Enhancing the MF-Swift Tyre Model for Inﬂation Pressure ... · Enhancing the MF-Swift Tyre Model...

Enhancing the MF-Swift TyreModel for Inflation Pressure

Changes

I.B.A. op het Veld

DCT 2007.144

Master’s thesis

Supervisor: Prof. Dr. H. Nijmeijer (Eindhoven University of Technology)

Coach(es): Dr. Ir. A.J.C. Schmeitz (TNO Automotive)Dr. Ir. I.J.M. Besselink (Eindhoven University of Technology / TNO Automotive)

Member of committee:Dr. Ir. J.A.W. van Dommelen (Eindhoven University of Technology)

Eindhoven University of TechnologyDepartment of Mechanical EngineeringDynamics and Control Group

Eindhoven, November, 2007

Preface

This Master thesis investigation was conducted from July 2006 till December 2007 at the section forIntegrated Safety of the Netherlands Organisation for Applied Scientific Research (TNO Automotive)in Helmond.

For the supervision and guidance during the investigation I would like to thank Prof. dr. H. Nijmeijerand Dr. Ir. I.J.M. Besselink of the Eindhoven University of Technology and Dr. Ir. A.J.C. Schmeitz ofTNO Automotive. Furthermore, I would like to thank my study colleagues, especially Thijs Spijkers,for the fruitful discussions and for their friendship.

Last but not least, I am thankful to my family and Annemieke for their non-technical support, patienceand encouragements throughout my entire study career.

i

AbstractEnhancing the MF-Swift Tyre Model for Inflation Pressure Changes - I.B.A. op het Veld

Tyres are a crucial part of the vehicle, they are the only contact between the road and the vehicle. Thetyre is the key link in the force transmission between the road surface and the vehicle. The forcesgenerated in the tyre are the result of tyre deflections due to vehicle support and interaction betweenthe tyre and the road. Every steering, braking or driving action is eventually transmitted through thetyre. Understanding the tyre properties is essential to the analysis and design of vehicles and vehiclecomponents. For this purpose different (mathematical) models are developed to analyse the behaviourof pneumatic tyres. At TNO two tyre models are developed: the Magic Formula (an empirical slipmodel) and the MF-Swift tyre model (a dynamic semi-empirical model, using a rigid ring approachwith the Magic Formula slip model). The current version of the MF-Swift tyre model can describethe tyre behaviour for one set of operating conditions. Current research projects aim to a clear, reliableand integrated solution to capture different operating conditions (inflation pressure, temperature, roadfriction, etc.).

In this study, the influence of the inflation pressure on the tyre behaviour is investigated. The subjectof this research project is to extend the MF-Swift model for inflation pressure changes. The objectivesof this research are: (1) investigating which parts of the MF-Swift tyre model must be made inflationpressure dependent; (2) deriving relations and parameter identification strategies for implementationof inflation pressure influences in the MF-Swift tyre model and (3) investigate the functionality andapplicability of the enhanced tyre model using a vehicle simulation model. To achieve these objectives,the MF-Swift tyre model is analysed and a literature study is performed on the inflation pressure de-pendency of the, in the MF-Swift model, modelled tyre characteristics. Where necessary, additionalexperiments are conducted and finite element method (FEM) analysis are used. Finally when the de-veloped inflation pressure dependent relations are implemented, a number of vehicle simulations areperformed with the enhanced MF-Swift tyre model.

The literature survey shows that the inflation pressure influence on the steady-state slip characteris-tics (Fx, Fy and Mz) is already implemented in the MF-Swift tyre model. Furthermore the literaturesurvey and the additional experiments, conducted with the Flatplank Tyre Test facility of EindhovenUniversity of Technology, show that the inflation pressure has the most influence on the characteristictyre stiffnesses, relaxation behaviour and camber effects (i.e. camber thrust and camber torque). Allthese tyre characteristics show a (almost) linear relation with the inflation pressure. FEM and ana-lytical analyses make clear that also a linear relation between the inflation pressure and the primarytyre eigenfrequencies (i.e. the first eigenmodes in which the tyre tread band almost retains its circularshape) exists. Next, the measurement and FEM results are used to derive inflation pressure, and if nec-essary vertical load, dependent relations for implementation in the MF-Swift tyre model. In addition,optimisation strategies are introduced to determine the inflation pressure and vertical load dependentlateral stiffness from measurements or lateral relaxation length experiments. Furthermore, strategiesare developed to describe and estimate the sidewall stiffnesses of the rigid ring model for changinginflation pressure. Finally, full vehicle simulations show that the enhanced MF-Swift tyre model canbe used to simulate the inflation pressure influence on ride comfort and handling behaviour. It isconcluded that the enhancements are a useful addition to the MF-Swift tyre model.

iii

SamenvattingHet aanpassen van het MF-Swift bandmodel voor bandspanningsveranderingen - I.B.A. op het Veld

Banden vormen een cruciaal onderdeel van een voertuig, ze zorgen voor het enige contact tussen vo-ertuig en wegdek. Bij de krachtoverdracht tussen wegdek en voertuig zijn banden dé koppeling. Deoptredende bandkrachten zijn het gevolg van de bandvervorming door het voertuiggewicht en de in-teractie tussen de band en het wegdek. Uiteindelijk wordt elke stuur, rem of accelereeractie via de bandovergedragen naar het wegdek. Voor het analyseren van het voertuiggedrag, is het van essentieel belangom het gedrag van een band te begrijpen. Om dit te bewerkstelligen zijn er verschillende (wiskundige)modellen ontwikkeld die het gedrag van banden analyseren/beschrijven. TNO heeft twee bandmod-ellen ontwikkeld: de Magic Formula (een empirisch slipmodel) en het MF-Swift bandmodel (een semi-empirisch dynamisch bandmodel, gebaseerd op een starre ring benadering en het Magic Formulaslipmodel). De huidige versie van het MF-Swift bandmodel kan het bandgedrag alleen beschrijvenvoor één set bedrijfscondities. Momenteel is het onderzoek gericht op een heldere, betrouwbare engeïntegreerde oplossing voor het beschrijven van verschillende bedrijfscondities. (bandenspanning,temperatuur, wegdekwrijving, enz.).

In dit onderzoek wordt de invloed van de bandenspanning op het bandgedrag onderzocht. Het on-derwerp van dit onderzoeksproject is het aanpassen/verbeteren van het MF-Swift bandmodel voorbandenspanningsveranderingen. De doelstellingen van dit onderzoek zijn: (1) het onderzoeken welkedelen van het MF-Swift bandmodel bandenspanningsafhankelijk moeten worden gemaakt; (2) hetafleiden van relaties en parameter identificatie strategieën voor de implementatie van bandenspan-ningsafhankelijkheid in het MF-Swift bandmodel en (3) het onderzoeken van de functionaliteiten toepasbaarheid van het nieuwe bandenspanningsafhankelijke MF-Swift bandmodel in voertu-igontwikkeling. Om deze doelstellingen te verwezenlijken, is het MF-Swift bandmodel geanalyseerden is een literatuurstudie uitgevoerd naar de invloed van bandenspanning op de, in het MF-Swiftbandmodel, gemodelleerde bandeigenschappen. Waar nodig zijn extra experimenten en eindige el-ementenmethode (FEM) analyses uitgevoerd. Uiteindelijk, na implementatie van de afgeleide ban-denspanningsafhankelijke relaties, is een aantal voertuigsimulaties uitgevoerd met het nieuwe MF-Swift bandmodel.

Uit het literatuuronderzoek is gebleken dat de bandenspanningsafhankelijkheid van de stationaireslip karakteristieken (Fx, Fy and Mz) reeds zijn geïmplementeerd in het MF-Swift bandmodel. Deresultaten van het literatuuronderzoek en de uitgevoerde experimenten met de Flatplank Tyre Testfacility van de Technische Universiteit Eindhoven tonen aan dat bandenspanning de meeste invloedheeft op de karakteristieke bandstijfheden, het relaxatiegedrag en de camber krachten en momenten.Al deze karakteristieke grootheden laten een (bijna) lineaire relatie zien met de bandenspanning. FEMen analytische analyses tonen aan dat er ook een lineaire relatie bestaat tussen de bandenspanningen de primaire eigenfrequenties van de band. De resultaten van de experimenten en de FEM analyseszijn gebruikt om bandspanningsafhankelijke, en zonodig verticale belastingsafhankelijke, relaties af teleiden. Daarnaast is er een optimalisatie strategie geïntroduceerd om de laterale stijfheid te bepalen uitstijfheidsmetingen of relaxatielengte metingen. Verder zijn er strategieën ontwikkeld om de zijwangstijfheden van het starre ring model voor bandenspanningsveranderingen te beschrijven. Tenslottelaten de voertuigsimulaties zien dat het nieuwe bandenspanningsafhankelijke MF-Swift bandmodel deinvloed van de bandenspanning op het rijcomfort en het handling gedrag goed kan simuleren. Hieruit

v

kan worden geconcludeerd dat de aanpassingen voor bandenspanningsveranderingen een bruikbareuitbreiding zijn van het MF-Swift bandmodel.

vi

Contents

Preface i

Abstract iii

Samenvatting v

List of Symbols xi

1 General Introduction 1

1.1 Motivation and background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Objectives and scope . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.3 Outline of thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2 The MF-Swift Tyre Model 5

2.1 The Magic Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.2 The MF-Swift tyre model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.2.1 General description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.2.2 Rigid ring dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.2.3 Contact model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.3 Summarising this chapter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

3 Literature Review 11

3.1 Force and moment characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

3.1.1 Longitudinal force . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

3.1.2 Lateral force . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

3.1.3 Aligning moment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

3.1.4 Vertical stiffness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

3.2 Empirical relations for aircraft tyres . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

3.2.1 Longitudinal stiffness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

3.2.2 Lateral stiffness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

3.2.3 Torsional Stiffness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

vii

viii CONTENTS

3.2.4 Tyre relaxation behaviour . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

3.3 Rolling Resistance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

3.4 Eigenfrequencies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19


4 Inflation Pressure Sensitivity Experiments 21

4.1 Stiffness measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

4.1.1 Tyre vertical stiffness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

4.1.2 Tyre longitudinal stiffness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

4.1.3 Tyre lateral stiffness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

4.1.4 Tyre torsional stiffness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

4.2 Lateral relaxation behaviour . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

4.2.1 Effective rolling radius . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

4.3 Camber thrust and camber torque . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

4.4 Eigenfrequencies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

4.4.1 FEM analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

4.4.2 Analytical results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37


5 Inflation Pressure Relations to Enhance the MF-Swift Tyre Model 39

5.1 Tyre vertical stiffness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

5.2 Tyre torsional stiffness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

5.3 Lateral relaxation behaviour . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

5.3.1 Current model implementation . . . . . . . . . . . . . . . . . . . . . . . . . . 42

5.3.2 Proposed enhancements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

5.4 Longitudinal relaxation behaviour . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

5.5 Rolling resistance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

5.6 Camber thrust and camber torque . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

5.7 Rigid ring dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

5.7.1 Rotta Membrane Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

5.7.2 Scaling the nominal pressure sidewall stiffnesses . . . . . . . . . . . . . . . . . 57

5.7.3 Stiffnesses rigid ring model, lateral components . . . . . . . . . . . . . . . . . 58

5.7.4 Stiffnesses rigid ring model, longitudinal components . . . . . . . . . . . . . . 59


6 Vehicle Behaviour Simulations 63

6.1 Steady-state circular test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

6.1.1 Sensitivity analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

6.1.2 Unloaded vs. loaded vehicle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

6.2 Random steering test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

6.3 Ride analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

CONTENTS ix

7 Conclusions and Recommendations 75

7.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

7.1.1 Inflation pressure influence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

7.1.2 Enhanced model equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

7.1.3 Applicability to the vehicle design . . . . . . . . . . . . . . . . . . . . . . . . . 77

7.1.4 Final conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

7.2 Recommendations for future research . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

Bibliography 78

A The Flatplank Tyre Tester 83

A.1 General description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

A.2 Forces and moments transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

B Measurement Results 87

B.1 Characteristic stiffnesses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

B.1.1 Tyre longitudinal stiffness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

B.1.2 Tyre lateral stiffness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

B.1.3 Tyre torsional stiffness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

B.2 Lateral relaxation behaviour . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

B.3 Camber thrust and camber torque . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

C Modal Analysis 95

C.1 FEM Simulations (ABAQUS) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

C.1.1 Free hanging tyre . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

C.1.2 non-rolling loaded tyre . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

C.2 Analytical analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

D Aspects of the introduced inflation pressure dependent relations 101

D.1 Tyre torsional stiffness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

D.2 Rolling resistance force . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

D.3 Stiffnesses of the rigid ring model extrapolation properties . . . . . . . . . . . . . . . . 103

D.3.1 Lateral situation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

D.3.2 Longitudinal situation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

D.4 Camber thrust and camber torque extrapolation . . . . . . . . . . . . . . . . . . . . . . 104

D.5 Overview implemented parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

E Simulation Parameters 109

x CONTENTS

List of Symbols

Symbol Description Unit

a half the contact length mB stiffness factor in Magic Formula -C shape factor in Magic Formula -cbγ torsional sidewall stiffness in x-direction Nm/radcbψ torsional sidewall stiffness in z-direction Nm/radcbθ rotational sidewall stiffness in y-direction Nm/radcbx translational sidewall stiffness in x-direction N/mcby translational sidewall stiffness in y-direction N/mcbz translational sidewall stiffness in z-direction N/mCFα cornering stiffness N/radCFκ longitudinal slip stiffness NCFx tyre longitudinal stiffness N/mCFy tyre lateral stiffness N/mCFz overall vertical stiffness N/mCMz total aligning moment stiffness Nm/radcpx longitudinal tread stiffness N/m2

crx residual longitudinal carcass stiffness N/mcry residual lateral carcass stiffness N/mcu tangential direction stiffness (Rotta) N/mcv lateral direction stiffness (Rotta) N/mcw radial direction stiffness (Rotta) N/mcx longitudinal carcass stiffness N/mcy total carcass stiffness N/mD peak factor in Magic Formula -d outside diameter of unloaded tyre mdx longitudinal deflection mdy lateral deflection mE curvature factor in Magic Formula -Fax longitudinal component of axle force NFaz vertical component of axle force NFa axle force Nfrot rotational eigenfrequency Hzftrans translational eigenfrequency HzFx,RR rolling resistance force NFxW longitudinal force in contact patch centre NFyW lateral force in contact patch centre NFy lateral force NFz0 nominal vertical load NFz vertical load N

xi

xii CONTENTS

G shear modulus tyre sidewall N/m2

Ibx moment of inertia of the belt about the x-axis kgm2

Iby moment of inertia of the belt about the y-axis kgm2

Ibz moment of inertia of the belt about the z-axis kgm2

Ir moment of inertia of the small residual (contact patch) mass kgm2

K stiffness matrix -k1 numerical constant longitudinal stiffness (aircraft tyre) -k2 numerical constant longitudinal stiffness (aircraft tyre) -Kα torsional stiffness Nm/radKλ lateral stiffness (aircraft tyre) N/mkbγ torsional sidewall damping in x-direction Nms/radkbψ torsional sidewall damping in z-direction Nms/radkbθ rotational sidewall damping in y-direction Nms/radkbx translational sidewall damping in x-direction Ns/mkby translational sidewall damping in y-direction Ns/mkbz translational sidewall damping in z-direction Ns/mKxκ longitudinal slip stiffness in Magic Formula -Kx longitudinal stiffness (aircraft tyre) N/mKyα cornering stiffness in Magic Formula N/radKyγ0 camber thrust stiffness in Magic Formula N/mls length of sidewall arc mM mass matrix -ma mass of parts that rotate with the rim kgmb belt mass kgMx overturning moment NmMy rolling resistance moment NmMzW aligning moment in contact patch center NmP inflation pressure barp fit parameter in Magic Formula -pi,r rated inflation pressure barpi0 nominal inflation pressure barpi inflation pressure barq fit parameter in Magic Formula -r0 unloaded tyre radius mRe effective rolling radius mrl loaded radius mSH horizontal shift in Magic Formula -SV vertical shift in Magic Formula -t sidewall thickness mV velocity m/sVx forward, longitudinal velocity m/sw undeflected tyre width mZ vertical load N

CONTENTS xiii

Greek Symbol Description Unit

α sideslip angle radβy effective forward slope radδ0 vertical deflection mδλ vertical sinking at lateral force (aircraft tyre) mγ angle about the x-axis, camber angle radλ scaling factor in Magic Formula radµ peak longitudinal friction coefficient -µy peak lateral friction coefficient -Ω tyre angular velocity about the y-axis rad/sω free vibration frequency, rotational speed rad/sψ steering angle radψr angle between wheel plane and contact patch radρz vertical deflection mσst static relaxation length mσx longitudinal relaxation length mσy lateral relaxation length mτλ lateral spring constant coefficient (aircraft tyre) -ϕs half the angle of tyre sidewall rad

xiv CONTENTS

Chapter 1

General Introduction

This chapter gives a general introduction to the work presented in this thesis. The background, objec-tives and scope of research are presented and a brief outline of the contents of this Master’s thesis isgiven.

1.1 Motivation and background

While the wheel may have been one of man’s first inventions, the pneumatic tyre is definitely not oneof the simplest components to analyse. The pneumatic tyre is a complex composite consisting mainlyof vulcanised rubber and layered reinforcing strands (plies), made of Nylon, Rayon or steel cords whichare oriented in a certain configuration.

The tyre is a crucial part of the vehicle. Not only to support the vehicle weight and to cushion roadirregularities, but also to generate and transmit forces needed to accelerate and decelerate the vehicleand to change the direction of motion of the vehicle. The tyre forces and moments are the result of tyredeflections due to interaction between wheel and road. Tyre vibrations arise through road irregularities,wheel axle motions, and tyre non-uniformities. The complex tyre structure with its compliance andinertia may give rise to attenuation from these irregularities in certain frequency ranges but also tomagnification at other frequencies.

Figure 1.1: Construction of radial ply tyre [9].

A modern radial (ply) tyre is characterised by parallel cords running directly across the tyre from onebead to the other, the so called carcass plies. Directional stability of the tyre is supplied by the enclosedpressurised air acting on the sidewall of the carcass and by the stiff belt of fabric or steel, that runsaround the circumference of the tyre. The soft carcass provides the tyre with a soft ride and the stiffbelt provides the radial tyre with good cornering proporties, by keeping the tread flat on the roaddespite horizontal deflections of the tyre. The contact between the tyre and the road is established and

1

2 General Introduction

maintained by the tread. The adhesion between tread and road is the main factor for the generation ofhorizontal forces. Not only the rubber compound and the orientation of the plies (radial or bias ply) canhave an effect on these tyre characteristics but also several external parameters, for instance; verticalload, inflation pressure, velocity and (ambient) temperature.

Understanding tyre properties is essential for the analysis and design of vehicles and vehicle com-ponents. For this purpose different (mathematical) models are developed to describe the behaviourof pneumatic tyres. These models can be distinguished between models based on the physical con-struction of the tyre (physical models) and models based on experimental data (empirical models).Combinations of both approaches are also used. The most essential requirements for a tyre model forvehicle analyses are:

• accurate prediction of forces and moments;

• practical in use (e.g. low computational effort);

• widely applicable (for many different operating conditions).

Through the years various modelling techniques have been developed, but it is still a long way to thecomplete description of all aspects of tyre behaviour.

At TNO two tyre models are developed: the Magic Formula and the Short WavelengthIntermediate Frequency Tyre model, SWIFT, of Pacejka [28]. The Magic Formula is a empiricaltyre handling model, first introduced in 1987, which is capable of dealing with the stationair slip char-acteristics and dynamics up to about 8 Hz. The SWIFT tyre model, or also called the MF-Swift tyremodel, is a dynamic semi-empirical model, based on a rigid ring tyre model combined with the MagicFormula. MF-Swift can describe tyre behaviour for in-plane (longitudinal and vertical) and out-of-plane(lateral and steering) motions up to about 60-100 Hz and it can deal with uneven road surfaces. Formore details it is referred to chapter 2. Through the years many revisions have been made. The currentresearch projects aim to a clear, reliable and integrated solution to capture the influence on the tyrebehaviour of different operating conditions (road friction, temperature, inflation pressure, etc.) Thecurrent model can describe the tyre behaviour for one set of operating conditions only. TNO Automo-tive already performed a number of research projects on these topics, namely:

• A validated tyre temperature model has been developed [38].

• The influence of pavement micro/macro texture on tyre forces and moments data has beeninvestigated.

• A brake performance model ("remvermogenmodel") has been developed to predict braking dis-tances and vehicle states during an emergency stop.

• Magic Formula parameters are extended with inflation pressure parameters, so that it is possibleto describe the stationary forces and moments for a range of tyre inflation pressures [12].

1.2 Objectives and scope

The subject of this Master’s thesis is to extend the MF-Swift model for inflation pressure changes.

Taking into account the structure of the model, the following objectives have been defined:

1. Investigate which parts of the model must be made inflation pressure dependent.

2. Derive relations and parameter identification strategies for the implementation of inflation pres-sure changes in the MF-Swift tyre model.

1.3 Outline of thesis 3

3. Investigate the functionality and applicability of the enhanced tyre model using a vehicle simu-lation model.

To achieve these objectives, a literature study is performed and previous measurements are analysed.Where necessary, additional experiments are conducted. When literature and measurement resultsdo not provide a clear answer, results from a finite element method (FEM) analysis are used. Finallya number of vehicle simulations are executed with the enhanced tyre model to analyse the extendedrelations on functionality and applicability in vehicle design applications.

1.3 Outline of thesis

The outline of this thesis is as follows. The TNO tyre models are the subject in Chapter 2. The MagicFormula and MF-Swift tyre model are discussed in more detail. In Chapter 3 a literature review onthe influence of inflation pressure on the different tyre characteristics is presented and possible trendsare identified. This knowledge is used during the analysis and development of the model in the sub-sequent chapters. Chapter 4 deals with the inflation pressure sensitivity experiments. An overview isgiven of the experimental investigations and of the tyre characteristics inflation pressure dependency.The subject of Chapter 5 is the enhancement of the MF-Swift tyre model. Inflation pressure dependingrelations are presented and validated with the results of Chapter 3 and 4. Furthermore methods arepresented to describe the influence of inflation pressure on the rigid ring dynamics. Chapter 6 illus-trates the influence of inflation pressure on vehicle handling and ride. A number of simulations areperformed with different inflation pressure settings to investigate the functionality and applicability ofthe enhanced relations in vehicle design. Finally, in Chapter 7, conclusions are given and recommen-dations for further research are formulated.

4 General Introduction

Chapter 2

The MF-Swift Tyre Model

As already discussed in Chapter 1, the subject of this Master’s thesis is to extend theMF-Swift model forinflation pressure changes. The present chapter gives an overview of the MF-Swift tyre model, that hasbeen designed to represent the tyre as a vehicle component in a multibody simulation environment.The MF-Swift tyre model is a combination of the Magic Formula and a rigid ring tyre model. The basicmodelling approach of the MF-Swift model is termed "semi-empirical", meaning the model is basedon measurement data but also contains parts that find their origin in physical models.

2.1 The Magic Formula

The so-called Magic Formula is a widely used empirical tyre model to calculate steady-state tyreforce and moment characteristics for the use in vehicle dynamics studies. The development of themodel was started in the mid-eighties, as a cooperative effort of the TU-Delft (Pacejka) and VolvoCar Corporation (Bakker et al.), resulting in a first version in 1987 [3]. In the following years severalenhancements were made. Michelin introduced in 1993 a purely empirical method using MagicFormula based functions to describe the tyre horizontal forces at combined slip conditions. Thisapproach was adopted by DVR (Delft Vehicle Dynamics Research Center, a joint venture of TU-Delftand TNO) resulting in the "Delft Tyre" tyre model.

All versions of the Magic Formula show the same basic form for pure slip characteristics: a sine of anarctangent. The general form of the Magic Formula reads:

y = D sin[C arctan(1− E)Bx+ E arctan(Bx)] (2.1)

Y (X) = y(x) + SV (2.2)

x = X + SH (2.3)

Y is the output variable: longitudinal force Fx(= f(κ)) or lateral force Fy(= f(α)). X is the inputvariable: sideslip angle α or longitudinal slip κ. Cosine of the arctangent versions are used for thedescription of the aligning torque Mz . The remaining variables of the Magic Formula describe thefollowing coefficients:

B : stiffness factorC : shape factorD : peak factorE : curvature factorSH : horizontal shiftSV : vertical shift

5

6 The MF-Swift Tyre Model

The coefficientsB,C,D,E and the offsets SH and SV characterise the shape of the slip characteristics.Figure 2.1 illustrates the meaning of some of the factors by means of a typical side force characteristic.Each coefficient represents a specific aspect of the slip characteristic. The product of the factors B,C and D determines the slope at the origin. The shape factor C determines the value of Y whenx → ∞. The peak factor D influences the maximum value of the characteristic. The curvature factorE influences the characteristic curvature around the peak value and controls the horizontal positionof the peak. The two shifts SH and SV make it possible to offset the curve in horizontal and verticaldirection with respect to the origin. The slip stiffness K influences the slip stiffness at small values ofslip and is determined by the product of BCD. This all results in a new set of coordinates Y (X), asshown in figure 2.1.

Figure 2.1: Curve produced by the Magic Formula; typical side force characteristic [28].

The offsets SH and SV appear to occur when ply-steer and conicity effects cause the lateral force Fycurve not to pass through the origin. Wheel camber may also result in considerable offsets of the Fyversus α curve.

