Road Vehicle Dynamics : Fundamentals and Modeling

363 Pages·2011·10.698 MB·English

by  Rill, Georg

Mechanical Engineering R I L L Ground Vehicle Engineering Series
In striving for optimal comfort and safety conditions in road vehicles,
today’s electronically controlled components provide a range of new
options. These are developed and tested using computer simulations
in software-in-the-loop or hardware-in-the-loop environments—an
advancement that requires the modern automotive engineer to be
able to build basic simulation models, handle higher level models,
and operate simulation tools effectively.
Combining the fundamentals of vehicle dynamics with the basics of
computer-simulated modeling, Road Vehicle Dynamics: Fundamen-
tals and Modeling draws on lecture notes from undergraduate and
graduate courses given by the author, as well as industry seminars and
symposiums, to provide practical insight on the subject. Requiring only
a first course in dynamics and programming language as a prerequisite,
this highly accessible book offers end-of-chapter exercises to reinforce
concepts as well as programming examples and results using G E O R G R I L L
MATLAB®.
The book uses SI-units throughout, and begins with an introduction
and overview of units and quantities, terminology and definitions,
multibody dynamics, and equations of motion. It then discusses the
road, highlighting both deterministic and stochastic road models; tire
handling including contact calculation, longitudinal and lateral forces,
vertical axis torques, and measurement and modeling techniques;
and drive train components and concepts such as transmission,
clutch, and power source.
Later chapters discuss suspension systems, including a dynamic
model of rack-and-pinion steering as well as double-wishbone
suspension systems; force elements such as springs, anti-roll bars,
and hydro-mounts; and vehicle dynamics in vertical, longitudinal, and
lateral directions using a simple model approach to examine the
effects of nonlinear, dynamic, and active force elements. Highlighting
useable knowledge, the book concludes with a three-dimensional
vehicle model and typical results of standard driving maneuvers
K11773
ISBN: 978-1-4398-3898-3
90000
9 781439 838983

Ground Vehicle Engineering Series
Series Editor
Dr. Vladimir V. Vantsevich
Professor and Director
Program of Master of Science in Mechatronic Systems Engineering
Lawrence Technological University, Michigan
Driveline Systems of Ground Vehicles: Theory and Design
Alexandr F. Andreev, Viachaslau Kabanau, Vladimir Vantsevich
Road Vehicle Dynamics: Fundamentals and Modeling
Georg Rill
G E O R G R I L L
Boca Raton London New York
CRC Press is an imprint of the
Taylor & Francis Group, an informa business
MATLAB® is a trademark of The MathWorks, Inc. and is used with permission. The MathWorks does not
warrant the accuracy of the text or exercises in this book. This book’s use or discussion of MATLAB® soft-
ware or related products does not constitute endorsement or sponsorship by The MathWorks of a particular
pedagogical approach or particular use of the MATLAB® software.
CRC Press
Taylor & Francis Group
6000 Broken Sound Parkway NW, Suite 300
Boca Raton, FL 33487-2742
© 2012 by Taylor & Francis Group, LLC
CRC Press is an imprint of Taylor & Francis Group, an Informa business
No claim to original U.S. Government works
Version Date: 20110823
International Standard Book Number-13: 978-1-4398-9744-7 (eBook - PDF)
This book contains information obtained from authentic and highly regarded sources. Reasonable efforts
have been made to publish reliable data and information, but the author and publisher cannot assume
responsibility for the validity of all materials or the consequences of their use. The authors and publishers
have attempted to trace the copyright holders of all material reproduced in this publication and apologize to
copyright holders if permission to publish in this form has not been obtained. If any copyright material has
not been acknowledged please write and let us know so we may rectify in any future reprint.
Except as permitted under U.S. Copyright Law, no part of this book may be reprinted, reproduced, transmit-
ted, or utilized in any form by any electronic, mechanical, or other means, now known or hereafter invented,
including photocopying, microfilming, and recording, or in any information storage or retrieval system,
without written permission from the publishers.
