The Science of Vehicle Dynamics Massimo Guiggiani

The Science of Vehicle Dynamics

Handling, Braking, and Ride of Road and Race Cars Massimo Guiggiani Dip. di Ingegneria Civile e Industriale Università di Pisa Pisa, Italy

ISBN 978-94-017-8532-7 ISBN 978-94-017-8533-4 (eBook) DOI 10.1007/978-94-017-8533-4 Springer Dordrecht Heidelberg New York London

Library of Congress Control Number: 2013958104

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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

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

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. Qualita- tive 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 Dy- namics. 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 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 for- mulate 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, understeer 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.

Acknowledgments Over the last few years I have had interactions and discussions with several engineers from Ferrari Formula 1. The problems they constantly have to face have been among the motivations for writing this book. Moreover, their deep knowledge of vehicle dynamics has been a source of inspiration. I would like to express my gratitude to Maurizio Bocchi, Giacomo Tortora, Carlo Miano, Marco Fainello, Tito Amato (presently at Mercedes), and Gabriele Pieraccini (presently at Bosch). I wish to thank Dallara Automobili and, in particular, Andrea Toso, Alessandro Moroni, and Luca Bergianti. They have helped me in many ways.

v vi Preface

At the Università di Pisa there are an M.S. degree course in Vehicle Engineering (where I teach Vehicle Dynamics) and a Ph. D. program in Vehicle Engineering and Transportation Systems. This very lively environment has played a crucial role in the development of some of the most innovative topics in this book. In particular, I wish to acknowledge the contribution of my colleague Francesco Frendo, and of my former Ph. D. students Antonio Sponziello, Riccardo Bartolozzi, and Francesco Bucchi. Francesco Frendo and Riccardo Bartolozzi have also reviewed part of this book. During the last six years I have been the Faculty Advisor of E-Team, the Formula Student team of the Università di Pisa. I thank all the team members. It has been a very interesting and rewarding experience, both professionally and personally. Testing real vehicles is essential to understand vehicle dynamics. I wish to thank Danilo Tonani, director of FormulaGuidaSicura, for having given me the opportu- nity of becoming a safe driving instructor. Every year, he organizes an excellent safe driving course for the M.S. students in Vehicle Engineering of the Università di Pisa. My collaborators and dear friends Alessio Artoni and Marco Gabiccini have care- fully reviewed this book. I am most grateful to them for their valuable suggestions to correct and improve the text. Pisa, Italy Massimo Guiggiani October 2013 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 ...... 8 2.2 Rim Position and Motion ...... 9 2.3 Carcass Features ...... 12 2.4 ContactPatch...... 13 2.5 FootprintForce...... 14 2.5.1PerfectlyFlatRoadSurface...... 16 2.6 Tire Global Mechanical Behavior ...... 17 2.6.1TireTransientBehavior...... 17 2.6.2 Tire Steady-State Behavior ...... 18 2.6.3 Rolling Resistance ...... 20 2.6.4 Speed Independence (Almost) ...... 21 2.6.5 Pure Rolling (not Free Rolling) ...... 21 2.7 TireSlips...... 26 2.7.1 Rolling Velocity ...... 27 2.7.2 Deﬁnition of Tire Slips ...... 27 2.7.3SlipAngle...... 30 2.8 GripForcesandTireSlips...... 31 2.9 TireTesting...... 33 2.9.1PureLongitudinalSlip...... 34 2.9.2PureLateralSlip...... 35 2.10MagicFormula...... 38 2.11 Mechanics of Wheels with Tire ...... 39 2.12Summary...... 43 2.13 List of Some Relevant Concepts ...... 44 References ...... 44

vii viii Contents

3 Vehicle Model for Handling and Performance ...... 47 3.1 MathematicalFramework...... 48 3.2 Vehicle Congruence (Kinematic) Equations ...... 48 3.2.1 Velocities ...... 48 3.2.2YawAngleandTrajectory...... 49 3.2.3VelocityCenter...... 51 3.2.4 Fundamental Ratios ...... 52 3.2.5 Accelerations and Radii of Curvature ...... 53 3.2.6 Acceleration Center ...... 54 3.2.7TireKinematics(TireSlips)...... 56 3.3 Vehicle Constitutive (Tire) Equations ...... 58 3.4 Vehicle Equilibrium Equations ...... 59 3.5 ForcesActingontheVehicle...... 59 3.5.1Weight...... 60 3.5.2 Aerodynamic Force ...... 60 3.5.3 Road-Tire Friction Forces ...... 61 3.5.4 Road-Tire Vertical Forces ...... 63 3.6 Vehicle Equilibrium Equations (more Explicit Form) ...... 63 3.7 LoadTransfers...... 65 3.7.1LongitudinalLoadTransfer...... 65 3.7.2LateralLoadTransfers...... 66 3.7.3 Vertical Loads on Each Tire ...... 66 3.8 Suspension First-Order Analysis ...... 67 3.8.1 Suspension Reference Conﬁguration ...... 67 3.8.2 Suspension Internal Coordinates ...... 68 3.8.3CamberVariation...... 69 3.8.4VehicleInternalCoordinates...... 70 3.8.5RollandVerticalStiffnesses...... 71 3.8.6 Suspension Internal Equilibrium ...... 73 3.8.7EffectsofaLateralForce...... 74 3.8.8No-rollCentersandNo-rollAxis...... 75 3.8.9ForcesattheNo-rollCenters...... 77 3.8.10 Suspension Jacking ...... 78 3.8.11RollAngleandLateralLoadTransfers...... 79 3.8.12ExplicitExpressionsofLateralLoadTransfers..... 81 3.8.13LateralLoadTransferswithRigidTires...... 82 3.9 Dependent Suspensions ...... 82 3.10SprungandUnsprungMasses...... 85 3.11 Vehicle Model for Handling and Performance ...... 86 3.11.1 Equilibrium Equations ...... 86 3.11.2 Constitutive (Tire) Equations ...... 88 3.11.3 Congruence (Kinematic) Equations ...... 88 3.11.4 Principles of Any Differential Mechanism ...... 90 3.12 The Structure of This Vehicle Model ...... 94 3.13Three-AxleVehicles...... 95 Contents ix