To explain the equations describing the Magic Formula coefficients B, C, D and E, the full set ofequations for pure lateral slip (pure cornering) will be considered. The Magic Formula for the lateralforce Fy reads:

Fy = Dy sin[Cy arctan(1− Ey)Byαy + Ey arctan(Byαy)], (2.4)

where the different coefficients are described by:

αy = α+ SHy (2.5)Cy = pCy1λCy (2.6)Dy = Fzµyλµy = Fz(pDy1 + pDy2dfz)(1− pDy3γ

2)λµy (2.7)Ey = (pEy1 + pEy2dfz)1− (pEy3 + pEy4γ)sign(αy)λEy (2.8)

Ky = ByCyDy = pKy1Fz0 sin(2 arctan

( FzpKy2Fz0

))(1− pKy3γ

2) (2.9)

SHy = (pHy1 + pHy2dfz)λHy + pHy3γλKyγ (2.10)SV y = Fz(pV y1 + pV y2dfz)λV y + (pV y3 + pV y4dfz)γλKyγ (2.11)

The dfz is the normalized load increment:

dfz =Fz − Fz0Fz0

, (2.12)

where Fz is the actual vertical load, Fz0 the nominal vertical load and γ the camber angle. The nom-inal vertical load is typically 80% of the load index (LI) of the tyre. Dimensionless parameters p are

2.2 The MF-Swift tyre model 7

introduced to describe the influence of the vertical load and camber angle on the coefficients. Theseparameters are determined by minimising the difference between the model and measurements usingnumerical optimisation techniques. Furthermore, a number of scaling factors λ are introduced to scalethe formula without changing the dimensionless parameter values. In this way, the tyre characteristicscan be adjusted or tuned (e.g. correction for different road types). The set of equations (2.5) till (2.11)shows the basic influence of the vertical load and camber angle on the lateral force. The equationspresented here correspond to the latest published version of the Magic Formula [28].

2.2 The MF-Swift tyre model

In this section the MF-Swift tyre model will be discussed in more detail. For a complete description ofthe model and the latest updates of the model it is referred to the work of Schmeitz [31]. The MF-Swifttyre model is based on the work of Zegelaar [40], Maurice [26] and finally Schmeitz. The research wasconducted at Delft University of Technology and supported by TNO Automotive and a consortium ofindustries. It has been developed first as a tyre model for in-plane dynamics. Later on, the model hasbeen extended for out of plane dynamics and road unevenesses.

2.2.1 General description

The MF-Swift tyre model uses the Magic Formula for the steady-state slip behaviour. This makesit possible to model tyre slip behaviour with empirical relations. To model high frequency dynamicbehaviour and obstacle response, a rigid ring is added to model the dynamic behaviour of the tyre(primary or rigid ring tyre modes). A schematic representation of the MF-Swift tyre model is shown infigure 2.2.

Figure 2.2: Schematic representation of the MF-Swift rigid ring tyre model [2].

The rigid ring, that represents the tyre belt, is attached to the rim by springs and dampers. Thesesprings and dampers represent the tyre sidewalls with pressurised air. Furthermore, residual stiffnessand damping elements are used to represent the quasi-static tyre stiffnesses correctly. Finally a contactmodel is added to describe the tyre slip behaviour and enveloping behaviour. The residual springand damper elements connect the contact model (a combination of a slip model and a model forenveloping obstacles) to the rigid ring. The slip model, based on the Magic Formula, describes the slipbehaviour of the contact patch and generates slip forces and moments according to the applied slip.The enveloping model takes the tyre flexibility into account when rolling over road surfaces with shortwavelength content. The total tyre mass is distributed between the rigid ring and the rim (typically:75% of the tyre mass is assigned to the belt and 25% is assigned to the rim according to [28]). Asmentioned before, the tyre tread band (belt) is modelled as a circular rigid body with mass mb andinertia Ibx, Iby and Ibz . Because the tyre tread band is modelled as a rigid body, the model is only ableto describe the dynamic behaviour for conditions where the tyre tread band retains its circular shape


(i.e flexible modes of the tyre can not be described). This limits the useable frequency range to 60-100Hz, depending on the tyre type to be modelled.

2.2.2 Rigid ring dynamics

For the rigid ring, the following equations of motions with respect to the non-rotating belt axis systemcan be derived. For the complete derivation of the equations of motion reference is made to the workof Schmeitz [31].

mb

(V aax − ωaazV

aay + xrb

)+ kbxxrb + cbxxrb − kbzΩzrb = F bbx (2.13)

mb

(V aay + ωaazV

aax − ωaaxV

aaz + yrb

)+ kby yrb + cbyyrb = F bby (2.14)

mb

(V aaz − ωaaxV

aay + zrb

)+ kbz zrb + cbzzrb − kbxΩxrb = F bbz (2.15)

Ibx(ωaax + γrb

)− Iby

(Ω + θrb

)(ωaaz + ψrb

)+ kbγ γrb + cbγγrb − kbψΩψrb = M b

bx (2.16)

Iby(θrb + Ω

)+ kbθ θrb = M b

by (2.17)

Ibz(ωaaz + ψrb

)− Iby

(Ω + θrb

)(ωaax + γrb

)+ kbψψrb + cbψψrb − kbγΩγrb = M b

bz (2.18)

mb is the belt mass, Ibx,y,z the belt inertia (x, y, z-direction), cbx,y,z the translational sidewall stiffness(x, y, z-direction) and kbx,y,z the translational sidewall damping (x, y, z-direction). Furthermore, cbγ,ψ isthe torsional sidewall stiffness (about x, z-axis), cbθ the rotational sidewall stiffness (about y-axis), kbγ,ψthe torsional sidewall damping (about x, z-axis) and kbθ the rotational sidewall stiffness (about y-axis).Finally, xrb, yrb, zrb, γrb, θrb and ψrb are relative motions of the belt with respect to the wheel centre.~F bb and ~M b

b are the forces and moments acting on the belt which are applied at the centre of the beltbody and are obtained from the forces and moments in the residual spring-damper elements and thecontact model.

2.2.3 Contact model

The contact model can be divided into an enveloping model and a slip model. When travelling onuneven road surfaces, the tyre acts as a geometric filter which smoothes the response at the axle (i.ethe unevenness of the actual road). This is the so-called enveloping behaviour of a tyre. When rollingover a (short) obstacle, the tyre always hits the obstacle at a certain distance before the centre of the axleis above the obstacle (so-called: lengthening). If the centre of the axle is above an obstacle, the (small)obstacle is fully or partially absorbed by the tyre (so-called: swallowing). In figure 2.3 the envelopingphenomena are depicted.

Figure 2.3: Tyre enveloping behaviour [2].

To represent the nonlinear enveloping behaviour, the concept of an effective road profile is used, asdeveloped by Zegelaar [40] and Schmeitz [31]. The wheel states and the effective road surface are theinput of the rigid ring tyre model.

2.2 The MF-Swift tyre model 9

The wheel states (position, orientation and velocity), consists of:

• Position vector of the wheel centre[xGw y

Gw zGw

]T(wrt. global position)

• Wheel yaw ψaa and camber angle γaa

• Absolute velocities of the wheel expressed in the local (axle) axis system:~Vwheel =

(~ea

)T [V aax V

aay V

aaz

]T~ωwheel =

(~ea

)T [ωaax Ω ωaaz

]TThe effective road surface is defined by the effective inputs (plane height w, effective forward slopeβy , effective road camber angle βx and road curvature dβy/dx) and is the result of the envelopmentbehaviour on the specific road profile, see figure 2.4. These effective inputs serve as an input for the restof the tyre model. The effective plane angle is defined as the quasi-statically (measured) longitudinalaxle force Fax divided by the vertical axle force Faz :

tanβy =FaxFaz

. (2.19)

When rolling over an obstacle with a constant vertical load, the effective height equals the difference inaxle height between the new height.

Figure 2.4: 2D effective road surface; tyre rolls quasi-statically over a step obstacle at a constant verticalload Faz [31].

Schmeitz [31] introduced the so-called tandem − cam model in 2004. The model consists of two el-liptical cams that move vertically when travelling over a road surface. The cams represent the outsidecontour in the area where contact with the obstacle occurs. The distance between the cams is aboutthe contact length ls of the tyre. From the heights of the cams the effective inputs w and βy are deter-mined, see figure 2.5. By using parallel tandem cam models the effective input for 3D road surfacescan be determined, see Schmeitz [31] for details.


Figure 2.5: The tandem model with elliptical cams [31].

To account for all other flexibilities (i.e. carcass construction), residual springs in longitudinal, lateral,vertical and yaw direction are introduced. They connect the slip model to the rigid ring, see figure 2.6.Furthermore, residual dampers are attached for computational stability reasons. The vertical load isdirectly applied to the residual vertical stiffness, calculated from the total vertical stiffness and the ringsidewall stiffness. The moments about the x and y-axis are also directly applied to the ring model.

Figure 2.6: MF-Swift contact model representation.

2.3 Summarising this chapter

In this chapter the MF-Swift tyre model and the tyre characteristics that are modelled are discussed inmore detail. The most important characteristics of the tyre model are:

• steady state slip characteristics (including camber effects),

• characteristic tyre stiffnesses (longitudinal, lateral, vertical and torsional),

• relaxation behaviour,

• eigenfrequencies of the tyre belt.

The inflation pressure dependency of these tyre characteristics has to be investigated.

Chapter 3

Literature Review

Based on the tyre characteristics that are modelled in the MF-Swift tyre model, a literature survey onthe influence of the inflation pressure is performed. In this chapter the results of the literature surveyare presented. For the investigation of the inflation pressure influence on passenger car tyres very oftenonly longitudinal force Fx, lateral force Fy and self-aligning moment Mz as result of slip (force andmoment characteristics) are taken into account. Because the literature on passenger car tyres does notprovide a clear answer for all the tyre characteristics that are modelled, literature on aircraft tyres isalso studied.

3.1 Force and moment characteristics

For the description of the force and moment characteristics the work of de Hoogh [12] and the workSchmeitz and de Hoogh [32] is important. In [12] the influence of inflation pressure and velocity effectson the common tyre characteristics are investigated and the Magic Formula is extended to incorporatethese effects. For his findings, measurement results are compared with TREADSIM (a discrete "brush"tyre model [28] with flexible carcass) simulations. To include inflation pressure pi, the inflation pres-sure increment dpi is introduced:

dpi =pi − pi0pi0

, (3.1)

where pi and pi0 are the current and nominal inflation pressure.

3.1.1 Longitudinal force

In [12] it is observed that the longitudinal slip stiffness and the peak longitudinal friction coefficientboth are depending on the inflation pressure. Since there is no carcass and belt compliance in longi-tudinal direction the longitudinal slip stiffness of TREADSIM depends solely on the contact length,described by the following equation from [28]:

CFκ =(∂Fx∂κ

)κ=0

= 2cpxa2, (3.2)

where, cpx is the tread stiffness in longitudinal direction and a is the half of the contact length. It canbe expected that higher inflation pressure results in a lower slip stiffness. With figure 3.1 this behaviourcan be confirmed. However, in [12] it is also concluded that other tyres exhibit a different behaviour.

11

12 Literature Review

0 2000 4000 6000 80000

5

10

15x 10

4

Fz [N]

Slip

Stif

fnes

s [N

]

Measurement TREADSIM

0 2000 4000 6000 80000

5

10

15x 10

4

Fz [N]

Slip

Stif

fnes

s [N

]

P = 1.8 [bar]P = 2.1 [bar]P = 2.4 [bar]

Figure 3.1: Longitudinal slip stiffness: 225/55 R16 tyre [12].

It appeared that a quadratic relation produces significantly smaller fit errors than a linear relation. Thefollowing extension for the longitudinal slip stiffnessKxκ is implemented in the Magic Formula:

Kxκ = Fz(pKx1 + pKx2dfz)epKx3dfz(1 + ppx1dpi + ppx2dp2i ), (3.3)

where the p’s are Magic Formula parameters and dfz is the vertical load increment. In [12] it is con-cluded that for the longitudinal friction coefficient in the measurements an optimal inflation pressureoccurs. Furthermore, it is concluded that the behaviour is similar to that of the peak lateral friction.The proposed equation for the longitudinal friction coefficient µx, is as follows:

µx = (pDx1 + pDx2dfz)(1 + ppx3dpi + ppx4dp2i ). (3.4)

The conclusions in [12] are supported by the findings of Marshek and Cuderman [25]. In [25] the effectof inflation pressure on deceleration for a series of emergency braking tests with different vehiclesequipped with anti-lock braking systems is investigated. The conclusion in [25] with respect to infla-tion pressure is that the effect on the braking performance is only slight for large tyres. For smallertyres there is an optimal inflation pressure, higher and lower inflation pressures lead to a decrease inlongitudinal friction coefficient.

3.1.2 Lateral force

In [12], for the lateral slip characteristic, the effect of inflation pressure changes on the cornering stiff-ness and the peak lateral friction coefficient is investigated. The conclusion with regard to the corner-ing stiffness is that an increase of inflation pressure has two counteracting effects; a lower corneringstiffness at low vertical loads and a higher cornering stiffness at high vertical loads. These effects areclearly visible in figure 3.2. The first effect is caused by the decreasing contact length as a result of theincreased vertical stiffness. A decrease of contact length results in a decrease of cornering stiffness.Secondly, an increase of inflation pressure leads to an increase in carcass stiffness, i.e. a decrease ofcarcass compliance. This leads to higher lateral force for the same slip angle, which results in a highercornering stiffness at high vertical loads. These effects have also been observed by van Erp and Verhoeff[14] in the TNO TEK-tyre project.

3.1 Force and moment characteristics 13

0 2000 4000 6000 80000

200

400

600

800

1000

1200

1400

Fz [N]

Cor

nerin

g S

tiffn

ess

[N/d

eg]


0 2000 4000 6000 80000

200

400

600

800

1000

1200

1400

Fz [N]

Cor

nerin

g S

tiffn

ess

[N/d

eg]

P = 2.0 [bar]P = 2.4 [bar]P = 2.8 [bar]

Figure 3.2: Inflation pressure influence on cornering stiffness: 185/60 R14 tyre [12].

To include the inflation pressure effects in the Magic Formula, two linear relations depending on ainflation pressure increment dpi are added:

Kyα = pKy1(1 + ppy1dpi)Fz0 sin[2 arctan

Fz

pKy2(1 + ppy2dpi)Fz0

], (3.5)

where Ky,α is the cornering stiffness, the p’s are Magic Formula parameters, Fz and Fz0 are theactual vertical load and the nominal vertical load respectively. With regard to the peak lateral frictioncoefficient µy , the force and moment measurements did not show a clear general relation for theinflation pressure. Some tyres show a minimum peak lateral friction coefficient at low vertical loadsthat shifts to an optimum at high vertical loads. Other tyres show an opposite behaviour. To be ableto fit the minimum/maximum a second order polynomial inflation pressure relation was proposed in[12]:

µy = (pDy1 + pDy2dfz)(1 + ppy3dpi + ppy4dp2i ). (3.6)

3.1.3 Aligning moment

The aligning moment shows a clear influence of the inflation pressure. A lower inflation pressureshows a higher peak in the aligningmoment characteristic, as can be observed in figure 3.3. An increaseof inflation pressure results in higher vertical stiffness, reflecting in a shorter contact length and ina smaller pneumatic trail. The pneumatic trail is the relation of aligning moment and lateral force,and is linearly dependent on the contact length [28]. The change in contact length is relatively largecompared to the changes of lateral force, which causes the aligning moment to decrease with theinflation pressure. In the Magic Formula, the aligning moment is calculated by multiplying the lateralforce with the pneumatic trail [28]. In [12] it is suggested to add a linear relation to the parameter Dt,which determines the magnitude of the pneumatic trail:

Dt = Fz( r0Fz0

)(qDz1 + qDz2dfz)(1− qpz1dpi)sgn(Vcx). (3.7)

In this equation the q’s are parameters, r0 the tyre free radius and Vcx the forward velocity of thecontact centre.


−20 −10 0 10 20−150

−100

−50

0

50

100

150

Alpha [deg]

Mz [N

m]

−20 −10 0 10 20−150

−100

−50

0

50

100

150

Alpha [deg]

Mz [N

m]


P = 2.0 [bar]P = 2.4 [bar]P = 2.8 [bar]

Figure 3.3: Aligning moment: 155/65 R15 tyre [12].

3.1.4 Vertical stiffness

In [32] an approach is presented to describe the inflation pressure effects on the vertical stiffness. Thevertical stiffness is obtained from the following empirical relation describing the vertical force Fz as afunction of the vertical deflection ρz and the inflation pressure increment dpi:

Fz = (1 + qFz3dpi)(qFz1ρz + qFz2ρ2z). (3.8)

The qFz ’s are fit-parameters. It is assumed that the relation between the inflation pressure and thevertical stiffness is linear. The parameter qFz3 indicates the ratio of the vertical stiffness due to inflationpressure changes with the nominal pressure. The expression for the vertical stiffness can be derivedusing (3.8):

CFz =dFzdρz

= (1 + qFz3dpi)(qFz1 + 2qFz2ρz). (3.9)

3.2 Empirical relations for aircraft tyres

In 1958 a study was conducted by Smiley and Horne [33] to derive empirical equations for pneumaticaircraft tyres for landing gear design. The variables used in these equations are in general the tyrewidth, tyre radius, tyre deflection and inflation pressure. The results give a relevant indication of theeffect of inflation pressure on the mechanical properties of pneumatic tyres. However, aircraft tyreshave a different construction and are used under different operating conditions than tyres in the au-tomotive industry; this should be taken into account when interpreting these results. It is reasonableto assume that the trends described below are not (completely) corresponding with the behaviour of apassenger car tyre, but still the results of the aircraft tyres give a good perception of the influence ofthe inflation pressure on the different characteristics of a pneumatic tyre. In [33] the inflation pressureinfluence is described for a lot of different tyre characteristics. In this section, only the characteristicsin which the literature on passenger car tyres does not provide a clear answer are presented. The char-acteristics described below are: the longitudinal stiffness, the lateral stiffness, the torsional stiffnessand the tyre relaxation behaviour.

3.2.1 Longitudinal stiffness

When a constant vertical force Fz is subjected to a longitudinal force Fx, the tyre experiences a cor-responding longitudinal deformation and an additional vertical sinking δx besides the initial vertical

3.2 Empirical relations for aircraft tyres 15

deflection δ0. Also a longitudinal shift of the vertical force centre of pressure location occurs. Tyre datashows that the longitudinal stiffness tends to increase with increasing vertical tyre deflection and toincrease only slightly with increasing inflation pressure, see figure 3.4. In according to [33], it seemsreasonable to expect that the longitudinal stiffness can be described by an equation of the type:

Kx = k1d(pi + k2pi,r)f(δ0/d), (3.10)

where k1 and k2 are numerical constants and d is the outside diameter of an free tyre.

0 1 2 3 4 5 6 70

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2x 10

4

Vert. deflection [in.]

Long

itudi

nal s

prin

g co

nsta

nt, K

x [lb/

in.]

pi/p

i,r=0.7

pi/p

i,r=0.8

pi/p

i,r=1.0

pi/p

i,r=1.1

Figure 3.4: Longitudinal stiffness variation with vertical deflection and inflation pressure [33].

It appears from experimental data that f(δ0/d) = 3√δ0/d and that the numerical values for k1=0.5...1.2

and k2=4 [33]. Upon application of a longitudinal force, a standing tyre sinks vertically through thesmall distance δx which is according to experimental data approximately 10 percent of the longitudinaldeflection.

δx = 0.1|λx|. (3.11)

3.2.2 Lateral stiffness

When a lateral force is applied to a vertically loaded tyre with a vertical force Fz , an inflation pressurepi and an initial vertical deflection δ0, the tyre experiences a corresponding lateral deformation λ0. Fur-thermore a vertical sinking δλ and a lateral shifting of the vertical force centre of pressure location willappear. The vertical force and the inflation pressure appear to be the primary variables influencing thevariation of lateral stiffness. Furthermore, the lateral stiffness is slightly influenced by the amplitudeof the force-deflection hysteresis loop. In figure 3.5 the lateral stiffness versus vertical deflection is de-picted for several inflation pressures. The lateral deflection can be predicted by and empirical equationof the type:

Kλ = τλw(pi + 0.24pi,r)[1− (0.7δ0w

)]. (3.12)

The quantity 0.24pi,r takes into account that, because of the tyre carcass stiffness, the lateral stiffnessat zero inflation pressure is not zero. This quantity may be regarded as an effective lateral carcasspressure. pi,r is the rated inflation pressure (1/4 of the bursting pressure) and w the width of theundeflected tyre. The lateral spring coefficient τλ is estimated from a large number of lateral stiffnessdata of aircraft tyres. The numerical value of coefficient τλ is approximately between 2 and 3. Whenapplying a lateral force, a tyre sinks vertically through a small distance δλ which can be representedas a fraction, according to experimental data approximately 15 - 21 percent, of the absolute value of thecorresponding lateral deflection λ0:

δλ = (0.15...0.21)|λ0|. (3.13)


0 0.5 1 1.5 2 2.5 3 3.5 40

500

1000

1500

2000

2500

3000

Vert. deflection, δ0 [in.]

Late

ral s

prin

g co

nsta

nt, K

λ [lb/

in.]

pi,r

=24 lb/in2

pi,r

=32 lb/in2

pi,r

=40 lb/in2

pi,r

=80 lb/in2

Figure 3.5: Lateral stiffness variation with vertical deflection and inflation pressure [33].

3.2.3 Torsional Stiffness

In figure 3.6 an illustration of the effects of vertical deflection and inflation pressure on the torsionalstiffness of aircraft tyres [33] is given. The dotted lines represent the inflation pressure trend lines forconstant vertical load. The figure indicates that the torsional stiffness increases approximately linearlywith both increasing inflation pressure and vertical deflection. In [33] the following type of empiricalequation is proposed to derive the torsional stiffness for aircraft tyres:

Kα = (pi + 0.8pi,r)w3250( δ0

2r0

)2 δ02r0

≤ 0.03

(3.14)

Kα = (pi + 0.8pi,r)w315[( δ0

2r0

)− 0.015

] δ02r0

≥ 0.03,

where Kα is the torsional spring constant, pi the current inflation pressure, δ0 the vertical deflectionof the pure vertical loading conditions and r0 the unloaded tyre radius.

Figure 3.6: Torsional stiffness aircraft tyre [33].

3.3 Rolling Resistance 17

3.2.4 Tyre relaxation behaviour

The static relaxation length (here: the lateral deformation of a non rolling tyre) is depending on theinflation pressure and vertical deflection. In [33] it is concluded that the static relaxation length de-creases with increasing inflation pressure and with increasing vertical deflection. For the aircraft tyresthe following type of empirical equation is derived:

σst = w(2.8− 0.8

pipi,r

)(1.0− 4.5

δ02r0

), (3.15)

in which σst is the static relaxation length, w is the width of the undeflected tyre, pi is the currentinflation pressure and pi,r is the rated inflation pressure (1/4 of the bursting pressure). Furthermoreδ0 is the vertical deflection of the pure vertical load conditions and r0 is the unloaded tyre radius. Thesame type of empirical equation holds for the unyawed rolling relaxation length.

For the yawed rolling relaxation length (i.e. the relaxation length of the tyre when a sideslip angle isapplied), the tyre builds up a lateral force which exponentially approaches an end-point condition forsteady yawed rolling. The lateral force Fy,r builds up with distance x rolled according to a relation ofthe form:

Fy,r = Fy,r,e −A1e−x/σ, (3.16)

where, Fy,r,e is the steady-state force, A1 is a constant which depends on the initial tyre deflection andσ is called the yawed rolling relaxation length (lateral relaxation length in the rest of the report). Therolling relaxation length can be defined with the same parameters as the static relaxation length, by thefollowing type of empirical equations:

σ = w(2.8− 0.8

pipi,r

)(11

δ02r0

) δ02r0

≤ 0.053

σ = w(2.8− 0.8

pipi,r

)(64

( δ02r0

)− 500

( δ02r0

)2

− 1.4045)

0.053 ≤ δ02r0

≤ 0.068 (3.17)

σ = w(2.8− 0.8

pipi,r

)((0.9075− 4

δ02r0

)) δ02r0

≥ 0.068

For aircraft tyres the relaxation length is approximately within the range of 2a to 3.5a, where a is halfthe contact length. For passenger car tyres this is quite different. In [19] it is indicated that for passengercar tyres the relaxation length is approximately five times the contact length or more (σ > 10a).

3.3 Rolling Resistance

Collier and Warchol (1980) [10] evaluated Bias, Bias-belted and Radial tyre performance on five dif-ferent inflation pressures with different vertical load conditions. The results, presented in figure 3.7,indicate that changing the inflation pressure has a lot of influence on the rolling resistance of a tyre.The nominal vertical load in figure 3.7 is Fz0 =6 kN. For a given vertical load condition, the tyre appearsto have a lower rolling resistance at increasing inflation pressure. Above 2.75 bar inflation pressure,the rolling resistance levels off rapidly.

SAE uses a standard (J2452) to model rolling resistance frommeasurements into vehicle models. Thismodel is developed and proposed by Grover [17]. The standard requires that the rolling resistance force,applied vertical load and tyre inflation pressure are measured at several velocities and for differentload/pressure cases. A minimum of six velocity steps at each load/pressure case is required. Withthe gathered measurement data, a rolling resistance force model is developed of the form Fx,RR =f(Load, Pressure, V elocity), which serves as a mathematical characterisation of the rolling resistance


1 1.5 2 2.5 3 3.570

80

90

100

110

120

130

140

Inflation pressure [bar]

% R

ollin

g R

esis

tanc

e

100% index = 1.65 bar

1.2F

z0

1.0Fz0

0.8Fz0

0.6Fz0

0.5 0.6 0.7 0.8 0.9 1 1.1 1.270

80

90

100

110

120

130

140

% nom. vertical load

% R

ollin

g R

esis

tanc

e

100% index = 1.65 bar

1.10 bar1.65 bar2.20 bar2.75 bar3.30 bar

Figure 3.7: Rolling resistance vs. vertical load and inflation pressure [10].

Figure 3.8: Rolling Resistance force: 195/70 R14 tyre (two different tyres) [18].

force in a range of operating conditions (load, pressure, velocity). Two relations, (3.18) and (3.19), canbe used with a difference in velocity dependency:

Fx,RR = K · PαZβV γ (3.18)

Fx,RR = PαZβ(a+ bV + cV 2), (3.19)

where Fx,RR is the rolling resistance force, K a constant, P the inflation pressure, Z the vertical loadand V the velocity. Furthermore α, β, γ and a, b, c are regression exponents and regression coefficientsrespectively. (3.18) and (3.19) can be seen as a signature of the rolling resistance performance of thetyre over the operating conditions. Besides that the models can be used for calculating the tyre rollingresistance at other load, pressure and velocity conditions they also can be used for fuel economy calcu-lations and measurements. Typical output is a set of Fx,RR as function of velocity curves for differentload/pressure conditions, as shown in figure 3.8. The figure shows a clearly visible influence of infla-tion pressure on the rolling resistance (see: (1) 536 kg, 2.0 bar and (2) 536 kg, 3.0 bar). It can be observedthat a decrease of inflation pressure results in an increase of the rolling resistance with velocity curve.

