Tai ngay!!! Ban co the xoa dong chu nay!!! 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 © Springer Science+Business Media Dordrecht 2014 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 Exempted from this legal reservation are brief excerpts in connection with reviews or scholarly analysis or material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work Duplication of this publication or parts thereof is permitted only under the provisions of the Copyright Law of the Publisher’s location, in its current version, and permission for use must always be obtained from Springer Permissions for use may be obtained through RightsLink at the Copyright Clearance Center Violations are liable to prosecution under the respective Copyright Law 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 While the advice and information in this book are believed to be true and accurate at the date of publication, neither the authors nor the editors nor the publisher can accept any legal responsibility for any errors or omissions that may be made The publisher makes no warranty, express or implied, with respect to the material contained herein 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 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 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, 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 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 opportunity 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 carefully reviewed this book I am most grateful to them for their valuable suggestions to correct and improve the text Pisa, Italy October 2013 Massimo Guiggiani Contents Introduction 1.1 Vehicle Definition 1.2 Vehicle Basic Scheme References Mechanics of the Wheel with Tire 2.1 The Tire as a Vehicle Component 2.2 Rim Position and Motion 2.3 Carcass Features 2.4 Contact Patch 2.5 Footprint Force 2.5.1 Perfectly Flat Road Surface 2.6 Tire Global Mechanical Behavior 2.6.1 Tire Transient Behavior 2.6.2 Tire Steady-State Behavior 2.6.3 Rolling Resistance 2.6.4 Speed Independence (Almost) 2.6.5 Pure Rolling (not Free Rolling) 2.7 Tire Slips 2.7.1 Rolling Velocity 2.7.2 Definition of Tire Slips 2.7.3 Slip Angle 2.8 Grip Forces and Tire Slips 2.9 Tire Testing 2.9.1 Pure Longitudinal Slip 2.9.2 Pure Lateral Slip 2.10 Magic Formula 2.11 Mechanics of Wheels with Tire 2.12 Summary 2.13 List of Some Relevant Concepts References 12 13 14 16 17 17 18 20 21 21 26 27 27 30 31 33 34 35 38 39 43 44 44 vii viii Contents Vehicle Model for Handling and Performance 3.1 Mathematical Framework 3.2 Vehicle Congruence (Kinematic) Equations 3.2.1 Velocities 3.2.2 Yaw Angle and Trajectory 3.2.3 Velocity Center 3.2.4 Fundamental Ratios 3.2.5 Accelerations and Radii of Curvature 3.2.6 Acceleration Center 3.2.7 Tire Kinematics (Tire Slips) 3.3 Vehicle Constitutive (Tire) Equations 3.4 Vehicle Equilibrium Equations 3.5 Forces Acting on the Vehicle 3.5.1 Weight 3.5.2 Aerodynamic Force 3.5.3 Road-Tire Friction Forces 3.5.4 Road-Tire Vertical Forces 3.6 Vehicle Equilibrium Equations (more Explicit Form) 3.7 Load Transfers 3.7.1 Longitudinal Load Transfer 3.7.2 Lateral Load Transfers 3.7.3 Vertical Loads on Each Tire 3.8 Suspension First-Order Analysis 3.8.1 Suspension Reference Configuration 3.8.2 Suspension Internal Coordinates 3.8.3 Camber Variation 3.8.4 Vehicle Internal Coordinates 3.8.5 Roll and Vertical Stiffnesses 3.8.6 Suspension Internal Equilibrium 3.8.7 Effects of a Lateral Force 3.8.8 No-roll Centers and No-roll Axis 3.8.9 Forces at the No-roll Centers 3.8.10 Suspension Jacking 3.8.11 Roll Angle and Lateral Load Transfers 3.8.12 Explicit Expressions of Lateral Load Transfers 3.8.13 Lateral Load Transfers with Rigid Tires 3.9 Dependent Suspensions 3.10 Sprung and Unsprung Masses 3.11 Vehicle Model for Handling and Performance 3.11.1 Equilibrium