Vehicle dynamics : theory and application

Vehicle Dynamics

Reza N. Jazar

Theory and Application

2nd Edition

Vehicle Dynamics

Vehicle Dynamics

Reza N. Jazar

Second Edition

Theory and Application

© Springer Science+Business Media New York 2014

Printed on acid-free paper

Springer is part of Springer Science+Business Media (www.springer.com)

ISBN 978-1-4614-

DOI 10.1007/978-1-4614-8544-5

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8543-8 ISBN 978-1-4614- eBook) 8544-5 (

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Reza N. Jazar

School of Aerospace, Mechanical and

RMIT University

Bundoora, VIC

Australia

Manufacturing Engineering

Library of Congress Control Number: 2013951659

Contents

Preface x

1 Tire and Rim Fundamentals 1

1.1 Tires and Sidewall Information . . . . . . . . . . . . . . . . 1

1.2 Tire Components . . . . . . . . . . . . . . . . . . . . . . . . 11

1.3 Radial and Non-Radial Tires . . . . . . . . . . . . . . . . . 15

1.4 Tread . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

1.5 Tireprint . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

1.6 Wheel and Rim . . . . . . . . . . . . . . . . . . . . . . . . . 21

1.7 Vehicle Classications . . . . . . . . . . . . . . . . . . . . . 26

1.7.1 ISO and FHWA Classication . . . . . . . . . . . . . 26

1.7.2 Passenger Car Classications . . . . . . . . . . . . . 29

1.7.3 Passenger Car Body Styles . . . . . . . . . . . . . . 31

1.8 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

1.9 Key Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . 34

Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

I Vehicle Motion 37

2 Forward Vehicle Dynamics 39

2.1 Parked Car on a Level Road . . . . . . . . . . . . . . . . . . 39

2.2 Parked Car on an Inclined Road . . . . . . . . . . . . . . . 45

2.3 Accelerating Car on a Level Road . . . . . . . . . . . . . . . 50

2.4 Accelerating Car on an Inclined Road . . . . . . . . . . . . 55

2.5 Parked Car on a Banked Road . . . . . . . . . . . . . . . . 66

2.6 F Optimal Drive and Brake Force Distribution . . . . . . . 70

2.7 F Vehicles With More Than Two Axles . . . . . . . . . . . 76

2.8 F Vehicles on a Crest and Dip . . . . . . . . . . . . . . . . 80

2.8.1 F Vehicles on a Crest . . . . . . . . . . . . . . . . . 81

2.8.2 F Vehicles on a Dip . . . . . . . . . . . . . . . . . . 86

2.9 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

2.10 Key Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . 90

Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

3 Tire Dynamics 99

3.1 Tire Coordinate Frame and Tire Force System . . . . . . . 99

3.2 Tire Stiness . . . . . . . . . . . . . . . . . . . . . . . . . . 103

vii

v

3.3 Eective Radius . . . . . . . . . . . . . . . . . . . . . . . . 107

3.4 F Tireprint Forces of a Static Tire . . . . . . . . . . . . . . 122

3.4.1 F Static Tire, Normal Stress . . . . . . . . . . . . . 123

3.4.2 F Static Tire, Tangential Stresses . . . . . . . . . . 126

3.5 Rolling Resistance . . . . . . . . . . . . . . . . . . . . . . . 128

3.5.1 Eect of Speed on the Rolling Friction Coe!cient . 131

3.5.2 Eect of In ation Pressure and Load on the Rolling

Friction Coe!cient . . . . . . . . . . . . . . . . . . . 135

3.5.3 F Eect of Sideslip Angle on Rolling Resistance . . 138

3.5.4 F Eect of Camber Angle on Rolling Resistance . . 138

3.6 Longitudinal Force . . . . . . . . . . . . . . . . . . . . . . . 139

3.7 Lateral Force . . . . . . . . . . . . . . . . . . . . . . . . . . 148

3.8 Camber Force . . . . . . . . . . . . . . . . . . . . . . . . . . 158

3.9 Tire Force . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164

3.10 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170

3.11 Key Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . 172

Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174

4 Driveline Dynamics 179

4.1 Engine Dynamics . . . . . . . . . . . . . . . . . . . . . . . . 179

4.2 Driveline and E!ciency . . . . . . . . . . . . . . . . . . . . 186

4.3 Gearbox and Clutch Dynamics . . . . . . . . . . . . . . . . 192

4.4 Gearbox Design . . . . . . . . . . . . . . . . . . . . . . . . . 200

4.4.1 Geometric Ratio Gearbox Design . . . . . . . . . . . 201

4.4.2 F Progressive Ratio Gearbox Design . . . . . . . . . 215

4.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218

4.6 Key Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . 220

Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222

II Vehicle Kinematics

5 F Applied Kinematics 233

5.1 Rotation About Global Cartesian Axes . . . . . . . . . . . . 233

5.2 Successive Rotation About Global Cartesian Axes . . . . . 238

5.3 Rotation About Local Cartesian Axes . . . . . . . . . . . . 239

5.4 Successive Rotation About Local Cartesian Axes . . . . . . 243

