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The Multibody Systems Approach to Vehicle Dynamics
The Multibody Systems Approach to Vehicle Dynamics
Michael Blundell, Damian Harty
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This is the first book to comprehensively bridge the gap between classical vehicle dynamics and the widelyused, computerbased technique of Multibody Systems analysis (MBS). MBS is firmly established as a key part of all modern vehicle design and development processes; any engineer working on problems involving vehicle ride or handling will use MBS to simulate vehicle motion. Suitable for use both as a teaching text and a professional reference volume, this book is an essential addition to the resources available to anyone working in vehicle design and development. Written by a leading academic in the field (who himself has considerable practical experience) and the chief dynamics engineer of Prodrive, the preeminent rally, race and road technology organization, the book has a unique blend of theory and practice that will be of immense value in this applications based field.
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Year:
2004
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Elsevier Limited
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english
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518 / 541
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0750651121
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9780080473529
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Multibody Systems Approach to Vehicle Dynamics This page intentionally left blank Multibody Systems Approach to Vehicle Dynamics Michael Blundell Damian Harty AMSTERDAM PARIS BOSTON SAN DIEGO HEIDELBERG SAN FRANCISCO LONDON NEW YORK SINGAPORE SYDNEY OXFORD TOKYO Elsevier ButterworthHeinemann 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; email: permissions@elsevier.co.uk. You may also complete your request online 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 ButterworthHeinemann publications visit our website at http://books.elsevier.com Composition by Charon Tec Pvt. Ltd, Chennai, India Printed and bound in MapleVail, Kirkwood, New York, USA Contents Prefac; e Acknowledgements Nomenclature xi xv xvi 1 Introduction 1.1 Overview 1.2 What is vehicle dynamics? 1.3 Why analyse? 1.4 Classical methods 1.5 Analytical process 1.6 Computational methods 1.7 Computerbased tools 1.8 Commercial computer packages 1.9 Benchmarking exercises 1 1 3 10 10 11 14 14 17 21 2 Kinematics and dynamics of rigid bodies 2.1 Introduction 2.2 Theory of vectors 2.2.1 Position and relative position vectors 2.2.2 The dot (scalar) product 2.2.3 The cross (vector) product 2.2.4 The scalar triple product 2.2.5 The vector triple product 2.2.6 Rotation of a vector 2.2.7 Vector transformation 2.2.8 Differentiation of a vector 2.2.9 Integration of a vector 2.2.10 Differentiation of the dot product 2.2.11 Differentiation of the cross product 2.2.12 Summary 2.3 Geometry analysis 2.3.1 Three point method 2.3.2 Vehicle suspension geometry analysis 2.4 Velocity analysis 2.5 Acceleration analysis 2.6 Static force and moment definition 2.7 Dynamics of a particle 2.8 Linear momentum of a rigid body 2.9 Angular momentum 2.10 Moments of inertia 2.11 Parallel axes theorem 2.12 Principal axes 2.13 Equations of motion 23 23 23 23 26 26 28 28 28 31 32 34 34 34 35 38 38 41 43 46 51 55 56 57 59 63 65 71 vi Contents 3 Multibody systems simulation software 3.1 Overview 3.2 Modelling features 3.2.1 Planning the model 3.2.2 Reference frames 3.2.3 Basic model components 3.2.4 Parts and markers 3.2.5 Equations of motion for a part 3.2.6 Basic constraints 3.2.7 Standard joints 3.2.8 Degrees of freedom 3.2.9 Force elements 3.2.10 Summation of forces and moments 3.3 Analysis capabilities 3.3.1 Overview 3.3.2 Solving linear equations 3.3.3 Nonlinear equations 3.3.4 Integration methods 3.4 Systems of units 3.5 Pre and postprocessing 75 75 78 78 79 85 85 86 90 95 98 102 114 115 115 116 119 121 126 127 4 Modelling and analysis of suspension systems 4.1 The need for suspension 4.1.1 Wheel load variation 4.1.2 Body isolation 4.1.3 Handling load control 4.1.4 Compliant wheel plane control 4.1.5 Kinematic wheel plane control 4.1.6 Component loading environment 4.2 Types of suspension system 4.3 Quarter vehicle modelling approaches 4.4 Determination of suspension system characteristics 4.5 Suspension calculations 4.5.1 Measured outputs 4.5.2 Suspension steer axes 4.5.3 Bump movement, wheel recession and half track change 4.5.4 Camber and steer angle 4.5.5 Castor angle and suspension trail 4.5.6 Steering axis inclination and ground level offset 4.5.7 Instant centre and roll centre positions 4.5.8 Calculation of wheel rate 4.6 The compliance matrix approach 4.7 Case study 1 – Suspension kinematics 4.8 Durability studies (component loading) 4.8.1 Overview 4.8.2 Case study 2 – Static durability loadcase 4.8.3 Case study 3 – Dynamic durability loadcase 4.9 Ride studies (body isolation) 4.9.1 Case study 4 – Dynamic ride analysis 4.10 Case study 5 – Suspension vector analysis comparison with MBS 131 132 133 137 139 145 145 147 149 152 158 160 160 162 163 163 165 165 166 171 172 175 180 180 184 187 190 191 202 Contents 4.10.1 4.10.2 4.10.3 4.10.4 4.10.5 4.10.6 Problem definition Velocity analysis Acceleration analysis Static analysis Dynamic analysis Geometry analysis 5 Tyre characteristics and modelling 5.1 Introduction 5.2 Tyre axis systems and geometry 5.2.1 The SAE and ISO tyre axis systems 5.2.2 Definition of tyre radii 5.2.3 Tyre asymmetry 5.3 The tyre contact patch 5.3.1 Friction 5.3.2 Pressure distribution in the tyre contact patch 5.4 Tyre force and moment characteristics 5.4.1 Components of tyre force and stiffness 5.4.2 Normal (vertical) force calculations 5.4.3 Longitudinal force in a free rolling tyre (rolling resistance) 5.4.4 Braking force 5.4.5 Driving force 5.4.6 Generation of lateral force and aligning moment 5.4.7 The effect of slip angle 5.4.8 The effect of camber angle 5.4.9 Combinations of camber and slip angle 5.4.10 Overturning moment 5.4.11 Combined traction and cornering (comprehensive slip) 5.4.12 Relaxation length 5.5 Experimental testing 5.6 Tyre modelling 5.6.1 Overview 5.6.2 Calculation of tyre geometry and velocities 5.6.3 Road surface/terrain definition 5.6.4 Interpolation methods 5.6.5 The ‘Magic Formula’ tyre model 5.6.6 The Fiala tyre model 5.6.7 Tyre models for durability analysis 5.7 Implementation with MBS 5.7.1 Virtual tyre rig model 5.8 Examples of tyre model data 5.9 Case study 6 – Comparison of vehicle handling tyre models 6 Modelling and assembly of the full vehicle 6.1 Introduction 6.2 The vehicle body 6.3 Measured outputs 6.4 Suspension system representation 6.4.1 Overview vii 202 202 214 226 234 242 248 248 249 249 249 253 254 254 256 257 257 258 260 264 267 269 269 272 276 277 278 281 284 291 291 295 299 299 301 306 308 314 315 318 320 326 326 327 330 331 331 viii Contents 6.5 6.6 6.7 6.8 6.9 6.10 6.11 6.12 6.13 6.14 6.15 6.4.2 Lumped mass model 6.4.3 Equivalent roll stiffness model 6.4.4 Swing arm model 6.4.5 Linkage model 6.4.6 The concept suspension approach Modelling of springs and dampers 6.5.1 Treatment in simple models 6.5.2 Modelling leaf springs Antiroll bars Determination of roll stiffness for the equivalent roll stiffness model Aerodynamic effects Modelling of vehicle braking Modelling traction Other driveline components The steering system 6.12.1 Modelling the steering mechanism 6.12.2 Steering ratio 6.12.3 Steering inputs for vehicle handling manoeuvres Driver behaviour 6.13.1 Steering controllers 6.13.2 A path following controller model 6.13.3 Body slip angle control 6.13.4 Twoloop driver model Case study 7 – Comparison of full vehicle handling models Summary 7 Simulation output and interpretation 7.1 Introduction 7.2 Case study 8 – Variation in measured data 7.3 A vehicle dynamics overview 7.3.1 Travel on a curved path 7.3.2 The classical treatment based on steady state cornering 7.3.3 Some further discussion of vehicles in curved path 7.3.4 The subjective/objective problem 7.3.5 Mechanisms for generating under and oversteer 7.4 Transient effects 7.5 Steering feel as a subjective modifier 7.6 Roll as an objective and subjective modifier 7.7 Frequency response 7.8 The problems imposed by … 7.8.1 Circuit racing 7.8.2 Rallying 7.8.3 Accident avoidance 7.9 The use of analytical models with a signaltonoise ratio approach 7.10 Some consequences of using signaltonoise ratio 8 Active systems 8.1 Introduction 8.2 Active systems 332 333 335 335 336 339 339 340 342 345 349 351 356 358 361 361 363 366 368 369 373 377 379 380 393 395 395 397 399 399 401 408 411 414 420 424 424 426 428 428 428 429 430 439 441 441 442 Contents ix 8.2.1 Active suspension and variable damping 8.2.2 Brakebased systems 8.2.3 Active steering systems 8.2.4 Active camber systems 8.2.5 Active torque distribution 8.3 Which active system? 443 447 448 449 449 450 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 This page intentionally left blank Preface This book is intended to bridge a gap between the subject of classical vehicle dynamics and the generalpurpose computerbased 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 computerbased analysis methodology (MBS) became established 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 development process. It is inevitable that the engineer working on problems involving 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 qualities. To address this problem the book has been coauthored 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 development 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 formulations in kinematics and dynamics required for a good understanding of multibody systems formulations. A threedimensional 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 postprocessing capabilities of a typical simulation software. There are many commercial programs to choose from including not only MSC.ADAMS but also other software packages 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 extensive 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 provided in Chapter 3 are based on a textbased 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 textbased 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 facility 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 textbased formulations translate more readily to book format and are also useful for demonstrating the underlying techniques in planning a model, preparing model schematics and establishing the degrees of freedom in a system model. These techniques 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 software at this fundamental level the dependence of the book on any one software 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 vectorbased 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 antiroll 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 performance 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 signalto 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 systems such as active suspension and variable damping, brakebased 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 advantage 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 (vehiclebodyreferenced) 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 everpresent 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 encouraging. 