Vehicle Dynamics

Vehicle Dynamics
Vehicle Dynamics
Compendium for Course MMF062
R3
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Rc
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L2=1m
𝛿1
R1
R2
L1=3m
2014
Bengt Jacobson et al, Vehicle Dynamics Group, Division of Vehicle and Autonomous Systems,
Department of Applied Mechanics, Chalmers University of Technology, www.chalmers.se
0
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, Last Modification: 2014-10-27 07:56
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:
Volvo Truck’s ETT project, Volvo’s new XC90 on winter test, Scania’s platooning project
This compendium is also available as pdf file at
http://pingpong.chalmers.se/public/courseId/4042/lang-en/publicPage.do.
© Copyright: Bengt Jacobson, Chalmers University of Technology
1
Contents
1
INTRODUCTION
1.1
9
Purpose of this compendium
9
1.2
Attributes and Functions
1.2.1
Attributes
1.2.2
Functions
1.2.2.1 Seeing versus Blind functions
1.2.2.1.1
Seeing functions
1.2.2.1.2
Blind functions
1.2.2.2 Dynamic versus Driver-informing functions
1.2.2.2.1
Dynamic functions
1.2.2.2.2
Driver-informing functions
1.2.2.2.3
Un-categorized
1.2.2.3 Other categorisations
1.2.3
1.2.4
Requirements
Models, methods and tools
1.3
Technical References
1.3.1
Engineering
1.3.2
General modelling, drawing and mathematics
1.3.3
Notations
1.3.3.1
Notation list
9
10
11
12
12
12
12
12
12
12
13
13
13
14
14
14
17
17
1.3.4
From general “Mechanical engineering”
21
1.3.5
Verification methods with real vehicles
23
1.3.6
Verification methods involving theoretical simulation
23
1.3.7
Tools & Methods
24
1.3.8
1.3.9
Coordinate Systems
Terms with special meaning
26
28
Architectures
30
1.3.4.1
1.3.4.2
1.3.5.1
1.3.5.2
1.3.6.1
1.3.6.2
1.3.6.3
1.3.6.4
1.3.6.5
1.3.7.1
1.3.7.2
1.3.7.3
1.3.7.4
1.3.7.5
1.3.9.1
1.3.9.2
1.3.9.3
1.3.9.4
1.3.9.5
1.3.9.6
1.3.9.7
1.3.10
1.3.10.1
Free-body diagrams
Operating conditions
Testing in real traffic
Testing on test track
HIL = Hardware in the loop simulation
SIL = Software in the loop simulation
MIL = Model in the loop simulation
Simulator = Driver in the loop simulation
AR = Augmented Reality
General mathematics tools
Block diagram based simulation tools
Vehicle dynamics specialized simulation tools
MBS tools
Modelica based modelling tools
Load levels
Open loop and Closed loop manoeuvres
Objective and subjective measures
Path, Path with orientation and Trajectory
Stable and Unstable
Subject and object vehicle
Active Safety and ADAS
Reference architecture of vehicle functionality
1.4
Heavy truck versus passenger cars
1.4.1
General differences
21
22
23
23
23
23
23
23
24
24
24
25
25
25
28
29
29
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29
29
30
30
32
32
2
1.4.2
2
Vehicle dynamics differences
32
1.5
Smaller vehicles
33
1.6
Automotive engineering
35
1.7
Typical numerical data
36
VEHICLE INTERACTIONS
2.1
39
Introduction
39
2.2
Introduction to Tyre Terminology
2.2.1
Wheel angles
39
40
2.3
41
2.2.1.1
2.2.1.2
2.2.1.3
2.2.1.4
2.2.1.5
Steering angle
Camber (Angle)
Caster Angle
Toe Angle
Kingpin inclination
40
40
40
41
41
Tyre Construction
2.4
Longitudinal Properties of Tyres
2.4.1
Tyre Rolling and Radii
2.4.2
The Rolling Resistance of Tyres
2.4.3
Longitudinal forces
43
44
45
50
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
55
56
58
58
61
61
2.6
Combined Longitudinal and Lateral Slip
63
2.7
Vertical Properties of Tyres
66
2.8
Tyre wear
67
2.4.3.1 Tyre brush model for longitudinal slip
2.4.3.2 Empirical tyre models
2.4.3.2.1
Magic Formula Tyre Model
2.4.3.2.2
TM-Easy Tyre Model
2.4.3.2.3
Other Tyre Models
2.4.3.3 Relaxation
2.5.5.1
2.5.5.2
2.5.5.3
2.6.1.1
50
53
53
54
54
55
Overturning moment
Tyre self-aligning moment
Camber force
Relaxation
61
61
62
66
2.9
Vehicle Aerodynamics
2.9.1
Longitudinal wind velocity
2.9.2
Lateral wind velocity
68
68
69
2.10 Driver interactions with vehicle dynamics
2.10.1
Open-loop and Closed-loop Manoeuvres
2.10.2
Pedals
2.10.3
Steering
69
70
71
71
2.11
75
2.10.3.1
2.10.3.2
Steering system angles
Steering system forces
Traffic interactions with vehicle dynamics
3
72
72
3
LONGITUDINAL DYNAMICS
77
3.1
Introduction
3.1.1
References for this chapter
77
77
3.2
Steady State Functions
3.2.1
Propulsion System
77
77
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
78
80
80
80
81
Traction diagram
Driving Resistance
Top speed *
Starting with slipping clutch
Steady state load distribution
Friction limit
Uphill performance
81
82
83
84
84
85
86
Start-ability *
Grade-ability *
Towing capacity *
86
87
87
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
Rotating inertia effects
Traction diagram with deceleration
Fuel or energy consumption *
Emissions *
Tyre wear *
Range *
Acceleration reserve *
Load Transfer due to acceleration, without considering suspension
Acceleration – simple analysis
Acceleration performance *
Solution using integration over time
Solution using integration over speed
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
3.4.3
3.4.4
3.4.5
3.4.6
3.4.7
3.4.8
3.4.8.1
3.4.8.2
3.4.8.3
3.4.8.4
88
88
90
90
90
91
93
94
96
96
96
97
98
99
99
99
101
101
102
102
Braking efficiency *
Braking Distance *
Stopping Distance *
(Friction) Brake system
Pedal Response *
Pedal Feel *
Brake proportioning
Body heave and pitch due to longitudinal wheel forces
Load Transfer including suspension linkage effects
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
4
102
102
103
103
104
104
104
106
109
110
110
111
112
3.4.8.5
3.4.9
3.4.10
3.4.11
3.4.12
3.4.13
Additional phenomena
112
Dive at braking *
Squat at propulsion *
Anti-dive and Anti-squat designs
Deceleration performance *
Acceleration performance *
113
113
113
114
116
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
3.5.3
4
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
116
116
116
116
117
117
118
118
119
119
121
123
4.1
Introduction
4.1.1
References for this chapter
123
123
4.2
Low speed manoeuvrability
4.2.1
Path with orientation
4.2.2
Vehicle and wheel orientations
4.2.3
Steering System
124
124
125
125
4.2.3.1
4.2.4
4.2.5
4.2.6
4.2.7
4.2.8
4.2.9
4.2.10
4.2.11
Steering geometry
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
4.3
Steady state cornering at high speed
4.3.1
Steady state driving manoeuvres
4.3.2
Steady state one-track model
4.3.3
Under-, Neutral- and Over-steering *
4.3.4
How to design for a desired understeer gradient
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
Tyre design, inflation pressure and number of tyres
Roll stiffness distribution between axles
Steering system compliance
Roll steer gradient
Side force steer gradient
Camber steer
Toe angle
Wheel Torque effects
Transient vehicle motion effects on yaw balance
Some other design aspects
125
126
128
129
132
132
133
134
134
137
138
139
143
145
145
146
146
146
147
147
147
147
147
148
4.3.5
Required Steering
148
4.3.6
Steady state cornering gains *
149
4.3.5.1
4.3.6.1
Critical and Characteristic speed *
148
Yaw Rate gain
150
5
4.3.6.2
4.3.6.3
4.3.6.4
Curvature gain
Lateral Acceleration gain
Side slip as function of speed
150
151
151
4.3.7
4.3.8
4.3.9
Manoeuvrability and Stability
Handling diagram
Steady state cornering at high speed, with Lateral Load Transfer
152
152
154
4.3.10
4.3.11
Steering feel *
Steady state cornering roll-over
166
166
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 within one axle
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
4.3.9.6 Lateral load transfer influence on steady state handling
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
4.4.2.2
4.4.3
167
168
169
169
169
170
170
170
171
171
Relation between accelerations in inertial system and velocity derivatives in vehicle fix system176
Validity of solution
177
Steering frequency response gains *
4.4.3.1 Single frequency response
4.4.3.1.1
Solution with Fourier transform
4.4.3.1.2
Solution with complex mathematics
4.4.3.2 Lateral Velocity and Yaw Rate response
4.4.3.3 Lateral Acceleration response
4.4.3.4 Other responses to oscillating steering
4.4.3.5 Random frequency response
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
154
155
157
157
157
158
162
163
165
Explicit form model
Additional phenomena
177
178
178
179
180
183
184
184
184
185
186
189
193
193
4.5.4
Step steering response *
194
4.5.5
Long heavy vehicle combinations manoeuvrability measures
195
4.5.4.1
4.5.4.2
4.5.5.1
4.5.5.2
4.5.5.3
Mild step steering response
Violent step steering response
Rearward Amplification *
Off-tracking *
Yaw Damping *
4.6
Lateral Control Functions
4.6.1
Lateral Control Design
4.6.2
Lateral Control Functions
194
195
196
197
197
198
198
198
4.6.2.1 Lane Keeping Aid, LKA *
4.6.2.2 Electronic Stability Control, ESC *
4.6.2.2.1
Over-steer Control
4.6.2.2.2
Under-steer Control
4.6.2.2.3
Over-speed Control
198
199
199
199
200
6
4.6.2.2.4
4.6.2.3
4.6.2.4
5
Other intervention than individual wheel brakes
200
4.6.2.2.4.1
Balancing with Propulsion per axle
200
4.6.2.2.4.2
Torque Vectoring
200
4.6.2.2.4.3
Steering guidance
200
Roll Stability Control, RSC *
Lateral Collision Avoidance, LCA *
201
201
VERTICAL DYNAMICS
203
5.1
Introduction
5.1.1
References for this chapter
203
203
5.2
204
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
5.3.2.2
Space domain and Space frequency domain
Spatial Mean Square (MS) and spatial Root Mean Square (RMS) of variable
Spatial Power Spectral Density and Frequency bands
206
207
207
208
209
209
210
211
212
5.4
Road models
5.4.1
One frequency road model
5.4.2
Multiple frequency road models
212
212
213
5.5
One-dimensional vehicle models
5.5.1
One-dimensional model without dynamic degree of freedom
216
216
5.4.2.1
Transfer function from road spectrum in spatial domain to system response in time domain215
5.5.1.1 Response to a single frequency excitation
5.5.1.1.1
Analysis of solution
5.5.1.2 Response to a multiple frequency excitation
217
217
219
5.5.2
One dimensional model with one dynamic degree of freedom
219
5.5.3
One dimensional model with two degrees of freedom
224
5.5.2.1 Response to a single frequency excitation
Solution with Fourier transform
5.5.2.1.1
Solution with trigonometry
5.5.2.1.2
Analysis of solution
5.5.3.1 Response to a single frequency excitation
5.5.3.1.1
Solution with Fourier transform
5.5.3.1.2
Analysis of solution
5.5.3.2 Simplified model
5.5.3.3 Variation of stiffness and damping
5.5.3.3.1
Varying suspension stiffness
5.5.3.3.2
Varying suspension damping
5.5.3.3.3
Varying unsprung mass
5.5.3.3.4
Varying tyre stiffness
220
220
220
221
225
225
225
228
229
230
230
231
231
5.6
Ride comfort *
5.6.1
Single frequency
5.6.2
Multiple frequencies
232
232
234
5.7
Fatigue life *
5.7.1
Single frequency
237
237
5.6.2.1
5.7.1.1
5.7.1.2
Certain combination of road, vehicle and speed
Loads on suspension spring
Fatigue of other components
235
237
237
7
5.7.2
Multiple frequencies
237
5.8
Road grip *
5.8.1
Multiple frequencies
238
238
5.9
Two dimensional oscillations
5.9.1
Bounce (heave) and pitch
239
239
5.9.1.1
5.9.2
5.10
5.11
Wheel base filtering
240
Bounce (heave) and roll
241
Transient vertical dynamics
242
Other excitation sources and functions
242
5.11.1.1
5.11.1.2
Other excitation sources
Other functions
242
243
BIBLIOGRAPHY
244
INDEX
246
8
INTRODUCTION
1 INTRODUCTION
1.1 Purpose of this compendium
This compendium is primarily written for studies of vehicle dynamics for the first time; which at
Chalmers University of Technology 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 used
as an introduction to more advance courses, which at Chalmers is the course “TME102 Vehicle
Dynamics Advanced”.
The overall purpose of the compendium is to contribute to education of automotive engineers with
good understanding of vehicle dynamics related complete vehicle requirements and how to design
for them and verify them.
The overall goal of the compendium is to introduce the underlying concepts that govern the
performance of road vehicles whether they are passenger cars or 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. The ambition for the course is to
give a platform for the design of vehicles with good characteristics. 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)
Lateral dynamics (e.g. steering)
Vertical dynamics (e.g. vertical comfort)
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. 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.
1.2 Attributes and Functions
Development of a vehicle is driven by the needs of the customers/users, legislation from the
authorities and constraints from the manufacturer’s platform/architecture on which is it to be built.
One way of describing most of those needs, 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.
9
INTRODUCTION
In this compendium, both attributes and functions implicitly concern the complete vehicle; not the
sublevel of the vehicles (systems or components) and not the superior level of the vehicle (which
would include such as road, road surface design).
Attributes and Functions are used to establish processes and structures for requirement setting and
verification. 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 customer/user.
If simplified to be explainable in a three-structure, each function can be seen as a sub-attribute an
attribute and each attribute can be seen as the superior function to a set of sub-functions. Such a
three-structure can be expanded to consist of many levels. The three-structure (one-to-many
relations) can also be developed to a network (many-to-many relations) since a function can be
motivated by many attributes.
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 engineering are listed in Table 1.1. The table is
much generalised and its attributes would typically need to be decomposed into more attributes
when used in the organisation of an OEM. Also, not mentioned in the table are 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 are, in turn, a way to group
requirements. Legal requirements are often, but not always, possible to trace back to a specific
attribute. Requirements arising from company-internal platform and architecture constraints are
probably even 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
Description
Environment
friendliness
Low usage of natural resources (mainly energy) and low pollution, while
performing the transportation. Requirements are defined over typical
vehicle usage pattern, which are longer driver events of typically 10 min to
10 hours. There are different ways to define such usages, e.g.
(Urban/Highway/Mixed) driving cycles (speed versus time) and Transport
tasks (speed & stops versus distance).
This attribute is to a large extent required and assessed by legislation.
Safety
Driving
Dynamics
This attribute can be extended to “Transport efficiency”, including
minimizing cost for fuel and transportation time while maximising pay
load. This is most clear for commercial vehicles, and the attribute then
becomes more required and assessed by the vehicle customers and users.
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 legislation. (In
some markets also important for vehicle customers and users.)
How the driver experiences driving, from relaxed transport driving
(comfort) to active driving (sensation).
Driving Dynamics contains sub-attributes as:
10
Abbreviation
E
S
D
INTRODUCTION
•
Driveability, Handling and Road-holding. How the vehicle
responds to inputs from driver and disturbances, and how driver
gets feedback from vehicle motion. (It can be noted that same
aspects are similarly important for a “virtual driver”, i.e. a control
algorithm doing some automatic/autonomous control.) Handling
and Road-holding most often refer to lateral manoeuvres.
Driveability often refers to longitudinal (acceleration, braking gear
shifting).
• Performance, which refers to how the vehicle can perform at the
limits of its capabilities; acceleration, deceleration or cornering.
• Comfort. Ride comfort often refers to vertical vibrations of the
occupants. Comfort in a wider sense can include NVH (noise,
vibrations and harshness) and handling related characteristics such
as steering feel.
This attribute is to a large extent required and assessed by the vehicle
customers and users.
1.2.2
Functions
A function supplies certain functionality to the customer/user of the vehicle. A function is a more
specific description which allows more quantitative requirements, see Section 1.2.3. So, the function
does not primarily stipulate any specific subsystem. However, the realisation of a function, in one
particular vehicle programme, does normally only engage a limited subset of all subsystems; 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.
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 algorithm (during 1990’s) it became much more visible as a function, sometimes
refered 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 subfunctions, e.g. better idle speed control.
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}”.
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”.
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.
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INTRODUCTION
The compendium defines functions on complete vehicle level, i.e. not cascaded to requirements on
specific sub-systems in the vehicles, which would require design decisions.
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
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.
12
INTRODUCTION
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 a the vehicle program or platform.
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 seconds 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
describing 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. But in order
to understand them, design for them and verify them, the engineers need knowledge in terms of
models, methods and tools.
13
INTRODUCTION
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.
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 1-1. 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 1-1: 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
14
INTRODUCTION
solving a particular problem, the models at least have to be able to reproduce the problem one is
trying to solve. Also, the models 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.
One can identify the modelling in different steps 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.
Analyse task is formulated, and a plan of how the result of that 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.
Mathematical model, in equations. 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:
𝒉(𝒛̇ , 𝒛, 𝒖, 𝒕) = 𝟎; 𝒖 = 𝒖(𝒕);
where z are the dependent variables, u are the input variables and t is time (independent
variable).
Note 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.
Explicit form model, in the form of an algorithm. This form typically occurs when solving
time sequence for dynamic systems using “ODE” (Ordinary Differential Equations) integration
methods. The general form is:
𝒙̇ = 𝒇(𝒙, 𝒖, 𝒕); 𝒚 = 𝒈(𝒙, 𝒖); 𝒖 = 𝒖(𝒕);
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
o
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:
𝒙̇ 𝒏𝒏𝒏 = 𝒇(𝒙(𝒕𝒏𝒏𝒏 ), 𝒖(𝒕𝒏𝒏𝒏 ), 𝒕𝒏𝒏𝒏 );
𝒙(𝒕𝒏𝒏𝒏 + 𝚫𝒕) = 𝒙(𝒕𝒏𝒏𝒏 ) + 𝚫𝒕 ∙ 𝒙̇ 𝒏𝒏𝒏 ;
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.
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, …
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INTRODUCTION
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.
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 [𝑣̇ = 𝐹 ⁄𝑚; 𝐹 = 𝑐 ∙ 𝑥; 𝑥̇ =
𝑣;] and [𝑣̇ = 𝐹 ⁄𝑚; 𝐹̇ = 𝑐 ∙ 𝑣; 𝑥 = 𝐹 ⁄𝑐 ;], respectively.
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 1-1.
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 1-2.
Any vector or Translation (translational speed or force)
Scalar flow Power
Along axis in plane of paper:
Perpendicular to paper, pointing out of paper:
Signal flow
Perpendicular to paper, pointing in to paper:
Rotation (rotational speed or moment/torque)
Physical Connection
Around axis in plane of paper:
(no prescribed causal direction):
Around axis perpendicular to paper:
Figure 1-2: 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
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.
16
INTRODUCTION
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(.).
Differentiation (of x) with respect to time (t):
𝑑𝑑
𝑑𝑑
= 𝑥̇ = 𝑑𝑑𝑑(𝑥).
Vectors exist of two kinds: Geometrical vector (or physical vector, or spatial vector), denoted by 𝑣⃗,
and mathematical vector, denoted [vx;vy]. The geometrical vector is the true vector, while the
mathematical is a representation of one value of the geometrical vector, 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. (𝑎⁄𝑏) ∙ 𝑐 or 𝑎⁄(𝑏 ∙ 𝑐), not 𝑎⁄𝑏 ∙ 𝑐.
Multiplication symbol (* or ∙) shall be used to avoid ambiguity, e.g. 𝑎 ∙ 𝑏𝑏, not 𝑎𝑎𝑎 (assuming 𝑏𝑏 is
one variable, not two).
An interval has a notation with double dots. Example: Interval between 𝑎 and 𝑏 is denoted 𝑎. . 𝑏.
An explanation, between two consecutively following steps in a derivation of equations is written
within {} brackets. Example: 𝑥 + 𝑦 = {𝑎 + 𝑏 = 𝑏 + 𝑎} = 𝑦 + 𝑥 or 𝑥 + 𝑦 = 7 ⇒ {𝑎 + 𝑏 = 𝑏 +
𝑎} ⇒ 𝑦 + 𝑥 = 7.
An inverse function is denoted with −1 as superscript, e.g. 𝑦 = 𝑓(𝑥); ⇔ 𝑥 = 𝑓 −1 (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 𝐹𝑟𝑟 or 𝐹𝑥𝑥 . The intention in this compendium
is to order subscript with the physical vehicle part (here rear axle) as first index and the specification
(longitudinal, x) as second: 𝐹𝑟𝑟 . If there is also further detailed specifications, such as coordinate
system, e.g. wheel coordinate (subscript w), it would be the third subscript: 𝐹𝑟𝑟𝑟 .
17
INTRODUCTION
Table 1.2: Notation
18
INTRODUCTION
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INTRODUCTION
20
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 1-3.
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 Fzf instead of Fzf0 in Figure 1-3, which would make it pointless to set up the
vertical force equilibrium for the front axle (Fzf-Fzf=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 1-3). These forces shall be opposite to the positive direction of
acceleration and act through the centre of gravity.
21
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
Fzr
air resistance
Fzf
Fxf
Figure 1-3: 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.
22
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
23
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 1-4.
24
INTRODUCTION
Figure 1-4: Graphical model of dynamic system 𝑥̇ = 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. There are
a number of tools which supports this format.
The model format itself is non-causal and symbolic. The equation part of an example model can be as
follows:
25
INTRODUCTION
m*der(v)=F;
v=der(x);
F=c*x;
Modelica is also declarative. So, with the example model, there could be this declaration part:
parameter Real m=1000;
parameter Real c=1000;
input Real F;
Real v (start=0);
Real x (start=0);
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.
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 1-5. Figure 1-5 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.
ground, road
or horizontal
plane
heave or
vertical
yaw
transversal
plane
symmetry
plane
Figure 1-5: Vehicle (motion) degrees of freedom and important planes.
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
26
INTRODUCTION
systems. Figure 1-6 shows the vehicle fixed coordinate systems for these two standards. This
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.
ISO 8855
z
SAEJ670
y
y
x
x
z
Figure 1-6: ISO and SAE coordinate systems
The distinction of vehicle fixed and inertial (= earth fixed = world fixed) coordinate systems is
important. Figure 1-7 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.
Vehicle fixed
coordinate system:
lateral= y
yaw rate=wz=wz
vy
CoG
v
x
β=β=(βody)
vx
side slip angle
n =course
𝜑𝑧 =pz=heading
y
y
Wheel corner
fixed coordinate
system:
x
z
=longitudinal
angle
angle
Speed of
wheel huβ
Wheel fixed
coordinate
system:
d=d=
(Road wheel)
Steering
angle
Inertial coordinate system:
z
x
βhTE: Above tyre side slip is positive for tyre sliding with
positive lateral speed. hpposite definition is sometimes used.
Figure 1-7: Coordinate systems and motion quantities in ground plane
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.
27
INTRODUCTION
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.
In ISO and Figure 1-7, 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 1-8 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
wheel 9 wheel 7
wheel 5 wheel 3
wheel 1
wheel 10 wheel 8
wheel 6 wheel 4
wheel 2
Fx1
Fx2
Fy2
axle 5 axle 4
axle 3
axle 2
axle 1
unit 1
unit 2
Figure 1-8: Proposed numbering of units, axles and wheels. Example shows a
truck with trailer.
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.
28
INTRODUCTION
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, 𝜑𝑧 (𝑥) or 𝜑𝑧 (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); 𝜑𝑧 (𝑡)]. But also other quantities, such
as steering angle or vehicle propulsion force can be called trajectory: 𝛿(𝑡) and 𝐹𝑥 (𝑡), respectively.
The word “trace” is sometimes used interchangeably with trajectory.
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.
29
INTRODUCTION
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.
1.3.10 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 1-9.
30
INTRODUCTION
Vehicle
Environment
V2V / V2I
Maps
…
Traffic Situation Layer
Corrward Direction
ACC
Traffic
Interpretation
LaneSteering
Vehicle Capability
Max/Min Acceleration
weq. 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
weq. e.g. Dlobal Corces
Coordination
Stability
Control
Status
weq. e.g. Wheel Torque
Motion Support Device Layer
Device
Cap./State
Estimator
EngAct
SteAct
FrntBrkAct
RearBrkAct
TransmAct
Vehicle
Motion
Sens
Figure 1-9: One example of reference architecture of vehicle motion functionality. Red
arrows: Requests, Blue arrow: Information, Black lines with dot-ends: physical connection.
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 1-9. 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.
• 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.
31
INTRODUCTION
• Signals, e.g. Request, Status and Capability.
• Parameters used in Functional blocks, e.g. Physical parameter and Tuning parameter.
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.
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 1-10. 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.
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INTRODUCTION
Figure 1-10: 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 1-11 is a good example. All vehicles in Figure 1-11 can be referred to as Urban
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.
33
INTRODUCTION
3 wheeled cambering
concept vehicle from BMW
Piaggio MP3
Twizy from Renault
Eco Electric Bicycle
from Monark
ZeeBee from Clean Motion
Figure 1-11: 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 1-11 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 1-11).
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
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
34
INTRODUCTION
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.
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
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INTRODUCTION
(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
Table 1.2 gives typical numerical data for a passenger car. The purpose of the table is that the
different parameters should be consistent with each other, as a kind of realistic example vehicle. The
table is balancing between being generic and specific, which is difficult. The aim for the table is a
normal sized and shaped sedan. Therefore, please consider the table as very approximate.
36
INTRODUCTION
Table 1.3: Typical data for a passenger car
Parameter
0. Mathematical
0. Physical
0. Physical
pi
Air density
Earth gravity
1. General
Length, bumper to bumper
Longitudinal distance from
CoG to front axle
Mass
lf
1.3 m
m
1600 kg
1. General
Moment of inertia for yaw
rotation
Jzz
1. General
Unsprung mass
mus
1. General
Wheel base
L
2. Systems
Engine inertia
2. Systems
Gear ratio, highest gear
2. Systems
Gear ratio, lowest gear
2. Systems
2. Systems
2. Systems
Road friction, at dry asphalt
Road friction, at wet ice
Rolling resistance
2. Systems
Steering ratio
2. Systems
2. Systems
Wheel radius
Wheel rotational inertia
3. Longitudinal
Projected area in a
transversal view
Afront
3. Longitudinal
Aerodynamic coefficients
cd, clf,
clr
4. Lateral
Cornering stiffness
1. General
1. General
Notation Typical
Value
pi
3.14159
roh
1
g
9.80665
Passeneger car, medium size
Unit
Note / Typical / Range /
Relation to other
<none>
kg/(m^3)
m/(s*s)
Group/Type
5.00 m
40-50% of wheel base:
lf=0.55*L;
Radius of gyration is sligthly
2700 kg*m*m less (0.9) than half wheel
base: =m*(0.9*L/2)^2
Sum of 4 wheels with
200 kg
suspensions
2.90 m
0.5 kg*m*m
rad/rad=
5.00
Engine to wheel. Possibly <5.
Nm/Nm
rad/rad= Engine to wheel. Possibly
10.00
Nm/Nm >10.
1.0 N/N
0.1 N/N
0.01 N/N
Steering wheel to Road
16 rad/rad
wheel
0.30 m
0.5 kg*m*m For one wheel
2.2 m^2
0.31, 0.10,
0.07
40 000
1
N/rad =
N/1
For calculation of air
resistance
For calculation of lift and
pitch moment
For one wheel
4. Lateral
5. Vertical
Suspension damping
damp
5. Vertical
Suspension stiffness
stif
5. Vertical
5. Vertical
Tyre vertical stiffness
12 000 N/(m/s)
142 000 N/m
200 000 N/m
37
Sum of 4 wheels. Some 40%
of critical damping:
d = 0.40*2*sqrt(c*m);
2..3 times softer in
compression than rebound.
Sum of 4 wheels. So that
bounce frequency f is
between 1 and 2 Hz:
sqrt(c/m)=f*2*pi;
For one wheel
INTRODUCTION
38
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
Air,
Weather,
…
Environment
Steering Wheel,
Pedals, Buttons,
Seat Accelerations,
Visual Instruments,
Sounds, …
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,
𝐹𝑥 , 𝐹𝑦 and 𝐹𝑧 , are obviously necessary for any free body diagrams of the loads on the vehicle.
Following these parameters, the term 𝑅𝑙 represents the loaded (deflected) or rolling radius of the
39
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: 𝐹𝑧
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 2.5.
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 2.5 and Section 2.10.3.2, and changes the camber angle when the wheel
is steered.
40
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
Kingpin inclination
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 2.5 and Section 2.10.3.2.
2.3 Tyre Construction
The tyres of a vehicle have the following tasks:
o
o
o
o
o
Carry the static load
Generate brake and traction forces
Generate lateral forces
Isolate vertical disturbances i.e. minimize dynamic forces
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.
41
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
42
VEHICLE INTERACTIONS
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
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)
2.4 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, 𝜔, 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,
𝑣𝑥 , and force, 𝐹ℎ𝑢𝑢,𝑥 . See Figure 2-6.
43
VEHICLE INTERACTIONS
x-direction =
= vehicle longitudinal forward
vehicle,
wheel
including torque source, excluding wheel
𝑻
𝝎
𝒗𝒙
𝒗𝒙
𝑭𝒉𝒉𝒉,𝒛
𝐹𝑔𝑔𝑔𝑔𝑔𝑔,𝑥
ground
𝑻
𝑭𝒉𝒉𝒉,𝒙
𝑅𝑙
𝑭𝒉𝒉𝒉,𝒙
𝑭𝒉𝒉𝒉,𝒛
𝐹𝑔𝑔𝑔𝑔𝑔𝑔,𝑥
𝐹𝑔𝑔𝑔𝑔𝑔𝑔,𝑧
𝐹𝑔𝑔𝑔𝑔𝑔𝑔,𝑧
Figure 2-6: A wheel as a transmission from rotational [𝜔; 𝑇] to translational �𝑣𝑥 ; 𝐹ℎ𝑢𝑢,𝑥 �.
2.4.1
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 2-7 a).
This relationship does not hold when the tyre is deflected as in Figure 2-7 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
rolling tyre looks like Figure 2-7 c), where the deformation at the leading edge also is drawn. This
means that 𝑣𝑥 is only ≈ 𝑅𝑙 ∙ 𝜔 and ≈ 𝑅𝑜 ∙ 𝜔 for limited slip levels.
a) Ideal rolling:
b) With vertical load Fz
and torque T≈0:
𝑭𝒛
ω
c) With vertical load Fz
and (significant) torque T>0:
𝑭𝒛
ω
vx
ω
𝑅0 = inflated unloaded radius
𝑅0 = 𝑅𝑙 and
𝑣𝑥 = 𝜔 ∙ 𝑅𝑙0 = 𝜔 ∙ 𝑅00
𝑅𝑙
vx
𝑅𝑙
𝑅𝑙
vx
𝑻
𝑅0 = inflated free radius
𝑅𝑙 = loaded radius
𝑅0 > 𝑅𝑙 and
𝑣𝑥 = 𝜔 ∙ 𝑅𝑙 ≠ 𝜔 ∙ 𝑅0
𝑅0 > 𝑅𝑙 and
𝑣𝑥 ≠ 𝜔 ∙ 𝑅𝑙 ≠ 𝜔 ∙ 𝑅0
Figure 2-7: Radius and speed relations of a rolling tyre 𝑅0 and 𝑅𝑙 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:
𝑠𝑥 =
44
R ∙ 𝜔 − 𝑣𝑥
;
|R ∙ 𝜔|
[2.1]
VEHICLE INTERACTIONS
For braked wheels it is often defined as:
𝑠𝑥 =
R ∙ 𝜔 − 𝑣𝑥
;
|𝑣𝑥 |
[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 502.4.3.1,:
𝑠𝑥 =
R ∙ 𝜔 − 𝑣𝑥
;
|R ∙ 𝜔 + 𝑣𝑥 |⁄2
[2.3]
Whether one should use 𝑅 ∙ 𝜔 or 𝑣𝑥 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. 𝑅0 or 𝑅𝑙 . However, this
compendium recommends 𝑅0, rather than 𝑅𝑙 , because 𝑅0 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 = (𝑣𝑥 ⁄𝜔)| 𝑇=0 , i.e. a speed ratio with
dimension length, between translator and rotating speeds, measured when the wheel is non-driven.
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. However, it is less useful for tyre
modelling as in this compendium. The Rolling radius is normally between 𝑅0 and 𝑅𝑙 .
The variable 𝑠𝑥 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, 𝑓𝑟 , is the rolling resistance force divided by the normal force, 𝐹𝑧 .
Assuming force equilibria in longitudinal and vertical direction, 𝐹ℎ𝑢𝑢,𝑧 = 𝐹𝑔𝑔𝑔𝑔𝑔𝑔,𝑧 = 𝐹𝑧 and
𝐹ℎ𝑢𝑢,𝑥 = 𝐹𝑔𝑔𝑔𝑔𝑔𝑔,𝑥 = 𝐹𝑥 , see Figure 2-6.
𝑇
𝑅𝑙 − 𝐹𝑥
𝑓𝑟 =
𝐹𝑧
[2.4]
𝐹𝑥 denotes the longitudinal force on the wheel, 𝑇 denotes the applied torque and 𝑅𝑙 denotes the
tyre radius. For a free rolling tyre, where 𝑇 = 0, 𝑓𝑟 becomes simply − 𝐹𝑥 ⁄𝐹𝑧 . One often see
definitions of 𝑓𝑟 which assumes free rolling tyre; but Eq [2.4] is more generally useful.
Regarding which radius to use, it should rather be 𝑅𝑙 than 𝑅0 , because 𝑅𝑙 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 2-8 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 2-8, is the main reason for this and it is not dependent on the longitudinal force.
45
VEHICLE INTERACTIONS
Same figure also shows the off-centre effect, which is directly influenced by the longitudinal force.
The force offset, e, explains 𝑓𝑟 . Additional to what is shown in Figure 2-8 one can motivate even
higher e, by including other effects, such as wheel bearing loss, brake disk drag and aerodynamic
torque loss.
ωesulting pressure and Equivalent forces
Pressure distribution components
ω
Fx
Fz
v
T
ω
Fx
ωl
Fz
v
T
Fx
ωl
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 2-8: Normal force distribution on a tyre. The measure e is the force offset.
Longitudinal and vertical force equilibria are already satisfied, due to assumptions above. However,
moment equilibrium around wheel hub requires:
𝑇 − 𝐹𝑥 ∙ 𝑅𝑙 − 𝐹𝑧 ∙ 𝑒 = 0 ⇒ 𝐹𝑥 =
𝑇
𝑒
− 𝐹𝑧 ∙ ;
𝑅𝑙
𝑅𝑙
[2.5]
An analysis of this result is that the force 𝐹𝑥 which pushes the vehicle body forward is the term 𝑇⁄𝑅𝑙
(arising from the applied propulsion or brake torque T), minus the term 𝐹𝑧 ∙ 𝑒⁄𝑅𝑙 . The term can be
seen as a force 𝐹𝑟𝑟𝑟𝑟 and referred to as the rolling resistance force. The dimensionless ratio
𝑒⁄𝑅 expression is the rolling resistance coefficient, 𝑓𝑟 :
𝑓𝑟 =
𝑒
;
𝑅𝑙
[2.6]
Eq [2.6] 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
which are measurable externally, see Eq [2.4]. Sometimes one see 𝑓𝑟 = − 𝐹𝑥 ⁄𝐹𝑧 as a definition, but
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 𝐹𝑥 ≠ −𝑓𝑟 ∙ 𝐹𝑧 except for the special case when undriven wheel (𝑇 = 0).
Figure 2-9 shows an un-driven wheel, which is a special case of a driven wheel.
46
VEHICLE INTERACTIONS
Driven
Undriven (T=0)
𝒆
𝑭𝒙 ≡ − ∙ 𝑭𝒛 ;
Fx has to be:
ω
T
Fz
v
ω
Fx
|Fx|
𝑅𝑙
Fx
e Fz
Fz
v
Fx
Fz
e
1
𝑅𝑙
Undriven
|Fx|
𝑹𝒍
𝑅𝑙
T
Driven, so that Fx=0
Figure 2-9: Driven wheel and its special case “Un-driven wheel” with
rolling resistance.
Coefficient of rolling resistance, f
Figure 2-10 shows how widely the rolling resistance changes due to different surface types 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 2-11. On soft ground, the ground is
deformed so that the wheel rolls in a “local uphill slope” with inclination angle 𝜑. Intuitively, a higher
inflation pressure will lead to more deformation of the ground, leading to a steeper 𝜑.
0.4
Sand
0.2
Medium Hard Soil
Concrete
0
0
20
Inflation Pressure [psi]
40
Figure 2-10 : Range of Coefficient of Rolling Resistance
for Different Surfaces
47
VEHICLE INTERACTIONS
Wheel rolling on hard ground:
ω
Wheel rolling on soft ground:
ω
v
e
v
ϕ
Figure 2-11 : Rolling resistance explanation model for hard and soft ground.
Several factors will affect the rolling resistance of tyres. Design factors:
•
•
•
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 2-10 and Figure 2-11.
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.
As an example, Figure 2-12 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.
48
VEHICLE INTERACTIONS
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 2-12: 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:
𝑓𝑟
𝑓𝑟
𝑓𝑟
𝑓𝑟
= 0.0136 + 0.04 ∙ 10−6 ∙ 𝑣 2
= 0.0169 + 0.19 ∙ 10−6 ∙ 𝑣 2
= 0.006 + 0.23 ∙ 10−6 ∙ 𝑣 2
= 0.007 + 0.45 ∙ 10−6 ∙ 𝑣 2
What is important to observe is that for passenger cars, a value of around 0.01 is normal and a
reasonable value to start an analysis. There is a weak variation with speed which can be ignored for
lower speeds. As seen in Figure 2-12, the influence of speed on rolling resistance is constant up to
about 100 km/h. Trucks tyres have a much lower rolling resistance (approximately half). Tyres have
developed in that way for trucks, because their fuel economy is so critical.
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 aerodynamics of the rotation of the wheel. These should be subtracted from
propulsion/brake torque when calculating T in Figure 2-9. 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.
49
VEHICLE INTERACTIONS
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
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
v
F
𝑅∙𝜔
N
N F
F
F
µstat*N
µpeak*N
slide
stick
µdyn*N
vx
T
F
N
N
F
µdyn*N
∆v=v
slip=∆v/vref=(R*w-vx)/vref
slide
• A stick µode where ∆v=0.
• In slide µode, the friction force, F,
is only depending on sign(∆v).
• No stick µode (if w>0).
• The friction force, F, is depending on
relative sliding speed, ∆v/vref.
Figure 2-13: 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.
50
VEHICLE INTERACTIONS
ω*ω
vx
Figure 2-14: 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 𝐹𝑧 1/2 for line contact or 𝐹𝑧 1/3 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 2-15.
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 [2.7].
𝜏 = 𝐺 ∙ 𝛾 ; 𝜏 = 𝑠ℎ𝑒𝑒𝑒 𝑠𝑠𝑠𝑠𝑠𝑠; 𝐺 = 𝑠ℎ𝑒𝑒𝑒 𝑚𝑚𝑚𝑚𝑚𝑚𝑚; 𝛾 = 𝑠ℎ𝑒𝑒𝑒 𝑎𝑎𝑎𝑎𝑎;
[2.7]
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 ∆𝑣 and the time 𝑡 it takes to reach that coordinate, 𝑡(𝜉):
∆𝑣 ∙ 𝑡(𝜉) (𝑅 ∙ 𝜔 − 𝑣𝑥 ) ∙ 𝑡(𝜉)
=
;
𝐻
𝐻
|𝑅 ∙ 𝜔 + 𝑣𝑥 |
𝑤ℎ𝑒𝑒𝑒 𝑡(𝜉) = 𝜉 ⁄𝑣𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇 𝑎𝑎𝑎 𝑣𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇 ≈
;
2
𝛾=
51
[2.8]
VEHICLE INTERACTIONS
rolling direction
L=length of contact patch
tyre
rubber
outlet end
leading edge of contact
(=rubber inlet end)
R*R
Ξ=Ξc
Ξ=L
Ξ=coordinate in contact
G=shear modulus
W=width (perpendicular to drawing)
road
Ξ=0
H=height of
rubber brushes
vx
(relative to wheel hub)
Ξ
pressure
Ξ
stick
slip
“mode”
Ξ
shear
stress
Figure 2-15: Physical model for deriving brush model for longitudinal slip. Speeds are
relative to hub translational motion (not relative to ground). Drawn for 𝑅 ∙ 𝜔 > 𝑣𝑥 .
Combining these equations yields the following, where the slip sx can be identified:
𝜏=
𝐺 (𝑅 ∙ 𝜔 − 𝑣𝑥 )
𝐺
∙
∙ 𝜉 = ∙ 𝑠𝑥 ∙ 𝜉;
|𝑅
|
⁄
𝐻
∙ 𝜔 + 𝑣𝑥 2
𝐻
[2.9]
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
𝐿
𝐺
𝐹𝑥 = 𝑊 ∙ � 𝜏 ∙ 𝑑𝑑 = 𝑊 ∙ � ∙ 𝑠𝑥 ∙ 𝜉 ∙ 𝑑𝑑 + 𝑊 ∙ � 𝜇 ∙ 𝑝 ∙ 𝑑𝑑 =
𝐻
0
0
𝜉𝑐
2
𝑊 ∙ 𝐺 ∙ 𝑠𝑥 𝜉𝑐
∙
+ 𝑊 ∙ 𝜇 ∙ 𝑝 ∙ (𝐿 − 𝜉𝑐 );
𝐻
2
𝜇∙𝑝∙𝐻
𝑤ℎ𝑒𝑒𝑒 𝜉𝑐 =
; 𝑝 ∙ 𝑊 ∙ 𝐿 = 𝐹𝑧
𝐺 ∙ 𝑠𝑥
=
[2.10]
Equation [2.10] is only valid for 𝑠𝑥 corresponding to 0 < 𝜉𝑐 < 𝐿. Otherwise, the simpler Equation
[2.9] can be used. This and further simplification of Equation [2.10] leads to:
𝐹𝑥 =
⎧=
⎨
⎊
𝐺 ∙ 𝑊 ∙ 𝐿2
∙ 𝑠𝑥 ;
2∙𝐻
𝜇 ∙ 𝐻 ∙ 𝐹𝑧
𝜇 ∙ 𝐹𝑧
⇔ |𝐹𝑥 | ≤
2
𝐺∙𝑊∙𝐿
2
𝜇 ∙ 𝐻 ∙ 𝐹𝑧
1
= 𝜇 ∙ 𝐹𝑧 ∙ �1 −
∙ � ; 𝑒𝑒𝑒𝑒
2 ∙ 𝐺 ∙ 𝑊 ∙ 𝐿2 𝑠𝑥
𝑓𝑓𝑓 |𝑠𝑥 | ≤
The shape of this curve becomes as shown in Figure 2-16.
