692262

Enhanced friction modeling forsteady-state rolling tires

The research reported in this thesis is supported by the CCAR project ‘FEM TyreModelling’, in cooperation with TNO Automotive, Helmond, the Netherlands andApollo Vredestein B.V., Enschede, the Netherlands.

René van der Steen (2010). Enhanced friction modeling for steady-state rolling tires.Ph.D. thesis, Eindhoven University of Technology, Eindhoven, the Netherlands.

A catalogue record is available from the Eindhoven University of Technology Library.ISBN: 978-90-386-2390-0

Cover design: Oranje Vormgevers, Eindhoven, the Netherlands.Reproduction: Ipskamp Drukkers B.V., Enschede, the Netherlands.

Copyright c© 2010 by René van der Steen. All rights reserved.

Enhanced friction modeling forsteady-state rolling tires

PROEFSCHRIFT

ter verkrijging van de graad van doctor aan deTechnische Universiteit Eindhoven, op gezag van derector magnificus, prof.dr.ir. C.J. van Duijn, voor een

commissie aangewezen door het College voorPromoties in het openbaar te verdedigen

op donderdag 9 december 2010 om 16.00 uur

door

René van der Steen

geboren te Borsele

Dit proefschrift is goedgekeurd door de promotor:

prof.dr. H. Nijmeijer

Copromotor:dr.ir. I. Lopez

Summary

Enhanced friction modeling for steady-state rolling tires

Tire modeling is nowadays a necessary tool in the tire industry. Car manufacturers, gov-ernments and consumers demand better traction under all circumstances, less wear andmore recently less noise and a lower rolling resistance. Therefore finite element analysisis adopted in the design process of new tires to cope with these, often conflicting, de-mands. Finite element tire modeling can increase the insight on specific properties of atire, decrease the development time and reduce development costs of new tires. Howeverin practice most finite element models are still not able to match outdoor experiments.Both the static deformation and the dynamic response of the tire rolling on the roadshould be accurately predicted. The cornering, braking and traction of a tire depend onthe generated friction forces. Friction depends not only on the tread properties of thetire, but also on the road surface and environmental conditions. The main goal of thisthesis is to develop a robust and numerically efficient friction model for finite elementtire simulations and to create a framework for the identification and implementation offriction related parameters.

The numerical modeling of a tire in combination with its environment is a challengingtask, since different physical phenomena play a role. Typically the mechanical, thermaland fluid domains contribute to the tire response. This research is restricted to the me-chanical domain, where a numerical modeling framework for steady-state rolling tiresimulations is defined. In future developments of the model other effects can be in-cluded using this framework as a base. One of the objectives of this thesis is to developand validate a tire friction model for finite element analysis, which captures observed ef-fects of dry friction on the handling characteristics of rolling tires. Friction by itself is ahighly complex interaction phenomenon between contacting materials and can be mod-eled on many different length scales, applying different numerical techniques. This canhowever lead to an enormous computational burden and as a result it can be impractical

v

vi SUMMARY

for an industrial application. To provide a numerically feasible and relatively fast solutiona phenomenological friction model is chosen, where the parameters are identified usinga two step experimental / numerical approach.

Firstly, friction experiments are performed on a laboratory abrasion and skid tester toinvestigate the influence of contact pressure on the frictional force. In this experimentalsetup a small solid tire, with adjustable side slip angle, is pressed on an abrasive disk. Thefriction present between the abrasive disk and solid tire drives the tire and the resultingforces are measured with a force sensor. Several experiments under different normalloads and side slip angles of the tire are conducted. These measurements, under lowrolling velocity, are used to identify contact pressure dependent friction parameters. Therelevant parts of this setup are modeled in the commercial finite element package ABAQUS

and the steady-state performance of the small tire under different slip angles is evaluatedand compared with experiments. It is shown that the present turn slip, which has greatimpact on the slip velocity field at the trailing edge of the contact area, is captured wellwith the model. Furthermore, the calculated cornering stiffness is in good agreementwith the experiments.

Secondly, outdoor braking experiments at different velocities with a full scale tire areconducted to obtain a velocity dependent parameter set for the tire friction model. Thederived friction model is then coupled to a finite element model of this full scale tire,which is also constructed in the software package ABAQUS. The finite element model isvalidated statically using measurements of the contact pressure distribution, contact areaand of the radial and axial stiffness of the tire. The steady-state transport approach inABAQUS is used to efficiently compute steady-state solutions at different forward velocitiesas used in the outdoor experiments.

Finally, the predictive capability of the finite element tire model in combination with theproposed friction model is assessed. The basic handling characteristics, such as purebraking, pure cornering, and combined slip under different loads, inflation pressuresand velocities are evaluated and validated with experiments. Based on this comparison, itcan be concluded that all three basic handling characteristics are adequately predicted.

Contents

Summary v

Nomenclature xi

1 Introduction 11.1 General introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Tire performance and modeling . . . . . . . . . . . . . . . . . . . . . . . . 21.3 Challenges in FE tire modeling . . . . . . . . . . . . . . . . . . . . . . . . 41.4 Objectives and research approach . . . . . . . . . . . . . . . . . . . . . . . 61.5 Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81.6 Outline of this thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2 Modeling framework for steady-state rolling tires with friction 92.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.2 Overview of friction models . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.2.1 Amontons-Coulomb . . . . . . . . . . . . . . . . . . . . . . . . . . 102.2.2 Rubber friction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.3 Tire research using FE Models . . . . . . . . . . . . . . . . . . . . . . . . 172.3.1 Implicit analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172.3.2 Explicit analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.4 Modeling framework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202.4.1 Numerical method . . . . . . . . . . . . . . . . . . . . . . . . . . . 202.4.2 Friction model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

2.5 Steady-state transport analysis . . . . . . . . . . . . . . . . . . . . . . . . . 222.5.1 Contact conditions for steady-state rolling . . . . . . . . . . . . . . 252.5.2 Frictional stress for steady-state rolling . . . . . . . . . . . . . . . . 26

2.6 Implementation of a friction law in steady-state rolling . . . . . . . . . . . 282.7 Validation of the implemented friction law . . . . . . . . . . . . . . . . . . 30

2.7.1 Cylindrical model . . . . . . . . . . . . . . . . . . . . . . . . . . . 302.7.2 Comparison with standard Coulomb friction law . . . . . . . . . . 30

vii

viii CONTENTS

2.7.3 Validation of friction law for varying friction coefficient . . . . . . 322.8 Steady-state free-rolling . . . . . . . . . . . . . . . . . . . . . . . . . . . . 332.9 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

3 Simulation procedure to compute handling characteristics with an FE model 373.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 373.2 Approach to identify parameters of the friction model . . . . . . . . . . . . 393.3 Design of the test tire used for experimental validation of the FE model . . 403.4 FE tire model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

