Multibody Systems Approach to Vehicle Dynamics P1

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Multibody Systems Approach to Vehicle Dynamics
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Multibody Systems Approach to Vehicle Dynamics Michael Blundell Damian Harty AMSTERDAM BOSTON HEIDELBERG LONDON NEW YORK OXFORD PARIS SAN DIEGO SAN FRANCISCO SINGAPORE SYDNEY TOKYO
Elsevier Butterworth-Heinemann Linacre House, Jordan Hill, Oxford OX2 8DP 200 Wheeler Road, Burlington, MA 01803 First published 2004 Copyright © 2004, Michael Blundell and Damian Harty. All rights reserved The right of Michael Blundell and Damian Harty to be identified as the authors of this work has been asserted in accordance with the Copyright, Designs and Patents Act 1988 No part of this publication may be reproduced in any material form (including photocopying or storing in any medium by electronic means and whether or not transiently or incidentally to some other use of this publication) without the written permission of the copyright holder except in accordance with the provisions of the Copyright, Designs and Patents Act 1988 or under the terms of a licence issued by the Copyright Licensing Agency Ltd, 90 Tottenham Court Road, London, England W1T 4LP. Applications for the copyright holder’s written permission to reproduce any part of this publication should be addressed to the publisher. Permissions may be sought directly from Elsevier’s Science and Technology Rights Department in Oxford, UK: phone: ( 44) (0) 1865 843830; fax: ( 44) (0) 1865 853333; e-mail: permissions@elsevier.co.uk. You may also complete your request on-line via the Elsevier Science homepage (http://www.elsevier.com), by selecting ‘Customer Support’ and then ‘Obtaining Permissions’. British Library Cataloguing in Publication Data A catalogue record for this book is available from the British Library ISBN 0 7506 5112 1 For information on all Elsevier Butterworth-Heinemann publications visit our website at http://books.elsevier.com Composition by Charon Tec Pvt. Ltd, Chennai, India Printed and bound in Maple-Vail, Kirkwood, New York, USA
Contents Preface xi Acknowledgements xv Nomenclature xvi 1 Introduction 1 1.1 Overview 1 1.2 What is vehicle dynamics? 3 1.3 Why analyse? 10 1.4 Classical methods 10 1.5 Analytical process 11 1.6 Computational methods 14 1.7 Computer-based tools 14 1.8 Commercial computer packages 17 1.9 Benchmarking exercises 21 2 Kinematics and dynamics of rigid bodies 23 2.1 Introduction 23 2.2 Theory of vectors 23 2.2.1 Position and relative position vectors 23 2.2.2 The dot (scalar) product 26 2.2.3 The cross (vector) product 26 2.2.4 The scalar triple product 28 2.2.5 The vector triple product 28 2.2.6 Rotation of a vector 28 2.2.7 Vector transformation 31 2.2.8 Differentiation of a vector 32 2.2.9 Integration of a vector 34 2.2.10 Differentiation of the dot product 34 2.2.11 Differentiation of the cross product 34 2.2.12 Summary 35 2.3 Geometry analysis 38 2.3.1 Three point method 38 2.3.2 Vehicle suspension geometry analysis 41 2.4 Velocity analysis 43 2.5 Acceleration analysis 46 2.6 Static force and moment definition 51 2.7 Dynamics of a particle 55 2.8 Linear momentum of a rigid body 56 2.9 Angular momentum 57 2.10 Moments of inertia 59 2.11 Parallel axes theorem 63 2.12 Principal axes 65 2.13 Equations of motion 71
vi Contents 3 Multibody systems simulation software 75 3.1 Overview 75 3.2 Modelling features 78 3.2.1 Planning the model 78 3.2.2 Reference frames 79 3.2.3 Basic model components 85 3.2.4 Parts and markers 85 3.2.5 Equations of motion for a part 86 3.2.6 Basic constraints 90 3.2.7 Standard joints 95 3.2.8 Degrees of freedom 98 3.2.9 Force elements 102 3.2.10 Summation of forces and moments 114 3.3 Analysis capabilities 115 3.3.1 Overview 115 3.3.2 Solving linear equations 116 3.3.3 Non-linear equations 119 3.3.4 Integration methods 121 3.4 Systems of units 126 3.5 Pre- and post-processing 127 4 Modelling and analysis of suspension systems 131 4.1 The need for suspension 132 4.1.1 Wheel load variation 133 4.1.2 