3.4 Eigenfrequencies 19

Michelin presented in 2005 their own rolling resistance force model [27], based on the SAE model(3.18). The model serves as the rolling resistance characterisation for two different operating condi-tions, namely; the inflation pressure pi and the vertical load Fz . The velocity effects are left out byMichelin. The model is described as follows:

Fx,RR = K · pαi (Fz)β , (3.20)

where Fx,RR is the rolling resistance force, K is a constant for a given tyre and α, β are regressionexponents. According to [27], the values of the regression exponents are approximately: α ≈ −0.4and β ≈ 0.85 for a passenger car tyre, and for a truck tyre designed for motorway use: α ≈ −0.2and β ≈ 0.9. Furthermore, the following representation of (3.18) is presented in [27] to determinethe rolling resistance force at different operating conditions according to the ISO 8767 measurementprotocol [20]:

Fx,RR = Fx,RR−ISO ·( pipi,ISO

)α( FzFz,ISO

)β, (3.21)

in which, Fx,RR−ISO is the rolling resistance force (measured) at the nominal inflation pressure andnominal vertical load according to ISO 8767, pi,ISO is the ISO nominal inflation pressure (2.1 bar)and Fz0 is the ISO nominal vertical load condition (80% of the maximum vertical load of the tyreFz,max). The terms pi and Fz are the actual inflation pressure and the vertical load.

3.4 Eigenfrequencies

Yam, Guan and Zhang [39] studied the three-dimensional mode shapes obtained under radial and tan-gential excitation. In their study they used a free suspension, a single-point excitation and multi-point(16) recorder to eliminate the influences and shortcomings of a fixed support. The natural frequencyof the free suspension (1 Hz) is much lower than the first natural frequency of a tyre with free rim(around 100 Hz). Therefore, the influence of the free suspension is negligible. The measurements areperformed for four inflation pressures and the tyre is excitated with a swept frequency of 80 - 380 Hz.For the responses in radial, tangential and lateral direction due to a radial excitation, the modes in thedifferent directions have nearly the same modal eigenfrequencies and mode shapes according to [39].The first obtained mode has an elliptical shape. The following modes have a (multiple) leaves shape,except for mode six which shows a translational mode with the rim moving in opposite direction tothe tyre movement. With respect to the inflation pressure influences in [39], it is concluded that for allmodes, except mode six, the natural frequencies increase with inflation pressure (figure 3.9). The sixthmode reflects the intrinsic characteristics of the tyre structure and is independent of inflation pressurechanges. Furthermore, the vibration amplitudes of mode six are about 25% of the amplitudes of theother modes and the modal damping ratio is less than 10% of the other modes [39]. As can be observedin figure 3.9 the first modal eigenfrequencies (mode 1, 2 and 3) are linearly related with the inflationpressure. The higher modes show a non-linear relation. For the modal damping a non linear relationwith the inflation pressure is visible. Furthermore, it can be seen that increasing the inflation pressureresults in a decrease of the modal damping. Again for the sixth mode no influence can be observed.

Because all the investigated modes are flexible modes, only the trend of the first mode (nearest tothe rigid ring modes) is interesting for this research project. As mentioned before, the MF-Swift tyremodel can only handle the modes where the tyre tread band retains its circular shape. This means thatno flexible modes can be analysed. From [39] it can be concluded that there probably will be a lineartrend in the rigid eigenmodes.


1.5 2 2.5 350

100

150

200

250

300

350

Inflat. pressure [bar]

Mod

al fr

eque

ncy

[Hz]

mode 1mode 2mode 3mode 4mode 5mode 6mode 7mode 8mode 9

1.5 2 2.5 30

0.5

1

1.5

2

2.5

3

3.5

4

Inflat. pressure [bar]

Dam

ping

[%]

Figure 3.9: Influence inflation pressure on modal frequency (left) and modal damping (right) [39].


In previous chapter the MF-Swift tyre model and the modelled tyre characteristics are discussed inmore detail. In this chapter the results are presented of a literature survey on the influence of the in-flation pressure on the tyre characteristics that are modelled. The literature survey shows that all theinvestigated tyre characteristics are inflation pressure dependent. Furthermore, it is shown that theinflation pressure influence on the steady-state slip characteristics (Fx, Fy and Mz) is already imple-mented in the MF-Swift tyre model. Although the results of the aircraft tyres give a good impression ofthe influence of the inflation pressure on different tyre characteristics, still it is reasonable to assumethat the trends of passenger car tyres are different due to construction and application area differences.

The data available in literature appears to be insufficient to describe the inflation pressure influence ofall the tyre characteristics that are modelled, the following characteristics need more investigation:

• characteristic tyre stiffnesses:

– vertical stiffness,

– lateral stiffness,

– longitudinal stiffness,

– torsional stiffness.

• sideslip relaxation behaviour;

• rolling resistance;

• camber effects;

• eigenfrequencies of the tyre belt.

These characteristics are investigated with an extensive measurement program with a common pro-duction passenger car tyre. The results of this measurement program are presented in the next chapter.

Chapter 4

Inflation Pressure SensitivityExperiments

In the previous chapter an overview of the published knowledge on the influence of inflation pres-sure is given. It is shown that the inflation pressure has significant influence on several (dynamic)tyre characteristics modelled in the MF-Swift tyre model. In order to get a better understanding ofthe influence of the inflation pressure on the different tyre characteristics, an elaborate measurementprogram has been conducted. The results of the measurements are presented in this chapter. In thefollowing chapter these results are used to derive inflation pressure dependent relations to improvethe MF-Swift tyre model. In order to be as consistent as possible the measurements within this thesishave been performed with one single type of tyre; a 225/50 R17 normal production passenger car tyre.The experiments are conducted with a relatively new tyre with no wear. TNO Automotive has executedrecently an extensive measurement program with this tyre on the new TNO Tyre Test Trailer in orderto characterise the steady-state force and moment characteristics. These results are also used withinthis research project. In addition to the high velocity measurements of the TNO Tyre Test Trailer ameasurement program for very low velocity has been set up and conducted on the TU/e Flatplank TyreTester. The Flatplank Tyre Tester has been chosen as it operates at very low velocities in comparison tothe TNO Tyre Test Trailer. Furthermore the forward velocity and the rotational velocity are measuredwith a higher accuracy compared to the TNO Tyre Test Trailer.

To determine to what extent the inflation pressure has influence on the different tyre characteristicsan expression is used to classify the difference between a certain inflation pressure and the nominalinflation pressure. The following expression for the influence level is used:

εA = 100

√√√√√√√√N∑i=1

(Apress,i −Apressnom,i)2

N∑i=1

(Apressnom,i)2. (4.1)

In this equation A is the tyre characteristic quantity (e.g. lateral stiffness) and N is the number ofmeasurement points. The subscripts press and pressnom are used to denote the measured inflationpressure and the nominal inflation pressure respectively. The nominal inflation pressure for the tyre is2.5 bar (cold tyre). The inflation pressure influence level expression (4.1) is a measure for the influenceof the inflation pressure on the specific tyre characteristic. A large influence expression implies a largeinfluence of the inflation pressure, a small influence expression implies little influence of the inflationpressure.

In general the experiments are performed at three different vertical load conditions (FzW= 3, 5, 7 kN)and five different inflation pressures (pi= 1.9, 2.2, 2.5, 2.7, 3.0 bar). These specific inflation pressures

21

22 Inflation Pressure Sensitivity Experiments

are chosen because earlier measurements performed on the TNO Tyre Test Trailer were carried outwith these inflation pressures. The inflation pressure influence levels presented in this chapter are themean values for all the load cases for an inflation pressure change of 0.5 bar.

The test tyre is marked at three (start)positions on its circumference, see figure4.1. Each measurementis performed at all start positions of the tyre to reduce the influence of tyre nonuniformities. TheFlatplank is marked so that the measurements are always started at the same point. Furthermore, adigital trigger unit is present at this point, which is used to start the sampling of all measurementchannels when the Flatplank is operated. More information on the Flatplank Tyre Tester is presentedin Appendix A.

Figure 4.1: Example start positions tyre and Flatplank.

4.1 Stiffness measurements

In the previous chapter it is shown that the literature does not provide an unambiguous answer to theinfluence of the inflation pressure on the stiffnesses of a passenger car tyre. In order to get a betterunderstanding of the inflation pressure influence, a number of experiments are performed on theFlatplank Tyre Tester. The following stiffnesses are investigated:

• tyre vertical stiffness CFz ;

• tyre lateral stiffness CFy ;

• tyre longitudinal stiffness CFx;

• tyre torsional stiffness CMz .

4.1.1 Tyre vertical stiffness

In order to determine the tyre vertical stiffness a measurement procedure has been set up. The tyre ispressed vertically against the Flatplank till a vertical deflection is reached that corresponds to a verticalload of 9 kN at the tested inflation pressure condition. The experiment is performed for five inflationpressure conditions (pi= 1.9, 2.2, 2.5, 2.7, 3.0 bar). For each inflation pressure, the measurement isrepeated three times to reduce the influence of tyre nonuniformities. The main steps in the measure-ment procedure are:

1. inflate the tyre to the prescribed pressure;

2. rotate the tyre to a start position and place it perpendicular to the start position on the flat plank;

4.1 Stiffness measurements 23

3. start moving the Flatplank;

4. apply maximum vertical load, with constant velocity, till the desired vertical deflection is reached;

5. decrease vertical load, with the same constant velocity, till zero vertical load remains.

To obtain a trend in the vertical stiffness, a second order polynomial is fitted through each separatemeasurement. Finally, the coefficients of the final polynomial are determined by interpolating the co-efficients of the three separated polynomials. A graphical representation of the vertical load and thevertical deflection is presented in figure 4.2.

0 5 10 15 20 25 30 350

1000

2000

3000

4000

5000

6000

7000

8000

9000

10000

Vertical deflection, ρz [mm]

Ver

tical

load

, Fz [N

]

p19p22p25p27p30

Figure 4.2: Vertical load vs. vertical deflection.

With the measured vertical load in the contact patch centre FzW and the vertical deflection ρz thevertical stiffness is obtained with:

CFz =∂FzW∂ρz

. (4.2)

To give a graphical representation of the vertical stiffness as function of the inflation pressure, thevertical stiffness is determined for the three general vertical load conditions (FzW= 3, 5 and 7 kN),see table 4.1. In figure 4.3 the results of table 4.1 are graphically presented, and a trend line is added tovisualise the relation between the inflation pressure and the vertical stiffness. It can be seen that there isa linear relation between the inflation pressure and the vertical stiffness, the vertical stiffness increaseswith increasing inflation pressure. The measurement results are in accordance with the findings in[32]; it is concluded that a linear relation between the inflation pressure and the vertical stiffness exists,see subsection 3.1.4 and relation(3.9).

Table 4.1: Tyre vertical stiffness for the different vertical load conditions.

Vertical stiffness CFz [N/m]Infl. Pressure [bar] FzW = 3 kN FzW = 5 kN FzW = 7 kN

1.9 198400 210700 2223002.2 217500 232100 2459002.5 238800 251700 2641002.7 257000 270400 2831003.0 271200 285000 298300

An inflation pressure change of 0.5 bar will lead to a change of approximately 13 percent in verticalstiffness value.


1.8 2 2.2 2.4 2.6 2.8 31.8

2

2.2

2.4

2.6

2.8

3

3.2x 10

5

Inflation pressure, pi [bar]

Ver

tical

stif

fnes

s, C

Fz [N

/m]

Fz=3000NFz=5000NFz=7000NTrend line

Figure 4.3: Vertical stiffness vs. inflation pressure.

4.1.2 Tyre longitudinal stiffness

To determine the tyre longitudinal stiffness CFx, the rotation of the test tyre is "locked-up", meaningno rotation is possible around the y-axis. The tyre is pressed vertically against the road surface with 0degree sideslip angle. The experiment is performed at three different vertical load conditions (FzW=2, 4, 6 kN) and for five inflation pressure conditions. The vertical load conditions differ from thegeneral load conditions to protect the measurement hub against overloading. The main steps in themeasurement procedure are:


2. rotate the tyre to a start position, place it perpendicular to the start position on the Flatplank andlock up the measuring hub;

3. apply prescribed vertical load (fix the axle height);

4. start moving the Flatplank.

The longitudinal stiffness is determined from the linear part of the relation between the longitudinalforce and longitudinal displacement. In figure 4.4 the results are presented for the nominal verticalload (4 kN). In the beginning of the experiments some starting phenomena can arise; the linear partis selected in a way that these starting phenomena are not taken into account. For determining thelongitudinal stiffness, a first order polynomial fit is used to characterise the linear part, see figure 4.4(right). For the results of the remaining vertical load conditions see Appendix B.

In general, the longitudinal stiffness is defined as the derivative of the longitudinal force versus thelongitudinal tyre deflection dx. Here, the longitudinal stiffness CFx represents the slope of the linearfit function FxW (dx):

CFx =∂FxW∂dx

∣∣∣∣dx=0

. (4.3)


0 0.01 0.02 0.03 0.04 0.05 0.06 0.070

1000

2000

3000

4000

5000

6000

7000

8000

9000

10000

Longitudinal displacement [m]

Long

itudi

nal f

orce

, Fx [N

]

p19p22p25p27p30

0 0.01 0.02 0.03 0.04 0.05 0.06 0.070

2000

4000

6000

8000

10000

12000

14000

16000

18000

Rel. longitudinal displacement [m]

Long

itudi

nal f

orce

, Fx [N

]

p19p22p25p27p30

Figure 4.4: Experimental results at nominal vertical load (left) and linear fits of the selected linear part(right).

Figure 4.4 shows that the slope of the fit increases with increasing inflation pressure. In figure 4.5the longitudinal stiffness versus the inflation pressure is depicted for the different vertical load con-ditions. It can be seen that a higher inflation pressure results in a higher longitudinal stiffness andthat according to the trend lines a linear relation between the inflation pressure and the longitudinalstiffness exists. This linear relation is also found in the literature of aircraft tyres, see subsection 3.2.2and relation (3.10).

1.8 2 2.2 2.4 2.6 2.8 31.7

1.8

1.9

2

2.1

2.2

2.3

2.4

2.5x 10

5


Long

itudi

nal s

tiffn

ess,

CF

x [N/m

]


Figure 4.5: Tyre longitudinal stiffness vs. inflation pressure.

An inflation pressure change of 0.5 bar will lead to a change of approximately 3.5 percent in longitudi-nal stiffness value. Again, The inflation pressure influence level expression is the mean influence levelof all the load cases.


Table 4.2: Tyre longitudinal stiffness for the different vertical load conditions.

Longitudinal stiffness CFx [N/m]Infl. Pressure [bar] FzW = 2 kN FzW = 4 kN FzW = 6 kN

1.9 173500 214300 2248002.2 179000 218200 2297002.5 183400 226600 2362002.7 188900 230800 2386003.0 190500 233500 244300

4.1.3 Tyre lateral stiffness

To determine the tyre lateral stiffness, the measuring hub is locked up and the tyre is placed perpen-dicular to the Flatplank, meaning that the sideslip angle α is 90 degrees. The tyre is pressed verticallyagainst the Flatplank, so no camber angle is applied. The experiment is performed for the general ver-tical load and inflation pressure conditions as defined at the beginning of this chapter. The steps in themeasurement procedure are mainly the same as the steps in the longitudinal stiffness procedure.

0 0.01 0.02 0.03 0.04 0.05 0.06 0.070

1000

2000

3000

4000

5000

6000

7000

8000

Lateral displacement [m]

Late

ral f

orce

, Fy [N

]

p19p22p25p27p30

0 0.01 0.02 0.03 0.04 0.05 0.06 0.070

1000

2000

3000

4000

5000

6000

7000

8000

Rel. lateral displacement [m]

Late

ral f

orce

, Fy [N

]

p19p22p25p27p30

Figure 4.6: Tyre lateral stiffness, experimental results at nominal vertical load (left) and linear fits of theselected linear part (right).

In figure 4.6 (left) the results of the measurements are presented for the nominal vertical load condi-tion (FzW=5 kN), the results of the remaining vertical load conditions are presented in Appendix B. Itcan be seen that at several inflation pressures, when the steady state value of the lateral force is reached,fluctuations occur. These fluctuations have no effect on the determination of the lateral stiffness andare the result of stick-slip situations that arose during several measurements. For the determinationof the lateral stiffness the linear part of the graphs is used leaving out starting phenomena. A linearpolynomial fit is used to characterise the linear part and eliminate possible small fluctuations in themeasurement data, see figure 4.6 (right).

The lateral stiffness is defined as the derivative of the lateral force versus the lateral tyre deflectiondy . This means that the tyre lateral stiffness CFy is represented by the slope of the linear fit functionFyW (dy);

CFy =∂FyW∂dy

∣∣∣∣dy=0

. (4.4)


Figure 4.6 (right) shows that the slope of the linear fit increases when increasing the inflation pressure.

1.8 2 2.2 2.4 2.6 2.8 31.15

1.2

1.25

1.3

1.35

1.4

1.45

1.5

1.55

1.6

1.65x 10

5


Late

ral s

tiffn

ess,

CF

y [N/m

]


Figure 4.7: Lateral stiffness vs. inflation pressure.

Table 4.3: Tyre lateral stiffness for the different vertical load conditions.

Lateral stiffness CFy [N/m]Infl. Pressure [bar] FzW = 3 kN FzW = 5 kN FzW = 7 kN

1.9 116900 124900 1292002.2 123400 136000 1391002.5 127200 143200 1474002.7 132600 148800 1528003.0 136700 156900 166000

The relation between the inflation pressure and the lateral stiffness is visualised in figure 4.7. Asobserved for the longitudinal stiffness, the lateral stiffness and the inflation pressure show a linearrelation and increasing the inflation pressure leads to an increase in lateral stiffness. This correspondswith the empirical relation for the lateral stiffness found in [33]. Table 4.3 presents the values of thelateral stiffnesses for the different vertical load conditions. An inflation pressure change of 0.5 bar willlead to a change of approximately 9.5 percent in lateral stiffness value.

4.1.4 Tyre torsional stiffness

To determine the tyre torsional stiffness, parking measurements are executed on the Flatplank. Theparking measurements are performed by turning the non-rolling wheel (Vx=0, plank is not moved)around the vertical axis of the tyre from 0 via 20 to -20 and back to 20 degrees.The linear part of the parking measurements can be used to determine the torsional stiffness of thetyre. The torsional stiffness CMz is the stiffness that can be derived from the derivative of the aligningmomentMzW versus the steer angle ψ:

CMz =∂MzW

∂ψ

∣∣∣∣ψ=0

. (4.5)

The parking experiments are performed for the five defined inflation pressures, but now for a widerange of vertical load conditions; FzW=1, 2, 3, 4, 5, 6 kN.


The development of the aligning moment Mz due to the steering motion is shown in figure 4.8, de-picted are the vertical load conditions FzW=2 kN and FzW=4 kN. For the remaining vertical load con-ditions, see Appendix B. The graphs are plotted at different y-axis scaling for readability and visibility ofthe differences between the separate inflation pressures. The slope of the linear part is representativefor the size of the torsional stiffness.

−25 −20 −15 −10 −5 0 5 10 15 20 25−60

−40

−20

0

20

40

60

Steer angle, ψ [deg]

Alig

ning

mom

ent,

Mz [N

m]

p19p22p25p27p30

−25 −20 −15 −10 −5 0 5 10 15 20 25−200

−150

−100

−50

0

50

100

150

200

Steer angle, ψ [deg]A

ligni

ng m

omen

t, M

z [Nm

]

p19p22p25p27p30

Figure 4.8: Parking behaviour, experimental results aligning moment at FzW =2 kN (left) and FzW =4 kN(right).

In figure 4.9 the determined torsional stiffness versus the inflation pressure is presented. It can beseen that a more or less linear relation between the inflation pressure and the aligning moment exists.At a vertical load of 1 kN the linear relation has a slightly increasing character. At increasing verticalload the character becomes more and more decreasing with increasing inflation pressure. Table 4.4shows the numerical values of the torsional stiffness for the different vertical load and inflation pres-sure conditions.

1.8 2 2.2 2.4 2.6 2.8 30

1000

2000

3000

4000

5000

6000

7000

8000

9000

10000


Tor

sion

al s

tiffn

ess,

CM

z [N

m/r

ad]

Fz=1 kNFz=2 kNFz=3 kNFz=4 kNFz=5 kNFz=6 kNTrend line

Figure 4.9: Torsional stiffness vs. inflation pressure.

In table 4.4 the influence of the inflation pressure on the torsional stiffness is presented. An infla-tion pressure change of 0.5 bar results in a change of approximately 12 percent. The accuracy of thesevalues can be questioned. It is expected that the torsional stiffness will decrease with increasing in-flation pressure, but for some vertical load conditions the measurement points do not always followthis decreasing trend. By combining the trends predicted in aircraft tyre literature and results obtained

4.2 Lateral relaxation behaviour 29

Table 4.4: Torsional stiffness for the different vertical load conditions

Torsional stiffness CMz [Nm/rad]Infl. Press [bar] 1 kN 2 kN 3 kN 4 kN 5 kN 6 kN

1.9 960 2320 4110 5160 8080 92402.2 950 2810 3960 5690 7200 88702.5 830 2120 3420 4780 6320 81702.7 880 2360 3690 4960 6750 79903.0 1180 2030 3300 4930 6970 7780

from the Flatplank experiments it is possible to make a verdict about the relation between the torsionalstiffness and the inflation pressure in the next chapter.

4.2 Lateral relaxation behaviour

In a physical model, the tyre lateral stiffness CFy and the cornering stiffness CFα of a tyre determinefor the lateral relaxation length σy , see (4.6). When performing a first order dynamic system identifi-cation, the relaxation length is defined as the distance travelled where 63% of the steady state lateralforce value is reached.

σy = −CFαCFy

. (4.6)

In order to determine the lateral relaxation length, measurements are performed with a constant axleheight and with 0 and 1 degree sideslip angle. The experiments are performed for the general verticalload and inflation pressure conditions. The main steps of the measurement procedure are:


2. rotate the tyre to a start position and place it perpendicular to the start position on the Flatplank;

3. apply the desired sideslip angle first and after that the vertical load (fix the axle height);

4. start moving the Flatplank.

The measurements with 0 degree sideslip angle are used as reference measurements to correct for ply-steer and conicity influences of the tyre. Furthermore, the forces andmoments due to non-uniformitiesin the tyre structure can be monitored. By subtracting the reference results from the 1 degree sideslipangle results, the tyre (structure) phenomena as mentioned above can be eliminated and the purerelaxation behaviour becomes visible. In figure 4.10 (left) the Flatplank results of the nominal verticalload condition are depicted.

The lateral relaxation length from the measurement results is determined using the dynamic systemidentification method (i.e. 63% of the steady state lateral force). In figure 4.10 (right) the resultinglateral relaxation lengths as function of the inflation pressure for three vertical load conditions aregraphically presented. It can be seen that the lateral relaxation length decreases with increasing infla-tion pressure. The relation of the inflation pressure and the lateral relaxation length can be describedwith a linear relation. This corresponds rather well with the lateral relaxation length relations (3.17)as described in [33]. Table 4.5 shows the determined relaxation lengths for the different vertical loadconditions. It is shown that a change of 0.5 bar in inflation pressure leads to a change of approximately12 percent in lateral relaxation length.


0 0.5 1 1.5 2 2.5−2500

−2000

−1500

−1000

−500

0

Track displacment, dx [m]

Late

ral f

orce

, Fy [N

]

p19p22p25p27p30

1.8 2 2.2 2.4 2.6 2.8 3

0.4

0.5

0.6

0.7

0.8

0.9

1


Late

ral r

elax

atio

n le

ngth

, σ y [m

]


Figure 4.10: Lateral relaxation behaviour, response lateral force to one degree sideslip angle at FzW =5 kN(left) and determined lateral relaxation length (right) for various vertical loads and inflation pressures.

Table 4.5: Lateral relaxation length for the different vertical load conditions.

Lateral relaxation length σy [m]Infl. Pressure [bar] FzW = 3 kN FzW = 5 kN FzW = 7 kN

1.9 0.51 0.78 0.892.2 0.44 0.71 0.832.5 0.43 0.69 0.822.7 0.38 0.61 0.773.0 0.36 0.58 0.74

4.2.1 Effective rolling radius

An important input parameter of the Magic Formula tyre model is the longitudinal slip. For a largepart, the determination of the longitudinal slip κ is governed by the effective rolling radius Re, see(4.7) and figure 4.11.

κ = −Vx −ReΩVx

. (4.7)

According to 4.8, the effective rolling radius determines the ratio between the forward velocity Vxand the rotational speed Ω for a freely rolling tyre. So, the effective rolling radius is important for anaccurate representation of the rotational speed.

Re =VxΩ. (4.8)

For automotive tyres it is shown, in [28], that the effective rolling radius depends on the vertical loadFzW . At low vertical load conditions, small increases in vertical load condition result in a decrease ofthe effective rolling radius. The vertical load will deform the tyre rubber and the circumference of thetyre, soRe, will decrease. As the vertical load further increases the tyre will be compressed, but the highcircumferential stiffness of the steel carcass will make that the resulting effect on the effective rollingradius is getting smaller. The tyre still deforms at increasing vertical load, but the circumference, andthus Re, remains almost constant.


Figure 4.11: Effective rolling radius and longitudinal slip [13].

To investigate the influence of the inflation pressure on the effective rolling radius, the reference mea-surements of the lateral relaxation length experiments are used. These measurements represent a freerolling tyre, so without longitudinal or lateral slip. The results are shown in figure 4.12. For all thevertical load conditions it can be seen that there is very little influence of the inflation pressure on theeffective rolling radius. When increasing the inflation pressure with 1 bar this results in an increase ofapproximately only 2 mm of the effective rolling radius, which is less than 1 percent. An increase of0.5 bar inflation pressure leads to an increase of approximately 0.25 percent in effective rolling radius.

1.8 2 2.2 2.4 2.6 2.8 30.31

0.312

0.314

0.316

0.318

0.32

0.322

0.324

0.326

0.328

0.33


Effe

ctiv

e ro

lling

rad

ius,

Re [m

]

Fz=3000[N]Fz=5000[N]Fz=7000[N]Trend line

Figure 4.12: Effective rolling radius vs. inflation pressure.