For permission to photocopy or use material electronically from this work, please access www.copyright.
com (http://www.copyright.com/) or contact the Copyright Clearance Center, Inc. (CCC), 222 Rosewood
Drive, Danvers, MA 01923, 978-750-8400. CCC is a not-for-profit organization that provides licenses and
registration for a variety of users. For organizations that have been granted a photocopy license by the CCC,
a separate system of payment has been arranged.
Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used
only for identification and explanation without intent to infringe.
Visit the Taylor & Francis Web site at
http://www.taylorandfrancis.com
and the CRC Press Web site at
http://www.crcpress.com
Contents
List of Listings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiii
List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xv
List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxi
Series Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxiii
Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxv
About the Author . . . . . . . . . . . . . . . . . . . . . . . . . . . xxix
1 Introduction 1
1.1 Units and Quantities . . . . . . . . . . . . . . . . . . . . . . 1
1.1.1 SI System . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1.2 Tire Codes . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2 Terminology . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2.1 Vehicle Dynamics. . . . . . . . . . . . . . . . . . . . . 3
1.2.2 Driver . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.2.3 Vehicle . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.2.4 Load . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.2.5 Environment . . . . . . . . . . . . . . . . . . . . . . . 5
1.3 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.3.1 Coordinate Systems . . . . . . . . . . . . . . . . . . . 5
1.3.2 Design Position of Wheel Center . . . . . . . . . . . . 7
1.3.3 Toe-In, Toe-Out . . . . . . . . . . . . . . . . . . . . . 8
1.3.4 Wheel Camber . . . . . . . . . . . . . . . . . . . . . . 8
1.3.5 Design Position of the Wheel Rotation Axis . . . . . . 9
1.3.6 Wheel Aligning Point . . . . . . . . . . . . . . . . . . 10
1.4 Multibody Dynamics Tailored to Ground Vehicles . . . . . . 11
1.4.1 Modeling Aspects . . . . . . . . . . . . . . . . . . . . 11
1.4.2 Kinematics . . . . . . . . . . . . . . . . . . . . . . . . 13
1.4.3 Equations of Motion . . . . . . . . . . . . . . . . . . . 15
1.5 A Quarter Car Model . . . . . . . . . . . . . . . . . . . . . . 17
1.5.1 Modeling Details . . . . . . . . . . . . . . . . . . . . . 17
1.5.2 Kinematics . . . . . . . . . . . . . . . . . . . . . . . . 18
1.5.3 Applied Forces and Torques . . . . . . . . . . . . . . . 21
1.5.4 Equations of Motion . . . . . . . . . . . . . . . . . . . 21
1.5.5 Simulation . . . . . . . . . . . . . . . . . . . . . . . . 22
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
v
vi
2 Road 27
2.1 Modeling Aspects . . . . . . . . . . . . . . . . . . . . . . . . 27
2.2 Deterministic Profiles . . . . . . . . . . . . . . . . . . . . . . 29
2.2.1 Bumps and Potholes . . . . . . . . . . . . . . . . . . . 29
2.2.2 Sine Waves . . . . . . . . . . . . . . . . . . . . . . . . 30
2.3 Random Profiles . . . . . . . . . . . . . . . . . . . . . . . . . 31
2.3.1 Statistical Properties . . . . . . . . . . . . . . . . . . . 31
2.3.2 Classification of Random Road Profiles . . . . . . . . 33
2.3.3 Sinusoidal Approximation . . . . . . . . . . . . . . . . 34