Body isolation 137 4.1.3 Handling load control 139 4.1.4 Compliant wheel plane control 145 4.1.5 Kinematic wheel plane control 145 4.1.6 Component loading environment 147 4.2 Types of suspension system 149 4.3 Quarter vehicle modelling approaches 152 4.4 Determination of suspension system characteristics 158 4.5 Suspension calculations 160 4.5.1 Measured outputs 160 4.5.2 Suspension steer axes 162 4.5.3 Bump movement, wheel recession and half track change 163 4.5.4 Camber and steer angle 163 4.5.5 Castor angle and suspension trail 165 4.5.6 Steering axis inclination and ground level offset 165 4.5.7 Instant centre and roll centre positions 166 4.5.8 Calculation of wheel rate 171 4.6 The compliance matrix approach 172 4.7 Case study 1 – Suspension kinematics 175 4.8 Durability studies (component loading) 180 4.8.1 Overview 180 4.8.2 Case study 2 – Static durability loadcase 184 4.8.3 Case study 3 – Dynamic durability loadcase 187 4.9 Ride studies (body isolation) 190 4.9.1 Case study 4 – Dynamic ride analysis 191 4.10 Case study 5 – Suspension vector analysis comparison with MBS 202
Contents vii 4.10.1 Problem definition 202 4.10.2 Velocity analysis 202 4.10.3 Acceleration analysis 214 4.10.4 Static analysis 226 4.10.5 Dynamic analysis 234 4.10.6 Geometry analysis 242 5 Tyre characteristics and modelling 248 5.1 Introduction 248 5.2 Tyre axis systems and geometry 249 5.2.1 The SAE and ISO tyre axis systems 249 5.2.2 Definition of tyre radii 249 5.2.3 Tyre asymmetry 253 5.3 The tyre contact patch 254 5.3.1 Friction 254 5.3.2 Pressure distribution in the tyre contact patch 256 5.4 Tyre force and moment characteristics 257 5.4.1 Components of tyre force and stiffness 257 5.4.2 Normal (vertical) force calculations 258 5.4.3 Longitudinal force in a free rolling tyre (rolling resistance) 260 5.4.4 Braking force 264 5.4.5 Driving force 267 5.4.6 Generation of lateral force and aligning moment 269 5.4.7 The effect of slip angle 269 5.4.8 The effect of camber angle 272 5.4.9 Combinations of camber and slip angle 276 5.4.10 Overturning moment 277 5.4.11 Combined traction and cornering (comprehensive slip) 278 5.4.12 Relaxation length 281 5.5 Experimental testing 284 5.6 Tyre modelling 291 5.6.1 Overview 291 5.6.2 Calculation of tyre geometry and velocities 295 5.6.3 Road surface/terrain definition 299 5.6.4 Interpolation methods 299 5.6.5 The ‘Magic Formula’ tyre model 301 5.6.6 The Fiala tyre model 306 5.6.7 Tyre models for durability analysis 308 5.7 Implementation with MBS 314 5.7.1 Virtual tyre rig model 315 5.8 Examples of tyre model data 318 5.9 Case study 6 – Comparison of vehicle handling tyre models 320 6 Modelling and assembly of the full vehicle 326 6.1 Introduction 326 6.2 The vehicle body 327 6.3 Measured outputs 330 6.4 Suspension system representation 331 6.4.1 Overview 331
viii Contents 6.4.2 Lumped mass model 332 6.4.3 Equivalent roll stiffness model 333 6.4.4 Swing arm model 335 6.4.5 Linkage model 335 6.4.6 The concept suspension approach 336 6.5 Modelling of springs and dampers 339 6.5.1 Treatment in simple models 339 6.5.2 Modelling leaf springs 340 6.6 Anti-roll bars 342 6.7 Determination of roll stiffness for the equivalent roll stiffness model 345 6.8 Aerodynamic effects 349 6.9 Modelling of vehicle braking 351 6.10 Modelling traction 356 6.11 Other driveline components 358 6.12 The steering system 361 6.12.1 Modelling the steering mechanism 361 6.12.2 Steering ratio 363 6.12.3 Steering inputs for vehicle handling manoeuvres 366 6.13 Driver behaviour 368 6.13.1 Steering controllers 369 6.13.2 A path following controller model 373 6.13.3 Body slip angle control 377 6.13.4 Two-loop driver model 379 6.14 Case study 7 – Comparison of full vehicle handling models 380 6.15 Summary 393 7 Simulation output and interpretation 395 7.1 Introduction 395 7.2 Case study 8 – Variation in measured data 397 7.3 A vehicle dynamics overview 399 7.3.1 Travel on a curved path 399 7.3.2 The classical treatment based on steady state cornering 401 7.3.3 Some further discussion of vehicles in curved path 408 7.3.4 The subjective/objective problem 411 7.3.5 Mechanisms for generating under- and oversteer 414 7.4 Transient effects 420 7.5 Steering feel as a subjective modifier 424 7.6 Roll as an objective and subjective modifier 424 7.7 Frequency response 426 7.8 The problems imposed by … 428 7.8.1 Circuit racing 428 7.8.2 Rallying 428 7.8.3 Accident avoidance 429 7.9 The use of analytical models with a signal-to-noise ratio approach 430 7.10 Some consequences of using signal-to-noise ratio 439 8 Active systems 441 8.1 Introduction 441 8.2 Active systems 442