Table 4.6: Effective rolling radius for different vertical load and inflation pressure conditions.

Effective rolling radius Re [m]Infl. Pressure [bar] FzW = 3 kN FzW = 5 kN FzW = 7 kN

1.9 0.3218 0.3205 0.32052.2 0.3217 0.3203 0.31992.5 0.3229 0.3220 0.32132.7 0.3227 0.3210 0.32053.0 0.3238 0.3225 0.3219

4.3 Camber thrust and camber torque

When a camber angle is applied to a free rolling tyre, a lateral force Fyγ (camber thrust), a momentaround the vertical axisMzγ (aligning camber torque) and a moment around the longitudinal axisMxγ

(overturning camber torque) arise, see figure 4.13.

Figure 4.13: Schematic representation of camber thrust and camber torque, rear view [24].

In earlier research on a flat-surface tyre-testing facility [24], it is shown that the contact length isstrongly depending on the amount of camber angle, while the contact patch area stays almost un-affected. The camber thrust and the camber torque are depending on the vertical load condition. In-creasing the vertical load results in an increase of the camber thrust and aligning camber torque, seefigure 4.14.

To determine to what extend the inflation pressure has influence on the camber thrust and aligningcamber torque, measurements are performed with a free rolling tyre for a wide range of camber angles(γ=-15...15 degrees) and for the general vertical load conditions. To eliminate ply-steer and conicityeffects, measurements are also performed with zero camber angle and subtracted from the cambermeasurements. The main steps of the measurement procedure are:


2. rotate the tyre to a start position and place it perpendicular to the start position on the Flatplank;

3. rotate the Flatplank to the desired camber angle;

4. apply vertical load, fix the axle height and start moving the Flatplank.

4.3 Camber thrust and camber torque 33

Figure 4.14: Camber thrust Fyγ and aligning camber torque Mzγ for different vertical load conditions[24].

In figure 4.15 the camber thrust and aligning camber torque versus camber angle for the nominalvertical load condition (FzW=5 kN) are depicted. Because the tested tyre has a symmetrical tread, theamount of camber thrust and aligning camber torque for positive and negative camber angles are ex-pected to be equal. However, figure 4.15 shows that a difference occurs when cambering with a positiveor a negative angle. For the camber thrust at nominal inflation pressure and nominal vertical load thedifference between +15 degrees camber angle and -15 degrees camber angle is approximately 150 N or17 percent, for the aligning camber torque this is approximately 5 Nm or 12 percent. When normalisingthe camber thrust and aligning camber torque with the vertical load the differences still exist, whichexcludes the influence of the vertical load. Further, conicity and ply-steer effects are eliminated usingthe reference measurements. The difference can be caused by a deviation or a wrong calibration of themeasuring hub or an asymmetry in the tyre carcass/construction. This is not further checked in thisresearch.

−15 −10 −5 0 5 10 15−1500

−1000

−500

0

500

1000

1500

Camber angle, γ [deg]

Cam

ber

thru

st, F

yγ [N

]

p19p22p25p27p30

−15 −10 −5 0 5 10 15−60

−40

−20

0

20

40

60

Camber angle, γ [deg]

Cam

ber

torq

ue, M

zγ [N

m]

p19p22p25p27p30

Figure 4.15: Camber thrust and aligning camber torque for the camber angle range: [-15 15] degrees.

Figure 4.16 and 4.17 show the camber thrust and aligning camber torque respectively as a function ofthe inflation pressure for a serie of positive and negative camber angles at the nominal vertical loadcondition. The trend lines show that a linear relation is visible between the inflation pressure and thecamber thrust / aligning camber torque. Again, it can be seen that the behaviour for a positive andnegative camber angle is not exactly the same. In figure 4.16, small negative camber angles (till -2.5degrees) show an opposite trend, caused by the measurement results at high inflation pressures (2.7and 3.0 bar).


The slope of the linear relations is depending on the size of the camber angle. For small camber angles(for instance: 2 degrees) the slope is significantly smaller than the slope for larger camber angles (forinstance: 15 degrees). This indicates that the camber thrust stiffness is decreasing with increasinginflation pressure. The same holds for the aligning camber torque stiffness.

1.8 2 2.2 2.4 2.6 2.8 3−1500

−1000

−500

0

500

1000

1500


Cam

ber

thru

st, F

yγ [N

]

−15.0 deg−10.0 deg−5.0 deg−2.5 deg−2.0 deg−1.5 deg−1.0 deg−0.5 deg0.5 deg1.0 deg1.5 deg2.0 deg2.5 deg5.0 deg10.0 deg15.0 degTrend line (neg)Trend line (pos)

Figure 4.16: Camber thrust vs. inflation pressure for various positive and negative camber angles for thenominal vertical load condition.

1.8 2 2.2 2.4 2.6 2.8 3−60

−40

−20

0

20

40

60


Cam

ber

torq

ue, M

zγ [N

m]


Figure 4.17: Aligning camber torque vs. inflation pressure for various positive and negative camber anglesfor the nominal vertical load condition.

Furthermore, when analysing the measurement results it appears that changing the inflation pressureof a cambered free rolling tyre has also influence on the overturning moment in the contact patch. Itis observed that when increasing the inflation pressure also the amount of overturning camber torqueincreases (here: further called the overturning camber torqueMxγ ), see figure 4.18. Again, the size ofoverturning camber torque for positive and negative camber angles is expected to be equal. However,figure 4.18 shows that a difference occurs when cambering with a positive or a negative angle. Noinfluence of the vertical load has been found and conicity and ply-steer effects are eliminated using thereference measurements.


1.8 2 2.2 2.4 2.6 2.8 3−400

−300

−200

−100

0

100

200

300

400


Cam

ber

torq

ue, M

xγ [N

m]


Figure 4.18: Overturning camber torque vs. inflation pressure for various positive and negative camberangles for the nominal vertical load condition.

To quantify the influence of the inflation pressure, table 4.7 shows the influence on the camber thrust,aligning camber torque and overturning camber torque. The influence is subdivided in two camberranges. The first range is limited to -5 and 5 degrees; this is the common range in which passenger cartyres are cambered. To quantify the influence also for the whole measurement range, the second rangeis limited to -15 to 15 degrees.

Table 4.7: Influence of the inflation pressure on camber effects for the nominal vertical load condition.

Influence inflation pressure εA [%]Camber range [deg] Fyγ Mzγ Mxγ

-5...5 12.5 8.0 7.0-15...15 15.0 15.0 7.0

When changing the inflation pressure with 0.5 bar the camber thrust increases/decreases approxi-mately with 13 percent of the thrust at nominal inflation pressure (2.5 bar). For the aligning cambertorque, 0.5 bar inflation pressure increase leads to a change of approximately 11.5 percent and for theoverturning camber torque to a change of approximately 7 percent. Note that these values are deter-mined at, and only hold for, the nominal vertical load condition. In Appendix B the results for theremaining vertical load conditions are presented. When comparing the results in Appendix B withthe results for the nominal vertical load condition it is observed that the progressiveness of the lineartrend for the camber thrust is decreasing when the vertical load increases, especially in the range [-55] degrees. In other words, the influence of the inflation pressure decreases with increasing verticalload. This observation can also be seen in the aligning and overturning camber torque, but to a smallerextent.

4.4 Eigenfrequencies

This section considers the influence of the inflation pressure on the tyre eigenfrequencies. At thestart of this project, no measurement data concerning the eigen modes of the measurement tyre wasavailable. Performing dynamic respons experiments is not an option due to costs and the availability ofa test facility. Instead, Finite Element Method (FEM) virtual experiments with a tyre model in ABAQUSare executed.


Furthermore to support the FEM analysis, the flexible ring model and the rigid ring model are analysedanalytically.

4.4.1 FEM analysis

In the FEM software ABAQUS, a three dimensional model of a passenger car tyre is available. The3D model is a 175 SR14 passenger car tyre. For more information about the used FEM tyre model it isreferred to Appendix C. With the FEM model two situations are investigated:

• a free hanging tyre with a fixed rim;

• a non-rolling loaded tyre (load: Fz=4 kN) with a fixed rim and fixed contact patch nodes.

The experiments are performed for six inflation pressures (pi= 1.5, 2.0, 2.2, 2.5, 2.7, 3.0 bar). Theeigenfrequencies are determined with the LANCZOS method [1]:(

− ω2MMN +KMN

)φN = 0, (4.9)

where, ω is the free vibration frequency, M is the mass matrix (symmetric and positive definite) andK is the stiffness matrix. Furthermore, φ is the eigenvector (mode shape) and superscripts M andN are the degrees of freedom. Because the MF-Swift tyre model consists of a rigid ring model, onlythe primary tyre modes (rigid belt/ring modes) are evaluated with the FEM experiments. Primary tyremodes are the first eigenmodes of the tyre in which the tyre tread band almost retains its circular shape.These modes can be described with the rigid ring model. Therefore these modes may also be calledrigid ring modes. The primary modes that are observed during the FEM experiments are: rotational,lateral, yaw, camber, vertical and longitudinal mode. See Appendix C for a visualisation of the differentprimary modes for both situations.

In figure 4.19 the primary modes as function of the inflation pressure for both situations are depicted.In the free hanging situation (left plot), clear linear trends are visible between the inflation pressure andthe eigenfrequencies. Note that, the yaw and camber modes and the vertical and longitudinal modesof the free hanging tyre occur at the same eigenfrequencies. The non-rolling loaded tyre situation isdepicted in the right plot of figure 4.19. Again linear trends can be observed. At high inflation pressures(pi=2.7 and 3.0 bar) some deviations occur. This is probably caused by variation in the number of nodesthat are in contact with the road.

1.5 2 2.5 340

45

50

55

60

65

70

75

80

85

Inflation presssure, pi [bar]

Nat

ural

freq

uenc

y [H

z]

rotationallateralyaw/cambervertical/longitudinal

1.5 2 2.5 340

45

50

55

60

65

70

75

80

85

90

Inflation presssure, pi [bar]

Nat

ural

freq

uenc

y [H

z]

rotationallateralyaw/camberverticallongitudinal

Figure 4.19: Primary eigenmodes free hanging tyre (left) and non-rolling loaded tyre (right).


Furthermore, it can be seen that the influence is approximately the same for each primary mode, withthe exception of the rotational mode. For the rotational mode of a free hanging tyre, the amount ofinflation pressure influence is a bit smaller compared to the other modes, see table 4.8. In table 4.8,the influence is presented for the different primary modes. In general, an increase of 0.5 bar leads toan increase of approximately 5 to 6 percent for both situations. The rotational mode of the free tyreshows little influence, because this mode mainly depends on the (torsional) carcass stiffness.

Table 4.8: The influence of the inflation pressure on the primary eigenmodes.

Influence level, increasing pi with 0.5 barAbaqus Primary eigen mode εA,0.5bar [%]

rotational 2.4Free tyre lateral 7.1

yaw, camber 6.8vertical, longitudinal 6.5

rotational 5.0Loaded tyre lateral 6.4

yaw/camber 5.0vertical 6.1

longitudinal 5.9

4.4.2 Analytical results

The analytical results of the flexible and rigid ring model are based on the findings of Zegelaar [40]and Gong [16]. In Appendix C.2, the method to determine the primary eigenmodes as a function of theinflation pressure of the rigid ring model using the flexible ring model is discussed in further detail.Only the main results are presented in this section. As set-up a free hanging tyre is analysed. Whenlooking at the plots in figure 4.20 it can be concluded that the rigid ring model shows the same trendas is obtained in the FEM results. Both models show a nearly linear relation between the inflationpressure and the eigenfrequency.

1.5 2 2.5 335

40

45

50

55

60

65

70


Nat

ural

freq

uenc

y [H

z]

ftrans

frot

Figure 4.20: Eigenfrequency versus inflation pressure for rigid ring model of a free hanging tyre.


In table 4.9 an overview is given of the influence of the inflation pressure for the rigid ring model.These results correspond reasonably well with the influence levels of table 4.8. It is concluded that theeigenfrequencies of the primary modes are linearly depending on the inflation pressure.

Table 4.9: The influence of the inflation pressure on the determined eigenfrequencies.

Influence level, increasing pi with 0.5 barEigenmode εA,0.5bar [%]

vertical, longitudinal 8.5rotational 4.3


In this chapter the measurement results are presented of an elaborate measurement program per-formed on the TU/e Flatplank Tyre Tester. This measurement program is conducted to get a betterunderstanding of the influence of the inflation pressure on the different tyre characteristics modelledin the MF-Swift tyre model. It is shown that for all the characteristic stiffnesses, i.e. vertical, longitu-dinal, lateral and torsional, a linear relation with the inflation pressure exists. The lateral relaxationlength and the effective rolling radius also show a linear relation with the inflation pressure. The cam-ber thrust and camber torque exhibit a linear relation with the inflation pressure of which the slopeis depending on the vertical load condition. Finally, it is concluded that the eigenfrequencies (primarymodes) also show a linear relation with the inflation pressure.

The influence of the inflation pressure on the different characteristics is determined. An overview ofthe influence when changing the inflation pressure with 0.5 bar is given in table 4.10 for increasingorder of influence. The quantity with the highest influence level will show the largest improvementwhen made inflation pressure dependent.

Table 4.10: Overview influence of the inflation pressure on the different quantities, when the inflationpressure is changed 0.5 [bar].

Influence inflation pressureQuantity εA,0.5bar [%]

Re 0.2CFx 3.5fnat (primary modes) 6.0Mxγ (FzW = 5 kN) 7.0CFy 9.5CMz 12.0σy 12.0CFz 13.0Mzγ (FzW = 5 kN) 15.0Fyγ (FzW = 5 kN) 15.0

Chapter 5

Inflation Pressure Relations toEnhance the MF-Swift Tyre Model

In this chapter relations will be derived to incorporate the inflation pressure influence in the MF-Swifttyre model. The relations are mainly based on the trends found in the previous chapter and supportedby trends described in literature.

To describe the influence of the inflation pressure in the relations, the inflation pressure will be ex-pressed in the inflation pressure increment:

dpi =pi − pi0pi0

, (5.1)

where pi is the current inflation pressure and pi0 is the nominal inflation pressure. If necessary, therelations will be extended with the vertical load increment.

dfz =Fz − Fz0Fz0

, (5.2)

where Fz and Fz0 are the current and nominal vertical load respectively. (5.1) and (5.2) are part ofthe latest MF-Swift version. As mentioned before, the nominal inflation pressure is 2.5 bar and thenominal vertical load condition is 5 kN in this thesis, unless otherwise indicated.

To assess the relations for each characteristic, a fit error is determined. The following expression isused to describe the fit error εA:

εA = 100

√√√√√√√√N∑i=1

(Arelation −Ameas,i)2

N∑i=1

(Ameas,i)2(5.3)

In this equation A is the tyre characteristic quantity (e.g. lateral stiffness) and N is the number ofconsidered measurement points for a certain inflation pressure. The subscripts relation andmeas areused to denote the proposed relations and the measurement data respectively. The error expression(5.3) is similar to the approach used in the MF-Swift tyre model. The error is determined per inflationpressure. Furthermore, a total error (the average of all the inflation pressure errors) is determined. Inthis way the quality of the fit can be determined, e.g. a low fit error equals a qualitative good fit (thedifference between the measurements and the fit is small).

In TNO’s Magic Formula software, the optimisation of the Magic Formula parameters is done byfinding a minimum of a constrained nonlinear multivariable function (fmincon routine from theMatlab Optimization Toolbox). This function is also used as optimising routine for the new pressuredependent relations.

39

40 Inflation Pressure Relations to Enhance the MF-Swift Tyre Model

5.1 Tyre vertical stiffness

As discussed in subsection 4.1.1, a clear linear relation between the inflation pressure and the tyrevertical stiffness exists. Furthermore, the vertical deflection shows a quadratic relation with the verticalforce. Schmeitz [32] derived a relation for the vertical load, depending on the vertical deflection ρz andthe inflation pressure increment dpi:

Fz = (1 + qFz3dpi)(qFz1ρz + qFz2ρ2z), (5.4)

where the q’s are theMagic Formula optimisation parameters. The relation (5.4) has been implementedin the MF-Swift tyre model in 2005. From this relation, the tyre vertical stiffness can be derived, read-ing:

CFz =dFzdρz

= (1 + qFz3dpi)(qFz1 + 2qFz2ρz). (5.5)

To assess the proposed relation, the vertical stiffness obtained with (5.5) is compared with the measure-ment results in figure 5.1. It can be seen that the proposed relation describes the measurements ratherwell. The total error is approximately 0.84 percent and per inflation pressure the error stays below 2.1percent. So by deriving the vertical load relation (5.4) the vertical stiffness can be described well and nofurther enhancements are necessary.

0 0.01 0.02 0.03 0.04 0.051.6

1.8

2

2.2

2.4

2.6

2.8

3

3.2

3.4

3.6x 10

5

Vertical deflection, ρz [m]

Ver

tical

stif

fnes

s, C

Fz [N

/m]

Error: 0.84215 %

measequationp19p22p25p27p30

Figure 5.1: Tyre vertical stiffness using (5.5) and measurements for 5 inflation pressures and the threegeneral vertical load conditions.

Error per inflation pressureInfl. press [bar] Error [%]

total 0.841.9 0.502.2 1.072.5 0.252.7 2.073.0 0.32

5.2 Tyre torsional stiffness

The torsional stiffness, shows a more or less linear relation with the inflation pressure. Furthermore, itis shown in Appendix D.1 that a linear relation is visible between the torsional stiffness and the verticalload. Therefore two linear relations are proposed to describe the torsional stiffness as a function of theinflation pressure and the vertical load:

CMz = CMz,nom(1 + pCMz1dpi)(1 + pCMz2dfz), (5.6)

where pCMzi are new Magic Formula parameters.When implementing this equation, it turns out that for small vertical load conditions, in this case

5.2 Tyre torsional stiffness 41

Fz < 500 N, the tyre torsional stiffness can become negative as a result of the linear vertical loadrelation. To solve this problem two solutions are proposed:

1) CMz = CMz,nom(1 + pCMz1dpi)FzFz0

; (5.7)

2) CMz = CMz,nom

(pCMz1

FzFz0

+ pCMz2FzFz0

2)(1 + pCMz3dpi). (5.8)

In figure 5.2 the torsional stiffness for (5.7) and (5.8) are compared with the measurement results.

0 1000 2000 3000 4000 5000 60000

20

40

60

80

100

120

140

160

180

Vertical load, Fz [N]

Tor

sion

al s

tiffn

ess,

CM

z [Nm

/deg

]

equationp19p22p25p27p30

0 1000 2000 3000 4000 5000 60000

20

40

60

80

100

120

140

160

180


Tor

sion

al s

tiffn

ess,

CM

z [Nm

/deg

]

equationp19p22p25p27p30

Figure 5.2: Tyre torsional stiffness using (5.7); left plot, and (5.8); right plot, compared with Flatplankmeasurement results.

It can be seen that the results of (5.8) describe the measurement results more accurately compared with(5.7). In table 5.1 an overview is given of the total error and the error per inflation pressure for bothproposed equations. Although, (5.8) shows better results, still (5.7) is chosen to be implemented in theMF-Swift model. The main reason is the more complex implementation of (5.8) (two additional para-meters) and the linear trend that is described in literature for aircraft tyres (figure 3.6). Furthermore,the results are based on Flatplank measurements. The friction coefficient of the Flatplank seems to beinsufficient to perform reliable torsion stiffness experiments. To solve this problem, the experimentsshould be repeated on the Flatplank on a high friction surface.

Table 5.1: Error results for the proposed torsional stiffness relations; (5.7) and (5.8).

Error level [%]Infl. press [bar] Equation (5.7) Equation (5.8)

total 9.58 5.051.9 10.48 5.152.2 8.47 3.502.5 7.97 7.122.7 8.16 3.073.0 12.80 6.46


5.3 Lateral relaxation behaviour

As mentioned in section 4.2, the tyre lateral stiffness CFy and the cornering stiffness CFα determinethe lateral (or sideslip) relaxation length σy . In this section the current strategy for calculating the lateralrelaxation length will be discussed and enhancements will be introduced to achieve more accuracy inthe determination of the lateral relaxation length and closely related quantities.

5.3.1 Current model implementation

In the current version of the MF-Swift tyre model, the lateral relaxation length is defined as the relationbetween the cornering stiffness and the total carcass stiffness cy increased with half the contact lengtha:

σy = −CFαCFy

= −CFαcy

+ a. (5.9)

The tyre lateral stiffness CFy reads:1

CFy=

1cy

+a

CFα. (5.10)

As mentioned in the literature survey, in [32] it is observed that the cornering stiffness depends onthe inflation pressure. To include the inflation pressure effects in the MF-Swift tyre model, two linearrelations to the cornering stiffness equation have been added:

Kyα = pKy1(1 + ppy1dpi)Fz0 sin[2 arctan

Fz

pKy2(1 + ppy2dpi)Fz0

], (5.11)

where the parameters ppyi are the inflation pressure dependent parameters. A linear inflation pressurerelation is added to the factor pKy1, which determines the maximum value of the stiffness (Kyα/Fz0),and to the factor pKy2, which determines the load at which the cornering stiffness reaches its maxi-mum value. Note that (5.11) only shows the enhanced parts of the cornering stiffness.

In the MF-Swift model the half contact length is defined as a function of the vertical deflection, usinga relation found by Besselink [6] in 2000. In [6] it is concluded that the contact length is not directlydepending on the vertical force, but is primarily a function of the vertical tyre deflection and reads:

a = pa1r0

(ρzr0

+ pa2

√ρzr0

), (5.12)

where ρz is the vertical tyre deflection, r0 the unloaded tyre radius and pai the fit parameters.The tyre lateral stiffness is assumed to be constant in the current version of the model, i.e. one averagestiffness for all the measured vertical load conditions is used:

CFy =1N

N∑i=1

CFy,i, (5.13)

N is the number of vertical load conditions measured. When no additional relaxation length mea-surements are available, the relaxation lengths can be determined with equation (5.9). If there areextra relaxation length measurements performed, it is possible to optimise the determined relaxationlengths to the measured relaxation lengths by adjusting the tyre lateral stiffness CFy in a way that theerror between the determined andmeasured relaxation lengths is minimised. Adjusting the tyre lateralstiffness automatically results in a change of the total carcass stiffness cy .


5.3.2 Proposed enhancements

The tyre lateral stiffness CFy is the quantity where the most progress can be made. Like mentionedin the previous section, the tyre lateral stiffness is currently the average stiffness value for all themeasured lateral stiffnesses. This is the best approximation of a vertical load range, when the lateralstiffness is assumed to be a constant value

Tyre lateral stiffnessIn chapter 4 it is shown that the tyre lateral stiffness depends on the inflation pressure and the verticalload. A linear relation between the tyre lateral stiffness and the inflation pressure exists. So it is obviousthat a linear relation has to be introduced to describe the relation with the inflation pressure.Currently there is also no relation that describes the influence of the vertical load on the tyre lateralstiffness. In figure 5.3 the lateral stiffness as function of the vertical load is depicted. It is observedthat no linear relation exists and that due to the limited measurement points available per vertical loadcondition only a second order polynomial can be proposed.

3000 3500 4000 4500 5000 5500 6000 6500 70001.15

1.2

1.25

1.3

1.35

1.4

1.45

1.5

1.55

1.6

1.65x 10

5


Late

ral s

tiffn

ess,

CF

y [N/m

]

p19p22p25p27p30Trend line

Figure 5.3: Measured tyre lateral stiffness vs. vertical load.

To describe both influences, the following equation is introduced:

CFy = CFy,nom(1 + pCFy1dfz + pCFy2df2z )(1 + pCFy3dpi), (5.14)

where CFy,nom is the tyre lateral stiffness at the nominal inflation pressure and nominal vertical loadcondition and pCFyi are the model coefficients.


In figure 5.4 the lateral stiffness from the proposed equation is compared with the measurements toassess the proposed relation. It can be seen that the proposed relation describes the measurementsrather well. The total error is approximately 0.82 percent and per inflation pressure the error staysbelow 2 percent.

3000 3500 4000 4500 5000 5500 6000 6500 70001.1

1.2

1.3

1.4

1.5

1.6

1.7x 10

5


Late

ral s

tiffn

ess,

CF

y [N/m

]

Error: 0.81764 %


Figure 5.4: Proposed lateral stiffness relation (5.14) compared with the Flatplank measurement results.


total 0.821.9 0.532.2 1.902.5 0.012.7 0.283.0 1.36

Relaxation lengthIn figure 5.5 the results are depicted when the lateral stiffness parameters of equation (5.14) are opti-mised so that themeasured relaxation lengths are described as good as possible. This strategy results inan optimised total error of 2.11 percent, so the optimised relaxation lengths correspond rather well withthe measurements. For the errors of the separate inflation pressure conditions see figure 5.5. Here, thefit errors stay within 3.5 percent.

3000 3500 4000 4500 5000 5500 6000 6500 7000

0.4

0.5

0.6

0.7

0.8

0.9

1


Late

ral r

elax

atio

n le

ngth

, σ y [m

]


Figure 5.5: Resulting relaxation lengths when optimising the parameters of the lateral stiffness using (5.14)compared with the measured relaxation lengths.


total 2.111.9 2.182.2 1.762.5 3.482.7 1.923.0 1.19

So when the parameters of (5.14) are optimised to the relaxation length, the relaxation length can bedescribed rather well. A strange trend occurs when the, for the relaxation length optimised, tyre lateral


stiffness is observed, see figure 5.6. It might be expected that the trend will be more or less the sameas the trend in figure 5.4, but figure 5.6 shows that the stiffness reaches a minimum at the nominalvertical load condition. This is physically hard to explain. The problem is caused by a sign change of pa-rameter pFCy2, which is the term of the quadratic vertical force influence. When optimising (5.14) forthe tyre lateral stiffness this parameter is negative, whereas optimising (5.14) for the relaxation lengthresults in a positive pFCy2. When constraining pFCy2 ≤ 0 during the relaxation length optimisationsequence, the resulting optimised value is 0.

Figure 5.6: Resulting tyre lateral stiffness when optimised for relaxation length data.

Therefore is has been dicided that the best fit is obtained when the vertical force influence on the tyrelateral stiffness is described by a linear relation. The following equation is proposed to describe the tyrelateral stiffness:

CFy = CFy,nom(1 + pCFy1dfz)(1 + pCFy2dpi) (5.15)

Performing the relaxation length optimising strategy with (5.15), results in a more plausible optimisedtyre lateral stiffness trend. The lateral stiffness increases with increasing vertical load, see figure 5.7.