2.3.4 Example . . . . . . . . . . . . . . . . . . . . . . . . . . 36
2.3.5 Shaping Filter . . . . . . . . . . . . . . . . . . . . . . 40
2.3.6 Two-Dimensional Model . . . . . . . . . . . . . . . . . 41
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
3 Tire 43
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
3.1.1 Tire Development . . . . . . . . . . . . . . . . . . . . 44
3.1.2 Tire Composites . . . . . . . . . . . . . . . . . . . . . 44
3.1.3 Tire Forces and Torques . . . . . . . . . . . . . . . . . 45
3.1.4 Measuring Tire Forces and Torques. . . . . . . . . . . 46
3.1.5 Modeling Aspects . . . . . . . . . . . . . . . . . . . . 48
3.1.6 Typical Tire Characteristics . . . . . . . . . . . . . . . 50
3.2 Contact Geometry . . . . . . . . . . . . . . . . . . . . . . . . 52
3.2.1 Basic Approach . . . . . . . . . . . . . . . . . . . . . . 52
3.2.2 Local Track Plane . . . . . . . . . . . . . . . . . . . . 54
3.2.3 Tire Deflection . . . . . . . . . . . . . . . . . . . . . . 56
3.2.4 Static Contact Point . . . . . . . . . . . . . . . . . . . 58
3.2.5 Length of Contact Patch. . . . . . . . . . . . . . . . . 59
3.2.6 Contact Point Velocity . . . . . . . . . . . . . . . . . . 60
3.2.7 Dynamic Rolling Radius . . . . . . . . . . . . . . . . . 61
3.3 Steady-State Forces and Torques . . . . . . . . . . . . . . . . 63
3.3.1 Wheel Load . . . . . . . . . . . . . . . . . . . . . . . . 63
3.3.2 Tipping Torque . . . . . . . . . . . . . . . . . . . . . . 65
3.3.3 Rolling Resistance . . . . . . . . . . . . . . . . . . . . 66
3.3.4 Longitudinal Force and Longitudinal Slip . . . . . . . 67
3.3.5 Lateral Slip, Lateral Force, and Self-Aligning Torque . 70
3.4 Combined Forces . . . . . . . . . . . . . . . . . . . . . . . . . 73
3.4.1 Combined Slip . . . . . . . . . . . . . . . . . . . . . . 73
3.4.2 Suitable Approximation . . . . . . . . . . . . . . . . . 75
3.4.3 Some Results . . . . . . . . . . . . . . . . . . . . . . . 78
3.5 Bore Torque . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
3.5.1 Modeling Aspects . . . . . . . . . . . . . . . . . . . . 81
3.5.2 Maximum Torque . . . . . . . . . . . . . . . . . . . . 82
3.5.3 Simple Approach . . . . . . . . . . . . . . . . . . . . . 83
3.5.4 Generalized Slip . . . . . . . . . . . . . . . . . . . . . 83
vii
3.6 Different Influences on Tire Forces and Torques . . . . . . . 84
3.6.1 Wheel Load . . . . . . . . . . . . . . . . . . . . . . . . 84
3.6.2 Friction . . . . . . . . . . . . . . . . . . . . . . . . . . 87
3.6.3 Camber . . . . . . . . . . . . . . . . . . . . . . . . . . 88
3.7 First-Order Tire Dynamics . . . . . . . . . . . . . . . . . . . 92
3.7.1 Simple Dynamic Extension . . . . . . . . . . . . . . . 92
3.7.2 Enhanced Force Dynamics . . . . . . . . . . . . . . . . 93
3.7.3 Enhanced Torque Dynamics . . . . . . . . . . . . . . . 96
3.7.3.1 Self-Aligning Torque . . . . . . . . . . . . . . 96
3.7.3.2 Bore Torque . . . . . . . . . . . . . . . . . . 96
3.7.3.3 Parking Torque . . . . . . . . . . . . . . . . 98
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
4 Drive Train 101
4.1 Components and Concepts . . . . . . . . . . . . . . . . . . . 101
4.1.1 Conventional Drive Train . . . . . . . . . . . . . . . . 101
4.1.2 Hybrid Drive . . . . . . . . . . . . . . . . . . . . . . . 102
4.1.3 Electric Drive . . . . . . . . . . . . . . . . . . . . . . . 103
4.2 Wheel and Tire . . . . . . . . . . . . . . . . . . . . . . . . . 104
4.2.1 Wheel Dynamics . . . . . . . . . . . . . . . . . . . . . 104