Contents ix 8.2.1 Active suspension and variable damping 443 8.2.2 Brake-based systems 447 8.2.3 Active steering systems 448 8.2.4 Active camber systems 449 8.2.5 Active torque distribution 449 8.3 Which active system? 450 Appendix A: Vehicle model system schematics and data sets 452 Appendix B: Fortran tyre model subroutines 472 Appendix C: Glossary of terms 487 References 502 Index 511
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Preface This book is intended to bridge a gap between the subject of classical vehi- cle dynamics and the general-purpose computer-based discipline known as multibody systems analysis (MBS). While there are several textbooks that focus entirely on the subject, and mathematical foundations, of vehicle dynamics and other more recent texts dealing with MBS, there are none yet that link the two subjects in a comprehensive manner. A book in this area is timely. The computer-based analysis methodology (MBS) became estab- lished as a tool for engineering designers during the 1980s in a similar manner to the growth in Finite Element Analysis (FEA) technology during the previous decade. A number of computer programs were developed and marketed to the engineering industry, the most well known being MSC.ADAMS™ (Automatic Dynamic Analysis of Mechanical Systems) which will form the basis for the examples provided here. During the 1990s MBS became firmly established as part of the vehicle design and develop- ment process. It is inevitable that the engineer working on problems involv- ing vehicle ride and handling in a modern automotive environment will be required to interface with the use of MBS to simulate vehicle motion. The book is aimed at a wide audience including not only undergraduate, postgraduate and research students working in this area, but also practising engineers in industry requiring a reference text dealing with the major relevant areas within the discipline. The book was originally planned as an individual effort on the part of Mike Blundell, drawing on past experience consulting on and researching into the application of MBS to solve a class of problems in the area of vehicle dynamics. From the start it was clear that a major challenge in preparing a book on this subject would be to provide meaningful comment on not only the modelling techniques but also the vast range of simulation outputs and responses that can be generated. Deciding whether a vehicle has good or bad handling characteristics is often a matter of human judgement based on the response or feel of the vehicle, or how easy the vehicle is to drive through certain manoeuvres. To a large extent automotive manufacturers still rely on track measurements and the instincts of experienced test engineers as to whether the design has produced a vehicle with the required handling qual- ities. To address this problem the book has been co-authored by Damian Harty who is the Chief Engineer, Dynamics at Prodrive. With experience not only in the area of computer simulation but also in the practical develop- ment and testing of vehicles on the proving ground, Damian has been able to help in documenting the realistic application of MBS in vehicle development. Chapter 1 is intended to document the emergence of MBS and provide an overview of its role in vehicle design and development. Previous work by contributors including Olley, Segel, Milliken, Crolla and Sharp is identified providing a historical perspective on the subject during the latter part of the twentieth century.