Equations 3.11.2 Constitutive (Tire) Equations 3.11.3 Congruence (Kinematic) Equations 3.11.4 Principles of Any Differential Mechanism 3.12 The Structure of This Vehicle Model 3.13 Three-Axle Vehicles 47 48 48 48 49 51 52 53 54 56 58 59 59 60 60 61 63 63 65 65 66 66 67 67 68 69 70 71 73 74 75 77 78 79 81 82 82 85 86 86 88 88 90 94 95 Contents ix 3.14 Summary 3.15 List of Some Relevant Concepts References Braking Performance 4.1 Pure Braking 4.2 Vehicle Model for Braking Performance 4.3 Equilibrium Equations 4.4 Longitudinal Load Transfer 4.5 Maximum Deceleration 4.6 Brake Balance 4.7 All Possible Braking Combinations 4.8 Changing the Grip 4.9 Changing the Weight Distribution 4.10 A Numerical Example 4.11 Braking Performance of Formula Cars 4.11.1 Equilibrium Equations 4.11.2 Longitudinal Load Transfer 4.11.3 Maximum Deceleration 4.11.4 Braking Balance 4.11.5 Typical Formula Braking Performance 4.12 Summary 4.13 List of Some Relevant Concepts References The Kinematics of Cornering 5.1 Planar Kinematics of a Rigid Body 5.1.1 Velocity Field and Velocity Center 5.1.2 Acceleration Field, Inflection Circle and Acceleration Center 5.2 The Kinematics of a Turning Vehicle 5.2.1 Fixed and Moving Centrodes of a Turning Vehicle 5.2.2 Inflection Circle 5.2.3 Variable Curvatures References Handling of Road Cars 6.1 Open Differential 6.2 Fundamental Equations of Vehicle Handling 6.3 Double Track Model 6.4 Single Track Model 6.4.1 Governing Equations of the Single Track Model 6.4.2 Axle Characteristics 6.5 Alternative State Variables 6.5.1 β and ρ as State Variables 6.5.2 β1 and β2 as State Variables 6.5.3 S and R as State Variables 97 97 98 99 99 100 101 101 102 103 103 105 106 106 107 107 108 108 109 109 109 110 111 113 113 113 115 119 119 123 126 130 131 131 132 136 137 138 140 144 145 147 149 x Contents 6.6 6.7 Inverse Congruence Equations Vehicle in Steady-State Conditions 6.7.1 The Role of the Steady-State Lateral Acceleration 6.7.2 Steady-State Analysis 6.8 Handling Diagram—The Classical Approach 6.9 Weak Concepts in Classical Vehicle Dynamics 6.9.1 Popular Definitions of Understeer/Oversteer 6.10 Map of Achievable Performance (MAP)—A New Global Approach 6.10.1 MAP Curvature ρ vs Steer Angle δ 6.10.2 MAP: Vehicle Slip Angle β vs Curvature ρ 6.11 Vehicle in Transient Conditions (Stability and Control Derivatives) 6.11.1 Steady-State Conditions (Equilibrium Points) 6.11.2 Linearization of the Equations of Motion 6.11.3 Stability 6.11.4 Forced Oscillations (Driver Action) 6.12 Relationship Between Steady State Data and Transient Behavior 6.13 New Understeer Gradient 6.14 Stability (Again) 6.15 The Single Track Model Revisited 6.15.1 Different Vehicles with Almost Identical Handling 6.16 Road Vehicles with Locked or Limited Slip Differential 6.17 Linear Single Track Model 6.17.1 Governing Equations 6.17.2 Solution for Constant Forward Speed 6.17.3 Critical Speed 6.17.4 Transient Vehicle Behavior 6.17.5 Steady-State Behavior: Steering Pad 6.17.6 Lateral Wind Gust 6.17.7 Banked Road 6.18 Compliant Steering System 6.18.1 Governing Equations 6.18.2 Effects of Compliance 6.19 Summary 6.20 List of Some Relevant Concepts References 159 161 165 169 170 171 173 173 175 179 180 180 184 186 186 187 188 190 191 193 194 198 198 199 200 201 201 201 Handling of Race Cars 7.1 Locked and Limited Slip Differentials 7.2 Fundamental Equations of Race Car Handling 7.3 Double Track Race Car Model 7.4 Tools for Handling Analysis 7.5 The Handling Diagram Becomes the Handling Surface 7.5.1 Handling with Locked Differential (no Wings) 7.6 Handling of Formula Cars 203 203 205 208 209 210 210 221 149 150 151 153 154 158 159 10.8 Brush Model Transient Behavior 341 precisely, the larger any of the ratios θx = wx , Kt θy = wy Kt (10.129) where Kt = 4abk is the tread stiffness, the more relevant the effect of the bristle deflection in