The multibody systems approach to vehicle dynamics

The Multibody

Systems Approach

to Vehicle Dynamics

Second Edition

Mike Blundell

Damian Harty

Faculty of Engineering and Computing,

Coventry University, Coventry, UK

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Preface

This book, the second edition, is intended to bridge a gap between the subject of

classical vehicle dynamics and the general-purpose computer-based discipline mul￾tibody systems (MBS) analysis. Whilst 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.

After 10 years a second edition of this book is indeed timely. Since the first edi￾tion there have been notable developments in the understanding and use of active

systems, tyre modelling and the use of MBS software.

MBS analysis became established as a tool for engineering designers during the

1980s in a similar manner to the growth in finite element analysis technology during

the previous decade. A number of computer programs were developed and marketed

to the engineering industry, such as MSC ADAMS
(Automatic Dynamic Analysis

of Mechanical Systems), which in this edition still forms the basis for many of the

examples provided. During the 1990s MBS became firmly established as part of

the vehicle design and development process. It is inevitable that the engineer work￾ing on problems involving vehicle ride and handling in a modern automotive envi￾ronment will be required to interface with the use of MBS to simulate vehicle

motion. During the last 10 years several other MBS programmes have become

more established, most notably SIMPACK which appropriately receives more

coverage in this edition.

The book is aimed at a wide audience including not only undergraduate, post￾graduate 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 hu￾man 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 man￾ufacturers 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 co-authored by

Damian Harty. At the time of writing the first edition Damian was the Chief

Engineer e Dynamics at Prodrive. In the 10 years since the first edition he

continued in that role and after a few years working as a Senior Research fellow at

Coventry University he moved to his current position with Polaris where he enjoys

xi

the additional challenge of modelling vehicles on wide ranging terrain. With experi￾ence not only in the area of computer simulation but also the in the practical develop￾ment and testing of vehicles on the proving ground Damian continues 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 over￾view 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.

Chapter 2 is included for completeness and covers the underlying formulations in

kinematics and dynamics required for a good understanding of MBS formulations.

A three-dimensional vector approach is used to develop the theory, this being the

most suitable method for developing the rigid body equations of motion and

constraint formulations described later.

Chapter 3 covers the modelling, analysis and 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 in the main on

MSC ADAMS; the main reason for this choice being that the two authors have be￾tween them 25 years of experience working with the software. The fact that the soft￾ware 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 avail￾able 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 sub￾ject and to provide a supporting reference for the vehicle modelling and analysis

methodologies described in the following chapters. As discussed, the emergence

of SIMPACK and its growing use by the automotive community has led to additional

examples to illustrate the modelling approaches with that software.

Existing users of MSC ADAMS will note that the modelling examples provided

in Chapter 3 are based on a text-based format of model inputs, known in MSC

ADAMS as solver data sets. This was the original method used to develop MSC

ADAMS models and has subsequently been replaced by a powerful graphical user

interface (GUI) known as ADAMS/View
that allows model parameterisation,

and design optimisation studies. The ADAMS/View environment is also the basis

for customised versions of MSC ADAMS such as ADAMS/Car
that are becoming

established in industry and are also discussed in Chapter 3. The use of text-based

data sets has been adopted here for a number of reasons. The first of these is that

the GUI of a modern simulation program such as MSC ADAMS is subject to exten￾sive and ongoing development. Any attempt to describe such a facility in a textbook

such as this would become outdated after a short passage of time. As mentioned, the

software developers provide their own user documentation covering this in any case.

It is also clear that the text-based formulations translate more readily to book format

and are also useful for demonstrating the underlying techniques in planning a model,

preparing model schematics and establishing the degrees of freedom in a system

xii Preface

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 soft￾ware at this fundamental level the dependence of the book on any one software pack￾age 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 suspen￾sion systems and the MBS 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 vehicle design process. The chapter concludes with

an extensive case study comparing a full set of analytical calculations, using the

vector-based methods introduced in Chapter 2, with the output produced from

MSC ADAMS. It is intended that this exercise will demonstrate to readers the

underlying computations in process when running an MBS simulation.

Chapter 5 addresses the tyre force and moment generating characteristics and the

subsequent modelling of these in an MBS simulation. As a major area of importance

it deserves to be the largest chapter in the book. 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. Since the first edition new tyre models such as the FTire

model from Gipser and the TAME Tire model from Michelin have become estab￾lished and therefore receive a more extended coverage in this edition.

Chapter 6 describes the modelling and assembly of the rest of the vehicle, including

the anti-roll bars and steering systems. Near the beginning a range of simplified sus￾pension 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 perfor￾mance of several full vehicle modelling strategies for a vehicle handling manoeuvre.