52
[2.11]
VEHICLE INTERACTIONS
1
𝑭Fx/(mu*Fz)
𝒙 ⁄𝝁 ∙ 𝑭𝒛
0.8
0.6
0.4
linear, up to𝑭𝒙 ⁄𝝁 ∙ 𝑭𝒛 = 𝟎. 𝟓
0.2
0
0
0.05
0.1
0.15
0.2
0.25
sx
𝒔
0.3
0.35
0.4
0.45
0.5
𝒙
Figure 2-16: Shape of force/slip relation derived with brush model (assuming
constant pressure distribution, etc).
Figure 2-25 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):
𝐹𝑥 = 𝐶𝑥 ∙ 𝑠𝑥
[2.12]
For the brush model, or any other model which describes Fx(sx), one can define the “Longitudinal
𝑚/𝑠
tyre stiffness” 𝐶𝑥 , which have the unit 𝑁 = 𝑁⁄1 = 𝑁/ � �. It is the derivative of force with respect
to slip. In many cases one means the derivative at sx=0:
𝑚/𝑠
𝜕
𝐶𝑥 = �
𝐹 ��
𝜕𝑠𝑥 𝑥 𝑠
𝑥 =0
[2.13]
Equation [2.11] 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, 𝐹𝑧 , 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.
2.4.3.2
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
fitted curve instead of Equation [2.11]. 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
53
VEHICLE INTERACTIONS
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 − 𝐸 ∙ (𝐵 ∙ 𝑥 − arctan(𝐵 ∙ 𝑥))�� ;
𝑌(𝑥) = 𝑦(𝑠) + 𝑆𝑉 ;
[2.14]
𝑥 = 𝑋 + 𝑆𝐻 ;
Where B is a stiffness factor, C is a shape factor, D is a peak value, and E is a curvature factor
describing the curve. The variable x is the tyre slip value. The variables SV and SH are shifting
constants for data that do not pass through the origin. The relationship between these coefficients
and the tyre slip/friction relation is shown in Figure 2-17.
Figure 2-17: Magic Formula Tyre Parameters, (Pacejka, 2005)
2.4.3.2.2 TM-Easy Tyre Model
Since Equation [2.14] 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 2-18.
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: 𝐹𝑥 = 𝐶𝑥 ∙ 𝑠𝑥
Linearized and saturated: 𝐹𝑥 = 𝑠𝑠𝑠𝑠(𝑠𝑥 ) ∙ 𝑚𝑚𝑚(𝐶𝑥 ∙ |𝑠𝑥 |; 𝜇 ∙ 𝐹𝑧 )
54
VEHICLE INTERACTIONS
Figure 2-18: TM-Easy Tyre Model, (Hirschberg, et al., 2002)
2.4.3.3
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 for the rubber in the
tyre. The transients between different steady state conditions involve finding a new steady state
deformation pattern of the rubber. The change of deformation pattern 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 Fx(sx) curve can be used. A driving situation when this definitely does
not apply is when a vehicle starts from stand-still with initially un-deformed tyres (no shear stress).
For vx=w=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 stress in an elastic part. A dynamic equation of this type is needed: 𝐹̇ = 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 ∙
(𝑅 ∙ 𝜔 − 𝑣𝑥 ). That would create the delay which occurs in practice, e.g. when starting from standstill.
A very common, but not completely physically motivated, way to model this phenomenon is to add a
first order delay of the force:
𝐹𝑥 = 𝑓�𝑠𝑥,𝑑𝑑𝑑𝑑𝑑𝑑𝑑 � =according to the stead state model, e.g. Equation [2.11]
𝑠̇𝑥,𝑑𝑑𝑑𝑑𝑑𝑑𝑑 = 𝐴 ∙ �𝑠𝑥 − 𝑠𝑥,𝑑𝑑𝑑𝑑𝑑𝑑𝑑 �
𝑣
𝑥
𝐴 = 𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅ℎ
or 𝐴 =
𝑅∙𝜔
𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅ℎ
[2.15]
RelaxationLength="fraction" (~25-50%) of tyre circumference
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 2-19.
55
VEHICLE INTERACTIONS
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
Îąs
Fs
Îąs
Fs
Îąs
Fs
Îąs
|FyÎą |
|FyÎą |
Figure 2-19: 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 1-7 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 2-21. The behaviour of the curve is similar to that exhibited for longitudinal forces Figure 2-17
and Figure 2-18. It becomes even more similar if lateral slip angle is replaced by lateral slip,
𝑠𝑦 = 𝑡𝑡𝑡(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 2-20 shows the model for lateral slip and should be compared to Figure
2-15. 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:
𝐹𝑦 =
2
⎧= − 𝐺 ∙ 𝑊 ∙ 𝐿 ∙ 𝑠 ;
𝑦
⎪
2∙𝐻
⎨
⎪
⎊
𝜇 ∙ 𝐻 ∙ 𝐹𝑧
𝜇 ∙ 𝐹𝑧
⇔ �𝐹𝑦 � ≤
2
𝐺∙𝑊∙𝐿
2
𝜇 ∙ 𝐻 ∙ 𝐹𝑧
1
= −𝜇 ∙ 𝐹𝑧 ∙ �1 −
∙ � ; 𝑒𝑒𝑒𝑒
2 ∙ 𝐺 ∙ 𝑊 ∙ 𝐿2 𝑠𝑦
𝑓𝑓𝑓 �𝑠𝑦 � ≤
where the lateral slip, 𝑠𝑦 , is defined as:
𝑠𝑦 =
56
𝑣𝑦
|𝑅 ∙ 𝜔 + 𝑣𝑥 |⁄2
[2.16]
[2.17]
VEHICLE INTERACTIONS
rolling direction
view from front,
showing rubber element
on one certain Ξ
L=length of contact patch
rubber
outlet end
Ξ = longitudinal
coordinate in
contact
tyre
R*R
Ξ=L
vy
leading edge of contact
(=rubber inlet end)
Ξ=Ξc
road
G=shear modulus
W=width (perpendicular to drawing)
Ξ=0
H=height
vx
(relative to wheel hub)
longitudinal
coordinate
Ξ
pressure
pressure
Ξ
stick
slip
“mode”
Ξ
shear stress
(lateral, i.e. perpendicular
to drawing)
Figure 2-20: 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 [2.17] is not consistent with the earlier mentioned
relation 𝑠𝑦 = 𝑡𝑡𝑡(𝛼) = 𝑣𝑦 ⁄𝑣𝑥 . One can identify two different lateral slip:
o “Lateral wheel slip”= 𝑠𝑦,𝑤ℎ𝑒𝑒𝑒 = 𝑣𝑦,𝑤ℎ𝑒𝑒𝑒 ⁄𝑣𝑥,𝑤ℎ𝑒𝑒𝑒 , which is defined by how wheel
hub moves over ground: 𝑠𝑦,𝑤ℎ𝑒𝑒𝑒 = 𝑠𝑦,𝑤ℎ𝑒𝑒𝑒 �𝑣𝑥,𝑉𝑉ℎ𝑖𝑖𝑖𝑖 , 𝑣𝑦,𝑉𝑉ℎ𝑖𝑖𝑖𝑖 , 𝜔𝑧,𝑉𝑉ℎ𝑖𝑖𝑖𝑖 �
o “Lateral tyre slip” = 𝑠𝑦,𝑡𝑡𝑡𝑡 = 𝑣𝑦 ⁄(|𝑅 ∙ 𝜔 + 𝑣𝑥 |⁄2), valid for the constitutive
relation: 𝐹𝑦 = 𝐹𝑦 �𝑠𝑦,𝑡𝑡𝑡𝑡 �.
These names are not well established, but invented in this compendium to facilitate
understanding. If 𝑅 ∙ 𝜔 = 𝑣𝑥 (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 [2.17] tells that the most correct way is that lateral force is 𝐹𝑦 ∝ 𝑠𝑦 ,
not 𝐹𝑦 ∝ α. Often one find 𝐹𝑦 ∝ α as starting point in the literature, but this compendium
uses 𝐹𝑦 ∝ 𝑠𝑦 .
(Note that when using 𝐹𝑦 ∝ α, it is usual to then approximate α with α = arctan�𝑠𝑦,𝑤ℎ𝑒𝑒𝑒 � ≈
𝑠𝑦,𝑤ℎ𝑒𝑒𝑒 = 𝑣𝑦 ⁄𝑣𝑥 to get the vehicle dynamics equations linear in 𝑣𝑦 and 𝜔𝑧 . Effectively, one
have then used 𝐹𝑦 ∝ 𝑠𝑦 but made a detour. 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 slip
(and constant normal load):
𝐹𝑦 = −𝐶𝑦 ∙ 𝑠𝑦 ;
57
or
𝐹𝑦 = −𝐶𝛼 ∙ 𝛼;
[2.18]
VEHICLE INTERACTIONS
For the brush model, or any other model which describes 𝐹𝑦 = 𝐹𝑦 �𝑠𝑦 �, one can define the “Lateral
tyre stiffness” or “Tyre Cornering Stiffness”, 𝐶𝑦 , which have the unit 𝑁. It is the derivative of force
with respect to slip. Reference (ISO8855) defines the cornering stiffness as this derivative for
arbitrary slip, but it is often enough to consider only the small slip and then the curve is almost linear.
Then, the cornering stiffness is the slop at zero slip:
𝜕
𝐶𝛼 = − � 𝐹𝑦 ��
𝜕𝜕
𝛼=0
=
𝐶𝑦 = − �
The minus is added to get positive values on 𝐶𝛼 and 𝐶𝑦 .
𝜕
𝐹 ��
𝜕𝑠𝑦 𝑦 𝑠
𝑦 =0
[2.19]
When using only small slip, it does not matter if the cornering stiffness is defined as the slope in an
𝐹𝑦 versus 𝛼 diagram or 𝐹𝑦 versus 𝑠𝑦 = 𝑡𝑡𝑡(𝛼) diagram. Therefore, the notation for cornering
stiffness varies between 𝐶𝛼 and 𝐶𝑦 . Cornering stiffness has the unit 𝑁 which can be interpreted as
𝑚/𝑠
� or 𝑁
𝑚/𝑠
𝑁 = 𝑁⁄1 = 𝑁/ �
= 𝑁⁄𝑟𝑟𝑟.
“Cornering Compliance” is sometimes used instead of “Cornering stiffness”. Cornering compliance is
simply the inverted quantity, the unit being 1⁄𝑁 or 𝑟𝑟𝑟 ⁄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 2-21, 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.
|Cornering Force| [N]
Adhesion
Adhesion Limit
Limit
2000
1000
0
4
8
12
|Slip Angle| [deg]
16
Figure 2-21: 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
58
VEHICLE INTERACTIONS
vertical load changes is presented in Figure 2-22 and Figure 2-23. 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 =
𝐹𝑧
𝐶𝛼
[2.20]
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 2-22: Influence of Vertical Load on Lateral Force, (Gillespie, 1992)
Figure 2-23: Cornering stiffness versus vertical load
59
VEHICLE INTERACTIONS
Figure 2-24: 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
contact theory, one can motivate 𝐹𝑧 ∝ 𝐿𝑎 = 𝐿2 (line contact) or 𝐹𝑧 ∝ 𝐿𝑎 = 𝐿3 (for point contact). An
extreme assumption, that the tyre is a membrane with an inside inflation pressure, gives 𝐹𝑧 ∝ 𝐿𝑎 =
𝐿1 . These can be inserted as 𝐿 = 𝑘 ∙ 𝑎�𝐹𝑧 in Equation [2.11]:
⎧ 𝐺 ∙ 𝑊 ∙ 𝑘 2 ∙ 𝐹𝑧
⎪=
2∙𝐻∙𝜇
2
� −1�
𝑎
|𝐹𝑥 |
1
∙ 𝑠𝑥 ; 𝑓𝑓𝑓
≤
𝐹𝑥
𝜇 ∙ 𝐹𝑧 2
=
𝜇∙𝐻
1
𝜇 ∙ 𝐹𝑧 ⎨
∙ ; 𝑒𝑒𝑒𝑒
2
⎪ =1−
� −1� 𝑠𝑥
2
𝑎
⎊
2 ∙ 𝐺 ∙ 𝑊 ∙ 𝑘 ∙ 𝐹𝑧
[2.21]
An equivalent reasoning for the lateral tyre force can be done starting from Equation [2.16].
In Figure 2-25, the tyre characteristics are plotted for a=2 and a=3 and varying vertical force. We see
that if 𝑎 > 2, the degressive characteristics, as in Figure 2-23 and Figure 2-24, 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.
60
VEHICLE INTERACTIONS
1
0.9
0.8
0.7
a=2;
a=2;
a=3;
a=3;
𝑭Fx/(mu*Fz)
𝒙 ⁄ 𝝁 ∙ 𝑭𝒛
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 to𝑭𝒙 ⁄ 𝝁 ∙ 𝑭𝒛 =
𝟎. 𝟓
0.2
0.1
0
0
0.05
0.1
0.15
𝒔𝒙
sx
0.2
0.25
0.3
0.35
Figure 2-25: 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 common, but not completely physical, way to model the
relaxation phenomena is to add a first order delay of the force:
𝐹𝑦 = 𝑓�𝑠𝑦,𝑑𝑑𝑑𝑑𝑑𝑑𝑑 � =according to the stead state model, e.g. Equation [2.11]
𝑠̇𝑦,𝑑𝑑𝑑𝑑𝑑𝑑𝑑 = 𝐴 ∙ �𝑠𝑦 − 𝑠𝑦,𝑑𝑑𝑑𝑑𝑑𝑑𝑑 �
𝑣
𝑥
𝐴 = 𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅ℎ
or 𝐴 =
𝑅∙𝜔
𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅ℎ
[2.22]
RelaxationLength="fraction" (~25-50%) of tyre circumference
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 2-19. 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.
2.5.5.2
Tyre self-aligning moment
(This section has large connection with Section 2.10.3.2 Steering system forces.)
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VEHICLE INTERACTIONS
In the lowest diagram in Figure 2-20, 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 2-19 b) it acts at a position 𝑡𝑝 behind the
wheel’s y axis. This distance is referred to as the pneumatic trail, see Figure 2-36, 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 2-8.
steering torque peaks
at lower slip than
lateral tyre force.
lateral tyre force = Fy (for a tyre with peak)
𝑐
total trail = caster trail + pneumatic trail = 𝑐 + 𝑡𝑝
𝐹𝑦 ∙ 𝑐
steering torque ∝ 𝐹𝑦 ∙ 𝑐 + 𝑡𝑝
side slip angle
Figure 2-26: General Response of Steering torque to Side slip angle. Tyre self-aligning
moment = 𝐹𝑦 ∙ 𝑡𝑝 is one part of the steering moment.
Figure 2-26 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 2-27. 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= 𝐹𝑦𝑦 = 𝐶𝛾 ∙ 𝛾. The camber stiffness, 𝐶𝛾 , is typically 5-10 % of the cornering stiffness and
opposite sign.
The camber force is superimposable for small lateral slip and small camber angle. The approximative
Eq [2.18], can then be developed to:
𝐹𝑦 = 𝐹𝑦,𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆 + 𝐹𝑦,𝐶𝐶𝐶𝐶𝐶𝐶 = 𝐶𝑦 ∙ 𝑠𝑦 + 𝐶𝛾 ∙ 𝛾;
62
or
𝐹𝑦 = 𝐶𝛼 ∙ 𝛼 + 𝐶𝛾 ∙ 𝛾
[2.23]
VEHICLE INTERACTIONS
𝛾
View from rear
or front:
𝐹𝑦𝑦
View from above:
inlet edge
𝐹𝑦𝑦
outlet edge
𝜔
𝑣𝑥
A point on tyre surface travels basically along the
edge of an ellipse, when the tyre has a camber
angle…
𝐹𝑦𝑦
…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.
Figure 2-27: Camber thrust
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 ∙ 𝐹𝑧 .
𝐹
2
𝐹
2
𝐹 2 = 𝐹𝑥 2 + 𝐹𝑦 2 ≤ (𝜇 ∙ 𝐹𝑧 )2 ⇒ � 𝑥 � + � 𝑦 � ≤ 𝜇2
𝐹
𝐹
𝑧
𝑧
[2.24]
Equation [2.24] 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”.
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.
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VEHICLE INTERACTIONS
1
maximum turning left without
propulsion and braking
Fy/(mu*Fz)
0.5
partial propulsion
0
max propulsion
-0.5
-1
-1
-0.5
0
Fx/(mu*Fz)
0.5
1
Figure 2-28: 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:
𝑣𝑦
𝐹𝑦
𝑙𝑙𝑙𝑙𝑙𝑙𝑙 𝑠𝑠𝑠𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠𝑠
=
=−
𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙 𝑠𝑠𝑠𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠𝑠 𝑅 ∙ 𝜔 − 𝑣𝑥
𝐹𝑥
[2.25]
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 2-29 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).
64
VEHICLE INTERACTIONS
Figure 2-29: Combined Longitudinal and Lateral Slip
Figure 2-30: Effects of combined longitudinal and lateral utilization of friction.
A simple but consistent combined slip model can be expressed using “effective slip, s”, as shown in
Equation [2.26]. 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.
65
VEHICLE INTERACTIONS
− sy

+ sx
⋅ F and Fy =
⋅F
 Fx =
s
s

s 2 = s x 2 + s y 2

Vy
R ⋅ ω − Vx

s
s
and
=
=
x
y

Vref
Vref


Vref = R ⋅ ω , Vref = Vx or Vref = R ⋅ ω + Vx / 2

 F = m ⋅ Fz ⋅ f (s )

p 
2

 f (s ) = {for example} = ⋅ arctan C0 ⋅ ⋅ s 
p
2 


C0 = {typically} = 20 [non − dimensional]


[2.26]
A simple combined model for cases when one knows 𝐹𝑥 without involving 𝑠𝑥 is shown in Equation
[2.27]. This is not completely physically motivated, but works relatively well. One can consider it as a
mathematical scaling inspired by the friction circle.
2.6.1.1
Relaxation
𝐹𝑥 2
ďż˝
𝐹𝑦 ≈ 1 − �
� ∙ 𝐹𝑦0 ; 𝑤ℎ𝑒𝑒𝑒 𝐹𝑦0 = 𝐹𝑦 �
𝑠𝑥 =0
𝜇 ∙ 𝐹𝑧
[2.27]
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 common, but not completely physical, way to model the
relaxation phenomena is to add a first order delay of the force:
�𝐹𝑥 ; 𝐹𝑦 � = 𝑓�𝑠𝑥,𝑑𝑑𝑑𝑑𝑑𝑑𝑑 ; 𝑠𝑦,𝑑𝑑𝑑𝑑𝑑𝑑𝑑 � =according to the stead state model
�𝑠̇𝑥,𝑑𝑑𝑑𝑑𝑑𝑑𝑑 ; 𝑠̇𝑦,𝑑𝑑𝑑𝑑𝑑𝑑𝑑 � = �𝐴𝑥 ∙ �𝑠𝑥 − 𝑠𝑥,𝑑𝑑𝑑𝑑𝑑𝑑𝑑 �; 𝐴𝑦 ∙ �𝑠𝑦 − 𝑠𝑦,𝑑𝑑𝑑𝑑𝑑𝑑𝑑 ��
𝑣
𝑥
𝐴𝑖 = 𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅ℎ
or 𝐴𝑖 =
𝑅∙𝜔
𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅ℎ
𝑤ℎ𝑒𝑒𝑒 𝑖 = 𝑥 𝑜𝑜 𝑖 = 𝑦
[2.28]
RelaxationLength="fraction" (~25-50%) of tyre circumference. 𝐴𝑥 can be different
from 𝐴𝑦 .
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 2-31.
66
VEHICLE INTERACTIONS
Figure 2-31: Vertical properties of a truck tyre.
Figure 2-32: Different tyre models which will filter road irregularities differently. Picture
from Peter Zegelaar, Ford Aachen.
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
𝑊𝑊𝑊𝑊𝑊𝑊𝑊𝑊 [in mass/s or mm tread depth/s], for one certain tyre at constant temperature, becomes
as follows:
67
VEHICLE INTERACTIONS
𝑊𝑊𝑊𝑊𝑊𝑊𝑊𝑊 ∝ 𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹 ∙ 𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆 ⇒
⇒ 𝑊𝑊𝑊𝑊𝑊𝑊𝑊𝑊 = 𝑘 ∙ 𝐹 ∙ ∆𝑣 ≈ 𝑘 ∙ (𝐶 ∙ 𝑠) ∙ �𝑠 ∙ 𝑣𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇 � ≈ 𝑘 ∙ 𝐶 ∙ 𝑠 2 ∙ 𝑣𝑥 ;
where 𝐹 = �𝐹𝑥 2 + 𝐹𝑦 2 ; 𝑠 = �𝑠𝑥 2 + 𝑠𝑦 2 ; 𝑣𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇 defined as in Eq [2.8] and
[2.29]
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 𝐹𝑥,𝑎𝑎𝑎 , which is proportional to the square of the longitudinal component of the
wind speed relative to the vehicle, 𝑣𝑥,𝑟𝑟𝑟 . For aerodynamic loads resisting forward motion of the
vehicle, the Equation [2.30] can be used. The parameters 𝑐𝑑 , 𝜌 𝑎𝑎𝑎 𝐴𝑓𝑓𝑓𝑓𝑓 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.
1
𝐹𝑥,𝑎𝑎𝑎 = − 2 ∙ 𝑐𝑑 ∙ 𝜌 ∙ 𝐴𝑓𝑓𝑓𝑓𝑓 ∙ 𝑣𝑥,𝑟𝑟𝑟 2 ;
[2.30]
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 it
dominates at highway speeds. For a normal sized and shaped modern sedan, 𝑐𝑑 ∙ 𝜌 ∙ 𝐴𝑓𝑓𝑓𝑓𝑓
Since a car structure moving through the air is not unlike an aircraft wing, there are other
aerodynamics loads that arise. 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.
1
𝐹𝑧,𝑎𝑎𝑎 = 2 ∙ 𝑐𝑙 ∙ 𝜌 ∙ 𝐴𝑓𝑓𝑓𝑓𝑓 ∙ 𝑣𝑥,𝑟𝑟𝑟 2 ;
1
𝑀𝑦,𝑎𝑎𝑎 = 2 ∙ 𝑐𝑝𝑝 ∙ 𝜌 ∙ 𝐴𝑓𝑓𝑓𝑓𝑓 ∙ 𝐿𝑐 ∙ 𝑣𝑥,𝑟𝑟𝑟 2 ;
[2.31]
The coefficient 𝑐𝑙 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,
𝐹𝑧,𝑎𝑎𝑎 , and the pitch moment, 𝑀𝑦,𝑎𝑎𝑎 , with lift forces at two different longitudinal positions, typically
over each axle:
68
VEHICLE INTERACTIONS
1
𝐹𝑎𝑎𝑎,𝑓𝑓 = 2 ∙ 𝑐𝑙𝑓 ∙ 𝜌 ∙ 𝐴𝑓𝑓𝑓𝑓𝑓 ∙ 𝑣𝑥,𝑟𝑟𝑟 2 ;
1
𝐹𝑎𝑎𝑎,𝑟𝑟 = 2 ∙ 𝑐𝑙𝑙 ∙ 𝜌 ∙ 𝐴𝑓𝑓𝑓𝑓𝑓 ∙ 𝑣𝑥,𝑟𝑟𝑟 2 ;
[2.32]
The coefficients 𝑐𝑙𝑙 and 𝑐𝑙𝑙 are lift coefficient over front and rear axle, respectively. Relations
between Equation [2.31] and Equation [2.32] are: 𝑐𝑙 = 𝑐𝑙𝑙 + 𝑐𝑙𝑙 ; and 𝑐𝑝𝑝 = �𝑐𝑙𝑙 − 𝑐𝑙𝑙 � ∙ 𝐿𝑐 ;.
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.
1
2
𝐹𝑦,𝑎𝑎𝑎 = ∙ 𝑐𝑠 ∙ 𝜌 ∙ 𝐴 ∙ 𝑣𝑦,𝑟𝑟𝑟 2
𝑀𝑧,𝑎𝑎𝑎 =
1
∙ 𝑐 ∙ 𝜌 ∙ 𝐴 ∙ 𝐿𝑐 ∙ 𝑣𝑦,𝑟𝑟𝑟 2
2 𝑦𝑦
[2.33]
[2.34]
The speed 𝑣𝑦,𝑟𝑟𝑟 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 [2.30]-[2.32]. The aerodynamic
coefficients, cs and cym, can typically be determined from wind tunnel testing or CFD calculations.
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 2-33, 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 2-34 and
Figure 2-1.
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 2-33: Commonly assumed causality (direction of cause and effect).
69
VEHICLE INTERACTIONS
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 2-34. 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
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.
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VEHICLE INTERACTIONS
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 2-34: 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.
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 DYNAMICS.
2.10.3 Steering
The driver interacts with steering wheel through two channels, steering wheel angle and steering
wheel torque.
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VEHICLE INTERACTIONS
A frequently used model is that the driver decides a steering wheel angle and expects a certain
steering wheel torque, 𝑇𝑠𝑠 , 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.
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.
2.10.3.1 Steering system 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.
2.10.3.2 Steering system forces
(This section has large connection with Section 2.5.5.2 Tyre self-aligning moment.)
The steering wheel torque, 𝑇𝑠𝑠 , 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
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 𝑇𝑠𝑠 (highway driving) than large 𝑇𝑠𝑠 (parking), see Figure 2-35. Here, the variation in
assistance is assumed to be hydraulic and follows a so called boost curve. At 𝑇𝑠𝑠 = 0, the assistance
is ≈0.45/0.55≈1 and for 𝑇𝑠𝑠 = 4 Nm, it is ≈0.9/0.1≈10.
Figure 2-35: 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 2-36. 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 “kingpin axis” (or steering axis) around which the wheel is rotated when
it is steered. The kingpin axis intersects with the ground plane at a point which normally has an offset
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VEHICLE INTERACTIONS
from the contact patch centre, both longitudinally and laterally. The offsets are called caster trail and
scrub radius, respectively, see Figure 2-36. This figure also defines kingpin offset. 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 offset at wheel
centre in side view, is called Caster offset.
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 caster trail 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 kingpin axis, turning towards increasing steering angle) is affected by tyre
forces on a steered axle as in Equation [2.35] .
Caster Trail 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 Caster Angle which gives a
beneficial higher Camber Angle gain at cornering. On a front wheel driven passenger vehicle a nonzero caster offset is not chosen due to drive axle lateral displacement.
− (Tb1 + Ts1 )
− Ts1
⋅ s ⋅ cos(KPI ) +
⋅ R ⋅ sin (KPI ) +
R
R
(T + Ts 2 ) ⋅ s ⋅ cos(KPI ) + Ts 2 ⋅ R ⋅ sin (KPI ) +
+ b2
R
R
+ (− 1) ⋅ (Fy1 ⋅ (c + t1 ) + Fy 2 ⋅ (c + t2 )) ⋅ cos(CA) =
Tsteer =
=
[2.35]
Tb 2 − Tb1
T − Ts1
⋅ s ⋅ cos(KPI ) + s 2
⋅ k − (Fy1 ⋅ (c + t1 ) + Fy 2 ⋅ (c + t2 )) ⋅ cos(CA)
R
R
(Notations in Equation [2.35] are defined in Figure 2-36.)
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 2-37 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 2-37 shows how KPI and scrub radius causes the vehicle body to
lift a distance 𝑠 ′′′ = 𝑠 ∙ cos(𝐾𝐾𝐼) ∙ (1 − cos(|𝜑𝑠𝑠𝑠𝑠𝑠 |)) ∙ sin(𝐾𝐾𝐾). This will require a work
𝑇𝑠𝑠𝑠𝑠𝑠 ∙ 𝜑𝑠𝑠𝑠𝑠𝑠 = 𝐹𝑖𝑖 ∙ 𝑠 ′′′ . This leads to an 𝑇𝑠𝑠𝑠𝑠𝑠 (additional to Eq [2.35]) as follows:
(𝑎𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑) 𝑇𝑠𝑠𝑠𝑠𝑠 =
=
ďż˝
𝑖=𝑙𝑙𝑙𝑙 𝑎𝑎𝑎 𝑟𝑟𝑟ℎ𝑡
𝐹𝑖𝑖 ∙ 𝑠 ′′′
=
𝜑𝑠𝑠𝑠𝑠𝑠
𝑠 ∙ cos(𝐾𝐾𝐾) ∙ (1 − cos(|𝜑𝑠𝑠𝑠𝑠𝑠 |)) ∙ sin(𝐾𝐾𝐾)
∙ 𝐹𝑎𝑎𝑎𝑎,𝑧 ;
𝜑𝑠𝑠𝑠𝑠𝑠
73
[2.36]
VEHICLE INTERACTIONS
kingpin inclination, KPI
View from
rear:
View from
right:
Caster
angle, CA
Wheel 1 with
(friction)
brake
torque, Tb:
brake
Tb
(Tb<0 if R>0)
R
bearing
forward
Fx1=Tb/R
Caster offset
(drawn positive)
Wheel 1 with
shaft torque, Ts:
Fx1
Scrub radius, s
(drawn positive)
R
Caster trail, c
(drawn positive)
Ts
Fy1
Pneumatic trail, t
(positive if “behind”)
Fy1 = resulting
lateral force
lateral force
distribution
Ts*sin(KPI)
Fx1=Ts/R
View from
rear:
𝐹𝑖𝑖
𝑇𝑠𝑡𝑡𝑡𝑡
𝜑𝑠𝑡𝑡𝑡𝑡
Contact point
tyre/ground
if 𝝋𝒔 = 𝟎
Contact point
for a certain 𝝋𝒔
s
KPI
𝑠 ′′′ = 𝑠 ′′ ∙ sin 𝐾𝐾𝐾
Figure 2-36: Caster trail, scrub radius and kingpin offset.
Fx1
Scrub radius, s
Figure 2-37: Lift effect due to steering angle and positive scrub radius.
It should be noted that Eq [2.36] is not complete with respect to all “returning effects”. There are
also effects from caster 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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VEHICLE INTERACTIONS
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 2-34. 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
76
LONGITUDINAL DYNAMICS
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
77
LONGITUDINAL DYNAMICS
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.
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,R
Transmission
mover (1)
P(1)
Wheels
T,R
(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, R
Prime mover
(ICE, internal
combustion
engine)
(1)
T, R
(on one
driven axle)
F, v
Wheels
Transmission (Clutch &
Body
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
78
LONGITUDINAL DYNAMICS
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.
T
Power TO Power FωOM
engine engine
T
ω
Power FωOM Power TO
engine engine
ω
Electric Motor
ICE
T
T
ω
ω
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]
Gasoline
6000
At wide open
throttle = max
torque
20
1500
2000 2500
SPEED [rpm]
3000
Diesel
Figure 3-3: Engine Characteristics for Gasoline and Diesel Engines. Typical values for
passenger car.
79
LONGITUDINAL DYNAMICS
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
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:
𝑇𝑜𝑜𝑜 =𝑟 ∙ 𝑇𝑖𝑖 ∙ 𝜂𝐶𝐶𝐶𝐶𝐶𝐶𝐶ℎ𝑒𝑒 − Δ𝑇𝐵𝐵𝐵𝐵𝐵𝐵𝐵𝐵 ∙ sign(𝜔𝑜𝑜𝑜 ) ;
𝜔𝑖𝑖
;
𝑟
𝑟 = 𝑟1 , 𝑟2 , ⋯ 𝑟𝑁 ; 𝑟 ≠ 0;
𝜔𝑜𝑜𝑜 =
[3.1]
Eq [3.1] is not valid for neutral gear (then there is no speed equation, but instead two torque
equations: 𝑇𝑜𝑜𝑜 = − Δ𝑇𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜 ∙ sign(𝜔𝑜𝑜𝑜 ) ; and 𝑇𝑖𝑖 =Δ𝑇𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 ∙ sign(𝜔𝑖𝑖 ) ;.
For any 1-dimensional transmission of rotational mechanical power between two rotating shafts, the
total efficiency, 𝜂𝑡𝑡𝑡𝑡𝑡 = 𝑃𝑜𝑜𝑜 ⁄𝑃𝑖𝑖 = 𝑇𝑜𝑜𝑜 ∙ 𝜔𝑜𝑜𝑜 ⁄(𝑇𝑖𝑖 ∙ 𝜔𝑖𝑖 ) ;, is depending on operating condition. If
assuming a nominal ratio, 𝑟, the total efficiency can be decomposed in 𝜂𝑡𝑡𝑡𝑡𝑡 = 𝜂 𝑇 ∙ 𝜂𝜔 ;, where
𝜂 𝑇 = 𝑇𝑜𝑜𝑜 ⁄(𝑟 ∙ 𝑇𝑖𝑖 ) ; and 𝜂𝜔 = 𝜔𝑜𝑜𝑜 ⁄(𝜔𝑖𝑖 ⁄𝑟) ;.
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:
𝜔𝑖𝑖 = 𝜔𝑙𝑙𝑙𝑙 ;
𝜔𝑖𝑖 = 𝜔𝑟𝑟𝑟ℎ𝑡 ;
𝑇𝑖𝑖 = 𝑇𝑙𝑙𝑙𝑙 + 𝑇𝑟𝑟𝑟ℎ𝑡 ;
[3.2]
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:
𝜔𝑙𝑙𝑙𝑙 + 𝜔𝑟𝑟𝑟ℎ𝑡
;
𝜔𝑖𝑖 =
2
𝑇𝑙𝑙𝑙𝑙 = 𝑇𝑟𝑟𝑟ℎ𝑡 ;
𝑇𝑖𝑖 = 𝑇𝑙𝑙𝑙𝑙 + 𝑇𝑟𝑟𝑟ℎ𝑡 ;
[3.3]
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.
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LONGITUDINAL DYNAMICS
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.
Part of the distribution transmissions are also shafts. If oscillations is to be studied, these has to be
modelled as energy storing compliances: 𝑇̇ = 𝑐 ∙ (𝜔𝑖𝑖 − 𝜔𝑜𝑜𝑜 ).
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:
= (𝑃𝑖𝑖 − 𝑃𝑜𝑜𝑜 ) ∙ 𝜂𝑐ℎ𝑎𝑎𝑎𝑎 ; 𝑓𝑓𝑓 𝑃𝑖𝑖 > 𝑃𝑜𝑜𝑜 ;
𝐸̇ = �
= (𝑃𝑖𝑖 − 𝑃𝑜𝑜𝑜 )/𝜂𝑢𝑢𝑢 ; 𝑒𝑒𝑒𝑒;
[3.4]
𝑤ℎ𝑒𝑒𝑒 𝑃𝑖𝑖 = 𝑇𝑖𝑖 ∙ 𝜔𝑖𝑖 ; 𝑎𝑎𝑎 𝑃𝑜𝑜𝑜 = 𝑇𝑜𝑜𝑜 ∙ 𝜔𝑜𝑜𝑜 ;
Including how the buffer is connected, one more equation can be found: Typically 𝜔𝑖𝑖 = 𝜔𝑜𝑜𝑜 ; or
𝑇𝑖𝑖 = 𝑇𝑜𝑜𝑜 ;.
The model uses stored energy, 𝐸. For batteries one often uses state of charge, 𝑆𝑆𝑆, instead.
Conceptually, 𝑆𝑆𝑆 = 𝐸/𝐸𝑚𝑚𝑚 ;, where 𝐸𝑚𝑚𝑚 is a nominal maximum charge level.
A first approximation of the efficiencies, can be 𝜂𝑐ℎ𝑎𝑎𝑎𝑎 = 𝜂𝑢𝑢𝑢 = 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 < 1, but typically the
efficiency is dependent of many things, such as 𝑃𝑖𝑖 − 𝑃𝑜𝑜𝑜 . The model above does not consider any
leakage when buffer is “resting”: 𝑃𝑖𝑖 = 𝑃𝑜𝑜𝑜 .
3.2.2
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:
𝐹 = 𝑟𝑟𝑟𝑟𝑟 ∙
𝑇
;
𝑊ℎ𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒
𝑎𝑎𝑎 𝑣 = 𝑊ℎ𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒 ∙
𝜔
;
𝑟𝑟𝑟𝑟𝑟
[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
𝑇
; 𝑤ℎ𝑒𝑒𝑒 𝜂 𝑇 ≤ 1;
; 𝑤𝑤𝑤ℎ 𝐹 = 𝜂 𝑇 ∙ 𝑟𝑟𝑟𝑟𝑟 ∙
𝑟𝑟𝑟𝑟𝑟𝑟
𝑟𝑟𝑟𝑟𝑟𝑟
𝜔
𝜔
; 𝑤𝑤𝑤ℎ 𝑣 = 𝜂𝜔 ∙ 𝑟𝑟𝑟𝑟𝑟𝑟 ∙
𝑅𝑅𝑅𝑅𝑅𝑅𝑅 𝑣 = 𝑟𝑟𝑟𝑟𝑟𝑟 ∙
; 𝑤ℎ𝑒𝑒𝑒 𝜂𝜔 ≤ 1;
𝑟𝑟𝑟𝑟𝑟
𝑟𝑟𝑟𝑟𝑟
𝑃𝑣𝑣ℎ𝑖𝑖𝑖𝑖 𝐹 ∙ 𝑣
𝑤ℎ𝑒𝑒𝑒 𝜂𝑡𝑡𝑡𝑡𝑡 =
=
= 𝜂 𝑇 ∙ 𝜂𝜔 ≤ 1
𝑃𝑒𝑒𝑒𝑒𝑒𝑒 𝑇 ∙ 𝜔
𝑅𝑅𝑅𝑅𝑅𝑅𝑅 𝐹 = 𝑟𝑟𝑟𝑟𝑟 ∙
This will basically move the curves in the first quadrant slightly downwards and to the left.
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[3.6]
LONGITUDINAL DYNAMICS
driven wheel
prime mover
𝑻
𝝎
𝑻
prime mover
characteristics:
𝑻
𝝎
𝒗=
𝝎⁄𝒓𝒓𝒓𝒓𝒓
transmission
𝒓𝒓𝒓𝒓𝒓 ∙ 𝑻
𝒓𝒓𝒓𝒓𝒓
∙𝑻
𝒓𝒓𝒓𝒓𝒓𝒓
𝒓𝒓𝒓𝒓𝒓𝒓
∙𝝎
𝒓𝒓𝒓𝒓𝒓
𝒗
𝑭
𝝎⁄𝒓𝒓𝒓𝒓𝒓
(vehicle) body
Traction Diagram
Multiply torque by
𝝎
𝑭=
𝒓𝒓𝒓𝒓𝒓 ∙ 𝑻
𝒓𝒓𝒓𝒓𝒓
𝒓𝒓𝒓𝒓𝒓𝒓
Multiply rotational speed by
𝒓𝒓𝒓𝒓𝒓𝒓
𝒓𝒓𝒓𝒓𝒓
𝑭
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
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LONGITUDINAL DYNAMICS
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
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 [2.30]. Notation f means rolling resistance coefficient.
2
1
2
𝐹𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟 = 𝑚 ∙ 𝑔 ∙ sin(𝜑) + 𝑓 ∙ 𝑚 ∙ 𝑔 ∙ cos(𝜑) + ∙ 𝑐𝑑 ∙ 𝜌 ∙ 𝐴 ∙ �𝑣𝑥 − 𝑣𝑤𝑤𝑤𝑤,𝑥 �
[3.7]
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.
F
v
air
available accele
(approximate!)
(to
air drag = (r*A*cdC2) ⋅v2
roll
ϕ
rolling resisPance =
=almosP consP.= f*m*g*cosϕ
F m*g
Pop speed
v
climbing resisPance
m*g*sinϕ
Figure 3-6: Top speed can be found from Traction diagram with Driving Resistance curves.
Head wind speed, 𝑣𝑤𝑤𝑤𝑤,𝑥 , is assumed to be zero.
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 DYNAMICS
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 DYNAMICS
m*g
-ϕ
Figure 3-8: Free Body Diagram for steady state vehicle. With ISO coordinate system, the
road gradient is negative when downhill.
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:
𝐹𝑧𝑧 ∙ �𝑙𝑓 + 𝑙𝑟 � − 𝑚 ∙ 𝑔 ∙ (𝑙𝑟 ∙ cos(−𝜑) − ℎ ∙ sin(−𝜑)) + 𝐹𝑎𝑎𝑎 ∙ ℎ𝑎𝑎𝑎 = 0 ⇒
⇒ 𝐹𝑧𝑧 = 𝑚 ∙ 𝑔 ∙
𝑙𝑟 ∙ cos(𝜑) + ℎ ∙ sin(𝜑)
ℎ𝑎𝑎𝑎
− 𝐹𝑎𝑎𝑎 ∙
𝑙𝑓 + 𝑙𝑟
𝑙𝑓 + 𝑙𝑟
⇒ 𝐹𝑧𝑧 = 𝑚 ∙ 𝑔 ∙
𝑙𝑓 ∙ cos(𝜑) − ℎ ∙ sin(𝜑)
ℎ𝑎𝑎𝑎
+ 𝐹𝑎𝑎𝑎 ∙
𝑙𝑓 + 𝑙𝑟
𝑙𝑓 + 𝑙𝑟
Moment equilibrium, around front contact with ground:
−𝐹𝑧𝑧 ∙ �𝑙𝑓 + 𝑙𝑟 � + 𝑚 ∙ 𝑔 ∙ �𝑙𝑓 ∙ cos(−𝜑) + ℎ ∙ sin(−𝜑)� + 𝐹𝑎𝑎𝑎 ∙ ℎ𝑎𝑎𝑎 = 0 ⇒
[3.8]
For most vehicles and reasonable gradients one can neglect ℎ ∙ sin(𝜑) since it is ≪ �𝑙𝑓 ∙ cos(𝜑)� ≈
|𝑙𝑟 ∙ cos(𝜑)|.