that direction Since wx wy , the transient behavior in the bristle deflection pattern has more influence when the wheel is subject to time-varying longitudinal slip For instance, with the data reported at p 302, we have θx = and θy = 0.25 10.8.3 Numerical Examples The proposed models for the transient behavior of tires are compared on a few numerical tests The goal is to show the range of applicability and to warn about employing a model without really understanding its capabilities In particular, three models of increasing complexity are compared All tests are performed with the data listed in (10.43), except for χ = 0, and under either pure longitudinal slip or pure lateral slip Moreover, a brush model with rectangular contact patch and parabolic pressure distribution is assumed The first model (semi-nonlinear single contact point) takes into account only the carcass compliance and employs a constant relaxation length si , with i = x, y According to (10.115), the model is defined by si ρ˙i + Vr ρi = Vr σi (10.130) ρi (0) = where si = Cσ /wi , with Cσ = 4ka b as in (10.74) Once the function ρi (t) has been obtained, the global tangential force is given by the nonlinear function |ρi | ρi Fi (ρi ) = −Cσ ρi − (10.131) + σs σs much like in (10.89) with (10.95) The second model (nonlinear single contact point) is similar, but employs a nonconstant relaxation length, as in (10.112) ⎧ ⎪ ⎨ − Fi (ρi ) ρ˙i + Vr ρi = Vr σi wi (10.132) ⎪ ⎩ ρi (0) = where (cf (10.96) with χ = 0) Fi (ρi ) = −Cσ 2 |ρi | ρi 1−2 + σs σs (10.133) 342 10 Tire Models A numerical solution is usually required Again, the function ρi (t) is then inserted into (10.131) The third model (nonlinear full contact patch) takes into account both the carcass and tread compliances, as in (10.128) ⎧ Vr ei,xˆ − ei,t = Vr ρi ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ k ei xˆs (t), t = μp xˆs (t) ⎪ ⎪ ⎪ ⎪ ⎨ wi σi (t) − 2bkei (xˆs (t), t) (10.134) ρi (t) = ⎪ wi + 2bk(a − xˆs (t)) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ei (a, t) = ⎪ ⎪ ⎪ ⎩ ei (x, ˆ 0) = To obtain a numerical solution, an iterative method can be employed First make an initial guess for ρi(0) (t) (for instance ρi(0) (t) = (σi (t) + ρis (t))/2, where ρis (t) is the (0) ˆ t), solution of (10.130)) By means of the first equation, numerically obtain ex (x, (0) and then, using the second equation, evaluate the function xˆs At this stage, the (1) first iteration can be completed by computing ρi (t) by means of the third equation The whole procedure has to be repeated (usually to 15 times) until convergence is attained ˆ t) and xˆs (t) (and also of ρi (t)) has been Once a good approximation of ei (x, computed, the tangential force can be obtained from the following integral over the contact patch Fi (t) = 2bk a xˆs (t) ei (x, ˆ t)dxˆ + μ sign ei xˆs (t), t xˆs (t) −a p(x)d ˆ xˆ (10.135) A step change in the input (forcing) function σi (t) works well to highlight the differences between the three models With the data of (10.43), except χ = 0, the static tangential force (10.131) has maximum magnitude for σ = 0.266 To test the models in both the (almost) linear and nonlinear ranges, a small (σi = −0.07) and a large (σi = −0.21) step have been selected Since wx = 4wy , both longitudinal and lateral numerical tests are performed In all cases, results are plotted versus the rolling distance s, instead of time, thus making Vr (t) irrelevant 10.8.3.1 Longitudinal Step Input The longitudinal force Fx (s), as obtained from the three tire models with step inputs σx = −0.07 and σx = −0.21, is shown in Fig 10.44 Because of the high value of the longitudinal carcass stiffness wx (equal to the tread stiffness Kt ), the transient phenomenon is quite fast Indeed, in the first model (dashed line) the relaxation length sx = 7.5 cm 10.8 Brush Model Transient Behavior 343 Fig 10.44 