The science of vehicle dynamics : Handling, braking, and ride of road and race cars

Massimo Guiggiani

The Science

of Vehicle

Dynamics

Handling, Braking, and Ride of Road and

Race Cars

Second Edition

The Science of Vehicle Dynamics

Massimo Guiggiani

The Science of Vehicle

Dynamics

Handling, Braking, and Ride of Road

and Race Cars

Second Edition

123

Massimo Guiggiani

Dipartimento di Ingegneria

Civile e Industriale

Università di Pisa

Pisa

Italy

ISBN 978-3-319-73219-0 ISBN 978-3-319-73220-6 (eBook)

https://doi.org/10.1007/978-3-319-73220-6

Library of Congress Control Number: 2018938357

1st edition: © Springer Science+Business Media Dordrecht 2014

2nd edition: © Springer International Publishing AG, part of Springer Nature 2018

This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part

of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations,

recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission

or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar

methodology now known or hereafter developed.

The use of general descriptive names, registered names, trademarks, service marks, etc. in this

publication does not imply, even in the absence of a specific statement, that such names are exempt from

the relevant protective laws and regulations and therefore free for general use.

The publisher, the authors and the editors are safe to assume that the advice and information in this

book are believed to be true and accurate at the date of publication. Neither the publisher nor the

authors or the editors give a warranty, express or implied, with respect to the material contained herein or

for any errors or omissions that may have been made. The publisher remains neutral with regard to

jurisdictional claims in published maps and institutional affiliations.

Disclaimer This book is not intended as a guide for designing, building or modifying vehicles, and

anyone who uses it as such does so entirely at his/her own risk. Testing vehicles may be dangerous.

The author and publisher are not liable for whatsoever damage arising from application of any

information contained in this book.

Printed on acid-free paper

This Springer imprint is published by the registered company Springer International Publishing AG

part of Springer Nature

The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

Preface to the Second Edition

This second edition pursues, even more than the first edition, the goal of approaching

vehicle dynamics as a scientific subject, with neat definitions, clearly stated

assumptions, sound mathematics, critical analysis of classical concepts, step-by-step

developments. This may sound theoretical, but it is actually very practical.

Indeed, some automotive companies have drastically changed their approach on

some topics according to some (apparently) theoretical results presented in the first

edition of this book.

These achievements, along with the willingness to better explain some issues,

have been the motivations for writing a new edition.

All chapters have been thoroughly revised, with the inclusion of some new

results. Several parts have been expanded, like the section on the differential

mechanism. Moreover, worked-out exercises have been included to help clarify the

matter, particularly for students.

In several parts, the book departs from commonly accepted explanations.

Somehow, the more you know (classical) vehicle dynamics, the more you will be

surprised.

Acknowledgements I wish to express my sincere gratitude and appreciation to

Gabriele Pieraccini, Maurizio Bocchi, Giacomo Tortora, Tito Amato, Francesco

Biral, Antonino Pizzuto, Andrea Quintarelli, Giuseppe Bandini, Alessandro

Moroni, Andrea Toso, Francesco Senni, Basilio Lenzo, Sandro Yemi Okutuga,

Andrea Ferrarelli, David Loppini, Carlo Rottenbacher, Claudio Ricci, Stylianos

Markolefas, Gene Lukianov.

My collaborators and dear friends Alessio Artoni and Marco Gabiccini have

carefully reviewed this book. I am most grateful to them for their valuable suggestions.

Pisa, Italy Massimo Guiggiani

February 2018

v

Preface to the First Edition

Vehicle dynamics should be a branch of dynamics, but, in my opinion, too often it

does not look like that. Dynamics is based on terse concepts and rigorous reasoning,

whereas the typical approach to vehicle dynamics is much more intuitive.

Qualitative reasoning and intuition are certainly very valuable, but they should be

supported and confirmed by scientific and quantitative results.

I understand that vehicle dynamics is, perhaps, the most popular branch of

dynamics. Almost everybody has been involved in discussions about some aspects

of the dynamical behavior of a vehicle (how to brake, how to negotiate a bend at

high speed, which tires give the best performance, etc.). At this level, we cannot

expect a deep knowledge of the dynamical behavior of a vehicle.

