motorcycel 5496

Improving a tyre model formotorcycle simulations

W.D. Versteden (s479891)

DCT 2005.65

Master's thesis

Supervisor and member of graduation committee:Dr.Ir. I.J.M. Besselink (Eindhoven University of Technology / TNO Automotive)Prof. Dr. H. Nijmeijer (Eindhoven University of Technology)

Member of graduation committee:Dr.Ir. W.J. Witteman (Eindhoven University of Technology)Ir. P.A.J. Ruijs (TNO Automotive)

Eindhoven University of TechnologyDepartment Mechanical EngineeringDynamics and Control Group

Eindhoven, June, 2005

Samenvatting

Met de rekencapaciteit van huidige computers wordt het simuleren van het gedrag van voertu-igen steeds belangrijker. Vooral bij motorfietssimulaties speelt het bandmodel een belangrijke rolin het gedrag van het voertuig. Door TNO Automotive is daarom een bandmodel voor motor-fietssimulaties ontwikkeld, MF-MCTyre. Ondanks het belang van een nauwkeurig bandmodel,zijn bepaalde onderdelen van het MF-MCTyre model afgeleid van een autobandmodel. Daarom isvooral de invloed van de camberhoek, die voornamelijk aanwezig is tijdens het rijden door bochten,op deze onderdelen onderbelicht gebleven. Het doel van dit onderzoek is dan ook om het stationairgedrag in bochten van het MF-MCTyre model te verbeteren.

Om vertrouwd te raken met het stationair gedrag van het bandmodel en een motorfiets ineen bocht, is een simulatiemodel van een motorfiets ontwikkeld aan de hand van het model vanCornelis Koenen. Door middel van simulaties met dit model en het bestaand bandmodel is hetstationair gedrag van beide geanalyseerd. Een literatuurstudie toont aan dat het model vanKoenen een van de meest complete modellen is. Tevens laat deze literatuurstudie zien dat deMagic Formula algemeen geaccepteerd wordt. Daarom wordt aangenomen dat het MF-MCTyremodel dat gebaseerd is op deze Magic Formula in staat moet zijn om het stationair gedrag vaneen motorfietsband in een bocht correct weer te geven.

Tijdens de analyse van het bandmodel komen er drie problemen naar voren die de nauwkeurigheidvan het model reduceren. Ten eerste worden er vereenvoudigde aannames gemaakt voor de ver-ticale stijfheid en de bandcontour. Dit leidt tot een onnauwkeurige hoogte van het wielcentrumen de belaste bandstraal gedurende simulaties. Omdat de belaste bandstraal ook tijdens metin-gen onnauwkeurig wordt bepaald zijn de transformaties van de momenten tussen het contactpunten het wielcentrum ook onnauwkeurig. Ten tweede wordt ook de effectieve rolstraal incorrectbepaald door het bandmodel. De bandcontour wordt buiten beschouwing gelaten en het effect vande verticale belasting wordt overgenomen van een autobandmodel. Daardoor zijn de rotatiesnel-heden van de wielen onnauwkeurig tijdens voertuigsimulaties. Tenslotte wordt er ook een sterkvereenvoudigde aanname gemaakt voor de rolweerstand. Als gevolg hiervan is weergave van delangskracht en het terugstelmoment onnauwkeurig tijdens simulaties.

Een uitgebreid meetprogramma is opgezet, om bepaalde specifieke aspecten van een motorfiets-band te bepalen. Door middel van deze metingen zijn de bandcontour, verticale stijfheid, effectieverolstraal en rolweerstand bepaald. Deze resultaten zijn gebruikt om de eerder genoemde problemenvan het bandmodel op te lossen en een verbeterd bandmodel te ontwikkelen. Uit tests blijkt datdit nieuwe model voornamelijk bij grote camber hoeken de krachten en momenten nauwkeurigerrepresenteert. Tevens is aangetoond dat het verbeterde bandmodel een duidelijke invloed heeft ofhet stationair gedrag van het motorfietsmodel. Parameters als de camberhoek, het stuurkoppel,de leunhoek van de bestuurder, de rotatiesnelheden van de wielen en de hoogte van de wielcentraveranderen significant als het nieuwe bandmodel wordt gebruikt in plaats van het oude.

i

Abstract

With the computational capacity of computers available nowadays, simulating the behaviour ofvehicles becomes more and more important. Especially for motorcycle simulations, the tyre modelhas a significant influence on the vehicle behaviour. TNO Automotive has therefore developeda tyre model for motorcycle simulations, MF-MCTyre. Although the importance of an accuratetyre model, several aspects of the MF-MCTyre model are derived from an automobile tyre model.Therefore, especially the influence of the camber angle on these aspects has been underexposed.The goal of this research is therefore to improve the steady state cornering behaviour of theMF-MCTyre model.

To get familiar with the tyre model and motorcycle steady state cornering behaviour, a mo-torcycle simulation model is developed for which the model of Cornelis Koenen is used as a basis.From simulations with the motorcycle model and the existing tyre model the steady state behaviourof the motorcycle model and its tyres is analyzed. A literature study shows that the model ofKoenen is one of the most complete models. Furthermore, this literature study shows that theMagic Formula is broadly accepted. Therefore, it is believed that the MF-MCTyre model, whichis based on this Magic Formula, should be able to represent the steady state cornering behaviourof a motorcycle tyre correctly.

If the tyre model is analyzed, three main problems are found which decrease the accuracy of thetyre model. First of all, simplified assumptions are made for the motorcycle tyre contour and thevertical tyre stiffness. Therefore the wheel center height and loaded radius during the simulationsare inaccurate. As the loaded radius is also inaccurately determined during measurements, thetransformation of the moments between the wheel center and the contact point are also inaccurate.Secondly, the effective rolling radius is incorrectly determined by the model. The tyre contour is nottaken into consideration and the effect of the vertical load is copied from an automobile tyre model.During vehicle simulations, this leads to inaccurate rotational velocities of the wheels. Finally, asimplified assumption is made for the rolling resistance. During simulations, this assumption leadsto an inaccurate representation of the longitudinal force and the self aligning moment.

An extensive measurement program is conducted, in order to determine the behaviour of specificmotorcycle tyre aspects. Moreover, the tyre contour, vertical stiffness, effective rolling radiusand rolling resistance are determined within this measurement program. The results of thesemeasurements are used to overcome the problems mentioned above and to develop an improvedtyre model. From simulations it is learned that this new tyre model represents the forces andmoments more accurately, especially under large camber angles. Furthermore, it is shown thatthe improved tyre model has a significant influence on the steady state cornering behaviour ofthe motorcycle model. Moreover, parameters as the camber angle, steer torque, rider lean angle,wheel rotational speeds and wheel center heights show significant differences if the improved tyremodel is used instead of the present one.

ii

Contents

Samenvatting i

Abstract ii

Sign conventions and symbols v

1 Introduction 11.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Problem statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.3 Research layout . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

2 Literature study 32.1 Motorcycle modelling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2.1.1 Model structure and features . . . . . . . . . . . . . . . . . . . . . . . . . . 32.1.2 Mass and inertial parameters . . . . . . . . . . . . . . . . . . . . . . . . . . 52.1.3 Stabilizing controller . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.2 Motorcycle tyre modelling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

3 A motorcycle simulation model 103.1 Model structure and features . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103.2 Stabilization and velocity controllers . . . . . . . . . . . . . . . . . . . . . . . . . . 123.3 Tyre model implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123.4 Simulation results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

3.4.1 Equilibria of forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163.4.2 Camber angle γ and steer torque Ms . . . . . . . . . . . . . . . . . . . . . . 17

4 Problems of the MF-MCTyre model 204.1 Explanation of the tyre model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204.2 Problems with the determination of the contact point and the vertical load, Fzw . 22

4.2.1 The wheel axle height, h . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224.2.2 The tyre loaded radius, Rl . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

4.3 Problems with the determination of the effective rolling radius, Re . . . . . . . . . 244.4 Errors in the rolling resistance assumption . . . . . . . . . . . . . . . . . . . . . . . 26

5 Motorcycle tyre measurements 295.1 Determination of the motorcycle tyre contour . . . . . . . . . . . . . . . . . . . . . 295.2 Vertical tyre stiffness measurements . . . . . . . . . . . . . . . . . . . . . . . . . . 305.3 The effect of the vertical load on the effective rolling radius . . . . . . . . . . . . . 335.4 Effective rolling radius reference measurements . . . . . . . . . . . . . . . . . . . . 335.5 Rolling resistance factor fr measurements . . . . . . . . . . . . . . . . . . . . . . . 35

iii

CONTENTS iv

6 Improvements to the MF-MCTyre model 376.1 Improving the contact point and vertical load determination . . . . . . . . . . . . . 37

6.1.1 The wheel axle height, h . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 406.1.2 The tyre loaded radius, Rl . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

6.2 Improving the effective rolling radius determination . . . . . . . . . . . . . . . . . . 416.3 Improving the rolling resistance description . . . . . . . . . . . . . . . . . . . . . . 43

6.3.1 Definition of the rolling resistance moment, Mrr . . . . . . . . . . . . . . . 436.3.2 The rolling resistance coefficient, fr . . . . . . . . . . . . . . . . . . . . . . 46

7 Analysis of the improved tyre model 477.1 Summary of the improvements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 477.2 Analysis by means of test rig simulations . . . . . . . . . . . . . . . . . . . . . . . 487.3 Analysis by means of motorcycle simulations . . . . . . . . . . . . . . . . . . . . . 52

8 Conclusions and recommendations 578.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 578.2 Recommendations for future research . . . . . . . . . . . . . . . . . . . . . . . . . . 58

A The MF-MCTyre model 62A.1 Contact routine . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

A.1.1 The contact point C and the normal load Fz . . . . . . . . . . . . . . . . . 62A.1.2 The effective rolling radius . . . . . . . . . . . . . . . . . . . . . . . . . . . 62A.1.3 Tyre slip quantities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

A.2 The Magic Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64A.2.1 Longitudinal force (pure slip) . . . . . . . . . . . . . . . . . . . . . . . . . . 65A.2.2 Lateral force (pure slip) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66A.2.3 Aligning moment (pure slip) . . . . . . . . . . . . . . . . . . . . . . . . . . 66A.2.4 Overturning moment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66A.2.5 Rolling resistance moment . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67A.2.6 Additional features . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

A.3 Tyre model parameter determination . . . . . . . . . . . . . . . . . . . . . . . . . . 69

B Motorcycle model parameters 70

C Processing measurement data 72C.1 Left measurement tower . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72C.2 Right measurement tower . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73C.3 Conversion from C-axis system to W-axis system . . . . . . . . . . . . . . . . . . . 74C.4 Conversion from W-axis system to C-axis system . . . . . . . . . . . . . . . . . . . 75

SIGN CONVENTIONS AND SYMBOLS v

Sign conventions

Throughout this report the ISO sign convention for force, moment and wheel slip of a tyre is used.This sign convention is depicted in figure 1.

axF x

y F y

V

M z

gF z

z

F yy

M xT o p v i e w R e a r v i e w

Figure 1: The ISO sign conventions

The MF-MCTyre model is in accordance with the standard TYDEX conventions. Two TYDEXcoordinate systems with ISO orientation are particularly important, the C- and W-axis systems asdepicted in figure 2. The C-axis system is fixed to the wheel carrier with the longitudinal xc-axis

XY

Z

Xw

C

Yc

Yw

Xc

nr

Zc Zw

Vc

Xw

C

nr

Zw

Zc

O

Xc

Vx

Yc

Yw

-γ

Figure 2: C- and W -axis systems used in MF-Tyre

parallel to the road and in the wheel plane (xc-zc-plane). The origin O of the C-axis system isthe wheel center. The origin of the W-axis system is the road contact point C defined by theintersection of the wheel plane, the plane through the wheel spindle and the road tangent plane.The xw-yw-plane is the tangent plane of the road in the contact point C, and it defines the camberangle γ together with the normal nr to the road plane (xw-yw-plane). Furthermore, in chapter 6the T -axis system is introduced. This axis system is defined in the contact and therefore it hasthe same origin as the W -axis system. However, the T -axis system rotates with the camber angleand is therefore parallel to the C-axis system.

SIGN CONVENTIONS AND SYMBOLS vi

Symbols

Symbol Description Unit

CapitalsA Actual contact pointB Magic Formula factor [−]C Magic Formula factor [−]C Fictitious tyre-road contact pointD Magic Formula factor [−]Cz Vertical tyre stiffness [N

m ]D Magic Formula factor [−]E Magic Formula factor [−]E Relative fitting error [−]F Magic Formula factor [−]Fa Centrifugal force [N ]Fd Aerodynamic drag force [N ]Fg Gravitational force [N ]Fl Aerodynamic lift force [N ]Fx Longitudinal force [N ]Fy Lateral force [N ]Fz Vertical force [N ]Fzo Nominal vertical load [N ]J Moment of inertia [kgm2]cmJ Inertia tensor with respect to the center of mass [kgm2]oJ Inertia tensor with respect to an arbitrary point O [kgm2]K Gain [−]Kz Vertical tyre damping [Ns

m ]Mrr Rolling resistance moment magnitude [Nm]Ms Steering torque [Nm]Mx Overturning moment [Nm]My Rolling resistance moment [Nm]Mz Self aligning moment [Nm]O Wheel center position [m]O Origin of motorcycle inertial frameR Cornering radius [m]Re Effective rolling radius [m]Reo Effective rolling radius of an undeformed tyre [m]Rl Loaded radius [m]Ro Unloaded radius [m]SH Horizontal shift in the Magic FormulaSV Vertical shift in the Magic FormulaV Velocity [m

s ]Vo Reference velocity [m

s ]Vr Linear rolling velocity [m

s ]Vsx Longitudinal slip speed [m

s ]Vsy Lateral slip speed [m

s ]Vx Longitudinal velocity [m

s ]Vy Lateral velocity [m

s ]Ym Measurement pointYf Description of the measurement pointYn Parameter obtained by simulating the improved tyre modelYo Parameter obtained by simulating the existing tyre model

SIGN CONVENTIONS AND SYMBOLS vii


Normala Ellipse parameter with respect to the x-direction [−]b Ellipse parameter with respect to the y-direction [−]d Relative difference [−]fr Rolling resistance coefficient [−]g Gravitational constant [ m

s2 ]g1x Distance from mass m1 to steer axis [m]g1ux Distance from mass m1u to steer axis [m]h Height of the wheel center [m]m Mass [kg]nr Normal to the roadpdz Vertical position of the aerodynamic application point [m]plx Horizontal position of the aerodynamic application point [m]qc Camber influence on vertical stiffness [−]qfcx Longitudinal force influence on vertical stiffness [−]qfcy Lateral force influence on vertical stiffness [−]qfz1 Vertical deflection influence on vertical stiffness [−]qfz2 Quadratic influence of vertical deflection on stiffness [−]qsy1 Initial rolling resistance moment Mrr [−]qsy5 Camber influence on Mrr [−]qsy6 Fzw influence on Mrr [−]qv2 Vertical stiffness increase with speed [−]~rcm Vector from an arbitrary point O to the center of mass [m]

rt Tyre cross section radius [m]t Time [s]t Mechanical trail [m]u Distance [m]yMx Distance between actual and fictitious contact point [m]

GreekΩ Tyre rotational velocity [ rad

s ]α Side slip angle [rad]β Twist angle [rad]δ Steerangle [rad]ε Rake angle [rad]γ Camber angle [rad]κ Longitudinal slip [−]λ Magic Formula scaling factor [−]ω Yaw velocity [ rad

s ]ρ Vertical tyre deformation [m]ρd Dimensionless vertical tyre deflection [−]ρFz0 Nominal tyre deformation [m]σκ Longitudinal relaxation length [m]σα Lateral relaxation length [m]ξ Rotational tyre deformation [rad]

SIGN CONVENTIONS AND SYMBOLS viii


Numbers1 Steerbody1s Twistbody1u Front unsprung part1w Frontwheel2 Mainbody2s Swingarm2w Rearwheel3 Rider upper body

Subscriptf Front tyre parameterr Rear tyre parameteru With respect to the u-directionv With respect to the v-directionx With respect to the x-directiony With respect to the y-directionz With respect to the z-directionc With respect to the c-axis systemt With respect to the t-axis systemw With respect to the w-axis system

Chapter 1

Introduction

1.1 Background

With the computational capacity of computers available nowadays, simulating the dynamic be-haviour of vehicles has become more and more important. During its development process, sim-ulations with a model of the vehicle make it possible to accurately predict its dynamic behaviourwithout building a prototype. With this so-called ’virtual prototyping’ both design times andcosts are reduced. Already in the early years of vehicle modelling it has been concluded thatthe behaviour of a vehicle is strongly depending on the tyre behaviour. This holds especially formotorcycles, as single track vehicles are inherent to instabilities which are partly governed by thetyre behaviour. Therefore, the quality of a motorcycle model strongly depends on the accuracy ofthe tyre model that is implemented.

In 1987 the first version of the Magic Formula has been presented [1], which is able to representthe stationary slip characteristics of an automobile tyre. However, from figure 1.1 it can be learnedthat motorcycles have different camber and slip angle ranges than automobiles. Therefore theMagic Formula has been adapted in 1997 [28], to be able to describe the stationary motorcycletyre characteristics for large camber angles. The improved description of the slip characteristics

M o t o r c y c l e s

A u t o m o b i l e s

S l i p a n g l e [ d e g ]

Camb

er ang

le [deg

]

5 1 0

1 5

3 0

4 5

Figure 1.1: Comparison between the slip and camber angle of automobiles and motorcycles

by the adapted Magic Formula is implemented in a tyre model (MF-MCTyre) which is used formotorcycle simulations. Although that the effect of large camber angles is introduced in the MagicFormula, it is insufficiently incorporated in other aspects of the MF-MCTyre model and in theprocessing of measurement data.

During tyre measurements forces and moments are determined in the wheel center. However,

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CHAPTER 1. INTRODUCTION 2

the Magic Formula is developed to describe the forces and moments in the tyre-road contactpoint. Therefore, the measured forces and moments are transformed from the wheel center to thecontact point. Furthermore, during simulations the Magic Formula is evaluated in the contactpoint and the forces and moments are applied to the vehicle in the wheel center, so again atransformation is required. Although the tyre loaded radius Rl plays an important role duringthis transformation, it is not unambiguously defined during the processing of the measurements andsimulations. Furthermore, also the rolling resistance moment plays a role in these transformations.Currently little is known about the rolling resistance of a motorcycle tyre, and it is thereforeassumed to be constant at any condition. Finally, during simulations the contact point, verticalload and the effective rolling radius are determined without taking into account motorcycle tyrefeatures as the typical tyre contour and a camber dependent vertical tyre stiffness. All thesefacts are expected to introduce considerable errors in the representation of the steady-state tyrebehaviour by the MF-MCTyre model.

1.2 Problem statement

The goal of this master thesis is to improve and validate the MF-MCTyre model. The improvedmodel should be able to correctly represent the behaviour of a motorcycle tyre for its completeworking area at steady-state conditions. A motorcycle simulation model is developed to get famil-iar with the steady-state cornering behaviour of both motorcycle and tyre models. Afterwards theMF-MCTyre model is thoroughly analysed to find its deficiencies. With the aid of measurementsthe following aspects of the tyre model are improved:

• The transformations of the forces and moments during both measurements and simulations

• The determination of the contact point and vertical load during simulations

• The determination of the effective rolling radius, Re, during simulations

• The description of the rolling resistance of a motorcycle tyre

1.3 Research layout

To get familiar with the tyre model and motorcycle cornering behaviour, a motorcycle simulationmodel is developed for which the comprehensive model of Koenen [10] is used as a basis. Inchapter 2 a compact literature study is presented, which has a two-sided goal. First of all thestudy is performed to get an idea of the quality of the motorcycle model of Koenen. Secondly,the developments within motorcycle tyre modelling are present to get a comparison with the MF-MCTyre model. The description of the motorcycle model is presented in chapter 3. The model ismade operational for computer simulations and an elementary controller is developed which makesit possible to simulate the steady-state cornering behaviour under large camber angles. Both themotorcycle and the tyre model are analysed for steady state cornering situations to get familiarwith their behaviour.

In chapter 4 the error sources of the tyre model and their origin are described. Furthermorepossible solutions to the present problems are discussed. In chapter 5, the results of an extensivemeasurement program are presented, with which typical motorcycle tyre features are determined.Moreover, these results are discussed in chapter 6 and used to elaborate and check the possiblesolutions to the problems presented earlier. With the improvements found from this research, anupdated tyre model is generated and presented in chapter 7. The model will be compared to theexisting MF-MCTyre model during simulations with two models. First of all, a tyre test-rig issimulated to test the model under extreme circumstances. With this test rig, the output of bothmodels can be compared with actual tyre measurements. Furthermore, the motorcycle modeldeveloped earlier is used to compare the models under ’real-life’ circumstances. With these resultsconclusions are drawn and recommendations for future research are given in chapter 8.