Longitudinal force response to small and large step changes in σx Comparison of three tire models: semi-nonlinear single contact point (dashed line), nonlinear single contact point (dot-dashed line), nonlinear full contact patch (solid line) Fig 10.45 Transient patterns of the tangential stress tx in the contact patch (third model) Quite remarkably, the three models provide very different results, thus showing that the selection of the transient tire model may be a crucial aspect in vehicle dynamics, particularly when considering vehicles equipped with ABS The behavior of the first model (dashed lines) is the same in both cases, except for a vertical scaling This is not the case for the second model (dot-dashed lines) because of the nonconstant generalized relaxation length The more detailed third model (solid lines) behaves in quite a peculiar way, thus confirming that the contribution of the transient tread deflection is far from negligible Figure 10.45 shows the transient pattern of the tangential longitudinal stress tx in the contact patch as provided by the third model with σx = −0.21 It is worth noting how greatly, in the adhesion region, the pattern departs from the linear behavior of the static case (Fig 10.10) 10.8.3.2 Lateral Step Input The lateral force Fy (s), as obtained from the three tire models with step inputs σy = −0.07 and σy = −0.21, is shown in Fig 10.46 Because of the low value of the lateral carcass stiffness wy (equal to one fourth of the tread stiffness Kt ), the transient phenomenon is not as fast as in the longitudinal case Indeed, in the first model the relaxation length sy = 30 cm In this case, the three models provide not very different results in the linear range, while they depart significantly in the nonlinear range, that is with σy = −0.21 Therefore, the selection of the transient tire model may be crucial in lateral dynamics as well 344 10 Tire Models Fig 10.46 Lateral force response to small and large step changes in σy Comparison of three tire models: semi-nonlinear single contact point (dashed line), nonlinear single contact point (dot-dashed line), nonlinear full contact patch (solid line) Fig 10.47 Transient patterns of the tangential stress ty in the contact patch (third model) It should be observed from Figs 10.44 and 10.46 that the first and second models have the same “formal” behavior Therefore, changing the carcass stiffness results only in a horizontal scaling This is not true for the third model Figure 10.47 shows the transient pattern of the tangential lateral stress in the contact patch as provided by the third model with σy = −0.21 There are still differences with respect to the static case, although not as much as in Fig 10.45 10.9 Summary In this chapter a relatively simple, yet significant, tire model has been developed It is basically a brush model, but with some noteworthy additions with respect to more common formulations For instance, the model takes care of the transient phenomena that occur in the contact patch A number of figures show the pattern of the local actions within the contact patch (rectangular and elliptical) 10.10 List of Some Relevant Concepts p 299 the skating slip takes into account both transient translational slip and spin slip; p 304 each bristle is undeformed when it enters the contact patch; References 345 p 312 the analysis of the steady-state behavior of the brush model is quite simple if there is no spin slip; p 318 full sliding does not imply wheel locking; p 322 the slip angle α is not a good parameter for a neat description of tire mechanics; p 323 tires have to be built in such a way to provide the maximum tangential force in any direction with small slip angles This is a fundamental requirement for a wheel with tire to have a directional capability; p 324 the tire action surface summarizes the tire