xii Preface Chapter 2 is included for completeness and covers the underlying formula- tions in kinematics and dynamics required for a good understanding of multibody systems formulations. A three-dimensional vector approach is used to develop the theory, this being the most suitable method for developing the rigid body equations of motion and constraint formulations described later. Chapter 3 covers the modelling, analysis and post-processing capabilities of a typical simulation software. There are many commercial programs to choose from including not only MSC.ADAMS but also other software pack- ages such as DADS and SIMPACK. The descriptions provided in Chapter 3 are based on MSC.ADAMS; the main reason for this choice being that the two authors have between them 25 years of experience working with the software. The fact that the software is also well established in automotive companies and academic institutions worldwide is also a factor. It is not intended in Chapter 3 to provide an MSC.ADAMS primer. There is exten- sive user documentation and training material available in this area from the program vendors MSC.Software. The information included in Chapter 3 is therefore limited to that needed to introduce a new reader to the subject and to provide a supporting reference for the vehicle modelling and analysis methodologies described in the following chapters. Existing users of MSC.ADAMS will note that the modelling examples pro- vided in Chapter 3 are based on a text-based format of model inputs, known in MSC.ADAMS as solver data sets. This was the original method used to develop MSC.ADAMS models and has subsequently been replaced by a powerful graphical user interface (GUI) known as ADAMS/View™ that allows model parameterization and design optimization studies. The ADAMS/View environment is also the basis for customized versions of MSC.ADAMS such as ADAMS/Car™ that are becoming established in industry and are also discussed in Chapter 3. The use of text-based data sets has been adopted here for a number of reasons. The first of these is that the GUI of a modern simulation program such as MSC.ADAMS is subject to extensive and ongoing development. Any attempt to describe such a facil- ity in a textbook such as this would become outdated after a short time. As mentioned the software developers provide their own user documentation covering this in any case. It is also clear that the text-based formulations translate more readily to book format and are also useful for demonstrating the underlying techniques in planning a model, preparing model schemat- ics and establishing the degrees of freedom in a system model. These tech- niques are needed to interpret the models and data sets that are described in later chapters and appendices. It is also hoped that by treating the soft- ware at this fundamental level the dependence of the book on any one soft- ware package is reduced and that the methods and principles will be adaptable for practitioners using alternative software. Examples of the later ADAMS/View command file format are included in Chapters 6 and 8 for completeness. Chapter 4 addresses the modelling and analysis of the suspension system. An attempt has been made to bridge the gap between the textbook treatment of suspension systems and the multibody systems approach to building and simulating suspension models. As such a number of case studies have been included to demonstrate the application of the models and their use in the
Preface xiii vehicle design process. The chapter concludes with an extensive case study comparing a full set of analytical calculations, using the vector-based methods introduced in Chapter 2, with the output produced from MSC.ADAMS. It is intended that this exercise will demonstrate to readers the underlying computations in process when running an MBS simulation. Chapter 5 addresses the tyre force and moment generating characteristics and the subsequent modelling of these in an MBS simulation. Examples are provided of tyre test data and the derived parameters for established tyre models. The chapter concludes with a case study using an MBS virtual tyre test machine to interrogate and compare tyre models and data sets. Chapter 6 describes the modelling and assembly of the rest of the vehicle, including the anti-roll bars and steering systems. Near the beginning a range of simplified suspension modelling strategies for the full vehicle is described. This forms the basis for subsequent discussion involving the representation of the road springs and steering system in simple models that do not include a model of the suspension linkages. The chapter includes a consideration of modelling driver inputs to the steering system using several control methodologies and concludes with a case study comparing the per- formance of several full vehicle modelling strategies for a vehicle handling manoeuvre. Chapter 7 deals with the simulation output and interpretation of results. An overview of vehicle dynamics for travel on a curved path is included. The classical treatment of understeer/oversteer based on steady state cornering is presented followed by an alternative treatment that considers yaw rate and lateral acceleration gains The