3.4.1 2D tire cross section . . . . . . . . . . . . . . . . . . . . . . . . . . 413.4.2 Rim mounting and inflation . . . . . . . . . . . . . . . . . . . . . 433.4.3 3D tire model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 433.4.4 Mesh effect on lateral force in 3D model . . . . . . . . . . . . . . . 453.4.5 Static loading of the tire . . . . . . . . . . . . . . . . . . . . . . . . 47

3.5 Simulation procedure to compute the braking characteristic of a rolling tire 473.5.1 Mesh effect on the force equilibrium in vertical direction . . . . . 483.5.2 Effect of penalty parameter in the friction model on the longitudi-

nal force . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 493.6 Simulation procedure to compute the cornering and combined slip char-

acteristics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 503.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

4 Friction parameter identification using a Laboratory Abrasion and skid Tester 554.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 554.2 Identification using lab scale experiments . . . . . . . . . . . . . . . . . . 564.3 Experimental setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 574.4 FE model of the setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

4.4.1 Material model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 584.4.2 2D model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 624.4.3 3D model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

4.5 Parameter identification . . . . . . . . . . . . . . . . . . . . . . . . . . . . 664.6 Comparison of numerical and experimental results . . . . . . . . . . . . . 674.7 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 724.8 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

5 Friction parameter identification using longitudinal slip characteristics 775.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 775.2 Identification using full scale experiments . . . . . . . . . . . . . . . . . . 785.3 Tire force and moment measurements . . . . . . . . . . . . . . . . . . . . 795.4 Friction parameter identification . . . . . . . . . . . . . . . . . . . . . . . 80

CONTENTS ix

5.5 Free-rolling rotational velocity: Comparison of FEM prediction and exper-iments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

5.6 Comparison between FEM and MF predictions . . . . . . . . . . . . . . . 865.7 Effect of inflation pressure on the longitudinal force . . . . . . . . . . . . 885.8 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 915.9 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

6 Predictive capability of the FE tire model 936.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 936.2 Prediction of the handling characteristics . . . . . . . . . . . . . . . . . . 94

6.2.1 Pure cornering characteristic . . . . . . . . . . . . . . . . . . . . . 946.2.2 Combined slip characteristic . . . . . . . . . . . . . . . . . . . . . 1006.2.3 Friction power distribution in the footprint . . . . . . . . . . . . . 105

6.3 Force and moment measurements and Magic Formula . . . . . . . . . . . 1076.4 Comparison of the FE model and the Magic Formula . . . . . . . . . . . . 111

6.4.1 Pure cornering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1116.4.2 Combined slip . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

6.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

7 Conclusions and recommendations 1217.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1217.2 Recommendations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

References 125

A Friction model implementation 133A.1 Stick . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133A.2 Slip . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135

B Mesh effect on the force equilibrium in vertical direction 137

C Computation of slip velocity field for the LAT 100 setup 141

Samenvatting 143

Dankwoord 145

Curriculum Vitae 147

Nomenclature

Acronyms and abbreviations

acronym description acronym description

ALE arbitrary Lagrangian-Eulerian MF Magic FormulaDoE design of experiments NASA national aeronautics and spaceFE(M) finite element (method) administrationFEA finite element analysis R radialISO international organisation for TNO Netherlands organization for applied

standardization scientific researchLAT laboratory abrasion and skid tester WLF Williams-Landel-Ferry equationL.I. load (capacity) index

Operations and notation

symbol description symbol description

a,A scalar x, X skew-symmetric matrix,a,A vector or matrix associated with x, X| · | absolute value × outer productX Lagrangian coordinate · inner productx Eulerian coordinate xT ,AT vector or matrix transposex, (x) (double) time derivative || · || magnitude∂ partial derivative

xi

xii NOMENCLATURE

Roman symbols and letters

symbol description unit

A0 cross sectional area m2

AExp experimental contact area m2

AMod calculated contact area m2

a line length mtuning parameter

a Eulerian acceleration vector m/s2

aT shift factorb line length m

tuning parameterC10 hyperelastic material coefficient PaCFα cornering stiffness N/degCMz aligning stiffness Nm/degc line length m

tuning parameterd tuning parameterD1 hyperelastic material parameter PaE Young’s modulus PaE′ elastic storage modulus PaE′′ elastic loss modulus Paerrorrel relative errorFfric frictional force NFx longitudinal force NFy lateral force NFz normal (vertical) force (load) Nf frequency rad/sG′ shear storage modulus PaG′′ shear loss modulus Pah tuning parameter friction model

penetration mI identity matrixJ Jacobian matrixk tuning parameter friction modelks slopeL1,2,3 length mL initial length ml current length mMx overturning moment NmMy moment or driving torque NmMz self-aligning moment NmMzr residual torque Nm

NOMENCLATURE xiii


n cornering axis in normal directionp contact pressure Pap0 tuning parameter friction model Pap1,2 tuning parameterplimit lower bound on contact pressure PaRα side slip rotation matrixRβ rotation matrixRc cornering rotation matrixRs spinning rotation matrixR radius m

parameter friction modelRinner inner tire radius mRouter outer tire radius mr radius mre effective rolling radius mrl loaded rolling radius mru unloaded rolling radius mT rigid axle positionT torque Nmt time s

pneumatic trail mti orthogonal unit vector, i ∈ 1, 2uc longitudinal carcass deflection mV velocity m/sVmax tuning parameter friction model m/sVsx longitudinal slip velocity m/sVsy lateral slip velocity m/sVx longitudinal (forward) velocity m/sVy lateral velocity m/sv (longitudinal) slip velocity m/svc lateral carcass deflection mvo lateral resultant force offset mvs slip velocity m/sv Eulerian velocity vector m/sWidth tire width mX Lagrangian coordinatex Eulerian coordinatex x-position mY Lagrangian coordinatey Eulerian coordinatey y-position m

xiv NOMENCLATURE

Greek symbols and letters


α side slip angle degβ angle degχ deformation map∆ increment in position mδ phase lag radεe nominal strainγ, γ slip velocity (vector) m/sγcrit critical slip velocity m/sκ longitudinal slipκmax tuning parameter friction modelλi wavelength i mµ friction coefficientµk kinetic friction coefficientµlock tuning parameter friction modelµMF Magic Formula dependent friction coefficientµm tuning parameter friction modelµs tuning parameter friction model

static friction coefficientΩ rotational velocity rad/sω rotational velocity rad/sωα free-rolling rotational velocity for nonzero

side slip anglesrad/s

ωfree free-rolling rotational velocity rad/sρ density kg/m3

σe nominal stress Paτ, τ frictional stress (vector) Paτcrit critical frictional stress Pa

Subscripts and indices

symbol description symbol description

0 center max maximum

c cornering n discrete time step

D deformable body p contact pressure dependent

disk abrasion disk R rigid foundation

eqv equivalent r reference frame

i index, i ∈ 1, 2 slip slip

j node j v slip velocity dependent

max maximum wheel wheel

n discrete time step

NOMENCLATURE xv

ISO sign conventions

Tire force and moment

V α

x

Fx

Fy

Mz

Fy

y

z

Fz

Mx

V α

x

y

Fx

Fy

Mz

Top view tire Rear view tire

Tire velocity and slip velocity

V α

Vy

Vx

VsxVs

Vsy

tan α = Vsy

Vx

κ = −VsxVx

Top view tire

longitudinal slip

side slip

CHAPTER ONE

Introduction

Abstract / In this chapter a general introduction of tire modeling is presented. The current chal-lenges in tire modeling using finite element methods are discussed, which form the motivation ofthe research objectives. Based on these objectives, a research approach is presented and the maincontributions of this thesis are stated. The chapter ends with the outline of the thesis.

