)
•
Steady State operation means that time history is irrelevant for the quantities studied. Seen
as a manoeuvre over time, the studied quantities are constant.
•
Transient manoeuvres means that time history is relevant; i.e. there are delays, represented
by “state variables” when simulated.
•
Stationary (oscillation) manoeuvre is a special case of transient, where cyclic variations
continues over long time with a constant pattern. This pattern is often assumed to be
harmonic, meaning that the variable varies in sinus and cosine with constant amplitudes and
phases. An example is sinusoidal steering, where also the vehicle response is harmonic if
steering amplitude is small enough to assume the vehicle response is governed by a linear
dynamic system.
•
Quasi-static and Quasi-steady state are terms with a more diffuse meaning, which refers
rather to analysis methods than the actual operation/manoeuvre. It is used when the
analysis neglects the dynamics of a variable which normally is a state variable. An example is
when constant non-zero deceleration is assumed, but speed is not changed; then the
dynamics “der(speed)=acceleration” is neglected, and speed is instead prescribed.
25
INTRODUCTION
1.3.5
Verification methods with real vehicles
Methods which are helpful and frequently used for vehicle dynamics are listed in this section.
1.3.5.1
Testing in real traffic
Driving on public roads in real traffic is the most realistic way to verify how a vehicle actually works. It
can be used for completely new vehicle models; or new systems, mounted on old models. The
drivers can be either ordinary drivers (FOT=Field Operational Test) or test drivers (expeditions). A
general existing vehicle population can also be studied by collecting data, e.g. as Accident Statistics
Databases.
1.3.5.2
Testing on test track
For vehicles and systems which are not yet allowed on public roads, or tests which are very severe or
need a high degree of repeatability, test are carried out at test tracks. There are specialized test
tracks for certain conditions, such as hot climate or slippery surfaces.
1.3.6 Verification methods involving
theoretical simulation
The following are examples of methods containing theoretical simulation to some extent:
1.3.6.1
HIL = Hardware in the loop simulation
The hardware is often one or several ECUs (Electronic Control Units). If several ECUs are tested, the
hardware can also contain the communication channel between them, e.g. a CAN bus. The hardware
is run with real-time I/O to simulation model of the remaining system (vehicle, driver and
environment).
In some cases, there is also mechanical hardware involved, such as if the ECU is the brake system
ECU, the actual hydraulic part of the brake system can also be included in the HIL set-up, a so called
“wet brake ECU HIL”.
1.3.6.2
SIL = Software in the loop simulation
The software is often one or several computer programs (intended for download in electronic control
units). The software is run with synchronized time-discrete I/O to a simulation model of the
remaining system (vehicle, driver and environment).
The software is often used in compiled format (black box format) so that the supplier of the software
can retain his intellectual property.
1.3.6.3
MIL = Model in the loop simulation
The model, or more correctly, a control algorithm, is a conceptual form of the computer programs
(intended for download in electronic control units). The control algorithm is run with I/O to a
simulation model of the remaining system (vehicle, driver and environment).
The control algorithms can appear in compiled format so that the supplier of the control algorithms
can retain his intellectual property. Then it is hard to tell the difference between MIL and SIL.
1.3.6.4
Simulator = Driver in the loop simulation
This is when a real human, not a driver model, uses real driver devices (pedals, steering wheel) to
influence a simulation model of the remaining system (vehicle and environment). The loop is closed
by giving the human feedback through display of what would be visible from driver seat, including
26
INTRODUCTION
views outside windscreen. Feedback can be further improved by adding a motion platform to the
driver’s seat, sound, vibrations in seat, steering wheel torque, etc.
1.3.6.5
AR = Augmented Reality
This is a new method. A typical example is: A real driver drives a real vehicle on a real road/test track.
Some additional (virtual/simulated) traffic objects are presented to driver, e.g. on a head-up display.
The same objects can be fed into the control functions, as if they were detected by the vehicles
camera/radar, which enables functions such as automatic braking to be triggered.
1.3.7
Tools & Methods
(Computer) Tools are helpful, and frequently used for vehicle dynamics. Some descriptions and
typical uses is also given.
1.3.7.1
General mathematics tools
Examples of tool: Matlab, Matrixx, Python
We will take Matlab as example. Matlab is a commercial computer program for general
mathematical analysis. It is developed by Mathworks Inc.
Solve linear systems of equations, A*x=b
>> x=inv(A)*b;
Solve non-linear systems of equations, f(x)=0
>> x=fsolve('f',...);
Parameter optimization (may be used for trajectory optimization, if trajectory is parameterised)
>> (x,fval) = fmincon('f',x0, …)
Solve ODE (=”systems of Ordinary Differential Equations”) as initial value problems, dx/dt=f(t,x),
given x(0)
>> x=ode23('f',x0,...);
Find Eigen vectors (V) and Eigen values (D) to linear systems: D*V=A*V
>> [V,D]=eig(A);
Matlab is mainly numerical, but also has a symbolic toolbox:
>> syms x a; Eq='a/x+x=0'; solve(Eq,x)
ans = (-a)^(1/2)
-(-a)^(1/2)
>> diff('a/x+x',x)
ans = 1 - a/x^2
>> int('x^3+log(x)',x)
ans = (x*(4*log(x) + x^3 - 4))/4
Some simple Matlab code will be used to describe models in this compendium.
1.3.7.2
Block diagram based simulation tools
Examples of tools: Simulink, Systembuild
We will take Simulink as example. Graphical definition of non-linear systems of ODE, if known on
explicit form dx/dt=f(t,x). An example of how the corresponding graphical model may look is given in
Figure 1-5.
27
INTRODUCTION
Figure 1-5: Graphical model of dynamic system 𝑥
𝑓(𝑡 𝑥); using Simulink.
Simulink is designed for designing/modelling signal processing and control design. It can also be used
for modelling the physics of the controlled systems.
There are no dedicated vehicle dynamics tools/libraries from Mathworks (but there are in-house
developed specific libraries in automotive companies).
From this type of tools it is often possible to automatically generate real time code, which is more
and more used instead of typing algorithms. It can be used for rapid prototyping of control functions,
or even for generation of executable code for production ECUs.
1.3.7.3
Vehicle dynamics specialized simulation tools
Examples of tools: CarMaker, veDYNA, CarSim.
These tools are specialized for vehicle dynamics. They contain purpose-built (not necessary complete
3D mechanics) and relatively advanced models of vehicles, drivers and scripts for test manoeuvres.
They often have an interface to Simulink, so that the user can add in their own models.
1.3.7.4
MBS tools
Examples of tools: Adams, Simpack, LMS Virtual Lab
These are general 3D mechanics modelling and simulation tools, so called MBS (Multi-Body
Simulation) tools. As one example, Adams contains libraries of general bodies, joints and force
elements. But there are toolboxes in Adams for vehicle dynamics, where template models and
special components (such as tyre models and driver models) are available for vehicles dynamics. The
models are very advanced and accurate for 3D mechanics, and there are import/export interfaces to
Simulink.
1.3.7.5
Modelica based modelling tools
Examples of tools: Dymola, Maplesim, System-Modeler, AMESim, Optimica Studio, Jmodelica,
OpenModelica
Modelica is not a tool but a globally standardized format for dynamic models on DAE form (see
Matematical/Declarative modelling in Section 1.3.2). There are a number of tools which supports this
format.
The model format itself is non-causal and symbolic. There is one equation part and one declaration
part in a model. An example of model is given in Figure 1-6.
28
INTRODUCTION
Physical model:
Simulation in Dymola:
𝑐
𝑐
𝑥
(Mathematical) Model in Modelica:
Figure 1-6: Example of model in Modelica forma (using the tool Dymola).
The model format is also object oriented, which means that libraries of model components are
facilitated. These are often handled with graphical representation, on top of the model code. There
are some open-source libraries for various physical domains, such as hydraulic, mechanics,
thermodynamics and control. There are also commercial libraries, where we find vehicle dynamics
relevant components: Vehicle Dynamics Library and Powertrain Library.
Some simple Modelica code will be used to describe models in this compendium.
In one way, Matematical modelling (Declarative modelling) is more efficient than Explicit form
modelling (Imperative modelling), since the engineer does not need to spend time on
symbolic/algebraic manipulation of the equations. In this compendium, most modelling is done with
Matematical modelling. One the other hand, an Explicit form model has the value of capturing the
causality, i.e. the cause-to-effect chain. The causality can sometimes facilitate the understanding and
in that way help the engineer, which is why at least one and rather complete model is shown as
Explicit form model, see Section 4.5.3.2.
1.3.8
Coordinate Systems
A vehicle’s (motion) degrees of freedom are named as in marine and aerospace engineering, such as
heave, roll, pitch and yaw, see Figure 1-7. Figure 1-7 also defines the 3 main geometrical planes, such
as transversal plane and symmetry plane. For ground vehicles, the motion in ground plane is often
treated as the primary motion, which is why longitudinal, lateral and yaw are called in-ground-plane
degrees of freedom. The remaining degrees of freedom are referred to as out-of-ground-plane.
The consistent use of parameters that describe the relevant positions, velocities, accelerations,
forces, and moments (torques) for the vehicle are critical. Unfortunately there are sometimes
disparities between the nomenclature used in different text books, scientific articles, and technical
reports. It is important to recognize which coordinate system is being applied and realize that all
conventions will provide the same results as long as the system is used consistently. Two
predominant approaches are encountered: the International Standards Organisation (ISO) and the
Society of Automotive Engineers (SAE). Both ISO, (ISO8855), and SAE, (SAEJ670), are right handed
systems. Figure 1-8 shows the vehicle fixed coordinate systems for these two standards. This
29
INTRODUCTION
compendium uses ISO, which also seems to be the trend globally. The new edition of (SAEJ670) now
also recognises an optional use of z-up.
ground, road
or horizontal
plane
heave or
vertical
yaw
transversal
plane
symmetry
plane
Figure 1-7: Vehicle (motion) degrees of freedom and important planes.
SAEJ670
ISO 8855
z
y
y
x
z
x
Figure 1-8: ISO and SAE coordinate systems
The distinction of vehicle fixed and inertial (= earth fixed = world fixed) coordinate systems is
important. Figure 1-9 depicts the four most relevant reference frames in vehicle dynamics: the
inertial, vehicle, wheel corner and wheel reference frames. All these different coordinate systems
allow for the development of equations of motion in a convenient manner.
The orientation of the axes of an inertial coordinate system is typically either along the vehicle
direction at the beginning of a manoeuvre or directed along the road or lane. Road or lane can also
be curved, which calls for curved longitudinal coordinate.
Origin for a vehicle fixed coordinate system is often centre of gravity of the vehicle, but other points
can be used, such as mid of front axle, mid of front bumper, outer edge of body with respect to
certain obstacle, etc. Positions often need to be expressed for centre of lane, road edge, other
moving vehicle, etc.
30
INTRODUCTION
Vehicle fixed
coordinate system:
lateral= y
yaw rate=wz=wz
v
x
b=b=(body)
vx
vy
CoG
=longitudinal
side slip angle
n =course
angle
Speed of
wheel hub
=pz=heading angle
Wheel fixed
coordinate
system:
y
y
Wheel corner
fixed coordinate
system:
x
d=d=
(Road wheel)
z
Steering
angle
Inertial coordinate system:
z
x
NOTE: Above tyre side slip is positive for tyre sliding with
positive lateral speed. Opposite definition is sometimes used.
Figure 1-9: Coordinate systems and motion quantities in ground plane
In ISO and Figure 1-9, tyre side-slip is defined so that it is positive for positive lateral speed. This
means that lateral forces on the wheel will be negative for positive side-slip angles. Some would
rather want to have positive force for positive angle. Therefore, one can sometimes see the opposite
definition of tyre side-slip angles, as e.g. in (Pacejka, 2005). It is called the “modified SAE” or
“modified ISO” sign convention. This compendium does not use the modified sign convention in
equations, but some diagrams are drawn with force-slip-curve in first quadrant. Which is preferable
is simply a matter of taste.
Often there is a need to number each unit/axle/wheel. The numbering in Figure 1-10 is proposed. It
should be noted that non-numeric notations are sometimes used, especially for two axle vehicles
without secondary units. Then front=f, rear=r. Also to differentiate between sides, l=left and r=right.
Fy1
Fx1
wheel 9 wheel 7
wheel 5 wheel 3
wheel 1
wheel 10 wheel 8
wheel 6 wheel 4
wheel 2
Fx2
Fy2
axle 5 axle 4
axle 3
axle 2
axle 1
unit 1
unit 2
Figure 1-10: Proposed numbering of units, axles and wheels. Example shows a
truck with trailer.
31
INTRODUCTION
1.3.9
Terms with special meaning
1.3.9.1
Load levels
The weight of the vehicle varies through usage. For many vehicle dynamic functions it is important to
specify the level, which is these definitions are important to know and use.
Kerb weight is the total weight of a vehicle with standard equipment, all necessary operating
consumables (e.g., motor oil and coolant), a full tank of fuel, while not loaded with either passengers
or cargo. Kerb weight definition differs between different governmental regulatory agencies and
similar organizations. For example, many European Union manufacturers include a 75 kilogram driver
to follow European Directive 95/48/EC.
Payload is the weight of carrying capacity of vehicle. Depending on the nature of the mission, the
payload of a vehicle may include cargo, passengers or other equipment. In a commercial context,
payload may refer only to revenue-generating cargo or paying passengers.
Gross Vehicle Mass (GVM) is the maximum operating weight/mass of a vehicle as specified by the
manufacturer including the vehicle's chassis, body, engine, engine fluids, fuel, accessories, driver,
passengers and cargo but excluding that of any trailers.
Other load definitions exist, such as:


“Design Weight” (typically Kerb with 1 driver and 1 passenger, 75 kg each, in front seats)
“Instrumented Vehicle Weight” (includes equipment for testing, e.g. out-riggers)
For vehicle dynamics it is often also relevant to specify where in the vehicle the load is placed.
1.3.9.2
Open loop and Closed loop manoeuvres
Depending on the drivers operation, one can differ between Open loop and Closed loop manoeuvres,
see Section 2.10.1.
1.3.9.3
Objective and subjective measures
Two main categories of finding measures are:


An Objective measure is a measure calculated in a defined and unique way from data which
can be logged in a simulation or from sensors in a real test.
A Subjective measure is a measure based on the test driver’s judgement (e.g. on a scale 1-10)
from a real test or driving simulator test.
One generally strives for objective measures, but many relevant functions are difficult to capture
objectively, such as Steering feel and Comfort in transient jerks.
1.3.9.4
Path, Path with orientation and Trajectory
A path can be 𝑥(𝑦) or 𝑦(𝑥) for centre of gravity where 𝑥 and 𝑦 are coordinates in the road plane. To
cope with all paths, it is often necessary to use a curved path coordinate instead, 𝑠, i.e. 𝑥(𝑠) and
𝑦(𝑠). The vehicle also has a varying orientation, (𝑥) or (𝑠), which often is often relevant, but
the term “path” does necessarily include this. In those cases, it might be good to use an expression
“path with orientation” instead.
A trajectory is a more general term than a path and it brings in the dependence of time, 𝑡. One
typical understanding is that trajectories can be [𝑥(𝑡); 𝑦(𝑡); (𝑡)]. But also other quantities, such
as steering angle or vehicle propulsion force can be called trajectory: 𝛿(𝑡) and (𝑡), respectively.
The word “trace” is sometimes used interchangeably with trajectory.
32
INTRODUCTION
1.3.9.5
Stable and Unstable
Often, in the automotive industry and vehicle dynamics, the words “stable” and “unstable” have a
broad meaning, describing whether high lateral slip on any axle is present or not. In more strict
physics/mathematical nomenclature, “unstable” would be more narrow, meaning only exponentially
increasing solution, which one generally finds only at high rear axle side-slip.
It is useful to know about this confusion. An alternative expression for the wider meaning is “loss of
control” or “loss of tracking”.
1.3.9.6
Subject and object vehicle
The subject vehicle is the vehicle that is studied. Often this is a relevant to have a name for it, since
one often study one specific vehicle, but it may interact with other in a traffic situation. Alternative
names are host vehicle, ego vehicle or simply studied vehicle.
If one particular other vehicle is studied, it can be called object vehicle or opponent vehicle. A special
case of object vehicle is lead vehicle which is ahead of, and travels in same direction as, subject
vehicle. Another special case is on-coming vehicle which is ahead of, and travels in opposite direction
as, subject vehicle.
1.3.9.7
Active Safety and ADAS
The expression Active Safety is used a lot in Automotive Engineering. There are at least two different
usages:


Active Safety can refer to the vehicle’s ability to avoid accidents, including both functions
where the driver is in control (such as ABS and ESC, but also steering response) and functions
with automatic interventions based on sensing of the vehicle surroundings (such as AEB and
LKA). See http://en.wikipedia.org/wiki/Active_safety. Active Safety can even include static
design aspects, such as designing the wind shield and head light for good vision/visibility.
Alternatively, Active Safety can refer to only the functions with automatic interventions
based on sensing of the vehicle surroundings. In those cases it is probably more specific to
use Advanced Driver Assistance Systems (ADAS) instead, see
http://en.wikipedia.org/wiki/Advanced_Driver_Assistance_Systems. ADAS does not only
contain safety functions, but also comfort functions like CC and ACC.
1.3.9.8
Autonomous driving
Functions that off-load the driver the direct tasks during driving can be sorted under Autonomous
driving. Fully autonomous driving, e.g. transportation without human driver on-board, is probably a
far future vision. On the other hand, it is already a reality that some portion of the driving tasks are
autonomous in the latest vehicle on the market, such as Adaptive cruise control, see Section 3.5.2.2,
and Lane Keeping Aid, see Section 4.6.2.1. If both those are active at the same time, we have already
autonomous driving in some sense. Today’s version of these systems normally have a way to handback responsibility to driver rather immediately in hazardous situations. Future autonomous driving
functions will need to always have a safe-stop function. The way to compete between vehicle
manufacturers will probably be to avoid hand-backs (maximize up-time) and to allow as long “handback times” as possible. So, vehicle dynamics will be important in the development, especially for
safety reasons for autonomous driving in higher speeds; hazardous situations where human driver
selects to take back the driving.
33
INTRODUCTION
1.3.10 Architectures
Vehicles are often designed in platforms, i.e. parts of the design solutions are reused in several
variants. Typical variants may be different model years or different propulsion system. To be able to
reuse solutions, the vehicles have to be built using the same architecture.
A mechanical architecture may include design decisions about certain type of wheel suspension on
front and rear axle. An electrical and electronic architecture may include design decisions about
electric energy supply system (battery voltage etc.) and electronic hardware for computers
(Electronic Control Units, ECUs) and how they are connected in networks, such as Controller Area
Network, CAN.
That the mechanical architecture influences vehicle dynamics functions is rather obvious. However, it
is noteworthy that also the electronic architecture also is very important for the vehicle dynamics,
through all electronic sensors, actuators and control algorithms. One example of this importance is
the ABS control of the friction brake actuators. Architectures for functions are therefore motivated,
see Section 1.3.10.1.
1.3.10.1 Reference architecture of vehicle functionality
As the number of electronically controlled functions increase, the need for some sort of reference
architecture for vehicle functionality, or “Functional Architecture”, becomes necessary to meet fast
introduction of new functionality and to manage different variety of vehicle configurations. A
reference architecture is a set of design decisions about how functions interact with each other (e.g.
signalling between control functions). An older expression which is related to functional architecture
is cybernetics. Examples (from Vehicle Dynamics functional domain) of modern expressions which
are related are Integrated Chassis Control (GM), Integrated Vehicle Dynamics Control (Ford),
Complete Vehicle Control (Volvo) and Vehicle Dynamics Integrated Management (Toyota). The
authors of the compendium claim that there is no exact and generally well accepted definition of
such architecture. However, it becomes more and more essential, driven by increasing content of
electronic control in vehicles. One way to visualize a reference architecture is given in Figure 1-11.
In order to be able to formulate design rules in reference architecture of functionality the following
are relevant questions:
• Which physical vehicle quantities should be represented on the interface between Sensors and
Actuators on one side and Vehicle Level Functionality on the other?
• Partitioning within a reference architecture of vehicle motion functionality could be realised as
shown in Figure 1-11. Different Layers/Domains are defined:
• Motion Support Device Layer: This includes the devices/actuators that can generate
vehicle motion. This layer is also consisting sensors which could include the capability
and status of each device e.g. max/min wheel torque.
• Vehicle Motion Layer: This includes the Energy management, powertrain coordination,
brake distribution, and vehicle stability such as ESC, ABS. This layer also estimates the
vehicle states e.g. vx, vy, wz. In addition this layer would be able to give vehicle level
capability of max/min acceleration and their derivative.
• Traffic Situation Layer: Interpret the immediate surrounding traffic which the vehicle is
in. Automated driving assistance functions such as adaptive cruise control, collision
avoidance, and lane steer support are typical functions.
• Vehicle Environment: Includes surrounding sensors mounted on vehicle but also
communication with other vehicles (V2V) and infrastructure (V2I) and map information.
34
INTRODUCTION
• Human Machine Interface: This includes the sensors/buttons which the driver use to
request services from the vehicle’s embedded motion functionality.
• Formalisation of different types of:
• Blocks, e.g. Controller, Signal Fusion, Arbitrator, and Coordinator.
• Signals, e.g. Request, Status and Capability.
• Parameters used in Functional blocks, e.g. Physical parameter and Tuning parameter.
Vehicle
Environment
V2V / V2I
Maps
…
Traffic Situation Layer
Forrward Direction
ACC
Traffic
Interpretation
LaneSteering
Vehicle Capability
Max/Min Acceleration
Req. speed / acceleration
Vehicle Motion and Coordination Layer
Vehicle Motion
Estimator
Device Capability
Max/Min Wheel Torque
ACCbtn
Arbitration
Status
Vehicle
Capability
Estimator
Driver
Interpretation
Dist btn
Collision
Avoidaice
FwdSens
Human
Machine
Interface
Driver
Interpretation
Vehicle Motion
Control
Energy
Management
SteWhl
AnSens
APed
BPed
Arbitration
Req. e.g. Global Forces
Coordination
Stability
Control
Status
Req. e.g. Wheel Torque
Motion Support Device Layer
Device
Cap./State
Estimator
EngAct
SteAct
FrntBrkAct
RearBrkAct
TransmAct
Vehicle
Motion
Sens
Figure 1-11: One example of reference architecture of vehicle motion functionality. Red arrows:
Requests, Blue arrow: Information, Black lines with dot-ends: physical connection.
The functionality is then allocated to ECUs, and signals allocated to network communication. The
reference architecture can be used for reasoning where the allocation should be done. Which
functionality is sensitive for e.g. time delay and thus should be allocated in the same ECU. Detailed
control algorithm design is not stipulated by a reference architecture. Instead the reference
architecture assists how detailed control algorithms be managed in the complete problem of
controlling the vehicle motion. Whether representation of solutions of Functional Safety (ISO 26262,
etc.) is represented in a reference architecture of functionality can vary.
1.4 Heavy truck versus passenger cars
Passenger cars are relatively well known to most people and they seldom appear in very complex
combinations. But heavy trucks are less well known and appear in vehicle combinations with many
more units. Hence the following overview of the differences and most common units is given.
35
INTRODUCTION
1.4.1
General differences
Trucks are normally bought by companies, not private persons. Each truck is bought for a specialized
transport task. Life, counted in covered distance, for trucks is typically 10 times passenger cars. The
life time cost of fuel is normally 5 times the vehicle cost, compared to passenger car where these
costs are about equal. The cost for driver salary is a part of mileage cost, typically same magnitude as
fuel cost. If investment cost for vehicle and repairs are distributed over travelled distance, these are
typically also of same magnitude. So, the cost for a transport typically comes from one third fuel, one
third driver salary and one third vehicle investment and repairs.
1.4.2
Vehicle dynamics differences
A truck has 5..10 times less power installed per vehicle weight. Trucks have their centre of gravity
much higher, meaning that roll-over occurs at typically 4 m/s2 lateral acceleration, as compared to
around 10 m/s2 for passenger cars. Trucks have centre of gravity far behind mid-point between axles,
where passenger cars have it approximately symmetrical between the axles. Trucks are often driven
with more units after, see Figure 1-12. The weight of the load in a truck can be up to 2..4 times the
weight of the empty vehicle, while the maximum payload in passenger cars normally are significantly
lower than the empty car weight. Trucks often have many steered axles, while passenger cars
normally are only steered at front axle.
Figure 1-12: An overview over conventional and longer combinations. From (Kharrazi , 2012).
1.5 Smaller vehicles
This section is about smaller vehicles, meaning bicycles, electric bicycles, motorcycles and 1-2 person
car-like vehicles. The latter group refers to vehicles which are rare today, but there is a logic
reasoning why they could become more common in the future: Increasing focus on energy
consumption and congestion in cities can be partly solved with such small car-like vehicles, of which
the Twizy in Figure 1-13 is a good example. All vehicles in Figure 1-13 can be referred to as Urban
36
INTRODUCTION
Personal Vehicle (UPVs), because they enable personalised transport in urban environments. The
transport can be done with low energy consumption per travelled person and distance, compared to
today’s passenger cars. The transport can be done with high level of flexibility and privacy for the
travelling persons, compared to today’s public transportation.
Piaggio MP3
3 wheeled cambering
concept vehicle from BMW
Toyota
iRoad
Twizy from Renault
ZeeBee from Clean Motion
Eco Electric Bicycle, Monark
Figure 1-13: Examples of Urban Personal Vehicles. Left: “Roll-stiff”. Right:
“Cambering“. From www.motorstown.com, www.cleanmotion.se, www.monarkexercise.se,
www.nycscootering.com.
New solutions as in Figure 1-13 may require some categorization.
 Climate and user type: Sheltered or open.
 Transport and user type: Short travels (typically urban, 5-10 km, 50 km/h) or long travels
(typically inter-urban, 10-30 km, 100 km/h).
 Chassis concept:
o Narrow (e.g. normal bicycles and motorcycles) or wide (at least one axle with 2 wheels,
resulting in 3-4 wheels on the vehicle).
Note that UPVs in both categories are typically still less wide than passenger cars.
o Roll moment during cornering carried by suspension roll stiffness or roll moment during
cornering avoided by vehicle cambering. The first concept can be called “Roll-stiff
vehicle”. The second concept can be called “Cambering vehicle” or “Leaning vehicle”. 1tracked are always Cambering vehicles. 2-tracked are normally Roll-stiff, but there are
examples of Cambering such (see upper right in Figure 1-13).
o (This compendium does only consider vehicles which are “Pitch-stiff”, i.e. such that can
take the pitch moment during acceleration and braking. Examples of vehicles not
considered, “Pitching vehicles”, are: one-wheeled vehicles as used at circuses and twowheeled vehicles with one axle, such as Segways.)
Note that also Roll-stiff Vehicles camber while cornering, but only slightly and outwards in curve,
while Cambering Vehicles above refer to significant cambering and inwards in curve.
Cambering Vehicles is more intricate to understand when it comes to how wheel steering is used. As
a reference: In Roll-stiff Vehicles, the wheel suspension takes the roll-moment (keeps the roll
balance), which means that driver can use wheel steering solely for making the vehicle steer (follow
an intended path). In Cambering Vehicles, the driver has to use the wheel steering for both keeping
37
INTRODUCTION
the roll balance and follow the intended path. This means one control for two purposes, which calls
for one more control. The additional control used comes from that the driver can move the CoG of
the vehicle (including driver) laterally. So, the driver of a Cambering Vehicle has to use a combination
of wheel steering and CoG moving for a combination of maintaining roll balance and following the
intended path.
Smaller vehicles might have significantly different vehicle dynamics in many ways:
 Influence of driver weight is larger than for other vehicle types. This especially goes for
bicycles, where driver weight can be typically 5 times larger than the vehicles own weight.
Drivers not only change the total inertia properties, but they can also actively use their weight
and move it during driving.
 Ratio between CoG height and wheel base is likely to be larger than passenger cars and trucks.
This gives a larger longitudinal load transfer during acceleration and braking. Some concepts
might even have the risk of “pitch-over”.
 For 2-tracked UPVs, the ratio between CoG height and track width is likely to be larger than
passenger cars, rather like trucks, because it is likely that one want to keep swept area low for
UPVs.
o If Roll-stiff, this gives a larger lateral load transfer and roll angles during cornering. The
risk for roll-over is also likely to be larger than passenger cars.
o If Cambering, this opens for the risk for slide (“roll-over inwards in curve”, so called “lowsider”), which is known from motorcycles.
 For future UPVs it is possible that the part costs need to be kept low, as compared to
passenger cars. This, and the fact that new inexperienced OEMs might show up on market,
might lead to low-cost and/or unproven solutions appear for the vehicle design. This might be
a challenge for driving experience and (driver and automated) active safety.
 Positive, for safety, is that future UPVs can have significantly lower top speed than passenger
cars. However, the acceleration performance up to this top speed might be high, e.g. due to
electric propulsion, which might cause new concerns for city traffic with surrounding.
1.6 Automotive engineering
This section is about a larger area than Vehicle Dynamics. It is about the context where Vehicle
Dynamics is mainly applied.
OEM means Original Equipment Manufacturer and is, within the automotive industry, used for a
company whose products are vehicles. OEM status is a legal identification in some countries.
In the automotive industry, the word Supplier means supplier to an OEM. There are Tier1 suppliers,
Tier2 suppliers, etc. A Tier1 supplies directly to an OEM. A Tier2 supplies to a Tier1 and so on.
Primarily, suppliers supply parts and systems to the OEMs, but suppliers can also supply competence,
i.e. consultant services.
From an engineering view it is easy to think that an OEM only does Product Development and
Manufacturing. But it is good to remember that there is also Marketing & Sales, After Sales, etc.
However, Product Development is the main area where the vehicles are designed. It is typically
divided into Powertrain, Chassis, Body, Electrical and Vehicle Engineering. Vehicle Dynamics
competence is mainly needed in Chassis and Powertrain.
On supplier side, Vehicle Dynamics competence is mainly needed for system suppliers that supplies
propulsion, brake, steering and suspension systems.
Except for OEMs and suppliers, Vehicle Dynamics competence is also needed at some governmental
legislation and testing as well as research institutes.
38
INTRODUCTION
There are engineering associations for automotive engineering. FISITA (Fédération Internationale des
Sociétés d'Ingénieurs des Techniques de l'Automobile, www.fisita.com) is the umbrella organisation
for the national automotive societies around the world. Examples of national societies are IMechE
(United Kingdom), JSAE (Japan), SAE (USA), SAE-C (Kina), SATL (Finland), SIA (France), SVEA (Sweden,
www.sveafordon.com) and VDI FVT (Germany). There is a European level association also, EAEC.
1.7 Typical numerical data
The purpose of the tables in this section is to give approximate numerical values of parameters.
Vehicle parameters are often dependent of eachother; changing one leads to that it is suitable to
change others. The parameters are given with the intention to be consistent with each other, for
each vehicle example. The tables balances between being generic and specific, which is difficult.
Therefore, please consider the table as very approximate.
Table 1.3: Typical data, common for typical passenger cars and heavy trucks
Parameter
Air density
Earth gravity
Road friction, at dry asphalt
Road friction, at wet asphalt
Road friction, at snow
Road friction, at wet ice
Tyre cornering stiffness,
normalized with vertical load
1.7.1
Notation Typical Value Unit
roh
1 kg/(m^3)
g
9.80665 m/(s*s)
1.0
0.6
mu
N/N
0.3
0.1
(N/rad)/N
C
10
= (N/1)/N
For passenger vehicle
Table 1.2 gives typical numerical data for a medium sized passenger car.
39
INTRODUCTION
Table 1.4: Typical data for a passenger vehicle
Group/Type
Parameter
1. General
Length, bumper to bumper
Longitudinal distance from
CoG to front axle
Mass
1. General
1. General
Notation Typical
Unit
Value
5.00 m
lf
1.3 m
m
1700 kg
1. General
Moment of inertia for yaw
rotation
Jzz
1. General
Unsprung mass
mus
1. General
1. General
Track width
Wheel base
W
L
2. Systems
Engine inertia
Transmission ratio, highest
gear
Transmission ratio, lowest
gear
Brake proportioning,
front/rear
Rolling resistance
Steering ratio
Wheel radius
Wheel rotational inertia
2. Systems
2. Systems
2. Systems
2. Systems
2. Systems
2. Systems
2. Systems
Projected area in a
transversal view
Afront
3. Longitudinal
Aerodynamic coefficients
cd, clf,
clr
5. Vertical
Suspension heave stiffness
(without roll)
cs
5. Vertical
Suspension roll stiffness,
only from anti-roll-bars
carb
5. Vertical
Suspension damping
ds
5. Vertical
Tyre vertical stiffness
ct
1.7.2
40-50% of wheel base:
lf=0.55*L;
Radius of gyration is sligthly
2900 kg*m*m less (0.9) than half wheel base:
=m*(0.9*L/2)^2
Sum of 4 wheels with
200 kg
suspensions
1.70 m
2.90 m
0.5 kg*m*m
rad/rad=
5.00
Nm/Nm
rad/rad=
10.00
Nm/Nm
N/N or
70/30
Nm/Nm
0.01 N/N
16 rad/rad
0.30 m
0.5 kg*m*m
3. Longitudinal
Note / Typical / Range /
Relation to other
2.2 m^2
0.31, 0.10,
0.07
1
100 000 N/m
Engine to wheel. In magnitude
of 5.
Engine to wheel. In magnitude
of 10.
Often tuned so that braking
both axles when braking 0.8*g
Steering wheel to Road wheel
For one wheel
For calculation of air resistance
For calculation of longitudinal
resistance and lift force over front
and rear axle
Vertical stiffness at wheel. Sum of 4
wheels. So that bounce frequency f
is between 1 and 2 Hz:
sqrt(c/ms)=f*2*pi;
14 000 N/m
Vertical stiffness at wheel. Sum of
both axles.
13 000 N/(m/s)
Wheel rate. Sum of 4 wheels. Some
40..60% of critical damping:
d = (0.4..0.6)*2*sqrt(c*m);
2..3 times softer in compression
than rebound.
250 000 N/m
For one wheel
For heavy vehicle
Compared to passenger cars, trucks differs much more in size and configuration, see Figure 1-12.
Also, a certain individual vehicle also differs much more between empty and fully loaded. So, Figure
1-14 and Table 1.5 are really to be seen only as an example. Globally, “Tractor with Semitrailer” is the
most common vehicle, which is why it is used as example.
40
INTRODUCTION
load length=13.6
wheel base=8
front
overhang
=1.5
overall height
=4 m
overall height =
3.7 m
front overhang
=1.5
forward
offset =4
wheel base=5
rear
axle spread overhang
=1.5
=3.6
rear
overhang
=1
vehicle overall length=16.5 m
Figure 1-14: Typical data for a heavy vehicle, exemplified with “Tractor with Semi-trailer”
Table 1.5: Typical data for a heavy vehicle, exemplified with “Tractor with Semi-trailer”
Group/Type
Parameter
Typical Value
Empty With max Unit
payload
Tractor
Semitrailer
Vehicle
Tractor
Semitrailer
Tractor
Semitrailer
Tractor
Semitrailer
Wheels
Wheels
Tyres
Trackwidth, centre-to-centre for
single tyres
CoG heigth
CoG heigth
CoG heigth
Mass (total), tractor
Mass (total), semi-trailer
Moment of yaw inertia
Moment of yaw inertia
Unsprung mass
Unsprung mass
Wheel radius
Wheel rotational inertia
Rolling resistance
Propulsion
Propulsion
Engine max power
Engine inertia
Propulsion
Transmission ratio, highest gear
3
Propulsion
Transmission ratio, lowest gear
30
Steering
Steering ratio
20
Vehicle
Projected area in a transversal
view
10.0
Vehicle
Aerodynamic coefficient
0.4
Trackwidth
2.25
1.00
1.00
1.00
2.00
2.00
7500
10000
32500
22500
150000
500000
1700
2400
0.50
10.0
0.005
370
5
41
Note / Typical / Range /
Relation to other
m
Outer tyre edge 2.55
m
m
m
kg
kg
kg*m*m
kg*m*m
kg
kg
m
kg*m*m
N/N
Around unit CoG
Around unit CoG
Sum of all axles at unit
Sum of all axles at unit
0.4-0.5
For one wheel, single tyre
Or less, 0.003
kNm/s=kW
kg*m*m
1=rad/rad= Engine to wheel
Nm/Nm
Engine to wheel
Steering wheel to Road
rad/rad
wheel
For calculation of air
m^2
resistance
c_d coefficient in aero
1
dynamic resistance formula
INTRODUCTION
42
VEHICLE INTERACTIONS
2 VEHICLE INTERACTIONS
2.1 Introduction
The study of vehicle dynamics starts with the interfaces between the vehicle and its environment,
see Figure 2-1. The vehicle tyres are the primary structures to transfer loads that produce desired
motions (vehicle weight, acceleration, steering, braking) and undesired disturbances (road vibrations,
road bumps, etc.). Additionally, the aerodynamic loads on the vehicle will create forces and moments
that are often undesirable (wind resistance, side gusts, etc.) but can sometimes be exploited for
better contact with the road (down-force). An example of extreme interactions to the vehicle is the
impact forces from a crash. An interaction of another kind, but still very influencing, is the driver.
The main focus of this chapter is to present the general characteristics of the tyres to start the
process of qualitatively and quantitatively studying the other vehicle systems (steering, propulsion,
suspension). Initial first order models will be presented with a brief interaction of combined loading.
General aerodynamic load calculations are presented but further discussion of aerodynamic loads
are outside the scope of this compendium. Driver interaction is briefly mentioned. Traffic
interactions with the vehicle are also briefly mentioned. Interaction with other road-users (collisions)
is not covered.
Driver
Environment
Steering Wheel,
Pedals, Buttons,
Seat Accelerations,
Visual Instruments,
Sounds, …
Air,
Weather,
…
Aerodynamic
Forces
Relative Positions,
Collision Forces,
Information,
(Object vehicles, Pedestrian,
Traffic signs, Infrastructure, …)
(Subject)
Vehicle
Tyre Forces
Ground/Road
Figure 2-1: Interactions of the vehicle
2.2 Introduction to Tyre Terminology
The tyre (or wheel) should be analysed in its own coordinate system, see Figure 2-2. There are many
different quantities and terms presented in Figure 2-2 and all are relevant for vehicle dynamics
analysis. Fortunately some terms dominate the analyses presented in this course. The three forces,
, and , are obviously necessary for any free body diagrams of the loads on the vehicle.
Following these parameters, the term 𝑅 represents the loaded (deflected) or rolling radius of the
43
VEHICLE INTERACTIONS
tyre needed to identify how the loads are applied relative to the spin axis of the vehicles. Terms
critical to the main topics in this course are:



Longitudinal Dynamics: Torques about the spin axis.
Lateral Dynamics: Slip angle: 𝛼
Vertical Dynamics: Vertical Load:
Other parameters presented in Figure 2-2 will have some influence on the theories developed in the
coming chapters and will be discussed as appropriate.
perpendicular
to road plane
Figure 2-2: Tyre Coordinate System according to ISO
2.2.1
Wheel angles
2.2.1.1
Steering angle
A wheel’s steering angle is the angle between the wheel plane and the longitudinal axis of the
vehicle. Steering angle can be defined for one wheel or one axle, using averaging of left and right
wheel.
2.2.1.2
Camber (Angle)
Camber or Camber angle is the angle between the wheel plane and the vertical, positive when the
top of the wheel leans outward. (This definition of sign is only applicable for axles with two wheels.
For instance, it is not applicable for motorcycles.) Camber generates lateral forces and gives tyre
wear. See also Section 0.
2.2.1.3
Caster Angle
The angle in a side-view between the steering axis and the vertical axis is called Caster angle. It is
positive if the top of the steering axis is inclined backwards. Caster angle provides an additional
aligning torque, see Section 0 and Section 4.2.3.2, and changes the camber angle when the wheel is
steered.
44
VEHICLE INTERACTIONS
2.2.1.4
Toe Angle
Toe angle (or toe-in) is defined for an axle with two wheels (not for a single wheel), as the difference
between right and left wheel’s steering angles. Toe angle is positive if front ends of the wheels are
pointing inward. Hence, toe can be called toe-in. Negative toe can be called toe-out. Toe angle
generates opposing lateral forces on each side.
The toe angles vary with the tyre forces, due to suspension linkage geometry and elasticity in
suspension bushings. This means that there is a static toe angle (when a vehicle parked in workshop,
with a certain load and not tyre forces in ground plane) has one value, while actual toe angle (in a
particular moment during a manoeuvre) has another value.
Zero toe gives low rolling resistance and low tire wear. Theoretically, toe-out on front axle and toe-in
on rear axle makes the vehicle most yaw stable (less over-steered). Tone-in on front axle makes
vehicle more yaw agile and improves on-centre steering feel. Normal design choice is toe-in on both
axles, and more on front axle.
2.2.1.5
Steering axis inclination
Steering axis inclination (or Kingpin inclination) is defined for a wheel with its suspension linkage (not
for an isolated wheel). The angle in a front view between the steering axis (or Kingpin) and the
vertical axis is called Kingpin inclination. See Section 4.2.3.2.
2.3 Tyre Construction
The tyres of a vehicle have the following tasks:
o
o
o
o
Carry the static load
Generate brake, traction and lateral forces
Isolate vertical disturbances, for comfort, road grip and reduction of suspension fatigue
Roll with minimum energy loss, particle emissions and noise emission
Before discussing the mechanics of tyre and road interactions, the physical structure of the wheel
assembly should be understood. Consisting of a steel rim and an inflated rubber toroid, pneumatic
tyres were invented and patented by Robert William Thomson in 1845 and are essentially the only
type of tyre found on motor vehicles today.
The physical construction of the tyre carcass affects the response of the tyre to different road
loadings. The carcass is a network of fabric and wire reinforcement that gives the tyre the mechanical
strength. The structure of the carcass can be divided into different types of tyres: BIAS-PLY, BIAS-PLY
BELTED, and RADIAL-PLY. Bias-ply tyres were the first types of pneumatic tyres to be used on motor
vehicles. Radial ply tyres followed in 1946 and became the standard for passenger car tyres. Figure
2-3 shows the main features of these tyre types.
45
VEHICLE INTERACTIONS
common on passenger
cars today
Figure 2-3: Common Tyre Constructions, from (Encyclopædia Britannica Online, 2007)
Note how the bias-ply constructions have textile structures oriented at an angle to the tyre
centreline along the x-z plane. This angle is referred to as the crown angle and is further illustrated in
Figure 2-4. Note the textile orientation for the bias-ply and radial tyres. Also note the difference in
crown angles between the two tyre constructions. This difference plays an important part in the
rolling resistance characteristics of the tyre which is Section 2.4.
Figure 2-4: Left: Carcass Construction, (Wong, 2001). Left top: Bias-ply construction. Left bottom:
Radial construction. Right: Radial Tyre Structure, (Cooper Tire & Rubber Co., 2007)
The tyre components have been constructed to provide the best tyre performance for different
loading directions. A trade-off must be made between handling performance and comfort, between
acceleration and wear, as well as between rolling resistance and desired friction for generating forces
in ground plane. Some of the tyre components have an important role in vehicle performance. For
example, the radial tyre components are presented in more detail in Figure 2-4. The rubber
46
VEHICLE INTERACTIONS
components and patterns incorporated in the tread are critical to the friction developed between the
tyre and road under all road conditions (wet, dry, snow, etc.). Friction is most relevant in longitudinal
and lateral vehicle dynamics. The belts define the circumferential strength of the tyre and thus
braking and acceleration performance. The sidewall and plies define the lateral strength of the tyre
and thus influence the lateral (cornering) performance of the vehicle. The sidewall as well as the
inflation pressure are also significant contributors to the vertical stiffness properties of the tyre and
affect how the tyre transmits road irregularities to the remainder of the vehicle. It becomes clear
that a tyre that has strong sidewalls will support cornering at the cost of vertical compliance –
reducing the comfort in the vehicle.
marking:
S/[mm] / 100*H/S construction Dr/[inch]
example, car: 165
/
65
R
14
example, truck: 315
/
80
R
22.5
Unladen radius R0 = (Dr*25.4+2*(H/S)*S)/2
for car example = (14*25.4+2*0.65*165)/2 = 285 mm
for truck example = (22.5*25.4+2*0.8*315)/2 = 538 mm
Figure 2-5: Tyre marking (radial tyre)
2.3.1
Tyre design and function
This section differs between design and function as well as between parameters and variables. It
assumes observations over time; i.e. time is the independent variable, i.e. variables vary with time. It
can be noted that, in another problem context, some of the quantities could switch between
parameters and variables. One example is a design optimization, where basically all design
parameters becomes variables (are varied) in the optimization.
Operational
parameters
Design
Parameters
Tyre
model
Driver
Physical
response
variables
Model of rest
of the vehicle
Operational
variables
Environment
Behavioural
response
variables
Figure 2-6: How tyre models can come into a model context.
2.3.1.1
Design parameters
We limit ourselves to today’s traditional pneumatic tyres. Then the design is captured by:

Carcass/Material: Rubber quality and plies arrangement.
47
VEHICLE INTERACTIONS



Tread/Grooves: Groove pattern, Groove depth, Tread depth, Spikes pattern (if spikes)
Main dimensions: Outer radius, Width, Aspect ratio
Installation parameters: Inflation pressure
2.3.1.2
Operational parameters
These are operating conditions which vary slowly, and in this description assumed to be constant
during one manoeuvre/driving cycle. These are:





Road surface (dry/wet, asphalt/gravel/snow/ice, …),
Road compliance and damping (hard/soft, …),
Wear state of tyre,
Age of tyre,
Temperature
2.3.1.3
Operational variables
These are operating conditions which vary quickly, and in this description assumed to be variable in
time during one manoeuvre/driving cycle. These are:




Longitudinal speed,
Tyre slip (𝑠 𝑠 , see Equation [2.2]),
Vertical tyre force, ( , see Figure 2-2)
Camber angle
Tyre forces in road plane (
) can be given instead of tyre slip. Another alternative is to give the
corresponding actuation, e.g. wheel shaft torque and wheel steering angle relative to wheel course
angle. With those setups, the reponse variables in Section 2.3.1.4.
Vertical tyre force can be modelled as arbitrarily varying in time or as an offset amplitude for
different frequencies where offset is from a mean value. The mean value would then be a parameter
instead of a variable. The latter alternative can be more efficient if simulating a longer driving cycle,
where following each road wave would be very computational inefficient.
2.3.1.4
Responses
Response refers to response to both Design changes and Operation changes.
2.3.1.4.1 Physical response (variables)
The variables from tyre model to model of rest of the vehicle is essential the forces and moments,
see Figure 2-2:




Longitudinal and lateral forces (
)
Roll moment or Overturning moment (
Rolling resistance moment ( )
Spin moment or Aligning moment ( )
)
Other responses are:

Wear rate [worn rubber mass or tread depth per time unit]
48
VEHICLE INTERACTIONS
2.3.1.4.2 Behavioural response (variables)
The Responses can be modelled as Behavioural variables instead:





Slip characteristics (𝐶𝑠 𝐶𝑠 , peak friction coefficient, … or other parameters in physical or
empirical tyre slip model, as in Section 2.4.3.2)
Rolling resistance coefficient
Vertical stiffness and damping
Wear coefficient
Relaxation length, …
When working with Behavioural instead of Physical, the distinction between variables and
parameters become less clear. So, we can see the same list of quantities as parameters instead of
variables.
2.4 Longitudinal Properties of Tyres
The forces in ground plane, in the tyre’s contact patch, depend on the tyre’s construction, the type of
road surface and the operating states (velocities and vertical force). This section treats the main
phenomena of tyre mechanics that influence the tyre-longitudinal force. The tyre can be under
braking, free rolling or traction.
The whole wheel, including the tyre, can be seen as a transmission from rotational mechanical
energy to translational mechanical energy. The shaft is on the rotating side; where we find rotational
speed, , and torque, 𝑇. The shaft torque T is then the sum of torque on the propulsion shaft and
torque on the brake disk or drum. The wheel hub is on the translator; were we find translating speed,
, and force, ℎ𝑢𝑏 . See Figure 2-7.
x-direction =
= vehicle longitudinal forward
vehicle,
wheel
including torque source, excluding wheel
𝒖 𝒙
𝒖 𝒛
𝒗𝒙
𝒗𝒙
𝒖 𝒙
𝒖 𝒛
𝑅
𝑢 𝑑
𝑢 𝑑
𝑢 𝑑
𝑢 𝑑
ground
Figure 2-7: A wheel as a transmission from rotational [ ; 𝑇] to translational [ ;
2.4.1
ℎ𝑢𝑏
].
Tyre Rolling and Radii
In all fundamental engineering mechanics, the radius of an object that rolls without sliding is a
compatibility link between the translational velocity and angular velocity as shown in Figure 2-8 a).
This relationship does not hold when the tyre is deflected as in Figure 2-8 b). Not even the deflected
radius can be assumed to be a proportionality constant between angular and translational velocity,
since the tyre contact surface slides, or slips, versus the ground. Then, an even truer picture of a
49
VEHICLE INTERACTIONS
rolling tyre looks like Figure 2-8 c), where the deformation at the leading edge also is drawn. This
means that is only ≈ 𝑅 ∙ and ≈ 𝑅 ∙ for limited slip levels.
a) Ideal rolling:
b) With vertical load Fz
and torque T0:
w
w
𝒛
vx
vx
∙𝑅
𝑅
𝑅
vx
inflated unloaded radius
𝑅 and
∙𝑅
𝒛
w
𝑅
𝑅
𝑅
c) With vertical load Fz
and (significant) torque T>0:
𝑅
𝑅
𝑅
inflated free radius
loaded radius
𝑅 and
∙𝑅
𝑅
𝑅 and
∙𝑅
∙𝑅
∙𝑅
Figure 2-8: Radius and speed relations of a rolling tyre 𝑅 and 𝑅 do not have exactly the same
numerical values across a), b) and c).
There is a difference speed between tyre and the road surface. The ratio between this speed and a
reference speed is defined as the “tyre slip”. The reference speed can be the translational speed of
the tyre or the circumferential speed of the tyre depending on the application. For a driven wheel,
the longitudinal (tyre) slip is often defined as:
𝑠
R∙
;
|R ∙ |
[2.1]
For braked wheels it is often defined as:
𝑠
R∙
| |
;
[2.2]
Averaging the reference speed gives the third variant in Eq [2.3]. This is actually possible to argue for
physically as we will see in Section 2.4.3.1.
𝑠
R∙
|R ∙
|⁄
;
[2.3]
With the definition in Equation [2.3], the following transformation falls out:
{
𝑠
∙ ; 𝑖𝑓 R ∙
𝑠 𝑅
𝑠
∙ ; 𝑖𝑓 R ∙
𝑠 𝑅
0 ⇒ 𝑖𝑓
0;
[2.4]
< 0 ⇒ 𝑖𝑓
< 0;
Whether one should use 𝑅 ∙ or or something else as reference speed (the denominators in
Equations [2.1]..[2.3]), is not obvious and is discussed in Section 2.4.3.
It is also not obvious which 𝑅 to use in Equations [2.1]..[2.3], e.g. 𝑅 or 𝑅 . However, this
compendium recommends 𝑅 , rather than 𝑅 , because 𝑅 is a better average value of the radius
around the tyre and the tyre’s circumference is tangentially stiff so speed has to be same around the
circumference.
Sometimes one defines a third radius, the Rolling Radius ( ⁄ )| 𝑇= , i.e. a speed ratio with
dimension length, between translator and rotating speeds, measured when the wheel is un-driven.
50
VEHICLE INTERACTIONS
This radius can be used for relating vehicle longitudinal speed to wheel rotational speed sensors, e.g.
for speedometer or as reference speed for ABS and ESC algorithms. The Rolling radius is normally
between 𝑅 and 𝑅 .
Yet another approach is to use the 𝑅
( ⁄ )|𝐹𝑥 = , i.e. the ratio when the wheel is pure rolling.
Using 𝑅 ( ⁄ )| 𝑇= or 𝑅 ( ⁄ )|𝐹𝑥 = in the slip definitions shifts
curve, see more in Section 2.4.3.3.
(𝑠 ) curve and 𝑇(𝑠 )
The variable 𝑠 is the longitudinal slip value, sometimes also denoted as 𝜅. When studying braking,
one sometimes uses the opposite sign definition, so that the numerical values of slip becomes
positive. It is used to model the longitudinal, traction or braking, tyre force in Section 2.4.3.
2.4.2
The Rolling Resistance of Tyres
The rolling resistance force is defined as the loss of longitudinal force on the vehicle body, as
compared to the longitudinal force, which would have been transferred with an ideal wheel. The
Rolling Resistance Coefficient, 𝑓 , is the rolling resistance force divided by the normal force, .
Assuming force equilibria in longitudinal and vertical direction, ℎ𝑢𝑏
and ℎ𝑢𝑏
𝑢 𝑑
, see Figure 2-7.
𝑢 𝑑
𝑓
𝑇
𝑅
[2.5]
denotes the longitudinal force on the wheel, 𝑇 denotes the applied torque and 𝑅 denotes the
⁄ . One often see
tyre radius. For a free rolling tyre, where 𝑇 0, 𝑓 becomes simply
definitions of 𝑓 which assumes free rolling tyre; but Eq [2.5] is more generally useful.
Regarding which radius to use, it should rather be 𝑅 than 𝑅 , because 𝑅 represents the lever for the
longitudinal tyre force around the wheel hub.
A free body diagram of the forces on the wheel can be used to explain the rolling resistance.
Consider Figure 2-9 which represents a free rolling wheel under steady state conditions. The inertia
of the wheel is neglected.
The main explanation model of rolling resistance for pneumatic tyres on hard flat surfaces is that the
pressure distribution is biased towards the edge towards which the wheel is rolling. Damping and
friction, see Figure 2-9, is the main reason for this and it is not dependent on the longitudinal force.
Same figure also shows the off-centre effect, which is directly influenced by the longitudinal force.
The force offset, e, explains 𝑓 . Additional to what is shown in Figure 2-9 one can motivate even
higher e, by including other effects, such as wheel bearing loss, brake disk drag and aerodynamic
torque loss.
Longitudinal and vertical force equilibria are already satisfied, due to assumptions above. However,
moment equilibrium around wheel hub requires:
𝑇
∙𝑅
∙𝑒
0 ⇒
𝑇
𝑅
∙
𝑒
;
𝑅
[2.6]
An analysis of this result is that the force which pushes the vehicle body forward is the term 𝑇⁄𝑅
(arising from the applied propulsion or brake torque T), minus the term ∙ 𝑒⁄𝑅 . The term can be
seen as a force
and referred to as the rolling resistance force. The dimensionless ratio
𝑒⁄𝑅 expression is the rolling resistance coefficient, 𝑓 :
51
VEHICLE INTERACTIONS
𝑒
;
𝑅
𝑓
Resulting pressure and Equivalent forces
Pressure distribution components
w
Fz
v
[2.7]
Fz
v
w
T
T
Fx
Fx
Rl
Fx
Rl
Fx
Pressure from elastic
radial deformation
e Fz
Pressure from radial deformation
speed (damping & friction)
Pressure
Contact patch offset, due to shear
of tyre walls (drawn for Fx>0)
equivalent
Figure 2-9: Normal force distribution on a tyre. The measure e is the force offset.
Eq [2.7] is a definition of rolling resistance coefficient based assumed physical mechanisms internally
in the tyre with road contact. Rolling resistance coefficient can also be defined based on quantities
⁄ as a definition, but
which are measurable externally, see Eq [2.5]. Sometimes one see 𝑓
that is not suitable since it assumes absence of torque.
It is important to refer to this phenomenon as rolling resistance as opposed to rolling friction. It is
not friction in the basic sense of friction, because
𝑓 ∙ except for the special case when undriven wheel (𝑇 0).
Figure 2-10 shows an un-driven wheel, which is a special case of a driven wheel.
Undriven (T=0)
Driven
Fx has to be:
w
Fz
v
T
w
Fx
𝑅
Fx
𝒙
Fz
v
|Fx|
Fx
1
e Fz
𝒛;
𝑹
|Fx|
𝑅
Fz
e
𝑅
T
Undriven
(T=0)
Pure rolling
(Fx=0)
Figure 2-10: Driven wheel with rolling resistance, special cases “Un-driven” and “Pure rolling”.
2.4.2.1
Variation of rolling resistance
Several factors will affect the rolling resistance of tyres. Design factors:
52
VEHICLE INTERACTIONS



Tyre material. Natural rubber often gives lower rolling resistance.
Radial tyres have more flexible sides, giving lower rolling resistance also bias ply have a
greater crown angle causing more internal friction within the tyre during deflection.
Geometry:
o Diameter. Large wheels often have lower coefficient of rolling resistance
o Width
o Groove depth
o Tread depth
Usage factors, long term varying:


Higher inflation pressure gives lower rolling resistance on hard ground but higher rolling
resistance on soft ground (and vice versa), see Figure 2-11 and Figure 2-12.
Wear. Worn tyres have lower rolling resistance than new ones (less rubber to deform).
Usage factors, short term varying:




Tyre loads (traction, braking and lateral forces)
Speed. Rolling resistance increases with vehicle speed due to rubber hysteresis and air drag.
High temperatures give low rolling resistance. Tyres need to roll approximately 30 km before
the rolling resistance drop to their lowest values.
Road/ground material. Asphalt, gravel, snow, etc.
Coefficient of rolling resistance, f
Figure 2-11 shows how widely the rolling resistance changes due to different road/ground material
and inflation pressures. As can be expected, a range of values exist depending on the specific tyre
and surface materials investigated. On hard ground, the rolling resistance decreases with increased
inflation pressure, which is in-line with the explanation model used above, since higher pressure
intuitively reduces the contact surface and hence reduces e. On soft ground the situation is reversed,
which requires a slightly different explanation model, see Figure 2-12. On soft ground, the ground is
deformed so that the wheel rolls in a “local uphill slope” with inclination angle . Intuitively, a higher
inflation pressure will lead to more deformation of the ground, leading to a steeper .
0.4
Sand
0.2
Medium hard
Hard soil
Soil
Medium
Concrete
0
0
20
Inflation Pressure [psi]
40
Figure 2-11 : Range of Coefficient of Rolling Resistance
for different road/ground material
53
VEHICLE INTERACTIONS
Wheel rolling on hard ground:
w
Wheel rolling on soft ground:
w
v
e
v
j
Figure 2-12 : Rolling resistance explanation model for hard and soft ground.
As an example, Figure 2-13 shows the influence of tyre construction and speed on rolling resistance.
The sudden increase in rolling resistance at high speed is important to note since this can lead to
catastrophic failure in tyres. The source of this increase in rolling resistance is a high energy standing
wave that forms at the trailing edge of the tyre/road contact.
Coeff. of Rolling Resistance
.05
.04
.03
.02
.01
0
Bias Ply
Bias-Belted
Radial
25
50
75
100 125 150 175
Speed [km/h]
Figure 2-13: Rolling Resistance Values for Tyre Type
and Operating Speed, (Gillespie, 1992)
There are some empirical relationships derived for the tyre's rolling resistance. It is advisable to refer
to the tyre manufacturer's technical specifications when exact information is required. This type of
information is usually very confidential and not readily available. Some general relationships have
been developed, from (Wong, 2001):
Radial-ply passenger car tyres:
Bias-ply passenger car tyres:
Radial-ply truck tyres:
Bias-ply truck tyres:
𝑓
𝑓
𝑓
𝑓
0.0136 0.04 ∙ 10−6 ∙ 2
0.0169 0.19 ∙ 10−6 ∙ 2
0.006 0. 3 ∙ 10−6 ∙ 2
0.007 0.45 ∙ 10−6 ∙ 2
There is a weak variation with speed which can be ignored for lower speeds. As seen in Figure 2-13,
the influence of speed on rolling resistance is constant up to about 100 km/h. Trucks tyres have a
54
VEHICLE INTERACTIONS
much lower rolling resistance (approximately half). Tyres have developed in that way for trucks,
because their fuel economy is so critical.
There is also a dependency on longitudinal force, see Figure 2-14. RRC increases when positive force,
since the shearing of the sidewall moves the contact forward and the shearing decreases the wheel
radius. For negative forces, these two effects have opposite influence, so the change is less for
negative force.
Rolling resistance
coefficient, 𝑅𝑅𝐶 𝑓
0.08
0.04
-0.4
-0.2
0
+0.2
+0.4
Normalised
Longitudinal
Force,
𝐹𝑥
⋅𝐹
Figure 2-14 : Rolling resistance dependency of longitudinal tyre force. Inspired by (Wong, 2001)
Rolling resistance is a torque loss. Other torque losses, which can be included or not in tyre rolling
resistance, are:




Losses associated with friction in gear meshes,
drag losses from oil in the transmission,
wheel bearing torque losses, and
drag losses from aerodynamic around the wheel.
These should, as rolling resistance, be subtracted from propulsion/brake torque when calculating 𝑇
in Figure 2-10. However, sometimes they are included as part of the tyres rolling resistance
coefficient, which is somewhat misleading even if it works if done correctly. The wheel bearing
torque losses are special in that they are rather proportional to vertical load on the wheel, and hence
it is rather natural to include them in rolling resistance coefficient. The aerodynamic losses due to
wheel rotation are special since they vary with wheel rotational speed, meaning that they (for
constant vertical load) are relevant to include when studying the variation of rolling resistance
coefficient with vehicle speed. A summarizing comment is that one has to be careful with where to
include different torque losses, so that they are included once and only once.
One should note that torque losses as discussed above are not the only losses which affect energy
efficiency of the vehicle. There are also speed losses: speed losses in the wheel slip and speed losses
in slipping clutch and torque converters. These cannot be modelled as rolling resistance.
2.4.3
Longitudinal forces
The longitudinal force (tangential to tyre circumference) between the tyre and road is critical to
vehicle propulsion and braking performance.
First compare the friction characteristics for a translating block of rubber and a rolling wheel of
rubber. Figure 2-2 shows the basic differences between classical dry-friction, or Coulomb friction,
models and the basic performance of a rolling tyre. Experiments have shown that the relative speed
between the tyre and the road produces a frictional force that has an initial linear region that builds
55
VEHICLE INTERACTIONS
to a peak value. After this peak is achieved, no further increase in the tangential (friction) force is
possible. There is not always a peak value, which is shown by the dashed curve in the figure.
The slope in the right diagram will be explained below, using the so called “tyre brush model”.
Sliding block
(or wheel with ω=0)
Rolling wheel of rubber
w
N
N
F
F
T
𝑅∙
v
F
N
F
F
mstat*N
mpeak*N
slide
stick
mdyn*N
F
N
N F
vx
N
F
mdyn*N
Dv=v
slip=Dv/vref=(R*w-vx)/vref
slide
• A stick mode where Dv=0.
• In slide mode, the friction force, F,
is only depending on sign(Dv).
• No stick mode (if w>0).
• The friction force, F, is depending on
relative sliding speed, Dv/vref.
Figure 2-15: Friction characteristics
2.4.3.1
Tyre brush model for longitudinal slip
The brush model is a physically-based model that uses shear stress and dry friction at a local level,
i.e. for each contact point in the contact patch.
56
VEHICLE INTERACTIONS
w*R
vx
displacemen
Figure 2-16: Picture of physical model of tyre, a so called “Brush model”. From Michelin.
Below shows a simplified version, using the following assumptions:
o
o
o
o
Sliding and shear stress only in longitudinal direction (as opposed to combined longitudinal
and lateral)
Constant and known pressure distribution. (As opposed to using a contact mechanics based
approach, which can calculate pressure distribution over the whole contact. For instance,
one can use Hertz contact theory, which would motivate parabolic pressure distribution and
contact length proportional to /2 for line contact or /3 for point contact.)
No difference between static and dynamic coefficient of friction
Only studying the steady state conditions, as opposed to including the transition between
operating conditions. Steady state as opposed to stationary variations, which would arise for
driving on undulated road.
These assumptions lead to a model as drawn in Figure 2-17.
The lines between tyre and road are sheared rubber elements, fixed to the wheel and having a
friction contact to the ground. The shear stress of the element develops as in Hooke’s law, see
Equation [2.8].
𝜏
𝐺 ∙𝛾;𝜏
𝑠 𝑒 𝑟 𝑠𝑡𝑟𝑒𝑠𝑠; 𝐺
𝑠 𝑒 𝑟
𝑜𝑑𝑢 𝑢𝑠; 𝛾
𝑠 𝑒 𝑟 𝑛𝑔 𝑒;
[2.8]
When a rubber element enters the contact patch, it lands un-deformed, 𝛾 0. The further into
contact, along coordinate 𝜉, we follow the element, the more sheared will it become. Since the
ground end of the element sticks to ground, the increase becomes proportional to the speed
difference ∆ and the time 𝑡 it takes to reach that coordinate, 𝑡(𝜉):
𝛾
𝑤 𝑒𝑟𝑒 𝑡(𝜉)
(𝑅 ∙
∆ ∙ 𝑡(𝜉)
𝐻
𝜉⁄
𝑇
𝑠𝑝
) ∙ 𝑡(𝜉)
𝐻
𝑡
𝑛𝑑
𝑇
57
𝑠𝑝
𝑡
≈
;
|𝑅 ∙
|
[2.9]
;
VEHICLE INTERACTIONS
Combining these equations yields the following, where the slip sx can be identified:
𝜏
𝐺 (𝑅 ∙
∙
|𝑅 ∙
𝐻 ⏟
)
∙ 𝜉;
|⁄
[2.10]
𝑠𝑥
This expression for shear stress only holds until the 𝜉 where the friction limit is reach, 𝜉𝑐 . At larger
𝜉 𝜉𝑐 , the rubber element will slide with a constant 𝜏 𝜇 ∙ 𝑝. Hence, the total force is the sum of
two integrals as follows:
𝜉𝑐
𝐿
𝐺
∙ ∫ ∙ 𝑠 ∙ 𝜉 ∙ 𝑑𝜉
𝐻
∙ ∫ 𝜏 ∙ 𝑑𝜉
2
∙ 𝐺 ∙ 𝑠 𝜉𝑐
∙
∙𝜇∙𝑝∙(
𝐻
𝜇∙𝑝∙𝐻
𝑤 𝑒𝑟𝑒 𝜉𝑐
; 𝑝∙ ∙
𝐺∙𝑠
rolling direction
L=length of contact patch
rubber
outlet end
tyre
road
x
∙ ∫ 𝜇 ∙ 𝑝 ∙ 𝑑𝜉
𝜉𝑐
x=xc
[2.11]
𝜉𝑐 );
G=shear modulus
W=width (perpendicular to drawing)
leading edge of contact
(=rubber inlet end)
R*w
x=L
x=coordinate in contact
𝐿
x=0
H=height of
rubber brushes
vx
(relative to wheel hub)
pressure
x
stick
slip
“mode”
x
shear
stress
Figure 2-17: Physical model for deriving brush model for longitudinal slip. Speeds are relative to
hub translational motion (not relative to ground). Drawn for 𝑅 ∙
.
Equation [2.11] is only valid for 𝑠 corresponding to 0 < 𝜉𝑐 < . Otherwise, the simpler Equation
[2.10] can be used. This and further simplification of Equation [2.11] leads to:
58
VEHICLE INTERACTIONS
𝐺∙
{
∙
∙𝐻
2
𝑓𝑜𝑟 |𝑠 | ≤
∙𝑠 ;
sign(𝑠 ) ∙ 𝜇 ∙
𝜇∙ ∙𝐻
∙𝐺∙ ∙
∙ (1
2
∙
1
) ; 𝑒 𝑠𝑒
|𝑠 |
𝜇∙
𝐺∙
∙𝐻
𝜇∙
|
|
⇔
≤
∙ 2
or, equivalent:
𝑓𝑜𝑟 |𝑠 | ≤
𝐶 ∙𝑠 ;
{
𝑤 𝑒𝑟𝑒 𝐶
sign(𝑠 ) ∙ 𝜇 ∙
𝐺∙
∙
∙𝐻
∙ (1
𝜇∙
𝜇∙
⇔| |≤
∙𝐶
[2.12]
;
𝜇∙
1
∙
) ; 𝑒 𝑠𝑒
4 ∙ 𝐶 |𝑠 |
2
;
The shape of this curve becomes as shown in Figure 2-18.
1
>> GWL2H=10; muN=1;
sx1=linspace(0,muN/
sx2=linspace(muN/(2
_div_muN1=GWL2H*sx1
Fx_div_muN2=1-muN./
plot(sx1,Fx_div_muN
r');, grid; xlabel(
ylabel('Fx/(mu*Fz)'
Fx/(mu*Fz)
𝒙⁄ ∙ 𝒛
0.8
0.6
0.4
0.2
0
linear, up to
0
0.05
0.1
𝒙⁄
0.15
∙
𝒛
0.2
.
0.25
sx
0.3
0.35
0.4
0.45
0.5
𝒙
Figure 2-18: Shape of force/slip relation derived with brush model (assuming
constant pressure distribution, etc.).
Figure 2-28 shows similar plots but with certain physically realistic assumptions about of parameters
in the brush model (W, H, G, etc.).With a brush height, H=2 cm, the slope of the curve becomes in
the correct magnitude, which makes it credible that the brush model actually models the most
significant mechanisms in a proper way.
The brush model can be extended as indicated above, see e.g. References (Pacejka, 2005) and
(Svendenius, 2007). Similar behaviour can also be derived assuming other physical models, e.g.
models using tangential stress instead of shear stress. Reference (Wong, 2001) shows such an
alternative to the brush model.
In summary for many models (and tests!) the following is a good approximation for small longitudinal
slip (and constant normal load):
𝐶 ∙𝑠
[2.13]
For the brush model, or any other model which describes Fx(sx), one can define the “Longitudinal
𝑚/𝑠
tyre (slip) stiffness” 𝐶 , which have the unit 𝑁 𝑁⁄1 𝑁/ ( ). It is the derivative of force with
𝑚/𝑠
respect to slip. In many cases one means the derivative at sx=0:
59
VEHICLE INTERACTIONS
𝐶
(
𝜕
𝜕𝑠
)|
[2.14]
𝑠𝑥 =
Note that 𝐶 is not a stiffness in the conventional sense, force/deformation. The tyre also have such
a conventional stiffness, defined by force and deformation of a not rotating wheel. Often, it is
obvious which stiffness is relevant, but to be unambiguous one can use the wording: “slip stiffness
[N/1]” and “deformation stiffness [N/m]”.
Equation [2.12] proposes that longitudinal stiffness (at sx=0) varies with G, W, L and H. For specified
tyre and inflation pressure, these are approximately constant, except for L. Length L vary with vertical
load, , which makes longitudinal stiffness vary with vertical load. This is true also for “lateral tyre
stiffness” and it is an important variation. This is further discussed in Section 2.5.3.
2.4.3.1.1 Brush model with parabolic pressure distribution
A parabolic pressure distribution can be assumed, instead of constant:
𝑝(𝑥)
3∙
∙
∙ (1
(
𝑥 2
) );
⁄
[2.15]
If we do the corresponding derivation, e.g. assuming a stick and slip zones, an equation for the force
function of slip can be derived:
{
sign(𝑠 ) ∙ (𝐶 ∙ |𝑠 |
sign(𝑠 ) ∙ 𝜇 ∙
where 𝐶
𝐶 2 ∙ |𝑠 |2
3∙𝜇∙
;
𝑓𝑢𝑛𝑐𝑡𝑖𝑜𝑛(𝐺
(𝐶 ∙ |𝑠 |)3
3∙𝜇∙
) ; 𝑓𝑜𝑟 |𝑠 | <
2
7 ∙ (𝜇 )
𝐶
𝑒 𝑠𝑒
⇔| |≤𝜇∙
𝐻);
1
>> C=10; muN=1;
sx1=linspace(0,3*muN
sx2=linspace(3*muN/C
uN1=(C*sx1C*C*sx1.*sx1.*sign(s
sx1.*sx1.*sx1/(27*mu
Fx_div_muN2=sign(sx2
plot(sx1,Fx_div_muN1
r');, grid; xlabel('
ylabel('Fx/(mu*Fz)')
0.8
∙ 𝒛
𝒙⁄
Fx/(mu*Fz)
[2.16]
0.6
0.4
0.2
>> axis([0 0.5 0 1.1
0
ear part
0
0.05
0.1
0.15
0.2
0.25
sx
0.3
0.35
0.4
0.45
0.5
𝒙
Figure 2-19: Shape of force/slip relation derived with brush model, assuming parabolic pressure
distribution.
It should be noted that this model will not have any linear part.
2.4.3.2
Empirical tyre models
The brush model is a physical model which explains the principles of how the forces are developed.
However, if a model is required which is numerically accurate to a certain tyre, one often uses a
60
VEHICLE INTERACTIONS
fitted curve instead of Equation [2.12]. To limit the number of parameters to fit, one often uses a
mathematical curve approximation, using trigonometric and exponential formulas.
2.4.3.2.1 Magic Formula Tyre Model
Probably the most well-known curve fit is called “Magic Formula” and it was proposed in (Bakker,
1987). This approach uses trigonometric functions to curve fit the experimental data. The curve fit
has the general form:
𝑦(𝑥)
𝐷 ∙ sin(𝐶 ∙ arctan(𝐵 ∙ 𝑥
𝑌(𝑥)
𝑦(𝑠)
𝑥
𝑋
𝐸 ∙ (𝐵 ∙ 𝑥
arctan(𝐵 ∙ 𝑥)))) ;
𝑆𝑉 ;
[2.17]
𝑆𝐻 ;
Where B is a stiffness parameter, C is a shape parameter, D is a peak value parameter, and E is a
curvature parameter describing the curve. The variable x is the tyre slip value. The parameters 𝑆𝑉
and 𝑆𝐻 simply shifts the curve so that it passes through the origin. The relationship between these
parameter and the tyre slip/friction relation is shown in Figure 2-20.
Figure 2-20: Magic Formula Tyre Parameters, (Pacejka, 2005)
2.4.3.2.2 TM-Easy Tyre Model
Since Equation [2.17] is a curve fit, it can be applied to any type of data with similar characteristics
and can thus be used to describe other loading behaviour of the tyre such as lateral stiffness and selfaligning torque. These parameters will be discussed in later sections. Other curve fitting approaches
to friction/slip modelling are possible such as the TM-Easy, (Hirschberg, et al., 2002), tyre model
shown in Figure 2-21.
2.4.3.2.3 Other Tyre Models
There are many more models with different degree of curve fitting to experimental data. However,
one can often have use for very simple curve fits, such as:


Linearized:
𝐶 ∙𝑠
Linearized and saturated:
𝑠𝑖𝑔𝑛(𝑠 ) ∙
61
𝑖𝑛(𝐶 ∙ |𝑠 |; 𝜇 ∙
)
VEHICLE INTERACTIONS
Figure 2-21: TM-Easy Tyre Model, (Hirschberg, et al., 2002)
2.4.3.3
Tyre with both rolling resistance and slip
So far, the models in Section 2.4.3 has not included rolling resistance. As we regard rolling resistance
as a torque, not a force, it does not affect the (𝑠 ) curve. But it does move the 𝑇(𝑠 ) curve
upwards, and if wheel is rolling rearwards, the curve moves downwards, see Figure 2-22. The (𝑠 )
curve is normally the suitable view for vehicle level studies, while the 𝑇(𝑠 ) curve is sometimes
needed for involving a model of the propulsion system. As mentioned in Section 2.4.1, one can select
𝑅 in slip definition so that the
0 for 𝑠
0, as shown in Figure 2-22. This compendium claims
this is the most appropriate if one does not know 𝑅. As also mentioned in Section 2.4.1, one can
alternatively select 𝑅 in slip definition so that the 𝑇 0 for 𝑠
0. This compendium claims that it is
(𝑠 ) ∙ 𝑅 sign( ) ∙ ∙ is better.
less appropriate and that using 𝑇
𝑅
𝑜𝑟𝑐𝑒 or
𝜇
𝑓
𝜇
𝑓
𝑇
𝑢
𝐿
𝑡ℎ
[N]
𝜇
𝑠
𝜇
𝑅
𝑅
𝑇/𝑅 (if
0)
𝑇/𝑅 (if
< 0)
⁄
𝑓
𝜇
𝜇
𝑓
Figure 2-22: Longitudinal tyre force ( ) and normalized wheel torque (𝑇/𝑅 ) including both rolling
resistance and slip.
62
VEHICLE INTERACTIONS
2.4.3.4
Relaxation
Both the physical and empirical tyre models discussed above are based on the assumption of steady
state condition in the contact patch, meaning steady state deformation pattern and macro-slip zone
for the rubber in the tyre. The transients between different steady state conditions involve finding a
new such steady state. The fact that the change takes some time is referred to as relaxation. In many
Vehicle Dynamics studies, the relaxation is such a quick process that it can be assumed to take place
instantaneously, i.e. the (𝑠 ) curve can be used. Generally one can need to model the transients in:
slip ( and/or ), vertical force ( ) and/or surface conditions (e.g. expressed in varying 𝜇).
A driving situation where slip varies a lot is when a vehicle starts from stand-still with initially undeformed tyres (no shear stress). For
0, the slip becomes undefined and the physical
explanation is that it take some degree rotation of the wheel before the rubber elements reach their
state of stress and deformation. During this transient there are other physical phenomena that are
involved, namely the dynamics of building up a force or stress in an elastic part.
A dynamic equation of this type is needed:
stiffness ∙ (𝑅 ∙
this phenomenon is to add a first order delay of the force:
𝐴 ∙ (𝑓(𝑠
𝜇 …)
). A common way to model
);
𝜇 … ) is the force according to a steady state model, e.g.Eq
where 𝑓(𝑠
|𝑅∙𝜔+ 𝑥 |⁄2
𝐿𝑟
𝑇𝑟𝑎𝑛𝑠𝑝𝑜𝑟𝑡
[2.12] and 𝐴
𝐿𝑟
𝑥
or 𝐴
𝐿𝑟
or 𝐴
𝑅∙𝜔
𝐿𝑟
and
is the
[2.18]
relaxation length, which is a fraction (≈ 5. .50%) of tyre circumference.
One can also express the delay in delaying the slip, as follows:
𝑓(𝑠
𝑠
𝑑
𝑑
𝜇 … );
𝑑
𝐴 ∙ (𝑠
𝑑
𝑠
𝑑 );
𝑑
[2.19]
where 𝑓(𝑠 𝑑
𝜇 … ) is the force according to a steady state model,
𝑑
e.g. Eq [2.12] and 𝐴 is as defined in Eq [2.18].
A physical motivation for this relaxation model is given in Reference (Rill, 2006). This identifies the
relaxation length as the relation between tyre slip stiffness [N/((m/s)/(m/s))] and tyre longitudinal
stiffness [N/m].
One non-physical behaviour of the model in Eq [2.18] is that it can give
𝜇 ∙ transiently, when
and/or 𝜇 is quickly decreasing. This calls for rather using Eq [2.9] or, if using Eq [2.8], limit the
integration of to to 𝜇 ∙ . Another way to capture the phenomena of varying is given in
Chapter 7 in Reference (Pacejka, 2005) and it is to add a term dependent of
𝜕𝐹𝑥
𝜕𝐹
∙
.
Introducing this delay can be motivated both for better description of fast transients and for avoiding
singularities, caused by 𝑇 𝑠𝑝 𝑡 close to zero. With 𝑓(𝑠
𝜇 … ) 𝑓(𝑠 ) 𝐶 ∙ 𝑠 ; as from Eq
[2.13], the singularity disappears as follows, starting from Eq [2.18]:
𝐴 ∙ (𝑓(𝑠 )
𝑇
𝑠𝑝
{𝑒. 𝑔. }
𝑡
∙ (𝐶 ∙
𝐶
∙ (𝑅 ∙
𝑇
)
𝑠𝑝
𝑡
𝑅∙
𝑇
𝑠𝑝
)
𝑡
∙ (𝐶 ∙ 𝑠
)
𝐶
|𝑅 ∙
)
∙ (𝑅 ∙
|⁄
63
∙
)
;
𝑇
𝑠𝑝
𝑡
∙
[2.20]
VEHICLE INTERACTIONS
2.5 Lateral Properties of Tyres
After a vehicle starts moving, controlling the direction of travel becomes a high priority for the
operator. For wheeled vehicles, the primary mode to control travel direction is to change the
orientation of the tyre, i.e. to apply a steering angle. Tyres generate a lateral force when they are
oriented at an angle different to the direction of the vehicle motion. The tyre typically deforms as in
Figure 2-23.
a) yz plane, view from rear
b) xy plane, view from above
rest of vehicle
left side wheel
vw
Fs
rest of vehicle
left side wheel
Ms
Fs
Ms
Fs
Ms
Fs
Ms
|Fya |
|Fya |
Figure 2-23: Deformation of a Cornering Tyre, (Clark, 1971)
It is essential to distinguish between the steering angle and (lateral or side) slip angle of the tyre.
Lower right part of Figure 1-9 shows this difference. The steering angle, 𝛿 or d, is the angle between
vehicle longitudinal direction and tyre longitudinal direction. The slip angle, 𝛼 or a, is the angle
between tyre longitudinal direction and the tyre translational velocity (=wheel hub velocity).
The relation between the lateral force of a tyre and the tyre side slip angle is typically as shown in
Figure 2-25. The behaviour of the curve is similar to that exhibited for longitudinal forces Figure 2-20
and Figure 2-21. It becomes even more similar if lateral slip angle is replaced by lateral slip, 𝑠
𝑡 𝑛(𝛼).
2.5.1
Tyre brush model for lateral slip
With corresponding simplification as in Section 2.4, we now use the brush model to also explain the
lateral properties. Figure 2-24 shows the model for lateral slip and should be compared to Figure
2-17. The differences are that the model for lateral slip has the deformation of the rubber elements
perpendicular to drawing.
The model equations become very similar, and the expression for how lateral force varies with lateral
slip becomes:
64
VEHICLE INTERACTIONS
𝐺∙
∙
∙𝐻
2
∙𝑠 ;
sign(𝑠 ) ∙ 𝜇 ∙
{
or, equivalent:
𝑓𝑜𝑟 |𝑠 | ≤
𝜇∙ ∙𝐻
∙𝐺∙ ∙
∙ (1
2
∙
1
∙𝐻
𝜇∙
⇔| |≤
2
∙
) ; 𝑒 𝑠𝑒
|𝑠 |
𝜇∙
𝜇∙
𝑓𝑜𝑟 |𝑠 | ≤
⇔| |≤
∙𝐶
𝐶 ∙𝑠 ;
{
𝜇∙
𝐺∙
sign(𝑠 ) ∙ 𝜇 ∙
[2.21]
;
𝜇∙
1
∙
) ; 𝑒 𝑠𝑒
4 ∙ 𝐶 |𝑠 |
∙ (1
𝑤 𝑒𝑟𝑒 𝐶
𝐺∙
∙
∙𝐻
2
;
where the lateral slip, 𝑠 is defined as:
𝑠
view from front,
rolling direction
showing rubber element
on one certain x
L=length of contact patch
rubber
outlet end
x = longitudinal
coordinate in
contact
tyre
R*w
x=L
|⁄
[2.22]
G=shear modulus
W=width (perpendicular to drawing)
vy
x=xc
road
|𝑅 ∙
leading edge of contact
(=rubber inlet end)
x=0
H=height
vx
(relative to wheel hub)
longitudinal
coordinate
x
pressure
pressure
x
stick
slip
“mode”
x
shear stress
(lateral, i.e. perpendicular
to drawing)
Figure 2-24: Physical model for deriving brush model for lateral slip.
Notes:



The slip is the sliding speed in lateral direction, divided by the same “transport speed” as for
longitudinal slip, i.e. the longitudinal transport speed.
There is a minus sign appearing, because of sign conventions.
The slip definition in Equation [2.22] is not consistent with the earlier mentioned
⁄ . One can identify two different lateral slip:
relation 𝑠
𝑡 𝑛(𝛼)
65
VEHICLE INTERACTIONS
o
o
“Lateral wheel slip” 𝑠 𝑤ℎ
hub moves over ground: 𝑠 𝑤ℎ
“Lateral tyre slip” 𝑠 𝑡
relation:
(𝑠 𝑡 ).
𝑤ℎ
⁄
𝑠 𝑤ℎ
⁄(|𝑅 ∙
, which is defined by how wheel
( 𝑉 ℎ𝑐
𝑉 ℎ𝑐
𝑉 ℎ𝑐 )
|⁄ ), valid for the constitutive
𝑤ℎ
These names are not well established, but invented in this compendium for writing about it.
If 𝑅 ∙
(no longitudinal slip), these two slips are the same. (This compendium does
almost not consider combined slip, why no further distinction is made between these two
slip definitions. Combined slip is briefly covered in Section 2.6.)

For a linearization, Eq [2.22] tells that the most correct way is that lateral force is ∝ 𝑠 ,
not ∝ α. Often one find ∝ α as starting point in the literature, but this compendium
uses ∝ 𝑠 .
(The advantage with using ∝ α is that the vehicle dynamics equations becomes linear in
and , e.g. see derivation of Equation [4.53]. However, the compendium proposes that
the tyre model itself should be linearized as ∝ 𝑠 and, only when linear dynamic model is
needed, one deliberately and consciously approximate 𝑠 ≈ 𝛼. The result is the same when
no longitudinal slip, but there is a risk that one don’t realize that longitudinal tyre slip has
been assumed small.)
In summary for many models (and tests!) the following is a good approximation for small lateral and
longitudinal slip (and constant normal load):
𝐶 ∙𝑠 ;
𝐶𝛼 ∙ 𝛼;
or
[2.23]
(α), one can define
For the brush model, or any other model which describes
(𝑠 ) or
the “Lateral tyre stiffness” or “Tyre Cornering Stiffness”, 𝐶 or 𝐶𝛼 , which have the unit N/1 or N/rad.
It is the derivative of force with respect to slip or slip angle. Reference (ISO8855) defines the
cornering stiffness as 𝐶𝛼 for slip angle=0:
𝐶𝛼
𝜕
(
𝜕𝛼
)|
𝛼=
𝐶
(
𝜕
𝜕𝑠
)|
[2.24]
𝑠𝑦 =
The minus is added to get positive values on 𝐶𝛼 and 𝐶 .
When using only small slip, it does not matter if the cornering stiffness is defined as the slope in an
versus 𝛼 diagram or versus 𝑠
𝑡 𝑛(𝛼) diagram. Therefore, the notation for cornering
stiffness varies between 𝐶𝛼 and 𝐶 . Cornering stiffness has the unit 𝑁 which can be interpreted as
𝑁
𝑁 ⁄1
𝑚/𝑠
𝑁/ (𝑚/𝑠) or 𝑁
𝑁 ⁄𝑟 𝑑 .
“Cornering Compliance” is sometimes used instead of “Cornering stiffness”. Cornering compliance is
simply the inverted quantity, the unit being 1⁄𝑁 or 𝑟 𝑑⁄𝑁.
2.5.2
Empirical tyre models
The cornering tyre forces initially exhibit a linear relation with the slip angle. A non-linear region is
then exhibited up to a maximum value. In Figure 2-25, the maximum slip angle is only 16 degrees (or
sy=tan(16 deg)=0.29) and one can expect that the tyre forces will drop as the slip angle approaches
90 degrees. The general form of the lateral force versus slip angle curve is also suitable for the Magic
Formula, TM-Easy, or similar curve fitting approach when sufficient test data is available.
66
VEHICLE INTERACTIONS
|Cornering Force| [N]
Adhesion
Adhesion Limit
Limit
2000
1000
0
4
8
12
|Slip Angle| [deg]
16
Figure 2-25: Cornering Forces of Tyres
2.5.3
Influence of vertical load
As discussed for longitudinal slip and the rolling resistance behaviour of tyres, the vertical load on the
tyre affects the force generation. The general behaviour of the tyre’s cornering performance as the
vertical load changes is presented in Figure 2-26. These figures show that the cornering stiffness is
influenced by vertical load. One term is used to describe the dependence of cornering stiffness on
vertical load: cornering coefficient, and is defined as:
𝐶𝐶𝛼
𝐶𝛼
[2.25]
The influence of vertical tyre force on the generated lateral forces is an important aspect of vehicle
performance in cornering and will be further discussed in 4.
Figure 2-26: Left: Influence of Vertical Load on Lateral Force, (Gillespie, 1992). Right: Cornering
stiffness versus vertical load
67
VEHICLE INTERACTIONS
Figure 2-27: Cornering stiffness versus vertical load
A variation of cornering stiffness with vertical load can be explained using the brush model and a
contact patch length, L, which varies with tyre normal load, Fz. From contact theories such as Hertz’s
2
3
contact theory, one can motivate ∝
(line contact) or ∝
(for point contact). An
extreme assumption, that the tyre is a membrane with an inside inflation pressure, gives ∝
. These can be inserted as
𝑘 ∙ 𝑎√ in Equation [2.12]:
𝐺∙
𝜇∙
∙ 𝑘2 ∙
∙𝐻∙𝜇
∙𝑠 ;
𝜇∙𝐻
1
{
2
( − )
∙𝐺∙
∙ 𝑘2 ∙
2
( −
𝑓𝑜𝑟
| |
1
≤
𝜇∙
1
∙ ;
) 𝑠
[2.26]
𝑒 𝑠𝑒
An equivalent reasoning for the lateral tyre force can be done starting from Equation [2.21].
In Figure 2-28, the tyre characteristics are plotted for a=2 and a=3 and varying vertical force. We see
that if
, the degressive characteristics, as in Figure 2-26 and Figure 2-27, can be explained. Since
a tyre is more like a line contact (width constant, a=2) or even a membrane (a=1), it is not likely that
the contact length variation is the whole explanation of the degressive stiffness.
Another mechanism which could influence the stiffness is that the pressure distribution is not
uniform (as assumed in equations above) and it most likely varies with vertical load. A distribution
which goes from triangular to uniform when Fz increases would explain a degressive stiffness. This
compendium does not derive the brush model for non-uniform pressure distribution, so this is left as
an unproven hypothesis.
68
VEHICLE INTERACTIONS
1
0.9
0.8
∙ 𝒛
𝒙⁄
Fx/(mu*Fz)
0.7
a=2;
a=2;
a=3;
a=3;
0.6
0.5
0.4
Fz=5 kN
Fz=10 kN
Fz=5 kN
Fz=10 kN
these 3
overlap
each other
0.3
linear, up to
.
0.2
𝒙⁄
∙
𝒛
0.1
0
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
sx𝒙
Figure 2-28: Shape of force/slip relation derived with brush model (assuming constant pressure
distribution, etc). Parameter a is varied between 2 to 3. Other data used: W=0.15 m, H=0.02 m,
mu=1, G=0.6e6 N/m2 and k tuned so that L=0.15 m when Fz=5 kN.
2.5.4
Relaxation
As for longitudinal, there is a delay in how fast the steady state conditions can be reached, which is
sometimes important to consider. A similar model, as for longitudinal is to add a first order delay of
the force:
𝑓(𝑠
𝑠
𝑑
𝑑
where 𝑓(𝑠
𝐴
𝑑 );
𝑑
𝐴 ∙ (𝑠
𝑑
𝑇𝑟𝑎𝑛𝑠𝑝𝑜𝑟𝑡
𝐿𝑟
𝑠
𝑑
𝑑 );
𝑑 ) is the force
|𝑅∙𝜔+ 𝑥 |⁄2
or 𝐴
𝐿𝑟
according to a steady state model, e.g.Eq [2.21] and
𝑥
𝐿𝑟
or 𝐴
𝑅∙𝜔
𝐿𝑟
and
[2.27]
is the relaxation length, which is
a fraction (≈ 5. .50%) of tyre circumference.
Most reasoning in Section 2.4.3.4 is applicable also for lateral relaxation.
2.5.5 Other effects than lateral force due to
lateral slip
The deformations of the tyre during cornering are quite complex when compared to the case of pure
longitudinal motion, see Figure 2-23. Hence, there are more effects than simply a lateral force. Some
of these will be discussed in the following.
2.5.5.1
Overturning moment
The contact patch is deflected laterally from the centre of the carcass. This creates an overturning
moment Mx due to the offset position of the normal force. The lateral force Fy also contributes to
the overturning moment.
69
VEHICLE INTERACTIONS
2.5.5.2
Tyre self-aligning moment
(This section has large connection with Section 4.2.3.2 Steering system forces.)
In the lowest diagram in Figure 2-24, one can see that shear stress is concentrated to the outlet side
of the contact patch for small slip angles. So the equivalent lateral force acts behind the centre of
wheel rotation for small slip angles. As seen in Figure 2-23 b) it acts at a position 𝑡𝑝 behind the
wheel’s y axis. This distance is referred to as the pneumatic trail, see Figure 4-5, and the resulting
moment of the lateral force acting on the ground and the reaction of wheel on the suspension
creates a moment, Mz, which is an aligning torque. More commonly the moment Mz is referred to
the as the self-aligning moment since it (with a positive caster offset) tends to align the tyre to zero
tyre slip angle. The reason why pneumatic trail can become slightly negative is because pressure
centre is in front of wheel centre, see Figure 2-9.
steering torque peaks
at lower slip than
lateral tyre force.
lateral tyre force = Fy (for a tyre with peak)
𝑐
𝑐
total trail = caster trail + pneumatic trail
𝑡𝑝
∙𝑐
steering torque ∝
∙ 𝑐
𝑡𝑝
side slip angle
Figure 2-29: General Response of Steering torque to Side slip angle. Tyre self-aligning moment =
∙ 𝑡𝑝 is one part of the steering moment.
Figure 2-29 shows the combined response of lateral force and slip angle. It is interesting to note that
the steering torque reaches a peak before the maximum lateral force capacity of the tyre is reached.
It can be used by drivers to find, via steering wheel torque, a suitable steering angle which gives a
large lateral force but still does not pass the peak in lateral force.
2.5.5.3
Camber force
Camber force (also called Camber thrust) is the lateral force caused by tha cambering of a wheel.
One explanation model is shown in Figure 2-30. It is that Camber force is generated when a point on
the outer surface of a leaning (cambered) and rotating tyre, that would normally follow a path that is
elliptical when projected onto the ground, is forced to follow a straight path due to friction with the
ground. This deviation towards the direction of the lean causes a deformation in the tyre tread and
carcass that is transmitted to the vehicle as a force in the direction of the lean. Another explanation
model is that the tyre “climbs” sideways as the inlet edge is directed, because there is more stick and
less slip in the inlet edge as compared to the outlet edge; see brush model.
Camber thrust is approximately linearly proportional to camber angle for small angles: Camber
thrust
𝐶𝛾 ∙ 𝛾. The camber stiffness, 𝐶𝛾 , is typically 5-10 % of the cornering stiffness and
𝑐
opposite sign.
The camber force is superimposable for small lateral slip and small camber angle. The approximative
Eq [2.23], can then be developed to:
70
VEHICLE INTERACTIONS
𝑆𝑑 𝑆 𝑝
𝐶 𝑚𝑏
𝐶 ∙𝑠
𝐶𝛾 ∙ 𝛾;
𝐶𝛼 ∙ 𝛼
or
𝐶𝛾 ∙ 𝛾
[2.28]
𝛾
View from rear
or front:
𝑐
A point on tyre surface travels basically along the
edge of an ellipse, when the tyre has a camber
angle…
View from above:
inlet edge
𝑐
𝑐
…but the ellipse is flattened in the contact patch
(if friction makes the rubber stick and wheel hub
is moving in pure longitudinal direction).
This is the path the point would take if not
flattened. Consequently, to flatten the ellipse, it
requires a force on the tyre, as arrow “camber
thrust” shows.
outlet edge
Figure 2-30: Camber thrust
There is also a rotation perpendicular to the ground due to the camber. The total rotation vector of
the wheel is directed along the wheel rotation axis. The component of this vector that is
perpendicular to ground creates friction moment which steers the wheel towards the side it tilts; on
a cambering vehicle like a bicycle, the wheel is steered “into the fall”, which counteracts falling.
2.6 Combined Longitudinal and Lateral
Slip
The operation of motor vehicles often involves a combination of turning, braking, and steering
actions. Hence, the performance of the tyre under combined (tyre) slip is important.
If the tyre has isotropic adhesion properties in the lateral and longitudinal direction, then one can
start an analysis assuming that the maximum force generated by the tyre is fixed by the maximum
resultant friction force, 𝜇 ∙ .
2
2
2
≤ (𝜇 ∙
𝐹
)2 ⇒ ( 𝑥 )
𝐹
2
𝐹
2
( 𝐹𝑦 ) ≤ 𝜇2
[2.29]
Equation [2.29] can be plotted as a circle, called the “Friction Circle”. Since the lateral and
longitudinal properties are not really isotropic (due to carcass deflection, tread patterns, camber, etc)
the shape may be better described as a “Friction Ellipse” or simply “Friction limit”.
71
VEHICLE INTERACTIONS
When not cornering, the tyre forces are described by a position between -1 (braking) and +1
(acceleration) along the Y-axis. Note that the scales of both axes are normalized to the maximum
value for friction.
1
maximum turning left without
propulsion and braking
Fy/(mu*Fz)
0.5
partial propulsion
0
>> p=0:360;plot(cosd(p),s
axis(1.1*[-1 1 -1 1]), ax
ylabel('Fy/(mu*Fz)'), gri
max propulsion
-0.5
-1
-1
-0.5
0
Fx/(mu*Fz)
0.5
1
Figure 2-31: Friction Circle with some examples of tyre friction utilization.
The “actuation” of the wheel means the propulsion, braking and steering (and sometimes suspension
control) of the wheel. An ideal actuation allows all conditions within the boundaries of the friction
circle to be achieved anytime during a vehicle manoeuvre. An example of limitation in actuation is a
wheel on a non-steered rear axle. They cannot access any of the lateral parts of the circle; unless the
vehicle slides laterally.
At the boundary of the friction circle, tyres become more sensitive to changes in slip. It is therefore
extra important to model the direction of the force in relation to shear deformation and relative slip
motion in the tyre contact patch. Here, isotropic shear and friction properties are assumed:
𝑡𝑒𝑟 𝑠 𝑖𝑑𝑖𝑛𝑔 𝑠𝑝𝑒𝑒𝑑
𝑜𝑛𝑔𝑖𝑡𝑢𝑑𝑖𝑛 𝑠 𝑖𝑑𝑖𝑛𝑔 𝑠𝑝𝑒𝑒𝑑
𝑅∙
[2.30]
Significant testing and modelling work have been undertaken to quantify the relationship between
tyre slip (sx and sy, or sx and 𝛼) and tyre forces. As an example, Figure 2-32 presents one method to
present the maximum lateral force for different longitudinal slips. Note that each line represents a
constant slip angle and that the maximum lateral force occurs for a slip angle between 10 and 18
degrees. Also the maximum value for longitudinal slip does not occur at zero, but for a slight braking
condition (around 0.05).
72
VEHICLE INTERACTIONS
Figure 2-32: Combined Longitudinal and Lateral Slip
Figure 2-33: Effects of combined longitudinal and lateral utilization of friction.
2.6.1.1
Physically consistent combined slip model
A simple but physically consistent combined slip model can be expressed using “effective slip, s”, as
shown in Equation [2.31]. The function f using arctan, given as example, does NOT then reflect a drop
in friction after macro slip has started in the contact patch.
73
VEHICLE INTERACTIONS
𝑠2
𝑠
∙
𝑠
𝑠2
;
𝑠2;
𝑅𝑤 ∙
𝑠
𝑇
𝑇
𝜇∙
𝑓(𝑠)
𝑠𝑝
𝑠𝑝
𝑡
𝑡
;
;
𝑛𝑑 𝑠
{𝑒. 𝑔. }
𝑇
|𝑅𝑤 ∙
𝑠𝑝
|
𝑡
;
;
∙ 𝑓(𝑠);
{𝑒. 𝑔. }
[2.31]
𝜋
∙ arctan (𝐶 ∙
𝜋
∙ 𝑠) ;
𝐶
min ( ∙ 𝑠; 1) ;
𝜇
𝐶 ≈ {𝑡𝑦𝑝𝑖𝑐 𝑦} ≈ 0 [non-dimensional];
𝐶
𝑜𝑟 𝑓(𝑠) {𝐸𝑞 [ .1 ] 𝑛𝑑 [ . 1]} min ( ∙ 𝑠; 1) ;
𝜇
𝐶 ≈ {𝑡𝑦𝑝𝑖𝑐 𝑦} ≈ 0 [non-dimensional];
𝑜𝑟 𝑓(𝑠)
{
𝑠
∙
𝑠
𝑛𝑑
{𝑒. 𝑔. }
In simulation models, it is often so that
are known, in which case the Equation [2.31] is
already on explicit form. However, the model becomes singular if 𝑇 𝑠𝑝 𝑡 is close to zero
(happens usually only at stand-still, or unusually other conditions with 𝑅𝑤 ∙
). Some other
model has to be switched in during these conditions. If 𝑠 becomes close to zero (no tyre forces in
road plane), the model in Equation [2.31] might also seem singular, but it is only a problem if one
calculate 𝑠 before dividing with 𝑠. Instead, one can express the tyre forces in ground plane as linear
𝐶
with sliding speeds (𝑅𝑤 ∙
and ). If using 𝑓(𝑠) {𝑒. 𝑔. } min ( 0 ∙ 𝑠; 1), Equation [2.31]
can be rewritten as Equation [2.32].
𝑠
∙
𝑠
{𝑠
𝑠⇒
𝑠
∙
𝑠
2.6.1.2
𝐶
𝜇
∙𝑠 <1⇔𝑠 <
}
𝜇
𝐶
{𝑠𝑖 𝑖 𝑟}
𝑠
∙𝜇∙
𝑠
∙
𝐶
∙𝑠
𝜇
∙𝐶 ∙𝑠 ;
[2.32]
∙𝐶 ∙𝑠 ;
Brush model for combined slip
The brush model can be applied also for combined slip. If doing that, the expression for
[2.32] can be used as
in Eq [2.31].
74
in Eq
VEHICLE INTERACTIONS
𝐺∙
{
∙
∙𝐻
𝜇∙
2
𝑓𝑜𝑟 |s| ≤
∙ s;
𝜇∙ ∙𝐻
∙𝐺∙ ∙
∙ (1
2
1
∙ ) ; 𝑒 𝑠𝑒
s
𝜇∙
𝐺∙
∙𝐻
⇔|
∙ 2
𝜇∙
|≤
Or, equivalent:
𝐶
{
𝑓𝑜𝑟 |s| ≤
∙ 𝑠;
𝜇∙
∙ (1
𝜇∙
4∙𝐶
𝜇∙
∙𝐶
⇔|
|≤
𝜇∙
[2.33]
1
∙ ) ; 𝑒 𝑠𝑒
s
Or, equivalent:
𝐶 ∙𝜇∙
{
𝜇∙
∙ (1
𝑓𝑜𝑟 |s| ≤
∙ 𝑠;
1
⇔|
∙𝐶
|≤
𝜇∙
1
1
∙ ) ; 𝑒 𝑠𝑒
4∙𝐶 s
In the same way, the brush model in Eq [2.16] in section 2.4.3.1.1, can be used in combined slip.
2.6.1.3
An approximate combined slip model
A combined model for cases when one knows without involving 𝑠 is shown in Equation [2.34].
This is not completely physically motivated, but works relatively well. One can consider it as a
mathematical scaling inspired by the friction circle.
2
≈ √1
2.6.1.4
(
𝜇∙
) ∙
; 𝑤 𝑒𝑟𝑒
|𝑠
𝑥=
[2.34]
Relaxation
The models above for combined slip assume steady state conditions. For separate longitudinal and
lateral slip, there is a delay in how fast the steady state conditions can be reached, which is
sometimes important to consider. . A similar model, as for longitudinal and lateral, is to add a first
order delay of the force:
[ ;
[𝑠
𝑑
𝑓(𝑠
]
𝑑
;𝑠
𝑑
𝑑
𝑑
𝑑]
;𝑠
𝑑
𝑑 );
[𝐴 ∙ (𝑠
𝑠
𝑑 );
𝑑
𝐴 ∙ (𝑠
𝑠
𝑑
𝑑 )];
[2.35]
where 𝑓(𝑠 𝑑
𝑑; 𝑠 𝑑
𝑑 ) is the force according to a steady state model, e.g.
Eq [2.33] and [𝐴 ; 𝐴 ] are relaxation lengths, as defined in Eq [2.18] and Eq [2.27].
Most reasoning in Section 2.4.3.4 is applicable also for combined slip relaxation.
Generally, the models become more stable the less state variables one uses, which can be a reason
to not model relaxation. For combined slip, there is also a possible to model with one instead of two
state variables. This is to model relaxation (delay) only of the force magnitude (or total slip) while
letting the force direction be without delay. This is shown in the following equation, using force as
state variable:
75
VEHICLE INTERACTIONS
𝐴 ∙ (𝑓(𝑠
[ ;
]
[
𝜇 …)
+𝑠𝑥 −𝑠𝑦
; 𝑠 ]∙
𝑠
);
;
𝜇 … ) is the force according to a steady state model, e.g. Eq [2.33]
where 𝑓(𝑠
and 𝐴 is a combined relaxation length, averaging the ones defined in Eq [2.18]
and Eq [2.27].
[2.36]
2.7 Vertical Properties of Tyres
The most important vertical property of a tyre is probably the stiffness. It mainly influences the
vertical dynamics, see 5. For normal operation, the vertical force of the tyre can be assumed to vary
linearly with vertical deflection. If comparing a tyre with different pressures, the stiffness increases
approximately linear with pressure. See Figure 2-34.
Figure 2-34: Vertical properties of a truck tyre.
Figure 2-35: Different tyre models which will filter road irregularities differently. Picture from Peter
Zegelaar, Ford Aachen.
76
VEHICLE INTERACTIONS
2.8 Tyre wear
There are many other aspects of tyres, for instance the wear. Wear models are often based around
the Archard’s (or Reye’s) wear hypothesis: worn material is proportional to work done by friction, i.e.
friction force times sliding distance. Wear rate (worn material per time) is therefore friction force
times sliding speed. Different approaches to apply this to tyres and expanding to temperature
dependency etc. is found for instance in Reference (Grosch, et al., 1961). A generalization of
𝑒 𝑟𝑅 𝑡𝑒 [in mass/s or mm tread depth/s], for one certain tyre at constant temperature, becomes
as follows:
𝑒 𝑟𝑅 𝑡𝑒 ∝ 𝑟𝑖𝑐𝑡𝑖𝑜𝑛 𝑜𝑟𝑐𝑒 ∙ 𝑆 𝑖𝑑𝑖𝑛𝑔𝑆𝑝𝑒𝑒𝑑 ⇒
⇒
where
𝑒 𝑟𝑅 𝑡𝑒
𝑘∙
2
2
√
∙ ∆ ≈ 𝑘 ∙ (𝐶 ∙ 𝑠) ∙ (𝑠 ∙
; 𝑠
√𝑠
2
𝑠 2;
𝑇
𝑇
𝑠𝑝
𝑠𝑝
𝑡
𝑡)
≈ 𝑘 ∙ 𝐶 ∙ 𝑠2 ∙
;
[2.37]
defined as in Eq [2.9]
and
𝑘 is a constant for a certain tyre with a certain temperature, rolling on a certain
road surface.
2.9 Vehicle Aerodynamics
The flow of air around the vehicle body produces different external forces and moments on the
vehicle. Considerable research and engineering has been carried out, and the fluid mechanics will not
be covered in this course. However practical first order models for aerodynamics forces have been
established and are presented here.
2.9.1
Longitudinal wind velocity
The most relevant force of interest in this course is the resistance force to forward motion,
aerodynamic drag
, which is proportional to the square of the longitudinal component of the
wind speed relative to the vehicle,
. For aerodynamic loads resisting forward motion of the
vehicle, the Equation [2.38] can be used. The parameters 𝑐𝑑 𝜌 𝑛𝑑 𝐴
𝑡 represent the drag
coefficient, the air density and a reference area, respectively. The reference area is the area of the
vehicle projected on a vehicle transversal plane.
2
∙ 𝑐𝑑 ∙ 𝜌 ∙ 𝐴
𝑡
∙
2
;
[2.38]
Typical values of drag coefficients (cd) for cars can be found from sources such as: (Robert Bosch
GmbH, 2004), (Barnard, 2010) and (Hucho, 1998). These coefficients are derived from coast down
tests, wind tunnel tests or CFD (Computational Fluid Dynamics) calculations. For light vehicles (cars,
SUVs, pick-up trucks) the air resistance can often be neglected for city traffic up to 30-50 km/h, but
not at highway speeds.
Since a car structure moving through the air is not unlike an aircraft wing, there are also aerodynamic
lift forces. Considering the forward motion only, a lift force and pitch moment will develop. This
affects the normal loads on front and rear axle, and consequently the tyre to road grip. Hence, it
affects the lateral stability.
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VEHICLE INTERACTIONS
2
∙𝑐 ∙𝜌∙𝐴
2
𝑡
2
∙
∙ 𝑐𝑝𝑚 ∙ 𝜌 ∙ 𝐴
𝑡∙
;
2
𝑐∙
;
[2.39]
The coefficient 𝑐 represents the lift characteristics of the vehicle. The lift force is here assumed to
act through the centre of gravity. The cpm (pitch moment coefficient) and Lc (characteristic length,
usually wheelbase) must be reported together.
With identical effect on the (rigid) vehicle body one can replace the lift force in centre of gravity,
, and the pitch moment,
, with lift forces at two different longitudinal positions, typically
over each axle:
2
∙𝑐 ∙𝜌∙𝐴
∙𝑐 ∙𝜌∙𝐴
2
∙
2
𝑡∙
2
𝑡
;
;
[2.40]
The coefficients 𝑐 and 𝑐 are lift coefficient over front and rear axle, respectively. Relations
between Equation [2.39] and Equation [2.40] are: 𝑐
𝑐
𝑐 ; and 𝑐𝑝𝑚 (𝑐
𝑐 ) ∙ 𝑐 ;.
2.9.2
Lateral wind velocity
When the wind comes from the side, there can be direct influences on the vehicle lateral dynamics.
Especially sensitive are large but light vehicles (such as buses or vehicles with unloaded trailers). The
problem can be emphasized by sudden winds (e.g. on bridges or exiting a forested area). Besides
direct effects on the vehicle lateral motion, side-winds can also disturb the driver through
disturbances in the steering wheel feel.
Similar expressions to the longitudinal loads are derived for lateral forces and from side-winds.
2
1
2
∙ 𝑐𝑠 ∙ 𝜌 ∙ 𝐴 ∙
∙𝑐
𝑚
∙𝜌∙𝐴∙
𝑐
∙
[2.41]
2
[2.42]
The speed
is the lateral component of the vehicle velocity relative to the wind. Note that A and
Lc may now have other interpretations and values than in Equations [2.38]-[2.40].
2.10 Driver interactions with vehicle
dynamics
The driver interacts with the vehicle mainly through steering wheel, accelerator pedal and brake
pedal. In addition to these, there are clutch pedal, gear stick/gear selector, in some vehicles, and
various buttons, etc., see Figure 2-36, but we focus here on the first 3 mentioned.
The driver is also a medium through which other road-users and traffic objects interact with a
vehicle. The driver sees and hears what happens in traffic and acts accordingly, see Figure 2-37 and
Figure 2-1.
78
VEHICLE INTERACTIONS
Acceleration Pedal Position, Brake Pedal Force,
Steering Wheel Angle
(and Clutch Pedal Force, Gear Stick/Selector Position,
Parking Brake Request, Direction indicators, HMI buttons, …)
Acceleration Pedal Force,
Brake Pedal Position, Steering Wheel Torque
(and HMI Lamps, Sounds, Haptics, …)
Figure 2-36: Commonly assumed causality (direction of cause and effect).
Driver’s control of vehicle dynamics, or vehicle motion including position, can be discussed in
longitudinal and lateral. The longitudinal is about how driver uses his vision to determine pedal
operation. Looming (optical expansion of an object in the driver’s field of view) is often used as a
cause for how driver pushes the pedals. The optical flow (the pattern of apparent motion of objects,
surfaces, and edges in the driver’s field of view) is often used as a cause for highest comfortable
speed.
2.10.1 Open-loop and Closed-loop Manoeuvres
Two common expressions used in analysing vehicle dynamics are "Open-loop" and "Closed-loop".
These concepts are used to describe how the vehicle control systems are manipulated in a "driving"
event.
An open-loop manoeuvre refers to the case where the driver controls (steering wheel, brake pedal
and accelerator pedal) are operated in a specific sequence, i.e. as functions of time. A typical case is a
sine wave excitation of the steering wheel. The time history of the steering wheel angle is defined as
a function that is independent of the road environment or driver input. This type of manoeuvre is a
practical design exercise to study the steering response of the system, but has little relation to a realworld driving case. Theoretical simulation and real testing with a steering robot are examples of how
such studies can be made.
A closed-loop manoeuvre refers to the case when (human) driver feedback via driver controls is
included. This represents real-world driving better. In real vehicle or driving simulator testing, a real
driver is used. This enables collection of the drivers’ subjective experience. In cases of theoretical
simulation, a "driver-model" is needed. A driver-model can have varying levels of complexity but in
all cases simulates the response of a human driver to different effects, such as lateral acceleration,
steering wheel torque, various objects appearing outside the vehicle, etc.
A test with real vehicle, carried out with a steering-robot (and/or pedal robot) can also be called
closed-loop if the robot is controlled with a control algorithm which acts differently depending on the
vehicle states, i.e. if the algorithm is a driver model.
The concepts of open- and closed-loop control are demonstrated in Figure 2-37. The important
difference to note is that in open-loop control there is no feedback to the driver. The figure also
shows a “Virtual driver” which represents today’s automatic functions based on environment
sensing, such as adaptive cruise control, which adapts speed to a vehicle ahead. Closed loop control
incorporates the different types of feedback to the driver. Drivers automatically adapt to the
different feedback. Understanding driver response to different feedback, and expressing it in
mathematical descriptions, is an active area of research, particularly for driver support functions.
The boxes “Suspension, Linkage, Wheels” and “Vehicle body” in the diagram are the main focus for
this course. The mechanics relating the wheels´ transfer of forces between the road and vehicle are
79
VEHICLE INTERACTIONS
the critical elements to study in introductory vehicle dynamics. In a first order analysis the influence
of the suspension and steering components can be ignored as long as the mechanical components
are not undergoing extensive deflections.
Position of road edges, lanes, traffic
signs, other road users, …
Visual
Environment
sensing & “Virtual
driver” system (radar,
Road irregularities, Road friction
camera, GPS, WiFi, …)
Road banking
and inclination,
Collision forces
Requests (signals)
Driver
Steering
wheel,
Pedals, …
Steering
Suspension,
system,
Surroundings /
Propulsion
Vehicle
Environment
Linkages,
system,
body
(air,
road, other
Position,
Forces,
Road wheel
Speed
Brake
road
users, …)
Moments
angles, Wheel
Wheels
system
torques, …
HMI (lamps,
Warnings, Info
sounds, …)
Sensed quantities
Noise, vibrations
Steering wheel torque
Inertial forces
Open-loop system:
Closed loop system: Whole diagram
Figure 2-37: Open and closed loop representations of vehicle dynamic systems
2.10.2 Pedals
Each pedal has both pedal position and pedal force as communication channels to the driver.
In broad terms, the accelerator pedal position is interpreted by a control system as a certain
requested engine torque, which means a certain wheel shaft torque on the propelled axle/axles. In
some advanced solutions, one could think of using accelerator pedal force as a feedback channel
from vehicle to driver.
In broad terms, the brake pedal force gives a certain friction brake torque application on each axle. In
some advanced solutions, one could think of changing brake pedal position versus force
characteristics as a feedback channel from vehicle to driver. But already with traditional (hydraulic)
brake systems for passenger cars, the brake pedal gives a feedback to driver in that ABS interventions
is felt.
For electric and hybrid vehicles the brake pedal can typically influence the propulsion system as well,
in that at low levels of pedal braking, the electric propulsion motors give negative wheel shaft
torque. This is for regenerating energy. Regenerative braking can even be engaged at low levels of
accelerator pedal position.
The propulsion and brake systems are to some extent also described in 3. LONGITUDINAL DYNAMIC.
80
VEHICLE INTERACTIONS
2.10.3 Steering
The driver interacts with steering wheel through two channels, steering wheel angle and steering
wheel torque.
A frequently used model is that the driver decides a steering wheel angle and expects a certain
steering wheel torque, 𝑇𝑠𝑤 , feedback. Only in extreme situations, the causality is modelled the other
way, e.g. studying what happens if driver takes hands-off, i.e. steering wheel torque =0. However, it
should be noted that, from neuromuscular point-of-view, it is not motivated to model a driver as
controlling the torque or forces, but only the position.
In a broad view, the angles of the road wheels are functions of steering wheel angle, independent of
forces and torques in the steering system. Further, the steering wheel torque is a function of the
lateral force on the front axle, possibly scaled down with assistance from the power steering or
steering servo. Some more detailed aspects on the two latter sentences are discussed below. See
more about steering system in Section 4.2.3.
2.11 Traffic interactions with vehicle
dynamics
Other road-users and traffic objects interact with a vehicle via the driver, as mentioned in Section
2.10, but also directly via environment sensors on-board the vehicle, see Figure 2-37. These sensors
give information to a “virtual driver”, which consists of control algorithms which can request basically
the same as a driver can (steering, brake, propulsion systems). Functions of this type were
categorized as both “Seeing” and “Dynamic” in Section 1.2.
As mentioned in Section 1.2, the same environment information is also the base for functions that
stimulates the driver to act, through functions categorized as both “Seeing” and “Driver informing”.
These do not have a direct interface to the vehicle, but interact via the driver.
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VEHICLE INTERACTIONS
82
LONGITUDINAL DYNAMIC
3 LONGITUDINAL
DYNAMICS
3.1 Introduction
The primary purpose of a vehicle is transportation, which requires longitudinal dynamics. The
chapter is organised with one group of functions in each section as follows:




3.2 Steady State Function
3.3 Functions over longer events
3.4 Functions in shorter events
3.5 Control functions
Most of the functions in “3.5 Control functions”, but not all, could be parts of ”3.4 Functions in
shorter events”. However, they are collected in one own section, since they are special in that they
partly rely on software algorithms.
There are a lot of propulsion related functions, originating from the attribute Driving dynamics.
Examples of such, not covered in this compendium are:



Off-road accessibility: Ability to pass obstacles of different kind, such as uneven ground,
extreme up- and down-hills, mud depth, snow depth, etc.
Shift quality: Quick and smooth automatic/automated gear shifts
Shunt & shuffle: Absence from oscillation for quick pedal apply, especially accelerator pedal.
3.1.1



References for this chapter
“Chapter 23. Driveline” in Reference (Ploechl, 2013).
“Chapter 24. Brake System Dynamics” in Reference (Ploechl, 2013).
“Chapter 27 Basics of Longitudinal and Lateral Vehicle Dynamics” in Reference (Ploechl,
2013).
3.2 Steady State Functions
Functions as top speed and grade-ability are relevant without defining a certain time period. For such
functions it is suitable to observe the vehicle in steady state, i.e. independent of time. Those
functions are therefore called steady state functions, in this compendium.
The main subsystems that influences here are the propulsion system, see Section 3.2.1, and the
(Friction) Brake system, see Section 3.4.3.
3.2.1
Propulsion System
A generalisation of the propulsion system is given in Figure 3-1, along with a specific example of a
conventional type. The component “Energy buffer” refers to an energy storage that can “buffer”
energy during vehicle operation. This means that an energy buffer can not only be emptied (during
propulsion), but also refilled by regenerating energy from the vehicle during deceleration. A fuel tank
is an energy storage, but not an energy buffer. Also, a battery which can only be charged from the
grid, and not from regenerating deceleration energy, is not an Energy buffer.
83
LONGITUDINAL DYNAMIC
Note that the approach in Figure 3-1 is one-dimensional: we consider neither the differential
between left and right wheel on the driven axle nor distribution between axles. Instead, we sum up
the torques at all wheels and assume same rotational speed.
Generalized propulsion system:
Transportation
task
Driver
Acceleration
Request (1)
other systems,
e.g. brake
Propulsion control
(1)
(2)
Prime T,w
Transmission
mover (1)
P(1)
Wheels
T,w
(as many
as driven
wheels)
(incl. road
contact)
F,v
(1)
Body
s
(1)
Traffic/Roads
Energy buffer (e.g.
battery in hybrid vehicle)
Conventional propulsion system:
Driver
other systems,
e.g. brake
e-throttle control
(1)
T, w
Prime mover
(ICE, internal
combustion
engine)
(1)
T, w
(on one
driven axle)
F, v
Wheels
Body
Transmission (Clutch &
Gearbox & Final gear)
Figure 3-1: Propulsion system
3.2.1.1
Prime movers
The conversion of stored energy to power occurs in the prime mover, see Figure 3-1. Details of the
conversion processes and transmission of power to the tyres are not covered in this compendium.
Some basic background is still necessary to describe the longitudinal performance of the vehicle. The
main information that is required is a description of the torque applied to the wheels over time
and/or as a function of speed. Sketches of how the maximum torque varies with speed for different
prime movers (internal combustion engine (ICE), electric motor or similar) are shown in Figure 3-2
and Figure 3-3. The torque speed characteristics vary dramatically between electric and internal
combustion engines. Also, gasoline and diesel engines characteristics vary.
The curve for electric motors in Figure 3-2 shows that the main speciality, compared to ICEs, is that
their operation range is nearly symmetrical for negative speeds and torques. However, the curve
should be taken as very approximate, since electric motors can work at higher torque for short
periods of time. The strong time duration dependency makes electric motors very different to ICEs
from a vehicle dynamics point of view. Other properties that makes them special are quick and
accurate response, well known actual torque and that it is much more realistic to divide them into
several smaller motors, which can operate on different wheels/axles.
84
LONGITUDINAL DYNAMIC
T
Power TO Power FROM
engine engine
T
w
Power FROM Power TO
engine engine
w
Electric Motor
ICE
T
T
w
w
Figure 3-2: Torque Characteristics of Prime Movers
100
TORQUE
80
200
180
POWER
40
SPECIFIC FUEL
CONSUMPTION
20
0
350
300
80
160
60
TORQUE
100
POWER [kW]
POWER [kW]
TORQUE
[Nm]
TORQUE
[Nm]
SPECIFIC FUEL
CONSUMPTION
[kg/kWh]
0.32
0.30
POWER
60
40
SPECIFIC FUEL
CONSUMPTION
SPECIFIC FUEL
CONSUMPTION
[kg/kWh]
0.250
0.225
0.28
2000
4000
SPEED [rpm]
20
1500
6000
Gasoline
2000 2500
SPEED [rpm]
3000
Diesel
Figure 3-3: Engine Characteristics for Gasoline and Diesel Engines, at maximum torque capability.
Typical values for passenger car.
3.2.1.2
Transmissions
In some contexts “transmission” means the 1-dimensional transmission of rotational mechanical
power from an input shaft to one output shaft. These are covered in Section 3.2.1.2.1
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LONGITUDINAL DYNAMIC
In other contexts, “transmission” means the system that distributes the energy to/from an energy
buffer and to/from multiple axles and/or wheels. These are covered in Section 3.2.1.2.2.
3.2.1.2.1 Main transmissions
An example of such transmission is a stepped gearbox, which can be modelled e.g. as:
𝑇
𝑢𝑡
𝑟 ∙ 𝑇 ∙ 𝜂𝐶
𝑀 𝑠ℎ 𝑠
Δ𝑇𝐵
𝑠
;
𝑟
𝑟 𝑟2 ⋯ 𝑟𝑁 ; 𝑟
𝑢𝑡 ) ;
∙ sign(
[3.1]
𝑢𝑡
𝑟
0;
Eq [3.1] is not valid for neutral gear (then there is no speed equation, but instead two torque
) ;.
equations: 𝑇 𝑢𝑡
Δ𝑇 𝑢𝑡𝐵
Δ𝑇 𝐵
𝑠 ∙ sign(
𝑢𝑡 ) ; and 𝑇
𝑠 ∙ sign(
For any 1-dimensional transmission of rotational mechanical power between two rotating shafts, the
) ;, is depending on operating condition. If
total efficiency, 𝜂𝑡 𝑡
𝑃 𝑢𝑡 ⁄𝑃
𝑇 𝑢𝑡 ∙ 𝑢𝑡 ⁄(𝑇 ∙
assuming a nominal ratio, 𝑟, the total efficiency can be decomposed in 𝜂𝑡 𝑡
𝜂 𝑇 ∙ 𝜂𝜔 ;, where 𝜂 𝑇
⁄
(𝑟
)
⁄
(
⁄
)
𝑇 𝑢𝑡
∙ 𝑇 ; and 𝜂𝜔
𝑟 ;.
𝑢𝑡
Clutches and torque converters can also be part of models of main transmissions.
3.2.1.2.2 Distribution transmissions
The distribution to energy buffer and multiple axles and/or wheels can basically be done in two ways:

Distribute in certain fractions of (rotational) speed. A (rotationally) rigid shaft between left
and right wheel is one example of this. We find this in special vehicles, such as go-carts, and
in more common vehicles when a differential lock is engaged. There are 3 shafts in such an
axle: input to the axle and two outputs (to left and right wheel). The equations will be:
𝑡;
𝑇