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 mentioned 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 allowing me to thrive defies any succinct description but I am deeply and continuously 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 {aJ}1 ax ay b c c c {dIJ}1 f h h k k k ks kw m m{g}1 mt n p qj r r1,r2,r3 {rI}1 {rJ}1 s1,s2,s3 tf tr vcog vx vy {xI}1 {yI}1 {xJ}1 {yJ}1 ys Unit vector at marker I resolved parallel to frame 1 (GRF) Unit vector at marker J resolved parallel to frame 1 (GRF) Longitudinal acceleration (Wenzel model) Lateral acceleration (Wenzel model) Longitudinal distance of body mass centre from front axle Damping coefficient Longitudinal distance of body mass centre from rear axle Specific heat capacity of brake rotor Position vector of marker I relative to J resolved parallel to frame 1 (GRF) Natural frequency (Hz) Brake rotor convection coefficient Height of body mass centre above roll axis Path curvature Radius of gyration Stiffness Spring stiffness Stiffness of equivalent spring at the wheel centre Mass of a body Weight force vector for a part resolved parallel to frame 1 (GRF) Mass of tyre Number of friction surfaces (pads) Brake pressure Set of part generalized coordinates Yaw rate Coupler constraint rotations Position vector of marker I relative to frame i resolved parallel to frame 1 (GRF) Position vector of marker J relative to frame j resolved parallel to frame 1 (GRF) Coupler constraint scale factors Front track Rear track Centre of gravity (Wenzel model) Longitudinal velocity (Wenzel model) Lateral velocity (Wenzel model) Unit vector along xaxis of marker I resolved parallel to frame 1 (GRF) Unit vector along yaxis of marker I resolved parallel to frame 1 (GRF) Unit vector along xaxis of marker J resolved parallel to frame 1 (GRF) Unit vector along yaxis of marker J resolved parallel to frame 1 (GRF) Asymptotic value at large slip (‘Magic Formula’) Nomenclature {zI}1 {zJ}1 A Ac [A1n] {An}1 Ap p {APQ}1 t {APQ }1 c {APQ }1 s {APQ }1 AyG B [B] BKid BM BT C [C] CF Cr CR CS Cp CP C C D DM(I,J) DX(I,J) DY(I,J) DZ(I,J) E E {FnA}1 {FnC}1 FFRC FRRC Fx Fy Fz xvii Unit vector along zaxis of marker I resolved parallel to frame 1 (GRF) Unit vector along zaxis of marker J resolved parallel to frame 1 (GRF) Area Convective area of brake disc Euler matrix for part n Acceleration vector for part n resolved parallel to frame 1 (GRF) Centripetal acceleration Centripetal acceleration vector P relative to Q referred to frame 1 (GRF) Transverse acceleration vector P relative to Q referred to frame 1 (GRF) Coriolis acceleration vector P relative to Q referred to frame 1 (GRF) Sliding acceleration vector P relative to Q referred to frame 1 (GRF) Lateral acceleration gain Stiffness factor (‘Magic Formula’) Transformation matrix from frame Oe to On Bottom Kingpin Marker Bump Movement Brake torque Shape factor (‘Magic Formula’) Compliance matrix Front axle cornering stiffness Rolling resistance moment coefficient Rear axle cornering stiffness Tyre longitudinal stiffness Process capability Centre of pressure Tyre lateral stiffness due to slip angle Tyre lateral stiffness due to camber angle Peak value (‘Magic Formula’) Magnitude of displacement of I marker relative to J marker Displacement in X direction of I marker relative to J marker parallel to GRF Displacement in Y direction of I marker relative to J marker parallel to GRF Displacement in Z direction of I marker relative to J marker parallel to GRF Young’s modulus of elasticity Curvature factor (‘Magic Formula’) Applied force vector on part n resolved parallel to frame 1 (GRF) Constraint force vector on part n resolved parallel to frame 1 (GRF) Lateral force reacted by front roll centre Lateral force reacted by rear roll centre Longitudinal tractive or braking tyre force Lateral tyre force Vertical tyre force xviii Nomenclature Fzc Fzk {FA}1{FB}1… [FE] FD FG G GC GO GRF {H}1 H() HTC I I ICY ICZ [In] J Jz K K K K Kz KT KTs KTr L L {L}1 LPRF LR MFRC {MnA}e {MnC}e Ms Mx My Mz MRF MRRC Nr [Nt] Nvy O1 Oe Oi Oj Vertical tyre force due to damping Vertical tyre force due to stiffness Applied force vectors at points A, B, … resolved parallel to frame 1 (GRF) Elastic compliance matrix (Concept suspension) Drag force Fixed Ground Marker Shear modulus Gravitational constant Ground Level Offset Ground Reference Frame Angular momentum vector for a body Transfer function Half Track Change Mass moment of inertia Second moment of area Y Coordinate of Instant Centre Z Coordinate of Instant Centre Inertia tensor for a part Polar second moment of area Vehicle body yaw inertia (Wenzel model) Drive torque controller constant Spring stiffness Stability factor Understeer gradient Tyre radial stiffness Torsional stiffness Roll stiffness due to springs Roll stiffness due to antiroll bar Length Wheelbase Linear momentum vector for a particle or body Local Part Reference Frame Tyre relaxation length Moment reacted by front roll centre Applied moment vector on part n resolved parallel to frame e Constraint moment vector on part n resolved parallel to frame e Equivalent roll moment due to springs Tyre overturning moment Tyre rolling resistance moment Tyre selfaligning moment Marker Reference Frame Moment reacted by rear roll centre Vehicle yaw moment with respect to yaw rate Norsieck vector Vehicle yaw moment with respect to lateral velocity Frame 1 (GRF) Euler axis frame Reference frame for part i Reference frame for part j Nomenclature On {Pnr}1 {Pnt}1 Pt QP QG R R1 R2 Rd Re {Ri}1 {Rj}1 Rl {Rn}1 {Rp}1 Ru {Rw}1 {RAG}n {RBG}n RCfront RCrear RCY RCZ Se Sh Sv SA SL SL* SN ST S SL S* S T T T Tenv T0 {TA}1{TB}1… xix Frame for part n Rotational momenta vector for part n resolved parallel to frame 1 (GRF) Translational momenta vector for part n resolved parallel to frame 1 (GRF) Constant power acceleration Position vector of a marker relative to the LPRF Position vector of a marker relative to the GRF Radius of turn Unloaded tyre radius Tyre carcass radius Radius to centre of brake pad Effective rolling radius Position vector of frame i on part i resolved parallel to frame 1 (GRF) Position vector of frame j on part j resolved parallel to frame 1 (GRF) Loaded tyre radius Position vector for part n resolved parallel to frame 1 (GRF) Position vector of tyre contact point P relative to frame 1, referenced to frame 1 Unloaded tyre radius Position vector of wheel centre relative to frame 1, referenced to frame 1 Position vector of point A relative to mass centre G resolved parallel to frame n Position vector of point B relative to mass centre G resolved parallel to frame n Front roll centre Rear roll centre Y Coordinate of Roll Centre Z Coordinate of Roll Centre Error variation Horizontal shift (‘Magic Formula’) Vertical shift (‘Magic Formula’) Spindle Axis reference point Longitudinal slip ratio Critical value of longitudinal slip Signaltonoise ratio Total variation Lateral slip ratio Comprehensive slip ratio Critical slip angle Variation due to linear effect Kinetic energy for a part Temperature Torque Environmental temperature Initial brake rotor temperature Applied torque vectors at points A, B, … resolved parallel to frame 1 (GRF) xx Nomenclature TK TR {Ur} {Us} UCF US V Va Ve Vg Vlowlimit {Vn}1 {Vp}1 Vs Vxc Vy Vz VR(I,J) WB WC WF WR XP {XSAE}1 Yr Yvy YRG {YSAE}1 {ZSAE}1 ZP {n}1 f r o i mean {n}e { }1 d Top Kingpin Marker Suspension Trail Unit vector normal to road surface at tyre contact point Unit vector acting along spin axis of tyre Units Consistency Factor Understeer Forward velocity Actual forward velocity Error variance Ground plane velocity Limiting velocity Velocity vector for part n resolved parallel to frame 1 (GRF) Velocity vector of tyre contact point P referenced to frame 1 Desired simulation velocity Longitudinal slip velocity of tyre contact point Lateral slip velocity of tyre contact point Vertical velocity of tyre contact point Radial line of sight velocity of I marker relative to J marker Wheel Base Marker Wheel Centre Marker Wheel Front Marker Wheel Recession Position vector of a point in a marker xz plane Unit vector acting at tyre contact point in XSAE direction referenced to frame 1 Vehicle side force with respect to yaw rate Vehicle side force with respect to lateral velocity Yaw rate gain Unit vector acting at tyre contact point in YSAE direction referenced to frame 1 Unit vector acting at tyre contact point in ZSAE direction referenced to frame 1 Position vector of a point on a marker zaxis Tyre slip angle Angular acceleration vector for part n resolved parallel to frame 1 (GRF) Front axle slip angle Rear axle slip angle Side slip angle Rate of change of side slip angle (beta dot) Steer or toe angle Steer angle of outer wheel Steer angle of inner wheel Average steer angle of inner and outer wheels Camber angle Set of Euler angles for part n Damping ratio Longitudinal slip (Pacjeka) Sensitivity of process 2nd Euler angle rotation Reaction force vector resolved parallel to frame 1 (GRF) Magnitude of reaction force for constraint d Nomenclature p 0 1 d d d err fns friction geom n {e}1 {n}1 0 D d x y {a}1 d p xxi Magnitude of reaction force for constraint p Magnitude of reaction force for constraint Friction coefficient Tyre to road coefficient of static friction Tyre to road coefficient of sliding friction Signaltonoise ratio Density Standard deviation Standard deviation of attribute d 3rd Euler angle rotation 1st Euler angle rotation Yaw rate (Wenzel model) Yaw rate Dampednatural frequency Demanded yaw rate Yaw rate error Front axle noslip yaw rate Yaw rate from limiting friction Yaw rate from geometry Undamped natural frequency Angular velocity vector for part n resolved parallel to frame e Angular velocity vector for part n resolved parallel to frame 1 (GRF) Angular velocity of free rolling wheel Angular velocity of driven wheel Allowable range for attribute d Change in longitudinal position of wheel (Concept suspension) Change in lateral position (half/track) of wheel (Concept suspension) Change in steer angle (toe in/out) of wheel (Concept suspension) Change in camber angle of wheel (Concept suspension) Vector constraint equation resolved parallel to frame 1 (GRF) Scalar constraint expression for constraint d Scalar constraint expression for constraint p Scalar constraint expression for constraint This page intentionally left blank 1 Introduction The most costeffective analysis activity is accurately recalling and comprehending what has gone before. 