1.1 General introduction

The development of pneumatic tires started with the patent by John Boyd Dunlop in 1888(Dunlop, 2010)1 and is still going on today. The first pneumatic tires had small cross sec-tions and high inflation pressures, mainly for bicycle applications. From the 1920s, largertires were introduced for the upcoming vehicle industry. Two major evolutions took placein the 1960s, the tubeless tire was introduced and bias ply tires were replaced with radialply tires, which improved the wear and handling properties significantly. The main dif-ference between the bias and radial ply tire is the orientation of the plies. In bias ply tires,the body ply cords are laid at angles substantially less than 90 to the tread centerline,extending from bead to bead. In radial tires, the body ply cords are laid radially frombead to bead, at 90 to the centerline of the tread. Two or more belts are laid diagonallyin the tread region to obtain the required strength and stability (Gent and Walter, 2005,chapter 1). In Figure 1.1, a typical layout of a radial tire, which is now the standard forpassenger car tires, is shown.The tire construction, such as aspect ratio and belt construction, depends on the size ofthe tire and the target market. This information is printed on the sidewall of every tire,e.g. 215/55 R16 97 H. The first number (215) is the nominal section width in mm, the

1The idea of a pneumatic tire was already patented by Thomson (1847). Dunlop’s patent was laterdeclared invalid on the basis of this patent, but is generally considered as the first practical pneumatic tire.

1

2 1 INTRODUCTION

Figure 1.1 / Cross section of a radial tire with the main components, Ghoreishy (2008)

second number (55) is the percentage of the height/width ratio of the cross section. TheR (radial) stands for the tire construction code. The rim diameter (16) is given in inchesand 97 is the load capacity index (L.I.), which is a reference to the maximum load capac-ity. The last symbol (H) is the speed symbol, which corresponds to a maximum allowablespeed (European Tyre and Rim Technical Organisation, 2010).More recent developments in the tire industry are the run-flat technology, which en-ables the vehicle to continue at reduced speeds after deflation of a tire, and the so-calledultralow-aspect tires, which have very short sidewalls (Rodgers, 2001).

1.2 Tire performance and modeling

All tires must meet the following fundamental set of performance factors (Rodgers,2001):

• Provide load carrying capacity.

• Provide cushioning, damping and minimum noise and vibration.

• Transmit forces and moments.

• Resist abrasion.

• Have a low rolling resistance.

• Be durable and safe throughout the expected lifespan.

The different components of the tire determine the tire overall characteristics in responseto the application of load, torque or steering input, resulting in the generation of forces

1.2 TIRE PERFORMANCE AND MODELING 3

and deflection of the tire. The mechanical properties are often interrelated, which meansthat a design change for one of the performance factors, can affect the other factors bothpositively or negatively (Rodgers, 2001). Besides the engineering aspects, also economi-cal factors, (limitations of) the manufacturing process (Walter, 2007b) and governmentregulations (Walter, 2007a) have to be taken into account. All these aspects are capturedin specific performance criteria, which are visualized in a performance chart in Figure 1.2.In this figure, a new tire design is compared with an existing reference tire for six perfor-mance criteria. It should be noted that these criteria are not defined unambiguously inliterature, e.g. (Gent and Walter, 2005, chapter 1).

noise and vibration

dry traction wear resistance

rolling resistancewet traction

95%100%

110%105%

ice and snow traction

Figure 1.2 / Example of a new tire design (dashed line) related to a reference tire for sev-eral performance functions. Based on the tread performance chart (Mundl et al., 2008).

Dry, wet, and, ice and snow traction are directly related to the handling properties of thetire and as such to safety issues, e.g. the braking distance at 100 km/h. Obviously, wearresistance is related to the tire’s lifespan, while rolling resistance has direct influence onvehicle fuel consumption. A tire rolling over a road generates undesired noise both at thesurroundings (exterior noise) and inside the vehicle itself (interior noise).An accurate model of the tire behavior enables the engineer to optimize the overall per-formance, while taking into account the different performance criteria. For this purpose,several modeling techniques have been developed during the last decades.

In the case of handling models, a distinction can be made between empirical models and(simple) physical models. Each type is developed for a specific purpose (Pacejka, 2006).The (semi)-empirical models are based on experimental data. These models have a spe-cific structure and the parameters are usually obtained by regression techniques. One ofthe most well-known and widely used tire handling models is the Magic Formula modelof Pacejka (Bakker et al., 1987). This model is very useful to reproduce and interpolate

4 1 INTRODUCTION

tire properties. The great advantage is of this model is the low computational cost. Thisis a very important requirement for tire models that are used in vehicle dynamics, wherethe tire is just one part of the total vehicle model. An overview of specific tire simulationchallenges in vehicle dynamics is presented by Rauh and Mössner-Beigel (2008). Thesimple physical models use analytical expressions to describe the forces and momentsand can produce realistic results, e.g. the stretched string model (Pacejka, 2006), if theparameters are assigned appropriate values. Usually the application field of these kind ofmodels is limited.The main drawback of these two types of models is that the parameters are experimen-tally determined from full scale tire tests and as such these models can not be used topredict the influence of tire construction design changes. For detailed analysis of a tireitself, the Finite Element Method (FEM) can be used. Such finite element models arecomplex, but allow to investigate the effects of tire design parameters on the generatedforces and moments. FE models are nowadays a standard tool in the tire industry, andtheir use opens the possibility of tire virtual prototyping.

1.3 Challenges in FE tire modeling

Finite element tire modeling can increase the insight on the relative influence of specifictire properties on the tire behavior, decrease the development time and eventually reducedevelopment costs of new tires. However in practice most finite element models are stillnot able to accurately match outdoor experiments.

INTERACTION MODEL

OUTPUTSINPUTS

FE Model

environment

contact model

road

Figure 1.3 / Schematic overview of the necessary components for every Finite Elementmodel of a rolling tire interacting with the environment.

1.3 CHALLENGES IN FE TIRE MODELING 5

The numerical modeling of a tire in combination with its environment, shown in Figure1.3, is a difficult task, since different physical phenomena play a role. Typically the me-chanical, thermal and fluid domains contribute to the tire response. These interrelateddomains and the wide range of operating conditions provide several challenges in themodeling process:

• Material modeling. A correct description of the different rubber and cord-rubberplies is required. Large deformations and large strains, as well as incompressibilityof rubber compounds should be accounted for.