subjective/objective problem is discussed with consideration of steering feel and roll angle as subjective modifiers. The chapter concludes with a consideration of the use of analytical models with a signal-to noise approach. Chapter 8 concludes with a review of the use of active systems to modify the dynamics in modern passenger cars. The use of electronic control in sys- tems such as active suspension and variable damping, brake-based systems, active steering systems, active camber systems and active torque distribution is described. A final summary matches the application of these systems with driving styles described as normal, spirited or the execution of emergency manoeuvres. Appendix A contains a full set of vehicle model schematics and a complete set of vehicle data that can be used to build suspension models and full vehicle models of varying complexity. The data provided in Appendix A was used for many of the case studies presented throughout the book. Appendix B contains example Fortran Tire subroutines to supplement the description of the tyre modelling process given in Chapter 5. A subroutine is included that uses a general interpolation approach using a cubic spline fit through measured tyre test data. The second subroutine is based on version 3 of the ‘Magic Formula’ and has an embedded set of tyre parameters based on the tyre data described in Chapter 5. A final subroutine ‘The Harty model’ was developed by Damian at Prodrive and is provided for readers who would like to experiment with a new tyre model that uses a reduced set of model parameters and can represent combined slip in the tyre contact patch.
xiv Preface In conclusion it seems to the authors there are two camps for addressing the vehicle dynamics problem. In one is the practical ride and handling expert. The second camp contains theoretical vehicle dynamics experts. This book is aimed at the reader who, like the authors, seeks to live between the two camps and move forward the process of vehicle design, taking full advan- tage of the widespread availability of convenient digital computing. There is, however, an enormous difficulty in achieving this end. Lewis Carroll, in Alice Through the Looking Glass, describes an encounter between Alice and a certain Mr H. Dumpty: ‘When I use a word’, Humtpy Dumpty said, in rather a scornful tone, ‘it means just what I choose it to mean – neither more nor less.’ ‘The question is’, said Alice, ‘whether you can make words mean so many different things.’ There is a similar difficulty between practical and theoretical vehicle dynamicists and even between different individuals of the same persuasion. The same word is used, often without definition, to mean just what the speaker chooses. There is no universal solution to the problem save for a thoughtful and attentive style of discussion and enquiry, taking pains to establish the meanings of even apparently obvious terms such as ‘camber’ – motorcycles do not have any camber by some definitions (vehicle- body-referenced) and yet to zero the camber forces in a motorcycle tyre is clearly folly. A glossary is included in Appendix C, not as some declaration of correctness but as an illumination for the text. Mike Blundell and Damian Harty February 2004
Acknowledgements Mike Blundell In developing my sections of this book I am indebted to my colleagues and students at Coventry University who have provided encouragement and material that I have been able to use. In particular I thank Barry Bolland and Peter Griffiths for their input to Chapter 2 and Bryan Phillips for his help with Chapter 5. I am also grateful to many within the vehicle dynamics community who have made a contribution including David Crolla, Roger Williams, Jim Forbes, Adrian Griffiths, Colin Lucas, John Janevic and Grahame Walter. Finally I thank the staff at Elsevier Science for their patience and help throughout the years it has taken to bring this book to print. Damian Harty Mike’s gracious invitation to join him and infectious enthusiasm for both the topic and this project has kept me buoyed. Robin Sharp, Doug and Bill Milliken keep me grounded and rigorous when it is tempting just to play in cars and jump to conclusions. David Crolla has been an ever-present voice of reason keeping this text focused on its raison d’être – the useful fusion of practical and theoretical vehicle dynamics. Professional colleagues who have used banter, barracking and sometimes even rational discussion to help me progress my thinking are too numerous to mention – apart from Duncan Riding, whom I have to single out as being exceptionally encour- aging. I hope I show my gratitude in person and on a regular basis to all of them and invite them to kick me if I don’t. Someone who must be men- tioned is Isaac Newton; his original and definitive brilliance at describing my world amazes me every day. As Mike, I thank the staff at Elsevier Science for their saintly patience. I owe the most thanks to the management of Prodrive. Their skill at allow- ing me to thrive defies any succinct description but I am deeply and con- tinuously both aware of and grateful for it. And finally, I’d just like to say I’m very sorry for all the cars I’ve damaged while ‘testing’ them. I really am.