3.2.7
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,
𝐹𝑧,𝑑𝑑𝑑𝑑𝑑𝑑 , that is the limiting factor:
𝐹𝑥 = min�𝐹𝑥,𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 ; 𝜇 ∙ 𝐹𝑧,𝑑𝑑𝑑𝑑𝑑𝑑 �
[3.9]
One realises, from Figure 2-9, that the rolling resistance on the driven axle works as a torque loss and
that the road friction limitation will be limiting 𝑇𝑒𝑒𝑒 ∙ 𝑟𝑟𝑟𝑟𝑟 − 𝑒 ∙ 𝐹𝑧,𝑑𝑑𝑑𝑑𝑑𝑑 rather than limiting
𝑇𝑒𝑛𝑛 ∙ 𝑟𝑟𝑟𝑟𝑟. Expressed using the rolling resistance coefficient, 𝑓𝑟𝑟𝑟𝑟 , gives:
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LONGITUDINAL DYNAMICS
𝐹𝑥 = min�𝐹𝑥,𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 ; 𝜇 ∙ 𝐹𝑧,𝑑𝑑𝑑𝑑𝑑𝑑 � =
𝑇𝑒𝑒𝑒 ∙ 𝑟𝑟𝑟𝑟𝑟
= min ďż˝
− 𝑓𝑟𝑟𝑟𝑟 ∙ 𝐹𝑧,𝑑𝑑𝑑𝑑𝑑𝑑 ; 𝜇 ∙ 𝐹𝑧,𝑑𝑑𝑑𝑑𝑑𝑑 �
𝑟𝑟𝑟𝑟𝑟𝑟
[3.10]
Propulsion
force, Fx
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 2-9.
Maximum
available 𝐹𝑥
𝑇𝑒𝑒𝑒 ∙ 𝑟𝑟𝑟𝑟𝑟 − 𝑒 ∙ 𝐹𝑧,𝑑𝑑𝑑𝑑𝑑𝑑
𝐹𝑥,𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 =
=
𝑟𝑟𝑟𝑟𝑟𝑟
𝑇𝑒𝑒𝑒 ∙ 𝑟𝑟𝑟𝑟𝑟
=
− 𝑓𝑟𝑟𝑟𝑟 ∙ 𝐹𝑧,𝑑𝑑𝑑𝑑𝑑𝑑
𝑟𝑟𝑟𝑟𝑟𝑟
Friction limit: 𝜇 ∙ 𝐹𝑧,𝑑𝑑𝑑𝑑𝑑𝑑
𝐹𝑟𝑟𝑟
0
𝐹𝑟𝑟𝑟,𝑎𝑎𝑎 = 0.5 ∙ 𝜌 ∙ 𝑐𝑑 ∙ 𝐴 ∙ 𝑣𝑥 2
𝐹𝑟𝑟𝑟,𝑟𝑟𝑟𝑟 = 𝑓𝑟𝑟𝑟𝑟 ∙ 𝐹𝑧,𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛
𝐹𝑟𝑟𝑟,𝑠𝑠𝑠𝑠𝑠 = 𝑚 ∙ 𝑔 ∙ sin −𝜑
0
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 slip 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 ∙ 𝜔𝑐𝑐𝑐𝑐𝑐ℎ and 𝑇𝑤ℎ𝑒𝑒𝑒 ∙ 𝜔𝑤ℎ𝑒𝑒𝑒 .
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LONGITUDINAL DYNAMICS
Friction limit
𝐹𝑟𝑟𝑟 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.
𝐹𝑟𝑟𝑟 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
(𝑔𝑔𝑔𝑔𝑒1 ) is the gradeability for and speed 𝑣2
Road slope of this curve
(𝑔𝑔𝑔𝑔𝑒2 ) is the gradeability for and speed 𝑣1
0
𝑣2
𝑣1
𝐹𝑟𝑟𝑟 for
varying
road grade
Speed, 𝑣
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 DYNAMICS
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, 𝑃𝑥 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
−ϕ
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.
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LONGITUDINAL DYNAMICS
80 km/h
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/ftptech.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 DYNAMICS
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 DYNAMICS
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/ldhwy/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:
𝑚 ∙ 𝑣̇ = 𝐹;
𝐽 ∙ 𝜔̇ = 𝑇 − 𝐹 ∙ 𝑅;
𝑣 = 𝑅 ∙ 𝜔 ⇒ 𝑣̇ = 𝑅 ∙ 𝜔̇ ;
T
ω
vx
m,J
F
Figure 3-17: Rolling wheel
91
[3.11]
LONGITUDINAL DYNAMICS
Assume that we know torque T and want to compare this with the situation without rotational
inertia:
𝑚 ∙ 𝑣̇ =
𝑇
;
𝑅
𝑎𝑎𝑎 𝐹 =
𝑇
;
𝑅
[3.12]
Then we use the compatibility equation to eliminate 𝜔 in Equation [3.11], which then becomes:
𝑇
1 𝑇
; 𝑎𝑎𝑎 𝐹 = ∙ ;
𝑅
𝑘 𝑅
𝑚 + 𝐽⁄𝑅 2 𝑚 + 𝐽𝑡𝑡𝑡𝑡𝑡𝑡
𝑤ℎ𝑒𝑒𝑒 𝑘 =
=
;
𝑚
𝑚
𝑘 ∙ 𝑚 ∙ 𝑣̇ =
[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: 𝐽𝑒
Rotating inertias after transmission, i.e. elements rotating with same speed as the wheel: 𝐽𝑤
The effective mass, 𝑘 ∙ 𝑚, will be dependent on the gear ratio as well:
𝑘∙𝑚 =𝑚+
𝐽𝑤 𝐽𝑒 ∙ 𝑟𝑟𝑟𝑟𝑟 2
+
;
𝑅2
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
𝐽𝑒 disappears from Equation [3.14]. If the wheel spins, both rotational terms disappear.
Propulsion
force, Fx
We can now learn how to determine acceleration from the Traction Diagram, see Figure 3-18.
Gear 1
𝑘3 ∙ 𝑚 ∙ 𝑎𝑥
𝑘4 ∙ 𝑚 ∙ 𝑎𝑥
𝑘1 ∙ 𝑚 ∙ 𝑎𝑥
𝑘2 ∙ 𝑚 ∙ 𝑎𝑥
Friction limit
Resistance
Gear 2
Speed, vx
• 𝑘1 = rotating in•rtia co•ffici•nt on g•ar 1,
without •ngin• int•rtia and without
wh••l in•rtia, sinc• slipping tyr•, i.•.
𝑘1 =1
• 𝑘2 = rotating in•rtia co•ffici•nt on g•ar 1,
• 𝑘3 = rotating in•rtia co•ffici•nt on g•ar 2,
without •ngin• int•rtia, sinc• slipping
clutch
• 𝑘4 = rotating in•rtia co•ffici•nt on g•ar 2
Figure 3-18: Acceleration in Traction Diagram, assuming the engine is run on its maximum
curve and the highest acceleration gear is selected.
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LONGITUDINAL DYNAMICS
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.
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 ∙ 𝑚 ∙ 𝑔
Gear 1
Propulsion friction limit,
𝜇∙𝑚∙𝑔
𝜇 ∙ 𝐹𝑧,𝑑𝑑𝑑𝑑𝑑𝑑 ≈
2
Gear 2
Speed, vx
Propulsion friction limit,
𝜇∙𝑚∙𝑔
− 𝜇 ∙ 𝐹𝑧,𝑑𝑑𝑑𝑑𝑑𝑑 ≈
2
Reverse gear
Braking friction limit, -𝜇 ∙ 𝑚 ∙ 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.
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LONGITUDINAL DYNAMICS
3.3.5
Fuel or energy consumption *
Function definition: Fuel (or Energy) Consumption is the amount of fuel [kg or liter] (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.
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
Figure 3-20: Left: Fuel consumption map. Curves show where specific fuel consumption
[g/(kW*h)] 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.
94
Propulsion
force, F
LONGITUDINAL DYNAMICS
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
dynamics” analysis. A summary of such algorithm for prediction of fuel or energy consumption is
given here:
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
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LONGITUDINAL DYNAMICS
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 dynamics”, i.e. where acceleration of inertial bodies is
prescribed and the required force is calculated.
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.
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 [2.29].
For a certain driving cycle we can integrate the 𝑊𝑊𝑊𝑊𝑊𝑊𝑊𝑊 [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, liter/km or
J/km], and multiplied with fuel tank size [kg or liter] 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,
96
LONGITUDINAL DYNAMICS
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
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
0
gear 1
…gear 1
…gear 2
Short term
Acceleration
Reserve
gear 2
0
Mass∙AccelerationReserve on…
0
Speed, v
gear 2
0
Speed, v
Figure 3-22: Acceleration reserves for different gears. Large dots mark assumed operating
points, each with its acceleration reserve shown.
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LONGITUDINAL DYNAMICS
3.3.10 Load Transfer due to acceleration,
without considering 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, 𝑎𝑥 .
m*g
−ϕ
Figure 3-23: Free Body Diagram for accelerating vehicle. (Rolling resistance is here drawn
as a longitudinal force, which is correct only on a non-driven axle. For a driven axle, it is
more accurate to see it as a torque on the wheel.)
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 𝑚 ∙ 𝑎𝑥 .
Moment equilibrium, around rear contact with ground:
𝐹𝑧𝑧 ∙ �𝑙𝑓 + 𝑙𝑟 � − 𝑚 ∙ 𝑔 ∙ (𝑙𝑟 ∙ cos(−𝜑) − h ∙ sin(−𝜑)) + 𝐹𝑎𝑎𝑎 ∙ ℎ𝑎𝑎𝑎 + 𝑚 ∙ 𝑎𝑥 ∙ ℎ = 0 ⇒
⇒ 𝐹𝑧𝑧 = 𝑚 ∙ �𝑔 ∙
𝑙𝑟 ∙ cos(𝜑) + h ∙ sin(𝜑)
ℎ
ℎ𝑎𝑎𝑎
− 𝑎𝑥 ∙
� − 𝐹𝑎𝑎𝑎 ∙
𝑙𝑓 + 𝑙𝑟
𝑙𝑓 + 𝑙𝑟
𝑙𝑓 + 𝑙𝑟
⇒ 𝐹𝑧𝑧 = 𝑚 ∙ �𝑔 ∙
𝑙𝑓 ∙ cos(𝜑) − h ∙ sin(𝜑)
ℎ
ℎ𝑎𝑎𝑎
+ 𝑎𝑥 ∙
� + 𝐹𝑎𝑎𝑎 ∙
𝑙𝑓 + 𝑙𝑟
𝑙𝑓 + 𝑙𝑟
𝑙𝑓 + 𝑙𝑟
Moment equilibrium, around front contact with ground:
−𝐹𝑧𝑧 ∙ �𝑙𝑓 + 𝑙𝑟 � + 𝑚 ∙ 𝑔 ∙ �𝑙𝑓 ∙ cos(−𝜑) + h ∙ sin(−𝜑)� + 𝐹𝑎𝑎𝑎 ∙ ℎ𝑎𝑎𝑎 + 𝑚 ∙ 𝑎𝑥 ∙ ℎ = 0 ⇒
[3.16]
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.
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LONGITUDINAL DYNAMICS
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.
Front wheels locked
Side forces on rolling
rear wheels tend to
return the vehicle to
straight-ahead motion.
Rear wheels locked
𝑣⃗
𝑣⃗
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 DYNAMICS
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;
[ax,gear_vec(i)]=max([ax0,ax1,ax2,ax3]);
vx_vec(i+1)=vx+ax*dt;
end
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 DYNAMICS
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
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]
25
35
30
40
Simulation result
14
12
10
8
vx [m/s]
gear (0=clutch slips on gear 2)
6
4
2
0
0
1
2
3
4
5
t [s]
6
7
8
9
10
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:
𝑚∙𝑎 =𝑚∙
𝑣
𝑑𝑑
= 𝐹(𝑣) − 𝐹𝑟𝑟𝑟 (𝑣); ⇒
𝑑𝑑
𝑡
𝑚 ∙ 𝑑𝑑
𝑚 ∙ 𝑑𝑑
⇒
= 𝑑𝑑 ⇒ �
= � 𝑑𝑑 ⇒
𝐹(𝑣) − 𝐹𝑟𝑟𝑟 (𝑣)
𝐹(𝑣) − 𝐹𝑟𝑟𝑟 (𝑣)
0
0
𝑣
𝑚 ∙ 𝑑𝑑
𝑡= �
;
𝐹(𝑣) − 𝐹𝑟𝑟𝑟 (𝑣)
[3.18]
0
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 𝐹(𝑣) and 𝐹𝑟𝑟𝑟 (𝑣) 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
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LONGITUDINAL DYNAMICS
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, 𝐹𝑥 , counter-acting the rotation of the wheel. 𝐹𝑥
is limited by the road friction: |𝐹𝑥 |𝑚𝑚𝑚 = 𝜇 ∙ 𝐹𝑧 ;. (NOTE: |𝐹𝑥 |𝑚𝑚𝑚 is not 𝜇 ∙ 𝐹𝑧 − 𝑓𝑟 ∙ 𝐹𝑧 , see Figure 2-9.)
Braking coefficient = −𝐹𝑥 ⁄𝐹𝑧 , is a property defined for an axle (or a single wheel) = brake
force/normal load. It can be seen as the utilized friction coefficient. A suitable notation can be 𝜇𝑢𝑢𝑢𝑢 .
The coefficient should not be mixed up with friction coeeficient, 𝜇; the relation between them is
𝜇𝑢𝑢𝑢𝑢 ≤ 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 = max
distribution of braking is optimal.
3.4.2.2
Braking Distance *
−𝑎𝑥⁄𝑔
;.
∀𝑎𝑎𝑎𝑎𝑎 𝜇𝑢𝑢𝑢𝑢
If Braking Efficiency=1=100%, the
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.
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).
102
LONGITUDINAL DYNAMICS
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.
103
LONGITUDINAL DYNAMICS
Driver pushes
on pedal
Front
Front
Front
HH
HH
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, vehicle speed and possibly gear on level ground without pressing the brake pedal. The
translation of pedal position to vehicle acceleration should be consistent, progressive and oscillation-free.
Function definition: Brake pedal response is how vehicle deceleration varies with brake pedal force,
vehicle speed on level ground without pressing the accelerator pedal. The translation of pedal force to vehicle
acceleration should be consistent, progressive and oscillation-free.
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.
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
104
LONGITUDINAL DYNAMICS
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.
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: 𝐹𝑓𝑓 /𝐹𝑓𝑓 = 𝐹𝑟𝑟 /𝐹𝑟𝑟 . This requirement gives the optimal 𝐹𝑓𝑓 and 𝐹𝑟𝑟 as:
𝑙𝑟
ℎ
𝐹𝑓𝑓
𝐹𝑓𝑓 𝐹𝑓𝑓 𝐹𝑓𝑓 𝑚 ∙ �𝑔 ∙ 𝑙𝑓 + 𝑙𝑟 − 𝑎𝑥 ∙ 𝑙𝑓 + 𝑙𝑟 � 𝑔 ∙ 𝑙𝑟 − 𝑎𝑥 ∙ ℎ
𝐹𝑟𝑟
=
⇒
=
⇒
=
=
;
𝑙𝑓
𝜇 ∙ 𝐹𝑓𝑓 𝜇 ∙ 𝐹𝑟𝑟
𝐹𝑟𝑟 𝐹𝑟𝑟
𝐹𝑟𝑟
𝑔 ∙ 𝑙𝑓 + 𝑎𝑥 ∙ ℎ
ℎ
𝑚 ∙ �𝑔 ∙
+ 𝑎𝑥 ∙
ďż˝
𝑙𝑓 + 𝑙𝑟
𝑙𝑓 + 𝑙𝑟
⇒ 𝐹𝑓𝑓 = 𝑚 ∙ 𝑎𝑥 ∙
𝐹𝑓𝑓 + 𝐹𝑟𝑟 = 𝑚 ∙ 𝑎𝑥 ;
𝑔 ∙ 𝑙𝑟 − 𝑎𝑥 ∙ ℎ
𝑔 ∙ �𝑙𝑓 + 𝑙𝑟 �
;
𝑎𝑎𝑎 𝐹𝑟𝑟 = 𝑚 ∙ 𝑎𝑥 ∙
𝑔 ∙ 𝑙𝑓 + 𝑎𝑥 ∙ ℎ
𝑔 ∙ �𝑙𝑓 + 𝑙𝑟 �
[3.19]
;
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
7000
6000
Fxr [N]
5000
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
10000
5000
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 or Electronic Brake Distribution (EBD). In principle, it bends down the straight line as
shown in Figure 3-27. EBD is the design used in today’s passenger cars, since it comes with ABS,
which is now a legal requirement on most markets.
105
LONGITUDINAL DYNAMICS
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. zfr=zrr=0.
𝐹𝑧𝑧 = 𝐹𝑧𝑧0 + 𝑐𝑓 ∙ �𝑧𝑓𝑓 − 𝑧𝑓 � 𝑎𝑎𝑎 𝐹𝑧𝑧 = 𝐹𝑧𝑧0 + 𝑐𝑟 ∙ (𝑧𝑟𝑟 − 𝑧𝑟 )
𝑤ℎ𝑒𝑒𝑒 𝐹𝑧𝑧0 + 𝐹𝑧𝑧0 = 𝑚 ∙ 𝑔 𝑎𝑎𝑎 𝐹𝑧𝑧0 ∙ 𝑙𝑓 − 𝐹𝑧𝑧0 ∙ 𝑙𝑟 = 0
106
[3.20]
LONGITUDINAL DYNAMICS
Suspension model with “trivial suspension”
ax
zf
z
Fair
m*ax
zr
x
hair
h
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 (𝑎⁄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 Fxf and Fxr always act together, so we rename
Fxf+Fxr=Fxw, where w refers to wheel. This and equilibrium give:
−𝐹𝑎𝑎𝑎 − 𝑚 ∙ 𝑎𝑥 + 𝐹𝑥𝑥 = 0;
𝑚 ∙ 𝑔 − 𝐹𝑧𝑧 − 𝐹𝑧𝑧 = 0;
𝐹𝑧𝑧 ∙ 𝑙𝑟 − 𝐹𝑧𝑧 ∙ 𝑙𝑓 − 𝐹𝑥𝑥 ∙ ℎ − 𝐹𝑎𝑎𝑎 ∙ (ℎ𝑎𝑎𝑟 − ℎ) = 0;
[3.21]
Compatibility, to introduce body displacements, z and py, gives:
𝑧𝑓 = 𝑧 − 𝑙𝑓 ∙ 𝑝𝑦 ;
𝑎𝑎𝑎 𝑧𝑟 = 𝑧 + 𝑙𝑟 ∙ 𝑝𝑦 ;
107
[3.22]
LONGITUDINAL DYNAMICS
b
cp*zp
Fzr
physical spring
with stiffness cp
zp
zrr
cp
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
_encyclopedia_suspsteer3.asp.
Combining constitutive relations, equilibrium, compatibility and renaming Fxf+Fxr=Fxw, where w
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:
108
[3.23]
LONGITUDINAL DYNAMICS
𝐹𝑥𝑥 − 𝐹𝑎𝑎𝑎
;
𝑚
𝑐𝑓 ∙ 𝑙𝑓 − 𝑐𝑟 ∙ 𝑙𝑟
(ℎ𝑎𝑎𝑎 − ℎ)�
𝑧 =−
2 ∙ �𝐹𝑥𝑥 ∙ ℎ + 𝐹𝑎𝑎𝑎 ∙
𝑐𝑓 ∙ 𝑐𝑟 ∙ �𝑙𝑓 + 𝑙𝑟 �
𝑐𝑓 + 𝑐𝑟
(ℎ𝑎𝑎𝑎 − ℎ)�
𝑝𝑦 = −
2 ∙ �𝐹𝑥𝑥 ∙ ℎ + 𝐹𝑎𝑖𝑖 ∙
𝑐𝑓 ∙ 𝑐𝑟 ∙ �𝑙𝑓 + 𝑙𝑟 �
𝑎𝑥 =
[3.24]
In agreement with intuition and experience the body dives (positive pitch) when braking (negative
Fxw). Further, the body centre of gravity is lowered (negative z) when braking and weaker
suspension front than rear (𝑐𝑓 ∙ 𝑙𝑓 < 𝑐𝑟 ∙ 𝑙𝑟 ), 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=PiPch CenPre
Figure 3-31: Models for including suspension effects in longitudinal load transfer
109
LONGITUDINAL DYNAMICS
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, zfr=zrr≡0. In free-body
diagram for the rear axle, Pxr and Pzr are reaction forces in the pivot point. Fsr is the force in the
elasticity, i.e. where potential spring energy is stored. The torque Tsr 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
zr
J*der(wy)=𝐽 ∙ 𝜔̇ 𝑦
m*ax
Fsr
Pzr
Pxr
m*g
h
Tsr
ef
zrr=0
zfr=0
lf
lr
er
Fxr
Fxf
Fzf
Fzr
gf
Fxr
Fzr
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 𝐹𝑠𝑠0 and 𝐹𝑠𝑠0 , respectively, and that
the compliances are linear. The total constitutive equations become:
𝐹𝑠𝑠 = 𝐹𝑠𝑠0 + 𝑐𝑓 ∙ �𝑧𝑓𝑓 − 𝑧𝑓 �
𝑎𝑎𝑎
𝐹𝑠𝑠 = 𝐹𝑠𝑠0 + 𝑐𝑟 ∙ (𝑧𝑟𝑟 − 𝑧𝑟 )
[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:
̇ = 0 + 𝑐𝑓 ∙ �𝑧̇𝑓𝑓 − 𝑧̇𝑓 � = −𝑐𝑓 ∙ 𝑣𝑧𝑧 𝑎𝑎𝑎 𝐹𝑠𝑠
̇ = 0 + 𝑐𝑟 ∙ (𝑧̇𝑟𝑟 − 𝑧̇𝑟 ) = −𝑐𝑟 ∙ 𝑣𝑧𝑧
𝐹𝑠𝑠
[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 𝐹𝑑𝑑 and 𝐹𝑑𝑑 . They will appear in the equilibrium equations quite similar
to 𝐹𝑠𝑠 and 𝐹𝑠𝑠 . Note that the damping coefficients, 𝑑𝑓 and 𝑑𝑟 , are the effective ones, i.e. the ones
110
LONGITUDINAL DYNAMICS
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.
𝐹𝑑𝑑 = 𝑑𝑓 ∙ �𝑧̇𝑓𝑓 − 𝑧̇𝑓 � = −𝑑𝑓 ∙ 𝑣𝑧𝑧 𝑎𝑎𝑎
𝐹𝑑𝑑 = 𝑑𝑟 ∙ (𝑧̇𝑟𝑟 − 𝑧̇𝑟 ) = −𝑑𝑟 ∙ 𝑣𝑧𝑧
[3.27]
Now, 3 equilibria for whole vehicle and one moment equilibria around pivot point for each axle gives:
−𝑚 ∙ 𝑣̇𝑥 + 𝐹𝑥𝑥 + 𝐹𝑥𝑥 = 0;
−𝑚 ∙ 𝑣̇𝑧 − 𝑚 ∙ 𝑔 + 𝐹𝑧𝑧 + 𝐹𝑧𝑧 = 0;
−𝐽 ∙ 𝜔̇ 𝑦 + 𝐹𝑧𝑧 ∙ 𝑙𝑟 − 𝐹𝑧𝑧 ∙ 𝑙𝑓 − �𝐹𝑥𝑥 + 𝐹𝑥𝑥 � ∙ ℎ = 0;
(𝐹𝑧𝑧 − 𝐹𝑠𝑠 − 𝐹𝑑𝑑 ) ∙ 𝑔𝑟 − 𝐹𝑥𝑥 ∙ 𝑒𝑟 + 𝑇𝑠𝑠 = 0;
�𝐹𝑠𝑠 + 𝐹𝑑𝑑 − 𝐹𝑧𝑧 � ∙ 𝑔𝑓 − 𝐹𝑥𝑥 ∙ 𝑒𝑓 + 𝑇𝑠𝑠 = 0;
[3.28]
Compatibility, to connect to body displacements, z and py, gives:
𝑧𝑓 = 𝑧 − 𝑙𝑓 ∙ 𝑝𝑦 ; 𝑎𝑎𝑎 𝑧𝑟 = 𝑧 + 𝑙𝑟 ∙ 𝑝𝑦 ;
𝑣𝑧𝑧 = 𝑣𝑧 − 𝑙𝑓 ∙ 𝜔𝑦 ; 𝑎𝑎𝑎 𝑣𝑧𝑧 = 𝑣𝑧 + 𝑙𝑟 ∙ 𝜔𝑦 ;
𝑧̇ = 𝑣𝑧 ; 𝑎𝑎𝑎 𝑝̇𝑦 = 𝜔𝑦 ;
[3.29]
By combining constitutive relations, equilibrium and compatibility we can find explicit function so
that:
𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆 = 𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓(𝑆𝑆𝑆𝑆𝑆𝑆, 𝐼𝐼𝐼𝐼𝐼𝐼);
𝑆𝑆𝑆𝑆𝑆𝑆 = [𝑣𝑥 𝑣𝑧 𝜔𝑦 𝐹𝑠𝑠 𝐹𝑠𝑠 𝑧 𝑝𝑦 ];
̇
̇
𝐹𝑠𝑠
𝑧̇
𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆 = �𝑣̇𝑥 𝑣̇𝑧 𝜔̇ 𝑦 𝐹𝑠𝑠
𝐼𝐼𝐼𝐼𝐼𝐼 = [𝐹𝑥𝑥 𝐹𝑥𝑥 𝑇𝑠𝑠 𝑇𝑠𝑠 ];
𝑝̇𝑦 �;
[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 𝑣̇𝑥 ) 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
𝑑𝑑𝑑(𝑣𝑥 ) = 𝑎𝑥 , in Equations [3.25] to [3.29]. The reasons why we keep ax at the “quasi-steady state”
assumption, which is that longitudinal acceleration (ax) 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
111
[3.31]
LONGITUDINAL DYNAMICS
% 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 Fxw=Fxf+Fxr is applied: both dependent on distribution between axles and dependent on
how much of the axle forces (Fxf and Fxr) that are actuated with shaft torques (Tsf and Tsr,
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.
112
LONGITUDINAL DYNAMICS
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, Fxf 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 Tsf, a good measure of the Anti-dive mechanism is
𝑒𝑓 /𝑔𝑓 . 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 Tsf 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 𝑇𝑠𝑠 = 𝐹𝑥𝑥 ∙ 𝑅𝑤 in the equation with 𝑇𝑠𝑠 in Equation [3.28]:
�𝐹𝑠𝑠 + 𝐹𝑑𝑑 − 𝐹𝑧𝑧 � ∙ 𝑔𝑓 − 𝐹𝑥𝑥 ∙ 𝑒𝑓 + 𝑇𝑠𝑠 = 0; 𝑤𝑤𝑤ℎ 𝑇𝑠𝑠 = 𝐹𝑥𝑥 ∙ 𝑅𝑤 ; ⇒
⇒ �𝐹𝑠𝑠 + 𝐹𝑑𝑑 − 𝐹𝑧𝑧 � ∙ 𝑔𝑓 − 𝐹𝑥𝑥 ∙ 𝑒𝑓 + 𝐹𝑥𝑥 ∙ 𝑅𝑤 = 0; ⇒
⇒ �𝐹𝑠𝑠 + 𝐹𝑑𝑑 − 𝐹𝑧𝑧 � ∙ 𝑔𝑓 − 𝐹𝑥𝑥 ∙ �𝑒𝑓 − 𝑅𝑤 � = 0;
[3.32]
We can then see that a good measure of the Anti-dive mechanism is �𝑒𝑓 − 𝑅𝑤 �/𝑔𝑓 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, Fxr 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 =𝑒𝑓 /𝑔𝑓 or
= �𝑒𝑓 − 𝑅𝑤 �/𝑔𝑓 and Anti-squat=𝑒𝑟 /𝑔𝑟 or = (𝑒𝑟 − 𝑅𝑤 )/𝑔𝑟 . Normal values are typically 0.05..0.15.
113
LONGITUDINAL DYNAMICS
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:
114
[3.33]
LONGITUDINAL DYNAMICS
zf = z zr = z +
vzf = vz
vzr = vz
lf*py;
lr*py;
- lf*wy;
+ 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
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].
115
LONGITUDINAL DYNAMICS
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.
116
LONGITUDINAL DYNAMICS
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, 2.5 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.
117
LONGITUDINAL DYNAMICS
Figure 3-36: ABS control
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
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.
118
LONGITUDINAL DYNAMICS
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.
Conceptually, an AEB algorithm can be assumed to know physical quantities as marked in Figure
3-37. The quantity time-to-collision, TTC, can then be defined as 𝑇𝑇𝑇 = 𝑥𝑜 ⁄(𝑣𝑥 − 𝑣𝑜𝑜 ), which means
the time within a collision will appear if no velocities changes. TTC can also be seen as the “time gap”
to the lead vehicle.
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
|𝑣̇𝑥 | < 𝑎𝑥𝑥 = 𝑒. 𝑔. 4 𝑚⁄𝑠 2 and �𝑣̇𝑦 � < 𝑎𝑦𝑦 = 𝑒. 𝑔. 6 𝑚⁄𝑠 2 . A physical model based algorithm can
start from the following simple models:
•
Normal deceleration (𝑣̇𝑥 = −𝑎𝑥𝑥 = −4 𝑚⁄𝑠 2 ) leads to collision if:
𝑡2
min
min
�𝑥𝑜 + 𝑣𝑜𝑜 ∙ 𝑡 − �𝑣𝑥 ∙ 𝑡 + 𝑎𝑥𝑥 ∙ 2 �� < 0 ⇒
�𝑥𝑜 (𝑡)� < 0 ⇒
𝑡>0
𝑡>0
𝑡2
⇒ �𝑥𝑜 + 𝑣𝑜𝑜 ∙ 𝑡 − �𝑣𝑥 ∙ 𝑡 + 𝑎𝑥𝑥 ∙ 2 ���
•
⇒ 𝑥𝑜 −
1
2∙(−𝑎𝑥𝑥 )
∙ (𝑣𝑥 − 𝑣𝑜𝑜 )2 < 0 ⇒
𝑣
−𝑣
𝑡= 𝑜𝑜 𝑥
𝑎𝑥𝑥
𝑥𝑜
=
𝑣𝑥 −𝑣𝑜𝑜
<0⇒
𝒗 −𝒗
𝒙
𝒐𝒐
𝑻𝑻𝑻 < 𝟐∙(−𝒂
=
)
𝒙𝒙
𝑣𝑥 −𝑣𝑜𝑜
;
8
Normal avoidance to the left (𝑣̇𝑦 = 𝑎𝑦𝑦 = 6 𝑚⁄𝑠 2 ) leads to collision if:
𝑤
�𝑦𝑜𝑜 (𝑡) + 2 ��
𝑥𝑜
•
⇒𝑣
•
⋯⇒𝑣
𝑥 −𝑣𝑜𝑜
𝑥𝑜 =0
< 0 ⇒ �𝑦𝑜𝑜 − 𝑎𝑦𝑦 ∙
= 𝑻𝑻𝑻 < �𝟐 ∙
𝒚𝒐𝒐 +𝒘/𝟐
𝒂𝒚𝒚
𝑡2
2
𝑤
2
+ ��
𝑡=
𝑥𝑜
𝑣𝑥 −𝑣𝑜𝑜
0.6+1.8/2
3
= {𝑒. 𝑔. } = �
<0 ⇒
≈ 0.4 𝑠;
Normal avoidance to the right (𝑣̇𝑦 = −𝑎𝑦𝑦 = −6 𝑚⁄𝑠 2 ) leads to collision if:
𝑥𝑜
𝑥 −𝑣𝑜𝑜
= 𝑻𝑻𝑻 < �𝟐 ∙
−𝒚𝒐𝒐 −𝒘/𝟐
𝒂𝒚𝒚
−𝑦𝑜𝑜 +𝑤/2
;
3
=ďż˝
Assuming that the AEB intervention decelerates the vehicle with −𝑣̇𝑥 = 𝑎𝑥𝑥𝑥𝑥 = −8 𝑚⁄𝑠 2 ,
a forward collision can be avoided if AEB intervenes AND if:
𝑥
𝒗𝒙 −𝒗𝒐𝒐
𝑣 −𝑣
= 𝑥 16 𝑜𝑜 ;
⋯ ⇒ 𝑣 −𝑣𝑜 = 𝑻𝑻𝑻 > 𝟐∙(−𝒂
)
𝑥
𝑜𝑜
𝒙𝒙𝒙𝒙
119
LONGITUDINAL DYNAMICS
Figure 3-37 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.
object vehicle
(or lead vehicle)
𝑥𝑜
= 𝑇𝑇𝑇
𝑣𝑥 − 𝑣𝑥𝑥
𝑣𝑥𝑥
subject vehicle
(or own vehicle)
𝑦𝑜𝑜
−𝑦𝑜𝑜
𝑤⁄2 𝑤⁄2
𝑣𝑥
𝑥𝑜
0.4 𝑠
Collision if normal
lateral avoidance
3.2 𝑚/𝑠
6.4 𝑚/𝑠
AEB triggered AND avoids collision
𝑣𝑥 − 𝑣𝑥𝑥
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 3-37: 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.
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. The A in the abbreviation AEB can be for either of “Automatic”, “Autonomous” or
“Advanced”. 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.
120
LONGITUDINAL DYNAMICS
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 3-38 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 3-38. 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.
Vehicle
Enivormnent
Traffic Situation Layer
Iuman
aachine
Interface
Corrward Direction
Dist
DistB Ctrl (ACC)
AEB
Maps
…
ACCbtn
Speed Ctrl (CC)
Dist btn
FwdSens
Arb(Min)
Vehicle aotion and Coordination Layer
Speed
APed
Arb(sum)
APedInterp
Arb (prio BPed,
else Max)
Energy Managament
Powertrain Coord
ABS &
EBD
BPedInterp
TC
ESC
BPed
RSC
Arb(min)
Arb
Arb(min)
Coord
Coord
Arb (switching)
Arb (switching)
aotion Support Device Layer
Vehicle
aotion
Sens
EngAct
Legend
GboxAct
FrntBrkAct
RearBrkAct
Border of Cunctional
Architecture
SW connection (signals,
here 2 scalar signals)
IW connection
Sensors & Actuators
End Customer Function
Arbitration
Coordination
Figure 3-38: Functional architecture for conventional front axle driven passenger car.
Mainly longitudinal functions (plus ESC, RSC) are shown, e.g. no steering. Cf Figure 1-9.
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.
121
LONGITUDINAL DYNAMICS
122
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).
123
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 (𝑀𝑀𝑀𝑀 ∙ 𝐿𝐿𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 = 𝑠𝑠𝑠 𝑜𝑜 𝐹𝐹𝐹𝐹𝐹𝐹) 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].
ψ
wz
vy
vx
ψ=pz
x
Figure 4-1: Model for connecting “path with orientation”
to speeds in vehicle coordinate system.
𝑥̇ = 𝑣𝑥 ∙ cos(𝜓) − 𝑣𝑦 ∙ sin(𝜓) ;
𝑦̇ = 𝑣𝑦 ∙ cos(𝜓) + 𝑣𝑥 ∙ sin(𝜓) ;
𝜓̇ = 𝜔𝑧
[4.1]
Knowing (𝑣𝑥 (𝑡), 𝑣𝑦 (𝑡), 𝜔𝑧 (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].
𝑣𝑦
𝑣𝑥
∙ cos(𝜓) − ∙ sin(𝜓) ;
𝑠̇
𝑠̇
𝑣𝑦
𝑣𝑥
′
𝑦 = ∙ cos(𝜓) + ∙ sin(𝜓) ;
𝑠̇
𝑠̇
𝜔𝑧
𝜓′ =
;
𝑠̇
𝑤ℎ𝑒𝑒𝑒 𝑝𝑝𝑝𝑝𝑒 𝑛𝑛𝑛𝑛𝑛 𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 𝑤𝑤𝑤ℎ 𝑟𝑟𝑟𝑟𝑟𝑟𝑟 𝑡𝑡 𝑠
𝑥′ =
Here, 𝑠̇ can be thought of like an arbitrary time scale, with which all speeds are scaled. One can
typically chose 𝑠̇ = 1 [𝑚⁄𝑠]. However, in this compendium we will keep notation t and the dot
notation for derivative.
124
[4.2]
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].
Velocity of body,
above this wheel
𝒗𝒙𝒙
𝛿
Force in road plane,
acting on this wheel
𝑭𝒙𝒙
𝛿
Components in vehicle
coordinates
𝒗𝒙𝒙
𝒗𝒙𝒙
𝒗𝒚𝒚
Components in wheel
coordinates
𝒗𝒙𝒙
Components in vehicle
coordinates
𝑭𝒙𝒙
𝑭𝒙𝒙
𝑭𝒚𝒚
𝒗𝒚𝒚
𝒗𝒙𝒙
Components in wheel
coordinates
𝑭𝒙𝒙
𝑭𝒚𝒚
𝑭𝒙𝒙
Figure 4-2: Transformation between forces and velocities in vehicle coordinate
system and wheel coordinate system.
Transformation from wheel coordinates to vehicle coordinates:
𝑣𝑥𝑥
𝐹𝑥𝑥
cos(𝛿) − sin(𝛿) 𝑣𝑥𝑥
cos(𝛿) − sin(𝛿) 𝐹𝑥𝑥
� ∙ �𝑣 � ; 𝑎𝑎𝑎 �𝐹 � = �
� ∙ �𝐹 � ;
�𝑣 � = �
𝑦𝑦
𝑦𝑦
sin(𝛿) cos(𝛿)
sin(𝛿) cos(𝛿)
𝑦𝑦
𝑦𝑦
Transformation from vehicle coordinates to wheel coordinates:
[4.3]
𝑣𝑥𝑥
𝑣𝑥𝑥
𝐹𝑥𝑥
𝐹𝑥𝑥
cos(𝛿) sin(𝛿)
cos(𝛿) sin(𝛿)
�𝑣 � = �
� ∙ � � ; 𝑎𝑎𝑎 �𝐹 � = �
� ∙ �𝐹 � ;
𝑦𝑦
− sin(𝛿) cos(𝛿) 𝑣𝑦𝑦
−
sin(𝛿)
cos(𝛿)
𝑦𝑦
𝑦𝑦
4.2.3
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
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:
125
LATERAL DYNAMICS
1
𝑅𝑟 − 𝑤⁄2 ⎫
=
;⎪
1
1
𝑤
tan(𝛿𝑖 )
𝐿
⇒
=
+ ;
1
𝑅𝑟 + 𝑤⁄2 ⎬ tan(𝛿𝑜 ) tan(𝛿𝑖 ) 𝐿
=
;⎪
tan(𝛿𝑜 )
𝐿
⎭
δi
[4.4]
δo
L
1 tan(δ o ) = 1 tan(δ i ) + w L
Common intersection of all
wheels’ axes of rotation
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
𝛿𝑖 = 𝛿𝑜 . 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 4-4. 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.
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 4-5. One-track model for
truck with trailer is exemplified in Figure 4-6.
126
LATERAL DYNAMICS
Ackermann
steering
(trapezoidal
geometry)
𝛿𝑠𝑠
(Erasmus Darwin 1758,
Rudolph Ackermann
1810.)
𝛿𝑟𝑟𝑟𝑟𝑟
𝛿𝑙𝑙𝑙𝑙
𝛿𝑙𝑙𝑙𝑙 = 𝑓𝑓𝑓𝑓 𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑟𝑟𝑟𝑟 =
= 𝑓𝑓𝑓𝑓 𝑟𝑟𝑟𝑟𝑟 ∙ 𝛿𝑠𝑠 ;
𝛿𝑟𝑟𝑟𝑟𝑟 = −𝑓𝑓𝑓𝑓 −𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑟𝑟𝑟𝑟 =
= −𝑓𝑓𝑓𝑓 −𝑟𝑟𝑟𝑟𝑟 ∙ 𝛿𝑠𝑠 ;
Figure 4-4: Example of Trapezoidal Steering. Left: Conceptual use of steering arms. Right:
More exact design, common today. From Gunnar Olsson, LeanNova.
δf δo
Common
intersection
of all wheels’
axes of
rotation
lr
L
lf
δi
Rr
Figure 4-5: Collapsing to one-track model.
127
LATERAL DYNAMICS
Figure 4-6: 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 2.5, lateral slip models for tyres were introduced. It is the constitutive equation 𝐹𝑦 = −𝐶𝑦 ∙
𝑠𝑦 . If the tyre force, 𝐹𝑦 , is small compared to cornering stiffness, 𝐶𝑦 , one would expect side slip, 𝑠𝑦 , 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 ∙ 𝐹𝑧 . 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.
128
LATERAL DYNAMICS
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.
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. 𝑚 ∙ 𝑎 = 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 4-5 this connection becomes:
𝐿
𝐿
𝐿
𝐿
;
⎞ ∙ sign(𝑅) ≈
𝑅𝑟 � ⇒ 𝛿𝑓 = arctan ⎛
∙ sign(𝑅) ≈ ;
𝑅
𝑅 2 = 𝑅𝑟 2 + 𝑙𝑟 2 ;
�𝑅2 − 𝑙𝑟 2
�𝑅 2 − 𝑙𝑟 2
⎝
⎠
𝑤ℎ𝑒𝑒𝑒 𝑅 > 0 𝑚𝑚𝑚𝑚𝑚 𝑡ℎ𝑎𝑎 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 𝑐𝑐𝑐𝑐𝑟𝑟 𝑜𝑜 𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟 𝑖𝑖 𝑙𝑙𝑙𝑙 𝑜𝑜 𝑡ℎ𝑒 𝑣𝑣ℎ𝑖𝑖𝑖𝑖
tan�𝛿𝑓 � =
[4.5]
To get a more complete model, where more variables can be extracted, we can set up the model in
Figure 4-7.
wz
vx
vrx
lr
vfyv
vry=0
vy
forces:
vfxv 𝑣⃗𝑓
Fry
Frx
δf
lf
L
Ffyv
speeds:
lr
δf
lf
L
𝑣⃗𝑓
Ffxv 𝐹⃗
𝑓
𝐹⃗𝑓
Figure 4-7: One-track model with ideally tracking axles. Lower view of front wheel shows
convertion between wheel and vehicle coordinate systems.