• Contact modeling. Contact models in normal and tangential direction need to bedefined. Especially, friction models for tangential contact depend on environmentalconditions.

• Geometric modeling. Modeling of geometric shapes is nowadays not an issue any-more, but in many cases the creation of valid elements for detailed tread patterns isdifficult.

• Temperature modeling. The temperature of the tire changes during operation,which affects the mechanical properties of the tire and hence its behavior.

• Steady-steady versus transient modeling. Different numerical algorithms are re-quired for steady-state rolling tires and the modeling of transient effects e.g. whenimpacts with obstacles occur.

• Measurements for validation purposes. This is not directly related to FE tire mod-eling, however necessary to validate FE models. Obtaining good measurementsof the forces and moments or other system quantities acting on rolling tires is adifficult task.

The latest overview of the current state of the art of FE modeling of rolling tires is givenin the paper of Ghoreishy (2008). In the summary, it is stated that despite the substantialprogress, achieved during the last decades, the analysis of the complicated tire structureis still a formidable task. Sofar, none of the current published works is capable of givinga full analysis of the tire under different loading conditions. Instead, each work tried tofocus on some critical aspect of the tire and investigated this specific feature as deep aspossible.Furthermore, the majority of tire related research in literature does not provide detailedinformation about the used models and parameter values are often lacking, which makesinterpretation of the results difficult. Finally, a validation of the used FE models for rollingsituations with full scale tire experiments is often missing.

6 1 INTRODUCTION

1.4 Objectives and research approach

This thesis focuses on the FE prediction of the steady-state handling characteristics ofrolling tires on dry roads. Although handling on wet roads is also important, the pres-ence of water or snow add significant complexity to the tire/road interface. Additionally,obtaining consistent measurements under controlled wet conditions is very challengingin practice. Therefore, wet roads are not considered in this research.

The handling characteristics considered in this thesis are pure braking, pure corneringand combined slip (Pacejka, 2006) and are defined as follows. In case of pure braking,the tire is braked from free-rolling up to wheel lock. For pure cornering, the free-rollingtire is steered up to ±12 side slip angle and in case of combined slip, the tire is brakedup to wheel lock when rolling at a constant slip angle. These conditions cover drivingsituations from normal driving to extreme manoeuvres.

To evaluate the handling performance of a tire, both the static deformation and thedynamic response of the tire rolling on the road should be accurately predicted. Thecornering, braking and traction of a tire depend on the generated friction forces. Frictiondepends not only on the tread properties of the tire, but also on the road surface and en-vironmental conditions. Frictional behavior for model parameterization can be acquiredby extensively testing tires under different conditions. Experimental characterizationof frictional properties of rubber compounds is cumbersome, since environmentalconditions influence these measurements. As a result, Coulomb’s friction law, with aconstant friction coefficient, is still often used in finite element simulations to predicthandling characteristics. It is however clear from experiments with elastomers thatrubber friction depends on various parameters like contact pressure, sliding velocity,temperature and surface roughness. Because of these dependencies Coulomb’s law isnot sufficient to accurately predict the handling characteristics over the desired range.Furthermore, numerical problems occur during steering at large side slip angles whenCoulomb’s law with realistic friction coefficients is used in finite element simulations.To overcome these limitations a different strategy is needed to capture observed effectsof dry friction on the handling characteristics of rolling tires. Therefore the essential goalof this thesis is:

The development of a robust and numerically efficient friction model for finite element tiresimulations and to create a framework for the identification and implementation of frictionrelated parameters.

Furthermore, this friction model should capture observed effects of dry friction and itshould be compatible with commercial FE codes. As mentioned above, rubber frictiondepends on several parameters. This thesis focuses on developing a friction model,

1.4 OBJECTIVES AND RESEARCH APPROACH 7

which includes the influence of contact pressure and sliding velocity.To achieve the above objective, the currently available state-of-the-art numerical methodsin the commercial finite element package ABAQUS are used. Emphasis is placed here onthe modeling of frictional contact of the tire with the road, the FE model of the tire itselfis provided by tire manufacturer Vredestein.

Lab scale experiments

3D Forces Experiments

3D Forces Simulations

Full scale experiments

Inputs

Frictionmodel

FE Tire

Virtual prototyping

Lab scale setupFEM 3D Forces Simulations

3D Forces ExperimentsInputs

Figure 1.4 / Schematic overview of the two step experimental / numerical approach toobtain friction information using both small scale and full scale experiments.

For the parameter identification of the friction model several measurements have beencarried out on two experimental setups on different scales, as shown in Figure 1.4. First,friction experiments are performed on a commercially available small scale lab setup(LAT 100). On this setup controlled experiments on a small solid tire, under low rollingvelocities, are performed. These measurements are used to identify the parameters of thefriction model, which are contact pressure dependent. The friction model is coupled toan FE model of this lab setup to simulate the hub forces. These forces are compared tothe experimentally found hub forces and used to validate the parameters of the frictionmodel. With the small scale setup it is only possible to conduct experiments at low slidingvelocities, since excessive wear of the small tire occurs at higher velocities. Thereforeoutdoor braking experiments with a full scale tire at different velocities are conducted toobtain a slip velocity dependent parameter set for the friction model. These experimentsare carried out with the TNO Tyre Test Trailer. Finally, the completely identified frictionmodel is validated by comparing simulated and measured cornering and combined slipcharacteristics.

8 1 INTRODUCTION

1.5 Contributions

The main contributions presented in this thesis are:

• The definition and implementation of a complex friction model in the commercialpackage ABAQUS, which is suitable for tire handling simulations in the full operatingrange.

• A procedure for the identification of the parameters of the friction model usinga two step experimental / numerical approach, combining small scale lab experi-ments with full scale outdoor experiments.

• Validation of the predicted handling characteristics with full scale outdoor experi-ments, covering the entire operating range.

1.6 Outline of this thesis

In Chapter 2, the modeling framework to obtain the handling characteristics of rollingtires, with friction, is described. The choice for the numerical approach and the frictionmodel, based on a literature overview, is made. The background of the numerical methodis reviewed and the implementation of the chosen friction model is described.After that, the strategy for identification of the parameters of the chosen friction model isexplained in more detail in Chapter 3. Furthermore, the design of the used test tire andthe corresponding layout of the FE model are presented. The different simulation stepsrequired to compute the pure braking, pure cornering and combined slip characteristicsare discussed.In Chapter 4, the identification of the contact pressure dependent part of the tire frictionmodel is given. Friction data is obtained using a commercial laboratory abrasion andskid tester. From this data the contact pressure related parameters are derived and imple-mented in an FE model for the tire/disk contact.Next, the identification of the slip velocity dependent part, using measured axle forces, ispresented in Chapter 5. The complete identified friction model is then coupled to the FEmodel of the tire under testing. The computed steady-state longitudinal slip characteris-tics are compared with the full scale outdoor experiments and a discussion of the resultsis given.In chapter 6, the fully identified friction model is used to compute the pure corneringand combined slip characteristics. The predicted characteristics are compared with ex-periments and it is shown that the handling performance of the tire can be adequatelypredicted with the identified friction model.Finally, in Chapter 7, the main conclusions are summarized and recommendations forfuture work are given.