Chapter 7 deals with the simulation output and interpretation of results. An over￾view of vehicle dynamics for travel on a curved path is included. The classical treat￾ment of understeer/oversteer based on steady state cornering is presented followed

by an alternative treatment that considers yaw rate and lateral acceleration gains.

The subjective/objective problem is discussed with consideration of steering feel

and roll angle as subjective modifiers. The chapter concludes with a consideration

of the use of analytical models with a signal-to-noise approach.

Chapter 8 concludes with a review of the use of active systems to modify the dy￾namics in modern passenger cars. The use of electronic control in systems such as

active suspension and variable damping, brake-based systems, active steering sys￾tems, active camber systems and active torque distribution is described. A final sum￾mary matches the application of these systems with driving styles described as

normal, spirited or the execution of emergency manoeuvres.

Preface xiii

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 were 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.

In conclusion it seems to the authors there still remain 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 for￾ward the process of vehicle design, taking full advantage of the widespread avail￾ability 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’, Humpty Dumpty said, in rather a scornful tone, ‘it means

just what I choose it to meandneither 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 discus￾sion and enquiry, taking pains to establish the meanings of even apparently obvious

terms such as ‘camber’ e motorcycles do not have any camber by some definitions

(vehicle-body-referenced) and yet to zero the camber forces in a motorcycle tyre is

clearly folly. A glossary is included in Appendix C, not as some declaration of cor￾rectness but as an illumination for the text. In this edition a new appendix has been

added. Appendix D lists some of the test procedures defined by the International

Standards Organisation that are used to validate the handling performance of a

new vehicle.

Mike Blundell, Damian Harty

April 2014

xiv Preface

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 Roger Williams, Jim Forbes, Adrian Griffiths, Colin Lucas, John Janevic

and Grahame Walter. I am especially grateful to the late David Crolla. He was an

inspiration to me as my career took me into the area of vehicle dynamics and my

mentor during the preparation of the first edition of this book. I will never forget

him. Finally I thank the staff at Elsevier Science for their patience and help

throughout the years it has taken to bring both editions of 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. At the time of the first edition I acknowledged

Robin Sharp, Doug and Bill Milliken for keeping me grounded and rigorous when it

is tempting just to play in cars and jump to conclusions. Bill Milliken in particular

made a significant contribution to the discipline for an astonishingly long period;

Bill passed away in 2012 after a fruitful and remarkable 101-year life that included

driving at speed up the hill at Goodwood in 2002 and again in 2007. For those un￾familiar with the clarity of his thinking and the vivacity of a life lived to the full, his

autobiography ‘Equations of Motion’ is an excellent read. Bill’s legacy persists with

his son Doug continuing to run Milliken Research Associates in Buffalo, NY, USA.

During our work on the first edition of this book the late David Crolla was an

ever-present voice of reason keeping this text focused on its raison d’etredthe use￾ful 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 mentiondapart 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 do not. Someone who must be mentioned is Isaac Newton; his original

and definitive brilliance at describing my world amazes me everyday. As Mike,

I thank the staff at Elsevier Science for their saintly patience.

Finally, I would just like to say I am very sorry for all the vehicles I have

damaged while ‘testing’ them. I really am.

xv

Nomenclature

a1, a2, a3 Distances for six-mass approximation

a, b Distance from CM to front and rear axles, respectively

a11.a22 Elements of a matrix (generic)

{aI}1 Unit vector at marker I resolved parallel to frame 1 (GRF)

{aJ}1 Unit vector at marker J resolved parallel to frame 1 (GRF)

ax Longitudinal acceleration (Wenzel model)

ay Lateral acceleration (Wenzel model)

b Longitudinal distance of body mass centre from front axle

c Damping coefficient

c Longitudinal distance of body mass centre from rear axle

c Specific heat capacity of brake rotor

d Wire diameter

dB

dFz

Variation in scaling factor with load (Harty Model)

{dIJ}1 Position vector of marker I relative to J resolved parallel to frame 1

(GRF)

e1 Path error

f Natural frequency (Hz)

g Gravitational acceleration

h Brake rotor convection coefficient

h Height of body mass centre above roll axis

i Square root of
1

k Path curvature

k Radius of gyration

k Stiffness

k Spring constant in hysteretic model

k Tyre spring constant

k1, k2 Front and rear ride rates, respectively

ks Spring stiffness

kw Stiffness of equivalent spring at the wheel centre

l Length of pendulum

m Mass of a body

m{g}1 Weight force vector for a part resolved parallel to frame 1 (GRF)

mt Mass of tyre

n Number of active coils

n Number of friction surfaces (pads)

p Brake pressure

qj Set of part generalised coordinates

r Yaw rate

r1, r2, r3 Coupler constraint rotations

{rI}1 Position vector of marker I relative to frame i resolved parallel to

frame 1 (GRF)