The “physical model” in Figure 4-7 gives the following “mathematical model”:
129
LATERAL DYNAMICS
Equilibrium (longitudinal, lateral and yaw-rotational):
0 = 𝐹𝑓𝑓𝑓 + 𝐹𝑟𝑟 ;
0 = 𝐹𝑓𝑓𝑓 + 𝐹𝑟𝑟 ;
0 = 𝐹𝑓𝑓𝑓 ∙ 𝑙𝑓 − 𝐹𝐹𝐹 ∙ 𝑙𝑙;
Transformation between vehicle and wheel coordinate systems:
𝐹𝑓𝑓𝑓
𝐹𝑓𝑓𝑓
𝑣𝑓𝑓𝑓
𝑣𝑓𝑓𝑓
Compatibility:
=
=
=
=
𝐹𝑓𝑓𝑓 ∙ cos�𝛿𝑓 �
𝐹𝑓𝑓𝑓 ∙ sin�𝛿𝑓 �
𝑣𝑓𝑓𝑓 ∙ cos�𝛿𝑓 �
𝑣𝑓𝑓𝑓 ∙ sin�𝛿𝑓 �
− 𝐹𝑓𝑓𝑓 ∙ sin�𝛿𝑓 � ;
+ 𝐹𝑓𝑓𝑓 ∙ cos�𝛿𝑓 � ;
− 𝑣𝑓𝑓𝑓 ∙ sin�𝛿𝑓 � ;
+ 𝑣𝑓𝑓𝑓 ∙ cos�𝛿𝑓 � ;
𝑣𝑓𝑓𝑓 = 𝑣𝑥 ;
𝑣𝑓𝑓𝑓 = 𝑣𝑦 + 𝑙𝑓 ∙ 𝜔𝑧 ;
𝑣𝑟𝑟 = 𝑣𝑥 ;
𝑣𝑟𝑟 = 𝑣𝑦 − 𝑙𝑙 ∙ 𝜔𝑧 ;
[4.6]
Ideal tracking (Constitutive relation, but without connection to forces):
𝑣𝑓𝑓𝑓 = 0;
𝑣𝑟𝑟 = 0;
Path with orientation:
𝑥̇ = 𝑣𝑥 ∙ cos(𝜑𝑧 ) − 𝑣𝑦 ∙ sin(𝜑𝑧 ) ;
𝑦̇ = 𝑣𝑦 ∙ cos(𝜑𝑧 ) + 𝑣𝑥 ∙ sin(𝜑𝑧 ) ;
𝜑̇ 𝑧 = 𝜔𝑧 ;
Prescription of steering angle operation:
= (35 ∙ 𝜋⁄180) ∙ sin(0.5 ∙ 2 ∙ 𝜋 ∙ 𝑡) ; 𝑖𝑖 𝑡 < 4.5;
𝛿𝑓 = �
= 35 ∙ 𝜋/180;
𝑒𝑒𝑒𝑒
Rear axle undriven, which gives drag from roll resistance:
𝐹𝑟𝑟 = −100;
Equation [4.6] is written in Modelica format in Equation [4.7]. 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 vfyw=vry=0. Global coordinates from Figure 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;
//Transformation between vehicle and wheel coordinate systems:
130
[4.7]
LATERAL DYNAMICS
Ffxv
Ffyv
vfxv
vfyv
=
=
=
=
Ffxw*cos(df)
Ffxw*sin(df)
vfxw*cos(df)
vfxw*sin(df)
+
+
Ffyw*sin(df);
Ffyw*cos(df);
vfyw*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 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 4-8. 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.
δf [raδ]
y [m]
35 deg
x[m]
time [s]
Figure 4-8: Simulation results of one-track model with ideal tracking.
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 steady state
model in section 4.3.2.
A driving resistance of 100 N is assumed on the rear axle (Frx=-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 a intersection point between the front and rear wheels rotational axes.
131
LATERAL DYNAMICS
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.
wall turning diameter
y [m]
curb turning diameter
The end of the simulation in Figure 4-8 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 4-9. 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 4-9. 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 4-9: Adding paths for wheels and body points, on top of result in Figure 4-8.
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 0.
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 4-9. It can occur as a kerb and wall version as well as turning
132
LATERAL DYNAMICS
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 4-9. 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.
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 4-10. 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 the such simulation
are the path coordinates with orientation (𝑥, 𝑦, 𝜑𝑧 ).
L
δf
Figure 4-10: Adding one-track model, on top of result in Figure 4-8.
From geometry in Figure 4-10 on can find an expression for off-tracking at low speed:
∆= 𝑅𝑓 − 𝑅𝑟 =
𝑅𝑟
cos�𝛿𝑓 �
− 𝑅𝑟 = 𝑅𝑟 ∙ �
1
cos�𝛿𝑓 �
− 1� = �𝑅 2 − 𝑙𝑟 2 ∙ �
133
1
cos�𝛿𝑓 �
− 1� ≈
[4.8]
LATERAL DYNAMICS
≈𝑅∙�
1
cos�𝛿𝑓 �
− 1� ≈ 𝑅 ∙ �1 − cos�𝛿𝑓 ��
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.
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 2-36. 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, 𝑀𝑍𝑍 in Figure 2-2. (𝑀𝑍𝑍 does not influence
very much, except for at very low vehicle speed, which is why quantitative models for 𝑀𝑍𝑍 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 4-7 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 4-11.
134
LATERAL DYNAMICS
Figure 4-11: Rigid Truck with 3 axles, whereof only the first is steered.
speeds:
vr2y
vr1y wz
vx
vr2x
vr1x
lr
forces:
Fr2y
Fr2x
vfxv 𝑣⃗𝑓
δf
Fr1y
Fr1x
b
Ffxv 𝐹⃗
𝑓
δf
𝐹⃗𝑓
lf
L
b
vfyv
vy
Ffyv
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 4-12. The measures appear in Figure 4-14, and you see that
it is not a truck, but a very unconventional passenger size car with two rear axles.
𝑣⃗𝑓
Figure 4-12: 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 [4.9]. 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 2.5.
//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;
[4.9]
//Compatibility:
vfxv = vx;
vfyv = vy + lf*wz;
vr1x = vx;
vr2x = vx;
vr1y = vy - (lr - b)*wz;
vr2y = vy - (lr + b)*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);
135
LATERAL DYNAMICS
//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 4-13, which should be compared to Figure 4-8. 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 4-14.
δf [δeg]
y [m]
35 deg
vx=2.778m/s
vy=1.036m/s
wz=0.652raδ/s
x[m]
time [s]
Figure 4-13: Simulation results of one-track model with ideal tracking.
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.
136
LATERAL DYNAMICS
• instantaneous centre for
3-axle vehicle
• instantaneous centre for
2-axle reference vehicle
b=1 b=1
δf
lr=1.5 lf=1.3
Figure 4-14: Instantaneous centre with a 3-axle vehicle, with the corresponding 2-axle
vehicle as reference.
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 0.
4.3 Steady state cornering at high speed
Steady state cornering refers to that all time derivatives of vehicle speeds (vx, vy, wz) are zero. The
physical understanding is then that the vehicle drives on a circle with constant yaw rate, see Figure
4-15Figure 4-1.
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 by 3 quantities, e.g. 𝑣𝑥 , 𝑣𝑦 , 𝜔𝑧
or 𝑣, 𝛽, 𝑅 .
Assuming a certain way of reaching this
state (e.g. only steering wheel angle and
accelerator pedal) there is only 2
quantities is needed.
Figure 4-15: Steady state cornering.
137
LATERAL DYNAMICS
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 4-5. 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.
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 4-16: An example of test track and some parts with special relevance to Vehicle
Dynamics. The example is Hällered Proving Ground, Volvo Car Corporation.
138
LATERAL DYNAMICS
=Vehicle Dynamics Area
Figure 4-17: 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).
4.3.2
Steady state one-track model
In steady state we have neither inertial effects from changing the total vehicle speed (𝑣 =
�𝑣𝑥 2 + 𝑣𝑦 2 is constant) nor changing the yaw rate (𝑤𝑧 is constant). However, the inertial
“centrifugal” effect of the vehicle must be modelled. The related acceleration is the centripetal
acceleration, 𝑎𝑐 = 𝑅 ∙ 𝜔𝑧 2 = 𝑣 2 /𝑅 = 𝜔𝑧 ∙ 𝑣.
A vehicle model for this is sketched in Figure 4-18. The model is a development of the model for lowspeed in Figure 4-7 and Equation [4.7], with the following changes:
•
•
Longitudinal and lateral accelerations are changed from zero to components of centripetal
acceleration, ac, as follows (see Figure 4-18):
o ax = -ac*sin(b) = -wz*v*sin(b) = -wz*vy;
o ay = +ac*cos(b) = +wz*v*cos(b) = +wz*vx;
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 2.5, but they can also capture steering system
compliance (see Section 4.3.4.3), roll steering (see Section 4.3.4.4) and side force steering
(see Section 4.3.4.5). The total mathematical relations can anyway be written as:
o 𝐹𝑓𝑓𝑓 = −𝐶𝑓 ∙ 𝑠𝑓𝑓 ; where 𝑠𝑓𝑓 = 𝑣𝑓𝑓𝑓 ⁄𝑣𝑓𝑓𝑓 ;
o 𝐹𝑟𝑟 = −𝐶𝑟 ∙ 𝑠𝑟𝑟 ; where 𝑠𝑟𝑟 = 𝑣𝑟𝑟 ⁄𝑣𝑟𝑟 ;
139
speeds:
vry wz
vx
vrx
lr
vfyv
vy
forces:
vfxv 𝑣⃗𝑓
δf
Fry
m*ax
Frx
lf
L
m*ay
lr
Ffxv 𝐹⃗
𝑓
δf
lf
L
𝑣⃗𝑓
Ffyv
LATERAL DYNAMICS
vehicle path
centre
𝐹⃗𝑓
-ax
ac=wz*v
β
-ax=ac*sin(β);
+ay=ac*cos(β);
ay
vx=v*cos(β) v
vy=v*sin(β)
β
Figure 4-18: One-track model. Dashed forces and moment are fictive forces.
The model in Figure 4-18 is documented in mathematical form in Equation [4.10] (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;
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;
[4.10]
//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:
140
LATERAL DYNAMICS
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 4-8. 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.
δf [raδ]
5 deg
y [m]
𝑣=
2
𝑣𝑥 2 + 𝑣𝑦 =
= 28.37 𝑚/𝑠
Ffxw
Ffyw
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 4-19: Simulation results of steady state one-track model. The vehicle sketched in
the path plot is not in scale, but correctly oriented.
Equation [4.10] is a complete model suitable for simulation, but it does not facilitate understanding
very well. We will reformulate it assuming small df (i.e. cos(df)=1 and sin(df)=df and df*df=0). The
two unknowns are basically the two vehicle states vy and wz. Solving Equation [4.10] for wz, yields:
𝐶𝑓 ∙ 𝐶𝑟 ∙ 𝐿2 − �𝐶𝑓 ∙ 𝑙𝑓 − 𝐶𝑟 ∙ 𝑙𝑟 � ∙ 𝑚 ∙ 𝑣𝑥 2 𝜔𝑧
𝛿𝑓 =
∙
≈
𝑣𝑥
𝐶𝑓 ∙ 𝐶𝑟 ∙ 𝐿 + �𝐶𝑟 ∙ 𝑙𝑓 + 𝐶𝑟 ∙ 𝑙𝑟 � ∙ 𝐹𝑓𝑓𝑓
𝐶𝑟 ∙ 𝑙𝑟 − 𝐶𝑓 ∙ 𝑙𝑓
𝜔𝑧
≈ �𝑢𝑢𝑢: 𝐹𝑓𝑓𝑓 = 0� ≈ �𝐿 +
∙ 𝑚 ∙ 𝑣𝑥 2 � ∙
=
𝐶𝑓 ∙ 𝐶𝑟 ∙ 𝐿
𝑣𝑥
141
[4.11]
LATERAL DYNAMICS
𝐶𝑟 ∙ 𝑙𝑟 − 𝐶𝑓 ∙ 𝑙𝑓
𝑙𝑓
𝑙𝑟
𝜔𝑧
=
−
+ 𝐾𝑢 ∙ 𝑚 ∙ 𝑎𝑦 ≈
�=𝐿∙
𝐶𝑓 ∙ 𝐶𝑟 ∙ 𝐿
𝐶𝑓 ∙ 𝐿 𝐶𝑟 ∙ 𝐿
𝑣𝑥
𝜔𝑧
= �𝑢𝑢𝑢: 𝑎𝑦 = 𝜔𝑧 ∙ 𝑣𝑥 � = 𝐿 ∙
+ 𝐾𝑢 ∙ 𝑚 ∙ 𝜔𝑧 ∙ 𝑣𝑥 ≈
𝑣𝑥
𝜔𝑧 1
𝐿
𝑚 ∙ 𝑣𝑥 2
≈ �𝑢𝑢𝑢:
≈ � ≈ + 𝐾𝑢 ∙
;
𝑣𝑥 𝑅
𝑅
𝑅
= �𝑢𝑢𝑢: 𝐾𝑢 =
The coefficient 𝐾𝑢 is the understeer gradient and it will be explained in next section.
A simpler way to reach the final expression is given in Figure 4-20. Here, the simplifications are
introduced earlier, already in physical model, which means e.g. that the influence of 𝐹𝑓𝑓𝑓 is not
identified.
Îąathematical model:
Physical model:
• Path radius >> the vehicle. Then, all forces
(and centripetal acceleration) are
approximately co0directed.
• 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 co0linear in figure.)
Fry
𝑚∙
𝛿𝑓 + 𝛼𝑓 = 𝛽𝑓 ; 𝛽𝑓 ≈
𝑣𝑥 2
𝑅
𝛼𝑟 = 𝛽𝑟 ≈
βf
Ffy
β
βr
Equilibrium:
𝑣𝑥 2
𝑚∙
≈ 𝐹𝑓𝑓 +𝐹𝑟𝑟 ; 0 ≈ 𝐹𝑓𝑓 ∙ 𝑙𝑓 − 𝐹𝑟𝑟 ∙ 𝑙𝑟 ;
𝑅
Constitution:
𝐹𝑓𝑓 = −𝐶𝑓 ∙ 𝑠𝑓𝑦 ; 𝐹𝑟𝑦 = −𝐶𝑟 ∙ 𝑠𝑟𝑦 ;
Compatibility:
Îąf
𝜔𝑧 ≈
𝑣𝑥
;
𝑅
𝑣𝑟𝑦
𝑣𝑥
𝑣𝑓𝑓
=
𝑣𝑦 +𝑙𝑓 ∙𝜔𝑧
𝑣𝑥
𝑣𝑥
𝑣𝑦 −𝑙𝑟 ∙𝜔𝑧
=
𝑣𝑥
;
⇒
𝛼𝑓 ≈ 𝑠𝑓𝑦 ; 𝛼𝑟 ≈ 𝑠𝑟𝑟 ;
⇒ 𝛿𝑓 + 𝛼𝑓 − 𝛼𝑟 ≈
δf
;
𝐿
;
𝑅
Eliminate 𝐹𝑓𝑓 , 𝐹𝑟𝑟 , 𝛼𝑓 , 𝛼𝑟 , 𝛽𝑓 , 𝛽𝑟 , 𝜔𝑧 yields:
𝐿
𝑚 ∙ 𝑣𝑥 2
𝛿𝑓 ≈ + 𝐾𝑢 ∙
;
𝑅
𝑅
wz
lr lf
L
𝐾𝑢 =
𝐶𝑟 ∙ 𝑙𝑟 − 𝐶𝑓 ∙ 𝑙𝑓
;
𝐶𝑓 ∙ 𝐶𝑟 ∙ 𝐿
Figure 4-20: Simpler derivation final step in Equation [4.11].
Solving Equation [4.10] for the other of the two degrees of freedom, vy, yields:
𝛿𝑓 =
𝐶𝑓 ∙ 𝐶𝑟 ∙ 𝐿2 − �𝐶𝑓 ∙ 𝑙𝑓 − 𝐶𝑟 ∙ 𝑙𝑟 � ∙ 𝑚 ∙ 𝑣𝑥 2
2
2� ∙
𝐶𝑓 ∙ 𝐶𝑟 ∙ 𝑙𝑟 ∙ 𝐿 − 𝐶𝑓 ∙ 𝑙𝑓 ∙ 𝑚 ∙ 𝑣𝑥 + �𝐶𝑟 ∙ 𝑙𝑟 ∙ 𝐿 − 𝑙𝑓 ∙ 𝑚 ∙ 𝑣𝑥
𝐹𝑓𝑓𝑓
2
2
𝐶𝑓 ∙ 𝐶𝑟 ∙ 𝐿 − �𝐶𝑓 ∙ 𝑙𝑓 − 𝐶𝑟 ∙ 𝑙𝑟 � ∙ 𝑚 ∙ 𝑣𝑥 𝑣𝑦
≈ �𝑢𝑢𝑢: 𝐹𝑓𝑓𝑓 = 0� ≈
∙
⇒
𝐶𝑓 ∙ 𝐶𝑟 ∙ 𝑙𝑟 ∙ 𝐿 − 𝐶𝑓 ∙ 𝑙𝑓 ∙ 𝑚 ∙ 𝑣𝑥 2
𝑣𝑥
𝐿 𝑣𝑦
⎧ 𝛿𝑓 �⎯⎯�
∙ ;
⎪
𝑣𝑥 →0 𝑙𝑟 𝑣𝑥
⇒
⎨𝛿 �⎯⎯� 𝐶𝑓 ∙ 𝑙𝑓 − 𝐶𝑟 ∙ 𝑙𝑟 ∙ 𝑣𝑦 = −𝐾 ∙ 𝐶 ∙ 𝐿 ∙ 𝑣𝑦 ;
𝑢
𝑟
⎪ 𝑓 𝑣𝑥 →∞
𝐶𝑓 ∙ 𝑙𝑓
𝑣𝑥
𝑙𝑓 𝑣𝑥
⎊
∙
𝑣𝑦
≈
𝑣𝑥
[4.12]
𝑣
We can see that there is a speed dependent relation between steering angle and side slip, 𝑣𝑦 . The
𝑣
𝑥
side slip can also be expressed as a side slip angle, 𝛽 = arctan � 𝑦 �. Since normally 𝐾𝑢 > 0, the side
𝑣
𝑥
slip changes sign, when increasing speed from zero to sufficient high enough. This should feel
142
LATERAL DYNAMICS
intuitively correct, if agreeing on the conceptually different side slip angles at low and high speed, as
shown in Figure 4-21. We will come back to this equation in context of Figure 4-27.
−β
-β
vehicle path
centre, at
high speed
v
path of
front axle
path of
rear axle
β
path of
front axle
path of
rear axle
v
vehicle path
centre, at
low speed
Figure 4-21: Body Slip Angle for Low and High Speed Steady State Curves
4.3.3
Under-, Neutral- and Over-steering *
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 [4.11] 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 [4.13]. (When defined as 𝐾𝑢3, one
can sometimes see the unit “rad/g” used, which present compendium recommended to not use.)
𝐿 𝐶𝑟 ∙ 𝑙𝑟 − 𝐶𝑓 ∙ 𝑙𝑓 𝑚 ∙ 𝑔 ∙ 𝑣𝑥 2
𝛿𝑓 = +
∙
=
𝑅
𝐶𝑓 ∙ 𝐶𝑟 ∙ 𝐿
𝑔∙𝑅
𝐶𝑟 ∙ 𝑙𝑟 − 𝐶𝑓 ∙ 𝑙𝑓
𝐿
𝑣𝑥 2
[1 𝑜𝑜 𝑟𝑟𝑟]� = + 𝐾𝑢2 ∙
= �𝑢𝑢𝑢: 𝐾𝑢2 = 𝑚 ∙ 𝑔 ∙
;
𝑅
𝐶𝑓 ∙ 𝐶𝑟 ∙ 𝐿
𝑔∙𝑅
𝐿 𝐶𝑟 ∙ 𝑙𝑟 − 𝐶𝑓 ∙ 𝑙𝑓 𝑚 ∙ 𝑣𝑥 2
∙
=
𝛿𝑓 = +
𝐶𝑓 ∙ 𝐶𝑟 ∙ 𝐿
𝑅
𝑅
𝐶𝑟 ∙ 𝑙𝑟 − 𝐶𝑓 ∙ 𝑙𝑓
1
𝑟𝑟𝑟
𝐿
𝑣𝑥 2
= �𝑢𝑢𝑢: 𝐾𝑢3 = 𝑚 ∙
𝑜𝑜
=
+
𝐾
∙
;
ďż˝
��
𝑢3
𝐶𝑓 ∙ 𝐶𝑟 ∙ 𝐿
𝑚 ⁄𝑠 2 𝑚 ⁄𝑠 2
𝑅
𝑅
143
[4.13]
LATERAL DYNAMICS
The first term in the final expression in Equation [4.11], 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, 𝛿𝑓𝑓 . It can also be defined for steering wheel angle, 𝛿𝑠𝑠𝑠 .
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 (𝑖𝑖 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, 𝐹𝑐 :
𝛿𝑓 = 𝛿𝑓𝑓 + 𝐾𝑢 ∙
𝛿𝑓 − 𝛿𝑓𝑓 ∆𝛿𝑓
𝑚 ∙ 𝑣𝑥 2
= 𝛿𝑓𝑓 + 𝐾𝑢 ∙ 𝐹𝑐 ; ⇒ 𝐾𝑢 =
=
;
𝑅
𝐹𝑐
𝐹𝑐
Or, using the steering ratio, 𝛿𝑓 = 𝛿𝑠𝑠 ⁄𝑟𝑠𝑠𝑠 :
𝛿𝑠𝑠 = 𝛿𝑠𝑠𝑠 + 𝐾𝑢 ∙
𝑟𝑠𝑠𝑠 ∙ 𝑚 ∙ 𝑣𝑥 2
= 𝛿𝑓𝑓 + 𝐾𝑢 ∙ 𝑟𝑠𝑠𝑠 ∙ 𝐹𝑐 ; ⇒
𝑅
𝛿𝑠𝑠 − 𝛿𝑠𝑠𝑠
∆𝛿𝑠𝑠
⇒ 𝐾𝑢 =
=
;
𝑟𝑠𝑠𝑠 ∙ 𝐹𝑐
𝑟𝑠𝑠𝑠 ∙ 𝐹𝑐
[4.14]
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 ∆𝛿𝑓 and
the 𝐹𝑐 should be small when using Eq [4.15]. However, if we accept that 𝐾𝑢 can change with 𝐹𝑐 , 𝐾𝑢
can be defined as a differential quantity:
𝐾𝑢 can also be understood as how much the additional steering angle, ∆𝛿𝑓 , has to increase per
increased centrifugal force, 𝐹𝑐 , or per lateral acceleration, 𝑎𝑦 :
𝐾𝑢 =
𝜕�∆𝛿𝑓 �
𝜕∆𝛿𝑓
𝜕�∆𝛿𝑓 �
𝜕∆𝛿𝑓
𝜕
𝜕
=
; 𝑜𝑜 𝐾𝑢3 =
=
;
�𝛿𝑓 − 𝛿𝐴 � =
�𝛿𝑓 − 𝛿𝐴 � =
𝜕𝐹𝑐
𝜕𝐹𝑐
𝜕𝐹𝑐
𝜕𝑎𝑦
𝜕𝑎𝑦
𝜕𝑎𝑦
[4.15]
Equation [4.15] 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 [4.15] is more relevant than Equation [4.11], because it reflects
that understeering gradient changes.
A third understanding of the word understeering is quite different and less strictly defined. It is to say
that a vehicle over-steers or understeers as an instantaneously varying state of the vehicle, during a
manoeuvre. For instance, a vehicle can be said to understeer if tyre side slip is larger on front axle
than on rear axle, �∝𝑓 � > |∝𝑟 |, and over-steer if opposite, |∝𝑟 | > �∝𝑓 �. 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.
A second look at Equation [4.11] tells us that we have to assume absence of propulsion and braking
on front axle, 𝐹𝑓𝑓𝑓 = 0, to get the relatively simple final expression. When propulsion on front axle
144
LATERAL DYNAMICS
(𝐹𝑓𝑓𝑓 > 0), the required steering angle, 𝛿𝑓 , will be smaller; the front propulsion pulls in the front end
of the vehicle. When braking on front axle (𝐹𝑓𝑓𝑓 < 0), the required steering angle, 𝛿𝑓 , 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.
4.3.4 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: 𝐶𝑖 =
𝑘𝑖 ∙ 𝐹𝑖𝑖 . 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:
𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙
𝑙𝑓
𝑙𝑓
𝐾𝑢2
𝑙𝑟
𝑙𝑟
=
−
=
−
= �𝑙𝑙𝑙𝑙 𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡� ⇒
𝑚 ∙ 𝑔 𝐶𝑓 ∙ 𝐿 𝐶𝑟 ∙ 𝐿 𝑘𝑓 ∙ 𝐹𝑓𝑓 ∙ 𝐿 𝑘𝑟 ∙ 𝐹𝑟𝑟 ∙ 𝐿
𝑚 ∙ 𝑎𝑥 ∙ ℎ/𝐿
⇒ 𝐾𝑢2 =
𝑚 ∙ 𝑔 ∙ 𝑙𝑓
𝑚 ∙ 𝑔 ∙ 𝑙𝑟
−
=
𝑙𝑟
ℎ
𝑙
𝑘𝑓 ∙ �𝑚 ∙ 𝑔 ∙ 𝐿 − 𝑚 ∙ 𝑎𝑥 ∙ 𝐿 � ∙ 𝐿 𝑘𝑟 ∙ �𝑚 ∙ 𝑔 ∙ 𝑓 + 𝑚 ∙ 𝑎𝑥 ∙ ℎ� ∙ 𝐿
𝐿
𝐿
=
0
𝑖𝑖
𝑎
𝑥
𝑙𝑓 ⁄𝐿
𝑙𝑟 ⁄𝐿
−
= � 𝑎𝑎𝑎 � = 0;
=
𝑙
𝑎 ℎ
𝑙
𝑘𝑓 = 𝑘𝑟
𝑘𝑓 ∙ � 𝐿𝑟 − 𝑔𝑥 ∙ 𝐿 � 𝑘𝑟 ∙ � 𝑓 + 𝑎𝑥 ∙ ℎ�
𝐿
𝑔 𝐿
[4.16]
Longitudinal load transfer was kept in the equations, in order to show that braking increases oversteering tendency. It is actually so, that the critical speed 𝑣𝑥,𝑐𝑐𝑐𝑐 = �𝐿⁄(−𝐾𝑢 ∙ 𝑚) (see Equation
145
LATERAL DYNAMICS
[4.18]) can come down to quite reachable levels when braking hard; i.e. hard braking at high speed
may cause instability. See Figure 4-22, inspired by Reference (Drenth, 1993).
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
100
80
60
-0.05
40
lf/L=0.25
lf/L=0.5
lf/L=0.75
-0.1
-10
120
-5
0
ax [m/(s*s)]
5
20
0
-1
10
-0.5
0
ax [g]
0.5
1
Figure 4-22: 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. 𝐶𝑖 = 𝑘𝑖𝑖 ∙ 𝐹𝑖𝑖 − 𝑘𝑖𝑖 ∙ 𝐹𝑖𝑖 2 . This is further
studied in Reference (Drenth, 1993).
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
Roll steer gradient
Roll steer gradient is how much the road wheel steering angle increases due to vehicle roll angle. It
can be influenced with linkage geometry. Roll stiffness distribution between front and rear axle
affects the understeer gradient. (But roll stiffness (sum of all axles) determines the roll gradient.)
146
LATERAL DYNAMICS
4.3.4.5
Side force steer gradient
Side force steer gradient is similar to roll steer gradient, but it is how much the road wheel steering
angle increases due to lateral force on the axle in question. It can be also on a non-steered axle, and
depends on the suspension linkage and the bushing compliance. The side force compliance often
represents 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.
If toe (=toe-in) is positive there are tyre-lateral forces on each tyre already when driving straight
𝑡𝑡𝑡
ahead, even if left and right cancel out each other: 𝐹𝑎𝑎 = �𝐶𝑙𝑙𝑙𝑙 − 𝐶𝑟𝑟𝑟ℎ𝑡 � ∙ 2 = 0;. Then, if the axle
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 [4.11].
Unsymmetrical wheel torques (left/right) will give a direct yaw moment in the yaw equilibrium in
Equation [4.10]. 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
147
LATERAL DYNAMICS
•
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.
4.3.5
Required Steering
From Equation [4.11], we can conclude:
𝑹𝑹𝑹𝑹𝑹𝑹𝑹𝑹 𝒔𝒕𝒕𝒕𝒕𝒕𝒕𝒕 =
𝛿𝑓 ∙ 𝑅
𝑚 ∙ 𝑣𝑥 2
= 1 + 𝐾𝑢 ∙
;
𝐿
𝐿
[4.17]
The required steering is plotted as function of speed, for different understeering gradients Figure
4-23.
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
/haracteristic speed (for
understeered vehicle)
0
0
5
10
15
20
Figure 4-23: Required steering (
4.3.5.1
/ritical speed (for
oversteered vehicle)
25
vx [m/s]
𝛿𝑓 ∙𝑅
𝐿
30
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. (Alternative definitions exist, such as the speed at which the lateral
acceleration gain per longitudinal speed reaches its highest value.)
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. Normal vehicles are
148
LATERAL DYNAMICS
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.
From Equation [4.11], we can find a formula for critical and characteristic speeds:
𝛿𝑓 =
𝐶𝑓 ∙ 𝐶𝑟 ∙ 𝐿2
𝐿
𝑚 ∙ 𝑣𝑥,𝑐𝑐𝑐𝑐 2
𝐿
+ 𝐾𝑢 ∙
= 0 ⇒ 𝑣𝑥,𝑐𝑐𝑐𝑐 = �
=ďż˝
;
𝑅
𝑅
−𝐾𝑢 ∙ 𝑚
�𝐶𝑓 ∙ 𝑙𝑓 − 𝐶𝑟 ∙ 𝑙𝑟 � ∙ 𝑚
𝐶𝑓 ∙ 𝐶𝑟 ∙ 𝐿2
𝐿
𝑚 ∙ 𝑣𝑥,𝑐ℎ𝑎𝑎 2
𝐿
𝐿
𝛿𝑓 = + 𝐾𝑢 ∙
=2∙
⇒ 𝑣𝑥,𝑐ℎ𝑎𝑎 = �
=ďż˝
;
𝑅
𝑅
𝑅
𝐾𝑢 ∙ 𝑚
�𝐶𝑟 ∙ 𝑙𝑟 − 𝐶𝑓 ∙ 𝑙𝑓 � ∙ 𝑚
4.3.6
[4.18]
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 [4.11], 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 𝐹𝑓𝑓𝑓 = 0, we get Equation [4.19].
𝒀𝒀𝒀 𝒓𝒓𝒓𝒓 𝒈𝒈𝒈𝒈 =
𝜔𝑧
𝜔𝑧 1
𝑣𝑥 ⁄𝑅
𝒗𝒙
= �𝑢𝑢𝑢:
≈ �≈
=
;
2
𝑚 ∙ 𝑣𝑥
𝐿
𝛿𝑓
𝑣𝑥 𝑅
𝑳 + 𝑲𝒖 ∙ 𝒎 ∙ 𝒗𝒙 𝟐
+ 𝐾𝑢 ∙
𝑅
𝑅
𝑪𝑪𝑪𝑪𝑪𝑪𝑪𝑪𝑪 𝒈𝒈𝒈𝒈 =
Îş
1⁄𝑅
1⁄𝑅
𝟏
=
=
=
;
2
𝐿
𝑚 ∙ 𝑣𝑥
𝛿𝑓
𝛿𝑓
𝑳 + 𝑲𝒖 ∙ 𝒎 ∙ 𝒗𝒙 𝟐
+
𝐾
∙
𝑢
𝑅
𝑅
𝑳𝑳𝑳𝑳𝑳𝑳𝑳 𝒂𝒂𝒂𝒂𝒂𝒂𝒂𝒂𝒂𝒂𝒂𝒂 𝒈𝒈𝒈𝒈 =
𝑢𝑢𝑢:
𝑣𝑥 2
𝑎𝑦 = 𝜔𝑧 ∙ 𝑣𝑥 ;
𝑎𝑦
𝒗𝒙 𝟐
𝑅
≈
=ďż˝
=
;
=
ďż˝
2
𝟐
𝜔𝑧 1
𝐿
𝑚
∙
𝑣
𝛿𝑓
𝑳
+
𝑲
∙
𝒎
∙
𝒗
𝑥
𝒖
𝒙
𝑎𝑎𝑎
≈ ;
𝑅 + 𝐾𝑢 ∙ 𝑅
𝑣𝑥 𝑅
=
[4.19]
Yaw rate gain is also derived for 𝐹𝑓𝑓𝑓 ≠ 0, and then we get Equation [4.20].
𝑾𝑾𝑾𝑾 𝑭𝒇𝒇𝒇 𝒕𝒕𝒕𝒕𝒕 𝒊𝒏𝒏𝒏 𝒂𝒂𝒂𝒂𝒂𝒂𝒂: 𝒀𝒀𝒀 𝒓𝒓𝒓𝒓 𝒈𝒈𝒈𝒈 =
=
𝐶𝑓 ∙ 𝐶𝑟 ∙ 𝐿 + �𝐶𝑟 ∙ 𝑙𝑓 + 𝐶𝑟 ∙ 𝑙𝑟 � ∙ 𝐹𝑓𝑓𝑓
𝜔𝑧
=
∙𝑣
𝛿𝑓 𝐶𝑓 ∙ 𝐶𝑟 ∙ 𝐿2 − �𝐶𝑓 ∙ 𝑙𝑓 − 𝐶𝑟 ∙ 𝑙𝑟 � ∙ 𝑚 ∙ 𝑣𝑥 2 𝑥
149
[4.20]
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]
/haracteristic
speed (for
understeered
vehicle)
15
10
Understeered
Ku = 2.525e-6 [1/N]
5
0
/ritical speed (for
oversteered vehicle)
0
5
10
15
20
25
vx [m/s]
30
35
40
45
50
Figure 4-24: Yaw rate gain (𝜔𝑧 /𝛿𝑓 ) for Steady State Cornering.
In each “cluster of 3 curves”: Mid curve shows 𝐹𝑓𝑓𝑓 = 0.
Upper shows 𝐹𝑓𝑓𝑓 = +0.25 ∙ 𝐹𝑓𝑓 . Lower shows 𝐹𝑓𝑓𝑓 = −0.25 ∙ 𝐹𝑓𝑓 .
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.
Curvature gain
3
Oversteered
Ku = -1.794e-6 [1/N]
2.5
curvature gain, (1/R)/df [1/(m*rad)]
4.3.6.2
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]
1⁄𝑅
30
35
40
45
50
Figure 4-25: Curvature gain ( 𝛿 ) for Steady State Cornering
𝑓
150
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 4-26 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 (�𝑎𝑦 ⁄𝛿𝑓 �⁄𝑣𝑥 ) reaches its highest value. This is an alternative definition of
characteristic speed, cf Section 4.3.5.1.
500
Oversteered
lateral acceleration gain, ay/df [(m/(s*s))/rad]
Neutral steered
Ku = -1.794e-6 [1/N]
450
Ku = 0e-6 [1/N]
400
350
300
250
Understeered
200
Ku = 2.525e-6 [1/N]
150
/haracteristic speed is
where gain curve has a
tangent though origin
100
50
0
0
5
10
15
20
25
vx [m/s]
30
35
40
45
50
𝑎𝑦
Figure 4-26: Lateral acceleration gain ( 𝛿 ) for Steady State Cornering
𝑓
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 as function of speed
All gains above can be found from solving wz from Equation [4.11]. If instead solving the other
unknown, vy, we can draw “side slip gain” instead. Equation [4.21] shows the formula for this.
𝑣𝑦
𝐶𝑓 ∙ 𝐶𝑟 ∙ 𝑙𝑟 ∙ 𝐿 − 𝐶𝑓 ∙ 𝑙𝑓 ∙ 𝑚 ∙ 𝑣𝑥 2
=
𝑣𝑥 ∙ 𝛿𝑓 𝐶𝑓 ∙ 𝐶𝑟 ∙ 𝐿2 − �𝐶𝑓 ∙ 𝑙𝑓 − 𝐶𝑟 ∙ 𝑙𝑟 � ∙ 𝑚 ∙ 𝑣𝑥 2
[4.21]
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 4-27.
151
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 4-27: “Side slip gain” (𝑣
𝑣𝑦
𝑥 ∙𝛿𝑓
) for Steady State Cornering.
All cases in Figure 4-27 goes from positive side slip to negative when speed increases. This is the
same as we expected already in Figure 4-21.
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 [4.15].
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 4-20, with the exception that we don’t assume linear tyre models.
152
LATERAL DYNAMICS
Equilibrium:
𝑙𝑓
𝑣𝑥 2
𝑙𝑟
= 𝑚 ∙ 𝑎𝑦 = 𝐹𝑓𝑓 + 𝐹𝑟𝑟 ; 0 = 𝐹𝑓𝑓 ∙ 𝑙𝑓 − 𝐹𝑟𝑟 ∙ 𝑙𝑟 ; � ⇒ 𝐹𝑓𝑓 = ∙ 𝑚 ∙ 𝑎𝑦 ; 𝐹𝑟𝑟 = ∙ 𝑚 ∙ 𝑎𝑦
�𝑚 ∙
𝑅
𝐿
𝐿
Constitution:
𝐹𝑓𝑓 = 𝐹𝑓𝑓 �𝛼𝑓 � ⇒ 𝛼𝑓 = 𝐹𝑓𝑓 −1 �𝐹𝑓𝑦 �; 𝐹𝑟𝑟 = 𝐹𝑟𝑟 (𝛼𝑟 ) ⇒ 𝛼𝑟 = 𝐹𝑟𝑟 −1 �𝐹𝑟𝑟 �;
Solving for 𝛼𝑓 − 𝛼𝑟 yields:
𝑙𝑓
𝑙𝑟
𝛼𝑟 − 𝛼𝑓 = 𝐹𝑟𝑟 −1 �𝐹𝑟𝑟 � − 𝐹𝑓𝑓 −1 �𝐹𝑓𝑓 � = 𝐹𝑟𝑟 −1 � ∙ 𝑚 ∙ 𝑎𝑦 � − 𝐹𝑓𝑓 −1 � ∙ 𝑚 ∙ 𝑎𝑦 � ;
𝐿
𝐿
So, we can plot 𝛼𝑟 − 𝛼𝑓 as function of 𝑎𝑦 . This relation is interesting because compatibility
𝐿
𝐿
(𝛿𝑓 + 𝛼𝑓 − 𝛼𝑟 = 𝑅 ;) yields 𝛼𝑟 − 𝛼𝑓 = 𝛿𝑓 − 𝑅 = 𝛿𝑓 − 𝛿𝐴 . And 𝛿𝑓 − 𝛿𝐴 is connected to one of the
understandings of 𝐾𝑢 in Equation [4.15], (𝐾𝑢 =
𝜕
�𝛿
𝜕𝑎𝑦 𝑓
− 𝛿𝐴 �;). If we plot 𝛼𝑟 − 𝛼𝑓 = 𝛿𝑓 − 𝛿𝐴 on
abscissa axis and 𝑎𝑦 on ordinate axis, we get the most common way of drawing the handling
diagram, see Figure 4-28. The axle’s constitutive relations can be used as graphical support to
𝐿
1
construct the diagram, but then the constitutive relations should be plotted as: 𝑎𝑦𝑦 (𝛼𝑖 ) = 𝐿−𝑙 ∙ 𝑚 ∙
𝑖
𝐹𝑖𝑖 (𝛼𝑖 );. The quantity 𝑎𝑦𝑦 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 4-28 shows the construction of a handling diagram from axle slip characteristics. Figure 4-29
show examples of handling diagrams constructed via tests with simulation tools. Handling diagrams
can be designed from real vehicle tests as well.
𝑎𝑦
front axle, 𝑎𝑦 = 𝑎𝑦𝑓 =
rear axle, 𝑎𝑦 = 𝑎𝑦𝑟 =
𝛼𝑖
𝐿
𝑙f
∙
1
𝑚
𝐿
𝑙𝑟
∙
1
𝑚
∙ 𝐹𝑟𝑟 𝛼𝑟
Handling diagram
𝑎𝑦
Linear axle models,
understeered vehicle
∙ 𝐹𝑓𝑦 𝛼𝑓
Using the axle
models above
Normal vehicle stays
understeered
𝛿𝑓 − 𝛿𝐴 =
= 𝛼𝑟 − 𝛼𝑓
Figure 4-28: 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.
153
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 4-29: Example of handling diagram.
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.
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 4-30: Free Body Diagram for cornering vehicle. The force m*ay is a
fictive force. Subscript l and r here means left and right.
154
LATERAL DYNAMICS
Moment equilibrium, around left contact with ground:
𝑤
𝑔
ℎ
𝑚 ∙ 𝑔 ∙ + 𝑚 ∙ 𝑎𝑦 ∙ ℎ − 𝐹𝑧𝑧 ∙ 𝑤 = 0 ⇒ 𝐹𝑧𝑧 = 𝑚 ∙ � + 𝑎𝑦 ∙ � ;
2
2
𝑤
[4.22]
𝑔
ℎ
Moment equilibrium, around right contact with ground: ⇒ 𝐹𝑧𝑧 = 𝑚 ∙ � 2 − 𝑎𝑦 ∙ 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.
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
y
zl
m*ay
zr
px
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 4-31: 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.
𝐹𝑧𝑧 = 𝐹𝑧𝑧0 + 𝑐𝑙 ∙ (𝑧𝑙𝑙 − 𝑧𝑙 ) 𝑎𝑎𝑎 𝐹𝑧𝑧 = 𝐹𝑧𝑧0 + 𝑐𝑟 ∙ (𝑧𝑟𝑟 − 𝑧𝑟 )
𝑤ℎ𝑒𝑒𝑒 𝐹𝑧𝑧0 + 𝐹𝑧𝑧0 = 𝑚 ∙ 𝑔 𝑎𝑎𝑎 𝐹𝑧𝑧0 ∙ 𝑤⁄2 − 𝐹𝑧𝑧0 ∙ 𝑤⁄2 = 0
[4.23]
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”.
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:
155
LATERAL DYNAMICS
𝐹𝑦𝑦 − 𝑚 ∙ 𝑎𝑦 = 0;
𝑚 ∙ 𝑔 − 𝐹𝑧𝑧 − 𝐹𝑧𝑧 = 0;
𝐹𝑧𝑧 ∙ (𝑤⁄2) − 𝐹𝑧𝑧 ∙ (𝑤⁄2) + 𝐹𝑦𝑦 ∙ ℎ + 𝑚 ∙ 𝑔 ∙ (−𝑦) = 0;
[4.24]
The term 𝑚 ∙ 𝑔 ∙ (−𝑦) is taken into account, but not corresponding for 𝑚 ∙ 𝑎𝑦 , 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, ℎ𝑅𝑅 ,we can expressed – 𝑦 = (ℎ − ℎ𝑅𝑅 ) ∙ 𝑝𝑥 . 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 ℎ𝑅𝑅 ≪ ℎ. This causes a significant “pendulum effect”, especially for heavy
trucks.
Compatibility, to introduce body displacements, z and px, gives:
𝑧𝑙 = 𝑧 + (𝑤⁄2) ∙ 𝑝𝑥 ;
𝑎𝑎𝑎 𝑧𝑟 = 𝑧 − (𝑤⁄2) ∙ 𝑝𝑥 ;
[4.25]
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-cl*zl', ...