ℎ𝑡 ;
𝑇
𝑇
𝑡
[3.2]
ℎ𝑡 ;
Distribute in certain fraction of torque. This requires some type of planetary gear
arrangement. A conventional (open) differential gear is one example of this, where the
equations will be:
𝑡
𝑇
𝑇
𝑡
𝑇
𝑇
ℎ𝑡
ℎ𝑡 ;
𝑡
𝑇
;
[3.3]
ℎ𝑡 ;
Generally speaking, the open differential is rather straight-forward to use in most vehicle dynamics
analysis: The speeds are given by vehicle motion (e.g. curve-outer wheel runs faster than curve-inner
wheel, defined by vehicle yaw rate and track width). The torques are defined by the differential, as
half of the propulsion torque at each side.
Also, a locked differential, it is generally more complex to analyse. Here, the wheels are forced to
have same rotational speed and, in a curve, that involves the tyre longitudinal slip characteristics.
The solution involves more equations with shared variables.
So, open/locked differential is the basic concept choice. But there are additions to those: One can
build in friction clutches which are either operated automatically with mechanical wedges or similar
or operated by control functions. One can also build in electric motors which moves torque from one
wheel to the other. However, the compendium does not intend to go further into these designs.
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LONGITUDINAL DYNAMIC
Part of the distribution transmissions are also shafts. If oscillations is to be studied, these has to be
modelled as energy storing compliances: 𝑇 𝑐 ∙ (
𝑢𝑡 ).
3.2.1.3
Energy buffers
Energy buffers in vehicles are today often electro-chemical batteries. However, other designs are
possible, such as flywheels and hydrostatic accumulators. A simple model of a buffer is as follows:
𝐸
{
(𝑃
(𝑃
𝑃
𝑃
𝑤 𝑒𝑟𝑒 𝑃
𝑢𝑡 ) ∙
𝜂𝑐ℎ
; 𝑓𝑜𝑟 𝑃
𝑢𝑡 )/𝜂𝑢𝑠 ; 𝑒 𝑠𝑒;
𝑇 ∙
; 𝑛𝑑 𝑃
𝑢𝑡
𝑇
𝑃
𝑢𝑡 ;
[3.4]
𝑢𝑡
∙
𝑢𝑡 ;
𝑢𝑡 ; or
Including how the buffer is connected, one more equation can be found: Typically
𝑇
𝑇 𝑢𝑡 ;.
The model uses stored energy, 𝐸. For batteries one often uses state of charge, 𝑆𝑜𝐶, instead.
Conceptually, 𝑆𝑜𝐶 𝐸/𝐸𝑚 ;, where 𝐸𝑚 is a nominal maximum charge level.
A first approximation of the efficiencies, can be 𝜂𝑐ℎ
efficiency is dependent of many things, such as 𝑃
leakage when buffer is “resting”: 𝑃
𝑃 𝑢𝑡 .
3.2.2
𝜂𝑢𝑠
𝑐𝑜𝑛𝑠𝑡 𝑛𝑡 < 1, but typically the
𝑃 𝑢𝑡 . The model above does not consider any
Traction diagram
The force generated in the prime mover is transmitted through a mechanical transmission to the
wheel which then generates the propulsive forces in the contact patch between tyre and road. In an
electric in-wheel motor, the transmission can be as simple as a single-step gear. In a conventional
vehicle it is a stepped transmission with several gear ratios (i.e. a gearbox). Then, the drivetrain can
be drawn as in Figure 3-4. The torque and rotational speed of the engine is transformed into force
and velocity curves via the mechanical drivetrain and driven wheel. The result is a Traction diagram.
The transformation follows the following formula, if losses are neglected:
𝑟 𝑡𝑖𝑜 ∙
𝑇
;
𝑒𝑒 𝑅 𝑑𝑖𝑢𝑠
𝑛𝑑
𝑒𝑒 𝑅 𝑑𝑖𝑢𝑠 ∙
𝑟 𝑡𝑖𝑜
;
[3.5]
A traction diagram for a truck is given in Figure 3-5, which also shows that there will be one curve for
each gear.
Losses in transmission can be included by using equations including loss models for transmission,
such as:
𝑇
; 𝑤𝑖𝑡
𝑟 𝑑𝑖𝑢𝑠
𝑅𝑒𝑝 𝑐𝑒
𝑟 𝑡𝑖𝑜 ∙
𝑅𝑒𝑝 𝑐𝑒
𝑟 𝑑𝑖𝑢𝑠 ∙
𝑟 𝑡𝑖𝑜
𝑤 𝑒𝑟𝑒 𝜂𝑡
𝑡
𝜂 𝑇 ∙ 𝑟 𝑡𝑖𝑜 ∙
; 𝑤𝑖𝑡
𝑃
𝑃
𝑇
; 𝑤 𝑒𝑟𝑒 𝜂 𝑇 ≤ 1;
𝑟 𝑑𝑖𝑢𝑠
𝜂𝜔 ∙ 𝑟 𝑑𝑖𝑢𝑠 ∙
ℎ𝑐
∙
𝑇∙
𝑟 𝑡𝑖𝑜
; 𝑤 𝑒𝑟𝑒 𝜂𝜔 ≤ 1;
𝜂 𝑇 ∙ 𝜂𝜔 ≤ 1
This will basically move the curves in the first quadrant slightly downwards and to the left.
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[3.6]
LONGITUDINAL DYNAMIC
prime mover
driven wheel
𝒕𝒐
∙
𝒖
𝒕 𝒐∙
⁄
𝒗
𝒕𝒐
𝒖
∙
𝒕𝒐
transmission
𝒗
(vehicle) body
𝒕 𝒐∙
prime mover
characteristics:
⁄
𝒕𝒐
Traction Diagram
𝒕𝒐
𝒖
Multiply torque by
Multiply rotational speed by
𝒖
𝒕𝒐
𝒗
Figure 3-4: Construction of Traction Diagram, i.e. diagram for transmission of torque to
longitudinal force on vehicle.
lowest gear
highest gear
Figure 3-5: Example of engine map and corresponding traction diagram map from a truck.
(D13C540 is an I6 diesel engine of 12.8-litre and 540 hp for heavy trucks.)
Tyre rolling friction is a loss mechanism, which on its own yields 𝜂𝜔 1 and ηT < 1. Tyre
longitudinal slip is a loss mechanism, which on its own yields 𝜂𝜔 < 1 and ηT 1. See Section 2.4.
A traction diagram is a kind of one degree of freedom graphical model. The diagram can be extended
with, e.g., the longitudinal force from the brake system.
3.2.3
Driving Resistance
From studying Figure 3-5 one could extrapolate that a very low transmission ratio, i.e., a very high
gear, we would enable almost infinite speed. We know that this is not possible. Something obviously
limits the top speed: the limit comes from “driving resistance”. As presented in Chapter 2, there are
tyre rolling-resistance forces and aerodynamic driving-resistance forces. Also, there can be an
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LONGITUDINAL DYNAMIC
additional resistance force to be considered when going uphill: the grade or gravitational load on the
vehicle. This resistance can also be negative, i.e. act as propulsion down-hill. The formula for driving
resistance becomes as follows, using Equation [2.38]. Notation f means rolling resistance coefficient.
𝑠 𝑠𝑡
∙ 𝑔 ∙ sin( )
𝑐
𝑓∙
∙ 𝑔 ∙ cos( )
2
∙ 𝑐𝑑 ∙ 𝜌 ∙ 𝐴 ∙ (
2
𝑤
𝑑
)
[3.7]
longitudinalF
forces
v
supplied force (from propulsion system)
available for acceleration
air
(total) driving resistance
Propulsion
Superimposing the external loads on the Traction Diagram allows us to identify when the “demands”
can be met by the “supply”. As seen in Figure 3-6, the combination of external resistance loads and
the internally generated drive forces can be presented on the same figure. The resulting intersection
of supply (prime mover) and demand (driving resistance) identifies the top speed of the vehicle.
These results hold only for steady state (no acceleration) conditions.
air drag = (*A*cd/2) v2
roll
rolling
j
rolling resistance =
=almost const.= f*m*g*cosj
F m*g
top speed
NOTE: We can now
predict the top speed!
𝒗v𝒙
climbing resistance
m*g*sinj
NOTE: The rolling resistance on the driven wheels can be
included in the transformation from the prime mover to the
traction force diagram.
Figure 3-6: Top speed can be found from Traction diagram with Driving Resistance curves. Head
wind speed, 𝑤 𝑑 , is assumed to be zero. Note that “rolling” actually is a moment loss on the
wheels, why it is more correct to let it influence the ”supplied force curve” for driven axles and the
driving resistance curve for the non-driven axles, see Figure 3-9.
As seen in Figure 3-6, the acceleration can be identified as a vertical measure in the traction diagram,
divided by the mass. However, one should be careful when using the traction diagram for more than
steady state driving. We will come back to acceleration performance later, after introducing the two
effects “Load transfer” and “Rotating inertia effect”.
There are more driving-resistance effects than covered in Equation [3.7]. One example is that lateral
tyre forces may have components in longitudinal direction, such as when the vehicle is making a turn
or if one axle have a significant toe-in.
3.2.4
Top speed *
Function definition: Top speed is the maximum longitudinal forward speed the vehicle can reach and
maintain on level and rigid ground without head-wind.
The top speed can be read out from a traction diagram with driving resistance, see Figure 3-6. Top
speed is the speed where the sum of all driving resistance terms is equal to the available propulsion
forces.
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LONGITUDINAL DYNAMIC
3.2.5
Starting with slipping clutch
As seen in previous traction diagrams, there is no available positive propulsion force at zero speed.
This means that the diagram can still not explain how we can start a vehicle from stand-still.
The concepts in Figure 3-4 were used to create the force-velocity diagram in Figure 3-7. It shows the
smooth curve of a Continuous Variable ratio Transmission (CVT) in comparison to the stepped
transmission. The CVT is the ideal situation for the engine since it can always let the engine work at a
maximum power or minimum fuel consumption (minimum for the momentarily required power). If
the CVT has unlimited high ratio, it can actually have a non-zero propulsion force at zero vehicle
speed. Without losses, this force would be infinite, but in reality it is limited, but still positive, so the
vehicle can start from stand-still.
Propulsion
force, F
A stepped transmission, as well as a CVT with limited ratio range, instead needs a clutch to enable
starting from vehicle stand-still. This is shown in Figure 3-7. The highest force level on each curve can
be reached at all lower vehicle speeds, because the clutch can slip. It requires the clutch to be
engaged carefully to the torque level just below the maximum the engine can produce. In traditional
automatic transmissions the slipping clutch is replaced with a hydrodynamic torque converter, to
enable start from stand-still.
ICE on continuously variable ratio
(can keep power at maximum ICE power, T=Pmax/v)
ICE on lowest gear (highest ratio)
slipping clutch
ICE on highest gear (lowest ratio)
fully engaged clutch
(start from v=0 is not possible)
0
0
Speed, v
Figure 3-7: Force/Speed Curves for a Multiple Gear Transmission and for CVT.
3.2.6
Steady state load distribution
The vehicle performance discussed previously does not rely on knowing the distribution of (vertical)
load between the axles. To be able to introduce limitations due to road friction, this distribution must
be known. Hence we set up the free-body diagram in Figure 3-8.
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LONGITUDINAL DYNAMIC
m*g
-j
Figure 3-8: Free Body Diagram for steady state vehicle. With ISO coordinate system, the road
gradient is negative when downhill. (Rolling resistance is included in .)
From the free-body diagram we can set up the equilibrium equations as follows and derive the
formula for load on front and rear axle:
Moment equilibrium, around rear contact with ground:
∙(
∙ 𝑔 ∙ ( ∙ cos( )
∙ sin( ))
)
⇒
∙𝑔∙
∙ cos( )
∙ sin( )
∙
∙
[3.8]
Moment equilibrium, around front contact with ground:
∙(
∙ 𝑔 ∙ ( ∙ cos( )
∙ sin( ))
)
⇒
∙𝑔∙
∙ cos( )
∙ sin( )
For most vehicles and reasonable gradients one can neglect
| ∙ cos( )|.
3.2.7
0⇒
∙
0⇒
∙
∙ sin( ) since it is ≪ | ∙ cos( )| ≈
Friction limit
With a high-powered propulsion system, there is a limitation to how much the vehicle can be
propelled, due to the road friction limit. It is the normal load and coefficient of friction, which limits
this. For a vehicle which is driven only on one axle, it is only the normal load on the driven axle,
, that is the limiting factor:
𝑑
min(
𝑃
𝑝𝑆 𝑠𝑡
; 𝜇∙
𝑑
)
[3.9]
One realises, from Figure 2-10, that the rolling resistance on the driven axle works as a torque loss
and that the road friction limitation will be limiting 𝑇 ∙ 𝑟 𝑡𝑖𝑜 𝑒 ∙ 𝑑
rather than limiting
𝑇 ∙ 𝑟 𝑡𝑖𝑜. Expressed using the rolling resistance coefficient, 𝑓 , gives:
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LONGITUDINAL DYNAMIC
min( 𝑃 𝑝𝑆 𝑠𝑡 ; 𝜇 ∙ 𝑑
𝑇 ∙ 𝑟 𝑡𝑖𝑜
min (
𝑓 ∙
𝑟 𝑑𝑖𝑢𝑠
)
; 𝜇∙
𝑑
[3.10]
)
𝑑
This is shown in the traction diagram in Figure 3-9, where it should also be noted that the rolling
resistance curve only consists of the rolling resistance on the non-driven axles. See also Figure 2-10.
Propulsion
force, Fx
𝑇
𝑃
𝑇
𝑟 𝑡𝑖𝑜 𝑒
𝑟 𝑑𝑖𝑢𝑠
𝑝𝑆 𝑠𝑡
𝑟 𝑡𝑖𝑜
𝑟 𝑑𝑖𝑢𝑠
𝑓
𝑑
𝑑
Friction limit: 𝜇
𝑑
𝑠
Maximum
available
𝑠
𝑠
0
𝑠𝑠 𝑝
0
0.5 𝜌 𝑐𝑑 𝐴
𝑓
2
𝑑
𝑔 sin
Speed, v
Figure 3-9: Traction diagram with Road Friction limitation and Driving Resistance curves.
3.2.8
Uphill performance
3.2.8.1
Start-ability *
Function definition: Start-ability is the maximum grade that a vehicle is capable to start in and
maintain the forward motion at a certain road friction level and a certain load. (Reference (Kati, 2013))
Figure 3-10 shows how we find the start-ability in the traction diagram.
To allow clutch or tyres to slip, is not very practical, because there will be a lot of wear and heat in
the slipping clutch or tyre. Hence, Figure 3-10 also shows also the case when we require absence of
slip in clutch and tyre in the end of the starting sequence. Then the start-ability is slightly reduced.
The reduction is however very small, since the resistance curves does not change very much during
this small speed interval (the resistance curves in the figure have exaggerated slope; the air
resistance can typically be neglected for start-ability).
However, in practice the energy loss (heat, wear) in clutch and tyre should be limited also during the
starting sequence. This can limit the start-ability more severely than the slope of the resistance
curves, but it is not easily shown in the traction diagram, since it is limited by energy losses in clutch
or tyre, which is a time integral of 𝑇 ∙ 𝑐 𝑢𝑡𝑐ℎ and 𝑇𝑤ℎ ∙ 𝑤ℎ .
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LONGITUDINAL DYNAMIC
Friction limit
𝑠 for
varying
road slope
Start-ability limited by road friction
Propulsion
force, Fx
Propulsion
force, Fx
Start-ability limited by propulsion system
Road slope of
this curve is the
start-ability with
rolling tyres.
Road slope of this
curve is the startability with spinning
tyres.
𝑠 for
varying
road slope
Road slope of this
curve is the startability with slipping
clutch.
Friction limit
lowest gear
0
lowest gear
Road slope of this curve is the
start-ability with engaged clutch.
0
0
Speed, 𝒗
0
Speed, 𝒗
Figure 3-10: How Start-ability is read-out from Traction diagram.
3.2.8.2
Grade-ability *
Function definition: Grade-ability is the maximum grade that a vehicle is capable to maintain the
forward motion on an uphill road at a certain constant speed, at a certain road friction level and with a
certain load. (from Reference (Kati, 2013))
Figure 3-11 shows how we find the start-ability in the traction diagram. For vehicles with high
installed propulsion power per weight, the road friction can be limiting, but this is not visualised in
Figure 3-11. Since the speeds are higher than for start-ability, the air resistance cannot be neglected.
Propulsion
force, Fx
Grade-ability limited by propulsion system
0
Road slope of this curve
(𝑔𝑟 𝑑𝑒 ) is the gradeability for and speed 2
Road slope of this curve
(𝑔𝑟 𝑑𝑒2 ) is the gradeability for and speed
𝑠 for
varying
road grade
0
2
Speed,
Figure 3-11: How Grade-ability is read-out from Traction diagram.
3.2.8.3
Towing capacity *
Function definition: Towing capacity is the maximum vehicle-external longitudinal force the vehicle
can have on its body and start and maintain a certain forward speed at a certain road friction and a certain
up-hill gradient.
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LONGITUDINAL DYNAMIC
The driving situation for defining towing capacity is similar to the one for defining grade-ability.
Towing capacity describes how much load the vehicle can tow, 𝑃 inFigure 3-12, on a certain up-hill
gradient. Since towing a load is more relevant as part of a longer transport mission, it is normally also
for a particular constant speed, typically in range 80 to 100 km/h. Since the speed is that large, the
air resistance may not be neglected. It is also important consider air resistance of the trailer and that
axle loads can change, which changes both friction limitation and rolling resistance. A free-body
diagram is shown in Figure 3-12. It is noticeable, that there can also be an additional air resistance of
the trailer which will influence in a test of Towing capacity.
For pure off-road vehicles and agriculture tractors, the term “towing” can mean the maximum pulling
force at very low forward speeds at level ground. This is related but different to the above described
towing capacity for road vehicles.
m*g
mt*g
-j
Figure 3-12: Towing Loads. The towing vehicle is front axle driven.
3.3 Functions over longer events
Functions as fuel or energy consumption and emissions are relevant only over certain but longer
periods of driving, typically some seconds (typically 10 s) to hours of driving. There are different
ways of defining such driving events.
3.3.1
Driving cycles
One way to define a longer event is a so called driving cycle; where the relevant variables are prescribed as function of time. At least on defines speed as a function of time. Examples of commonly
used driving cycles are given in Figure 3-13 and Figure 3-14. In addition, it can also be relevant to give
road inclination as function of time. Engine temperature and selected gear as functions of time may
also be defined. For hybrid vehicles, the possibility to regenerate energy via electric machines is
limited in curves, so curvature radius can also be prescribed as function of time. For heavy vehicles,
the weight of transported goods can be another important measure to prescribe.
80 km/h
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LONGITUDINAL DYNAMIC
Figure 3-13: New European Driving Cycle (NEDC). From (Boerboom, 2012)
80 km/h
20 min
80 km/h
10 min
Figure 3-14: Top: FTP cycle from http://www.epa.gov/oms/regs/ld-hwy/ftp-rev/ftp-tech.pdf.
Bottom: HFTP cycle from http://www.epa.gov/nvfel/methods/hwfetdds.gif.
FTP and HFTP are examples of cycles derived from logging actual driving, mainly used in North
America. NEDC is an example of a “synthetically compiled” cycle, mainly used in Europe. Worldwide
harmonized Light duty driving Test Cycle (WLTC) is a work with intention to be used world-wide, see
Figure 3-15. WLTC exists in different variants for differently powered vehicle [power/weight].
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LONGITUDINAL DYNAMIC
Figure 3-15: WLTC cycle from http://www.unece.org.
3.3.2
Other ways of defining the driving event
Alternative ways of defining the driving event are:
3.3.2.1
Driving pattern
A driving cycle can be condensed into a 2-dimensional function of speed and acceleration, as shown
in Figure 3-16. This figure is simply a scatter plot showing which combinations are most common, but
can also be converted to durations (in seconds or fractions of total time) for different intervals of
combination of speed and acceleration. This is a less advanced and less realistic description of the
transport task. With this method one does not need any dynamic vehicle model at all and there will
be no influence of the time history from the original driving cycle.
Driving patterns can use more than 2 dimensions, such as [speed, acceleration, road gradient]. In
principle, they can also use less than 2 dimensions, maybe only [speed].
3.3.2.2
Transport task/Operating cycle
Desired speed, road inclination, weight of transported goods etc are defined in position rather than
time. For stops along the route, the stop duration or departure time has to be separately defined.
This is a more advanced and realistic description of the vehicle usage. It can be extended to include
operation at stand-still, e.g. idle and loading/unloading cargo.
Theoretical analyses of such vehicle usage require some kind of driver model. A consequence is that
different driver models will give different results, e.g., different fuel consumption due to different
driver preferred acceleration. Hence, the driver model itself can be seen as a part of the vehicle
usage definition.
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LONGITUDINAL DYNAMIC
Figure 3-16: FTP cycle converted to a Driving pattern, i.e. a distribution of operating duration in
speed and acceleration domain. From http://www.epa.gov/oms/regs/ld-hwy/ftp-rev/ftp-tech.pdf.
3.3.3
Rotating inertia effects
In Figure 3-6 it was shown that the acceleration cannot be found directly as a force difference
(distance between curves) divided by vehicle mass. This is because the Traction Diagram does not
contain any dynamics, and dynamics are more complicated than simply accelerating the vehicle
mass. The phenomenon that occurs is referred to here as “rotating inertia effect”. The rotating part
of the propulsion system, e.g. engine and wheels, must be synchronically accelerated with the
vehicle mass. This “steals” some of the power from the propulsion system. This affects the required
propulsion force when following accelerations in a driving cycle.
Consider a wheel rolling which is ideally rolling (no slip), with a free-body diagram and notations as in
Figure 3-17. Setting up 2 equilibrium equations and 1 compatibility equation gives:
𝐽∙
∙
𝑇
𝑅∙
T
;
⇒
∙ 𝑅;
[3.11]
𝑅∙ ;
w
vx
m,J
F
Figure 3-17: Rolling wheel
Assume that we know torque T and want to compare this with the situation without rotational
inertia:
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LONGITUDINAL DYNAMIC
𝑇
;
𝑅
∙
Then we use the compatibility equation to eliminate
𝑘∙
𝑇
;
𝑅
𝑛𝑑
[3.12]
in Equation [3.11], which then becomes:
∙
𝑇
;
𝑅
𝑤 𝑒𝑟𝑒 𝑘
𝑛𝑑
𝐽 ⁄𝑅 2
1 𝑇
∙ ;
𝑘 𝑅
𝐽𝑡
𝑠
[3.13]
;
Thus, we can see the effect of the rotating inertia as making the mass a factor k larger and making
the reaction force correspondingly lower. We call the factor k the “rotational inertia coefficient”.
In a vehicle propulsion system, there are rotational inertias mainly at two places:


Rotating inertias before transmission, i.e. elements rotating with the same speed as the
engine: 𝐽
Rotating inertias after transmission, i.e. elements rotating with same speed as the wheel: 𝐽𝑤
The effective mass, 𝑘 ∙
, will be dependent on the gear ratio as well:
𝑘∙
𝐽𝑤
𝑅2
𝐽 ∙ 𝑟 𝑡𝑖𝑜 2
;
𝑅2
[3.14]
Typically for a passenger car, k=1.4 in the first gear and k=1.1 in the highest gear. So, in the first gear,
approximately one third of the engine torque is required to rotationally accelerate the driveline, and
only two thirds is available for accelerating the vehicle!
When the clutch is slipping, there is no constraint between engine speed and vehicle. The term with
𝐽 disappears from Equation [3.14]. If the wheel spins, both rotational terms disappear.
We can now learn how to determine acceleration from the Traction Diagram, see Figure 3-18.
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Propulsion
force, Fx
LONGITUDINAL DYNAMIC
• 𝑘 = rotating inertia coefficient on gear 1,
without engine intertia and without
wheel inertia, since slipping tyre, i.e.
𝑘 =1
Gear 1
Friction limit
Gear 2
𝑘2
𝑘
Resistance
• 𝑘2 = rotating inertia coefficient on gear 1,
• 𝑘3 = rotating inertia coefficient on gear 2,
without engine intertia, since slipping
clutch
𝑘
𝑘3
• 𝑘 = rotating inertia coefficient on gear 2
Speed, vx
Figure 3-18: Acceleration in Traction Diagram. Rotating inertia effects are shown assuming that
the engine is run on its maximum curve and the gear (or slipping clutch) for highest acceleration is
selected.
3.3.4
Traction diagram with deceleration
The cases 1-4 in Figure 3-18 assumes that the engine is running on its maximum torque curve, which
is not generally necessary during a certain driving cycle. Instead, the accelerations are often more
modest, meaning that two gears are often used. Then one can generally find an engine operating
point between the maximum and minimum curve of the engine, see Figure 3-2.
When the driving cycle shows a deceleration which is larger than can be achieved with resistance
force, we need to brake with a combination of engine braking and friction brakes. If only friction
braking is used, it can be with engaged or disengaged clutch. That influences the rotating inertia
coefficient by using or not using the Je term in Equation [3.14], respectively.
The traction diagram can be extended to also cover engine braking and friction braking. However,
the friction brake system is seldom limiting factor for how negative the longitudinal force can be. But
the road friction is, see Figure 3-19.
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LONGITUDINAL DYNAMIC
Propulsion
force, Fx
This linetype marks upper and lower
limits between where operating points
can be found using propulsion system
and friction brake system, including
slipping clutch and spinning driven axle.
Braking friction
limit, 𝜇 ∙ ∙ 𝑔
Gear 1
Propulsion friction limit,
∙𝑚∙
𝜇∙ 𝑑
≈
2
Gear 2
Speed, vx
Propulsion friction limit,
∙𝑚∙
𝜇∙ 𝑑
≈
2
Reverse gear
Braking friction limit, -𝜇 ∙
∙𝑔
Figure 3-19: Traction Diagram including both maximum and minimum propulsion. One of two
axles is assumed to be driven, which limits propulsion to approximately half of braking friction
limit. Up-hill slope is assumed, which is seen as asymmetric resistance.
3.3.5
Fuel or energy consumption *
Function definition: Fuel (or Energy) Consumption is the amount of fuel [kg or litre] (or energy
[ J ]) consumed by the vehicle per performed transportation amount. Transportation amount can e.g. be
measured in km, person*km, ton*km or m3*km. The vehicle operation has to be defined, e.g. with a
certain driving cycle (speed as function of time or position), including road gradient, cargo load, road surface
conditions, etc.
The consumption arises in the prime mover. The consumption is dependent on the operating point in
the speed vs torque diagram, or map, for the prime mover. Each point have a consumption rate [g/s]
or [Nm/s=W], but it is often given as a specific consumption or an efficiency. Specific consumption is
the consumption rate divided with output power=speed*torque; for a fuel consuming prime mover
this leads to the unit [(g/s)/(Nm/s)] or [g/J] or [g/(Ws)] or [g/(kWh)]. Efficiency is, conceptually, the
inverse of specific consumption; “out/in” instead of “in/out”. The units for efficiency becomes
[W/W=1]. An example of a map for an ICE is given in Figure 3-20. Maps with similar function can be
found for other types of prime movers, such as the efficiency map for an electric motor, see Figure
3-21.
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LONGITUDINAL DYNAMIC
Propulsion
force, F
Figure 3-20 and Figure 3-21 also show that the efficiencies can be transformed to the traction
diagram. The maps for different gears partly overlap each other, which show that an operating point
of the vehicle can be reached using different gears. The most fuel or energy efficient way to select
gear is to select the gear which gives the lowest specific fuel consumption, or highest efficiency. Such
a gear selection principle is one way of avoiding specifying the gear selection as a function over time
in the driving cycle. For vehicles with automatic transmission, that principle can be programmed into
the control algorithms for the transmission.
gear 1
gear 2
0
0
Speed, v
Propulsion
force, F
Figure 3-20: Left: Fuel consumption map. Curves show where specific fuel consumption
[𝑔⁄(𝑘 ⋅ )] is constant. For a specific fuel, this is proportional to 1/efficiency. Right: Specific fuel
consumption curves can also be transformed in Traction Diagram, for a given gear.
gear 1
gear 2
0
0
Speed, v
Figure 3-21: Left: Efficiency map for a typical brushless DC motor, from (Boerboom, 2012). Elliptic
curves show where efficiency is constant. Right: The efficiency curves can also be transformed in
Traction Diagram, for a given gear.
How to predict the consumption for a vehicle during a certain driving cycle is rather straight-forward
using what has been presented earlier in this chapter. Since a driving cycle is a prediction of how the
vehicle is moving, it actually stipulates the acceleration of a mass, which calls for an “inverse
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LONGITUDINAL DYNAMIC
dynamics” analysis. In such analysis one assumes that the driving cycle is met exactly, which means
that both required wheel speed and required wheel torque can be calculated for each time instant.
Then, via a propulsion and brake system model, the corresponding fuel consumption in the engine
can also be found for each time instant. A summary of such an inverse dynamic algorithm for
prediction of fuel or energy consumption is given in Equation [3.15].
For each time step in the driving cycle:






Calculate operating point for vehicle (speed and acceleration)
from driving cycle. Acceleration is found as slope of v(t) curve.
Other quantities, such as road slope, also needs to be
identified;
Select gear (and clutch state, tyre spin, friction brake state,
etc) to obtain this operating point. Select also friction brake,
especially for operating points which can be reached using only
friction brake. If the vehicle has an energy buffer, regenerative
braking is also an option;
Calculate required actuation from propulsion system on the driven
wheel, i.e. rotational speed and shaft torque;
Calculate backwards through propulsion system, from wheel to
prime motor. It gives the operating point for prime mover
(rotational speed and torque);
Read prime mover consumption [in kg/s or W=J/s, not specific
consumption, not efficiency] from prime mover consumption map;
Sum up consumption [in kg or J] with earlier time steps, e.g.
using the Euler forward integration method:
AccumulatedConsumption = AccumulatedConsumption +
Consumption*TimeStepLength;
[3.15]
end;
The final accumulated consumption [in kg or J] is often divided by the total covered distance in the
driving cycle, which gives a value in kg/km or J/km. If the fuel is liquid, it is also convenient to divide
by fuel density, to give a value in litre/(100*km). It can also be seen as a value in m2, which is the
area of the “fuel pipe” which the vehicle “consumes” on the way.
We should note that the calculation scheme in Equation [3.15] does not always guarantee a solution.
An obvious example is if the driving cycle prescribes such high accelerations at such high speeds that
the propulsion system is not enough, i.e. we end up outside maximum torque curve in engine
diagram. This is often the case with “inverse dynamic” analysis, i.e. when acceleration of inertial
bodies is prescribed and the required force is calculated. An alternative is to do a (fully) dynamic
analysis, which means that a driver model calculates the pedals in order to follow the driving cycle
speed as closely as possible, but not exactly. Inverse dynamic analysis is more computational
efficient, but limits what can phenomena that can be modelled in the propulsion and brake system.
Inverse dynamics and dynamic simulations are sometimes referred to as backward and forward
analysis, respectively; see Reference (Wipke, o.a., 1999).
For hybrid vehicles with energy buffers the same driving cycle can be performed but with quite
different level of energy in the buffer after completing the driving cycle. Also, the level when starting
the driving cycle can be different. This can make it unfair to compare energy efficiency only by
looking at fuel consumption in kg/km or litre/km.
If the driving event is given in some other way than a driving cycle, see Section 3.3.2, the prediction
method have to be somewhat different.
Driving cycles are used for legislation and rating for passenger cars. For commercial vehicles, the
legislation is done for the engine alone, and not for the whole vehicle.
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LONGITUDINAL DYNAMIC
3.3.6
Emissions *
Function definition: As Energy consumption, but amount of certain substance instead of amount of energy.
There are emission maps where different emission substances (NOx, HC, etc) per time or per
produced energy can be read out for a given speed and torque. This is conceptually the same as
reading out specific fuel consumption or efficiency from maps like in Figure 3-20 and Figure 3-21. A
resulting value can be found in mass of the emitted substance per driven distance.
Noise is also sometimes referred to as emissions. It is not relevant to integrate noise over the time
for the driving cycle, but maximum or mean values can have relevance. Noise emissions are very
peripheral to vehicle dynamics.
3.3.7
Tyre wear *
Function definition: Tyre wear is the worn out tyre tread depth on a vehicle per performed transportation
amount. Transportation amount can be measured as for Energy consumption. Tyre wear as a vehicle
function has to consider all tyres on the vehicle, e.g. as maximum over the wheels (assuming that all tyres are
changed when one is worn out) or average (assuming that single tyres are exchanged when worn out).
There are models for tyre wear (e.g. outputting “worn tread depth per time”), see Equation [2.37].
For a certain driving cycle we can integrate the 𝑒 𝑟𝑅 𝑡𝑒 [in mass/s or mm tread depth/s], over
time similar to energy consumption rate, which becomes worn material [in mass or mm tread depth].
The wear rate per wheel is a function of the total slip, so it can include both longitudinal slip
(propulsion and brake) and lateral slip (from cornering and toe angles).
Generally, the worn material will be different for different axles, or wheels, so a tyre change strategy
might be necessary to assume to transform the worn material on several axles into one cost. The cost
will depend on whether one renews all tyres on the vehicle at once or if one change per axle. The
tyre wear is a cost which typically sums up with energy cost and cost of transport time (e.g. driver
salary, for commercial vehicles).
3.3.8
Range *
Function definition: Range [km] is the inverted value of Energy consumption [kg/km, litre/km or
J/km], and multiplied with fuel tank size [kg or litre] or energy storage size [J].
The range is how far the vehicle can be driven without refilling the energy storage, i.e. filling up fuel
tank or charging the batteries from the grid. This is in principle dependent on how the vehicle is used,
so the driving cycle influences the range. In principle, the same prediction method as for energy
consumption and substance emissions can be used. In the case of predicting range, you have to
integrate speed to distance, so that you in will know the travelled distance.
3.3.9
Acceleration reserve *
Function definition: Acceleration reserve is the additional acceleration the vehicle will achieve within a
certain time (typically 0.1..1 s) without manual gear-shifting by pressing accelerator pedal fully, when
driving in a certain speed on level ground without head-wind. For vehicles with automatic transmissions or
CVTs the certain time set can allow automatic gear-shift (or ratio-change) or not. The reserve can also be
measured in propulsion force.
In general terms, the lowest consumption is found in high gears. However, the vehicle will then tend
to have a very small reserve in acceleration. It will, in practice, make the vehicle less comfortable and
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LONGITUDINAL DYNAMIC
less safe to drive in real traffic, because one will have to change to a lower gear to achieve a certain
higher acceleration. The gear shift gives a time delay. Figure 3-22 shows one way of defining a
momentary acceleration reserve. The reserve becomes generally larger the lower gear one selects.
A characteristic of electric propulsion systems is that an electric motor can be run at higher torque
for a short time than stationary, see Figure 3-2. On the other hand, the stationary acceleration
reserve is less gear dependent, since an electric motor can work at certain power levels in large
portions of its operating range.
One can calculate the acceleration reserve at each time instant over a driving cycle. However,
integration of acceleration reserve, as we did with fuel, emissions and wear, makes less sense.
Instead, a mean value of acceleration reserve tells something about the vehicle’s driveability.
Minimum or maximum values can also be useful measures.
Acceleration reserve was above described as limited by gear shift strategy. Other factors can be
limiting, such as energy buffer state of charge for parallel hybrid vehicles or how much overload an
electric machine can take short term, see right part of Figure 3-22.
gear 1
Vehicle with electric
propulsion system
Propulsion
force, F
Propulsion
force, F
Vehicle with conventional
propulsion system
Mass∙AccelerationReserve on…
…gear 1
…gear 2
Mass∙AccelerationReserve on…
gear 1
…gear 1
…gear 2
Short term
Acceleration
Reserve
gear 2
gear 2
0
0
0
0
Speed, v
Speed, v
Figure 3-22: Acceleration reserves for different gears. Large dots mark assumed operating points,
each with its acceleration reserve shown.
3.3.10 Load Transfer, without delays from
suspension
It is useful to understand longitudinal load transfer because unloading one axle often limits the
traction and braking. This is because the propulsion and brake systems are normally designed such
that axle torques cannot always be ideally distributed. We start with the free-body diagram in Figure
3-23, which includes acceleration, .
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LONGITUDINAL DYNAMIC
m*g
-j
Figure 3-23: Free Body Diagram for accelerating vehicle. (Rolling resistance included in
.)
Note that the free-body diagram and the following derivation is very similar to the derivation of
Equation [3.8], but we now include the fictive force ∙ .
Moment equilibrium, around rear contact with ground:
∙(
∙ 𝑔 ∙ ( ∙ cos( ) h ∙ sin( ))
)
⇒
∙ (𝑔 ∙
∙ cos( )
h ∙ sin( )
∙
∙
Moment equilibrium, around front contact with ground:
∙(
∙ 𝑔 ∙ ( ∙ cos( ) h ∙ sin( ))
)
⇒
∙ (𝑔 ∙
∙ cos( )
h ∙ sin( )
∙
∙
)
∙
0⇒
∙
[3.16]
∙
)
∙
∙
0⇒
∙
These equations confirm what we know from experience, the front axle is off-loaded under
acceleration with the load shifting to the rear axle. The opposite occurs under braking.
The load shift has an effect on the tyre’s grip. If one considers the combined slip conditions of the
tyre (presented in Chapter 2), a locked braking wheel limits the amount of lateral tyre forces. The
same is true for a spinning wheel. This is an important problem for braking as the rear wheels
become off-loaded. This can cause locking of the rear wheels if the brake pressures are not adjusted
appropriately.
Consider the different conditions for a vehicle where either the front axle or rear axle is locked.
Figure 3-24 shows the vehicle response of the vehicle with a small disturbance and one locked axle.
105
LONGITUDINAL DYNAMIC
Front wheels locked
Rear wheels locked
⃗
⃗
Side forces on rolling
rear wheels tend to
return the vehicle to
straight-ahead motion.
Side forces on rolling
front wheels tend to
increase the
perturbation – the
vehicle “stumbles”.
Unstable motion!
Stable motion!
Friction forces are counter-directed
the sliding motion for locked wheels
Figure 3-24: Locked Axle Braking. Same reasoning works for acceleration, if replacing “locked“
with “large longitudinal tyre slip”.
Thus:


A vehicle with locked front wheels will have a stable straight-ahead motion. However,
steering ability is lost.
A vehicle with locked rear wheels will be unstable. It turns around and ends up in stable
sliding, i.e. with the rear first.
Usually, it is preferred that the front wheels lock first. But it is of course also important that both
axles are used as much as possible during braking, to improve braking efficiency. Hence, there are
trade-offs when designing the brake torque distribution.
3.3.11 Acceleration – simple analysis
3.3.11.1 Acceleration performance *
Function definition: Acceleration performance is the time needed to, with fully applied accelerator
pedal, increase speed from a certain speed to another certain higher speed, at certain road friction on level
ground without head-wind and certain load.
Acceleration events will be considered in Section 3.4, as being shorter events, where more dynamic
aspects become important. If we are only interested in an approximate analysis of acceleration
performance, and typically over a longer acceleration (0-100 km/h over 5..10 s), we can do a simpler
analysis.
3.3.11.2 Solution using integration over time
A front-wheel-drive passenger car with a stepped gearbox should accelerate from 0 to 100 km/h. A
Matlab code is given in Equation [3.17], which simulates the acceleration uphill from stand-still, using
simple numerical integration. The code calculates the possible acceleration in each of the gears, and
one mode with slipping clutch. In each time step it selects that which gives the highest acceleration.
The numerical data and results are not given in the code, but some diagrams are shown in Figure
3-25. The code is not fully documented, only using equations so far presented in this compendium.
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LONGITUDINAL DYNAMIC
dt=0.1; t_vec=[0:dt:10]; vx_vec(1)=0;
for i=1:length(t_vec)
vx=vx_vec(i);
Fres=m*g*sin(p)+froll*m*g*cos(p)+0.5*roh*A*cd*vx*vx;
%if gear 1 (clutch engaged)
ratio=ratios(1);
we=vx*ratio/radius;
Te=interp1(Engine_w,Engine_T,we);
Fx=Te*ratio/radius;
ax=(Fx-Fres)/(m+(Jw+Je*(ratio^2))/(radius^2));
Fzf=m*(g*lr/(lf+lr)-ax*h/(lf+lr));
if Fx>mu*Fzf
Fx=mu*Fzf;
ax=(Fx-Fres)/m;
end
ax1=ax;
%if gear 2 (clutch engaged)
ratio=ratios(2);
… then similar as for gear 1
ax2=ax;
%if gear 3 (clutch engaged)
ratio=ratios(3);
… then similar as for gears 1 and 2
ax3=ax;
[3.17]
%if clutch slipping on gear 2
ratio=ratios(2);
wc=vx*ratio/radius; %speed of output side of clutch
Te=max(Engine_T);
we=Engine_w(find(Engine_T>=Te)); %engine runs on speed where max
torque
Fx=Te*ratio/radius;
ax=(Fx-Fres)/(m+Jw/(radius^2));
Fzf=m*(g*lr/(lf+lr)-ax*h/(lf+lr));
if Fx>mu*Fzf
Fx=mu*Fzf;
ax=(Fx-Fres)/m;
end
if wc>we %if vehicle side (wc) runs too fast, we cannot slip on
clutch
ax=-inf;
end
ax0=ax;
end
[ax,gear_vec(i)]=max([ax0,ax1,ax2,ax3]);
vx_vec(i+1)=vx+ax*dt;
Phenomena that are missing in this model example are:




Gear shifts are assumed to take place instantly, without any duration
The option to use slipping clutch on 1st and 3rd gear is not included in model
The tyre slip is only considered as a limitation at a strict force level, but the partial slip is not
considered. The code line “we=vx*ratio/radius;” is hence not fully correct. Including the slip,
the engine would run at somewhat higher speeds, leading to that it would lose its torque
earlier, leading to worse acceleration performance. The calculation scheme in Equation
[3.17] would be much less explicit.
Load transfer is assumed to take place instantly quick; Suspension effects as described in
Section 3.3 are not included.
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LONGITUDINAL DYNAMIC
Engine
4
2
Torque[Nm]
Power[kW]
160
Traction diagram
x 10
gear 1
gear 2
gear 3
resistance
slip limit
1.8
1.6
120
1.4
Longitudinal force [N]
140
100
80
60
max achievable
1.2
1
0.8
0.6
40
0.4
20
0
0.2
0
200
100
400
300
rotational speed [rad/s]
0
700
600
500
15
10
5
0
20
speed [m/s]
35
30
25
40
Simulation result
14
12
10
8
vx [m/s]
gear (0=clutch slips on gear 2)
6
4
2
0
1
0
2
10
9
8
7
6
5
t [s]
4
3
Figure 3-25: Example of simulation of acceleration, using the code in Equation [3.17].
3.3.11.3 Solution using integration over speed
An alternative way to calculate how speed varies with time is to separate the differential equation as
follows:
∙
⇒
∙𝑑
( )
𝑠(
)
𝑑𝑡 ⇒ ∫
∙
𝑑
𝑑𝑡
( )
); ⇒
𝑡
∙𝑑
( )
𝑠(
𝑠(
)
∫ 𝑑𝑡 ⇒
𝑡
∫
[3.18]
∙𝑑
( )
𝑠(
)
;
Now, the time is calculated by means of integration over speed, as opposed to integration over time.
If simple mathematic functions are used to describe ( ) and 𝑠 ( ) the solution can potentially be
mathematically explicit, but the previous integration over time is more general and works for more
advanced models.
3.4 Functions in shorter events
This section targets models and methods to define and predict functions in a certain and shorter time
frame, typically 0.5 to 5 seconds. It can be both acceleration and deceleration. (Friction) Brake
108
LONGITUDINAL DYNAMIC
system and phenomena as load transfer then becomes important, why these are presented early.
But first, some typical driving manoeuvres are presented.
3.4.1
Typical test manoeuvres
When applying the longitudinal actuator systems (propulsion system and brake system) there are a
couple of different situations which are typical to consider:




Straight line maximum braking from, typically 100 km/h to stand-still for passenger cars.
Braking in curve with significant lateral acceleration, see References (ISO, 2006) and (ISO,
2011).
Straight line acceleration, typically 0 to 100 km/h and 80-100 km/h.
Accelerating in curve with significant lateral acceleration.
For these four main situations, one can also vary other, typically:






At high road friction and at low friction, often called “hi-mu” and “lo-mu”
At different road friction left and right, often called “split-mu”
At sudden changes in road friction, called “step-mu”
At high speed, typically 200 km/h, for checking stability
At different up-hill/down-hill gradients
At different road banking (slope left to right)
Braking of heavy trucks is special in that their load varies a lot and that they often have towed units.
Depending on brake distribution between the units, the longitudinal forces in the couplings may push
or pull. When braking in a curve, this also leads to an extra influence on the lateral dynamics.
A braked wheel or axle develops a longitudinal force, , counter-acting the rotation of the wheel.
is limited by the road friction: | |𝑚
𝜇 ∙ ;. (NOTE: | |𝑚 is not 𝜇 ∙
𝑓 ∙ , see Figure
2-10.)
⁄ , is a property defined for an axle (or a single wheel) = brake
Braking coefficient
force/normal load. It can be seen as the utilized friction coefficient. A suitable notation can be 𝜇𝑢𝑡 .
The coefficient should not be mixed up with friction coeeficient, 𝜇; the relation between them is
𝜇𝑢𝑡 ≤ 𝜇.
3.4.2
Deceleration performance
There are some different functions that measures braking performance or deceleration performance.
3.4.2.1
Braking efficiency *
Function definition: Braking Efficiency is the ratio of vehicle deceleration and the best brake-utilized
axle (or wheel), while a certain application level of the brake pedal at a certain speed straight ahead, at
certain road friction on level ground without head-wind and certain load at a certain position in the vehicle.
In equation form, the Braking Efficiency becomes
− 𝑥⁄
;.
max 𝑢𝑡𝑖𝑙
If Braking Efficiency=1=100%, the
∀𝑎𝑥𝑙𝑒𝑠
distribution of braking is optimal.
3.4.2.2
Braking Distance *
Function definition: Braking Distance is the distance travelled during braking with fully applied brake
pedal from a certain speed straight ahead to another certain lower speed, at certain road friction on level
ground without head-wind and certain load at a certain position in the vehicle.
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LONGITUDINAL DYNAMIC
For passenger cars one typically brakes fully from 100 km/h and then the braking distance is typically
around 40 m (average deceleration 9.65 m/s2). For a truck it is typically longer, 51..55 m (7.5..7 m/s2).
3.4.2.3
Stopping Distance *
Function definition: Stopping Distance is the distance travelled from that an obstacle becomes visible to
driver have taken the vehicle to stand-still. Certain conditions, as for Braking Distance, have to be specified,
but also a certain traffic scenario and a certain driver to be well defined.
Stopping Distance is the braking distance + the “thinking/reaction distance”, which depends on the
speed and the reaction time. The reaction time of a driver is typically 0.5..2 seconds.
3.4.3
(Friction) Brake system
Generally speaking, there are several systems that can brake a vehicle:






Service brake system (brake pedal and ABS/ESC controller, which together applies brake pads
to brake discs/drums)
Parking brake (lever/button that applied brake pads to brake discs/drums, normally on rear
axle)
Engine braking (ICE run at high speeds generate a negative propulsion)
Electric machines (machines can be used symmetrical, i.e. both for positive and negative
torques)
Retarders, or auxiliary brakes, for heavy vehicles
Large steering angles will actually decelerate the vehicle
This section is about Friction brakes, meaning Service brakes and Parking brake. In vehicle dynamics
perspective these have the following special characteristics:



In practice, Friction brakes are unlimited in force since they can lock the wheels for most
driving situations and road friction (ICE and electric motors are often limited by their
maximum power, since it is often smaller than available road friction.)
Friction brakes can only give torque in opposite direction to wheel rotation. (Electric motors
can brake so much that wheel spins rearwards.)
Friction brakes can hold the vehicle at exact standstill. (If using electric machines for holding
stand-still in a slope, a closed loop control would be necessary, resulting in that vehicle
“floats” a little.)
The basic design of a passenger car brake system is a hydraulic system is show in Figure 3-26. Here,
the brake pedal pushes a piston, which causes a hydraulic pressure (pressure = pedal force/piston
area), The hydraulic pressure is then connected to brake callipers at each wheel, so that a piston at
each wheel pushes a brake pad towards a brake disc (disc force = pressure * piston area). The brake
torque on each wheel is then simply: torque = number of friction surfaces * disc coefficient of friction
* disc force * disc radius. (Normally, there are 2 friction surfaces, since double-acting brake callipers.)
By selecting different piston area and disc radii at front and rear, there is a basic hydro mechanical
brake distribution ratio between front and rear axle. There is normally two circuits for redundancy.
Brake systems for heavy trucks are generally based on pneumatics, as opposed to hydraulics.
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LONGITUDINAL DYNAMIC
Driver pushes
on pedal
Front
Front
Front
II
HI
Front
Front
LL
HH
Hydraulics
Pistons with
brake pad
Brake
disc
A more advanced
functional model,
would show hydraulic
pump for ABS/ESC
and hydraulic valves.
Rear
X
(common)
Figure 3-26: Principles for a hydraulic brake system, which is conventional on passenger cars.
3.4.4
Pedal Response *
Function definition: Accelerator pedal response is how vehicle acceleration varies with accelerator
pedal position, for a certain vehicle speed and possibly certain gear, on level ground without pressing the
brake pedal.
Function definition: Brake pedal response is how vehicle deceleration varies with brake pedal force, for
a certain vehicle speed, on level ground without pressing the accelerator pedal.
These functions, together with the functions in Section 3.4.5, enable the driver to operate the vehicle
longitudinally with precision and in an intuitive and consequent way. The requirements based on
above function definitions, are typically that the translation of pedal position (or force) to vehicle
acceleration (or deceleration) should be consistent, progressive and oscillation-free.
For accelerator pedal steps, there should be enough acceleration, but also absence of “shunt and
shuffle” (driveline oscillations). When accelerator pedal is suddenly lifted off, there shall be certain
deceleration levels, depending on vehicle speed and gear selected.
3.4.5
Pedal Feel *
Function definition: Accelerator pedal feel is the pedal’s force response to pedal position.
Function definition: Brake pedal feel is the pedal’s position response to pedal force.
These functions, together with the functions in Section 3.4.4, enable the driver to operate the vehicle
longitudinally with precision and in an intuitive and consequent way.
3.4.6
Brake proportioning
The basic function of a brake system is that brake pressure (hydraulic on passenger cars and
pneumatic on trucks) is activated so that it applies brake pads towards brake discs or drums. In a first
approximation, the pressure is distributed with a certain fraction on each axle. For passenger cars
this is typically 60..70% front and the remaining rear, where percentage is calculated in axle torque.
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LONGITUDINAL DYNAMIC
If neglecting air resistance and slope in Equation [3.16], the vertical axle loads can be calculated as
function of deceleration (=-ax). An ideal brake distribution would be if each axle always utilize same
fraction of friction:
/
/ . This requirement gives the optimal
and
as:
𝜇∙
𝜇∙
⇒
∙ (𝑔 ∙
∙
)
∙ (𝑔 ∙
∙
)
⇒
∙
⇒
∙
∙
𝑔∙
𝑔∙(
∙
)
;
𝑔∙
𝑔∙
∙
;
∙
[3.19]
;
𝑛𝑑
∙
∙
𝑔∙
𝑔∙(
∙
)
;
Equation [3.19] is plotted for some variation in centre of gravity height and longitudinal position in
Figure 3-27.
The proportioning is done by selecting pressure areas for brake callipers, so the base proportioning
will be a straight line, marked as “Hydrostatic brake proportioning”. For passenger cars, one typically
designs this so that front axle locks first for friction below 0.8 for lightest vehicle load and worst
variant. For heavier braking than 0.8*g, or higher (or front-biased) centre of gravity, rear axle will
lock first, if only designing with hydrostatic proportioning..
8000
clear; clf; m=
lr=1.5; h=0.4;
7000
Fxf=m*ax.*(g*l
Fxr=m*ax.*(g*l
6000
plot(-Fxf, -Fx
ylabel('Fxr [N
Fxr [N]
5000
h=0.5; Fxf=m*a
Fxr=m*ax.*(g*l
Fxf, -Fxr,'--'
4000
3000
2000
lf=1.25; lr=1.5; h=0.4;
lf=1.25; lr=1.5; h=0.5; (higher load)
lf=1.15; lr=1.6; h=0.4; (front-biased load)
1000
0
0
5000
10000
15000
Fxf [N]
Figure 3-27: Brake Proportioning diagram. The curved curves mark optimal distribution for some
variation in position of centre of gravity.
To avoid rear axle lock up, one restricts the brake pressure to the rear axle. This is done by pressure
limiting valve, brake pads with pressure dependent friction coefficient or Electronic Brake
Distribution (EBD). In principle, it bends down the straight line as shown in Figure 3-27. With pressure
dependent values one gets a piece-wise linear curve, while pressure dependent friction coefficient
gives a continuously curved curve. EBD is an active control using same mechatronic actuation as ABS.
EBD is the design used in today’s passenger cars, since it comes with ABS, which is now a legal
requirement on most markets.
112
h=0.4; lf=1.15
ax*h)/(g*(lf+l
Fxr=m*ax.*(g*l
Fxf, -Fxr,':')
legend(' lf=1.
lr=1.5; h=0.5;
lr=1.6; h=0.4;
plot([0 m*0.3*
plot([0 m*0.8*
axis([0 15 0 8
LONGITUDINAL DYNAMIC
Ideal curve
ECE regulation
limits to this region
[data from BMW 320i E46]
Figure 3-28: Brake Proportioning. From (Boerboom, 2012).
3.4.7 Body heave and pitch due to longitudinal
wheel forces
Additional to that the axle vertical loads change due to acceleration ax, there are also change in
displacement (heave and pitch). In the following section, a model assuming “quasi-steady state” will
be derived, see Figure 3-29. This assumption means that the accelerations and speeds are constant,
except for in longitudinal direction. The model is approximately valid for heave and pitch during such
a constant braking and acceleration. The model differs between the “unsprung mass” (wheels and
the part of the suspension that does not heave) and the “sprung mass” or “body” (parts that heaves
and pitches as one rigid body).
It is especially noted that the model in Figure 3-29 models the suspension in a very trivial way; purely
vertical. It can be seen as a very trivial (suspension) linkage. The model will be further developed in
Section 3.4.8, to model linkage in the suspension better, which allows validity for transients braking
and acceleration. It will also generally improve the validity also for quasi-steady state acceleration or
braking.
There is no damping included in model, because their forces would be zero, since there is no
displacement velocity, due to the “quasi-steady-state” assumption. As constitutive equations for the
compliances (springs) we assume that displacements are measured from a static conditions and that
the compliances are linear. The road is assumed to be smooth, i.e. 𝑧
𝑧
0.
𝑐 ∙ (𝑧
𝑤 𝑒𝑟𝑒
𝑧 )
𝑛𝑑
∙ 𝑔 𝑛𝑑
113
∙
𝑐 ∙ (𝑧
∙
0
𝑧 )
[3.20]
LONGITUDINAL DYNAMIC
Suspension model with “trivial suspension”
ax
zf
z
Fair
m*ax
zr
x
-Fair-m
m*g=Fzf
Fzr*lr-
h
hair
py
m*g
Fxf
zfr=0
lf
Fxr
Fzf
zrr=0
Fzr
lr
Quasi steady-state assumed, so that
longitudinal acceleration (ax) may
be non-zero, but vertical and pitch
acceleration are zero.
z,x,py,zf,zr, are displacements from
a static stand-still position.
zfr=zrr=0 means that road is smooth.
Figure 3-29: Model for quasi-steady state heave and pitch due to longitudinal accelerations.
Suspension design is briefly discussed at these places in this compendium: Section 3.4.7, Section
4.3.9.5 and Section 5.2.
The stiffnesses cf and cr are the effective stiffnesses at each axle. The physical spring may have
another stiffness, but its effect is captured in the effective stiffness. An example of different physical
and effective stiffness is given in Figure 3-30. Note that the factor ( ⁄𝑏)2 is not the only difference
between effective and physical stiffness, but the effective can also include compliance from other
parts than just the spring, such as bushings and tyre. A common alternative name for the effective
stiffness is axle rate, or wheel (spring) rate if measured per wheel. There will also be a need for a
corresponding effective damping coefficient, see Section 3.4.8.2, or wheel (damping) rate.
We see already in free-body diagram that
and
always act together, so we rename
𝑤 , where w refers to wheel. This and equilibrium give:
∙
∙𝑔
∙
𝑤
0;
∙
0;
∙(
𝑤∙
)
0;
[3.21]
Compatibility, to introduce body displacements, 𝑧 and 𝑝 , gives:
𝑧
𝑧
∙𝑝 ;
114
𝑛𝑑 𝑧
𝑧
∙𝑝 ;
[3.22]
LONGITUDINAL DYNAMIC
b
cp*zp
Fzr
physical spring
with stiffness cp
cp
zp
zrr
zp
a
Fzr
b
Fzr
Moment equilibrium of arm:
cp*zp*a=Fzr*b
 Fzr=cp*zp*a/b;
Effective suspension design
virtual spring
with stiffness cr
Compatibility: zp/a=zrr/b;
cr
Equivalence in stiffness:
Fzr=cr*zrr;
zrr
Together:
 cp*(zrr*a/b)*a/b=cr*zrr 
 cr=cp*(a/b)2;
Fzr
Figure 3-30: From physical suspension design to effective stiffness. Upper left: From
http://www.procarcare.com/icarumba/resourcecenter/encyclopedia/icar_resourcecenter_encyclo
pedia_suspsteer3.asp.
Combining constitutive relations, equilibrium, compatibility and renaming
refers to wheel), gives, as Matlab script:
clear, syms zf zr Fzf Fzr Fzf0 Fzr0 ax z py
sol=solve( ...
'Fzf=Fzf0-cf*zf', ...
'Fzr=Fzr0-cr*zr', ...
'Fzf0+Fzr0=m*g', ...
'Fzf0*lf-Fzr0*lr=0', ...
'-Fair-m*ax+Fxw=0', ...
'm*g-Fzf-Fzr=0', ...
'Fzr*lr-Fzf*lf-Fxw*h-Fair*(hair-h)=0', ...
'zf=z-lf*py', ...
'zr=z+lr*py', ...
zf, zr, Fzf, Fzr, Fzf0, Fzr0, ax, z, py);
The solution from Matlab script in Equation [3.23] becomes:
115
𝑤,
where w
[3.23]
LONGITUDINAL DYNAMIC
𝑤
;
𝑐 ∙
𝑧
𝑐 ∙
𝑐 ∙𝑐 ∙(
𝑐
𝑝
𝑐
𝑐 ∙𝑐 ∙(
)
∙(
2
2
)
∙(
𝑤
∙(
∙
𝑤
))
∙(
∙
[3.24]
))
In agreement with intuition and experience the body dives (positive pitch) when braking (negative
𝑤 ). Further, the body centre of gravity is lowered (negative z) when braking and weaker suspension
front than rear (𝑐 ∙ < 𝑐 ∙ ), which is normally the chosen design for cars.
The air resistance force is brought into the equation. It can be noted that for a certain deceleration,
there will be different heave and pitch depending on how much of the decelerating force that comes
from air resistance and from longitudinal wheel forces. But, as already noted, heave and pitch does
not depend on how wheel longitudinal force is distributed between the axles.
3.4.8 Load Transfer including suspension
linkage effects
In previous model of load transfer, see Section 3.3.10, we assumed nothing about the transients of
transfer (vertical wheel) loads between front and rear axle. So, if we study longer events when the
wheel force is applied and then kept constant for a longer time (1-5 s), it is often a good enough
model. But if the wheel forces vary more, we need to capture the transients better. Then it is
important to consider that the linkage can transfer some of the wheel longitudinal forces. When
studying the transients, it is also relevant to consider the damping.
Another reason for doing a better model than in Section 3.3.10 can be that one is interested in the
displacements, heave and pitch, which are not covered in the model in Section 3.3.10.
There are basically two modelling ways to include the suspension in the load transfer: through a
pitch centre or through a pivot point for each axle, see Figure 3-31.
Model with pitch center
Model with Axle Pivot Points
vz
vx
h-hPC
h-hPC
PC
wx
lPCf
lf
lr
lPCf
PC=Pitch Centre
Figure 3-31: Models for including suspension effects in longitudinal load transfer
116
LONGITUDINAL DYNAMIC
3.4.8.1
Load Transfer model with Pitch Centre
This model will not be deeply presented in this compendium. It has drawbacks in that it has only one
suspended degree of freedom. Also, it does not take the distribution of longitudinal wheel forces
between the axles into account. These short comings is not very important for studying dive and
squat, but they are essential when studying rapid individual wheel torque changes in time frames of
0.1 s, such as studying ABS or traction control. So, since the model with axle pivot points is more
generally useful and not much more computational demanding (and probably more easy intuitively),
that model is prioritized in this compendium.
3.4.8.2
Load Transfer model with Axle Pivot Points
Behold the free-body diagram in Figure 3-32. The road is assumed to be flat, 𝑧
𝑧
0. In freebody diagram for the rear axle, 𝑃 and 𝑃 are reaction forces in the pivot point. 𝑠 is the force in
the elasticity, i.e. where potential spring energy is stored. The torque 𝑇𝑠 is the shaft torque, i.e. from
the propulsion system. No torque from friction brake is visible in this free-body diagram, since such
appears as internal torque between brake pad and brake calliper, which both are within the free laid
rear axle.
ax
vz
zf vx
wy,py
m*az
J*der(wy)=𝐽 ∙
zr
m*ax
Fsr
Pzr
Pxr
m*g
h
ef
lr
lf
er
Fxr
Fxf
Fzf
zrr=0
zfr=0
Tsr
Fxr
Fzr
Fzr
gf
gr
z,x,py,zf,zr, are displacements from a static stand-still position.
zfr=zfr=0 means that road is smooth.
Figure 3-32: Free-body diagram for model with Axle Pivot Points
We assume that displacements are measured from the forces 𝑠 and
the compliances are linear. The total constitutive equations become:
𝑠
𝑠
𝑐 ∙ (𝑧
𝑧 )
𝑛𝑑
𝑠
𝑠
𝑐 ∙ (𝑧
, respectively, and that
𝑠
𝑧 )
[3.25]
Now, there are two ways of representing the dynamics in spring-mass systems: Either as second
order differential equations in position or first order differential equations in velocity and force. We
select the latter, because it is easier to select suitable initial values. Then we need the differentiated
versions of the compliances constitutive equations:
𝑠
0
𝑐 ∙ (𝑧
𝑧 )
𝑐 ∙
𝑛𝑑
𝑠
0
𝑐 ∙ (𝑧
𝑧 )
𝑐 ∙
[3.26]
Even if damping is not drawn in Figure 3-32, for graphical clarity, we will include it in the model. The
damper forces are denoted 𝑑 and 𝑑 . They will appear in the equilibrium equations quite similar
to 𝑠 and 𝑠 . Note that the damping coefficients, 𝑑 and 𝑑 , are the effective ones, i.e. the ones
117
LONGITUDINAL DYNAMIC
defined at the wheel contact point with ground, as opposed to the physical ones defined for the
actual physical damper. C.f. effective stiffness in Section 3.4.7.
𝑑
𝑑 ∙ (𝑧
𝑧 )
𝑑 ∙
𝑛𝑑
𝑑 ∙ (𝑧
𝑑
𝑧 )
𝑑 ∙
[3.27]
Now, 3 equilibria for whole vehicle and one moment equilibria around pivot point for each axle gives:
∙
∙
0;
∙𝑔
∙
𝐽∙
(
(
𝑠
𝑠
𝑑
𝑑
∙
)∙𝑔
)∙𝑔
0;
(
∙𝑒
𝑇𝑠
∙𝑒
𝑇𝑠
)∙
0;
0;
0;
[3.28]
Compatibility, to connect to body displacements, z and py, gives:
𝑧
𝑧
𝑧
;
∙𝑝 ;
𝑛𝑑 𝑧
∙ ;
𝑛𝑑
𝑛𝑑 𝑝
;
𝑧
∙𝑝 ;
∙
;
[3.29]
By combining constitutive relations, equilibrium and compatibility we can find explicit function so
that:
𝑆𝑡 𝑡𝑒𝐷𝑒𝑟𝑖 𝑡𝑖 𝑒𝑠
𝑆𝑡 𝑡𝑒𝑠 [
𝑆𝑡 𝑡𝑒𝐷𝑒𝑟𝑖 𝑡𝑖 𝑒𝑠
𝐼𝑛𝑝𝑢𝑡𝑠 [
𝑓𝑢𝑛𝑐𝑡𝑖𝑜𝑛(𝑆𝑡 𝑡𝑒𝑠 𝐼𝑛𝑝𝑢𝑡𝑠);
𝑧 𝑝 ];
𝑠
𝑠
𝑧
[
𝑠
𝑠
𝑇𝑠 𝑇𝑠 ];
𝑝 ];
[3.30]
Equation [3.30] can be solved with well-established methods for numerical ode solution, if Inputs are
known as functions of time and initial values of States are known. Such simulation of Equation [3.30]
is shown in Section 3.4.12.
3.4.8.3 Steady state heave and pitch due to longitudinal
wheel forces
If we study long term quasi-steady state (all derivatives zero, except ) for the model described in
Section 3.4.8.2 we find a model comparable with the model in Section 3.4.7. but also correct steady
state displacements, heave and pitch. For this reason we set all time derivatives equal to zero except
𝑑𝑒𝑟( )
, in Equations [3.25] to [3.29]. The reasons why we keep ax at the “quasi-steady state”
assumption, which is that longitudinal acceleration ( ) may be non-zero, but vertical and pitch
acceleration are zero. This means that damper forces are zero. We also neglect air resistance force
for clarity of equations. Equation [3.25] becomes, in Matlab format:
clear, syms zf zr Fzf Fzr Fsf0 Fsr0 ax Fsf Fsr z py
sol=solve( ...
'Fsf=Fsf0-cf*zf',
'Fsr=Fsr0-cr*zr', ...
'Fsf0+Fsr0=m*g',
'Fsf0*lf-Fsr0*lr=0', ...
'-m*ax+Fxf+Fxr=0', ...
'-m*0-m*g+Fzf+Fzr=0', ...
'-J*0+Fzr*lr-Fzf*lf-(Fxf+Fxr )*h=0', ...
'(Fzr-Fsr-0 )*gr-Fxr*er+Tsr=0', '(Fsf+0-Fzf )*gf-Fxf*ef+Tsf=0', ...
'zf=z-lf*py',
'zr=z+lr*py', ...
zf, zr, Fzf, Fzr, Fsf0, Fsr0, ax, Fsf, Fsr, z, py);
%results:
% ax = (Fxf + Fxr)/m
118
[3.31]
LONGITUDINAL DYNAMIC
% z = -(Tsr*cf*gf*lf^2 - Tsf*cr*gr*lr^2 + Tsr*cf*gf*lf*lr Tsf*cr*gr*lf*lr - Fxr*cf*er*gf*lf^2 + Fxf*cr*ef*gr*lr^2 +
Fxf*cf*gf*gr*h*lf + Fxr*cf*gf*gr*h*lf - Fxf*cr*gf*gr*h*lr Fxr*cr*gf*gr*h*lr - Fxr*cf*er*gf*lf*lr +
Fxf*cr*ef*gr*lf*lr)/(cf*cr*gf*gr*(lf + lr)^2)
% py = -(Tsr*cf*gf*lf + Tsr*cf*gf*lr + Tsf*cr*gr*lf + Tsf*cr*gr*lr +
Fxf*cf*gf*gr*h + Fxr*cf*gf*gr*h + Fxf*cr*gf*gr*h + Fxr*cr*gf*gr*h Fxr*cf*er*gf*lf - Fxf*cr*ef*gr*lf - Fxr*cf*er*gf*lr Fxf*cr*ef*gr*lr)/(cf*cr*gf*gr*(lf + lr)^2)
The solution should be compared with corresponding solution in Equation [3.24]. One can see that ax
is in complete alignment between the two models. Then, a general reflection is that the
displacement, z and py, in Equation [3.31] follows a complex formula, but that they are dependent
on how the 𝑤
is applied: both dependent on distribution between axles and
dependent on how much of the axle forces (
and
) that are actuated with shaft torques (𝑇𝑠
and 𝑇𝑠 , respectively). In Figure 3-35, dashed lines show the quasi-steady state solutions from
Equation [3.24].
3.4.8.4
Examples of real suspension design
In Figure 3-32, some kind of “trailing arm” is drawn both for front and rear axle. For rear axle, that is
a realistic design even if other designs are equally common. However, for front axle a so called
McPherson suspension is much more common, see Figure 3-33.
ef
90 deg
gf
ef
gf
Figure 3-33: Example of typical front axle suspension, and how pivot point is found. The example
shows a McPherson suspension. From Gunnar Olsson, LeanNova.
3.4.8.5
Additional phenomena
It is relevant to point out the following, which are not modelled in this compendium:


Stiffness and damping may be dependent of wheel (vertical) displacement and wheel
steering angle. One way of inserting this in the model is to make the coefficients varying with
spring force, which is a measure of how much compressed the suspension is. Here, nonlinearities within spring working range, as well as bump stops, can be modelled. Also,
position of pivot points (or pitch and roll centres) can be dependent of wheel displacement
steering angle.
Dampers are often deformation direction dependent, i.e. different damping coefficients are
suitable to use for compression and rebound. Typical is 2..4 times softer (smaller d [N/(m/s)])
in compression than in rebound.
119
LONGITUDINAL DYNAMIC
3.4.9
Dive at braking *
Function definition: Dive at braking is pitch angle change that the vehicle body experience when brake
pedal is applied to a certain deceleration level, from driving straight ahead. Either the peak or steady state
pitch angle can be addressed.
Now, study the suspension at front axle in Figure 3-32. When the axle is braked,
will be negative
and push the axle rearwards, i.e. in under the body. The front of the vehicle will then be lifted as in
pole jumping. This means that this design counter-acts the (transient) dive of the front. (Only the
transient dive will be reduced, while the dive after a longer time of kept braking is dependent only on
the stiffnesses according to Equation [3.24].) The design concept for front axle suspension to place
the pivot point behind axle and above ground is therefore called “anti-dive”.
If the braking is applied without shaft torque 𝑇𝑠 , a good measure of the Anti-dive mechanism is
𝑒 /𝑔 . This is the normal way for braking, since both the action and reaction torque acts on the axle.
For in-board brakes, or braking via propulsion shaft, the reaction torque is not taken within the axle,
but the reaction torque is taken by the vehicle body. The action torque 𝑇𝑠 then appears in the
equilibrium equation for the axle, as shown in Equation [3.28]. If we neglect the wheel rotational
dynamics for a while, we can insert 𝑇𝑠
∙ 𝑅𝑤 in the equation with 𝑇𝑠 in Equation [3.28]:
( 𝑠
⇒(
⇒(
𝑑
𝑠
𝑑
𝑠
𝑑
)∙𝑔
)∙𝑔
)∙𝑔
∙𝑒
𝑇𝑠
∙𝑒
∙ (𝑒
0; 𝑤𝑖𝑡 𝑇𝑠
∙ 𝑅𝑤 0; ⇒
𝑅𝑤 ) 0;
We can then see that a good measure of the Anti-dive mechanism is (𝑒
∙ 𝑅𝑤 ; ⇒
[3.32]
𝑅𝑤 )/𝑔 instead.
3.4.10 Squat at propulsion *
Function definition: Squat at propulsion is pitch angle change that the vehicle body experience when
accelerator pedal is applied to a certain acceleration level, from driving straight ahead. Either the peak or
steady state pitch angle can be addressed.
Now, study the suspension at rear axle in Figure 3-32. When the axle is propelled,
will push the
axle in under the body. This means that this design reduces the rear from squatting (transiently). The
design concept for rear axle suspension to place the pivot point ahead of axle and above ground is
therefore called “anti-squat”.
3.4.11 Anti-dive and Anti-squat designs
With Anti-dive front and Anti-squat rear, we avoid front lowering at braking and rear lowering at
acceleration, respectively. But how will the designs influence the parallel tendencies: that rear tend
to lift at braking and front then to lift at propulsion? Well, they will luckily counteract also these:
Braking at rear axle will stretch the rear axle rearwards and upwards relative to the body. When
propelling the front axle, the propulsion force will stretch the front axle forwards and upwards
relative to the body. (If one brakes at one axle and propels at the other, the reasoning is not valid.
This mode may seem irrelevant, but could be desired for a hybrid vehicle with ICE on front axle and
electric motor on rear axle, if one would like to charge batteries “via the road”. This is an example of
that novel designs may rise new issues.)
In summary: Anti-dive and anti-squat refer to the front diving when braking and the rear squatting
when acceleration. Anti-dive and anti-squat can be measured in fractions: Anti-dive for =𝑒 /𝑔 or
𝑅𝑤 )/𝑔 and Anti-squat=𝑒 /𝑔 or (𝑒
𝑅𝑤 )/𝑔 . Normal values are typically 0.05..0.15.
(𝑒
120
LONGITUDINAL DYNAMIC
The fractions, or percentages, can be seen as slope of lines from wheel contact to pivot points, as
shown in Figure 3-34.
Figure 3-34: Anti-Dive Geometry, (Gillespie, 1992)
3.4.12 Deceleration performance *
Function definition: See Section 3.4.2.2.
Deceleration performance can now be predicted, including the suspension mechanisms. It is a very
important function, and every decimetre counts when measuring braking distance in standard tests
like braking from 100 to 0 km/h. The active control of the brake torques (ABS function) is then very
important, and this is so fast dynamics that the suspension mechanisms of Anti-lift and Anti-dive
influences. The position of the load in the vehicle will influence, since it influences the load transfer.
We will now set up a mathematical model, see Equation [3.33], which shows how the normal forces
change during a braking event. It is based on the physical model in Figure 3-32. Driving resistance
contributes normally with a large part of the deceleration, but we will neglect this for simplicity, just
to show how the suspension mechanism works. The equations in the model are presented in the
dynamic modelling standardized format “Modelica”, and are hence more or less identical to Equation
[3.26] to [3.29].
Fxf = if
Fxr = if
Tsf/Rw =
Tsr/Rw =
1 < time and time < 3 then -0.4*m*g else 0;
3 < time and time < 7 then -0.4*m*g else 0;
0;
if 5 < time and time < 7 then -0.4*m*g else 0;
// Motion equations:
der(z) = vz;
der(py) = wy;
// Consitutive equations for the springs:
der(Fsf) = -cf*vzf;
der(Fsr) = -cr*vzr;
// Consitutive equations for the dampers:
Fdf = -df*vzf;
Fdr = -dr*vzr;
//(Dynamic) Equilibrium equations:
-m*der(vx) + Fxf + Fxr = 0;
-m*der(vz) - m*g + Fzf + Fzr = 0;
-Jy*der(wy) + Fzr*lr - Fzf*lf - (Fxf + Fxr)*h = 0;
(Fzr - Fsr - Fdr)*gr - Fxr*er + Tsr = 0;
(Fsf + Fdf - Fzf)*gf - Fxf*ef + Tsf = 0;
//Compatibility:
zf = z - lf*py;
121
[3.33]
LONGITUDINAL DYNAMIC
zr = z + lr*py;
vzf = vz - lf*wy;
vzr = vz + lr*wy;
The simulation results are shown in Figure 3-35. It shows a constant deceleration, but it is changed
how the decelerating force is generated. At time=3 s, there is a shift from braking solely on front axle
to solely on rear axle. The braking is, so far, only done with friction brakes, i.e. generating torque by
taking reaction torque in the axle itself. At time=5 s, there is a shift from braking with friction brakes
to braking with shaft torque. It should be noted that if we shift axle or shift way to take reaction
torque, gives transients even if the deceleration remains constant.
One can also see, at time=1 s, that the normal load under the braked axle first changes in a step. This
is the effect of the Anti-dive geometry. Similar happens when braking at rear axle, due to the Antisquat geometry. Since brake performance is much about controlling the pressure rapidly, the
transients are relevant and the plots should make it credible that it is a control challenge to reach a
high braking efficiency.
zr
Body changes vertical position over front and rear axle
zf
zr_SS = 0.
zf_SS = -0.0
Fzf
Same steady state load shift, but transiently different
Fzr
Braking on front axle
with friction brakes
Braking on rear axle
with friction brakes
Braking on rear axle
with shaft torque
Figure 3-35: Deceleration sequence with constant vehicle deceleration, but changing between
different ways of actuation. Dashed lines are solutions from Equation [3.24].
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LONGITUDINAL DYNAMIC
3.4.13 Acceleration performance *
Function definition: See Section 3.3.11.1.
The model presented in Equation [3.33] can also be used to predict acceleration performance in a
more accurate way compared to Section 3.3.11. Especially, the more accurate model is needed when
propelling or braking on the limit of tyre to road adhesion, since the normal load of each tyre then is
essential. It is a challenge to control the propulsion and brake wheel torques to utilize the varying
normal loads under each axle.
3.5 Control functions
Some control functions will be presented briefly. There are more, but the following are among the
most well-established ones. First, some general aspects on control are given.
3.5.1
Longitudinal Control
Some of the most important sensors available and used for longitudinal control are:
 Wheel Speed Sensors, WSS. For vehicle control design, one can often assume that “sensorclose” software also can supply information about longitudinal vehicle speed.
 Vehicle body inertial sensors. There is generally a yaw rate gyro and a lateral accelerometer
available, but sometimes also a longitudinal accelerometer. The longitudinal accelerometer is
useful for longitudinal control and longitudinal velocity estimation.
 Pedal sensors. Accelerator pedal normally has a position sensor and brake pedal force can be
sensed via brake system main pressure sensor. Heavy vehicles often have both a brake pedal
position and brake pressure sensors.
 High specification modern vehicles have environment sensors (camera, radar, GPS with
electronic map, etc.) that can give information (relative distance and speed, etc.) about objects
ahead of subject vehicle. It can be both fixed objects (road edges, curves, hills, …) and moving
objects (other road users, animals, …).
 Information about what actuation that is actually applied at each time instant is available, but
it should be underlined that the confidence in that information often is questionable.
Information about axle propulsion torque is generally present, but normally relies on imprecise
models of the whole combustion process and torque transmission, based on injected amount
of fuel and gear stick position. (Electric motors can typically give better confidence in
estimation, especially if motor is close to the wheel without too much transmission in
between.) Wheel individual friction brake torque is available, but normally rely on imprecise
models of the brake systems hydraulic/pneumatic valves and disc friction coefficient, based on
brake main cylinder pressure.
 Information about what actuation levels that are possible upon request (availability or
capability) is generally not so common. It is difficult to agree of general definitions of such
information, because different functions have so different needs, e.g. variations in accepted
time delay for actuation.
3.5.2
Longitudinal Control Functions
3.5.2.1
Pedal driving *
Function definition: See Section 3.4.4.
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LONGITUDINAL DYNAMIC
These functions, Pedal Response *, are often not seen as comparable with other control functions,
but they become more and more relevant to define as such, since both accelerator and brake pedals
tend to be electronically controlled, and hence they become increasingly tuneable. Also, more and
more functions, such as those below in section 3.5.2, will have to be arbitrated with the pedals.
In modern passenger vehicles, Accelerator pedal is normally electronically controlled but the Brake
pedal is basically mechanical. In modern heavy commercial vehicles, both functions are electronically
controlled.
The functions in Section “3.4.5 Pedal Feel *” are normally not actively controlled, but in there are
concept studies with active pedals, where also the pedal feel can be actively changed to give
feedback to driver.
3.5.2.2 Cruise Control and Adaptive Cruise Control (CC and
ACC) *
Function definition: Cruise Control, CC, controls the vehicle’s longitudinal speed. Driver can activate
the function and decide desired speed.
Function definition: Adaptive Cruise Control, ACC, is an addition to CC. ACC controls the
vehicle’s time gap to a lead vehicle. Driver can activate the function and decide desired gap. When there is no
lead vehicle, CC controls the vehicle’s speed.
The purpose of CC is to keep the vehicle at a driver selected longitudinal speed, while driver not
pushes the accelerator pedal. The actuator used is the propulsion system. In heavy vehicles also the
braking system (both retarders and service brakes) is used to maintain or regulate the vehicle speed.
ACC is an addition to CC. The purpose of ACC is to keep a safe distance to the lead vehicle (vehicle
ahead of subject vehicle). ACC uses also friction brake system as actuator, but normally limited to a
deceleration of 2..4 m/s2.
CC is normally only working down to 30..40 km/h. ACC can have same limitation, but with good
forward looking environment sensors, brake actuators and speed sensing, it can be allowed all the
way down to stand-still.
3.5.2.3
Anti-lock Braking System, ABS *
Function definition: Anti-lock Braking System, ABS, prohibits driver to lock the wheels while
braking. The wheel brake torques requested are limited by ABS in a way that each individual wheel’s
longitudinal slip stays above a certain (negative) value. An extended definition of ABS also includes vehicle
deceleration requested by other functions than pedal braking, such as AEB. ABS only uses friction brake
as actuator.
The purpose of ABS is to avoid losing vehicle brake force due to that the tyre force curve drops at
high slips AND to leave some friction for steering and cornering, see Sections 2.4, 0 and 2.6. ABS is a
wheel slip closed loop control, active when driver brakes via brake pedal. It keeps the slip above a
certain value, typically -15..-20 %. ABS uses the friction brakes as actuator.
Each wheel is controlled individually, but all wheels speed sensors contribute to calculation of vehicle
longitudinal speed (which is needed to calculate actual slip value). In the ABS function, it may be
included how slip are distributed between the wheels, such as normally the front axle is controlled to
a slip closer to locking than the rear axle. Also, a sub function called “select-low” which means that
the wheel closest to locking decides the pressure also for the other wheel at the same axle. Selectlow is typically used at rear axles.
124
LONGITUDINAL DYNAMIC
Figure 3-36: ABS control. Principle and control sequence
"DataLog_Braking.xls","ABS straight"
0.05
0
-0.05
-0.1
sxfl/[1]
sxfr/[1]
sxrl/[1]
sxrr/[1]
-0.15
-0.2
-0.25
1
1.5
2
2.5
3
3.5
"DataLog_Braking.xls","ABS straight"
4
4.5
4
4.5
5
140
120
100
80
60
pfl/[bar]
pfr/[bar]
prl/[bar]
prr/[bar]
vx/[km/h]
40
20
0
1
1.5
2
2.5
3
3.5
time [s] 5
Figure 3-37: ABS control, Data log from passenger car test.
3.5.2.4
Electronic Brake Distribution, EBD *
Function definition: Electronic Brake Distribution prohibits driver to over-brake the rear axle
while braking. An extended definition of EBD also includes vehicle deceleration requested by other
functions than pedal braking, such as AEB. EBD only uses friction brake as actuator.
With a fix proportioning between front and rear axle braking, there is a risk to over-brake rear axle
when friction is very high, since rear axle is unloaded so much then. Before electronic control was
available, it was solved by hydraulic valves, which limited the brake pressure to rear axle when pedal
force became too high. In today’s cars, where electronic brake control is present thanks to legislation
of ABS, the software base function EBD fulfils this need. In heavy vehicles, pneumatic valves are used
125
LONGITUDINAL DYNAMIC
that limits the brake pressure in relation to the rear axle load (deflection of mechanical spring
suspension or air pressure in air suspension).
There are other side functions enabled by having ABS on-board. Such are “select low”, which means
that the brake pressure to both wheels on an axel is limited by the one with lowest pressure allowed
from ABS. So, if one wheel comes into ABS control, the other gets the same pressure. This is most
relevant on rear axle (to reduce risk of losing side grip) but one tries to eliminate the need of it
totally, because it reduces the brake efficiency when braking in curve or on different road friction
left/right.
It is often difficult to define strict border between functions that is a part of ABS and which is part of
EBD, which is why sometimes one say ABS/EBD as a combined function.
3.5.2.5
Traction Control, TC *
Function definition: Traction Control prohibits driver to spin the driven axle(s) in positive direction
while accelerating. An extended definition of TC also includes vehicle acceleration requested by other
functions than pedal braking, such as CC. TC uses both friction brakes and propulsion system as
actuators.
The purpose of Traction Control is to maximise traction AND to leave some friction for lateral forces
for steering and cornering. Traction control is similar to ABS, but for keeping slip below a certain
value, typically +(15..20)%.
Traction control can use different ways to control slip, using different actuators. One way is to reduce
engine torque, which reduces slip on both wheels on an axle if driven via differential. Another way is
to apply friction brakes, which can be done on each wheel individually. Vehicles with propulsion on
several axles can also redistribute propulsion from one axle to other axles, when the first tends to
slip. Vehicles with transversal differential clutch or differential lock can redistribute between left and
right wheel on one axle.
3.5.2.6
Engine Drag Torque Control, EDC *
Function definition: Engine Drag Torque Control prohibits over-braking of the driven axle(s) while
engine-braking. EDC uses both friction brakes and propulsion system as actuators.
The purpose of Engine Drag Torque Control is as the purpose of ABS, but the targeted driving
situation is when engine braking at low road friction, when engine drag torque otherwise can force
the wheels to slip too much negative. Similarly to ABS, it keeps the slip above a certain negative
value. However, it does it by increasing the engine torque from negative (drag torque) to zero (or
above zero for a short period of time).
3.5.2.7
Automatic Emergency Brake, AEB *
Function definition: Automatic Emergency Brake decelerates vehicle without driver having to use
brake pedal when probability for forward collision is predicted as high.
The purpose of AEB is to eliminate or mitigate collisions where subject vehicle collides with a lead
vehicle. AEB uses friction brake system as actuator, up to full brake which would be typically 10 m/s2.
An AEB system is often limited by that it cannot be designed to trigger too early, because driver
would be disturbed or it could actually cause accidents. Therefore, in many situations, AEB will rather
mitigate than avoid collisions. (The A in the abbreviation AEB can be for either of “Automatic”,
“Autonomous” or “Advanced”.)
Conceptually, an AEB algorithm can be assumed to know physical quantities as marked in Figure
), which means
3-38. The quantity time-to-collision, TTC, can then be defined as 𝑇𝑇𝐶 𝑥 ⁄(
the time within a collision will appear if no velocities changes. TTC can also be seen as the “time gap”
to the lead vehicle.
126
LONGITUDINAL DYNAMIC
AEB function shall, continuously, decide whether or not to trigger AEB braking. AEB shall intervene by
braking when driver can be assumed to collide without intervention. If no other information, this can
be predicted as when driver can NOT avoid by normal driving. Avoidance manoeuvres that have to be
considered are (normal) deceleration and (normal) lateral avoidance to the left and to the right.
What to assume as normal driving is a question of tuning; here the following limits are used | | <
𝑒. 𝑔. 4 ⁄𝑠 2 and | | <
𝑒. 𝑔. 6 ⁄𝑠 2 . A physical model based algorithm can start from
the following simple models:

⇒ (𝑥
∙𝑡
⇒ 𝑥

2∙(− 𝑥𝑛 )
⇒
𝑡2
2
∙𝑡
(
∙ ))|
)2
∙(
𝑤
)|
2
𝑜=
𝑜
𝑣 −𝑣
𝑡= 𝑜𝑥 𝑥
𝑜
<0⇒
𝑪<
𝑥 − 𝑜𝑥
∙
𝑡2
𝑤
)|
2 𝑡=
2
{𝑒. 𝑔. }
𝒚𝒏
−𝒚𝒐 −𝒘/
𝒚𝒏
√
𝑥𝑜
𝑣𝑥 −𝑣𝑜𝑥
.6+ .8/2
3
6
√
𝒗𝒙 −𝒗𝒐𝒙
∙(− 𝒙𝒏 )
𝑜𝑥
8
;
<0 ⇒
≈ 0.4 𝑠;
⁄𝑠 2 ) leads to collision if:
− 𝑜𝑟 +𝑤/2
;
3
Assuming that the AEB intervention decelerates the vehicle with
a forward collision can be avoided if AEB intervenes AND if:
𝒗𝒙 −𝒗𝒐𝒙
𝑥 − 𝑜𝑥
⋯ ⇒ −𝑜
𝑪
;
)
∙(−
6
𝑥
𝑥 − 𝑜𝑥
⁄𝑠 2 ) leads to collision if:
6
𝒚𝒐 +𝒘/
𝑪<√ ∙
∙ 2 )) < 0 ⇒
<0⇒
𝑜
Normal avoidance to the right (
𝑥 − 𝑜𝑥
𝑡2
∙𝑡
𝑎𝑥𝑛
< 0 ⇒ (𝑦
𝑪<√ ∙
𝑥 − 𝑜𝑥
⋯⇒

∙𝑡
Normal avoidance to the left (
(𝑦 (𝑡)

(
⁄𝑠 2 ) leads to collision if:
4
Normal deceleration (
min
min
(𝑥 (𝑡)) < 0 ⇒
(𝑥
𝑡 0
𝑡 0
𝐴𝐸𝐵
8
⁄𝑠 2 ,
𝒙𝑨𝑬𝑩
Figure 3-38 shows a diagram where different condition areas are marked. The sectioned area shows
where AEB will be triggered, using above rules. The smaller of the sectioned areas shows where it
also will be possible to trigger AEB so timely that a collision is actually avoided; with the assumed
numbers, this is for speeds up to 6.4 m/s≈23 km/h.
The reasoning above is very simplified. Additional information can improve effectiveness of AEB,
such as knowing if a lateral avoidance on one side of object vehicle is blocked, whether the object
vehicle is decelerating and what the road friction is. Also, the reasoning is only valid for vehicles
driving in straight path with no lateral relative velocity. The vehicle dynamics model used is simply a
point mass with predicted constant velocity and certain assumed acceleration capability, which of
course can be extended a lot; both with taking actual accelerations into account and more advanced
vehicle dynamics models.
AEB is on market and legal requirement for both passenger vehicles and heavy vehicles.
Related functions are, e.g. extra force assistance in brake pedal when driver steps quickly onto brake.
Another related function is automatic braking triggered by a first impact and intended to mitigate or
avoid secondary accident events, starts to appear at market, see Reference (Yang, 2013). In semantic
meaning this could be seen as AEB, but they are normally not referred to as AEB; AEB normally refers
to functions that use environment sensors (forward directed radar, camera, etc).
When designing and evaluating AEB, it is important to also know about the function Forward
Collision Warning, FCW. FCW is a function that warns the driver via visual and/or audio signals when
a forward collision is predicted. FCW is typically triggered earlier than AEB.
127
LONGITUDINAL DYNAMIC
object vehicle
(or lead vehicle)
𝑥
𝑇𝑇𝐶 <
𝑇𝑇𝐶
∙
𝑇𝑇𝐶 <
𝑥
𝑇𝑇𝐶
0.4 𝑠
Collision if normal
lateral avoidance
𝑦
𝑦
subject vehicle
(or own vehicle)
3.
𝑤⁄ 𝑤⁄
/𝑠
6.4
/𝑠
AEB triggered AND avoids collision
AEB triggered AND mitigates (not avoids) collision
“Mild AEB” might be triggered, since too late for
normal lateral avoidance
“Mild Automatic Lane Change” might be triggered, since
too late for normal longitudinal deceleration
Figure 3-38: Left: Quantities known for an AEB algorithm in the subject vehicle, assuming forward
directed camera or radar. Right: Model based decision of triggering AEB and effectiveness of AEB
if triggered.
3.5.3 Longitudinal Motion Functionality shown
in a reference architecture
All control functions controls have to cooperate and they have to be transferable between platforms
and vehicle variants. It is very complex to take all functions into consideration, but with a scope
limited to the longitudinal Motion functionality Figure 3-39 can be drawn as a solution within the
reference architecture.
By using a reference architecture it can be illustrated that Adaptive Cruise control and cruise control
can be seen as part of Traffic Situation Layer (ACC=CC if no vehicle ahead). The Traffic Situation Layer
has the purpose and scope to understand the ego vehicle’s surrounding traffic by looking at e.g.
Forward Sensors. The forward looking sensor is in this case part of Vehicle Environment sensors.
Vehicle Motion and Coordination Layer would include the arbitration of Driver’s Acceleration and
Brake pedal input and Traffic functionality, see Figure 3-39. In addition, Vehicle Motion and
Coordination Layer would perform the powertrain coordination and brake distribution. The
coordinated requests are then sent to Motion Support Device Layer.
The Human Machine Interface would include the services available for the driver to activate or
request, E.g. ACC activation to Traffic Situation or Deceleration by pressing the brake pedal.
128
∙
∙
𝑦
𝐴
LONGITUDINAL DYNAMIC
Vehicle
Enivormnent
Traffic Situation Layer
Human
Machine
Interface
Forrward Direction
Dist
Dist. Ctrl (ACC)
AEB
Maps
…
APed
BPe
ACCbtn
Speed Ctrl (CC)
Dist btn
FwdSens
Arb(Min)
Vehicle Motion and Coordination Layer
Speed
APed
Arb(sum)
APedInterp
Arb (prio BPed,
else Max)
Energy Managament
Powertrain Coord
ABS &
EBD
BPedInterp
TC
ESC
BPed
Ener
RSC
Arb(min)
Arb
Arb(min)
Coord
Coord
Arb (switching)
Arb (switching)
Motion Support Device Layer
Vehicle
Motion
Sens
EngAct
Legend
GboxAct
FrntBrkAct
RearBrkAct
EngAct G
Border of Functional
Architecture
SW connection (signals,
here 2 scalar signals)
HW connection
Sensors & Actuators
End Customer Function
Arbitration
Coordination
System Safety
Figure 3-39: Functional architecture for conventional front axle driven passenger car. Mainly
longitudinal functions (plus ESC, RSC) are shown, e.g. no steering. Cf Figure 1-11.
If a reference architecture is used, it can assist function developers from OEM’s Electrical,
Powertrain, and Chassis departments and suppliers to have a common view of how vehicle’s
embedded motion functionality is intended to be partitioned and to understand how different
functions relate and interact with each other and what responsibilities they have.
129
LONGITUDINAL DYNAMIC
130
LATERAL DYNAMICS
4 LATERAL DYNAMICS
4.1 Introduction
The lateral motion of a vehicle is needed to follow the roads’ curves and select route in intersections
as well as to laterally avoid obstacles that appear. The vehicle needs to be steerable. With some
simplification, one can say that lateral dynamics is about how steerable the vehicle is for different
given longitudinal speeds. Vehicle steering is studied mainly through the vehicle degrees of freedom:
yaw rotation and lateral translation.
A vehicle can be steered in different ways:



Applying steering angles on, at least one, road wheel. Normally both of front wheels are
steered.
Applying longitudinal forces on road wheels. Either unsymmetrical between left and right
side of vehicle, e.g. one sided braking, or deliberately use up much friction longitudinally on
one axle in a curve, so that that axle loses lateral force.
Articulated steering, where the axles are fixed mounted on the vehicle but the vehicle itself
can bend.
The chapter is organised with one group of functions in each section as follows:





4.2 Low speed manoeuvrability
4.3 Steady state cornering at high speed
4.4 Stationary oscillating steering
4.5 Transient handling
4.6 Lateral Control Functions
Most of the functions in “4.6 Lateral Control Functions”, but not all, could be parts of ”4.5 Transient
handling”. However, they are collected in one own section, since they are special in that they partly
rely on software algorithms.
The lateral dynamics of vehicles is often experienced as the most challenging for the new automotive
engineer. Longitudinal dynamics is essentially motion in one plane and rectilinear. Vertical dynamics
may be 3 dimensional, but normally the displacements are small and in this compendium the vertical
dynamics is mainly studied in one plane as rectilinear. However, lateral dynamics involves analysis of
motion in the vehicle coordinate system which introduces curvilinear motion since the coordinate
system is rotating as the vehicle yaws.
The turning manoeuvres of vehicles encompass two concepts. Handling is the driver’s perception of
the vehicle’s response to the steering input. Cornering is usually used to describe the physical
response (open-loop) of the vehicle independent of how it influences the driver.
4.1.1


References for this chapter
“Chapter 25 Steering System” in Reference (Ploechl, 2013).
“Chapter 27 Basics of Longitudinal and Lateral Vehicle Dynamics” in Reference (Ploechl,
2013).
131
LATERAL DYNAMICS
4.2 Low speed manoeuvrability
This section is about operating vehicles in low speeds, including stand-still and reverse. Specific for
low speed is that inertial effects can be neglected, i.e. one can assume that left hand side in motion
equation ( 𝑠𝑠 ∙ 𝑡𝑒𝑟 𝐴𝑐𝑐𝑒 𝑒𝑟 𝑡𝑖𝑜𝑛 𝑠𝑢 𝑜𝑓 𝑜𝑟𝑐𝑒𝑠) is zero.
In low speed, one often needs to find the path with orientation and understand the steering system
and how tyres can be modelled to track ideally. This, and the resulting one-track model for low
speeds, is described in Sections 4.2.1, …,4.2.6.
4.2.1
Path with orientation
The path and path with orientation was introduced in Section 1.3. The path, in global coordinate
system, is related to vehicle speeds, in vehicle fix coordinates, as given in Figure 4-1 and Equation
[4.1].
y
wz
vy
vx
y=pz
x
Figure 4-1: Model for connecting “path with orientation”
to speeds in vehicle coordinate system.
𝑥
[ ]
𝑦
cos(𝜓)
[
sin(𝛿𝜓)
sin(𝜓)
]∙[
cos(𝛿𝜓)
];
[4.1]
𝜓
Knowing ( (𝑡) (𝑡) (𝑡)), we can determine “path with orientation” (𝑥(𝑡) 𝑦(𝑡) 𝜓(𝑡)), by time
integration of the right hand side of the equation. Hence, the positions are typically “state variables”
in lateral dynamics models.
It should be noted that in some problems, typically manoeuvring at low speed, the real time scale is
of less interest. Then, the problem can be treated as time independent, e.g. by introducing a
coordinate, s, along the path, as in Equation [4.2].
𝑥′
𝑦′
𝜓
𝑠
𝑠
∙ cos(𝜓)
∙ cos(𝜓)
𝑠
𝑠
∙ sin(𝜓) ;
∙ sin(𝜓) ;
[4.2]
′
;
𝑠
𝑤 𝑒𝑟𝑒 𝑝𝑟𝑖 𝑒 𝑛𝑜𝑡𝑒𝑠 𝑑𝑖𝑓𝑓𝑒𝑟𝑒𝑛𝑡 𝑡𝑖𝑜𝑛 𝑤𝑖𝑡 𝑟𝑒𝑠𝑝𝑒𝑐𝑡 𝑡𝑜 𝑠
Here, 𝑠 can be thought of like an arbitrary time scale, with which all speeds are scaled. One can
typically chose 𝑠 1 [ ⁄𝑠]. However, in this compendium we will keep notation t and the dot
notation for derivative.
132
LATERAL DYNAMICS
4.2.2
Vehicle and wheel orientations
For steered wheels, there are often reason to translate forces and velocities between vehicle
coordinate system and wheel coordinate system, see Figure 4-2 and Equation [4.3].
Components in vehicle
coordinates
Velocity of body,
above this wheel
𝒗𝒙𝒚
𝛿
𝒗𝒙𝒚
𝒗𝒙𝒗
Components in wheel
coordinates
𝒗𝒚𝒗
Force in road plane,
acting on this wheel
𝒙𝒚
𝒗𝒚𝒘
Components in vehicle
coordinates
𝛿
𝒗𝒙𝒘
𝒗𝒙𝒚
𝒙𝒚
Components in wheel
coordinates
𝒙𝒚
𝒙𝒗
𝒚𝒗
𝒙𝒘
𝒚𝒘
Figure 4-2: Transformation between forces and velocities in vehicle coordinate
system and wheel coordinate system.
Transformation from wheel coordinates to vehicle coordinates:
[
]
[
cos(𝛿)
sin(𝛿)
sin(𝛿)
]∙[
cos(𝛿)
𝑤
𝑤
];
𝑛𝑑 [
]
[
cos(𝛿)
sin(𝛿)
sin(𝛿)
]∙[
cos(𝛿)
𝑤
𝑤
];
[4.3]
Transformation from vehicle coordinates to wheel coordinates:
[
𝑤
𝑤
]
4.2.3
[
cos(𝛿) sin(𝛿)
]∙[
sin(𝛿) cos(𝛿)
];
𝑛𝑑 [
𝑤
𝑤
]
[
cos(𝛿)
sin(𝛿)
sin(𝛿)
]∙[
cos(𝛿)
];
Steering System
The steering system is here referred to the link between steering wheel and the road wheel’s
steering, on the steered axle. It is normally the front axle that is steered. Driver’s interaction is twofolded, both steering wheel angle and torque, which is introduced in Section 2.10. In present section,
we will focus on how wheel steering angles are distributed between the wheels.
4.2.3.1
Chassis steering geometry
The most basic intuitive relation between the wheels steering angles is probably that all wheels
rotation axes always intersect in one point. This is called Ackermann geometry and is shown Figure
4-3. The condition for having Ackermann geometry is, for the front axle steered vehicle that:
133
LATERAL DYNAMICS
1
tan(𝛿 )
1
tan(𝛿 )
𝑅
𝑅
di
𝑤⁄
𝑤⁄
;
⇒
;
}
do
1
tan(𝛿 )
1
tan(𝛿 )
𝑤
;
[4.4]
1 turn centre
a Ackermann error, front.
b Ackermann error, rear.
L
1 tan(d o )  1 tan(d i )  w L
Common intersection of all
wheels’ axes of rotation
Rr
w
Figure 4-3: Ackermann steering geometry. Left: One axle steered. Right: Both axles steered and
including “Ackermann errors”. From (ISO8855).
The alternative to Ackermann steering geometry is parallel steering geometry, which is simply that
𝛿
𝛿 . Note that Ackermann geometry is defined for a vehicle, while parallel steering is defined for
an axle. This means that, for a vehicle with 2 axles, each axle can be parallel steered, which means
that the vehicle is non-Ackermann steered. However, the vehicle can still be seen as Ackermann
steered with respect to mean steering angles at each axle.
For low-speed, Ackermann gives best manoeuvrability and lowest tyre wear. For high-speed, Parallel
is better in both aspects. This is because vehicles generally corner with a drift outwards in curves,
which means that the instantaneous centre is further away than Ackermann geometry assumes, i.e.
more towards optimal for parallel. Hence the chosen geometry is normally somewhere between
Ackermann and parallell.
Practical arrangement to design the steering geometry is shown in Figure 4-7. The design of linkage
will also make the transmission from steering wheel angle to road wheel steering angle non-linear.
This can lead to different degrees of Ackerman steering for small and large steering wheel angles.
In traditional steering systems, the steering wheel angle has a monotonically increasing function of
the steering angle of the two front axle road wheels. This relation is approximately linear with a
typical ratio of 15..17 for passenger cars. For trucks the steering ratio is typically 18..22. Differences
between left and right wheels are discussed in 4. In some advanced solutions, steering on other axles
is also influenced (multiple-axle steering, often rear axle steering). There are also advanced solutions
for adding an additional steering angle to the basic steering ratio gives, so called AFS=Active Front
Steering.
4.2.3.2
Steering system forces
(This section has large connection with Section 2.5.5.2 Tyre self-aligning moment.)
The steering wheel torque, 𝑇𝑠𝑤 , should basically be a function of the tyre/road forces, mainly the
wheel-lateral forces. This gives the driver a haptic feedback of what state the vehicle is in. The
134
LATERAL DYNAMICS
torque/force transmission involves a servo actuator, which helps the driver to turn the steering
system, typically that assists the steering wheel torque with a factor varying between 1 and 10, but
less for small 𝑇𝑠𝑤 (highway driving) than large 𝑇𝑠𝑤 (parking), see Figure 4-4. Here, the variation in
assistance is assumed to be hydraulic and follows a so called boost curve. At 𝑇𝑠𝑤 0, the assistance
is ≈0.45/0.55≈1 and for 𝑇𝑠𝑤 4 Nm, it is ≈0.9/0.1≈10.
Figure 4-4: Left: Boost Curve with different working areas depending on the driving envelope.
Right: Torque distribution between manual torque, FM, and assisting torque, FA, depending on
applied steering wheel torque. From Reference (Rösth, 2007).
For vehicle dynamics, one important effect of a steered axle, is that the lateral force on the axle tries
to align the steering in the direction that the body (over the steered axle) moves, i.e. towards a zero
tyre side slip. This is designed in via the sign of the caster trail, see Figure 4-5. Also, asymmetry in
longitudinal tyre forces (wheel shaft torques and/or brake torques) affects the steering wheel
torque. This is analysed in the following.
On a steered axle, there is a Steering axis (or Kingpin axis) around which the wheel is rotated when it
is steered. The steering axis intersects with the ground plane at a point which normally has an offset
from the contact patch centre, both longitudinally and laterally. The offsets are called Castor offset at
ground and Steering axis offset at ground, respectively, see Figure 4-5. (Steering axis offset at ground
is sometimes called Scrub radius, but that is also used for the resulting distance of both offsets, so it
is an ambiguous name.) This figure also defines Normal steering axis offset at wheel centre . We will
use the 3 latter measures to explain why steering is affected by differences in shaft torque left/right,
differences in brake torque left/right, and lateral wheel forces. The steering axis distance from wheel
centre in side view, is called Castor offset at wheel centre .
Often, the actual forces between tyre and ground are not in the exact centre of the nominal contact
patch, which also creates “effective” version of caster offset and scrub radius. For instance, the
Pneumatic trail adds to the Castor offset at ground due to the lateral force distribution being
longitudinally offset from nominal contact point; so that the lever becomes not only s, but s+t. The
moment, Tsteer, on steering system (around Steering axis, turning towards increasing steering angle) is
affected by tyre forces on a steered axle as in Equation [4.5].
Castor offset at ground is, on passenger vehicles, 15-20 mm (at motorcycles approximately 100 mm).
On rear wheel driven passenger vehicles it can typically be 5 mm, due to higher Castor Angle which
gives a beneficial higher Camber angle gain at cornering. On a front wheel driven passenger vehicle a
non-zero caster offset is not chosen due to drive axle lateral displacement.
(Notations in Equation [4.5] are defined in Figure 4-5.)
135
LATERAL DYNAMICS
𝑇𝑠𝑡
(𝑇𝑏2
(𝑇𝑏
𝑇𝑠2 )
𝑅
( 1) ⋅ (
𝑇𝑏2 𝑇𝑏
𝑅
𝑇𝑠 )
𝑅
∙ 𝑠 ⋅ cos(𝐾𝑃𝐼)
𝑇𝑠
⋅ 𝑅 ⋅ sin(𝐾𝑃𝐼)
𝑅
𝑇𝑠2
⋅ 𝑅 ⋅ sin(𝐾𝑃𝐼)
𝑅
⋅ (𝑐 𝑡 )
𝑡2 )) ⋅ cos(𝐶𝐴)
2 ⋅ (𝑐
𝑇𝑠2 𝑇𝑠
⋅ 𝑠 ⋅ cos(𝐾𝑃𝐼)
⋅𝑘 (
⋅ (𝑐
𝑅
⋅ 𝑠 ⋅ cos(𝐾𝑃𝐼)
[4.5]
𝑡 )
2
⋅ (𝑐
𝑡2 )) ⋅ cos(𝐶𝐴) ;
The equation shows that difference in both brake torque and shaft torque affects steering. So does
the sum of lateral forces. For reducing torque steer and disturbances from one-sided longitudinal
forces due to road irregularities, kingpin offset, scrub radius different road friction should be as small
as possible, but it is limited by geometrical conflicts between brake disc, bearing, damper, etc.
Positive scrub radius contributes to self-centring, thanks to lifting the car body, see below. Negative
scrub radius compensates for split-𝜇 braking, or failure in one of the brake circuits. Hence, the scrub
is a balance between these two objectives. Scrub radius is often slightly negative on modern
passenger cars. Scrub radius is often positive on trucks, maybe 10 cm, due to packaging.
The geometry in Figure 4-6 shows one part of the lifting effect. This is that if steering angle is
changed from zero, it lifts the vehicle slightly which requires a steering torque. One can see this as a
kind of “return spring effect”. Figure 4-6 shows how KPI and scrub radius causes the vehicle body to
lift a distance 𝑠 ′′′ 𝑠 ∙ cos(𝐾𝑃𝐼) ∙ (1 cos(| 𝑠𝑡 |)) ∙ sin(𝐾𝑃𝐼). This will require a work 𝑇𝑠𝑡 ∙
∙ 𝑠 ′′′ . This leads to an 𝑇𝑠𝑡 (additional to Eq [4.5]) as follows:
𝑠𝑡
( 𝑑𝑑𝑖𝑡𝑖𝑜𝑛 ) 𝑇𝑠𝑡
=
𝑠 ∙ cos(𝐾𝑃𝐼) ∙ (1
∙ 𝑠 ′′′
∑
cos(|
𝑠𝑡
𝑡
𝑑
𝑠𝑡
ℎ𝑡
𝑠𝑡
|)) ∙ sin(𝐾𝑃𝐼)
[4.6]
∙
;
It should be noted that Eq [4.6] is not complete with respect to all “returning effects”. There are also
effects from Castor angle and Caster trail as well as that the tyre has a width and radius. However, in
total, these give rise to a returning steering torque which is depending on the steering angle.
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LATERAL DYNAMICS
Steering-axis inclination angle,
KPI
View from
rear:
View from
right:
Caster
angle, CA
Wheel 1 with
(friction)
brake
torque, Tb:
brake
Tb
(Tb<0 if w>0)
w
bearing
forward
Fx1=Tb/R
Castor offset at
wheel center
(drawn positive)
Wheel 1 with
shaft torque, Ts:
Fx1
Castor offset at ground, c,
(drawn positive)
Steering axis offset
at ground, s
(drawn positive)
(sometimes called Scrub radius)
R
Ts
Fy1
Fy1 = resulting
lateral force
Pneumatic trail, t
(positive if “behind”)
Ts*sin(KPI)
Fx1=Ts/R
lateral force
distribution
𝑇𝑠𝑡
View from
rear:
𝑠𝑡
Contact point
tyre/ground
for
𝒕
Contact point
for
𝒕
s
𝑠 ′′′
KPI
𝑠 ′′ sin 𝐾𝑃𝐼
Figure 4-5: Steering geometry and offsets.
Fx1
Steering axis offset
at ground, s
Figure 4-6: Lift effect due to steering angle and positive Steering axis offset at ground.
4.2.4
One-track models
When conducting analyses of the vehicle cornering response, it is useful to combine the effects of all
tyres on the axle into one virtual tyre. This assumption, referred to as the one-track model (or singletrack model or bicycle model), facilitates understanding but can also capture most important
phenomena. A one-track model of a two-axle vehicle is shown in Figure 4-8. One-track model for
truck with trailer is exemplified in Figure 4-9.
137
LATERAL DYNAMICS
Ackermann
steering
(trapezoidal
geometry)
𝛿𝑠𝑤
(Erasmus Darwin 1758,
Rudolph Ackermann
1810.)
𝛿
ℎ𝑡
𝛿
𝛿
𝛿
𝑓𝑢𝑛𝑐 𝑝𝑜𝑠𝑖𝑡𝑖𝑜𝑛
𝑡
𝑓𝑢𝑛𝑐 𝑟 𝑐 ∙ 𝛿𝑠𝑤 ;
𝑡
𝑐
𝑓𝑢𝑛𝑐 𝑝𝑜𝑠𝑖𝑡𝑖𝑜𝑛
𝑓𝑢𝑛𝑐 𝑟 𝑐 ∙ 𝛿𝑠𝑤 ;
ℎ𝑡
𝑐
Figure 4-7: Example of Trapezoidal Steering. Left: Conceptual use of steering arms. Right: More
exact design, common today. From Gunnar Olsson, LeanNova.
df do
Common
intersection
of all wheels’
axes of
rotation
lr
L
lf
di
Rr
Figure 4-8: Collapsing to one-track model.
138
LATERAL DYNAMICS
Figure 4-9: Upper: Two-track model of truck with trailer. Lower: One-track model of a truck with a
trailer (not exactly same vehicle combination as above). In SAE coordinate system. From (Kharrazi ,
2012).
Phenomena which one-track models not capture are, e.g.:



Large deviations from Ackerman geometry within an axle.
Varying axle cornering stiffness due to lateral load shift and axle propulsion/braking, see
Section 4.3.
Additional yaw moment due to non-symmetrical wheel torque interventions, such as ESC
interventions.
These effects can, however, be approximately captured with a one-track vehicle model, extended
with sub-models for the axles. The axle sub-models, can propagate the varying cornering stiffnesses
and yaw moments to the superior one-track vehicle model.
4.2.5
Ideally tracking wheels and axles
In Section 0, lateral slip models for tyres were introduced. It is the constitutive equation
𝐶 ∙
𝑠 . If the tyre force, , is small compared to cornering stiffness, 𝐶 , one would expect side slip, 𝑠 , to
be very small. If this assumption is taken to its extreme, one get something we can call “ideal
tracking” for a tyre or for an axle, which is that side slip is zero and side force can be any (finite)
value. For low speed this is often a good enough model of the tyre. The validity of this assumption is
limited by when lateral force becomes large with respect to available lateral force, 𝜇 ∙ . Hence, the
speed border between low speed and high speed is lowered on low road friction.
Models using ideally tracking wheels are sometimes referred to as “kinematic models”. However, a
strictly kinematic model does not contain the forces at all, why validity cannot be checked.
The assumption about ideally tracking wheel or axle is a constitutive assumption, although that the
equation (tyre lateral speed=tyre side slip=0) does not couple force and speed, but only stipulates
speed. If forces should be calculated, it has to be done using the equilibrium.
139
LATERAL DYNAMICS
4.2.6 One-track model for low speeds, with
Ackerman geometry
Low speed manoeuvres are characterised by that the inertial forces are neglected, i.e. ∙
0. If
the geometry is according to Ackermann, it is reasonable to assume ideal tracking axles. This means
that the intersection point of the wheels rotational axes coincides with the instantaneous centre of
vehicle rotation in ground plane. We can there through connect steering angles and path radius. For
the model in Figure 4-8 this connection becomes:
tan(𝛿 )
𝑅
2
𝑅
𝑅
2
2
𝑤 𝑒𝑟𝑒 𝑅
;
0
}⇒𝛿
;
arctan
∙ sign(𝑅) ≈
2
√𝑅 2
∙ sign(𝑅) ≈
2
√𝑅 2
𝑅
;
[4.7]
(
)
𝑡 𝑖𝑛𝑠𝑡 𝑛𝑡 𝑛𝑒𝑜𝑢𝑠 𝑐𝑒𝑛𝑡𝑟𝑒 𝑜𝑓 𝑟𝑜𝑡 𝑡𝑖𝑜𝑛 𝑖𝑠 𝑒𝑓𝑡 𝑜𝑓 𝑡 𝑒 𝑒 𝑖𝑐 𝑒
𝑒 𝑛𝑠 𝑡
To get a more complete model, where more variables can be extracted, we can set up the model in
Figure 4-10.
wz
vx
vrx
lr
vfyv
vry=0
vy
forces:
vfxv ⃗
Fry
Frx
df
lf
lr
L
Ffxv ⃗
Ffyv
speeds:
lf
L
⃗
df
⃗
Figure 4-10: One-track model with ideally tracking axles. Lower view of front wheel shows
conversion between wheel and vehicle coordinate systems.
The “physical model” in Figure 4-10 gives the following “mathematical model”:
Equilibrium (longitudinal, lateral and yaw-rotational):
0
0
0
;
;
∙
∙ ;
Transformation between vehicle and wheel coordinate systems:
∙ cos(𝛿 )
𝑤 ∙ sin(𝛿 )
𝑤 ∙ cos(𝛿 )
𝑤 ∙ sin(𝛿 )
∙ sin(𝛿 ) ;
𝑤 ∙ cos(𝛿 ) ;
𝑤 ∙ sin(𝛿 ) ;
𝑤 ∙ cos(𝛿 ) ;
𝑤
𝑤
140
[4.8]
LATERAL DYNAMICS
Compatibility:
;
∙
;
𝑟∙
;
;
Ideal tracking (Constitutive relation, but without connection to forces):
0;
0;
𝑤
Path with orientation:
𝑥
𝑦
∙ cos(
∙ cos(
;
)
)
∙ sin(
∙ sin(
);
);
Prescription of steering angle operation:
𝛿
{
(35 ∙ 𝜋⁄180) ∙ sin(0.5 ∙ ∙ 𝜋 ∙ 𝑡) ; 𝑖𝑓 𝑡 < 4.5;
35 ∙ 𝜋/180;
𝑒 𝑠𝑒
Rear axle undriven, which gives drag from roll resistance:
100;
The “Compatibility” in Eq [4.8] neglects the influence of steering axis offsets at ground, see Section
4.2.3.2. The terms neglected are of the type 𝑡𝑒𝑟 𝑂𝑓𝑓𝑠𝑒𝑡 ∙ 𝛿 ; in the equation for
and
𝑜𝑛𝑔𝑖𝑡𝑢𝑑𝑖𝑛 𝑂𝑓𝑓𝑠𝑒𝑡 ∙ 𝛿 in the equation for
. This is generally well motivated for normal road
vehicles, except for quick steering when vehicle is stand-still.
Equation [4.8] is written in Modelica format in Equation [4.9]. Comments are marked with //. The
subscript v and w refers to vehicle coordinate system and wheel coordinate system, respectively. The
actual assumption about ideal tracking lies in that
0. Global coordinates from Figure
𝑤
4-1 is also used.
//Equilibrium:
0 = Ffxv + Frx;
0 = Ffyv + Fry;
0 = Ffyv*lf - Fry*lr;
//Ideal tracking (Constitutive relation, but without connection to
forces):
vfyw = 0;
vry = 0;
//Compatibility:
vfxv = vx;
vfyv = vy + lf*wz;
vrx = vx;
vry = vy - lr*wz;
[4.9]
//Transformation between vehicle and wheel coordinate systems:
Ffxv = Ffxw*cos(df) - Ffyw*sin(df);
Ffyv = Ffxw*sin(df) + Ffyw*cos(df);
vfxv = vfxw*cos(df) - vfyw*sin(df);
vfyv = vfxw*sin(df) + vfyw*cos(df);
//Path with orientation:
der(x) = vx*cos(pz) - vy*sin(pz);
der(y) = vy*cos(pz) + vx*sin(pz);
der(pz) = wz;
141
LATERAL DYNAMICS
// Prescription of steering angle:
df = if time < 4.5 then (35*pi/180)*sin(0.5*2*pi*time) else 35*pi/180;
//Rear axle undriven, which gives drag from roll resistance:
Frx = -100;
The longitudinal speed is a parameter, vx=10 km/h. A simulation result from the model is shown in
Figure 4-11. It shows the assumed steering angle function of time, which is an input. It also shows the
resulting path, y(x). The variables x,y,pz are the “state variables” of this simulation.
df [rad]
y [m]
35 deg
x[m]
time [s]
Figure 4-11: Simulation results of one-track model with ideal tracking.
The variables 𝑥 𝑦 𝑝 are the only “state variables” of this simulation. If not including the path model
(Equation [4.1]), the model would actually not be a differential equation problem at all, just an
algebraic system of equations. That system of equations could be solved isolated for any value of
steering angle without knowledge of time history. These aspects are the same for the steady state
model in section 4.3.2.
A driving resistance of 100 N is assumed on the rear axle (
100;). This is to exemplify that
forces does not need to be zero, even if forces normally are not so interesting for low speed
manoeuvres. Anyway, one should note that the speeds and forces are not coupled, since there are
no inertial forces modelled. The modelling of forces is really a preparation for the case where we
don’t have Ackerman geometry. For a two-axle one-track model we always have Ackermann
geometry, because there is always an intersection point between the front and rear wheels
rotational axes.
4.2.7
Turning circle *
Function definition: Turning diameter is the diameter of the smallest possible circular path obtained
steady state at low speed, measured to a certain point at the vehicle. The certain point can be either outermost point on wheel (Kerb Turning diameter) or outer-most point on body (Wall Turning diameter).
The turning circle (or turning diameter) is the diameter of the smallest circular turn that the vehicle is
capable of making at low speed. The value should be as low as possible to enable best
manoeuvrability in e.g. U-turns.
142
LATERAL DYNAMICS
wall turning diameter
y [m]
curb turning diameter
The end of the simulation in Figure 4-11 is made with constant steering angle. If we assume that it is
the maximum steering angle, the circle actually shows the turning circle (diameter) for centre of
gravity. If we add the path for the outermost wheels, we get the kerb or kerb-to-kerb turning circle
diameter, see Figure 4-12. If we add the path for the outermost point at the vehicle body we get the
wall or wall-to-wall turning circle diameter, also shown in Figure 4-12. The outermost point at the
vehicle body is normally the front outer corners of the vehicle body, i.e. the outer front bumper end.
• CoG path diameter=8.54 m
• front right wheel path diameter=11.19 m
• real left wheel path diameter=6.29 m
x[m]
Figure 4-12: Adding paths for wheels and body points, on top of result in Figure 4-11.
Turning circle can also be defined for high speed, but it is more conventional to talk about curvature
gain, which is not curvature at maximum steering angle, but the curvature per steering angle, see
Section 4.2.12.
4.2.8
Swept path width and Swept Area *
Function definition: Swept path width is the distance between the outermost and innermost trajectories
of wheels (or body edges). The trajectories are then from a certain turning or lane change manoeuvre at a
certain speed.
Function definition: Swept path area is the area between the outermost and innermost trajectories in the
definition of Swept path width.
For manoeuvrability, there is a function which is complementary to turning radius diameter. It is
“Swept path width” (SPW), see Figure 4-12. It can occur as a kerb and wall version as well as turning
circle radius can. It is the distance between the outermost and innermost wheel/point on the vehicle.
The SPW should be as small as possible for improving manoeuvrability.
Another way of measuring approximately the same is “Swept Area”, which is the area between the
outer and inner circle in Figure 4-12. Again, one can differ between kerb and wall version of this
measure.
“Swept path width” and “Swept Area” is often used for low vehicle speeds. For higher speeds it is
more common to talk about off-tracking, see next section.
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LATERAL DYNAMICS
4.2.9
Off-tracking *
Function definition: Off-tracking is the distance between the outermost and innermost trajectories of
centre points of the axles. The trajectories are then from a certain turning or lane change manoeuvre at a
certain speed.
Another measure of manoeuvrability is “Off-tracking”, see Figure 4-13. It is like swept area width, but
for the centre point of the axles. It is also used for higher speeds, and then the rear axle can track on
a larger circle than front axle.
Off-tracking is most relevant for vehicles with several units, such as truck with trailer. Off-tracking for
driving several rounds in a circle is well defined since it is a steady state, but seldom the most
relevant. Often, it is a more relevant to set requirements on off-tracking when driving from straight,
via curve with certain radius, to a new straight in a certain angle from the first straight, e.g. 90
degrees. A way to predict off-tracking for this is to simulate with time integration, even if no
traditional dynamics (mass inertia times acceleration) is modelled. The states in such simulation are
the path coordinates with orientation (𝑥 𝑦 ).
L
df
Figure 4-13: Adding one-track model, on top of result in Figure 4-11.
From geometry in Figure 4-13 on can find an expression for off-tracking at low speed:
∆
𝑅
≈𝑅∙(
𝑅
1
cos(𝛿 )
𝑅
cos(𝛿 )
𝑅
1) ≈ 𝑅 ∙ (1
𝑅 ∙(
1
cos(𝛿 )
1)
√𝑅 2
2
∙(
1
cos(𝛿 )
1) ≈
[4.10]
cos(𝛿 ))
4.2.10 Steering effort *
Function definition: Steering effort at low speed is the steering wheel torque needed to turn the
steering wheel a certain angle at a certain angular speed at vehicle stand-still on high road friction.
144
LATERAL DYNAMICS
Function definition: Steering effort at high speed is the steering wheel torque (or subjectively assessed
effort) needed to perform a certain avoidance manoeuvre at high road friction.
At low or zero vehicle speed, it is often difficult to make steering wheel torque low enough. This has
mainly two reasons:
See the Caster offset in Figure 4-5. It gives the wheel a side slip when steering and hence a tyre
lateral force is developed. Tyre lateral forces times the caster offset is one part that steering efforts
has to overcome.
Additionally, there is a spin moment in the contact patch, 𝑍𝑇 in Figure 2-2. ( 𝑍𝑇 does not influence
very much, except for at very low vehicle speed, which is why quantitative models for 𝑍𝑇 are not
presented in this compendium.) The spin moment also has to be overcome by steering efforts.
A critical test for steering effort at low speed is to steer a parked vehicle with a certain high steering
wheel rotational speed, typically some hundred deg/s. The steering wheel torque is then required to
stay under a certain design target value, normally a couple of Nm. The torque needed will be
dependent on lateral force, spin moment and steering geometry (which is not steering speed
dependent) and dependent on the capability of the power steering system (which is dependent on
steering, due to delays in the steering assistance actuator). A failure in this test is called “catch-up”,
referring to that driver catches up with the power steering system. It can be felt as a soft stop and
measured as a step in steering wheel torque.
At higher vehicle speeds, the steering effort is normally less of a problem since unless really high
steering wheel rate. Hence, steering wheel torque in avoidance manoeuvres in e.g. 70 km/h can be a
relevant requirement. In these situations, the subjective assessment of steering effort can also be the
measure. Then, steering effort is probably assessed based on both steering wheel rate and steering
wheel torque.
4.2.11 One-track model for low speeds, with
non-Ackerman geometry
The model in Figure 4-10 can rather easily be extended to more axles, more units and two-track
model, if Ackermann geometry. However, if not Ackermann geometry, one has to do a structural
change in the model, because ideal tracking is no longer consistent with geometry. One example is a
two-track model of a two-axle vehicle which has parallel steering on one axle. Another example is a
one-track model of a truck with three axles, whereof the two last are non-steered, see Figure 4-14.
Figure 4-14: Rigid Truck with 3 axles, whereof only the first is steered.
We will go through model changes needed in the latter example. In order to compare the models as
closely as possible, we simply split the rear axle into two rear axles, in the example in Section 4.2.6.
The physical model becomes as in Figure 4-15. The measures appear in Figure 4-17, and you see that
it is not a truck, but a very unconventional passenger size car with two rear axles.
145
LATERAL DYNAMICS
vx
vr2x
vr1x
lr
df
Fr1y
Fr1x
Ffxv ⃗
df
⃗
lf
L
b
forces:
Fr2y
Fr2x
vfxv ⃗
Ffyv
vr1y wz
vfyv
vy
speeds:
vr2y
⃗
b
Figure 4-15: One-track model. Not Ackermann geometry, due to un-steered rear axles.
The changes we have to do in the model appear as underlined in Equation [4.11]. There has to be
double variables for vrx, vry, Frx, Fry, denoted 1 and 2 respectively. Also, we cannot use vfyw=vry=0
anymore, but instead we have to introduce a lateral tyre force model, as described in Section 0.
//Equilibrium:
0 = Ffxv + Fr1x + Fr2x;
0 = Ffyv + Fr1y + Fr2y;
0 = Ffyv*lf - Fr1y*(lr - b) - Fr2y*(lr + b);
//Constitutive relation, i.e.
Lateral tyre force model (instead of Ideal tracking):
Ffyw = -Cf*sfy;
Fr1y = -Cr1*sr1y;
Fr2y = -Cr2*sr2y;
sfy = vfyw/vfxw;
sr1y = vr1y/vr1x;
sr2y = vr2y/vr2x;
//Compatibility:
vfxv = vx;
vfyv = vy + lf*wz;
vr1x = vx;
vr2x = vx;
vr1y = vy - (lr - b)*wz;
vr2y = vy - (lr + b)*wz;
[4.11]
//Transformation between vehicle and wheel coordinate systems:
Ffxv = Ffxw*cos(df) - Ffyw*sin(df);
Ffyv = Ffxw*sin(df) + Ffyw*cos(df);
vfxv = vfxw*cos(df) - vfyw*sin(df);
vfyv = vfxw*sin(df) + vfyw*cos(df);
//Path with orientation:
der(x) = vx*cos(pz) - vy*sin(pz);
der(y) = vy*cos(pz) + vx*sin(pz);
der(pz) = wz;
// Prescription of steering angle:
df = if time < 4.5 then (35*pi/180)*sin(0.5*2*pi*time) else 35*pi/180;
//Rear axles undriven, which gives drag from roll resistance:
Fr1x = -100/2;
Fr2x = -100/2;
The new result is shown in Figure 4-16, which should be compared to Figure 4-11. The radius of the
final path radius increases a little. If we read out more carefully, we can draw the different locations
of the instantaneous centre for both cases. This is shown, in scale, in Figure 4-17.
146
LATERAL DYNAMICS
y [m]
df [deg]
35 deg
vx=2.778m/s
vy=1.036m/s
wz=0.652rad/s
x[m]
time [s]
Figure 4-16: Simulation results of one-track model with ideal tracking. Non-Ackermann geometry
due to two non-steered rear axles.
We could tune the steering angle required to reach exactly the same path radius as for the 2-axle
reference vehicle. Then, we would have to steer a little more than the 35 degrees used, and we could
find a new instantaneous centre, and we could identify a so called Equivalent wheelbase. This leads
us to a definition: The equivalent wheel base of a multi-axle vehicle is the wheel base of a
conventional two-axle vehicle which would exhibit the same turning behaviour as exhibited by the
multi-axle vehicle, given same steering angle and similar axle cornering stiffnesses.
• instantaneous centre for
3-axle vehicle
• instantaneous centre for
2-axle reference vehicle
b=1 b=1
df
lr=1.5 lf=1.3
Figure 4-17: Instantaneous centre with a 3-axle vehicle, with the corresponding 2-axle vehicle as
reference.
147
Le=equivalent wheelbas
Lg=geometric wheelbas
LATERAL DYNAMICS
Section 4.2.11 shows that the lateral tyre force model, which is a constitutional relation, can
sometimes be needed also at low speed analysis. However, it is always needed in next section about
high speed cornering, Section 4.2.12.
4.2.12 One-track model for low speeds, with
trailer
When the vehicles has an articulation point, also the low speed case (disregarding inertias) has
transients in the sense that the articulation angles changes transiently. Consider the case of
instantaneous step steer. For a vehicle without articulation point, a steady state is reached directly,
since inertia is not considered. But for an articulated vehicle it takes some travelled distance (not
time, since it can be studied independent of speed and time) before a steady state articulation angle
is achieved.
So, for articulated vehicles, a scalar requirement on turning radius is not so relevant as for two axle
vehicles. But, the function “4.2.8 Swept path width and Swept Area *” and “4.2.9 Off-tracking *” is, if
road geometry is reflected, e.g. through outer radius and total angle of turning.
4.3 Steady state cornering at high speed
Steady state cornering refers to that all time derivatives of vehicle speeds (
) are zero. The
physical understanding is then that the vehicle drives on a circle with constant yaw rate, see Figure
4-18Figure 4-1. Alternatively, this can be described as driving with constant tangential speed ( ), on a
constant path radius (𝑅) with and with a constant side slip angle (𝛽).
𝜷
v
Steady state cornering can typically be
regarded as defined by either 3 or 2
quantities, depending on assumptions:
• A general vehicle operating state is fully
defined only by 3 quantities, e.g.
or
𝛽 𝑅.
•
Assuming a certain vehicle and
constraints on how to reach this state
(e.g. brake pedal 0, use only steering
wheel angle and accelerator pedal) there
is only 2 quantities is needed.
Figure 4-18: Steady state cornering.
4.3.1
Steady state driving manoeuvres
When testing steady state function, one usually runs on a so called “skid-pad” which appears on
most test tracks, see Figure 4-8. It is a flat circular surface with typically 100 m diameter and some
concentric circles marked. A general note is that tests in real vehicles are often needed to be
performed in simulation also, and normally earlier in the product development process.
Typical steady state tests are:


Constant path radius. Driven for different longitudinal speeds.
Constant longitudinal speed. Driven for different path radii.
148
LATERAL DYNAMICS

Constant steering wheel angle. Then increase accelerator pedal (or apply brake pedal) gently.
(If too quick, the test would fall under transient handling instead.)
Vehicle
Dynamics
Area
Handling
track
Skid pad
High speed
track
Figure 4-19: An example of test track and some parts with special relevance to Vehicle Dynamics.
The example is Hällered Proving Ground, Volvo Car Corporation.
=Vehicle Dynamics Area
Figure 4-20: An example of test track. The example is AstaZero (Active Safety Test Arena), SP
Technical Research Institute of Sweden and Chalmers University of Technology.
Standards which are relevant to these test manoeuvres are, e.g. References (ISO 4138) and (ISO
14792).
149
LATERAL DYNAMICS
4.3.2
Steady state one-track model
In steady state we have neither inertial effects from changing the total vehicle speed (
√
2
2
is constant) nor from changing the yaw rate (𝑤 is constant). However, the inertial
“centrifugal” effect of the vehicle must be modelled. The related acceleration is the centripetal
2
acceleration, 𝑐 𝑅 ∙ 2
/𝑅
∙ .
A vehicle model for this is sketched in Figure 4-21. The model is a development of the model for lowspeed in Figure 4-10 and Equation [4.9], with the following changes:
speeds:
vy
vry wz
vx
vrx
lr
forces:
vfxv ⃗
df
Fry
m*ax
Frx
lf
L
m*ay
lr
Ffxv ⃗
Ffyv

Longitudinal and lateral accelerations are changed from zero to components of centripetal
acceleration, ac, as follows (see Figure 4-21):
o
⋅ ⋅ sin(𝛽)
⋅ ;
𝑐 ⋅ sin(𝛽)
o
⋅
cos(𝛽)
⋅
⋅
cos(𝛽)
⋅ ;
𝑐
The constitutive assumptions for the axles are changed from ideal tracking to a (linear)
relation between lateral force and lateral slip. The relations should capture the slip
characteristics for the tyres, see Section 0, but they can also capture steering system
compliance (see Section 4.3.4.3), roll steering (see Section 4.3.4.5) and side force steering
(see Section 4.3.4.4). The total mathematical relations can anyway be written as:
o
𝐶 ∙ 𝑠 ; where 𝑠
𝑤
𝑤⁄
𝑤;
⁄ ;
o
𝐶 ∙ 𝑠 ; where 𝑠
vfyv

lf
L
⃗
df
vehicle path
centre
⃗
-ax
ac=wz*v
b
-ax=ac*sin(b);
+ay=ac*cos(b);
ay
vx=v*cos(b) v
vy=v*sin(b)
b
Figure 4-21: One-track model. Dashed forces and moment are fictive forces.
The model in Figure 4-21 is documented in mathematical form in Equation [4.12] (in Modelica
format). The subscript v and w refers to vehicle coordinate system and wheel coordinate system,
respectively.
//Equilibrium:
m*ax = Ffxv + Frx;
m*ay = Ffyv + Fry;
//Air and grade resistance neglected
150
[4.12]
LATERAL DYNAMICS
J*0 = Ffyv*lf - Fry*lr;
-ax = wz*vy;
+ay = wz*vx;
// der(wz)=0
//Constitutive relation, i.e. Lateral tyre force model:
Ffyw = -Cf*sfy;
Fry = -Cr*sry;
sfy = vfyw/vfxw;
sry = vry/vrx;
//Compatibility:
vfxv = vx;
vfyv = vy + lf*wz;
vrx = vx;
vry = vy - lr*wz;
//Transformation between vehicle and wheel coordinate systems:
Ffxv = Ffxw*cos(df) - Ffyw*sin(df);
Ffyv = Ffxw*sin(df) + Ffyw*cos(df);
vfxv = vfxw*cos(df) - vfyw*sin(df);
vfyv = vfxw*sin(df) + vfyw*cos(df);
//Path with orientation:
der(x) = vx*cos(pz) - vy*sin(pz);
der(y) = vy*cos(pz) + vx*sin(pz);
der(pz) = wz;
// Prescription of steering angle:
df = if time < 2.5 then (5*pi/180)*sin(0.5*2*pi*time) else 5*pi/180;
// Rear axle undriven, which gives drag from roll resistance:
Frx = -100; // =10000; //
//Ffxw=0;
The longitudinal speed is a parameter, vx=100 km/h. A simulation result from the model is shown in
Figure 4-11. It shows the assumed steering angle function of time, which is an input. It also shows the
resulting path, y(x).
A driving resistance of 100 N is assumed on the rear axle (Frx=-100;). This is to exemplify that
longitudinal forces does not need to be zero, even if longitudinal forces normally are not so
interesting for steady state high speed manoeuvres.
The variables x,y,pz are the only “state variables” of this simulation. If not including the path model
(Equation [4.1]), the model would actually not be a differential equation problem at all, just an
algebraic system of equations. That system of equations could be solved isolated for any value of
steering angle without knowledge of time history. These aspects are the same for the low speed
model in section 4.2.6.
151
LATERAL DYNAMICS
df [rad]
5 deg
y [m]
2
8.37
2
=
/𝑠
Ffyw
Ffxw
Frx-100 Nm
vx = 100 km/h=27.78 m/s
wz = 0.3995 rad/s
vy = -5.761 m/s
x[m]
time [s]
Figure 4-22: Simulation results of steady state one-track model. The vehicle sketched in the path
plot is not in scale, but correctly oriented.
Equation [4.12] is a complete model suitable for simulation, but it does not facilitate understanding
very well. We will reformulate it assuming small 𝛿 (i.e. cos(𝛿 ) 1, sin(𝛿 ) 0, and 𝛿 2 0).
Eliminate slip, all forces that are not wheel longitudinal, and all velocities that are not CoG velocities:
⋅
𝑤
⋅
⋅ ⋅(
⋅(
(
⋅ ⋅(
𝐶 ⋅(
𝐶 ⋅(
(
(
⋅𝛿
⋅𝛿
⋅ )⋅𝛿 )
)⋅𝛿 ) 𝐶 ⋅(
⋅ )⋅𝛿 )
⋅
(
⋅
(
⋅
))
)) ⋅
⋅
𝑤
⋅𝛿 ⋅
𝑤
⋅𝛿 ⋅
)⋅𝛿
𝐶 ⋅
⋅
⋅(
𝑥
⋅(
𝐶 ⋅
𝑥
(
⋅
(
⋅
⋅
⋅(
) ⋅ 𝛿 );
[4.13]
) ⋅ 𝛿 );
(
⋅
) ⋅ 𝛿 );
Equation [4.13] is a complete model, which we can see as a dynamic system without state variables.