But there are people who could greatly benefit from mastering vehicle dynamics,

from having clear concepts in mind, from having a deep understanding of the main

phenomena. This book is intended for those people who want to build their

knowledge on sound explanations, who believe equations are the best way to

formulate and, hopefully, solve problems, of course along with physical reasoning

and intuition.

I have been constantly alert not to give anything for granted. This attitude has led

to criticize some classical concepts, such as self-aligning torque, roll axis, under￾steer gradient, handling diagram. I hope that even very experienced people will find

the book interesting. At the same time, less experienced readers should find the

matter explained in a way easy to absorb, yet profound. Quickly, I wish, they will

feel not so less experienced any more.

Pisa, Italy Massimo Guiggiani

October 2013

vii

Contents

1 Introduction .......................................... 1

1.1 Vehicle Definition ................................. 2

1.2 Vehicle Basic Scheme .............................. 3

References ............................................ 6

2 Mechanics of the Wheel with Tire ......................... 7

2.1 The Tire as a Vehicle Component ..................... 9

2.2 Carcass Features .................................. 9

2.3 Contact Patch .................................... 10

2.4 Rim Position and Motion............................ 12

2.4.1 Reference System .......................... 13

2.4.2 Rim Kinematics ........................... 13

2.5 Footprint Force ................................... 16

2.5.1 Perfectly Flat Road Surface ................... 18

2.6 Global Mechanical Behavior ......................... 20

2.6.1 Tire Transient Behavior...................... 20

2.6.2 Tire Steady-State Behavior ................... 20

2.6.3 Simplifications Based on Tire Tests ............. 21

2.7 Rolling Resistance Moment .......................... 23

2.8 Definition of Pure Rolling for Tires .................... 25

2.8.1 Zero Longitudinal Force ..................... 26

2.8.2 Zero Lateral Force ......................... 28

2.8.3 Zero Vertical Moment ....................... 28

2.8.4 Zero Lateral Force and Zero Vertical Moment ..... 28

2.8.5 Pure Rolling Summary ...................... 29

2.8.6 Rolling Velocity and Rolling Yaw Rate .......... 31

2.9 Definition of Tire Slips ............................. 33

2.9.1 Theoretical Slips ........................... 34

2.9.2 The Simple Case (No Camber) ................ 35

2.9.3 From Slips to Velocities ..................... 35

ix

2.9.4 (Not So) Practical Slips...................... 36

2.9.5 Tire Slips Are Rim Slips Indeed ............... 36

2.9.6 Slip Angle ............................... 37

2.10 Grip Forces and Tire Slips........................... 38

2.11 Tire Tests ....................................... 39

2.11.1 Tests with Pure Longitudinal Slip .............. 41

2.11.2 Tests with Pure Lateral Slip .................. 42

2.12 Magic Formula ................................... 45

2.12.1 Magic Formula Properties .................... 46

2.12.2 Fitting of Experimental Data .................. 47

2.12.3 Vertical Load Dependence ................... 47

2.12.4 Horizontal and Vertical Shifts ................. 50

2.12.5 Camber Dependence ........................ 50

2.13 Mechanics of the Wheel with Tire ..................... 50

2.13.1 Braking/Driving ........................... 51

2.13.2 Cornering ................................ 51

2.13.3 Combined ............................... 53

2.13.4 Camber ................................. 55

2.13.5 Grip .................................... 56

2.13.6 Vertical Moment ........................... 57

2.14 Exercises ....................................... 58

2.14.1 Pure Rolling .............................. 58

2.14.2 Theoretical and Practical Slips ................. 58

2.14.3 Tire Translational Slips and Slip Angle .......... 58

2.14.4 Tire Spin Slip and Camber Angle .............. 59

2.14.5 Motorcycle Tire ........................... 59

2.14.6 Finding the Magic Formula Coefficients.......... 60

2.15 Summary ....................................... 63

2.16 List of Some Relevant Concepts ...................... 63

2.17 Key Symbols .................................... 63