'Fzr=Fzr0-cr*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);
[4.26]
The results from the Matlab script in Equation [4.26]:
𝐹𝑦𝑦 = 𝑚 ∙ 𝑎𝑦 ;
𝑚 ∙ 𝑎𝑦 ∙ ℎ ∙ 𝑤
𝑐𝑙 − 𝑐𝑟
∙
;
2
2
−𝑤 ∙ 𝑐𝑙 ∙ 𝑐𝑟 + 𝑚 ∙ 𝑔 ∙ (ℎ − ℎ𝑅𝑅 ) ∙ (𝑐𝑙 + 𝑐𝑟 )
𝑐𝑙 + 𝑐𝑟
;
𝑝𝑥 = 𝑚 ∙ 𝑎𝑦 ∙ ℎ ∙
2
+𝑤 ∙ 𝑐𝑙 ∙ 𝑐𝑟 − 𝑚 ∙ 𝑔 ∙ (ℎ − ℎ𝑅𝑅 ) ∙ (𝑐𝑙 + 𝑐𝑟 )
𝑧=
1
⎛𝑔 𝑎𝑦 ∙ ℎ
⎞
∙�
𝐹𝑧𝑧 = 𝑚 ∙ ⎜ −
�⎟ ;
ℎ
−
ℎ
𝑐 +𝑐
𝑤
2
𝑅𝑅
1 + 𝑚 ∙ 𝑔 ∙ 𝑐𝑙 ∙ 𝑐 𝑟 ∙
𝑤2
𝑙
𝑟
⎝
⎠
1
⎛𝑔 𝑎𝑦 ∙ ℎ
⎞
𝐹𝑧𝑧 = 𝑚 ∙ ⎜ +
∙�
�⎟ ;
𝑐 + 𝑐 ℎ − ℎ𝑅𝑅
2
𝑤
1 + 𝑚 ∙ 𝑔 ∙ 𝑐𝑙 ∙ 𝑐 𝑟 ∙
𝑤2
𝑙
𝑟
⎝
⎠
[4.27]
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) if cl=cr, which is
reasonable for most vehicles due to symmetry left / right. The formula uses ℎ𝑅𝑅 which we cannot
estimate without analysis of the suspension. Since front and rear axle normally are different, we
could expect that ℎ𝑅𝑅 is expressed in some similar quantities for each of front and rear axle, which
also is the case in our further analysis, see Equation [4.34].
156
LATERAL DYNAMICS
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 within one axle
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 4-31). 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 dynamics is,
that because of the non-symmetry, we need to introduce this already in steady state analysis.
The two modelling ways to include the suspension in the load transfer are shown in Figure 4-32.
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:
•
•
It has both heave and roll degree of freedom. (Roll centre model is restricted to roll around
roll centre.)
lt 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.
157
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 4-32: 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 4-33. 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.
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.
158
LATERAL DYNAMICS
forces, on body:
displacements, speeds, accelerations:
ay
m*g
z
zfl=zrl=0
y
m*ay
zfr=zrr
px
Pry
Pfy
Fsfl+Fsrl
zflr=zrlr=0
zfrr=zrrr=0
forces,
on vehicle:
forces, on front axle:
m*g
m*ay
Fsfl
Fflz+Frlz
forces, on rear axle:
Fsfr
Fsrl
Pfz
Ffry+Frry
Ffrz+Frrz
Prz
Ffly
Fflz
Ffry
Ffrz
Fsrr
Pry
Pfy
h
Ffly+Frly
Fsfr+Fsrr
Pfz+Prz
Frly
Frlz
Frry
Frrz
Figure 4-33: 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.
𝐹𝑠𝑠𝑠 = 𝐹𝑠𝑠𝑠0 + 𝑐𝑓𝑓 ∙ �𝑧𝑓𝑓𝑓 − 𝑧𝑓𝑓 �;
𝐹𝑠𝑠𝑠 = 𝐹𝑠𝑠𝑠0 + 𝑐𝑓𝑓 ∙ �𝑧𝑓𝑓𝑓 − 𝑧𝑓𝑓 �;
𝐹𝑠𝑠𝑠 = 𝐹𝑠𝑠𝑠0 + 𝑐𝑟𝑟 ∙ (𝑧𝑟𝑟𝑟 − 𝑧𝑟𝑟 );
𝐹𝑠𝑠𝑠 = 𝐹𝑠𝑠𝑠0 + 𝑐𝑟𝑟 ∙ (𝑧𝑟𝑟𝑟 − 𝑧𝑟𝑟 );
𝑚 ∙ 𝑔 ∙ 𝑙𝑓
𝑚 ∙ 𝑔 ∙ 𝑙𝑟
𝑤ℎ𝑒𝑒𝑒 𝐹𝑠𝑠𝑠0 = 𝐹𝑠𝑠𝑠0 =
; 𝑎𝑎𝑎 𝐹𝑠𝑠𝑠0 = 𝐹𝑠𝑠𝑟0 =
;
2∙𝐿
2∙𝐿
[4.28]
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):
159
LATERAL DYNAMICS
𝐹𝑓𝑓𝑓 + 𝐹𝑓𝑓𝑓 + 𝐹𝑟𝑟𝑟 + 𝐹𝑟𝑟𝑟 = 𝑚 ∙ 𝑔;
𝑚 ∙ 𝑎𝑦 = 𝐹𝑓𝑓 + 𝐹𝑟𝑟 ;
0 = 𝐹𝑓𝑓 ∙ 𝑙𝑓 − 𝐹𝑟𝑟 ∙ 𝑙𝑟 ;
−�𝐹𝑓𝑓𝑓 + 𝐹𝑓𝑓𝑓 � ∙ 𝑙𝑓 + (𝐹𝑟𝑟𝑟 + 𝐹𝑟𝑟𝑟 ) ∙ 𝑙𝑟 = 0;
𝑤
𝑤
�𝐹𝑓𝑓𝑓 + 𝐹𝑟𝑟𝑟 � ∙ − �𝐹𝑓𝑓𝑓 + 𝐹𝑟𝑟𝑟 � ∙ + �𝐹𝑓𝑓 + 𝐹𝑟𝑟 � ∙ ℎ = 0;
2
2
[4.29]
Equilibrium for each axle (roll, around roll centre):
𝑤
𝑤
− �𝐹𝑓𝑓𝑓 − 𝐹𝑠𝑠𝑠 � ∙ + 𝐹𝑓𝑓 ∙ ℎ𝑅𝑅𝑅 = 0;
2
2
𝑤
𝑤
(𝐹𝑟𝑟𝑟 − 𝐹𝑠𝑠𝑠 ) ∙ − (𝐹𝑟𝑟𝑟 − 𝐹𝑠𝑠𝑠 ) ∙ + 𝐹𝑟𝑟 ∙ ℎ𝑅𝑅𝑅 = 0;
2
2
�𝐹𝑓𝑓𝑓 − 𝐹𝑠𝑠𝑠 � ∙
[4.30]
Compatibility, to introduce body displacements, z, px and py, gives:
𝑤
∙ 𝑝 − 𝑙𝑓 ∙ 𝑝𝑦 ;
2 𝑥
𝑤
𝑧𝑓𝑓 = 𝑧 − ∙ 𝑝𝑥 − 𝑙𝑓 ∙ 𝑝𝑦 ;
2
𝑤
𝑧𝑟𝑟 = 𝑧 + ∙ 𝑝𝑥 + 𝑙𝑟 ∙ 𝑝𝑦 ;
2
𝑤
𝑧𝑟𝑟 = 𝑧 − ∙ 𝑝𝑥 + 𝑙𝑟 ∙ 𝑝𝑦 ;
2
𝑧𝑓𝑓 + 𝑧𝑓𝑓 = 0;
𝑧𝑟𝑟 + 𝑧𝑟𝑟 = 0;
𝑧𝑓𝑓 = 𝑧 +
[4.31]
The measure ∆h is redundant and can be connected to the other geometry measures as follows. The
geometrical interpretation is given in Figure 4-34.
∆ℎ = ℎ −
𝑙𝑟 ∙ ℎ𝑅𝑅𝑅 + 𝑙𝑓 ∙ ℎ𝑅𝑅𝑅
;
𝐿
[4.32]
Combining Equations [4.28] to [4.32] 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)', ...
160
[4.33]
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 [4.33], but in a prettier writing format:
𝑙𝑟
;
𝐿
𝑙𝑓
𝐹𝑟𝑟 = 𝑚 ∙ 𝑎𝑦 ∙ ; ;
𝐿
𝑧 = 0;
𝑚 ∙ 𝑎𝑦 ∙ ∆ℎ �𝐹𝑓𝑓 + 𝐹𝑟𝑟 � ∙ ∆ℎ
=
;
𝑝𝑥 =
𝑐𝑟𝑟𝑟𝑟,𝑣𝑣ℎ𝑖𝑖𝑖𝑖
𝑐𝑟𝑟𝑟𝑟,𝑣𝑣ℎ𝑖𝑖𝑖𝑖
𝑝𝑦 = 0;
ℎ𝑅𝑅𝑅 ∙ 𝑙𝑟 ∆ℎ
𝑐𝑓,𝑟𝑟𝑟𝑟
𝑔 ∙ 𝑙𝑟
𝐹𝑓𝑓𝑓 = 𝑚 ∙ �
− 𝑎𝑦 ∙ �
+
∙
�� ;
2∙𝐿
𝐿∙𝑤
𝑤 𝑐𝑟𝑟𝑟𝑟,𝑣𝑣ℎ𝑖𝑖𝑖𝑖
𝐹𝑓𝑓 = 𝑚 ∙ 𝑎𝑦 ∙
𝐹𝑓𝑓𝑓 = 𝑚 ∙ �
𝐹𝑟𝑟𝑟
ℎ𝑅𝑅𝑅 ∙ 𝑙𝑟 ∆ℎ
𝑐𝑓,𝑟𝑟𝑟𝑟
𝑔 ∙ 𝑙𝑟
+ 𝑎𝑦 ∙ �
+
∙
�� ;
2∙𝐿
𝐿∙𝑤
𝑤 𝑐𝑟𝑟𝑟𝑟,𝑣𝑣ℎ𝑖𝑖𝑖𝑖
𝑔 ∙ 𝑙𝑓
ℎ𝑅𝑅𝑅 ∙ 𝑙𝑓 ∆ℎ
𝑐𝑟,𝑟𝑟𝑟𝑟
= 𝑚∙�
− 𝑎𝑦 ∙ �
+
∙
�� ;
2∙𝐿
𝐿∙𝑤
𝑤 𝑐𝑟𝑟𝑟𝑟,𝑣𝑣ℎ𝑖𝑖𝑖𝑖
[4.34]
𝑔 ∙ 𝑙𝑓
ℎ𝑅𝑅𝑅 ∙ 𝑙𝑓 ∆ℎ
𝑐𝑟,𝑟𝑟𝑟𝑟
+ 𝑎𝑦 ∙ �
+
∙
�� ;
2∙𝐿
𝐿∙𝑤
𝑤 𝑐𝑟𝑟𝑟𝑟,𝑣𝑣ℎ𝑖𝑖𝑖𝑖
𝐹𝑟𝑟𝑟 = 𝑚 ∙ �
where, roll stiffnesses in [moment/angle], are:
𝑤 2
𝑐𝑓,𝑟𝑟𝑟𝑟 = 2 ∙ 𝑐𝑓𝑓 ∙ � �
2
𝑤 2
𝑐𝑟,𝑟𝑟𝑟𝑟 = 2 ∙ 𝑐𝑟𝑟 ∙ � �
2
𝑁𝑁
ďż˝;
𝑟𝑟𝑟
𝑁𝑁
ďż˝;
ďż˝
𝑟𝑟𝑟
ďż˝
𝑐𝑟𝑟𝑟𝑟,𝑣𝑣ℎ𝑖𝑖𝑖𝑖 = 𝑐𝑓,𝑟𝑟𝑟𝑟 + 𝑐𝑟,𝑟𝑟𝑟𝑟
ďż˝
𝑁𝑁
ďż˝;
𝑟𝑟𝑟
We should compare Equation [4.34] with Equation [4.27]. Equation [4.27] considers the “pendulum
effect”, but not the differentiation between front and rear suspension. Equation [4.34] does the
opposite.
Assume ℎ = ℎ𝑅𝑅 and look at the sum of vertical force on one side, 𝐹𝑧𝑧 in Equation [4.27]. Compare
𝐹𝑧𝑧 in Equation [4.27] and 𝐹𝑓𝑓𝑓 + 𝐹𝑟𝑟𝑟 in Equation [4.34]; the equations agree if:
ℎ𝑅𝑅𝑅 ∙ 𝑙𝑟 + ℎ𝑅𝑅𝑅 ∙ 𝑙𝑓 ∆ℎ
𝑔
𝑔
ℎ
𝐹𝑓𝑓𝑓 + 𝐹𝑟𝑟𝑟 = 𝐹𝑧𝑧 ⇒ 𝑚 ∙ � − 𝑎𝑦 ∙ �
+ �� = 𝑚 ∙ � − 𝑎𝑦 ∙ � = 𝐹𝑧𝑧 ⇒
2
𝐿∙𝑤
2
𝑤
𝑤
ℎ𝑅𝑅𝑅 ∙ 𝑙𝑟 + ℎ𝑅𝑅𝑅 ∙ 𝑙𝑓 ∆ℎ ℎ
⇒
+
= ⇒ ℎ𝑅𝑅𝑅 ∙ 𝑙𝑟 + ℎ𝑅𝑅𝑅 ∙ 𝑙𝑓 = (ℎ − ∆ℎ) ∙ 𝐿;
𝐿∙𝑤
𝑤
𝑤
This is exactly in agreement with the definition of the redundant geometric parameter ∆h, see
Equation [4.32]. This means that a consistent geometric model of the whole model is as drawn in
Figure 4-34. Here the artefact roll axis is also defined.
161
LATERAL DYNAMICS
front axle
roll centre
CoG
hRCf
∆h
front axle
h
rear axle
roll centre
hRCr
rear axle
∆ℎ = ℎ −
𝑙𝑟 ∙ ℎ𝑅𝑅𝑅 + 𝑙𝑓 ∙ ℎ𝑅𝑅𝑅
𝐿
Figure 4-34: Roll Axis for 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
ℎ𝑅𝑅𝑅 ∙𝑙𝑗
𝐿∙𝑤
in Equation [4.34] can be seen as the part of the lateral tyre forces that goes
via the stiff linkage. The terms of type
∆ℎ
𝑐
∙ 𝑖𝑖
𝑤 𝑐𝑖𝑖 +𝑐𝑗𝑗
in Equation [4.34] 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 4-35 shows the physical model which
includes anti-roll bars. It is an update of Figure 4-33.
Eq [4.28] will remain, and constitutive relation for anti-roll bars is added as in Eq [4.35]. 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, 𝑐𝑖𝑖 𝑤ℎ𝑒𝑒𝑒 𝑖 = 𝑓, 𝑟.
𝐹𝑎𝑎 = 𝑐𝑓𝑓 ∙ ��𝑧𝑓𝑓𝑓 − 𝑧𝑓𝑓 � − �𝑧𝑓𝑓𝑓 − 𝑧𝑓𝑓 �� ;
𝐹𝑎𝑎 = 𝑐𝑟𝑟 ∙ �(𝑧𝑟𝑟𝑟 − 𝑧𝑟𝑟 ) − (𝑧𝑟𝑟𝑟 − 𝑧𝑟𝑟 )�;
162
[4.35]
LATERAL DYNAMICS
forces, on body:
displacements, speeds, accelerations :
m*g
ay
z
zfl=zrl=0
y
Faf*w
Far*w
zfr=zrr
px
zflr=zrlr=0
zfrr=zrrr=0
Anti-roll bar
torsional spring
m*ay
Pry
Pfy
hnly one anti-roll bar spring
drawn, but there can be one
anti-roll bar on each axle.
Fsfl+Fsrl
Fsfr+Fsrr
Pfz+Prz
w
forces,
on vehicle:
forces, on front axle:
m*g
m*ay
Fsfl
Ffry+Frry
Fflz+Frlz
Fsfr
Fsrl
Pfz
h
Ffly+Frly
forces, on rear axle:
Ffrz+Frrz
Faf
Ffly
Fflz
Pfy Faf
Ffry
Ffrz
Prz
Fsrr
Pry
Far
Far
Frly
Frlz
Frry
Frrz
Figure 4-35: 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.
The equilibrium for whole vehicle, Eq [4.29], does not change. The equilibrium for each axle, Eq
[4.30], changes to the following:
Equilibrium for each axle (pitch, around roll centre):
𝑤
𝑤
�𝐹𝑓𝑓𝑓 − 𝐹𝑠𝑠𝑠 + 𝐹𝑎𝑎 � ∙ − �𝐹𝑓𝑓𝑓 − 𝐹𝑠𝑠𝑠 − 𝐹𝑎𝑎 � ∙ + 𝐹𝑓𝑓 ∙ ℎ𝑅𝑅𝑅 = 0;
2
2
𝑤
𝑤
(𝐹𝑟𝑟𝑟 − 𝐹𝑠𝑠𝑠 + 𝐹𝑎𝑎 ) ∙ − (𝐹𝑟𝑟𝑟 − 𝐹𝑠𝑠𝑠 − 𝐹𝑎𝑎 ) ∙ + 𝐹𝑟𝑟 ∙ ℎ𝑅𝑅𝑅 = 0;
2
2
[4.36]
The compatibility Equation, Eq [4.31], does not change and the definition of Δℎ, Eq [4.32], can also
be kept. It shows that Eq [4.34] is still valid, but with new definitions of the roll stiffnesses as follows:
4.3.9.5
𝑤 2
𝑁𝑁
𝑐𝑓,𝑟𝑟𝑟𝑟 = 2 ∙ �𝑐𝑓𝑓 + 2 ∙ 𝑐𝑎𝑎 � ∙ � �
ďż˝
ďż˝;
2
𝑟𝑟𝑟
𝑁𝑁
𝑤 2
𝑐𝑟,𝑟𝑟𝑟𝑟 = 2 ∙ (𝑐𝑟𝑟 + 2 ∙ 𝑐𝑎𝑎 ) ∙ � �
ďż˝;
ďż˝
𝑟𝑟𝑟
2
𝑁𝑁
𝑐𝑟𝑟𝑟𝑟,𝑣𝑣ℎ𝑖𝑖𝑖𝑖 = 𝑐𝑓,𝑟𝑟𝑟𝑟 + 𝑐𝑟,𝑟𝑟𝑟𝑟 �
ďż˝;
𝑟𝑟𝑟
[4.37]
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 4-32 , 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
4-32. For these, there are no (rigid) connections between left and right wheel.
163
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 4-36 and Figure 4-37 show design of two axles with independent wheel suspensions. Figure
4-38 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 4-37, one should mention
that the strut is designed to take bending moments. For the rigid axle in Figure 4-38, 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 4-36: 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
Figure 4-37: Example of how to appoint the pivot point for one wheel and roll centre for
axle with double McPherson suspension.
164
LATERAL DYNAMICS
Figure 4-38: 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).
Generally, a ”rigid axle” gives roll centre height on approximately the same magnitude as wheel
radius, see Figure 4-38. With individual wheel suspension one have much larger flexibility, and typical
chosen designs are 30..90 mm front and 90..120 mm rear.
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 2-23. 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.
165
LATERAL DYNAMICS
3.7 ∙ 105 𝑁/𝑟𝑟𝑟
3.1 ∙ 105 𝑁/𝑟𝑟𝑟
2.1 ∙ 105
𝑁/𝑟𝑟𝑟
Both wheels, if no load transfer.
Axle cornering stiffness ≈
≈ 2 ∙ 3.1 = 6.2 ∙ 105 𝑘𝑁/𝑟𝑟𝑟
Left and right wheel, if load transfer ±25 𝑘𝑘.
Axle cornering stiffness ≈ 2.1 + 3.7 = 5.8 ∙ 105 𝑁/𝑟𝑟𝑟
±25 𝑘𝑘
𝜕
±25 𝑘𝑘
Figure 4-39: The wheels cornering stiffness (�𝜕𝜕 𝐹𝑦 ��
𝑠𝑦 =0
) 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 Steady state cornering roll-over
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:
166
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.
In the following, 4-wheeled vehicles will be assumed. The 2nd threshold will be used.
167
LATERAL DYNAMICS
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:
SSF =
𝐻𝐻𝐻𝐻 𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇ℎ 𝑤⁄2
=
;
𝐻𝐻𝐻𝐻𝐻ℎ𝑂𝑂𝑂𝑂𝑂
ℎ
[4.38]
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 lif 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
4-40.
view from rear, when turning left
ay
m*g
m*ay
h
Fy
Fiz≥0
(=roll-over
threshold)
w/2
Foz
w
Figure 4-40: Model for verification of requirement based on Static Stability Factor, SSF.
The derivation of the SSF based requirement looks as follows:
𝑀𝑀𝑀𝑀𝑀:
⎧
⎫
𝑤 ⎫
⎪𝐹𝑖𝑖 ∙ 𝑤 + 𝑚 ∙ 𝑎𝑦 ∙ ℎ = 𝑚 ∙ 𝑔 ∙ ;⎪
1 ℎ∙𝜇 ⎪
2
⇒ 𝐹𝑖𝑖 = 𝑚 ∙ 𝑔 ∙ � −
� ;⎪ 1 ℎ ∙ 𝜇
𝑤
2
𝑤
⎨
𝐹𝑖𝑖 + 𝐹𝑜𝑜 = 𝑚 ∙ 𝑔;
⎬
⇒ >
⇒
= 𝑆𝑆𝑆 > 𝜇;
⎪
⎪
2
𝑤
2
∙ℎ
⎬
⎩ 𝑚 ∙ 𝑎𝑦 = 𝐹𝑦 = 𝜇 ∙ (𝐹𝑖𝑖 + 𝐹𝑜𝑜 ); ⎭
⎪
𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅:
⎪
ďż˝
ďż˝
𝐹𝑖𝑖 ≥ 0;
⎭
[4.39]
Maximum road friction, µ, is typically 1, which is why SSF>µ=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.
168
LATERAL DYNAMICS
4.3.11.3 Steady-state cornering roll-over
4.3.11.3.1
Model with fore/aft symmetry
The model in Figure 4-40 and Equation [4.39] assumes fore/aft symmetry. One can derive this
𝑤
> 𝜇, and this interpretation of performance (𝑎𝑦 ) limitation due to rollrequirement on design:
over:
𝑎𝑦
𝑔
<
𝑤
.
2∙ℎ
𝑎𝑦
affects this:
Figure 4-41.
•
𝑔
2∙ℎ
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
The tyre will take the vertical load on its outer edge in a roll-over situation. This suggests a
𝑎
𝑤+𝑤𝑡𝑡𝑡𝑡
change of performance and requirement to: 𝑦 <
> 𝜇. This effect is accentuated
𝑔
•
relative to the tyre. This could motivate
•
•
•
2∙ℎ
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,
𝑎𝑦
𝑔
<
𝑤−𝐷𝐷𝐷𝑦
2∙ℎ
> 𝜇. This effect increases the risk for
roll-over.
Due to suspension linkage and compliances, the body will roll. Since the CoG height above
𝑎
𝑤−∆ℎ∙𝜑𝑥
roll axis, ∆ℎ, normally is positive, this could motivate 𝑦 <
> 𝜇. This effect increases
𝑔
2∙ℎ
the risk for roll-over.
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
𝑎
𝑤
into account. The heave is normally positive. This could motivate 𝑦 <
> 𝜇. The effect
𝑔
2∙(ℎ+𝑧)
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
𝜑𝑥
ay
h
∆ℎ
m*g
m*ay
𝑧
𝐷𝐷𝐷𝑦
Fiz=0
(=roll-over
threshold)
w/2
w
Fy
Foz
wtyre/2
Figure 4-41: Steady-state roll-over model, with fore/aft symmetry. The measures
𝑤𝑡𝑡𝑡𝑡 , 𝐷𝐷𝐷𝑦 , ∆ℎ ∙ 𝜑𝑥 𝑎𝑎𝑎 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 [4.34]. 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:
169
LATERAL DYNAMICS
𝑔∙𝑙
𝐹𝑓𝑓𝑓 = 𝑚 ∙ � 2∙𝐿𝑟 − 𝑎𝑦 ∙ �
𝑔∙𝑙𝑓
ℎ𝑅𝑅𝑅 ∙𝑙𝑟
𝐿∙𝑤
𝑐
∆ℎ
∙ 𝑓𝑓 ��
𝑤 𝑐𝑓𝑓 +𝑐𝑟𝑟
ℎ𝑅𝑅𝑅 ∙𝑙𝑓
𝐹𝑟𝑟𝑟 = 𝑚 ∙ � 2∙𝐿 − 𝑎𝑦 ∙ �
⇒ 𝑎𝑦 >
+
𝐿∙𝑤
+
< 0; 𝐴𝐴𝐴⎫
⎪
∆ℎ
𝑐
∙ 𝑟𝑟 ��
𝑤 𝑐𝑓𝑓 +𝑐𝑟𝑟
< 0;
⎬
⎪
⎭
⇒
[4.40]
𝑙𝑓
𝑔
𝑙𝑟
⎞;
∙ max ⎛
;
ℎ𝑅𝑅𝑅 ∙ 𝑙𝑟 ∆ℎ
𝑐𝑓𝑓
ℎ𝑅𝑅𝑅 ∙ 𝑙𝑓 ∆ℎ
2∙𝐿
𝑐𝑟𝑟
+
∙
+
∙
𝑤 𝑐𝑓𝑓 + 𝑐𝑟𝑟
𝐿∙𝑤
𝑤 𝑐𝑓𝑓 + 𝑐𝑟𝑟 ⎠
⎝ 𝐿∙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
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 4.3.11.3.2 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 4-26.
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.
170
LATERAL DYNAMICS
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 [4.19].
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 4-5, 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 4-5, 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
The response will be very dependent of the vehicle longitudinal speed, why the same tests are
typically done at different such speeds.
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.
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 [4.45], and a less general derivation of this
equation in Figure 4-45.)
The vehicle model is sketched in Figure 4-43. The model is a development of the model for steady
state cornering in Figure 4-18, 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:
𝑎𝑥 = 𝑑𝑑𝑑(𝑣𝑥 ) − 𝜔𝑧 ∙ 𝑣𝑦 = 𝑣̇𝑥 − 𝜔𝑧 ∙ 𝑣𝑦 ;
𝑎𝑦 = 𝑑𝑒𝑒�𝑣𝑦 � + 𝜔𝑧 ∙ 𝑣𝑥 = 𝑣̇𝑦 + 𝜔𝑧 ∙ 𝑣𝑥 ;
[4.41]
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 𝑎⃗ = �𝑎𝑥 ; 𝑎𝑦 � and [der(vx); der(vy)]=�𝑣̇𝑥 ; 𝑣̇𝑦 � can be confusing. A
way to understand those could be to think of 𝑎⃗ as the (geometric vector) acceleration in the inertial
coordinate system and �𝑣̇𝑥 ; 𝑣̇𝑦 � as a (mathematical) vector, with derivatives of the scalar
mathematical variables 𝑣𝑥 and 𝑣𝑦 , which are the velocities in the vehicle fix (and moving) coordinate
system. A graphical derivation of Equation [4.41] is found below, in Figure 4-46.
171
speeds:
wz
vx
vrx
lr
vfyv
vry
vy
forces:
vfxv 𝑣⃗𝑓
δf
Frx
m*ay
Fry
m*ax
Ffyv
LATERAL DYNAMICS
Ffxv 𝐹⃗
𝑓
δf
J*δer(wz)
lf
L
lr
lf
L
𝑣⃗𝑓
ax
ay
𝒂
β
𝐹⃗𝑓
𝒗
vy=v*sin(β)
Figure 4-43: One-track model for transient dynamics. Dashed forces and moment are
fictive forces. Compare to Figure 4-43.
The model in Figure 4-43 is documented as mathematical model in Equation [4.42]. 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
[4.10], with the changes marked with underlined text:
172
LATERAL DYNAMICS
//(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;
[4.42]
//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.
The initial longitudinal speed is a parameter, vx=100 km/h. A simulation result from the model is
shown in Figure 4-44. The manoeuvre selected is same steering wheel function of time as in Figure
4-19, for better comparison of the different characteristics of the models.
173
LATERAL DYNAMICS
wz [deg/s]
y [m]
δf [δeg]
vx [km/h]
beta [deg]
x[m]
time [s]
Figure 4-44: 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 [4.42] is a complete model suitable for simulation. Eliminating some variable and rewrite in
prettier format gives:
Equilibrium:
𝑚 ∙ �𝑣̇𝑥 − 𝜔𝑧 ∙ 𝑣𝑦 � = 𝐹𝑓𝑓𝑓 ∙ cos�𝛿𝑓 � − 𝐹𝑓𝑓𝑓 ∙ sin�𝛿𝑓 � + 𝐹𝑟𝑟 ;
𝑚 ∙ �𝑣̇𝑦 + 𝜔𝑧 ∙ 𝑣𝑥 � = 𝐹𝑓𝑓𝑓 ∙ sin�𝛿𝑓 � + 𝐹𝑓𝑓𝑓 ∙ cos�𝛿𝑓 � + 𝐹𝑟𝑟 ;
𝐽 ∙ 𝜔̇ 𝑧 = �𝐹𝑓𝑓𝑓 ∙ sin�𝛿𝑓 � + 𝐹𝑓𝑓𝑓 ∙ cos�𝛿𝑓 �� ∙ 𝑙𝑓 − 𝐹𝑟𝑟 ∙ 𝑙𝑟 ;
Constitution:
𝐹𝑓𝑓𝑓 = −𝐶𝑓 ∙ 𝑠𝑓𝑓 ; 𝑎𝑎𝑎
Compatibility:
𝑠𝑓𝑓 =
𝐹𝑟𝑟 = −𝐶𝑟 ∙ 𝑠𝑟𝑟 ;
[4.43]
𝑣𝑓𝑓𝑓
𝑣𝑦 − 𝑙𝑟 ∙ 𝜔𝑧
; 𝑎𝑎𝑎 𝑠𝑟𝑟 =
;
𝑣𝑓𝑓𝑓
𝑣𝑥
Transformation from vehicle to wheel coordinate system on front axle:
𝑣𝑓𝑓𝑓 = �𝑣𝑦 + 𝑙𝑓 ∙ 𝜔𝑧 � ∙ sin�𝛿𝑓 � + 𝑣𝑥 ∙ cos�𝛿𝑓 � ;
𝑣𝑓𝑓𝑓 = �𝑣𝑦 + 𝑙𝑓 ∙ 𝜔𝑧 � ∙ cos�𝛿𝑓 � − 𝑣𝑥 ∙ sin�𝛿𝑓 � ;
Typically, this model is used for simulation, where 𝛿𝑓 , 𝐹𝑓𝑓𝑓 and 𝐹𝑟𝑟 are input variables. Suitable state
variables are then 𝑐, 𝑣𝑦 and 𝜔𝑧 . 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
174
LATERAL DYNAMICS
slip model. It is also non-linear with respect to that the states appear as multiplied with each other,
e.g. 𝜔𝑧 ∙ 𝑣𝑥 .
For the stationary oscillation events, this compendium will be limited to manoeuvres where both
small angles (steering angle and side slip angles ⇒ sin(𝑎𝑎𝑎𝑎𝑎) = 𝑎𝑎𝑎𝑎𝑎, cos(𝑎𝑎𝑎𝑎𝑎) = 1, 𝑎𝑎𝑎𝑎𝑎 2 =
0) and far from saturating tyre grip and at constant 𝑣𝑥 . Then the model becomes fully linear, which is
suitable for frequency sweep studies. The model then becomes as follows. Note that one state is
eliminated by modelling 𝑣𝑥 is constant, 𝑣̇𝑥 = 0.
𝑚 ∙ �0 − 𝜔𝑧 ∙ 𝑣𝑦 � ∙ 𝑣𝑥 = 𝐹𝑓𝑓𝑓 ∙ 𝑣𝑥 + 𝐶𝑓 ∙ �𝑣𝑦 + 𝑙𝑓 ∙ 𝜔𝑧 � ∙ 𝛿𝑓 + 𝐹𝑟𝑟 ∙ 𝑣𝑥 ;
𝑚 ∙ �𝑣̇𝑦 + 𝜔𝑧 ∙ 𝑣𝑥 � ∙ 𝑣𝑥
= 𝐹𝑓𝑓𝑓 ∙ 𝛿𝑓 ∙ 𝑣𝑥 − 𝐶𝑓 ∙ �𝑣𝑦 + 𝑙𝑓 ∙ 𝜔𝑧 − 𝑣𝑥 ∙ 𝛿𝑓 � − 𝐶𝑟 ∙ �𝑣𝑦 − 𝑙𝑟 ∙ 𝜔𝑧 �;
[4.44]
𝐽 ∙ 𝜔̇ 𝑧 ∙ 𝑣𝑥 = �𝐹𝑓𝑓𝑓 ∙ 𝛿𝑓 ∙ 𝑣𝑥 − 𝐶𝑓 ∙ �𝑣𝑦 + 𝑙𝑓 ∙ 𝜔𝑧 − 𝑣𝑥 ∙ 𝛿𝑓 �� ∙ 𝑙𝑓 + 𝐶𝑟 ∙ �𝑣𝑦 − 𝑙𝑟 ∙ 𝜔𝑧 � ∙ 𝑙𝑟 ;
This equation is written on matrix form (for the two state variables, 𝑣𝑦 and 𝜔𝑧 ) and one algebraic
equation for the longitudinal equilibrium. Also, the small angle assumption is used once again. Then:
𝐶𝑓 + 𝐶𝑟
𝐶𝑓 ∙ 𝑙𝑓 − 𝐶𝑟 ∙ 𝑙𝑟
⎡
+ 𝑚 ∙ 𝑣𝑥 ⎤
𝑣𝑥
𝑣𝑥
𝑣̇𝑦
𝑚 0
⎢
⎥ 𝑣𝑦
�∙� �+⎢
ďż˝
2
2
⎥ ∙ �𝜔𝑧 �
0 𝐽
𝜔̇ 𝑧
𝐶𝑓 ∙ 𝑙𝑓 − 𝐶𝑟 ∙ 𝑙𝑟
𝐶𝑓 ∙ 𝑙𝑓 + 𝐶𝑟 ∙ 𝑙𝑟
⎢
⎥
⎣
𝑣𝑥
𝑣𝑥
⎦
𝐶𝑓 + 𝐹𝑓𝑓𝑓
=ďż˝
� ∙ 𝛿𝑓 ;
�𝐶𝑓 + 𝐹𝑓𝑓𝑓 � ∙ 𝑙𝑓
𝑣𝑦 + 𝑙𝑓 ∙ 𝜔𝑧
𝐹𝑓𝑓𝑓 + 𝐹𝑟𝑟 = −𝑚 ∙ 𝜔𝑧 ∙ 𝑣𝑦 − 𝐶𝑓 ∙ �
� ∙ 𝛿𝑓 ≈ �𝛽𝑓 ∙ 𝛿𝑓 ≈ 0� ≈ −𝑚 ∙ 𝜔𝑧 ∙ 𝑣𝑦 ;
𝑣𝑥
[4.45]
The longitudinal equilibrium (the 3rd, algebraic equation) is often not mentioned, because the
longitudinal dynamics is simply a prescribed constant speed, 𝑣𝑥 . However, this equation falls out and
can be used to calculate required propulsion on the axles to maintain the constant longitudinal
speed.
Neglecting the longitudinal equilibrium and assuming that propulsion forces can be neglected for
lateral and yaw equilibrium:
ďż˝
𝑚
0
𝐶𝑓 + 𝐶𝑟
⎡
𝑣𝑥
𝑣̇𝑦
0
⎢
�∙� �+⎢
𝐽
𝜔̇ 𝑧
𝐶 ∙ 𝑙 − 𝐶𝑟 ∙ 𝑙𝑟
⎢ 𝑓 𝑓
⎣
𝑣𝑥
𝐶𝑓 ∙ 𝑙𝑓 − 𝐶𝑟 ∙ 𝑙𝑟
+ 𝑚 ∙ 𝑣𝑥 ⎤
𝑣𝑥
1
⎥ 𝑣𝑦
∙ � � = �𝑙 � ∙ 𝐶𝑓 ∙ 𝛿𝑓 ;
𝑓
𝐶𝑓 ∙ 𝑙𝑓 2 + 𝐶𝑟 ∙ 𝑙𝑟 2 ⎥ 𝜔𝑧
⎥
𝑣𝑥
⎦
[4.46]
A simpler way to reach this final expression is given in Figure 4-45. Here, the simplifications are
introduced earlier, already in physical model, which means e.g. that the influence of 𝐹𝑓𝑓𝑓 is not
found.
175
LATERAL DYNAMICS
Physical model:
• Path radius >> the vehicle. Then, all forces
(and centripetal acceleration) are
approximately co0directed.
• 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 co0linear in figure.)
Fry
𝑚 ∙ 𝑎𝑦
𝐽 ∙ 𝜔̇ 𝑧
Constitution: 𝐹𝑓𝑓 = −𝐶𝑓 ∙ 𝑠𝑦𝑓 ; 𝐹𝑟𝑟 = −𝐶𝑟 ∙ 𝑠𝑦𝑟 ;
Compatibility:
𝑣𝑓𝑓 𝑣𝑦 + 𝑙𝑓 ∙ 𝜔𝑧
=
;
𝑣𝑥
𝑣𝑥
𝑣𝑟𝑦 𝑣𝑦 − 𝑙𝑟 ∙ 𝜔𝑧
𝛼𝑟 = 𝛽𝑟 ≈
=
;
𝑣𝑥
𝑣𝑥
𝛼𝑓 ≈ 𝑠𝑦𝑓 ; 𝛼𝑟 ≈ 𝑠𝑦𝑦 ; ;
𝛿𝑓 + 𝛼𝑓 = 𝛽𝑓 ; 𝛽𝑓 ≈
βf
Ffy
Îąf
β
βr
Îąathematical model:
Equilibrium:
𝑚 ∙ 𝑣̇𝑥 − 𝜔𝑧 ∙ 𝑣𝑦 ≈ 𝐹𝑓𝑥 + 𝐹𝑟𝑥 ; 𝑤𝑤𝑤𝑤𝑤 𝑣̇𝑥 = 0;
𝑚 ∙ 𝑣̇𝑦 + 𝜔𝑧 ∙ 𝑣𝑥 ≈ 𝐹𝑓𝑓 + 𝐹𝑟𝑟 ;
𝐽 ∙ 𝜔̇ 𝑧 ≈ 𝐹𝑓𝑓 ∙ 𝑙𝑓 − 𝐹𝑟𝑟 ∙ 𝑙𝑟 ;
𝑚 ∙ 𝑎𝑥
δf
wz
lr lf
L
𝑎𝑥 = 𝑣̇𝑥 − 𝜔𝑧 ∙ 𝑣𝑦 ;
𝑎𝑦 = 𝑣̇𝑦 + 𝜔𝑧 ∙ 𝑣𝑥 ;
Eliminate 𝑭𝒇𝒇 , 𝑭𝒓𝒓 , 𝜶𝒇 , 𝜶𝒓 , 𝜷𝒇 , 𝜷𝒓 yields:
𝐶𝑓 + 𝐶𝑟
𝐶𝑓 ∙ 𝑙𝑓 − 𝐶𝑟 ∙ 𝑙𝑟
𝑚 ∙ 𝑣̇𝑦 +
∙ 𝑣𝑦 +
+ 𝑚 ∙ 𝑣𝑥 ∙ 𝜔𝑧 ≈
𝑣𝑥
𝑣𝑥
≈ 𝐶𝑓 ∙ 𝛿𝑓 ;
𝐶𝑓 ∙ 𝑙𝑓 − 𝐶𝑟 𝑙𝑟
𝐶𝑓 ∙ 𝑙𝑓 2 + 𝐶𝑟 ∙ 𝑙𝑟 2
𝐽 ∙ 𝜔̇ 𝑧 +
∙ 𝑣𝑦 +
∙ 𝜔𝑧 ≈
𝑣𝑥
𝑣𝑥
≈ 𝐶𝑓 ∙ 𝑙𝑓 ∙ 𝛿𝑓 ;
𝐹𝑓𝑥 + 𝐹𝑟𝑥 ≈ −𝑚 ∙ 𝜔𝑧 ∙ 𝑣𝑦 ;
Figure 4-45: Simpler derivation final step in Equation [4.46].
4.4.2.1 Relation between accelerations in inertial system
and velocity derivatives in vehicle fix system
See Equation [4.41]. The relation between accelerations in inertial system, 𝑎⃗ = �𝑎𝑥 ; 𝑎𝑦 � , and velocity
derivatives in vehicle fix system, [der(vx); der(vy)]=�𝑣̇𝑥 ; 𝑣̇𝑦 �, is further explained in Figure 4-46.
yglobal
y
∆wz
∆vy
vy
time=t:
wz
vx
wz
vx
y
vy
x
time=t+∆t:
∆vx
∆y
y
x
y
xglobal
Figure 4-46: Vehicle motion in ground plane
We can express the velocity in direction of the x axis at time t, at the two time instants:
176
LATERAL DYNAMICS
Velocity at time = 𝑡:
𝒗𝒙
Velocity at time = 𝑡 + ∆𝑡: (𝑣𝑥 + ∆𝑣𝑥 ) ∙ cos(∆𝜓) − �𝑣𝑦 + ∆𝑣𝑦 � ∙ sin(∆𝜓) =
(𝑣𝑥 ∙ cos(∆𝜓) + ∆𝑣𝑥 ∙ cos(∆𝜓)) − �𝑣𝑦 ∙ sin(∆𝜓) + ∆𝑣𝑦 ∙ sin(∆𝜓)� ≈ {∆𝜓 𝑠𝑠𝑠𝑠𝑠} ≈
(𝑣𝑥 + ∆𝑣𝑥 ) − �𝑣𝑦 ∙ ∆𝜓 + ∆𝑣𝑦 ∙ ∆𝜓� ≈ �∆𝑣𝑦 ∙ ∆𝜓 𝑠𝑠𝑠𝑠𝑠� ≈ (𝒗𝒙 + ∆𝒗𝒙 ) − 𝒗𝒚 ∙ ∆𝝍
Using these two expressions, we can express 𝑎𝑥 as the change of that speed per time unit:
�(𝑣𝑥 +∆𝑣𝑥 )−𝑣𝑦 ∙∆𝜓�−𝑣𝑥
Change per time = 𝒂𝒙 =
≈ 𝑣𝑥̇ − 𝑣𝑦 ∙ 𝜓̇ = 𝒗𝒙̇ − 𝒗𝒚 ∙ 𝝎𝒛
∆𝑡
=
∆𝑣𝑥 −𝑣𝑦 ∙∆𝜓
∆𝑡
=
∆𝑣𝑥
∆𝑡
− 𝑣𝑦
∆𝜓
∆𝑡
≈
[4.47]
[4.48]
Corresponding for the lateral direction gives, in total, the Equation [4.41].