{rJ}1 Position vector of marker J relative to frame j resolved parallel to

frame 1 (GRF)

xvii

ru Unladen radius

rl Laden radius

rw Wheel radius

s1, s2, s3 Coupler constraint scale factors

tf Front track

tr Rear track

vcog Centre of gravity (Wenzel model)

vx Longitudinal velocity (Wenzel model)

vy Lateral velocity (Wenzel model)

x Generic variable for describing tanh function

xi, yi, zi Coordinates of each of the six masses in the six-mass

approximation

xi, yi, zi Components of the ith eigenvector

x(t) Function of time (generic)

xCM, yCM, zCM Coordinates of body centre of mass

{xI}1 Unit vector along x-axis of marker I resolved parallel to frame 1

(GRF)

{yI}1 Unit vector along y-axis of marker I resolved parallel to frame 1

(GRF)

{xJ}1 Unit vector along x-axis of marker J resolved parallel to frame 1

(GRF)

{yJ}1 Unit vector along y-axis of marker J resolved parallel to frame 1

(GRF)

ys Asymptotic value at large slip (Magic Formula)

z Auxiliary state variable

z Heave displacement variable

{zI}1 Unit vector along z-axis of marker I resolved parallel to frame 1

(GRF)

{zJ}1 Unit vector along z-axis of marker J resolved parallel to frame 1

(GRF)

A Area

A Linear acceleration

A, B, C Intermediate terms in a cubic equation

A Scaling for solution form of a differential equation (generic)

A Step height

Ac Convective area of brake disc

[A1n] Euler matrix for part n

{An}1 Acceleration vector for part n resolved parallel to frame 1 (GRF)

Ap Centripetal acceleration

{Ap

PQ}1 Centripetal acceleration vector P relative to Q referred to frame 1

(GRF)

{At

PQ}1 Transverse acceleration vector P relative to Q referred to frame 1

(GRF)

{Ac

PQ}1 Coriolis acceleration vector P relative to Q referred to frame 1

(GRF)

{As

PQ}1 Sliding acceleration vector P relative to Q referred to frame 1 (GRF)

Avehicle Acceleration of vehicle

xviii Nomenclature

AX Longitudinal curvature factor

Ay Lateral acceleration

AyG Lateral acceleration gain

B Load scaling factor (Harty Model)

B Stiffness factor (Magic Formula)

[B] Transformation matrix from frame Oe to On

BKid Bottom kingpin marker

BM Bump movement

BT Brake torque

C Shape factor (Magic Formula)

[C] Compliance matrix

CD0 Drag coefficient at zero aerodynamic yaw angle

CDb Drag coefficient sensitivity to aerodynamic yaw angle

CF Front axle cornering stiffness

Cg Camber coefficient

CL0 Coefficient of lift at zero angle of attack

CLa Variation in coefficient of lift with angle of attack

CMX Overturning moment coefficient

Cr Rolling resistance moment coefficient

CR Rear axle cornering stiffness

CS Tyre longitudinal stiffness

Cp Process capability

CP Centre of pressure

Ca Tyre lateral stiffness due to slip angle

Caf Front tyre lateral stiffness due to slip angle

Car Rear tyre lateral stiffness due to slip angle

Cg Tyre lateral stiffness due to camber angle

D Clipped camber scale constant

D Mean coil diameter

D Peak value (Magic Formula)

DZ Displacement variable (generic)

DM(I,J) Magnitude of displacement of I marker relative to J marker

DX(I,J) Displacement in X-direction of I marker relative to J marker parallel

to GRF

DY(I,J) Displacement in Y-direction of I marker relative to J marker parallel

to GRF

DZ(I,J) Displacement in Z-direction of I marker relative to J marker parallel

to GRF

E Camber clip curvature constant

E Young’s modulus of elasticity

E Curvature factor (Magic Formula)

F Aerodynamics force

F Applied force

F Force generated by hysteretic model

F Spring force

Fhyst Amplitude of hysteretic force

Fhyst Final outcome from sequence of hysteretic calculations

{FnA}1 Applied force vector on part n resolved parallel to frame 1 (GRF)

Nomenclature xix

{FnC}1 Constraint force vector on part n resolved parallel to frame 1 (GRF)