1.1 Overview In 1969, man travelled to the moon and back, using maths invented by Kepler, Newton and Einstein to calculate trajectories hundreds of thousands of miles long and spacecraft with less onboard computing power than today’s pocket calculator. With today’s computing power and the mathematical frameworks handed down to us by Newton and Lagrange, it is scarcely credible that the motor car, itself over 100 years old, can exercise so many minds and still show scope for improvement. Yet we are still repeating errors in the dynamic design of our vehicles that were made in the 1960s. Some recent car designs, not named for legal reasons, exist that reproduce with distressing familiarity the abrupt rearward transfer of roll moment prototyped by the early Triumph Vitesse and Chevrolet Corvair. Vehicle manufacturers are not currently forced by legislation to achieve a measurable standard of vehicle handling and stability. International standards exist that outline procedures for proving ground tests with new vehicles but these are nothing more than recommendations. Vehicle manufacturers make use of many of the tests but in the main will develop and test vehicles using incompany experience and knowledge to define the test programme. In the absence of legislated standards, vehicle manufacturers are driven by market forces. Journalists report favourably on vehicles they enjoy driving – whether or not these are safe in the hands of the general public – and the legal profession seeks every opportunity to blur the distinction between bad driving and poor vehicle design. Matters are further complicated by market pressures driving vehicle designs to be too tall for their width – city cars and sportutility vehicles have this disadvantage in common. The growth in media attention and reporting to the public is undoubtedly significant. In recent years the most wellpublicized example of this was the reported rollover of the Mercedes AClass during testing by the motoring press (Figure 1.1). The test involves a slalom type manoeuvre and became popularly known as the ‘elk test’. Criticisms of the manoeuvre being overly severe were addressed by suggesting that it represents a panic swerve around a fictional elk on Scandinavian roads. The media attention was little short of disastrous for Mercedes as the car was already in production and thousands were recalled and modified. The fact that following modification the car passed the test and that Mercedes were also able to demonstrate that competitive vehicles in that class also failed the test was barely noticed. By that stage the damage had already been done. 2 Multibody Systems Approach to Vehicle Dynamics It seems to the authors there are two camps for addressing the vehicle dynamics problem. In one is the practical ride and handling expert. Skilled at the driving task and able to project themselves into the minds of a variety of different possible purchasers of the vehicle, they are able to quickly take an established vehicle design and adjust its character to make it acceptable for the market into which it will be launched. Rarely, though, are experts from this camp called upon to work in advance on the concept or detail of the vehicle design. The second camp contains theoretical vehicle dynamics experts. They are skilled academics in the mould of Leonard Segel who in 1956 published his ‘Theoretical prediction and experimental substantiation of the responses of the automobile to steering control’ (Segel, 1956). Typical of the vehicles available at that time is the Chevrolet BelAir convertible shown in Figure 1.2. They were rarely commended for their dynamic qualities. Segel’s work, and that of others from this era including the earlier work ‘Road manners of the modern motor car’ (Olley, 1945), made way for all Fig. 1.1 Rollover of the Mercedes AClass (courtesy of Auto Motor und Sport) Fig. 1.2 1957 Chevrolet BelAir convertible Introduction 3 subsequent ‘classical’ vehicle dynamic analysis and forms a firm foundation upon which to build. It is rare for an expert from this camp to be part of the downstream vehicle development process. 1.2 What is vehicle dynamics? The field known as vehicle dynamics is concerned with two aspects of the behaviour of the machine. The first is isolation and the second is control. Figure 1.3 is the authors’ subjective illustration of the intricacy and interconnection of the tasks to be approached; it is not claimed to be authoritative or complete but is rather intended as a thoughtstarter for the interested reader. Isolation is about separating the driver from disturbances occurring as a result of the vehicle operation. This, too, breaks into two topics: disturbances the vehicle generates itself (engine vibration and noise, for example) and those imposed upon it by the outside world. The former category is captured by the umbrella term ‘refinement’. The disturbances in the latter category are primarily road undulations and aerodynamic interaction of the vehicle with its surroundings – crosswinds, wakes of structures and wakes of other vehicles. The behaviour of the vehicle in response to road undulations is referred to as ‘ride’ and could conceivably be grouped with refinement, though it rarely is in practice. There is some substantial crossover in aerodynamic behaviour between isolation and control, since control implies the rejection of disturbances Vehicle dynamics Isolation Control Speed Externally generated disturbances Internally generated disturbances Path Aerodynamic Stability Agility Fidelity Road surface Vertical Longitudinal Linearity Lateral Refinement Ride Subjective Fig. 1.3 Vehicle dynamics interactions Handling Objective Performance 4 Multibody Systems Approach to Vehicle Dynamics (‘fidelity’) and an absence of their amplification (‘stability’). Similarly, one response to road disturbances is a change in the vertical load supported by the tyre; this has a strong influence on the lateral force the tyre is generating at any given instant in time and is thus crucial for both fidelity and stability. It can be seen with some little reflection that one of the difficulties of vehicle dynamics work is not the complexity of the individual effects being considered but rather the complexity of their interactions. Control is concerned largely with the behaviour of the vehicle in response to driver demands. The driver continuously varies both path curvature and speed, subject to the limits of the vehicle capabilities, in order to follow an arbitrary course. Speed variation is governed by vehicle mass and tractive power availability at all but the lowest speed, and is easily understood. Within the performance task, issues such as unintended driveline oscillations and tractive force variation with driver demand may interact strongly with the path of the vehicle. The adjustment of path curvature at a given speed is altogether more interesting. In a passenger car, the driver has a steering wheel, which for clarity will be referred to as a handwheel1 throughout the book. The handwheel is a ‘yaw rate’ demand – a demand for rotational velocity of the vehicle when viewed from above. The combination of a yaw rate and a forward velocity vector that rotates with the vehicle gives rise to a curved path. There is a maximum path curvature available in normal driving, which is the turning circle, available only at the lowest speeds. In normal circumstances (that is to say in daytoday road use) the driver moves the handwheel slowly and is well within the limits of the vehicle capability. The vehicle has no difficulty responding to the demanded yaw rate. If the driver increases yaw rate demand slightly then the vehicle will increase its yaw rate by an appropriate amount. This property is referred to as ‘linearity’; the vehicle is described as ‘linear’ (Figure 1.4). For the driver, the behaviour of the vehicle is quite instinctive. A discussion of the analysis and interpretation of vehicle linearity and departure from linearity is given in Chapter 7. In the linear region, the behaviour of the vehicle can be represented as a connected series of ‘steady state’ events. Steady state is the condition in which, if the handwheel remains stationary, all the vehicle states – speed, yaw rate, path curvature and so on – remain constant and is more fully defined in Chapter 7. The steady state condition is easy to represent using an equilibrium analogy, constructed with the help of socalled ‘centrifugal force’. It should be noted that this fictitious force is invented solely for convenience of calculation of an analogous equilibrium state, or the calculation of forces in an accelerating frame of reference. When a vehicle is travelling on a curved path it is not in equilibrium. The curved path of the vehicle requires some lateral acceleration. Correctly, the lateral acceleration on a cornering vehicle is a centripetal 1 ‘Steering wheel’ could mean a roadwheel which is steered, or a wheel held by the driver. In generic discussions including vehicles other than fourwheeled passenger cars (motorcycles, tilting tricycles, etc.) then ‘steering wheel’ contains too much ambiguity; therefore, ‘handwheel’ is preferred since it adds precision. Introduction 5 Fig. 1.4 Linearity: more handwheel input results in proportionally more yaw rate (vehicle on left) 2 3 1 Fig. 1.5 A thought experiment comparing centripetal acceleration with linear acceleration acceleration – ‘centre seeking’. Note that speed is not the same as velocity; travelling in a curved path with a constant speed implies a changing direction and therefore a changing velocity. The centripetal acceleration definition causes some problems since everyone ‘knows’ that they are flung to the outside of a car if unrestrained and so there is much lax talk of centrifugal forces – ‘centre fleeing’. To clarify this issue, a brief thought experiment is required (Figure 1.5). Imagine a bucket of water on a rope being swung around by a subject. If the subject looks at the bucket then the water is apparently pressed into the bucket by the mythical ‘centrifugal force’ (presuming the bucket is being swung fast enough). If the swinging is halted and the bucket simply suspended by the rope then the water is held in the bucket by the downward gravitational field of the earth – the weight of the water pulls it into the bucket. Imagine now a different scenario in which the bucket (on a frictionless plane) is pulled horizontally towards the observer at a constant acceleration in a linear fashion. It’s best not to 6 Multibody Systems Approach to Vehicle Dynamics complicate the experiment by worrying about what will happen when the bucket reaches the subject. It is this third scenario and not the second that is useful in constructing the cornering case. If both the first and third cases are imagined in a zero gravity environment, they still work – the water will stay in the bucket. Note that for the third scenario – what we might call the ‘inertial’ case as against the gravitational case in the second scenario – the acceleration is towards the open end of the bucket. This is also true for the first scenario, in which the bucket is swung; the acceleration is towards the open end of the bucket and is towards the subject – i.e. it is centripetal. That the water stays in the bucket is simply a consequence of the way the bucket applies the centripetal force to the water. Thus the tyres on a car exert a force towards the centre of a turn and the body mass is accelerated by those forces centripetally – in a curved path. An accelerometer in the car is effectively a load cell that would be between the bucket and the water in the scenarios here and so it measures the centripetal force applied between the calibrated mass within the accelerometer (the water) and its support in the casing (the bucket). The socalled centrifugal force is one half of an action–reaction pair within the system but a freebody diagram of the bucket and rope in all three cases