4.4.2.2
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 �𝑎𝑦 � in the same magnitude as 𝜇 ∙ 𝑚 ∙ 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
𝑔∙𝑤
.
circumstance is wheel lift, which can be approximately checked by checking that �𝑎𝑦 � ≪ 𝑆𝑆𝑆 =
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.
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 [4.46] (or resulting eq in Figure 4-45) can be seen as a state-space-model:
𝑣𝑦
𝑣̇𝑦
� � = 𝑨 ∙ �𝜔 � + 𝑩 ∙ 𝛿𝑓 ;
𝜔̇
𝑧
𝑆𝑆𝑆𝑆𝑆 𝑠𝑠𝑠𝑠𝑠 𝑓𝑓𝑓𝑓: � 𝜔𝑧
𝑣𝑦
𝑧
�𝑎 � = 𝑪 ∙ �𝜔 � + 𝑫 ∙ 𝛿𝑓 ;
𝑚
where 𝑨 = − �
0
𝑚
and 𝑩 = �
0
𝐶𝑓 +𝐶𝑟
𝑣𝑥
0 −1
� ∙ �𝐶 ∙𝑙 −𝐶 ∙𝑙
𝐽
𝑓 𝑓
𝑟 𝑟
0 −1 1
� ∙ �𝑙 � ∙ 𝐶𝑓 ;
𝐽
𝑓
𝑣𝑥
0 1
0 0
and 𝑪 = �
�∙𝑨+�
ďż˝;
0 𝑣𝑥
1 0
and 𝑫 = 𝑩;
When studying “stationary oscillating steering”:
177
𝑦
𝐶𝑓 ∙ 𝑙𝑓 −𝐶𝑟 ∙ 𝑙𝑟
𝑣𝑥
2
𝑧
+ 𝑚 ∙ 𝑣𝑥
𝐶𝑓 ∙ 𝑙𝑓 +𝐶𝑟 ∙ 𝑙𝑟 2
𝑣𝑥
ďż˝;
[4.49]
LATERAL DYNAMICS
ďż˝
𝑣𝑦
𝑣𝑦
𝑣̇𝑦
� = 𝐴 ∙ �𝜔 � + 𝐵 ∙ 𝛿(𝑡) = 𝐴 ∙ �𝜔 � + 𝐵 ∙ 𝛿̂ ∙ cos(𝜔 ∙ 𝑡)
𝜔̇ 𝑧
𝑧
𝑧
(Note similar notation for vehicle yaw rate, 𝜔𝑧 , and steering angular frequency, 𝜔.) Knowing 𝛿̂ and
�𝑧 , 𝜑𝑦 and 𝜑𝑧 :
𝜔, it is possible to calculate the responses 𝑣�𝑦 , 𝜔
𝜑𝑣𝑦
𝑣𝑦
𝑣�𝑦
�𝜔 � = � � ∙ cos �𝜔 ∙ 𝑡 − �𝜑 �� ;
𝜔𝑧
𝑧
𝜔
�𝑧
𝜑𝜔𝑧
𝜔𝑧
𝜔
�𝑧
�𝑎 � = � � ∙ cos �𝜔 ∙ 𝑡 − �𝜑𝑎 �� ;
𝑎�𝑦
𝑦
𝑦
Different methods are available for calculation of the responses:
•
•
•
Real trigonometry, using cos(𝜔 ∙ 𝑡 + 𝑝ℎ𝑎𝑎𝑎) , sin(𝜔 ∙ 𝑡 + 𝑝ℎ𝑎𝑎𝑎) or cos(𝜔 ∙ 𝑡) + sin(𝜔 ∙ 𝑡).
Complex mathematics, using 𝑒 𝑗∙𝜔∙𝑡
Fourier transform
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 [4.49] can be transformed to the frequency domain (where the Fourier transform of a
∞
function, 𝜉(𝑡), is denoted ℱ�𝜉(𝑡)� = ∫0 𝑒−𝑗∙𝜔∙𝑡 ∙ 𝜉(𝑡) ∙ 𝑑𝑑):
𝑣𝑦
𝑣𝑦
j ∙ ω ∙ ℱ ��𝜔 �� = 𝑨 ∙ ℱ ��𝜔 �� + 𝑩 ∙ ℱ�𝛿𝑓 �;
ďż˝
Solving for states and outputs:
𝑧
𝑧
𝜔𝑧
𝑣𝑦
ℱ �� 𝑎 �� = 𝑪 ∙ ℱ ��𝜔 �� + 𝑫 ∙ ℱ�𝛿𝑓 �;
𝑦
𝑧
⎧� 𝑣𝑦 � = ℱ−1 �ℱ �� 𝑣𝑦 ��� = ℱ−1 �(j ∙ ω ∙ I − 𝑨)−1 ∙ 𝑩 ∙ ℱ�𝛿𝑓 �� ;
𝜔𝑧
⎪ 𝜔𝑧
𝜔𝑧
𝑣𝑦
⎨ 𝜔𝑧
−1
−1
⎪ � 𝑎𝑦 � = ℱ �ℱ �� 𝑎𝑦 ��� = ℱ �𝑪 ∙ ℱ ��𝜔𝑧 �� + 𝑫 ∙ ℱ�𝛿𝑓 �� ;
⎊
[4.50]
[4.51]
Expressed as transfer functions:
𝑣𝑦
ℱ ��𝜔 �� (j ∙ ω ∙ I − 𝑨)−1 ∙ 𝑩 ∙ ℱ�𝛿 �
⎧
𝑓
𝑧
𝑣𝑦 =
=
= (j ∙ ω ∙ I − 𝑨)−1 ∙ 𝑩;
⎪
⎪𝐻𝛿𝑓→�ω
ďż˝
ℱ
ℱ
�𝛿
ďż˝
�𝛿
ďż˝
𝑓
𝑓
𝑧
𝜔𝑧
⎨
ℱ ��𝑎 ��
𝑦
⎪
⎪𝐻
𝜔𝑧 =
𝑣𝑦 + 𝑫 = 𝑪 ∙ (j ∙ ω ∙ I − 𝑨)−1 ∙ 𝑩 + 𝑫;
=𝑪∙𝐻
𝛿𝑓 →�ω �
𝛿
→�
ďż˝
𝑓
ℱ�𝛿𝑓 �
𝑎𝑦
⎊
𝑧
[4.52]
We have derived the transfer functions. The subscript tells that the transfer function is for the vehicle
𝜔𝑧
𝑣𝑦
operation with excitation=input= 𝛿𝑓 and response=output= �𝜔 � and output= �𝑎 �. The transfer
𝑦
𝑧
functions has dimension 2x1 and it is complex. They operate as follows:
178
LATERAL DYNAMICS
Amplitudes:
Phase delays:
𝑣�
⎧� 𝑦 � = �𝐻
𝑣𝑦 � ∙ 𝛿̂𝑓 ;
𝛿𝑓 →�𝜔 �
⎪𝜔
�𝑧
𝑧
�𝑧
⎨�𝜔
⎪ 𝑎�𝑦 � = �𝐻𝛿𝑓→�𝜔𝑧� � ∙ 𝛿̂𝑓 ;
𝑎𝑦
⎊
𝜑
⎧� 𝑣𝑦 � = �arg�𝑣𝑦 �� − �1� ∙ arg�𝛿𝑓 � = �arg�𝛿𝑓 �� = arg �𝐻
𝑣𝑦 � ;
𝛿𝑓 →�𝜔 �
⎪ 𝜑𝜔𝑧
1
arg(𝜔 )
=0
𝑧
[4.53]
𝑧
⎨ 𝜑𝜔𝑧 = �arg(𝜔𝑧 )� − 1 ∙ arg�𝛿 = arg�𝛿𝑓 � = arg
𝜔𝑧 � ;
ďż˝ ďż˝
ďż˝
ďż˝
�𝐻
𝑓�
⎪� 𝜑𝑎𝑦 �
𝛿𝑓 →�𝑎 �
arg�𝑎𝑦 �
1
=0
⎊
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 [4.58] and Equation [4.57]. With Fourier Transform, the expression for the
Transfer Function, H, can be reached with less algebra. Knowing H, it can be used in Equation [4.57].
The fundamental situation for steering frequency response is that the excitation is: 𝛿𝑓 = 𝛿̂𝑓 ∙
cos(2 ∙ 𝜋 ∙ 𝑓 ∙ 𝑡) = �𝑒 𝑗∙𝑎 = cos(𝑎) + 𝑗 ∙ sin(𝑎)� = Re�𝛿̂𝑓 ∙ 𝑒 𝑗∙2∙𝜋∙𝑓∙𝑡 �, where f is the (time) frequency
in Hz and 𝑗 is the imaginary unit. We rewrite 2 ∙ 𝜋 ∙ 𝑓 as 𝜔 (angular frequency), which has to be
carefully distinguished from 𝜔𝑧 (yaw rate). Insert this in Equation [4.45] and neglecting the
longitudinal force 𝐹𝑓𝑓𝑓 . The full (complex) equation is used:
⎡
⎢𝑚
Re ⎢�
0
⎢
⎣
𝐶𝑓 + 𝐶𝑟
⎡
𝑣𝑥
𝑣̇𝑦𝑦
0
⎢
�∙� �+⎢
𝐽
𝜔̇ 𝑧𝑧
𝐶𝑓 ∙ 𝑙𝑓 − 𝐶𝑟 ∙ 𝑙𝑟
⎢
⎣
𝑣𝑥
𝐶𝑓 ∙ 𝑙𝑓 − 𝐶𝑟 ∙ 𝑙𝑟
⎤
+ 𝑚 ∙ 𝑣𝑥 ⎤
𝐶𝑓
⎥
𝑣𝑥
⎥ 𝑣𝑦𝑦
∙� �=�
� ∙ 𝛿̂𝑓 ∙ 𝑒 𝑗𝑗∙𝑡 ⎥ ;
𝐶𝑓 ∙ 𝑙𝑓
𝐶𝑓 ∙ 𝑙𝑓 2 + 𝐶𝑟 ∙ 𝑙𝑟 2 ⎥ 𝜔𝑧𝑧
⎥
⎥
𝑣𝑥
⎦
⎦
[4.54]
We intend to solve the complex equation, and then find the solutions as real parts: 𝑣𝑦 = Re�𝑣𝑦𝑦 �;
and 𝜔𝑧 = Re[𝜔𝑧𝑧 ];. (Subscript c means complex.)
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.
𝑣𝑦𝑦
𝑣�𝑦𝑦
𝑣̇𝑦𝑦
𝑣�𝑦𝑦
�𝜔 � = � � ∙ 𝑒 𝑗∙𝜔∙𝑡 ⇒ �
� = 𝑗 ∙ 𝜔 ∙ � � ∙ 𝑒 𝑗∙𝜔∙𝑡 ;
𝜔̇ 𝑧𝑧
𝑧𝑧
𝜔
�𝑧𝑧
𝜔
�𝑧𝑧
[4.55]
Inserting the assumption in the differential equation gives:
𝑣�𝑦𝑦
0
�∙𝑗∙𝜔∙�
� ∙ 𝑒 𝑗∙𝜔∙𝑡 +
𝐽
𝜔
�𝑧𝑧
𝐶𝑓 + 𝐶𝑟
𝐶𝑓 ∙ 𝑙𝑓 − 𝐶𝑟 ∙ 𝑙𝑟
⎡
+ 𝑚 ∙ 𝑣𝑥 ⎤
𝑣𝑥
𝑣𝑥
1
⎢
⎥ 𝑣�𝑦𝑦
+⎢
∙�
� ∙ 𝑒 𝑗∙𝜔∙𝑡 = �𝑙 � ∙ 𝐶𝑓 ∙ 𝛿̂𝑓 ∙ 𝑒 𝑗∙𝜔∙𝑡 ⇒
2
2
⎥
𝜔
�𝑧𝑧
𝑓
𝐶 ∙ 𝑙 − 𝐶𝑟 ∙ 𝑙𝑟
𝐶𝑓 ∙ 𝑙𝑓 + 𝐶𝑟 ∙ 𝑙𝑟
⎢ 𝑓 𝑓
⎥
⎣
𝑣𝑥
𝑣𝑥
⎦
𝑣�𝑦𝑦
⇒�
ďż˝=
𝜔
�𝑧𝑧
ďż˝
𝑚
0
179
[4.56]
LATERAL DYNAMICS
𝐶𝑓 + 𝐶𝑟
⎡
𝑣𝑥
⎛𝑚 0
⎢
= ⎜�
�∙𝑗∙𝜔+⎢
0 𝐽
𝐶 ∙ 𝑙 − 𝐶𝑟 ∙ 𝑙𝑟
⎢ 𝑓 𝑓
⎣
𝑣𝑥
⎝
𝑣𝑦 ∙ 𝛿̂𝑓 ;
=𝐻
𝛿𝑓 →�𝜔 �
𝑧
𝐶𝑓 ∙ 𝑙𝑓 − 𝐶𝑟 ∙ 𝑙𝑟
+ 𝑚 ∙ 𝑣𝑥 ⎤
𝑣𝑥
⎥⎞
2
2
⎥⎟
𝐶𝑓 ∙ 𝑙𝑓 + 𝐶𝑟 ∙ 𝑙𝑟
⎥
𝑣𝑥
⎦⎠
−1
1
∙ �𝑙 � ∙ 𝐶𝑓 ∙ 𝛿̂𝑓 =
𝑓
Then, we can assume we know 𝑣�𝑦𝑦 and 𝜔
�𝑧𝑧 from Equation [4.56], and consequently we know 𝑣𝑦𝑦
𝑣𝑦 . The subscript tells
and 𝜔𝑧𝑧 from Equation [4.55]. We have derived the transfer function, 𝐻
𝛿𝑓 →�𝜔 �
𝑧
that the transfer function is for the vehicle operation with excitation=input= 𝛿𝑓 and
𝑣𝑦
response=output= �𝜔 � case. This transfer function has dimension 2x1 and it is complex. It operates
𝑧
as follows:
𝑣�𝑦
�𝑣 �
𝑣𝑦 � ∙ �𝛿𝑓 �;
� = � 𝑦 � = �𝐻
𝛿𝑓 →�𝜔 �
𝜔
�𝑧
|𝜔𝑧 |
𝑧
arg�𝑣𝑦 �
1
𝑣𝑦 �
Phase delays: ďż˝
� − � � ∙ arg�𝛿𝑓 � = �arg�𝛿𝑓 � = 0� = arg �𝐻
𝛿𝑓 →�𝜔 �
1
arg(𝜔𝑧 )
𝑧
Amplitudes: ďż˝
[4.57]
However, we can derive expressions for 𝑣𝑦𝑦 and 𝜔𝑧𝑧 on a real (non-complex) form, 𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎 ∙
cos(𝑎𝑎𝑎𝑎𝑎𝑎𝑎 𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓 ∙ 𝑡 − 𝑝ℎ𝑎𝑎𝑎 𝑑𝑑𝑑𝑑𝑑), without involving transfer function. That is done in the
following:
𝑣𝑦 = Re�𝑣𝑦𝑦 � = Re�𝑣�𝑦𝑦 ∙ 𝑒 𝑗∙𝜔∙𝑡 � = ⋯ = �𝑣�𝑦𝑦 � ∙ cos�𝜔 ∙ 𝑡 + arg�𝑣�𝑦𝑦 �� ;
The same rewriting can be done with 𝜔𝑧 , so that in total:
�𝑣�𝑦𝑦 � ∙ cos �𝜔 ∙ 𝑡 − �−arg�𝑣�𝑦𝑦 ���
𝑣𝑦
𝑣� ∙ cos�𝜔 ∙ 𝑡 − 𝜑𝑣𝑣 �
ďż˝=ďż˝
�𝜔 � = � 𝑦
ďż˝
𝑧
𝜔
�𝑧 ∙ cos(𝜔 ∙ 𝑡 − 𝜑𝜔𝜔 )
|𝜔
�𝑧𝑧 | ∙ cos�𝜔 ∙ 𝑡 − (− arg(𝜔
�𝑧𝑧 ))�
[4.58]
Equation [4.56] and Equation [4.58] now gives us the possibility to find vehicle response amplitude
and phase delay.
The ratios between amplitude of responses and amplitude of excitation, 𝑣�𝑦 ⁄𝛿̂𝑓 and 𝜔
�𝑧 ⁄𝛿̂𝑓 , 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 [4.56], Equation [4.58] and Figure 4-47. The curves show response for
same vehicle, but different speed.
See Figure 4-47. 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 [4.18] 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 4-47 to Figure 4-50, one can find these special speeds in another appearance:
180
LATERAL DYNAMICS
•
For an understeered vehicle, speeds above the characteristic speed gives a negative yaw rate
delay for low steering frequencies will be negative.
For an over-steered vehicle, speeds above the critical speed gives a yaw rate delay which is
larger than 180 deg and yaw rate amplitudes which are very large for low steering
frequencies.
•
Yaw Rate Response
Lateral Velocity 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
-0.5
-2
10
2
10
-1
10
0
10
frequency [Hz]
1
10
2
10
Figure 4-47: 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).
From Equation [4.18] we can calculate that characteristic speed for the vehicle is 120 km/h.
In Figure 4-48 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.
181
LATERAL DYNAMICS
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 4-48: Vehicle response to harmonically oscillating steering angle. Same vehicle data
as in Figure 4-47, except varying Cf and Cr but keeping understeering gradient Ku
constant. Vehicle speed = 100 km/h.
In Figure 4-49 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.
182
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
10
0
10
1
10
30
20
10
0
-2
10
2
10
2
-1
10
1
0
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
10
0
-2
10
2
10
-1
10
0
10
frequency [Hz]
1
10
2
10
Figure 4-49: Vehicle response to harmonically oscillating steering angle. Same vehicle data
as in Figure 4-47, 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):
𝑣̇𝑦 + 𝜔𝑧 ∙ 𝑣𝑥
𝑗 ∙ 𝜔 ∙ 𝑣𝑦𝑦 + 𝜔𝑧𝑧 ∙ 𝑣𝑥
ďż˝=ďż˝
ďż˝=
𝑚
𝑚
𝛿𝑓
𝑣𝑥 ] ∙ 𝐻
𝑣𝑦 ∙ 𝛿𝑓 � = �[𝑗 ∙ 𝜔 𝑣𝑥 ] ∙ 𝐻
𝑣𝑦 � ∙
;
𝛿𝑓 →�𝜔 �
𝛿𝑓 →�𝜔 � 𝑚
𝑧
𝑧
Amplitude: 𝑎�𝑦 = �𝑎𝑦𝑦 � = �𝑢𝑢𝑢: 𝑎𝑦 =
1
= � ∙ [𝑗 ∙ 𝜔
𝑚
Phase delay: arg�𝑎𝑦 � − arg�𝛿𝑓 � = �𝑢𝑢𝑢: arg�𝛿𝑓 � = 0� = arg �[𝑗 ∙ 𝜔
[4.59]
𝑣𝑥 ] ∙ 𝐻
𝑣𝑦 � ;
𝛿𝑓 →�𝜔 �
Lateral acceleration response is plotted for different vehicle speeds in Figure 4-50.
183
𝑧
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
1
150
ay delay [rad]
ay gain [(m/(s*s))/rad]
200
1.5
100
0.5
0
-0.5
-1
50
-1.5
0
-2
10
-1
10
0
10
frequency [Hz]
1
10
-2
-2
10
2
10
-1
10
0
10
frequency [Hz]
1
10
2
10
Figure 4-50: 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.
4.4.3.5
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.
184
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 4-5. 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
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 4-52: Cone track for one standardized Double lane change
185
LATERAL DYNAMICS
Figure 4-53: Cone track for one standardized test of Over-speeding into curve
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.
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 4-43 and Equations [4.42] and [4.43]. 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 [4.60].
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
𝑀𝑎𝑎𝑎,𝑧 in yaw equilibrium in Equation [4.60].
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.
186
LATERAL DYNAMICS
Equilibrium in road plane (longitudinal, lateral, yaw):
𝑚 ∙ �𝑣̇𝑥 − 𝜔𝑧 ∙ 𝑣𝑦 � = 𝐹𝑓𝑓𝑓 ∙ cos�𝛿𝑓 � − 𝐹𝑓𝑓𝑓 ∙ sin�𝛿𝑓 � + 𝐹𝑟𝑟 ;
𝑚 ∙ �𝑣̇𝑦 + 𝜔𝑧 ∙ 𝑣𝑥 � = 𝐹𝑓𝑓𝑓 ∙ sin�𝛿𝑓 � + 𝐹𝑓𝑓𝑓 ∙ cos�𝛿𝑓 � + 𝐹𝑟𝑟 ;
𝐽 ∙ 𝜔̇ 𝑧 = �𝐹𝑓𝑓𝑓 ∙ sin�𝛿𝑓 � + 𝐹𝑓𝑓𝑓 ∙ cos�𝛿𝑓 �� ∙ 𝑙𝑓 − 𝐹𝑟𝑟 ∙ 𝑙𝑟 + 𝑀𝑎𝑎𝑎,𝑧 ;
Equilibrium out of road plane (vertical, pitch):
𝐹𝑓𝑓 + 𝐹𝑟𝑟 − 𝑚 ∙ 𝑔 = 0;
−𝐹𝑓𝑓 ∙ 𝑙𝑓 + 𝐹𝑟𝑟 ∙ 𝑙𝑟 − �𝐹𝑓𝑓𝑓 ∙ cos�𝛿𝑓 � − 𝐹𝑓𝑓𝑓 ∙ sin�𝛿𝑓 � + 𝐹𝑟𝑟 � ∙ ℎ = 0;
Constitution:
𝐹𝑓𝑓𝑓 = −sign�𝑠𝑓𝑓 � ∙ min�𝐶𝑓 ∙ �𝑠𝑓𝑓 �; 𝜇 ∙ 𝐹𝑓𝑓 � ;
𝐹𝑟𝑟 = −sign�𝑠𝑟𝑟 � ∙ min�𝐶𝑟 ∙ �𝑠𝑟𝑟 �; 𝜇 ∙ 𝐹𝑟𝑟 � ;
[4.60]
Compatibility:
𝑠𝑓𝑓 =
𝑣𝑓𝑓𝑓
𝑣𝑦 − 𝑙𝑟 ∙ 𝜔𝑧
; 𝑎𝑎𝑎 𝑠𝑟𝑟 =
;
𝑣𝑓𝑓𝑓
𝑣𝑥
Transformation from vehicle to wheel coordinate system on front axle:
𝑣𝑓𝑓𝑓 = �𝑣𝑦 + 𝑙𝑓 ∙ 𝜔𝑧 � ∙ sin�𝛿𝑓 � + 𝑣𝑥 ∙ cos�𝛿𝑓 � ;
𝑣𝑓𝑓𝑓 = �𝑣𝑦 + 𝑙𝑓 ∙ 𝜔𝑧 � ∙ cos�𝛿𝑓 � − 𝑣𝑥 ∙ sin�𝛿𝑓 � ;
The model in Modelica format is given in Equation [4.61]. Changes compared to Equations [4.42] 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;
[4.61]
//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 4-44. 𝑀𝑎𝑎𝑎,𝑧 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
187
LATERAL DYNAMICS
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 𝑀𝑎𝑎𝑎,𝑧 could be of interest. However,
it is beyond the scope of this compendium.
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
4-44 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]
δf=δf [δeg]
x[m]
yz=pz [δeg]
x[m]
time [s]
Figure 4-54: 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.
188
LATERAL DYNAMICS
4.5.3 Transient model, with Lateral load
transfer
In previous model of load transfer in Section 0, we assumed nothing about transients in the process
of building up the load transfer. In present chapter we need to capture the transients better, why we
add the following to the model presented in Figure 4-33 and Equations [4.28]..[4.34]:
•
•
•
Inertial term for roll rotation, i.e. J*der(wx).
Inertial term for lateral acceleration need to capture the whole lateral acceleration,
𝑎𝑦 = 𝑣̇𝑦 + 𝑣𝑥 ∙ 𝜔𝑧 , not only the centripetal acceleration, 𝑎𝑦 = 𝑣𝑥 ∙ 𝜔𝑧 .
Damping forces in parallel to each spring force, i.e adding Fdfl, Fdfr, Fdrl, and Fdrr.
Due to the axle roll centre model, as opposed to wheel pivot point model, the heave acceleration is
zero. Hence, no inertial force m*az needs to be introduced.
The free-body diagrams are given in Figure 4-55, which should be compared to Figure 4-33.
forces, on body:
displacements, speeds, accelerations:
ay
m*g
z
zfl=zrl
y
J*der(wx)
zfr=zrr
m*ay
px
Pry
Pfy
Fsfl+Fsrl+
+Fdfl+Fdrl
zflr=zrlr=0
zfrr=zrrr=0
forces,
on vehicle:
J*der(wx)
forces, on front axle:
m*g
m*ay
Fsfl+Fdfl
Fflz+Frlz
Fsfr+Fdfr
Pfz
Ffry+Frry
Ffrz+Frrz
forces, on rear axle:
Fsrl+Fdrl Fsrr+Fdrr
Prz
Pry
Pfy
h
Ffly+Frly
Fsfr+Fsrr+
Pfz+Prz +Fdfr+Fdrr
Ffly
Fflz
Ffry
Ffrz
Frly
Frlz
Frry
Frrz
Figure 4-55: 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 [4.27]. Note
that the anti-roll bars are not modelled. Note that we differentiate, since we will later use the spring
forces as state variables in a simulation.
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LATERAL DYNAMICS
̇ = −𝑐𝑓𝑓 ∙ 𝑧̇𝑓𝑓 = −𝑐𝑓𝑓 ∙ 𝑣𝑓𝑓𝑓 ;
𝐹𝑠𝑠𝑠 = 𝐹𝑠𝑠𝑠0 + 𝑐𝑓𝑓 ∙ �𝑧𝑓𝑓𝑓 − 𝑧𝑓𝑓 �; ⇒ 𝐹𝑠𝑠𝑠
̇ = −𝑐𝑓𝑓 ∙ 𝑧̇𝑓𝑓 = −𝑐𝑓𝑓 ∙ 𝑣𝑓𝑓𝑓 ;
𝐹𝑠𝑠𝑠 = 𝐹𝑠𝑠𝑠0 + 𝑐𝑓𝑓 ∙ �𝑧𝑓𝑓𝑓 − 𝑧𝑓𝑓 �; ⇒ 𝐹𝑠𝑠𝑠
̇ = −𝑐𝑟𝑟 ∙ 𝑧̇𝑟𝑟 = −𝑐𝑟𝑟 ∙ 𝑣𝑟𝑟𝑟 ;
𝐹𝑠𝑠𝑠 = 𝐹𝑠𝑠𝑠0 + 𝑐𝑟𝑟 ∙ (𝑧𝑟𝑟𝑟 − 𝑧𝑟𝑟 ); ⇒ 𝐹𝑠𝑠𝑠
̇ = −𝑐𝑟𝑟 ∙ 𝑧̇𝑟𝑟 = −𝑐𝑟𝑟 ∙ 𝑣𝑟𝑟𝑟 ;
𝐹𝑠𝑠𝑠 = 𝐹𝑠𝑠𝑠0 + 𝑐𝑟𝑟 ∙ (𝑧𝑟𝑟𝑟 − 𝑧𝑟𝑟 ); ⇒ 𝐹𝑠𝑠𝑠
𝑚 ∙ 𝑔 ∙ 𝑙𝑓
𝑚 ∙ 𝑔 ∙ 𝑙𝑟
𝑤ℎ𝑒𝑒𝑒 𝐹𝑠𝑠𝑠0 = 𝐹𝑠𝑠𝑠0 =
; 𝑎𝑎𝑎 𝐹𝑠𝑠𝑠0 = 𝐹𝑠𝑠𝑠0 =
;
2∙𝐿
2∙𝐿
[4.62]
The constitutive equations for the dampers have to be added:
𝐹𝑑𝑑𝑑 = −𝑑𝑓𝑓 ∙ 𝑣𝑓𝑓𝑓 ;
𝐹𝑑𝑑𝑑 = −𝑑𝑓𝑓 ∙ 𝑣𝑓𝑓𝑓 ;
𝐹𝑑𝑑𝑑 = −𝑑𝑟𝑟 ∙ 𝑣𝑟𝑟𝑟 ;
𝐹𝑑𝑑𝑑 = −𝑑𝑟𝑟 ∙ 𝑣𝑟𝑟𝑟 ;
[4.63]
As comparable with Equation [4.28], we get the next equation to fulfil the equilibrium. The change
compared to Equation [4.28] 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].
Equilibrium for whole vehicle (longitudinal, lateral, yaw):
𝑚 ∙ 𝑎𝑥 = 𝑚 ∙ �𝑣̇𝑥 − 𝜔𝑧 ∙ 𝑣𝑦 � = 𝐹𝑓𝑓 + 𝐹𝑟𝑟 ;
𝑚 ∙ 𝑎𝑦 = 𝑚 ∙ �𝑣̇𝑦 + 𝜔𝑧 ∙ 𝑣𝑥 � = 𝐹𝑓𝑓 + 𝐹𝑟𝑟 ;
𝐽𝑧 ∙ 𝜔̇ 𝑧 = 𝐹𝑓𝑓 ∙ 𝑙𝑓 − 𝐹𝑟𝑟 ∙ 𝑙𝑟 ;
Equilibrium for whole vehicle (vertical, pitch, roll):
𝐹𝑓𝑓𝑓 + 𝐹𝑓𝑓𝑓 + 𝐹𝑟𝑟𝑟 + 𝐹𝑟𝑟𝑟 − 𝑚 ∙ 𝑔 = 0;
−�𝐹𝑓𝑓𝑓 + 𝐹𝑓𝑓𝑓 � ∙ 𝑙𝑓 + (𝐹𝑟𝑟𝑟 + 𝐹𝑟𝑟𝑟 ) ∙ 𝑙𝑟 − �𝐹𝑓𝑓 + 𝐹𝑟𝑟 � ∙ ℎ = 0;
𝑤
𝑤
𝐽𝑥 ∙ 𝜔̇ 𝑥 = �𝐹𝑓𝑓𝑓 + 𝐹𝑟𝑟𝑟 � ∙ − �𝐹𝑓𝑓𝑓 + 𝐹𝑟𝑟𝑟 � ∙ + �𝐹𝑓𝑓 + 𝐹𝑟𝑟 � ∙ ℎ;
2
2
Equilibrium for each axle (pitch, around roll centre):
𝑤
𝑤
�𝐹𝑓𝑓𝑓 − �𝐹𝑠𝑠𝑠 + 𝐹𝑑𝑑𝑑 �� ∙ − �𝐹𝑓𝑓𝑓 − �𝐹𝑠𝑠𝑠 + 𝐹𝑑𝑑𝑑 �� ∙ + 𝐹𝑓𝑦 ∙ ℎ𝑅𝑅𝑅 = 0;
2
2
𝑤
𝑤
�𝐹𝑟𝑟𝑟 − (𝐹𝑠𝑠𝑠 + 𝐹𝑑𝑑𝑑 )� ∙ − �𝐹𝑟𝑟𝑟 − (𝐹𝑠𝑠𝑠 + 𝐹𝑑𝑑𝑑 )� ∙ + 𝐹𝑟𝑟 ∙ ℎ𝑅𝑅𝑅 = 0;
2
2
[4.64]
Compatibility gives, keeping in mind that 𝜔𝑥 is the only non-zero out-of road plane velocity (i.e.
𝑧̇ = 𝜔𝑦 = 0):
𝑤
𝑤
∙ 𝜔𝑥 ; 𝑎𝑎𝑎 𝑧̇𝑓𝑓 = 𝑣𝑓𝑓𝑓 = − ∙ 𝜔𝑥 ;
2
2
𝑤
𝑤
= + ∙ 𝜔𝑥 ; 𝑎𝑎𝑎 𝑧̇𝑟𝑟 = 𝑣𝑟𝑟𝑟 = − ∙ 𝜔𝑥 ;
2
2
𝑧̇𝑓𝑓 = 𝑣𝑓𝑓𝑓 = +
𝑧̇𝑟𝑟 = 𝑣𝑟𝑟𝑟
[4.65]
Equations [4.62]..[4.65] give a model very similar to the one in Equation [4.61]. Only the additional
equations from this new model are shown in Equation [4.66]. 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.
190
LATERAL DYNAMICS
//Equilibrium, roll
Jx*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;
[4.66]
//Compatibility, out of road plane:
vflz = +w/2*wx;
vfrz = -w/2*wx;
vrlz = +w/2*wx;
vrrz = -w/2*wx;
A simulation of this model should be compared with the simulation in Figure 4-44. When comparing
these, we see a slight difference, which is that the axles saturate gradually during 3<time<3.5,
instead of all 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 4-57. 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 (𝐹𝑠𝑠𝑠 ), damper
(𝐹𝑑𝑑𝑑 ) and linkage (𝐹𝑙𝑙𝑙𝑙 , 𝑓𝑙,𝑧 = 𝐹𝑓𝑓𝑓 − 𝐹𝑠𝑠𝑠 − 𝐹𝑑𝑑𝑑 ).
191
LATERAL DYNAMICS
wz [deg/s]
y [m]
δf=δf [δeg]
x[m]
yz=pz [δeg]
x[m]
time [s]
Figure 4-56: 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.
Frrz [N]
Fsrr [N]
Ffrz [N]
Frrz [N]
Flink,rr,z [N]
Fflz [N]
Fdrr [N]
Frlz [N]
time [s]
time [s]
Figure 4-57: Suspension vertical force plots from simulation with one-track model for
transient dynamics with lateral load transfer dynamics (same simulation as in Figure
4-56). Left: Road contact forces for all wheels. Right: Different forces for one wheel, rear
left.
192
LATERAL DYNAMICS
4.5.3.1
Explicit form model
The above implementation of the model is automatically handled by the Modelica-tool. If modelling
in a tool using data flow diagrams, such as Simulink, the model has to be manually converted into an
“Explicit form model”, see Section 1.3.2. Figure 4-58 shows one possible explicit form model for
(approximately) the model described in Eq [4.61] and the additions in Eq [4.66]. Some changes are
done to reduce complexity of data flow, e.g. wheel rotations are modelled as states (𝐽𝑤ℎ𝑒𝑒𝑒 ∙ 𝜔̇ =
𝑇 − 𝐹𝑥 ∙ 𝑅 for each of the N wheels). Note that, there is still one “algebraic loop”. An algebraic loop is
when the calculation of a variable requires the same variable as input. The algebraic loop in the
model in Figure 4-58 can be resolved by inserting a “memory block” where marked “*”. This means
that the variable value from last time step is used, which normally works satisfactory if small enough
time steps are used but it can also lead to instable numerical solutions.
transf. to
wheel corners
𝑣𝑥 , 𝑣𝑦 , 𝜔𝑧
𝑧, 𝜑𝑥 , 𝜑𝑦
3 integrators
for in-plane
velocities
𝑁×
transf. to wheel corners
vehicle in-plane
motion
equations
vehicle out-of-plane
motion equations
*
𝑁 × đšđ‘§
𝑁 × integrator for
wheel rotations
3 integrators for outof-plane velocities
𝑣̇𝑧 , 𝜔̇ 𝑥 , 𝜔̇ 𝑦
𝑁 × đ‘Łđ‘§đ‘§ , 𝑣̇ 𝑧𝑧
𝑁×𝜔
𝑣𝑧 , 𝜔𝑥 , 𝜔𝑦
𝑎𝑥 , 𝑎𝑦 , 𝜔̇ 𝑧
𝑁×
transf. to wheel coord.
𝑁 × đ‘Łđ‘Ľđ‘Ľ , 𝑣𝑦𝑦
𝑁×
Suspension
3 integrators for outof-plane positions
𝑁×𝛿
𝑁 × đ‘Łđ‘Ľđ‘¤ , 𝑣𝑦𝑤
𝑁 × Tyre model
𝑁 × đšđ‘Śđ‘Ś
𝑁 × đœ”Ě‡
𝑁 ×T
𝑁 × wheel
rotational motion
equation
𝑁 × đšđ‘Ľđ‘Ľ , 𝐹𝑦𝑦
𝑁 × đšđ‘Ľđ‘Ľ
𝑁×
transf. to vehicle coord.
Figure 4-58: One possible “Explicit form model” for (approximately) the model in Eq [4.61]
and Eq [4.66]. The figure shows a generalization to a vehicle with N wheels. Wheel
steering angle (𝛿) and Wheel torques (𝑇) on each of these N wheels are inputs to model.
4.5.3.2
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.
193
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 Step steering response, 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 [4.49] allows an easy analysis of stationary oscillating steering, but also an easy analysis of
step response:
Start from Equation [4.49]: ďż˝
With initial conditions: ďż˝
𝑣𝑦
𝑣̇𝑦
� = 𝐀 ∙ �𝜔 � + 𝐁 ∙ 𝛿𝑓 ;
𝜔̇ 𝑧
𝑧
𝑣𝑦0
𝑣𝑦0
𝑣𝑦 (0)
� = �𝜔 � ; or �𝜔 � = −𝐀−1 ∙ 𝐁 ∙ 𝛿𝑓0 ; where 𝛿𝑓0 is before step.
𝑧0
𝑧0
𝜔𝑧 (0)
𝜆1 ∙𝑡
𝑣𝑦∞
𝑣𝑦
𝑣�𝑦1
𝑣�𝑦2
0 � ∙ �𝑎1 � ; ⇒
Assume: �𝜔 � = �𝜔 � + ��
ďż˝ ďż˝
�� ∙ �𝑒
𝜆2 ∙𝑡
𝑎2
𝑧
𝑧∞
𝜔
�𝑧1
𝜔
�𝑧2
0
𝑒
𝜆
∙𝑡
1
𝑎1
𝑣�𝑦1
𝑣�𝑦2
𝑣̇𝑦
0
𝜆 ∙𝑒
� ∙ �𝑎 � ;
⇒ � � = ��
ďż˝ ďż˝
�� ∙ � 1
𝜆2 ∙𝑡
𝜔̇ 𝑧
2
𝜔
�𝑧1
𝜔
�𝑧2
0
𝜆2 ∙ 𝑒
𝑎1
𝑣�𝑦2
𝑣�𝑦1
0
𝜆 ∙ 𝑒 𝜆1 ∙𝑡
� ∙ �𝑎 � =
Insert: ��
ďż˝ ďż˝
�� ∙ � 1
𝜆2 ∙𝑡
2
𝜔
�𝑧1
𝜔
�𝑧2
0
𝜆2 ∙ 𝑒
𝜆1 ∙𝑡
𝑣𝑦∞
𝑣�𝑦1
𝑣�𝑦2
= 𝐀 ∙ ��𝜔 � + ��
ďż˝ ďż˝
�� ∙ �𝑒
𝑧∞
𝜔
�𝑧1
𝜔
�𝑧2
0
𝑒
0 � ∙ �𝑎1 �� + 𝐁 ∙ 𝛿 ;
𝑓
𝑎
𝜆2 ∙𝑡
Solve for each time function term (constant, 𝑒 𝜆1 ∙𝑡 and 𝑒 𝜆2 ∙𝑡 terms):
𝑣𝑦∞
𝑣�𝑦2
𝑣�𝑦1
𝜆
ďż˝ ďż˝
�� , � 1
�𝜔 � = −𝐀−1 ∙ 𝐁 ∙ 𝛿𝑓 ; 𝑎𝑎𝑎 ���
0
𝑧∞
𝜔
�𝑧1
𝜔
�𝑧2
2
0
�� = eig(𝐀) ;
𝜆2
The function ”eig” is identical to function ”eig” in Matlab. It is is defined as eigenvalues and
eigenvectors for the matrix input argument.
𝑣𝑦0
𝑣𝑦∞
𝑎1
𝑣�𝑦1
𝑣�𝑦2
Initial conditions: �𝜔 � = �𝜔 � + ��
ďż˝ ďż˝
�� ∙ �𝑎 � ; ⇒
𝑧0
𝑧∞
2
𝜔
�𝑧1
𝜔
�𝑧2
−1
𝑣𝑦0
𝑎1
𝑣𝑦∞
𝑣�𝑦1
𝑣�𝑦2
⇒ �𝑎 � = ��
ďż˝ ďż˝
�� ∙ ��𝜔 � − �𝜔 �� ;
2
𝑧0
𝑧∞
𝜔
�𝑧1
𝜔
�𝑧2
The solution in summary:
𝜆1 ∙𝑡
𝑣𝑦
𝑣𝑦∞
𝑣�𝑦1
𝑣�𝑦2
0 � ∙ �𝑎1 � ;
�𝜔 � = �𝜔 � + ��
ďż˝ ďż˝
�� ∙ �𝑒
𝜆
𝑧
𝑧∞
𝜔
�𝑧1
𝜔
�𝑧2
0
𝑒 2 ∙𝑡 𝑎2
ďż˝
𝑎𝑦 = 𝑣̇𝑦 + 𝑣𝑥 ∙ 𝜔𝑧 = 𝜆1 ∙ 𝑣�𝑦1 ∙ 𝑒 𝜆1 ∙𝑡 ∙ 𝑎1 + 𝜆2 ∙ 𝑣�𝑦2 ∙ 𝑒 𝜆2 ∙𝑡 ∙ 𝑎2 + 𝑣𝑥 ∙ 𝜔𝑧 ;
𝑣𝑦∞
where: �𝜔 � = −𝐀−1 ∙ 𝐁 ∙ 𝛿𝑓 ;
𝑧∞
194
[4.67]
LATERAL DYNAMICS
and ���
𝑣�𝑦1
𝑣�𝑦2
𝜆
ďż˝ ďż˝
�� , � 1
0
𝜔
�𝑧1
𝜔
�𝑧2
0
�� = eig(𝐀) ;
𝜆2
𝑣𝑦0
𝑎1
𝑣𝑦∞
𝑣�𝑦1
𝑣�𝑦2 −1
and �𝑎 � = ��
ďż˝ ďż˝
�� ∙ ��𝜔 � − �𝜔 �� ;
2
𝑧0
𝑧∞
𝜔
�𝑧1
𝜔
�𝑧2
Results from this model for step steer to +3 deg are shown in Figure 4-59. 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 4-59: Steering step response. Simulation with model from Equation [4.65].
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 [4.62]..[4.66] 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
[4.62]..[4.66], we can simulate as in Figure 4-61.
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.
Some typical measures for multi-unit vehicle combination are given in Figure 4-61, Figure 4-62 and
Figure 4-63.
195
LATERAL DYNAMICS
vx
28
vx[m/s]
26
0
1
df
2
3
sfy
4
5
sry
df[rad]
0.0
-0.2
0
1
2
3
sfy[1]
4
beta_deg
15
sry[1]
5
wz_dps
ay
10
5
Ffyw
8000
Fry
Ffyw[N]
4000
0
Fry[N]
-5
0
0
1
2
ti
3
[ ]
4
5
-10
0
1
2
3
4
5
Figure 4-60: Step steer with 2 deg on road wheels at vx=100 km/h. Simulated with model
in Equations [4.62]..[4.66].
