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vehicle dynamics.pdf
Dieter Schramm · Manfred Hiller
Roberto Bardini
Vehicle
Dynamics
Modeling and Simulation
Vehicle Dynamics
Dieter Schramm • Manfred Hiller
Roberto Bardini
Vehicle Dynamics
Modeling and Simulation
123
Dieter Schramm
Manfred Hiller
Universität Duisburg-Essen
Duisburg
Germany
Roberto Bardini
München
Germany
ISBN 978-3-540-36044-5
DOI 10.1007/978-3-540-36045-2
Springer Heidelberg New York Dordrecht London
ISBN 978-3-540-36045-2
(eBook)
Library of Congress Control Number: 2014942274
Ó Springer-Verlag Berlin Heidelberg 2014
This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of
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Preface
The main focus of this book is on the fundamentals of ‘‘Vehicle Dynamics’’ and
the mathematical modeling and simulation of motor vehicles. The range of
applications encompasses basic single track models as well as complex, spatial
multibody systems. The reader will be enabled to develop own simulation models,
supported to apply successfully commercial programs, to choose appropriate
models and to understand and assess simulation results. The book describes in
particular the modeling process from the real vehicle to the mathematical model as
well as the validation of simulation results by means of selected applications.
The book is aimed at students and postgraduates in the field of engineering
sciences who attend lectures or work on their thesis. To the same extent it
addresses development engineers and researches working on vehicle dynamics or
apply associated simulation programs.
The modeling of Vehicle Dynamics is primarily based on mathematical
methods used throughout the book. The reader should therefore have a basic
understanding of mathematics, e.g., from the first three semesters’ study course in
engineering or natural sciences.
This edition of the book is the English version of the second German edition.
The authors thank all persons who contributed to this edition of the book.
Amongst all persons who contributed by giving hints and sometimes simply asking
the right questions we want to highlight in particular the indispensable contributions
of Stephanie Meyer, Lawrence Louis and Michael Unterreiner who contributed with
translation and proof reading of some chapters. We also thank Frederic Kracht for
diligent proofreading and the solution of unsolvable problems incident to the secrets
of contemporary word processor software.
Duisburg, May 2014
Dieter Schramm
Manfred Hiller
Roberto Bardini
v
Contents
1
2
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.1
Problem Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Modeling Technical Systems . . . . . . . . . . . . . . . . .
1.1.1
Definition of a System . . . . . . . . . . . . . . . . . . . . .
1.1.2
Simulation and Simulation Environment . . . . . . . . .
1.1.3
1.1.4
Vehicle Models . . . . . . . . . . . . . . . . . . . . . . . . . .
Complete Vehicle Model . . . . . . . . . . . . . . . . . . . . . . . . . . .
Vehicle Models and Application Areas . . . . . . . . . .
1.2.1
1.2.2
Commercial Vehicle Simulation Systems. . . . . . . . .
Outline of the Book . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.3
1.4 Webpage of the Book . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2
Fundamentals of Mathematics and Kinematics . . . . . . . . . . . . . .
Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.1
Elementary Algorithms for Vectors. . . . . . . . . . . . .
2.1.1
2.1.2
Physical Vectors . . . . . . . . . . . . . . . . . . . . . . . . . .
Coordinate Systems and Components . . . . . . . . . . . . . . . . . .
2.2.1
Coordinate Systems. . . . . . . . . . . . . . . . . . . . . . . .
Component Decomposition . . . . . . . . . . . . . . . . . .
2.2.2
Relationship Between Component
2.2.3
Representations . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2.4
Properties of the Transformation Matrix . . . . . . . . .
Linear Vector Functions and Second Order Tensors . . . . . . . .
Free Motion of Rigid Bodies . . . . . . . . . . . . . . . . . . . . . . . .
General Motion of Rigid Bodies. . . . . . . . . . . . . . .
2.4.1
2.4.2
Relative Motion . . . . . . . . . . . . . . . . . . . . . . . . . .
2.4.3
Important Reference Frames. . . . . . . . . . . . . . . . . .
Rotational Motion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.5.1
2.3
2.4
Spatial Rotation and Angular Velocity
in General Form . . . . . . . . . . . . . . . . . . . . . . . . . .
Parameterizing of Rotational Motion. . . . . . . . . . . .
The Rotational Displacement Pair and Tensor
of Rotation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2
2.5
2.5.2
2.5.3
1
1
3
5
5
6
9
11
11
13
14
14
17
17
17
18
19
19
19
20
22
22
24
24
28
30
31
32
32
34
vii
viii
Contents
2.5.4
Rotational Displacement Pair and Angular
Velocity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
CARDAN (BRYANT) Angles . . . . . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.5.5
3.2
3.1
3 Kinematics of Multibody Systems . . . . . . . . . . . . . . . . . . . . . . . .
Structure of Kinematic Chains . . . . . . . . . . . . . . . . . . . . . . .
3.1.1
Topological Modelling . . . . . . . . . . . . . . . . . . . . .
3.1.2
Kinematic Modelling. . . . . . . . . . . . . . . . . . . . . . .
Joints in Kinematic Chains . . . . . . . . . . . . . . . . . . . . . . . . .