Figure 5.7: Optimising (5.15) for relaxation length; resulting tyre lateral stiffness.


In figure 5.8 the results of the relaxation lengths are depicted. The total fit error of the relaxation lengthis approximately 2.93 percent; this is an increase of 0.8 percent compared to optimising with (5.14).The table in figure 5.8, gives an overview of the relaxation length errors per inflation pressure.

3000 3500 4000 4500 5000 5500 6000 6500 7000

0.4

0.5

0.6

0.7

0.8

0.9

1


Late

ral r

elax

atio

n le

ngth

, σ y [m

]


Figure 5.8: Optimising (5.15) for relaxation length; relaxation length results and overview of the error perinflation pressure.


total 2.931.9 3.372.2 3.442.5 4.212.7 2.143.0 1.48

Finally, to assess the new proposed relation purely for the tyre lateral stiffness, figure 5.9 shows theresults of (5.15) optimised for lateral stiffness measurements compared with the measurement results.It is obvious that (5.15) describes the measured tyre lateral stiffnesses less accurate than (5.14), seefigure 5.4. The total error has increased from approximately 0.81 percent to 3.44 percent. However,(5.15) gives the best compromise for the tyre lateral stiffness of a rolling and non-rolling tyre.

3000 3500 4000 4500 5000 5500 6000 6500 70001.1

1.2

1.3

1.4

1.5

1.6

1.7

1.8x 10

5


Late

ral s

tiffn

ess,

CF

y [N/m

]

Error: 3.4445 %


Figure 5.9: Linear tyre lateral stiffness relation (5.15) compared with the Flatplank measurement results.


total 3.441.9 2.692.2 1.962.5 3.452.7 3.883.0 5.23

5.4 Longitudinal relaxation behaviour 47

5.4 Longitudinal relaxation behaviour

Currently, the tyre longitudinal stiffness is assumed to be constant in the tyre model, being the averagestiffness for all the measured vertical load conditions:

CFx =1N

N∑i=1

CFx,i, (5.16)

where N is the number of vertical load conditions measured. With regard to inflation pressure andvertical load influences, the tyre longitudinal stiffness and the tyre lateral stiffness show a similar be-haviour as already indicated in section 4.1. It is shown that a linear relation with regard to the inflationpressure exists. As can be seen in figure 5.10, a non-linear relation with the vertical load is visible. Dueto the limited measurement points (only three points), the proposed relation is restricted to a secondorder polynomial.

2000 2500 3000 3500 4000 4500 5000 5500 60001.7

1.8

1.9

2

2.1

2.2

2.3

2.4

2.5x 10

5


Long

itudi

nal s

tiffn

ess,

CF

x [N/m

]

p19p22p25p27p30

Figure 5.10: Measured tyre longitudinal stiffness vs. vertical load.

To describe the inflation pressure and vertical load influences on the tyre longitudinal stiffness, thefollowing equation is introduced:

CFx = CFx,nom(1 + pCFx1dfz + pCFx2df2z )(1 + pCFx3dpi). (5.17)

Here, CFx,nom is the tyre longitudinal stiffness at nominal inflation pressure and nominal verticalload condition. pCFxi are the optimisation parameters. To assess the proposed relation, in figure 5.11the longitudinal stiffness for the proposed equation is compared with the Flatplank measurements.It can be seen that the proposed relation describes the measurements rather well. The total error isapproximately 0.54 percent and per inflation pressure the error stays below 1.0 percent.


2000 2500 3000 3500 4000 4500 5000 5500 60001.7

1.8

1.9

2

2.1

2.2

2.3

2.4

2.5x 10

5


Long

itudi

nal s

tiffn

ess,

CF

x [N/m

]

Error: 0.54253 %


Figure 5.11: Proposed tyre longitudinal stiffness relation (5.17) compared with the Flatplank measure-ments.


total 0.541.9 0.342.2 0.752.5 0.012.7 0.753.0 0.88

The longitudinal relaxation length is determined by:

σx = −CFκCFx

=CFκcx

+ a, (5.18)

where CFκ is the longitudinal slip stiffness (according to (3.2)) and cx is the longitudinal carcass stiff-ness. Because the tyre longitudinal stiffness shows similar behaviour as the tyre lateral stiffness, thesame problems are likely to occur for the relaxation length as is described in the previous subsection.Furthermore, when extrapolating (5.17) for higher vertical load conditions, it is possible that the stiff-ness decreases at a higher vertical load condition. This is caused by the second order polynomial thatis optimised on the measurements. To overcome this problem, a linear relation is used to describe thevertical load influence instead of the second order polynomial. Therefore, the following equation isfinally proposed to describe the tyre longitudinal stiffness:

CFx = CFx,nom(1 + pCFx1dfz)(1 + pCFx2dpi). (5.19)

To assess the new proposed relation, figure 5.12 shows the results of (5.19) compared with the mea-surements. Obviously, it can be seen that (5.19) describes the measured tyre longitudinal stiffness lessaccurate than (5.17), figure 5.4. The total error is increased from approximately 0.54 percent to 6.35percent. The influence of equation 5.19 on the longitudinal relaxation length is not further evaluated,because no data of the longitudinal relaxation length σx for the different general inflation pressures isavailable. the longitudinal relaxation length cannot be measured with the Flatplank.

5.5 Rolling resistance 49

2000 2500 3000 3500 4000 4500 5000 5500 60001.7

1.8

1.9

2

2.1

2.2

2.3

2.4

2.5

2.6

2.7x 10

5


Long

itudi

nal s

tiffn

ess,

CF

x [N/m

]

Error: 6.3538 %


Figure 5.12: Linear tyre longitudinal stiffness relation (5.19) compared with the Flatplank measurements.


total 6.351.9 4.382.2 5.342.5 6.352.7 6.933.0 8.77

5.5 Rolling resistance

Although a lower inflation pressure results in less compression of the tread blocks in the contact patch,it results in an increase in tread bending and shearing. This results in an increase in rolling resistance.The rolling resistance increases rapidly as the inflation pressure decreases. In the MF-Swift tyre modelcurrently the rolling resistance force Fx,RR is determined by the following expression:

Fx,RR =My

r0= −FzλMy

qsy1+qsy2

FxFz0

+qsy3

∣∣∣∣ VxVref

∣∣∣∣+qsy4( VxVref

)4

+qsy5γ2+qsy6FzFz0

γ2

. (5.20)

Here, the influence of the vertical load Fz , the longitudinal velocity Vx and the camber angle γ aretaken into account by the qsy ’s in this equation.

As discussed in section 3.3, Michelin introduced in 2005 a rolling resistance force model based on SAEstandard J2452. The Michelin model is described in (3.21) as follows:

Fx,RR = Fx,RR−ISO ·( pipi,ISO

)α( FzFz,ISO

)β.

The nominal conditions are defined according to the ISO 8767 standard. Therefore the nominal condi-tions are identified by pi,ISO and Fz,ISO and the rolling resistance force at these conditions is identifiedby Fx,RR−ISO. When implementing the model of Michelin in the MF-Swift tyre model nominal sub-script 0 is used instead of the ISO subscript. The new expression (enhancement underlined) for therolling resistance force calculation in the MF-Swift model reads:

Fx,RR =My

r0= −Fz0λMy

qsy1 + qsy2

FxFz0

+ qsy3

∣∣∣∣ VxVref

∣∣∣∣ + qsy4

(VxVref

)4

+ qsy5γ2 + ... (5.21)

qsy6FzFz0

γ2

(FzFz0

)qsy7(pipi0

)qsy8,

where the new Magic Formula parameters qsy7 and qsy8 refer to the coefficients α and β of (3.21) andcover the influence of vertical load change and inflation pressure change respectively. Furthermore, Fz


in the original expression (5.20) is replaced by Fz0 to determine the rolling resistance force at nominalvertical load condition with velocity and camber compensation. Note that by using qsy7=1 and qsy8=0as default values in (5.20), the original expression (5.20) is obtained again.

Because no suitable rolling resistance data at different inflation pressures is available from the mea-sured tyre in this thesis or another tyre in the TNO database, a selection of the data found in literature[18] will be used to evaluate the quality of the new formula. The measurements to obtain this data wereperformed according to the ISO 8767 international standard. For an overview of the selected rollingresistance data it is referred to Appendix D.2. In figure 5.13 the results are depicted of the "old" relation(5.20) when the Magic Formula parameters are optimised to the measurement data. In the left plot,the rolling resistance is illustrated as a function of the vertical load. At a constant vertical load con-dition, the lowest rolling resistance value corresponds to the lowest forward velocity and the highestrolling resistance value corresponds to the highest forward velocity. At the highest vertical load condi-tion (Fz=5360 N) only four values are obtained with (5.20), while eight measurement points exist (2inflation pressures at 4 velocities). Because (5.20) does not take inflation pressure into account, onlyfour velocity optimised values are generated.

In the right plot of figure 5.13 the rolling resistance force versus the velocity is depicted. At a certainvelocity, the highest rolling resistance force corresponds with the highest vertical load. The resultingerror of (5.20) of all data points is approximately 8.5 percent.

1500 2000 2500 3000 3500 4000 4500 5000 5500−60

−55

−50

−45

−40

−35

−30

−25

−20

−15

−10

Vertical force, Fz [N]

Rol

ling

resi

stan

ce fo

rce,

Fx,

RR

[N

]

Old relationMeasurement

5 10 15 20 25 30 35−60

−55

−50

−45

−40

−35

−30

−25

−20

−15

−10

Forward velocity, V [m/s]

Rol

ling

resi

stan

ce fo

rce,

Fx,

RR

[N

]

Old relationMeasurement

Figure 5.13: "Old" rolling resistance force relation (5.20) optimised for the measurement data, as functionof Fz (left) and as function of V (right).

When comparing the results of figure 5.13 with the results of the "new" relation (5.21) in figure 5.14,it is seen that relation (5.21) significantly improves the fit accuracy compared to the old relation. Byimplementing the two exponential terms, the fit error is approximately 6.7 percent improved to 1.8percent error over all data points, see table 5.2. The optimised Magic Formula parameters for bothsituations are given in Appendix D.2.

Table 5.2: Total error of the "old" and "new" rolling resistance force relation, (5.20) and (5.21) respec-tively.

Error of all data pointsEquation Figure Total Error [%]

(5.20) 5.13 8.50(5.21) 5.14 1.80


1500 2000 2500 3000 3500 4000 4500 5000 5500−60

−55

−50

−45

−40

−35

−30

−25

−20

−15

−10

Vertical force, Fz [N]

Rol

ling

resi

stan

ce fo

rce,

Fx,

RR

[N

]

New relationMeasurement

5 10 15 20 25 30 35−60

−55

−50

−45

−40

−35

−30

−25

−20

−15

−10

Forward velocity, V [m/s]

Rol

ling

resi

stan

ce fo

rce,

Fx,

RR

[N

]

New relationMeasurement

Figure 5.14: "New" rolling resistance force relation (5.21) optimised for the measurement data, as functionof Fz (left) and as function of V (right).

To create relations as unambiguous as possible, it is tried to replace the vertical load and inflationpressure divisions by the vertical load increment dfz and inflation pressure increment dpi. This createsa problem when Fz = Fz0 and/or pi = pi0; the exponential relation becomes zero and eliminates therolling resistance force. So implementing the vertical load increment and inflation pressure is not anoption.

5.6 Camber thrust and camber torque

In section 4.3 it is shown that the inflation pressure has some influence on the camber thrust, thealigning camber torque and overturning camber torque for a large camber angle range. Therefore,it is desirable to implement an inflation pressure dependency. Camber is one of the effects that isdeeply embedded in the Magic Formula tyre model. It is not present at one specific position in theequations, but it is present at many different places. Currently, all the implemented inflation pressuredependencies, like in the cornering stiffness Cfα, have very little or no influence on camber effects.

Camber thrustAs mentioned in section 4.3, the inflation pressure has influence on the camber thrust stiffnessKyγ0.Currently, the camber thrust stiffness is implemented in the Magic Formula with only a vertical loaddependency. To implement inflation pressure dependency in this quantity, the following (underlined)enhancement is proposed:

Kyγ0 = (pKy6 + pKy7dfz)FzλKyγ(1 + ppy5dpi) (5.22)

The camber thrust stiffness has influence on different parts of the lateral force calculation. (5.23) showsthe Magic Formula relation of the lateral force for pure slip conditions. Where for example Kyγ0 hasinfluence on the stiffness factor By .

Fyp = Dy sin[Cy arctanByαy − Ey(Byαy − arctan(Byαy))

]+ SV y, (5.23)


To assess the enhancement, (5.22) is implemented in the Magic Formula and compared with Flatplankmeasurements. In figure 5.15, the results are depicted for the nominal vertical load condition (Fz=5kN); the left plot shows the results for negative camber angles and the right plot shows the results forpositive camber angles. The lines represent the Magic Formula optimisation results and the markersare the Flatplank measurement results.

1.8 2 2.2 2.4 2.6 2.8 30

500

1000

1500


Cam

ber

thru

st, F

yγ [N

]

−15.0 deg−10.0 deg−5.0 deg−2.5 deg−2.0 deg−1.5 deg−1.0 deg−0.5 deg

1.8 2 2.2 2.4 2.6 2.8 3−1600

−1400

−1200

−1000

−800

−600

−400

−200

0


Cam

ber

thru

st, F

yγ [N

]

0.5 deg1.0 deg1.5 deg2.0 deg2.5 deg5.0 deg10.0 deg15.0 deg

Figure 5.15: Implemented camber thrust enhancement (lines), equation (5.22), compared with the mea-surements (markers) for various camber angles at the nominal vertical load condition: negative camberangles (left) and positive camber angles (right).

The results in figure 5.15 are manually tuned to describe the common passenger car camber range (i.e.till γ ≤+/- 5.0 degrees) well. A result of this is that for camber angles larger than +/- 5 degrees a largerdeviation occurs. This larger deviation is caused by the Magic Formula. At nominal inflation pressure(pi=2.5 bar) the camber thrust generated by the Magic Formula for large camber angles is too high.The Magic Formula can perfectly optimise pure camber effects, but when slip measurement data isadded to the Magic Formula optimisation routine larger deviation occurs for camber angles γ >+/- 5degrees. Furthermore, the camber measurements are performed on the Flatplank, while the slip datacomes from the TNO Tyre Test Trailer. In earlier research, [35], it is already shown that the results ofindoor test facilities (e.g. The Flatplank) do not always correspond well with the results of outdoor testfacilities (e.g. the TNO Tyre Test Trailer). To exclude this possible cause, the camber measurementshave to be repeated on the TNO Tyre Test Trailer.

Overturning camber torqueThe lateral force, determined with (5.23), is used in the determination of the overturning momentMx. In the Magic Formula, the overturning moment Mx is determined according to the followingexpression:

Mx = R0Fz

qsx1 − qsx2γy − qsx12γy|γy|+ qsx3

FyFz0

+ ...

+R0Fy

qsx13 + qsx14γy

, (5.24)

where Fy is the lateral force determined in the Magic Formula. Because the inflation pressure influ-ence on the lateral force shows almost no influence on the overturning moment, an enhancement ofthe overturning moment equation is necessary. To implement inflation pressure dependency in theoverturning camber torque, the following (underlined) enhancement of (5.24) is proposed:

Mx = R0Fz

qsx1 − qsx2γy(1 + ppMx1dpi)− qsx12γy|γy|+ qsx3

FyFz0

+ ...

+ ... (5.25)

To assess the influence of the enhancement, (5.25) is implemented in the Magic Formula. In figure5.16 the results of the enhanced relation are compared with Flatplank camber measurements; the left


plot shows the results for negative camber angles and the right plot shows the results for the positivecamber angles. The lines represent the Magic Formula results and the markers are the Flatplank mea-surements. It can be seen that the overturning camber torque describes all measurements rather well.

1.8 2 2.2 2.4 2.6 2.8 30

50

100

150

200

250

300

350

400


Cam

ber

torq

ue, M

xγ [N

m]


1.8 2 2.2 2.4 2.6 2.8 3−400

−350

−300

−250

−200

−150

−100

−50

0


Cam

ber

torq

ue, M

xγ [N

m]


Figure 5.16: Overturning camber torque enhancement relation compared with the measurements for vari-ous camber angles at the nominal vertical load condition: negative camber angles (left) and positive camberangles (right).

Aligning camber torqueIn the Magic Formula model the aligning moment due to camber is determined by the lateral force Fy ,the pneumatic trail t [28] and the residual aligning momentMzr. The enhanced relation of the lateralforce, (5.22), appears to have almost no influence on the aligning momentMz . In the Magic Formulathe aligning moment is determined by the following expression:

Mz = Fyt+Mzr, (5.26)Mzr = Dr cos

arctan(Brαr,eq)

cos(αM ). (5.27)

To implement inflation pressure influence of the aligning camber torque Mzγ , the peak factor Dr in(5.27) is extended in the following (underlined) way:

Dr = FzR0

[(qDz6 + qDz7dfz)λr + (qDz8 + qDz9dfz)γλKzγ(1 + ppz2dpi) ... (5.28)

+(qDz10 + qDz11dfz)γ | γ |]+ 1.

Again, the enhanced relation, (5.28), is implemented in the Magic Formula to assess the influenceof the enhancement. In figure 5.17 the results are depicted for the nominal vertical load conditionand compared with the Flatplank measurements; left plot: negative camber angles, right plot: positivecamber angles. The aligning camber torque is manually tuned to describes the common passenger carcamber angle range (γ ≤+/- 5 degrees) well. It can be seen in figure 5.17 that at larger camber anglesa significant difference occurs. As already mentioned for the camber thrust, the cause of this lies inthe by the Magic Formula generated too high aligning camber torque values for large camber anglesat nominal inflation pressure and the difference between Flatplank measurements and TNO Tyre TestTrailer results.

Finally, note that the proposed enhancements in (5.22), (5.25) and (5.28) show that it is possible todescribe the influence of inflation pressure on camber effects (Fyγ , Mzγ , Mxγ ). Figures 5.15 till 5.17are used to illustrate the influence of the enhancements and to confirm the useability of the proposedenhancements. The depicted results are manually tuned for the common passenger car camber anglerange (γ ≤+/- 5 degrees), this may result in a bigger deviation for γ >+/- 5 degrees. Furthermore,no optimisation strategy for camber effects is developed yet. In Appendix D.4 the Magic Formula re-sults are presented for two extreme vertical load conditions to assess the robustness of the proposedenhancements.


1.8 2 2.2 2.4 2.6 2.8 30

10

20

30

40

50

60

70

80

90


Cam

ber

torq

ue, M

zγ [N

m]


1.8 2 2.2 2.4 2.6 2.8 3−90

−80

−70

−60

−50

−40

−30

−20

−10

0

10


Cam

ber

torq

ue, M

zγ [N

m]


Figure 5.17: Aligning camber torque enhancement relation compared with the measurements for variouscamber angles at the nominal vertical load condition: negative camber angles (left) and positive camberangles (right).

5.7 Rigid ring dynamics

As already described in chapter 2, the rigid ring model consists of a rigid ring that represents the tyrebelt. The ring is elastically suspended to the rim by means of spring-damper elements, representingthe tyre sidewalls with pressurised air. The contact model consist of two models that are used fordescribing the contact of the tyre with the road surface, i.e. an enveloping model and a slip model.In addition, residual stiffnesses between the contact model and the rigid ring are used to obtain thecorrect (quasi-static) tyre stiffness. The rigid ring model can only describe the primary tyre modes,i.e. those motions of the tyre where the shape of the belt remains circular. The flexible eigenmodesof the belt are neglected. In section 4.4, the primary eigenmodes of the tyre are analysed with FEMexperiments to determine the influence of the inflation pressure.

Currently MF-Swift can handle one inflation pressure condition only. In this section, strategies areproposed to implement the inflation pressure influence in the MF-Swift model, in such a way thatthe rigid ring tyre dynamics for a certain inflation pressure range can be parameterised. Two possibleconcepts are discussed based on: 1) the Rotta Membrane Theory and 2) scaling the nominal sidewallstiffnesses.

5.7.1 Rotta Membrane Theory

Rotta introduced The Membrane Theory for Cylindrical Tyres in 1949 [30]. The theory of Rotta isbased on neglecting the circumference curvature of the membrane. In [30] characteristic stiffnesses inthe tangential, lateral and radial deformation directions are introduced, see figure 5.18. These tangentialcu, lateral cv and radial cw direction stiffnesses depend on the dimensions of the "membrane" and theinflation pressure:

cu =Gt

ls+

(1

tanϕs

)pi; (5.29)

cv =(

sin γ1 + cos γ − π−γ

2 sin γ

)pi; (5.30)

cw =(

cosϕs + ϕs sinϕssinϕs − ϕs cosϕs

)pi; (5.31)

where G is the shear modulus of the tyre sidewall, t is the sidewall thickness and ls is the length of thesidewall arc. Furthermore, ϕs is half the angle of the tyre sidewall, pi is the inflation pressure and γ is

5.7 Rigid ring dynamics 55

the angle between the tangent line of the tyre sidewall swell near the rim and the horizontal surface.

Figure 5.18: Sidewall deformations [5].

The relation between the rigid ring model sidewall stiffnesses and the Rotta directional stiffnessesreads [31]:

cbx,z = πr(cu + cw); (5.32)cby = 2πrcv; (5.33)

cbγ,ψ = πr3cv; (5.34)cbθ = 2πr3cu. (5.35)

To describe the sidewall stiffnesses as a function of the inflation pressure, the Rotta direction stiffnessesare implemented in equations (5.32) - (5.35). Assuming that ϕs and γ are constant and not influenced bythe inflation pressure, the inflation pressure pi is the only variable in the direction stiffness equations(5.29) - (5.31). Knowing this, the directional stiffnesses can be rewritten in a constant part and aninflation pressure dependent part:

cu = cu1 + cu2pi; (5.36)cv = cv1pi; (5.37)cw = cw1pi; (5.38)

where cu1, cu2, cv1 and cw1 are constants. If (5.36) - (5.38) are implemented in the sidewall stiffnessrelations (5.32) - (5.35), the sidewall stiffnesses read:

cbx,z = πr[cu1 + (cu2 + cw1)pi

]; (5.39)

cby = 2πrcv1pi; (5.40)cbγ,ψ = πr3cv1pi; (5.41)cbθ = 2πr3

[cu1 + cu2pi

]. (5.42)

In these equations cbx is the overall sidewall stiffness in longitudinal direction, cbz in vertical directionand cby in lateral direction. Furthermore, cbγ is the overall torsion stiffness about the x-axis, cbψ aboutthe z-axis and cbθ about the y-axis.

The equations of motion of a free hanging rigid ring model with fixed rim for the in-plane dynamicsread:

mbxb + kbxxb + cbxxb = 0 (5.43)mbzb + kbz zb + cbzzb = 0 (5.44)Iby θb + kbθ θb + cbθθb = 0 (5.45)

and for the out-of-plane dynamics:

mbyb + kby yb + cbyyb = 0 (5.46)Ibxγb + kbγ γb + cbγγb = 0 (5.47)

Ibzψb + kbψψb + cbψψb = 0 (5.48)


Using the equations of motion and the inflation pressure depending sidewall stiffnesses (5.39) - (5.42),the eigenfrequencies for a free hanging tyre can be determined with:

ωtrans =√cbx,y,zmb

; (5.49)

ωrot =√cbγ,θ,ψIbx,z

. (5.50)

When the primary eigenfrequencies of a free hanging tyre with fixed rim at the nominal inflationpressure pi0 are known, the nominal sidewall stiffnesses cb0 can be determined using (5.49) and (5.50).With the determined nominal sidewall stiffnesses, cb0x,z and cb0θ, the direction stiffnesses cu and cwcan be determined and subsequently the tyre sidewall angle ϕs can be derived using (5.31). Finally, theconstants (i.e. cu1, cu2, cv1 and cw1) can be determined using equations (5.29) and (5.30).

In figure 5.19 the eigenfrequencies of a free hanging tyre with fixed rim determined with the FEMsimulations and estimated with the Rotta equations (5.39) - (5.42) are depicted for the pressure range0 - 3.0 bar. The with Rotta estimated results can only be compared in the pressure range 1.5 - 3.0 bar,because FEM simulations are only performed in this pressure range. Note that in this case the nominalinflation pressure pi0=2.2 bar.

0 0.5 1 1.5 2 2.5 30

10

20

30

40

50

60

70

80

90


Nat

ural

freq

uenc

y [H

z]

AbaqusRottarotationallateralyaw/cambervertical/radial

Figure 5.19: FEM primary eigenfrequencies compared with the Rotta estimations.

Error at 1.5 bar inflation pressurePrimary eigenmode Error [%]vertical, radial 5.22

lateral 5.04rotational 1.90camber, yaw 5.99

The eigenfrequencies correspond rather well, showing a maximum difference of approximately 3 Hz(i.e. error of 6 percent) at an inflation pressure of 1.5 bar. Below 1.5 bar the Rotta estimations are robustfor both the sidewall stiffnesses and the eigenfrequencies. At pi=0 bar, the sidewall stiffnesses (i.e. cby ,cbψ , cbγ ) and the corresponding primary eigenfrequencies become zero. This is caused by the fact thatthe Rotta theory neglects the influence of the tyre material (rubber) flexibility/stiffness.


5.7.2 Scaling the nominal pressure sidewall stiffnesses

A second approach is to scale the nominal inflation pressure sidewall stiffnesses cb0. I.e. the sidewallstiffnesses are made inflation pressure dependent by scaling the nominal sidewall stiffnesses with alinear inflation pressure relation, meaning:

cbx,z = cbx0,z0(1 + fx,zdpi); (5.51)

cby = cby0(1 + fydpi); (5.52)

cbγ,ψ = cbγ0,ψ0(1 + fγ,ψdpi); (5.53)

cbθ = cbθ0(1 + fθdpi). (5.54)

Here the fi’s are the inflation pressure fit parameters. In this case no distinction is made betweenparts that are inflation pressure depending and parts that are not depending on the inflation pressure,like in the Rotta approach. This gives the advantage that it is not necessary to determine certain tyre-construction and inflation pressure depending parameters first, like the tyre sidewall angle ϕs. Thefit parameters fi are determined by optimising (5.51) - (5.54) towards the FEM experiments. In figure5.20, the FEM eigenfrequency results and the eigenfrequencies estimated with the equations (5.51) -(5.54) are depicted for the pressure range 0 - 3.0 bar.