characteristics under a constant vertical load; p 326 the tire action surface summarizes the steady-state behavior of a tire; p 332 good wheel directional capability means small slip angles References Clark SK (ed) (1971) Mechanics of pneumatic tires National Bureau of Standards, Washington Deur J, Asgari J, Hrovat D (2004) A 3D brush-type dynamic tire friction model Veh Syst Dyn 42:133–173 Deur J, Ivanovic V, Troulis M, Miano C, Hrovat D, Asgari J (2005) Extension of the LuGre tyre friction model related to variable slip speed along the contact patch length Veh Syst Dyn 43(Supplement):508–524 Hüsemann T (2007) Tire technology: simulation and testing Tech rep, Institut für Kraftfahrwesen, Aachen http://www.ika.rwth-aachen.de/lehre/kfz-labor/2_tires_en.pdf Kreyszig E (1999) Advanced engineering mathematics, 8th edn Wiley, New York Lugner P, Pacejka H, Plöchl M (2005) Recent advances in tyre models and testing procedures Veh Syst Dyn 43:413–436 Milliken WF, Milliken DL (1995) Race car vehicle dynamics SAE International, Warrendale Pacejka HB (2002) Tyre and vehicle dynamics Butterworth–Heinemann, Oxford Pacejka HB, Sharp RS (1991) Shear force development by pneumatic tyres in steady state conditions: a review of modelling aspects Veh Syst Dyn 20:121–176 10 Savaresi SM, Tanelli M (2010) Active braking control systems design for vehicles Springer, London 11 Truesdell C, Rajagopal KR (2000) An introduction to the mechanics of fluids Birkhäuser, Boston 12 Zwillinger D (ed) (1996) CRC standard mathematical tables and formulae, 30th edn CRC Press, Boca Raton Index Mathematical Symbols A, 27 A, 171 A1 , 119 A2 , 119 a, 117, 293 a1 , 4, 39 a1b , 65, 74 a2 , 4, 39 a2b , 65, 74 a3 , 39 a4 , 39 aij , 174 an , 54 at , 54 ax , 53 ay , 53 aG , 53 aK , 54 a˜ y , 53, 150 b, 4, 239, 293 B, 38 b1 , 237 b2 , 237 bi , 174 C, 8, 23, 38, 51, 114, 294 c, 239 C1 , 184, 186 c1 , 236 C2 , 184, 186 c2 , 236 Cγ , 311 cφ , 287 cφ1 , 287 cφ2 , 287 cθ , 287 Ci , 186 CMσ , 311 CMϕ , 311 cr , 23 Cx , 60 Cy , 60 Cz , 60 Cz1 , 61 Cz2 , 61 C , 12 ˆ 115 C, C, 238 C˜ , 200 Cα , 35 α , 321 C Cσ , 311 σ , 320 C Cϕ , 311 copt , 248 Cκx , 34 κx , 321 C D, 38, 292 d, 116, 125, 275 d1 , 258 d2 , 258 Di , 207 dr , 27 dt , 15 D, 123 E, 38 e, 117, 195 E1 , 126 E2 , 126 ex , 15, 297 ey , 15, 297 e, 292, 297 e˙ , 299 M Guiggiani, The Science of Vehicle Dynamics, DOI 10.1007/978-94-017-8533-4, © Springer Science+Business Media Dordrecht 2014 347 348 e , 303 e0 , 301 ea , 303 ef , 306 e,s , 302 es , 303 e˜ s , 306 eˆ s , 307 e,t , 300 e,xˆ , 300 f , 22, 117 f1 , 245 f2 , 245 fβ , 154 Fβ , 174 fρ , 154 Fρ , 174 Fa , Fi , 338 Fl , 195 fr , 140 Ft , 315 fv , 140 Fx , 14 Fxn , 307 p Fx , 33 Fy , 14 Fy1 , 138 Fy2 , 138 Fyi , 140 Fyn , 307 p Fy , 33 Fz , 14 F, 14 F, 18 Fp , 15 Fp , 18 Fp , 21 Ft , 15 Ft , 18 Ft , 21 Fnt , 307 Ftmax , 319 G, g, 60, 252 Gs , 236 h, 4, H , 239 h, 239 Hi , 82 i, 9, 48, 49 i0 , 49 Index Jy , 236 Jyz , 59 Jz , 59 Jzx , 59 j, 9, 48, 49 jc , 10 j0 , 49 J˜zx , 286 J˜z , 286 K, 52, 116, 154, 158 k, 239, 298 k1 , 236 k2 , 236 Kβδ , 153, 176, 209 Kβy , 153, 176, 182, 209 kφ , 80 p kφ2 , 72 kφ1 , 72, 80 p kφ1 , 72 kφs , 72 kφ2 , 72, 80 kφs , 72 Kρy , 153, 176, 182, 209 Kρδ , 153, 176, 209 kθ , 287 kθz , 266 kθθ , 266 kb , 259 kp , 259 ks1 , 199 Kt , 336 kz1 , 73 p kz1 , 73 kzs1 , 73 kz2 , 73 p kz2 , 73 kzs2 , 73 kzθ , 266 kzz , 266 k, 9, 48, 49 K, 239 k0 , 49 l, L, 59, 240 Lb , 78 M, 59, 281 M1 , 277 M2 , 278 mb , 259 Mf , 91 Mh , 90 mui , 86 Index Ml , 90 mn , 239 mn1 , 236 mn2 , 236 mp , 259 MpO , 16 Mr , 90 ms , 85, 236 Ms , 91 MtO , 17 mu , 85 Mx , 14 My , 14 Mz , 14 p Mz , 33 M, 238 m, 305 MO , 14 O , 18 M O M p , 18 MO p , 21 O M t , 18 O M t , 21 N , 59, 126 N0 , 171 Nβ , 171 Nδ , 173 Nρ , 171 Nd , 206 ni , 178 Np , 195 Nu , 173 NX , 64 NY , 64 n, 53 O, p, 16, 239, 275, 295 p0 , 295 p1 , 236 p2 , 236 pa , P , 13 Q, 9, 77 