FFRC Lateral force reacted by front roll centre

FRRC Lateral force reacted by rear roll centre

Fx Frictional force

Fx Longitudinal tractive or braking tyre force

Fx1 Friction moderated longitudinal load in moderate slip

Fx2 Friction moderated longitudinal load in deep slip

Fy Lateral tyre force

FY1 Friction moderated lateral load at moderate slip angles

FY2 Friction moderated lateral load at deep slip angles

F0

y Lagged (relaxed) side force

Fya Lateral load due to slip angle

Fya0 Friction moderated side force due to slip angle

Fyg Lateral load due to camber/inclination angle

Fyg0 Friction moderated side force due to camber/inclination angle

F

_

y

mFz Lateral capacity fraction

Fz Normal force

Fz Vertical tyre force

Fz Time varying tyre load

Fz0 Static corner load

Fzc Vertical tyre force due to damping

Fzk Vertical tyre force due to stiffness

{FA}1 {FB}1. Applied force vectors at points A, B,. resolved parallel to frame

1 (GRF)

[FE] Elastic compliance matrix (concept suspension)

FD Drag force

FG Fixed ground marker

G Shear modulus

GC Gravitational constant

GO Ground level offset

GRF Ground reference frame

{H}1 Angular momentum vector for a body

H(u) Transfer function

HTC Half track change

I Mass moment of inertia

I Second moment of area

I2 Pitch inertia of vehicle

I1, I2, I3 Principal mass moments of inertia of a body

Iwheel Mass moment of inertia of road wheel in the rolling direction

Ixx, Iyy, Izz, Ixy, Iyz, Ixz Components of inertia tensor

ICY Y-coordinate of instant centre

ICZ Z-coordinate of instant centre

[In] Inertia tensor for a part

J Polar second moment of area

Jz Vehicle body yaw inertia (Wenzel model)

K Drive torque controller constant

K Spring stiffness

xx Nomenclature

K Stability factor

K Understeer gradient

Kz Tyre radial stiffness

KT Torsional stiffness

KTs Roll stiffness due to springs

KTr Roll stiffness due to anti-roll bar

L Contact patch length

L Length

L Wheelbase

{L}1 Linear momentum vector for a particle or body

LPFZ2 Pneumatic lead scaling factor with load squared

LPFZ Pneumatic lead scaling factor with load

LPC Pneumatic lead at reference load

LPRF Local part reference frame

LR Tyre relaxation length

MFRC Moment reacted by front roll centre

{MnA}e Applied moment vector on part n resolved parallel to frame e

{MnC}e Constraint moment vector on part n resolved parallel to frame e

Ms Equivalent roll moment due to springs

Mx Tyre overturning moment

MXgk Overturning moment due to longitudinal forces

My Moment about y-axis

My Tyre rolling resistance moment

Mz Tyre self aligning moment

Mza Friction moderated side force due to slip angle

Mzg Friction moderated side force due to camber/inclination angle

MZgk Aligning moment due to longitudinal forces

MRF Marker reference frame

MRRC Moment reacted by rear roll centre

Nr Vehicle yaw moment with respect to yaw rate

[Nt] Norsieck vector

Nvy Vehicle yaw moment with respect to lateral velocity

O1 Frame 1 (GRF)

Oe Euler axis frame

Oi Reference frame for part i

Oj Reference frame for part j

On Frame for part n

OP Lateral offset of contact patch

P0 Initial tyre pressure at zero load

P Average footprint pressure

{Pnr}1 Rotational momenta vector for part n resolved parallel to frame 1

(GRF)

{Pnt}1 Translational momenta vector for part n resolved parallel to frame

1 (GRF)

Pt Constant power acceleration

PDz Change in nominal pressure

PDz Pressure due to tyre vertical deflection

QG Position vector of a marker relative to the GRF

Nomenclature xxi

QP Position vector of a marker relative to the LPRF

R Radius (generic)

R Radius of turn

R Fraction of roll moment distributed between front and rear axles

R1 Unloaded tyre radius

R2 Tyre carcass radius

Rd Radius to centre of brake pad

Re Effective rolling radius

{Ri}1 Position vector of frame i on part i resolved parallel to frame 1

(GRF)

{Rj}1 Position vector of frame j on part j resolved parallel to frame 1

(GRF)

Rl Loaded tyre radius

{Rn}1 Position vector for part n resolved parallel to frame 1 (GRF)

{Rp}1 Position vector of tyre contact point P relative to frame 1, referenced

to frame 1

Ru Unloaded tyre radius

{Rw}1 Position vector of wheel centre relative to frame 1, referenced to

frame 1

{RAG}n Position vector of point A relative to mass centre G resolved parallel

to frame n

{RBG}n Position vector of point B relative to mass centre G resolved parallel

to frame n

RCfront Front roll centre

RCrear Rear roll centre

RCY Y-coordinate of roll centre

RCZ Z-coordinate of roll centre

RZ Reference load (Harty Model)