References ............................................ 64

3 Vehicle Model for Handling and Performance ................ 67

3.1 Mathematical Framework............................ 68

3.1.1 Vehicle Axis System........................ 68

3.2 Vehicle Congruence (Kinematic) Equations .............. 69

3.2.1 Velocity of G, and Yaw Rate of the Vehicle ...... 69

3.2.2 Yaw Angle of the Vehicle, and Trajectory of G .... 70

3.2.3 Velocity Center C .......................... 72

3.2.4 Fundamental Ratios b and q .................. 73

3.2.5 Acceleration of G and Angular Acceleration

of the Vehicle ............................. 73

3.2.6 Radius of Curvature of the Trajectory of G ....... 76

x Contents

3.2.7 Radius of Curvature of the Trajectory

of a Generic Point ......................... 78

3.2.8 Telemetry Data and Mathematical Channels ....... 78

3.2.9 Acceleration Center K ....................... 79

3.2.10 Inflection Circle ........................... 80

3.3 Tire Kinematics (Tire Slips).......................... 81

3.3.1 Translational Slips ......................... 84

3.3.2 Spin Slips ............................... 85

3.4 Steering Geometry (Ackermann) ...................... 85

3.4.1 Ackermann Steering Kinematics ............... 87

3.4.2 Best Steering Geometry ..................... 89

3.4.3 Position of Velocity Center and Relative

Slip Angles .............................. 89

3.5 Vehicle Constitutive (Tire) Equations ................... 90

3.6 Vehicle Equilibrium Equations........................ 91

3.6.1 Inertial Terms ............................. 92

3.6.2 External Force and Moment .................. 92

3.7 Forces Acting on the Vehicle ......................... 93

3.7.1 Weight .................................. 93

3.7.2 Aerodynamic Force ......................... 93

3.7.3 Road–Tire Friction Forces .................... 95

3.7.4 Road–Tire Vertical Forces.................... 99

3.8 Vehicle Equilibrium Equations (More Explicit Form) ....... 100

3.9 Vertical Loads and Load Transfers..................... 102

3.9.1 Longitudinal Load Transfer ................... 102

3.9.2 Lateral Load Transfers ...................... 103

3.9.3 Vertical Load on Each Tire ................... 103

3.10 Suspension First-Order Analysis....................... 104

3.10.1 Suspension Reference Configuration ............ 105

3.10.2 Suspension Internal Coordinates ............... 106

3.10.3 Kinematic Camber Variation .................. 107

3.10.4 Kinematic Track Width Variation .............. 108

3.10.5 Vehicle Internal Coordinates .................. 109

3.10.6 Definition of Roll and Vertical Stiffnesses ........ 109

3.10.7 Suspension Internal Equilibrium ............... 113

3.10.8 Effects of a Lateral Force .................... 113

3.10.9 No-Roll Centers and No-Roll Axis ............. 115

3.10.10 Suspension Jacking ......................... 118

3.10.11 Roll Moment ............................. 118

3.10.12 Roll Angles and Lateral Load Transfers .......... 120

3.10.13 Explicit Expressions of the Lateral Load

Transfers ................................ 122

3.10.14 Lateral Load Transfers with Rigid Tires .......... 124

Contents xi

3.11 Sprung and Unsprung Masses ........................ 124

3.12 Dependent Suspensions (Solid Axle) ................... 125

3.13 Linked Suspensions................................ 128

3.14 Differential Mechanisms ............................ 128

3.14.1 Relative Angular Speeds ..................... 130

3.14.2 Torque Balance ........................... 130

3.14.3 Internal Efficiency and TBR .................. 131

3.14.4 Locking Coefficient ......................... 135

3.14.5 Rule of Thumb ............................ 136

3.14.6 A Simple Mathematical Model ................ 138

3.14.7 Alternative Governing Equations ............... 138

3.14.8 Open Differential .......................... 139

3.14.9 Limited-Slip Differentials .................... 139

3.14.10 Geared Differentials ........................ 140

3.14.11 Clutch-Pack Differentials..................... 141

3.14.12 Spindle Axle ............................. 144

3.14.13 Differential–Tire Interaction ................... 144

3.14.14 Informal Summary About the Differential

Behavior ................................ 150