Joints in Spatial Kinematic Chains . . . . . . . . . . . . .
3.2.1
Joints in Planar Kinematic Chains. . . . . . . . . . . . . .
3.2.2
3.2.3
Joints in Spherical Kinematic Chains . . . . . . . . . . .
3.2.4
Classification of Joints . . . . . . . . . . . . . . . . . . . . .
Degrees of Freedom and Generalized Coordinates . . . . . . . . .
Degrees of Freedom of Kinematic Chains . . . . . . . .
3.3.1
3.3.2
Examples from Road Vehicle
Suspension Kinematics . . . . . . . . . . . . . . . . . . . . .
3.3.3
Generalized Coordinates . . . . . . . . . . . . . . . . . . . .
Basic Principles of the Assembly of Kinematic Chains . . . . . .
3.4.1
3.3
3.4
3.5
3.4.3
3.4.2
Sparse-Methods: Absolute Coordinates
Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Vector Loop Methods
(‘‘LAGRANGE’’ Formulation) . . . . . . . . . . . . . . . .
Topological Methods: Formulation
of Minimum Coordinates . . . . . . . . . . . . . . . . . . . .
Kinematics of a Complete Multibody System . . . . . . . . . . . .
Basic Concept . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.5.1
3.5.2
Block Wiring Diagram and Kinematic Networks . . .
Relative Kinematics of the Spatial
3.5.3
Four-Link Mechanism . . . . . . . . . . . . . . . . . . . . . .
Relative, Absolute and Global Kinematics . . . . . . . .
Example: Double Wishbone Suspension . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.5.4
3.5.5
4
Equations of Motion of Complex Multibody Systems . . . . . . . . . .
4.1
Fundamental Equation of Dynamics for Point
Mass Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
JOURDAIN’S Principle. . . . . . . . . . . . . . . . . . . . . . . . . . . .
LAGRANGE Equations of the First Kind
for Point Mass Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . .
LAGRANGE Equations of the Second Kind
for Rigid Bodies. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
D’ALEMBERT’s Principle . . . . . . . . . . . . . . . . . . . . . . . . .
4.2
4.3
4.4
4.5
36
36
40
43
43
43
45
46
46
47
48
50
50
50
53
53
55
55
58
59
62
62
63
64
66
68
71
73
73
75
75
76
78
Contents
4.6
Computer-Based Derivation of the Equations of Motion . . . . .
Kinematic Differentials of Absolute Kinematics . . . .
4.6.1
Equations of Motion . . . . . . . . . . . . . . . . . . . . . . .
4.6.2
4.6.3
Dynamics of a Spatial Multibody Loop . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.1
5.2
5 Kinematics and Dynamics of the Vehicle Body . . . . . . . . . . . . . .
Vehicle-Fixed Reference Frame . . . . . . . . . . . . . . . . . . . . . .
Kinematical Analysis of the Chassis . . . . . . . . . . . . . . . . . . .
5.2.1
Incorporation of the Wheel Suspension
Kinematics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Equations of Motion . . . . . . . . . . . . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.2.2
6.1
6.2
6 Modeling and Analysis of Wheel Suspensions . . . . . . . . . . . . . . .
Function of Wheel Suspension Systems. . . . . . . . . . . . . . . . .
Different Types of Wheel Suspension . . . . . . . . . . . . . . . . . .
Beam Axles . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.2.1
Twist-Beam Suspension. . . . . . . . . . . . . . . . . . . . .
6.2.2
Trailing-Arm Axle . . . . . . . . . . . . . . . . . . . . . . . .
6.2.3
6.2.4
Trailer Arm Axle . . . . . . . . . . . . . . . . . . . . . . . . .
Double Wishbone Axles . . . . . . . . . . . . . . . . . . . .
6.2.5
Wheel Suspension Derived from the MacPherson
6.2.6
Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Multi-Link Axles . . . . . . . . . . . . . . . . . . . . . . . . .
6.2.7
Characteristic Variables of Wheel Suspensions. . . . . . . . . . . .
One Dimensional Quarter Vehicle Models. . . . . . . . . . . . . . .
Three-Dimensional Model of a MacPherson
Wheel Suspension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Kinematic Analysis . . . . . . . . . . . . . . . . . . . . . . . .
6.5.1
6.5.2
Explicit Solution. . . . . . . . . . . . . . . . . . . . . . . . . .
Three-Dimensional Model of a Five-Link Rear
Wheel Suspension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.6.1
Kinematic Analysis . . . . . . . . . . . . . . . . . . . . . . . .
Implicit Solution. . . . . . . . . . . . . . . . . . . . . . . . . .
6.6.2
Simulation Results of the Three Dimensional
6.6.3
Quarter Vehicle Model . . . . . . . . . . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.3
6.4
6.5
6.6
7 Modeling of the Road-Tire-Contact. . . . . . . . . . . . . . . . . . . . . . .
Tire Construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Forces Between Wheel and Road . . . . . . . . . . . . . . . . . . . . .
7.1
7.2
ix
80
80
83
84
92
93
93
96
96
99
100
101
101
103
104
105
106
108
108
110
111
113
116
119
120
124
129
129
132
137
141
143
144
145
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