0 0.5 1 1.5 2 2.5 320

30

40

50

60

70

80

90


Nat

ural

freq

uenc

y [H

z]

AbaqusScalingrotationallateralyaw/cambervertical/radial

Figure 5.20: FEM eigenfrequencies compared with optimised scaling estimation.

Optimisation sidewall stiffnessesPres. parameter V alue [−] Error [%]

fx, fz 0.65 0.21fy 0.74 0.27

fγ , fψ 0.49 0.08fθ 0.69 0.25

Again, comparison between the FEM results and the scaling approach can only bemade in the pressurerange 1.5 - 3.0 bar, because FEM simulation are only performed in this pressure range. The resultscorrespond rather well, with an error of approximately 0.2 percent over the 1.5 - 3.0 bar pressure range.The trend below 1.5 bar seems plausible, but cannot be verified.

When comparing the results of the Rotta approach (figure 5.19) and the scaling approach (figure 5.20),it is assessed that the scaling approach results in more accurate estimations. The scaling approach iseasier to implement in the tyre model, and therefore the preference is given to this strategy. However,the eigenfrequencies of the tyre have to be determined experimentally (or with FEM analyses) for atleast two inflation pressures.


5.7.3 Stiffnesses rigid ring model, lateral components

To fully understand what the influence of the inflation pressure is on the stiffnesses of the rigid ringtyre model in lateral direction, an overview of the lateral stiffness components is depicted in figure 5.21.

Figure 5.21: Schematic view of lateral rigid ring model stiffness components.

The tyre lateral stiffness CFy is a relation of the total lateral carcass stiffness cy and the corneringstiffness CFα and reads:

1CFy

=1cy

+a

CFα, (5.55)

where a is the half contact length. Following figure 5.21, the total carcass stiffness can be described asfollows:

1cy

=1cry

+1cby

+r2lcbγ

. (5.56)

In this equation, cry is the residual lateral carcass stiffness, cby the lateral sidewall stiffness and cbγ thetorsional sidewall stiffness about the x-axis. Furthermore, rl is the loaded radius.

In figure 5.22 the lateral stiffnesses are depicted as a function of the inflation pressure for the mea-sured passenger car tyre. The tyre lateral stiffness and the cornering stiffness are determined out of theFlatplank measurements; for pi < 1.9 bar the data is obtained with extrapolation. The sidewall stiff-nesses cby and cbγ at the nominal inflation pressure are determined using a MF-Swift tyre property file.The sidewall stiffnesses at the remaining inflation pressures are determined using the scaling strategy.The total lateral carcass stiffness cy is known from (5.55) and the residual lateral carcass stiffness cry isused to balance equation (5.56).

The trends in figure 5.22 show that the different lateral stiffness components increase when increasingthe inflation pressure. Except for the cornering stiffness, which shows only little influence of the infla-tion pressure compared to the other lateral stiffness components. The results for pi < 1.9 bar seemplausible but should be handled with some reservations. The sidewall stiffnesses cby and cbγ show themost influence of the inflation pressure.


0 0.5 1 1.5 2 2.5 30

2

4

6

8

10

12x 10

5


Stif

fnes

s [N

/m]

CFy

CFα/a

cy

cbγ/rl

2

cby

cry

Trend line

Figure 5.22: Lateral stiffnesses as function of the inflation pressure at nominal vertical load condition.

5.7.4 Stiffnesses rigid ring model, longitudinal components

An overview of the longitudinal stiffness components is depicted in figure 5.23.

Figure 5.23: Schematic view of longitudinal rigid ring model, stiffness components.

The longitudinal relaxation length is described by the following relation:

σκ =CFκCFx

=CFκcx

+ σc (5.57)

whereCFκ is the longitudinal slip stiffness,CFx the tyre longitudinal stiffness, cx the total longitudinalcarcass stiffness and σc the contact patch relaxation length. When knowing that σc ' a [28], theoverall longitudinal stiffness can be expressed as a relation of the longitudinal carcass stiffness and thelongitudinal slip stiffness as follows:

1CFx

=1cx

+a

CFκ, (5.58)


where a is the half contact length.Following figure 5.23, the total longitudinal carcass stiffness can be described as follows:

1cx

=1crx

+1cbx

+r2lcbθ

(5.59)

In this equation, crx is the residual longitudinal carcass stiffness, cbx the longitudinal sidewall stiffnessand cbθ the rotational sidewall stiffness about the y-axis. Furthermore, rl is the loaded radius.

In figure 5.24 the longitudinal stiffnesses are depicted as a function of the inflation pressure for themeasured passenger car tyre. The tyre longitudinal stiffness is determined out of the Flatplank mea-surements; for pi < 1.9 bar the data is obtained with extrapolation. The longitudinal slip stiffness isdetermined using measurement data of the TNO Tyre Test Trailer and processed with MF-Tool thathas the equation (3.2) of de Hoogh implemented. The sidewall stiffnesses cbx and cbθ at the nominalinflation pressure are determined using a MF-Swift dataset. The sidewall stiffnesses at the remaininginflation pressures are determined using the scaling strategy. The total longitudinal carcass stiffnesscx is known from (5.58). The residual longitudinal carcass stiffness crx is used to balance (5.59).

0 0.5 1 1.5 2 2.5 30

0.5

1

1.5

2

2.5x 10

6


Stif

fnes

s [N

/m]

C

Fx

CFκ/a

cx

cbθ/r

l2

cbx

crx

Trend line

Figure 5.24: Overview longitudinal (carcass) stiffnesses as function of the inflation pressure.

The trends in figure 5.24 show that the different longitudinal stiffness components increase more orless with increasing inflation pressure. The longitudinal slip stiffness shows a optimum at approx-imately 2.0 bar. The stiffnesses for pi < 1.9 bar seem plausible but should be handled with somereservations. The sidewall stiffnesses cbx and cbθ show the most influence of the inflation pressure.

In Appendix D.3 an overview of the longitudinal and lateral (carcass) stiffnesses for 2 extreme verticalload conditions, Fz=0.1 and 10 kN, is presented. The stiffness trends in the appendix show that nonegative values occur for extreme vertical load conditions, i.e. the rigid ring model with the inflationpressure enhanced stiffness relations produces robust results for a wide vertical load and inflationpressure range.

5.8 Summarising this chapter 61


In this chapter relations are derived to describe the inflation pressure influence on different quantities.The relations are mainly based on the trends as found in the previous chapter and supported bytrends described in literature. It is shown that not always the equations that describe the inflationpressure influences most accurately (smallest errors) are the most robust solutions for the MF-Swifttyre model fit strategies (e.g. tyre lateral stiffness and torsional stiffness). In the overview below, thefinal equations that are implemented in the MF-Swift tyre model are presented.

Tyre lateral stiffness:

CFy = CFy,nom(1 + pCFy1dfz + pCFy2df2z )(1 + pCFy3dpi)

Tyre longitudinal stiffness:

CFx = CFx,nom(1 + pCFx1dfz + pCFx2df2z )(1 + pCFx3dpi)

Note that for the tyre lateral and longitudinal stiffness a quadratic relation is included instead of amore reliable linear relation, as described in this chapter. To create a linear relation, the quadraticparameters pCFy2 and pCFx2 are fixed to zero in normal use. When higher accuracy is desired thequadratic parameters can be used in the optimisation, but results have to be checked carefully.

Tyre torsional stiffness:

CMz = CMz,nom(1 + pCMz1dpi)FzFz0

Rolling Resistance force:

Fx,RR = Fx,RR0

(FzFz0

)qsy7(pipi0

)qsy8

Fx,RR0 = −Fz0λMy

qsy1 + qsy2

FxFz0

+ qsy3

∣∣∣∣ VxVref

∣∣∣∣ + qsy4

(VxVref

)4

+ qsy5γ2 + qsy6

FzFz0

γ2

Camber thrust:Kyγ0 = (pKy6 + pKy7dfz)FzλKyγ(1 + ppy5dpi)

Aligning camber torque:

Dr = FzR0

[(qDz6 + qDz7dfz)λr + (qDz8qDz9dfz)γλKzγ(1 + ppz2dpi) ...

Overturning camber torque:

Mx = R0Fz

qsx1 − qsx2γy(1 + ppMx1dpi)− qsx12γyγy + qsx3

FyFz0

+ ...

+R0Fy

qsx13 + qsx14γy


Translational sidewall stiffness:cbt = cbt0(1 + ftdpi)

where, subscript t is the translation direction x, y or z.

Rotational sidewall stiffness:cbr = cbr0(1 + frdpi)

where, subscript r is the rotation about x, y or z axis, corresponding with the angles θ, γ and ψrespectively.

In Appendix D.5 an overview is given of the new implemented parameters, with proposed correspond-ing initial and boundary conditions.

Chapter 6

Vehicle Behaviour Simulations

In this chapter vehicle simulations are performed with the enhanced TNO tyre models to analysethe extended relations on functionality and applicability in vehicle design. To investigate the handlingbehaviour, a steady-state circular test according to ISO 4138 [21] and a random steering test accordingto ISO 7401 [23] are executed. The ride comfort is analysed with a vehicle behaviour test on an unevenroad. The vehicle model used to analyse the vehicle behaviour is a SimMechanics model of a two trackvehicle model with a roll axis, see figure 6.1.

Figure 6.1: Two track vehicle model with a roll axis (ISO).

When going through a curve, the body rolls around the roll axis. This axis is a virtual axis that goesthrough the front roll centre rc1 and the rear roll centre rc2. The roll stiffness and damping (whichresult from suspension springs, dampers and anti-roll bars) are modelled with torsional springs anddampers in the roll centres. Furthermore, translational stiffness and damping elements are used, in theroll centres, to model the vertical stiffness and damping of the suspension when driving over unevenroad surfaces. The modelled vehicle is an upper class rear wheel driven sedan with a total unloadedvehicle mass of 1659.5 kg (distribution front-rear 50-50%). In Appendix E an overview is given of theparameters that are used to parameterise the vehicle model.

63

64 Vehicle Behaviour Simulations

6.1 Steady-state circular test

This test is standardised in ISO 4138 with the aim to determine the steady-state behaviour of a vehicle(i.e. no transient effects are considered). The vehicle is driving over a circle with a fixed radiusR (R=100m). This is done with different constant forward velocities V . The steering wheel angle δ is adjusted tomaintain the constant radius. To perform a steady-state circular test with the two track model, a steercontroller is used to drive over a circle and a cruise control controller is necessary to drive at constantvelocities. Within the framework of applicability in vehicle design, two types of analyses are performed:a sensitivity analysis and a loaded vs. unloaded vehicle analysis.

6.1.1 Sensitivity analysis

The inflation pressure sensitivity analyses are performed to investigate what the influence is of theinflation pressure on the handling behaviour of the vehicle. Three inflation pressure approaches aremade, with as base inflation pressure pi,front = pi,rear=2.3 bar, namely:

1. front and rear tyres have same inflation pressure, with pi,front = pi,rear=[ 1.8 2.3 2.8] bar;

2. changing inflation pressure front tyres pi,front=[1.9 2.3 2.7] bar, constant inflation pressure reartyres pi,rear=2.3 bar;

3. changing inflation pressure rear tyres pi,rear=[1.9 2.3 2.7] bar, constant inflation pressure fronttyres pi,front=2.3 bar.

Table 6.1 gives an overview of the different inflation pressure settings.

Table 6.1: Inflation pressure settings.

Inflation pressure [bar]Setting Front tyres Rear tyres1 1.8 1.8

2.3 2.32.8 2.8

2 1.9 2.32.3 2.32.7 2.3

3 2.3 1.92.3 2.32.3 2.7

To illustrate the influence of changing the inflation pressure on the steady-state handling behaviourof a vehicle, the lateral vehicle acceleration ay as function of the axle sideslip angle and the handlingdiagram are assessed. In the handling diagram, the lateral vehicle acceleration ay is described as a func-tion of the difference between the sideslip angle of the rear axle α2 and the front axle α1. A handlingdiagram illustrates if a vehicle is understeered, neutral or oversteered. In figure 6.2, the simulationresults of the first inflation pressure setting (setting 1) are presented. The handling diagram in figure6.2 shows that the vehicle has an understeered behaviour. For lateral accelerations below 4.5 m/s2,the vehicle shows a linear response. For higher lateral accelerations the vehicle shows a nonlinear be-haviour, caused by the nonlinear tyre behaviour. The tyres are starting to reach their maximum lateralforce.

6.1 Steady-state circular test 65

0 2 4 6 8 100

1

2

3

4

5

6

7

8

9

10

αaxle

[deg]

Late

ral a

ccel

erat

ion,

ay [m

/s2 ]

front axle, α1

rear axle, α2

p18p23p28

−4.5 −4 −3.5 −3 −2.5 −2 −1.5 −1 −0.5 00

1

2

3

4

5

6

7

8

9

10

Late

ral a

ccel

erat

ion,

ay [m

/s2 ]

α2 − α

1 [deg]

p18p23p28

Figure 6.2: Lateral acceleration vs. axle sideslip angle (left) and handling diagram (right) for inflationpressure setting 1; front and rear tyres have the same inflation pressure 1.8, 2.3 and 2.8 bar.

Lowering the inflation pressure results in a decrease of the axle sideslip angles, due to a change in theaxle cornering stiffness CFα,axle, see figure 6.3. Lowering the inflation pressure results in an increaseof the axle cornering stiffness. So assuming that 2.3 bar is the nominal inflation pressure, increasingthe inflation pressure leads to a decrease in axle cornering stiffness and decreasing the pressure leadsto an increase. Furthermore, the lateral axle force necessary for a certain lateral acceleration ay isdescribed by:

Fy1 + Fy2 = may, (6.1)

where m is the vehicle mass and Fy1 and Fy2 the lateral force of the front and rear axle respectively.The lateral axle force is determined by:

Fyi = CFα,iαi, (6.2)

where the index i represents the front or the rear axle. Considering (6.1) and (6.2), it is obvious thatwhen the axle cornering stiffness CFα decreases, the axle sideslip angle α has to increase in order toretain the same lateral force. Changing the inflation pressure results, for lateral accelerations below 4.5m/s2, in a more or less equal decrease of the front axle sideslip angle and the rear axle sideslip angle(see figure 6.2). Almost no change is visible in the handling diagram. For higher lateral accelerations,the rear axle sideslip angle changes more than the front axle sideslip angle, resulting in a more (forinflation pressure decrease) or lesser (for inflation pressure increase) understeer behaviour.


1.8 2 2.2 2.4 2.6 2.81400

1600

1800

2000

2200

2400

2600

2800

3000

3200

3400


Cor

nerin

g st

iffne

ss (

axle

), C

Fα,

axle

[N/d

eg]

Front axleRear axleUnloadedLoaded

Figure 6.3: Axle cornering stiffness vs. inflation pressure.

In figure 6.4 and 6.5 the simulation results of settings 2 and 3 respectively are presented. Note that thep-terms in the legend represent the inflation pressure setting (e.g p1923 means the inflation pressureof the front tyres is 1.9 bar and the rear tyres is 2.3 bar). When comparing the simulation results forsettings 2 and 3 with the results of setting 1, it can be seen that changing the tyre inflation pressure ofone axle has more influence on the steady-state handling behaviour of the vehicle. This is confirmedby the understeer coefficients η, presented in table 6.2. The understeer coefficient, only valid for thelinear part of the handling behaviour, is described by the following expression:

η = −d(α2 − α1)day

∣∣∣∣ay=0

. (6.3)

Table 6.2 makes clear that changing the inflation pressure of the front tyres has slightly more influenceon the steady-state handling behaviour of the vehicle than changing the pressure of the rear tyres.

0 2 4 6 8 100

1

2

3

4

5

6

7

8

9

10

αaxle

[deg]

Late

ral a

ccel

erat

ion,

ay [m

/s2 ]

front axle, α1

rear axle, α2

p1923p2323p2723

−4.5 −4 −3.5 −3 −2.5 −2 −1.5 −1 −0.5 00

1

2

3

4

5

6

7

8

9

10

Late

ral a

ccel

erat

ion,

ay [m

/s2 ]

α2 − α

1 [deg]

p1923p2323p2723

Figure 6.4: Axle sideslip angles (left) and handling diagram (right) for inflation pressure setting 2; con-stant inflation pressure rear tyres 2.3 bar, front tyres 1.9, 2.3 and 2.7 bar.


0 2 4 6 8 100

1

2

3

4

5

6

7

8

9

10

αaxle

[deg]

Late

ral a

ccel

erat

ion,

ay [m

/s2 ]

front axle, α1

rear axle, α2

p2319p2323p2327

−4.5 −4 −3.5 −3 −2.5 −2 −1.5 −1 −0.5 00

1

2

3

4

5

6

7

8

9

10

Late

ral a

ccel

erat

ion,

ay [m

/s2 ]

α2 − α

1 [deg]

p2319p2323p2327

Figure 6.5: Axle sideslip angles (left) and handling diagram (right) for inflation pressure setting 3; con-stant inflation pressure front tyres 2.3 bar, rear tyres 1.9, 2.3 and 2.7 bar.

Table 6.2: Understeer coefficient for the different inflation pressure settings.

Understeer coefficient η [deg/m/s2]Setting Inflation pressure1 p18 0.33

p23 0.34p28 0.35

2 p1923 0.31p2323 0.34p2723 0.37

3 p2319 0.36p2323 0.34p2327 0.32

6.1.2 Unloaded vs. loaded vehicle

In table 6.3 a small selection of the vehicle specifications distributed by the car manufacturer is given.In this table also the inflation pressures, as recommended by the manufacturer, are presented. As canbe seen the loaded vehicle has a different weight distribution than the unloaded vehicle. A changein weight distribution has influence on the (steady-state) handling behaviour of the vehicle. For theloaded vehicle the inflation pressure of the front tyres is increased to 2.3 bar and the inflation pressureof the rear tyres is increased to 2.8 bar. It is expected that the inflation pressures for the loaded vehicleare chosen in such a way that the handling behaviour, compared to the unloaded vehicle, remains thesame. This is analysed in this section. To assess the (steady-state) handling behaviour for the loadedvehicle and the unloaded vehicle, a steady state circular test is performed with the following load andinflation pressure situations:

• unloaded vehicle with the recommended inflation pressure, pi,front=1.9 bar and pi,rear=2.3 bar;

• loaded vehicle with the inflation pressure of the unloaded vehicle, pi,front=1.9 bar and pi,rear=2.3bar;

• loaded vehicle with the recommended inflation pressure, pi,front=2.3 bar and pi,rear=2.8 bar.


Table 6.3: Vehicle model parameters (unloaded and loaded).

Vehicle model parametersQuantity Unloaded LoadedTotal vehicle weight [kg] 1659.5 1954Weight distribution (F-R) [%] 50− 50 42.5− 57.5Inflation pressure (F-R) [bar] 1.9− 2.3 2.3− 2.8

In figure 6.6, the lateral acceleration as function of the axle sideslip angle and the handling diagramfor the three load and inflation pressure situations is depicted.

0 2 4 6 8 100

1

2

3

4

5

6

7

8

9

10

αaxle

[deg]

Late

ral a

ccel

erat

ion,

ay [m

/s2 ]

front axle, α1

rear axle, α2

p1923 (unloaded)p1923 (loaded)p2328 (loaded)

−6 −5 −4 −3 −2 −1 00

1

2

3

4

5

6

7

8

9

10

Late

ral a

ccel

erat

ion,

ay [m

/s2 ]

α2 − α

1 [deg]


Figure 6.6: Axle sideslip angles (left) and handling diagram (right) for the three situations.

The loaded vehicle shows larger axle sideslip angles, caused by the different weight distribution. Thehandling diagram makes clear that the loaded vehicle shows approximately the same linear vehiclebehaviour (i.e. lateral acceleration below 4.5 m/s2) as the unloaded vehicle. This is confirmed by theundersteer coefficients in table 6.4. The understeer coefficient of the loaded vehicle with unchangedinflation pressure setting (p1923) is a fraction decreased (i.e. less understeer), increasing the inflationpressure to the recommended level (p2328) results in the same understeer coefficient as an unloadedvehicle. For higher lateral accelerations (i.e. lateral acceleration above 4.5 m/s2) the loaded vehicleshows a more understeered behaviour, which means that the front axle sideslip angle is increasingmore than the rear axle sideslip angle. By increasing the inflation pressure to the recommended levela part of this more understeer behaviour is eliminated.

Table 6.4: Understeer coefficient for unloaded and loaded vehicle.

Understeer coefficient η [deg/m/s2]V ehicle Inflation pressureUnloaded p1923 0.31Loaded p1923 0.30Loaded p2328 0.31

For the loaded vehicle the rear axle vertical load increases significantly and consequently the rear axlecornering stiffness increases, see figure 6.3. When increasing the inflation pressure, the axle cornering


stiffness decreases. This causes the axle sideslip angles to increase for the loaded vehicle with therecommended inflation pressure (p2328).

Furthermore, the influence of the load and the inflation pressure situation on the vehicle sideslip angleβ is depicted in figure 6.7. The vehicle sideslip angle β, see figure 6.1, is defined as:

β =v

u, (6.4)

in which, u is the forward velocity of the vehicle centre of gravity and v is the lateral velocity of thevehicle centre of gravity. The sideslip angle of the rear angle α2 is defined as:

α2 =v + br

u= β +

br

u, (6.5)

where b is the position of the vehicle centre of gravity in relation to the rear axle and r is the yawvelocity. When driving over a circle with a fixed radius R, rR = u, than the sideslip angle of the rearangle α2 reads:

α2 = β +b

R, (6.6)

thus:

β = − b

R+ α2. (6.7)

Following (6.2), the sideslip angle of the rear angle α2 can also be described by:

α2 =Fy2CFα,2

=1

CFα,2may

a

l, (6.8)

where a is the position of the vehicle centre of gravity in relation to the front axle, l is the wheel base,m is the total vehicle mass and CFα,2 is the rear axle cornering stiffness. Finally, when implementing(6.8) in (6.7) the vehicle sideslip angle β reads:

β = − b

R+

am

CFα,2lay, (6.9)

As can be seen in figure 6.7, the vehicle sideslip angle increases when the vehicle is loaded. When alsothe inflation pressure is increased to the recommended level of the loaded vehicle, the vehicle sideslipangle increases even more. This is caused by the increase of the distance a (15%) and mass m (18%)in relation (6.9), whereas the rear axle cornering stiffness CFα,2 for the loaded vehicle increases just23% (for p1923) and 15% (for p2328), see figure 6.3.

Against all expectations, the vehicle sideslip angle increases with the recommended inflation pressureof the loaded vehicle. It would be expected that the vehicle sideslip angle remains constant (i.e. samevehicle sideslip angle as the unloaded vehicle) with the recommended inflation pressure. To maintainthis, in the here presented simulations, the inflation pressure has to be decreased. Note that similarresults are observed in measurements with the vehicle but equipped with other tyres (205/60 R15)[37]and [36]. Furthermore note that wheel orientation changes (toe, camber, etc.) due to change in verticalload are not taken into account in the vehicle model. This has also influence on the handling behaviourof the loaded vehicle.


0 1 2 3 4 5 6 7 8 9−4

−3.5

−3

−2.5

−2

−1.5

−1

−0.5

0

0.5

Lateral acceleration, ay [m/s2]

Veh

icle

sid

e sl

ip a

ngle

, β [d

eg]


Figure 6.7: Vehicle sideslip angle versus lateral acceleration for the unloaded and loaded situations.

6.2 Random steering test

The random steering test is standardised in ISO 7410 and ISO/TR 8726. With this test it is possibleto identify the transfer function between the steering input and various response variables (here: thelateral acceleration and the yaw velocity). To estimate the transfer function of a dynamic system, thesystem can be subjected to a random input signal that contains all relevant frequencies, see figure 6.8.The transfer function can be estimated by the quotient of the cross power spectral density (CPSD) of theinput and the output and the power spectral density (PSD) of the input. This procedure is implementedin Matlab in a single function, tfestimate, and works only for linear systems. The test is performedat a constant forward velocity V =100 km/h. The (sinus shaped) steering input varies from 0.1 Hz to5.0 Hz. To avoid nonlinear behaviour, the amplitude of the input has to be small to make sure that thelateral acceleration stays well below 4.5 m/s2.

Figure 6.8: Example of random steering input.

The same inflation pressure settings are used as in the sensitivity analysis of the steady-state circulartest, table 6.1. The simulation results from the three different settings are given in figure 6.9, 6.10 and6.11.

Changing the inflation pressure of both the front and rear tyres, has influence on the magnitude andthe phase delay of the lateral acceleration and the yaw velocity transfer function, see figure 6.9. Whenlowering the inflation pressure of all four tyres (i.e. pi=1.8 bar), themagnitude in the lateral accelerationand the yaw velocity transfer function increases slightly and the peak in magnitude of the yaw velocityoccurs at a higher frequency. Furthermore, less phase delay occurs in the lateral acceleration transfer

6.2 Random steering test 71

function for frequencies below 1 Hz. This is not in accordance to the inflation pressure effects on therelaxation behaviour of a tyre, see Section 4.2. Lowering the inflation pressure results in an increaseof the lateral relaxation length (i.e. more phase delay). On the other hand, in figure 6.3 it is shown thatthe axle cornering stiffness increases when the inflation pressure is lowered. This results in a decreasein phase delay. Because the decrease in phase delay as result of the axle cornering stiffness increasehas more effect than the phase delay increase as result of the increased lateral relaxation length, lessphase delay occurs for frequencies below 1Hz.

Figures 6.10 and 6.11 make clear that changing the inflation pressure of the rear tyres has more in-fluence on the lateral acceleration and yaw velocity response and magnitude level, than changing theinflation pressure of the front tyres. Changing the inflation pressure of the front tyres only shows slightchange in magnitude up to 1 Hz and no influence on the phase delay of the lateral acceleration andyaw velocity transfer functions.

10−1

100

102

103

lateral acceleration to δs

mag

nitu

de [m

/s2 ⋅

deg.

−1 ]

10−1

100

−100

−50

0

frequency [Hz]

phas

e [d

eg.]