q, 275 Q1 , 75 Q2 , 75 qi , 75 Qi , 75 qx , 294 qy , 294 Qz , 278 q b , 78 349 q, 294 q , 302 r, 49, 275 R, 51 r0 , 25 RG , 54 ri , 57 rp , 150 rr , 23, 85 R, Ri , 275 S, 51 s, 298 Sa , 60 si , 178, 338 sr , 25 sx , 337 sxx , 337 sxy , 337 sy , 337 syx , 337 syy , 337 S, 10, 48 S, 259 ˆ , 292 S s, 305 S0 , 49 sϕ , 337 Sf , 10 Si , 273 T , 15 t1 , t2 , ta , 314 tc , 311 tc1 , 199 ts , 314 ts1 , 199 tx , 16 ty , 16 T, 15 t, 16 tˆ, 50 t, 53 ta , 307 ts , 307 u, 49 uβ , 194 uchar , 179 ucr , 180 ua , 170 ut , 173, 192 v, 49 Vμ , 297 350 Vcx , 27 Vcy , 27 Vi , 82 vp , 150 Vr , 27 Vμ , 297, 300 Va , 60 Vc , 27, 299 Vd , 294, 298 VG , 48 Vo , 11 Vr , 27 Vs , 28, 29 Vt , 299 Vox , 11 Voy , 11 W , 60 Wd , 90 Wi , 91 wi , 336 Wo , 91 wx , 295 wy , 295 W, 294 wo , 188, 244 wp , 188 x, 4, 48 X, 59 x0 , 49 x0G , 50 X1 , 61 x1 , 96 X10 , 105 X2 , 61 x2 , 96 X20 , 105 x3 , 96 Xa , 60 xf , 10, 123 Xij , 61 xm , 38 xN , 65 x, 244 x, ˆ 292 xˆ0 , 292 ΔX1 , 61 ΔX2 , 61 xˆb , 299 xˆs , 305 y, 48 Y , 59, 74, 242 y0 , 49 Y0 , 171 Index y0G , 50 Y1 , 61, 138 Y2 , 61, 138 Yβ , 171 Yδ , 173 Yρ , 171 ya , 38 Ya , 60 yf , 10, 123 Yiu , 85 Yij , 61 ym , 38 Yu , 173 y, ˆ 292 ΔY1 , 61 Yˆ1 , 152 ΔY2 , 61 Yˆ2 , 152 z, 48 Z, 59, 126, 242 z0 , 49 Z1 , 63 Z10 , 65 Z1a , 60 Z1b , 73 Z2 , 63 Z20 , 65 Z2a , 60 Z2b , 73 Z30 , 96 Za , 60 zb , 259 zf , 10 p zi , 71 zis , 71 Zij , 63 zp , 259 ΔZ, 65 zˆ , 292 ΔZ1 , 63 zˆ , 73 ΔZ2 , 63 zˆ , 73 ΔZiY , 78 ΔZiu , 85 ΔZiL , 78 ΔZi , 78 α, 30, 256 α1 , 137, 149 α˜ , 200 α2 , 137, 149 Index αij , 56 β, 52, 146, 256 ˆ 52 β, β1 , 147 β2 , 147 βd , 150 βˆij , 56 βij , 56 βP , 103 βp , 153 βs , 150 βt , 171 γ , 9, 58 γi10 , 57 δij0 , 89 γi20 , 57 δv , 4, 89 δ, 157, 160 δ11 , 47 δ1 , 139, 149, 160 δ12 , 47 δ2 , 139, 149, 160 δij , 47 δva , 170 δvt , 173 ε, 200 1 , 105 ε, 299 i , 57 εr , 25 ζ , 10 ζ1 , 108 ζ2 , 108 η, 260 η1 , 133, 206 η2 , 133, 206 ηh , 92 θ , 10, 273 θx , 341 θy , 341 κ, 30 κ, 29 κx , 29 κy , 29 λ1 , 172 λ2 , 172 μ, 101, 244, 297 μ0 , 297 μ1 , 297 μxp , 34, 101 μp , 39, 87 y μp , 35 351 ν1 , 206 ν2 , 206 ξ , 55, 108, 115 ξij , 81 φ, 70, 75, 80, 273 φis , 75 p φi , 75 ϕ, 28, 58, 93, 206 ϕt , 29 ρ, 52, 146, 260, 336 ρ, 299 p ρ1 , 133 ρ1s , 133 ρ , 301 ρ2s , 133 p ρ2 , 133 ρG , 54 ρi , 130, 341 ρis , 342 ρp , 153 ρt , 171 ρx , 336 ρy , 336 σ , 313 σ , 29 σf , 114 σi , 338 σm , 114 σp , 319 σs , 317 σx , 28, 58 σy , 28, 58 ς , 93, 206 τ1 , 137, 198 τ2 , 137, 198 τij , 89 χˆ , 160 χ , 181, 297 χi , 83, 135 ψ , 50, 273 ωc , 246 ωdi , 244 ωc , 10 ωf , 91 ωˆ f , 91 ωh , 90 ωij , 57 ωˆ l , 90 ωl , 90 ωˆ r , 90 ωr , 90 ωˆ s , 91 ωs , 91 352 ωz , 10 Φ1 , 181 Φ2 , 181 Υij , 89, 134 , 9, 48, 275 Ω, 241 Ωr , 27 Ωsz , 28 Ωz , 11 A ABS, 324, 334, 343 Acceleration angular, 53 center, 52 lateral, 53 steady-state, 53, 150, 151 longitudinal, 53 of the velocity center, 52 Acceleration center, 54, 116 Achievable region, 160, 161, 170 Ackermann angle, 52 Adhesion, 297, 300, 304, 313, 314, 339 Aerodynamic downforce, 60, 151, 210, 225, 227, 228 Aerodynamic force, 60 Aerodynamic moment, 60 Alternative state variable, 144 Angular momentum, 59 Angular velocity, 275 Anti-roll bar, 268 Apparent slip angle, 140, 152, 154, 210 Assumptions, 4, 47 Autocorrelation function, 252 Axle characteristic, 138, 140, 152, 181, 209 Axle lateral slip stiffness, 186 B Bicycle model, 136 Body roll angle, 68 Body vertical displacement, 68 Bounce, 258, 264 Brake balance, 103 bias, 103 Brake balance, 99, 103 Brake bias, 99 Braking, 100 best performance, 103 efficiency, 105 of Formula car, 107 Braking efficiency, 105 Bristle stiffness, 298 Index Brush model, 23, 291, 297, 303 transient, 301 C Camber, 57, 100, 142, 308 Camber angle, 9–11, 32, 57, 90, 135, 327 variation, 69 Camber force, 311, 328, 330 Camber reduction factor, 25, 28, 90 Camber stiffness, 311 Camber variation, 135, 143 Carcass, 12 compliance, 23, 294, 308, 323 