5.5 General Transformation . . . . . . . . . . . . . . . . . . . . 251

5.6 Local and Global Rotations . . . . . . . . . . . . . . . . . . 258

5.7 Axis-angle Rotation . . . . . . . . . . . . . . . . . . . . . . 259

5.8 Rigid Body Motion . . . . . . . . . . . . . . . . . . . . . . . 264

5.9 Angular Velocity . . . . . . . . . . . . . . . . . . . . . . . . 267

5.10 F Time Derivative and Coordinate Frames . . . . . . . . . 275

5.11 Rigid Body Velocity . . . . . . . . . . . . . . . . . . . . . . 284

5.12 Angular Acceleration . . . . . . . . . . . . . . . . . . . . . . 288

vi

231

Contents

5.13 Rigid Body Acceleration . . . . . . . . . . . . . . . . . . . . 293

5.14 F Screw Motion . . . . . . . . . . . . . . . . . . . . . . . . 296

5.15 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . 309

5.16 Key Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . 312

Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313

6 Applied Mechanisms 319

6.1 Four-Bar Linkage . . . . . . . . . . . . . . . . . . . . . . . . 319

6.2 Slider-Crank Mechanism . . . . . . . . . . . . . . . . . . . . 339

6.3 Inverted Slider-Crank Mechanism . . . . . . . . . . . . . . . 346

6.4 Instant Center of Rotation . . . . . . . . . . . . . . . . . . . 352

6.5 Coupler Point Curve . . . . . . . . . . . . . . . . . . . . . . 364

6.5.1 Coupler Point Curve for Four-Bar Linkages . . . . . 364

6.5.2 Coupler Point Curve for a Slider-Crank Mechanism . 366

6.5.3 Coupler Point Curve for Inverted Slider-Crank Mech￾anism . . . . . . . . . . . . . . . . . . . . . . . . . . 370

6.6 F Universal Joint . . . . . . . . . . . . . . . . . . . . . . . 371

6.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . 380

6.8 Key Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . 381

Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 382

7 Steering Dynamics 387

7.1 Kinematic Steering . . . . . . . . . . . . . . . . . . . . . . . 387

7.2 Vehicles with More Than Two Axles . . . . . . . . . . . . . 404

7.3 F Vehicle with Trailer . . . . . . . . . . . . . . . . . . . . . 407

7.4 Steering Mechanisms . . . . . . . . . . . . . . . . . . . . . . 411

7.5 F Four wheel steering. . . . . . . . . . . . . . . . . . . . . . 417

7.6 F Road Design . . . . . . . . . . . . . . . . . . . . . . . . . 434

7.7 F Steering Mechanism Optimization . . . . . . . . . . . . . 461

7.8 F Trailer-Truck Kinematics . . . . . . . . . . . . . . . . . . 469

7.9 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . 483

7.10 Key Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . 484

Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 486

8 Suspension Mechanisms 497

8.1 Solid Axle Suspension . . . . . . . . . . . . . . . . . . . . . 497

8.2 Independent Suspension . . . . . . . . . . . . . . . . . . . . 508

8.3 Roll Center and Roll Axis . . . . . . . . . . . . . . . . . . . 513

8.4 F Car Tire Relative Angles . . . . . . . . . . . . . . . . . . 524

8.4.1 F Toe . . . . . . . . . . . . . . . . . . . . . . . . . . 527

8.4.2 F Caster Angle . . . . . . . . . . . . . . . . . . . . . 529

8.4.3 F Camber . . . . . . . . . . . . . . . . . . . . . . . 530

8.4.4 F Thrust Angle . . . . . . . . . . . . . . . . . . . . 532

8.5 F Suspension Requirements and Coordinate Frames . . . . 533

8.5.1 Kinematic Requirements . . . . . . . . . . . . . . . . 533

Contents vii

8.5.2 Dynamic Requirements . . . . . . . . . . . . . . . . 534

8.5.3 Wheel, wheel-body, and tire Coordinate Frames . . . 534

8.6 F Caster Theory . . . . . . . . . . . . . . . . . . . . . . . . 544

8.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . 554

8.8 Key Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . 557

Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 559

III Vehicle Dynamics 567

9 F Applied Dynamics 569

9.1 Elements of Dynamics . . . . . . . . . . . . . . . . . . . . . 569

9.1.1 Force and Moment . . . . . . . . . . . . . . . . . . . 569

9.1.2 Momentum . . . . . . . . . . . . . . . . . . . . . . . 570

9.1.3 Vectors . . . . . . . . . . . . . . . . . . . . . . . . . 571

9.1.4 Equation of Motion . . . . . . . . . . . . . . . . . . 573

9.1.5 Work and Energy . . . . . . . . . . . . . . . . . . . . 573

9.2 Rigid Body Translational Dynamics . . . . . . . . . . . . . 579

9.3 Rigid Body Rotational Dynamics . . . . . . . . . . . . . . . 582

9.4 Mass Moment Matrix . . . . . . . . . . . . . . . . . . . . . 593

9.5 Lagrange’s Form of Newton’s Equations of Motion . . . . . 603

9.6 Lagrangian Mechanics . . . . . . . . . . . . . . . . . . . . . 610

9.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . 620

9.8 Key Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . 623

Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 624

10 Vehicle Planar Dynamics 631

10.1 Vehicle Coordinate Frame . . . . . . . . . . . . . . . . . . . 631