Chapter 2

Literature study

In order to get a better idea of the quality of the motorcycle model developed by Koenen and ofthe MF-MCTyre model, the history of both multi-body motorcycle modelling and motorcycle tyremodelling is investigated. First of all the history of motorcycle models is discussed chronologically.It becomes clear that, especially the last ten years, a lot of motorcycle research is being conductedat the department of Mechanical Engineering of the University of Padua. Furthermore also asection at the Imperial College of Science in Londen under leadership of Prof. Robin S. Sharphas made important contributions. The basis of the motorcycle model described in chapter 3 isformed by the PhD thesis of Cornelis Koenen [10]. Both the degrees of freedom (DoF’s) and themass and inertial parameters used in other models are compared with this model of Koenen. Theneed for a stabilizing controller is explained and the controller types used in history are listed.Secondly the modelling of motorcycle tyres over the years is investigated. The behaviour of avehicle strongly depends on the performance of the tyres. Therefore a lot of research has beenconducted to describe the behaviour of an automobile tyre, but surprisingly the motorcycle tyrealways has been underexposed, until the last decade.

2.1 Motorcycle modelling

2.1.1 Model structure and features

One of the main researchers on the subject of motorcycle dynamics is Prof. Robin S. Sharp. Al-ready in 1971 he developed a pioneering work [17] in which a relatively simple motorcycle model isdescribed. This model only comprises two rigid bodies that are joined via a conventional steeringjoint and two wheels. In particular, this model has been used to investigate the sensitivity ofthe dynamic behaviour with respect to several design parameters as steering damping, mechanicaltrail and other geometrical parameters. The work of Sharp [20] and Verma [26] from 1980 showindependently that frame compliance should be included in the multi-body description of a mo-torcycle. This in order to correctly describe the eigenmodes of the motorcycle in straight running.One year later, this has been confirmed for the compliance of the front frame, by the researchof Spierings [22] at the University of Eindhoven. During his PhD work, Koenen has published apaper [11] in which the importance of frame elasticity and simple rider body dynamics on the freevibrations are described. Next to the two DoF’s representing the front frame elasticity and therider upper body lean, Koenen has also introduced aerodynamic effects and front and rear suspen-sion in the motorcycle model. The final motorcycle model with its degrees of freedom can be seenin figure 2.1. The most remarkable about this model is the method with which the rear suspensionis has been included. While almost all others use a swing arm and a translational spring/dampersystem, Koenen uses a joint around which the mainbody can rotate with respect to the rear wheel(see figure 2.2). By using this joint, no swing arm is needed which makes the system somewhatless complicated which is favorable as Koenen has analysed the motorcycle model without the aidof a computer. From his complicated model he has determined the dynamic behaviour both when

3

CHAPTER 2. LITERATURE STUDY 4

P i t c h

B o d y l e a nS t e e r

T w i s t

g

P i t c h

x

zy

O

Figure 2.1: The degrees of freedom of the motorcycle model of Koenen

running straight ahead and when cornering. Especially the analysis of the dynamic behaviour ofthe cornering motorcycle has given new insights. Moreover, it has been concluded that duringcornering the in-plane and out-of-plane modes of motion start to influence each other. Therefore,the in- and out of plane motion cannot be decoupled and the eigenmodes during cornering containboth. Next to this, Koenen he has also investigated the influence of several design parameterson this dynamic behaviour. In 1985 Sharp has written a paper [18] with a review of the state

P i t c h P i t c h

O t h e r s K o e n e n

Figure 2.2: Different methods of introducing a rear suspension

of knowledge and understanding of the steering behaviour of single-track vehicles. This with themain accent on vehicle design, vehicle design analysis and behaviour prediction. In this paper, themodel of Koenen is referred to as the most comprehensive motorcycle model, which represents astep change in the technology. As the model of Koenen is so comprehensive, little to no changesin the modelling of motorcycles can be seen until nowadays.

The major breakthrough after this period has been the development of computers with moreand more memory and computational capacity. This leaded to the development of software pack-ages with which it is possible to analyse complicated multi-body models with relatively little effort.In recent history, a lot of research is conducted at the department of Mechanical Engineering of theUniversity of Padua. Next to modelling of motorcycles also a lot of measurements are performedto confirm the results gained by modelling. In 2000 a paper [3] has been presented in which amultibody simulation model of a motorcycle is present. The model contains the same DoF’s asKoenen, only again a swing arm is modelled for the rear suspension and no frame compliance


was taken into account. With this model, the motorcycle handling is evaluated and the resultsobtained by simulation show very good similarity with measurements. Two years later, an im-proved model has been presented by Cossalter [6]. In this paper also tyre modelling has beenimproved, as will be discussed later. The results of slalom simulations have again been comparedwith measurement results, and show very good correspondence. Recently Cossalter [7] has shownthat this model (with again some changes) can also be used for modal analysis of a motorcycle.The results for the frequency of the eigenmodes are similar to those obtained from measurements,however the damping is of worse quality, for which no exact reason has been found.

In 2001 again the confidence of Sharp in Koenen’s work has been shown as the motorcyclemodel has been used to develop a motorcycle model for computer simulations. In this research [21]the Koenen model is build with the automated model-building platform AutoSim, with the onlydifference that a swing arm is used. The striking part of this research is that the results obtainedby the AutoSim model only partly match the results of Koenen’s research. The explanation forthis is two-folded. By checking the AutoSim code of Sharp, it appears that the inertial parametersof Koenen are generally misinterpret. Secondly, Koenen has analyzed the 28th order motorcyclemodel completely by hand, which is almost impossible to do without minor mistakes. In the sameyear Sharp has presented a complete sensitivity and stability analysis of the same model [19]. Inthis paper, also the power of present multibody software has been discussed.

Concluding, it can be said that the model of Koenen is still one of the most comprehensivemotorcycle models. Frame compliance is taken into account by a twist degree of freedom, the riderbehaviour is partly modelled as well as the aerodynamic effects. The only doubtful point is theway in which the rear suspension is modelled. The results of Koenen show good comparison to theresults of others, especially for the dynamic behaviour when running straight ahead. Some doubtremains about the dynamic behaviour while cornering, but also some errors might be present inhis work because the complexity of the analysis.

2.1.2 Mass and inertial parameters

The parameters that are used by Koenen are measured from a motorcycle, except for the rider’sparameters. The masses are to be trusted on their validity, but the moments of inertia aremuch harder to measure. Especially the moments of inertia of bodies that are introduced in themodel because of frame compliance are difficult to determine. Therefore, the parameters usedby Koenen are compared to the parameters used in the work of others. The main problem withsuch a comparison are the differences between both the measured motorcycles and the motorcyclemodels used. First of all, data is found in a time span of 23 years in which motorcycles haveundergone significant changes. Secondly, there exists a large variety of motorcycles which all havetheir specific geometrical and inertial properties. Finally, different researchers have built theirown models which all have their own DoF’s. In table 2.1, the masses used in several models arelisted. If models divide certain bodies into submasses, those are specified. Later those submassesare added to make a correct comparison. As can be concluded from table 2.1, the masses used

Masses [kg] Koenen [10] [8] [9] [5] [22] [20] [20] [20] [20]mmain 209.6 170.3 154.5mrider 44.5 50.0 44.0

mrearwheel 25.6 25.0 16.5mmaintotal 279.7 245.3 215.0 153.4 355.0 245.3 218.0 221.5 273.9

mfrontsprung 13.1 15.5 18.0mfrontunsprung 17.5 10.4 28.0

mfronttotal 30.6 40.6 25.9 18.0 46.0 40.6 30.7 35.1 36.9

Table 2.1: Comparison of body masses used in several motorcycle models


in the model of Koenen are in good agreement with the masses of other models. For a modernmotorcycle, the entire motorcycle mass would be somewhat large, but the model is already 20 yearsold. Especially the distribution of masses over the separate bodies seems good in comparison withother models.

Next to the masses, also the moments of inertia of the bodies have an important role in thedynamic behaviour of the motorcycles. The inertias of the models mentioned above are foundin table 2.2. All inertias are given in the local center of mass of each body, with respect to thelocal axis system. Furthermore, the axis system is defined as in figure 2.1, with the x-axis inlongitudinal direction, the y-axis in lateral direction and the z-axis in vertical direction. In thenominal situation all bodies are assumed to be symmetrical with respect to the vehicle centerplane. Because the local y-axis of each body is directed perpendicular to this plane, the productof inertia Jxy, Jyx, Jyz and Jzy are all zero. The models of [10], [8] and [9] all assume the rider

Inertias [kgm2] Koenen [10] [8] [9] [5] [22] [20] [20] [20] [20]JmainX 37.75 38.40 23.00 22.46 30.61 34.20 31.20 32.35 34.34JmainY 42.65 32.90 18.90 36.24 0.00 0.00 0.00 0.00 0.00JmainZ 22.69 8.26 26.00 17.36 19.78 22.53 21.00 19.84 22.81

JmainXZ -7.89 -3.00 -3.70 -4.40 0.00 0.00 0.00 0.00 0.00

JriderX 1.30 0.00 2.90JriderY 2.10 11.60 1.50JriderZ 1.40 7.40 1.00

JriderXZ 0.30 3.97 0.00

JsteerX 4.80 3.97 4.00 1.80 1.58 3.97 3.70 3.45 3.58JsteerY 4.10 0.00 2.31 1.77 0.00 0.00 0.00 0.00 0.00JsteerZ 0.49 0.00 0.65 0.35 1.00 0.36 0.36 0.30 0.41

JsteerXZ -0.29 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00

JfrontwheelY 0.58 0.58 0.46 0.49 1.04 0.58 0.72 0.57 0.68JrearwheelY 0.74 0.41 1.67 0.71 1.04 1.06 1.06 1.13 1.25

Table 2.2: Comparison of body inertias used in several motorcycle models

to be non-rigidly attached to the mainframe. Therefore, the moments of inertia of the rider areseparately defined. As the other models assume a rigidly attached rider, the moments of inertiaof the main-bodies of those three models need to be compensated in order to make a correctcomparison. For this comparison the rider upper bodies are assumed to be rigidly attached forthose models. From [25] it is learned that the formula of Huygens-Steiner can be used to writethe inertia tensor of a certain mass m with respect to an arbitrary point O if the vector betweenboth points ~rcm is known.

oJ =cm J + m( ~rcm. ~rcmI− ~rcm ~rcm) (2.1)

This equation tells us that the inertia matrix with respect to an arbitrary reference point oJ equalsthe inertia matrix with respect to the center of mass cmJ plus the inertia matrix with respect tothe arbitrary point of the the mass m concentrated at the center of mass. With this known itis possible to translate the inertia tensor of the rider upper body to the center of mass of themain-body. At this point it is added to the inertia tensor of the main-body. The same has beendone for the moments of inertia of the steer body of the models of [10] and [9]. Those modelsdefine separate bodies that rotate around the steering axis, which have been defined as if theywere one body to make a fair comparison. With some exceptions, also the moments of inertia arein good correspondence with other models. In general the moments of inertia used are somewhathigher than those used by others, which is in agreement with the higher masses in the motorcycle


of Koenen. Some doubt is present towards the parameters of the rider upper body. The momentsof inertia of [9] are in the same order as those of Koenen, those of [8] are much higher. Theproblem with the rider upper body is of course how to define the division between the lower andupper body of a rider and how to determine the moments of inertia of the upper body.

2.1.3 Stabilizing controller

From both literature and practice it becomes clear that motorcycles are inherent to stabilityproblems at different velocities. In order to be able to simulate a motorcycle a controller needs tobe developed to remove those instabilities. In motorcycle dynamics a lot of eigenmodes are present,but three main instability modes are distinguished. First of all there is the non-vibrational capsizemode, where the motorcycle rolls over and falls down to the ground. Secondly, a so-called wobblemode is present at velocities between approximately 40-100 [km/h], dependent on the design ofthe motorcycle and the front tyre. This mode comprises a high frequent (8-10 [Hz]) vibration ofthe front frame around the steering axis. Finally, also a weave mode is present at very low (≤20 [km/h]) and at very high (≥ 170 [km/h]) velocities. In this mode the complete motorcycle isvibrating in the sideways direction.

To be able to simulate the motorcycle under several circumstances a controller needs to bedeveloped, which removes the instability of these three modes. This controller should thereforetake over some of the tasks that are normally fulfilled by the driver. Moreover, motorcyclesare typically steered by prescribing the steer torque which is in contradiction to automobiles,for which the steer angle is prescribed. As the modes are different for different speeds and fordriving straight out or cornering, the controller differs from situation to situation. In the literatureonly little information can be found on controlling a motorcycle, but some references show somethoughts about stabilization and path control. First of all Berritta [3] uses a so-called ’look ahead’path follower in order to follow a certain trajectory. Starting from the motorcycle’s actual positionand velocity, the controller calculates in which position with respect to the prescribed trajectorythe motorcycle should be after a certain time t. This distance u, its derivative, the camber angleγ and the roll speed are then used to calculated the steering torque that is needed to follow thetrajectory. This is being done with the proportional gains K1, K2, K3 and K4:

Tδ = K1γ + K2γ + K3u + K4u (2.2)

Both Sharp [21] and Kamata [9] only use the camber angle of the mainframe to stabilize themotorcycle. Where Sharp uses a hand-tuned PID-controller, Kamata uses a more sophisticatedmethod. System identification is applied in order to design a stabilizing H∞-controller. Finally,also Ruijs [16] conducted research to the stability of motorcycles. As he is the successor of Koenen,his research is based on the motorcycle model of Koenen. To stabilize this model, not only thefeedback of again camber and its time-derivative are needed but also the time derivative of thesteer angle. With Nyquist plots it is explained that feedback of the camber angle is needed tostabilize the capsize mode, its time-derivative is needed to stabilize the weave mode and the timederivative of the steering angle is needed to stabilize the wobble mode.

2.2 Motorcycle tyre modelling

For the history of tyre modelling before 1985 is again referred to the review of Sharp [18]. Inthe very first beginning (1869) tyres are not viewed as producers of forces and moments. Thetyre/road contact is treated as constraining the bicycle to move in the direction in which thewheels pointed [14]. Much later, in 1942, the knowledge of tyre behaviour is greatly improved andthe rolling wheel is regarded as a force producer [27]. In this time it has also been concluded thatfor motorcycle tyres sideslip angles are small and cornering is mainly possible by camber thrust.It has been usual to describe tyres giving side forces linearly dependent on slip angle and camberangle. In 1971 Sharp [17] has introduced lag in the tyre side force by a first order relaxation model,


which seems of large influence on the dynamic behaviour of a motorcycle. Only two years later,also non-linear tyre forces have been introduced in motorcycle models [15].

Next to the comprehensive motorcycle model Koenen has also developed a sophisticated tyremodel. The tyres are modelled as radially flexible, and their cross-sectional shape is accounted forto some extent. Inputs for the tyre model are the slip-angle, camber angle, turn-slip and verticalload. As the vertical load is depending on the lateral force and the lateral force on its turn isdepending on the vertical load, the equations are solved with a Newton-Raphson iteration. Alsothe non-stationary tyre behaviour is taken into account with a first order relaxation model.

In 1987 the first version of the Magic Formula tyre model has been presented [1]. This modelwith an empirical background contains a set of mathematical formulae, which are partly basedon a physical background. Those formulae are able to accurately approximate the typical forcecharacteristics that are generated by a pneumatic tyre. At first, the Magic Formula only considerspassenger car and truck tyres, i.e. tyres with relatively large slip angles and camber angles whichonly exceptionally exceed 10 degrees. First developments of a Magic Formula tyre model applicablefor motorcycle tyres by De Vries are seen in 1997 [28]. The results show that the adapted modelis able to accurately describe the stationary lateral force and aligning torque. Some doubts arepresent about the correctness of the first-order relaxation model at high (≥ 100 [km

h ]) velocities.At high velocities the model is not capable of taking into account the gyroscopic effects on the tyrebelt. A year later De Vries has investigated the influence of tyre modelling on the stability analysisof a motorcycle [29]. The root loci of a motorcycle model which is equipped with different tyremodels are analyzed. First of all, the motorcycle is equipped with a front and a rear tyre whichhave a constant relaxation length. Secondly, the rear tyre is replaced with a rigid-ring tyre model,while the front tyre is left unchanged. The first impression is that the implementation of the rigidring model does not give large differences in the weave mode of a motorcycle in comparison tothe constant relaxation length model. At high speeds stabilizing contributions of the gyroscopiccouple can be found in this mode if the rigid ring tyre model is used. At moderate speeds (80[km/h]) this tyre model gives little less damping than the constant relaxation length model. It ismentioned that the effect of the rigid ring tyre model of a front tyre still has to be studied, andthat its influence on the wobble mode is expected to be larger. The general acceptance of theMagic Formula tyre model is shown by the use of the model by several researchers [23] [3].

C

S p i n d l e a x i s

T y r e c o n t o u r

R o a d s u r f a c ey M x

A

Figure 2.3: Different contact point definitions

In the meanwhile, a tyre model is developed at the University of Padua to overcome severalproblems of the first-order relaxation model. The first signs of this model are seen in the work ofCossalter in 1999 [5]. The Magic Formula is adapted in such a way that it is suitable to act inthe ’actual’ contact point (A). The original Magic Formula is used in the ’fictitious’ contact point(C) defined by the wheel-plane and the road surface, as can be seen in figure 2.3. One of theadvantages of moving the contact point is that the overturning moment Mx in this point can beneglected. This model is further extended by Cossalter [4] with the elastic modelling of the tyrecarcass. In order to describe the contact point accurately two linear models are developed. The


simpler model has two DoF’s, the lateral and radial displacements of the tyre with respect to thehub. The tyre stiffness is modelled by linear springs, which are aligned with the radial and lateraldirection. The second model has one extra DoF with respect to the first model; the rotation of thetyre. By measurements all parameters needed for the description of such models were gained. Thegraphical representation of those models can be seen in figure 2.4. In 2002 Cossalter has extended

Figure 2.4: Different tyre models developed by Cossalter

the two DoF model with rotational deformation in the wheel spin direction (See figure 2.5) andimplemented it in a multi-body motorcycle model. It is proven that such a model represents thedynamic tyre behaviour in a way equivalent to the relaxation tire models. However, this approachhas several advantages. First of all, it explains the physical behaviour of the tyre in a morerealistic manner. Furthermore, only static and steady state experimental tests are required tocharacterize the tyre behaviour in both static and dynamic conditions. The latest developments

V

t i r er i m

x

Figure 2.5: Rotational tyre deformation with respect to the rim

to this tyre model are recently presented by Lot [12]. Again the three elastic deformations of thetyre are taken into account. However one should interpret the determined tyre stiffnesses critically,as the measurement range of the applied force is limited (0-300 [N]). In this model a very goodapproximation of the tyre geometry is introduced. The tyre cross section is photographed andthe contour of the tyre is fitted. Finally, the presented model is compared with the relaxationmodel finding that both models are equivalent in absence of camber angle. When a camber angleis introduced the new model is able to properly fit experimental tests, while the relaxation modelis not. The phase lag of the relaxation model is too large in comparison with the new model.

Chapter 3

A motorcycle simulation model

In the year 1983 Cornelis Koenen has published his PhD work in the report ’The dynamic be-haviour of a motorcycle’ [10]. In this report a physical model of a motorcycle is described witha multibody approach. In order to get a better understanding of the dynamics of both a motor-cycle and its tyres, Koenen’s work is made operational by developing a model in the multibodytoolbox of Matlab/Simulink, SimMechanics. The model of Koenen is explained in this chapter,the parameters used in this model can be found in Appenix B.

3.1 Model structure and features

The multibody model is build with respect to an orthogonal axis system (O,x,y,z). The originO of this axis system lies in the contact point between the rear tyre and the ground plane.The gravity g is pointing in the −z direction. The multibody model is composed of eight rigidparts, interconnected by kinematic constraints. This model, together with its sign conventions, isdepicted in figure 3.1. All the joints in the model are one degree of freedom revolute joints, exceptfor the front suspension which is a one degree of freedom translational joint.