Actuation: Steering and wheel torque on each axle: 𝛿
Motion quantities:
.
𝑤
For example, we can chose 𝛿
𝑤 𝑥 and use the 3 equations to calculate
𝑦
𝑧 . In steady state
st
analysis one often disregard the longitudinal equilibrium, which means to skip 1 equation and the
unknown :
𝐶 ⋅𝐶 ⋅
𝐶 ⋅𝐶 ⋅
𝐶 ⋅ (𝐶 ⋅
⋅
(𝐶 ⋅
2
⋅
𝐶 ⋅𝐶 ⋅
(𝐶 ⋅
⋅
2
2
)
(𝐶 ⋅
𝐶 ⋅ )⋅
𝐶 ⋅ )⋅
⋅
(𝐶 ⋅ ⋅
𝐶 ⋅ )⋅
152
𝑤
2
⋅
⋅
⋅
⋅
2
⋅𝛿 ;
2
)⋅
[4.14]
𝑤
⋅
⋅𝛿 ;
LATERAL DYNAMICS
4.3.2.1
Relation 𝜹𝒇 𝒗𝒙 and 𝑹
Solving the first equation in Equation [4.14] yields:
2
𝐶 ∙𝐶 ∙
𝛿
(𝐶 ∙
𝐶 ∙ )∙
2
∙
∙
≈
𝐶 ∙ )∙
𝑤
𝐶 ∙
𝐶 ∙
≈ { 𝑠𝑠𝑢 𝑒:
∙ ∙
𝑤 ≈ 0} ≈ (
𝐶 ∙𝐶 ∙
𝐶 ∙
𝐶 ∙
{𝑑𝑒𝑓𝑖𝑛𝑒: 𝐾𝑢
}
𝐶 ∙𝐶 ∙
𝐶 ∙
𝐶 ∙
𝐶 ∙𝐶 ∙
(𝐶 ∙
{𝑢𝑠𝑒 𝑠𝑡𝑒 𝑑𝑦 𝑠𝑡 𝑡𝑒:
≈ { 𝑠𝑠𝑢 𝑒 𝑠
∙
}
1
≈ }≈
𝑅
𝑅
⇒
2
)∙
∙
∙
𝐾𝑢 ∙
𝐾𝑢 ∙
∙
𝑅
∙
𝐾𝑢 ∙
∙
∙
[4.15]
≈
≈
2
;
The coefficient 𝐾𝑢 is the understeer gradient and it will be explained in next section.
4.3.2.2
Relation 𝜹𝒇 𝒗𝒙 and 𝜷
Solving the second equation in Equation [4.14] yields:
𝐶 ∙𝐶 ∙
𝛿
2
(𝐶 ∙
𝐶 ∙ )∙
2
∙
∙ ∙ 2 ) (𝐶 ⋅ ⋅
⋅ ⋅
2
𝐶 ∙𝐶 ∙
𝐶 ∙ )∙ ∙
(𝐶 ∙
0} ≈
𝐶 ∙𝐶 ∙ ∙
𝐶 ∙ ∙ ∙ 2
𝐶 ∙ (𝐶 ∙ ⋅
≈ {𝑢𝑠𝑒:
𝑤
𝛿 →
𝑥→
⇒
𝛿 →
𝑥 →∞
{
∙
2
)∙
2
∙
∙
≈
𝑤
≈
[4.16]
;
𝐶 ∙
𝐶 ∙
𝐶 ∙
∙
𝐾𝑢 ∙ 𝐶 ∙
∙
;
We can see that there is a speed dependent relation between steering angle and side slip,
side slip can also be expressed as a side slip angle, 𝛽
arctan ( 𝑦 ). Since normally 𝐾𝑢
𝑥
𝑦
𝑥
. The
0, the side
slip changes sign, when increasing speed from zero to sufficient high enough. This should feel
intuitively correct, if agreeing on the conceptually different side slip angles at low and high speed, as
shown in Figure 4-23. We will come back to this equation in context of Figure 4-33.
low speed
𝜷
path of
front axle
path of
rear axle
high speed
𝜷<
𝜷
v
𝜷
v
vehicle
path
centre
vehicle
path
centre
path of
front axle
path of
rear axle
Figure 4-23: Body Slip Angle for Low and High Speed Steady State Curves
153
LATERAL DYNAMICS
4.3.2.3
Relation 𝒗𝒙 𝑹 and 𝜷
If we approximate
𝐶 ⋅ ⋅
𝑤
⋅
0 and use both equations in Equation [4.14] to eliminate 𝛿 we get:
⋅
2
⋅
𝐶 ⋅
⇒
tan(𝛽)
2
⋅
⋅𝐶
⋅
⋅
(
2
⋅
⋅𝐶
≈ 𝑅 ⋅ tan(𝛽) ; ⇒
[4.17]
1
)⋅ ;
𝑅
Kus=2.6042e-06 /[1/N]
0.01
0.01
vx=0 [km/h]
-0.01
-0.02
-0.03
-0.04
-0.05
-0.06
-0.07
R= [m]
0
tan(beta)=vy/vx /[1=(m/s)/(m/s)]
tan(beta)=vy/vx /[1=(m/s)/(m/s)]
0
-0.01
-0.02
-0.03
-0.04
-0.05
-0.06
0
0.5
1
1.5
curvature=1/R /[1/m]
-0.07
2
-3
x 10
0
50
100
vx /[km/h]
150
200
Figure 4-24: How side slip varies with curvature and longitudinal vehicle speed.
4.3.2.4
Simpler model
A simpler way to reach the final expression is given in Figure 4-25. Here, the simplifications are
introduced earlier, already in physical model, which means e.g. that the influence of
𝑤 is not
identified.
154
LATERAL DYNAMICS
Physical model:
Mathematical model:
• Path radius >> the vehicle. Then, all forces
(and centripetal acceleration) are
approximately co-directed.
• Small tyre and vehicle side slip. Then,
angle=sin(angle)=tan(angle).
Equilibrium:
2
𝑥
Compatibility:
𝛿
2
𝛼
𝛽 ; 𝛽 ≈
𝛼
𝛽 ≈
𝑅
𝛽
𝛽
𝛽
𝛼
≈
𝑥
𝑅
𝑟𝑦
𝜔
𝑦+
𝑦
𝑥
𝑦− 𝑟
𝜔
𝑥
𝑥
;
𝑥
⇒
;
; 𝛼 ≈𝑠 ; 𝛼 ≈𝑠 ;
𝛼 ≈
𝛼
⇒𝛿
𝛿
𝑠 ;
𝐶
𝑠 ;
𝐶
(Angles are not drawn small, which is the reason
why the forces not appear co-linear in figure.)
;
; 0≈
≈
𝑅
Constitution:
𝛽
𝛼 𝛼 𝛽
Eliminate
;
𝑅
yields:
2
𝛿 ≈
𝐾𝑢
𝑅
𝐾𝑢
𝑅
𝐶
𝐶
𝐶
𝐶
;
;
Figure 4-25: Simpler derivation final step in Equation [4.15].
The validity of the steady state model described in this section, section 4.3.2, is of course limited by if
the manoeuvre is transient which would mean that steady state is not reached. But it is also limited
by if the assumption of linear tyre characteristics,
𝐶 ∙ 𝑠 ;, is violated. Therefore, one should
check if some axle (or wheel) is calculated to use more than some certain fraction of available
friction,
√
2
2
𝑓𝑟 𝑐𝑡𝑖𝑜𝑛 ∙ 𝜇 ∙
;. In many cases, one can argue for this limiting fraction to
be in the order of 0.5.
4.3.3
Under-, Neutral- and Over-steering *
4.3.3.1
Understeering as built-in measure
Function definition: Understeering (gradient) is the additional steering angle needed per increase of
lateral force (or lateral acceleration) when driving in high speed steady state cornering on level ground and
high road friction. Additional refers to low speed. The gradient is defined at certain high speed steady state
cornering conditions, including straight-line driving. Steering angle can be either road wheel angle or steering
wheel angle.
𝐾𝑢 in Equation [4.15] is called “understeer gradient” and has hence the unit rad/N or 1/N.
Sometimes one can see slightly other definitions of what to include in definition of understeer
gradient, which have different units, see 𝐾𝑢2 and 𝐾𝑢3 in Equation [4.18]. (When defined as 𝐾𝑢3, one
can sometimes see the unit “rad/g” used, which present compendium recommended to not use.)
𝛿
𝑅
𝐶 ∙
𝐶 ∙
𝐶 ∙𝐶 ∙
{𝑢𝑠𝑒: 𝐾𝑢2
∙𝑔∙
𝑔∙𝑅
𝐶 ∙
𝐶 ∙
∙𝑔∙
𝐶 ∙𝐶 ∙
2
∙
155
[1 𝑜𝑟 𝑟 𝑑]}
[4.18]
2
𝑅
𝐾𝑢2 ∙
𝑔∙𝑅
;
LATERAL DYNAMICS
𝛿
2
𝐶 ∙
𝐶 ∙
∙
∙
𝑅
𝐶 ∙𝐶 ∙
𝑅
𝐶 ∙
𝐶 ∙
∙
{𝑢𝑠𝑒: 𝐾𝑢3
𝐶 ∙𝐶 ∙
[
1
𝑟 𝑑
𝑜𝑟
]}
⁄𝑠 2
⁄𝑠 2
2
𝐾𝑢3 ∙
𝑅
;
𝑅
The first term in the final expression in Equation [4.15], L/R, can be seen as a reference steering
angle. This is referred to as either “low speed steering angle” or “Ackermann steering angle”. The
understanding of this steering angle can be expanded to cover more general vehicles: It is the angle
required for low speed turning. For a general vehicle, e.g. with other steering geometry than
Ackermann, the angle cannot be calculated as simple as L/R, but with some specific derived formula,
or it can be measured on a real vehicle. It is common to use the subscript A. It can be defined for
front axle road wheel steering angle, 𝛿 𝐴 . It can also be defined for steering wheel angle, 𝛿𝑠𝑤𝐴 .
The understeering gradient, 𝐾𝑢 , is normally positive, which means that most vehicles require more
steering angle for a given curve, the higher the speed is. Depending on the sign of 𝐾𝑢 a vehicle is said
to be oversteered (if 𝐾𝑢 < 0), understeered (if 𝐾𝑢 0) and neutral steered (𝑖𝑓 𝐾𝑢 0). In practice,
all vehicles are designed as understeered, because over steered vehicle would become instable very
easily.
The understeering gradient 𝐾𝑢 can be understood as how much additionally to the Ackermann
steering angle one have to steer, in relation to centrifugal force, 𝑐 :
𝛿
𝛿
𝐴
𝐾𝑢 ∙
∙
𝑅
𝛿𝑠𝑤𝐴
𝛿
𝐾𝑢 ∙
𝐴
𝑐;
⇒ 𝐾𝑢
𝛿
𝛿
∆𝛿
𝐴
𝑐
𝑐
;
𝛿𝑠𝑤 ⁄𝑟𝑠𝑡 :
Or, using the steering ratio, 𝛿
𝛿𝑠𝑤
2
𝐾𝑢 ∙
𝑟𝑠𝑡 ∙
∙
𝑅
[4.19]
2
𝛿
𝐴
𝐾𝑢 ∙ 𝑟𝑠𝑡 ∙
⇒ 𝐾𝑢
𝑐;
⇒
𝛿𝑠𝑤 𝛿𝑠𝑤𝐴
𝑟𝑠𝑡 ∙ 𝑐
∆𝛿𝑠𝑤
;
𝑟𝑠𝑡 ∙ 𝑐
4.3.3.2 Understeering as varying with steady state lateral
acceleration
So far, the understeering gradient is presented as a fix built-in vehicle parameter. There is nothing
that says that a real vehicle behaves linear, so in order to get a well-defined value of 𝐾𝑢 , the ∆𝛿 and
the 𝑐 should be small when using Eq [4.20]. However, if we accept that 𝐾𝑢 can change with 𝑐 , 𝐾𝑢
can be defined as a differential quantity:
𝐾𝑢 can also be understood as how much the additional steering angle, ∆𝛿 , has to increase per
increased centrifugal force, 𝑐 , or per lateral acceleration, :
𝐾𝑢
𝜕(∆𝛿 )
𝜕 𝑐
𝜕
(𝛿
𝜕 𝑐
𝛿𝐴 )
𝜕∆𝛿
; 𝑜𝑟 𝐾𝑢3
𝜕 𝑐
𝜕(∆𝛿 )
𝜕
𝜕
(𝛿
𝜕
𝛿𝐴 )
𝜕𝛿
;
𝜕
[4.20]
Equation [4.20] shows the understeering gradient as a (mathematical) function, rather than a scalar
parameter. But it is still fix and built-in in the vehicle. If assessing understeering for a lateral forces up
to near road friction limit, Equation [4.20] is more relevant than Equation [4.15], because it reflects
that understeering gradient changes.
156
LATERAL DYNAMICS
4.3.3.3 Understeering as variable during a transient
manoeuvre
A third understanding of the word understeering is quite different and less strictly defined. It is to see
the understeering as a variable during a transient manoeuvre. For instance, a vehicle can be said to
understeer if tyre side slip is larger on front axle than on rear axle, |∝ | |∝ |, and over-steer if
opposite, |∝ | |∝ |. This way of defining understeering and oversteering is not built in, but varies
over time through a (transient) manoeuvre. E.g., when braking in a curve a vehicle loses grip on rear
axle due to temporary load transfer from rear to front. Then the rear axles can slide outwards
significantly and the vehicle can be referred to as over-steering at this time instant, although the
built-in understeering gradient is >0. This “instantaneous” under-/over-steering can be
approximately found from log data with this simple approximation:
𝛿
𝑢𝑡
𝑅
≈{
≈𝑅∙
}≈
∙
≈{
≈
∙
}≈
∙
2
;
[4.21]
If the actual vehicle has|𝛿| |𝛿 𝑢𝑡 | the vehicle oversteers (|𝛼 | |𝛼 |), and vice versa. This is
often very practical since it only requires simply logged data, 𝛿
and . Note that when 𝛿 and
have different signs, neither understeer or oversteers is suitable as classification, but it can
sometimes be called “counter-steer”.
150
100
50
0
-50
dfront /[10*deg]
-100
-150
ay /[10*m/s 2]
1
1.5
2
2.5
3
t
3.5
4
vx4.5
/[km/h]
5
wz /[deg/s]
beta /[10*deg]
betaRear /[10*deg]
10
df /[deg]
dneutral /[deg]
5
US /[boolean]
OS /[boolean]
0
-5
-10
1
1.5
2
2.5
3
t
3.5
4
4.5
5
Figure 4-26: Upper: Log data from passenger car with ESC system in a double lane change. Lower:
𝛿 𝑢𝑡 can be used to approximately assess “instantaneous” under-/over-steering (US/OS).
157
LATERAL DYNAMICS
A second look at Equation [4.15] tells us that we have to assume absence of propulsion and braking
on front axle,
0, to get the relatively simple final expression. When propulsion on front axle
𝑤
( 𝑤 0), the required steering angle, 𝛿 , will be smaller; the front propulsion pulls in the front end
of the vehicle. When braking on front axle ( 𝑤 < 0), the required steering angle, 𝛿 , will be larger;
the front braking hinders the front end to turn in. To keep vx constant, which is required within
definition of steady state, one have to propel the vehicle because there will always be some driving
resistance to overcome. Driving fast on a small radius is a situation where the driving resistance from
tyre lateral forces becomes significant, which is a part of driving resistance which was only briefly
mentioned in Section 3.2.
Above reasoning was made for steady-state both longitudinally and laterally. If longitudinal
acceleration, we can still assume lateral steady-state and study under-steering gradient. This is
discussed in Section 4.3.4.1.
4.3.3.4
Neutral steering point
An alternative measure to understeering coefficient is the longitudinal position of the neutral
steering point. The point is defined for lateral force disturbance during steady state straight-ahead
driving, as opposed to steady state cornering without lateral force disturbance. The point is where a
vehicle-external lateral force, such as wind or impact, can be applied on the vehicle without causing a
yaw velocity (
0), i.e. only causing lateral velocity (
0). From this definition we can derive a
formula for calculating the position of the neutral steering point, see Figure 4-27.
Physical model:
•
•
•
•
Steady state (
0)
Straight ahead driving (
0)
No steering
Small tyre and vehicle side slip. Then,
angle=sin(angle)=tan(angle).
Equilibrium:
0
;
0
Constitution:
𝑠;
𝐶
𝐶
𝑠 ;
𝑠 ;
Compatibility:
Fey
Fry
Mathematical model:
Ffy
𝑠
b
𝑠
𝑠
𝑠
Eliminate
𝐶
;
𝑠
yields:
𝐶
;
𝐶
𝐶
Express with understeering gradient,
𝑠
0
𝐾𝑢
L
𝐶𝑟 𝑟 −𝐶
𝐶 𝐶𝑟 𝐿
:
𝑠
𝐾𝑢
;
Figure 4-27: Model for definition and calculation of neutral steering point.
The result is condensed in Eq [4.22].
𝐶 ∙
𝑠
𝐶
𝐶 ∙
𝐶
𝐾𝑢 ∙ ; 𝑤 𝑒𝑟𝑒 𝐾𝑢
158
𝐶 ∙
𝐶 ∙
𝐶 ∙𝐶 ∙
;
[4.22]
LATERAL DYNAMICS
We can see that the understeer gradient from steady state cornering model appears also in the
formula for neutral steering point position, 𝑠 . This is why 𝑠 (or rather 𝐿𝑠) and 𝐾𝑢 can be said to be
alternative measures for the same vehicle function/character, the yaw balance. It also proposes an
alternative understanding of the understeering gradient; if 𝐾𝑢 0.1, the neutral steering point is
located 0.1 ∙ rear of CoG.
4.3.4 How to design for a desired understeer
gradient
The values of cornering stiffness for each axle are design parameters to influence understeer
gradient. However, cornering stiffness is an abstract design parameter, in the sense that one cannot
put it on a drawing. Instead, cornering stiffness is a combined effect from various, more concrete,
design parameters. Such more concrete design parameters are briefly presented in the following.
4.3.4.1
Tyre design, inflation pressure and number of tyres
(This section discusses also influence of longitudinal acceleration, which is no tyre design parameter.
However, it is logical in the sense that longitudinal acceleration influence yaw balance via how the
tyres cornering stiffness varies with tyre normal load, which is directly influenced by the longitudinal
acceleration.)
The cornering stiffness of each tyre is an obvious parameter which influences the axle stiffness. The
cornering stiffness of an axle is influence by the sum of cornering stiffness for all tyres. There are
normally two tyres per axle, but there are also vehicles with one tyre (e.g. bicycles) or 4 (typically 2
double mounted on each side in heavy trucks).
Tyre design influences, which is geometrical dimensions and material selection. Inflation pressure is
in this context very close to a design parameter.
In a first approximation, tyre cornering stiffness is approximately proportional to vertical load: 𝐶
𝑘 ∙ . For a vehicle with equally many and same tyres front and rear, this means that it will be
neutral steered. This is because, in steady state cornering, vertical loads distributes in same relation
as lateral loads. Using definition of understeer gradient:
𝐾𝑢2
∙𝑔
⇒ 𝐾𝑢2
𝐶 ∙
𝐶 ∙
𝑘 ∙
∙
𝑘 ∙
∙
𝑜𝑛𝑔𝑖𝑡𝑢𝑑𝑖𝑛
{ 𝑜 𝑑 𝑡𝑟 𝑛𝑠𝑓𝑒𝑟} ⇒
∙ ∙ /
∙𝑔∙
∙𝑔∙
𝑘 ∙(
∙𝑔∙
∙
∙ )∙
𝑘 ∙(
𝑔
∙ )
𝑘 ∙(
∙
{
𝑔 ∙ )
[4.23]
∙ )∙
𝑖𝑓
⁄
⁄
𝑘 ∙(
∙𝑔∙
𝑛𝑑
𝑘
𝑘
0
}
0;
Longitudinal load transfer (influence of in the equations) show that braking increases oversteering tendency. It is actually so, that the critical speed 𝑐 𝑡 √ ⁄( 𝐾𝑢 ∙ ) (see Equation
[4.27]) can come down to quite reachable levels when braking hard; i.e. hard braking at high speed
may cause instability. This is especially so for front biased CoG location. See Figure 4-28, inspired by
Reference (Drenth, 1993).
159
LATERAL DYNAMICS
Cf/Ffz=kf=13.4831 and Cr/Frz=kr=13.4831
Solid=vxCrit, Dashed=vxChar
200
0.1
180
160
140
vx [km/h]
Ku2 [1 or rad]
0.05
0
120
100
60
-0.05
40
lf/L=0.25
lf/L=0.5
lf/L=0.75
-0.1
-10
80
-5
0
ax [m/(s*s)]
5
20
0
-1
10
-0.5
0
ax [g]
0.5
1
Figure 4-28: Left: Under-steering gradient as function of longitudinal acceleration, ax, and static
load distribution, lf/L. Right: Critical and characteristic velocity as function of acceleration and load
distribution.
However, the cornering stiffness varies degressively, e.g. 𝐶
studied in Reference (Drenth, 1993).
𝑘𝑝∙
𝑘𝑑∙
2
. This is further
If taking the degressiveness of tyre cornering stiffness into account, the weight distribution plays a
role also without longitudinal load transfer; front biased weight distribution gives under-steered
vehicles and vice versa. Also, the number of wheel per axle influence stronger; single wheel front (or
double-mounted rear) gives under-steered vehicles and vice versa.
4.3.4.2
Roll stiffness distribution between axles
During cornering, the vertical load is shifted towards the outer wheels. Depending on the roll
stiffness of each axle, the axles take differently much of this load shift. Then, the lateral load transfer
also influences the yaw balance, see Section 4.3.4.2.
The more roll stiff an axle is, the more of the lateral load shift it takes. Tyre cornering stiffness varies
degressively with vertical load. Together, this means that increasing the roll stiffness on the front
axle, leads to less front cornering stiffness and consequently more understeered vehicle. Increasing
roll stiffness on rear axle makes the vehicle less understeered.
One can change the roll stiffness of an axle by changing roll centre height, wheel stiffness rate and
anti-roll bar stiffness. This effect is further described in Section 4.3.9.6.
4.3.4.3
Steering system compliance
A compliant steering system will reduce the effective cornering stiffness on the steered axle. With
normal front axle steering, a compliant steering means a more understeered vehicle.
4.3.4.4
Side force steer gradient
Side force steer gradient, 𝑐 𝑆𝐹𝑆 , is defined for an axle and it is how much the wheels on an axle steers
[deg] per lateral force [N]. Also a non-steered axle steers due to side force steering, which depends
on the compliance of the suspension bushings.
It can be modelled as an extra compliance, with the constitutional equations:
𝑐 𝑆𝐹𝑆 ∙ Δ𝛿 ;
and
𝑐 𝑆𝐹𝑆 ∙ Δ𝛿 ;, where the Δ marks additional steering angle due to the lateral force. These
160
LATERAL DYNAMICS
extra compliances comes into play as series connected with the tyre cornering compliances. If we
update Equation [4.15] with side force steering it becomes:
𝛿
𝑤 𝑒𝑟𝑒 𝐾𝑢
𝑤 𝑒𝑟𝑒 𝐶
4.3.4.5
𝐶
𝑡 𝑡
𝐾𝑢 ∙
𝑅
∙
𝐶
𝑡 𝑡
∙
1
𝑡 𝑡
1
𝐶
𝑐
∙ 2
;
𝑅
𝐶 𝑡 𝑡∙
𝐶 𝑡 𝑡∙
𝐶 𝑡 𝑡∙𝐶 𝑡 𝑡∙
𝑛𝑑 𝐶
1
1
𝑡 𝑡
𝑆𝐹𝑆
1
𝐶
𝑐
1
;
[4.24]
;
𝑆𝐹𝑆
Roll steer gradient
Roll steer gradient, 𝑐 𝑅𝑆 , is defined for an axle and it is how much the wheels on an axle steers [deg]
per vehicle roll angle [deg]. Also a non-steered axle steers due to roll-steering, which depends on the
suspension linkage geometry. The added steering angle can hence be expressed: Δ𝛿
𝑐 𝑅𝑆 ∙ ;
If one assumes steady state cornering, the vehicle roll angle depends on the vehicle’s roll
gradient, 𝑐
as:
𝑐
∙ ;.For steady state, it is also possible to express
∙
∙
𝑐 𝑅𝑆 ∙ 𝑐
∙
∙ ⁄( ∙ (
𝑐 𝑅𝑆 𝐹 ∙ ;. The parameter
(
)⁄ ;. In total: Δ𝛿
))
𝑐 𝑅𝑆 𝐹 comes into the equations in exactly the same way as side force steering gradient, 𝑐 𝑆𝐹𝑆 , so:
𝛿
𝑤 𝑒𝑟𝑒 𝐾𝑢
𝑤 𝑒𝑟𝑒 𝐶
𝐶
𝑡 𝑡
𝑅
∙
𝐶
𝐾𝑢 ∙
𝑡 𝑡
1
𝑡 𝑡
1
𝐶
𝑐
1
∙
∙ 2
;
𝑅
𝐶 𝑡 𝑡∙
𝐶 𝑡 𝑡∙
𝐶 𝑡 𝑡∙𝐶 𝑡 𝑡∙
𝑛𝑑 𝐶
𝑅𝑆 𝐹
1
𝑡 𝑡
1
𝐶
𝑐
1
;
[4.25]
;
𝑅𝑆 𝐹
Side force steering and roll-steering are very similar. Side force steering does not require a roll angle
change, so side-force steering has basically no delay (or at least same magnitude of delay as the tyre
cornering forces). However, roll-steering requires that sprung body changes roll angle which takes
significantly longer time, typically roll eigen-frequency for a passenger vehicle is around 1.5 Hz. Rollsteering also come into play for one-sided road unevenness, i.e. also without cornering.
Side force steering and roll-steering compliances often represent a significant part of the front axle
cornering compliance, e.g. 20-50% on a passenger car.
4.3.4.6
Camber steer
Negative camber (wheel top leaning inwards) increases the cornering stiffness. One explanation to
this is that curve outer wheel gets more vertical load than the curve inner wheel. Hence, the inwards
directed camber force from outer wheel dominates over outwards directed camber force from the
inner wheel. Negative camber is often used at rear axle at passenger cars. Drawback with non-zero
camber is tyre wear.
4.3.4.7
Toe angle
Toe has some, but limited effect on an axles cornering stiffness. Non-zero toe increases tyre wear.
Toe-angle: When rolling ahead, tyre side forces pre-tension bushes.
161
LATERAL DYNAMICS
If toe (=toe-in) is positive there are tyre-lateral forces on each tyre already when driving straight
𝑡
ahead, even if left and right cancel out each other:
0;. Then, if the axle
(𝐶 𝑡 𝐶 ℎ𝑡 ) ∙ 2
takes a side force, the vertical load of the wheels are shifted between left and right wheel, which also
changes the tyre cornering stiffnesses. The outer wheel will get more cornering stiffness. Due to
positive toe, it will also have the largest steering angle. So, the axle will generate larger lateral force
than with zero toe. For steady-state cornering vehicle models, this effect comes in as an increased
axle cornering stiffness, i.e. a linear effect.
4.3.4.8
Wheel Torque effects
Wheel torque give tyre longitudinal force, directed as the wheel is pointing. If the wheel is steered,
the wheel longitudinal forces can influence the yaw balance, see also Ffxw in Equation [4.15].
Unsymmetrical wheel torques (left/right) will give a direct yaw moment in the yaw equilibrium in
Equation [4.12]. The actuated yaw moment around CoG is then of the magnitude of wheel
longitudinal wheel force times half the track width. ESC and Torque vectoring interventions have
such effects.
High longitudinal utilization of friction on an axle leads to that lateral grip is reduced on that axle. The
changed yaw moment, compared to what one would have without using friction longitudinally, can
be called an indirect yaw moment. The actuated change in yaw moment around CoG is then of the
magnitude of change in wheel lateral wheel force times half the wheel base. It influences the yaw
balance. That is the reason why a front axle driven vehicle may be more understeered than a rear
axle driven one. On the other hand, the wheel-longitudinal propulsion force on the front axle does
also help the turn-in, which acts towards less understeering.
4.3.4.9
Transient vehicle motion effects on yaw balance
The effects presented here are not so relevant for steady state understeering coefficient. However,
they affect the yaw balance in a more general sense, why it is relevant to list them in this section.


Longitudinal load transfer changes normal forces. E.g. strong deceleration by wheel forces
helps against under-steering, since front axle gets more normal load. This effect has some
delay. Also, it vanishes after the transient.
(This effect can be compared with the effect described in Section 4.3.4.2, which is caused by
tyre cornering stiffness varying degressively with vertical load, while the longitudinal load
transfer effect can be explained solely with the proportional variation.)
Change of longitudinal speed helps later in manoeuvre. E.g. deceleration early in a
manoeuvre makes the vehicle easier to manoeuvre later in the manoeuvre. It is the effect of
the term wz*vx that is reduced.
4.3.4.10 Some other design aspects
High cornering stiffness is generally desired.
Caster offset gives a self-aligning steering moment, which generally improves the steering feel.
Longer wheel base (with unchanged yaw inertia and unchanged steering ratio) improves the
transient manoeuvrability, because the lateral forces have larger levers to generate yaw moment
with.
162
LATERAL DYNAMICS
4.3.5
Normalized required steering angle
A fundamental property of the vehicle is what steering angle that is required to negotiate a certain
curvature (=1/path radius 1/𝑅𝑝 ). This value can vary with longitudinal speed and it can be
normalized with wheel base, . From Equation [4.15], we can conclude:
𝑵𝒐
𝒛
𝒒𝒖
𝒕
1
𝒏
[4.26]
2
∙
𝐾𝑢 ∙
𝛿 ∙𝑅
𝒏
;
The normalized required steering angle is plotted for different understeering gradients Figure 4-29. It
is the same as the inverted and normalized curvature gain, see 4.3.6.2.
2.5
Ku = 2.525e-6 [1/N]
Understeered
required steering, df*R/L [rad]
2
1.5
Neutral steered
Ku = 0e-6 [1/N]
1
Oversteered
Ku = -1.794e-6 [1/N]
0.5
Characteristic speed (for
understeered vehicle)
0
0
5
10
15
20
Critical speed (for
oversteered vehicle)
25
vx [m/s]
30
𝛿 ∙𝑅
Figure 4-29: Normalized steering angle (
4.3.5.1
𝐿
35
40
45
50
) for Steady State Cornering
Critical and Characteristic speed *
Function definition: Critical speed is the speed above which the vehicle becomes instable in the sense that
the yaw rate grows significantly for a small disturbance in, e.g., steering angle.
Function definition: Characteristic speed is the speed at which the vehicle requires twice as high
steering angle for a certain path radius as required at low speed (Figure 4-29). (Alternative definitions: The
speed at which the yaw rate gain reaches maximum (Figure 4-30). The speed at which the lateral
acceleration gain per longitudinal speed reaches its highest value. (Figure 4-32).)
We can identify that zero steering angle is required for the over-steered vehicle at 28 m/s. This is the
so called Critical Speed, which is the speed where the vehicle becomes unstable. It should be noted
here, that there are stable conditions also above critical speed, but one has to steer in the opposite
direction then. Normal vehicles are built understeered, which is why a Critical speed is more of a
theoretical definition. However, if studying (quasi-steady state) situations where the rear axle is
heavily braked, the cornering stiffness rear is reduced, and a critical speed can be relevant.
For understeered vehicles, we can instead read out another measure, the Characteristic Speed. The
understanding of Characteristic Speed is, so far just that required steering increases to over twice
what is needed for low speed at the same path radius. A better feeling for Characteristic Speed is
easier suggested in a later section, see Section 4.3.6.3.
163
LATERAL DYNAMICS
From Equation [4.15], we can find a formula for critical and characteristic speeds:
𝛿
𝛿
𝑅
𝑅
∙
𝐾𝑢 ∙
2
0 ⇒
𝑅
∙
𝐾𝑢 ∙
4.3.6
𝑐 𝑡
√
𝑐 𝑡
√
(𝐶 ∙
𝐾𝑢 ∙
∙
𝑅
𝑅
⇒
√
𝐾𝑢 ∙
𝑐ℎ
;
𝐶 ∙ )∙
2
𝑐ℎ
2
𝐶 ∙𝐶 ∙
√
(𝐶 ∙
𝐶 ∙𝐶 ∙
2
𝐶 ∙ )∙
[4.27]
;
Steady state cornering gains *
Function definition: Steady state cornering gains are the amplification from steering angle to certain
vehicle response measures for steady state cornering at a certain longitudinal speed.
From Equation [4.15], we can derive some interesting ratios. We put steering angle in the
denominator, so that we get a gain, in the sense that the ratio describes how much of something we
get “per steering angle”. If we assume
0, we get Equation [4.28].
𝑤
𝒀 𝒘
𝒕
𝒏
𝛿
𝑪𝒖 𝒗 𝒕𝒖
𝒄𝒄
𝛿
1⁄𝑅
𝛿
𝒕 𝒐𝒏
𝑢𝑠𝑒:
{
∙
𝑅
𝐾𝑢 ∙
𝑅
𝒗𝒙
;
𝑲𝒖 ∙ ∙ 𝒗𝒙
2
𝑳
𝑳
𝑲𝒖 ∙
∙ 𝒗𝒙
𝑳
𝒗𝒙
𝑲𝒖 ∙
𝒇𝒙𝒘
1⁄ 𝑅
𝑅
𝒏
∙
𝑛𝑑
𝑅
[4.28]
2
∙ 𝒗𝒙
;
0, and then we get Equation [4.29].
𝐶 ∙𝐶 ∙
𝐶 ∙𝐶 ∙
∙
𝑅
𝐾𝑢 ∙
𝒕 𝒌 𝒏 𝒏𝒕𝒐 𝒄𝒄𝒐𝒖𝒏𝒕: 𝒀 𝒘
𝛿
;
2
;
𝑤
2
∙
𝑅
𝐾𝑢 ∙
1 }≈
≈ ;
𝑅
𝑅
Yaw rate gain is also derived for
𝑾𝒕
⁄𝑅
𝒏
κ
𝛿
𝑳 𝒕
1
≈ }≈
𝑅
{𝑢𝑠𝑒:
(𝐶 ∙
2
(𝐶 ∙
164
𝒕
𝒏
𝐶 ∙ )∙
𝐶 ∙ )∙
𝑤
∙
∙
2
[4.29]
LATERAL DYNAMICS
4.3.6.1
Yaw Rate gain
30
Oversteered
yaw rate gain, wz/df [(rad/s)/rad]
25
Ku = -1.794e-6 [1/N]
20
Neutral steered
Ku = 0e-6 [1/N]
Characteristic
speed (for
understeered
vehicle)
15
10
Understeered
Ku = 2.525e-6 [1/N]
5
0
Critical speed (for
oversteered vehicle)
0
5
10
15
20
25
vx [m/s]
30
35
40
45
50
Figure 4-30: Yaw rate gain ( /𝛿 ) for Steady State Cornering.
In each “cluster of 3 curves”: Mid curve shows
0.
𝑤
Upper shows
0. 5 ∙
. Lower shows
0. 5 ∙
.
𝑤
𝑤
The yaw rate gain gives us a way to understand Characteristic Speed. Normally one would expect the
yaw rate to increase if one increases the speed along a circular path. However, the vehicle will also
increase its path radius when speed is increased. At the Characteristic Speed, the increase in radius
cancel out the effect of increased speed, so that yaw rate in total decrease with increased speed.
One can find the characteristic speed as the speed where one senses or measures the highest value
of yaw rate increase for a given steering angle step.
The curves for
0. 5 ∙
and
𝑤
first stages in Eq [4.15], before assuming
𝑤
0. 5 ∙
≈ 0.
in Figure 4-30 are generated using the
Curvature gain
3
Oversteered
Ku = -1.794e-6 [1/N]
2.5
curvature gain, (1/R)/df [1/(m*rad)]
4.3.6.2
𝑤
2
1.5
1
Neutral steered
0.5
Ku = 0e-6 [1/N]
Ku = 2.525e-6 [1/N] Understeered
0
0
5
10
15
20
25
vx [m/s]
⁄𝑅
30
35
40
45
50
Figure 4-31: Curvature gain ( 𝛿 ) for Steady State Cornering
165
LATERAL DYNAMICS
If driving on a constant path radius, and slowly increase speed from zero, an understeered vehicle
will require more and more steering angle (“steer-in”), to stay at the same path radius. For an oversteered vehicle one has to steer less (“open up steering”) when increasing the speed.
4.3.6.3
Lateral Acceleration gain
Figure 4-32 shows the lateral acceleration gain as function of vehicle speed. The characteristics speed
is once again identified in this diagram, and now as the speed when lateral acceleration gain per
longitudinal speed (( ⁄𝛿 )⁄ ) reaches its highest value. This is an alternative definition of
characteristic speed, cf Section 4.3.5.1.
500
Oversteered
Ku = -1.794e-6 [1/N]
lateral acceleration gain, ay/df [(m/(s*s))/rad]
450
Neutral steered
Ku = 0e-6 [1/N]
400
350
300
250
Understeered
200
Ku = 2.525e-6 [1/N]
150
Characteristic speed is
where gain curve has a
tangent through origin
100
50
0
0
5
10
15
20
25
vx [m/s]
30
35
40
45
50
Figure 4-32: Lateral acceleration gain ( 𝑦 ) for Steady State Cornering
𝛿
From the previous figures the responsiveness of the vehicle can be identified for different understeer
gradients. In all cases the vehicle which is understeered is the least responsive of the conditions.
Both the yaw rate and lateral acceleration cannot achieve the levels of the neutral steered or oversteered vehicle. The over-steered vehicle is seen to exhibit instability when the critical speed is
reached since small changes in the input result in excessive output conditions. In addition, the oversteered vehicle will have a counter-intuitive response for the driver. To maintain a constant radius
curve, an increase in speed requires that the driver turns the steering wheel opposite to the direction
of desired path. The result of these characteristics leads car manufacturers to produce understeered
vehicles that are close to neutral steering to achieve the best stability and driver feedback.
4.3.6.4
Side slip gain as function of speed
All gains above can be found from solving wz from Equation [4.15]. If instead solving the other
unknown, vy, we can draw “side slip gain” instead. Equation [4.30] shows the formula for this.
𝐶 ∙𝐶 ∙
∙𝛿
𝐶 ∙𝐶 ∙
2
∙
(𝐶 ∙
∙
2
𝐶 ∙ )∙
∙
𝐶 ∙
∙
2
[4.30]
It is not solely the understeering gradient that sets the curve shape, but we can still plot for the some
realistic numerical data, which are under-, neutral and over-steered, see Figure 4-33.
166
LATERAL DYNAMICS
m = 2000 [kg]; lf = 1.3 [m]; lr = 1.5 [m]
1
"side slip gain", (vy/vx)/df [1/rad]
understeered
neutral steered
oversteered
lr/L
0.5
0
-0.5
-1
Understeered
-1.5
-2
Oversteered
Neutral steered
-2.5
-3
0
5
10
15
20
25
vx [m/s]
Figure 4-33: “Side slip gain” (
𝑦
𝑥 ∙𝛿
) for Steady State Cornering.
All cases in Figure 4-33 goes from positive side slip to negative when speed increases. This is the
same as we expected already in Figure 4-23.
4.3.7
Manoeuvrability and Stability
The overall conclusion of previous section is that all gains become higher the more over-steered (or
less understeered) the vehicle is. Higher gains are generally experienced as a sportier vehicle and it
also improves safety because it improves the manoeuvrability. A higher manoeuvrability makes it
easier for the driver to do avoidance manoeuvres. This would motivate a design for low
understeering gradient.
However, there is also the effect that a vehicle with too small understeer gradient becomes very
sensitive to the steering wheel angle input. In extreme, the driver would not be able to control the
vehicle. This limits how small the understeering gradient one can design for.
The overall design rule today is to make the understeering as small as possible, but the neutral steer
is a limit which cannot be passed.
It is not impossible for a driver to keep an unstable vehicle (Ku<0 and vx>Critical Speed) on an
intended path, but it requires an active compensation with steering wheel. If adding support
systems, such as yaw damping by steering support or differentiated propulsion torques, it can be
even easier. If one could rely on a very high up time for such support systems, one could move
today’s trade-off between manoeuvrability and stability. This conceptual design step has been taken
for some airplanes, which actually are designed so that they would be instable without active
control.
4.3.8
Handling diagram
This section gives a brief introduction to handling diagram, which is a useful tool for discussing yaw
stability during different steady state cornering situations, as initiated with Equation [4.20].
There are many frequently used graphical tools or diagrams to represent vehicle characteristics. One
is the “handling diagram”, which is constructed as follows. Same simplifying assumptions are done as
in Figure 4-25, with the exception that we don’t assume linear tyre models.
167
LATERAL DYNAMICS
Equilibrium:
2
{
∙
∙
𝑅
; 0
∙
Constitution:
−
(𝛼 ) ⇒ 𝛼
Solving for 𝛼
−
𝛼
So, we can plot 𝛼
𝛼
(
∙
∙
;
−
(𝛼 ) ⇒ 𝛼
);
(
∙
∙
);
𝛼 yields:
𝛼
𝛼
∙ ;} ⇒
𝐿
;)
𝑅
(
−
)
𝛼 as function of
yields 𝛼
𝛼
𝛿
𝐿
𝑅
(
−
)
( ∙
∙
−
)
( ∙
∙
);
. This relation is interesting because compatibility (𝛿
𝛿
𝛿𝐴 . And 𝛿
𝜕
(𝛿
𝜕 𝑦
understandings of 𝐾𝑢 in Equation [4.20], (𝐾𝑢
𝛿𝐴 is connected to one of the
𝛿𝐴 );). If we plot 𝛼
𝛼
𝛿
𝛿𝐴 on
abscissa axis and
on ordinate axis, we get the most common way of drawing the handling
diagram, see Figure 4-34. The axle’s constitutive relations can be used as graphical support to
𝐿
(𝛼 )
construct the diagram, but then the constitutive relations should be plotted as:
∙
𝐿− 𝑖 𝑚
∙
(𝛼 );. The quantity
can be seen as the lateral force on the axle, but scaled so that both axles’
values correspond to the same vehicle lateral acceleration.
Figure 4-34 shows the construction of a handling diagram from axle slip characteristics. Figure 4-35
show examples of handling diagrams constructed via tests with simulation tools. Handling diagrams
can be designed from real vehicle tests as well. The slope in the handling diagram corresponds to
understeering gradient 𝐾𝑢3 in Equation [4.18].
𝐿
front axle,
𝑟
𝐿
rear axle,
∙
𝑚
∙
∙
𝑚
∙
𝛼
𝛼
𝛼
Handling diagram
ay
g
H
Using the axle
models above
Linear axle models,
understeered vehicle
Normal vehicle stays
understeered
𝛼
𝛼
𝛿
𝛼
𝛿𝐴
𝛼
Figure 4-34: Construction of the “Handling diagram”. The axle’s slip characteristics (solid) are
chosen so that vehicle transits from understeer to over-steer with increased longitudinal speed, vx.
The dashed shows two other examples.
168
𝛼
𝛼
Try to get some
measured as well.
From VCC, Saab?
LATERAL DYNAMICS
From: Daniel A. Fittanto, et al. “Passenger Vehicle Steady-State Directional Stability Analysis
Utilizing EDVSM and SIMON”, Copyright 2004 by Engineering Dynamics Corporation
Figure 4-35: Example of handling diagram.
Using a one-track model of a two axle vehicle, there is such curve as drawn in handling diagram, but
from a more advanced model, a vehicle with non-Ackermann steering or a real test, the curve
becomes an area. This can be understood as
if a certain lateral acceleration can be reached at several
(EDVSM and SIMON are two different simulation tools.)
different longitudinal speeds, using different path radii. One can therefore see handling diagrams
with a set of curves, each for a certain longitudinal speed. This is typical for handling diagrams for
heavy trucks with several non-steered axles.
4.3.9 Steady state cornering at high speed,
with Lateral Load Transfer
In the chapter about longitudinal dynamics we studied (vertical tyre) load transfer between front and
rear axle. The corresponding issue for lateral dynamics is load transfer between left and right side of
the vehicle. Within the steady state lateral dynamics, we will cover some of the simpler effects, but
save the more complex suspension linkage dependent effects to Sections 4.5.
The relevance of analysing the load transfer is the function of limited roll angle in cornering (for
comfort) and yaw balance (understeering gradient, see 4.3.9.6). Additionally, the load transfer can
influence the transient handling; see Section 4.4 and Section 4.5.
4.3.9.1
Load transfer between vehicle sides
The total load transfer can be found without involving suspension effects.
Moment equilibrium, around left contact with ground:
𝑤
∙𝑔∙
∙
∙
∙𝑤 0⇒
Moment equilibrium, around right contact with ground: ⇒
𝑔
∙(
∙ );
𝑤
∙ (2
[4.31]
ℎ
∙ 𝑤)
These equations confirm what we know from experience, the left side if off-loaded if turning left.
Generally, curve inner side is off-loaded.
169
LATERAL DYNAMICS
View from rear:
curve inner side
(if ay>0, i.e.
turning left)
m*g
m*ay
h
Fyl
Fzr
Fzl
curve outer side
(if ay>0 , i.e.
turning left)
Fyr
w/2 w/2
Figure 4-36: Free Body Diagram for cornering vehicle. The force m*ay is a
fictive force. Subscript l and r here means left and right.
4.3.9.2
Body heave and roll due to lateral wheel forces
Now, we shall find out how much the vehicle rolls and heaves during steady state cornering. First, we
decide to formulate the model in “effective stiffnesses”, in the same manner as for longitudinal load
transfer in previous chapter.
-y
z
ay
zl
y
px
m*ay
zr
h
Fyl
zlr=0
h
m*g
zrr=0
Fzl
w/2 w/2
Fyr
hRC
Fzr
w/2 w/2
z,y,px,zl,zr, are displacements from
a static stand-still position.
zlr=zrr=0 means that road is smooth.
Steady-state assumed, so that
lateral acceleration (ay) may be
non-zero, but vertical and roll
acceleration are zero.
Figure 4-37: Model for steady state heave and roll due to lateral acceleration. Suspension model is
no linkage (or “trivial linkage”) and without difference front and rear.
There is no damping included in model, because their forces would be zero, since there is no
displacement velocity, due to the “quasi-steady-state” assumption. As constitutive equations for the
compliances (springs) we assume that displacements are measured from a static conditions and that
the compliances are linear. The road is assumed to be smooth, i.e. zlr=zrr=0.
𝑤 𝑒𝑟𝑒
𝑐𝑠 𝑑 ∙ (𝑧
𝑧)
𝑛𝑑
∙ 𝑔 𝑛𝑑
∙ 𝑤⁄
𝑐𝑠 𝑑 ∙ (𝑧
∙ 𝑤⁄
0
𝑧 )
[4.32]
The stiffnesses cf and cr are the effective stiffnesses at each axle. The physical spring may have
different values of stiffness, but its effect is captured in the effective stiffness. Some examples of
different physical design are given in Section “Axle suspension system”.
170
LATERAL DYNAMICS
It is especially noted that this model is further developed in next section, to include linkage in the
suspension, which allows validity for transient lateral dynamics. Also, the development gives better
reflection of steady state levels of displacements, z and px.
We see already in free-body diagram that Fyl and Fyr always act together, so we rename
Fyl+Fyr=Fyw, where w refers to wheel. This and equilibrium gives:
∙
𝑤
0;
∙𝑔
∙ (𝑤⁄ )
0;
∙ (𝑤⁄ )
∙ 𝑔 ∙ ( 𝑦)
𝑤∙
0;
[4.33]
The term ∙ 𝑔 ∙ ( 𝑦) is taken into account, but not corresponding for ∙ , since symmetry of the
vehicle motivates that the roll takes place around a point on the vertical symmetry axis. Therefore 𝑦
is significant but corresponding displacement in vertical direction is not. If we assume a height for the
point where the roll takes place, 𝑅𝐶 ,we can expressed – 𝑦 (
𝑅𝐶 ) ∙ 𝑝 . We don’t know the
value of it, until below where we study the suspension design, but it can be mentioned already here
that most vehicles have an 𝑅𝐶 ≪ . This causes a significant “pendulum effect”, especially for heavy
trucks.
Compatibility, to introduce body displacements, z and px, gives:
𝑧
(𝑤⁄ ) ∙ 𝑝 ;
𝑧
𝑛𝑑 𝑧
𝑧
(𝑤⁄ ) ∙ 𝑝 ;
[4.34]
Combining constitutive relations, equilibrium and compatibility, gives, as Matlab script:
clear, syms zl zr Fzl Fzr Fzl0 Fzr0 Fyw z px
sol=solve( ...
'Fzl=Fzl0-cside*zl', ...
'Fzr=Fzr0-cside*zr', ...
'Fzl0+Fzr0=m*g', ...
'Fzl0*w/2-Fzr0*w/2=0', ...
'Fyw-m*ay=0', ...
'm*g-Fzl-Fzr=0', ...
'Fzl*(w/2)-Fzr*(w/2)+Fyw*h+m*g*(h-hRC)*px=0', ...
'zl=z+(w/2)*px', ...
'zr=z-(w/2)*px', ...
zl, zr, Fzl, Fzr, Fzl0, Fzr0, Fyw, z, px);
[4.35]
The results from the Matlab script in Equation [4.35]:
∙
𝑤
𝑧
𝑝
;
0;
⋅
𝑤2
∙ 𝑐𝑠 𝑑
𝑔
∙
∙
⋅
∙
𝑤
∙
∙𝑔∙(
∙(
1
(
𝑔
∙
∙
𝑤
∙(
1
(
171
𝑅𝐶 )
⋅
𝑐𝑠 𝑑
1
∙𝑔
;
∙
1
⋅ ∙𝑔
𝑐𝑠 𝑑 ∙
𝑤2
𝑅𝐶
𝑤2
) ;
)
𝑅𝐶
) ;
)
[4.36]
LATERAL DYNAMICS
In agreement with intuition and experience the body rolls with positive roll when steering to the left
(positive Fyw). Further, the body centre of gravity is unchanged in heave (vertical z). The formula
uses 𝑅𝐶 which we cannot estimate without analysis of the suspension. Since front and rear axle
normally are different, we could expect that 𝑅𝐶 is expressed in some similar quantities for each of
front and rear axle, which also is the case in our further analysis, see Equation [4.43].
4.3.9.2.1 Steady-state roll-gradient *
Function definition: Steady state roll-gradient is the body roll angle per lateral acceleration for the
vehicle during steady state cornering with a certain lateral acceleration and certain path radius on level
ground.
4.3.9.3
Lateral load transfer
For longitudinal load transfer, during purely longitudinal dynamic manoeuvres, the symmetry of the
vehicle makes it reasonable to split vertical load on each axle equally between the left and right
wheel of the axle. For lateral dynamics it is not very realistic to assume symmetry between front and
rear axle. Hence, the suspension has to be considered separately for front and rear axle. The
properties that are important to model for each axle is not only left and right elasticity (as we
modelled the whole vehicle in Figure 4-37). It is also how the lateral tyre forces are transmitted from
road contact patches to the vehicle body. We end up with conceptually the same two possible
linkage modelling concepts as we found for longitudinal load transfer, see Figure 3-31. Either we can
introduce roll centre heights for each axle (c.f. pitch centre in Section 3.4) or we can introduce the
pivot point for each wheel (c.f. axle pivot points in Section 3.4). A difference for lateral load transfer,
compared to longitudinal load transfer, is that it is significant also at steady state (due to centrifugal
force).
The two modelling ways to include the suspension in the load transfer are shown in Figure 4-38.
Generally speaking, they can be combined, so that one is used on front axle and the other on rear
axle. However, in this compendium we will not combine, but select one of the concepts to describe.
4.3.9.3.1 Load Transfer model with Wheel Pivot Points
This model will not be deeply presented in this compendium. However, it should be mentioned as
having quite a few advantages:


The model has both heave and roll degree of freedom. (Roll centre model is restricted to roll
around roll centre.)
The model does take the distribution of longitudinal wheel forces between left and right side
into account. (Roll centre model only uses the sum of lateral forces per axle.)
Generally spoken, this model is more accurate and not much more computational demanding and
probably more easy to intuitively understand. (For non-individual, rigid axles or beam axles, the roll
centre model is accurate enough and probably more intuitive.)
Cases when this model is recommended as opposed to the model with roll centres are:



Steady state and transient analysis where heave displacement is important.
When large differences between lateral load on left and right wheels are present, such as:
Large load transfer, i.e. high CoG and large lateral accelerations. One example is when
studying wheel lift and roll-over tendencies.
o Large differences between longitudinal slip, while axle skids sideways. Then one
wheel might have zero lateral force, due to that friction is used up longitudinally,
while the other can have a large lateral force.
o If individual steering within an axle would be studied. One could think of an extreme
case if actuating a sudden toe-in or toe-out, which would causes large but counterdirected lateral forces on left and right wheel.
172
LATERAL DYNAMICS
Model with wheel pivot points
Model with axle roll centres
Transversal sections from rear over front axle:
h
h
hRCf
ef
gf
w/2 w/2
w/2 w/2
Transversal sections from rear over rear axle:
h
h
hRCr
er
gr
w/2 w/2
w/2 w/2
Figure 4-38: Two alternative models for including suspension effects in lateral load transfer
4.3.9.3.2 Load Transfer model with Axle Roll Centres
The model with axle roll centres has some drawback as listen before. To mention some advantages, it
is somewhat less computational demanding. However, the main reason why using the roll centre
based model in this compendium is that the compendium then cover two different concepts with
longitudinal and lateral load transfer.
Behold the free-body diagrams in Figure 4-39. The road is assumed to be flat, zflr=zfrr=zrlrzrrr=0. In
free-body diagram for the front axle, Pfz and Pfy are the reaction force in the rear roll-centre.
Corresponding reaction forces are found for rear axle. Note that roll centres are free of roll moment,
which is the key assumption about roll centres! Fsfl, Fsfr, Fsrl and Fsrr are the forces in the
compliances, i.e. where potential spring energy is stored. One can understand the roll-centres as also
unable to take vertical force, as opposed to constraining vertical motion (as drawn). Which of
vertically force-free or vertically motion-free depends on how one understand the concept or rollcentre, and it does not influence the equations.
Note carefully that the “pendulum effect” is NOT included here, in section 4.3.9.3, as it was in section
4.3.9.2. The motivation is to get simpler equations for educational reasons.
173
LATERAL DYNAMICS
forces, on body:
displacements, speeds, accelerations:
ay
m*g
z
zfl=zrl=0
y
m*ay
zfr=zrr
px
Pry
Pfy
zflr=zrlr=0
Fsfl+Fsrl
zfrr=zrrr=0
forces,
on vehicle:
forces, on front axle:
m*g
m*ay
Fsfl
Ffry+Frry
Fflz+Frlz
𝑃𝑟𝑧
Fsfr+Fsrr
forces, on rear axle:
Fsrl
Fsfr
𝑃𝑓𝑧
Ffrz+Frrz
Ffly
Fflz
Fsrr
𝑃𝑟𝑧
Pry
Pfy
h
Ffly+Frly
𝑃𝑓𝑧
Ffry
Ffrz
Frly
Frlz
Frry
Frrz
Figure 4-39: Model for steady state heave and roll due to lateral acceleration, using roll centres,
which can be different front and rear.
There is no damping included in model, because their forces would be zero, since there is no
displacement velocity, due to the steady-state assumption. As constitutive equations for the
compliances (springs) we assume that displacements are measured from a static conditions and that
the compliances are linear. For simplicity, the anti-roll bars are NOT modelled, but these are
generally important and will be added in Section 4.3.9.4. The road is assumed to be smooth, i.e.
zflrzfrrzrlrzrrr0.
𝑠
𝑠
𝑠
𝑠
𝑠
𝑠
𝑠
𝑠
𝑤 𝑒𝑟𝑒
𝑠
𝑐 𝑤 ∙ (𝑧
𝑐 𝑤 ∙ (𝑧
𝑐 𝑤 ∙ (𝑧
𝑐 𝑤 ∙ (𝑧
𝑠
𝑧 );
𝑧 );
𝑧 );
𝑧 );
∙𝑔∙
;
∙
[4.37]
𝑛𝑑
𝑠
𝑠
∙𝑔∙
∙
;
The stiffnesses cfw and crw are the effective stiffnesses per wheel at front and rear axle,
respectively. The physical spring may have a different value of stiffness, but its effect is captured in
the effective stiffness. Some examples of different physical design are given in Section “Axle
suspension system”.
It is especially noted that this model is further developed in Section 4.5, to include non-steady state
phenomena (damping as well as heave and roll inertial effects), which is needed for model validity
for more violent transients lateral dynamics.
We see already in free-body diagram that Ffly and Ffry always act together, so we rename
Ffly+Ffry=Ffy and Frly+Frry=Fry.
Equilibrium for whole vehicle (vertical, lateral, yaw, pitch, roll):
174
LATERAL DYNAMICS
∙ 𝑔;
∙
;
0
∙
)∙
𝑤
)∙
(
(
∙ ;
(
)∙
𝑤
)∙
(
[4.38]
0;
)∙
(
0;
Equilibrium for each axle (roll, around roll centre):
(
𝑠
)∙
(
𝑠
)∙
𝑤
𝑤
(
𝑠
(
𝑠
𝑤
)∙
𝑤
)∙
∙
∙
𝑅𝐶
𝑅𝐶
0;
0;
[4.39]
Compatibility, to introduce body displacements, z, px and py, gives:
𝑧
𝑧
𝑧
𝑧
𝑧
𝑧
𝑧
𝑧
𝑧
𝑧
𝑧
𝑧
𝑤
𝑤
𝑤
𝑤
∙𝑝
∙𝑝 ;
∙𝑝
∙𝑝 ;
∙𝑝
∙𝑝 ;
∙𝑝
∙𝑝 ;
[4.40]
0;
0;
The measure Dh is redundant and can be connected to the other geometry measures as follows. The
geometrical interpretation is given in Figure 4-40.
∆
∙
𝑅𝐶
∙
𝑅𝐶
;
[4.41]
Combining Equations [4.37] to [4.41] gives, as Matlab script and solution:
clear, syms zfl zfr zrl zrr Fflz Ffrz Frlz Frrz Fsfl Fsfr Fsrl
Fsrr Fsfl0 Fsfr0 Fsrl0 Fsrr0 Ffy Fry z px py h
sol=solve( ...
'Fsfl=Fsfl0-cfw*zfl', ...
'Fsfr=Fsfr0-cfw*zfr', ...
'Fsrl=Fsrl0-crw*zrl', ...
'Fsrr=Fsrr0-crw*zrr', ...
'Fsfl0=(1/2)*m*g*lr/L', ...
'Fsfr0=(1/2)*m*g*lr/L', ...
'Fsrl0=(1/2)*m*g*lf/L', ...
'Fsrr0=(1/2)*m*g*lf/L', ...
'Fflz+Ffrz+Frlz+Frrz=m*g', ...
'm*ay=Ffy+Fry', ...
'0=Ffy*lf-Fry*lr', ...
'-(Fflz+Ffrz)*lf+(Frlz+Frrz)*lr=0', ...
'(Fflz+Frlz)*w/2-(Ffrz+Frrz)*w/2+(Ffy+Fry)*h=0', ...
'(Fflz-Fsfl)*w/2-(Ffrz-Fsfr)*w/2+Ffy*hRCf=0', ...
'(Frlz-Fsrl)*w/2-(Frrz-Fsrr)*w/2+Fry*hRCr=0', ...
'zfl=z+(w/2)*px-lf*py', ...
'zfr=z-(w/2)*px-lf*py', ...
'zrl=z+(w/2)*px+lr*py', ...
'zrr=z-(w/2)*px+lr*py', ...
'zfl+zfr=0', ...
'zrl+zrr=0', ...
'dh=h-(lr*hRCf+lf*hRCr)/(lf+lr)', ...
175
[4.42]
LATERAL DYNAMICS
zfl,
zfr, zrl,
zrr, ...
Fsfl, Fsfr, Fsrl, Fsrr, ...
Fsfl0, Fsfr0, Fsrl0, Fsrr0, ...
Fflz, Ffrz, Frlz, Frrz, ...
Ffy, Fry, z, px, py, h);
The result from the Matlab script in Equation [4.42], but in a prettier writing format:
𝑧
∙
∙
;
∙
∙
;;
0;
∙
𝑝
∙∆
𝑐
0;
𝑝
)∙∆
(
𝑐
ℎ𝑐
ℎ𝑐
;
∙
∙𝑤
𝑐
∆
∙
𝑤 𝑐
ℎ𝑐
∙
∙𝑤
𝑐
∆
∙
𝑤 𝑐
ℎ𝑐
𝑔∙
∙(
∙
𝑅𝐶 ∙
∙(
∙𝑤
∆
𝑐
∙
𝑤 𝑐
ℎ𝑐
𝑔∙
∙(
∙
∙(
∙
∙𝑤
∆
𝑐
∙
𝑤 𝑐
ℎ𝑐
𝑔∙
∙(
∙
∙(
𝑅𝐶
𝑔∙
∙(
∙
∙(
𝑅𝐶
𝑅𝐶
)) ;
)) ;
[4.43]
)) ;
)) ;
where, roll stiffnesses in [moment/angle], are:
𝑐
∙𝑐
𝑤 2
∙
(
)
𝑤
𝑐
∙𝑐
𝑤
𝑐
𝑤 2
∙( )
𝑐
ℎ𝑐
𝑁
];
𝑟 𝑑
𝑁
[
];
𝑟 𝑑
[
𝑐
[
𝑁
];
𝑟 𝑑
We should compare Equation [4.43] with Equation [4.36]. Equation [4.36] considers the “pendulum
effect”, but not the differentiation between front and rear suspension. Equation [4.43] does the
opposite.
Assume
in Equation [4.36]. Compare
𝑅𝐶 and look at the sum of vertical force on one side,
in Equation [4.36] and
in Equation [4.43]; the equations agree if:
⇒
⇒
𝑅𝐶
∙
𝑅𝐶
∙𝑤
∙
𝑔
∙(
∆
𝑤
∙(
𝑤
⇒
𝑅𝐶
∙
𝑅𝐶
∙
∆
))
𝑤
𝑅𝐶
∙
(
∙𝑤
𝑅𝐶
∙
𝑔
∙(
∙ )
𝑤
⇒
∆ )∙ ;
This is exactly in agreement with the definition of the redundant geometric parameter Dh, see
Equation [4.41]. This means that a consistent geometric model of the whole model is as drawn in
Figure 4-40. Here the artefact roll axis is also defined.
176
LATERAL DYNAMICS
front axle
roll centre
CoG
hRCf
Dh
front axle
h
rear axle
roll centre
hRCr
rear axle
𝑅𝐶
∆
𝑅𝐶
Figure 4-40: Roll axis for a two axle vehicle. (Note that the picture may indicate that the roll
centres and roll axis are above wheel centre, but this is normally not the case.)
The terms of type
ℎ𝑅𝐶𝑖 ∙ 𝑗
𝐿∙𝑤
in Equation [4.43] can be seen as the part of the lateral tyre forces that goes
via the stiff linkage. The terms of type
∆ℎ
𝑐
∙ 𝑖𝑤
𝑤 𝑐𝑖𝑤 +𝑐𝑗𝑤
in Equation [4.43] can be seen as the part of the
lateral tyre forces that goes via the compliance. The latter part is distributed in proportion to roll
stiffness of the studied axle, as a fraction of the vehicle roll stiffness. This should be in agreement
with intuition and experience from other preloaded mechanical systems (load distributes as
stiffness).
Body rolls with positive roll when steering to the left, as long as CoG is above roll axle. Further, the
body centre of gravity is unchanged in heave (vertical z) because the model does not allow any
vertical displacements, which is a drawback already mentioned.
4.3.9.4
Influence of anti-roll bars
Anti-roll bars will be added to model in this section. Figure 4-41 shows the physical model which
includes anti-roll bars. It is an update of Figure 4-39.
Eq [4.37] will remain, and constitutive relation for anti-roll bars is added as in Eq [4.44]. Also here,
the road is assumed to be smooth. Note that anti-roll bars are assumed to have no pre-tension. Antiroll bar stiffnesses for each axle are design parameters, 𝑐 𝑤 𝑒𝑟𝑒 𝑖 𝑓 𝑟.
𝑐
∙ ((𝑧
𝑧 )
(𝑧
𝑧 )) ;
𝑐
∙ ((𝑧
𝑧 )
(𝑧
𝑧 ));
[4.44]
The equilibrium for whole vehicle, Eq [4.38], does not change. The equilibrium for each axle, Eq
[4.39], changes to the following:
177
LATERAL DYNAMICS
Equilibrium for each axle (pitch, around roll centre):
𝑤
𝑤
(
)∙
(
)∙
𝑠
𝑠
𝑤
𝑤
(
)∙
(
)∙
𝑠
𝑠
∙
∙
0;
𝑅𝐶
[4.45]
0;
𝑅𝐶
The compatibility Equation, Eq [4.40], does not change and the definition of Δ , Eq [4.41], can also
be kept. It shows that Eq [4.43] is still valid, but with new definitions of the roll stiffnesses as follows:
𝑐
∙ (𝑐
𝑤
𝑤 2
∙𝑐 )∙( )
𝑐
∙ (𝑐
𝑤
𝑤 2
∙𝑐 )∙( )
𝑐
𝑐
ℎ𝑐
y
zfrr=zrrr=0
forces,
on vehicle:
Faf*w
Far*w
zfr=zrr
px
zflr=zrlr=0
Pry
Pfy
Only one anti-roll bar spring
drawn, but there can be one
anti-roll bar on each axle.
Fsfl+Fsrl
forces, on front axle:
Fsfl
h
Ffry+Frry
Fflz+Frlz
m*ay
Anti-roll bar
torsional spring
m*g
m*ay
Ffly+Frly
𝑁
];
𝑟 𝑑
m*g
z
zfl=zrl=0
[
[4.46]
forces, on body:
displacements, speeds, accelerations :
ay
𝑐
𝑁
];
𝑟 𝑑
𝑁
[
];
𝑟 𝑑
[
Ffrz+Frrz
Faf
Ffly
Fflz
Pfz
Fsfr+Fsrr
Pfz+Prz
forces, on rear axle:
Fsfr
Pfy Faf
Ffry
Ffrz
Fsrl
Prz
Fsrr
Pry
Far
Frly
Frlz
Far
Frry
Frrz
Figure 4-41: Model for steady state heave and roll due to lateral acceleration, using roll centres,
which can be different front and rear. Including anti-roll bar on each axle.
4.3.9.5
Axle suspension system
Suspension design is briefly discussed at these places in this compendium: Section 3.4.7, Section
4.3.9.5 and Section 5.2.
There are axles with dependent wheel suspensions, which basically look as the roll centre axle model
in Figure 4-38 , i.e. that left right wheel are rigidly connected to each other. Then, there are axles
with dependent wheel suspensions, which look more like the model with wheel pivot points in Figure
4-38. For these, there are no (rigid) connections between left and right wheel.
178
LATERAL DYNAMICS
Many axles have a so called anti-roll bar, which is a elastic connection between left and right side. It
is connected such that if the wheel on one side is lifted, it lifts also the wheel on the other side. Note
that, if an anti-roll bar is added to an independent wheel suspension it is still called independent,
because the connection is not rigid.
Figure 4-42 and Figure 4-43 show design of two axles with independent wheel suspensions. Figure
4-44 shows an axle with dependent wheel suspension. In all these figures it is shown how to find
wheel pivot points and roll centre. In the McPherson suspension in Figure 4-43, one should mention
that the strut is designed to take bending moments. For the rigid axle in Figure 4-44, one should
mention that the leaf spring itself takes the lateral forces. Symmetry between left and right wheel
suspension is a reasonable assumption and it places the roll centre symmetrically between the
wheels, which is assumed in the previous models and equations regarding roll centre. (Symmetry is
not reasonable to assume for pitch centre.)
motion for points
moving with hub
Pivot point
for wheel
90 deg
motion for a point
moving with hub, where
the wheel which is in
contact with ground
90 deg
Roll centre
for axle
From the other
wheel on same
axle
90 deg
Figure 4-42: Example of how to appoint the pivot point for one wheel and roll centre for axle with
double wishbone suspension.
motion for
points moving
with hub
90 deg
Pivot point
for wheel
motion for a
point moving
with hub, where
the wheel which
is in contact
with ground
90 deg
Roll centre
for axle
From the other
wheel on same
axle
90 deg
general
(which, infor
theaxle with
Figure 4-43: Example
of how to
the pivot
point for =one wheel
andmotion
roll centre
radial motion
+ appoint
angular
motion
pivot&slide point has to be
double McPherson suspension. perpendicular to the slide).
Generally, a ”rigid axle” gives roll centre height on approximately the same magnitude as wheel
radius, see Figure 4-44. With individual wheel suspension one have much larger flexibility, and typical
chosen designs are 30..90 mm front and 90..120 mm rear.
179
LATERAL DYNAMICS
Figure 4-44: Example of how to appoint the pivot point for one wheel and roll centre for axle with
rigid axle suspended in leaf springs. From (Gillespie, 1992).
The design of roll centre height is a trade-off. On one side, high roll centre is good because it reduces
roll in steady state cornering. On the other side, low roll centre height is good because it gives small
track width variations due to vehicle heave variations. Track width variations are undesired, e.g.
because it makes the left and right tyre lateral force fight against each other, leaving less available
friction for longitudinal and lateral grip. Roll centre is normally higher rear than front. One reason for
that is that the main inertia axis leans forward, and parallelism between roll axis and main inertia axis
is desired.
4.3.9.6 Lateral load transfer influence on steady state
handling
The lateral load transfer will influence the steady state cornering in some different ways. A very
fundamental view of a tyre is maybe that forces are proportional to vertical load, and then the
cornering stiffness should also vary in this way. This is a good first approximation, but there are
actually mechanism that makes the increase degressive, see Figure 2-26. This means that the axle
cornering stiffness decrease with increased load transfer.
The total roll stiffness of the vehicle does not influence the understeering gradient, but the
distribution of roll stiffness between front and rear axle does. Normally one makes the front axle
more roll stiff than the rear axle. This means that vehicle becomes more and more understeered for
increased lateral acceleration, e.g. more steering angle is needed to maintain a certain path radius if
speed increases.
180
LATERAL DYNAMICS
3.7 ∙ 10 𝑁/𝑟 𝑑
3.1 ∙ 10 𝑁/𝑟 𝑑
Both wheels, if no load transfer.
Axle cornering stiffness ≈
≈ ∙ 3.1 6. ∙ 10 𝑘𝑁/𝑟 𝑑
.1 ∙ 10
𝑁/𝑟 𝑑
Left and right wheel, if load transfer
Axle cornering stiffness ≈ .1 3.7
5 𝑘𝑁
𝜕
Figure 4-45: The wheels cornering stiffness ((𝜕𝛼
)|
5 𝑘𝑁.
5.8 ∙ 10 𝑁/𝑟 𝑑
5 𝑘𝑁
𝑠𝑦 =
) changes degressively with vertical load.
The axle cornering stiffness therefore decreases with load transfer.
4.3.10 Steering feel *
Function definition: Steering feel is the steering wheel torque response to steering wheel angle. The
function is used in a very wide sense; on a high level, it is a measure of steering wheel torque, or its variation,
for certain driving situations. Often, it can only be subjectively assessed.
At steady state driving at high speed, there are basically three aspects of steering feel:



Lateral steering feel feedback at cornering. The steering wheel torque is normally desired to
increase monotonously with lateral forces on the front axle. This is basically the way the
mechanics work due to caster trail. Some specifications on steering assistance system is
however needed to keep the steering wheel torque low enough for comfort.
Steering torque drop when cornering at low-friction. It is built into the mechanics of the
caster trail and the pneumatic trail that steering wheel torque drops slightly when one
approaches the friction limit on front axle. This is normally a desired behaviour because it
gives driver feedback that the vehicles is approach the limits.
On-centre feel in straight line driving. When the vehicle is driven in straight line, the steering
wheel is normally desired to return to centre position after small perturbations. This is a
comfort function, which OEMs works a lot with and it is often rather subjectively assessed.
4.3.11 Roll-over in steady state cornering
When going in curves, the vehicle will have roll angles of typically some degrees. At that level, the roll
is a comfort issue. However, there are manoeuvres which can cause the vehicle to roll-over, which
basically means that it rolls at least 90 degrees. So, this is an actual accident event.
Roll-over can be seen as a special event, but if sorting into the chapters of this compendium it
probably fits best in present chapter, about lateral dynamics.
One can categorize roll-overs in e.g. 3 different types:
181
LATERAL DYNAMICS


Tripped roll-over. This is when the car skids sideways and hits an edge, which causes the rollover. It can be an uprising edge, e.g. pavement or refuge. It can be the opposite, a ditch or
loose gravel outside road. In both these cases, it is strong lateral forces on the wheels on one
side of the vehicle that causes the roll-over.
Tripped roll-over can also be when the vehicle is exposed to large one-sided vertical wheel
forces, e.g. by running over a one-sided bump.
A third variant of tripped roll-over is when the vehicle is hit by another vehicle so hard that it
rolls over.
Un-tripped roll-over or on-road roll-overs. These happen on the road and triggered by high
tyre lateral forces. This is why they require high road friction. For sedan passenger cars these
event are almost impossible, since road friction seldom is higher than approximately 1. For
SUVs, un-tripped roll-overs can however occur but require dry asphalt roads, where friction
is around 1. For trucks, un-tripped roll-over, can happen already at very moderate friction,
like 0.4, due to their high CoG in relation to track width. Within un-tripped roll-overs, one can
differ between:
o Steady state roll-over. If lateral acceleration is slowly increased, e.g. as running with
into a hairpin curve or a highway exit, the vehicle can slowly lift off the inner wheels
and roll-over. This is the only case of roll-over for which an analysis model is given in
this compendium.
o Transient roll-over. This is when complex manoeuvres, like double lane changes or
sinusoidal steering, are made at high lateral accelerations. This can trigger roll eigenmodes, which can be amplified due to unlucky timing between the turns. Analysis
models from Section 4.5 can be used as a start, but it is required that load transfer is
modelled carefully and includes wheel lifts, suspension end-stops and bump stops.
4.3.11.1 Roll-over threshold definitions
An overall requirement on a vehicle is that the vehicle should not roll-over for certain manoeuvres.
Heavy trucks will be possible to roll-over on high-mu conditions. The requirement for those are based
on some manoeuvres which not utilize the full road friction. For passenger cars, it is often the
intended design that they should be impossible to roll-over, even at high mu. Any requirement need
a definition of what exactly roll-over is, i.e. a Roll over threshold definition. Candidates for Roll over
threshold definition are:



One wheel lift from ground
All wheels on one side lift from ground
Vehicle CoG passes its highest point
Note that:



It is the 3rd threshold which really is the limit, but other can still be useful in requirement
setting. To use the 3rd for requirement setting makes the verification much more complex, of
course in real vehicles but also in simulation.
The 1st is not a very serious situation for a conventional vehicle with 4 wheels. However, for
a 3-wheeled vehicle, such as small “tuc-tucs” or a 3-wheel moped, it is still a relevant
threshold.
The 2nd threshold is probably the most useful threshold for two-tracked vehicles, because it
defines a condition from which real roll-over is an obvious risk, and still it is relatively easy to
test and simulate. For 3-wheeled vehicle, 2nd and 3rd threshold generally coincide.
Figure 4-46 shows how the inner wheels lift off subsequently during a slowly increasing lateral force
(or lateral acceleration) build-up. Before any wheel is lifted, the load transfer is proportional to rollcentre heights and roll stiffnesses, as shown in Equation [4.43]. But everytime a wheel lifts, the
distribution changes, so that a “knee” on the curves appears, see Figure 4-46. So, the relation of type
182
LATERAL DYNAMICS
Vertical force on
inner wheels
last inner
wheel lift
⋅𝑔
second inner
wheel lift
first inner
wheel lift
as Equation [4.43] is no longer valid. For instance, it is not physically motivated to keep the rollcentre model for an axle which has lifted one side. So, the analysis is not trivial, especially for heavy
vehicles which has many axles, and often also a fifth wheel which can transfer roll-moment to a
certain extent. There are approximate standards for how to calculate steady state roll-over
thresholds for such vehicle, e.g. UN ECE 111
(http://www.unece.org/fileadmin/DAM/trans/main/wp29/wp29regs/r111e.pdf).
Lateral force
(or lateral acceleration)
Figure 4-46: Example of 3-axle vehicle steady state roll-over wheel lift diagram.
In the following, 2-axled and 4-wheeled vehicles will be assumed. The 2nd threshold will be used.
4.3.11.2 Static Stability Factor, SSF
One very simple measure of the vehicles tendency to roll-over is the Static Stability Factor, SSF. It is
proposed by NHTSA, http://www.nhtsa.gov/cars/rules/rulings/roll_resistance/, and it is simply
defined as:
𝐻 𝑓 𝑇𝑟 𝑐𝑘 𝑖𝑑𝑡
𝐻𝑒𝑖𝑔𝑡 𝑂𝑓𝐶𝑜𝐺
SSF
𝑤⁄
;
[4.47]
A requirement which requires SSF>number cannot be directly interpreted in terms of certain
manoeuvre and certain roll-over threshold. It is not a performance based requirement, but a design
based requirement. However, one of many possible performance based interpretations is that the
vehicle shall not roll-over for steady-state cornering on level ground with a certain friction
coefficient, using one-sided wheel lift as threshold. Since the requirement is not truly performance
based, each interpretation will also stipulate a certain verification method; here it would be
theoretical verification using a rigid suspension model. Such model and threshold is shown in Figure
4-47.
The derivation of the SSF based requirement looks as follows:
∙𝑤
∙
∙
𝑜𝑑𝑒 : {
∙𝑔∙
𝑤
}⇒
∙ 𝑔;
∙
𝑅𝑒𝑞𝑢𝑖𝑟𝑒 𝑒𝑛𝑡:
𝜇∙(
;
∙𝑔∙(
1
);
≥ 0;
∙𝜇
);
⇒
𝑤
}
⇒ 𝑅𝑒𝑞𝑢𝑖𝑟𝑒 𝑒𝑛𝑡:
1
183
∙𝜇
𝑤
⇒
𝑤
∙
𝑆𝑆
𝜇;
[4.48]
LATERAL DYNAMICS
Maximum road friction, m, is typically 1, which is why SSF>m1 would be a reasonable. However,
typical values of SSF for passenger vehicles are between 0.95 and 1.5. For heavy trucks, it can be
much lower, maybe 0.3..0.5, much depending on how the load is placed. There are objections to use
SSF as a measure, because SSF ignores suspension compliance, handling characteristics, electronic
stability control, vehicle shape and structure.
view from rear, when turning left
ay
m*g
m*ay
h
Fy
Fiz≥0
(=roll-over
threshold)
w/2
Foz
w
Figure 4-47: Model for verification of requirement based on Static Stability Factor, SSF.
4.3.11.3 Steady-state cornering roll-over
A function defined for requirement setting can be:
Function definition: Steady state roll-over resistance is the maximum lateral acceleration the
vehicle can take in steady state cornering without lifting all inner wheels. On level ground with enough road
friction and certain weight and position of payload.
For a long combination-vehicle with several articulation points, one often need to drive a long
distance after a steering angle change before steady state values on articulation angles are reached.
Hence, it can be more relevant to formulate a corresponding roll-over function in terms of curvature
to follow, total yaw angle for turn and longitudinal speed. A common way is also the, somewhat
artificial, tilt-table test, which means that one measure steady-state roll-over with a (real or virtual)
tilt-table where the maximum road pitch angle before wheel lift on one side is the measure to set
requirement on.
4.3.11.3.1
Model with fore/aft symmetry
The model in Figure 4-47 and Equation [4.48] assumes fore/aft symmetry. One can derive this
𝑤
requirement on design: 2∙ℎ 𝜇, and this interpretation of performance ( ) limitation due to rollover:
𝑦
<
𝑤
.
2∙ℎ
affects this:
𝑦
In the following, we will elaborate with 4 additional effects and derive how they
<
𝑤
2∙ℎ
𝜇. The 4 effects are each connected to one measure, which is marked in
Figure 4-48.


The tyre will take the vertical load on its outer edge in a roll-over situation. This suggests a
𝑤+𝒘
change of performance and requirement to: 𝑦 < 2∙ℎ𝒕𝒚
𝜇. This effect is accentuated
when low tyre profile and/or high inflation pressure. This effect decreases the risk for rollover.
Due to suspension and tyre lateral deformation, the body will translate laterally outwards,
relative to the tyre. This could motivate
𝑦
<
roll-over.
184
𝑤−𝑫 𝒇𝒚
2∙ℎ
𝜇. This effect increases the risk for
LATERAL DYNAMICS

Due to suspension linkage and compliances, the body will roll. Since the CoG height above
𝑦
roll axis, ∆ , normally is positive, this could motivate


<
𝑤−∆ ∙ 𝒙
2∙ℎ
𝜇. This effect increases
the risk for roll-over. At heavy vehicle this “pendulum effect” is large.
Due to suspension linkage and compliances, the body will also heave. This requires a
suspension model with pivot points per wheel, as opposed to roll-centre per axle, to be taken
𝑤
into account. The heave is normally positive. This could motivate 𝑦 < 2∙(ℎ+ ) 𝜇. The effect
is sometimes called “jacking” and it increases the risk for roll-over.
Road leaning left/right (road banking), or driving with one side on a different level (e.g. outside road or on pavement) also influence the roll-over performance.
view from rear, when turning left
𝑥
ay
m*g
m*ay
𝑧
∆
h
Fiz=0
(=roll-over
threshold)
Fy
𝐷𝑒𝑓𝑦
Foz
wtyre/2
w/2
w
Figure 4-48: Steady-state roll-over model, with fore/aft symmetry. The measures 𝑤𝑡
𝐷𝑒𝑓 ∆ ∙
𝑛𝑑 𝑧 mark effects additional to what is covered with a simple SSF approach.
4.3.11.3.2
Model without fore/aft symmetry
As steady state roll-over threshold which takes into account the differences between front and rear
suspension is shown in Equation [4.43]. If assuming roll-over in left curve, the wheels to observe
would be the front left and rear left. Hence, the vehicle will roll-over when:
∙
∙ ( 2∙𝐿𝑟
∙(
∙
ℎ𝑅𝐶𝑟 ∙
∙ ( 2∙𝐿
⇒
ℎ𝑅𝐶 ∙ 𝑟
𝐿∙𝑤
∙(
𝑔
∙ max
∙
𝑅𝐶
(
𝐿∙𝑤
∙
∙𝑤
𝑐 𝑤
∆ℎ
∙
))
𝑤 𝑐 𝑤 +𝑐𝑟𝑤
< 0; 𝐴𝑁𝐷
∆ℎ
𝑐
∙ 𝑟𝑤 ))
𝑤 𝑐 𝑤 +𝑐𝑟𝑤
𝑐𝑤
∆
∙
𝑤 𝑐𝑤 𝑐
;
𝑤
⇒
< 0;
}
∙
∙𝑤
𝑅𝐶
[4.49]
∆
𝑐𝑤
∙
𝑤 𝑐𝑤 𝑐
;
𝑤)
It should be noted that this roll-over condition is approximate, in that the effects in section
4.3.11.3.1, are NOT taken into account. Furthermore, the end-stop of the inner wheel which lifts first
is NOT taken into account, i.e. it is NOT modelled that the wheel that have lifted takes no more
vertical forces.
4.3.11.3.3
Using a transient model for steady-state roll-over
Another work-around to avoid complex algebra is to run a fully transient model, including
suspension, and run it until a steady state cornering conditions occur. If then, the lateral acceleration
is slowly increased, one can identify when or if the roll-over threshold is reached. Lateral acceleration
185
LATERAL DYNAMICS
increase can be through either increase of longitudinal speed or steering angle. It should be noted
that the model should reasonably be able to manage at least lift of one wheel from the ground. This
way of verifying steady state cornering roll-over requirements has the advantage that, if using tyre
models with friction saturation, the limitation discussed in Section 0 does not have to be checked
separately.
4.3.11.4 Roll-over and understeering/propulsion
With the above formulas for roll-over there will always be a certain lateral acceleration that leads to
roll-over, because neither limitation due to road friction nor propulsion power modelled yet. Since
vehicles generally are understeered, they are limited to develop lateral acceleration, see Figure 4-32.
For propulsion-weak vehicles, there is also the limitation of lateral acceleration due to limited
longitudinal speed, which in turn is due to driving resistance from the steered wheels (=wheel lateral
force * sin(steering angle)) and loosing propulsion power due to longitudinal wheel slip. However,
one should take into account that the propulsion limitation is less in down-hill driving, which
increases the roll-over risk again. Also, if the vehicles goes relatively quickly into steady state
cornering, the longitudinal speed will not have time to decelerate to its real (longitudinal) steady
state value.
For heavy trucks, the critical lateral acceleration is typically 0.3..0.4 g, which is quite possible to reach
during normal road conditions, because road friction is there around 1. For passenger cars, the
critical lateral acceleration is typically in the region of 1, so it is not obvious that it is possible to reach
the roll-over-critical lateral acceleration. This is also the case for heavy trucks on low road friction.
4.4 Stationary oscillating steering
In between steady state and transient manoeuvres, one can identify stationary oscillations as an
intermediate step.
Generally, a mechanical system can be excited with a stationary oscillating disturbance. The response
of the system is, after possible transients are damped out, a stationary oscillation. If staying within
the linear region for the system and the excitation is harmonic (sinus and cosine), the ratio between
the response amplitude and the excitation amplitude is only dependent of the frequency. The ratio is
called transfer function.
For lateral vehicle dynamics, the excitation is typically steering wheel angle and the response is
amplitudes of yaw rate, curvature or lateral acceleration. The corresponding transfer functions are a
frequency version of the gains defined in Equation [4.28].
Also, there will be a delay between excitation and response. This is another important measure,
beside the amplitude ratio.
4.4.1
Stationary oscillating steering tests
When testing Stationary oscillating steering functions, one usually drives on a longer part of the test
track. It might be a high speed track, see Figure 4-8, because one generally need to find the response
at high speeds, rather than driving close to lateral limits. So the track rather needs to be long than
wide. If the available Vehicle Dynamics Area, see Figure 4-8, is long enough this can be a safer option.
A Vehicle Dynamics Area is a flat surface with typically 100..300 m diameter. It normally has entrance
roads for accelerating up to a certain speed.
Typical tests in this part of lateral vehicle dynamics are:


Sweeping frequency and/or amplitude
Random frequency and amplitude
186
LATERAL DYNAMICS
There are ISO standards for both these tests. The response will be very dependent of the vehicle
longitudinal speed, why the same tests are typically done at different such speeds.
4.4.2
Transient one-track model
The model needed for stationary oscillation is only a linearization of the model needed for fully
transient handling, in Section 4.5. However, a rather complete model will be derived already in
present section, to capture the physical assumptions in a proper way. (If the reader is satisfied with
the linearized model, it can be found directly in Equation [4.54], and a less general derivation of this
equation in Figure 4-53.)
The vehicle model is sketched in Figure 4-51. The model is a development of the model for steady
state cornering in Figure 4-21, with the following changes:

Longitudinal and lateral accelerations don’t only have the components of centripetal
acceleration (wz*vy and wz*vx), but also the derivatives of vx and vy:
𝑑𝑒𝑟( )
𝑑𝑒𝑟( )


∙
∙
∙
∙
;
;
[4.50]
The yaw acceleration, der(wz), is no longer zero.
The speed vx is no longer defined as a parameter, but a variable. Then, one more
prescription is needed to be a consistent model. For this purpose, an equation that sets front
axle propulsion torque to 1000 Nm is added.
The difference between acceleration ⃗ [ ; ] and [der(vx); der(vy)]=[ ; ] can be confusing. A
way to understand those could be to think of ⃗ as the (geometric vector) acceleration in the inertial
coordinate system and [ ; ] as a (mathematical) vector, with derivatives of the scalar
mathematical variables and , which are the velocities in the vehicle fix (and moving) coordinate
system. A graphical derivation of Equation [4.50] is found below, in Figure 4-54.
The model in Figure 4-51 is documented as mathematical model in Equation [4.51]. The equation is
given in Modelica format. The subscript v and w refers to vehicle coordinate system and wheel
coordinate system, respectively. The model is a development of the model for low-speed in Equation
[4.12], with the changes marked with underlined text:
187
LATERAL DYNAMICS
wz
vx
vrx
lr
vfyv
vry
vy
forces:
vfxv ⃗
Frx
df
m*ay
Ffxv ⃗
Ffyv
speeds:
Fry
m*ax
df
J*der(wz)
lf
L
lr
lf
L
⃗
⃗
ax
𝒗
ay
b
vy=v*sin(b)
Figure 4-51: One-track model for transient dynamics. Dashed forces and moment are fictive forces.
Compare to Figure 4-21.
//(Dynamic) Equilibrium:
m*ax = Ffxv + Frx;
m*ay = Ffyv + Fry;
J*der(wz) = Ffyv*lf - Fry*lr;
ax=der(vx)-wz*vy;
ay=der(vy)+wz*vx;
//Constitutive relation (Lateral tyre force model):
Ffyw=-Cf*sfy;
Fry=-Cr*sry;
sfy=vfyw/vfxw;
sry=vry/vrx;
//Compatibility:
vfxv = vx;
vfyv = vy + lf*wz;
vrx = vx;
vry = vy - lr*wz;
[4.51]
//Transformation between vehicle and wheel coordinate systems:
Ffxv = Ffxw*cos(df) - Ffyw*sin(df);
Ffyv = Ffxw*sin(df) + Ffyw*cos(df);
vfxv = vfxw*cos(df) - vfyw*sin(df);
vfyv = vfxw*sin(df) + vfyw*cos(df);
//Path with orientation:
der(x) = vx*cos(pz) - vy*sin(pz);
der(y) = vy*cos(pz) + vx*sin(pz);
der(pz) = wz;
// Prescription of steering angle:
df = if time < 2.5 then (5*pi/180)*sin(0.5*2*pi*time) else 5*pi/180;
//Shaft torques:
Ffxw = +1000; // Front axle driven.
Frx = -100; // Rolling resistance on rear axle.
188
LATERAL DYNAMICS
The initial longitudinal speed is a parameter, vx=100 km/h. A simulation result from the model is
shown in Figure 4-52. The manoeuvre selected is same steering wheel function of time as in Figure
4-22, for better comparison of the different characteristics of the models.
y [m]
df [deg]
wz [deg/s]
vx [km/h]
beta [deg]
x[m]
time [s]
Figure 4-52: Simulation results of one-track model for transient dynamics.
Both position variables x, y, pz and speed variables vx, vy and wz are “state variables” of this
simulation. For each continuation from one time instant, the future solution requires knowledge of
the previous states. This means that in the beginning, initial values were needed on the state
variable. The only non-zero initial value was for vx, which was given to 100 km/h.
Equation [4.51] is a complete model suitable for simulation. Eliminating some variable and rewrite in
prettier format gives:
Equilibrium:
∙(
∙(
𝐽∙
∙
∙
)
)
𝑤 ∙ sin(𝛿 )
(
∙ cos(𝛿 )
𝑤 ∙ sin(𝛿 )
𝑤 ∙ sin(𝛿 )
𝑤 ∙ cos(𝛿 )
∙ ;
𝑤 ∙ cos(𝛿 )) ∙
𝑤
;
;
Constitution:
𝐶 ∙𝑠 ;
𝑤
𝑛𝑑
𝐶 ∙𝑠 ;
Compatibility:
𝑤
𝑠
𝑤
;
𝑛𝑑 𝑠
∙
;
Transformation from vehicle to wheel coordinate system on front axle:
𝑤
𝑤
(
(
∙
∙
) ∙ sin(𝛿 )
) ∙ cos(𝛿 )
189
∙ cos(𝛿 ) ;
∙ sin(𝛿 ) ;
[4.52]
LATERAL DYNAMICS
Typically, this model is used for simulation, where 𝛿 ,
are input variables. Suitable state
𝑤 and
variables are then , and . It is a model suitable for arbitrary transient manoeuvres and we will
come back to this in Section 4.5. It is non-linear with respect to angles, but linear with respect to tyre
slip model. It is also non-linear with respect to that the states appear as multiplied with each other,
e.g.
∙ .
For the stationary oscillation events, this compendium will be limited to manoeuvres where far from
saturating tyre grip and at constant (⇒
0). We assume:


Low usage of lateral tyre force, i.e. small tyre slip angle (𝛼 ): 𝑠
Low level of cornering, i.e.
o small steering angle: sin(𝛿 ) ≈ 𝛿 ; cos(𝛿 ) ≈ 1 ;.
o small front body slip angle (𝛽 ): 𝛼
𝛽
𝛿;
where 𝛽
𝑟𝑐𝑡 𝑛 (
𝑦𝑣
𝑥𝑣
)≈
𝑦+
𝑦𝑣
𝑥𝑣
∙𝜔
𝑡 𝑛(𝛼 ) ≈ 𝛼 ;
;
𝑥
We then rewrite the two last equations in Equation [4.52]:
∙(
{
∙
𝐽∙
(
)
𝑤
𝑤
∙𝛿
∙𝛿
∙
𝐶 ∙(
∙
𝐶 ∙(
𝛿 )
𝛿 )) ∙
∙
𝐶 ∙
∙
𝐶 ∙
;
∙ ;
}
⇒
∙
𝐶
∙
{
∙
𝐶
⋅
𝐶 ∙
𝐽∙
𝑤
(
𝐶 ∙
𝐶 ∙
𝐶 ∙
𝐶 ∙
∙
2
∙
𝐶 ∙
[4.53]
∙
𝐶 ∙(
𝛿 )∙𝛿 ;
)∙
[
0
0
]∙[
𝐽
]
𝐶 ∙
[
[
(𝐶
𝐶
∙
𝐶 ∙
𝐶 ∙
𝛿 ;
2
𝑤) ∙
∙𝛿 ;
and
) and one algebraic
(𝐶
This equation is written on matrix form (for the two state variables,
equation for the longitudinal equilibrium. Then:
𝐶
𝑤) ∙
(𝐶
𝐶 ∙
𝐶 ∙
}
∙
2
𝐶 ∙
∙[
2
]
[4.54]
]
𝐶
𝑤
𝑤) ∙
]∙𝛿 ;
Neglecting the longitudinal equilibrium and assuming that propulsion forces can be neglected for
lateral and yaw equilibrium:
𝐶
[
0
0
]∙[
𝐽
]
𝐶 ∙
𝐶
𝐶 ∙
𝐶 ∙
𝐶 ∙
𝐶 ∙
2
[
∙
𝐶 ∙
∙[
2
]
1
[ ]∙𝐶 ∙𝛿 ;
[4.55]
]
It should be noted that the assumption about small front body slip angle (𝛽 ) makes the linearization
questionable at high curvature, i.e. at small path radii.
190
LATERAL DYNAMICS
The longitudinal equilibrium (first equation in Equation [4.52]) is often not mentioned, because the
longitudinal dynamics is prescribed as a constant speed, . However, this equation can be used to
calculate required propulsion on the axles to maintain the constant longitudinal speed. Note that it is
complemented with air and grade resistance force.
𝑤
∙ cos(𝛿 )
⋅
𝑑
∙
𝐶 ∙
(
∙
) ∙ cos(𝛿 )
∙ sin(𝛿 )
(
∙
) ∙ sin(𝛿 )
∙ cos(𝛿 )
∙ sin(𝛿 ) ;
[4.56]
If 𝛿 𝛼 𝛽 are small and 𝛽 ∙ 𝛿 ≈ 0; 𝛿 ∙ 𝛿 ≈ 0;:
𝑤
4.4.2.1
𝑑
∙
∙
;
Simpler model
A simpler way to reach this final expression is given in Figure 4-53. Here, the simplifications are
introduced earlier, already in physical model, which means e.g. that the influence of
𝑤 is not
found.
Physical model:
• Path radius >> the vehicle. Then, all forces
(and centripetal acceleration) are
approximately co-directed.
• Small tyre and vehicle side slip. Then,
angle=sin(angle)=tan(angle).
(Angles are not drawn small, which is the reason
why the forces not appear co-linear in figure.)
Equilibrium:
∙
∙
𝐽∙
≈
∙
∙
∙
≈
≈
; 𝑤 𝑒𝑟𝑒
;
∙
𝐶 ∙𝑠 ;
Constitution:
𝐽∙
bf
Ffy
Fry
𝛿
af
b
∙
wz
lr lf
df
𝛼
𝛽; 𝛽 ≈
𝛼
𝛽 ≈
0;
;
𝐶 ∙𝑠 ;
Compatibility:
∙
br
Mathematical model:
∙
∙
;
;
𝛼 ≈ 𝑠 ; 𝛼 ≈ 𝑠 ;;
Eliminate
𝐶
∙
𝒇𝒚
𝐶
𝒚
𝜷𝒇 𝜷 yields:
𝐶 ∙
𝐶 ∙
𝒇
∙
∙
∙
≈
≈𝐶 ∙𝛿 ;
L
𝐽∙
∙
∙
𝐶 ∙
𝐶
𝐶 ∙
∙
2
𝐶 ∙
2
∙
≈
≈𝐶 ∙
;
;
≈
∙
∙
∙𝛿 ;
;
Figure 4-53: Simpler derivation final step in Equation [4.55].
4.4.2.2 Relation between accelerations in inertial system
and velocity derivatives in vehicle fix system
See Equation [4.50]. The relation between accelerations in inertial system, ⃗ [ ; ] , and velocity
derivatives in vehicle fix system, [der(vx); der(vy)]=[ ; ], is further explained in Figure 4-54.
We can express the velocity in direction of the x axis at time t, at the two time instants:
191
LATERAL DYNAMICS
𝑡:
Velocity at time
𝒗𝒙
Velocity at time 𝑡 ∆𝑡: (
∆ ) ∙ cos(∆𝜓) (
∆ ) ∙ sin(∆𝜓)
}≈
( ∙ cos(∆𝜓) ∆ ∙ cos(∆𝜓)) ( ∙ sin(∆𝜓) ∆ ∙ sin(∆𝜓)) ≈ {∆𝜓 𝑠
(
∆ ) ( ∙ ∆𝜓 ∆ ∙ ∆𝜓) ≈ {∆ ∙ ∆𝜓 𝑠
} ≈ (𝒗𝒙 ∆𝒗𝒙 ) 𝒗𝒚 ∙ ∆𝝍
Using these two expressions, we can express
Change per time
≈
∙𝜓
𝒙
𝒗𝒙
{( 𝑥 +∆ 𝑥 )− 𝑦 ∙∆𝜓}− 𝑥
∆𝑡
𝒗𝒚 ∙
[4.57]
as the change of that speed per time unit:
∆ 𝑥 − 𝑦 ∙∆𝜓
∆𝑡
∆ 𝑥
∆𝑡
∆𝜓
∆𝑡
≈
[4.58]
𝒛
Corresponding for the lateral direction gives, in total, the Equation [4.50].
yglobal
y
Dwz
Dvy
vy
time=t:
wz
vx
wz
vx
y
vy
x
time=t+Dt:
Dvx
Dy
y
x
y
xglobal
Figure 4-54: Vehicle motion in ground plane
4.4.2.3
Validity of solution
When only studying the response as yaw rate and curvature response, it is easy to forget that one
very easily comes into manoeuvres where road friction is limited, i.e. where the linear tyre model is
not valid. Hence it is good to look at lateral acceleration response, because we can roughly say that
for | | in the same magnitude as 𝜇 ∙ ∙ 𝑔, it is doubtful if the model is valid. If the wheel torques
are significant, the validity limit is even lower. For high CoG vehicles, another invalidating
∙𝑤
circumstance is wheel lift, which can be approximately checked by checking that | | ≪ 𝑆𝑆
.
2∙ℎ
If one really wants to include nonlinear tyre models in stationary oscillation response, one can
simulate using time integration (same method as usually used for transient handling) over several
excitation cycles, until the response shows a clear stationary oscillation. This consumes more
computational efforts and the solutions become approximate and numerical.
192
LATERAL DYNAMICS
4.4.3
Steering frequency response gains *
Function definition: Steering frequency response gains are the amplifications from steering angle
amplitude to certain vehicle response measure’s amplitudes for stationary oscillating harmonic steering at a
certain longitudinal speed.
Equation [4.55] (or resulting eq in Figure 4-53) can be seen as a state-space-model:
[
𝑆𝑡 𝑡𝑒 𝑠𝑝 𝑐𝑒 𝑓𝑜𝑟 : {
[
where 𝑨
[
0−
] ∙ [𝐶
𝐽
0
]
𝑨∙[
]
𝑩∙𝛿 ;
]
𝑪∙[
]
𝑫∙𝛿 ;
𝐶 +𝐶𝑟
𝐶 ∙
−𝐶𝑟 ∙ 𝑟
𝑥
𝑥
∙ −𝐶𝑟 ∙ 𝑟
2
𝐶 ∙
𝑥
and 𝑩
[
and 𝑪
0
[
1
and 𝑫
𝑩;
0
∙
+𝐶𝑟 ∙ 𝑟
2
];
𝑥
[4.59]
1
0−
] ∙[ ]∙𝐶 ;
𝐽
0
]∙𝑨
0
[
0
0
1
];
When studying “stationary oscillating steering”:
[
]
𝐴∙[
]
𝐵 ∙ 𝛿(𝑡)
𝐴∙[
𝐵 ∙ 𝛿̂ ∙ cos( ∙ 𝑡)
]
(Note similar notation for vehicle yaw rate, , and steering angular frequency, .) Knowing 𝛿̂ and
, it is possible to calculate the responses ̂ , ̂ ,
and :
[
]
[
̂
] ∙ cos ( ∙ 𝑡
̂
[
[
]
̂
[ ̂ ] ∙ cos ( ∙ 𝑡
[
𝑦
𝜔
𝜔
𝑦
]) ;
]) ;
Different methods are available for calculation of the responses:
•
Real trigonometry, using cos( ∙ 𝑡
•
Complex mathematics, using 𝑒 𝑗∙𝜔∙𝑡
•
Fourier transform
𝑝
𝑠𝑒) sin( ∙ 𝑡
𝑝
𝑠𝑒) or cos( ∙ 𝑡)
sin( ∙ 𝑡).
For each of these, one can also work with scalar or matrix algebra. Matrix based Fourier transform is
generally the most efficient, especially if Matlab is available and numerical solutions can be accepted.
4.4.3.1
Single frequency response
4.4.3.1.1 Solution with Fourier transform
Equation [4.59] can be transformed to the frequency domain (where the Fourier transform of a
∞
function, 𝜉(𝑡), is denoted ℱ(𝜉(𝑡)) ∫0 𝑒 𝑗∙ ∙𝑡 ∙ 𝜉(𝑡) ∙ 𝑑𝑡):
193
LATERAL DYNAMICS
j ∙ ω ∙ ℱ ([
{
ℱ ([
𝑦
𝑨 ∙ ℱ ([
])
𝑪 ∙ ℱ ([
])
𝑩 ∙ ℱ(𝛿 );
])
[4.60]
𝑫 ∙ ℱ(𝛿 );
])
Solving for states and outputs:
[
]
ℱ
1
(ℱ ([
]))
ℱ
1
𝑨)− ∙ 𝑩 ∙ ℱ(𝛿 )) ;
((j ∙ ω ∙ I
[4.61]
[
{
𝑦
]
1
ℱ
(ℱ ([
𝑦
]))
ℱ
1
(𝑪 ∙ ℱ ([
𝑫 ∙ ℱ(𝛿 )) ;
])
Expressed as transfer functions:
ℱ ([
𝐻
𝑦
𝛿 →[ω ]
{
𝜔
𝛿 →[ ]
𝑦
𝑨)− ∙ 𝑩 ∙ ℱ(𝛿 )
(j ∙ ω ∙ I
ℱ(𝛿 )
ℱ ([
𝐻
])
])
𝑪∙𝐻
𝑦
𝛿 →[ω ]
ℱ(𝛿 )
𝑨)− ∙ 𝑩;
(j ∙ ω ∙ I
ℱ(𝛿 )
[4.62]
𝑪 ∙ (j ∙ ω ∙ I
𝑫
𝑨)− ∙ 𝑩
𝑫;
We have derived the transfer functions. The subscript tells that the transfer function is for the vehicle
operation with excitation=input
𝛿 and response=output
[
] and output
[
]. The transfer
functions has dimension 2x1 and it is complex. They operate as follows:
̂
]
̂
Amplitudes:
̂
[̂ ]
{
[
[
Phase delays:
[
{
𝑦
𝜔
𝜔
𝑦
]
]
arg(
[
arg(
arg(
[
arg(
)
]
)
)
]
)
|𝐻
𝑦
𝛿 →[𝜔 ]
|𝐻
| ∙ 𝛿̂ ;
𝜔 |∙
𝛿 →[ ]
𝛿̂ ;
𝑦
[4.63]
1
[ ] ∙ arg(𝛿 )
1
arg(𝛿 )
{
}
0
arg (𝐻
1
[ ] ∙ arg(𝛿 )
1
arg(𝛿 )
{
}
0
arg (𝐻
𝑦
𝛿 →[𝜔 ]
);
𝜔 );
𝛿 →[ ]
𝑦
4.4.3.1.2 Solution with complex mathematics
This section avoids requiring skills in Fourier transform. This makes the derivation quite long to reach
the final results Equation [4.68] and Equation [4.67]. With Fourier Transform, the expression for the
Transfer Function, H, can be reached with less algebra. Knowing H, it can be used in Equation [4.67].
The fundamental situation for steering frequency response is that the excitation is: 𝛿
𝛿̂ ∙
𝑗∙
𝑗∙2∙𝜋∙
∙𝑡
cos( ∙ 𝜋 ∙ 𝑓 ∙ 𝑡) {𝑒
cos( ) 𝑗 ∙ sin( )} Re(𝛿̂ ∙ 𝑒
), where f is the (time) frequency
in Hz and 𝑗 is the imaginary unit. We rewrite ∙ 𝜋 ∙ 𝑓 as (angular frequency), which has to be
carefully distinguished from
(yaw rate). Insert this in Equation [4.54] and neglecting the
longitudinal force
.
The
full
(complex) equation is used:
𝑤
194
LATERAL DYNAMICS
𝐶
Re [
0
]∙[
𝐽
0
𝑐
𝑐
]
𝐶
𝐶 ∙
[
[
𝐶 ∙
𝐶 ∙
𝐶 ∙
2
𝐶 ∙
∙
∙[
2
𝐶 ∙
𝑐
𝑐
]
[
𝐶
] ∙ 𝛿̂ ∙ 𝑒 𝑗𝜔∙𝑡 ;
𝐶 ∙ 𝑓
]
[4.64]
]
We intend to solve the complex equation, and then find the solutions as real parts:
and
Re[ 𝑐 ];. (Subscript c means complex.)
Re[
𝑐 ];
If only interested in the stationary solution, which is valid after possible initial value dependent
transients are damped out, we can assume a general form for the solution.
[
𝑐
𝑐
]
[
̂𝑐
] ∙ 𝑒 𝑗∙𝜔∙𝑡 ⇒ [
̂𝑐
𝑐
𝑐
𝑗∙
]
∙[
̂𝑐
] ∙ 𝑒 𝑗∙𝜔∙𝑡 ;
̂𝑐
[4.65]
Inserting the assumption in the differential equation gives:
[
̂𝑐
0
]∙𝑗∙ ∙[
] ∙ 𝑒 𝑗∙𝜔∙𝑡
𝐽
̂𝑐
𝐶
𝐶
𝐶 ∙
𝐶 ∙
0
𝐶 ∙
[
̂𝑐
⇒[
]
̂𝑐
𝐶 ∙
𝐶 ∙
0
(
𝐻
𝐶 ∙
∙[
2
̂𝑐
] ∙ 𝑒 𝑗∙𝜔∙𝑡
̂𝑐
1
[ ] ∙ 𝐶 ∙ 𝛿̂ ∙ 𝑒 𝑗∙𝜔∙𝑡 ⇒
]
[4.66]
𝐶
[
2
∙
0
]∙𝑗∙
𝐽
𝑦
𝛿 →[𝜔 ]
𝐶 ∙
[
∙ 𝛿̂ ;
𝐶
𝐶 ∙
𝐶 ∙
𝐶 ∙
−
𝐶 ∙
2
∙
𝐶 ∙
1
∙ [ ] ∙ 𝐶 ∙ 𝛿̂
2
])
Then, we can assume we know ̂ 𝑐 and ̂ 𝑐 from Equation [4.66], and consequently we know 𝑐
and 𝑐 from Equation [4.65]. We have derived the transfer function, 𝐻
𝑦 . The subscript tells
𝛿 →[𝜔 ]
𝛿 and
that the transfer function is for the vehicle operation with excitation=input
response=output
[
] case. This transfer function has dimension 2x1 and it is complex. It operates
as follows:
̂
| |
] |𝐻
] [
𝑦 | ∙ |𝛿 |;
𝛿 →[𝜔 ]
̂
| |
arg( )
1
Phase delays: [
] [ ] ∙ arg(𝛿 ) {arg(𝛿 )
1
arg( )
Amplitudes: [
[4.67]
0}
arg (𝐻
𝑦
𝛿 →[𝜔 ]
)
However, we can derive expressions for 𝑐 and 𝑐 on a real (non-complex) form,
𝑝 𝑖𝑡𝑢𝑑𝑒 ∙
cos( 𝑛𝑔𝑢 𝑟 𝑓𝑟𝑒𝑞𝑢𝑒𝑛𝑐𝑦 ∙ 𝑡 𝑝 𝑠𝑒 𝑑𝑒 𝑦), without involving transfer function. That is done in the
following:
Re(
𝑐)
Re( ̂
The same rewriting can be done with
𝑐
∙ 𝑒 𝑗∙𝜔∙𝑡 )
⋯
| ̂ 𝑐 | ∙ cos( ∙ 𝑡
, so that in total:
195
arg( ̂ 𝑐 )) ;
LATERAL DYNAMICS
[
[
]
̂ ∙ cos( ∙ 𝑡
̂ ∙ cos( ∙ 𝑡
𝜔
)
]
)
| ̂ 𝑐 | ∙ cos ( ∙ 𝑡
[
| ̂ 𝑐 | ∙ cos( ∙ 𝑡
( arg( ̂ 𝑐 )))
]
( arg( ̂ 𝑐 )))
[4.68]
Equation [4.66] and Equation [4.68] now gives us the possibility to find vehicle response amplitude
and phase delay. The ratios between amplitude of responses and amplitude of excitation, ̂ ⁄𝛿̂ and
̂ ⁄𝛿̂ , are called gains. The difference in argument is the phase delay.
4.4.3.2
Lateral Velocity and Yaw Rate response
The frequency response for the two states, Lateral Velocity and Yaw Rate, can be plotted for given
vehicle data, see Equation [4.66], Equation [4.68] and Figure 4-55. The curves show response for
same vehicle, but different speed.
See Figure 4-55. The yaw rate gain curve has a knee at 0.5..1 Hz. The decrease after that is a measure
of yaw damping. The curve for high speed actually has a weak peak just before the knee. This is not
desired, because the vehicle might feel a bit nervous. Yaw damping can also be how fast yaw rate
decays after a step response, see Section 4.5.
From Equation [4.27] we can calculate that characteristic speed for the vehicle is 120 km/h. With
another understeering coefficient, we could have calculated a critical speed. With analysis as in
Figure 4-55 to Figure 4-58, one can find these special speeds in another appearance:


For an understeered vehicle, speeds above the characteristic speed gives a negative yaw rate
delay for low steering frequencies will be negative.
For over-steered vehicles, speeds above the critical speed gives a yaw rate delay larger than
180 deg and yaw rate amplitudes which are very large for low steering frequencies.
Lateral Velocity Response
Yaw Rate Response
100
8
wz gain [(rad/s)/rad]
10
vy gain [(m/s)/rad]
120
80
60
40
20
-1
10
0
10
1
10
4
2
0
-2
10
2
10
2
2
1
1.5
wz delay [rad]
vy delay [rad]
0
-2
10
6
0
-1
-2
0
10
1
10
2
10
vx=150 km/h
vx=100 km/h
vx=50 km/h
vx=1 km/h
1
0.5
0
-3
-4
-2
10
-1
10
-1
10
0
10
frequency [Hz]
1
10
2
10
-0.5
-2
10
-1
10
0
10
frequency [Hz]
1
10
2
10
Figure 4-55: Vehicle response to harmonically oscillating steering angle. Vehicle data: m=2000 kg;
J=3000 kg*m2; lf=1.3 m; lr=1.5 m; Cf=81400 N/rad; Cr=78000 N/rad; (Ku= 1.26 rad/MN).
196
LATERAL DYNAMICS
From Equation [4.27] we can calculate that characteristic speed for the vehicle is 120 km/h.
In Figure 4-56 the curves are for same speed and constant understeering gradient, but they show the
response for different sums of cornering stiffness (Cf+Cr). Increasing the stiffness increases the yaw
rate gain (agility) at high frequencies.
Lateral Velocity Response
Yaw Rate Response
8
Cf+Cr=119 kN/rad
Cf+Cr=159 kN/rad
Cf+Cr=199 kN/rad
60
wz gain [(rad/s)/rad]
vy gain [(m/s)/rad]
80
40
20
0
-2
10
-1
10
0
10
1
10
6
4
2
0
-2
10
2
10
2
-1
10
0
10
1
10
2
10
2
1
wz delay [rad]
vy delay [rad]
1.5
0
-1
-2
1
0.5
-3
-4
-2
10
-1
10
0
10
frequency [Hz]
1
10
0
-2
10
2
10
-1
10
0
10
frequency [Hz]
1
10
2
10
Figure 4-56: Vehicle response to harmonically oscillating steering angle. Same vehicle data as in
Figure 4-55, except varying Cf and Cr but keeping understeering gradient Ku constant. Vehicle
speed = 100 km/h.
In Figure 4-57 the curves are for same speed and constant sum of cornering stiffness (Cf+Cr), but
they show the response for different values of understeering gradient (Ku). Increasing understeer
gradient decreases the yaw rate gain (agility) at low frequencies.
197
LATERAL DYNAMICS
Lateral Velocity Response
Yaw Rate Response
40
Ku=
Ku=
Ku=
Ku=
300
-1.26 rad/kN
-0.01 rad/kN
0.01 rad/kN
1.26 rad/kN
wz gain [(rad/s)/rad]
vy gain [(m/s)/rad]
400
200
100
0
-2
10
-1
0
10
1
10
20
10
0
-2
10
2
10
30
10
2
-1
0
10
1
10
10
2
10
2
1
wz delay [rad]
vy delay [rad]
1.5
0
-1
-2
1
0.5
-3
-4
-2
10
-1
10
0
10
frequency [Hz]
1
0
-2
10
2
10
10
-1
10
0
10
frequency [Hz]
1
10
2
10
Figure 4-57: Vehicle response to harmonically oscillating steering angle. Same vehicle data as in
Figure 4-55, except varying understeering gradient Ku but keeping Cf+Cr constant. Vehicle speed =
100 km/h.
4.4.3.3
Lateral Acceleration response
The lateral acceleration response is another useful response to study and set requirements on.
Actually, yaw rate and lateral acceleration are the most frequently used response variables, since
they are easily measured, e.g. from ESC sensors in most vehicles.
The transfer function is found (here using Fourier transform and previous results):
Amplitude: ̂
|
1
∙ [𝑗 ∙
|
𝑐|
]∙𝐻
Phase delay: arg(
𝑦
𝛿 →[𝜔 ]
)
∙
{𝑢𝑠𝑒:
∙𝛿 |
arg(𝛿 )
arg ([𝑗 ∙
|[𝑗 ∙
}
𝑗∙
]∙𝐻
∙
𝑦
𝛿 →[𝜔 ]
{𝑢𝑠𝑒: arg(𝛿 )
]∙𝐻
|
𝑦
𝛿 →[𝜔 ]
|∙
𝑐
𝛿
𝑐
∙
|
;
0}
);
Lateral acceleration response is plotted for different vehicle speeds in Figure 4-58.
198
[4.69]
LATERAL DYNAMICS
Lateral Acceleration Response
Lateral Acceleration Response
250
2
vx=150 km/h
vx=100 km/h
vx=50 km/h
vx=1 km/h
150
1.5
1
ay delay [rad]
ay gain [(m/(s*s))/rad]
200
100
0.5
0
-0.5
-1
50
-1.5
0
-2
10
-1
10
0
10
frequency [Hz]
1
10
2
10
-2
-2
10
-1
10
0
10
frequency [Hz]
1
10
2
10
Figure 4-58: Vehicle lateral acceleration response to harmonically oscillating steering angle.
Vehicle data: m=2000 kg; J=3000 kg*m2; lf=1.3 m; lr=1.5 m; Cf=81400 N/rad; Cr=78000 N/rad;
(Ku= 1.26 rad/MN).
4.4.3.4
Other responses to oscillating steering
In principle, it is possible to study a lot of other responses, such as Path Curvature response, Side slip
response and Lateral Path Width response etc. These are not very standardized in requirement
setting, but can be used as such or generally used for comparing different designs during the
development work. For combination-vehicles it is common to plot Rearward amplification (RWA) as
̂
max ( ̂ 𝑖 ) ; where 𝑖 denotes vehicle units or
function of frequency. RWA can be either 𝑅 𝐴
axles, or 𝑅 𝐴
4.4.3.5
̂ 𝑖
𝜔
max (𝜔
);
̂
=2 3 …
1
=2 3 …
1
where 𝑖 denotes vehicle units.
Random frequency response
Solutions to harmonic excitation of linear dynamic systems are superimposable. This is also why the
response from a mixed frequency excitation can be spliced into separate frequencies, e.g. using
Fourier transformation. Hence, a common way to measure the frequency response diagrams is to log
data from a random steering excitation. The frequency response diagram can then be extracted from
this test.
4.5 Transient handling
Transient handling in general vehicle dynamics is when the vehicle is manoeuvred in an arbitrary, but
not constant or cyclic, way. Generally, this can be turning and braking/accelerating at the same time
through a manoeuvre. In this chapter, we only cover transient handling within lateral dynamics. This
should be understood as that we assume a reasonably constant longitudinal vehicle speed and
modest longitudinal forces on the wheels. The latter assumptions make it possible to base most of
the content in this chapter on the model derived in Section 4.4.
199
LATERAL DYNAMICS
4.5.1
Transient driving manoeuvres
When testing Transient driving manoeuvres, the typical part of the test track is the Vehicle Dynamics
Area or a Handling Track, see Figure 4-8. A Handling Track is a normal width road, intentionally
curved and with safety areas beside the curves for safety in case of run-off road during tests.
Typical transient tests are:





Step steer, where one can measure transient versions of
o Yaw rate response
o Lateral acceleration response
o Curvature response
o Yaw damping
Lateral avoidance manoeuvres:
o Single lane change in cone track
o Double lane change in cone track, see example in Figure 4-26
o Lane change while full braking
o Sine with dwell
o Steering effort in evasive manoeuvres
Tests from steady state cornering
o Brake or accelerate in curve
o Lift throttle and steer-in while cornering
o Over-speeding into curve
Handling type tests
o Slalom between cones
o Handling track, general driving experience
Roll-over tests
Figure 4-60: Cone track for one standardized Double lane change
Standards which are relevant to these test manoeuvres are, e.g. References (ISO 3888), (ISO 7401),
(ISO 7975, 2006), (ISO 11026), (ISO 14791), (ISO 14793), (ISO 14794, 2011) and (NHTSA).
A general note is that tests in real vehicles are often needed to be performed in simulation also, and
normally earlier in the product development process.
200
LATERAL DYNAMICS
Figure 4-61: Cone track for one standardized test of Over-speeding into curve
4.5.2 Transient one-track model, without
Lateral load transfer
Transient handling within lateral vehicle dynamics should be understood as that we assume a
reasonably constant longitudinal vehicle speed and modest longitudinal forces on the wheels. The
latter assumptions makes it possible to base most of the content in this chapter on the model
derived in Figure 4-51 and Equations [4.51] and [4.52]. However, in the context of transient dynamics
it is more relevant to use the model for more violent manoeuvres, and also active control such as ESC
interventions. Hence, we extend the model in two ways:



The constitutive relation is saturated, to reflect that each axle may reach friction limit,
friction coefficient times normal load on the axle. See max functions in Equation [4.70].
To be able to do mentioned limitation, the load transfer is modelled, but only in the simplest
possible way using stiff suspension models. Basically it is the same model as given in
Equation [3.16].
A yaw moment representing (left/right) unsymmetrical braking/propulsion. See the term
𝑐𝑡 in yaw equilibrium in Equation [4.70].
It should be noted that the model still lacks some: Transients in load transfer and the reduced
cornering stiffness and reduced max friction due to load transfer and utilizing friction for wheel
longitudinal forces.
Equilibrium in road plane (longitudinal, lateral, yaw):
∙(
∙(
𝐽∙
∙
∙
(
)
)
𝑤 ∙ sin(𝛿 )
∙ cos(𝛿 )
𝑤 ∙ sin(𝛿 )
𝑤 ∙ sin(𝛿 )
𝑤 ∙ cos(𝛿 )
∙
𝑤 ∙ cos(𝛿 )) ∙
𝑤
Equilibrium out of road plane (vertical, pitch):
∙𝑔
0;
201
;
;
[4.70]
𝑐𝑡
;
LATERAL DYNAMICS
∙
∙
(
𝑤
∙ cos(𝛿 )
𝑤
∙ sin(𝛿 )
)∙
0;
Constitution:
𝑤
sign(𝑠 ) ∙ min(𝐶 ∙ |𝑠 |; 𝜇 ∙
);
sign(𝑠 ) ∙ min(𝐶 ∙ |𝑠 |; 𝜇 ∙ ) ;
Compatibility:
𝑤
𝑠
𝑤
;
𝑛𝑑 𝑠
∙
;
Transformation from vehicle to wheel coordinate system on front axle:
𝑤
𝑤
(
(
∙
∙
) ∙ sin(𝛿 )
) ∙ cos(𝛿 )
∙ cos(𝛿 ) ;
∙ sin(𝛿 ) ;
The model in Modelica format is given in Equation [4.71]. Changes compared to Equations [4.51] are
marked as underlined code.
//Equilibrium, in road plane:
m*(der(vx)-wz*vy) = Ffxv + Frx;
m*(der(vy)+wz*vx) = Ffyv + Fry;
J*der(wz) = Ffyv*lf - Fry*lr + Mactz;
//Equilibrium, out of road plane:
Ffz + Frz - m*g = 0;
-Ffz*lf + Frz*lr -(Ffxv + Frx)*h = 0;
//Compatibility:
vfxv = vx;
vfyv = vy + lf*wz;
vrx = vx;
vry = vy - lr*wz;
//Lateral tyre force model:
Ffyw = -sign(sfy)*min(Cf*abs(sfy), mu*Ffz);
Fry = -sign(sry)*min(Cr*abs(sry), mu*Frz);
sfy = vfyw/vfxw;
sry = vry/vrx;
[4.71]
//Transformation between vehicle and wheel coordinate systems:
Ffxv = Ffxw*cos(df) - Ffyw*sin(df);
Ffyv = Ffxw*sin(df) + Ffyw*cos(df);
vfxv = vfxw*cos(df) - vfyw*sin(df);
vfyv = vfxw*sin(df) + vfyw*cos(df);
//Shaft torques
Ffxw = +1000; // Front axle driven.
Frx = -100; // Rolling resistance on rear axle.
Mactz=0;
A simulation of this model, with same steering input as used in Figure 4-52. 𝑐𝑡 is zero. Cornering
stiffnesses are chosen so that the vehicle is understeered in steady state. Road friction coefficient is
1. We can see that the vehicle now gets instable and spins out with rear to the right. This is mainly
because longitudinal load transfer unloads the rear axle, since the kept steering angle decelerates
the vehicle. In this manoeuvre, it would have been reasonable to model also that the rear cornering
stiffness decreases with the decreased rear normal load, and opposite on front. Such addition to the
model would make the vehicle spin out even more. On the other hand, the longitudinal load shift is
modelled to take place immediately. With a suspension model, this load shift would require some
more time, which would calm down the spin-out. In conclusion, the manoeuvre is violent enough to
trigger a spin-out, so a further elaboration with how to control 𝑐𝑡 could be of interest. However,
it is beyond the scope of this compendium.
202
LATERAL DYNAMICS
The vehicle reaches zero speed already after 7 seconds, because the wide side slip decelerates the
vehicle a lot. The simulation is stopped at time=7 seconds, because the model cannot handle zero
speed. That vehicle models become singular at zero speed is very usual, since the slip definition
becomes singular due to zero speed in the denominator. The large difference compared to Figure
4-52 is due to the new constitutive equation used, which shows the importance of checking validity
region for any model one uses.
wz [deg/s]
y [m]
df=df [deg]
x[m]
yz=pz [deg]
x[m]
time [s]
Figure 4-62: Simulation results of one-track model for transient dynamics. The vehicle drawn in the
path plot is not in proper scale, but the orientation is approximately correct.
4.5.3 Transient model, with Lateral load
transfer
In model of longitudinal load transfer in Section 3.3.10, we assumed instantaneous load transfer. In
present chapter we aim to capture the transients better.
4.5.3.1
(Example of) Mathematical model
Compared to the model presented in Figure 4-39 and Equations [4.37]..[4.43], we add the following:



Inertial term for roll rotation, i.e. 𝐽𝑠 ⋅ . Note that sprung body roll inertia (𝐽𝑠 ), not whole
vehicle roll inertia (𝐽 ), is used in roll equilibrium because the unsprung parts does not
participate in roll motion.
Inertial term for lateral acceleration need to capture the lateral acceleration,
∙
, not only the centripetal acceleration,
∙ .
Damping forces in parallel to each spring force, i.e. adding 𝑑
and 𝑑 .
𝑑
𝑑
Due to the axle roll centre model, as opposed to wheel pivot point model, the heave acceleration is
zero. Hence, no inertial force ⋅ needs to be introduced.
203
LATERAL DYNAMICS
The free-body diagrams are given in Figure 4-63, which should be compared to Figure 4-39.
forces, on body:
displacements, speeds, accelerations:
ay
m*g
z
zfl=zrl
y
m*ay
𝐽𝑠 ⋅
zfr=zrr
px
Pry
Pfy
zflr=zrlr=0
Fsfl+Fsrl+
+Fdfl+Fdrl
zfrr=zrrr=0
forces,
on vehicle:
forces, on front axle:
m*g
m*ay
𝐽𝑠 ⋅
Fsfr+Fdfr
Fsfl+Fdfl
Fsrl+Fdrl Fsrr+Fdrr
Prz
Ffrz+Frrz
Pry
Pfy
Ffry+Frry
Fflz+Frlz
forces, on rear axle:
Pfz
h
Ffly+Frly
Fsfr+Fsrr+
Pfz+Prz +Fdfr+Fdrr
Ffly
Ffry
Ffrz
Fflz
Frly
Frlz
Frry
Frrz
Figure 4-63: Model for transient load transfer due to lateral acceleration, using axle roll centres.
The constitutive equations for the compliances (or springs) are as follows, cf Equation [4.36]. Note
that the anti-roll bars are not modelled. Note also that we differentiate, since we will later use the
spring forces as state variables in a simulation.
𝑠
𝑠
𝑠
𝑠
𝑠
𝑠
𝑠
𝑠
𝑤 𝑒𝑟𝑒
𝑠
𝑐 𝑤 ∙ (𝑧
𝑐 𝑤 ∙ (𝑧
𝑐 𝑤 ∙ (𝑧
𝑐 𝑤 ∙ (𝑧
𝑠
𝑧 ); ⇒ 𝑠
𝑧 ); ⇒ 𝑠
𝑧 ); ⇒ 𝑠
𝑧 ); ⇒ 𝑠
∙𝑔∙
; 𝑛𝑑
∙
𝑐
∙𝑧
𝑐 𝑤∙𝑧
𝑐 𝑤∙𝑧
𝑐 𝑤∙𝑧
𝑤
𝑠
𝑠
𝑐
𝑐
𝑤
∙
𝑐
𝑤
;
∙
;
𝑤∙
𝑐𝑤∙
;
∙𝑔∙
;
∙
;
[4.72]
The constitutive equations for the dampers have to be added:
𝑑
𝑑
𝑑
𝑑
𝑑 𝑤∙
𝑑𝑤∙
𝑑 𝑤∙
𝑑 𝑤∙
;
;
;
;
[4.73]
As comparable with Equation [4.37], we get the next equation to fulfil the equilibrium. The change
compared to Equation [4.37] is that we also have a roll and lateral inertia terms and 4 damper forces,
acting in parallel to each of the 4 spring forces. Actually, when setting up equations, we also
understand that a model for longitudinal load transfer is needed, which is why the simplest possible
such, which is the stiff suspension on in Equation [3.16].
204
LATERAL DYNAMICS
In-road-plane: Equilibrium for vehicle (longitudinal, lateral, yaw):
∙
∙
𝐽 ∙
∙(
∙(
∙
∙ )
∙ )
∙ ;
;
;
Out-of-road-plane: Equilibrium for vehicle (vertical, pitch, roll):
∙𝑔
𝐽𝑠 ∙
(
)∙
(
)∙
(
𝑤
0;
)∙
(
)∙
(
Equilibrium for each axle (pitch, around roll centre):
𝑤
(
(
( 𝑠
( 𝑠
𝑑 )) ∙
𝑤
( 𝑠
( 𝑠
(
(
𝑑 )) ∙
Compatibility gives, keeping in mind that
0):
𝑧
𝑧
𝑤
𝑑
𝑑
)∙
(
𝑤
)) ∙
𝑤
)) ∙
[4.74]
0;
)∙ ;
∙
∙
0;
𝑅𝐶
0;
𝑅𝐶
is the only non-zero out-of road plane velocity (i.e. 𝑧
𝑤
𝑤
∙
∙
;
;
𝑛𝑑 𝑧
𝑛𝑑 𝑧
𝑤
𝑤
∙
;
∙
;
[4.75]
Equations [4.72]..[4.75] give a model very similar to the one in Equation [4.71]. Only the additional
equations from this new model are shown in Equation [4.76]. Note especially the new lateral tyre
force model equations, which now have one term per wheel, because one wheel on an axle can
saturate independent of the other on the same axle.
//Equilibrium, roll
Jsx*der(wx) = (Fflz + Frlz)*w/2 - (Ffrz + Frrz)*w/2 + (Ffyv + Fry)*h;
//Equilibrium for each axle (pitch, around roll centre):
(Fflz - (Fsfl + Fdfl))*w/2 - (Ffrz - (Fsfr + Fdfr))*w/2 + Ffy*hRCf = 0;
(Frlz - (Fsrl + Fdrl))*w/2 - (Frrz - (Fsrr + Fdrr))*w/2 + Fry*hRCr = 0;
//Constitutive relation for tyres (Lateral tyre force model):
Ffyw = sign(sfy)*(min((Cf/2)*abs(sfy), mu*Fflz) + min((Cf/2)*abs(sfy), mu*Ffrz));
Fry = sign(sry)*(min((Cr/2)*abs(sry), mu*Frlz) + min((Cr/2)*abs(sry), mu*Frrz));
sfy = vfyw/vfxw;
sry = vry/vrx;
//Constitution for springs:
der(Fsfl) = -cfw*vflz;
der(Fsfr) = -cfw*vfrz;
der(Fsrl) = -crw*vrlz;
der(Fsrr) = -crw*vrrz;
//Constitution for dampers:
Fdfl = -dfw*vflz;
Fdfr = -dfw*vfrz;
Fdrl = -drw*vrlz;
Fdrr = -drw*vrrz;
//Compatibility, out of road plane:
vflz = +w/2*wx;
vfrz = -w/2*wx;
vrlz = +w/2*wx;
205
[4.76]
LATERAL DYNAMICS
vrrz = -w/2*wx;
A simulation of this model should be compared with the simulation in Figure 4-52. When comparing
these, we see a slight difference, which is that the axles saturate gradually during 3<time<3.5,
instead of both at once at time=3.25.
Even if the load transfer model does not influence the vehicle path a lot in this case, it may be
important to include it to validity check the model through checking wheel lift. Wheel lift can be
identified as negative vertical wheel forces, which are why we plot some vertical wheel forces, see
Figure 4-65. In this case we see that we have no wheel lift (which would disqualify the simulation). In
the right part of the figure we can also see the separate contribution from spring ( 𝑠 ), damper
( 𝑑 ) and linkage (
𝑓
𝑠
𝑑 ).
wz [deg/s]
y [m]
df=df [deg]
x[m]
yz=pz [deg]
x[m]
time [s]
Figure 4-64: Simulation results of one-track model for transient dynamics with lateral load transfer
dynamics. The vehicle drawn in the path plot is not in proper scale, but the orientation is
approximately correct.
206
LATERAL DYNAMICS
Frrz [N]
Fsrr [N]
Ffrz [N]
Frrz [N]
Flink,rr,z [N]
Fflz [N]
Fdrr [N]
Frlz [N]
time [s]
time [s]
Figure 4-65: Suspension vertical force plots from simulation with one-track model for transient
dynamics with lateral load transfer dynamics (same simulation as in Figure 4-64). Left: Road
contact forces for all wheels. Right: Different forces for one wheel, rear left.
4.5.3.1.1 Additional phenomena
It is relevant to point out the following, which are not modelled in this compendium:


Same as pointed out as missing for longitudinal load transfer, see Section 3.4.8.5.
Additionally, anti-roll arrangements (elastic connections between left and right wheel on one
axle, often built as torsion bar) are not modelled in this compendium. With same modelling
concept as used above, each such would be treated as a separate spring with one state
variable, e.g. Faf (Force-antiroll-front). This force will act in parallel with Fsf and Fdf on each
side. Note that it will be added on one side and subtracted on the other.
4.5.3.2
(Example of) Explicit form model
A disclaimer to readers which only read the compendium, without following the course: This section
might appear incomplete, while the course lectures and hand-out of model is more complete. Still, the
description in compendium is regarded as relevant to give general feeling for an Explicit form model.
The implementation of the model in Section 4.5.3 is done in Modelica. A Modelica tool (such as
Dymola) automatically transforms the model to explicit form and automatically generates simulation
code, which is very efficient. But, as mentioned in Section 1.3.7.5, explicit form models can
sometimes facilitate the understanding of the vehicle’s dynamics. This is why the following model is
presented. It is implemented in the data flow diagram tool Simulink. This section explains how the
states (or state variables) together with inputs (or input variables), influence the derivatives (or state
derivatives).
The example model in this section is similar, but not identical, to the model in Section 4.5.3. The aim
is to model in-road-plane motion, due to transient actuation (wheel torques and wheel steering
angles). Limitations in this example model are:




Influence from vertically uneven road is NOT modelled.
Influence from varying road friction is NOT modelled.
Wheel lift from ground is NOT modelled
Hitting bump stop is NOT modelled
207
LATERAL DYNAMICS




Control functions (such as ABS, TC and ESC) are NOT modelled
The pendulum effect is NOT modelled
Wheel camber angle change with suspension travel is NOT modelled (Camber
Wheel steering angle change with suspension travel is NOT modelled (change
0).
0).
The model is a typical two axle vehicle with interface to driver and environment, see Figure 4-66. The
driver interface is the normal, 2 pedals and one steering wheel. The additional discrete request from
driver, LongDir (=-1 or +1), is also needed to reflect whether driver requests forward or reverse
longitudinal actuation. The interface to environment is motion (variable position) in surrounding
world. To try out the vehicle model, driver and environment is also modelled. This includes also the
interface between them, which is motion of obstacles in environment, relative to subject vehicle. The
suspension is exemplified with wheel-individual suspension on both axles.
Driver
Vehicle
Environment
21 states
Figure 4-66: Top level of model with model tree structure. The Environment is a track test with
cone walls to go left and right around. Notation “irp” and “oorp” refers to in-road-plane and outof-road-plane, respectively.
As an initial overview, the states are presented. There are 21 states in total, and distributed:



Driver: 0 states
Vehicle:
o Vehicle Control & Actuation: 0 states
o Wheels, Tyres & Suspension: 12 states (4 wheel rotational speed, 4 Elastic part of
vertical wheel forces, 4 Magnitude of tyre force in road plane)
o Vehicle Motion: 9 states (6 velocities and 3 positions)
Environment: 0 states
One should also mention that 4 “memories” (on vertical tyre forces) are added in one part of the
Tyre model to resolve an algebraic loop. Using memory blocks are generally a dangerous way of
modelling, see more in Section 4.5.3.2.4.
4.5.3.2.1 Submodel Driver
The driver model is here very sparsely described and the intention is not that it should be a state of
the art model. However, in a nut-shell: Driver model is divided into two parts:

Driver planning (perception): Based on position of obstacles relative to the vehicle, one of
the objects is selected to be aim. Basically, the nearest obstacle ahead of vehicle is selected
as aim.
208
LATERAL DYNAMICS

Driver operation (motorics): This should output the steering wheel angle. In the example, it is
simply an inverse model of an ideally tracking two-axle vehicle which calculates which
constant steering wheel angle that would be needed to make front axle run over the aim
obstacle.
4.5.3.2.2 Submodel Environment
The environment model is here very sparsely described and the intention is not that it should be a
state of the art model. It captures only stand-still point obstacles with pre-determined whether it
should be passed as obstacle left or right of the vehicle. Inputs to environment model is the vehicles
position, including orientation in global coordinates. Outputs are the relative position (x,y) to each
obstacle, in vehicle coordinate system.
4.5.3.2.3 Submodel Vehicle
9 states
12 states
Figure 4-67: Vehicle submodel.
The figure shows the decomposition of the vehicle into 3 parts:



Submodel “Vehicle Control & Actuation” models the actuators (Propulsion system, Brake
system and Steering system, including “control functions”) that respond on requests from
the driver with wheel torques (T_s and T_us) and wheel steering angles (delta_w). The
notation ending “4” refers to that the quantities are vectors with 4 components, one per
wheel. T_s is shaft torque and T_us is torque applied from unsprung parts, e.g. friction
brake torque from brake calliper. One can think of very advanced models of these actuator
systems, including e.g. propulsion system dynamics and control functions (ABS, ESC, TC, …).
However, in this example model it is only modelled very simple:
o Propulsion system outputs a fraction (determined by APed) of a certain maximum
power, distributed equally on front left and front right wheel. If brake pedal is
applied, the propulsion system outputs zero torque.
o Brake system outputs a fraction (determined by BPed) of a maximum brake torque
(𝜇 ⋅ ⋅ 𝑔/𝑅𝑤 ), distributed in a certain fix fraction between front and rear axle
(70/30). The distribution within each axle is equal on left and right wheel.
o There are no states modelled in the vehicle Control & Actuator submodel.
Submodel “Wheels, Tyre & Suspension” models the part which pushes the tyres towards the
ground and consequently transforms the wheel torques and wheel steering angles, via the
tyre, to forces on the whole vehicle. F_xyv8 is the x and y forces in each of the 4 wheels,
2x4=8. F_z4 is the 4 vertical forces under each wheel. Submodel “Wheels, Tyre &
Suspension” is further explained in Section 4.5.3.2.4.
Submodel “Vehicle Motion” models motion of whole vehicle in-road-plane and motion of
sprung body out-of-road-plane. The inertial effects (mass∙acceleration) of the unsprung parts
209
LATERAL DYNAMICS
are taken into account for in-road-plane but not for out-of-road-plane. This submodel
includes integrators for the 3+3+3=9 states:
o Velocities in-road-plane,
: v_irp3 (which is transformed to x- and yvelocities of each wheel and then fed back as v_irpv8)
o Position in-road-plane, 𝑥 𝑦 : pos_irp3 (which is only fed forward to
“Environment”)
o Velocities out-of-road-plane,
: v_oorp3 (which is transformed to zvelocities of sprung body over each wheel and then fed back as v_z4)
4.5.3.2.4 Submodel Wheels, Tyres & Suspension
• state
• memory
Figure 4-68: Wheels, Tyres & Suspension submodel.