3.14Summary...... 97 3.15 List of Some Relevant Concepts ...... 97 References ...... 98 4 Braking Performance ...... 99 4.1 PureBraking...... 99 4.2 Vehicle Model for Braking Performance ...... 100 4.3 Equilibrium Equations ...... 101 4.4 LongitudinalLoadTransfer...... 101 4.5 Maximum Deceleration ...... 102 4.6 Brake Balance ...... 103 4.7 AllPossibleBrakingCombinations...... 103 4.8 Changing the Grip ...... 105 4.9 Changing the Weight Distribution ...... 106 4.10ANumericalExample...... 106 4.11 Braking Performance of Formula Cars ...... 107 4.11.1 Equilibrium Equations ...... 107 4.11.2LongitudinalLoadTransfer...... 108 4.11.3 Maximum Deceleration ...... 108 4.11.4 Braking Balance ...... 109 4.11.5 Typical Formula 1 Braking Performance ...... 109 4.12Summary...... 109 4.13 List of Some Relevant Concepts ...... 110 References ...... 111 5 The Kinematics of Cornering ...... 113 5.1 Planar Kinematics of a Rigid Body ...... 113 5.1.1VelocityFieldandVelocityCenter...... 113 5.1.2 Acceleration Field, Inﬂection Circle and Acceleration Center...... 115 5.2 TheKinematicsofaTurningVehicle...... 119 5.2.1 Fixed and Moving Centrodes of a Turning Vehicle .... 119 5.2.2 Inﬂection Circle ...... 123 5.2.3VariableCurvatures...... 126 References ...... 130 6 Handling of Road Cars ...... 131 6.1 OpenDifferential...... 131 6.2 Fundamental Equations of Vehicle Handling ...... 132 6.3 Double Track Model ...... 136 6.4 Single Track Model ...... 137 6.4.1 Governing Equations of the Single Track Model ..... 138 6.4.2AxleCharacteristics...... 140 6.5 AlternativeStateVariables...... 144 6.5.1 β and ρ asStateVariables...... 145 6.5.2 β1 and β2 asStateVariables...... 147 6.5.3 S and R asStateVariables...... 149 x Contents

6.6 Inverse Congruence Equations ...... 149 6.7 Vehicle in Steady-State Conditions ...... 150 6.7.1 The Role of the Steady-State Lateral Acceleration .... 151 6.7.2 Steady-State Analysis ...... 153 6.8 Handling Diagram—The Classical Approach ...... 154 6.9 Weak Concepts in Classical Vehicle Dynamics ...... 158 6.9.1 Popular Deﬁnitions of Understeer/Oversteer ...... 159 6.10 Map of Achievable Performance (MAP)—A New Global Approach ...... 159 6.10.1 MAP Curvature ρ vs Steer Angle δ ...... 161 6.10.2 MAP: Vehicle Slip Angle β vs Curvature ρ ...... 165 6.11 Vehicle in Transient Conditions (Stability and Control Derivatives) 169 6.11.1 Steady-State Conditions (Equilibrium Points) ...... 170 6.11.2 Linearization of the Equations of Motion ...... 171 6.11.3 Stability ...... 173 6.11.4 Forced Oscillations (Driver Action) ...... 173 6.12 Relationship Between Steady State Data and Transient Behavior 175 6.13 New Understeer Gradient ...... 179 6.14 Stability (Again) ...... 180 6.15 The Single Track Model Revisited ...... 180 6.15.1 Different Vehicles with Almost Identical Handling . . . 184 6.16RoadVehicleswithLockedorLimitedSlipDifferential..... 186 6.17 Linear Single Track Model ...... 186 6.17.1GoverningEquations...... 187 6.17.2 Solution for Constant Forward Speed ...... 188 6.17.3 Critical Speed ...... 190 6.17.4TransientVehicleBehavior...... 191 6.17.5 Steady-State Behavior: Steering Pad ...... 193 6.17.6LateralWindGust...... 194 6.17.7BankedRoad...... 198 6.18CompliantSteeringSystem...... 198 6.18.1GoverningEquations...... 199 6.18.2 Effects of Compliance ...... 200 6.19Summary...... 201 6.20 List of Some Relevant Concepts ...... 201 References ...... 201 7 Handling of Race Cars ...... 203 7.1 LockedandLimitedSlipDifferentials...... 203 7.2 Fundamental Equations of Race Car Handling ...... 205 7.3 Double Track Race Car Model ...... 208 7.4 ToolsforHandlingAnalysis...... 209 7.5 The Handling Diagram Becomes the Handling Surface ..... 210 7.5.1HandlingwithLockedDifferential(noWings)...... 210 7.6 HandlingofFormulaCars...... 221 Contents xi