3.15 Vehicle Model for Handling and Performance ............ 150

3.15.1 Equilibrium Equations....................... 150

3.15.2 Camber Variations ......................... 152

3.15.3 Roll Angles .............................. 153

3.15.4 Steer Angles.............................. 153

3.15.5 Tire Slips ................................ 154

3.15.6 Tire Constitutive Equations ................... 155

3.15.7 Differential Mechanism Equations .............. 156

3.15.8 Summary ................................ 156

3.16 The Structure of This Vehicle Model ................... 157

3.17 Three-Axle Vehicles ............................... 157

3.18 Exercises ....................................... 160

3.18.1 Center of Curvature QG of the Trajectory of G ..... 160

3.18.2 Track Variation ........................... 160

3.18.3 Camber Variation .......................... 160

3.18.4 Power Loss in a Self-locking Differential ......... 161

3.18.5 Differential–Tires Interaction .................. 161

3.19 Summary ....................................... 164

3.20 List of Some Relevant Concepts ...................... 165

3.21 Key Symbols .................................... 165

References ............................................ 167

xii Contents

4 Braking Performance ................................... 169

4.1 Pure Braking .................................... 170

4.2 Vehicle Model for Braking Performance ................. 170

4.3 Equilibrium Equations .............................. 171

4.4 Longitudinal Load Transfer .......................... 172

4.5 Maximum Deceleration ............................. 172

4.6 Brake Balance .................................... 173

4.7 All Possible Braking Combinations .................... 173

4.8 Changing the Grip ................................ 175

4.9 Changing the Weight Distribution ..................... 176

4.10 A Numerical Example .............................. 176

4.11 Braking Performance of Formula Cars .................. 177

4.11.1 Equilibrium Equations....................... 177

4.11.2 Vertical Loads ............................ 178

4.11.3 Maximum Deceleration ...................... 179

4.11.4 Brake Balance ............................ 180

4.11.5 Speed Independent Brake Balance .............. 181

4.11.6 Typical Formula 1 Braking Performance ......... 181

4.12 Braking, Stopping, and Safe Distances .................. 183

4.13 Exercises ....................................... 183

4.13.1 Minimum Braking Distance ................... 183

4.13.2 Braking with Aerodynamic Downforces .......... 185

4.13.3 GP2 Brake Balance ......................... 185

4.13.4 Speed Independent Brake Balance .............. 186

4.14 Summary ....................................... 187

4.15 List of Some Relevant Concepts ...................... 187

4.16 Key Symbols .................................... 188

References ............................................ 188

5 The Kinematics of Cornering ............................. 189

5.1 Planar Kinematics of a Rigid Body .................... 189

5.1.1 Velocity Field and Velocity Center ............. 190

5.1.2 Acceleration Field and Acceleration Center ....... 192

5.1.3 Inflection Circle and Radii of Curvature .......... 193

5.2 The Kinematics of a Turning Vehicle ................... 196

5.2.1 Moving and Fixed Centrodes of a Turning

Vehicle ................................. 197

5.2.2 Inflection Circle of a Turning Vehicle ........... 201

5.2.3 Tracking the Curvatures of Front and Rear

Midpoints................................ 205

5.2.4 Evolutes ................................. 210

Contents xiii

5.3 Exercises ....................................... 210

5.3.1 Front and Rear Radii of Curvature .............. 210

5.3.2 Drawing Centrodes ......................... 211

5.4 Key Symbols .................................... 211

References ............................................ 212

6 Handling of Road Cars .................................. 213

6.1 Additional Simplifying Assumptions for Road

Car Modeling .................................... 214

6.1.1 Negligible Vertical Aerodynamic Loads .......... 214

6.1.2 Almost Constant Forward Speed ............... 214

6.1.3 Open Differential .......................... 215

6.2 Mathematical Model for Road Car Handling.............. 215