1

1 s

1 u2 s

2 w1 w

2

3

P i t c h

B o d y l e a nS t e e r

T w i s t

g

x

z y

O

Figure 3.1: Motorcycle model components

10

CHAPTER 3. A MOTORCYCLE SIMULATION MODEL 11

All parts that are shown in figure 3.1 are assumed to be infinitely stiff. The most relevantelasticity property of the frame is accounted for in the ’twist’ degree of freedom. The main frame(2) of the motorcycle forms the basis part of the model. In the SimMechanics model, the connectionto the ground plane is made with this body by means of a 6 DoF joint i.e. the motorcycle canfreely with respect to the inertial frame. In some studies concerning motorcycle dynamics therider body is assumed to be rigid and rigidly connected to the main frame, which gives a poorrepresentation of the reality. To avoid too large differences between the model and reality, in thiscase the rider body is split up in two parts. The lower segment of the rider body is assumedto be rigidly attached to the main frame (2), the upper part (3) to rotate about an axis whichis horizontal in the initial condition, see figure 3.1. This rotation is both sprung and damped.Furthermore, the rear wheel (2) is connected to the main mass with a sprung and damped swingarm. This massless swing arm makes it possible for the rear wheel to rotate around a point onthe main body and in the plane of symmetry, the ’pitch’ movement. This is the only point atwhich the SimMechanics model differs from the Koenen model because Koenen uses a differentjoint for the pitch motion, as explained in section 2.1.1. As this joint is difficult to model in theSimMechanics environment the choice is made to use a swing arm, which is normally used inmotorcycle models. The rear wheel (2w) is of course also given a DoF in such a way that it isable to rotate around its own axle. At the front end of the main mass the steer pivot is located.The steer body (1), twist body (1s), front unsprung mass (1u) and front wheel (1w) togetherrotate as a whole relative to the main mass, about an inclined steering axis. As said, the mainelastic property of the frame has been accounted for in the twist degree of freedom. The twistaxis, which is perpendicular to the steering axis, allows the twist body (1s), front unsprung mass(1a) and front wheel (1w) to rotate out of the plane of symmetry of the motorcycle. Also thisrotation is sprung and damped. The front suspension is modelled as a translatory movement ofthe front unsprung mass (1u) and front wheel (1w) perpendicular to the steering axis if no twistangle is present. Again this movement is both sprung and damped. Finally, the front wheel (1w)is given one DoF, to be able to rotate around its spindle. All parameters considering this modelare documented in Appendix B, together with a geometrical figure of the model.

P l x

P d z

F l

F d

Figure 3.2: Aerodynamic forces acting on the motorcycle

Additional to the parts of the model that are depicted in figure 3.1, the environment of themotorcycle needs to be modelled. This comprises the road surface and the air through whichthe vehicle moves. The road surface is assumed to be a flat and even plane perpendicular to thedirection of the local gravitational field. The air surrounding the vehicle is assumed to be initialllystill relative to this ground plane. The motion of the vehicle will give rise to both stationaryand non-stationary forces acting on it. From these forces only two components are regarded, the


stationary drag and lift forces. The direction and lines of application of these forces can be seenin figure 3.2. The motorcycle is modelled in such a way that the aerodynamic forces act at aspecified point of the main mass (2).

3.2 Stabilization and velocity controllers

In the literature survey (Chapter 2) it is stated that two-track vehicles are inherent to stabilityproblems. In real life, the driver of the motorcycle acts as a controller, in order to remove theinstabilities of the motorcycle. This is done by both moving the upper body in order to move thecenter of gravity and by giving a certain torque on the handle bars. As no driver model whichsteers the motorcycle is present in the model of Koenen also this two-track vehicle will be unstable,so a stabilizing controller needs to be developed.

In section 2.1.3 it is explained that the three main instabilities of a motorcycle are capsize,weave and wobble. It is also shown that Ruijs [16] has developed a stabilizing controller formotorcycles, based on the model of Koenen [10]. From this research it can be learned that threeseparate feedback loops are required and each feedback signal needs its own gain. First of all, theangular velocity with which the front fork rotates around the steering axis is needed, to stabilizethe wobble mode. Secondly, the capsize mode is stabilized with the feedback of the camber angleof the motorcycle. Finally, the time derivative of this camber angle is used to stabilize the weavemode. The schematic representation of this controller is depicted in figure 3.3.

M o t o r c y c l e m o d e lS t e e r t o r q u e ggd

KK d

Kg

g

0

0r e f

Figure 3.3: Stabilizing feedback controller

In order to steer the motorcycle into a corner, the reference of the camber angle γ is changedfrom 0 [rad] to a certain value. The references of the camber velocity and the steer angle velocityremain 0 [rad/s]. The feedback gains of the controller are depending on the forward speed of themotorcycle. For several forward velocities, these gains are determined empirically.

Next to the stabilizing controller, also a controller needs to be developed to keep the forwardvelocity constant during a simulation. The controller therefore needs to compensate for the rollingresistance of the tyres and the aerodynamic drag. To overcome this problem a PD-controller isimplemented, with the feedback of the actual forward velocity of the motorcycle. The output ofthis controller is a driving torque that is applied at the rear wheel.

3.3 Tyre model implementation

In order to describe the behaviour of the motorcycle tyres, of course the MF-MCTyre model isused in the motorcycle model. A short description of the implementation is given here, moreinformation about the model can be found in Appendix A. The tyre behaviour is implementedby means of the Standard Tire Interface which needs eight input signals, as can be seen in figure3.4. In this figure only a wheel of the vehicle model is depicted. This wheel body is connected


Figure 3.4: Implementation of the MF-MCTyre model in the motorcycle simulation model

to the rest of the vehicle by means of the ’Vehicle model connection’. The first input of thetyre model (road) defines the road profile during the simulation. The height of the road andthe slope of the road in both the x and y direction are specified. As only flat road surfaces areconsidered, all three input variables will remain 0. The second input dis is defined as the x, yand z position of the wheel in the global axis system. This position is measured by means of aSimMechanics element; the wheel sensor. This sensor also defines the input tramat, which is a9 component transformation matrix defined in the global plane. Both vel and omega are alsodefined by a sensor connected to the wheel center. This sensor defines the global velocity (vel)and the global angular velocity (omega) of the wheel center with respect to the wheel’s localaxis system. The signals angtwc and omegar give the rotation angle and the relative rotationalvelocity between the tyre belt and the wheel body. As this motion is not taken into considerationwithin the simulations within this research, both signals remain 0. Finally, seven coefficients toscale several Magic Formula coefficients are introduced. For example the longitudinal and lateralpeak friction coefficients and the pneumatic trail coefficients can be adapted. The output of theinterface consists of the forces (force) and torques (torques) defined in the local axis system ofthe wheel. By means of the actuators those forces and moments are fed back to the center of thewheel. The third output varinf contains 40 signals representing the tyre behaviour. In this signalfor example the forces and moments in the contact center, the longitudinal slip κ, the side slipangle α and the camber angle γ are given.

3.4 Simulation results

The main purpose of this research is to investigate the steady-state behaviour of both the motor-cycle and tyre model. More specifically, cornering at large camber angles is of particular interestas this operation area is typical for motorcycles. Therefore, several simulations are conducted atdifferent cornering conditions. In this section the results of a simulation with a forward velocityof 100 [km/h] are presented. Within the first five seconds the motorcycle is running in a straightline at a camber angle of 0 degrees. At exactly five seconds, the camber angle reference of thecontroller is gradually changed to 48 degrees. Therefore, the motorcycle drives into a right-handcorner where it reaches a steady state. To make sure that a steady state is reached, the motorcy-cle remains cornering until the end of the simulation at 100 [s]. The trajectory of the motorcycleduring this simulation is depicted in figure 3.5. Although the fact the steady state behaviour is ofinterest during in this research, first of all a short remark is made about the non-stationary steer


50 100 150 200 250

−180

−160

−140

−120

−100

−80

−60

−40

−20

0

X−distance [m]

Y−

dist

ance

[m]

Figure 3.5: Trajectory of the motorcycle model during a cornering simulation

torque. In figure 3.6 the camber angle and the steering torque during the simulation are depicted.If one examines the steering torque while the motorcycle is driving into the corner, one sees the

0 10 20 30 40 50 60 70 80 90 100−10

0

10

20

30

40

50

Cam

ber

angl

e, γ

[deg

]

0 10 20 30 40 50 60 70 80 90 100−50

−40

−30

−20

−10

0

10

Time [s]

Ste

er to

rque

[Nm

]

Figure 3.6: Camber angle and steering torque response during a simulation

typical paradox in the motorcycle steering behaviour. When the motorcycle needs to drive intoa right hand corner (γ ≥ 0), first a steering impulse to the left is needed. The origin of thisbehaviour, called gyroscopic precession, lies in the inertia of the rotating front wheel. A spinning


wheel has a very stable axis of rotation, i.e. a strong tendency to maintain its plane of rotation.If a twisting moment is applied which tries to change this plane of rotation (as happens when thewheel is steered) this leads to a tilting moment around an axis at 90 degrees to that of the twistingmoment. If, for example, a spinning wheel is steered to the left this results in a moment whichtilts the wheel to the right. Therefore, a large steering torque to the left is needed for an instantto steer the motorcycle into a right hand corner.

Once the motorcycle is cornering in a steady state, several equilibriums of forces and momentsare present which can be analysed. The signals used for the analysis of these equilibriums areaveraged over the last 10 seconds of the simulation. The tyre in- and output quantities can befound in table 3.1 and the variables measured from the motorcycle are presented in table 3.2.Note that the tyre parameters are given an extra index f of r to define the difference betweenthe front and rear tyre respectively. Furthermore, note that the front and rear suspension have apretension, so the forces and moments cannot be directly derived from the deformation and thestiffness. The pretension is of such magnitude that when the motorcycle is in an upright positionwith no forward velocity, the geometrical parameters of the motorcycle that are given in AppendixB are present. Moreover, the suspension deflection and the pitch angle are given with respect tothis initial state.

Front tyreκf [-] 0.003αf [deg] 0.27γf [deg] 48.46Fxwf [N] -101.7Fywf [N] -1126.4Fzwf [N] 1310.3Mxwf [Nm] -67.8Mywf [Nm] -5.9Mzwf [Nm] -38.4

Rear tyreκr [-] 0.004αr [deg] 1.37γr [deg] 48.26Fxwr [N] 332.0Fywr [N] -1635.9Fzwr [N] 1758.2Mxwr [Nm] -197.3Mywr [Nm] -8.6Mzwr [Nm] -27.6

Table 3.1: The tyre in- and output for the front and rear tyre during steady state cornering

Camber angle, γ [deg] 48.26Aerodynamic drag, Fd [N] -187.1Aerodynamic lift, Fl [N] -43.7Pitch angle [deg] 0.62Pitch torque [Nm] -714.6Suspension deflection [m] -0.02Suspension force [N] 1359.6Rider lean angle [deg] 0.81Rider lean torque [Nm] -14.12Twist angle [deg] 0.21Twist torque [Nm] -113.2Steer angle [deg] 0.03Steer torque [Nm] -4.9

Table 3.2: Motorcycle behaviour during steady state cornering

In figure 3.7 the deformation of the motorcycle during steady state cornering is graphicallyrepresented. As can be seen from the rear view, the positive rider lean angle leads to the riderupper body leaning into the right hand corner. Furthermore, also the twist angle is positive dueto which the front fork and the front wheel are rotated with respect to the main body. Therefore,the camber angle of the front wheel is larger than the camber angle of the rear wheel. From the


gb

1

1 s

1 u

2

P i t c h

S t e e r

T w i s t

3

1 w2 w

2 s

B o d y l e a n

r e a r t y r e

g rB o d y l e a n a x i s

T w i s t a x i s

f r o n t t y r e

3

Figure 3.7: Rear view(left) and side view(right) of motorcycle deformations during steady statecornering

side view the deformation of the rear suspension (pitch) and the front suspension can be seen. Thefront suspension is somewhat compressed and therefore the front suspension deflection is negative.Finally, the pitch angle is positive, so the wheel is rotated towards the main body and the rearsuspension is compressed.

3.4.1 Equilibria of forces

The equilibria of forces in the global x, y and z direction are analysed. With this analysis it canbe explained which tyre forces are present and why they have their specific magnitude duringthis steady state cornering situation. The equilibrium in the longitudinal direction is formed bythe tyre forces and the aerodynamic drag. The only force which is driving the motorcycle is thelongitudinal force of the rear wheel. The longitudinal force of the front wheel (due to the rollingresistance) and the drag force are negative. Furthermore, the combination of the negative lateralforces and the positive slip angles also lead to a negative longitudinal force. Therefore it shouldhold that:

Fxwr + Fd + Fxwf + Fywr sin αr + Fywf sin αf = 0 (3.1)

If this equilibrium is computed a residual force of -1.3 [N] is present most probably due to averagingand rounding. With respect to the 332 [N] of the rear tyre, this residual force is only 0.4% whichis negligible.

The overall vertical force of both tyres balances the overall gravitational force and the compo-nent of the aerodynamic lift in the z-direction.

Fzwf + Fzwr + Fl cos γ + g(m1 + m1u + m2 + m2u + m3) = 0 (3.2)

When the equilibrium is determined with the averaged forces, a residual force of 4.6 [N] is present.Again this is a negligible 0.15% with respect to the overall vertical tyre force of 3068.5 [N].

The lateral tyre forces are negative (pointing inwards the corner) and for an equilibrium in thelateral direction they need to compensate for the centrifugal forces (pointing outwards the corner)that are acting on each body. Moreover, these centrifugal forces Fa can be calculated for each


individual body.

Fa = mω2R (3.3)

with m the mass of the body, ω the rotational velocity and R the cornering radius of the centerof gravity of the body. Therefore the equilibrium is formed by:

Fyf + Fyr +n∑

i=1

Fai = 0 (3.4)

Fyf + Fyr +n∑

i=1

ω2miRi = 0 (3.5)

If this equilibrium is determined, a residual force of -26.85 [N] is present, which means that thetotal of the tyre forces is 0.97% larger than the total centrifugal force.

3.4.2 Camber angle γ and steer torque Ms

During steady state cornering, several other equilibria exist within the simulation model. For ex-ample, equilibria of the rider upper body, the equilibrium around the pitch axis and the equilibriumaround the twist axis are present. Two parameters that are also determined by such equilibriaare the camber angle γ and the steer torque Ms. The specific equilibria that determine theseparameters are analysed. With this analysis, it is possible to determine which specific motorcycletyre parameters have a significant influence on them.

M x w

gF g

F a

S

S

S

H

RD

mS

Figure 3.8: Equilibrium of moments of the complete motorcycle around its local x-axis

The camber angle that is present during steady state cornering can be obtained from theequilibrium of the moment around the local x-axis. This equilibrium is schematically depicted inthe left part of figure 3.8. The center of gravity of the motorcycle is depicted in which the sumof the centrifugal (

∑Fa) and gravitational forces (

∑Fg) act. Together with the height of the

centers of gravity of the bodies, the centrifugal forces lead to a moment which tries to decreasethe camber angle γ. Due to the camber angle, the bodies are cornering on a different radius thanthe contact points of the tyres. Therefore, the gravitational forces lead to a moment which tries toincrease the camber angle. Finally, there is an overturning moment Mxw in the contact point ofeach tyre which has already been explained with the aid of figure 2.3. The equilibrium is formedby:

Mxwf + Mxwr +n∑

i=1

Fgi∆Ri =n∑

i=1

FaiHi (3.6)


Mxwf + Mxwr +n∑

i=1

gmi(Rf −Ri)−n∑

i=1

ω2miRiHi = 0 (3.7)

For the determination of this equilibrium, the cornering radius of the front tyre Rf is used. Withthis assumption, the residual torque is 2.6 [Nm]. Again, this is relatively small as the total momentaround the local x-axis is around 1135 [Nm]. It is shown that the camber angle is determined byhe overturning moments of the front and the rear wheel. Therefore, if these moments would bealtered also the camber angle will change.

g 1 xm 1

e g 1 u xm 1 u

R o

t

M s

g

m 1

m 1 u

F y c M z c

F a 1 u

F g 1 u

F g 1

F a 1

w h e e l c e n t e r

t w i s t a x i ss t e e r a x i s

Figure 3.9: A side view(left) and a rear view(right) of the equilibrium of moments around thesteer axle

The equilibrium of the steer torque is somewhat more complicated. As can be seen fromfigure 3.9, several components are present which have an influence on the equilibrium. For thedetermination of the equilibrium the twist angle and the steer angle are not taken into considerationas they are negligibly small (See table 3.2). Furthermore, the tyre forces and moments are firstlytransformed to the C-axis system which has its origin in the wheel center. With the forces andmoments in this axis system, the equilibrium is somewhat easier to explain and understand. Infigure 3.10 the forces and moments in the tyre road contact point O and the wheel center C aredepicted. As the forces in the tyre road contact point are known from table 3.1, the forces in thewheel center can be derived with this figure.

Fyc = Fyw cos γ + Fzw sin γ = 234.1[N ] (3.8)

Fzc = −Fyw sin γ + Fzw cos γ = 1712.0[N ] (3.9)

It is important to notice that the lateral force at the wheel center Fyc is relatively small. Thetyre forces are balanced in such a way that the major part of the force is present in the planeof symmetry of the wheel, namely in Fzw. As will be shown, this force has no influence on theequilibrium of the steer torque. Furthermore, the front wheel is freely rolling and therefore themoment around the spindle axle Myc remains zero. The moment Mzc can again be derived withthe aid of figure 3.10.

Mzc = −Myw sin γ + Mzw cos γ = −21.0 (3.10)


F y w = - 1 1 2 6 . 4 [ N ]

F z w = 1 3 1 0 . 3 [ N ]

F z c = 1 7 1 2 . 0 [ N ]F y c = 2 3 4 . 1 [ N ]

R lg

O

CM y w = - 5 . 9 [ N m ]

M z w = - 3 8 . 4 [ N m ]

M z c = - 2 1 . 0 [ N m ]M y c = 0 [ N m ]

R lg

C

O

Figure 3.10: The forces (left) and moments (right) in the W and C-axis system of the front wheel

Next to the tyre forces and moments, also the gravitational (Fg1 and Fg1u) and centrifugal(Fa1 and Fa1u) forces of the masses m1 and m1u influence the equilibrium. The force of each massthat effectively influences the steer torque equilibrium are calculated by:

Fm1 = R1m1ω2 cos γ − gm1 sin γ (3.11)

Fm1u = R1um1uω2 cos γ − gm1u sin γ (3.12)

for masses m1 and m1u respectively.With this known, the equilibrium can be derived.

Ms − Fyct− Fm1g1x − Fm1ug1ux −Mzc cos ε + Mxc sin ε = 0 (3.13)

If the equilibrium is determined a residual torque of 0.3 [Nm] is present which is again negligible.Most probably, this residual torque is caused by the fact that the twist angle and the steer angleare neglected. Moreover, it is remarked that the largest contribution in this equilibrium are themoment caused by the force Fyc in combination with the mechanical trail t and the moment causedby Mzc in combination with the rake angle ε. With -15.4 [Nm] and 18.0 [Nm] these torques arerelatively large and opposite to each other. Therefore if one of both would be altered, a significantchange in steer torque would be seen. Moreover, with this equilibrium it is also proven that themechanical trail t and the rake angle ε have a significant influence on the steering behaviour of amotorcycle.

Chapter 4

Problems of the MF-MCTyremodel

The basis of the MF-MCTyre model is the Magic Formula. Research [28] proves that the mo-torcycle version of this formula is able to accurately describe the characteristics of the forces andmoments generated by a motorcycle tyre. The model surrounding the Magic Formula has beenderived from an automobile tyre model, which is therefore suited especially for the slip and cam-ber angle range of these tyres. As the camber angle range is significantly larger for motorcycletyres than for automobile tyres, the model is known to be less accurate when it is evaluated underlarge camber angles. Furthermore, for the processing of the measurements several assumptions aremade which negatively influence the accuracy. In this chapter the weak points and error sourcesof the MF-MCTyre model are revealed and possible solutions are mentioned. In this chapter theresults of an elaborate measurement program are presented. These measurement results are usedto elaborate the solutions presented in this chapter and to evaluate their correctness. In thischapter only several aspects of the MF-MCTyre model are discussed. For more information aboutthe tyre model the reader is referred to Appendix A.

4.1 Explanation of the tyre model

In the present and the following chapters the axis systems of the tyre model play an importantrole. Therefore the definitions of the most important axis systems are again given, with the aidof figure 4.1. The C-axis system is fixed to the wheel carrier with the longitudinal xc-axis parallel

XY

Z

Xw

C

Yc

Yw

Xc

nr

Zc Zw

Vc

Xw

C

nr

Zw

Zc

O

Xc

Vx

Yc

Yw

-γ

Figure 4.1: C- and W -axis systems used in MF-MCTyre

to the road and in the wheel plane (xc-zc-plane). The origin O of the C-axis system is the wheel

20

CHAPTER 4. PROBLEMS OF THE MF-MCTYRE MODEL 21

center. The origin of the W-axis system is the road contact point C defined by the intersection ofthe wheel plane, the plane through the wheel spindle and the road tangent plane. The xw-yw-planeis the tangent plane of the road in the contact point C, and it defines the camber angle γ togetherwith the normal nr to the road plane (xw-yw-plane). Furthermore, the forces and moments aredescribed in both axis systems. Therefore all forces and moments have a c or w index, whichpoints out with respect to which reference axis system they are defined.