7.6.1HandlingSurface...... 223 7.6.2 Map of Achievable Performance (MAP) ...... 225 7.7 Summary...... 231 7.8 List of Some Relevant Concepts ...... 233 References ...... 233 8 Ride Comfort and Road Holding ...... 235 8.1 Vehicle Models for Ride and Road Holding ...... 236 8.2 Quarter Car Model ...... 239 8.2.1 The Inerter as a Spring Softener ...... 243 8.2.2 Quarter Car Natural Frequencies and Modes ...... 244 8.3 Shock Absorber Tuning ...... 247 8.3.1ComfortOptimization...... 247 8.3.2RoadHoldingOptimization...... 248 8.3.3TheInerterasaToolforRoadHoldingTuning..... 251 8.4 RoadProﬁles...... 252 8.5 FreeVibrationsofRoadCars...... 254 8.5.1GoverningEquations...... 254 8.5.2 Proportional Viscous Damping ...... 256 8.5.3 Vehicle with Proportional Viscous Damping ...... 257 8.6 Tuning of Suspension Stiffnesses ...... 262 8.6.1 Optimality of Proportional Damping ...... 263 8.6.2ANumericalExample...... 264 8.7 Non-proportional Damping ...... 265 8.8 Interconnected Suspensions ...... 265 8.9 Summary...... 268 8.10 List of Some Relevant Concepts ...... 269 References ...... 269 9 Handling with Roll Motion ...... 271 9.1 Vehicle Position and Orientation ...... 271 9.2 Yaw,PitchandRoll...... 272 9.3 Angular Velocity ...... 275 9.4 Angular Acceleration ...... 277 9.5 VehicleLateralVelocity...... 277 9.5.1TrackInvariantPoints...... 277 9.5.2VehicleInvariantPoint(VIP)...... 279 9.5.3 Lateral Velocity and Acceleration ...... 281 9.6 Three-Dimensional Vehicle Dynamics ...... 282 9.6.1 Velocity and Acceleration of G ...... 282 9.6.2 Rate of Change of the Angular Momentum ...... 284 9.6.3CompletingtheTorqueEquation...... 285 9.6.4 Equilibrium Equations ...... 285 9.6.5IncludingtheUnsprungMass...... 286 9.7 HandlingwithRollMotion...... 287 9.7.1 Equilibrium Equations ...... 287 xii Contents

9.7.2LoadTransfers...... 287 9.7.3 Constitutive (Tire) Equations ...... 288 9.7.4 Congruence (Kinematic) Equations ...... 288 9.8 Steady-State and Transient Analysis ...... 289 9.9 Summary...... 289 9.10 List of Some Relevant Concepts ...... 289 References ...... 289 10 Tire Models ...... 291 10.1 Brush Model Deﬁnition ...... 291 10.1.1 Roadway and Rim ...... 292 10.1.2 Shape of the Contact Patch ...... 292 10.1.3 Force-Couple Resultant ...... 293 10.1.4 Position of the Contact Patch ...... 294 10.1.5PressureDistribution...... 295 10.1.6Friction...... 297 10.1.7 Constitutive Relationship ...... 297 10.1.8Kinematics...... 298 10.2 General Governing Equations of the Brush Model ...... 300 10.2.1DataforNumericalExamples...... 302 10.3 Brush Model Steady-State Behavior ...... 302 10.3.1GoverningEquations...... 303 10.3.2 Adhesion and Sliding Zones ...... 303 10.3.3 Force-Couple Resultant ...... 307 10.4 Adhesion Everywhere (Linear Behavior) ...... 308 10.5 Wheel with Pure Translational Slip (σ = 0, ϕ = 0)...... 312 10.5.1 Rectangular Contact Patch ...... 317 10.5.2 Elliptical Contact Patch ...... 325 10.6 Wheel with Pure Spin Slip (σ = 0, ϕ = 0)...... 326 10.7 Wheel with Both Translational and Spin Slips ...... 328 10.7.1 Rectangular Contact Patch ...... 328 10.7.2 Elliptical Contact Patch ...... 331 10.8 Brush Model Transient Behavior ...... 334 10.8.1 Transient Model with Carcass Compliance only ..... 336 10.8.2 Transient Model with Carcass and Tread Compliance . . 338 10.8.3NumericalExamples...... 341 10.9Summary...... 344 10.10 List of Some Relevant Concepts ...... 344 References ...... 345 Index ...... 347