6.2.1 Global Equilibrium ......................... 216

6.2.2 Approximate Lateral Forces................... 217

6.2.3 Lateral Load Transfers and Vertical Loads ........ 218

6.2.4 Roll Angles .............................. 220

6.2.5 Camber Angle Variations .................... 220

6.2.6 Steer Angles.............................. 222

6.2.7 Tire Slips ................................ 223

6.2.8 Simplified Tire Slips ........................ 224

6.2.9 Tire Lateral Forces ......................... 226

6.3 Double Track Model ............................... 227

6.3.1 Governing Equations of the Double

Track Model.............................. 227

6.3.2 Dynamical Equations of the Double

Track Model.............................. 228

6.3.3 Alternative State Variables (b and q) ............ 228

6.4 Vehicle in Steady-State Conditions..................... 229

6.5 Single Track Model................................ 231

6.5.1 From Double to Single ...................... 231

6.5.2 “Forcing” the Lateral Forces .................. 234

6.5.3 Axle Characteristics ........................ 235

6.5.4 Governing Equations of the Single

Track Model.............................. 244

6.5.5 Dynamical Equations of the Single

Track Model.............................. 246

6.5.6 Alternative State Variables (b and q) ............ 247

6.5.7 Inverse Congruence Equations ................. 248

6.5.8 b1 and b2 as State Variables .................. 248

6.5.9 Driving Force ............................. 250

xiv Contents

6.5.10 The Role of the Steady-State Lateral

Acceleration .............................. 251

6.5.11 Slopes of the Axle Characteristics .............. 252

6.6 Double Track, or Single Track? ....................... 252

6.7 Steady-State Maps................................. 253

6.7.1 Steady-State Gradients ...................... 255

6.7.2 Alternative Steady-State Gradients .............. 256

6.7.3 Understeer and Oversteer .................... 256

6.7.4 Handling Diagram ......................... 259

6.8 Map of Achievable Performance (MAP) ................. 261

6.8.1 MAP Fundamentals ........................ 262

6.8.2 MAP Curvature q Versus Steer Angle d ......... 268

6.8.3 Other Possible MAPs ....................... 273

6.9 Weak Concepts in Classical Vehicle Dynamics............ 274

6.9.1 The Understeer Gradient ..................... 275

6.9.2 Popular Definitions of Understeer/Oversteer ....... 276

6.10 Double Track Model in Transient Conditions ............. 276

6.10.1 Equilibrium Points ......................... 277

6.10.2 Free Oscillations (No Driver Action) ............ 277

6.10.3 MAP for Transient Behavior .................. 281

6.10.4 Stability of the Equilibrium ................... 282

6.10.5 Forced Oscillations (Driver Action) ............. 282

6.11 Relationship Between Steady-State Data and Transient

Behavior........................................ 284

6.11.1 Stability Derivatives from Steady-State

Gradients ................................ 285

6.11.2 Equations of Motion ........................ 287

6.11.3 Estimation of the Control Derivatives ........... 288

6.11.4 Objective Evaluation of Car Handling ........... 288

6.12 Stability (Again) .................................. 290

6.13 New Understeer Gradient ........................... 291

6.14 The Nonlinear Single Track Model Revisited ............. 292

6.14.1 Different Vehicles with Identical Handling ........ 295

6.15 Linear Single Track Model .......................... 298

6.15.1 Governing Equations........................ 299

6.15.2 Solution for Constant Forward Speed ............ 301

6.15.3 Critical Speed ............................. 303

6.15.4 Transient Vehicle Behavior ................... 303

6.15.5 Steady-State Behavior: Steering Pad ............ 306

6.15.6 Lateral Wind Gust ......................... 307

6.15.7 Banked Road ............................. 311

Contents xv

6.16 Compliant Steering System .......................... 312

6.16.1 Governing Equations........................ 313

6.16.2 Effects of Steer Compliance .................. 314

6.17 Road Vehicles with Locked or Limited Slip Differential ..... 315

6.18 Exercises ....................................... 315

6.18.1 Camber Variations ......................... 315

6.18.2 Ackermann Coefficient ...................... 315