Nomenclature {aI}1 Unit vector at marker I resolved parallel to frame 1 (GRF) {aJ}1 Unit vector at marker J resolved parallel to frame 1 (GRF) ax Longitudinal acceleration (Wenzel model) ay Lateral acceleration (Wenzel model) b Longitudinal distance of body mass centre from front axle c Damping coefficient c Longitudinal distance of body mass centre from rear axle c Specific heat capacity of brake rotor {dIJ}1 Position vector of marker I relative to J resolved parallel to frame 1 (GRF) f Natural frequency (Hz) h Brake rotor convection coefficient h Height of body mass centre above roll axis k Path curvature k Radius of gyration k Stiffness ks Spring stiffness kw Stiffness of equivalent spring at the wheel centre m Mass of a body m{g}1 Weight force vector for a part resolved parallel to frame 1 (GRF) mt Mass of tyre n Number of friction surfaces (pads) p Brake pressure qj Set of part generalized co-ordinates r Yaw rate r1,r2,r3 Coupler constraint rotations {rI}1 Position vector of marker I relative to frame i resolved parallel to frame 1 (GRF) {rJ}1 Position vector of marker J relative to frame j resolved parallel to frame 1 (GRF) s1,s2,s3 Coupler constraint scale factors tf Front track tr Rear track vcog Centre of gravity (Wenzel model) vx Longitudinal velocity (Wenzel model) vy Lateral velocity (Wenzel model) {xI}1 Unit vector along x-axis of marker I resolved parallel to frame 1 (GRF) {yI}1 Unit vector along y-axis of marker I resolved parallel to frame 1 (GRF) {xJ}1 Unit vector along x-axis of marker J resolved parallel to frame 1 (GRF) {yJ}1 Unit vector along y-axis of marker J resolved parallel to frame 1 (GRF) ys Asymptotic value at large slip (‘Magic Formula’)
Nomenclature xvii {zI}1 Unit vector along z-axis of marker I resolved parallel to frame 1 (GRF) {zJ}1 Unit vector along z-axis of marker J resolved parallel to frame 1 (GRF) A Area Ac Convective area of brake disc [A1n] Euler matrix for part n {An}1 Acceleration vector for part n resolved parallel to frame 1 (GRF) Ap p Centripetal acceleration {APQ}1 Centripetal acceleration vector P relative to Q referred to frame 1 (GRF) t {APQ}1 Transverse acceleration vector P relative to Q referred to frame 1 (GRF) c {APQ}1 Coriolis acceleration vector P relative to Q referred to frame 1 (GRF) s {APQ}1 Sliding acceleration vector P relative to Q referred to frame 1 (GRF) AyG Lateral acceleration gain B Stiffness factor (‘Magic Formula’) [B] Transformation matrix from frame Oe to On BKid Bottom Kingpin Marker BM Bump Movement BT Brake torque C Shape factor (‘Magic Formula’) [C] Compliance matrix CF Front axle cornering stiffness Cr Rolling resistance moment coefficient CR Rear axle cornering stiffness CS Tyre longitudinal stiffness Cp Process capability CP Centre of pressure C Tyre lateral stiffness due to slip angle C Tyre lateral stiffness due to camber angle D Peak value (‘Magic Formula’) DM(I,J) Magnitude of displacement of I marker relative to J marker DX(I,J) Displacement in X direction of I marker relative to J marker parallel to GRF DY(I,J) Displacement in Y direction of I marker relative to J marker parallel to GRF DZ(I,J) Displacement in Z direction of I marker relative to J marker parallel to GRF E Young’s modulus of elasticity E Curvature factor (‘Magic Formula’) {FnA}1 Applied force vector on part n resolved parallel to frame 1 (GRF) {FnC}1 Constraint force vector on part n resolved parallel to frame 1 (GRF) FFRC Lateral force reacted by front roll centre FRRC Lateral force reacted by rear roll centre Fx Longitudinal tractive or braking tyre force Fy Lateral tyre force Fz Vertical tyre force