In section 3.3 the implementation of the MF-MCTyre model by means of the Standard TyreInterface (STI) is described. Each time step, several kinematic parameters are retrieved fromthe wheel center (the C-axis system) and used as an input for this interface. Furthermore theinterface returns the forces and moments in the C-axis system as a feedback to the wheel center.The processing of the input parameters within the STI is schematically depicted in figure 4.2.First of all the contact routine uses the position and orientation of the wheel and the road profile

S t a n d a r d T y r e I n t e r f a c e

M e a s u r e m e n t s ( D T T )

M - t y r e

M F - t o o l

C o n t a c t r o u t i n e C o n v e r s i o n

C - a x i s C - a x i s

C - a x i s

W - a x i s

W - a x i sW - a x i s M a g i c F o r m u l a

V e h i c l e m o d e l

T y r e p r o p e r t y f i l e

F o r c e s & M o m e n t sK i n e m a t i c s

C o n v e r s i o n

Figure 4.2: Schematic representation of the MF-MCTyre model

to determine the exact position of the contact point. This point is used as the origin of the W-axis system. In this origin the input variables of the Magic Formula, the vertical load Fzw, thelongitudinal slip κ, the side slip angle α and the camber angle γ are determined by this routine.With these input parameters known, the Magic Formula is evaluated and the forces and momentsin the contact point are determined. As these forces and moments are applied to the vehicle modelat the wheel center, they are converted from the W - to the C-axis system. As said, the loop withthe vehicle model and the STI is evaluated at each time step of the vehicle simulation.

Next to the simulation loop, also the measurement line is depicted in figure 4.2. In contra-diction to the simulation loop this line is only evaluated once. First of all, the tyre behaviour isexperimentally determined by the Delft-Tyre Test Trailer (DTT). During tests the test trailer hasa certain forward velocity and the tyre is pushed against the road at various loads, orientationsand motion conditions. During these tests, all forces and moments are measured at the wheelcenter (the C-axis system). As the Magic Formula is evaluated in the W -axis system, the M-Tyresoftware is used to convert the measurements to the contact center. Finally, the MF-Tool softwareis used to fit the Magic Formula parameters to the measured forces and moments. This leads toa set of parameters that represents one single tyre. This set of parameters and some general tyreparameters are captured in a tyre property file which is used by the STI during each time step ofa vehicle simulation, in order to describe the momentary tyre behaviour.


4.2 Problems with the determination of the contact pointand the vertical load, Fzw

Within the STI, first of all the fictitious contact point C of the tyre with the road is determined.To do this, the road profile is approximated by its tangent plane at the point on the road belowthe wheel center O (See figure 4.3). This approximation is justified by the assumption that theradius of curvature of the road profile is considered large as compared to the radius of the tyre.Therefore, the difference between the actual road and its approximation remains small. With thisroad tangent plane, the wheel plane and the plane through the wheel spindle form one intersectionpoint which is used as the origin of the W system of axis, point C. As the contact point is now

g

R o

Cr

h

r o a d t a n g e n t p l a n e

r o a d p r o f i l e

O

r t

R l

Figure 4.3: Determination of the contact point C and the vertical compression ρ

known, also the vertical deformation ρ of the tyre can be determined. In order to do this thetyre contour is approximated by a circle with radius rt. With the aid of figure 4.3, the normalcompression of the tyre can be determined.

ρ′ = rt + (R0 − rt) cos γ − h (4.1)

ρ = max(0, ρ′) (4.2)

As the tyre is assumed to have a constant vertical stiffness Cz and damping Kz, the normal loadFzw can then be calculated as:

Fzw = Czρ + Kz ρ (4.3)

with ρ the deflection velocity. If during a time step the vertical load is for example higher than thecalculated Fzw, the axle height will slightly decrease. In the next time step the vertical compressionand therefore also the vertical load will increase and the vehicle load is again supported. Withthis ’loop’ the equilibrium between tyre load and calculated vertical force is constantly balancedduring a simulation.

4.2.1 The wheel axle height, h

The determination of both the contact point and the vertical load calculation have a significantinfluence on the correctness of the tyre model. First of all, they determine the height of thewheel axle h during a simulation, which plays an important role in the dynamic behaviour of thesimulated vehicle. If the axle height is incorrect, also the positions of the centers of gravity of


parts of the vehicle will be incorrect. The axle height is especially critical for motorcycles as thesevehicles are very sensitive to mass distribution. Moreover, motorcycles are inherent to instabilitieswhich are partly governed by the positions of the centers of gravity of several parts. Despite theimportance of an accurate axle height, the contact routine contains assumptions which decreasethe accuracy of the axle height during simulations. First of all, the approximation of the road bya tangent plane at the point on the road below the wheel center introduces an error. Especiallywhen the tyre model is evaluated under a large camber angle the distance between the point belowthe wheel center and the contact point becomes rather large. Therefore, the road profile at thecontact point may be different from the estimated tangent plane. Furthermore, the tyre contouris assumed circular which is not the case in practice. Each tyre has its individual shape whichshould be taken into consideration if a correct axle height is requested. Finally, the vertical tyrestiffness Cz is modelled as a linear spring that is independent of the camber angle and the verticalcompression ρ, which is not validated.

To overcome these problems a new contact routine should be developed. First off all, thisroutine should contain a more accurate determination of the contact point. Therefore, the actualcontact point should be determined without the approximation of the tangent road plane. Fur-thermore, also an accurate description of the tyre contour is needed. Finally the assumption of aconstant vertical tyre stiffness should be replaced with an accurate description of the vertical loadas a function of the vertical compression. To be able to make these improvements, first of all thevertical tyre contour and the vertical tyre stiffness should be measured.

4.2.2 The tyre loaded radius, Rl

In the previous section it has been shown that the forces and moments are converted several timesbetween both axis system during measurements and simulations. In Appendix C the processingof the measurement data and the conversions between the axis systems are described. For theconversion of the moments the distance between the axis systems, the loaded radius Rl, of courseplays an important role. To obtain correct conversions it is important to accurately determinethe loaded radius during both measurements and simulations. Moreover, if the circumstancesare similar also similar loaded radii should be present during measurements and simulations toguarantee equal conversions of the moments.

As can be concluded from figure 4.3, the loaded radius Rl during simulations is determinedby the contact point and the vertical force calculations. Due to the incorrect assumptions of thetangent road plane, circular tyre cross section and constant vertical tyre stiffness the loaded radiusis inaccurate. Furthermore, also the loaded radius that is used to convert measurement data isincorrect. The measurement truck with which tyre measurements are performed contains twomeasurement towers, each with its own measurement hub. For the right tower it is impossible tomeasure the loaded radius and a rough estimation is made. For the left tower it is possible tomeasure the loaded radius, but with an ultrasonic sensor which has a very poor accuracy.

The overturning moment Mxw is especially sensitive to the loaded radius, which can be shownwith the measurement signals depicted in figure 4.4. The minimum sideslip angle α is around −7.5degrees and is reached at approximately 55 seconds. At this point the force Fyc is approximately1750 [N] and the moment Mxc is approximately 460 [Nm]. With the conversion described inAppendix C and the loaded radius determined during this measurement, the overturning momentis determined as:

Mxw = Mxc − FycRl = 460− 1750× 0.302 = −68.5[Nm] (4.4)

During measurements, the loaded radius is determined with an accuracy of approximately 5 [mm]and relatively much noise is present. Combined with the force Fyc, the loaded radius error leadsto an absolute error of 8.75 [Nm], which is more than 12% of the actual overturning momentcomputed with (4.4). Due to this error and the large amount of noise, the Magic formula fit is ofa low quality.

Furthermore, the loaded radius used during the conversion of the measurements differs signifi-cantly from the loaded radius used during simulations. As a result, different conversions between


0 10 20 30 40 50 60 70−10

0

10

α [d

eg]

0 10 20 30 40 50 60 70−1000

0

1000

2000

Fyc

[N]

0 10 20 30 40 50 60 70−500

0

500

Time [s]

Mxc

[Nm

]

Figure 4.4: Measured signals during an α-sweep under 50 degrees camber

the axis systems are present and the the tyre model will produce incorrect moments on the wheelaxle during a simulation. The obvious solution to this problem is to use similar parameters whichdetermine the loaded radius during both the processing of the measurements and simulations. Ifthis is done, the loaded radius can be unambiguously defined and is always equal during equalcircumstances.

4.3 Problems with the determination of the effective rollingradius, Re

Within the contact routine another input parameter of the Magic Formula is determined, namelythe longitudinal slip κ. From the signals that are retrieved from the vehicle model the linear rollingvelocity of a tyre is defined by the rotational speed of the tyre Ω and the effective rolling radiusRe (See also figure 4.5).

Vr = ΩRe (4.5)

The longitudinal slip of a tyre is defined as the relative difference between this linear rolling velocityand the actual longitudinal velocity Vx.

κ =Vr − Vx

Vx(4.6)

As becomes clear from (4.5) and (4.6), the effective rolling radius Re of a tyre is of considerableimportance for the correct determination of this slip. This radius determines the ratio betweenthe forward velocity Vx and the rotational speed Ω when no longitudinal slip is present. Therefore,if the effective rolling radius is incorrectly implemented the rotational speed of the wheel will beincorrect during a simulation.

For automobile tyres the effective rolling radius has a specific dependency on the vertical loadFzw which is depicted in figure 4.6. At first, an increase in vertical load will decrease the effective


R o

V x

R eR l

r

W

Figure 4.5: Effective rolling radius and longitudinal slip

rolling radius in an almost similar way as the loaded radius Rl. The vertical load will deform thetyre rubber, the circumference of the tyre will decrease and with that also Re decreases. As thevertical load further increases the tyre will be compressed but the effect on the rolling radius isgetting smaller. This is explained by the fact that the steel carcass of a tyre is radially flexible,but in the circumferential direction very stiff. Therefore at increasing vertical load the tyre stilldeforms, but its circumference remains almost equal. As the circumference does not alter, also theeffective rolling radius will not decrease significantly. Although this effect is thoroughly checkedfor automobile tyres, little information is present on this effect for motorcycle tyres. In the presenttyre model the behaviour of an automobile tyre is therefore implemented.

0 500 1000 1500 2000 2500286

288

290

292

294

296

298

300

302

304

Vertical force, Fzw

[N]

Rad

ius

[mm

]

Effective rolling radius Re

Loaded radius Rl

Figure 4.6: The effective rolling radius and loaded radius as a function of the vertical load

For the estimation of the effective radius Re as a function of the vertical load Fzw a MagicFormula approach is used.

Re = Ro − ρFz0(D arctan(Bρd) + Fρd) (4.7)

In this formula, Ro is the unloaded free radius when no camber is present. The nominal tyredeflection ρFz0 is defined by the vertical tyre stiffness Cz and the nominal wheel load Fz0.

ρFz0 =Fz0

Cz(4.8)


Finally the dimensionless radial tyre deflection ρd can be calculated with the momentary tyredeflection ρ, which is calculated with (4.1)-(4.2) in the previous section.

ρd =ρ

ρFz0

(4.9)

The factor B in (4.7) determines the slope at Fzw = 0, the factor D defines the height of the asymp-tote at high wheel loads and the factor F defines the ratio between the tyre radial deformationand the effective tyre deformation.

It has been proven that this method describes the effective rolling radius of an automobiletyre with sufficient accuracy. Therefore it has also been implemented in the present motorcycletyre model, without any adaptations. However, several assumptions that are made specifically forautomobile tyres negatively affect the accuracy. First of all, it has already been said that littleinformation is known on the effect of the vertical load on the effective rolling radius for motorcycletyres. Copying the parameters B, D and F from automobile tyre measurements is doubtful as theconstruction of a motorcycle tyre is completely different. Therefore, motorcycle tyre measurementsare needed to determine these parameters accurately. Furthermore, no attention is paid to the tyrecontour and its effect on the effective rolling radius when the wheel is cambered. The unloaded freeradius Ro is used in (4.7) instead of the effective rolling radius of a cambered and undeformed tyre,Reo. In figure 4.7 an idea is given of the impact of this assumption. Especially under large camber

R e o

R o

=R e o R o g

Figure 4.7: Effect of camber on the effective rolling radius

angles the effect is considerable. The effective rolling radius is estimated too large and thereforethe longitudinal slip is determined too small during vehicle simulations under high camber angles,which affect the rotational velocity of the wheel. Furthermore, the vertical stiffness used in (4.8)is again assumed as being independent from the camber angle.

Summarizing, several types of measurements are needed to correctly implement the effectiverolling radius in the tyre model. The Magic Formula coefficients that describe the influence of thevertical load need to be determined from measurements at different vertical loads. Furthermorea correct description of both the tyre contour and the vertical tyre stiffness are needed to includethe camber dependency of the effective rolling radius. When these changes are implemented, anextensive measurement program is needed to check the correctness of the new description of theeffective rolling radius.

4.4 Errors in the rolling resistance assumption

When the MF-MCTyre model is evaluated under a large camber angle, it shows an unexpectedbehaviour. If a so-called α-sweep is simulated, significant fluctuations of the longitudinal forceFxw are present. This is in contradiction with measurements where fluctuations of this force are


0 10 20 30 40 50 60 70−10

−5

0

5

10

Slip

ang

le, α

[deg

]

0 10 20 30 40 50 60 70

−300

−200

−100

0

100

Time [s]

Fxw

[N]

MF−MCTyreMeasurements

Figure 4.8: Fluctuating Fxw during an α-sweep under 50 degrees camber

negligible. The difference between the output of the MF-MCTyre model and measurements resultsis depicted in figure 4.8.

The cause of this problem can be found in the processing of the measurement data. Duringtesting the moment around the wheel axle (Myc) cannot be measured and therefore an assumptionhas to be made. To overcome this problem an estimation is made for the rolling resistance momentMyw. In the past, tests have been performed with a non-cambered freely rolling automobile tyre.As the wheel is freely rolling the rotational velocity Ω is almost constant, which leads to theconclusion that there is no effective moment around the wheel axle (2nd law of Newton). Withthis assumption, the equilibrium of a non-cambered wheel can be formulated.

Myc = Myw − FxwRl = 0 (4.10)

A schematic representation of this equilibrium can be seen in the left part of figure 4.9. As the

M y c

MF x w

R l

g

y w

z w

y w

z w

y c z c

z cy c

R l

F M

M

F

FF

M M

x cF

C C y w

OO

Figure 4.9: Equilibrium of a freely rolling wheel

longitudinal force Fxw and the loaded radius Rl can be determined from test results an estimation


of the rolling resistance Myw has been made. This estimation has been defined as:

Myw = −FzwfrRl (4.11)

with fr the rolling resistance coefficient of the tyre. Despite the good results with this assumptionfor automobile tyres, problems arise with motorcycle tyres. More specifically, the problems shownin figure 4.8 arise due to the large camber angle range of a motorcycle tyre. The problem canbe explained with the equilibrium of moments around the spindle axle for a freely rolling andcambered wheel. In the right part of figure 4.9 the forces and moments of a cambered wheel inthe (y-z)-plane are depicted, from which the equilibrium is derived.

Myc = Myw cos γ + Mzw sin γ − FxwRl (4.12)

From an α-sweep, the specific characteristic of the self aligning moment Mzw as a function of theslip angle is determined and fitted with the Magic Formula. It is therefore known that this selfaligning moment fluctuates during an α-sweep. In contradiction, the measured longitudinal forceand the estimated rolling resistance do remain almost constant during the same measurement. Ifthe forces and moments in the contact point are determined with the current rolling resistanceassumption and the equilibrium of 4.12 is determined, the calculated Myc cannot remain constant.In figure 4.10 the fluctuations of the calculated Myc are shown for a measurement that is processedwith the current rolling resistance assumption and a rolling resistance coefficient fr of 1.5 %.However, in practice Myc ≈ 0 as the wheel has an almost constant rotational speed during anα-sweep.

0 10 20 30 40 50 60 70−10

−5

0

5

10

Slip

ang

le, α

[deg

]

0 10 20 30 40 50 60 70−120

−100

−80

−60

−40

−20

0

Time [s]

Myc

[Nm

]

Figure 4.10: Myc during an α-sweep with the present rolling resistance estimation

As all measurements are processed with this incorrect rolling resistance assumption, also thefluctuations in the longitudinal force Fxw of figure 4.8 can be explained. The MF-MCTyre modelcontains information about the forces and moments in the W -axis system which lead to a fluctu-ation Myc during simulations. Due to this moment, the wheel is accelerated while the rest of thevehicle has a constant forward velocity. Therefore, longitudinal slip is generated which affects thelongitudinal force Fxw. To solve these problems, measurements of a cambered and freely rollingmotorcycle wheel need to be analysed. With this analysis a new axis system can be introduced inthe contact point in which the rolling resistance can be defined correctly. Furthermore, the camberangle is expected to have an effect on the rolling resistance factor fr which is nowadays consideredto be constant. Therefore, the measurements are also used to describe the camber dependance ofthe rolling resistance coefficient fr.

Chapter 5

Motorcycle tyre measurements

In the previous chapter some deficiencies of the MF-MCTyre model are revealed. It is shown thatseveral important motorcycle tyre parameters should be measured to be able to include them in thetyre model. Therefore, an elaborate measurement program is conducted in order to get a betterunderstanding of the tyre contour, vertical tyre stiffness, effective rolling radius and the rollingresistance of motorcycle tyres. Especially the effect of the camber angle on these parameters willbe investigated. The results of the measurements are presented in this chapter and in the followingchapter they are used to improve the tyre model.

In order to be as consistent as possible the research within this master thesis has been performedwith one single type of motorcycle tyres. Of this type a front and a rear tyre are taken and theirproperties are determined. TNO Automotive has recently executed an extensive measurementprogram with the Delft Tyre Test trailer in order to determine the Magic Formula parametersof these tyres. The results of these measurements are also used within this research, but in alater stadium. Additional to the measurements conducted by TNO a measurement program wasset-up for this master thesis. Vertical stiffness and effective rolling radius measurements have beenconducted on the Flatplank Tyre Tester at the Eindhoven University of Technology. The FlatplankTyre Tester has been chosen for these measurements as the road surface disturbances are negligiblein comparison to the disturbances measured with the Delft Tyre Test trailer. Furthermore theforward velocity and rotational velocity are measured with a higher accuracy compared to the testtrailer, which is especially favourable for the effective rolling radius measurements.

5.1 Determination of the motorcycle tyre contour

In the previous chapter the importance of an accurate representation of the tyre contour wasemphasized. The tyre contour is needed for an accurate determination of the contact point andthe effective rolling radius during simulations. Each tyre has its individual shape and to representthe tyre behaviour with a tyre model, this shape needs to be measured and described. To determinethe tyre contour a strip of lead is used. This strip can be bent over the tyre and as it easily deformsplastically it takes the form of the tyre. From the lead strip a print is made on cardboard and thecontour is cut out of this cardboard. Finally, the cardboard contour is scanned and from this scanthe contour can be determined digitally. In figure 5.1 the measured tyre contour of a front and arear tyre are depicted. Furthermore, they are compared to the approximation of the round tyrecontour which is presently implemented in the tyre model for the determination of the verticalload. It can be seen that the round approximation is only moderately correct for these tyres.In order to correctly represent different tyre shapes a better and more flexible solution is neededwhich is developed in the next chapter.

29

CHAPTER 5. MOTORCYCLE TYRE MEASUREMENTS 30

0 10 20 30 40 50 600

10

20

30

40

50

60

Width [mm]

Hei

ght [

mm

]

Round contourMeasured contour

0 20 40 60 80 1000

10

20

30

40

50

60

70

80

90

100

Width [mm]

Hei

ght [

mm

]

Round contourMeasured contour

Figure 5.1: The round and measured tyre contour of a front(left) and a rear(right) tyre

5.2 Vertical tyre stiffness measurements

In order to determine the vertical tyre stiffness a measurement procedure has been set up for botha rear and a front tyre. To exclude tyre damping effects the choice has been made to perform allmeasurements with a constant wheel center height. At several camber angles the tyre is pushedagainst the road with three different vertical deformations of 3, 6 and 10 [mm]. For each differentcombination of camber angle and vertical displacement the measurement is repeated three timesto reduce the influence of random errors. The tyre is marked at one point on its circumference, sothat the measurement are always started at an equal point. A trigger is present on the flatplankroad surface which is used to start recording all measurement channels. Furthermore, the mainsteps of the measurement procedure are:

• Inflate the tyre to the prescribed pressure

• Apply a prescribed camber angle by a combination of road and wheel camber

• Rotate the tyre to the marked point on its circumference

• Place the tyre exactly against the road so that there remains zero vertical load

• Remove all offsets by zeroing the measurement channels

• Apply a prescribed vertical displacement perpendicular to the road

• Start moving the road surface and the trigger starts the measurement

The vertical tyre stiffness is desired during steady state conditions. Tyres are known to showa relaxation effect and only reach a steady state condition after a certain travelled distance. Toexclude this transient tyre behaviour measurements are conducted over approximately two tyrerotations. As three different vertical displacements are measured for each camber angle and eachmeasurement is repeated three times, nine measurements are conducted for each camber angle.With the measured Fzw and ρ the vertical stiffness is defined as:

Cz =Fzw

ρ(5.1)

The stiffness measurements are conducted from 0 to 50 degrees with intervals of 5 degrees, for thefront tyre. For the rear tyre, the same procedure is followed but now the largest camber angle is45 degrees due to limitations of the measurement equipment.