CHAPTER TWO

Modeling framework for steady-staterolling tires with friction

Abstract / A numerical modeling framework to obtain the handling characteristics of rolling tires,with friction, is defined in this chapter. Based on a literature overview of friction models and tiresimulation methods the choice for the numerical approach and the friction model is motivated.The background of the numerical method is presented together with the implementation of thechosen friction model. The validation of this implementation is presented at the end of the chap-ter.

2.1 Introduction

In this chapter the numerical approach to obtain the handling characteristics of rollingtires including friction is described. First a short literature review of friction is presentedin Section 2.2, which shows the difference of rubber friction compared to other solids andin Section 2.3 Finite Element Analysis (FEA) related to tires is discussed. Based on thesefindings a numerical method is chosen, which is able to efficiently compute the handlingcharacteristics of steady-state rolling tires. Furthermore, the choice of the friction modelis motivated in Section 2.4. The background of the numerical method is reviewed in Sec-tion 2.5 and the implementation of the friction model is discussed in Section 2.6. Thisimplementation is validated using different test models, which are presented in Section2.7. In Section 2.8 a description of an algorithm to obtain the solution, which corre-sponds to a so-called free-rolling tire is discussed. The chapter ends with conclusions inSection 2.9.

9

10 2 MODELING FRAMEWORK FOR STEADY-STATE ROLLING TIRES WITH FRICTION

2.2 Overview of friction models

2.2.1 Amontons-Coulomb

Although Leonardo da Vinci (1452-1519) is generally credited as being the first to developthe basic concepts of friction (Bowden and Tabor, 1964), Amontons formulated the firsttwo laws of friction and Coulomb the last two. These classic laws are summarized byMoore (1972) as follows:

• The friction force is proportional to load.

• The coefficient of friction is independent of the apparent contact area.

• The static coefficient is greater than the kinetic coefficient.

• The coefficient of friction is independent of the sliding velocity.

This can be formulated as

Ffric = µFz, (2.1)

where Ffric is the frictional force, µ the constant friction coefficient and Fz the appliedload and (2.1) is usually referred to as the Coulomb friction model.

Deviations from Coulomb friction

Experiments often show deviations from the basic Coulomb friction model. Variations ofthe basic model are the difference between a static µs and a kinetic µk friction coefficient,see Figure 2.1 for some examples. This transition can be discontinuous or continuous,e.g. using an exponential decaying function between the static and kinetic coefficient.In general, the friction coefficient varies for increasing sliding velocity especially in the

µ µ µ

µs µs µsµk

µk µk

v v v

µ

µs

µk

v

fric

tion

coeffi

cien

t

sliding velocity

Figure 2.1 / Coulomb friction model and possible variations.

2.2 OVERVIEW OF FRICTION MODELS 11

presence of lubricants, e.g. the Stribeck curve (Stribeck, 1902). First the coefficient de-creases to a minimum value and after that it increases with higher sliding velocity. Thereview paper of Olsson et al. (1998) describes several static friction models, which focusonly on sliding velocity, and dynamic friction models. The dynamic models also describethe stick-slip transitions around zero sliding velocity. Examples are the Dahl model andtwo variants of this model, the LuGre model and the model of Bliman and Sorine. Thesemodels are often used in control systems, in such a way that they can be used for frictioncompensation in mechanical systems.

2.2.2 Rubber friction

Rubber friction differs from friction of most other solids. According to Moore (1972) thesecond friction law appears to be valid only for materials possessing a definite yield pointand it is does not apply to elastic or viscoelastic materials. The third law is not obeyed byany viscoelastic material and the fourth law is not valid for any material (Moore, 1972).It is also clear from experiments with elastomers that rubber friction depends on variousparameters like contact pressure, sliding velocity, temperature and surface roughness.Because of these dependencies Coulomb’s law is not sufficient to model the frictionalresponse of an elastomer.

The friction force between rubber and a rough surface has two contributions commonlydescribed as the adhesion and hysteretic components (Moore, 1972). The hysteretic com-ponent results from internal friction of the rubber; during sliding asperities of a roughsurface exert oscillating forces on the rubber surface. This leads to cyclic deformations ofthe rubber and to energy dissipation caused by the internal damping of the rubber (Pers-son, 2001). The adhesion component is caused by the intermolecular attractive forcesbetween the contacting bodies (Wriggers and Reinelt, 2009).The article of Grosch (1963) is one of the early reports, which describes the analogy be-tween the friction coefficient as function of sliding velocity v and the energy dissipatedper cycle (tan δ) as function of frequency f , see Figure 2.2. The phase angle is givenby tan δ = E ′′/E ′, with δ the phase lag between stress and strain; E ′ and E ′′ are thestorage and loss modulus respectively. In this work experimental results of different vul-canized rubbers are presented. The experiments are conducted at different temperaturesand shifted, with shift factor aT , to a master curve using the WLF equation, developed byWilliams et al. (1955).

Grosch indicates that friction is due to energy dissipated when rubber is compressedand released by asperities. Friction for dry sliding on a smooth surface is due to energydissipated as rubber sticks and slips on a molecular scale. In the case of sliding on a roughdry surface the dissipative process appears at different speeds, corresponding to differentlength scales: asperities (hysteretic component) and molecules (adhesion component).


Several contributions to the friction coefficient µ can arise at the same sliding speed fromstick-slip processes occurring at different length scales, see Figure 2.3.An overview of models to describe the frictional interaction between tire and road is givennext.

Figure 2.2 / Grosch’s interpretation of the equivalence between master curves of rubberfriction versus sliding velocity and tan δ versus frequency f , Gent (2007).

Figure 2.3 / Combined effect of sliding on a rough and dry surface, Gent (2007).

Savkoor proposed a model, based on results of Grosch, where he described the shape ofan isothermal master curve with an empirical relation (Savkoor, 1966, 1987)

µ(v) = µs + (µm − µs) exp

−h2 log2

(v

Vmax

), (2.2)

where µs is a static coefficient of friction, µm the peak value of the function (which occursat |v| = Vmax) and h is a dimensionless parameter reflecting the width of the speed rangein which friction varies, as shown in Figure 2.4. According to Savkoor (1966), the valuesof Vmax and µm depend on the viscoelastic properties of the rubber. At higher tempera-


tures Vmax increases and the friction curve is shifted significantly towards higher speeds.The advantage of this function is that the parameters are related directly to the shape ofthe friction curve, which makes a physical interpretation possible. The disadvantage ofthis model is that only the influence of sliding velocity on the friction coefficient is con-sidered, at constant temperature, and that the influence of contact pressure is not takeninto account.