shows tension in the rope as an externally applied force when considering the rope as a separate free body. Only in case 2 is the bucket actually in equilibrium, with the addition of the gravitational force on the bucket and water. Therefore an accelerometer (or an observer) in the vehicle apparently senses a centrifugal force while theoretical vehicle dynamicists talk always of centripetal acceleration. Changing the sign on the inertial force, so that it is now a d’Alembert force, appears to solve the apparent confusion. This can be misleading as we now have the impression that the analysis of the cornering vehicle is a static equilibrium problem. The water is not in equilibrium when travelling in a curved path, and neither is a car. Centripetal forces accelerate the vehicle towards the centre of the turn. This acceleration, perpendicular to the forward velocity vector, is often referred to as ‘lateral’ acceleration, since the vehicle generally points in the direction of the forward velocity vector (see Chapter 7 for a more precise description of the body attitude). It can be seen that the relationship between centripetal acceleration, Ap, yaw rate, , forward velocity, V, and radius of turn, R, is given by: Ap V2/R V 2R (1.1) The absolute limit for lateral acceleration, and hence yaw rate, is the friction available between the tyres and the road surface. Competition tyres (‘racing slicks’) have a coefficient of friction substantially in excess of unity and, together with large aerodynamic downforces, allow a lateral acceleration in the region of 30 m/s2, with yaw rates correspondingly over 40 deg/s for a speed of 40 m/s (90 mph). For more typical road vehicles, limit lateral accelerations rarely exceed 9 m/s2, with yaw rates correspondingly down at around 12 deg/s at the same speed. However, for the tyre behaviour to remain substantially linear, for a road car the lateral accelerations must be generally less than about 3 m/s2, so yaw rates are down to a mere 4 deg/s at the same speed. Introduction 7 While apparently a small fraction of the capability of the vehicle, there is much evidence to suggest that the driving population as a whole rarely exceed the linearity limits of the vehicle at speed and only the most confident exceed them at lower speeds (Lechmer and Perrin, 1993). Accident investigators rarely presume a lateral acceleration of greater than 4 m/s2 when reconstructing road traffic accidents even on dry roads unless there is strong evidence from the witness marks on the road surface. When racing or during emergency manoeuvres on the road – typically attempting to avoid an accident – the vehicle becomes strongly ‘nonlinear’. The handwheel is moved rapidly and the vehicle generally has difficulty in responding accurately to the handwheel. This is the arena called ‘transient handling’ and is correctly the object of many studies during the product design process. In contrast to the steady state condition, all the vehicle states fluctuate rapidly and the expressions above are modified. Steady state and transient behaviour are connected. While good steady state behaviour is connected with good transient behaviour, it is not in itself sufficient (Sharp, 2000). Transient handling studies concentrate on capturing, analysing and understanding the yaw moments applied to the vehicle and its response to them. Those moments are dominated by the lateral and longitudinal forces from the tyres. For road cars, additional aerodynamic contributions are a small modifier but for racing the aerodynamic behaviour rises in importance. The generation of tyre forces is frequently the biggest source of confusion in vehicle dynamics, since both lateral and longitudinal mechanisms are neither obvious nor intuitive. Tyres are dealt with in some depth by Pacejka (2002) in a companion volume in this series and also have some further coverage in Chapter 5. The tyres generate lateral forces by two mechanisms, ‘camber’ and ‘slip angle’. Camber is the angle at which the tyre is presented to the road when viewed from the front. There exists some confusion when referring to and measuring camber angle; for clarity within this text camber angle is measured with respect to the road unless explicitly defined as being relative to the vehicle body. It is the angle with respect to the road that generates a side force. Thus a motorcycle runs a large camber angle when cornering but runs no camber angle with respect to the vehicle body. Slip angle is the angle at which the moving tyre is presented to the road when viewed in plan. It is important to note that slip angle only exists when the vehicle is in motion. At a standstill (and at speeds under about 10 mph) the lateral stiffness of the tyres generates the forces that constrain the vehicle to its intended path. As speed rises above walking pace, the tyres have a falling static lateral stiffness until above about 5 m/s (12 mph) they have effectively none; an applied lateral force, such as a wind load, will move the vehicle sideways from its intended path. It is important to note that the presence of a slip angle does not necessarily imply sliding behaviour at the contact patch. Slip angle forces are typically more than 20 times camber forces for a particular angle, and are thus the more important aspect for vehicle dynamics. 8 Multibody Systems Approach to Vehicle Dynamics The lateral forces induced by the angles are strongly modified by the vertical loads on the tyres at each moment. The tyres generate longitudinal forces by spinning at a speed different to their ‘freerolling’ speed. The freerolling speed is the speed at which the wheel and tyre would spin if no brake or drive forces are applied to them. The difference in speed is described as ‘slip ratio’, which is unfortunate since it is confusingly similar to slip angle. It is expressed as a percentage, so, for example, a tyre turning with a 5% slip ratio will perform 105 revolutions to travel the same distance as a freerolling tyre performing 100 revolutions. In doing so it will impart a tractive force to the vehicle. A 5% slip ratio would imply 95 revolutions of the same wheel and the presence of a braking force. Managing lateral tyre forces by controlling slip and camber angles is the work of the suspension linkage. For the front wheels, the driver has the ability to vary the slip angle using the handwheel. Managing the vertical loads on the tyres is the function of the suspension ‘calibration’ (springs, dampers and any active devices which may be present). Chapter 4 deals with suspension analysis in some detail. Management of longitudinal forces is the role of the vehicle driveline and braking system, including ABS or brake intervention systems. A vehicle travelling in a straight line has a yaw velocity of zero and a centripetal acceleration of zero. When travelling in a steady curve, the centripetal acceleration is not zero and the yaw rate is not zero; both are constant and are related as described in equation (1.1). In performing the transition from straight running to a curved path there must be a period of yaw acceleration in order to acquire the yaw velocity that matches the centripetal acceleration. The yaw acceleration is induced and controlled by yaw moments acting on the vehicle yaw inertia. Transient handling therefore implies the variation of yaw moments applied to the vehicle. Those moments are applied by aerodynamic behaviour and the forcegenerating qualities of the tyres at a distance from the vehicle centre of mass. No other mechanisms exist for generating a meaningful yaw moment on the vehicle; while gyroscopic torques exist associated with camber changes, they are small. For road vehicles, the aerodynamic modifications are generally small. Multibody system methods allow the convenient exploration of aspects of the vehicle design that influence those qualities of the tyres. Chapter 5 addresses different methods of modelling those aspects of the vehicle and their relative merits. In order to progress from travelling in a straight line to travelling in a curved path, the following sequence of events is suggested: 1. The driver turns the handwheel, applying a slip angle at the front wheels. 2. After a delay associated with the front tyre relaxation lengths, side force is applied at the front of the vehicle. Lateral and yaw accelerations exist. Introduction 9 3. The body yaws (rotates in plan), applying a slip angle at the rear wheels. 4. After a delay associated with the rear tyre relaxation lengths, side force is applied at the rear of the vehicle. Lateral acceleration is increased, yaw acceleration is reduced to zero. In the real world, the driver intervenes and the events run into one another rather than being discrete as suggested here, but it is a useful sequence for discussion purposes. A similar sequence of events describes the return to straightline travel. Any yaw rate adjustments made by the driver follow similar sequences, too. During the period of yaw acceleration (stages 2 and 3 above) there exists the need for an excess of lateral forces from the front tyres when compared to the rear in order to deliver the required yaw moment. At the end of this period, that excess must disappear. Side force requirements for the rear tyre are thus increasing while those for the front tyres are steady or decreasing. To understand the significance of this fact, some further understanding of tyre behaviour is necessary. To a first approximation, camber forces may be neglected from tyre behaviour for vehicles that do not roll (lean) freely. Slip angle is the dominant side force generation mechanism. It is important to note that a tyre will adjust its slip angle to support the required side force, and not the other way around. This is a frequent source of difficulty in comprehending vehicle dynamic behaviour. All tyres display a slip angle at which the maximum side force is generated, referred to as the ‘critical slip angle’. If a force is required which is greater than that which can be generated at the critical slip angle, the tyre will run up to and then beyond the critical slip angle. Beyond the critical slip angle the side force falls off with slip angle, and so an increasing amount of lateral force is available to accelerate the growth of slip angle once the critical slip angle is passed. Returning to the vehicle, the side force requirement for the rear tyres is increasing while that for the front tyres is steady or decreasing. The rear tyres will be experiencing a growing slip angle, while the front tyres experience a steady or reducing one. If at this time the rear tyres exceed their critical slip angle, their ability to remove yaw moment is lost. The only possible way for yaw moment to be removed is by a reduction in the front tyre forces. If the yaw moment persists then yaw acceleration persists. With increasing yaw velocity, the slip angle at the front axle is reduced while that at the rear axle is increased further, further removing the rear tyres’ ability to remove yaw moment from the vehicle. If the front tyres are past their critical slip angle, too, the normal stabilization mechanism is reversed. The result is an accelerating spin that departs rapidly from the driver’s control. The modelling and interpretation of such events is dealt with in Chapter 7. The behaviour of the driver is important to the system performance as a whole. The driver is called on to act as a yaw rate manager, acting on the vehicle controls as part of a closedloop feedback system to impart the yaw moments required to control the yaw rate of the vehicle. At critical times, the workload of the driver may exceed his or her capability, resulting in a loss of control. 