The two submodels “Coord Transf …” are straight-forward coordinate transformations, using
Eq [4.3].
The submodel “Wheels” is also relatively straight-forward. For each wheel, the rotational
equilibrium is used as model: 𝐽 ∙
𝑇𝑢𝑠 𝑇𝑠
∙ 𝑅𝑤 sign( ) ∙ 𝑅𝑅𝐶 ∙ ∙ 𝑅𝑤 ; where 𝑅𝑤
is wheel radius and 𝑅𝑅𝐶 is rolling resistance coeeficient. The submodel will hence contain
the 4 states:
o Rotational speeds of each wheel: w_w4.
The submodel “Springs, Dampers & Linkage” models the springs (incl. anti-rollbars) and
dampers and the linkage. For each wheel:
o Four states: Elastic part of vertical tyre force under each wheel: F_s4
o The derivatives are governed by the differentiated constitution os the springs:
Conceptually 𝑠
𝑐 ⋅ ; but involving both wheel spring and anti-roll-bar.
o The force in damper is governed by the dampers constitutive relation: 𝑑
𝑑⋅ ;
o The contact forces are calculated in submodel “Suspension Equilibrium” in Figure
4-69. They are calculated from moment equilibrium of unsprung parts around a 3dimensional pivot axis. The pivot axis is defined by two points, the pivot point in
longitudinal load transfer (see Figure 3-31) and the pivot point in longitudinal load
transfer (see Figure 4-38). The scalar equilibrium equation for one wheel can be
210
LATERAL DYNAMICS

expressed, with vector (cross) and scalar (dot) products, in
and
𝑠 𝑑 𝑇𝑠
point coordinates. From this, can be solved.
The submodel “Tyres” models the tyre mechanics, very much like the combined slip model in
Equation [2.31] and the relaxation model in Equation [2.36]. For each wheel:
o Four states: Magnitudes of tyre forces in ground plane ( ) for each wheel:
F_xyw4
o Unfortunately, the tyre mechanics are such that and depends on . This
creates an algebraic loop, see Figure 4-69. The loop can almost be resolved by using
as state, but there is a weak dependency left because is needed for small total
slip (𝑠 ) to find out the direction of
. This dependency of is seen in Equation
[2.32]. Since this dependency is weak, we allow ourselves to resolve it using
“memories”, mentioned in Section 4.5.3.2.
o The algebraic loop is resolved by introducing 4 so called “memories”. “Memories”
are such that value from last time instant is used to calculate derivatives in present
time instant. This is generally NOT a recommended way of modelling, but it works
relatively well in this case. To involve full two-dimensional relaxation (both and
as states) would avoid the algebraic loop without using memories, but it seems to
result in smaller time steps, meaning slower or non-converging simulations.
Figure 4-69: Submodel “Tyres” on top and submodel “Springs, Dampers & Linkage” below. The
arrows between shows the algebraic loop which appears when they are connected.
211
LATERAL DYNAMICS
4.5.3.2.5 Submodel Vehicle Motion
• state
30
Figure 4-70: Vehicle Motion.
20
4.5.3.2.6 Simulation example
A double lane change between cones is used as simulation example, see Figure 4-71. The cones are
run over and even passed on the wrong side because the driver model is very simple.
10
cone
cone
0
7
6.5
6
tyre force in ground plane
(vertical tyre force is radius
of circle)
5.5
-10
5
4.5
wheel hub translational
velocity
4
3.5
cone
3
-20
0.3
2.5
1
187
188
0
-1
0
1502
4
6
160
8
10
170
12
14
fl
fr
rl
rr
191
0
-0.1
-0.2
180 16
2
4
2006
8
10
210
12
14
220
16
0
2
4
6
8
10
12
14
16
0
2
4
6
8
t /[s]
10
12
14
16
0
190
0.3
40
0.2
s /[1=(m/s)/(m/s)]
30
0.1
0
-0.1
y
20
-0.2
10
0
8000
-10
vx /[m/s]
-20
beta /[deg]
wz /[deg/s]
6000
F /[N]
ay /[m/(s*s)]
z
-30
190
0.1
x
APed /[0..1]
-BPed /[0..-1]
0.01*[SteWhlAn /[deg]]
Longitudinal Direction request
-0.5
189
0.2
s /[1=(m/s)/(m/s)]
186
0.5
-30
-40
4000
2000
0
2
4
6
8
t /[s]
10
12
14
0
16
Figure 4-71: Simulation results of a double lane change between cones.
212
LATERAL DYNAMICS
4.5.4
Step steering response *
Function definition: Step steering response is the response to a step in steering wheel angle measured
in certain vehicle measures. The step is made from a certain steady state cornering condition to a certain
steering wheel angle. The response can be the time history or certain measures on the time history, such as
delay time and overshoot.
4.5.4.1
Mild step steering response
This section is to be compared with section 4.5.4.2, which uses a more advanced model. In present
section a less advanced model will be used, which is enough for small steering steps.
The model used for single frequency stationary oscillating steering can also be used for other
purposes, as long as limited lateral accelerations.
Most common interpretation is to make the steering step from an initial straight line driving. In
reality, the step will be a quick ramp. In simulations an ideal step can be used.
Equation [4.59] allows an easy analysis of stationary oscillating steering, but also an easy analysis of
step response:
Start from Equation [4.59]: [
(0)
]
(0)
With initial conditions: [
Assume: [
Insert: [[
]
[
𝐀∙[
]
[
]
𝐁∙𝛿 ;
] ; or [
]
𝐀− ∙ 𝐁 ∙ 𝛿 ; where 𝛿
𝜆1 ∙𝑡
̂
̂ 2
0 ] ∙ [ ]; ⇒
] [
]] ∙ [𝑒
∞
2
̂
̂ 2
0
𝑒 𝜆2 ∙𝑡
𝜆1 ∙𝑡
̂
̂ 2
𝜆 ∙𝑒
0
⇒ [ ] [[
]∙[
] [
]] ∙ [
̂
̂ 2
0
𝜆2 ∙ 𝑒 𝜆2 ∙𝑡
∞
]
is before step.
[[
̂
̂ 2
𝜆 ∙ 𝑒 𝜆1 ∙𝑡
] [
]] ∙ [
̂
̂ 2
0
𝐀 ∙ ([
∞
∞
]
0
]∙[ ]
2
𝜆2 ∙ 𝑒 𝜆2 ∙𝑡
𝜆1 ∙𝑡
̂
̂ 2
[[
] [
]] ∙ [𝑒
̂
̂ 2
0
0 ]∙[
𝑒 𝜆2 ∙𝑡
Solve for each time function term (constant, 𝑒 𝜆1 ∙𝑡 and 𝑒 𝜆2 ∙𝑡 terms):
̂
̂ 2
𝜆
∞
𝐀− ∙ 𝐁 ∙ 𝛿 ; 𝑛𝑑 [[[
[
]
] [
]] [
0
∞
̂
̂ 2
2
])
0
]]
𝜆2
2
];
𝐁∙𝛿 ;
eig(𝐀) ;
The function ”eig” is identical to function ”eig” in Matlab. It is defined as eigenvalues and
eigenvectors for the matrix input argument.
Initial conditions: [
]
[
∞
⇒[
∞
]
2
]
̂
̂ 2
] [
]] ∙ [ ] ; ⇒
2
̂
̂ 2
̂
̂ 2 −
[[
] [
]] ∙ ([
]
̂
̂ 2
[[
The solution in summary:
213
[
∞
∞
]) ;
LATERAL DYNAMICS
[
{
]
[
∙
where: [
∞
∞
𝜆1 ∙𝑡
̂
̂ 2
] [
]] ∙ [𝑒
∞
̂
̂ 2
0
𝜆1 ∙𝑡
𝜆 ∙ ̂ ∙𝑒
∙
𝜆2 ∙ ̂
∞
]
[[
2
]
[[
2
];
∙
2
;
𝐀− ∙ 𝐁 ∙ 𝛿 ;
]
̂
̂ 2
𝜆
and [[[
] [
]] [
0
̂
̂ 2
and [
0 ]∙[
𝑒 𝜆2 ∙𝑡
𝜆2 ∙𝑡
∙
2∙𝑒
0
]]
𝜆2
[4.77]
eig(𝐀) ;
̂
̂ 2 −
] [
]] ∙ ([
̂
̂ 2
]
[
∞
∞
]) ;
Results from this model for step steer to +3 deg are shown in Figure 4-72. Left diagram shows step
steer from straight line driving, while right diagram shows a step from steady state cornering with -3
deg steering angle.
steering step response, passenger car in vx=70 km/h
steering step response, passenger car in vx=70 km/h
20
20
15
15
10
10
5
5
0
0
-5
-5
beta=arctan(vy/vx) [deg]
wz [deg/s]
ay [m/(s*s)]
df [deg]
-10
-15
-20
-0.5
0
0.5
1
1.5
-10
beta=arctan(vy/vx) [deg]
wz [deg/s]
ay [m/(s*s)]
df [deg]
-15
2
-20
-0.5
0
0.5
1
1.5
2
time [s]
time [s]
Figure 4-72: Steering step response. Simulation with model from Equation [4.77].
4.5.4.2
Violent step steering response
This section is to be compared with section 4.5.4.1 Mild step steering response, which uses a model
with linear tyre models without saturation. In present section a more advanced model will be used,
which might be needed when the step steering is more violent.
Most common interpretation is to make the steering step from an initial straight line driving. In
reality, the step will be a quick ramp. In simulations an ideal step can be used.
The transients can easily be that violent that a model as Equations [4.72]..[4.76] is needed. If ESC is to
be simulated, even more detailed models are needed (full two-track models, which are not
presented in this compendium). Anyway, if we apply a step steer to the model in Equations
[4.72]..[4.76], we can simulate as in Figure 4-73.
4.5.5 Long heavy vehicle combinations
manoeuvrability measures
It is sometimes irrelevant (or difficult) to apply functions/measures from two axle vehicles on
combinations of units. This can be the case for passenger cars with a trailer, but it is even more
obvious for long combinations of heavy vehicles.
214
df
sfy
sry
0.0
LATERAL DYNAMICS
-0.2
1 in Figure
2
3 Figure 44-75 and
5
Some typical measures for multi-unit vehicle combination0 are given
4-74,
Figure 4-76.
28
0
1
2
3
4
5
0
0
df
sfy
2
3
4
5
df[rad]
-0.2
2
3
sfy[1]
4
beta_deg
15
sry[1]
1
1
sry
0.0
0
Fry
4000
vx[m/s]
26
Ffyw
8000
vx
5
wz_dps
ay
wz[deg/s]
10
ay[m/(s*s)]
5
Ffyw
8000
Fry
Ffyw[N]
0
Fry[N]
4000
beta[deg]
-5
0
0
1
2
3
time[s]
4
5
-10
0
1
2
3
time[s]
4
5
Figure
4-73:beta_deg
Step steer withwz_dps
2 deg on roadaywheels at vx=100 km/h. Simulated with model in
15
Equations [4.72]..[4.76].
10
4.5.5.1
5
Rearward Amplification *
Function definition: Rearward Amplification for long heavy vehicle combinations is the
0
ratio
of the maximum value of the motion variable of interest (e.g. yaw rate or lateral acceleration of the
centre of gravity) of the worst excited following vehicle unit to that of the first vehicle unit during a specified
-5
manoeuvre at a certain friction level and constant speed. From Reference [ (Kati, 2013)].
Figure-104-74 illustrates Rearward Amplification, RWA. RWA is defined for a special manoeuvre, e.g. a
0
1
2
3
4
5
certain lane change or step steer. RWA is the ratio of the peak value of yaw rate or lateral
acceleration for the rearmost unit to that of the lead unit. This performance measure indicates the
increased risk for a swing out or rollover of the last unit compared to what the driver is experiencing
in the lead unit.
Figure 4-74: Illustration of rearward amplification, P denotes peak value of the motion variable of
interest. From (Kharrazi , 2012).
215
LATERAL DYNAMICS
4.5.5.2
Off-tracking *
Function definition: High speed transient off-tracking for long heavy vehicle
combinations is the overshoot in the lateral distance between the paths of the centre of the front axle and
the centre of the most severely off-tracking axle of any unit in a specified manoeuvre at a certain friction level
and a certain constant longitudinal speed. From Reference [ (Kati, 2013)].
Function definition: High speed steady-state off-tracking for long heavy vehicle
combinations is the lateral offset between the paths of the centre of the front axle and the centre of the
most severely off-tracking axle of any unit in a steady turn at a certain friction level and a certain constant
longitudinal speed. From Reference [ (Kati, 2013)].
Figure 4-75 illustrates Off-tracking. Off-tracking was also mentioned in Section 4.2. The measure is, as
RWA, a comparison between the lead and last unit, but in terms of the additional road space
required for the last unit manoeuvring. High speed Off-tracking, which is an outboard off-tracking,
can be either determined in a steady state turn or in a transient manoeuvre such as lane change; the
latter is termed as high speed transient off-tracking.
Figure 4-75: Illustration of high speed steady-state off-tracking. From (Kharrazi , 2012).
4.5.5.3
Yaw Damping *
Function definition: Yaw Damping for long heavy vehicle combinations is the damping ratio of the least
damped articulation joint’s angle of the vehicle combination during free oscillations excited by actuating the
steering wheel with a certain pulse or a certain sine-wave steer input at a certain friction level. From
Reference [ (Kati, 2013)].
Figure 4-76 illustrates Yaw Damping. It is the damping ratio of the least damped articulation joint of
the vehicle combination during free oscillations. Yaw damping ratio of an articulation joint is
determined from the amplitudes of the articulation angle of subsequent oscillations.
216
LATERAL DYNAMICS
Figure 4-76: Illustration of yaw damping for multi-unit vehicle combination, DR denotes damping
ratio of the articulation joint. From (Kharrazi , 2012).
4.6 Lateral Control Functions
Some control functions involving lateral vehicle dynamics will be presented briefly. There are more,
but the following are among the most well-established ones. But initially, some general aspects of
lateral control is given.
4.6.1
Lateral Control Design
Sensors available and used for lateral control are, generally those mentioned as available for
Longitudinal Control, see Section 3.5 plus some more:
Steering wheel sensors gives at least steering wheel angle, if the vehicle is equipped with ESC (which
is a legal requirement on many markets). Additionally, if the steering assistance is electrical, the
steering wheel torque can be sensed.
High specification modern vehicles also have environment sensors (camera, radar, etc) that can give
laterally interesting information, such as: Subject vehicle lateral position versus lane markers ahead
and other vehicles to the side or rear of subject vehicle.
As general considerations for actuators, one can mention that interventions with friction brake
normally have to have thresholds, because interventions are noticed by driver and also generate
energy loss. Interventions with steering are less sensitive, and can be designed without thresholds.
4.6.2
Lateral Control Functions
4.6.2.1
Lane Keeping Aid, LKA *
Function definition: Lane Keeping Aid steers the vehicle without driver having to steer, when
probability for lane departure is predicted as high. It is normally actuated as an additional steering wheel
torque. Conceptually, it can also be actuated as a steering wheel angle offset.
Lane Keeping Aid (or Lane Departure Prevention) has the purpose to guide the driver to keep in the
lane. Given the lane position from a camera, the function detects whether vehicle tends to leave the
lane. If so, the function requests a mild steering wheel torque (typically 1..2 Nm) in appropriate
direction. Driver can easily overcome the additional torque. Function does not intervene if too low
speed or turning indicator (blinker) is used. There are different concepts whether the function
continuously should aim at keeping the vehicle in centre of lane, or just intervene when close to
leaving the lane, see Figure 4-77.
217
LATERAL DYNAMICS
added torque on
steering wheel
Two concepts:
• Keep in centre of lane
• Intervene when about
to leave lane
lateral
position
in lane
Figure 4-77: Two concepts for Lane Keeping Aid.
4.6.2.2
Electronic Stability Control, ESC *
Function definition: Electronic Stability Control directs the vehicle to match a desired yaw
behaviour, when the deviation from desired behaviour becomes above certain thresholds. ESC typically
monitors vehicle speed, steering angle and yaw rate to calculate a yaw rate error and uses friction brakes as
actuator to reduce it.
There are 3 conceptual parts of ESC: Over-steer control, Under-steer control, Over-speed control.
The actual control error that the vehicle reacts on is typically the yaw rate error between sensed yaw
rate and desired yaw rate. Desired yaw rate is calculated from a so called reference model. Some of
today’s advanced ESC also intervenes on sensed/estimated side slip.
Desired yaw rate (and side-slip) is calculated using a reference model. The reference model requires
at least steering angle and longitudinal velocity as input. The reference model can be either of steady
state type (approximately as Equation [4.28]) or transient (approximately as Equation [4.55]). The
vehicle modelled by the reference model should rather be the desired vehicle than the actually
controlled vehicle. But, in order to avoid too much friction brake interventions; the reference model
cannot be too different. Also, in order to avoid that vehicle yaws more than its path curvature; the
reference model has to be almost as understeered as the controlled vehicle, which typically can be
fixed by saturating lateral tyre forces on the front axle of the reference model. This requires some
kind of friction estimation, especially for low-mu driving.
When controlling yaw via wheel torques, one can identify some different concepts such as direct and
in-direct yaw moment, see Section 4.3.4.8. For ESC there is also a “pre-cautious yaw control” which
aims at reducing speed, see Section 4.6.2.2.3.
4.6.2.2.1 Over-steer Control
Over-steer control was the first and most efficient concept in ESC. When a vehicle over-steers, ESC
will brake the outer front wheel. It can brake to deep slip levels (typically -50%) since losing side grip
on front axle is positive in an over-steer situation. More advanced ESC variants also brakes outer
rear, but less and not to same deep slip level.
For multi-unit vehicle combinations with trailers that have controllable brakes, also the trailer is
braked to avoid jack-knife effect, see Figure 4-75, or swing-out of the towed units.
4.6.2.2.2 Under-steer Control
Under-steer control means that inner rear is braked when vehicle under-steers. This helps the vehicle
turn-in. This intervention is most efficient on low mu, because on high mu the inner rear wheel
normally has very little normal load. Also, the slip levels are not usually as deep as corresponding
218
LATERAL DYNAMICS
over-steer intervention, but rather -10%. This is because there is always a danger in braking too
much on rear axle, since it can cause over-steering. More advanced ESC variants also brakes inner
front, which makes the function very similar to the function in Section 4.6.2.2.3.
Figure 4-78: ESC when over-steer and understeer, on a truck with trailer
4.6.2.2.3 Over-speed Control
Over-speed control is not always recognised as a separate concept, but a part of under-steer control.
It means that more than just inner rear wheels are braked. In this text, we identify this as done to
decrease speed, which has a positive effect later in the curve.
4.6.2.2.4 Wheel-level control
A pre-requisite for all controls mentioned above in Section 4.6.2.2 is that the wheel torque actuator
respond to a torque request, but it is a normal design that it is also responding to a longitudinal slip
request. The slip request is generally used as a sort of “safety net” to not lock-up the wheel more
than 10-20%, but at RSC interventions (see Section 4.6.2.3) the lateral grip should be braked away, so
a deeper slip requet is then used.
TorqueRequest,
𝑇
Torque
controller
ESC & RSC
controllers
Fax
Min
LongSlipRequest,
𝑠
vehicle state
estimator
Fx
Closed loop
slip controller
𝑠
hi-mu
Fax
Actuator
lo-mu (or large sy)
Wheel rotational
speed sensor
Calculate
slip
sx
normal request
RSC request
sx-max = 10..20%
sx-max = 50..60%
Figure 4-79: Individual wheel control by friction brakes for ESC-type functions
219
LATERAL DYNAMICS
4.6.2.2.5 Other intervention than individual wheel brakes
4.6.2.2.5.1 Balancing with Propulsion per axle
For vehicles with controllable distribution of propulsion torque between the axles, ESC can intervene
also with a request for redistribution of the propulsion torque. If over-steering, the propulsion should
be redistributed towards front and opposite for understeering.
4.6.2.2.5.2 Torque Vectoring
For vehicles with controllable distribution of propulsion torque between the left and right, ESC can
intervene also with a request for redistribution of the propulsion torque. If over-steering, the
propulsion should be redistributed towards inner side and opposite for understeering.
4.6.2.2.5.3 Steering guidance
For vehicles with controllable steering wheel torque, ESC can intervene also with a request for
additional steering wheel torque. The most obvious functions is to guide driver to open up steering
(counter-steer) when the vehicle over-steers. Such functions are on market in passenger cars today.
Less obvious is how to guide the driver when vehicle is under-steering.
4.6.2.3
Roll Stability Control, RSC *
Function definition: Roll Stability Control, RSC, prohibits vehicle to roll-over due to lateral wheel
forces from road friction. RSC uses friction brake as actuator.
The purpose of RSC is to avoid un-tripped roll-overs. The actuator used is the friction brake system.
When roll-over risk is detected, via lateral acceleration sensor (or in some advanced RSC
implementations, also roll rate sensor), the outer front wheel is braked. RSC can brake to deep slip
levels (typically -70%..-50%) since losing side grip on front axle is positive in this situation.
On heavy vehicles, RSC intervenes earlier and similar to function described in Section 4.6.2.2.3 Overspeed Control.
In future, RSC could be developed towards also using steering.
4.6.2.4
Lateral Collision Avoidance, LCA *
Function definition: Lateral Collision Avoidance support the driver when he has to do late lateral
obstacle avoidance, when probability for forward collision is predicted as high.
There are systems on the market for Automatic Emergency Brake, see Section 3.5. If these are seen
as Longitudinal Collision Avoidance, one could also think of Lateral Collision Avoidance functions,
which would automatically steer away laterally from an obstacle ahead of subject vehicle. There are
not yet any such functions on market. One possible first market introduction could be that triggering
requires driver to initiate steering. Another would be to trigger on a first collision impact, see
Reference (Yang, 2013).
220
VERTICAL DYNAMICS
5 VERTICAL DYNAMICS
5.1 Introduction
The vertical dynamics are needed since vehicles are operated on real roads, and real roads are not
perfectly smooth. Also, vehicle scan be operated off-road, where the ground unevenness is even
larger.
The irregularities of the road can be categorized. A transient disturbance, such as a pothole or object
on the road, can be represented as a step input. Undulating surfaces like grooves across the road
may be a type of sinusoidal or other stationary (or periodic) input. More natural input like the
random surface texture of the road may be a random noise distribution. In all cases, the same
mechanical system must react when the vehicle travels over the road at varying speeds including
doing manoeuvres in longitudinal and lateral directions.
The chapter is organised with one group of functions in each section as follows:











5.2 Suspension System
5.3 Stationary oscillations theory
5.4 Road models
5.5 One-dimensional vehicle models
5.6 Ride comfort
5.7 Fatigue life
5.8 Road grip
5.9 Variation of stiffness and damping
5.10 Two dimensional oscillations
5.11 Transient vertical dynamics
5.12 Other excitation sources
This chapter is, to a larger extent than Chapters 3 and 4, organised so that all theory (knowledge)
comes first and the vehicle functions comes after. In Figure 5-1 shows the 3 main functions, Ride
comfort, Fatigue life and Road grip. It is supposed to explain the importance of the vehicle’s dynamic
structure. The vehicle’s dynamic structure calls for a pretty extensive theory base, described mainly
in Section 5.3.
Models in this chapter focus the disturbance from vertical irregularities from the road, i.e. only the
vertical forces on the tyre from the road and not the forces in road plane. This enables the use of
simple models which are independent of exact wheel and axle suspension, such as pivot axes and roll
centres. Only the wheel stiffness rate (effective stiffness) and wheel damping rate (effective
damping), see Figure 3-30, has to be given. This has the benefit that the chapter becomes relatively
independent of previous chapters, but it has the drawback that the presented models are not really
suitable for studies of steep road irregularities (which have longitudinal components) and sudden
changes in wheel torque or tyre side forces.
A vehicle function which is not covered in this compendium is noise.
5.1.1


References for this chapter
“Chapter 21 Suspension Systems” in Reference (Ploechl, 2013).
“Chapter 29 Ride Comfort and Road Holding” in Reference (Ploechl, 2013).
221
VERTICAL DYNAMICS
𝒛
𝒛 ⋯
𝒛
𝒛
Stresses in
vehicle
structure
Road surface
irregularities
Vehicle
created
disturbances
𝒛
𝒛
Fatigue Life
𝒛
Material
fatigue
Ride Comfort
Vehicle’s
dynamic
structure
Vehicle
response to
excitation
Compression
of tyre
𝒛
Human
perception of
vibrations
𝒛
Contact
between tyres
and road
Road Grip
Figure 5-1: Different types of knowledge and functions in the area of vertical vehicle dynamics,
organised around the vehicle’s dynamic structure.
5.2 Suspension System
Suspension design is briefly discussed at these places in this compendium: Section 3.4.7, Section
4.3.9.5 and Section 5.2.
Suspension design influence ride comfort, load on suspension and road grip through how vertical
forces and camber and steering angles on each wheel changes with body motion (heave, roll, pitch),
road unevenness (bumps, potholes, waviness) and wheel forces in ground plane (from Propulsion,
Braking and Steering subsystems).
Suspension in a vehicle may refer to suspension wheels (or axles), suspension of sub-frame and
drivetrain and suspension of cabin (for heavy trucks). In present section, only wheel (or axle)
suspension is considered.
A wheel suspension has the purpose to constrain the wheel from 6 degrees of freedom, dofs, relative
the body, to 2 or 3 dofs. A steered wheel needs 3 dofs (pitch rotation, vertical translation and yaw
rotation). For an un-steered wheel, also the yaw rotation should be constrained. For most steering,
left and right wheels on an axle have dependent steering angles, which could be seen as 2.5
dofs/wheel.
222
VERTICAL DYNAMICS
(sprung) body
driver,
traffic
motion of body
above each wheel
prop, brk, ste
systems (HW&SW)
Tq, SteAn
RotSpeed, SteForces
forces on body
from each wheel
suspension
wheel
& tyre
multiple
vertical
displacement
Fy
Fz
Fx
road
Figure 5-2: Individual wheel suspension described as one modular sub-model per wheel. It may be
noted that a both wheel model (main geometry such as wheel radius) and tyre model (how Fx and
Fy vary with tyre slip and Fz) is a part of each such sub-model.
An (axle) suspension system mainly consists of:



Linkage, which has the purpose to constrain the relative motion between wheel and body.
An alternative way to express this is that the linkage defines how longitudinal and lateral
wheel forces are brought into the body (body = sprung mass). Effective pivot points and
roll/pitch centres, mentioned earlier in this compendium, are defined through the linkage. In
the real pivot points in the linkage, there are bushings with stiffness and damping. The
bushings stiffness is much larger than the stiffness of the compliances mentioned below. For
steered wheels, the linkage also has the purpose to allow steering.
There are two main concepts: Individual wheel suspension and rigid axle suspension.
Compliances (or springs), which has the purpose to allow temporary vertical displacement of
the wheels relative to the body. There are often one spring per wheel but also a spring per
axle. The second is called anti-roll bar and connects left and right wheel to each other to
reduce body roll. Compliances often have a rather linear relation between the vertical
displacement and force of each wheel, but there are exceptions:
o Anti-roll bars make two wheels dependent of each other (still linear). Anti-roll bars
can be used on both individual wheel suspensions and rigid axle suspensions.
o The compliances can be intentionally designed to be non-linear in the outer end of
the stroke, e.g. bump stops.
o The compliances can be non-linear during the whole stroke, e.g. air-springs and leafsprings.
o The compliances can be intentionally designed to be controllable during operation of
the vehicle. This can be to change the pre-load level to adjust for varying roads or
varying weight of vehicle cargo or to be controllable in a shorter time scale for
compensating in each oscillation cycle. The latter is very energy consuming and no
such “active suspension” is available on market.
Dampers, which has the purpose to dissipate energy from any oscillations of the vertical
displacement of the wheel relative to the body. Dampers often has a rather linear relation
between the vertical deformation speed and force of each wheel, but there are exceptions:
223
VERTICAL DYNAMICS
o
o
o
o
The dampers can be intentionally designed to be different in different deformation
direction. This is actually the normal design for dampers of hydraulic piston type,
and it means that damping coefficient is different in compression and rebound.
Damping in leaf springs is non-linear since they work with dry friction.
Damping in air-springs is non-linear due to the nature of compressing gas.
The dampers can be intentionally designed to be controllable during operation of
the vehicle. This can be to change the damping characteristics to adjust for varying
roads or varying weight of vehicle cargo or to be controllable in a shorter time scale
for compensating in each oscillation cycle. The latter is called “semi-active
suspension” and is available on some high-end vehicles on market.
The simplest view we can have of a suspension system is that there is an individual suspension
between the vehicle body and each wheel. Each such suspension is a parallel arrangement of one
linear spring and one linear damper. Chapter 5 uses this simple view for analysis models, because it
facilitates understanding and it is enough for a first order evaluation of the functions studied
(comfort, road grip and fatigue load) during normal driving on normal roads.
5.3 Stationary oscillations theory
Many vehicle functions in this chapter will be studied using stationary oscillations (cyclic repeating),
as opposed to transiently varying. An example of transiently varying quantity is a single step function
or single square pulse. A stationary oscillation can always be expressed as a sum of several harmonic
terms, a so called multiple frequency harmonic stationary oscillation. The special case that only one
frequency is contained can be called single frequency harmonic stationary oscillation. See Figure 5-3
and Equation [5.1].
m
zz1
zz1
z1
c
F1
xt
step up, as one example
of transient (nonestationary oscillating)
zz1
tx
harmonic stationary
oscillation, single
frequency
other examples can be
step down, square pulse,
ramp etc.
xt
none-harmonic
stationary oscillation,
single frequency
(or multiple-frequency
harmonic stationary
oscillation)
tx
random noise
(or multiple- and
random-frequency
harmonic stationary
oscillation)
Figure 5-3: Different types of variables, both transient and stationary oscillating. The independent
variable 𝜉 can, typically, be either time or distance.
𝑯
𝒐𝒏 𝒄 𝒕 𝒕 𝒐𝒏 𝒚 𝒐 𝒄
𝒕 𝒐𝒏 :
𝑆𝑖𝑛𝑔 𝑒 𝑓𝑟𝑒𝑞𝑢𝑒𝑛𝑐𝑦 ∶ 𝑧(𝜉) 𝑧̂ ∙ cos( ∙ 𝜉
𝑁
𝑢 𝑡𝑖𝑝 𝑒 𝑓𝑟𝑒𝑞𝑢𝑒𝑛𝑐𝑖𝑒𝑠 ∶ 𝑧(𝜉)
𝑤 𝑒𝑟𝑒 𝜉 𝑖𝑠 𝑡 𝑒 𝑖𝑛𝑑𝑒𝑝𝑒𝑛𝑑𝑒𝑛𝑡
224
∑ 𝑧̂ ∙ cos(
=
𝑟𝑖 𝑏 𝑒.
);
∙𝜉
);
[5.1]
VERTICAL DYNAMICS
The most intuitive is probably to think of time as the independent variable, i.e. that the variation
takes place as function of time. This would mean that 𝜉 𝑡 in Equation [5.1]. However, for one
specific road, the vertical displacement varies with longitudinal position, rather than with time. This
is why we can either do analysis in time domain (𝜉 𝑡) and space domain (𝜉 𝑥).
Since the same oscillation can be described either as a function of 𝜉 (𝑧 𝑧(𝜉)) or as a function of
frequency (𝑧̂ 𝑧̂ ( )), we can do analysis either in the independent variable domain (𝜉) or in
frequency domain ( ).
The four combinations of domains are shown in Figure 5-4.
time
domain, t
𝒙
space
domain, x
Fourier Transform
(time) frequency
domain, 𝒐 𝒇
𝒇
𝒗𝒙 ∙ 𝒕;
Fourier Transform
𝒇⁄𝒗𝒙 ;
⁄𝒗𝒙 ;
spatial frequency
domain,
𝒐 𝒇
Figure 5-4: Four combinations of domains
Time and space domains are treated in section 5.3.1 and 0. In addition to the domains, we also need
to differ between discrete and continuous representations in the frequency domains, see 5.3.1.2 and
5.3.2.2.
5.3.1
Time domain and time frequency domain
In time domain, the frequency has the common understanding of “how often per time”. Even so,
there are two relevant ways to measure frequency: angular (time) frequency, and (time) frequency.
∙ 𝜋 ∙ 𝑓;
𝑟 𝑑
𝑤 𝑒𝑟𝑒
𝑛𝑔𝑢 𝑟 𝑓𝑟𝑒𝑞𝑢𝑒𝑛𝑐𝑦;
[
]
𝑠
1 𝑜𝑠𝑐𝑖 𝑡𝑖𝑜𝑛𝑠
𝑛𝑑 𝑓 [
] (𝑡𝑖 𝑒)𝑓𝑟𝑒𝑞𝑢𝑒𝑛𝑐𝑦;
𝑠
𝑠
[5.2]
The time for one oscillation is called the period time. It is denoted 𝑇:
𝑇
1
𝑓
∙𝜋
;
[5.3]
5.3.1.1 Mean Square (MS) and Root Mean Square (RMS) of
variable
For a variable, z, we can define MS and RMS values as follows:
225
VERTICAL DYNAMICS
𝑉 𝑟𝑖 𝑏 𝑒:
𝑧
𝑒 𝑛𝑆𝑞𝑢 𝑟𝑒:
𝑧(𝑡);
𝑡
∫ 𝑒𝑛𝑑 𝑧 2 ∙ 𝑑𝑡
𝑆(𝑧)
𝑅𝑜𝑜𝑡 𝑒 𝑛𝑆𝑞𝑢 𝑟𝑒:
𝑡
√
𝑅 𝑆(𝑧)
𝑑
𝑡
∫ 𝑒𝑛𝑑 𝑧 2
𝑡
;
[5.4]
∙ 𝑑𝑡
𝑑
;
If the variable is written as a single frequency harmonic stationary oscillation, these values becomes
as follows:
);
𝑧̂ ∙ cos( ∙ 𝑡
𝑡𝑒𝑛𝑑 2
𝑡𝑒𝑛𝑑
(𝑧̂ ∙ cos( ∙ 𝑡
𝑧 ∙ 𝑑𝑡 ∫
∫
𝑆(𝑧)
𝑡 𝑑
𝑡 𝑑
∙ 𝑡) 𝑡=𝑡𝑒𝑛𝑑
𝑡 𝑑 sin( ∙ ∙ 𝑡 𝑑 )
2
] 𝑡=
𝑧̂ ∙ (
)
4∙
𝑉 𝑟𝑖 𝑏 𝑒:
𝑧
𝑒 𝑛𝑆𝑞𝑢 𝑟𝑒:
𝑧̂ 2 ∙ [
𝑡
sin( ∙
4∙
𝑡 𝑑
𝑅𝑜𝑜𝑡 𝑒 𝑛𝑆𝑞𝑢 𝑟𝑒:
𝑡
√
𝑅 𝑆(𝑧)
∫
𝑡𝑒𝑛𝑑 2
𝑧
𝑡
∙ 𝑑𝑡
𝑧̂ 2
→
𝑡𝑒𝑛𝑑 →∞
𝑑
|𝑧̂ |
√ 𝑆(𝑧)
𝑑
))2 ∙ 𝑑𝑡
√
[5.5]
;
;
If the variable is written as a multiple frequency harmonic stationary oscillation, these values
becomes as follows:
𝑁
𝑉 𝑟𝑖 𝑏 𝑒:
𝑧
𝑒 𝑛𝑆𝑞𝑢 𝑟𝑒:
∫
𝑡𝑒𝑛𝑑
(∑𝑁=
⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗
𝑡 𝑑→∞
∫
𝑡𝑒𝑛𝑑
∑𝑧
∫
𝑆(𝑧)
𝑧̂ ∙ cos(
𝑡
∑𝑁= 𝑧̂
𝑁
=
𝑡𝑒𝑛𝑑
∑ 𝑧̂ ∙ cos(
=
𝑧 2 ∙ 𝑑𝑡
𝑡
2
𝑑
2
∙ (cos(
𝑡
∙𝑡
𝑅 𝑆(𝑧)
𝑑
⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗
𝑡 𝑑→∞
𝑁
))2 ∙ 𝑑𝑡
∑
𝑑
𝑁
𝑆(𝑧 )
=
√ 𝑆(𝑧)
√∑
=
𝑁
2
(∑𝑁= 𝑧 ) ∙ 𝑑𝑡
𝑡
𝑁
𝑅𝑜𝑜𝑡 𝑒 𝑛𝑆𝑞𝑢 𝑟𝑒:
𝑡𝑒𝑛𝑑
𝑑
)) ∙ 𝑑𝑡
∙𝑡
∫
);
∙𝑡
𝑧̂
2
∑
𝑧̂
2
;
[5.6]
=
𝑁
√∑
𝑆(𝑧 )
=
2
√∑(𝑅 𝑆(𝑧 )) ;
=
5.3.1.2
Power Spectral Density and Frequency bands
So far, the frequency has been a discrete number of frequencies,
2 ⋯
𝑁 . There are reasons
to treat the frequency as a continuous variable instead. The discrete amplitudes, 𝑧̂ 𝑧̂2 ⋯ 𝑧̂𝑁 , should
then be thought of as integrals of a “continuous amplitude curve”, 𝑧̂𝑐 , where the integration is done
over a small frequency interval, centred around a each mid frequency, :
226
VERTICAL DYNAMICS
𝜔𝑖 +𝜔𝑖+1
2
𝑧̂
∫
𝑧̂𝑐 ∙ 𝑑
𝑧̂𝑐 (
)∙
+
−
𝑧̂𝑐 (
)∙∆
; ⇒ 𝑧̂𝑐 (
𝜔 +𝜔
𝜔= 𝑖−12 𝑖
)
𝑧̂
;
∆
[5.7]
We realize that the unit of 𝑧̂𝑐 has to be same as for 𝑧, but per [rad/s]. So, if z is a displacement in [m],
𝑧̂𝑐 has the unit [m/(rad/s)]. Now, 𝑧̂𝑐 is a way to understand the concept of a spectral density. A
similar value, but more used, is the Power Spectral Density, PSD (also called Mean Square Spectral
Density).
𝑃𝑆𝐷( ) is a continuous function, while 𝑧̂ is a discrete function. That means that 𝑃𝑆𝐷( ) is fully
determined by a certain measured or calculated variable 𝑧(𝑡), while 𝑧̂ depends on which
discretization (which or which ∆ ) that is chosen.
𝑃𝑆𝐷(𝑧(𝑡)
𝑆(𝑓𝑖 𝑡𝑒𝑟(𝑧(𝑡) ∆ ))
𝐺( );
∆
𝑏 𝑛𝑑𝑝 𝑠𝑠 𝑓𝑖 𝑡𝑒𝑟 𝑐𝑒𝑛𝑡𝑒𝑟𝑒𝑑 𝑟𝑜𝑢𝑛𝑑
𝑛𝑑 𝑤𝑖𝑡 𝑏 𝑛𝑑 𝑤𝑖𝑑𝑡 ∆ ;
∆ )
𝑤 𝑒𝑟𝑒 filter𝑖𝑠
[5.8]
PSD can also be defined with band width in time frequency instead of angular frequency. The
Equation [5.8] is the same but replacing ∆ with ∆𝑓.
When the variable to study (z) is known and the band width is known, one often write simply
𝑃𝑆𝐷( ) or 𝐺( ). G has the same unit as 𝑧 2 , but per [rad/s] or per [oscillations/s]. So, if z is a
displacement in [m], G has the unit [
𝑚2
]
𝑑 ⁄𝑠
or [
𝑚2
⁄𝑠
2
∙ 𝑠].
The usage of the PSD is, primarily, to easily obtain the RMS via integration:
𝑁
𝑅 𝑆(𝑧)
√∑
∞
𝑁
𝑆(𝑧 )
=
√∑ 𝐺(
)∙∆
√ ∫ 𝐺( ) ∙ 𝑑 ;
=
[5.9]
𝜔=
RMS is square root of the area under the PSD curve.
5.3.1.2.1 Differentiation of PSD
Knowing the PSD of a variable, we can easily obtain the PSD for the derivative of the same variable:
𝐺 ( )
5.3.1.3
2
∙ 𝐺 ( );
[5.10]
Transfer function
In a minimum model for vertical dynamics there is at least one excitation, often road vertical
displacement, 𝑧 , and one response, e.g. vertical displacement of sprung mass (=vehicle body), 𝑧𝑠 . A
Transfer function, 𝐻 𝐻(𝑗 ∙ ), is the function which we can use to find the response, given the
excitation:
𝑍𝑠 ( ) 𝐻( ) ∙ 𝑍 ( ); ⟺ ℱ(𝑧𝑠 (𝑡)) 𝐻( ) ∙ ℱ(𝑧 (𝑡));
𝑤 𝑒𝑟𝑒 ℱ 𝑖𝑠 𝑡 𝑒 𝑜𝑢𝑟𝑖𝑒𝑟 𝑜𝑝𝑒𝑟 𝑡𝑜𝑟: 𝑍( ) ℱ(𝑧(𝑡))
∞
[5.11]
∫ 𝑒 −𝑗∙𝜔∙𝑡 ∙ 𝑧(𝑡) ∙ 𝑑𝑡 ;
227
VERTICAL DYNAMICS
√(Re(𝐻))2
𝐻 is complex, which is why it has a magnitude, |𝐻|
arg(𝐻( )) arctan(Im(𝐻)⁄Re(𝐻)).
(Im(𝐻))2 , and phase,
𝐴 𝑝 𝑖𝑡𝑢𝑑𝑒: 𝑧̂𝑠 ( ) |𝐻( )| ∙ 𝑧̂ ( );
( ) arg(𝐻( )) ;
𝑃 𝑠𝑒: 𝑠 ( )
𝑁
∑ 𝑧̂ (
𝑤 𝑒𝑟𝑒 𝑧
) ∙ cos(
∙𝑡
[5.12]
);
=
Since there can be different excitations and responses in a system, there are several transfer
functions. To distinguish between those, a subscripting of 𝐻 is often used: 𝐻 𝑐 𝑡 𝑡 → 𝑠𝑝 𝑠 ,
which would be 𝐻 𝑟 → 𝑠 𝐻 𝑑 𝑑 𝑠𝑝 𝑐 𝑚 𝑡→𝑠𝑝 𝑢 𝑚 𝑠𝑠 𝑑 𝑠𝑝 𝑐 𝑚 𝑡 in the example above. Other
examples of relevant transfer functions in vertical vehicle dynamics are:

𝐻
𝑑 𝑑 𝑠𝑝 𝑐 𝑚
𝑡→𝑠𝑝 𝑢

𝐻
𝑑 𝑑 𝑠𝑝 𝑐 𝑚
𝑡→𝑠𝑢𝑠𝑝

𝐻
𝑑 𝑑 𝑠𝑝 𝑐 𝑚
𝑡→𝑡
𝑚 𝑠𝑠 𝑐𝑐
𝑠
𝑑
𝑁
𝑐 [𝑚]
𝑡
𝑚 𝑡
𝑚⁄ 𝑠 2
], see Section 5.6
𝑚
𝑚
[𝑚] , see Section 5.7
[
, see Section 5.8
When transfer function for one derivative is found, it is often easy to convert it to another:
𝐻
𝐻
𝐻
𝑗∙
𝑗∙
1→ 2
1→ 2
1→ 2− 3
∙ 𝐻 1→ 2 ;
∙ 𝑗 ∙ ∙ 𝐻 1→ 2
2
∙ 𝐻 1→ 2 ;
𝐻 1→ 2 𝐻 1→ 3 ;
[5.13]
Note that these relations are valid for the complex transfer function. They corresponding relations
can be used with |𝐻| replacing 𝐻, in cases where the phase is same for all involve variables, which
often can be the case when damping is neglected.
The usage of the transfer function is, primarily, to easily obtain the response from the excitation, as
shown in Equation [5.12]. Also, the transfer function can operate on the Power Spectral Density,
PSD=G, as shown in the following:
𝐺 𝑠( )
|𝐻
𝑆(𝑧𝑠 (𝑡)
∆
2
𝑟→ 𝑠
( )| ∙
2
)
2
(𝑧̂𝑠 ( )) ⁄
∆
(|𝐻( )| ∙ 𝑧̂ ( )) ⁄
∆
2
(𝑧̂ ( )) ⁄
∆
|𝐻
[5.14]
2
𝑟→ 𝑠
( )| ∙ 𝐺 𝑟 ( );
Using Equation [5.9], we can then calculate 𝑅 𝑆(𝑧𝑠 ) from 𝐺 𝑠 ( ).
5.3.2 Space domain and Space frequency
domain
All transformations, in this compendium, between time domain and space domain requires a
constant longitudinal speed, , so that:
𝑥
228
∙𝑡
𝑥 ;
[5.15]
VERTICAL DYNAMICS
The offset (𝑥 ) is the phase (spatial) offset (𝑥 ) is the correspondence to the phase angle ( ).
The corresponding formulas as given in Equations [5.2]..[5.13] can be formulated when changing to
space domain, or spatial domain. It is generally a good idea to use a separate set of notations for the
spatial domain. Hence the formulas are repeated with new notations, which is basically what will be
done in present section.
There are not relevant to define transfer functions for the spatial domain before a road model is
given, see Section 5.4.2.1.
In space domain, the frequency has the common understanding of “how often per distance”. Even
so, there are two relevant ways to measure frequency: spatial angular frequency and spatial
frequency.
Ω
∙ 𝜋 ∙ 𝑓𝑠 ;
𝑟 𝑑
𝑤 𝑒𝑟𝑒 Ω [
]
𝑛𝑑 𝑓𝑠 [
1
𝑠𝑝 𝑡𝑖
𝑜𝑠𝑐𝑖
𝑡𝑖𝑜𝑛𝑠
𝑛𝑔𝑢 𝑟 𝑓𝑟𝑒𝑞𝑢𝑒𝑛𝑐𝑦;
]
𝑠𝑝 𝑡𝑖
[5.16]
𝑓𝑟𝑒𝑞𝑢𝑒𝑛𝑐𝑦;
The correspondence to period time is wave length, denoted 𝜆:
𝜆[ ]
1
𝑓𝑠
∙𝜋
;
Ω
[5.17]
Now, the basic assumption in Equation [5.15] and definitions of frequencies gives:
∙ Ω;
𝑛𝑑 𝑓
∙ 𝑓𝑠 ;
[5.18]
The relation between the phase (spatial) offset (𝑥 ) and the phase angle ( ) is:
𝜆∙
;
∙𝜋
𝑥
[5.19]
5.3.2.1 Spatial Mean Square (MS) and spatial Root Mean
Square (RMS) of variable
In space domain, a variable, z, varies with distance, x. We can define Mean Square and Root Mean
Square values also in space domain. We subscript these with s for space.
𝑉 𝑟𝑖 𝑏 𝑒:
𝑧
𝑆𝑠 (𝑧)
𝑒 𝑛𝑆𝑞𝑢 𝑟𝑒:
𝑅𝑜𝑜𝑡 𝑒 𝑛𝑆𝑞𝑢 𝑟𝑒:
𝑅 𝑆𝑠 (𝑧)
𝑧(𝑥);
∫ 𝑒𝑛𝑑 𝑧 2 ∙ 𝑑𝑥
;
𝑥 𝑑
∫
√
𝑒𝑛𝑑
𝑥
𝑧2
𝑑
∙ 𝑑𝑥
[5.20]
;
Because 𝑥 is constant, the Mean Square and Root Mean Square will be the same in time and space
domain. If the variable is written as a single frequency harmonic stationary oscillation, these values
becomes as follows:
229
VERTICAL DYNAMICS
𝑉 𝑟𝑖 𝑏 𝑒:
𝑧̂ ∙ cos(Ω ∙ 𝑥 𝑥 ) ;
𝑧̂ 2
⋯
𝑆(𝑧);
𝑧
𝑆𝑠 (𝑧)
𝑒 𝑛𝑆𝑞𝑢 𝑟𝑒:
𝑅 𝑆𝑠 (𝑧)
𝑅𝑜𝑜𝑡 𝑒 𝑛𝑆𝑞𝑢 𝑟𝑒:
⋯
|𝑧̂ |
[5.21]
𝑅 𝑆(𝑧);
√
If the variable is written as a multiple frequency harmonic stationary oscillation, these values
becomes as follows:
𝑁
𝑉 𝑟𝑖 𝑏 𝑒:
𝑧
∑𝑧
=
𝑁
𝑆𝑠 (𝑧)
𝑒 𝑛𝑆𝑞𝑢 𝑟𝑒:
𝑁
∑
𝑧̂
𝑥 );
∑ 𝑧̂ ∙ cos(Ω ∙ 𝑥
=
2
𝑆(𝑧);
[5.22]
=
𝑁
𝑅𝑜𝑜𝑡 𝑒 𝑛𝑆𝑞𝑢 𝑟𝑒:
𝑅 𝑆𝑠 (𝑧)
√ 𝑆𝑠 (𝑧)
2
√∑(𝑅 𝑆(𝑧 ))
=
𝑅 𝑆(𝑧);
5.3.2.2 Spatial Power Spectral Density and Frequency
bands
A correspondence to Power Spectral Density in space domain is denoted 𝑃𝑆𝐷𝑠 in the following:
𝑃𝑆𝐷𝑠 (𝑧(𝑥) Ω ∆𝜆)
𝑀𝑆(
𝑡
( ( ) Ω ∆𝜆))
Φ(Ω);
∆𝜆
[5.23]
where "filter" is a band pass filter centred around ω and with band width ∆𝑓;
When the variable to study (z) is known and the band width is known, one often write simply
𝑃𝑆𝐷𝑠 (Ω) or Φ(Ω). Φ has the same unit as 𝑧 2 , but per [rad/m] or per [oscillations/m]. So, if z is a
displacement in [m], Φ has the unit [
𝑚2
𝑑 ⁄𝑚
𝑚3
]
𝑑
or [
𝑚2
⁄𝑚
3
].
5.4 Road models
In general, a road model can include ground properties such as coefficient of friction,
damping/elasticity of ground and vertical position. The independent variable is either one, along an
assumed path, or generally two, x and y in ground plane. In vertical dynamics in this compendium,
we only assume vertical displacement as function of a path. We use x as independent variable along
the path, meaning that the road model is: 𝑧
𝑧 (𝑥). The function 𝑧 (𝑥) can be either of the types
in Figure 5-3. We will concentrate on stationary oscillations, which by Fourier series, always can be
expressed as multiple (spatial) frequency harmonic stationary oscillation. This can be specialized to
either single (spatial) frequency or random (spatial) frequency. Hence, the general form of the road
model is multiple (spatial) frequencies:
𝑁
𝑧
𝑧 (𝑥)
∑ 𝑧̂ ∙ cos(Ω ∙ 𝑥
=
230
𝑥 );
[5.24]
VERTICAL DYNAMICS
5.4.1
One frequency road model
For certain roads, such as roads built with concrete blocks, a single (spatial) frequency can be a
relevant approximation to study a certain single wave length. Also, the single (spatial) frequency road
model is good for learning the different concepts. A single (spatial) frequency model is the same as a
single wave length model (𝜆
∙ 𝜋⁄Ω, from Equation [5.16]) and it can be described as:
𝑧
5.4.2
𝑧 (𝑥)
𝑧̂ ∙ cos(Ω ∙ 𝑥
𝑥 );
[5.25]
Multiple frequency road models
Based on the general format in Equation [5.24], we will now specialise to models for different road
qualities. In Figure 5-5, there are 4 types of road types defined. Approximately, the 3 upper of those
are roads and they are also defined as PSD-plots in Figure 5-6. The mathematical formula is given in
Equation [5.26] and numerical parameter values are given in Equation [5.27].
𝛷
𝑤
w
𝛺
𝛺
𝛺 −𝑤
𝛷 ∙( )
𝛺
𝑆𝑠 (𝑧 𝛺)
;
∆𝛺
m2
𝑒𝑟𝑒 𝛷
road severity [
];
rad⁄m
road waviness [1];
spatial angular frequency [rad⁄m];
1 [rad/m];
𝛷(𝛺)
[5.26]
Typical values are
m2
1 ∙ 10−6 [
];
rad⁄m
m2
Bad road ∶
𝛷
10 ∙ 10−6 [
];
rad⁄m
m2
Very bad road ∶ 𝛷
100 ∙ 10−6 [
];
rad⁄m
𝑇 𝑒 𝑤 𝑖𝑛𝑒𝑠𝑠 𝑖𝑠 𝑛𝑜𝑟
𝑦 𝑖𝑛 𝑡 𝑒 𝑟 𝑛𝑔𝑒 𝑜𝑓 𝑤
𝑤 𝑒𝑟𝑒 𝑠 𝑜𝑜𝑡 𝑟𝑜 𝑑𝑠
𝑒 𝑟𝑔𝑒𝑟 𝑤 𝑖𝑛𝑒𝑠𝑠 𝑡
Very good road: 𝛷
[5.27]
. .3 [1]
𝑛 𝑏 𝑑 𝑟𝑜 𝑑𝑠.
The decreasing amplitude for higher (spatial) frequencies (i.e. for smaller wave length) can be
explained by that height variation over a short distance requires large gradients. On micro-level, in
the granular level in the asphalt, there can of course be steep slopes on the each small stone in the
asphalt. These are of less interest for vehicle vertical dynamics, since the wheel dimensions filter out
wave length << tyre contact length, see Figure 2-35.
A certain road can be described with:



𝛺 ⋯ 𝛺𝑁
𝑧̂ ⋯ 𝑧̂𝑁
𝑥
⋯ 𝑥 𝑁
231
VERTICAL DYNAMICS
Figure 5-5: Four typical road types, whereof the upper 3 can be considered as road types. From (AB
Volvo, 2011).
232
VERTICAL DYNAMICS
PHI=PHI0.*((OMEGA/OMEGA0).^(-waviness))
Φ Ω
Φ ∙ Ω⁄Ω −
-2
10
very rough
rough
smooth
-3
𝑚2
]
𝑑 ⁄𝑚
10
very bad road
bad road
very good road
-4
𝑃𝑆𝐷𝑠PHI
Ω /[m*m/(rad/m)]
Φ Ω in [
10
-5
10
-6
10
-7
10
∙
very rough
∙
rough
-8
10
smooth
∙
m2
a ⁄m
m2
−
;
a ⁄m
2
m
−
;
a ⁄m
−
; 𝒘
;
𝒘
.
;
𝒘
;
rad⁄m for all;
-9
10
-1
0
10
1
10
Ω
in
[𝑟
𝑑⁄ ]
OMEGA /[rad/m]
50
0
10
5
10
1
𝜆
Figure 5-6: PSD spectra for the three typical roads in Figure 5-5.
Number of frequency components, N, to select is a matter of accuracy or experience. The offsets,
𝑥
⋯ 𝑥 𝑁 , can often be assumed to be zero. If phase is to be studied, as in Figure 5-6, a random
generation of offsets is suitable. See also Reference (ISO 8608).
If we generate the actual 𝑧 (𝑥) curves for the 3 road types in Figure 5-6, we can plot as shown in
Figure 5-7. To generate those plots, we have to assume different number of harmonic components
(N in Equation [5.24]) and also randomly generate the phase for each component (each 𝑥 ) .
233
VERTICAL DYNAMICS
rough
very
RoadQuality=1
zr /[m]
0.05
N=10
N=20
N=100
0
-0.05
10
9
8
7
6
5
x/[m]
4
3
2
1
0
rough
RoadQuality=2
zr /[m]
0.05
N=10
N=20
N=100
0
-0.05
10
9
8
7
6
5
x/[m]
4
3
2
1
0
smooth
RoadQuality=3
zr /[m]
0.05
N=10
N=20
N=100
0
-0.05
10
9
8
7
6
5
x/[m]
4
3
2
1
0
Figure 5-7: Road profiles, 𝑧 (𝑥), for the three typical roads in Figure 5-5.
5.4.2.1 Transfer function from road spectrum in spatial
domain to system response in time domain
Since we assume constant longitudinal velocity, vx, the road spectrum can be transformed to the
time-frequency domain:
)
𝑆(𝑧
∆
𝐺 𝑟( )
{𝑢𝑠𝑒: 𝛷 ∙ (
−𝑤
𝛷 ∙𝛺
𝛺 −𝑤 ∙
𝛺 −𝑤
)
𝛺
𝛷0
𝛺
−𝑤
𝑆𝑠 (𝑧 𝛺)
∙ ∆𝛺
𝛺 −𝑤
𝛷 ∙ (𝛺 )
{𝑢𝑠𝑒:
∙ 𝛺}
𝑆𝑠 (𝑧 𝛺)
}
∆𝛺
[5.28]
−𝑤
∙
( )
𝛷0
𝛺
−𝑤
∙
𝑤−
−𝑤
∙
;
Then, we can use Equation [5.14] to obtain the response𝑧𝑠 :
𝐺 𝑠( )
|𝐻
2
𝑟→ 𝑠
( )| ∙ 𝐺 𝑟 ( )
|𝐻
2
𝑟→ 𝑠
( )| ∙
𝛷0
𝛺
−𝑤
∙
Then we can use Equation [5.9] to obtain the RMS of the response 𝑧𝑠 :
234
𝑤−
∙
−𝑤
;
[5.29]
VERTICAL DYNAMICS
𝑁
√∑ 𝐺 𝑠 (
𝑅 𝑆(𝑧𝑠 )
)∙∆
=
√
𝛷0
𝛺
−𝑤
𝑁
∙
𝑥
𝑤 1
2
∙ ∑|𝐻 𝑟→ 𝑠 ( )| ∙
𝑤
∙∆ ;
[5.30]
=
or
∞
𝑅 𝑆(𝑧𝑠 )
√ ∫ 𝐺 𝑠( ) ∙ 𝑑
0
√
𝛷0
𝛺
−𝑤
∞
∙
𝑥
𝑤 1
2
𝑤∙𝑑
∙ ∫|𝐻 𝑟→ 𝑠 ( )| ∙
;
0
5.5 One-dimensional vehicle models
“One-dimensional” refers to pure vertical motion, i.e. that the vehicle heaves without pitch and
without roll. The tyre is stiff and massless.
This can be seen as that the whole vehicle mass, m, is modelled as suspended by the sum of all
wheels’ vertical forces,
. However, the model can sometimes be
referred to as a “quarter-car-model”. That is because one can see the model as a quarter of the
vehicle mass, /4, which is suspended by one of the wheel’s vertical force, 𝑗 . The exact physical
interpolation of a quarter car is less obvious, since one can argue whether the fraction ¼ of the
vehicle mass is the proper fraction or from which point of view it is proper. Using the fraction ¼ is as
least debatable if the vehicle is completely symmetrical, both left/right and front/rear.
z2
rr=0
h
r
5.5.1 One-dimensional model without dynamic
degree of freedom
“Without dynamic degree of freedom” refers to that the (axle) suspension is modelled as ideally stiff.
The model can be visualised as in Figure 5-8.
real vehicle
model
zs
m
z
m
x
∙ 𝑧𝑠
py
∙ 𝑧𝑠
∙𝑔
Pz
∙𝑔
Pz
zr
cle
cf
zs
Frz
zr
Frz
Figure 5-8: One-dimensional model without dynamic degree of freedom
235
Z
m
VERTICAL DYNAMICS
Probably, we could find the equations without very much formalism ( ∙ 𝑧𝑠
; 𝑛𝑑 𝑧𝑠
𝑧 (𝑡);), but the following equations exemplifies a formalism which will be useful when we expand
the model later.
𝑬𝒒𝒖
𝒖 :
∙ 𝑧𝑠
𝑪𝒐 𝒑 𝒕
𝒕𝒚: 𝑧
𝑬𝒙𝒄 𝒕 𝒕 𝒐𝒏:
𝑧
∙𝑔
𝑃
0;
𝑧𝑠 ;
𝑧 (𝑡);
𝑃;
[5.31]
Note that an alternative model would be to distribute the mass in the model; 𝑠 above 𝑃 and
under 𝑃 . However, it would not influence the transfer functions derived later in 5.5.1.1.
5.5.1.1
𝑢
Response to a single frequency excitation
Assume that the road has only one (spatial) frequency, i.e. one wave length. Then the excitation is as
follows:
𝑧 (𝑥)
𝑧
{
𝑧̂ ∙ cos(Ω ∙ 𝑥
𝑥 )
𝑧̂ ∙ cos (
∙π
∙𝑥
𝜆
𝑥 );
}⇒
𝑥
∙ 𝑡;
𝐴𝑠𝑠𝑢 𝑒 𝑥
0;
∙π∙
𝑧̂ ∙ cos (
∙ 𝑡) 𝑧̂ ∙ cos(ω ∙ 𝑡) ; ⇒
𝜆
∙π∙
∙π∙
∙ 𝑧̂ ∙ sin (
∙ 𝑡)
ω ∙ 𝑧̂ ∙ sin(ω ∙ 𝑡) ; ⇒
𝜆
𝜆
2
∙π∙
∙π∙
(
) ∙ 𝑧̂ ∙ cos (
∙ 𝑡)
ω2 ∙ 𝑧̂ ∙ cos(ω ∙ 𝑡) ;
𝜆
𝜆
⇒ 𝑧 (𝑡)
⇒ 𝑧 (𝑡)
⇒ 𝑧 (𝑡)
[5.32]
Insertion in the model in Equation [5.31] gives directly the solution:
(𝑡)
∙ 𝑔 ∆ (𝑡)
∙ 𝑔 ̂ ∙ cos(ω ∙ 𝑡) ;
𝑧 (𝑡) 𝑧𝑠 (𝑡) 𝑧̂ ∙ cos (ω ∙ 𝑡);
𝑧 (𝑡) 𝑧𝑠 (𝑡) ̂ ∙ cos(ω ∙ 𝑡) ;
𝑃 (𝑡)
{
𝑤 𝑒𝑟𝑒 {
̂
̂
[5.33]
∙ ω2 ∙ 𝑧̂ ;
ω2 ∙ 𝑧̂ ;
5.5.1.1.1 Analysis of solution
We can identify the magnitude of the transfer functions. The negative sign in Equation [5.33] means
180 degrees phase shift:
ℱ(𝑧𝑠 )
} 1 𝑗 ∙ 0;
ℱ(𝑧 )
𝐻 𝑟 → 𝑟 − 𝑠 {𝐻 𝑟 → 𝑟 − 𝑠 𝐻 𝑟 → 𝑠 𝐻 𝑠 → 𝑠 𝐻 𝑟 → 𝑠 1} 0 𝑗 ∙ 0;
𝐻 𝑟 → 𝑠 {𝐻 𝑟 → 𝑠 (𝑗 ∙ )2 ∙ 𝐻 𝑟 → 𝑠
ω ∙ 𝐻 𝑟→ 𝑠 }
ω
𝑗 ∙ 0;
𝐻 𝑟 →∆ 𝑟𝑧 {𝐻 𝑟 →∆ 𝑟𝑧
∙ 𝐻 𝑟→ 𝑠 }
∙ω
𝑗 ∙ 0;
𝐻
𝑟→ 𝑠
{𝐻
𝑟→ 𝑠
[5.34]
The motivation to choose exactly those transfer functions is revealed later, in Section5.6, 5.7 and 5.8.
For now, we simply conclude that various transfer functions can be identified and plotted. The plots
are found in Figure 5-9. Numerical values for m and 𝜆 has been chosen.
236
VERTICAL DYNAMICS
Driving at road with wave-length, lambda = 5 [m].
7
10
6
10
5
10
4
abs(H)
10
H_zr_zs (==1)
H_zr_zr-zs (==0)
H_zr_derderzs
H_zr_Frz
3
10
2
10
m = ms+mu = 1600[kg];
1
10
f=1 Hz at vx=5[m/s]; f=10 Hz at vx=50[m/s]
0
10
-1
10
0
1
10
2
10
3
10
10
vx [m/s]
Figure 5-9: Transfer functions from model in Figure 5-8, excited with single frequencies.
So, for example, we can use the transfer function diagram as follows: If we have a road displacement
amplitude of 1 cm (𝑧̂
0.01 ) and a speed of 50 km/h ( ≈ 14 /𝑠 ̂≈ .8𝐻𝑧), we can read
out, e.g.:



( )| ≈ 305; ⇒ | ̂| 310 ∙ 𝑧̂𝑟 305 ∙ 0.01 3.05 ⁄𝑠 2 ;. From this we can
calculate 𝑅 𝑆(𝑧𝑠 ) |3.05|⁄√ ≈ .16 ⁄𝑠 2 . The RMS value of acceleration will later be
|𝐻
𝑟→ 𝑠
related to ride comfort, see Section 5.6.
|𝐻 𝑟 → 𝑟 − 𝑠 ( )| 0;, i.e. no deformation, which is not strange, since model is stiff.
The deformation of suspension will later be related to fatigue life, see Section 5.7.
|𝐻 𝑟 →∆ 𝑟𝑧 ( )| ≈ 487000; ⇒ | ̂ | 487000 ∙ 𝑧̂𝑟 487000 ∙ 0.01 4870 𝑁;. If | ̂ | had
been
∙ 𝑔 ≈ 16000 𝑁, the model would have been outside its validity region, because it
would require pulling forces between tyre and road, which is not possible. The variation in
tyre road contact force will be related to road grip, see Section 5.8.
The phases for the studied variables are seen from the complex transfer functions in Equation [5.34].
5.5.1.2
Response to a multiple frequency excitation
Using Equation [5.30], Equation [5.45] and values for road type “rough” in Figure 5-6, we can
conclude:
𝑅 𝑆(𝑧𝑠 )
√
𝛷0
𝛺
−𝑤
∞
∙
𝑥
𝑤 1
2
∙ ∫|𝐻 𝑟→ 𝑠 ( )| ∙
𝑤∙𝑑
0
10 ∙ 10
√
1
[5.35]
∞
6
∙
𝑥
.5 1
2
∙ ∫|𝐻 𝑟→ 𝑠 ( )| ∙
0
237
.5 ∙ 𝑑
;
VERTICAL DYNAMICS
For now, we simply note that it is possible to calculate this (scalar) RMS value for each vehicle speed
over the assumed road. In corresponding way, an RMS value can be calculated for any of the
oscillating variables, such as 𝑧𝑠 , 𝑧
𝑧𝑠 and . We will come back to Equation [5.35] in Section
5.6.2.
5.5.2 One dimensional model with one
dynamic degree of freedom
“With one dynamic degree of freedom” refers to that the axle suspension is modelled as a linear
spring and linear (viscous) damper in parallel. The tyre is still stiff and massless. The model can be
visualised as in Figure 5-10.
z2
The corresponding mathematical model becomes as follows:.
r
𝑬𝒒𝒖
𝒖 :
h
𝑪𝒐𝒏 𝒕 𝒕𝒖𝒕 𝒐𝒏:
𝑬𝒙𝒄 𝒕 𝒕 𝒐𝒏:
∙ 𝑧𝑠
𝑃
𝑧
∙𝑔
𝑃
0;
𝑐 ∙ (𝑧
𝑧𝑠 )
𝑧 (𝑡);
real vehicle
rr=0
zs
py
x
𝑑 ∙ (𝑧
𝑧𝑠 )
m
∙ 𝑧𝑠
Z
∙𝑔
∙ 𝑧𝑠
∙𝑔
c
[5.36]
∙ 𝑔;
zs
model
z
m
𝑃;
Pz
Pz
zs
d
Pz
cle
zr
Frz
zr
zr
Pz
Frz
Figure 5-10: One-dimensional model with one dynamic degree of freedom
cf
Note that if we measure 𝑧𝑢 and 𝑧𝑠 from the static equilibrium, the static load,
when constitution is inserted in equilibrium.
∙ 𝑔, disappears
Note also that an alternative model would be to distribute the mass in the model; 𝑠 above 𝑃 and
𝑢 under 𝑃 . model lower part as having a mass. Such change would also affect the transfer
functions.
5.5.2.1
Response to a single frequency excitation
Eliminating 𝑃 and 𝑧 gives:
238
m
VERTICAL DYNAMICS
∆
∙ 𝑧𝑠
where
𝑐 ∙ (𝑧 (𝑡) 𝑧𝑠 ) 𝑑 ∙ (𝑧 (𝑡) 𝑧𝑠 );
𝑐 ∙ (𝑧 (𝑡) 𝑧𝑠 ) 𝑑 ∙ (𝑧 (𝑡) 𝑧𝑠 );
(𝑡)
∙𝑔
[5.37]
(𝑡);
∆
Assume that the road has only one (spatial) frequency, i.e. one wave length. Then the excitation is as
in Equation [5.25], in which we assume 𝑥
0. Insertion in Equation [5.36] yields:
∙ 𝑧𝑠
∙ 𝑧𝑠
𝑐 ∙ (𝑧̂ ∙ cos(ω ∙ 𝑡) 𝑧𝑠 ) 𝑑 ∙ ( ω ∙ 𝑧̂ ∙ sin(ω ∙ 𝑡) 𝑧𝑠 );
𝑑 ∙ 𝑧𝑠 𝑐 ∙ 𝑧𝑠 𝑧̂ ∙ (𝑐 ∙ cos(ω ∙ 𝑡) 𝑑 ∙ ω ∙ sin(ω ∙ 𝑡));
[5.38]
From here, the system of differential equations can be solved with trigonometry or Fourier
transform.
5.5.2.1.1 Solution with Fourier transform
Fourier transform of Equation [5.37] yields:
∙ ( ω2 ∙ ℱ(𝑧𝑠 )) 𝑐 ∙ (ℱ(𝑧 ) ℱ(𝑧𝑠 )) 𝑑 ∙ (𝑗 ∙ ∙ ℱ(𝑧 ) 𝑗 ∙
ℱ(∆ 𝑟𝑧 ) 𝑐 ∙ (ℱ(𝑧𝑟 ) ℱ(𝑧𝑠 )) 𝑑 ∙ (𝑗 ∙ ∙ ℱ(𝑧𝑟 ) 𝑗 ∙ ∙ ℱ(𝑧𝑠 ));
∙ ℱ(𝑧𝑠 ));
[5.39]
From this, we can then solve the transfer functions:
𝐻
𝑟→ 𝑠
𝐻
𝑟 →∆𝐹𝑟
ℱ(𝑧𝑠 )
𝑐
ℱ(𝑧 )
(𝑐
(𝑐
𝑗∙𝑑∙
∙ ω2 ) 𝑗 ∙ 𝑑 ∙
𝑗 ∙ 𝑑 ∙ ) ∙ (1
𝐻
𝑟→ 𝑠
;
[5.40]
);
5.5.2.1.2 Solution with trigonometry
One way to solve the mathematical model in Equations [5.36] and [5.38]) is to assume a real solution,
insert it in Equation [5.38]. Then the parameters be solved for. Assume 𝑥
0 in Equation [5.25]:
∙ 𝑧𝑠
𝑑 ∙ 𝑧𝑠
𝑐 ∙ 𝑧𝑠
𝑧̂ ∙ (𝑐 ∙ cos( ∙ 𝑡)
𝑑∙
∙ sin( ∙ 𝑡));
[5.41]
Assumed solution:
𝑧𝑠
⇒
⇒
⇒
⇒
); ⇒
𝑧̂𝑠 ∙ cos( ∙ 𝑡
{𝑢𝑠𝑒: cos(
𝑏) cos ∙ cos 𝑏
𝑧𝑠 𝑧̂𝑠 ∙ [cos( ∙ 𝑡) ∙ cos
sin(
𝑧𝑠 𝑧̂𝑠 ∙ ∙ [ sin( ∙ 𝑡) ∙ cos
𝑧𝑠
𝑧̂𝑠 ∙ 2 ∙ [cos( ∙ 𝑡) ∙ cos
Insertion:
239
sin ∙ sin 𝑏} ⇒
∙ 𝑡) ∙ sin ]; ⇒
cos( ∙ 𝑡) ∙ sin ]; ⇒
sin( ∙ 𝑡) ∙ sin ];
[5.42]
VERTICAL DYNAMICS
∙ 𝑧̂𝑠 ∙ 2 ∙ [cos( ∙ 𝑡) ∙ cos
sin( ∙ 𝑡) ∙ sin ]
𝑑 ∙ 𝑧̂𝑠 ∙ ∙ [ sin( ∙ 𝑡) ∙ cos
cos( ∙ 𝑡) ∙ sin ]
𝑐 ∙ 𝑧̂𝑠 ∙ [cos( ∙ 𝑡) ∙ cos
sin( ∙ 𝑡) ∙ sin ]
𝑧̂ ∙ (𝑐 ∙ cos( ∙ 𝑡) 𝑑 ∙ ∙ sin( ∙ 𝑡)); ⇒
𝒄𝒐 𝒕
:
∙ 𝑧̂𝑠 ∙ 2 ∙ cos
𝑑 ∙ 𝑧̂𝑠 ∙ ∙ sin
𝑐 ∙ 𝑧̂𝑠 ∙ cos
⇒ {
2
𝒏𝒕
:
∙ 𝑧̂𝑠 ∙
∙ sin
𝑑 ∙ 𝑧̂𝑠 ∙ ∙ cos
𝑐 ∙ 𝑧̂𝑠 ∙ sin
⇒
∙𝑑∙ 3
arctan ( 2
);
𝑐
∙ 𝑐 ∙ 2 𝑑2 ∙ 2
⇒
𝑧̂𝑠
𝑑∙
|𝐻𝑧𝑟 →𝑧𝑠 |;
2
( ∙
𝑐) ∙ sin
𝑑 ∙ ∙ cos
{𝑧̂
∙π∙
𝑤 𝑒𝑟𝑒
𝑥
𝜆
𝑧̂ ∙ 𝑐;
𝑧̂ ∙ 𝑑 ∙ ;
[5.43]
;
We have identified |𝐻 𝑟 → 𝑠 ( )|, which can be compared to |𝐻 𝑟 → 𝑠 ( )| in Equation [5.34]. The
other transfer functions in Equation [5.34] are more difficult to express using the method with real
algebra. We leave them to next section.
5.5.2.1.3 Analysis of solution
We can elaborate further with Equation [5.40]:
Amplitude:
𝑧̂𝑠
𝑧̂
|𝐻𝑧𝑟 →𝑧𝑠 |
𝐴𝑠𝑠𝑢 𝑒
|
𝑐
(𝑐
𝑐
(𝑐
𝑗∙𝑑∙ω
𝑅𝑒𝐻
∙ ω2 ) 𝑗 ∙ 𝑑 ∙ ω
𝑆𝑜 𝑒 𝑓𝑜𝑟 𝑅𝑒𝐻 𝑛𝑑 𝐼 𝐻;
|𝐻𝑧𝑟 →𝑧𝑠 |
{
𝑗∙𝑑∙ω
|
∙ ω2 ) 𝑗 ∙ 𝑑 ∙ ω
𝑗 ∙ 𝐼 𝐻;
⋯
𝐼 𝐻2 ;
√𝑅𝑒𝐻 2
}
√
𝑐
(𝑐
𝑑 ∙ω
∙ω )
𝑑 ∙ω
;
Phase:
𝑠(
[5.44]
)
( )
arg (
𝑗∙𝑑∙ω
𝑅𝑒𝐻 𝑗 ∙ 𝐼 𝐻;
∙ ω2 ) 𝑗 ∙ 𝑑 ∙ ω
𝑆𝑜 𝑒 𝑓𝑜𝑟 𝑅𝑒𝐻 𝑛𝑑 𝐼 𝐻;
tan(arg(𝐻𝑧𝑟→𝑧𝑠 )) 𝐼 𝐻⁄𝑅𝑒𝐻 ;
}
𝐴𝑠𝑠𝑢 𝑒
{
𝑐 𝑗∙𝑑∙ω
)
∙ ω2 𝑐 𝑗 ∙ 𝑑 ∙ ω
arctan (
𝑐
(𝑐
∙𝑑∙
∙𝑐∙ 2
𝑐2
⋯
3
𝑑2
∙
2)
;
where
∙𝜋∙
𝜆
;
Equation [5.13] now allows us to get the magnitudes of the other transfer functions as well:
𝐻
𝐻
𝐻
𝐻
𝑟→ 𝑠
𝑟→ 𝑟− 𝑠
𝑟→ 𝑠
𝑟 →∆ 𝑟𝑧
from Equation [5.40];
𝐻
𝐻 𝑟→ 𝑠 1 𝐻 𝑟→ 𝑠 ;
∙ 𝐻 𝑟→ 𝑠 ;
{∆ 𝑟𝑧 𝑐 ∙ (𝑧𝑟 𝑧𝑠 ) 𝑑 ∙ (𝑧𝑟 𝑧𝑠 )}
𝑐 ∙ (𝐻 𝑟 → 𝑟 𝐻 𝑟 → 𝑠 ) 𝑑 ∙ (𝐻 𝑟 →𝑧𝑟
(𝑐 𝑗 ∙ 𝑑 ∙ ) ∙ (𝐻 𝑟 → 𝑟 𝐻 𝑟 → 𝑠 )
(𝑐 𝑗 ∙ 𝑑 ∙ ) ∙ (1 𝐻 𝑟 → 𝑠 );
2
𝑟→ 𝑟
240
[5.45]
𝐻
𝑟 →𝑧𝑠
)
VERTICAL DYNAMICS
The motivation to choose exactly those transfer functions is revealed later, in Section5.6, 5.7 and 5.8.
Some of those magnitudes are easily expressed in reel (non-complex) mathematics using Equation
[5.44]:
|𝐻
|𝐻
𝑟→ 𝑠
𝑟→ 𝑠
|
√
|
𝑐
(𝑐
2
𝑑 ∙ω
∙ω )
∙√
𝑐
(𝑐
𝑑 ∙ω
;
[5.46]
𝑑 ∙ω
∙ω )
𝑑 ∙ω
;
The transfer functions in Equation [5.44] are plotted in Figure 5-11 and Figure 5-12. Numerical values
for m and 𝜆 has been chosen.
Driving at road with wave-length, lambda = 5 [m].
7
10
6
10
5
10
m = ms = 1415[kg]; c = cs = 68 [kN/m]; d = ds = 8[kN/(m/s)];
4
abs(H)
10
mu = 0 [kg]; ct = inf [kN/m];
f=1 Hz at vx=5[m/s]; f=10 Hz at vx=50[m/s]
3
10
2
10
1
H_zr_zs
H_zr_zr-zs
H_zr_derderzs
H_zr_Frz
10
0
10
-1
10
0
10
1
2
10
10
3
10
vx [m/s]
Figure 5-11: Transfer functions for amplitudes from model in Figure 5-10, excited with single
frequencies. Thin lines are without damping. Notation: 𝐻 →𝑏 is denoted H_a_b.
For example, we can use Figure 5-11 as follows, cf. end of Section 5.5.1.1.1: If we have a certain road,
with displacement amplitude of 1 cm (𝑧̂
0.01 ) and the vehicle should drive on it at a speed of
50 km/h ( ≈ 14 /𝑠 ̂≈ .8𝐻𝑧), we can read out, e.g.:



Ride comfort related: |𝐻 𝑟 → 𝑠 ( )| ≈ 1 0; ⇒ | ̂| 1 0 ∙ 𝑧̂𝑟 1 0 ∙ 0.01 1. 0 ⁄𝑠 2 ;.
From this we can calculate 𝑅 𝑆(𝑧𝑠 ) |1. 0|⁄√ ≈ 0.8485 ⁄𝑠 2 .
Fatigue life related: |𝐻 𝑟 → 𝑟 − 𝑠 ( )| ≈ 1.11; ⇒ |𝑧̂
𝑧̂𝑠 | 1.11 ∙ 𝑧̂𝑟 1.11 ∙ 0.01
0.0111
1.11 𝑐 ;.
Road grip related: |𝐻 𝑟 →∆ 𝑟𝑧 ( )| ≈ 59795; ⇒ | ̂ 𝑟𝑧 | 59795 ∙ 𝑧̂𝑟 59795 ∙ 0.01
598 𝑁;.
241
VERTICAL DYNAMICS
200
H_zr_zs
H_zr_zr-zs
H_zr_derderzs
H_zr_Frz
phase delay=arg(H) [deg]
150
100
50
0
-50
-100
0
10
1
2
10
10
3
10
vx [m/s]
Figure 5-12: Transfer functions for phase delays from model in Figure 5-10, excited with single
frequencies.
We compare these numbers with the corresponding numbers for the simpler model, see Sections
5.5.1.1.1. The comfort is has been better. The fatigue life and road grip have become more realistic.
Figure 5-11 also shows the curves for the undamped system (d=0). The highest peaks appear at
approximately
5. .6 m/s. This corresponds to the speed where the natural (=undamped) eigen
frequency appears ( 𝑐 𝑡 𝜆 ∙ 𝑓𝑐 𝑡 𝜆 ∙ 𝑐 𝑡 ⁄( ∙ 𝜋) 𝜆 ∙ √𝑐⁄ ⁄( ∙ 𝜋) ≈ 5.5 /𝑠).
Figure 5-12 shows the phase angles for the different responses.
5.5.3 One dimensional model with two degrees
of freedom
The expression “two dynamic degree of freedom” in section the heading refers to that both elasticity
between road and wheel as well as between wheel and sprung mass is modelled. Alternatively, one
can describe the two degrees of freedom as the suspension spring deformation and the tyre spring
deformation.
The model can be visualised as in Figure 5-13. No damping is modelled in tyre (in parallel with
elasticity 𝑐𝑡 ) because it is generally relatively low.
242
VERTICAL DYNAMICS
model
real vehicle
zs
𝑠
∙ 𝑧𝑠
ms
𝑠
∙𝑔
Fsz
Fsz
zs
z
ds
cs
Fsz
x
py
zu
mu
ct
𝑢
dt
∙ 𝑧𝑢
zu
𝑢
∙𝑔
Frz
zu
zr
zr
Fsz
Frz
Frz
Figure 5-13: One-dimensional model with two dynamic degrees of freedom
The corresponding mathematical model becomes as follows:
Equilibrium:
𝑠
∙ 𝑧𝑠
𝑢
∙ 𝑧𝑢
𝑠
𝑠
∙ 𝑔;
𝑠
𝑢
∙ 𝑔;
Constitution (displacements counted from static equilibrium):
𝑠
𝑐𝑠 ∙ (𝑧𝑢
𝑧𝑠 )
𝑑𝑠 ∙ (𝑧𝑢
𝑧𝑠 )
𝑐𝑡 ∙ (𝑧
𝑧𝑢 )
𝑑𝑡 ∙ (𝑧
𝑧𝑢 )
𝑠
(
∙ 𝑔;
𝑠
𝑢) ∙
[5.47]
𝑔;
Excitation:
𝑧 (𝑡);
𝑧
The same can be formulated with matrices and Fourier transforms:
[
𝑐𝑠
𝑐𝑠
𝑧𝑠
𝑑𝑠
𝑧
] ∙ [ 𝑠] [ 𝑐 𝑐
] ∙ [𝑧 ]
𝑐
𝑧
𝑑𝑠 𝑑𝑡
𝑢
𝑠
𝑠
𝑡
𝑢
0
0
[ ]∙𝑧
[ ]∙𝑧 ;
𝑑𝑡
𝑐𝑡
𝑧𝑠
𝑧
𝑧
⇒ 𝑴 ∙ [ 𝑠 ] 𝑫 ∙ [ 𝑠 ] 𝑪 ∙ [𝑧 ] 𝑫 ∙ 𝑧
𝑪 ∙𝑧 ;⇒
𝑧𝑢
𝑧𝑢
𝑢
0
𝑧
] ∙ [ 𝑠]
𝑧
𝑢
𝑢
𝑠
0
⇒𝑴∙(
⇒(
2
∙ 𝑍)
2
[
𝑑𝑠
𝑑𝑠
𝑫 ∙ (𝑗 ∙
∙𝑴
𝑤 𝑒𝑟𝑒 𝑍
𝑗∙
∙ 𝑍)
∙𝑫
𝑍
[ 𝑠]
𝑍𝑢
𝑫 ∙ (𝑗 ∙
𝑪∙𝑍
𝑪) ∙ 𝑍
[
(𝑗 ∙
∙𝑫
ℱ(𝑧𝑠 )
] ; 𝑛𝑑 𝑍
ℱ(𝑧𝑢 )
∙𝑍 )
[5.48]
𝑪 ∙ 𝑍 ;⇒
𝑪 )∙𝑍 ;
ℱ(𝑧 );
The matrix 𝑫 is zero, but marks a more general form, typical for modelling damping also in the tyre.
243
VERTICAL DYNAMICS
5.5.3.1
Response to a single frequency excitation
5.5.3.1.1 Solution with Fourier transform
We can find the transfer functions via Fourier transform, starting from Equation [4.64]:
[
𝐻 𝑟→ 𝑠
]
𝐻 𝑟→ 𝑢
[
𝑍𝑠 1
]∙
𝑍𝑢 𝑍
𝑍∙
1
𝑍
(
2
∙𝑴
𝑗∙
∙𝑫
𝑪)− ∙ (𝑗 ∙
∙𝑫
𝑪 );
[5.49]
This format is very compact, since it includes both transfer functions for amplitude and phase. For
numerical analyses, the expression in Equation [5.52] is explicit enough, since there are tools, e.g.
Matlab, which supports matrix inversion and complex mathematics.
For analytic solution one would need symbolic tools, e.g. Mathematica or Matlab with symbolic
toolbox, or careful manual algebraic operations. With 𝑑𝑡 0, the explicit forms becomes as follows:
𝐻
𝑍𝑠
𝑍
𝑟→ 𝑠
𝑢
𝐻
(
∙ ω2
𝑟→ 𝑢
∙π∙ 𝑥
𝜆
where
∙ ω2
𝑐𝑠 ) ∙ 𝑐𝑡
𝑐𝑡
(𝑐𝑠 𝑗 ∙ 𝑑𝑠 ∙ ω)2
𝑗 ∙ 𝑑𝑠 ∙ ω 𝑐𝑡 𝑐𝑠
2
𝑗 ∙ 𝑑𝑠 ∙ ω 𝑐𝑠
𝑠∙ω
;
𝑐𝑠 𝑗 ∙ 𝑑𝑠 ∙ ω
𝑐𝑡
(𝑐𝑠 𝑗 ∙ 𝑑𝑠 ∙ ω)2
2
𝑗 ∙ 𝑑𝑠 ∙ ω 𝑐𝑡 𝑐𝑠
𝑢∙ω
2
𝑗 ∙ 𝑑𝑠 ∙ ω 𝑐𝑠
𝑠∙ω
𝑢
𝑗 ∙ 𝑑𝑠 ∙ ω
𝑐𝑡
[5.50]
;
5.5.3.1.2 Analysis of solution
Equation [5.13] now allows us to get the magnitudes of the other transfer functions as well:
𝐻
𝐻
𝐻
𝐻
𝐻
𝐻
𝐻
𝑟→ 𝑠
𝑟→ 𝑢
see Equation [5.50];
see Equation [5.50];
𝑟→ 𝑟− 𝑢
𝑟→ 𝑢− 𝑠
𝑟→ 𝑠
𝑟 →Δ𝐹𝑠
𝑟 →Δ𝐹𝑟
𝐻𝑧𝑟 →𝑧𝑟 𝐻𝑧𝑟 →𝑧𝑢 1
𝐻𝑧𝑟 →𝑧𝑢 𝐻𝑧𝑟 →𝑧𝑠 ;
∙ 𝐻𝑧𝑟 →𝑧𝑠 ;
{Δ
{Δ
𝑠
𝐻𝑧𝑟 →𝑧𝑢 ;
𝑐𝑠 ∙ (𝑧𝑢 𝑧𝑠 ) 𝑑𝑠 ∙ (𝑧𝑢 𝑧𝑠 )}
𝑐𝑠 ∙ (𝐻 𝑟 → 𝑢 𝐻 𝑟 → 𝑠 ) 𝑑𝑠 ∙ 𝑗 ∙ ∙ (𝐻 𝑟 → 𝑢 𝐻 𝑟 → 𝑠 )
(𝑐𝑠 𝑗 ∙ 𝑑𝑠 ∙ ) ∙ (𝐻 𝑟 → 𝑢 𝐻 𝑟 → 𝑠 );
𝑐𝑡 ∙ (𝑧
𝑧𝑢 )} 𝑐𝑡 ∙ (𝐻𝑧𝑟 →𝑧𝑟 𝐻𝑧𝑟 →𝑧𝑢 ) 𝑐𝑡 ∙ (1 𝐻𝑧𝑟 →𝑧𝑢 );
[5.51]
Expression in real can be derived, see Equation [5.52]. Note that it is more general than Equation
[5.51], having a tyre damper modelled, 𝑑𝑡 0. The derivation is not documented in present
compendium.
244
VERTICAL DYNAMICS
2
|𝐻
𝑟→ 𝑠
|
|𝐻
|
𝑟→ 𝑢− 𝑠
|𝐻
𝑟→ 𝑟− 𝑢
√(
𝐴
𝐵
𝑠
3
∙
∙(
∙
√(𝑐𝑠 ∙ 𝑐𝑡
𝑠
∙ √(𝑐𝑡 ∙
2 )2
√𝐴2
|
∙
𝑢
𝑠
𝑑𝑠 ∙ 𝑑𝑡 ∙
∙
𝑠
∙
𝑢
∙ 𝑑𝑡
2
∙(
2 )2
2
( ∙ (𝑑𝑠 ∙ 𝑐𝑡
𝑑𝑡 ∙ 𝑐𝑠 ))
√𝐴2 𝐵2
(𝑑𝑡 ∙ 3 )2
;
;
𝐵2
𝑠
[5.52]
2
𝑢 ) ∙ 𝑐𝑠 )
(
3
∙(
𝑠
𝑢) ∙
√𝐴2 𝐵2
2 (
∙ 𝑠 ∙ 𝑐𝑡
𝑑𝑠 ∙ 𝑑𝑡 𝑐𝑠 ∙ 𝑢 )
𝑠 ∙ 𝑐𝑠
∙ (𝑑𝑠 ∙ 𝑐𝑡 𝑑𝑡 ∙ 𝑐𝑠 );
𝑠 ∙ 𝑑𝑠
𝑢 ∙ 𝑑𝑠 )
𝑑𝑠 )2
;
𝑐𝑠 ∙ 𝑐𝑡 ;
The transfer functions in Equation [5.51] are plotted in Figure 5-14. If we have a certain road, with
displacement amplitude of 1 cm (𝑧̂
0.01 ) and the vehicle should drive on it at a speed of 50
𝑚
km/h ( ≈ 14 𝑠 ̂≈ .8𝐻𝑧), we can read out, e.g.:



Ride comfort related: |𝐻 𝑟 → 𝑠 ( )| ≈ 1 3; ⇒ | ̂| 1 3 ∙ 𝑧̂𝑟 1 3 ∙ 0.01
1. 3 ⁄𝑠 2 ;. From this we can calculate 𝑅 𝑆(𝑧𝑠 ) |1. 3|⁄√ ≈ 0.8697 ⁄𝑠 2 .
Fatigue life related: |𝐻 𝑟 → 𝑢 − 𝑠 ( )| ≈ 1.14; ⇒ |𝑧̂𝑢 𝑧̂𝑠 | 1.14 ∙ 𝑧̂𝑟 1.14 ∙ 0.01
0.0114
1.14 𝑐 ;.
Road grip related: |𝐻 𝑟 →Δ 𝑟𝑧 ( )| ≈ 177470; ⇒ | ̂ 𝑟𝑧 | 177470 ∙ 𝑧̂𝑟 177470 ∙
0.01 1775 𝑁;.
This analysis can be compared with the analysis in Section 5.5.2.1.3. Ride comfort and fatigue does
not change a lot, but road grip does. This indicates that the more advanced model is only needed for
road grip evaluation.
Figure 5-15 shows the phase angles for the different responses.
Figure 5-16, shows the amplitude gains for the corresponding un-damped system. We identify
natural frequencies at around 5 m/s and 50 m/s. These two speeds correspond to
frequencies 𝑐 𝑡 /𝜆, i.e. approximately 1 Hz and 10 Hz. The lower frequency is an oscillation mode
where the both masses move in phase with each other, the so called “bounce mode”. The higher
frequency comes from the mode where the masses are in counter-phase to each other, the so called
“wheel hop mode”. In the wheel hop mode, the sprung mass is almost not moving at all. We will
come back to these modes in Section 5.5.3.2.
245
VERTICAL DYNAMICS
Driving at road with wave-length, lambda = 5 [m].
7
10
6
10
5
10
ms = 1415[kg]; cs = 68 [kN/m]; ds = 8[kN/(m/s)];
4
abs(H)
10
mu = 185[kg]; ct = 676 [kN/m];
f=1 Hz at vx=5[m/s]; f=10 Hz at vx=50[m/s]
3
10
2
10
1
10
H_zr_zs
H_zr_zu-zs
H_zr_derderzs
H_zr_Frz
0
10
-1
10
0
10
1
2
10
3
10
10
vx [m/s]
Figure 5-14: Transfer functions for amplitudes from model in Figure 5-13, excited with single
frequencies. Notation: 𝐻 →𝑏 is denoted H_a_b.
200
150
100
arg(H)
50
0
-50
-100
-150
-200
0
10
H_zr_zs
H_zr_zu-zs
H_zr_derderzs
H_zr_Frz
1
2
10
10
3
10
vx [m/s]
Figure 5-15: Transfer functions for phase delays. Same model and data as in Figure 5-14.
246
VERTICAL DYNAMICS
7
10
6
10
5
10
4
abs(H)
10
H_zr_zs
H_zr_zu-zs
H_zr_derderzs
H_zr_Frz
3
10
2
10
1
10
0
10
-1
10
0
10
1
2
10
3
10
10
vx [m/s]
Figure 5-16: Un-damped transfer functions for amplitudes. Same model and data as in Figure 5-14,
except 𝑐𝑠 0.
5.5.3.2
Simplified model
A more practical way to approach the system is to consider the properties of the system. The sprung
mass is typically an order of magnitude greater than the unsprung mass and the suspension spring is
usually an order of magnitude lower than the tyre stiffness. Then, one can split the model in to
models, which explains one mode each, see Figure 5-17. The modes were identified already in
Section 5.5.3.1.1.
Modes. Arrows means displacement amplitudes.
model for
bounce mode
model for wheel
hop mode
zs
m
bounce
m
bounce
ms
cs
bounce
(sprung & unsprung
wheel hop
(sprung & unsprung
cs
ds
ct
zu
ds
mu
ct
Figure 5-17: Modes and approximate models.
We will now derive the natural frequencies for the two models, and compared with the natural
frequencies (ds=0) found for the combined model, in Figure 5-16. Both models are one degree of
freedom models with mass and spring, why the Eigen frequency is √stiffness/mass.
For the bounce model, the mass is ms. Stiffnesses cs and ct are series connected, which means that
the total stiffness=1/(1/cs+1/ct).
247
VERTICAL DYNAMICS
For the wheel hop model, the mass is mu. Stiffnesses cs and ct are parallel connected, which means
that the total stiffness=cs+ct.
𝐵 𝑢 𝑐
𝑊ℎ
𝐻 𝑝
1
1⁄(
𝑐
√
𝑠
1
)
𝑐𝑡
𝑠
√
𝑐𝑠
𝑐𝑡
𝑢
15.3
86.0
𝑟 𝑑
;
𝑠
[5.53]
𝑟 𝑑
;
𝑠
We see that the numbers match well with the more advanced model, which gave 15.2 rad/s and 86.7
rad/s, respectively.
Bounce refers to the mode where the sprung mass has the greatest amplitude and wheel hop is
related to the case when the unsprung mass exhibits the greatest amplitude. For a passenger car, the
spring mass has the lowest frequency, typically around 1 Hz while tyre hop is more prevalent at
frequencies around 10 Hz.
5.6 Ride comfort *
Function definition: (Stationary) Ride comfort is the comfort that vehicle occupants experience from
stationary oscillations when the vehicle travels over a road with certain vertical irregularity in a certain speed.
The measure is defined at least including driver (or driver seat) vertical acceleration amplitudes.
Ride comfort is sometimes divided into:
•
Primary Ride – the vehicle body on its suspension. Bounce(Heave), Pitch and Roll 0..4 Hz
•
Secondary Ride – same but above body natural frequencies, i.e. 4..25 Hz
5.6.1
Single frequency
It is generally accepted for stationary vibrations, that humans are sensitive to the RMS value of the
acceleration. However, the sensitivity is frequency dependent, so that highest discomfort appears for
a certain range of frequencies. Some human tolerance curves are shown in Figure 5-18 and Figure
5-19.
The curves can be considered a threshold for acceptance where everything above the line is
unacceptable and points below the curve are acceptable. Discomfort is a subjective measure, and
this is why the different diagrams cannot be directly compared to each other. The SAE has suggested
that frequencies from 4 to 8 Hz are the most sensitive and the accepted accelerations for these are
no higher than 0.025 g (RMS).
The curves in Figure 5-18 mostly represent an extended exposure to the vibration. As one can expect,
a human can endure exposure to more severe conditions for short periods of time. The SAE limits
presented are indicative of 8 hours of continuous exposure. Curves for different exposure times can
also be obtained from ISO, (ISO2631). The ISO curves are from the first version of ISO 2631 and were
later modified, see Figure 5-19.
248
VERTICAL DYNAMICS
How to use the diagram
RMS
acceleration
not
OK
OK
4 Hz
8 Hz
Frequency
Figure 5-18: Various Human Sensitivity Curves to Vertical Vibration, (Gillespie, 1992)
Figure 5-19: ISO 2631 Tolerance Curves
5.6.2
Multiple frequencies
The curves in Figure 5-18 and Figure 5-19 can be interpreted as a filter, where the response of the
human is influenced by the frequencies they are exposed to. This leads to the concept of a Human
249
VERTICAL DYNAMICS
(𝑓). (
Filter Function
refers to vertical whole human body vibration sensitivity, while there are
other for sensitivities for other directions and human parts.) This approach uses the concept of a
transfer function from driver seat to somewhere inside the drivers brain, where discomfort is
perceived.
ISO2631
1
Asymptotic approximations of ISO 2631 weighting curves
/2
0
0
𝑢𝑚𝑏
)
10
-5
10
Wk in dB
-1
10
-2
10
(number= 0 ∙ log Gain (dB)
⇒
Wk [1]
10
-10
-15
-20
-25
Wd (horizontal)
Wk (vertical)
-30
-3
10
-1
10
0
10
1
10
f [Hz]
2
10
10
3
10
0
1
10
Frequency (Hz)
Figure 5-20: Human Filter Function. From (ISO2631). Right: Asymptotic approximation
Figure 5-21: Human Filter Function for vertical vibrations. Table from (ISO2631).
With formulas from earlier in this chapter we can calculate an RMS value of a signal with multiple
frequencies, see Equation [5.6]. Consequently, we can calculate RMS of multiple frequency
acceleration. Since humans are sensitive to acceleration, it would give one measure of human
discomfort. However, to get a measure which is useful for comparing accelerations with different
frequency content, the measure have to take the human filter function into account. The Weighted
RMS Acceleration, aw, in the following formula is such measure:
250
VERTICAL DYNAMICS
𝑁
𝑤 (𝑧(𝑡))
𝑤
𝑢𝑠𝑒: 𝑅 𝑆(𝑧(𝑡))
√∑
=
{
𝑁
2
𝑧̂
√∑
}
(
(
2
) ∙ 𝑧̂ )
;
=
or
∞
𝑤 (𝑧(𝑡))
𝑤
{𝑢𝑠𝑒: 𝑅 𝑆(𝑧(𝑡))
[5.54]
√ ∫ 𝐺 ( )∙𝑑 }
𝜔=
∞
√ ∫(
2
( )) ∙ 𝐺 ( ) ∙ 𝑑 ;
𝜔=
and 𝐺 . If vibrations in
Equation [5.54] is written for a case with only vertical vibrations, hence
several directions, a total 𝑤 can still be calculated, see (ISO2631).
In (ISO2631) one can also find the following equation, which weights together several time periods,
with different vibrations spectra. The 𝑤 is the time averaged whole-body vibration exposure value.
𝑤
∑
√
𝑤
2
∑ 𝑇
∙𝑇
;
[5.55]
The 𝑤 in Eq [5.55] is used both for vehicle customer requirement setting at OEMs and governmental
legislation. One example of legislation is (DIRECTIVE 2002/44/EC, 2002). This directive stipulates that
2
𝑤 in Eq [5.55] in any direction, normalized to 8 hours, may not exceed 1.15 m/s , and if the value
exceeds 0.5 m/s2 action must be taken.
5.6.2.1
Certain combination of road, vehicle and speed
Now we can use Equation [5.35] without assuming road type. However, we have to identify 𝑧̂𝑠 and
( ), according to Equation [5.54]. Then we get [5.56].
multiply it with
Using Equation [5.56], we can calculate the weighted RMS value for the different models in Sections
5.5.1, 5.5.2 and 5.5.3. For each model, it will vary with speed, vx. A plot, assuming a certain road type
(“Rough” from Figure 5-6) is shown in Figure 5-28. We can see that the simplest model “stiff tyre, no
unsprung mass” gives much different comfort value than the two other, so the simplest is not goof to
estimate comfort. However, the two other models give approximately same result, which indicates
that the medium model, “stiff tyre, no unsprung mass”, is enough for comfort evaluation. This is no
general truth but an indication that the most advanced model, “two masses, elastic tyre”, is not
needed for comfort on normal roads. The advanced model is more needed for road grip.
We can also see that the comfort decreases, the faster the vehicle drives. If we read out at which
speed we reach 𝑤 1 ⁄𝑠 2 (which is a reasonable value for long time exposure) we get around
≈ 70 ⁄𝑠 ≈ 50 𝑘 / on this road type (“Rough”) with the medium (and advanced) model.
With the simplest model we get ≈ 1 ⁄𝑠 ≈ 3. .4 𝑘 / .
251
VERTICAL DYNAMICS
𝑅 𝑆(𝑧𝑠 )
√
√
⇒
𝛷0
𝛺
−𝑤
−𝑤
√
𝛷0
𝛺
−𝑤
∙
𝑥
𝑤 1
2
𝑤∙𝑑
∙ ∫|𝐻 𝑟→ 𝑠 ( )| ∙
{
𝑢𝑠𝑒: 𝐻
2
0
𝑟→ 𝑠
∙𝐻
𝑟→ 𝑠
}
∞
∙
𝑥
𝑤 1
∙ ∫|
2
∙𝐻
2
𝑟→ 𝑠
𝑤∙𝑑
( )| ∙
∞
∙
𝑥
𝑤 1
2
∙ ∫|𝐻 𝑟→ 𝑠 ( )| ∙
𝑢𝑠𝑒:
Equation
;⇒ {
}⇒
[5.54]
4 𝑤∙𝑑
0
𝑤
√
𝛺
∞
0
𝛷0
𝛺
√
𝛷0
−𝑤
𝛷0
𝛺
−𝑤
[5.56]
∞
∙
𝑥
𝑤 1
∙ ∫(
𝑘(
)) ∙ |𝐻
2
𝑟→ 𝑠
( )| ∙
4 𝑤∙𝑑
0
∞
∙
𝑥
𝑤 1
∙ ∫(
𝑘(
)) ∙ |𝐻
2
𝑟→ 𝑠
( )| ∙
4 𝑤∙𝑑 ;
0
Ride Comfort. For road type "Rough"
2
10
1
Weigthed RMS value, aw [m/(s*s)]
10
0
10
-1
10
-2
10
-3
10
-2
10
-1
10
0
10
vx [m/s]
1
10
2
10
Figure 5-22: Weighted RMS values for road type “Smooth” from Figure 5-6. The 3 curves show 3
different models: Simplest (from Sections 1.5.1), Medium (from Section 1.5.2) and Most advanced
(from Section 1.5.3).
252
VERTICAL DYNAMICS
5.7 Fatigue life *
Function definition: (Vehicle) Fatigue life is the life that the vehicle, mainly suspension, can reach due
to stationary oscillations when vehicle travels over a road with certain vertical irregularity in a certain speed.
The measure is defined at least including suspension vertical deformation amplitudes.
Beside human comfort, the fatigue of the vehicle structure itself is one issue to consider in vertical
vehicle dynamics.
5.7.1
Single frequency
5.7.1.1
Loads on suspension spring
In particular, the suspension spring may be subject to fatigue. The variation in spring material stress
is dimensioning, which is why the force variation or amplitude in the springs should be under
observation. Since spring force is proportional to deformation, the suspension deformation
amplitude is proposed as a good measure. This is the explanation to why the amplitude of 𝑧𝑢 𝑧𝑠 is
plotted in Figure 5-11.
Beside fatigue loads, 𝑧𝑢 𝑧𝑠 is also relevant for judging whether suspension bump-stops become
engaged or not. At normal driving, that limit should be far from reached, except possibly at maximum
loads (many persons and much luggage).
5.7.1.2
Fatigue of other components
Fatigue of other parts may require other amplitudes.
One other relevant example can be the damper fatigue. Damper fatigue would be more relevant to
judge from amplitude of 𝑧𝑢 𝑧𝑠 , which determines the force level and hence the stress level.
Another example is the load of the road itself. For heavy trucks it is important to consider how much
they wear the road. At some roads with legislated maximum (static) axle load, one can be allowed to
exceed that limit if the vehicle has especially road friendly suspensions. For these judgements, it is
the contact force between tyre and road, ̂ , which is the most relevant variable to study.
5.7.2
Multiple frequencies
If the excitation is of one single frequency, the stress amplitude can be used when comparing two
designs. However, for spectra of multiple frequencies, one cannot look at amplitudes solely,
[𝑧̂ 𝑧̂2 ⋯ 𝑧̂𝑁 ], because the amplitudes will depend on how the discretization is done, i.e. the
number N. Some kind of integral of a spectral density is more reasonable. In this compendium it is
proposed that a very approximate measure of fatigue load is calculated as follows, exemplified for
the case of fatigue of the spring:
253
VERTICAL DYNAMICS
𝑅 𝑆(𝑧𝑢 (𝑡)
𝑒 𝑠𝑢𝑟𝑒 𝑓𝑜𝑟 𝑠𝑝𝑟𝑖𝑛𝑔 𝑓 𝑡𝑖𝑔𝑢𝑒 𝑖𝑓𝑒
√
√
𝛷0
𝛺
−𝑤
∞
∙
𝑥
𝑤 1
∙ ∫|𝐻 𝑟→𝑧𝑢
2
𝑧𝑠 (
𝑤∙𝑑
)| ∙
{
𝑢𝑠𝑒: 𝐻
𝐻
0
𝛷0
𝛺
𝑧𝑠 (𝑡))
−𝑤
𝑟 →𝑧𝑢 𝑧𝑠
𝑟 →𝑧𝑢
𝐻
𝑟 →𝑧𝑠
}
[5.57]
∞
∙
𝑥
𝑤 1
2
𝑤∙𝑑 ;
∙ ∫|𝐻 𝑟→𝑧𝑢 ( ) 𝐻 𝑟→𝑧𝑠 ( )| ∙
0
Equation is written for application to a known road spectra (𝛷 𝑤 ) and vehicle dynamic structure
(𝐻 𝑟 →𝑧𝑢 𝐻 𝑟 →𝑧𝑠 ), but the first expression (𝑅 𝑆(𝑧𝑢 (𝑡) 𝑧𝑠 (𝑡))) is applicable on a measured or
simulated time domain solution.
Road grip *
Function definition: Road grip (on undulated roads) is how well the longitudinal and lateral grip
between tyres and road is retained due to stationary oscillations when the vehicle travels over a road with
certain vertical irregularity in a certain speed.
Sections 3.3 and 3.4 show the brush model explain how the tyre forces in the ground plane appears.
It is a physical model where the contact length influences how stiff the tyre is for longitudinal and
lateral slip. There is also a brief description of relaxation models for tyres. This together motivates
that a tyre have more difficult to build up forces in ground plane if the vertical force varies. We can
understand it as when contact length varies, the shear stress build up has to start all over again. As
an average effect, the tyre will lose more and more grip, the more the vertical force varies.
5.8.1
Multiple frequencies
If the excitation is of one single frequency, the stress amplitude can be used when comparing two
designs. However, for spectra of multiple frequencies, one cannot look at amplitudes solely,
[𝑧̂ 𝑧̂2 ⋯ 𝑧̂𝑁 ], because the amplitudes will depend on how the discretization is done, i.e. the
number N. Some kind of integral of a spectral density is more reasonable. In this compendium it is
proposed that a very approximate measure of road grip is calculated as follows:
𝑒 𝑠𝑢𝑟𝑒 𝑓𝑜𝑟 (𝑏 𝑑)𝑟𝑜 𝑑 𝑔𝑟𝑖𝑝
√
𝛷0
𝛺
−𝑤
∙
𝑥
𝑤 1
2
𝑤∙𝑑
∙ ∫|𝐻 𝑟→Δ 𝑟𝑧 ( )| ∙
0
𝛷0
𝛺
𝑟𝑧 (𝑡))
∞
{𝑢𝑠𝑒: 𝐻 𝑟 →𝑧𝑢
√
𝑅 𝑆(Δ
−𝑤
𝑧𝑠
𝑐𝑡 ∙ (1
𝐻
𝑟→ 𝑢
[5.58]
)}
∞
∙
𝑥
𝑤 1
2
∙ ∫|𝑐𝑡 ∙ (1 𝐻 𝑟→ 𝑢 )| ∙
𝑤∙𝑑 ;
0
Equation is written for application to a known road spectra (𝛷 𝑤 ) and vehicle dynamic structure
(𝐻 𝑟 →𝑧𝑢 ), but the first expression (𝑅 𝑆(Δ 𝑟𝑧 (𝑡))) is applicable on a measured or simulated time
domain solution.
254
VERTICAL DYNAMICS
5.9 Variation of stiffness and damping
The influence of design parameters on vehicle functions Ride comfort, Suspension fatigue and Road
grip can now be made. E.g., it is important to not only use the transfer function, but also take the
road and human sensitivity into account, which calls for different weighting for different frequencies.
Transfer function for the model in Section 5.5.3 is shown as dashed lines in Figure 5-23. Same figure
also shows the Road- and Human-weighted versions. Studying how these curves change with design
parameters gives a quantitative understanding of how different suspension design parameters
influence. Such variations will be done in Sections 5.9.1 to 5.9.4.
Comfort as der(der(zs))
3
Suspension Fatigue as zu-zs
1
Road Grip as zr-zu
1
10
10
10
0
2
10
10
0
10
-1
0
10
amplification /[m/m]
10
amplification /[m/m]
amplification /[(m/(s*s))/(m)]
1
10
-2
10
-1
10
-2
10
-3
-1
10
10
-3
amplification /[m/m]
Road-weighted
-4
-2
10
10
10
amplification /[(m/(s*s))/(m)]
Road-weighted
Road- and Human-weighted
amplification /[m/m]
Road-weighted
-4
-5
-3
10
-1
10
1
0
10
10
f /[Hz]
2
10
10
-1
10
1
0
10
10
2
10
10
-1
10
f /[Hz]
1
0
10
10
2
10
f /[Hz]
Figure 5-23: For a passenger car with 𝑠
1600 [𝑘𝑔] 𝑢
00 [𝑘𝑔] 𝑐𝑠 76 [𝑘𝑁⁄ ] 𝑑𝑠
9 [𝑘𝑁⁄( ⁄𝑠)] 𝑐𝑡 764 [𝑘𝑁⁄ ] 𝑑𝑡 0. Left is vertical acceleration (amplitude) of sprung mass
for Ride Comfort. Middle is relative displacement (amplitude) between sprung and unsprung mass
for Suspension Fatigue. Right is deformation (amplitude) of tyre spring for Road Grip. Weightings
for typical road and for human sensitivity is shown.
There are two particular frequency intervals of the graphs to observe. These are the 2 peaks around
the two the natural frequencies of the sprung and unsprung masses, the peak at lower frequency is
mainly a resonance in bounce mode, while the higher one is in wheel hop mode.
5.9.1
Varying suspension stiffness
In Figure 5-24 the benefits of the low suspension stiffness (1 Hz) is seen for suspension travel and
comfort without much change in the road grip performance.
255
VERTICAL DYNAMICS
Comfort as der(der(zs))
2
Suspension Fatigue as zu-zs
1
Road Grip as zr-zu
0
10
10
10
0
10
-1
10
Road-weighted amplification /[m/m]
10
Road-weighted amplification /[m/m]
Road- and Human-weighted amplification /[(m/(s*s))/(m)]
0
10
1
-1
10
-2
10
-3
10
-1
10
-2
10
-2
10
-4
10
smaller cs = 38 [kN/m]
reference cs = 76 [kN/m]
larger cs = 153 [kN/m]
-3
10
-5
-1
10
0
1
10
10
f /[Hz]
2
10
10
-3
-1
10
0
1
10
10
2
10
10
-1
10
f /[Hz]
0
1
10
10
2
10
f /[Hz]
Figure 5-24: Result from varying suspension stiffness, cs
Regarding Figure 5-24 and Figure 5-25 we see that there is a large influence of the acceleration gain
at low frequencies with little change at the wheel hop and higher frequencies. The suspension
stiffness and damping was seen to have little influence on the ride comfort / road grip response
around 10 Hz.
5.9.2
Varying suspension damping
In Figure 5-25, we see that the changes in suspension damping have opposite effects for the bounce
and wheel hop frequency responses. High damping is good for reducing bounce, but not so effective
for wheel hop.
256
VERTICAL DYNAMICS
Comfort as der(der(zs))
2
Suspension Fatigue as zu-zs
1
Road Grip as zr-zu
0
10
10
10
0
10
-1
10
Road-weighted amplification /[m/m]
Road-weighted amplification /[m/m]
Road- and Human-weighted amplification /[(m/(s*s))/(m)]
0
10
1
10
-1
10
-2
10
-3
10
-1
10
-2
10
-2
10
-4
10
smaller ds = 4 [kN/m]
reference ds = 9 [kN/m]
larger ds = 18 [kN/m]
-3
10
-5
-1
10
0
1
10
10
f /[Hz]
2
10
10
-3
-1
10
0
1
10
10
2
10
10
-1
10
f /[Hz]
0
1
10
10
2
10
f /[Hz]
Figure 5-25: Result from varying suspension damping, ds
5.9.3
Varying unsprung mass
In Figure 5-26, we see that if the response around the wheel hop frequency is to be changed, the
unsprung mass is one of the most influential parameters. The unsprung mass is usually in the range
of 10% of the sprung mass. Opposite to the suspension parameters, the unsprung mass influences
frequencies around the wheel hop frequency with little influence around the bounce frequency.
In Figure 5-26, the case with 𝑢 0 is added. This is to demonstrate what a model with neglected
mass gives and can be nearly compared with the model in Section 5.5.2.
5.9.4
Varying tyre stiffness
In Figure 5-27 a general observation is that low sprung mass natural frequencies are preferred for
comfort considerations. Another parameter that has a strong affect near the wheel hop frequency is
the tyre stiffness. The strongest response is noticed for the road grip function. (Note that, since 𝑐𝑡 is
now varying, we have to express road grip as 𝑐𝑡 ∙ (𝑧
𝑧𝑢 ); only 𝑧
𝑧𝑢 does not give a fair
comparison.)
257
VERTICAL DYNAMICS
Comfort as der(der(zs))
2
Road Grip as zr-zu
0
10
0
10
1
0
10
-1
10
Road-weighted amplification /[m/m]
10
Road-weighted amplification /[m/m]
Road- and Human-weighted amplification /[(m/(s*s))/(m)]
Suspension Fatigue as zu-zs
1
10
10
-1
10
-2
10
-3
10
-1
10
-2
10
-2
10
-4
10
-3
10
smaller mu = 100 [kg]
reference mu = 200 [kg]
larger mu = 400 [kg]
zero mu = 0 [kg]
-5
-1
10
0
1
10
10
10
2
10
-3
-1
10
0
1
10
f /[Hz]
10
10
2
10
-1
10
0
1
10
f /[Hz]
10
2
10
f /[Hz]
Figure 5-26: Result from varying unsprung mass, mu
Comfort as der(der(zs))
2
Road Grip as ct*(zr-zu)
6
10
0
10
1
0
10
-1
10
Road-weighted amplification /[m/m]
10
Road-weighted amplification /[m/m]
Road- and Human-weighted amplification /[(m/(s*s))/(m)]
Suspension Fatigue as zu-zs
1
10
10
-1
10
-2
10
-3
10
5
10
4
10
-2
10
-4
10
-3
10
smaller ct = 382 [kN/m]
reference ct = 764 [kN/m]
larger ct = 1529 [kN/m]
-5
-1
10
0
1
10
10
f /[Hz]
2
10
10
3
-1
10
0
1
10
10
2
10
10
-1
10
f /[Hz]
0
1
10
10
2
10
f /[Hz]
Figure 5-27: Result from varying Tyre Stiffness, ct
5.10 Two dimensional oscillations
The one-dimensional model is useful for analysing the response of one wheel/suspension assembly.
Some phenomena do connect other vehicle body motions than the vertical translation, especially
pitch and roll. Here, other models are needed, such as Figure 5-28 and Figure 5-32.
258
VERTICAL DYNAMICS
5.10.1 Bounce (heave) and pitch
A model like in Figure 5-28 is proposed. We have been studying bounce and pitch before, in Section
3.4.8. Hence compare with corresponding model in Figure 3-32. In Chapter 3, the excitation was
longitudinal tyre forces, while the vertical displacement of the road was assumed to be zero. In
vertical vehicle dynamics, it is the opposite. The importance of model with linkage geometry (pitch
centre or axle pivot points above ground level) is that tyre forces are transferred correctly to the
body. That means that the linkage geometry is not so relevant for vertical vehicle dynamics in
Chapter 5. So the model can be somewhat simpler.
m*az
vz
zf
wx
m*g
h
zrr
zfr
Fzf
lr
lf
J*der(wy)=𝐽 ∙
zr
Fzr
Figure 5-28: Bounce (heave) and pitch model.
No equations will be formulated
for this model. The model would typically show two different
Bounce Mode
Pitch Mode
modes, see Figure 5-29. The bounce eigen-frequency
is
typically
1-1.5
Hz
for
a
passenger
car.
The
o
o
z
pitch frequency

z is somewhat higher.
We should reflect on that if we are analysis the same vehicle with the models in Section 5.5 and
l01 models will most likely give
5.10.1, we are actually talking about the same bounce mode. But the
different numbers of, e.g. Eigen frequency. A total model, with all motion degrees of freedom, would
l02
align those values, but
the larger a model is the more data it produces which often leads to less easy
design decisions.
Motion node
for Pitch mode
Pitch
mode
Bounce
mode
Motion node for
Bounce mode
Figure 5-29: Oscillation modes of a Bounce and Pitch model.
259
VERTICAL DYNAMICS
5.10.1.1 Wheel base filtering
The response of the vehicle to road irregularities can be envisioned as in Figure 5-30. The vehicle can
be excited in a pure bounce motion if the road wavelength is exciting each axle identically. In a
similar analysis, the pure pitch motion can be excited with the road displacing the axles out of phase
with each other. So, at some frequencies, one mode is not excited at all, which is called “wheel base
filtering”.
Figure 5-31 gives an example of spectrum for vertical and pitch accelerations. It compares the
accelerations for the correlated and uncorrelated excitation.
Figure 5-30: Response of the Vehicle to Different Road Wavelengths. (Gillespie, 1992)
Figure 5-31: Example of wheelbase filter effect
260
VERTICAL DYNAMICS
5.10.2 Bounce (heave) and roll
A model like in Figure 5-32 is proposed. We have been studying bounce and pitch before, in Sections
4.3.9 and 4.5.3. Hence compare with corresponding model in Figure 4-63. In Chapter 4, the
excitation was lateral tyre/axle forces, while the vertical displacement of the road was assumed to be
zero. In vertical vehicle dynamics, it is the opposite. That means that the linkage geometry (roll
centre or wheel pivot points) is not so relevant here. So the model can be somewhat simpler.
m*az
z
zl
zr
m*g
J*der(wx)
px
zrl
zrl
x
zrr
zrr
x
Flz
Frrz
Figure 5-32: Bounce (heave) and roll model.
No equations will be formulated for this model.
The model will typically show two different modes, the bounce and roll. Bounce Eigen frequency is
typically 1-1.5 Hz for a passenger car, as mentioned before. The roll frequency is similar or somewhat
higher.
5.11 Transient vertical dynamics
The majority of the chapter you read now, considers normal driving which is how the vehicle is driven
for during long time periods. Functions are then suitably analysed using theory for stationary
oscillations.
Vertical vehicle dynamics also have transient disturbances to consider. Test cases can be one-sided or
two sides road bumps or pot-holes. Two sided bump is envisioned in Figure 5-33. It can represent
driving over a speed bump or an object/low animal on a road.
Models from earlier in this chapter are all relevant for two-sided bumps/pot-holes, but one might
need to consider non-linearities such as bump stops or wheel lift as well as different damping in
compression and rebound. For one-sided bumps/potholes, the models from earlier in this chapter
are generally not enough.
The evaluation criteria should be shifted somewhat:




Human comfort for transients are often better described as time derivative of acceleration
(called “jerk”).
The material loads are more of maximum load type than fatigue life dimensioning, i.e. higher
material stress but fewer load cycles during vehicle life time.
Road grip studies over road bumps and pot-holes are challenging. Qualitatively, the tyre
models have to include relaxation, because that is the mechanism which reduces road grip
when vertical load shifts. To get quantitatively correct tyre models is beyond the goal of the
compendium you presently read.
Roll-over can be tripped by large one sided bumps. This kind of roll-overs is unusual and
requires complex models.
261
VERTICAL DYNAMICS
Figure 5-33: Response of Vehicle for Front and Rear Axle Impulses, (Gillespie, 1992)
5.12 Other excitation sources and
functions
5.12.1.1 Other excitation sources
The chapter you read now have analysed the influence of excitation from vertical displacement of
the road. Examples of other, but often co-operating, excitation sources are:



Powertrain vibrations, non-uniform rotation in engine. Frequencies will be proportional to
engine speed
Wheel vibrations, e.g. due to non-round wheels or otherwise unbalanced wheels.
Frequencies will be proportional to vehicle speed.
Special machineries mounted on vehicles (e.g. climate systems or concrete mixers)
5.12.1.2 Other functions
The chapter you read now have analysed some functions. Examples of other, but related, functions
are:





An area of functions that encompasses the vertical dynamics is Noise, Vibration, and
Harshness – NVH. It is similar to ride comfort, but the frequencies are higher, stretching up
to sound which is heard by humans.
Ground clearance (static and dynamic) between vehicle body and ground. Typically
important for off-road situations.
Longitudinal comfort, due to drive line oscillations and/or vertical road displacements.
Especially critical when driver cabin is separately suspended to the body. This is the case for
heavy trucks.
Disturbances in steering wheel feel, due to one-sided bumps. Especially critical for rigid
steered axles. This is often the design of the front axle in heavy trucks.
There are of course an infinite amount of combined manoeuvres, in which functions with
requirements can be found. Examples can be bump during strong cornering (possibly
destabilizing vehicle) or one-sided bump (exciting both bounce, pitch and roll modes).
262
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Index
(discrete) state diagram ......................... 20
Cornering Compliance ........................... 66
Ackermann steering angle .................... 156
counter-steer ....................................... 157
Ackermann steering geometry ............. 133
Critical Speed ....................................... 163
Active safety ........................................... 33
data flow diagram .................................. 20
actual toe angle ...................................... 45
Differential-Algebraic system of
Equations, DAE....................................... 18
actuation........................................... 15, 72
direct yaw moment.............................. 162
Advanced Driver Assitance Systems
(ADAS)..................................................... 33
effective damping ........................ 114, 117
aerodynamic drag ................................... 77
effective stiffness ................................. 114
agility .................................................... 197
Electronic Brake Distribution, EBD ...... 125
aligning torque ....................................... 70
Electronic Control Units, ECUs ............... 34
angular (time) frequency ...................... 225
Electronic Stability Control, ESC .......... 218
Anti-lock Braking System, ABS .............. 124
Engine Drag Torque Control, EDC ........ 126
architecture ............................................ 34
Equivalent wheelbase .......................... 147
Attributes................................................ 12
flow charts ............................................. 20
Automatic Emergency Brake, AEB ........ 126
Forward Collision Warning (FCW).......... 15
axle rate ................................................ 114
Friction Circle ......................................... 71
bicycle model ........................................ 137
Functions ............................................... 14
Blind Spot Detection (BSD) ..................... 15
handling diagram ................................. 167
bounce mode........................................ 245
Human Filter Function ......................... 250
brush model............................................ 56
ideal tracking ....................................... 139
Cambering Vehicle.................................. 37
indirect yaw moment........................... 162
castor offset at ground ......................... 135
inertial coordinate system ..................... 30
Castor offset at wheel center ............... 135
internal combustion engine (ICE) .......... 84
Characteristic Speed ............................. 163
kinematic models................................. 139
Closed-loop Manoeuvres........................ 79
Kingpin axis .......................................... 135
coefficient of rolling resistance .............. 51
Lane Departure Prevention ................. 217
combined (tyre) slip................................ 71
Lane Departure Warning (LDW) ............ 15
Continuous Variable ratio Transmission,
CVT.......................................................... 90
Lane Keeping Aid, LKA ......................... 217
Lateral Collision Avoidance, LCA .......... 220
Controller Area Network, CAN ............... 34
lateral slip .............................................. 64
Cornering .............................................. 131
Leaning vehicle ...................................... 37
cornering coefficient .............................. 67
longitudinal (tyre) slip ............................ 50
266
low speed steering angle ...................... 156
Scrub radius ......................................... 135
Magic Formula ........................................ 61
Secondary Ride .................................... 248
Mean Square Spectral Density ............. 227
side slip .................................................. 64
neutral steered vehicle ......................... 156
single-track model ............................... 137
Normal steering axis offset at wheel
centre ................................................... 135
slip angle ................................................ 64
spectral density.................................... 227
object vehicle.......................................... 33
sprung mass ......................................... 113
Off-tracking........................................... 216
static toe angle ...................................... 45
one-track model ................................... 137
Steering axis ......................................... 135
Open-loop Manoeuvres ......................... 79
steering axis offset at ground .............. 135
Ordinary Differential Equations, ODE..... 27
subject vehicle ....................................... 33
oversteered vehicle .............................. 156
TM-Easy ................................................. 61
overturning moment .............................. 69
Traction Control, TC ............................. 126
parallell steering geometry .................. 134
Traction diagram.................................... 87
pendulum effect ................................... 185
turning circle ........................................ 142
physical stiffness................................... 114
turning diameter .................................. 142
platform .................................................. 34
tyre slip .................................................. 50
pneumatic trail ....................................... 70
understeer gradient ..................... 153, 155
Power Spectral Density, PSD ................ 227
understeered vehicle ........................... 156
Primary Ride ......................................... 248
unsprung mass ..................................... 113
Rearward Amplification, RWA .............. 215
vehicle fixed coordinate systems........... 29
roll axis.................................................. 176
Weighted RMS Acceleration ................ 250
Roll Stability Control, RSC ..................... 220
wheel base filtering ............................. 260
rolling radius ........................................... 43
wheel hop mode .................................. 245
rolling resistance .................................... 51
wheel rate ............................................ 114
Roll-stiff Vehicle...................................... 37
yaw damping........................................ 216
267
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