4.2.2 Eigen-Dynamics . . . . . . . . . . . . . . . . . . . . . 104
4.2.2.1 Steady-State Tire Forces . . . . . . . . . . . 104
4.2.2.2 Dynamic Tire Forces . . . . . . . . . . . . . 106
4.2.3 Simple Vehicle Wheel Tire Model . . . . . . . . . . . . 108
4.2.3.1 Equations of Motion . . . . . . . . . . . . . . 108
4.2.3.2 Driving Torque . . . . . . . . . . . . . . . . . 109
4.2.3.3 Braking Torque . . . . . . . . . . . . . . . . 110
4.2.3.4 Simulation Results . . . . . . . . . . . . . . . 111
4.3 Differentials . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
4.3.1 Classic Design . . . . . . . . . . . . . . . . . . . . . . 115
4.3.2 Active Differentials . . . . . . . . . . . . . . . . . . . . 118
4.4 Generic Drive Train . . . . . . . . . . . . . . . . . . . . . . . 119
4.5 Transmission . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
4.6 Clutch . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
4.7 Power Sources . . . . . . . . . . . . . . . . . . . . . . . . . . 124
4.7.1 Combustion Engine . . . . . . . . . . . . . . . . . . . 124
4.7.2 Hybrid Drive . . . . . . . . . . . . . . . . . . . . . . . 125
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
5 Suspension System 127
5.1 Purpose and Components . . . . . . . . . . . . . . . . . . . . 127
5.2 Some Examples . . . . . . . . . . . . . . . . . . . . . . . . . 128
5.2.1 Multipurpose Systems . . . . . . . . . . . . . . . . . . 128
5.2.2 Specific Systems . . . . . . . . . . . . . . . . . . . . . 129
5.2.3 Steering Geometry . . . . . . . . . . . . . . . . . . . . 130
viii
5.3 Steering Systems . . . . . . . . . . . . . . . . . . . . . . . . . 132
5.3.1 Components and Requirements . . . . . . . . . . . . . 132
5.3.2 Rack-and-Pinion Steering . . . . . . . . . . . . . . . . 132
5.3.3 Lever Arm Steering System . . . . . . . . . . . . . . . 133
5.3.4 Toe Bar Steering System . . . . . . . . . . . . . . . . 133
5.3.5 Bus Steering System . . . . . . . . . . . . . . . . . . . 134
5.3.6 Dynamics of a Rack-and-Pinion Steering System . . . 135
5.3.6.1 Equation of Motion . . . . . . . . . . . . . . 135
5.3.6.2 Steering Forces and Torques . . . . . . . . . 137
5.3.6.3 Parking Effort . . . . . . . . . . . . . . . . . 139
5.4 Kinematics of a Double Wishbone Suspension . . . . . . . . 144
5.4.1 Modeling Aspects . . . . . . . . . . . . . . . . . . . . 144
5.4.2 Position and Orientation. . . . . . . . . . . . . . . . . 145
5.4.3 Constraint Equations . . . . . . . . . . . . . . . . . . 146
5.4.3.1 Control Arms and Wheel Body . . . . . . . . 146
5.4.3.2 Steering Motion . . . . . . . . . . . . . . . . 148
5.4.4 Velocities . . . . . . . . . . . . . . . . . . . . . . . . . 149
5.4.5 Acceleration. . . . . . . . . . . . . . . . . . . . . . . . 152
5.4.6 Kinematic Analysis. . . . . . . . . . . . . . . . . . . . 153
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159
6 Force Elements 161
6.1 Standard Force Elements . . . . . . . . . . . . . . . . . . . . 161
6.1.1 Springs . . . . . . . . . . . . . . . . . . . . . . . . . . 161
6.1.2 Anti-Roll Bar . . . . . . . . . . . . . . . . . . . . . . . 163
6.1.3 Damper . . . . . . . . . . . . . . . . . . . . . . . . . . 165
6.1.4 Point-to-Point Force Elements . . . . . . . . . . . . . 167
6.1.4.1 Generalized Forces . . . . . . . . . . . . . . . 167
6.1.4.2 Example . . . . . . . . . . . . . . . . . . . . 170
6.1.5 Rubber Elements . . . . . . . . . . . . . . . . . . . . . 174
6.2 Dynamic Force Elements . . . . . . . . . . . . . . . . . . . . 175
6.2.1 Testing and Evaluating Procedures . . . . . . . . . . . 175