10 Multibody Systems Approach to Vehicle Dynamics The goal of vehicle dynamics work is to maintain the vehicle behaviour within the bounds that can be comprehended by and controlled by the driver. 1.3 Why analyse? In any real productengineering programme, particularly in the ground vehicle industry, there are always time constraints. The need to introduce a new product to retain market share or to preserve competitive advantage drives increasingly tight timetabling for product engineering tasks. In the western world, and to a growing extent in the developing world, tastes are becoming ever more refined such that the demand for both quality of design and quality of construction is increasing all the while. Unlike a few decades ago, there are few genuinely bad products available. It seems, therefore, that demands for better products are at odds with demands for compressed engineering timetables. This is true; the resolution of this conflict lies in improving the efficiency of the engineering process. It is here that predictive methods hold out some promise. Predictive methods notionally allow several good things: ● Improved comprehension and ranking of design variables ● Rapid experimentation with design configurations ● Genuine optimization of numerical response variables Therefore the use of predictive methods is crucial for staying ‘ahead of the game’ in vehicle engineering. 1.4 Classical methods These methods are taught formally in universities as part of the syllabus. While they can be daunting at first sight, they are elegant and can prove tremendously illuminating in forming a holistic framework for what can easily be a bewildering arena. The best practitioners of the art (Crolla, Sharp, Hales, Hemingway and the Millikens to name some) recommend the use of a bodycentred state–space formulation. While full of simplifications, useful insights can be gained by studying a two degreeoffreedom model for typical passenger cars. With a reasonable increase in sophistication but well worth the effort is the elaboration to three degrees of freedom (four states) to include the influence of suspension roll. Such classical models help the analyst discern ‘the wood for the trees’ – they easily bring forth, for example, the influence of suspension steer derivatives on straightline stability. In this they contrast strongly with ‘literal’ linkage models, in which all the problems of real vehicles (the lack of isolation of single effects) can sometimes preclude their ranking and comprehension. Although the task of deriving the equations of motion and arranging the terms for subsequent solution is laborious and may be errorprone, the proponents of the method point quite correctly to the increased comprehension of the problem to which it leads. Introduction Aspiration 11 Review Definition Confirmation Analysis Simulation Composition Decomposition Synthesis Fig. 1.6 ‘V’ process for product design 1.5 Analytical process It is clear that the tasks undertaken have expanded to fit the time available. Despite increases in computing power of a factor of 100 or more over the last two decades, analysis tasks are still taking as long to complete as they always have done. The increased computing power available is an irresistible temptation to add complexity to predictive models (Harty, 1999). Complex models require more data to define them. This data takes time to acquire. More importantly, it must exist. Early in the design cycle, it is easy to fall into the ‘paralysis of analysis’; nothing can be analysed accurately until it is defined to a level of accuracy matching the complexity of the modelling technique. More than the model itself, the process within which it fits must be suited to the tasks at hand. There is nothing new in the authors’ observations; Sharp (1991) comments: Models do not possess intrinsic value. They are for solving problems. They should be thought of in relation to the problem or range of problems which they are intended to solve. The ideal model is that with minimum complexity which is capable of solving the problems of concern with an acceptable risk of the solution being ‘wrong’. This acceptable risk is not quantifiable and it must remain a matter of judgement. However, it is clear that diminishing returns are obtained for model elaboration. Any method of analysis must be part of a structured process if it is to produce useful results in a timely manner. Interesting results that are too late to influence product design are of little use in modern concurrent2 engineering practice. Rapid results that are so flawed as to produce poor engineering decisions are also of little use. The use of predictive methods within vehicle design for addressing dynamic issues with the vehicle should follow a pattern not dissimilar to that in Figure 1.6, whatever the problem or the vehicle. 2 Concurrent taking place at the same time. ‘Concurrent engineering’ was a fashionable phrase in the recent past and refers to the practice of considering functional, cost and manufacturing issues together rather than the historically derived ‘sequential’ approach. It was also referred to as ‘simultaneous engineering’ for a while, though the segmented connotations of simultaneous were considered unhelpful and so the ‘concurrent’ epithet was adopted. 12 Multibody Systems Approach to Vehicle Dynamics Aspiration: The method properly starts with the recognition of the end goals. In some organizations, confusion surrounds this part of the process, with obfuscation between targets, objectives and goals; the terms are used differently between organizations and frequently with some differences between individuals in the same organization. Cutting through this confusion requires time and energy but is vital. Definition: After definition, a clear description of ‘success’ and ‘failure’ must exist; without it the rest of the activities are, at best, wasteful dissipation. Aspirations are frequently set in terms of subjective comparisons – ‘Ride comfort better than best in class’. To usefully feed these into an analytical process, they must be capable of being quantified – i.e. of having numbers associated with them. Without numbers, it is impossible to address the task using analytical methods and the analytical process should be halted. This is not to say that product development cannot continue but that to persist with a numerical process in the absence of numbers becomes folly. Analysts and development staff must be involved closely with each other and agree on the type of numerical data that defines success, how it is calculated predictively and how it is measured on a real vehicle. Commonly, some form of ‘benchmarking’ study – a measurement exercise to quantify the current best performers – is associated with this stage. The activity to find the benchmark is a useful shakedown for the proposed measurement processes and is generally a fruitful education for those involved. Analysis: When success and failure have been defined for the system as a whole, the individual parts of the system must be considered. There is generally more than one way to reach a system solution by combining individual subsystems or elements. During this stage of the process, some decisions must be taken about what combination is preferred. It may be, for example, that in seeking a certain level of vehicle performance there is a choice between increasing power output and reducing weight in order to achieve a given powertoweight ratio. That choice will be influenced by such simple things as cost (saving weight may be more expensive than simply selecting a larger engine from the corporate library) or by more abstruse notions (the need to be seen to be ‘environmentally friendly’, perhaps). The task of analysis is to illuminate that choice. The analysis carried out must be sufficiently accurate but not excessively so. ‘Simple models smartly used’ is the order of the day for analysis work. The analysis may consider many possible combinations in order to recommend a favoured combination. This activity is sometimes referred to by Introduction 13 the authors as ‘mapping the design space’ – producing guidance for those who wish to make design decisions based on wider considerations and who wish to comprehend the consequences of their decisions. The most costeffective activity at this stage is accurately recalling and comprehending what has gone before. Decomposition: Once the analytical stage is complete, it is time for design decisions to be made. The whole entity must be decomposed into its constituent parts, each of which has design goals associated with it – cost, performance, weight, etc. It is at this time the first real design decisions are made that shape the product. Those decisions are to be made in the light of the preceding analysis. Synthesis: Once the design is decomposed into manageable portions, the task of synthesizing (creating) the design begins. During this phase, analytical tools are used to support individual activities and verify the conformance of the proposed design with the intended design goals. An example of this might be the use of kinematic simulation to verify that the suspension geometry characteristics are those required. Discerning the requirement itself is the function of the earlier decomposition phase. Composition: The reassembly of the separate portions of the design, each of which by now has a high level of confidence at reaching its individual design goals. Simulation: Might also be referred to as ‘virtual prototyping’. High fidelity models that have been a long time in preparation are used to assess predictively, in some detail, the behaviour of the whole design. Models prepared during the synthesis activity are taken and reused. It is in this arena that great strides have been made in terms of processing power, model reuse and interpackage integration over the last decade. Unfortunately, in the minds of some, these super elaborate models are all that is possible and anything less is simply worthless, passé and old fashioned. These are valuable models and have a crucial part to play in the process, but without a wellshaped concept design they are unwieldy white elephants. Also included in the simulation activity should be prototype vehicles, produced from nonrepresentative tools and/or processes, that are physical simulations instead of mathematical ones. The increasing use of ‘virtual’ prototyping obviates these physical prototypes except for those where an understanding of the man/machine interaction is necessary. One of several arenas where this remains true is the dynamics task. Confirmation: Signoff testing is to be carried out on real vehicles. This stage should reveal no surprises, as changes at this stage are expensive. 