During a measurement significant fluctuations in the measured vertical force Fzw are presentalthough the vertical deformation ρ is almost constant. This can be declared by the un-roundnessof the rolling tyre which is confirmed by the fact that the vertical force reaches its minimum andmaximum at the same points of the tyre circumference. This is shown in figure 5.2 where themeasured vertical displacement and force are depicted of a front tyre which is cambered over 25degrees. To exclude the influence of this fluctuation both Fzw and ρ the signals measured overthe last complete tyre rotation are used.

0 100 200 300 400 500 600 7005.96

5.97

5.98

5.99

6

6.01

Def

orm

atio

n, ρ

[mm

]

0 100 200 300 400 500 600 700750

800

850

900

950

1000

Ver

tical

forc

e, F

zw [N

]

Rotation angle [deg]

Figure 5.2: Fluctuating Fzw over the tyre circumference of a front tyre

For the graphical representation of the vertical tyre stiffness, the vertical load Fzw and thevertical deformation ρ are averaged over the last complete tyre rotation. From these averagesan average vertical tyre stiffness can be determined for each measurement, which is depicted infigure 5.3. The results of the stiffness measurements of both tyres generally show a consistentbehaviour for the influences of the camber angle. As the camber angle rises, the overall verticalstiffness is decreasing which is in consistence with the known behaviour of automobile tyres. Forautomobile tyres it is known that without camber the lateral stiffness is significantly smallerthan the vertical stiffness. Therefore, the overall vertical tyre stiffness perpendicular to the roaddecreases with an increasing camber angle. For both the front and rear motorcycle tyres this effectis also present. Moreover, also the effect of the vertical deformation ρ on the vertical tyre stiffnessCz is in consistence with the behaviour of automobile tyres. Especially for the front tyre, thevertical stiffness increases with an increase of the vertical deformation. However, for the rear tyrethis effect is not consistent over the camber angle range and overall it can be concluded that ishas less influence than on the stiffness of the front tyre.

The vertical load Fzw during simulations is defined from the calculated vertical deformation ρ.From (5.1) it is learned that the relationship between both is given by the vertical tyre stiffness.Therefore, the vertical tyre stiffness measurements presented in this section are used in chapter 6to describe the relationship between the vertical deformation and the vertical tyre force. As theresults show, this description should be a function of at least the camber angle γ and the verticaldeformation ρ.


0 5 10 15 20 25 30 35 40 45 50110

120

130

140

150

160

170

Camber angle, γ [deg]

Ver

tical

stif

fnes

s, C

z [kN

/m]

ρ = 3 [mm]ρ = 6 [mm]ρ = 10 [mm]

0 5 10 15 20 25 30 35 40 45175

180

185

190

195

200

205

210

215

220

225


Ver

tical

stif

fnes

s, C

z [kN

/m]

ρ = 3 [mm]ρ = 6 [mm]ρ = 10 [mm]

Figure 5.3: Measured vertical stiffnesses for a front(top) and a rear(bottom) tyre


5.3 The effect of the vertical load on the effective rollingradius

0 500 1000 1500 2000 2500288

290

292

294

296

298

300

302

Vertical force, Fzw

[N]

Effe

ctiv

e ro

lling

rad

ius

[mm

]

MeasurementsMF−MCTyre

Figure 5.4: Effect of the vertical load on the effective rolling radius of a front tyre

In the previous chapter it has been explained that the influence of the vertical load on theeffective rolling radius is nowadays copied from automobile tyres. To correctly introduce the effectof the vertical load in the new motorcycle tyre model, this effect needs to be measured. Thereforemeasurements have been performed on the Flatplank Tyre Tester with an un-cambered freelyrolling wheel. The procedure for these measurements is similar to the procedure used for thevertical stiffness measurements. During these measurements both the forward velocity Vx and therotational velocity Ω are determined, and as no driving or braking torque is applied the effectiverolling radius Re can be determined as:

Re =Vx

Ω(5.2)

Furthermore, the measurements are again performed over two complete tyre rotations and themeasured signals during the last complete tyre rotation are used. For the graphical representation,the results are obtained by averaging the signals over this last complete tyre rotation. Figure 5.4shows the measured relationship between the rolling radius and vertical load and the relationshipwhich is presently implemented in the tyre model. As one can see large differences are presentbetween the model and the measurement results. The results of these measurements will thereforebe used in the next chapter to correctly implement the effect of the vertical load on the effectiverolling radius.

5.4 Effective rolling radius reference measurements

In a further stage of this thesis, a new description of the effective rolling radius for motorcycletyres will be generated and needs to be thoroughly checked by comparing it with measurements.Therefore a set of reference measurements are conducted at various orientations and verticalloads. On the Flatplank Tyre Tester an air spring is present, which makes it possible to performmeasurements with an almost constant vertical load Fzw. For each camber angle two differentvertical loads of approximately 750 [N ] and 1500 [N ] are applied. For each combination of the


camber angle and vertical load the measurements are again repeated three times, in order to reducethe influence of random errors. Furthermore, the main steps of the measurement procedure are:

• Inflate the tyre to the prescribed pressure

• Apply a prescribed camber angle

• Place the tyre exactly against the road so that there remains zero vertical load

• Remove all offsets by zeroing the measurement channels

• Apply a prescribed vertical load with the air spring

• Start the measurement

0 100 200 300 400 500 600 700720

740

760

780

Ver

tical

load

, Fzw

[N]

0 100 200 300 400 500 600 7000.27

0.28

0.29

0.3

0.31

0.32

Effe

ctiv

e ro

ll.ra

d., R

e [m]

Rotation angle [deg]

MeasurementAverage

Figure 5.5: Effective rolling radius measurement of a front tyre under 30 degrees camber

Similar to the measurements described earlier, the measurements are performed over two com-plete tyre rotations. In figure 5.5 the measured vertical load Fzw and the effective rolling radiuscalculated with (5.2) are depicted for one measurement. From this figure it can be concluded thatespecially the vertical load fluctuates with the tyre circumference. Therefore, the results are againaveraged over the last complete tyre rotation, from which the results are also shown.

Equal measurements have been performed for the front and rear tyre from 0 to 50 degreescamber with intervals of 5 degrees. The results of these measurements and therefore the effect ofcamber on the effective rolling radius are depicted in figure 5.6. Furthermore, also the effectiverolling radius that is calculated by the present tyre model is depicted and as one can see, largedifferences are present between both. These differences can be clarified by the fact that the tyrecontour is not taken into account by the tyre model. The effect that cannot be explained are thelarge differences between the effective rolling radius at low and high vertical load. At high camberangles (45 and 50 degrees) the difference between both is suddenly much larger than it is a lowercamber angles. Most probably, the unreliable results at high camber angles are caused by the lackof grip on the Flatplank surface. As the camber angle of the tyre increases, also the lateral forceof the tyre increases. At a certain point, the grip in the contact patch of the tyre is not sufficient


anymore and the tyre partly begins to slide over the surface. With the sliding, a lot of stick-slipappears which can be noticed during the measurements by its specific sound. Most probably, thisstick slip effect negatively influences the measurements and therefore the results at large camberangles are not completely trusted on their validity.

0 10 20 30 40 50265

270

275

280

285

290

295

300


Effe

ctiv

e ro

lling

rad

ius,

Re [m

m]

MeasuredMF−MCTyre

0 10 20 30 40 50270

280

290

300

310

320

330


Effe

ctiv

e ro

lling

rad

ius,

Re [m

m]

MeasuredMF−MCTyre

Figure 5.6: Measured and calculated effective rolling radius for a front(top) and a rear(bottom)tyre

5.5 Rolling resistance factor fr measurements

Little is known about the camber dependence of the rolling resistance coefficient fr of motorcycletyres. As the rolling resistance can best be measured with a freely rolling wheel, the effective rollingradius measurements are perfectly suited. In section 6.3 a new description of the rolling resistance


will be defined, with which the measurements described in this section are already processed. Itwill be explained that with this new definition for a freely rolling wheel it follows that:

Fxw = Fzwfr (5.3)

With this known, the measured longitudinal force Fxw and vertical force Fzw can be used toobtain the rolling resistance coefficient fr at different circumstances. For each measurement, theaverage rolling resistance coefficient is determined to give an impression of its dependence onthe vertical force and camber angle. If the measurement results that are depicted in figure 5.7are studied, the strong camber dependence of the rolling resistance becomes clear. For both thefront and rear tyre, the rolling resistance is approximately quadrupled over the camber range.Furthermore, also the vertical load plays an important role. The influence of the vertical load alsoincreases with the camber angle, especially for the rear tyre. With these results it is shown thatthe present assumption of a constant rolling resistance factor fr of 1.5% should be replace witha more sophisticated description. Therefore, in section 6.3, these measurement results are usedto generate an accurate description of the rolling resistance which takes the effects of camber andvertical load into account.

0 10 20 30 40 500.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09


Rol

l. re

sist

ance

coe

ffici

ent [

−]

Fzw

= 750 [N]F

zw = 1500 [N]

0 10 20 30 40 500.01

0.02

0.03

0.04

0.05

0.06

0.07


Rol

l. re

sist

ance

coe

ffici

ent [

−]

Fzw

= 750 [N]F

zw = 1500 [N]

Figure 5.7: Rolling resistance coefficients for both a front and a rear tyre

Chapter 6

Improvements to the MF-MCTyremodel

In chapter 4 the deficiencies of the MF-MCTyre model have been discussed, and possible solutionsare proposed. These solutions are further elaborated in this chapter and their correctness ischecked. To do so, the results of the measurements presented in the previous chapter are used.Moreover, they are used to improve the determination of the contact point and the vertical load,the description of the effective rolling radius and the description of the rolling resistance moment.

Throughout the present and the following chapter it is attempted to describe the results ofseveral measurements. The accuracy of these descriptions is indicated with the relative error E inpercentages. For each measurement and its description, this relative error is determined as:

E = 100

√∑ni=1(Yfi − Ymi)2∑n

i=1(Ymi)2(6.1)

with n the number of measurement points, Yf the description and Ym the measurement.

6.1 Improving the contact point and vertical load determi-nation

For a correct determination of the contact point and the vertical load, the tyre contour is needed.As this contour has been measured, it can now be described. To represent the tyre contourmathematically an ellipse with parameters a and b can be used.

u2

a2+

v2

b2= 1 (6.2)

In this case the ellipse is described with respect to its local axis system, with axis u representingthe direction of the tyre width and the v axis describing the tyre height direction. Within this axissystem the tyre contour is described independent from its orientation, due to which it is easier toimplement the tyre contour in the contact routine. The ellipse equation is used to describe the tyrecontour and the error definition of (6.1) is used to calculate the error made by this description.This error can be minimized for the ellipse parameters a and b, which is done with the Matlabbuilt-in function fminunc.m. In figure 6.1 a fit of the rear tyre contour is depicted from which itcan be seen that an elliptic fit represents the tyre contour very well. The minimal fitting errorsare 2.14% and 3.46% for respectively the front and rear tyre contour. Moreover, these errors are21.24% and 22.58% if the tyre contours are describe with a circle, which used to be done. Largeimprovements are therefore made and the major advantage is that this fit is described only by thetwo ellipse parameters a and b. As the tyre contour is accurately described it is possible to correctly

37

CHAPTER 6. IMPROVEMENTS TO THE MF-MCTYRE MODEL 38

−100 0 1000

100

200

300

400

500

Width [mm]

Hei

ght [

mm

]

0 20 40 60 80 100

0

10

20

30

40

50

60

70

80

Width [mm]

Hei

ght [

mm

]

Measured curveFitted ellipse

Figure 6.1: Ellipse fitted through the rear tyre contour

determine the contact point between an undeformed tyre and the road. For simplicity, the contactroutine developed in this research is limited to simulations with a flat road surface. With a flatroad surface, the problem with the inaccurate assumption of the tangent wheelplane beneath thewheel center is avoided. In a further stage, the new contact routine should be improved in such away that it can handle non-flat road surfaces.

During a simulation the [3× 3] rotation matrix which defines the orientation of the wheel withrespect to the global axis system is known. At each simulation time-step, the camber angle γ isdefined from this matrix. Next to the camber angle, the tyre contour is described in the alreadyintroduced (u, v)-axis system which rotates with the camber angle. With these parameters knownit is possible to calculate the actual contact point A, which is the point at which the tyre touchesthe road if no deformation is present, see figure 6.2. This point is computed with the derivative

A

y

z

R o

z gR o

A

y

u vu

v

b

a

r o a d

rFigure 6.2: Rotated undeformed tyre ellipse


of the ellipse equation in the (u,v)-axis system:

u2

a2+

v2

b2= 1 (6.3)

v2 = ±b

√1− u2

a2(6.4)

du

dv= − b2

a2

u

b√

1− u2

a2

= −u

v(b

a)2 (6.5)

The point at which this derivative equals tan γ is the contact point A(uA,vA) of the undeformedtyre.

−uA

vA(b

a)2 = tan γ (6.6)

uA = − a(ab ) tan γ√

1 + (ab )2 tan2 γ

(6.7)

vA = − b√1 + (a

b )2 tan2 γ(6.8)

As the height of the wheel center h is also retrieved from the vehicle model, also the verticaldeformation of the tyre at point A can be determined. With the aid of figure 6.2 it follows that:

ρ = −uA sin γ + (Ro − b− vA) cos γ − h (6.9)

Now that the vertical deformation ρ of the tyre is known, also the vertical force Fzw can bedetermined. The relationship between these parameters is known as the vertical tyre stiffness,which has been measured. Therefore, an accurate description of these measurements would besufficient to describe the vertical tyre force during a simulation. At TNO automotive, also theMF-Swift tyre model is developed with which the tyre dynamic behaviour can be simulated.Within this model, the relationship between the vertical tyre force and the vertical compressionis described as:

Fz = 1 + qv2|ΩRo

Vo| − (

qFcxFx

Fzo)2 − (

qFcyFy

Fzo)2.qFz1

ρ

Ro+ qFz2(

ρ

Ro)2.Fzo (6.10)

From this formula it can be observed that:

• a parabolic approximation is used to describe the load-deflection curve

• the vertical stiffness increases almost linearly with the angular velocity

• a reduction of the vertical stiffness appears when longitudinal and/or lateral forces are ap-plied in the contact patch

In order to make the different tyre models as consistent as possible, this formulation is also usedwithin this research to describe the vertical stiffness measurements presented in the previouschapter. When the parameters are fitted to the measurements, it appears that the effect of thelongitudinal and lateral force on the vertical stiffness are negligible. Furthermore, the flatplankmeasurements are performed at a very low and constant velocity, so also the effect of the rotationalvelocity is negligible. As appears from the measurements the camber angle does have a significantinfluence on the vertical stiffness which should be included in the description. Therefore (6.10) isrewritten to:

Fz = (1− qc|γ|).(qfz1ρ

Ro+ qfz2(

ρ

Ro)2).Fzo (6.11)


with which the flatplank measurements are described. Following the error description of (6.1)the fitting errors for the front and rear tyre are 7.3% and 5.0%. If the camber dependence isneglected during the fit procedure, the fit errors increase to 7.9% and 5.2% for the front and reartyre respectively. Although this effect is smaller then expected, it is still significant and thereforethe choice is made to remain the inclusion of the camber dependence by means of factor qc. Inthe tyre property file the vertical stiffness Cz at the nominal load is specified instead of qFz1. Itcan be calculated as:

Cz =Fzo

Ro

√q2fz1 + 4qfz2 (6.12)

With this description of the vertical force the contact routine fulfills the demands. The contactpoint is determined as a function of the tyre contour, and the vertical load is accurately determined.Therefore, the effects of this method on the wheel axle height h and the tyre loaded radius Rl canbe discussed.

6.1.1 The wheel axle height, h

In chapter 4 it has been pointed out that the axle height h is of large importance, especiallyduring motorcycle simulations. With the new contact routine, the problems causing inaccuracyare overcome and the axle height is correct over the complete camber range. First of all thetyre contour is taken into account by the ellipse. Therefore, the camber dependence of the axleheight caused by the tyre contour is correct. Secondly, also the camber dependency of the verticalstiffness is taken into consideration. As the vertical tyre stiffness is decreasing with the camberangle, the axle height will be smaller at large camber angles.

6.1.2 The tyre loaded radius, Rl

In section 4.2.2 the importance of the loaded radius Rl during the conversions of forces andmoments has been pointed out. The loaded radius determined during measurements has a pooraccuracy which especially has a large effect on the accuracy of the Magic Formula fit of theoverturning moment Mxw. Furthermore, it has been stated that the transformations during bothsimulations and the processing of the measurements should be unambiguously defined. At first,it has been suggested to determine the loaded radius with the tyre contour, the camber angleand the vertical load Fzw. During measurements, the camber angle and the vertical load aremeasured and together with a predefined vertical stiffness and tyre contour, the loaded radiuscan be defined. During a simulation the same approach can be used. If equal tyre contour andstiffness parameters are used, the loaded radius is also equal under similar circumstances. Theproblem with this method is that both the vertical stiffness and the tyre contour should be definedbefore the measurements can be processed. This brings along a relatively large effort to be able toprocess the measurements and therefore this method is not used. To overcome all problems withthe loaded radius without too large effort, the choice is made to define the free rolling radius Ro asthe distance between the axis system. With this conversion the loaded radius measurements areavoided and the conversion is unambiguously defined. The only complication with this assumptionis that the overturning moment Mxw is relatively sensitive to the length of the loaded radius, andtherefore its shape as a function of the slip angle may change. As this shape changes, it is possiblethat the Magic Formula is not suited anymore to be fitted to the moment Mxw. Therefore theoverturning moments of front and rear tyre measurements have been fitted with the loaded radiusand the free rolling radius. Due to the large spreading on the loaded radius Rl, the errors inthe Mxw fits following the definition of (6.1) are 20.5% and 29.4% for the front and rear tyrerespectively. With the free rolling radius Ro these errors are reduced to 14.9% and 11.9%. Theuse of Ro for the transformation of the forces and moments is therefore trusted on its validity andwill be used in further research.


6.2 Improving the effective rolling radius determination

For a large part, the determination of the longitudinal slip κ is governed by the effective rollingradius Re. In section 4.3 it has been pointed out that the method of describing this radius needssome adaptations to overcome several problems. First of all, the effect of the vertical load Fzw

needs to be introduced correctly. Therefore, the measurements presented in section 5.3 have beenconducted and the results are introduced in the new description of the effective rolling radius. Todo this, the Magic Formula approach presented in the previous chapter is being used.

Re = Ro − ρFz0(D arctan(Bρd) + Fρd) (6.13)

With the actual measurements as an input, (6.13) is used to calculated the effective rolling radiusat each measurement point. The relative error defined by (6.1) is determined and minimized for theparameters B, D and F with the Matlab built-in function fminunc.m. In this way, the effectiverolling radius can be represented accurately with as can be seen for a front tyre in figure 6.3.

0 500 1000 1500 2000 2500 30000.288

0.29

0.292

0.294

0.296

0.298

0.3

0.302

0.304

Vertical force, Fzw

[N]

Effe

ctiv

e ro

lling

rad

ius,

Re [m

m]

FitMeasurements

Figure 6.3: Measurements and fit of the effect of the vertical load on the effective rolling radius

Next to the correct Magic Formula parameters, also the effective rolling radius of an undeformedtyre Reo is needed to determine the actual effective rolling radius. In section 6.1 the coordinatesof the actual contact point A have been determined with the tyre contour described by an ellipse.With these coordinates, the free rolling radius Ro and the aid of figure 6.4, Reo can be defined.

Reo = Ro − b− vA (6.14)

The effect of the tyre contour can now be taken into account by using Reo instead of the freerolling radius Ro in 6.13. Furthermore, the vertical tyre deformation only partly affects theeffective rolling radius because of the camber angle of the wheel. This effect is also taken intoaccount by multiplying the deformation term with cos(γ).

Re = Reo − ρFz0(D arctan(Bρd) + Fρd) cos(γ) (6.15)


z gR o

RA

y

u v

R e o

b

Figure 6.4: Effect of camber on the effective rolling radius of an undeformed tyre

For the determination of the nominal tyre deflection ρFz0 the constant stiffness described in (6.12)is used.

ρFz0 =Fz0

Cz(6.16)

Finally the dimensionless radial tyre deflection ρd is calculated with the momentary tyre deflectionρ. With the improvements in the determination of the contact point and the vertical load Fzw thathave been shown in the previous section, also the momentary tyre deflection is determined moreaccurately. Of course, this also positively influences the accuracy of the effective rolling radius.

ρd =ρ

ρFz0

(6.17)

As the determination of the effective rolling radius is now expected to be correct, its accuracycan be checked. The effective rolling radius of both the front and rear tyre have been determinedwith measurements, from which the results have been presented in section 5.4. These measure-ments are now used as a reference, and will be compared with the new description of the effectiverolling radius. The camber angle γ and the vertical load Fzw are averaged over the last completetyre rotation and used as input parameters for the calculation method. By subtracting the cal-culated average radius from the measured one, the error of the calculation method is determined.In figure 6.5 the absolute errors made by the new effective rolling radius calculation method aredepicted for both the front and rear tyre.