10.2 Rigid Vehicle Newton-Euler Dynamics . . . . . . . . . . . . 637

10.3 Force System Acting on a Rigid Vehicle . . . . . . . . . . . 644

10.3.1 Tire Force and Body Force Systems . . . . . . . . . 644

10.3.2 Tire Lateral Force . . . . . . . . . . . . . . . . . . . 648

10.3.3 Two-wheel Model and Body Force Components . . . 649

10.4 Two-wheel Rigid Vehicle Dynamics . . . . . . . . . . . . . . 659

10.5 Steady-State Turning . . . . . . . . . . . . . . . . . . . . . . 670

10.6 F Linearized Model for a Two-Wheel Vehicle . . . . . . . . 695

10.7 F Transient Response . . . . . . . . . . . . . . . . . . . . . 699

10.8 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . 727

10.9 Key Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . 729

Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 731

11 F Vehicle Roll Dynamics 741

11.1 F Vehicle Coordinate and DOF . . . . . . . . . . . . . . . . 741

11.2 F Equations of Motion . . . . . . . . . . . . . . . . . . . . 742

11.3 F Vehicle Force System . . . . . . . . . . . . . . . . . . . . 746

viii Contents

11.3.1 F Tire and Body Force Systems . . . . . . . . . . . 746

11.3.2 F Tire Lateral Force . . . . . . . . . . . . . . . . . . 749

11.3.3 F Body Force Components on a Two-wheel Model . 752

11.4 F Two-wheel Rigid Vehicle Dynamics . . . . . . . . . . . . 759

11.5 F Steady-State Motion . . . . . . . . . . . . . . . . . . . . 763

11.6 F Time Response . . . . . . . . . . . . . . . . . . . . . . . 767

11.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . 782

11.8 Key Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . 783

Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 786

IV Vehicle Vibration 795

12 Applied Vibrations 797

12.1 Mechanical Vibration Elements . . . . . . . . . . . . . . . . 797

12.2 Newton’s Method and Vibrations . . . . . . . . . . . . . . . 805

12.3 Frequency Response of Vibrating Systems . . . . . . . . . . 812

12.3.1 Forced Excitation . . . . . . . . . . . . . . . . . . . 813

12.3.2 Base Excitation . . . . . . . . . . . . . . . . . . . . . 823

12.3.3 Eccentric Excitation . . . . . . . . . . . . . . . . . . 835

12.3.4 F Eccentric Base Excitation . . . . . . . . . . . . . 841

12.3.5 F Classication for the Frequency Responses of One￾DOF Forced Vibration Systems . . . . . . . . . . . . 847

12.4 Time Response of Vibrating Systems . . . . . . . . . . . . . 852

12.5 Vibration Application and Measurement . . . . . . . . . . . 864

12.6 F Vibration Optimization Theory . . . . . . . . . . . . . . 869

12.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . 880

12.8 Key Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . 882

Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 885

13 Vehicle Vibrations 891

13.1 Lagrange Method and Dissipation Function . . . . . . . . . 891

13.2 F Quadratures . . . . . . . . . . . . . . . . . . . . . . . . . 901

13.3 Natural Frequencies and Mode Shapes . . . . . . . . . . . . 908

13.4 Bicycle Car and Body Pitch Mode . . . . . . . . . . . . . . 915

13.5 Half Car and Body Roll Mode . . . . . . . . . . . . . . . . . 920

13.6 Full Car Vibrating Model . . . . . . . . . . . . . . . . . . . 925

13.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . 933

13.8 Key Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . 934

Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 936

14 Suspension Optimization 939

14.1 Mathematical Model . . . . . . . . . . . . . . . . . . . . . . 939

14.2 Frequency Response . . . . . . . . . . . . . . . . . . . . . . 945

14.3 RMS Optimization . . . . . . . . . . . . . . . . . . . . . . . 949

Contents ix

14.4 F Time Response Optimization . . . . . . . . . . . . . . . . 972

14.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . 978

14.6 Key Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . 979

Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 981

15 F Quarter Car Model 985

15.1 Mathematical Model . . . . . . . . . . . . . . . . . . . . . . 985

15.2 Frequency Response . . . . . . . . . . . . . . . . . . . . . . 987

15.3 F Natural and Invariant Frequencies . . . . . . . . . . . . . 994

15.4 F RMS Optimization . . . . . . . . . . . . . . . . . . . . 1006

15.5 F Optimization Based on Natural Frequency and Wheel

Travel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1016

15.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . 1022

15.7 Key Symbols . . . . . . . . . . . . . . . . . . . . . . . . . 1023

Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1025

References 1027

A Frequency Response Curves 1031

B Trigonometric Formulas 1037

C Unit Conversions 1041

Index 1045

x Contents

Dedicated to

my son, Kavosh,

my daughter, Vazan,

and my wife, Mojgan.

Nature is not perfect, not even optimum.