14 Multibody Systems Approach to Vehicle Dynamics Review: Once the design is successfully signed off, a stage that is frequently omitted is the review. What was done well? What could have been better? What technology do we wish we had then that might be available to us now? Since the most costeffective analysis activity is to recall accurately and comprehend what has gone before, a welldocumented review activity saves time and money in the next vehicle programme. The process described is not definitive, nor is it intended to be prescriptive. It should, however, illustrate the difference between ‘analysis’ and ‘simulation’ and clearly differentiate between them. 1.6 Computational methods Whether the equations of motion have been derived by hand or delegated to a commercial software package, the primary goal when considering vehicle dynamics is to be able to predict the timedomain solution to those equations. Once the equations of motion have been assembled, they are integrated numerically. This is a specialized field in its own right. There are many publications in the field and it is an area rife with difficulties and pitfalls for the unwary. However, in order to successfully use the commercially available software products, some comprehension of the difficulties involved are necessary for users. This chapter deals with some of the more common difficulties with some examples for the student. By far the most dangerous type of difficulty is the ‘plausible but wrong’ solution. Commercial analysts must studiously guard against the ‘garbage in, gospel out’ mentality that pervades the engineering industry at present. The equations can be solved in a fairly direct fashion as assembled by the commercial package preprocessor or they can be subject to further symbolic manipulation before numerical solution. Socalled ‘symbolic’ codes, discussed later, offer some tremendous computational efficiency benefits and are being hailed by many as the future of multibody system analysis since they allow realtime computation of reasonably complex models without excessive computing power. The prospect of a realtime multibody system of the vehicle solved onboard in order to generate reference signals for the generationafternext vehicle control systems seems genuine. 1.7 Computerbased tools Multibody systems analysis software has become so easy to use that many users lack even a basic awareness of the methods they are using. This chapter charts the background and development to the current generation of multibody systems analysis programs. While the freedom from the purgatory of formulating one’s own equations of motion is a blessing, it is partly this purgatory that aids the analyst’s final understanding of the problem. Chapter 2, Kinematics and dynamics of rigid bodies, is intended as a reference and also as a ‘launch pad’ for the enthusiastic readers to be able to teach themselves the process of socalled ‘classical’ modelling. Introduction 15 The best multibody system codes now include, specifically for the vehicle handling task, a ‘concept level’ model that has no literal detail but instead a ‘wheel trajectory map’. In the near future, it seems that reverse engineering tools for deducing the required wheel trajectory map will become available. Crolla (1995) identifies the main types of computerbased tools which can be used for vehicle dynamic simulation and categorizes these as: (i) Purposedesigned simulation codes (ii) Multibody simulation packages that are numerical (iii) Multibody simulation packages that are algebraic (symbolic) (iv) Toolkits such as MATLAB One of the major conclusions that Crolla draws is that it is still generally the case that the ride and handling performance of a vehicle will be developed and refined mainly through subjective assessments. Most importantly he suggests that in concentrating on sophistication and precision in modelling, practising vehicle dynamicists may have got the balance wrong. This is an important issue that reinforces the main approach in this book, which is to encourage the application of models that lead to positive decisions and inputs to the vehicle design process. Crolla’s paper also provides an interesting historical review that highlights an important meeting at IMechE headquarters in 1956, ‘Research in automobile stability and control and tyre performance’. The author states that in the field of vehicle dynamics the papers presented at this meeting are now regarded as seminal and are referred to in the USA as simply ‘The IME Papers’. One of the authors at that meeting, Segel, can be considered to be a pioneer in the field of vehicle dynamics. His paper (Segel, 1956) is one of the first examples where classical mechanics has been applied to an automobile in the study of lateral rigid body motion resulting from steering inputs. The paper describes work carried out on a Buick vehicle for General Motors and is based on transferable experience of aircraft stability gained at the Flight Research Department, Cornell Aeronautical Laboratory (CAL). The main thrust of the project was the development of a mathematical vehicle model that included the formulation of lateral tyre forces and the experimental verification using instrumented vehicle tests. Another author at the meeting, Milliken (Milliken and Whitcomb, 1956), has also continued to make a significant contribution to the discipline. In 1993, almost 40 years after embarking on this early work in vehicle dynamics, Segel again visited the IMechE to present a comprehensive review paper (Segel, 1993), ‘An overview of developments in road vehicle dynamics: past, present and future’. This paper provides a historical review that considers the development of vehicle dynamics theory in three distinct phases: Period 1 – Invention of the car to early 1930s Period 2 – Early 1930s to 1953 Period 3 – 1953 to present 16 Multibody Systems Approach to Vehicle Dynamics In describing the start of Period 3 Segel references his early ‘IME paper’ (Segel, 1956). In terms of preparing a review of work in the area of vehicle dynamics there is an important point made in the paper regarding the rapid expansion in literature that makes any comprehensive summary and critique difficult. This is highlighted by the example of the 1992 FISITA Congress where a total of seventy papers were presented under the general title of ‘Total Vehicle Dynamics’. Following Segel’s historical classification of the vehicle dynamics discipline to date, the authors of this text suggest that we have now entered a fourth era that may be characterized by the use of engineering analysis software as something of a ‘commodity’, bought and sold and often used without a great deal of formal comprehension. In these circumstances there is a need for the software to be absolutely watertight (currently not possible to guarantee) or else for a small number of experts – ‘champions’ – within organizations to ensure the ‘commodity’ users aren’t drifting off the rails, to use a horribly mixed metaphor. This mode of operation is already becoming established within the analysis groups of large automotive companies where analysts make use of customized software programs such as ADAMS/Car. These programs have two distinct types of usage. At one level the software is used by an ‘expert’ with the experience, knowledge and skill to customize the models generated, the types of simulation to be performed and the format in which selected results will be presented. A larger group of ‘standard’ users are then able to use the program to carry out suspension or full vehicle simulations assuming little or no knowledge of multibody systems formulations and solution methods. The authors in Hogg et al. (1992) give further insights into how computer models and simulation programs are used by industry in the field of road vehicle dynamics. In this case the company is Lotus. In this paper the authors describe how simulation tools can be used at various stages in the design process. This includes the manner in which MSC.ADAMS is used to ‘tune’ a suspension design during development to produce, for example, very low but accurately controlled levels of steer change during suspension stroke. Hogg et al. (1992) continue to describe how for vehicle handling they used their own Simulation and Analysis Model (SAM). This functional model required a minimum of design information and used input parameters that can be obtained by measurement of suspension characteristics using a static test rig. The SAM model had 17 rigid body degrees of freedom (DOF). The paper identified that the vehicle body contributed 6 of these DOF and that each corner suspension unit had 2 DOF, one of which was the rotation of the road wheel and another that allowed vertical movement relative to the vehicle body. The suspensions were modelled to pivot about an instant centre. This is the same approach used with the swing arm full vehicle model described later in Chapter 6. The SAM model also had 3 DOF associated with steering which suggests steering torque inputs and the modelling of compliance in the steering system. The SAM model used an early version of the tyre model proposed by Pacejka and his associates (Bakker et al., 1986). The use of MSC.ADAMS by Lotus for handling simulations is also described in this paper (Hogg et al., 1992). In this case an example output Introduction 17 shows good correlation between MSC.ADAMS and test measurements when comparing yaw rate for an 80 kph lane change manoeuvre. It is also stated, however, that this model had over 200 DOF and used the Pacejka tyre model that required up to 50 parameters. Pilling (1995) also gives information about the work at Lotus in the field of vehicle dynamics and simulation with an emphasis on the role of the tyre. 1.8 Commercial computer packages General purpose programs such as MSC.ADAMS have been developed with a view to commercial gain and as such are able to address a much larger set of problems across a wide range of engineering industries. In addition to the automotive industry MSC.ADAMS is an established tool within the aerospace, large construction, electromechanical and the general mechanical engineering industries. The general nature of the program means that within any one industry the class of applications may develop and extend over a broad range. The MSC.ADAMS program is typical of the range of multibody analysis programs described as numeric where the user is concerned with assembling a physical description of the problem rather than writing equations of motion. A comprehensive overview of MSC.ADAMS is provided by Ryan (1990), although since the date of that publication the development of the software has moved on considerably, particularly in the area of graphical pre and postprocessing. Blundell (1999; 2000a,b) published a series of four IMechE papers with the aim of summarizing typical processes involved with using MSC.ADAMS to simulate full vehicle handling manoeuvres. The first paper provided an overview of the usage of multibody systems analysis in vehicle dynamics. The second paper described suspension modelling and analysis methodologies. The third paper covered tyre modelling and provided example routines used with MSC.ADAMS for different tyre models and data. The fourth and final paper brought the series together with a comparative study of full vehicle models, of varying complexity, simulating a double lane change manoeuvre. Results from the simulation models were compared with measured test data from the proving ground. The overall emphasis of the series of papers was to demonstrate the accuracy of simple efficient models based on parameters amenable to design sensitivity study variations rather than blindly modelling the vehicle ‘as is’. Before the evolution of programs like MSC.ADAMS, engineers analysed the behaviour of mechanisms such as camfollowers and four bar linkages on the basis of pure kinematic behaviour. Graphical methods were often used to obtain solutions. Chace (1985) summarizes the early programs that led to the development of the MSC.ADAMS program. One of the first programs (Cooper et al., 1965) was KAM (Kinematic Analysis Method) capable of performing displacement, velocity and acceleration analysis and solving reaction forces for a limited set of linkages and suspension models. Another early program (Knappe, 1965) was COMMEND (ComputerOrientated Mechanical Engineering Design) which was used for planar problems. 