The results obtained by the new calculation method are very promising from 0 up to 30degrees camber. Within this range the absolute error remains around 1 [mm], which correspondsto relative errors within 0.5 % for both the front and rear tyre. However, at higher camber anglesthe error strongly increases to an unacceptable magnitude. The reason for this rising error canmost probably be found in the measurement results. In section 5.4 it has already been observedthat the measurements at larger camber angles were inconsistent and not completely trusted ontheir validity. Unexpected sudden jumps of the effective rolling radius are seen, although thedeformation and the vertical force remain almost constant. Therefore, the calculation method istrusted on its validity and the increasing error at larger camber angle is expected to be caused bythe measurements.


0 10 20 30 40 50−7

−6

−5

−4

−3

−2

−1

0

1

2


Err

or [m

m]

Fzw

= 1500 [N]F

zw = 750 [N]

0 10 20 30 40 50−20

−15

−10

−5

0

5


Err

or [m

m]

Fzw

= 1500 [N]F

zw = 750 [N]

Figure 6.5: Error in calculated Re for a front(top) and a rear(bottom) tyre

6.3 Improving the rolling resistance description

In section 4.4 the problems with the rolling resistance definition for motorcycles have been ex-plained. When a freely rolling wheel is cambered the equilibrium around the spindle axle cannotbe guaranteed. To overcome these problems a new description of the rolling resistance moment isgenerated, which is done in two steps. In first instance, the magnitude of the rolling resistance isleft unchanged but the direction in which it is defined is altered. When its direction is correctlydefined, also a new description of the magnitude of the rolling resistance is developed.

6.3.1 Definition of the rolling resistance moment, Mrr

First, the magnitude of the rolling resistance moment is assumed to be correct. The problem istherefore reduced to the fact in which direction the rolling resistance should be defined. In orderto solve this question, first of all the magnitude of the rolling resistance moment Mrr is definedindependent from any direction. It is shown that in the present model this magnitude is describedby the vertical load Fzw, the loaded radius Rl and the rolling resistance coefficient fr. As the


loaded radius Rl for the conversion of the moments is changed to the free tyre radius Ro, this isalso done for the magnitude of the rolling resistance. Moreover, the rolling resistance coefficientis assumed as a contant 1.5% which is also done for automobile tyres.

Mrr = −FzwfrRo (6.18)

To define the direction of this moment a new axis system is generated in the tyre-road contactpoint, the T -axis system. In contradiction to the W -axis system the T -axis system is rotated withthe camber angle with respect to the road plane. Therefore, this new axis system is perpendicularto the C-axis system. All three axis systems are depicted in figure 6.6. In chapter 4 it has been

g

y w

z w

y w

z w

y c z c

z cy c

R l

F M

M

F

FF

M M

M y t M z t

C

O

Figure 6.6: Rolling resistance defined in the wheel-plane

shown that for automobile tyres the assumption has been made to define the rolling resistanceMrr in the W axis system, moreover to define that:

Myw = Mrr = −FzwfrRo (6.19)

For motorcycle tyres this assumption has been copied, and it has been shown that this leads tothe fluctuating moment around the spindle axis of a freely rolling wheel. As the camber anglesremain small for automobile tyres, the difference between the W - and T -axis system remainssmall. For motorcycles though, camber angles become significant and large differences are presentbetween both axis systems. To overcome the problems it is therefore suggested to define the rollingresistance moment Mrr in the T -axis system. With this definition, the moments in the T -axissystem are defined as:

Myt = Mrr = −FzwfrRo (6.20)

Mzt = Mzc (6.21)

With this assumption, the moments in the W-axis are derived by rotating the T -axis system overthe camber angle.

Myw = Myt cos γ −Mzt sin γ (6.22)

Mzw = Myt sin γ + Mzt cos γ (6.23)

Now both definitions of the rolling resistance can be compared on their correctness. To do this,first of all the equilibrium around the spindle axle of a freely rolling motorcycle wheel is stated.

Myc = Myw cos γ + Mzw sin γ − FxwRl = 0 (6.24)


In the previous chapter it has been shown that assuming the rolling resistance in the W -axis systemleads to fluctuations in the moment Myc. This is because of the fact that with this assumption afluctuating Mzw is present during measurements, while all other variables are almost constant. Ifthe rolling resistance definition in the T -axis system is now examined, it appears from equations(6.22)-(6.23) that the fluctuating moment Mzc is present in both Myw and Mzw. Therefore, thismight lead to a more constant Myc, which is expected from the 2nd Law of Newton. With bothrolling resistance definitions a large number of test trailer measurements with a freely rolling wheelunder several camber angles have been analyzed. Significant improvements were found with thenew definition, with the difference between both definitions increasing with the camber angle. Theresult of a representative α-sweep under a camber angle of 50 degrees is shown in figure 6.7. This

0 10 20 30 40 50 60 70−10

−5

0

5

10

Sid

e sl

ip a

ngle

, α[d

eg]

0 10 20 30 40 50 60 70

−100

−50

0

50

100

Time [s]

Myc

[Nm

]

Mrr defined in T

Mrr defined in W

Figure 6.7: Difference in Myc for different definitions of the rolling resistance

figure clearly shows the large differences between both methods. A large and fluctuating residualmoment on the wheel axle is present when the rolling resistance is defined in the W -axis system.When the rolling resistance is defined in the T -axis system the moment Myc is very close to zero,as expected with a freely rolling wheel. Therefore, the rolling resistance is from now on defined inthis axis system.

However, the definition of the rolling resistance in the T -axis system also affects the descriptionof the moment Mz in the contact point. This moment is nowadays fitted in the W -axis system,but with the definition of the rolling resistance in the T -axis system it would be more convenientto fit the moment Mzt. If this is possible no transformation of the moment Mz is needed anymore,following (6.21). Therefore the measurements of a front and a rear tyre are processed with the newrolling resistance definition and the moments Mzw and Mzt are fitted with the Magic Formula. Forthe rear tyre, the fit error of Mzw is 7.6% which decreases to 6.7% for the moment Mzt. For thefront tyre though, the fit of the moment Mzw is more accurate than the fit of the moment Mzt asthe fitting errors are 7.1% and 8.0% respectively. Although the fit of the front tyre is less accuratein the T -axis system, the choice is made to fit all moments in this axis system. The increase in theerror percentage is relatively small, and for the rear tyre even a smaller fit percentage is achievedwith fits in the T -axis system. Moreover, the advantage of a more convenient definition of themoments in the T -axis system is decisive.


6.3.2 The rolling resistance coefficient, fr

For automobile tyres the rolling resistance coefficient fr has always been assumed to be 1.5%. Formotorcycles, the idea exists that this resistance is a function of at least the camber angle γ. Fora freely rolling wheel this can be checked with (6.24). With the definition of the rolling resistancemoment in the T -axis system, this equilibrium can be reduced to:

Mrr − FxwRo = FzwRofr − FxwRo = 0 (6.25)

Fzwfr = Fxw (6.26)

As the forces Fzw and Fx are registered during measurements, the rolling resistance coefficientfr can be determined for each measurement point. In section 5.5 the average rolling resistancecoefficient has been determined for a set of Flatplank measurements that have been processed withthe rolling resistance defined in the T -axis system. The results of these measurements showed anincreasing rolling resistance over the camber range. Moreover, the influence of the camber angle isgetting larger as the camber angle increases which suggests at least a quadratic camber dependence.Furthermore, a larger vertical load Fzw also leads to a larger rolling resistance, especially for therear tyre. Where the difference in rolling resistance at high or low loads is negligible at smallcamber angles, the effect of the vertical load is strongly increasing with the camber angle. Withthis known, a formula has been derived that describes the rolling resistance magnitude and takesthe effects of the camber angle γ and vertical load Fzw into account.

Mrr = −FzwRoλmyfr (6.27)

Mrr = −FzwRoλmyqsy1 + qsy5γ2 + qsy6(

Fzw

Fz0)γ2 (6.28)

The factor λmy is introduced to give the tyre model users the freedom to scale the rolling resistancewith little effort. The Flatplank measurements have been used to determine the correct parametersqsy1, qsy5 and qsy6 for the front and rear tyre. With (6.28) it is possible to fit the rolling resistancemoment of the front tyre measurements with an error percentage of 13.28%, following the definitionof (6.1). The error when fitting the rolling resistance of the rear tyre is 11.08%. With an optimalbut constant rolling resistance coefficient these percentages are 44.84% and 42.28%. Therefore,it is now possible to describe the rolling resistance of a tyre sufficiently accurate with only threeparameters.

Chapter 7

Analysis of the improved tyremodel

In the previous chapter a number of improvements to the tyre model are developed. Theseimprovements are implemented in an updated version of the MF-MCTyre model which can beused during simulations. In this chapter, several simulations are used to show the accuracy of theimproved tyre model and the effect of the improvements. To give an idea of the differences andtheir cause, first of all a short summary of the improvements to the tyre model are given. Thenthe improved and the old tyre model will be used in a SimMechanics simulation which mimicsthe tyre measurements performed with the Delft Tyre Test trailer. With this model it is possibleto compare accuracy of the models with respect to the actual measurements. Finally, also themotorcycle simulation model of chapter 3 is used. The results of simulations with this model areused to determine the effect of the improvements to the tyre model on the steady state corneringbehaviour of a motorcycle.

7.1 Summary of the improvements

To give a correct impression of the results gained with the improved tyre model, first of all theimprovements that are made are listed below.

• The free tyre radius Ro is used for the conversions between the contact point and the wheelcenter:

– The unreliable loaded radius measurements of the Delft Tyre Test trailer are avoided– The loaded radius is unambiguously defined during the processing of the measurements

and simulations

• The contact routine is made robust for large camber angles:

– An ellipse is used to correctly describe the tyre contour– The vertical load is described as a function of the deflection ρ and the camber angle γ

• The effective rolling radius Re is accurately determined for the complete camber range:

– The effect of the vertical load Fzw on the effective rolling radius is correctly implemented– The effect of the tyre contour on the effective rolling radius is implemented

• The moments My and Mz in the contact point are defined in the T -axis system:

– The moment Mz is fitted by the Magic Formula in the T -axis system.– The influences of the camber angle and the vertical load Fzw are introduced in the

description of the rolling resistance coefficient fr

47

CHAPTER 7. ANALYSIS OF THE IMPROVED TYRE MODEL 48

7.2 Analysis by means of test rig simulations

Figure 7.1: The SimMechanics model which mimics the Delft Tyre Test trailer measurements

First of all, a model made in SimMechanics is being used to evaluate the different tyre models.This simulation model (which is depicted in figure 7.1 has the same DoF’s as the testing towerof the Delft Tyre Test trailer. As can be seen, the forward velocity Vx, the side slip angle α, thelongitudinal slip κ, the camber angle γ and the vertical load Fzw are input parameters for thesimulation. These parameters are also determined during measurements and these signals can beused as an input for the simulation. In this way, a tyre measurement can be accurately representedby the simulation model.

The forces and moments during measurements and simulations used to be compared in theW -axis system. However, both the output of the forces and moments by the tyre model asmeasured forces and moments are in the C-axis system. The representation of the forces andmoments in this axis system by the tyre model therefore determines its accuracy. Therefore, inthis section the results of the simulations are compared with the measurements in the C-axissystem, which is much more logical. It should be remarked that for these specific simulations,the rolling resistance moment measured with the Delft Tyre Test trailer is used. It is knownthat during measurements an incorrect amplification factor is used for the longitudinal force Fxw.However, the signal cannot be corrected as the the correct amplification factor is not known.Although the longitudinal force is incorrect in magnitude, the effects of the camber angle are stillseen in the rolling resistance. In order to compare the simulation results with the measurements,therefore the incorrect measurements are used in the improved tyre model.

First of all, an α-sweep at 5 degrees of camber and a nominal load Fzw of 2000 [N] is usedto check the correctness of the tyre models. In figure 7.2 the forces and moments in the C-axissystem during this simulation are depicted. Moreover, in table 7.1 the relative errors made byboth models during this simulation can be found. For the determination of these errors, again thedefinition of (6.1) is used.

The error percentage of the forces Fyc and Fzc made by the existing and improved tyre modelare small and only differ tenths of percents. These forces are determined by the lateral andvertical forces in the W -axis system (Fyw and Fzw), which are rotated over the camber angle. Asthe vertical force Fzw is prescribed for the simulation and the description of the lateral force issimilar for both tyre models, these forces are also expected to show large correspondence. As thesimulation results are almost equal and the differences between these results and the measurementsare very small, the differences can hardly be seen in the graphical representation of figure 7.2. Itis clear that the major difference lies in the longitudinal force Fxc. In paragraph 4.4 it has beenexplained that the old rolling resistance assumption leads to the large and fluctuating longitudinalforce which can also be seen in figure 7.2. Moreover, these fluctuations are known to increasewith the camber angle which will also be shown in this paragraph. With the definition of themoments in the T -axis system and the improved description of the rolling resistance factor fr

these problems are overcome. The results presented in figure 7.2 show that the longitudinal force


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Figure 7.2: Measured and simulated forces and moments in the C-axis system during an α-sweepunder 5 degrees camber


is much more constant, which is also the case during measurements. The noise on the longitudinalforce generated by the improved model is declared by the fact that the new rolling resistancecoefficient fr is dependent on the vertical force Fzw. The vertical force is used as an input forthe simulation and the noise level on this signal is significant. As the longitudinal force of afreely rolling wheel is determined by the rolling resistance, the noise can also be seen on thisforce. Therefore, the relative error is still 57.0% for the improved model. In comparison with therelative error of 148.9% with the existing model, this is a significant improvement but if the noiselevel is lower during the simulation (which is normally the case), the relative error will decreasesignificantly.

Next to the forces, also the moments Mxc and Mzc are depicted in figure 7.2. Due to the smallcamber angle, the differences between both models and the differences with the measurementsremain relatively small. The moment Mxc is determined by the moment Mxw, the lateral forceFyc and the distance between the wheel center and the contact point. It is already explainedthat the difference in Fyc is negligible. Furthermore, the moment Mxw generated by the tyremodels under a small camber angle is comparable. As the loaded radius of the existing modelis reasonably accurate under a small camber angle and Ro is used by the improved tyre model,the difference in the relative error in moment Mxc is only 0.1%. The moment Mzc in the contactpoint is fitted in the T -axis system by the improved tyre model. With this definition no conversionbetween the wheel center and the contact point is needed anymore, which increases the accuracyof the improved tyre model. The error made in Mzc therefore decreases from 11.1% to 8.7% forthe existing and improved tyre model respectively, as depicted in table 7.1.

Existing model Improved modelFxc 148.9 % 57.0 %Fyc 4.5 % 4.4 %Fzc 1.1 % 1.1 %Mxc 5.9 % 6.0 %Mzc 11.1 % 8.7 %

Table 7.1: Errors made by the existing and improved tyre model during an α-sweep simulationunder 5 degrees camber, following the definition of (6.1)

The forces and moments during a measurement and simulations with the two tyre modelsduring an α-sweep under a camber angle of 50 degrees can be found in figure 7.3. As expected,the difference in the longitudinal force between both models strongly increases. As the camberangle is much larger, also the difference between the W - and T -axis system increases. Therefore,the error made by the old rolling resistance assumption is increasing. As the tyre model uses thelongitudinal force Fxw to remain the equilibrium around the spindle axle of a freely rolling wheel,the error of this force is increasing with the camber angle. The error made in Fxc by the existingtyre model is now 582.4%, while the error of the improved model is only 36.2%. The error madeby the improved model decreases with respect to the error made with the simulation at 5 degreescamber, which can be declared by two facts. First of all, the noise level on the vertical load Fzw isrelatively smaller than during the measurement at 5 degrees camber. Also less noise is thereforepresent on the rolling resistance coefficient fr, which also leads to less noise on the longitudinalforce Fxw. Furthermore, the rolling resistance coefficient increases quadratically with the camberangle. As the influence of the vertical load on the rolling resistance coefficient increases onlylinearly with the camber angle, the relative influence of the vertical load is decreasing with thecamber angle. Therefore, the relative effect of the noise on the vertical load is also decreasing withthe camber angle and the relative error decreases.

During the simulation with the existing tyre model, large differences arise in the moment Mxc ascan also be seen in figure 7.3. The cause of this can be found in the inaccurately and unambiguouslydefined loaded radius Rl. By substituting the loaded radius with the undeformed tyre radius Ro,these problems are overcome. The conversions are ambiguously defined and accurate, which also


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leads to an accurate representation of the overturning moment by the Magic Formula fit. Asa result, the relative error of 41.7% made by the existing model decreases to 8.8% with theimproved model. Next to the improvements in Mxc, also significant improvements are made withthe representation of the moment Mzc. The new rolling resistance assumption has a significantinfluence, especially when the camber angle increases. Furthermore, it seemed that the MagicFormula is better suited to fit the moment Mzt than the moment Mzw. Therefore, the errorpercentage made by the improved tyre model is 8.7% instead of 11.1% with the existing model.

Existing model Improved modelFxc 582.4 % 36.2 %Fyc 8.8 % 8.9 %Fzc 3.8 % 3.8 %Mxc 41.7 % 8.8 %Mzc 12.0 % 8.5 %

Table 7.2: Errors made by the existing and improved tyre model during an α-sweep simulationunder 50 degrees camber, following the definition of (6.1)

7.3 Analysis by means of motorcycle simulations

Next to the simulations with the SimMechanics model of the test rig, also the motorcycle simulationmodel presented in chapter 3 is used. With this model it is possible to determine the effects ofthe improvements to the tyre model on the steady state behaviour of a motorcycle. To do this,the motorcycle is turned into a right hand corner with a constant velocity of 100 [km/h]. As thesteady state behaviour is of interest, the motorcycle is cornering for a relatively long period inorder to make sure that a steady state is reached. Moreover, the controller is tuned in such a waythat the motorcycle is cornering on an equal radius for both the existing and the improved tyremodel. As the forward velocity is also constant during both simulations, the motorcycle is in asimilar state and the results can be compared to each other. In figure 7.4, the trajectories of thesimulations with both tyre models are depicted. Although the trajectories are not exactly equal,the steady state cornering radius during both simulations is 47.5 [m].

During steady state cornering several parameters of both simulations are compared. To be surethat no transient behaviour is present anymore, the signals are averaged over the last 10 secondsof the simulation. To make the comparison, a relative difference is calculated with the existingtyre model as a reference. This relative difference d percentage is defined as:

d = 100Yn − Yo

Yo(7.1)

with Yn and Yo the parameters obtained from the simulation with the improved and the existingtyre model respectively. First of all, the tyre input and output parameters for the front and reartyre are compared. In table 7.3 these parameters and their relative differences are depicted. Firstof all, it should be said that the forces are compared in the contact point (W -axis system). Boththe improved and existing tyre model determine the forces in the W -axis system, so they can easilybe compared. As the existing model uses the W -axis system and the improved model uses theT -axis system to determine the moments, it is hard to make a correct comparison. The momentsare therefore compared in the wheel center (C-axis system).

As the motorcycle is driving on the same cornering radius with the same velocity, the lateralacceleration is almost equal during both simulations. Therefore, the lateral force Fyw and verticalforce Fzw of the front and rear wheel only differ tenths of percents between both simulations. Themain differences can be seen in the moments Mxc and Mzc. The negative moment Mxc is tryingto get the motorcycle in an upright position. As the overal moment Mxc generated by both the


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Existing model Improved model Rel. diff. dFrontwheelκ [-] -0.003 -0.003 0.9%α [deg] 0.265 0.097 -63.5%γ [deg] 48.46 50.36 3.9%Fxw [N] -101.72 -102.06 0.3%Fyw [N] -1125.40 -1123.30 -0.3%Fzw [N] 1310.30 1317.20 0.5%Mxc [Nm] 7.97 -0.42 -111.4%Myc [Nm] 0 0 0%Mzc [Nm] -20.74 -27.48 32.5%

Rearwheelκ [-] 0.004 0.004 -2.3%α [deg] 1.37 1.35 -1.5%γ [deg] 48.26 50.16 3.9%Fxw [N] 331.98 328.59 1.0%Fyw [N] -1635.90 -1639.00 -0.2%Fzw [N] 1758.20 1749.60 0.5%Mxc [Nm] -115.96 -144.02 -24.2%Myc [Nm] 146.38 152.54 -4.2%Mzc [Nm] -17.51 -15.78 9.9%

Table 7.3: Comparison between tyre in- and output for the front and rear wheel


front and rear tyre is larger with the improved tyre model, the camber angle is also larger toobtain a steady state cornering situation. The slip angle of the front wheel is mainly determinedby the moment Mzc. The negative Mzc which is present is turning the wheel into the corner. Apositive side slip angle is turning the wheel out of the corner. As the moment Mzc is 32.5% smallerwith the existing tyre model, the tyre sideslip angle is larger during the simulation with this tyremodel. Finally, the rolling resistance of the existing and the new front tyre are comparable asthe longitudinal forces are almost similar. As the front wheel is freely rolling the moment Myc ofthe front wheel is zero during the simulations. The driving torque Myc of the rear tyre is 4.2%larger during the simulation with the improved tyre model. This moment is present in order toovercome the rolling resistance of both tyres and the aerodynamic drag. The forward velocity isequal during both simulations, and therefore also the aerodynamic drag is equal. As the rollingresistance of the front tyre is also almost similar, the difference is caused by the rolling resistanceof the rear tyre.