µ

v

hµs

µm

Vmax

Figure 2.4 / Model proposed by Savkoor to describe an isothermal master curve.

Based on the original Schallamach 2D model (Schallamach, 1971) a generalized theorythat predicts the buckling effects, known as Schallamach waves (Schallamach, 1952; Bar-quins, 1992) which occur at very smooth surfaces, is presented by Berger and Heinrich(2000). In contrast to the original Schallamach model, where Coulomb friction is used,this generalized theory considers a friction coefficient that depends on the local normalpressure over the contact line,

µ(p) = µ0

( pE

)n(2.3)

with E the linear elastic modulus and µ0 and n fitted parameters. A similar pressuredependent model is used in the work of Trinko (2007), where µ0 and E are replaced witha single constant.

The phenomenological models proposed by Dorsch et al. (2002) are fully empirically,where a velocity v and pressure dependency p for the friction coefficient is assumed.

µ(v, p) = c1pc2vc3 (2.4)

µ(v, p) = c1p+ c2p2 + c3v + c4v

2 + c5pv (2.5)

The friction coefficient is described by a power law or a linear approximation, in Dorschet al. (2002) a so-called full quadratic model is presented. Although these are very simplemodels, several experiments are required to identify the coefficients ci. Design of Experi-ments (DoE) techniques can be used to determine the number and range of pressure and


sliding velocity values needed (Montgomery and Runger, 1999).A velocity and pressure dependent model (Wriggers, 2006), which is proposed by Nack-enhorst (2000) is given by

µ(v, p) = c1

[p

c2

]c3+ c4 ln

v

c5

− c6 lnv

c7

, (2.6)

where all seven parameters must be determined using experiments. In this model theinfluence of contact pressure and sliding velocity is decoupled and summed to obtain thetotal friction coefficient, which indicates that this model also originates from statisticaltechniques. All these three models are able to accurately fit experimental data, but theobtained parameters of these models have no straightforward physical interpretation.

The model proposed by Huemer et al. (2001a) is developed for sliding rubber blocks onice and concrete. The phenomenological friction law is given by

µ(v, p) =α|p|n−1 + β

a+ b|v|1/m + c

|v|2/m

(2.7)

and is developed for a macroscopic model. In this approach the coefficient of friction de-pends on normal pressure p, sliding velocity v and temperature. The friction coefficientitself (2.7) is only a function of normal pressure and sliding velocity. Temperature effectsare incorporated using the WLF transformation, i.e., if the current temperature is differ-ent from a reference temperature, an equivalent new sliding velocity for the referencetemperature is calculated.

The friction law in (2.7) requires seven parameters (a, b, c, n, m, α, β), which must beidentified using experimental data. They account for the dependence of the friction coef-ficient on friction surface, rubber compound, and dimensions and geometric shape of therubber block. The identification is based on a least square error method and performediteratively. First the coefficients a, b and c related to the sliding velocity are fitted and thenα and β related to the contact pressure, this process is repeated until a defined error cri-terium is reached. The whole process is done for every value of n and m. Furthermore, inthe identification procedure the contact pressure is replaced with the averaged pressureon the contact surface. An evaluation of the model is shown in Figure 2.5.

This work has been continued by Hofstetter et al. (2003), where a thermo-mechanicalcoupling has been introduced. Energy dissipation during sliding is converted intoheat and this heat flux causes a temperature rise of the rubber and road. Simulationsof abrasion of the rubber block are considered in Hofstetter et al. (2006), where theobtained numerical results are also compared with experimental data. Material lossoccurs only at the front edge and is captured qualitatively with the proposed friction andabrasion models. However the model also predicts material loss in the middle of the


0 0.5 1 1.5 20.65

0.7

0.75

0.8

0.85

0.9

log(v) [mm/s]

µ [−

]

Figure 2.5 / Friction coefficient as function of sliding velocity using (2.7), the parametersare extracted from Hofstetter et al. (2006) and a chosen contact pressure of 0.5 MPa.

bottom surface, but this is not observed experimentally.

A different approach is used by Persson, who has published many papers (Persson, 1993,1995, 1998, 1999, 2002; Persson and Tosatti, 2000; Persson and Volokitin, 2000, 2002;Persson et al., 2002) on the subject of rubber friction and the role of the surface, whichis in contact with the rubber. This theory is a continuation of the early studies of Grosch.Persson states that the friction force is related to the internal friction of the rubber, whichis a bulk property of the material. The hysteretic friction component is determined bysliding of the rubber over asperities of a rough surface. These oscillating forces lead toenergy dissipation. The contribution of a every asperity size can be described with afractal description of the rough surface. Every length scale λ, up to the largest particlesof asphalt, can be related to a frequency: f ∼ v/λ. The friction coefficient is based onanalytical expressions, which limits the rubber models to linear elastic theory.A similar method is described by Klüppel and Heinrich (2000), but this theory is basedon a cylindrical rubber block undergoing only a one-dimensional deformation duringsliding contact with a rough surface. A comparison of experiments and the predictivecapabilities of the physical theories of Klüppel and Heinrich (2000) and Persson (2001)is presented by Westermann et al. (2004). It is concluded that each theory has one openparameter, which needs to be fitted on experimental data to obtain a good agreementbetween theory and data. However this match only holds for a sliding speed interval upto a few cm/s. Towards higher sliding speeds, systematic deviations appear, which arerelated to flash temperature effects in the contact patch.This flash temperature effect is described in a recent article of Persson (2006), where


he also takes the local heating of the rubber into account. At very low sliding velocitiesthe temperature increases are negligible because of heat diffusion, but already forvelocities in the order of 0.01 m/s local heating plays a role. He shows that in a typicalcase the temperature increase results in a decrease in rubber friction with increasingsliding velocity, if the sliding velocity is above 0.01 m/s. The advanced model, with flashtemperature effects included, is able to predict the friction coefficient for slightly highersliding speeds compared to the model without flash temperature. However validatedmodels for the range of sliding velocities that appear in tire/road contact are not availableyet.

An FE multi-scale approach for frictional contact is proposed by Wriggers and Reinelt(2009). The proposed model is based on the analytical models of Klüppel and Heinrich(2000) and Persson (2001), however it is well-known that large deformations occur whena tire is in contact with a rough road surface and therefore the numerical simulationsare based on finite deformation models instead of linear elastic theory. The numericalcalculation of a rough surface demands a very detailed model to include all length scalesand a model of a tire tread block also requires a very fine mesh to accurately describethe contact surface. This is not possible with current computer resources and thereforethe rough surface is modeled with only a few superimposed harmonic functions and thetread is replaced with a small rubber block. On each length scale several simulations forvarying sliding velocity are carried out, where the following function is fitted though theobtained data points

µ(v, p) =

(2vv

v2 + v2

)cµmax, (2.8)

in which v denotes the sliding velocity, where the maximum point of the friction curveµmax is reached. The dependency of the applied normal pressure is included by the func-tions

v = ap, (2.9)

µmax =b

parctan(dp), (2.10)

which describe the effect that for increasing global pressure the maximum friction valuedecreases and is shifted to larger velocities. The friction law on micro-scale requiresa fit of four parameters a, b, c and d, which are obtained by a nonlinear least-squaremethod. The obtained homogenized friction law is then applied within each so-calledrepresentative contact element at the next larger scale. This method is able to predict thequalitative frictional behavior, but is not suitable for complete tires due to computationallimitations.