Carcass stiffness, 336, 338 Caster angle, 47 Center of zero acceleration, 55 of zero velocity, 51 Center of acceleration, 116 Center of curvature, 117 Center of gravity, Center of velocity, 114 Centrode fixed, 114 moving, 114 Characteristic speed, 179 Coefficient of kinetic friction, 297 Coefficient of static friction, 297 Comfort, 235, 247 Compliant steering system, 198 Congruence equations, 48, 136, 237, 254 Constant steering wheel test, 179 Constitutive equations, 48, 238, 255 Constitutive relation, 297 Contact patch, 7, 8, 13, 14, 17, 19, 20, 30, 33, 292, 312 Contractive suspensions, 72 Control derivatives, 170, 173, 175, 177, 181 Convergence, 100 Cornering force, 311 Cornering stiffness, 35 generalized, 321 Critical speed, 157, 180, 190 Curvature, 54 radius, 54 Curvature factor, 38 D Damping coefficient, 236, 287 factor, 244 matrix, 238, 241 ratio, 244 Deflection, 297 Index Dependent suspension, 82 Differential, 90 limited slip, 93 locked, 93 open, 93, 132 Directional capability, 324, 332 Double track model, 136 Drag coefficient, 60 Driveable road vehicle, Dynamic index, 260 E Elemental rotation, 272 Empirical tire models, 38 Equilibrium equations, 48, 238, 255 global, 63 Ergodic process, 252 Euler, 113 Euler angles, 273 Evolute, 130 F Footprint, see contact patch, Force-couple system, 14 Formula 1, 109 Forward velocity, 27, 49, 281 Fourier transform, 252 Free rolling, 22, 328 Frequency spectrum, 252 Friction circle, 321 Friction coefficient global, 34 local, 297 Frontal area, 60 G Global friction coefficient, 39, 319 lateral, 35 longitudinal, 34 Global longitudinal friction coefficient, 101 Gough plot, 324 Gradient, 176, 179 Grip, 13, 152 coefficient, 105 H Handling, curve, 156, 211 diagram, 157, 210, 225 map, 160 surface, 210, 211, 223, 225 Hop, 241 353 I Inclination angle, see camber angle Inertance, 237 Inerter, 235, 243, 251 Inertia tensor, 59 Inflection circle, 55, 115, 123 Instantaneous center of rotation, 114 Instantaneous center of zero acceleration, 116 Instantaneous center of zero velocity, 51 Interconnected suspension, 267 Internal efficiency, 92 Invariant, 281 Invariant point, 289 J J-Damper, 243 Jacking, 79, 278 K Kingpin inclination angle, 47 L Lateral acceleration, 53, 150 steady-state, 153 Lateral force, 14, 24, 35, 59, 315, 328, 331 Lateral load transfer, 66, 288 Lateral slip stiffness, 35 Lateral velocity, 24, 27, 49, 281 Leading edge, 292, 298, 301 Limited slip differential, 93 Line of nodes, 273 Load transfer lateral, 66, 79, 288 longitudinal, 65, 102, 288 Local friction coefficient, 297 Locked wheel, 30 Longitudinal acceleration, 53 Longitudinal force, 14, 22, 34, 59, 315 Longitudinal load transfer, 65, 102, 288 Longitudinal slip stiffness, 34 M MacPherson strut, 75 Macroroughness, 13 Magic Formula, 38, 320 Magic numbers, 178 MAP, 160, 170, 210, 225, 228, 230, 231 Map of Achievable Performance, 160, 225 Mass sprung, 85, 282 unsprung, 85, 286 Mass matrix, 238, 241 Maximum deceleration, 102, 108 Microroughness, 13 354 Moment pitching, 59 rolling, 59 yawing, 59 Motion center, 258 N Net steer angle, 160 Neutral steer point, 195 Nitrogen, No-roll axis, 77, 95 center, 76, 83 triangle, 97 No-roll axis, 271 No-roll center, 82, 277 No-roll center for a MacPherson strut, 76 Node, 258 Normal force, 14 Normalized lateral force, 307, 308, 327, 328 longitudinal force, 307, 308 Normalized axle characteristics, 152 Nose-in, 151 Nose-out, 151 O Open differential, 93, 132, 133 Optimal damping, 246 Optimal damping coefficient, 248 Oscillation center, 258 Oversteer, 154, 164 Oversteer behavior, 157 Overturning, 102, 106 Overturning moment, 14 P Panhard rod, 82 Parallel steering, 210 Peak value, 38, 319 Performance, map of achievable, 225 Pitch, 236, 258, 264, 272, 287 angle, 272 Pitching moment, 59 Pneumatic tire, Pneumatic trail, 199, 311, 312, 317, 323 Power spectral density, 253 