xviii Nomenclature Fzc Vertical tyre force due to damping Fzk Vertical tyre force due to stiffness {FA}1{FB}1… Applied force vectors at points A, B, … resolved parallel to frame 1 (GRF) [FE] Elastic compliance matrix (Concept suspension) FD Drag force FG Fixed Ground Marker G Shear modulus GC Gravitational constant GO Ground Level Offset GRF Ground Reference Frame {H}1 Angular momentum vector for a body H( ) Transfer function HTC Half Track Change I Mass moment of inertia I Second moment of area ICY Y Co-ordinate of Instant Centre ICZ Z Co-ordinate of Instant Centre [In] Inertia tensor for a part J Polar second moment of area Jz Vehicle body yaw inertia (Wenzel model) K Drive torque controller constant K Spring stiffness K Stability factor K Understeer gradient Kz Tyre radial stiffness KT Torsional stiffness KTs Roll stiffness due to springs KTr Roll stiffness due to anti-roll bar L Length L Wheelbase {L}1 Linear momentum vector for a particle or body LPRF Local Part Reference Frame LR Tyre relaxation length MFRC Moment reacted by front roll centre {MnA}e Applied moment vector on part n resolved parallel to frame e {MnC}e Constraint moment vector on part n resolved parallel to frame e Ms Equivalent roll moment due to springs Mx Tyre overturning moment My Tyre rolling resistance moment Mz Tyre self-aligning moment MRF Marker Reference Frame MRRC Moment reacted by rear roll centre Nr Vehicle yaw moment with respect to yaw rate [Nt] Norsieck vector Nvy Vehicle yaw moment with respect to lateral velocity O1 Frame 1 (GRF) Oe Euler axis frame Oi Reference frame for part i Oj Reference frame for part j
Nomenclature xix On Frame for part n {Pnr}1 Rotational momenta vector for part n resolved parallel to frame 1 (GRF) {Pnt}1 Translational momenta vector for part n resolved parallel to frame 1 (GRF) Pt Constant power acceleration QP Position vector of a marker relative to the LPRF QG Position vector of a marker relative to the GRF R Radius of turn R1 Unloaded tyre radius R2 Tyre carcass radius Rd Radius to centre of brake pad Re Effective rolling radius {Ri}1 Position vector of frame i on part i resolved parallel to frame 1 (GRF) {Rj}1 Position vector of frame j on part j resolved parallel to frame 1 (GRF) Rl Loaded tyre radius {Rn}1 Position vector for part n resolved parallel to frame 1 (GRF) {Rp}1 Position vector of tyre contact point P relative to frame 1, referenced to frame 1 Ru Unloaded tyre radius {Rw}1 Position vector of wheel centre relative to frame 1, referenced to frame 1 {RAG}n Position vector of point A relative to mass centre G resolved parallel to frame n {RBG}n Position vector of point B relative to mass centre G resolved parallel to frame n RCfront Front roll centre RCrear Rear roll centre RCY Y Co-ordinate of Roll Centre RCZ Z Co-ordinate of Roll Centre Se Error variation Sh Horizontal shift (‘Magic Formula’) Sv Vertical shift (‘Magic Formula’) SA Spindle Axis reference point SL Longitudinal slip ratio SL* Critical value of longitudinal slip SN Signal-to-noise ratio ST Total variation S Lateral slip ratio SL Comprehensive slip ratio S * Critical slip angle S Variation due to linear effect T Kinetic energy for a part T Temperature T Torque Tenv Environmental temperature T0 Initial brake rotor temperature {TA}1{TB}1… Applied torque vectors at points A, B, … resolved parallel to frame 1 (GRF)