6.18.3 Toe-In .................................. 316

6.18.4 Steering Angles ........................... 316

6.18.5 Axle Characteristics ........................ 316

6.18.6 Playing with Linear Differential Equations ........ 317

6.18.7 Static Margin ............................. 317

6.18.8 Banked Road ............................. 317

6.18.9 Rear Steer ............................... 318

6.18.10 Wind Gust ............................... 318

6.19 Summary ....................................... 319

6.20 List of Some Relevant Concepts ...................... 320

6.21 Key Symbols .................................... 320

References ............................................ 322

7 Handling of Race Cars .................................. 323

7.1 Assumptions for Race Car Handling ................... 323

7.1.1 Aerodynamic Downloads .................... 324

7.1.2 Limited-Slip Differential ..................... 324

7.2 Vehicle Model for Race Car Handling .................. 325

7.2.1 Equilibrium Equations....................... 326

7.2.2 Lateral Forces for Dynamic Equilibrium.......... 328

7.2.3 Tire Forces............................... 328

7.2.4 Tire Slips ................................ 329

7.2.5 Camber Angles............................ 330

7.2.6 Steer Angles.............................. 331

7.2.7 Vertical Loads on Each Wheel ................ 332

7.2.8 Lateral Load Transfers ...................... 333

7.2.9 Roll Angles .............................. 334

7.2.10 Behavior of the Limited-Slip Differential ......... 334

7.2.11 Reducing the Number of Equations ............. 335

7.3 Double Track Race Car Model ....................... 337

7.3.1 Single Track? ............................. 337

7.4 Basics for Steady-State Handling Analysis ............... 338

7.5 The Handling Diagram Becomes the Handling Surface ...... 339

7.5.1 Handling with Locked Differential

(and No Wings) ........................... 339

xvi Contents

7.6 Handling of Formula Cars ........................... 352

7.6.1 Handling Surface .......................... 353

7.6.2 Map of Achievable Performance (MAP) ......... 354

7.7 Exercises ....................................... 363

7.7.1 Vehicle Kinematic Equations.................. 363

7.7.2 Spin Slip Contributions ...................... 367

7.7.3 Acceleration Center K and Acceleration

of the Velocity Center C ..................... 368

7.7.4 Aerodynamic Downforces .................... 368

7.7.5 Roll Stiffnesses in Formula Cars ............... 369

7.7.6 Lateral Load Transfers in Formula Cars .......... 370

7.7.7 Centrifugal Force not Applied at the Center

of Mass ................................. 371

7.7.8 Global Aerodynamic Force ................... 371

7.8 Summary ....................................... 372

7.9 List of Some Relevant Concepts ...................... 373

7.10 Key Symbols .................................... 373

References ............................................ 375

8 Map of Achievable Performance (MAP) ..................... 377

8.1 MAP Fundamental Idea ............................. 377

8.2 Achievable Regions................................ 378

8.2.1 Input Achievable Region ..................... 378

8.2.2 Output Achievable Regions ................... 382

8.2.3 Mixed I/O Achievable Regions ................ 384

8.3 Achievable Performances on Input Regions .............. 384

8.4 Achievable Performances on Output Regions ............. 386

8.5 Achievable Performances on Mixed I/O Regions .......... 387

8.6 MAP from Slowly Increasing Steer Tests ................ 388

8.7 MAP from Constant Steer Tests....................... 390

8.8 Concluding Remarks ............................... 392

8.9 Key Symbols .................................... 392

9 Handling with Roll Motion ............................... 393

9.1 Vehicle Position and Orientation ...................... 393

9.2 Yaw, Pitch and Roll ............................... 394

9.3 Angular Velocity.................................. 397

9.4 Angular Acceleration ............................... 399

9.5 Vehicle Lateral Velocity ............................ 399

9.5.1 Track Invariant Points ....................... 399

9.5.2 Vehicle Invariant Point (VIP) ................. 403

9.5.3 Lateral Velocity and Acceleration .............. 404

Contents xvii