10−1

100

10−1

yaw velocity to δs

mag

nitu

de [1

/s]

10−1

100

−120

−100

−80

−60

−40

−20

0

frequency [Hz]

phas

e [d

eg.]

p18p23p28

Figure 6.9: Transfer functions of the simulation model, lateral acceleration (left) and yaw velocity (right),inflation pressure setting 1; front and rear tyres have the same inflation pressure 1.8, 2.3 and 2.8 bar.


10−1

100

102

103


mag

nitu

de [m

/s2 ⋅

deg.

−1 ]

10−1

100

−100

−50

0

frequency [Hz]

phas

e [d

eg.]

10−1

100

10−1

yaw velocity to δs

mag

nitu

de [1

/s]

10−1

100

−120

−100

−80

−60

−40

−20

0

frequency [Hz]

phas

e [d

eg.]

p1923p2323p2723

Figure 6.10: Transfer functions of the simulation model, lateral acceleration (left) and yaw velocity (right),inflation pressure setting 2; constant inflation pressure rear tyres 2.3 bar, front tyres 1.9, 2.3 and 2.7 bar.

10−1

100

102

103


mag

nitu

de [m

/s2 ⋅

deg.

−1 ]

10−1

100

−100

−50

0

frequency [Hz]

phas

e [d

eg.]

10−1

100

10−1

yaw velocity to δs

mag

nitu

de [1

/s]

10−1

100

−120

−100

−80

−60

−40

−20

0

frequency [Hz]

phas

e [d

eg.]

p2319p2323p2327

Figure 6.11: Transfer functions of the simulation model, lateral acceleration (left) and yaw velocity (right),inflation pressure setting 3; constant inflation pressure front tyres 2.3 bar, rear tyres 1.9, 2.3 and 2.7 bar.

6.3 Ride analysis 73

6.3 Ride analysis

In this section the vehicle behaviour on an uneven road surface is analysed using the vehicle model offigure 6.1. The vehicle model is driving over an uneven road surface of a motorway with a constant ve-hicle velocity of 120 km/h. Again, two analyses are performed: an inflation pressure sensitivity analysisand an unloaded vs. loaded vehicle analysis. For these analyses the same inflation pressure settings areused as for the steady-state circular tests, see table 6.1 and table 6.3. The ride analysis will be assessedby means of change in ISO ride comfort index (corresponding ISO 2631 [22]), vertical acceleration ofthe vehicle body az and pitch acceleration of the body. The results for the sensitivity analysis are givenin table 6.5. In this table the inflation pressure setting front and rear 2.3 bar (p2323) is the base config-uration.

Table 6.5: Ride simulation results.

Influence inflation pressureInflation pressure setting p2323 p1818 p2828 p1923 p2723 p2319 p2327ISO ride comfort [m/s2] 0.180 −9.0% +3.5% −5.8% +1.1% −2.0% +0.7%body CG az [m/s2] 0.265 −1.0% +6.4% −4.8% +2.4% −3.3% +2.8%body pitch acc. [deg./s2] 33.7 −0.9% +1.7% +4.9% −1.3% −4.0% +3.8%

Table 6.5 makes clear that the ride comfort gets worse when increasing the inflation pressure. Themost influence occurs when increasing the inflation pressure of both the front and rear tyres, a changeof +3.5% is visible. Decreasing the inflation pressure leads to a better ride comfort, increasing the infla-tion pressure of both the front and rear tyres leads to a change of -9.0% of the ISO ride comfort index.Note that in previous research it is concluded that a 5% change in ISO ride comfort index is noticeableby an average driver [7]. Furthermore, it can be seen that lowering the inflation pressure of the fronttyres (i.e. p1923) leads to the largest decrease in vertical acceleration level az . Table 6.5 also shows thatchanging the inflation pressure of all four tyres has far less impact on the pitch acceleration of thebody.

The results of the unloaded vs. loaded vehicle analysis are presented in table 6.6. In this case therecommended inflation pressure setting for an unloaded vehicle (p1923) is the base configuration.

Table 6.6: Ride simulation results, unloaded vs. loaded vehicle.

Influence inflation pressureInflation pressure setting p1923 (unloaded) p1923 (loaded) p2328 (loaded)ISO ride comfort [m/s2] 0.169 −13.9% +7.8%body CG az [m/s2] 0.252 −13.2% +6.0%body pitch acc. [deg./s2] 35.4 −11.5% −11.5%

The results in table 6.6 make clear that the ride comfort gets better when the sprung mass of thevehicle is increased (loaded vehicle), also the vertical and pitch accelerations decrease. Furthermoretable 6.6 shows that increasing the inflation pressure of the loaded vehicle has a negative effect on theride comfort and corresponding vertical acceleration. The pitch acceleration shows no change whenthe inflation pressure of the loaded vehicle is increased.

Chapter 7

Conclusions and Recommendations

This Chapter presents the main conclusions with regard to this Master’s thesis. In addition, somerecommendations for further research are formulated.

7.1 Conclusions

The conclusions are split into three parts. The first part comprehends the influence of the inflationpressure on the tyre behaviour. The second part encompasses the enhancements and extension of theMF-Swift tyre model. The third part deals with some example cases of the applicability of the enhancedtyre model in vehicle design.

7.1.1 Inflation pressure influence

The following conclusions have been drawn with regard to the influence of the inflation pressure onthe investigated tyre behaviour:

• Characteristic stiffnessesIt is shown that for all characteristic stiffnesses, an almost linear relation with the inflation pres-sure exists. The tyre lateral, longitudinal and vertical stiffnesses show an increase with increasinginflation pressure, while the tyre torsional stiffness shows a decrease with increasing inflationpressure.

• Lateral relaxation lengthA linear decreasing relation between the lateral relaxation length and the inflation pressure ex-ists, i.e. the relaxation length becomes shorter when the inflation pressure increases.

• Effective rolling radiusThe influence of the inflation pressure on the effective rolling radius can be described with alinear relation. The influence is however very small.

• Camber effectsThe camber thrust, overturning and aligning camber torque exhibit a linear relation with theinflation pressure. The slope of this linear relation is depending on the vertical load conditionand the camber angle.

• Eigenfrequencies (primary tyre modes)It is shown that for all primary tyre modes, the influence of the inflation pressure on the eigen-frequencies can be described with a linear relation. The eigenfrequencies increase when theinflation pressure is increased.

75

76 Conclusions and Recommendations

The quantity that shows the highest inflation pressure influence is the camber thrust and the tyrevertical stiffness. The effective rolling radius shows the least influence.

7.1.2 Enhanced model equations

Regarding the inflation pressure enhanced equations, the following conclusions have been drawn:

• Tyre vertical stiffnessNo additional parameter identification work is required to implement inflation pressure depen-dency of the tyre vertical stiffness. The already implemented inflation pressure dependent verti-cal force relation can be used to describe the inflation pressure dependency of the tyre verticalstiffness.

• Tyre torsional stiffnessThe enhanced tyre torsional stiffness relation is based on the nominal tyre torsional stiffnessand a linear relation that depends on the normalised change in inflation pressure. The enhancedtyre torsional stiffness relation is made vertical load dependent by implementing a normalisedvertical load term.

• Lateral relaxation behaviourA pressure and load dependent relation for the lateral stiffness has been introduced. In addition,optimisation strategies are introduced to determine this pressure and vertical load dependentlateral stiffness from direct measurements or relaxation length experiments. Considerably moreaccurate results are achieved.

• Longitudinal relaxation behaviourA pressure and load dependent relation for the longitudinal stiffness has been introduced. In ad-dition, optimisation strategies are introduced based on the strategies as described for the lateralrelaxation behaviour.

• Rolling resistance forceThe rolling resistance force can be described more accurately by implementing two exponentialrelations that describe the inflation pressure and vertical load change in combination with thenominal rolling resistance force parameterisation (i.e. the rolling resistance force at nominalinflation pressure and vertical load condition).

• Camber thrust and camber torqueBy implementing inflation pressure dependency in the camber thrust stiffness equation, theoverturning moment equation and the aligning moment equation of the Magic Formula model,the influence of the inflation pressure on the camber thrust and camber torque can be describedmore accurately. For camber angles larger than the common passenger car camber angle rangethe results are less accurate. The cause is that for a combination of Flatplank pure camber dataand TNO Tyre Test Trailer slip data the Magic Formula fitting is less accurate. This can be causedby the deviations between Flatplank data, measured at low velocity on a steel surface, and TNOTyre Test Trailer data measured at relatively high velocity on an asphalt road.

• Tyre dynamicsThe different sidewall stiffnesses of the rigid ring model can be described and estimated ratheraccurately for a certain inflation pressure range (i.e. pi,nom ±1 bar) using the Rotta MembraneTheory. It has been shown that the primary eigenmodes of a free hanging tyre can be estimatedrather well. At pi=0 bar this approach gives implausible results, because Rotta neglects the influ-ence of the tyre material flexibility.

Scaling the pi,nom sidewall stiffnesses with the inflation pressure describes the sidewall stiff-nesses for a certain inflation pressure range more accurately. Also outside the mentioned infla-tion pressure range (e.g. 0 bar), this approach seems to give plausible results.

7.2 Recommendations for future research 77

7.1.3 Applicability to the vehicle design

Within the framework of applicability to the vehicle design it is shown that with the implementation ofinflation pressure dependency in the MF-Swift tyre model, the vehicle behaviour (i.e. handling and ridecomfort) can be analysed in an accurate way. The inflation pressure sensitivity of the tyre behaviour(i.e. change in: stiffnesses, transient behaviour, enveloping behaviour, etc.) on the ride comfort and(steady-state) handling behaviour can be predicted and the inflation pressure selection for a certainloading condition of the vehicle can be estimated. The inflation pressure enhancements are a usefuladdition to the MF-Swift tyre model.

7.1.4 Final conclusion

Considering the objectives of this thesis, it is concluded that all objectives formulated at the beginningof this research project have been achieved in a satisfying way. The tyre models have undergone signif-icant improvements, although several topics will require further investigations. These topics are listedin the next section where recommendations for further research are formulated.

7.2 Recommendations for future research

In future research on the area of the inflation pressure influence on tyre behaviour, the following topicsrequire further investigations:

• Camber effectsFor a symmetric tyre, the same camber thrust / torque level is expected when applying a posi-tive or negative camber angle. The Flatplank measurements show a dissimilarity between posi-tive and negative camber angles. Furthermore, the Flatplank results show some deviation whencomparing with TNO Tyre Test Trailer results, especially the cornering stiffness shows significantdeviations. This was also observed in previous research projects. The camber effects experimentsas performed on the Flatplank have to be executed with the TNO Tyre Test Trailer to check theFlatplank measurements and examine if the discovered camber thrust / torque trends can bereproduced.

• Flatplank Tyre TesterWithin the measurement results of the tyre lateral stiffness and the tyre torsional stiffness someirregularities can be seen. Most probably, these irregularities are stick-slip effects caused by alack of grip on the Flatplank Tyre Tester. To solve this problem, the road surface on the Flatplanktyre tester can be adapted such that a higher level of friction is obtained.

Furthermore, within the measurement results of the pure camber effects (i.e. camber thrust andcamber torque) some deviance occurs between positive and negative camber angles. This can becaused by a wrong or off calibration of the measuring hub. To exclude this option, a calibrationof the measurement hub is advisable.

• Rigid ring tyre dynamicsThe two proposed approaches to estimate the sidewall stiffnesses (i.e. Rotta membrane theoryand scaling nominal sidewall stiffnesses) are only validated for inflation pressures above 1.5 bar,because FEM analyses are only performed for the inflation pressure range 1.5 - 3.0 bar. To inves-tigate the accuracy of the two approaches for inflation pressures below 1.5 bar, it is recommendedto perform FEM analyses for inflation pressures below 1.5 bar.

The proposed approaches are only validated with FEM simulation data of a free hanging andnon-rolling tyre. To create a wider validation support, high speed cleat experiments have to beperformed with the tyre used in this thesis. It should be checked that the eigenfrequencies found

78 Conclusions and Recommendations

in the cleat experiments at different inflation pressures can be described with the proposed rela-tions.

The nominal sidewall stiffnesses scaling strategy is optimised to describe the FEM results asaccurately as possible. Currently fixed values for these parameters have been selected and imple-mented in the MF-Swift tyre model. However, these parameters have to be implemented in theMF-Swift optimisation routine in such a way that they are optimised for the tyre that is beingprocessed.

• Validation of the enhanced relationsIn this thesis the enhanced relations are validated with one single type of tyre. It is recommendedto conduct more measurements with different type of tyres and a wider inflation pressure rangeto investigate if the enhanced relations still hold and to create a wider validation base.

• Enhanced tyre modelsNow all the necessary inflation pressure enhancements are separately validated and imple-mented in the MF-Swift tyre model, it is interesting to investigate if the accuracy of the modelis increased when analysing experiments (e.g. ABS braking on an uneven road) where the com-plete (dynamical) behaviour of a tyre is important. With the enhanced tyre models it should bepossible to describe the tyre behaviour more accurately for a certain inflation pressure range.

Initially, the investigation of the turnslip behaviour was also a part of the research project. Later on, thispart is left out because of the extensive inflation pressure part. However, a lead for the investigationof tyre turnslip behaviour will be given. Several tests have already been executed on the FlatplankTyre Tester, which seem to confirm that improvements to the MF-Swift tyre model are possible. Thefollowing topics can be a lead for the investigation:

• Parameter identification methodThe goal is to develop a simple and reliable fitting process, with a focus on an automated para-meter identification method and a critical review of the required tests.

• Standstill situationThe representation of the turnslip forces and moments at standstill and the transition to lowforward velocity need to be analysed.

Bibliography

[1] ABAQUS. Example Problems Manual. ABAQUS Documentation, Version 6.6, Section 3.1,ABAQUS inc., 2006.

[2] TNO Automotive. Tyre models users manual; Using the MF-Tyre Model. TNO Automotive, May2002.

[3] E. Bakker, H.B. Pacejka, and L. Lidner. A New Tire Model with an Application in Vehicle DynamicsStudies. SAE Paper No. 890087, 1989.

[4] P. Bandel and C. Monguzzi. Simulation Model of the Dynamic Behaviour of a Tire Running Over anObstacle. Tire Science and Technology, TSTCA, Volume 16, No. 2, 1988.

[5] F. Bohm. Zur Mechanic des Luftreifens. Technische Hochschule Stuttgart, 1966.

[6] I.J.M. Besselink. Shimmy of Aircraft Main Landing Gears. Dissertation, TU Delft, 2000.

[7] I.J.M. Besselink, L.W.L. Houben, and I.B.A. op het Veld. Run-Flat versus Conventional Tyres: Anexperimental and model based comparison. VDI Berichte, No. 2014, pp. 185-202, 2007.

[8] M. Biancolini and R. Baudille. Integrated Multibody / FEM Analysis of Vehicle Dynamic Behaviour.The 29th FISITA World Automotive Congress, Helsinky, 2002.

[9] S.K. Clark. Mechanics of Pneumatic Tires. U.S. Dept. of Transportation, National Highway TrafficSafety Administration, Washington D.C., 1982.

[10] B. Collier and J. Warchol. The Effect of Inflation Pressure on Bias, Bias-Belted and Radial Tire Perfor-mance. SAE Paper No. 800087, 1980.

[11] R. Cremers. Investigating Dynamic Tyre Behaviour. Master’s thesis, Technical University Eind-hoven, DCT 2005.37, 2005.

[12] J. de Hoogh. Implementing Inflation Pressure and Velocity Effects into The Magic Formula Tyre Model.Master’s thesis, Eindhoven University of Technology,2005.46, 2005.

[13] N. Dodge, D. Orne, and S. Clark. Fore-and-aft Stiffness Characteristics of Pneumatic Tyres. NASAContractor Report, CR-900, Washington D.C., 1967.

[14] D.J. Erp and L. Verhoeff. The Development of Tyre Modeling at Various Direction Input. TNO Auto-motive Report, 04.OR.AC.016.1/LV, Helmond, 2004.

[15] M. Gipser. BRIT version 2. Documentation, Reading Massachusetts, 1980.

[16] S. Gong. A Study of In-plane Dynamics of Tires. Dissertation, TU Delft, 1993.

[17] P.S. Grover. Modeling of Rolling Resistance Test Data. SAE Paper No. 980251, 1998.

[18] P.S. Grover and S.H. Bordelon. New Parameters for comparing Tire Rolling Resistance. SAE PaperNo. 1999-01-0787, 1999.

79

80 BIBLIOGRAPHY

[19] A. Higuchi. Transient Response of Tyres at Large Wheel Slip and Camber. Dissertation, TU Delft,2000.

[20] ISO. ISO 8767: Passenger car tyres, Methods of measuring rolling resistance. International Organiza-tion for Standardization, 1992.

[21] ISO. ISO 4138: Passenger Cars, Steady-state Circular Driving Behaviour, Open-loop Test Procedure.International Organization for Standardization, 1996.

[22] ISO. ISO 2631: Mechanical vibration and shock, Evaluation of human exposure to whole-body vibration.International Organization for Standardization, 1997.

[23] ISO. ISO 7401: Road Vehicles, Lateral Transient Response Test Methods, Open-loop Test Methods. In-ternational Organization for Standardization, 2003.

[24] I. Kageyama and S. Kuwahara. A study on Tire Modeling on Camber Thrust and Camber Torque.JSAE Review, No. 23, pp. 325-331, 2002.

[25] K.M. Marshek and J.F. Cuderman. Performance of Anti-Lock Braking system Equipped PassengerVehicles-Part III: Braking as a function of Tire Inflation Pressure. SAE Paper No. 2002-01-0306,2002.

[26] J.P. Maurice. Short Wavelength and Dynamic Tyre Behaviour under Lateral and Compined Slip Con-ditions. Dissertation, TU Delft, 2000.

[27] Michelin. Der Reifen. Michelin Reifenwerke KGaA, [ISBN 2-06-711658-4], 2005.

[28] H.B. Pacejka. Tyre and Vehicle Dynamics. Butterworth-Heinemann, 2002.

[29] A. Popov, D. Cole, D. Cebon, and C. Winkler. Laboratory Measurement of Rolling Resistance inTruck Tyres under Dynamic Vertical Load. Proceedings of the Institution of Mechanical Engineers,Volume 217, No. 12, pp. 1071-1079, 2002.

[30] J. Rotta. Zur Statik des Luftreifens. Ingenieur-Archive, Volume 17, pp. 12-141, Berlin, 1949.

[31] A.J.C. Schmeitz. A Semi-Emperical Three-Dimensional Model of the Pneumatic Tyre Rolling overArbitrarily Uneven Road Surfaces. Dissertation, TU Delft, 2004.

[32] A.J.C. Schmeitz, J. de Hoogh, I.J.M. Besselink, andH. Nijmeijer. Extending TheMagic Formula andSWIFT Tyre Models for Inflation Pressure Changes. In: 10th International VDI Congress, Hannover,2005.

[33] R. Smiley and W. Horne. Mechanical properties of Pneumatic Tires with Special Reference to ModernAircraft Tires. Technical Report NASA, No. R-64, Langley Research Center, USA, 1960.

[34] H.J. Unrau and J. Zamow. TYDEX; Description and Reference Manual. Release 1.1, InternationalTire Working Group, 1995.

[35] P. van der Jagt. The Road to Virtual Vehicle Prototyping. Dissertation, TU Eindhoven, 2000.

[36] J. van Honk, C.W. Klootwijk, and F.W. Laméris. Results of handling behaviour test program, Maxi-mum load condition. TNO Automotive Report, 96.MR.VD.038.1/JVH, 1996.

[37] J. van Honk, C.W. Klootwijk, and F.W. Laméris. Results of handling behaviour test program, Mini-mum load condition. TNO Automotive Report,96.MR.VD.039.1/JVH, 1996.

[38] V.C. Vlad. THERMAL TYRE: An Analysis Tool for Tire Temperature Modeling. TNO AutomotiveReport, 2001.

[39] L.H. Yam, D.H. Guan, and A.Q. Zhang. Three-dimensional Mode Shapes of a Tire Using Experimen-tal Modal Analysis. Experimental Mechanics, Volume 40, No. 4, pp. 369-375, 2000.

BIBLIOGRAPHY 81

[40] P.W.A Zegelaar. The Dynamic Response of Tyres to Brake Torque Variations and Road Unevennesses.Dissertation, TU Delft, 1998.

82 BIBLIOGRAPHY

Appendix A

The Flatplank Tyre Tester

A.1 General description

With the Flatplank Tyre Tester measurements can performed to investigate tyre characteristics: (com-bined) slip characteristics, relaxation lengths, stiffnesses and tyre response to road irregularities.

In figure A.1 the Flatplank Tyre Tester is depicted. The Flatplank consists of a flat steel road surface,measuring hub and a turn table. Furthermore, it consists of mechanisms to adjust the axle height, thelateral position and the camber angle. The road surface is positioned upside down and can be movedwith a constant velocity. To create a slip angle, the tyre and the measuring hub can be rotated aboutthe vertical axis of the turn table. To change the lateral position of the tyre with respect to the roadsurface, the measuring hub can be moved in lateral direction. The axle height can be adjusted by ajack (constant axle height) or by an air spring system (constant vertical load). The camber angle can beadjusted by rotating the road surface around the longitudinal axis on the middle of its surface and/or byrotating the measuring hub around its longitudinal axis. Different obstacles (cleats) with a maximumheight of 30 mm can be mounted on the road surface to create road irregularities.

road cambermechanism obstacle

strain gaugemeasuring hub

wheel carrier

road surface (plank)

air spring system

Figure A.1: The Flatplank Tyre Tester [31].

The steel surface of the plank is covered with 3M Safetywalk grid 80 to give certain roughness tothe road surface. The road surface velocity can be varied between 0 and 4.75 cm/s. Furthermore, themeasuring hub measures reaction forces and moments in the wheel axle. The following forces andmoments are measured: longitudinal force Fx, lateral force Fy , vertical force Fz , overturning momentMx and aligning momentMz . The measuring hub can measure forces (up to 10 kN) and moments byusing five strain gauge bridges, see next section. The angle and velocity of rotation of the wheel, and thedisplacement and velocity of the road surface are measured with incremental encoders. Furthermore,the slip angle and the vertical displacement of the measuring hub are measured. The measured signalsare sent to the signal conditioning system. This system contains an amplifier, power supply, low-pass

83

84 The Flatplank Tyre Tester

filter and an A/D converter. After the A/D conversion the data is sampled by a data acquisition program(in this case LABView) which is triggered by a digital triggering signal.

A.2 Forces and moments transformations

Figure A.2 depicts the measuring hub. Asmentioned before, the measuring hubmeasures the reactionforces andmoments by using five strain gauge bridges. The strain gauge bridges sense the longitudinalforces (Gx1 and Gx2), the lateral force (Gy) and the vertical forces (Gz1 andGz2). To calculate themoments at the wheel centre the distance a between the forcesGx1 andGx2 and the distance b betweenGx2 and the wheel centre plane are used.

Figure A.2: Measuring hub [19].

The forcesK and moments T at the wheel centre are defined as follows:

Kx = Gx1 +Gx2 (A.1)Ky = Gy (A.2)Kz = Gz1 +Gz2 (A.3)Tx = Gz1(a+ b) +Gz2b (A.4)Ty = 0 (A.5)Tz = Gx1(a+ b) +Gx2b (A.6)

The coordinate system of the measuring hub and at the contact centre is depicted in figure A.3. Theforces and moments measured at the measuring hub are transformed to forces and moments at thecontact patch (ISO coordinate system). According to figure A.3, the transformations are as follows:

α = ψ (A.7)γ = φwheel + φplank (A.8)Fx = Kx (A.9)Fy = −Ky cos γ −Kz sin γ (A.10)Fz = −Ky sin γ +Kz cos γ (A.11)Mx = −Tx +Kyrl (A.12)My = −Tz sin γ +Kxrl cos γ (A.13)Mz = Tz cos γ +Kxrl sin γ (A.14)

A.2 Forces and moments transformations 85

Note that rl is the loaded radius and consequently has a positive value in these equations. Furthermoreφwheel and φplank are the inclination angle of the wheel and the steel surface plank respectively.

Figure A.3: Coordinate systems of measuring hub (no ISO) and contact patch centre (ISO).

In general when studying tyre dynamics, two different types of coordinate systems are used. The coor-dinate systems are the C-axis system and the W-axis system. In figure A.4 the orientation of these axissystems is depicted.

Figure A.4: C- and W-axis system [2].

These axis systems are in accordance with the standard TYDEX conventions [34]. The C-axis systemis the centre axis system with its origin fixed in the wheel centre. The W-system is the standard ISObased axis system with its origin at the wheel intersection point with the road, shown in figure A.3.

To transform the measuring hub forces and moments to the C-axis system, the following transforma-tions hold:

FxC = Kx (A.15)FyC = −Ky (A.16)FzC = Kz (A.17)MxC = −Tx (A.18)MyC = 0 (A.19)MzC = Tz (A.20)

86 The Flatplank Tyre Tester

Appendix B

Measurement Results

As described in chapter 4, the measurements are carried out at three vertical load conditions. Theresults of the nominal vertical load conditions are discussed in the different sections of chapter 4.In this Appendix the result of the other vertical load conditions are presented. First the results ofthe characteristic stiffnesses are discussed, after that the remaining results of the lateral relaxationbehaviour and the camber influence experiments are presented.

B.1 Characteristic stiffnesses

B.1.1 Tyre longitudinal stiffness

This subsection presents the results of the tyre longitudinal stiffness measurements for the followingvertical load conditions: 2 and 6 kN. In figure B.1, the experimental results are presented. The linearfits of the selected linear part are depicted in figure B.2.

0 0.01 0.02 0.03 0.04 0.05 0.06 0.070

1000

2000

3000

4000

5000

6000

7000

8000

9000

10000


Long

itudi

nal f

orce

, Fx [N

]

p19p22p25p27p30

0 0.01 0.02 0.03 0.04 0.05 0.06 0.070

1000

2000

3000

4000

5000

6000

7000

8000

9000

10000


Long

itudi

nal f

orce

, Fx [N

]

p19p22p25p27p30

Figure B.1: Tyre longitudinal stiffness; experimental results for Fz=2 kN (left) and Fz=6 kN (right).