Next to the tyre in- and output, also the response of the motorcycle can be compared. Theresults of this comparison can be found in table 7.3. It has already been explained that the camber

Existing model Improved model Rel. diff. dCamber, γ [N] 48.26 50.16 -3.9%Steer torque [Nm] -5.24 -3.21 38.7%Steer angle [deg] 0.03 -0.04 225.8%Twist angle [deg] 0.21 0.25 -20.1%Suspension deflection [m] -0.02 -0.02 1.2%Rider lean angle [deg] 0.81 1.04 -29.0%Pitch angle [deg] 0.62 0.57 9.0%Front wheel rot. speed [deg/s] 5357.10 5860.00 -9.4%Rear wheel rot. speed [deg/s] 4968.30 5788.80 -16.5%Front wheel axle height [m] 0.2129 0.2013 5.5%Rear wheel axle height [m] 0.2407 0.2338 2.9%

Table 7.4: Comparison between the geometrical response of the motorcycle

angle of both wheels is 3.9% larger due to the larger moment Mxc. The overall camber angle istherefore of course also 3.9% larger. The steer torque on the other hand is much smaller withthe improved tyre model than with the existing one. The systems which cause a torque aroundthe steering axle have been explained in section 3.4. As the lateral force and the mechanical trailare similar during both simulations, the smaller steering torque is mainly caused by the momentMzc. Although the steering torque is significantly larger, the steer angle is almost unaltered.Although the fact that a relatively large difference is present (225.8%), the absolute steer angleis too small to be significant. As the camber angle is larger with the improved tyre model, theequilibrium around the twist angle also changes. With a larger camber angle the vertical forceFzw gets more influence on the twist equilibrium, while the lateral force Fyw gets less influence.As the vertical force is larger than the lateral force, the twist angle increases with the improvedtyre model. Within table 7.3 it has been shown that the overal moment Mxc and therefore alsothe camber angle are larger with the improved tyre model. The equilibrium of the motorcyclearound its local x-axis therefore changes which leads to a larger rider lean angle.

The introduction of the tyre contour and the correct effect of the vertical load, should lead tosignificant changes in the effective rolling radius during simulations. If the front wheel is taken intoconsideration, it is learned from table 7.3 that the longitudinal slip κ does not change significantly.This is also expected as there is no effective driving or braking torque around the wheel axle.In figure 7.5, the effective rolling radius of the front wheel during both simulations is depicted.Already in the first simulation seconds, there is a significant difference between the effective rollingradius of the existing and improved tyre model. Research shows that the difference, which is inthe order of 10 [mm], is caused by the different Magic Formula parameters that introduce the


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effect of the vertical load. Furthermore, after 10 seconds the motorcycle drives into the cornerand the difference increases significantly. This can be explained by the fact that the tyre contouris implemented in the improved model, while this contour was absent in the determination of theeffective rolling radius in the existing model. In section 4.3 the definition of linear rolling velocityVr and the longitudinal slip κ has already been presented:

Vr = ΩRe (7.2)

κ =Vr − Vx

Vx(7.3)

As the longitudinal slip κ and the forward velocity Vx are almost constant during the simulation,also the linear rolling velocity Vr has to be almost constant. As the effective rolling radius stronglydecreases with the camber angle, the rotational velocity needs to increase to obtain an almostconstant linear rolling velocity. From table 7.3 it is learned that the rotational velocity of thefront wheel is 9.4% larger during steady state cornering with the improved tyre model. For therear tyre this increase is even larger, namely 16.5%. Moreover the rotational velocity of the wheelsis known to have a significant influence on the handling behaviour of motorcycles. Together withthe inertia of the wheel around the spindle axis, the rotational velocity of the wheel determines thegyroscopic precession which has been explained in chapter 3. This effect not only has a significantinfluence on the steering behaviour of a motorcycle, it also partly defines the instabilities of themotorcycle.

Finally also the wheel center height is investigated. In table 7.3 it is shown that the difference inwheel center height during steady state cornering is 5.5% for the front and 2.9% for the rear wheel.However, those differences are also caused by the different camber angles during the simulationswith the existing and improved tyre model. To get the effect of the changes in the tyre model onthe wheel center height in the right perspective, also a simulation with a constant camber angle of50 degrees is conducted. The axle height of the front wheel during these simulations can be seenin figure 7.6. With the implementation of the improved description of the tyre contour and thevertical tyre stiffness, the axis height is known to be more accurate in the improved tyre model.Due to the tyre contour implementation and the decreasing vertical stiffness with an increasing


camber angle, a smaller axle height is present with the improved tyre model than with the existingtyre model. The difference during steady state cornering is around 6 [mm], which is in the order of3.1% of the total axis height. In section 4.2.1, it has been explained that the motorcycle behaviouris partly governed by the height of the wheel axles. Therefore, an accurate axle height is of largeimportance and a difference of 3.1% is significant.

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Chapter 8

Conclusions and recommendations

8.1 Conclusions

The goal of this master thesis is to improve and validate the motorcycle tyre model, MF-MCTyre.An improved model should be able to correctly represent the behaviour of a motorcycle tyre for itscomplete working area at steady-state conditions. Therefore, deficiencies of the model have beenlocalized and solutions are developed. It appears that the MF-MCTyre model has to be improvedin three areas:

1. Forces and moments representationThe accuracy of the representation of the forces and moments has been improved, for whichtwo steps are taken. First of all, the free rolling radius Ro is used to convert the moments be-tween the contact point and the wheel center. Therefore, the conversions are unambiguouslyand accurately defined. During simulations with the tyre model, this leads to a decreasein the relative error in the moment Mxc. Secondly, the description of the rolling resistanceis improved. It is more convenient to define the rolling resistance moment Mrr in the newT -axis system. Furthermore, the influence of the camber and vertical load dependence of therolling resistance coefficient fr are measured and described. Both rolling resistance adap-tations lead to a more accurate force Fxc and moment Mzc during simulations. During asimulation with a motorcycle model it is shown that these improvements influence the equi-librium state. Moreover, important parameters as the camber angle and the steer torqueshow relatively large differences.

2. Wheel center heightThe tyre contour and the vertical tyre stiffness are measured and new descriptions are devel-oped which represent them more accurately than before. A new contact routine is generatedwhich uses the tyre contour to determine the contact point between the tyre and the road.With the assumption of a flat road surface and the new vertical tyre stiffness definition, thevertical tyre deformation and vertical load can be determined. Due to these improvements,the wheel center height during motorcycle simulations is more accurate. With the wheel cen-ter height also the positions of the centers of gravity of the motorcycle components change,which influences the dynamic behaviour of the model.

3. Effective rolling radiusThe adapted description of the effective rolling radius Re leads to correct rotational velocitiesof the wheels during simulations. The effect of the vertical load on the effective rollingradius is measured and described. Furthermore, the effect of the tyre contour is takeninto consideration, which leads to significant improvements. Reference measurements areconducted and although these measurements seem unreliable at large camber angles, thenew effective rolling radius description is trusted on its validity. A steady state corneringsimulation with the motorcycle model shows a large increase in rotational velocity of the

57

CHAPTER 8. CONCLUSIONS AND RECOMMENDATIONS 58

wheels. Not only does this have an impact on the dynamic behaviour of the motorcycle, therotational speed of the wheels also play an important role if for example the driveline of themotorcycle is modelled.

As an overall conclusion it can be said that the motorcycle tyre model MF-MCTyre has un-dergone significant improvements. Due to these improvements, this tyre model is now able toaccurately represent the steady state cornering behaviour of a motorcycle tyre.

8.2 Recommendations for future research

For future research on the area of motorcycle tyre modelling and motorcycle simulations, thefollowing recommendations are given:

• Contact routineThe contact routine which is presently implemented in the tyre model only works with a flatroad surface. This routine should be expanded to a 3-dimensional routine, which can takean uneven road into consideration. Within the old contact routine, the road is approximatedby its tangent plane underneath the wheel center. Especially at large camber angles thisassumption introduces an error which leads to an incorrect contact point determination.This can be overcome by developing a new routine which iteratively determines the contactpoint at each time step.

• Overturning moment Mxw

In section 4.2.2 it is shown that the overturning moment is very sensitive to the transforma-tion from the C- to the W -axis system. Due to the fact that the loaded radius Rl used intransformations is substituted with the free rolling radius Ro, the overturning moment Mxw

changes significantly if the camber angle increases. The overturning moment of the tyremeasurements that are used in this research can be represented accurately by the presentMagic Formula. Though, it is possible that if other tyres are measured the shape of the over-turning moment curve changes in such a way that it cannot be represented by the presentMagic Formula. Therefore, further research should be conducted to identify the robustnessof the present overturning moment description in the Magic Formula.

• MeasurementsWithin the measurement results of the vertical tyre stiffness and effective rolling radius,unexpected jumps are seen. Most probably, these jumps are caused by a lack of grip on theFlatplank tyre tester due to which stick-slip effects are present. To overcome this problem,two possible solutions are present. First of all, the road surface on the Flatplank tyretester can be adapted in such a way that it represents an asphalt road. This would not onlypositively influence the measurements performed in this research, also all other measurementswould represent the behaviour of a tyre on an asphalt road more accurate. The secondsolution would be to conduct the measurements with the Delft Tyre Test trailer on asphalt.Recently, the trailer has been adapted and the accuracy of the measured wheel rotationalspeed and wheel center height are increased. The only problem which is left is that thelevel of noise on the measured signals is much higher than during measurements with theFlatplank tyre tester.

• Tyre dynamicsWithin this research, the steady state cornering behaviour of a motorcycle tyre has beeninvestigated and modelled. In the literature study presented in chapter 2, it has alreadybeen explained that a motorcycle is inherent to stability problems at different velocities.Research has shown that these instabilities are partly governed by the tyres. Therefore, thelogical next step in the development of the tyre model is to determine and model the dynamicbehaviour of a motorcycle tyre. In resent research Lot [12] has presented a tyre model whichcontains aspects that incorporate parts of the dynamic tyre behaviour. Moreover, TNO

CHAPTER 8. CONCLUSIONS AND RECOMMENDATIONS 59

Automotive has developed a model which is able to represent the dynamic behaviour of anautomobile tyre up to 100 [Hz]. It is therefore recommended that the present knowledgeon tyre dynamics is extended and combined with the improved MF-MCTyre model. Thiscombination can lead to a model which is able to correctly represent both the dynamic andsteady state behaviour of a motorcycle tyre.

• Motorcycle model parametersThe motorcycle simulation model which is developed in this research is derived from themodel of Koenen [10]. His research has been completed in 1983 and since then motorcycleshave undergone significant developments. Within the literature study presented in chapter 2the correctness of the range of parameters used in the Koenen model is shown. However, theseparameters should be updated in such a way that they represent a nowadays motorcycle.To do so, too little information is present in the available literature. Therefore, an elaboratemeasurement program is needed to find the geometrical, mass and inertial parameters of arecent motorcycle.

• Motorcycle dynamics full scale testingMotorcycle simulations that contain a combination of a sophisticated dynamic tyre modeland an elaborate motorcycle model have not been found in the literature research. However,with a tyre model that correctly incorporates motorcycle tyre dynamics and an update ofthe motorcycle model of Koenen, it will be possible to simulate the dynamic behaviour of amotorcycle. It is therefore suggested that such research can lead a more accurate predictionof the dynamic behaviour of a motorcycle. Moreover, it is strongly recommended to validatethe results of such research with dynamic motorcycle measurements. However, this validationwould require a motorcycle that is equipped with measurement devices for the registrationof its dynamic behaviour.

Bibliography

[1] E. Bakker, L. Nyborg, and H.B. Pacejka. Tyre modelling for use in vehicle dynamic studies.SAE paper, (870421), 1987.

[2] P. Bayle, J.F. Forissier, and S. Lafon. A new tire model for vehicle dynamics simulations.Automotive Technology International, pages 193–198, 1993.

[3] R. Berritta, F. Biral, and S. Garbin. Evaluation of motorcycle handling with multibodymodelling and simulation. High tech engines and cars, 6th international conference, May25th-26th 2000.

[4] V. Cossalter and A. Doria. Model simulation: The latest dynamic simulation developmentsfor motorcycle tires. Tire Technology International, pages 38–41, September 2001.

[5] V. Cossalter, A. Doria, and R. Lot. Steady turning of two-wheeled vehicles. Vehicle SystemDynamics, 31:157–181, 1999.

[6] V. Cossalter and R. Lot. A motorcycle multi-body model for real time simulations based onthe natural coordinates approach. Vehicle System Dynamics, 37(6):423–447, 2002.

[7] V. Cossalter, R. Lot, and F. Maggio. The modal analysis of a motorcycle in straight runningand on a curve. Meccanica, 39(1):1–16, 2004.

[8] S. Evangelou and D.J.N. Limebeer. Lisp programming of the ’sharp 1994’ motorcycle model,March 28th 2004. Department of Electrical and Electronic Engineering, Imperial College ofScience, Technology and Medicine.

[9] Y. Kamata and H. Nishimura. System identification and attitude control of motorcycle bycomputer aided analysis. JSAE Review, 24(4):411–416, October 2003.

[10] C. Koenen. The dynamic behaviour of a motorcycle when running straight ahead and whencornering. PhD thesis, Delft University of Technology, 1983.

[11] C. Koenen and H.B. Pacejka. The influence of frame elasticity, simple rider body dynamicsand tyre moments on free vibrations of motorcycles in curves. Vehicle System Dynamics,pages 53–65, 1982. Proceedings of IAVSD symposium.

[12] R. Lot. A motorcycle tire model for dynamic simulations: Theoretical and experimentalaspects. Meccanica, 39:207–220, 2004.

[13] H.B. Pacejka. Tyre and vehicle dynamics. Butterworth Heinemann, 2002.

[14] J.S. Rankine. Dynamical principles of the motion of velocipedes. The Engineer, 28, 1869.

[15] R.D. Roland. Computer simulation of bicycle dynamics. ASME symposium on mechanicsand sport, pages 1115–1121, 1973.

[16] P.A.J. Ruijs and H.B. Pacejka. Recent research in lateral dynamics of motorcycles. VehicleSystem Dynamics, 15:467–480, 1985. Proceedings of IAVSD symposium.

60

BIBLIOGRAPHY 61

[17] R.S. Sharp. The stability and control of motorcycles. Journal of Mechanical EngineeringScience, 13(5):316–329, 1971.

[18] R.S. Sharp. The lateral dynamics of motorcycles and bicycles. Vehicle System Dynamics,14:265–283, 1985.

[19] R.S. Sharp. Stability, control and steering response of motorcycles. Vehicle System Dynamics,35(4-5):291–318, 2001.

[20] R.S. Sharp and C.J. Alstead. The influence of structural flexibilities on the straight-runningstability of motorcycles. Vehicle System Dynamics, 9:327–357, 1980.

[21] R.S. Sharp and D.J.N. Limebeer. A motorcycle model for stability and control analysis.Multibody System Dynamics, 6(2):123–142, September 2001.

[22] P.T.J. Spierings. The effects of lateral front fork flexibility on the vibrational modes of straightrunning single-track vehicles. Vehicle System Dynamics, 10:21–35, 1981.

[23] Y. Tezuka, H. Ishii, and S. Kiyota. Application of the magic formula tire model to motorcyclemanoevrability analysis. JSAE Review, 22(3):305–310, July 2001.

[24] TNO Automotive. Tyre models users manual; Using the MF-MCTyre model, May 2002.

[25] N. van de Wouw. Multibody dynamics, lecture notes, 2003. page 25.

[26] M.K. Verma, R.A. Scott, and L. Segel. Effect of frame compliance on the lateral dynamicsof motorcycles. Vehicle System Dynamics, 9(4):181–205, 1980.

[27] B. von Schlippe and R. Dietrich. Zur mechanik des luftreifens. Zentrale fur wissenschaftlichesBerichtwesen Berlin Adlershof, 1942.

[28] E.J.H. De Vries and H.B. Pacejka. Motorcycle tyre measurements and models. In Proceedingsof the 15th IAVSD Symposium, pages 280–298, 1997.

[29] E.J.H. De Vries and H.B. Pacejka. The effect of tire modelling on the stability analysis of amotorcycle. In Proceedings of the 4th international symposium on advanced vehicle control,pages 063/1–063/6, 1998.

Appendix A

The MF-MCTyre model

This master thesis has been devoted to improve the MF-MCTyre model. In this appendix, its mainfeatures will be explained on the basis of the user manual of the model [24]. For the descriptionof the Magic Formula also the book ”Tyre and vehicle dynamics” of Pacejka [13] is used. For amore comprehensive description of the tyre model and the Magic Formula the reader is thereforereferred to those works. The implementation of the tyre model into a vehicle model has beendiscussed in paragraph 3.3.

A.1 Contact routine

As said, the model uses the Magic Formula for the description of the stationary slip. For theMagic Formula the longitudinal and/or lateral slip (α,κ), wheel camber (γ) and the vertical force(Fz) are used as input quantities. Those quantities are derived from the vehicle model by meansof a contact routine.

A.1.1 The contact point C and the normal load Fz

The radius of curvature of the road profile is considered large as compared to the radius of the tyre.The tyre is assumed to have only a single contact point (C) with the road profile. Furthermore,for calculating the motion of the tyre relative to the road, the road is approximated by its tangentplane at the point on the road below the wheel center (See the left side of figure A.1). The tangentplane is an accurate approximation of the road, as long as the road radius of curvature is not toosmall (≥ 2[m]). With the aid of the right side of figure A.1, the normal compression ρ of the tyreon the road can be derived by the tyre free radius R0, the cross section tyre radius rt and the axleheight h to the road tangent plane;

ρ′ = rt + (R0 − rt) cos γ − h (A.1)

The normal load Fz is then calculated with the tyre vertical stiffness Cz and damping Kz;

Fz = Czρ + Kz ρ (A.2)

with ρ the deflection velocity.

A.1.2 The effective rolling radius

The effective rolling radius Re (at free rolling of the tyre) is defined by the forward velocity Vx

and the rotational velocity of the wheel Ω;

Re =Vx

Ω(A.3)

62

APPENDIX A. THE MF-MCTYRE MODEL 63

XY

Z

Xw

C

Yc

Yw

Xc

nr

ZcZw

Vx

ha

tyre crosssection

road tangentplane

Fz

Fy

R0 Rcs

γ

O

ρ

Mx

rt

C

Figure A.1: The contact point C and tyre compression ρ in this point

0 500 1000 1500 2000 2500286

288

290

292

294

296

298

300

302

304

Vertical force, Fzw

[N]

Rad

ius

[mm

]

Effective rolling radius Re

Loaded radius Rl

Figure A.2: The effective rolling radius as a function of the load

This radius decreases with increasing vertical load at low loads, but around its nominal load theinfluence of the vertical load is small as can be seen in figure A.2 for a tyre with a nominal loadof 1475 [N]. When assuming a constant vertical tyre stiffness Cz, the radial tyre deflection can becalculated with;

ρ =Fz

Cz(A.4)

For the estimation of the effective radius Re a Magic Formula approach is chosen;

Re = R0 − ρFz0(D arctan(Bρd) + Fρd) (A.5)

in which R0 is the unloaded free radius and the nominal tyre deflection ρFz0 is defined by thevertical tyre stiffness Cz and the nominal wheel load Fz0;

ρFz0 =Fz0

Cz(A.6)

and the dimensionless radial tyre deflection ρd can be calculated with;

ρd =ρ

ρFz0

(A.7)


Factor B in equation A.5 determines the slope at Fz = 0, factor D defines the height of theasymptote at high wheel loads and factor F defines the ratio between the tyre radial deformationand the effective tyre deformation.

A.1.3 Tyre slip quantities

VsyVsVrV

α

VsxVx

Vy

Figure A.3: Slip quantities of a tyre

With figure A.3 the tyre slip quantities can be derived. The longitudinal slip speed is definedas;

Vsx = Vx − ΩRe (A.8)

and the lateral slip speed;

Vsy = Vy (A.9)

With these slip speeds the practical slip quantities (κ and α) that are used as an input for theMagic Formula are defined as;

κ = −Vsx

Vx(A.10)

α = arctan(Vsy

|Vx| ) (A.11)

With Vsx and Vsy the components of the slip speed that may be defined as the velocity of pointS in the W -axis system (see figure A.3). With Ω denoting the rotational speed of the tyre, thelinear rolling speed becomes;

Vr = ReΩ (A.12)

A.2 The Magic Formula

For a given pneumatic tyre and road condition, the tyre forces due to slip follow a typical charac-teristic. These characteristics can be accurately approximated by a mathematical formula knownas the ’Magic Formula’. The parameters in the Magic Formula depend on the type of the tyreand the road conditions. These parameters can be derived from experimental data obtained fromtests. The tyre is rolled over a road at various loads, orientations and motion conditions.