Power-off, 92, 217, 231 Power-on, 92, 217, 231 Practical slip, 29, 33, 311 Pressure distribution, 295 Principal coordinates, 261 Proportional damping, 256 Index Pure rolling, 8, 21, 22, 26, 27, 58, 132, 292, 294, 299 Pure rolling radius, 23 Pure slip conditions, 33 Q Quarter car model, 239, 253 R Radius of curvature, 54, 117 Random process, 252 Rear steering, 160, 168 Reference configuration, 47 Relaxation length, 337 Ride, 3, 235 Rigid body, 59, 113 Rigid tire, 82 Rim kinematics, 11 position, 11 Rim angular velocity, 10 Road holding, 235, 247 Road profile, 237, 252 Roll, 272 angle, 70, 79, 272 axis, 75, 95 center, 75 stiffness, 71 Roll angle, 75, 80, 87 body, 68 Roll axis, 75, 271, 278 Roll center, 68, 75 Roll steer, 89, 134, 140, 186 Rolling free, 21 pure, 21 Rolling distance, 298, 301, 342 Rolling moment, 59 Rolling radius, 23, 25, 28, 57 Rolling resistance, 20, 22, 63 Rolling resistance coefficient, 22 Rolling resistance moment, 14 Rolling spin velocity, 27 Rolling velocity, 27, 30, 57, 132, 298, 301 Rotation elemental, 272 Roughness, 253 S Scrub radius, 47 Self-aligning torque, see vertical moment, 14 Shape factor, 38, 39 Shifted coordinates, 177 Single track model, 131, 136, 158, 181, 199 Index Skating slip, 299, 302, 305 Skating velocity, 299 Sliding, 297, 305, 314, 318, 325, 339 Sliding velocity, 297, 300 Slip, 33 practical, 29 skating, 299 spin, 28, 299 theoretical, 28, 89 translational, 28, 89, 299 transient, 299, 337 turn, 29 Slip angle, 30, 33, 56, 321, 332 apparent, 140, 210 Slip functions, 154 Slip ratio, 30 Slip spin velocity, 28, 298, 299 Slip stiffness, 311, 312, 321 generalized, 320 Slip velocity, 28 Slowly increasing steer, 209 Speed of travel, 27, 30, 299 Spin slip, 29, 56, 89, 90, 326 Spin stiffness, 311 Spin velocity, 11, 25 Spinning wheel, 30 Sprung mass, 4, 85, 236, 282 Stability boundary, 164 Stability derivatives, 170, 171, 175, 177, 181 Static condition, Static margin, 195 Steady-state conditions, 53 Steer angle, 160 step input, 33 Steering angle, 47 Steering axis, 4, 15, 47 Step steering input, 33, 177, 184 Stick region, see adhesion Stiffness, 236 roll, 72 vertical, 72 Stiffness factor, 38 Stiffness matrix, 239, 241 Suspension deflection, 47 dependent, 3, 82 double wishbone, 6, 74 first order analysis, 67 independent, interconnected, 267 internal coordinates, 67 jacking, 79 355 reference configuration, 67 swing arm, 6, 74 Suspension deflection, Suspension jacking, 79 Swing axle suspension, 68 T Tangent speed, 194 Tangential force, 293, 307 normalized, 307 TBR, 92, 204 Theoretical lateral slip, 29 Theoretical longitudinal slip, 29 Theoretical slip, 29 Three-axle vehicle, 95 Tire action surface, 324 lateral slip, 134 mechanics, 39 rigid, 82 slips, 26, 27, 56 steady-state behavior, 18 stiffness, 67 testing, 33 transient behavior, 17 vertical stiffness, 236 Toe-in, 100, 134, 142 Toe-out, 142 Torque with respect to wheel axis, 15, 20, 328 Torque Bias Ratio, 92 Track, Track invariant point, 278 Track variation, 88 Trail, 47 Trailing edge, 292 Trajectory, 50, 117 Translational slip, 56 Transport equation, 339 Tread, 12 Tread pattern, 14 Tread stiffness, 336, 338, 341 Trim conditions, 170 Truck, 95 Turn slip, 29 U Understeer, 154, 161 Understeer behavior, 157 Understeer gradient, 154, 179 new, 179 Unsprung mass, 4, 85, 236, 286 356 V Vehicle internal coordinates, 70 Vehicle definition, Vehicle invariant point, 281 Vehicle slip angle, 52, 147 Velocity center, 51, 114 Vertical force, 59 Vertical load, 14, 19 Vertical moment, 14, 15, 24, 37, 293, 308, 312, 323, 324, 332 VIP, 281 W Weak concept, 95, 158, 179, 180 Weight, 60 Wheelbase, 4, 52, 153, 158 Index Willis formula, 91 Wrench, 15 Y Yaw, 272 angle, 50, 272 rate, 49 Yaw rate, 11, 49, 311 of the wheel, 10, 32 Yawing moment, 59, 64, 176, 206, 214 y(x), 38 Z Zero lateral force, 24 longitudinal force, 23 vertical moment, 24