6.2.1.1 Simple Approach . . . . . . . . . . . . . . . . 175
6.2.1.2 Sweep Sine Excitation . . . . . . . . . . . . . 177
6.2.2 Spring Damper in Series . . . . . . . . . . . . . . . . . 179
6.2.2.1 Modeling Aspects . . . . . . . . . . . . . . . 179
6.2.2.2 Linear Characteristics . . . . . . . . . . . . . 180
6.2.2.3 Nonlinear Damper Topmount Combination . 182
6.2.3 General Dynamic Force Model . . . . . . . . . . . . . 185
6.2.4 Hydro-Mount . . . . . . . . . . . . . . . . . . . . . . . 186
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189
ix
7 Vertical Dynamics 191
7.1 Goals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191
7.2 From Complex to Simple Models . . . . . . . . . . . . . . . . 192
7.3 Basic Tuning . . . . . . . . . . . . . . . . . . . . . . . . . . . 196
7.3.1 Natural Frequency and Damping Ratio . . . . . . . . 196
7.3.2 Minimum Spring Rate . . . . . . . . . . . . . . . . . . 199
7.3.3 Example . . . . . . . . . . . . . . . . . . . . . . . . . . 199
7.3.4 Undamped Eigenfrequencies . . . . . . . . . . . . . . . 200
7.3.5 Influence of Damping . . . . . . . . . . . . . . . . . . 200
7.4 Optimal Damping . . . . . . . . . . . . . . . . . . . . . . . . 202
7.4.1 Disturbance Reaction Problem . . . . . . . . . . . . . 202
7.4.2 Optimal Safety . . . . . . . . . . . . . . . . . . . . . . 204
7.4.3 Optimal Comfort . . . . . . . . . . . . . . . . . . . . . 205
7.4.4 Example . . . . . . . . . . . . . . . . . . . . . . . . . . 207
7.5 Practical Aspects . . . . . . . . . . . . . . . . . . . . . . . . 208
7.5.1 General Remarks . . . . . . . . . . . . . . . . . . . . . 208
7.5.2 Quarter Car Model on Rough Road . . . . . . . . . . 209
7.6 Nonlinear Suspension Forces . . . . . . . . . . . . . . . . . . 212
7.6.1 Progressive Spring . . . . . . . . . . . . . . . . . . . . 212
7.6.2 Nonlinear Spring and Nonlinear Damper . . . . . . . . 214
7.6.3 Some Results . . . . . . . . . . . . . . . . . . . . . . . 215
7.7 Sky Hook Damper . . . . . . . . . . . . . . . . . . . . . . . . 217
7.7.1 Modeling Aspects . . . . . . . . . . . . . . . . . . . . 217
7.7.2 Eigenfrequencies and Damping Ratios . . . . . . . . . 218
7.7.3 Technical Realization . . . . . . . . . . . . . . . . . . 220
7.7.4 Simulation Results . . . . . . . . . . . . . . . . . . . . 220
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224
8 Longitudinal Dynamics 225
8.1 Dynamic Wheel Loads . . . . . . . . . . . . . . . . . . . . . 225
8.1.1 Simple Vehicle Model . . . . . . . . . . . . . . . . . . 225
8.1.2 Influence of Grade . . . . . . . . . . . . . . . . . . . . 227
8.1.3 Aerodynamic Forces . . . . . . . . . . . . . . . . . . . 227
8.2 Maximum Acceleration . . . . . . . . . . . . . . . . . . . . . 228
8.2.1 Tilting Limits . . . . . . . . . . . . . . . . . . . . . . . 228
8.2.2 Friction Limits . . . . . . . . . . . . . . . . . . . . . . 229
8.3 Driving and Braking . . . . . . . . . . . . . . . . . . . . . . . 230
8.3.1 Single Axle Drive . . . . . . . . . . . . . . . . . . . . . 230
8.3.2 Braking at Single Axle . . . . . . . . . . . . . . . . . . 231
8.3.3 Braking Stability . . . . . . . . . . . . . . . . . . . . . 232
8.3.4 Optimal Distribution of Drive and Brake Forces . . . 234
8.3.5 Different Distributions of Brake Forces . . . . . . . . . 236
8.3.6 Braking in a Turn . . . . . . . . . . . . . . . . . . . . 236
8.3.7 Braking on µ-Split . . . . . . . . . . . . . . . . . . . . 238
8.3.8 Anti-Lock System . . . . . . . . . . . . . . . . . . . . 239