18 Multibody Systems Approach to Vehicle Dynamics The origin of MSC.ADAMS can be traced back to a program of research initiated by Chace at the University of Michigan in 1967. By 1969 Chace (1969, 1970) and Korybalski (Chace and Korybalski, 1970) had completed the original version of DAMN (Dynamic Analysis of Mechanical Networks). This was historically the first general program to solve time histories for systems undergoing large displacement dynamic motion. This work led in 1971 to a new program DRAM (Dynamic Response of Articulated Machinery) that was further enhanced by Angel (Chace and Angel, 1977). The first program forming the basis of MSC.ADAMS was completed by Orlandea in 1973 and published in a series of two ASME papers (Orlandea et al., 1976a, b). This was a development of the earlier twodimensional programs to a threedimensional code but without some of the impact capability contained in DRAM at that time. Blundell (1991) describes how the MSC.ADAMS software is used to study the behaviour of systems consisting of rigid or flexible parts connected by joints and undergoing large displacement motion and in particular the application of the software in vehicle dynamics. The paper also lists a number of other systems based on MSC.ADAMS that had at that time been developed specifically for automotive vehicle modelling applications. Several of the larger vehicle manufacturers have at some time integrated MSC.ADAMS into their own inhouse vehicle design systems. Early examples of these were the AMIGO system at Audi (Hudi, 1988), and MOGESSA at Volkswagen (Terlinden et al., 1987). The WOODS system based on user defined worksheets was another system at that time in this case developed by German consultants for Ford in the UK (Kaminski, 1990). Ford’s global vehicle modelling activities have since focused on inhouse generated linear models and the ADAMS/Chassis™ (formerly known as ADAMS/Pre™) package, a layer over the top of the standard MSC.ADAMS pre and postprocessor that is strongly tailored towards productivity and consistency in vehicle analysis. Another customized application developed by the automotive industry is described in Scapaticci et al. (1992). In this paper the authors describe how MSC.ADAMS has been integrated into a system known as SARAH (Suspension Analyses Reduced ADAMS Handling). This inhouse system for the automotive industry was developed by the Fiat Research Centre Handling Group and used a suspension modelling technique that ignored suspension layout but focused on the final effects of wheel centre trajectory and orientation. At Leeds University a vehiclespecific system was developed under the supervision of Crolla. In this case all the commonly required vehicle dynamics studies have been embodied in their own set of programs (Crolla et al., 1994) known as VDAS (Vehicle Dynamics Analysis Software). Examples of the applications incorporated in this system included ride/handling, suspensions, natural frequencies, mode shapes, frequency response and steady state handling diagrams. The system included a range of models and further new models could be added using a preprocessor. Crolla et al. (1994) also define two fundamental types of MBS program, the first of which is that such as MSC.ADAMS where the equations are Introduction 19 generated in numerical format and are solved directly using numerical integration routines embedded in the package. The second and more recent type of MBS program identified formulates the equations in symbolic form and often uses an independent solver. The authors also describe toolkits as collections of routines that generate models, formulate and solve equations, and present results. The VDAS system is identified as falling into this category of computer software used for vehicle dynamics. Other examples of more recently developed codes formulate the equations algebraically and use a symbolic approach. Examples of these programs include MESA VERDE (Wittenburg and Wolz, 1985), AUTOSIM (Sayers, 1990), and RASNA Applied Motion Software (Austin and Hollars, 1992). Crolla (Crolla et al., 1992) provides a summary comparison of the differences between numeric and symbolic code. As stated MBS programs will usually automatically formulate and solve the equations of motion although in some cases such as with the work described by Costa (1991) and Holt (Holt and Cornish, 1992; Holt, 1994) a program SDFAST has been used to formulate the equations of motion in symbolic form and another program ACSL (Automatic Continuous Simulation Language) has been used to generate a solution. Specialpurpose programs are designed and developed with the objective of solving only a specific set of problems. As such they are aimed at a specific group of problems. A typical example of this type of program would be AUTOSIM described by Sayers (1990, 1992), Sharp (1997) and Mousseau et al. (1992) which is intended for vehicle handling and has been developed as a symbolic code in order to produce very fast simulations. Programs such as this can be considered to be special purpose as they are specifically developed for a given type of simulation but do, however, allow flexibility as to the choice and complexity of the model. An extension of this is where the equations of motion for a fixed vehicle modelling approach are programmed and cannot be changed by the user such as the HVOSM (HighwayVehicleObject Simulation Model) developed at the University of Michigan Transport Research Institute (UMTRI) (Sayers, 1992). The program includes tyre and suspension models and can be used for impact studies in addition to the normal ride and handling simulations. Crolla et al. (1992) indicate that the University of Missouri has also developed a light vehicle dynamics simulation (LVDS) program that runs on a PC and can produce animated outputs. In the mid1980s Systems Technology Inc. developed a program for vehicle dynamics analysis nonlinear (VDANL) simulation. This program is based on a 13 degree of freedom, lumped parameter model (Allen et al., 1987) and has been used by researchers at Ohio State University for sensitivity analysis studies (Tandy et al., 1992). The modelling of the tyre forces and moments at the tyre to road contact patch is one of the most complex issues in vehicle handling simulation. The models used are not predictive but are used to represent the tyre force and moment curves typically found through laboratory or roadbased rig testing of a tyre. Examples of tyre models used for vehicle handling discussed in this book include: (i) A sophisticated tyre model known as the ‘Magic Formula’. This tyre model has been developed by Pacejka and his associates (Bakker et al., 20 Multibody Systems Approach to Vehicle Dynamics 1986, 1989; Pacejka and Bakker, 1993) and is known to give an accurate representation of measured tyre characteristics. The model uses modified trigonometric functions to represent the shape of curves that plot tyre forces and moments as functions of longitudinal slip or slip angle. In recent years the work of Pacejka has become well known throughout the vehicle dynamics community. The result of this is a tyre model that is now widely used both by industry and academic institutions and is undergoing continual improvement and development. The complexity of the model does, however, mean that well over 50 or more parameters may be needed to define a tyre model and that software must be obtained or developed to derive the parameters from measured test data. (ii) The second model considered here is known as the Fiala tyre model (Fiala, 1954) and is provided as the default tyre model in MSC.ADAMS. This is a much simpler model that also uses mathematical equations to represent the tyre force and moment characteristics. Although not so widely recognized as Pacejka’s model the fact that this model is the default in MSC.ADAMS and is simpler to use led to its inclusion. The advantage of this model is that it only requires 10 parameters and that the physical significance of each of these is easy to comprehend making this a good starting point for students and newcomers to the discipline. The parameters can also be quickly and easily derived from measured test data without recourse to special software. It should also be noted, however, that this model unlike Pacejka’s is not suitable for combined braking and cornering and can only be used under pure slip conditions. The Fiala formulation also has some limitations at high slip angles and is thus unsuitable for limit manoeuvres even when pure slip conditions are included. (iii) The fourth modelling approach is to use a straightforward interpolation model. This was the original tyre modelling method used in MSC.ADAMS (Ryan, 1990). This methodology is still used by some companies but has, to a large extent, been superseded by more recent parameterbased models. The method is included here as a useful benchmark for the comparison of other tyre models in Chapter 5. Another tyre model is provided for readers as a source listing in Appendix B. This model (Blundell, 2003) has been developed by Harty and as with the Fiala model has the advantage of requiring only a limited number of input parameters. The implementation is more complete, however, than the Fiala model and includes representation of the following: ● Comprehensive slip ● Load dependency ● Camber thrust ● Post limit It has been found that the Harty model is robust when modelling limit behaviour including, for example, problems involving low grip or prolonged wheelspin. Introduction 21 1.9 Benchmarking exercises In addition to the software discussed so far other multibody systems analysis programs such as DADS, SIMPACK, and MADYMO are commercially available and used by the engineering community. A detailed description of each of these is beyond the scope of this text and in any case the rapid development of commercial software means any such assessment here would rapidly become outdated. In broad terms DADS and SIMPACK appear to be comparable with MSC.ADAMS although at the time of writing MSC.ADAMS is the most widely used, particularly by the vehicle dynamics community. MADYMO is a program recognized as having a multibody foundation with an embedded nonlinear finite element capability. This