4.5.5.1
Rearward Amplification *
Function definition: Rearward Amplification for long heavy vehicle combinations is the
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
manoeuvre at a certain friction level and constant speed. From Reference [ (Kati, 2013)].
Figure 4-61 illustrates Rearward Amplification, RWA. RWA is defined for a special manoeuvre, e.g. a
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 4-61: Illustration of rearward amplification, P denotes peak value of the motion
variable of interest. From (Kharrazi , 2012).
196
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 4-62 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 4-62: 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 4-63 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.
197
LATERAL DYNAMICS
Figure 4-63: 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 llane 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 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 4-64.
198
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 4-64: 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 [4.19]) or transient (approximately as Equation [4.46]). 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-catious 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 4-62, 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
199
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 4-65: 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 Other intervention than individual wheel brakes
4.6.2.2.4.1 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.
4.6.2.2.4.2 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.
4.6.2.2.4.3 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.
200
LATERAL DYNAMICS
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).
201
LATERAL DYNAMICS
202
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 Two dimensional
5.10 Transient vertical dynamics
5.11 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).
203
𝒛𝒔
Road surface
irregularities
Vehicle
created
disturbances
𝒛𝒓
𝑭𝒓𝒓
Stresses in
vehicle
structure
Vehicle’s
dynamic
structure
𝒛𝒓 − 𝒛𝒔 ,
𝒛̇ 𝒓 − 𝒛̇ 𝒔 , ⋯
VERTICAL DYNAMICS
Fatigue Life
Material
fatigue
Vehicle
response to
excitation
𝒛̈ 𝒔
Compression
of tyre
𝑭𝒓𝒓
Ride Comfort
Human
perception of
vibrations
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.
204
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
Fz
Fx
Fy
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:
205
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].
zz1
zz1
Ξt
z1
z1
tΞ
t
t
Figure 5-3: Different types of variables, both transient and stationary oscillating. The
independent variable 𝜉 can, typically, be either time or distance.
𝑯𝑯𝑯𝑯𝑯𝑯𝑯𝑯 𝒔𝒔𝒔𝒔𝒔𝒔𝒔𝒔𝒔𝒔 𝒐𝒐𝒐𝒐𝒐𝒐𝒐𝒐𝒐𝒐𝒐𝒐:
𝑆𝑆𝑆𝑆𝑆𝑆 𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓 ∶ 𝑧(𝜉) = 𝑧̂ ∙ cos(𝜔 ∙ 𝜉 + 𝜑) ;
𝑁
𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀 𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓 ∶ 𝑧(𝜉) = � 𝑧̂𝑖 ∙ cos(𝜔𝑖 ∙ 𝜉 + 𝜑𝑖 ) ;
𝑖=1
𝑤ℎ𝑒𝑒𝑒 𝜉 𝑖𝑖 𝑡ℎ𝑒 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑒.
206
[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 𝜔 (𝑧̂ = 𝑧̂ (𝜔)), we can do analysis either in the independent variable domain (𝜉) or in
frequency domain (𝜔).
The four combinations of domains are shown in Figure 5-4.
time
domain, t
Fourier Transform
𝒙 = 𝒗𝒙 ∙ 𝒕;
space
domain, x
Fourier Transform
(time) frequency
domain, 𝛚 𝒐𝒐 𝒇
𝒇𝒔 = 𝒇⁄𝒗𝒙 ;
𝝎𝒔 = 𝝎⁄𝒗𝒙 ;
spatial frequency
domain, 𝝎𝒔 𝒐𝒐 𝒇𝒔
Figure 5-4: Four combinations of domains
Time and space domains are treated in section 5.3.1 and 5.3.2. 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.
𝜔 = 2 ∙ 𝜋 ∙ 𝑓;
𝑟𝑟𝑟
𝑤ℎ𝑒𝑒𝑒 𝜔 �
� = 𝑎𝑎𝑎𝑎𝑎𝑎𝑎 𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓;
𝑠
1 𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜
𝑎𝑎𝑎 𝑓 � =
� = (𝑡𝑡𝑡𝑡)𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓;
𝑠
𝑠
The time for one oscillation is called the period time. It is denoted 𝑇:
𝑇=
1 2∙𝜋
=
;
𝑓
𝜔
[5.2]
[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:
207
VERTICAL DYNAMICS
𝑉𝑉𝑉𝑉𝑉𝑉𝑉𝑉:
𝑧
𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀:
𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅:
= 𝑧(𝑡);
𝑡
∫0 𝑒𝑒𝑒 𝑧 2 ∙ 𝑑𝑑
𝑀𝑀(𝑧) =
;
𝑡𝑒𝑒𝑒
𝑅𝑅𝑅(𝑧) =
𝑡
∫ 𝑒𝑒𝑒 𝑧 2
�0
𝑡𝑒𝑒𝑒
∙ 𝑑𝑑
[5.4]
;
If the variable is written as a single frequency harmonic stationary oscillation, these values becomes
as follows:
𝑉𝑉𝑉𝑉𝑉𝑉𝑉𝑉:
𝑧
= 𝑧̂ ∙ cos(𝜔 ∙ 𝑡 + 𝜑) ;
𝑡𝑒𝑒𝑒
𝑡𝑒𝑒𝑒
∫0 𝑧 2 ∙ 𝑑𝑑 ∫0 (𝑧̂ ∙ cos(𝜔 ∙ 𝑡 + 𝜑))2 ∙ 𝑑𝑑
𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀:
𝑀𝑀(𝑧) =
=
=
𝑡𝑒𝑒𝑒
𝑡𝑒𝑒𝑒
𝑡 sin(2 ∙ 𝜔 ∙ 𝑡) 𝑡=𝑡𝑒𝑒𝑒
𝑡
sin(2 ∙ 𝜔 ∙ 𝑡𝑒𝑒𝑒 )
𝑧̂ 2 ∙ �2 +
� 𝑡=0
ďż˝
𝑧̂ 2 ∙ � 𝑒𝑒𝑒
𝑧̂ 2
4∙𝜔
2 +
4∙𝜔
=
=
�⎯⎯⎯⎯� ;
𝑡𝑒𝑒𝑒 →∞ 2
𝑡𝑒𝑒𝑒
𝑡𝑒𝑒𝑒
𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑟𝑟:
𝑅𝑅𝑅(𝑧) = �
[5.5]
𝑡
∫0 𝑒𝑒𝑒 𝑧 2 ∙ 𝑑𝑑
|𝑧̂ |
= �𝑀𝑀(𝑧) =
;
𝑡𝑒𝑒𝑒
√2
If the variable is written as a multiple frequency harmonic stationary oscillation, these values
becomes as follows:
𝑉𝑉𝑉𝑉𝑉𝑉𝑉𝑉:
𝑁
= � 𝑧𝑖 = � 𝑧̂𝑖 ∙ cos(𝜔𝑖 ∙ 𝑡 + 𝜑𝑖 ) ;
𝑧
𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀:
𝑁
𝑖=1
𝑧 2 ∙ 𝑑𝑑
𝑡𝑒𝑒𝑒
∍0
=
𝑡𝑒𝑒𝑒
2
𝑡𝑒𝑒𝑒
∫0 �∑𝑁
𝑖=1 𝑧̂𝑖 ∙ cos(𝜔𝑖 ∙ 𝑡 + 𝜑𝑖 )� ∙ 𝑑𝑑
𝑡�������������������⃗
=
𝑒𝑒𝑒 → ∞
𝑡𝑒𝑒𝑒
𝑡�������������������⃗
𝑒𝑒𝑒 → ∞
𝑀𝑀(𝑧) =
𝑖=1
𝑡𝑒𝑒𝑒
∍0
𝑡𝑒𝑒𝑒
∍0
2
2
∑𝑁
𝑖=1 𝑧̂𝑖 ∙ (cos(𝜔𝑖 ∙ 𝑡 + 𝜑𝑖 )) ∙ 𝑑𝑑
𝑡𝑒𝑒𝑒
𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅:
𝑁
𝑁
2
�∑𝑁
𝑖=1 𝑧𝑖 � ∙ 𝑑𝑑
𝑡𝑒𝑒𝑒
=
𝑁
𝑁
𝑖=1
𝑖=1
𝑧̂𝑖 2
= � 𝑀𝑀(𝑧𝑖 ) = �
;
2
𝑁
[5.6]
𝑧̂𝑖 2
= �� 𝑀𝑀(𝑧𝑖 ) =
𝑅𝑅𝑅(𝑧) = �𝑀𝑀(𝑧) = ��
2
𝑖=1
𝑖=1
2
= ���𝑅𝑅𝑅(𝑧𝑖 )� ;
5.3.1.2
𝑖=1
Power Spectral Density and Frequency bands
So far, the frequency has been a discrete number of frequencies, 𝜔1 , 𝜔2 , ⋯ , 𝜔𝑁 . There are reasons
to treat the frequency as a continuous variable instead. The discrete amplitudes, 𝑧̂1 , 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, 𝜔𝑖 :
208
VERTICAL DYNAMICS
𝑧̂𝑖 =
𝜔𝑖 +𝜔𝑖+1
2
ďż˝
𝜔 +𝜔
𝜔= 𝑖−12 𝑖
𝑧̂𝑐 ∙ 𝑑𝑑 = 𝑧̂𝑐 (𝜔𝑖 ) ∙
𝜔𝑖+1 − 𝜔𝑖−1
𝑧̂𝑖
= 𝑧̂𝑐 (𝜔𝑖 ) ∙ ∆𝜔𝑖 ; ⇒ 𝑧̂𝑐 (𝜔𝑖 ) =
;
2
∆𝜔𝑖
[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).
𝑃𝑃𝑃(𝜔) is a continuous function, while 𝑧̂𝑖 is a discrete function. That means that 𝑃𝑃𝑃(𝜔) is fully
determined by a certain measured or calculated variable 𝑧(𝑡), while 𝑧̂𝑖 depends on which
discretization (which 𝜔𝑖 or which ∆𝜔) that is chosen.
𝑀𝑀�𝑓𝑓𝑓𝑓𝑓𝑓(𝑧(𝑡), 𝜔, ∆𝜔)�
= 𝐺(𝜔);
∆𝜔
𝑤ℎ𝑒𝑒𝑒 filter𝑖𝑖 𝑎 𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏 𝑓𝑓𝑓𝑓𝑓𝑓 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 𝑎𝑎𝑎𝑎𝑎𝑎 𝜔 𝑎𝑎𝑎 𝑤𝑤𝑤ℎ 𝑏𝑏𝑏𝑏 𝑤𝑤𝑤𝑤ℎ ∆𝜔;
𝑃𝑃𝑃(𝑧(𝑡), 𝜔, ∆𝜔) =
[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 ∆𝜔 with ∆𝑓.
When the variable to study (z) is known and the band width is known, one often write simply
𝑃𝑃𝑃(𝜔) or 𝐺(𝜔). G has the same unit as 𝑧 2 , but per [rad/s] or per [oscillations/s]. So, if z is a
𝑚2
𝑚2
displacement in [m], G has the unit [𝑟𝑟𝑟⁄𝑠] or [1⁄𝑠 = 𝑚2 ∙ 𝑠].
The usage of the PSD is, primarily, to easily obtain the RMS via integration:
𝑁
𝑁
𝑖=1
𝑖=1
∞
𝑅𝑅𝑅(𝑧) = �� 𝑀𝑀(𝑧𝑖 ) = �� 𝐺(𝜔𝑖 ) ∙ ∆𝜔𝑖 = � � 𝐺(𝜔) ∙ 𝑑𝑑 ;
RMS is square root of the area under the PSD curve.
[5.9]
𝜔=0
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:
5.3.1.3
Transfer function
𝐺𝑧̇ (𝜔) = 𝜔2 ∙ 𝐺𝑧 (𝜔);
[5.10]
In a minimum model for vertical dynamics there is at least one excitation, often road vertical
displacement, 𝑧𝑟 , and one response, e.g. vertical displacement of sprung mass (=vehicle body), 𝑧𝑠 . A
Transfer function, 𝐻 = 𝐻(𝑗 ∙ 𝜔), is the function which we can use to find the response, given the
excitation:
𝑍𝑠 (𝜔) = 𝐻(𝜔) ∙ 𝑍𝑟 (𝜔);
⟺ ℱ�𝑧𝑠 (𝑡)� = 𝐻(𝜔) ∙ ℱ�𝑧𝑟 (𝑡)�;
∞
𝑤ℎ𝑒𝑒𝑒 ℱ 𝑖𝑖 𝑡ℎ𝑒 𝐹𝐹𝐹𝐹𝐹𝐹𝐹 𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜: 𝑍(𝜔) = ℱ�𝑧(𝑡)� = � 𝑒 −𝑗∙𝜔∙𝑡 ∙ 𝑧(𝑡) ∙ 𝑑𝑑 ;
0
209
[5.11]
VERTICAL DYNAMICS
𝐻 is complex, which is why it has a magnitude, |𝐻| = �(Re(𝐻))2 + (Im(𝐻))2 , and phase,
arg�𝐻(𝜔)� = arctan(Im(𝐻)⁄Re(𝐻)).
𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴: 𝑧̂𝑠 (𝜔) = |𝐻(𝜔)| ∙ 𝑧̂𝑟 (𝜔);
𝑃ℎ𝑎𝑎𝑎: 𝜑𝑠 (𝜔) − 𝜑𝑟 (𝜔) = arg�𝐻(𝜔)� ;
[5.12]
𝑁
𝑤ℎ𝑒𝑒𝑒 𝑧 = � 𝑧̂ (𝜔𝑖 ) ∙ cos(𝜔𝑖 ∙ 𝑡 + 𝜑𝑖 ) ;
𝑖=1
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: 𝐻𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒→𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟 ,
which would be 𝐻𝑧𝑟 →𝑧𝑠 = 𝐻𝑟𝑟𝑟𝑟 𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑→𝑠𝑠𝑠𝑠𝑠𝑠 𝑚𝑚𝑚𝑚 𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 in the example above. Other
examples of relevant transfer functions in vertical vehicle dynamics are:
•
•
•
𝑚⁄ 𝑠 2
], see Section 5.6
𝑚
𝑚
𝐻𝑟𝑟𝑟𝑟 𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑→𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 [ ] , see Section 5.7
𝑚
𝑁
𝐻𝑟𝑟𝑟𝑟 𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑→𝑡𝑡𝑡𝑡 𝑓𝑓𝑓𝑓𝑓 [ ] , see Section 5.8
𝑚
𝐻𝑟𝑟𝑟𝑟 𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑→𝑠𝑠𝑠𝑠𝑠𝑠 𝑚𝑚𝑚𝑚 𝑎𝑎𝑎𝑎𝑎𝑎𝑟𝑟𝑟𝑟𝑟𝑟 [
When transfer function for one derivative is found, it is often easy to convert it to another:
𝐻𝑧1 →𝑧̇2 = 𝑗 ∙ 𝜔 ∙ 𝐻𝑧1 →𝑧2 ;
𝐻𝑧1 →𝑧̈2 = 𝑗 ∙ 𝜔 ∙ 𝑗 ∙ 𝜔 ∙ 𝐻𝑧1 →𝑧2 = −𝜔2 ∙ 𝐻𝑧1 →𝑧2 ;
𝐻𝑧1 →𝑧2 −𝑧3 = 𝐻𝑧1 →𝑧2 − 𝐻𝑧1 →𝑧3 ;
[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:
2
2
𝑀𝑀(𝑧𝑠 (𝑡), 𝜔) �𝑧̂𝑠 (𝜔)� ⁄2 �|𝐻(𝜔)| ∙ 𝑧̂𝑟 (𝜔)� ⁄2
𝐺𝑧𝑠 (𝜔) =
=
=
=
∆𝜔
∆𝜔
∆𝜔
2
2 �𝑧̂𝑟 (𝜔)� ⁄2
2
= �𝐻𝑧𝑟 →𝑧𝑠 (𝜔)� ∙
= �𝐻𝑧𝑟 →𝑧𝑠 (𝜔)� ∙ 𝐺𝑧𝑟 (𝜔);
∆𝜔
[5.14]
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, 𝑣𝑥 , so that:
𝑥 = 𝑣𝑥 ∙ 𝑡 + 𝑥0 ;
The offset (𝑥0 ) is the phase (spatial) offset (𝑥0 ) is the correspondence to the phase angle (𝜑).
[5.15]
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
210
VERTICAL DYNAMICS
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 = 2 ∙ 𝜋 ∙ 𝑓𝑠 ;
𝑟𝑟𝑟
𝑤ℎ𝑒𝑒𝑒 Ω �
� = 𝑠𝑠𝑠𝑠𝑠𝑠𝑠 𝑎𝑎𝑎𝑎𝑎𝑎𝑎 𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓;
𝑚
1 𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜
𝑎𝑎𝑎 𝑓𝑠 � =
� = 𝑠𝑠𝑠𝑠𝑠𝑠𝑠 𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓;
𝑚
𝑚
The correspondence to period time is wave length, denoted 𝜆:
𝜆[𝑚] =
1 2∙𝜋
=
;
𝑓𝑠
Ί
[5.16]
[5.17]
Now, the basic assumption in Equation [5.15] and definitions of frequencies gives:
𝜔 = 𝑣𝑥 ∙ Ω; 𝑎𝑎𝑎 𝑓 = 𝑣𝑥 ∙ 𝑓𝑠 ;
The relation between the phase (spatial) offset (𝑥0 ) and the phase angle (𝜑) is:
𝑥0 =
𝜆∙𝜑
;
2∙𝜋
[5.18]
[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.
𝑉𝑉𝑉𝑉𝑉𝑉𝑉𝑉:
𝑧
𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀:
= 𝑧(𝑥);
𝑥𝑒𝑒𝑒 2
𝑧 ∙ 𝑑𝑑
∍0
𝑀𝑀𝑠 (𝑧) =
;
𝑥𝑒𝑒𝑒
𝑥
∫0 𝑒𝑒𝑒 𝑧 2
ďż˝
∙ 𝑑𝑑
[5.20]
𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅:
𝑅𝑅𝑅𝑠 (𝑧) =
𝑉𝑉𝑉𝑉𝑉𝑉𝑉𝑉:
= 𝑧̂ ∙ cos(Ω ∙ 𝑥 + 𝑥0 ) ;
𝑧̂ 2
𝑀𝑀𝑠 (𝑧) = ⋯ = = 𝑀𝑀(𝑧);
2
|𝑧̂ |
𝑅𝑅𝑅𝑠 (𝑧) = ⋯ =
= 𝑅𝑅𝑅(𝑧);
√2
𝑥𝑒𝑒𝑒
;
Because 𝑣𝑥 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:
𝑧
𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀:
𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅:
211
[5.21]
VERTICAL DYNAMICS
If the variable is written as a multiple frequency harmonic stationary oscillation, these values
becomes as follows:
𝑉𝑉𝑉𝑉𝑉𝑉𝑉𝑉:
𝑁
𝑁
= � 𝑧𝑖 = � 𝑧̂𝑖 ∙ cos(Ω𝑖 ∙ 𝑥 + 𝑥0𝑖 ) ;
𝑧
𝑖=1
𝑁
2
𝑖=1
𝑧̂𝑖
= 𝑀𝑀(𝑧);
2
𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀:
𝑀𝑀𝑠 (𝑧) = �
𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅:
𝑅𝑅𝑅𝑠 (𝑧) = �𝑀𝑀𝑠 (𝑧) = ���𝑅𝑅𝑅(𝑧𝑖 )� = 𝑅𝑅𝑅(𝑧);
𝑖=1
𝑁
𝑖=1
[5.22]
2
5.3.2.2 Spatial Power Spectral Density and Frequency
bands
A correspondence to Power Spectral Density in space domain is denoted 𝑃𝑃𝑃𝑠 in the following:
𝑃𝑃𝑃𝑠 (𝑧(𝑥), Ω, ∆𝜆) =
𝑀𝑀�𝑓𝑓𝑓𝑓𝑓𝑓(𝑧(𝑥),Ω,∆𝜆)�
∆𝜆
= Ό(Ί);
where "filter" is a band pass filter centred around ω and with band width ∆𝑓;
[5.23]
When the variable to study (z) is known and the band width is known, one often write simply
𝑃𝑃𝑃𝑠 (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
𝑚2
𝑚3
𝑚2
displacement in [m], Φ has the unit [𝑟𝑟𝑟⁄𝑚 = 𝑟𝑟𝑟] or [1⁄𝑚 = 𝑚3].
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: 𝑧𝑟 = 𝑧𝑟 (𝑥). The function 𝑧𝑟 (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
𝑧𝑟 = 𝑧𝑟 (𝑥) = � 𝑧̂𝑖 ∙ cos(Ω𝑖 ∙ 𝑥 + 𝑥0𝑖 ) ;
[5.24]
𝑖=1
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 = 2 ∙ 𝜋⁄Ω, from Equation [5.16]) and it can be described as:
212
VERTICAL DYNAMICS
𝑧𝑟 = 𝑧𝑟 (𝑥) = 𝑧̂ ∙ cos(Ω ∙ 𝑥 + 𝑥0 ) ;
5.4.2
[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].
Figure 5-5: Four typical road types, whereof the upper 3 can be considered as road types.
From (AB Volvo, 2011).
213
VERTICAL DYNAMICS
PHI=PHI0.*((OMEGA/OMEGA0).^(-waviness))
Φ Ω = Φ ∙ Ω⁄Ω −w
0
-2
10
0
very rough
rough
smooth
-3
𝑚2
]
𝑟𝑟𝑟⁄𝑚
10
very bad road
bad road
very good road
-4
𝑃𝑃𝑃𝑠PHI
Ί /[m*m/(rad/m)]
= Ό Ί in [
10
-5
10
-6
10
-7
10
rough
-8
10
smooth
-9
10
-1
10
m2
rad⁄m
m2
𝜱𝟎 = 𝟏𝟏 ∙ 𝟏𝟏−𝟔 rad⁄m ;
m2
𝜱𝟎 = 𝟏 ∙ 𝟏𝟏−𝟔 rad⁄m ;
very rough 𝜱𝟎 = 𝟏𝟏𝟏 ∙ 𝟏𝟏−𝟔
𝜴𝟎 = 𝟏 rad⁄m for all;
50
20
; 𝒘=𝟐 𝟏;
𝒘 = 𝟐. 𝟓 𝟏 ;
𝒘=𝟑 𝟏;
0
10
⁄𝑚 ]
Ί
in
[𝑟𝑟𝑟
OMEGA /[rad/m]
10
5
1
10
2
Figure 5-6: PSD spectra for the three typical roads in Figure 5-5.
Typical values are
𝛺 −𝑤
𝑀𝑀𝑠 (𝑧𝑟 , 𝛺)
𝛷 = 𝛷(𝛺) = 𝛷0 ∙ � �
=
;
𝛺0
∆𝛺
m2
𝑤ℎ𝑒𝑒𝑒 𝛷0 = road severity �
ďż˝;
rad⁄m
w = road waviness [1];
𝛺 = spatial angular frequency [rad⁄m];
𝛺0 = 1 [rad/m];
1
𝜆 𝑚
[5.26]
m2
Very good road: 𝛷0 = 1 ∙ 10−6 �
ďż˝;
rad⁄m
m2
Bad road ∜
𝛷0 = 10 ∙ 10−6 �
ďż˝;
rad⁄m
[5.27]
2
m
ďż˝;
Very bad road ∶ 𝛷0 = 100 ∙ 10−6 �
rad⁄m
𝑇ℎ𝑒 𝑤𝑤𝑤𝑤𝑤𝑤𝑤𝑤 𝑖𝑖 𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛 𝑖𝑖 𝑡ℎ𝑒 𝑟𝑟𝑟𝑟𝑟 𝑜𝑜 𝑤 = 2. .3 [1],
𝑤ℎ𝑒𝑒𝑒 𝑠𝑠𝑠𝑠𝑠ℎ 𝑟𝑟𝑟𝑟𝑟 ℎ𝑎𝑎𝑎 𝑙𝑙𝑙𝑙𝑙𝑙 𝑤𝑤𝑤𝑤𝑤𝑤𝑤𝑤 𝑡ℎ𝑎𝑎 𝑏𝑏𝑏 𝑟𝑟𝑟𝑟𝑟.
214
VERTICAL DYNAMICS
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 2-32.
A certain road can be described with:
•
•
•
𝛺1 , ⋯ , 𝛺𝑁
𝑧̂1 , ⋯ , 𝑧̂𝑁
𝑥01 , ⋯ , 𝑥0𝑁
Number of frequency components, N, to select is a matter of accuracy or experience. The offsets,
𝑥01 , ⋯ , 𝑥0𝑁 , 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 𝑧𝑟 (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 𝑥0𝑖 ) .
RoadQuality=1
zr /[m]
0.05
N=10
N=20
N=100
0
-0.05
0
1
2
3
4
5
x/[m]
6
7
8
9
10
RoadQuality=2
zr /[m]
0.05
N=10
N=20
N=100
0
-0.05
0
1
2
3
4
5
x/[m]
6
7
8
9
10
smooth
RoadQuality=3
zr /[m]
0.05
N=10
N=20
N=100
0
-0.05
0
1
2
3
4
5
x/[m]
6
7
8
9
10
Figure 5-7: Road profiles, 𝑧𝑟 (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:
215
VERTICAL DYNAMICS
𝑀𝑀(𝑧𝑟 , 𝜔)
𝑀𝑀𝑠 (𝑧𝑟 , 𝛺)
= {𝑢𝑢𝑢: 𝜔 = 𝑣𝑥 ∙ 𝛺} =
=
∆𝜔
𝑣𝑥 ∙ ∆𝛺
𝛺 −𝑤
−𝑤
𝛷
∙
ďż˝
0
𝛺
𝑀𝑀𝑠 (𝑧𝑟 , 𝛺)
𝛺0 �
= �𝑢𝑢𝑢: 𝛷0 ∙ � � =
ďż˝=
=
𝛺0
𝑣𝑥
∆𝛺
𝜔 −𝑤
ďż˝
−𝑤
𝛷0 ∙ 𝛺
𝛷0 𝑣 �
𝛷0
= −𝑤
= −𝑤 ∙ 𝑥
= −𝑤 ∙ 𝑣𝑥 𝑤−1 ∙ 𝜔−𝑤 ;
𝑣𝑥
𝛺0 ∙ 𝑣𝑥 𝛺0
𝛺0
𝐺𝑧𝑟 (𝜔) =
Then, we can use Equation [5.14] to obtain the response𝑧𝑠 :
2
2
𝐺𝑧𝑠 (𝜔) = �𝐻𝑧𝑟 →𝑧𝑠 (𝜔)� ∙ 𝐺𝑧𝑟 (𝜔) = �𝐻𝑧𝑟 →𝑧𝑠 (𝜔)� ∙
𝛷0
𝛺0 −𝑤
∙ 𝑣𝑥 𝑤−1 ∙ 𝜔−𝑤 ;
[5.28]
[5.29]
Then we can use Equation [5.9] to obtain the RMS of the response 𝑧𝑠 :
𝑁
𝑅𝑅𝑅(𝑧𝑠 ) = �� 𝐺𝑧𝑠 (𝜔𝑖 ) ∙ ∆𝜔 = �
or
𝑖=1
∞
𝛺0
𝑅𝑅𝑅(𝑧𝑠 ) = � � 𝐺𝑧𝑠 (𝜔) ∙ 𝑑𝑑 = �
𝜔=0
𝛷0
−𝑤
𝛷0
𝛺0
−𝑤
∙ 𝑣𝑥
𝑤−1
𝑁
2
∙ ��𝐻𝑧𝑟→𝑧𝑠 (𝜔𝑖 )� ∙ 𝜔𝑖 −𝑤 ∙ ∆𝜔 ;
𝑖=1
∞
[5.30]
2
∙ 𝑣𝑥 𝑤−1 ∙ ��𝐻𝑧𝑟→𝑧𝑠 (𝜔)� ∙ 𝜔−𝑤 ∙ 𝑑𝑑 ;
𝜔=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, 𝐹𝑧 = 𝐹𝑓𝑓𝑓 + 𝐹𝑓𝑓𝑓 + 𝐹𝑟𝑟𝑟 + 𝐹𝑟𝑟𝑟 . 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, 𝑚/4, which is suspended by one of the wheel’s vertical force, 𝐹𝑖𝑖𝑖 . 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.
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.
216
VERTICAL DYNAMICS
real vehicle
model
zs
m
z
m
x
𝑚 ∙ 𝑧̈𝑠
py
zr
zs
𝑚 ∙ 𝑧̈𝑠
𝑚∙𝑔
Pz
Frz
𝑚∙𝑔
Pz
zr
Frz
Figure 5-8: One-dimensional model without dynamic degree of freedom
Probably, we could find the equations without very much formalism (𝑚 ∙ 𝑧̈𝑠 = 𝐹𝑟𝑟 ; 𝑎𝑎𝑎 𝑧𝑠 =
𝑧𝑟 (𝑡);), but the following equations exemplifies a formalism which will be useful when we expand
the model later.
𝑬𝑬𝑬𝑬𝑬𝑬𝑬𝑬𝑬𝑬𝑬:
5.5.1.1
𝑚 ∙ 𝑧̈𝑠 + 𝑚 ∙ 𝑔 = 𝑃𝑧 ;
𝐹𝑟𝑟 − 𝑃𝑧 = 0;
𝑪𝑪𝑪𝑪𝑪𝑪𝑪𝑪𝑪𝑪𝑪𝑪𝑪: 𝑧𝑟 = 𝑧𝑠 ;
𝑬𝑬𝑬𝑬𝑬𝑬𝑬𝑬𝑬𝑬:
𝑧𝑟 = 𝑧𝑟 (𝑡);
[5.31]
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:
2∙π
𝑧𝑟 = 𝑧𝑟 (𝑥) = 𝑧̂ ∙ cos(Ω ∙ 𝑥 + 𝑥0 ) = 𝑧̂ ∙ cos �
∙ 𝑥 + 𝑥0 � ;
𝜆
ďż˝
�⇒
𝑥 = 𝑣𝑥 ∙ 𝑡;
𝐴𝐴𝐴𝐴𝐴𝐴 𝑥0 = 0;
2 ∙ π ∙ 𝑣𝑥
∙ 𝑡� = 𝑧̂𝑟 ∙ cos(ω ∙ 𝑡) ; ⇒
⇒ 𝑧𝑟 (𝑡) = 𝑧̂𝑟 ∙ cos �
𝜆
2 ∙ π ∙ 𝑣𝑥
2 ∙ π ∙ 𝑣𝑥
∙ 𝑧̂𝑟 ∙ sin �
∙ 𝑡� = −ω ∙ 𝑧̂𝑟 ∙ sin(ω ∙ 𝑡) ; ⇒
⇒ 𝑧̇𝑟 (𝑡) = −
𝜆
𝜆
2 ∙ π ∙ 𝑣𝑥 2
2 ∙ π ∙ 𝑣𝑥
⇒ 𝑧̈𝑟 (𝑡) = − �
� ∙ 𝑧̂𝑟 ∙ cos �
∙ 𝑡� = −ω2 ∙ 𝑧̂𝑟 ∙ cos(ω ∙ 𝑡) ;
𝜆
𝜆
[5.32]
Insertion in the model in Equation [5.31] gives directly the solution:
𝑃𝑧 (𝑡) = 𝐹𝑟𝑟 (𝑡) = 𝑚 ∙ 𝑔 + ∆𝐹(𝑡) = 𝑚 ∙ 𝑔 + 𝐹� ∙ cos(ω ∙ 𝑡) ;
ďż˝
𝑧𝑟 (𝑡) = 𝑧𝑠 (𝑡) = 𝑧̂𝑟 ∙ cos (ω ∙ 𝑡);
𝑧̈𝑟 (𝑡) = 𝑧̈𝑠 (𝑡) = 𝑎� ∙ cos(ω ∙ 𝑡) ;
ďż˝
𝑤ℎ𝑒𝑒𝑒 �𝐹 = −𝑚 ∙ ω ∙ 𝑧̂𝑟 ;
𝑎� = −ω2 ∙ 𝑧̂𝑟 ;
2
5.5.1.1.1 Analysis of solution
217
[5.33]
VERTICAL DYNAMICS
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(𝑧𝑟 )
𝐻𝑧𝑟 →𝑧𝑟 −𝑧𝑠 = �𝐻𝑧𝑟 →𝑧𝑟 −𝑧𝑠 = 𝐻𝑧𝑟 →𝑧𝑠 − 𝐻𝑧𝑠 →𝑧𝑠 = 𝐻𝑧𝑟 →𝑧𝑠 − 1� = 0 + 𝑗 ∙ 0;
𝐻𝑧𝑟 →𝑧̈𝑠 = �𝐻𝑧𝑟 →𝑧̈𝑠 = (𝑗 ∙ 𝜔)2 ∙ 𝐻𝑧𝑟 →𝑧𝑠 = −ω2 ∙ 𝐻𝑧𝑟 →𝑧𝑠 � = −ω2 + 𝑗 ∙ 0;
𝐻𝑧𝑟 →∆𝐹𝑟𝑟 = �𝐻𝑧𝑟 →∆𝐹𝑟𝑟 = 𝑚 ∙ 𝐻𝑧𝑟 →𝑧̈𝑠 � = −𝑚 ∙ ω2 + 𝑗 ∙ 0;
𝐻𝑧𝑟 →𝑧𝑠 = �𝐻𝑧𝑟 →𝑧𝑠 =
[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.
m = 1600 kg; lambda = 5 m;
7
10
6
10
5
10
4
abs(H)
10
3
10
H_zr_zs (==1)
H_zr_zr-zs (==0)
H_zr_derderzs
H_zr_Frz
2
10
1
10
0
10
-1
10
0
10
1
2
10
10
3
10
vx [m/s]
Figure 5-9: Transfer functions from model in Figure 5-8, excited with single frequencies.
218
VERTICAL DYNAMICS
So, for example, we can use the transfer function diagram as follows: If we have a road displacement
� ≈ 2.8𝐻𝐻), we can read
amplitude of 1 cm (𝑧̂𝑟 = 0.01 𝑚) and a speed of 50 km/h (𝑣𝑥 ≈ 14 𝑚/𝑠 =
out, e.g.:
•
•
•
�𝐻𝑧𝑟 →𝑧̈𝑠 (𝑣𝑥 )� ≈ 305; ⇒ |𝑎� | = 310 ∙ 𝑧�𝑟 = 305 ∙ 0.01 = 3.05 𝑚⁄𝑠 2 ;. From this we can
calculate 𝑅𝑅𝑅(𝑧̈ 𝑠 ) = |3.05|⁄√2 ≈ 2.16 𝑚⁄𝑠 2 . The RMS value of acceleration will later be
related to ride comfort, see Section 5.6.
�𝐻𝑧𝑟 →𝑧𝑟 −𝑧𝑠 (𝑣𝑥 )� = 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.
�𝐻𝑧𝑟 →∆𝐹𝑟𝑟 (𝑣𝑥 )� ≈ 487000; ⇒ |𝐹� | = 487000 ∙ 𝑧�𝑟 = 487000 ∙ 0.01 = 4870 𝑁;. If �𝐹� � had
been > 𝑚 ∙ 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:
𝑅𝑅𝑅(𝑧𝑠 ) = �
𝛷0
𝛺0
−𝑤
=ďż˝
∞
2
∙ 𝑣𝑥 𝑤−1 ∙ ��𝐻𝑧𝑟→𝑧𝑠 (𝜔)� ∙ 𝜔−𝑤 ∙ 𝑑𝑑 =
10 ∙ 10
1
−6
𝜔=0
∞
[5.35]
2
∙ 𝑣𝑥 2.5−1 ∙ ��𝐻𝑧𝑟→𝑧𝑠 (𝜔)� ∙ 𝜔−2.5 ∙ 𝑑𝑑 ;
𝜔=0
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 𝑧̈𝑠 , 𝑧𝑟 − 𝑧𝑠 and 𝐹𝑟𝑟 . 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.
The corresponding mathematical model becomes as follows:.
𝑬𝑬𝑬𝑬𝑬𝑬𝑬𝑬𝑬𝑬𝑬:
𝑪𝑪𝑪𝑪𝑪𝑪𝑪𝑪𝑪𝑪𝑪𝑪:
𝑬𝑬𝑬𝑬𝑬𝑬𝑬𝑬𝑬𝑬:
𝑚 ∙ 𝑧̈𝑠 + 𝑚 ∙ 𝑔 = 𝑃𝑧 ;
𝐹𝑟𝑟 − 𝑃𝑧 = 0;
𝑃𝑧 = 𝑐 ∙ (𝑧𝑟 − 𝑧𝑠 ) + 𝑑 ∙ (𝑧̇𝑟 − 𝑧̇𝑠 ) + 𝑚 ∙ 𝑔;
𝑧𝑟 = 𝑧𝑟 (𝑡);
219
[5.36]
VERTICAL DYNAMICS
real vehicle
model
zs
z
m
m
𝑚 ∙ 𝑧̈𝑠
x
py
c
d
zs
𝑚 ∙ 𝑧̈𝑠
𝑚∙𝑔
Pz
𝑚∙𝑔
Pz
zs
Pz
zr
zr
zr
Pz
Frz
Frz
Figure 5-10: One-dimensional model with one dynamic degree of freedom
Note that if we measure 𝑧𝑢 and 𝑧𝑠 from the static equilibrium, the static load, 𝑚 ∙ 𝑔, disappears
when constitution is inserted in equilibrium.
5.5.2.1
Response to a single frequency excitation
Eliminating 𝑃𝑧 and 𝑧𝑟 gives:
𝑚 ∙ 𝑧̈𝑠 = 𝑐 ∙ (𝑧𝑟 (𝑡) − 𝑧𝑠 ) + 𝑑 ∙ (𝑧̇𝑟 (𝑡) − 𝑧̇𝑠 );
∆𝐹𝑟𝑟 = 𝑐 ∙ (𝑧𝑟 (𝑡) − 𝑧𝑠 ) + 𝑑 ∙ (𝑧̇𝑟 (𝑡) − 𝑧̇𝑠 );
[5.37]
where 𝐹𝑟𝑟 (𝑡) = 𝑚 ∙ 𝑔 + ∆𝐹𝑟𝑧 (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 𝑥0 = 0. Insertion in Equation [5.36] yields:
𝑚 ∙ 𝑧̈𝑠 = 𝑐 ∙ (𝑧̂𝑟 ∙ cos(ω ∙ 𝑡) − 𝑧𝑠 ) + 𝑑 ∙ (−ω ∙ 𝑧̂𝑟 ∙ sin(ω ∙ 𝑡) − 𝑧̇𝑠 );
𝑚 ∙ 𝑧̈𝑠 + 𝑑 ∙ 𝑧̇𝑠 + 𝑐 ∙ 𝑧𝑠 = 𝑧̂𝑟 ∙ (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.
Solution with Fourier transform
Fourier transform of Equation [5.37] yields:
𝑚 ∙ �−ω2 ∙ ℱ(𝑧𝑠 )� = 𝑐 ∙ �ℱ(𝑧𝑟 ) − ℱ(𝑧𝑠 )� + 𝑑 ∙ �𝑗 ∙ 𝜔 ∙ ℱ(𝑧𝑟 ) − 𝑗 ∙ 𝜔 ∙ ℱ(𝑧𝑠 )�;
ℱ(∆𝐹𝑟𝑟 ) = 𝑐 ∙ �ℱ(𝑧𝑟 ) − ℱ(𝑧𝑠 )� + 𝑑 ∙ �𝑗 ∙ 𝜔 ∙ ℱ(𝑧𝑟 ) − 𝑗 ∙ 𝜔 ∙ ℱ(𝑧𝑠 )�;
[5.39]
From this, we can then solve the transfer functions:
𝐻𝑧𝑟 →𝑧𝑠 =
ℱ(𝑧𝑠 )
ℱ(𝑧𝑟 )
=
𝑐+𝑗∙𝑑∙𝜔
;
(𝑐 − 𝑚 ∙ ω2 ) + 𝑗 ∙ 𝑑 ∙ 𝜔
𝐻𝑧𝑟 →∆𝐹𝑟𝑟 = (𝑐 + 𝑗 ∙ 𝑑 ∙ 𝜔) ∙ �1 − 𝐻𝑧𝑟 →𝑧𝑠 �;
5.5.2.1.1 Solution with trigonometry
220
[5.40]
VERTICAL DYNAMICS
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 𝑥0 = 0 in Equation [5.25]:
Assumed solution:
𝑚 ∙ 𝑧̈𝑠 + 𝑑 ∙ 𝑧̇𝑠 + 𝑐 ∙ 𝑧𝑠 = 𝑧̂𝑟 ∙ (𝑐 ∙ cos(𝜔 ∙ 𝑡) − 𝑑 ∙ 𝜔 ∙ sin(𝜔 ∙ 𝑡));
Insertion:
𝑧𝑠 = 𝑧̂𝑠 ∙ cos(𝜔 ∙ 𝑡 − 𝜑) ; ⇒
⇒ {𝑢𝑢𝑢: cos(𝑎 − 𝑏) = cos 𝑎 ∙ cos 𝑏 + sin 𝑎 ∙ sin 𝑏} ⇒
⇒ 𝑧𝑠 = 𝑧̂𝑠 ∙ [cos(𝜔 ∙ 𝑡) ∙ cos 𝜑 + sin(𝜔 ∙ 𝑡) ∙ sin 𝜑]; ⇒
⇒ 𝑧̇𝑠 = 𝑧̂𝑠 ∙ 𝜔 ∙ [− sin(𝜔 ∙ 𝑡) ∙ cos 𝜑 + cos(𝜔 ∙ 𝑡) ∙ sin 𝜑]; ⇒
⇒ 𝑧̈𝑠 = −𝑧̂𝑠 ∙ 𝜔2 ∙ [cos(𝜔 ∙ 𝑡) ∙ cos 𝜑 + sin(𝜔 ∙ 𝑡) ∙ sin 𝜑];
−𝑚 ∙ 𝑧̂𝑠 ∙ 𝜔2 ∙ [cos(𝜔 ∙ 𝑡) ∙ cos 𝜑 + sin(𝜔 ∙ 𝑡) ∙ sin 𝜑] +
+𝑑 ∙ 𝑧̂𝑠 ∙ 𝜔 ∙ [− sin(𝜔 ∙ 𝑡) ∙ cos 𝜑 + cos(𝜔 ∙ 𝑡) ∙ sin 𝜑] +
+𝑐 ∙ 𝑧̂𝑠 ∙ [cos(𝜔 ∙ 𝑡) ∙ cos 𝜑 + sin(𝜔 ∙ 𝑡) ∙ sin 𝜑] =
= 𝑧̂𝑟 ∙ (𝑐 ∙ cos(𝜔 ∙ 𝑡) − 𝑑 ∙ 𝜔 ∙ sin(𝜔 ∙ 𝑡)); ⇒
𝒄𝒄𝒄 𝒕𝒕𝒕𝒕𝒕: − 𝑚 ∙ 𝑧̂𝑠 ∙ 𝜔2 ∙ cos 𝜑 + 𝑑 ∙ 𝑧̂𝑠 ∙ 𝜔 ∙ sin 𝜑 + 𝑐 ∙ 𝑧̂𝑠 ∙ cos 𝜑 = 𝑧̂𝑟 ∙ 𝑐;
⇒
⇒ �
𝒔𝒔𝒔 𝒕𝒕𝒕𝒕𝒕: − 𝑚 ∙ 𝑧̂𝑠 ∙ 𝜔2 ∙ sin 𝜑 − 𝑑 ∙ 𝑧̂𝑠 ∙ 𝜔 ∙ cos 𝜑 + 𝑐 ∙ 𝑧̂𝑠 ∙ sin 𝜑 = −𝑧̂𝑟 ∙ 𝑑 ∙ 𝜔;
𝑚 ∙ 𝑑 ∙ 𝜔3
⎧
𝜑
=
arctan
ďż˝;
ďż˝
⎪
𝑐 2 − 𝑚 ∙ 𝑐 ∙ 𝜔 2 + 𝑑2 ∙ 𝜔 2
⇒
𝑑∙𝜔
⎨𝑧̂𝑠
= �𝐻𝑧𝑟 →𝑧𝑠 �;
⎪ = (𝑚 2
∙ 𝜔 − 𝑐) ∙ sin 𝜑 + 𝑑 ∙ 𝜔 ∙ cos 𝜑
⎩𝑧̂𝑟
𝑤ℎ𝑒𝑒𝑒 𝜔 =
2 ∙ π ∙ 𝑣𝑥
𝜆
;
[5.41]
[5.42]
[5.43]
We have identified �𝐻𝑧𝑟 →𝑧𝑠 (𝑣𝑥 )�, which can be compared to �𝐻𝑧𝑟 →𝑧𝑠 (𝑣𝑥 )� 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.