2.3 TIRE RESEARCH USING FE MODELS 17

2.3 Tire research using FE Models

The use of FEM in tire development started together with the development of nonlinearFEM. All four types of nonlinearities (Kouznetsova, 2006) in solid mechanics are presentin pneumatic tires. Geometric nonlinearity occurs due to the large displacements androtations involved, while material nonlinearity is present for the almost incompressible(visco)elastic rubbers. Force boundary conditions nonlinearities arise as pressure loadinside the tire and displacement boundary conditions nonlinearities are present due tocontact with a foundation. This creates changes in the boundary conditions during asimulation.There is a vast literature, sometimes related to tires, on computational developments forcontact problems in FE and quite some work is also incorporated into ABAQUS. Examplesare the works of Oden on friction and rolling contact (Oden and Pires, 1983, 1984; Odenand Martins, 1985; Oden and Lin, 1986; Oden et al., 1988; Faria et al., 1989), Laursen andSimo in the field of contact problems with friction (Simo and Laursen, 1992; Laursen andSimo, 1993b,a) and Padovan on rolling viscoelastic cylinders (Zeid and Padovan, 1981;Padovan, 1987; Kennedy and Padovan, 1987; Nakajima and Padovan, 1987; Padovan etal., 1992). More recent are the works of Wriggers (Wriggers et al., 1990; Zavarise etal., 1992; Haraldsson and Wriggers, 2000; Bandeira et al., 2004; Wriggers, 2006) onconstitutive interface laws with friction.Literature specifically related to tires does usually not provide detailed information. Themain reason is that most tire manufacturers use own (in-house) finite element codes andspecific implementations are kept confidential. However the literature provides insight inthe trends and developments of tire modeling throughout the past decades. Besides thejournal of Tire Science and Technology the reader is referred to the two overview papersof Mackerle (1998, 2004) about rubber and rubber-like materials, finite element analysesand simulations for an extensive reference list.The following overview focuses on methods to obtain the cornering and braking forcesacting on a rolling tire with friction. A distinction is made between implicit and explicitmethods.

2.3.1 Implicit analysis

Static and quasi-static analysis

Static analyses are used when inertia effects can be neglected and time-dependent mate-rial effects are not included. In this case the time increments are then simply fractionsof the total period of the step, which are used as increments in the analysis. If time-dependent material effects are taken into account, such as viscoelastic materials, the ap-proach is called quasi-static. An implicit analysis is solved using an incremental-iterative


procedure, which requires matrix inversion (SIMULIA, 2009c).One of the first overview papers, where FE models and the contact problem for tires aredescribed, is the paper of Noor and Tanner (1985). In this article, for a NASA researchprogram for the space shuttle tires, the current status and developments of computationalmodels for tires are summarized. This review has been made again by Danielson et al.(1996). Basically all the tire models are axisymmetric, with no tread pattern or only cir-cumferential groves due to the limits of computational resources. The effect of friction,using the Coulomb model, is investigated on the vertical load versus vertical deflectioncurve.Another popular way to reduce the number of degrees of freedom is the application ofthe global-local analysis. In this approach, an analysis of the complete structure is firstperformed with a coarse mesh. After that a part of the structure is meshed finer and in-terpolated displacements are applied at the boundaries of this region. Such an approachcan be used to compute the forces in the contact area of a deflected tire (Gall et al., 1995;Meschke et al., 1997). Although a local model is able to give detailed numerical resultscorresponding to a tread block, the accuracy is strongly influenced by the simple globalmodel.With the ever increasingly computational power it is nowadays possible to mesh a partor even the whole tread of the tire with a detailed pattern, e.g. Cho et al. (2004), andperform very detailed static analyses including footprint shapes as function of axle loadand inflation pressure.

There are three possibilities to obtain the cornering and braking forces acting on a rollingtire, which follow on a (quasi)static analysis. The first possibility is an implicit dynam-ical analysis, the second one is the arbitrary Lagrangian-Eulerian method and the lastpossibility is an explicit analysis.

Dynamic analysis

An implicit dynamical analysis is not often used for rolling tires, since it is well-knownthat this type of analysis is not efficient in solving changing contact conditions. Thenonlinear equation solving process is expensive due to the Newton iterations, and if theequations are very nonlinear, as in the case of changing contact, it may be difficult toobtain a solution (SIMULIA, 2009c).

Arbitrary Lagrangian-Eulerian method

The arbitrary Lagrangian-Eulerian (ALE) method is developed for numerical analysis ofrolling contact problems, see the articles of Nackenhorst (2004); Ziefle and Nackenhorst(2005) and Laursen and Stanciulescu (2006). This method converts the steady state mov-ing contact problem into a pure spatially dependent simulation, where the mesh is fixed

2.3 TIRE RESEARCH USING FE MODELS 19

in space and the material flows trough the mesh. Thus the mesh needs to be refined onlyin the contact region, which leads to a computational time reduction.This type of modeling, for simulating tire spindle force and moment response under sideslip angles is described by Darnell et al. (2002). The simulation model is composed ofshell elements, which model the tread deformation, coupled to special purpose finite ele-ments that model the deformation of the sidewall and contact between the tread and theground, where Coulomb friction is used. Despite the seemingly simple model the resultscorrespond quite well with experiments.FE simulations with ABAQUS, in which the effect of tire design parameters on lateralforces and moments is studied, are presented in a paper of Olatunbosun and Bolarinwa(2004). Parametric studies are performed on a simplified tire (no tread, negligible rimcompliance and viscoelastic properties, shear forces modeled with Coulomb friction withconstant µ) and a comparison with literature is made to show the computational time re-duction of the method compared to explicit analyses. The effects of variations in stiffnessand geometry on the nonuniformity of tires is investigated by Jeong et al. (2007) usingABAQUS, where also a Coulomb friction model is used for the tire-road interaction.