program has been developed by TNO in the Netherlands and complements their established crash test work with dummies. Recent developments in MADYMO have included the development of biofidelic humanoid models to extend the simulation of crash test dummies to ‘real world’ pedestrian impact scenarios. A detailed comparison between the various codes is beyond the capability of most companies when selecting an MBS program. In many ways the use of multibody systems has followed on from the earlier use of finite element analysis, the latter being approximately 10 years more mature as applied commercial software. Finite element codes were subject to a rigorous and successful series of benchmarks under the auspices of NAFEMS (National Agency for Finite Elements and Standards) during the 1980s. The published results provided analysts with useful comparisons between major finite element programs such as NASTRAN and ANSYS. The tests performed compared results obtained for a range of analysis methods with various finite elements. For the vehicle dynamics community Kortum et al. (1991) recognized that with the rapid growth in available multibody systems analysis programs a similar benchmarking exercise was needed. This exercise was organized through the International Association for Vehicle System Dynamics (IAVSD). In this study the various commercially available MBS programs were used to benchmark two problems. The first was to model the Iltis military vehicle and the second a fivelink suspension system. A review of the exercise is provided by Sharp (1994) where some of the difficulties involved with such a wideranging study are discussed. An example of the problems involved would be the comparison of results. With different investigators using the various programs at widespread locations a simple problem occurred when the results were sent in plotted form using different size plots and inconsistent axes making direct comparisons between the codes extremely difficult. It was also very difficult to ensure that a consistent modelling approach was used by the various investigators so that the comparison was based strictly on the differences between the programs and not the models used. An example of this with the Iltis vehicle would be modelling a leaf spring for which in many programs there were at the time no standard elements within the main code. Although not entirely successful the exercise was useful in being the only known attempt to provide a comparison between all the main multibody programs at the time. It should also be recognized that in the period since the exercise programs such as MSC.ADAMS have been extensively developed to add a wide range of capability. 22 Multibody Systems Approach to Vehicle Dynamics Anderson and Hanna (1989) have carried out an interesting study where they have used two vehicles to make a comparison of three different vehicle simulation methodologies. They have also made use of the Iltis, a vehicle of German design, which at that time was the current small utility vehicle used by the Canadian military. The Iltis was a vehicle that was considered to have performed well and had very different characteristics to the M151 Jeep that was the other vehicle in this study. The authors state that the M151 vehicle, also used by the Canadian military, had been declared unsafe due to a propensity for rolling over. Work has been carried out at the University of Bath (RossMartin et al., 1992) where the authors have compared MSC.ADAMS with their own hydraulic and simulation package. The results for both programs are compared with measured vehicle test data provided in this case by Ford. The Bath model is similar to the Roll Stiffness Model described later in this book but is based on a force roll centre as described by Dixon (1987). This requires the vehicle to actually exist so that the model can use measured inputs obtained through static rig measurements, using equipment of the type described by Whitehead (1995). The rollcentre model described in this book is based on a kinematic roll centre derived using a geometric construction as described in Chapter 4, though there is little to preclude a forcebased prediction by modelling the test rig on which the real vehicle is measured. As a guide to the complexity of the models discussed in RossMartin et al. (1992), the Bath model required 91 pieces of information and the MSC.ADAMS model, although not described in detail, needed 380 pieces of information. It is also stated in this paper that the MSC.ADAMS model used 150 sets of nonlinear data pairs that suggests detailed modelling of all the nonlinear properties of individual bushes throughout the vehicle. 2 Kinematics and dynamics of rigid bodies 2.1 Introduction The application of a modern multibody systems computer program requires a good understanding of the underlying theory involved in the formulation and solution of the equations of motion. Due to the threedimensional nature of the problem the theory is best described using vector algebra. In this chapter the starting point will be the basic definition of a vector and an explanation of the notation that will be used throughout this text. The vector theory will be developed to demonstrate, using examples based on suspension systems, the calculation of new geometry and changes in body orientation, such as the steer change in a road wheel during vertical motion relative to the vehicle body. This will be extended to show how velocities and accelerations may be determined throughout a linked threedimensional system of rigid bodies. The definition of forces and moments will lead through to the definition of the full dynamic formulations typically used in a multibody systems analysis code. 2.2 Theory of vectors 2.2.1 Position and relative position vectors Consider the initial definition of the position vector that defines the location of point P in Figure 2.1. In this case the vector that defines the position of P relative to the reference frame O1 may be completely described in terms of its components with magnitude Px, Py and Pz. The directions of the components are defined by Z1 P {RP}1/1 Pz Y1 O1 X1 Px Py Fig. 2.1 Position vector 24 Multibody Systems Approach to Vehicle Dynamics attaching the appropriate sign to their magnitudes: Px {RP}1/ 1 Py Pz (2.1) The use of brackets { } here is a shorthand representation of a column matrix and hence a vector. Note that it does not follow any quantity that can be expressed as the terms in a column matrix is also a vector. In writing the vector {RP}1/1 the upper suffix indicates that the vector is measured relative to the axes of reference frame O1. In order to measure a vector it is necessary to determine its magnitude and direction relative to the given axes, in this case O1. It is then necessary to resolve it into components parallel to the axes of some reference frame that may be different from that used for measurement as shown in Figure 2.2. In this case we would write {RP}1/2 where the lower suffix appended to {RP}1/2 indicates the frame O2 in which the components are resolved. We can also say that in this case the vector is referred to O2. Note that in most cases the two reference frames are the same and we would abbreviate {RP}1/1 to {RP}1. It is now possible in Figure 2.3 to introduce the concept of a relative position vector {RPQ}1. The vector {RPQ}1 is the vector from Q to P. It can also be described as the vector that describes the position of P relative to Q. Z1 X2 P {RP}1/2 Px2 Y1 Z2 O1 O2 Y2 X1 Py2 Pz2 Fig. 2.2 Resolution of position vector components Z1 {RPQ}1 P {RP}1 {RQ}1 O1 Y1 X1 Fig. 2.3 Relative position vector Q Kinematics and dynamics of rigid bodies 25 These vectors obey the triangle law for the addition and subtraction of vectors, which means that {RPQ}1 {RP}1 {RQ}1 {RP}1 {RQ}1 {RPQ}1 or (2.2) It also follows that we can write {RQP}1 {RQ}1 {RP}1 {RQ}1 {RP}1 {RQP}1 or (2.3) Application of Pythagoras’ theorem will yield the magnitude RP of the vector {RP}1 as follows: RP Px 2 Py 2 Px 2 (2.4) Similarly the magnitude RPQ of the relative position vector {RPQ}1 can be obtained using RPQ ( Px Qx )2 ( Py Qy)2 ( Pz Qz)2 (2.5) Consider now the angles X, Y and Z which the vector {RP}1 makes with each of the X, Y and Z axes of frame O1 as shown in Figure 2.4. This gives the direction cosines lx, ly and lz of vector {RP}1 where Px  RP  Py ly cos y  RP  Pz lz cos z  RP  lx cos x (2.6) These direction cosines are components of the vector {lP}1 where {l p}1 {RP}  RP  (2.7) It can be seen that {lP}1 has unit magnitude and is therefore a unit vector. Z1 P {RP}1 θz O1 Pz θy θx Y1 X1 Px Py Fig. 2.4 Direction cosines 26 Multibody Systems Approach to Vehicle Dynamics {B}1 θ {A}1 Fig. 2.5 Vector dot product 2.2.2 The dot (scalar) product The dot, or scalar, product {A}1 • {B}1 of the vectors {A}1 and {B}1 yields a scalar C with magnitude equal to the product of the magnitude of each vector and the cosine of the angle between them. Thus: {A}1 • {B}1 C A B cos (2.8) The calculation of {A}1 • {B}1 requires the solution of {A}1 • {B}1 {A}1T{B}1 AxBx AyBy AzBz where Bx {B}1 By and {A}1T [ Ax Bz Ay (2.9) Az ] (2.10) The T superscript in {A}1T indicates that the vector is transposed. Clearly {A}1 • {B}1 {B}1 • {A}1 and the dot product is a commutative operation. The physical significance of the dot product will become apparent later but at this stage it can be seen that the angle between two vectors {A}1 and {B}1 can be obtained from cos {A}1 • {B}1  A B (2.11) A particular case which is useful in the formulation of constraints representing joints and the like is the situation when {A}1 and {B}1 are perpendicular making cos 0. As can be seen in Figure 2.6 the equation that enforces the perpendicularity of the two spindles in the universal joint can be obtained from {A}1 • {B}1 0 (2.12) 2.2.3 The cross (vector) product The cross, or vector, product of two vectors, {A}1 and {B}1, is another vector {C}1 given by {C}1 {A}1 {B}1 (2.13) The vector {C}1 is perpendicular to the plane containing {A}1 and {B}1 as shown in Figure 2.7. Kinematics and dynamics of rigid bodies 27 {A}1 {B}1 Fig. 2.6 Application of the dot product to enforce perpendicularity {C}1 {B}1 θ {A}1 Fig. 2.7 Vector cross product The magnitude of {C}1 is defined as C A B sin (2.14) The direction of {C}1 is defined by a positive rotation about {C}1 rotating {A}1 into line with {B}1. The calculation of {A}1 {B}1 requires {A}1 to be arranged in skewsymmetric form as follows: Az Ay 0 {C}1 {A}1 {B}1 [ A]1{B}1 Az 0 Ax Ay Ax 0 Bx By Bz (2.15) Multiplying this out would give the vector {C}1: AzBy AyBz {C}1 AzBx AxBz AyBx AxBy (2.16) Exchange of {A}1 and {B}1 will show that the cross product operation is not commutative and that {A}1 {B}1 {B}1 {A}1 (2.17) 28 Multibody Systems Approach to Vehicle Dynamics 2.2.4 The scalar triple product The scalar triple product D of the vectors {A}1, {B}1 and {C}1 is defined as D {{A}1 {B}1} • {C}1 (2.18) 2.2.5 The vector triple product The vector triple product {D}1 of the vectors {A}1, {B}1 and {C}1 is defined as {D}1 {A}1 {{B}1 {C}1} (2.19) 2.2.6 Rotation of a vector In multibody dynamics bodies may undergo motion which involves rotation about all three axes of a given reference frame. The new components of a vector {A}1, shown in Figure 2.8, ma