From the graph of Fz=2 kN n figure B.1, it can be seen that at an inflation pressure of 1.9 bar fluctua-tions occur when reaching the steady state longitudinal force level. These fluctuations are the result ofstick-slip due to the insufficient friction level of the Flatplank combined with the low inflation pressure

87

88 Measurement Results

and the low vertical load condition. The stick-slip situation does not occur during the build up of thelongitudinal force, so there is no influence on the determination of the longitudinal stiffness.

0 0.01 0.02 0.03 0.04 0.05 0.06 0.070

2000

4000

6000

8000

10000

12000

14000

16000

18000


Long

itudi

nal f

orce

, Fx [N

]

p19p22p25p27p30

0 0.01 0.02 0.03 0.04 0.05 0.06 0.070

2000

4000

6000

8000

10000

12000

14000

16000

18000


Long

itudi

nal f

orce

, Fx [N

]

p19p22p25p27p30

Figure B.2: Tyre longitudinal stiffness; linear fit results for the selected linear part for Fz=2 kN (left) andFz=6 kN (right).

B.1.2 Tyre lateral stiffness

Following section 4.1.3, in this section the tyre lateral stiffness measurement results for the verticalload conditions of 3 and 7 kN are presented. Figure B.3 shows the results of the lateral force versusthe lateral displacement. The linear fits for the determination of the tyre lateral stiffness are depictedin figure B.4. Again, the fluctuations that occur in several measurements are the result of stick-slip,which occurs due to the low friction level of the Flatplank.

0 0.01 0.02 0.03 0.04 0.05 0.06 0.070

1000

2000

3000

4000

5000

6000

7000

8000


Late

ral f

orce

, Fy [N

]

p19p22p25p27p30

0 0.01 0.02 0.03 0.04 0.05 0.06 0.070

1000

2000

3000

4000

5000

6000

7000

8000


Late

ral f

orce

, Fy [N

]

p19p22p25p27p30

Figure B.3: Tyre lateral stiffness; experimental results for Fz=3 kN (left) and Fz=7 kN (right).

B.1 Characteristic stiffnesses 89

0 0.01 0.02 0.03 0.04 0.05 0.06 0.070

1000

2000

3000

4000

5000

6000

7000

8000


Late

ral f

orce

, Fy [N

]

p19p22p25p27p30

0 0.01 0.02 0.03 0.04 0.05 0.06 0.070

1000

2000

3000

4000

5000

6000

7000

8000


Late

ral f

orce

, Fy [N

]

p19p22p25p27p30

Figure B.4: Tyre lateral stiffness; linear fit results for the selected linear part for Fz=3 [kN] (left) andFz=7 [kN] (right).

B.1.3 Tyre torsional stiffness

The parking experiments are performed for the five generally defined inflation pressures and for awider range of vertical load conditions; FzW=1, 2, 3, 4, 5 and 6 kN. In section 4.1.4 the results arepresented for the vertical load conditions: FzW=2 and 4 kN. Here, an overview of the remaining Mz

results is given in figure B.5. From left to right and from top to bottom the plots refer to the follow-ing vertical load conditions: 1, 3, 5 and 6 kN respectively. To improve readability and visibility of thedifferences in the separate inflation pressures, the graphs are plotted at different y-axis scaling.


−25 −20 −15 −10 −5 0 5 10 15 20 25−25

−20

−15

−10

−5

0

5

10

15

20

25


Alig

ning

mom

ent,

Mz [N

m]

p19p22p25p27p30

−25 −20 −15 −10 −5 0 5 10 15 20 25−150

−100

−50

0

50

100

150


Alig

ning

mom

ent,

Mz [N

m]

p19p22p25p27p30

−25 −20 −15 −10 −5 0 5 10 15 20 25−250

−200

−150

−100

−50

0

50

100

150

200

250


Alig

ning

mom

ent,

Mz [N

m]

p19p22p25p27p30

−25 −20 −15 −10 −5 0 5 10 15 20 25−300

−200

−100

0

100

200

300


Alig

ning

mom

ent,

Mz [N

m]

p19p22p25p27p30

Figure B.5: Parking behaviour; experimental results for different vertical load conditions: 1 kN (top left),3 kN (top right), 5 kN (bottom left) and 6 kN (bottom right).

B.2 Lateral relaxation behaviour 91

B.2 Lateral relaxation behaviour

This section presents the results of the lateral relaxation behaviourmeasurements for the following ver-tical load conditions: 3 and 7 kN. In figure B.6 the experimental results are presented for the 1 degreesideslip lateral relaxation length experiment. Unfortunately afterwards it appeared that the measure-ment duration of the 7 kN vertical load condition is pretty tight in order to reach a clear steady stateFy,ss level. An extrapolation of the exponential function, used to fit the measurement data, is made toestimate the correct Fy,ss level.

−0.5 0 0.5 1 1.5 2 2.5−2500

−2000

−1500

−1000

−500

0


Late

ral f

orce

, Fy [N

]

p19p22p25p27p30

0 0.5 1 1.5 2 2.5 3−2500

−2000

−1500

−1000

−500

0


Late

ral f

orce

, Fy [N

]

p19p22p25p27p30

Figure B.6: Lateral relaxation length vs. inflation pressure for FzW =3 kN (left) and FzW =7 kN (right).

B.3 Camber thrust and camber torque

In section 4.4 the camber thrust Fyγ , aligning camber torque Mzγ and overturning camber torqueMxγ in relation to the inflation pressure for the nominal vertical load condition are presented. Here,the results of the remaining vertical load conditions, FzW=3 and 7 kN, are presented. Figure B.7 showsthe camber thrust for FzW= 3 and 7 kN respectively.

1.8 2 2.2 2.4 2.6 2.8 3−600

−400

−200

0

200

400

600

800


Cam

ber

thru

st, F

yγ [N

]


1.8 2 2.2 2.4 2.6 2.8 3−2000

−1500

−1000

−500

0

500

1000

1500

2000


Cam

ber

thru

st, F

yγ [N

]


Figure B.7: Camber thrust vs. inflation pressure for various positive and negative camber angles forFzW =3 kN (left) and FzW =7 kN (right).


1.8 2 2.2 2.4 2.6 2.8 3−30

−20

−10

0

10

20

30

40


Cam

ber

torq

ue, M

zγ [N

m]


1.8 2 2.2 2.4 2.6 2.8 3−100

−80

−60

−40

−20

0

20

40

60

80

100


Cam

ber

torq

ue, M

zγ [N

m]


Figure B.8: Aligning camber torque vs. inflation pressure for various positive and negative camber anglesfor FzW =3 kN (left) and FzW =7 kN (right).

The resulting aligning camber torque for the remaining vertical load conditions is depicted in figureB.8. Taking a closer look at the aligning camber torque trend for the camber range [-5 5] degrees in fig-ure B.8 an opposite trend is visible compared to figure B.8(right) and figure 4.17. In figure B.8(right)the aligning camber torque increases with increasing inflation pressure while in figure B.8(left) and4.17 a decreasing trend is visible. As mentioned in section 4.4, the tested tyre has a symmetrical treadso the amount of camber thrust and aligning camber torque for positive and negative camber angleswould be expected to be equal. However, figures B.7 till B.8 show that a difference occurs when cam-bering with a positive or a negative angle. The difference can be caused by a deviation or a wrongcalibration of the measuring hub.

In figure B.9 the overturning camber torque is presented for FzW= 3 and 7 kN. It is shown that whenincreasing the inflation pressure also the amount of overturning camber torque increases. Again, anasymmetric behaviour is visible between positive and negative camber angles.

1.8 2 2.2 2.4 2.6 2.8 3−250

−200

−150

−100

−50

0

50

100

150

200

250


Cam

ber

torq

ue, M

xγ [N

m]


1.8 2 2.2 2.4 2.6 2.8 3−500

−400

−300

−200

−100

0

100

200

300

400

500


Cam

ber

torq

ue, M

xγ [N

m]


Figure B.9: Overturning camber torque vs. inflation pressure for various positive and negative camberangles for FzW =3 kN (left) and FzW =7 kN (right).

Finally, to quantify the influence of the inflation pressure, table 4.7 shows the amount of influence onthe camber thrust, aligning camber torque and overturning camber torque. The influence is subdividedin two camber ranges. The first range is limited to -5 and 5 degrees (common range for passenger cartyres) and the second range is limited to -15 to 15 degrees (complete measurement range).

B.3 Camber thrust and camber torque 93

Table B.1: The influence of the inflation pressure on camber effects.

Influence inflation pressure εA [%]FzW [kN] Camber range [deg] Fyγ Mzγ Mxγ

3 -5...5 34.0 13.0 9.0-15...15 15.0 18.0 1.0

7 -5...5 14.0 23.0 9.5-15...15 17.0 16.5 2.0

Appendix C

Modal Analysis

Finite Element Method (FEM) analysis and analytical analysis of the flexible ring model and the rigidringmodel are performed to get a clear view on the influence of the inflation pressure. In this Appendixthe FEM simulations with ABAQUS and the analytical analysis are discussed in more detail.

C.1 FEM Simulations (ABAQUS)

In chapter 4, it is shown that the inflation pressure influence is analysed by means of Finite ElementAnalysis. In the FEM software ABAQUS a 3 dimensional model of a 175 SR14 passenger car tyre isavailable . In this Appendix a short explanation of the model structure is given, which is derived fromthe ABAQUS problems manual [1], and the eigenmodes of the free hanging tyre and standing loadedtyre are presented.

Figure C.1: Construction sequence in ABAQUS of the 3D tyre model [1].

95

96 Modal Analysis

The construction sequence of the 3 dimensional model is illustrated in figure C.1. The tyre tread andsidewalls are made of rubber, and the belts and carcass are constructed from reinforced rubber compos-ites. The rubber is modeled as an incompressible hyperelastic material, and the reinforced compositesare modeled as a linear elastic material. First a 2 dimensional axisymmetic model is created of the side-wall, tread, reinforced carcass and belts. The 2 dimensional model is rotated 360 degrees in steps of10 degrees, to create a partial 3 dimensional model (half tyre). Finally the partial 3 dimensional modelis mirrored to a full 3 dimensional model. The air inside the tyre is modeled with acoustic elements,which obviously are made of air. In figure C.2 air elements as used in the tyre model are depicted.

Figure C.2: Air elements inside the tyre model [1].

C.1.1 Free hanging tyre

The primary eigenmodes of a free hanging tyre with fixed rim are discussed in chapter 5. In thissection a visualisation of the mode shapes is given. Figure C.3 and C.4 show the in-plane andout-of-plane primary mode shapes respectively.

Figure C.3: In-plane primary mode shapes of a free hanging tyre: rotational(left) and radial(right).

C.1 FEM Simulations (ABAQUS) 97

Figure C.4: Out-of-plane primary mode shapes of a free hanging tyre: lateral(left), yaw and camber(right).

C.1.2 non-rolling loaded tyre

In this section a visualisation of the primary mode shapes is given for a non-rolling loaded tyre withFz=4 kN. Figure C.5 and C.6 show the in-plane and out-of-plane primary mode shapes respectively.Note that the contact patch nodes are modelled to be welded/fixed to the road.

Figure C.5: In-plane primary mode shapes of a non-rolling loaded tyre: rotational(left), vertical(middle)and longitudinal(right).

Figure C.6: Out-of-plane primary mode shapes of a non-rolling loaded tyre: lateral/camber(left) andyaw(right).

98 Modal Analysis

C.2 Analytical analysis

The rigid ring model and the flexible ring model are used to determined the influence of the inflationpressure on the primary eigenmodes of a tyre in an analytical way. The analytical analysis is basedon the findings of Zegelaar [40] and Gong [16]. The following situation is used: a free hanging tyre(flexible ring model) with a fixed rim, to analyse the primary modes (i.e. eigenmodes in which the tyretread band almost retains its circular shape). The flexible ring model is based on the findings of Gong[16], see figure C.7. The parameters used in the analysis are derived from Zegelaar [40] and depictedin table C.1.

Figure C.7: Flexible ring model and carcass cross section [40].

Table C.1: Used parameters derived from [40].Parameters

Quantity V alue Descriptionmb [kg] 7.1 mass of tyre ringIby [kgm2] 0.636 moment of inertia that moves with the tyre ringr [m] 0.300 tyre ring radiusls [m] 0.121 length of sidewall arct [m] 0.010 thickness of tyre sidewallG [N/m2] 1.6e6 shear modules of tyre sidewallϕs [deg] 62.3 half the angle of tyre sidewall

Following Gong and Zegelaar, the tangential sidewall stiffness cbv and the radial sidewall stiffness cbwof the flexible ring model can be determined by:

cbv =Gt

ls+ pi

1tanϕs

(C.1)

cbw = picosϕs + ϕs sinϕssinϕs − ϕs cosϕs

(C.2)

Because only the primary modes are analysed, the flexible ring model can be seen as a rigid ring model,see figure C.8.

C.2 Analytical analysis 99

The translational cb and rotational cbθ sidewall stiffness of the rigid ring model, see figure C.8, arerelated to cbv and cbw of the flexible ring model as follows [40]:

cb = πr(cbv + cbw) (C.3)cbθ = 2πcbvr3 (C.4)

Figure C.8: Rigid ring model [40].

The translational eigenfrequency ftrans and the rotational eigenfrequency frot of the free hangingrigid ring model can be determined with the following expressions:

ftrans =12π

√cbmb

(C.5)

frot =12π

√cθbIby

(C.6)

In figure C.9 the resulting eigenfrequencies are depicted as function of the inflation pressure. Again,a linear trend can be observed between the primary eigenmodes and the inflation pressure.

1.5 2 2.5 335

40

45

50

55

60

65

70


Nat

ural

freq

uenc

y [H

z]

ftrans

frot

Figure C.9: Eigenfrequency versus inflation pressure for rigid ring model of a free hanging tyre.

100 Modal Analysis

Appendix D

Aspects of the introduced inflationpressure dependent relations

In this Appendix important aspects of the proposed enhancements and introduced relations aretreated. First, the relation between the torsional stiffness and the vertical load as derived from theFlatplank measurements will be presented. After that, the rolling resistance force data, which is usedto assess the proposed rolling resistance force enhancement, is treated and the robustness of the (car-cass)stiffnesses equations of the rigid ring model are presented. Finally, the extrapolation properties ofthe camber thrust and torque equations are treated.

D.1 Tyre torsional stiffness

In this section, the relation between the determined torsional stiffness and the vertical force load ofthe Flatplank measurements is presented. The torsional stiffness values are derived from the parkingbehaviour Flatplank experiments. The results are depicted in figure D.1. It can be seen that a more orless linear relation exists between the torsional stiffness and the vertical load. At the different verticalload conditions, the inflation pressures do not show an unambiguous trend; the maxima and minimafluctuate. For example the highest torsional stiffness at Fz = 4 kN is when pi = 1.9 bar, while atFz = 2 kN the highest torsional stiffness is shown at pi = 2.2.

Figure D.1: Torsional stiffness as function of the vertical load, Flatplank measurements and linear trendlines.

101

102 Aspects of the introduced inflation pressure dependent relations

D.2 Rolling resistance force

As mentioned in section 5.2, no suitable rolling resistance data at different inflation pressures is avail-able from the measured tyre in this thesis or another tyre in the TNO tyre database. Therefore, aselection of data found in literature [18] will be used to evaluate the quality of the new formula. Themeasurements in this publication are performed according to the ISO 8767 international standard[20].

Table D.1: Rolling resistance measurement data of a 195/70R14 tyre [18].

Rolling resistance conditionsVx [m/s] pi [bar] Fz [N] Fx,RR [N]

2.5 1800 142.75 2380 17

5.6 2.0 3560 273.0 5360 352.0 5360 412.5 1800 15

2.75 2380 189.7 2.0 3560 29

3.0 5360 372.0 5360 432.5 1800 16

2.75 2380 1915.3 2.0 3560 31

3.0 5360 392.0 5360 462.5 1800 21

2.75 2380 2530.6 2.0 3560 40

3.0 5360 482.0 5360 59

An overview of the optimised Magic Formula parameters of the "old" relation, equation (5.20), and the"new" relation, equation (5.21), are presented in table D.2.

Table D.2: Overview of the Magic Formula parameter values for the "old" rolling resistance relation (5.20)and the "new" rolling resistance relation (5.21).

Rolling resistance Magic Formula parametersParameter equation (5.20) equation (5.21)

qsy1 0.00673 0.00730qsy2 0.0 0.0qsy3 0.00144 0.00157qsy4 0.00008 0.00009qsy5 0.0 0.0qsy6 0.0 0.0qsy7 − 0.9qsy8 − −0.409

D.3 Stiffnesses of the rigid ring model extrapolation properties 103

D.3 Stiffnesses of the rigid ring model extrapolation proper-ties

In this section an overview of the lateral and longitudinal (carcass) stiffnesses of the rigid ring modelfor two extreme vertical load conditions, Fz =0.1 and 10 kN, is given. Note that these stiffnessesare estimated using the, in [32] and in this thesis, introduced inflation pressure relations and usingextrapolation. The results presented in this section are for robustness interpretation only.

D.3.1 Lateral situation

In figure D.2 an overview of the stiffnesses for the lateral rigid ring model is depicted. The stiffnessesfor the extreme load conditions are estimated using MF-Tool, equation (5.15) and the strategy as de-scribed in sections 5.7.2 and 5.7.3. The stiffnesses show robust results for the extreme load conditions,no negative values arise.

0 0.5 1 1.5 2 2.5 30

1

2

3

4

5

6x 10

6


Stif

fnes

s [N

/m]

C

Fy

CFα/a

cy

cbγ/rl

2

cby

cry

Trend line

0 0.5 1 1.5 2 2.5 30

2

4

6

8

10

12

14x 10

5


Stif

fnes

s [N

/m]

C

Fy

CFα/a

cy

cbγ/rl

2

cby

cry

Trend line

Figure D.2: Overview lateral (carcass) stiffnesses as function of the inflation pressure for Fz =0.1 kN(left) and Fz =10.0 kN (right).


D.3.2 Longitudinal situation

In figure D.3 an overview of the stiffnesses for the longitudinal rigid ring model is depicted. The stiff-nesses for the extreme load conditions are estimated using MF-Tool, equation (5.19) and the strategyas described in sections 5.7.2 and 5.7.4. In accordance with the lateral situation, the longitudinal stiff-nesses show robust results for the extreme load conditions.

0 0.5 1 1.5 2 2.5 30

0.5

1

1.5

2

2.5x 10

6


Stif

fnes

s [N

/m]

C

Fx

CFκ/a

cx

cbθ/r

l2

cbx

crx

Trend line

0 0.5 1 1.5 2 2.5 30

0.5

1

1.5

2

2.5

3x 10

6


Stif

fnes

s [N

/m]

C

Fx

CFκ/a

cx

cbθ/r

l2

cbx

crx

Trend line

Figure D.3: Overview longitudinal (carcass) stiffnesses as function of the inflation pressure for Fz =0.1kN (left) and Fz =10.0 kN (right).

D.4 Camber thrust and camber torque extrapolation

In this section an overview is given of the camber thrust and both the aligning and overturning cambertorque for two extreme vertical load conditions (Fz =0.1 and 10 kN). In table D.3 an overview is givenof the manually tuned parameters of the in Section 5.6 proposed equations to describe the camberthrust and camber torque. The parameters are tuned for the nominal vertical load condition (Fz,nom=5 kN) and the common passenger car camber range (i.e. till γ ≤+/- 5.0 degrees).

In the figures D.4, D.5 and D.6 the Magic Formula results are presented for various positive and nega-tive camber angles; the markers indicate the evaluated camber angles (i.e. in this case the markers donot represent measurement points). The enhancements, proposed in section 5.6, show robust resultsfor the extreme load conditions. Note that the results for Fz =0.1 and 10 kN are produced with thecamber parameters as presented in table D.3.

Table D.3: Overview of the manually tuned camber parameters.

Parameter V alue Equationppy5 −0.15 (5.22)ppMx1 −0.10 (5.25)ppz2 −0.60 (5.27)

D.4 Camber thrust and camber torque extrapolation 105

1.8 2 2.2 2.4 2.6 2.8 3−40

−30

−20

−10

0

10

20

30


Cam

ber

thru

st, F

yγ [N

]

−15.0 deg−10.0 deg−5.0 deg−2.5 deg−2.0 deg−1.5 deg−1.0 deg−0.5 deg0.0 deg0.5 deg1.0 deg1.5 deg2.0 deg2.5 deg5.0 deg10.0 deg15.0 degTrend line (neg)Trend line (pos)

1.8 2 2.2 2.4 2.6 2.8 3−4000

−3000

−2000

−1000

0

1000

2000

3000

4000


Cam

ber

thru

st, F

yγ [N

]


Figure D.4: Camber thrust vs. inflation pressure for various positive and negative camber angles at twoextreme vertical load conditions; Fz=0.1 kN (top) and Fz=10 kN (bottom).


1.8 2 2.2 2.4 2.6 2.8 3−60

−40

−20

0

20

40

60


Cam

ber

torq

ue, M

xγ [N

m]


1.8 2 2.2 2.4 2.6 2.8 3−400

−300

−200

−100

0

100

200

300

400


Cam

ber

torq

ue, M

xγ [N

m]


Figure D.5: Overturning camber torque vs. inflation pressure for various positive and negative camberangles at two extreme vertical load conditions; Fz=0.1 kN (top) and Fz=10 kN (bottom).

D.4 Camber thrust and camber torque extrapolation 107

1.8 2 2.2 2.4 2.6 2.8 3−2

−1.5

−1

−0.5

0

0.5

1

1.5

2


Cam

ber

torq

ue, M

zγ [N

m]


1.8 2 2.2 2.4 2.6 2.8 3−150

−100

−50

0

50

100

150


Cam

ber

torq

ue, M

zγ [N

m]


Figure D.6: Aligning camber torque vs. inflation pressure for various positive and negative camber anglesat two extreme vertical load conditions; Fz=0.1 kN (top) and Fz=10 kN (bottom).


D.5 Overview implemented parameters

In this section an overview is given of the implemented parameters with the proposed initial andboundary values, see table D.4.

Table D.4: Overview of the implemented parameters with the proposed boundary conditions.

Parameter Low Initial UpperCFx pCFx1 −1.0 0.23 1.0

pCFx2 −0.6 0.0 0.0pCFx3 0.0 0.21 1.0

CFy pCFy1 −1.0 0.17 1.0pCFy2 −0.6 0.0 0.0pCFy3 0.0 0.50 1.0

Fx,RR qsy7 0.5 0.85 1.0qsy8 −1.0 −0.40 0.0

CMz pCMz1 −1.0 −0.42 1.0Fyγ ppy5 −1.0 −0.15 0.0Mxγ ppMx1 −1.0 −0.10 0.0Mzγ ppz2 −1.0 −0.60 0.0

The sidewall stiffnesses are made inflation pressure dependent by implementing the scaling approachof the nominal pressure sidewall stiffnesses (5.51) - (5.54). In table D.5 an overview is given of the fitparameters fi for the different sidewall stiffnesses. The fit parameters are optimised towards the FEMexperiments and are currently implemented as fixed parameters.

Table D.5: Overview of the implemented parameters of the sidewall stiffnesses.

Parameter Initialfx 0.65fy 0.74fz 0.65fγ 0.49fθ 0.69fψ 0.49

Appendix E

Simulation Parameters

In chapter 6 a two track vehicle model with a roll axis is used to analyse the vehicle handling behaviour.The roll stiffness and damping are modelled with torsional springs and dampers in the roll centres,translational stiffness and damping elements are used to model the vertical stiffness and damping ofthe suspension. In this Appendix an overview is given of the parameters that are used to parameterisethe vehicle model. The parameters of the unloaded vehicle are depicted in table E.1. Table E.2 showsthe parameters that change when the vehicle is fully loaded. For confidentiality reasons the numericalvalues are omitted.

109

110 Simulation Parameters

Table E.1: Vehicle model parameters.Parameters unloaded vehicle

Quantity V alue Descriptionl [m] vehicle wheelbasehc [m] height of vehicleis [-] steer ratiomFL [kg] weight front leftmFR [kg] weight front rightmRL [kg] weight rear leftmRR [kg] weight rear rightmunFL [kg] unsprung mass front leftmunFR [kg] unsprung mass front rightmunRL [kg] unsprung mass rear leftmunRR [kg] unsprung mass rear rightm [kg] vehicle body massIxx [kgm2] vehicle body inertia about world x-axisIyy [kgm2] vehicle body inertia about world y-axisIzz [kgm2] vehicle body inertia about world z-axish‘ [m] vertical body centre of gravity z-coordinatemtyre [kg] weight tyremrim [kg] weight rimIxx [kgm2] rim inertia about x-axisIyy [kgm2] rim inertia about y-axism1 [kg] weight front axles1 [m] front axle track widthKPI [deg] king pin inclinationcaster [deg] caster angleγ1 [deg] camber angleψtoe,1 [deg] toe anglerc1 [m] front roll centre heightcz,1 [N/m] vertical stiffnessdz,1 [Ns/m] vertical dampingcϕ,1 [Nm/rad] roll stiffnessdϕ,1 [Nms/rad] roll dampingccomplFy,1 [deg/N] steering compliance due to FyccomplMz,1 [deg/Nm] steering compliance due to Mz

m2 [kg] weight rear axles2 [m] rear axle track widthγ2 [deg] camber angleψtoe,2 [deg] toe anglerc2 [m] rear roll centre heightcz,2 [N/m] vertical stiffnessdz,2 [Ns/m] vertical dampingcϕ,2 [Nm/rad] roll stiffnessdϕ,2 [Nms/rad] roll dampingccomplFy,2 [deg/N] steering compliance due to FyccomplMz,2 [deg/Nm] steering compliance due to Mz

Pengine [kW] max. engine powerTengine [Nm] max. engine torquenengine [rpm] max. engine revolutions

Simulation Parameters 111

Table E.2: Changed vehicle model parameters when vehicle is loaded.Parameters loaded vehicle

Quantity V alue DescriptionmFL [kg] weight front leftmFR [kg] weight front rightmRL [kg] weight rear leftmRR [kg] weight rear rightm [kg] vehicle body massIxx [kgm2] vehicle body inertia about world x-axisIyy [kgm2] vehicle body inertia about world y-axisIzz [kgm2] vehicle body inertia about world z-axis

Enhancing the MF-Swift Tyre Model for Inﬂation Pressure ... · Enhancing the MF-Swift Tyre Model...

Documents

Transcript of Enhancing the MF-Swift Tyre Model for Inﬂation Pressure ... · Enhancing the MF-Swift Tyre Model...