S Distance travelled

SA Spindle axis reference point

SCX Critical slip ratio

Se Error variation

Sh Horizontal shift (Magic Formula)

Sv Vertical shift (Magic Formula)

SL Longitudinal slip ratio

SL* Critical value of longitudinal slip

SN Signal-to-noise ratio

ST Total variation

Sa Lateral slip ratio

SLa Comprehensive slip ratio

Sa* Critical slip angle

Sk Variation due to linear effect

T Camber clipping threshold fraction

T Kinetic energy for a part

T Temperature

T Torque

TB Brake torque

Tenv Environmental temperature

xxii Nomenclature

TPFZ2 Pneumatic trail scaling factor with load squared

TPFZ Pneumatic trail scaling factor with load

TPC Pneumatic trail scaling constant

TS Spin up torque

T0 Initial brake rotor temperature

{TA}1 {TB}1. Applied torque vectors at points A, B,. resolved parallel to frame

1 (GRF)

TK Top kingpin marker

TR Suspension trail

{Ur} Unit vector normal to road surface at tyre contact point

{Us} Unit vector acting along spin axis of tyre

UCF Units consistency factor

US Understeer

V Forward velocity

V0 Initial tyre volume at zero load

Va Actual forward velocity

Ve Error variance

Vg Ground plane velocity

Vlowlimit Limiting velocity

{Vn}1 Velocity vector for part n resolved parallel to frame 1 (GRF)

{Vp}1 Velocity vector of tyre contact point P referenced to frame 1

Vs Desired simulation velocity

Vx Sliding velocity

Vxc Longitudinal slip velocity of tyre contact point

Vy Lateral slip velocity of tyre contact point

Vz Vertical velocity of tyre contact point

Vref Reference velocity in hysteretic model

VR(I,J) Radial line of sight velocity of I marker relative to J marker

VZ Velocity variable (generic)

VDz Reduced tyre cavity volume

W Tyre width

WB Wheelbase marker

WC Wheel centre marker

WF Wheel front marker

WR Wheel recession

XP Position vector of a point in a marker xz-plane

{Xsae}1 Unit vector acting at tyre contact point in Xsae direction referenced

to frame 1

Yr Vehicle side force with respect to yaw rate

Yvy Vehicle side force with respect to lateral velocity

YRG Yaw rate gain

{Ysae}1 Unit vector acting at tyre contact point in Ysae direction referenced

to frame 1

{Zsae}1 Unit vector acting at tyre contact point in Zsae direction referenced

to frame 1

ZP Position vector of a point on a marker z-axis

a Angle of attack

a Tyre slip angle

Nomenclature xxiii

aCY Critical slip angle (Harty Model)

{an}1 Angular acceleration vector for part n resolved parallel to frame 1

(GRF)

af Front axle slip angle

ar Rear axle slip angle

b Aerodynamic yaw angle (or body slip angle surrogate)

b Side slip angle

b_ Rate of change of side slip angle (Beta Dot)

d Steer or toe angle

do Steer angle of outer wheel

di Steer angle of inner wheel

dmean Average steer angle of inner and outer wheels

g Camber angle

{gn}e Set of Euler angles for part n

z Damping ratio

k Longitudinal slip (Pacjeka)

k Sensitivity of process

q 2nd Euler angle rotation

q Pendulum displacement variable

q Pitch displacement variable

q1 Orientation of the first principal axis within a plane of symmetry

l Eigenvalue (generic)

{l}1 Reaction force vector resolved parallel to frame 1 (GRF)

ld Magnitude of reaction force for constraint d

lp Magnitude of reaction force for constraint p

la Magnitude of reaction force for constraint a

m Friction coefficient

mo Tyre to road coefficient of static friction

m1 Tyre to road coefficient of sliding friction

h Signal-to-noise ratio

h Hysteresis constant/loss factor

r Density

s Standard deviation

sd Standard deviation of attribute d

F 3rd Euler angle rotation

j 1st Euler angle rotation

j Compass heading angle

j_ Yaw rate (Wenzel model)

u Angular frequency (rads s
1

)

u Yaw rate

ud Damped natural frequency

ud Demanded yaw rate

uerr Yaw rate error

ufns Front axle no-slip yaw rate

ufriction Yaw rate from limiting friction

ugeom Yaw rate from geometry

un Undamped natural frequency

{ue}1 Angular velocity vector for part n resolved parallel to frame e

xxiv Nomenclature