The Magic Formula tyre model is mainly of an empirical nature and contains a set of mathemat-ical formula, which are partly based on a physical background. The Magic Formula generates theforces (Fx,Fy) and moments (Mx,My,Mz) acting on the tyre at pure and combined slip conditions,using longitudinal and/or lateral slip (α,κ), wheel camber (γ) and the vertical force (Fz) as inputquantities. An extension has been provided that describes transient and oscillatory tyre behaviourfor limited frequencies lower than 8 [Hz] and wavelengths larger than the tyre circumference.


The original form of the formula that holds for given values of vertical load and camber anglereads;

y(x) = D sin[C arctanBx− E(Bx− arctanBx)] (A.13)

with

Y (X) = y(x) + SV (A.14)

x = X + SH (A.15)

This original form is still being used for the representation of both the longitudinal (Fx) andlateral forces (Fy) of automobile tyres. For motorcycle tyres the original formula is only used forthe longitudinal force. The Magic Formula y(x) typically produces a curve that passes through theorigin, reaches a maximum and subsequently tends to a horizontal asymptote. For given valuesof the coefficients B, C, D and E the curve shows an anti-symmetric shape with respect to theorigin. To allow the curve to have an offset with respect to the origin, two shifts SH and SV areintroduced. A new set of coordinates Y (X) arises as shown in figure A.4. The formula is capableof producing characteristics that closely match measured curves for the longitudinal force as afunction of the longitudinal slip κ with the effect of load Fz and camber angle γ included in theparameters.

Figure A.4 illustrates the meaning of some of the factors used in the formula. Obviously,coefficient D represents the peak value (for C ≥ 1) and the product BCD corresponds to theslope at the origin. The shape factor C controls the limits of the range of the sine functionappearing in formula A.13. Thereby it determines the shape of resulting curve as also the heightof the horizontal asymptote ya. The factor B is left to determine the slope at the origin and iscalled the stiffness factor. The factor E is introduced to control the curvature at the peak and atthe same time the horizontal position of the peak, xm.

y Y

xX

S V

S H

D

X m

a r c t a n ( B C D )y a

Figure A.4: Curve produced by the original Magic Formula

A.2.1 Longitudinal force (pure slip)

As said, the original form of the Magic Formula is still used for motorcycles to represent thelongitudinal force, Fx for which the input κx is the longitudinal slip.

Fx0 = Dx sin[Cx arctanBxκx − Ex(Bxκx − arctan(Bxκx))+ SV x (A.16)


Next to κ, also the momentary vertical load Fz and camber angle γ are used as an input as thecoefficients are determined dependently on those parameters.

A.2.2 Lateral force (pure slip)

In 1997 de Vries and Pacejka adapted the Magic Formula in order to make it suitable for use at largecamber ranges [28]. In the automobile Magic Formula, the camber contribution was introducedat the horizontal and vertical shift. The disadvantage of this method is the strong interactionbetween camber angle and side slip angle which will not be included directly. To account for thisinteraction, the camber angle appears in the coefficients of the slip angle for the automobile tyremodel. Next to this, also a camber force introduced in the vertical shift will generate extra forceseven if the peak value of the slip angle characteristic has already been reached. The peak coefficientD then loses its physical and quantitative meaning. To properly cover the characteristics also atlarge camber angles therefore a new approach was chosen. The maximum possible side force isthe leading factor Dy in the basic Magic Formula. If all force contributions appear within thebrackets of the sine function, they will all respect the friction limit. Introduction of a completelyindependent arctangent function for the camber contribution in the formula, with its own stiffnessfactor Bγ , shape factor Cγ , and curvature factor Eγ provides sufficient freedom to describe thepure camber characteristics accurately. The camber force will respect the friction limit, and stronginteraction between camber and slip angle contributions are inherently build in.

Fy0 = Dy sin(Cα arctan(Bαα− Eα(Bαα− arctan(Bαα)))+Cγ arctan(Bγγ − Eγ(Bγγ − arctan(Bγγ)))) (A.17)

A.2.3 Aligning moment (pure slip)

In the nowadays motorcycle Magic Formula a part of the aligning moment Mz is obtained bymultiplying the pneumatic trail t with the side force that is attributed to the side slip and not tothe camber angle, Fy,γ=0. This part of the side force is obtained form the side force calculationsby setting γ = 0. The other part of the aligning moment consists of the residual torque;

Mz0 = −t.Fy,γ=0 + Mzr0 (A.18)

The pneumatic trail on its turn is represented by the cosine version of the magic formula;

t0 = Dt cos(Ct arctan(Btαt − Et(Btαt − arctan(Btαt)))) cos(α) (A.19)

With;

αt = α + SHt (A.20)

This version of the Magic Formula is able to produce the characteristic hill shaped curve, whichcan be seen in figure A.5. In this figure the basic properties of the cosine based curve have beenindicated. Again, D is the peak value, C is a shape factor determining the level ya of the horizontalasymptote and now B influences the curvature at the peak (illustrated with the inserted parabola).Factor E modifies the shape at larger values of slip and governs the location x0 of the point wherethe curve intersects the x-axis.

A.2.4 Overturning moment

The overturning moment for a motorcycle tyre has contributions of both the lateral force Fy andthe camber angle γ. Therefore it is implemented as;

Mx = R0Fz(qsx1λV mx + (−qsx2γ + qsx3Fy

Fz0)) (A.21)


D

X x

yY- S h

- y a

- x 0B C2

Figure A.5: Curve produced by the cosine version of the Magic Formula

C

S p i n d l e a x i s

T y r e c o n t o u r

R o a d s u r f a c ey M x

A

Figure A.6: Difference between actual and used contact point

with qsx1 the factor with which a vertical offset can be introduced at 0 sideslip and camber.Furthermore qsx2 and qsx3 are the factors introducing the overturning couple induced by camberand lateral force respectively. Both λ values are scale factors. The main contribution of theoverturning moment while cornering will be induced by the camber factor. This is because of thefact that the forces and moments are not exactly applied in the contact center A, but in the pointC, as can be seen in figure A.6. The moment introduced by this lateral shift, Fz.yMx, thereforeneeds to be taken into account. As can be seen in figure A.6, this moment is considerable at largecamber angles.

A.2.5 Rolling resistance moment

With the measurement equipment which is presently used it is not possible to measure the momentaround the y-axis in the wheel center (Myc). As therefore only two moments are measured andthree moments need to be known in the contact center, an assumption is made for the rollingresistance;

My = −FzRlfr (A.22)

The vertical load and the loaded radius are determined in the contact routine, the rolling resistancecoefficient is determined as a function of the longitudinal force Fx and the longitudinal velocityVx. This in contradiction to the processing of the measurements, where fr is taken as 1.5%.


v2 v1

Mz

Fy

elastic foundation

stretched stringpath ofcontact points

σαa

a

φ V

Figure A.7: Stretched string model for transient tyre behaviour

A.2.6 Additional features

The effect of combined slip was introduced into the Magic Formula tyre model in a purely empiricway by Bayle [2]. This method describes the effect of combined slip on the lateral force and onthe longitudinal force characteristics. Weighting functions G have been introduced which, whenmultiplied with the original pure slip functions, produce the interaction effects of κ on Fy and ofα on Fx. The weighting functions have a hill shape. They take the value 1 in the special case ofpure slip (κ or α equal to zero). When, for example, at a given slip angle a from zero increasingbrake slip is introduced, the relevant weighting function for Fy may first show a slight increase inmagnitude (becoming larger than 1) but will soon reach its peak after which a continuous decreasefollows. The cosine version of the Magic Formula (which is also used in equation A.19) is used torepresent this hill shaped function for the longitudinal and lateral force Fx and Fy and the aligningmoment Mz. The rolling resistance moment My is not affected by any slip, so no combined slipeffect needs to be introduced. The overturning moment Mx is only indirectly affected by theweighing functions, as it depends on the lateral force Fy.

Next to the combined slip an extension has been provided that describes transient and oscilla-tory tyre behaviour for limited frequencies lower than 8 [Hz] and wavelengths larger than the tyrecircumference. First-order lag of tyre longitudinal and lateral deformations u and v are introducedthrough relaxation lengths σκ and σα, as depicted in figure A.7:

σκdu

dt+ |Vx|u = −σκVsx (A.23)

σαdν

dt+ |Vx|ν = −σαVsy (A.24)

These differential equations are based on the assmuption that the contact points near the leadingedge remain in the adhesion with the road surface (no sliding). The relaxation lengths are functionsof the vertical load and camber angle. The practical tyre deformation are defined as;

κ′ =u

σκsign(Vx) (A.25)

α′ = arctanν

σα(A.26)

The longitudinal and lateral force and the self-aligning moment are now determined with κ′ andα′ instead of the longitudinal and lateral wheel slip quantities κ and α.

Fx = Fx(α′, γ, κ′, Fz) (A.27)

Fy = Fy(α′, γ, κ′, Fz) (A.28)

Mz = Mz(α′, γ, κ′, Fz) (A.29)


A.3 Tyre model parameter determination

As seen in the previous paragraph, several Magic Formula parameters need to be defined to obtaina correct representation of the tyre behaviour. These parameters can be derived from experimentaldata obtained from tests. The tyre is rolled over a road at various loads, orientations and motionconditions. During these tests the forces (Fx, Fy, Fz) and moments (Mx, Mz) are measuredwith respect to the C-axis system in the wheel center. Next to the forces and moments alsoseveral other parameters are being measured, for example the camber angle, slip angle, forwardvelocity, rotational velocity and wheel center height. All data is recorded in a Matab *.mat file.The process from measurement data to a so-called tyre property file which contains all MagicFormula parameters is depicted in figure A.8. With the measurement data in the *.mat files,

F o r c e s & M o m e n t s i n C * . m a t f i l e s

F o r c e s & M o m e n t s i n W * . t d x f i l e s

M a g i c F o r m u l a p a r a m e t e r s * . t i r f i l e s

M - t y r e M F - t o o l

Figure A.8: Determination of the Magic Formula parameters

the program M-tyre generates TYDEX (*.tdx) files. These files contain the forces and momentsthat are converted to the contact point C. Next to these forces and moments also other variablesderived from the measurements are present, for example the loaded radius, longitudinal slip etc.On their turn all TYDEX files are loaded in MF-Tool, which is a fitting program. In this programall Magic Formula parameters are fitted to the measurement data and saved in a *.tir file. Thisfile is used by the MF-MCTyre model to read the Magic Formula parameters during simulations.

Appendix B

Motorcycle model parameters

To be able to implement the Koenen motorcycle model in SimMechanics all parameter valuesneed to be specified. In ’The dynamic behaviour of a motorcycle’ [10] all parameters are specifiedin Appendix G (p. 162-163). These parameters are listed in this Appendix, together with ageometrical figure of the motorcycle.

R

s x ms x j f x

m 2 w m 2 u

m 2

m 3d x

g 2 x

g 2 zd zr

g 3 zf z

g 1 x m 1 g 1 z

g 1 u za 1 z

g 1 s xm 1 s

g 1 s z

e g 1 u xm 1 u

m 1 wR o f

a 1 x

t w i s t a x i s

s t e e r a x i s

p i t c h a x i s

r i d e r l e a n a x i s

o r

Figure B.1: Motorcycle model components

In figure A.1 eight masses can be distinguished which are;m1 = 13.1 [kg] m2 = 209.6 [kg]m1s = 0 [kg] m2u = 25.6 [kg]m1u = 17.5 [kg] m2w = 0 [kg]m1w = 0 [kg] m3 = 44.5 [kg]

For these eight masses the moments of inertia that are being used are;Jx1 = 0.46 [kgm2] Jx1s = 0.0 [kgm2]Jy1 = 1.2 [kgm2] Jy1s = 0.0 [kgm2]

70

APPENDIX B. MOTORCYCLE MODEL PARAMETERS 71

Jz1 = 0.21 [kgm2] Jz1s = 0.0 [kgm2]Jxz1 = 0.0 [kgm2] Jxz1s = 0.0 [kgm2]

Jx1u = 0.29 [kgm2] Jx1w = 0.0 [kgm2]Jy1u = 0.0 [kgm2] Jy1w = 0.58 [kgm2]Jz1u = 0.29 [kgm2] Jz1w = 0.0 [kgm2]Jxz1u= 0.0 [kgm2] Jxz1w = 0.0 [kgm2]

Jx2 = 15.23 [kgm2] Jx2u = 0.37 [kgm2]Jy2 = 32.0 [kgm2] Jy2u = 0.0 [kgm2]Jz2 = 19.33 [kgm2] Jz2u = 0.37 [kgm2]Jxz2 = -1.4 [kgm2] Jxz2u = 0.0 [kgm2]

Jx2w = 0.0 [kgm2] Jx3 = 1.3 [kgm2]Jy2w = 0.74 [kgm2] Jy3 = 2.1 [kgm2]Jz2w = 0.0 [kgm2] Jz3 = 1.4 [kgm2]Jxz2w= 0.0 [kgm2] Jxz3 = -0.3 [kgm2]

The following geometrical parameters hold for the motorcycle model;a1x = 0.066 [m] g1uz = 0.632 [m]a1z = 0.632 [m] g2x = 0.680 [m]dx = 0.600 [m] g2z = 0.211 [m]dz = 0.679 [m] g3z = 0.190 [m]fx = 1.168 [m] Rof = 0.319 [m]fz = 0.513 [m] Ror = 0.321 [m]g1x = 0.015 [m] sxj = 0.400 [m]g1z = 0.032 [m] sxm = 0.100 [m]g1sx = 0.000 [m] ε = 0.520 [rad]g1sz = 0.000 [m] pLx = 0.770 [m]g1ux = 0.066 [m] pDz = 0.900 [m]

The front suspension vertical spring stiffness and damping are;Cs = 9000 [N

m ] Ks = 550 [Nsm ]

The rear suspension (pitch movement) rotational stiffness and damping are;Cp = 10280 [Nm

rad ] Kp = 440 [Nmsrad ]

The twist movement rotational stiffness and damping are;Cβ = 34100 [Nm

rad ] Kβ = 99.7 [Nmsrad ]

The rider upper body lean rotational stiffness and damping are;Cr = 10000 [Nm

rad ] Kr = 85.2 [Nmsrad ]

Finally, an aerodynamic drag force FD and lift force FL are acting on the motorcycle. These forcesare determined as:

FD =12ρCDAv2 (B.1)

FL =12ρCLAv2 (B.2)

with ρ the density of air, v the forward velocity and CDA and CLA the effective drag and lift areasrespectively. These areas are;

CDA = 0.488 [m2] CLA = 0.114 [m2]

Appendix C

Processing measurement data

With the Delft Tyre Test Trailer, the forces and moments that are generated by a tyre can bemeasured. This can be done at two different measurement towers on the right and left side of thetest trailer. The tyre forces and moments are measured in the hub of the wheel (in the C-axissystem) and transformed to the forces and moments in the tyre road contact point (the W -axissystem). Furthermore, the tyre model converts the forces and moments from the contact point tothe wheel center. Therefore, in this Appendix the conversions will be given.

C.1 Left measurement tower

The measurement hub of the left tower consists of two measurement crosses at a defined distancethat measure the forces in xc and zc direction. The forces in yc direction are measured with anaxial support, see figure C.1.

b aG G

GG

G

x 1x 2

z 1 z 2

y

Figure C.1: Measurement hub with five strain gauges

With the output of the strain gauges, the forces and moments in the C-axis system can bedetermined. With the distances a and b known, the forces (Fc) and moments (Mc) are defined as:

Fxc = Gx1 + Gx2 (C.1)

Fyc = Gy (C.2)

72

APPENDIX C. PROCESSING MEASUREMENT DATA 73

Fzc = Gz1 + Gz2 (C.3)

Mxc = −Gz1(a + b)−Gz2b (C.4)

Mzc = Gx1(a + b) + Gx2b (C.5)

The moment Myc cannot be measured by the measurement hub. To be able to describe allmoments in the C- and W -axis systems, some assumptions will be made later on.

C.2 Right measurement tower

In the right measurement tower, the forces and moments are measured with piezo electric trans-ducers. When looking at the measurement hub from the wheel, the piezo electric transducers arepositioned as depicted in figure C.2.

3 2

4 1

Figure C.2: Positioning of the piezo electric transducers

All four of these cells measure a force in the x-, y- and z-direction. These forces are sometimescombined and all measurement data is given in 7 measurement channels. The output of thesechannels are defined as:

Fx1234 = Fx1 + Fx2 + Fx3 + Fx4 (C.6)

Fy1 = Fy1 (C.7)

Fy2 = Fy2 (C.8)

Fy3 = Fy3 (C.9)

Fy4 = Fy4 (C.10)

Fz12 = Fz1 + Fz2 (C.11)

Tz34 = Fz3 + Fz4 (C.12)

In figure C.3 two views of the wheel connected to the measurement hub are depicted. On theleft side the rear view of the wheel is shown, on the right side the top view is shown. With theknown distances b, h and v the forces and moments in the C-axis system can be computed. Itshould be mentioned that also with the right tower, it is impossible to measure the moment Ty.

As was done for the left measurement hub, the equilibria of forces and moments are examinedfrom which the forces and moments can be defined. One last fact should be taken into considerationbefore the forces and moments can be determined. The zero measurement that removes all biasesfrom the measurement signals is performed with the hub at the zero camber position. At thatposition, the measured wheel weight is zeroed. When the measurement is performed at a predefinedcamber angle, the effect of the hanging mass must be compensated. With this compensation, theforces and moments become:

Fxc = −Fx1234 (C.13)

Fyc = −Fy1 − Fy2 − Fy3 − Fy4 + mwheelg sin(γ) (C.14)


F zF y

M x

F z

F y

F y

1 2 3 4

2 3

1 4

b

v C

C

C F xF y

M z

F x

F y

F y

1 2 3 4

1 2

3 4

b

h

1 2 3 4

C

C

C

Figure C.3: Positioning of the piezo electric transducers

Fzc = −Fz1 − Fz2 −mwheelg(1− cos(γ)) (C.15)

Mxc =12v((Fy2 + Fy3)− (Fy1 + Fy4)) + Fzcb (C.16)

Mzc =12h((Fy3 + Fy4)− (Fy1 + Fy2))− Fxcb (C.17)

C.3 Conversion from C-axis system to W-axis system

With the forces in the center of the axle known, the forces in the tyre road contact point canalso be derived. The easiest way to do this is to evaluate the equilibrium of forces. The forcesand moments in the y-z plane are depicted in figure C.4. The x-axis of both systems of axes areperpendicular, the y-axis and z-axis of the C-axis system are rotated around the x-axis by theinclination angle γ.

g

F y w

F z w

M y w

M z w

C

F y cF z c

M z cM y c

R l o a d

Figure C.4: Forces and moments in the y-z plane


The equilibrium of forces is formulated with figure C.4 and from this equilibrium the forces inthe W -axis system are defined as:

Fxw = Fxc (C.18)

Fyw = Fyc cos(γ)− Fzc sin(γ) (C.19)

Fzw = Fyc sin(γ) + Fzc cos(γ) (C.20)

From the equilibrium of moments, the moments in the W -axis system are defined as:

Mxw = Mxc − FycRl (C.21)

Myw = Myc cos(γ)−Mzc sin(γ) + FxcRl cos(γ) (C.22)

Mzw = Myc sin(γ) + Mzc cos(γ) + FxcRl sin(γ) (C.23)

As the moment Myc is unknown, an assumption is required to be able to describe all themoments in the W -axis system. With measurement results and experience, the assumption ismade for the rolling resistance to be:

Myw = −FzwfrRl (C.24)

The friction factor fr is estimated to be 1.5 %. With this assumption, the moments in the W -axissystem become:

Mxw = Mxc − FycRl (C.25)

Myw = −FzwfrRl (C.26)

Mzw =Mzc

cos(γ)+ Myw tan(γ) (C.27)

C.4 Conversion from W-axis system to C-axis system

In the MF-MCTyre model, the Magic Formula is evaluated in the W -axis system. The tyreforces and moments in a vehicle model though are requested in the C-axis system. This meansthat after the Magic Formula evaluation another conversion needs to take place. Therefore boththe equilibrium of forces and the equilibrium of moments around point O are used. With theseequilibria, the forces (Fc) and moments (Mw) in the W -axis system are given by:

Fxc = Fxw (C.28)

Fyc = Fyw cos(γ) + Fzw sin(γ) (C.29)

Fzc = −Fyw sin(γ) + Fzw cos(γ) (C.30)

Mxc = Mxw + FywRl cos(γ) + FzwRl sin(γ) = Mxw + FycRl (C.31)

Myc = Myw cos(γ) + Mzw sin(γ)− FxwRl (C.32)

Mzc = −Myw sin(γ) + Mzw cos(γ) (C.33)

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