5.5.2.1.2 Analysis of solution
We can elaborate further with Equation [5.40]:
221
VERTICAL DYNAMICS
Amplitude:
�𝐻𝑧𝑟 →𝑧𝑠 � =
𝑧̂𝑠
𝑐+𝑗∙𝑑∙ω
=ďż˝
ďż˝=
(𝑐 − 𝑚 ∙ ω2 ) + 𝑗 ∙ 𝑑 ∙ ω
𝑧̂𝑟
𝑐+𝑗∙𝑑∙ω
⎧𝐴𝐴𝐴𝐴𝐴𝐴
= 𝑅𝑅𝑅 + 𝑗 ∙ 𝐼𝐼𝐼;⎫
2
⎪
⎪
(𝑐 − 𝑚 ∙ ω2 ) + 𝑗 ∙ 𝑑 ∙ ω
𝑐2 + 𝑑 ∙ ω2
=
=⋯=�
;
𝑆𝑆𝑆𝑆𝑆 𝑓𝑓𝑓 𝑅𝑅𝑅 𝑎𝑎𝑎 𝐼𝐼𝐼;
2
2
⎨
⎬
�𝑐 − 𝑚 ∙ ω2 � + 𝑑 ∙ ω2
⎪
⎪
2
2
�𝐻𝑧𝑟 →𝑧𝑠 � = �𝑅𝑅𝑅 + 𝐼𝐼𝐼 ;
⎊
⎭
Phase:
[5.44]
𝑐+𝑗∙𝑑∙ω
𝜑𝑠 (𝜔) − 𝜑𝑟 (𝜔) = arg �
ďż˝=
−𝑚 ∙ ω2 + 𝑐 + 𝑗 ∙ 𝑑 ∙ ω
𝑐+𝑗∙𝑑∙ω
⎧𝐴𝐴𝐴𝐴𝐴𝐴
= 𝑅𝑅𝑅 + 𝑗 ∙ 𝐼𝐼𝐼;⎫
⎪
⎪
(𝑐 − 𝑚 ∙ ω2 ) + 𝑗 ∙ 𝑑 ∙ ω
=
=⋯=
𝑆𝑆𝑆𝑆𝑆
𝑓𝑓𝑓
𝑅𝑅𝑅
𝑎𝑎𝑎
𝐼𝐼𝐼;
⎨
⎬
⎪
⎪
tan�arg�𝐻𝑧𝑟→𝑧𝑠 �� = 𝐼𝐼𝐼⁄𝑅𝑅𝑅 ;
⎊
⎭
= arctan ďż˝
𝑚 ∙ 𝑑 ∙ 𝜔3
ďż˝;
𝑐 2 − 𝑚 ∙ 𝑐 ∙ 𝜔 2 + 𝑑2 ∙ 𝜔 2
where 𝜔 =
2 ∙ 𝜋 ∙ 𝑣𝑥
;
𝜆
Equation [5.13] now allows us to get the magnitudes of the other transfer functions as well:
𝐻𝑧𝑟 →𝑧𝑠 = from Equation [5.40];
𝐻𝑧𝑟 →𝑧𝑟 −𝑧𝑠 = 𝐻𝑧𝑟 →𝑧𝑟 − 𝐻𝑧𝑟 →𝑧𝑠 = 1 − 𝐻𝑧𝑟 →𝑧𝑠 ;
𝐻𝑧𝑟 →𝑧̈𝑠 = −𝜔2 ∙ 𝐻𝑧𝑟 →𝑧𝑠 ;
𝐻𝑧𝑟 →∆𝐹𝑟𝑟 = {∆𝐹𝑟𝑟 = 𝑐 ∙ (𝑧𝑟 − 𝑧𝑠 ) + 𝑑 ∙ (𝑧̇ 𝑟 − 𝑧̇ 𝑠 )} =
= 𝑐 ∙ �𝐻𝑧𝑟 →𝑧𝑟 − 𝐻𝑧𝑟 →𝑧𝑠 � + 𝑑 ∙ �𝐻𝑧𝑟 →𝑧̇𝑟 − 𝐻𝑧𝑟 →𝑧̇𝑠 � =
= (𝑐 + 𝑗 ∙ 𝑑 ∙ 𝜔) ∙ �𝐻𝑧𝑟 →𝑧𝑟 − 𝐻𝑧𝑟 →𝑧𝑠 � =
= (𝑐 + 𝑗 ∙ 𝑑 ∙ 𝜔) ∙ �1 − 𝐻𝑧𝑟 →𝑧𝑠 �;
[5.45]
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]:
�𝐻𝑧𝑟 →𝑧𝑠 � = �
𝑐2 + 𝑑2 ∙ ω2
(𝑐 − 𝑚 ∙ ω2 )2 + 𝑑2 ∙ ω2
�𝐻𝑧𝑟 →𝑧̈𝑠 � = 𝜔2 ∙ �
𝑐2
2
+𝑑 ∙
ω2
;
(𝑐 − 𝑚 ∙ ω2 )2 + 𝑑2 ∙ ω2
[5.46]
;
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.
222
VERTICAL DYNAMICS
m = 1600[kg]; c = 568000 [N/m]; d = 48000[N/(m/s)]; lambda = 5 m;
7
10
6
10
H_zr_zs
H_zr_zr-zs
H_zr_derderzs
H_zr_Frz
5
10
4
abs(H)
10
3
10
2
10
1
10
0
10
-1
10
0
10
1
2
10
3
10
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: 𝐻𝑎→𝑏 is denoted H_a_b.
m = 1600[kg]; c = 568000 [N/m]; d = 48000[N/(m/s)]; lambda = 5 m;
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.
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 (𝑧̂𝑟 = 0.01 𝑚) and the vehicle should drive on it at a speed of
50 km/h (𝑣𝑥 ≈ 14 𝑚/𝑠 =
� ≈ 2.8𝐻𝐻), we can read out, e.g.:
223
VERTICAL DYNAMICS
•
•
•
Ride comfort related: �𝐻𝑧𝑟 →𝑧̈𝑠 (𝑣𝑥 )� ≈ 366; ⇒ |𝑎
� | = 366 ∙ 𝑧�𝑟 = 366 ∙ 0.01 = 3.66 𝑚⁄𝑠 2 ;.
From this we can calculate 𝑅𝑅𝑅(𝑧̈ 𝑠 ) = |3.66|⁄√2 ≈ 2.59 𝑚⁄𝑠 2 .
Fatigue life related: �𝐻𝑧𝑟 →𝑧𝑟 −𝑧𝑠 (𝑣𝑥 )� ≈ 0.579; ⇒ |𝑧̂𝑟 − 𝑧̂𝑠 | = 0.579 ∙ 𝑧�𝑟 = 0.579 ∙ 0.01 =
0.00579 𝑚;.
Road grip related: �𝐻𝑧𝑟 →∆𝐹𝑟𝑟 (𝑣𝑥 )� ≈ 487000; ⇒ |𝐹� 𝑟𝑟 | = 487000 ∙ 𝑧�𝑟 = 487000 ∙
0.01 = 4870 𝑁;.
If we compare with the corresponding numbers for the model, see Sections 5.5.1.1.1 and 0, we find
that going from the stiff model to the model with one dynamic degree of freedom: For the frequency
content from the speed of 50 km/h, the sprung mass acceleration variation has become larger
(probably increased discomfort), the deformation of the suspension has become larger (probably
shorter fatigue life) and the road contact force variation has become larger (probably worse road
grip).
Figure 5-11 also shows the curves for the undamped system (d=0). The highest peaks appear at
approximately vx=15 m/s. This corresponds to the speed where the natural (=undamped) eigen
frequency appears (𝑣𝑥,𝑐𝑐𝑐𝑐 = 𝜆 ∙ 𝑓𝑐𝑐𝑐𝑐 = 𝜆 ∙ 𝜔𝑐𝑐𝑐𝑐 ⁄(2 ∙ 𝜋) = 𝜆 ∙ �𝑐⁄𝑚�(2 ∙ 𝜋) ≈ 14.7 𝑚/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.
real vehicle
model
𝑚𝑠 ∙ 𝑧̈𝑠
zs
ms
𝑚𝑠 ∙ 𝑔
z
cs
zu
Frz
zs
Fsz
zu
mu
ct
zr
Fsz
ds
x
py
Fsz
𝑚𝑠 ∙ 𝑔
𝑚𝑢 ∙ 𝑧̈𝑢 𝑚𝑢 ∙ 𝑔
Frz
Fsz
Frz
zu
zr
Frz
Figure 5-13: One-dimensional model with two dynamic degrees of freedom
224
VERTICAL DYNAMICS
The corresponding mathematical model becomes as follows:
𝑬𝑬𝑬𝑬𝑬𝑬𝑬𝑬𝑬𝑬𝑬: 𝑚𝑠 ∙ 𝑧̈𝑠 + 𝑚 ∙ 𝑔 = 𝐹𝑠𝑠 ;
𝑚𝑢 ∙ 𝑧̈𝑢 = 𝐹𝑟𝑟 − 𝐹𝑠𝑠 ;
𝑪𝑪𝑪𝑪𝑪𝑪𝑪𝑪𝑪𝑪𝑪𝑪: 𝐹𝑠𝑠 = 𝑐𝑠 ∙ (𝑧𝑢 − 𝑧𝑠 ) + 𝑑𝑠 ∙ (𝑧̇𝑢 − 𝑧̇𝑠 ) + 𝑚 ∙ 𝑔;
𝐹𝑟𝑟 = 𝑐𝑡 ∙ (𝑧𝑟 − 𝑧𝑢 ) + 𝑚 ∙ 𝑔;
𝑬𝑬𝑬𝑬𝑬𝑬𝑬𝑬𝑬𝑬:
𝑧𝑟 = 𝑧𝑟 (𝑡);
[5.47]
The same can be formulated with matrices and and Fourier transforms:
ďż˝
𝑚𝑠
0
𝑐𝑠
−𝑐𝑠
𝑧𝑠
−𝑑𝑠
𝑧̇
0
0
� ∙ � 𝑠 � + �−𝑐 𝑐 + 𝑐 � ∙ �𝑧 � = � � ∙ 𝑧̇𝑟 + � � ∙ 𝑧𝑟 ; ⇒
𝑐𝑡
𝑧̇𝑢
𝑑𝑠
𝑢
0
𝑠
𝑠
𝑡
𝑧𝑠
𝑧̈
𝑧̇
⇒ 𝑴 ∙ � 𝑠 � + 𝑫 ∙ � 𝑠 � + 𝑪 ∙ �𝑧 � = 𝑫𝒓 ∙ 𝑧̇𝑟 + 𝑪𝒓 ∙ 𝑧𝑟 ; ⇒
𝑧̈𝑢
𝑧̇𝑢
𝑢
0
𝑧̈
𝑑
� ∙ � 𝑠� + � 𝑠
𝑚𝑢 𝑧̈𝑢
−𝑑𝑠
[5.48]
⇒ 𝑴 ∙ (−𝜔2 ∙ 𝑍) + 𝑫 ∙ (𝑗 ∙ 𝜔 ∙ 𝑍) + 𝑪 ∙ 𝑍 = 𝑫𝒓 ∙ (𝑗 ∙ 𝜔 ∙ 𝑍𝑟 ) + 𝑪𝒓 ∙ 𝑍𝑟 ; ⇒
⇒ (−𝜔2 ∙ 𝑴 + 𝑗 ∙ 𝜔 ∙ 𝑫 + 𝑪) ∙ 𝑍 = (𝑗 ∙ 𝜔 ∙ 𝑫𝒓 + 𝑪𝒓 ) ∙ 𝑍𝑟 ;
𝑤ℎ𝑒𝑒𝑒 𝑍 = �
𝑍𝑠
ℱ(𝑧𝑠 )
ďż˝= ďż˝
� ; 𝑎𝑎𝑎 𝑍 = ℱ(𝑧𝑟 );
𝑍𝑢
ℱ(𝑧𝑢 )
The matrix 𝑫𝒓 is zero, but marks a more general form, typical for modelling damping also in the tyre.
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 [4.54]:
ďż˝
𝐻𝑧𝑟 →𝑧𝑠
1
𝑍
𝑍
=
=
� = � 𝑠� ∙
𝐻𝑧𝑟 →𝑧𝑢
𝑍𝑢 𝑍𝑟 𝑍𝑟
[5.49]
= (−𝜔2 ∙ 𝑴 + 𝑗 ∙ 𝜔 ∙ 𝑫 + 𝑪)−1 ∙ (𝑗 ∙ 𝜔 ∙ 𝑫𝒓 + 𝑪𝒓 );
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:
(−𝑚𝑢 ∙ ω2 + 𝑗 ∙ 𝑑𝑠 ∙ ω + 𝑐𝑡 + 𝑐𝑠 ) ∙ 𝑐𝑡
− 𝑐𝑡
(𝑐𝑠 + 𝑗 ∙ 𝑑𝑠 ∙ ω)2
2+𝑗∙𝑑 ∙ω+𝑐 +𝑐 −
−𝑚
∙
ω
𝑢
𝑠
𝑡
𝑠
𝑍𝑠
−𝑚𝑠 ∙ ω2 + 𝑗 ∙ 𝑑𝑠 ∙ ω + 𝑐𝑠
𝐻𝑧𝑟 →𝑧𝑠 =
=
;
𝑍𝑟
𝑐𝑠 + 𝑗 ∙ 𝑑𝑠 ∙ ω
𝑐𝑡
𝐻𝑧𝑟 →𝑧𝑢 =
(𝑐𝑠 + 𝑗 ∙ 𝑑𝑠 ∙ ω)2
−𝑚𝑢 ∙ ω2 + 𝑗 ∙ 𝑑𝑠 ∙ ω + 𝑐𝑡 + 𝑐𝑠 −
−𝑚𝑠 ∙ ω2 + 𝑗 ∙ 𝑑𝑠 ∙ ω + 𝑐𝑠
where 𝜔 =
2∙π∙𝑣𝑥
𝜆
;
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:
225
[5.50]
VERTICAL DYNAMICS
𝐻𝑧𝑟 →𝑧𝑠 = see Equation [5.50];
𝐻𝑧𝑟 →𝑧𝑢 = see Equation [5.50];
𝐻𝑧𝑟 →𝑧𝑟 −𝑧𝑢 = 𝐻𝑧𝑟 →𝑧𝑟 − 𝐻𝑧𝑟 →𝑧𝑢 = 1 − 𝐻𝑧𝑟 →𝑧𝑢 ;
𝐻𝑧𝑟 →𝑧𝑢 −𝑧𝑠 = 𝐻𝑧𝑟 →𝑧𝑢 − 𝐻𝑧𝑟 →𝑧𝑠 ;
𝐻𝑧𝑟 →𝑧̈𝑠 = −𝜔2 ∙ 𝐻𝑧𝑟 →𝑧𝑠 ;
𝐻𝑧𝑟 →Δ𝐹𝑠𝑠 = {Δ𝐹𝑠𝑠 = 𝑐𝑠 ∙ (𝑧𝑢 − 𝑧𝑠 ) + 𝑑𝑠 ∙ (𝑧̇𝑢 − 𝑧̇𝑠 )} =
= 𝑐𝑠 ∙ �𝐻𝑧𝑟 →𝑧𝑢 − 𝐻𝑧𝑟 →𝑧𝑠 � + 𝑑𝑠 ∙ 𝑗 ∙ 𝜔 ∙ �𝐻𝑧𝑟 →𝑧𝑢 − 𝐻𝑧𝑟 →𝑧𝑠 � =
= (𝑐𝑠 + 𝑗 ∙ 𝑑𝑠 ∙ 𝜔) ∙ �𝐻𝑧𝑟 →𝑧𝑢 − 𝐻𝑧𝑟 →𝑧𝑠 �;
𝐻𝑧𝑟 →Δ𝐹𝑟𝑟 = {Δ𝐹𝑟𝑧 = 𝑐𝑡 ∙ (𝑧𝑟 − 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 . The derivation is not documented in present
compendium.
2
�𝐻𝑧𝑟 →𝑧̈𝑠 � = 𝜔 ∙
2
�(𝑐𝑠 ∙ 𝑐𝑡 − 𝑑𝑠 ∙ 𝑑𝑡 ∙ 𝜔 2 )2 + �𝜔 ∙ (𝑑𝑠 ∙ 𝑐𝑡 + 𝑑𝑡 ∙ 𝑐𝑠 )�
;
√𝐴2 + 𝐵2
𝑚𝑠 ∙ �(𝑐𝑡 ∙ 𝜔 2 )2 + (𝑑𝑡 ∙ 𝜔 3 )2
;
�𝐻𝑧𝑟 →𝑧𝑢 −𝑧𝑠 � =
√𝐴2 + 𝐵2
�(−𝑚𝑠 ∙ 𝑚𝑢 ∙ 𝜔 4 + 𝜔 2 ∙ (𝑚𝑠 + 𝑚𝑢 ) ∙ 𝑐𝑠 )2 + (𝜔 3 ∙ (𝑚𝑠 + 𝑚𝑢 ) ∙ 𝑑𝑠 )2
;
�𝐻𝑧𝑟 →𝑧𝑟 −𝑧𝑢 � =
√𝐴2 + 𝐵2
𝐴 = 𝜔4 ∙ 𝑚𝑠 ∙ 𝑚𝑢 − 𝜔2 ∙ (𝑚𝑠 ∙ 𝑐𝑡 + 𝑚𝑠 ∙ 𝑐𝑠 + 𝑑𝑠 ∙ 𝑑𝑡 + 𝑐𝑠 ∙ 𝑚𝑢 ) + 𝑐𝑠 ∙ 𝑐𝑡 ;
𝐵 = 𝜔3 ∙ (𝑚𝑠 ∙ 𝑑𝑡 + 𝑚𝑠 ∙ 𝑑𝑠 + 𝑚𝑢 ∙ 𝑑𝑠 ) − 𝜔 ∙ (𝑑𝑠 ∙ 𝑐𝑡 + 𝑑𝑡 ∙ 𝑐𝑠 );
[5.52]
The transfer functions in Equation [5.51] are plotted in Figure 5-14.
So, for example, we can use the transfer function diagram as follows, cf. end of Section 0: If we have
a certain road, with displacement amplitude of 1 cm (𝑧̂𝑟 = 0.01 𝑚) and the vehicle should drive on it
𝑚
� ≈ 2.8𝐻𝐻), we can read out, e.g.:
at a speed of 50 km/h (𝑣𝑥 ≈ 14 =
•
•
•
𝑠
Ride comfort related: �𝐻𝑧𝑟 →𝑧̈𝑠 (𝑣𝑥 )� ≈ 827; ⇒ |𝑎
� | = 827 ∙ 𝑧�𝑟 = 827 ∙ 0.01 =
2
8.27 𝑚⁄𝑠 ;. From this we can calculate 𝑅𝑅𝑅(𝑧̈ 𝑠 ) = |8.27|⁄√2 ≈ 5.85 𝑚⁄𝑠 2 .
Fatigue life related: �𝐻𝑧𝑟 →𝑧𝑢 −𝑧𝑠 (𝑣𝑥 )� ≈ 1.16; ⇒ |𝑧̂𝑢 − 𝑧̂𝑠 | = 1.16 ∙ 𝑧�𝑟 = 1.16 ∙ 0.01 =
0.0116 𝑚;.
Road grip related: �𝐻𝑧𝑟 →Δ𝐹𝑟𝑟 (𝑣𝑥 )� ≈ 1290000; ⇒ |𝐹� 𝑟𝑟 | = 1290000 ∙ 𝑧�𝑟 = 1290000 ∙
0.01 = 12900 𝑁;.
The reason why 𝑅𝑅𝑅(𝑧̈ 𝑠 ) becomes so much larger now, compared to in Section 0 is that there is a
small resonance which is better modelled with the two degree of freedom model.
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 12.1 m/s = 15.2 rad/s = 2.42 Hz and 69.0 m/s = 86.7 rad/s = 13.8 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.
226
VERTICAL DYNAMICS
ms = 1415[kg]; cs = 568 [kN/m]; ds = 48[kN/(m/s)]; lambda = 5 m;
7
10
mu = 185[kg]; ct = 800 [kN/m];
6
10
5
10
H_zr_zs
H_zr_zu-zs
H_zr_derderzs
H_zr_Frz
4
abs(H)
10
3
10
2
10
1
10
0
10
-1
10
0
10
1
2
10
10
3
10
vx [m/s]
Figure 5-14: Transfer functions for amplitudes from model in Figure 5-13, excited with
single frequencies. Thin lines are without damping. Notation: 𝐻𝑎→𝑏 is denoted H_a_b.
ms = 1415[kg]; cs = 568 [kN/m]; ds = 48[kN/(m/s)]; lambda = 5 m;
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 from model in Figure 5-13, excited with
single frequencies. Thin lines are without damping.
227
VERTICAL DYNAMICS
ms = 1415[kg]; cs = 568 [kN/m]; ds = 0[kN/(m/s)]; lambda = 5 m;
7
10
mu = 185[kg]; ct = 800 [kN/m];
6
10
5
H_zr_zs
H_zr_zu-zs
H_zr_derderzs
H_zr_Frz
10
4
abs(H)
10
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 from model in Figure 5-13,
excited with single frequencies. The corresponding plot with realistic damping values is
found in Figure 5-14.
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.
228
VERTICAL DYNAMICS
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).
For the wheel hop model, the mass is mu. Stiffnesses cs and ct are parallel connected, which means
that the total stiffness=cs+ct.
𝜔𝐵𝐵𝐵𝐵𝐵𝐵
1 1
1��𝑐 + 𝑐 �
𝑟𝑟𝑟
𝑠
𝑡
=ďż˝
= 15.3
;
𝑚𝑠
𝑠
𝜔𝑊ℎ𝑒𝑒𝑒𝑒𝑒𝑒 = �
[5.53]
𝑐𝑠 + 𝑐𝑡
𝑟𝑟𝑟
= 86.0
;
𝑚𝑢
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.
5.5.3.3
Variation of stiffness and damping
A typical response for the model in Section 5.5.3 is shown in Figure 5-18. It should be carefully noted
that this is only the vehicle transfer function. The road influence (see Section 5.4) and sensitivity for
comfort, fatigue and road grip (see Sections 5.6, 5.7 and 5.8, respectively) are all treated as ≡ 1. With
such assumptions the curves would be amplified differently in different frequency ranges. Anyway,
the vehicle transfer function alone, as plotted, gives a qualitative impression of how different
suspension design parameters influence.
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.
3
Road Grip
Suspension Travel
Ride Comfort
1 .10
10
10
1
100
10
Gain
Gain
Gain
1
0.1
0.1
0.01
1
0.1
0.1
1
10
Frequency (Hz)
100
0.01
0.1
1
10
Frequency (Hz)
100
1 .10
3
0.1
1
10
Frequency (Hz)
100
Figure 5-18: Transfer Functions for a passenger car. Ride Comfort is for vertical
acceleration of sprung mass, Suspension Travel is for displacement difference between
sprung and unsprung mass, and Road Grip is for force between tyre and road.
The same three curves are shown again in Figure 5-19 to Figure 5-22, varying one design parameter
per figure.
229
VERTICAL DYNAMICS
Regarding Figure 5-19 and Figure 5-20 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.
5.5.3.3.1 Varying suspension stiffness
In Figure 5-19 the benefits of the low suspension stiffness (1 Hz) is seen for suspension travel and
comfort without much change in the road grip performance.
Ride Comfort
Suspension Travel
60
Road Grip
10
10
0
0
10
10
50
20
Gain [dB]
30
Gain [dB]
Gain [dB]
40
20
20
30
30
10
0
10
0.1
1
10
Frequency (Hz)
40
100
0.1
1 Hz
1.5 Hz
2 Hz
1
10
Frequency (Hz)
40
100
0.1
1 Hz
1.5 Hz
2 Hz
1
10
Frequency (Hz)
100
1 Hz
1.5 Hz
2 Hz
Figure 5-19: Result from varying suspension stiffness, cs
5.5.3.3.2 Varying suspension damping
In Figure 5-20, 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.
Road Grip
Suspension Travel
Ride Comfort
60
10
10
0
0
Gain [dB]
40
Gain [dB]
Gain [dB]
50
10
10
30
20
20
20
10
0.1
100
1
10
Frequency (Hz)
0.8Cs
Cs
1,2 Cs
1
30
0.1
1
10
Frequency (Hz)
100
30
0.1
1
10
Frequency (Hz)
0.8Cs
Cs
1.2Cs
0.8Cs
Cs
1.2Cs
Figure 5-20: Result from varying suspension damping, ds
230
100
VERTICAL DYNAMICS
5.5.3.3.3 Varying unsprung mass
In Figure 5-21, 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.
Road Grip
Suspension Travel
Ride Comfort
60
10
10
0
0
10
10
50
20
Gain [dB]
30
Gain [dB]
Gain [dB]
40
20
20
30
30
10
0
10
0.1
1
10
Frequency (Hz)
40
100
0.1
1
10
Frequency (Hz)
40
100
0.1
100
Mu=0.05Ms
Mu=0.1Ms
Mu=0.2Ms
Mu=0.05Ms
Mu=0.1Ms
Mu=0.2Ms
Mu=0.05Ms
Mu=0.1Ms
Mu=0.2Ms
1
10
Frequency (Hz)
Figure 5-21: Result from varying unsprung mass, mu
5.5.3.3.4 Varying tyre stiffness
In Figure 5-22 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 as expected.
Road Grip
Suspension Travel
Ride Comfort
60
10
10
0
0
10
10
50
20
Gain [dB]
30
Gain [dB]
Gain [dB]
40
20
20
30
30
10
0
10
0.1
1
10
Frequency (Hz)
Kt=160kN/m
Kt=200kN/m
Kt=240kN/m
100
40
0.1
1
10
Frequency (Hz)
100
40
0.1
Kt=160kN/m
Kt=200kN/m
Kt=240kN/m
Figure 5-22: Result from varying Tyre Stiffness, ct
231
1
10
Frequency (Hz)
Kt=160kN/m
Kt=200kN/m
Kt=240kN/m
100
VERTICAL DYNAMICS
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 it’s 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 5-23 and Figure
5-24.
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 5-23 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 5-24.
232
VERTICAL DYNAMICS
How to use the diagram
waS
acceleration
not
OK
OK
4 Hz
8 Hz
Frequency
Figure 5-23: Various Human Sensitivity Curves to Vertical Vibration, (Gillespie, 1992)
Figure 5-24: ISO 2631 Tolerance Curves
233
VERTICAL DYNAMICS
5.6.2
Multiple frequencies
The curves in Figure 5-23 and Figure 5-24 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
Filter Function 𝑊𝑘 (𝑓). (𝑊𝑘 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
(number=20 ∙ log10Gain
𝑊𝑘 (dB)
⇒ 𝑊𝑘 = 10𝑛𝑛𝑛𝑛𝑛𝑛/20 )
10
0
-5
0
10
Wk in dB
Wk [1]
-10
-1
10
-2
-15
-20
-25
10
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 5-25: Human Filter Function. From (ISO2631). Right: Asymptotic approximation
Figure 5-26: 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:
234
VERTICAL DYNAMICS
𝑁
𝑁
2
2
⎧
𝑧̈̂𝑖 ⎫
�𝑊𝑘 (𝜔𝑖 ) ∙ 𝑧̈̂𝑖 �
ďż˝
ďż˝
𝑎𝑤 = 𝑎𝑤 �𝑧̈ (𝑡)� = 𝑢𝑢𝑢: 𝑅𝑅𝑅�𝑧̈ (𝑡)� = �
= ďż˝
;
2⎬
2
⎨
𝑖=1
𝑖=1
⎊
⎭
or
∞
[5.54]
𝑎𝑤 = 𝑎𝑤 �𝑧̈ (𝑡)� = �𝑢𝑢𝑢: 𝑅𝑅𝑅�𝑧̈ (𝑡)� = � � 𝐺𝑧̈ (𝜔) ∙ 𝑑𝑑 � =
∞
𝜔=0
2
= � � �𝑊𝑘 (𝜔)� ∙ 𝐺𝑧̈ (𝜔) ∙ 𝑑𝑑 ;
𝜔=0
Equation [5.54] is written for a case with only vertical vibrations, hence 𝑊𝑘 and 𝐺𝑧̈ . If vibrations in
several directions, a total 𝑎𝑤 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 𝑎𝑤 is the time averaged whole-body vibration exposure value.
∑𝑖 𝑎𝑤𝑖 2 ∙ 𝑇𝑖
𝑎𝑤 = �
;
∑𝑖 𝑇𝑖
[5.55]
The 𝑎𝑤 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
𝑎𝑤 in Eq [5.55] in any direction, normalized to 8 hours, may not exceed 1.15 m/s2, 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 𝑧̈̂𝑠 and
multiply it with 𝑊𝑘 (𝜔), according to Equation [5.54]. Then we get [5.56].
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 road type “rough”
from Figure 5-6, is shown in Figure 5-28.
We can see that the models differ a lot, which tells us that at least “two masses, elastic tyre” is
needed to judge comfort.
We can also see that the comfort decreases a lot, the faster the vehicle drives. Using the model “two
masses, elastic tyre”, we can read out that 𝑎𝑤 = 1 𝑚⁄𝑠 2 (which is a reasonable value for long time
exposure) is reached at 𝑣𝑥 = 8.5 𝑚⁄𝑠 ≈ 31 𝑘𝑘/ℎ on this road type.
235
VERTICAL DYNAMICS
𝑅𝑅𝑅(𝑧̈𝑠 ) = �
=ďż˝
=ďż˝
𝛷0
𝛺0
−𝑤
𝛷0
𝛺0 −𝑤
⇒ 𝑎𝑤 = �
=ďż˝
𝛷0
𝛺0
−𝑤
𝛷0
𝛺0
−𝑤
∞
2
∙ 𝑣𝑥 𝑤−1 ∙ ��𝐻𝑧𝑟→𝑧̈𝑠 (𝜔)� ∙ 𝜔−𝑤 ∙ 𝑑𝑑 = �
𝜔=0
∞
= −𝜔2 ∙ 𝐻𝑧𝑟 →𝑧𝑠
𝜔=0
∙ 𝑣𝑥
𝛷0
𝛺0
−𝑤
∞
∙ ��𝐻𝑧𝑟→𝑧𝑠 (𝜔)�
𝜔=0
ďż˝=
2
∙ 𝑣𝑥 𝑤−1 ∙ � �𝜔 2 ∙ 𝐻𝑧𝑟→𝑧𝑠 (𝜔)� ∙ 𝜔−𝑤 ∙ 𝑑𝑑 =
𝑤−1
𝑢𝑢𝑢: 𝐻𝑧𝑟 →𝑧̈𝑠 =
∞
2
𝑢𝑢𝑢:
∙ 𝜔4−𝑤 ∙ 𝑑𝑑 ; ⇒ �Equation �
[5.54]
[5.56]
⇒
2
2
∙ 𝑣𝑥 𝑤−1 ∙ ��𝑊𝑘 (𝜔)� ∙ �𝐻𝑧𝑟→𝑧𝑠 (𝜔)� ∙ 𝜔4−𝑤 ∙ 𝑑𝑑 =
∞
𝜔=0
2
2
∙ 𝑣𝑥 𝑤−1 ∙ ��𝑊𝑘 (𝜔)� ∙ �𝐻𝑧𝑟→𝑧𝑠 (𝜔)� ∙ 𝜔4−𝑤 ∙ 𝑑𝑑 ;
𝜔=0
Ride Comfort. For road type "rough"
2
Weigthed RMS value, aw [m/(s*s)]
10
1
10
0
10
-1
10
-2
10
-2
10
-1
10
0
10
vx [m/s]
1
10
2
10
Figure 5-27: Weighted RMS values for models in Sections 1.5.1, 1.5.2 and 1.5.3 and road
type “rough” from Figure 5-6. The 3 curves show 3 different models.
236
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 𝑧̇𝑢 − 𝑧̇𝑠 , 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, 𝐹�𝑟𝑟 , 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,
[𝑧̂1 , 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:
237
VERTICAL DYNAMICS
𝑀𝑀𝑀𝑀𝑀𝑀𝑀 𝑓𝑓𝑓 𝑠𝑠𝑠𝑠𝑠𝑠 𝑓𝑓𝑓𝑓𝑓𝑓𝑓 𝑙𝑙𝑙𝑙 = 𝑅𝑅𝑅�𝑧𝑢 (𝑡) − 𝑧𝑠 (𝑡)� =
=ďż˝
=ďż˝
𝛷0
𝛺0
−𝑤
𝛺0
−𝑤
𝛷0
∞
𝑢𝑢𝑢: 𝐻𝑧𝑟 →𝑧𝑢 −𝑧𝑠 =
ďż˝=
𝑧𝑟 →𝑧𝑢 − 𝐻𝑧𝑟 →𝑧𝑠
2
∙ 𝑣𝑥 𝑤−1 ∙ ��𝐻𝑧𝑟→𝑧𝑢−𝑧𝑠 (𝜔)� ∙ 𝜔−𝑤 ∙ 𝑑𝑑 = �= 𝐻
𝜔=0
∞
[5.57]
2
∙ 𝑣𝑥 𝑤−1 ∙ ��𝐻𝑧𝑟→𝑧𝑢 (𝜔) − 𝐻𝑧𝑟→𝑧𝑠 (𝜔)� ∙ 𝜔−𝑤 ∙ 𝑑𝑑 ;
𝜔=0
Equation is written for application to a known road spectra (𝛷0 , 𝑤 ) and vehicle dynamic structure
(𝐻𝑧𝑟 →𝑧𝑢 , 𝐻𝑧𝑟 →𝑧𝑠 ), but the first expression (𝑅𝑅𝑅�𝑧𝑢 (𝑡) − 𝑧𝑠 (𝑡)�) is applicable on a measured or
simulated time domain solution.
5.8 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,
[𝑧̂1 , 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:
𝑀𝑀𝑀𝑀𝑀𝑀𝑀 𝑓𝑓𝑓 (𝑏𝑏𝑏)𝑟𝑟𝑟𝑟 𝑔𝑔𝑔𝑝 = 𝑅𝑅𝑅�Δ𝐹𝑟𝑟 (𝑡)� =
=ďż˝
𝛷0
𝛺0
−𝑤
∞
2
∙ 𝑣𝑥 𝑤−1 ∙ ��𝐻𝑧𝑟→Δ𝐹𝑟𝑟 (𝜔)� ∙ 𝜔−𝑤 ∙ 𝑑𝑑 =
𝜔=0
= �𝑢𝑢𝑢: 𝐻𝑧𝑟 →𝑧𝑢 −𝑧𝑠 = 𝑐𝑡 ∙ �1 − 𝐻𝑧𝑟 →𝑧𝑢 �� =
=ďż˝
𝛷0
𝛺0
−𝑤
∞
[5.58]
2
∙ 𝑣𝑥 𝑤−1 ∙ ��𝑐𝑡 ∙ �1 − 𝐻𝑧𝑟→𝑧𝑢 �� ∙ 𝜔−𝑤 ∙ 𝑑𝑑 ;
𝜔=0
Equation is written for application to a known road spectra (𝛷0 , 𝑤 ) and vehicle dynamic structure
(𝐻𝑧𝑟 →𝑧𝑢 ), but the first expression (𝑅𝑅𝑅�Δ𝐹𝑟𝑟 (𝑡)�) is applicable on a measured or simulated time
domain solution.
238
VERTICAL DYNAMICS
5.9 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.
5.9.1
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
zr
m*g
h
zrr
zfr
lf
J*der(wy)=𝐽 ∙ 𝜔̇ 𝑦
Fzf
lr
Fzr
Figure 5-28: Bounce (heave) and pitch model.
No equations will be formulated for this model.
The model will typically show two different modes, see Figure 5-29. The bounce eigen-frequency is
typically 1-1.5 Hz for a passenger car. The pitch frequency is somewhat higher.
We should reflect on that if we are analysis the same vehicle with the models in Section 5.5 and
5.9.1, we are actually talking about the same bounce mode. But the models will most likely give
different numbers of, e.g. Eigen frequency. A total model, with all motion degrees of freedom, would
align those values, but the larger a model is the more data it produces which often leads to less easy
design decisions.
239
VERTICAL DYNAMICS
Motion node
for titch mode
titch
mode
Bounce
mode
Motion node for
Bounce mode
Figure 5-29: Oscillation modes of a Bounce and Pitch model.
5.9.1.1
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)
240
VERTICAL DYNAMICS
Figure 5-31: Example of wheelbase filter effect
5.9.2
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 4-55. 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.
241
VERTICAL DYNAMICS
5.10 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.
Figure 5-33: Response of Vehicle for Front and Rear Axle Impulses, (Gillespie, 1992)
5.11 Other excitation sources and
functions
5.11.1.1 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
242
VERTICAL DYNAMICS
•
•
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)
5.11.1.2 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).
243
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245
Index
(discrete) state diagram ......................... 16
Critical Speed ....................................... 148
Ackermann steering angle .................... 144
data flow diagram .................................. 16
Ackermann steering geometry ............. 125
Differential-Algebraic system of
Equations, DAE....................................... 15
Active safety ........................................... 30
direct yaw moment.............................. 147
actual toe angle ...................................... 41
effective damping ........................ 107, 110
actuation........................................... 12, 64
effective stiffness ................................. 107
Advanced Driver Assitance Systems
(ADAS)..................................................... 30
Electronic Brake Distribution, EBD ...... 118
aerodynamic drag ................................... 68
Electronic Control Units, ECUs ............... 30
agility .................................................... 181
Electronic Stability Control, ESC .......... 199
aligning torque ....................................... 62
Engine Drag Torque Control, EDC ........ 119
angular (time) frequency ...................... 207
Equivalent wheelbase .......................... 136
Anti-lock Braking System, ABS .............. 117
flow charts ............................................. 16
architecture ............................................ 30
Forward Collision Warning (FCW).......... 12
Attributes................................................ 10
Friction Circle ......................................... 63
Automatic Emergency Brake, AEB ........ 119
Functions ............................................... 11
axle rate ................................................ 107
handling diagram ................................. 152
bicycle model ........................................ 126
Human Filter Function ......................... 234
Blind Spot Detection (BSD) ..................... 12
ideal tracking ....................................... 128
bounce mode........................................ 226
indirect yaw moment........................... 147
brush model............................................ 50
inertial coordinate system ..................... 27
Cambering Vehicle.................................. 34
internal combustion engine (ICE) .......... 78
Caster offset ........................................... 73
kinematic models................................. 128
caster trail ............................................... 73
kingpin offset ......................................... 73
Characteristic Speed ............................. 149
Lane Departure Prevention ................. 198
Closed-loop Manoeuvres........................ 70
Lane Departure Warning (LDW) ............ 12
coefficient of rolling resistance .............. 46
Lane Keeping Aid, LKA ......................... 198
combined (tyre) slip................................ 63
Lateral Collision Avoidance, LCA .......... 201
Continuous Variable ratio Transmission,
CVT.......................................................... 84
lateral slip .............................................. 56
Leaning vehicle ...................................... 34
Controller Area Network, CAN ............... 30
longitudinal (tyre) slip ............................ 44
Cornering .............................................. 123
low speed steering angle ..................... 144
cornering coefficient .............................. 59
Magic Formula ....................................... 54
Cornering Compliance ............................ 58
Mean Square Spectral Density ............ 209
246
neutral steered vehicle ......................... 144
side slip .................................................. 56
object vehicle.......................................... 29
single-track model ............................... 126
Off-tracking........................................... 197
slip angle ................................................ 56
one-track model ................................... 126
spectral density.................................... 209
Open-loop Manoeuvres ......................... 70
sprung mass ......................................... 106
Ordinary Differential Equations, ODE..... 24
static toe angle ...................................... 41
oversteered vehicle .............................. 144
subject vehicle ....................................... 29
overturning moment .............................. 61
TM-Easy ................................................. 54
parallell steering geometry .................. 126
Traction Control, TC ............................. 118
physical stiffness................................... 107
Traction diagram.................................... 81
platform .................................................. 30
turning circle ........................................ 132
pneumatic trail ....................................... 62
turning diameter .................................. 132
Power Spectral Density, PSD ................ 209
tyre slip .................................................. 44
Primary Ride ......................................... 232
understeer gradient ..................... 142, 143
Rearward Amplification, RWA .............. 196
understeered vehicle ........................... 144
roll axis.................................................. 161
unsprung mass ..................................... 106
Roll Stability Control, RSC ..................... 201
vehicle fixed coordinate systems........... 27
rolling radius ........................................... 39
Weighted RMS Acceleration ................ 234
rolling resistance .................................... 45
wheel base filtering ............................. 240
Roll-stiff Vehicle...................................... 34
wheel hop mode .................................. 226
scrub radius ............................................ 73
wheel rate ............................................ 107
Secondary Ride ..................................... 232
yaw damping........................................ 197
247
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