2.3.2 Explicit analysis

In an explicit analysis the dynamic response problems are solved using an explicit directintegration procedure. The displacements and velocities are calculated in terms of quan-tities that are known at the beginning of an increment and no iterations and no tangentstiffness matrix are required, which is an advantage compared to the implicit method.However due to the explicit time integration a very small time-step, which depends onthe highest frequency present in the model, is usually required. Therefore this approachis ideal to simulate transient behavior in a short time span, such as impact of a tire with acleat. It can also be used to compute handling characteristics, but longer time spans arerequired to reach the steady-state situation. Furthermore for these longer time spans therisk of error accumulation is present as shown by Tönük and Ünlüsoy (2001).Explicit simulations to predict tire cornering forces with a maximum side slip angle ofthree degrees, using the package PAM-SHOCK, are presented by Koishi et al. (1998). Be-sides a comparison with experiments, parametric studies on the effect of inflation pres-sure, belt angle and rubber modulus are performed. Koishi et al. (1998) use the Coulombfriction model, with coefficient equal to one.A prediction of tire cornering forces on a drum is given in a paper of Tönük and Ünlü-soy (2001), where the FE package MARC is used. A comparison with experiments is alsopresented. With the model a maximum side slip angle of five degrees is obtained, higherangles created problems caused by the used Coulomb friction model and error accumu-lation dominates the model results before a steady-state is reached, which occurs evenearlier with higher normal loads. In the region below five degrees, when the cornering


force is almost linear for increasing side slip angle, the results show a good comparisonwith experimental data.A cleat test, with a treadless tire model, can be found in a paper of Olatunbosun andBurke (2002), where the package NASTRAN has been used. A comparison with experi-ments shows deviations as the traverse speed increases above 20 km/h.Rao et al. (2003) simulated the dynamic behavior of a pneumatic tire usingABAQUS/EXPLICIT to replace the extensive measurement program test to fit Magic For-mula parameters. A study on a passenger car radial tire to simulate cornering behavior,braking behavior, and combined cornering & braking behavior is presented. Furthermorethe effect of camber angle and grooved tread on tire cornering behavior is studied. To re-duce computational load two tire models are used, one treadless tire and a tire with fivecircumferential groves. Again the Coulomb friction model with coefficient equal to oneis used. The simulations are however unstable for side slip angles above five degreesand longitudinal slip larger than 12%, which is explained due to the use of an inadequatefriction model.In an article of Cho et al. (2005) the dynamic response of a fully patterned tire rollingover a cleat is presented, using ABAQUS/EXPLICIT. A constant friction coefficient of oneis used for the tire-cleat contact. To decrease the computational load the fiber-reinforcedrubber is modeled with composite shell elements and mass lumping is used to decreasecomputation time (SIMULIA, 2009c). Kerchman (2008) conducted an analysis for rideand harshness analysis with a detailed tire-wheel model coupled with a suspension andattached to a simplified vehicle model, again with the Coulomb friction model.

2.4 Modeling framework

2.4.1 Numerical method

Based on the literature overview it is clear that there are only two methods suitable to ob-tain the cornering and braking forces of a rolling tire: the ALE method or the dynamicalexplicit approach. The explicit method does not require the inversion of the global massand stiffness matrices, which is an advantage. However due to the explicit time integra-tion a very small time-step is required to obtain a stable solution. Therefore it is idealto simulate transient behavior in a very short time span, such as impact of a tire with acleat. It is however not ideal to compute handling characteristics, since longer time spansare required and this results in unacceptable computation times. To overcome this theALE method can be used, which is also available in ABAQUS, and is developed to com-pute the steady-state response of rolling structures. This is an effective method to obtainthe global force and moment characteristics of a tire under different driving conditions,such as braking or side slip and camber angles (SIMULIA, 2000). This so-called steady-

2.4 MODELING FRAMEWORK 21

state transport analysis (SIMULIA, 2009a, chapter 6.4) uses a moving reference framein which rigid body rotation is described in an Eulerian manner and the deformation isdescribed in a Lagrangian manner (the ALE-formulation). Furthermore frictional effects,inertia effects, and history effects in the material can all be accounted for.In an article of Kabe and Koishi (2000) a comparison between the steady-state transportmethod and ABAQUS/EXPLICIT and experiments for steady-state cornering tires is made.Although the results of both methods are closer to each other than to the experiments,the steady-state transport method is significantly faster, 6 hours and 8 days respectively.This, together with the possibility to reduce the number of elements outside the contactarea, clearly shows the benefits of the steady-state transport analysis and therefore thismethod is chosen to compute the steady-state characteristics of a rolling tire.

2.4.2 Friction model

From the literature review it follows that Coulomb friction is still often used in numeri-cal simulations. Deviations from the experiments are often attributed to this model andthis indicates that a different model should be used to improve the results. One of therestrictions of the steady-state transport approach is that the underlying surface must be(rigid) flat, convex or concave (SIMULIA, 2009a, chapter 6.4). This means that it is notpossible to incorporate a rough road surface as used by Wriggers and Reinelt (2009).So the possible surface effects should be captured in the parameters of a friction model,since a flat surface can not exert oscillating forces on the rubber surface. Although themodels of Klüppel and Heinrich (2000) and Persson (2001) incorporate the effect of sur-face roughness directly in their model description, the models are not suitable for highsliding velocities yet, which occur when tires are tested at high velocities.If the sliding velocity influence in the friction models (2.2), (2.4), (2.5), (2.6) and (2.7) iscompared, it follows that all models are able describe the shape as observed by Grosch.The models given by (2.4), (2.5) and (2.6) provide however no direct insight in the fric-tional properties and are therefore not preferred. The model of Huemer et al. (2001a) isdeveloped for a rubber block with a specific geometry and dimension, which means that(2.7) describes the combined effect of contact and block geometry on the friction force.Therefore the model of Savkoor is chosen to describe the sliding velocity dependence,since the parameters in this model are directly related to the shape of the friction curve.It is shown by Lupker et al. (2004) that this model can also be used in situations withvery high sliding velocities.Furthermore the pressure dependence in most models show a decreasing friction coeffi-cient for increasing pressure, which is not included in the original model of Savkoor. Toincorporate this effect (2.2) is extended with a pressure term (Lupker et al., 2004), to give


a friction law, which depends on the contact pressure and the slip velocity as follows

µ(p, v) =

(p

p0

)−k [µs + (µm − µs) exp

−h2 log2

(v

Vmax

)], (2.11)

where the parameters p0 and k are related to the contact pressure and the parametersµs, µm, h and Vmax are related to the sliding velocity. This model can also be found in thework of Smith et al. (2008), where the steady-state approach of ABAQUS is used to predictthe wear of a tread profile.

2.5 Steady-state transport analysis

In this section the relevant parts of the ALE approach are presented, which are basedon the documentation of SIMULIA (2009c). The kinematics of the rolling problem aredescribed in terms of a coordinate frame that moves along with the ground motion ofthe body. In this moving frame the rigid body rotation is described in a spatial or Eule-rian manner and the deformation in a material of Lagrangian manner. This kinematicdescription converts the steady-state moving contact field problem into a purely spatiallydependent simulation. In the following derivation Lagrangian coordinates are denoted inuppercase (e.g. X) and the Eulerian coordinates are denoted in lowercase (e.g. x).A deformable body is rotating with a constant angular rolling velocity ω around a rigidaxle T at X0, which in turn rotates with a constant angular velocity Ω around the fixedcornering axis n, which is normal to the rigid surface, through point Xc, see Figure 2.6.Hence, the motion of a particle X at time t consists of a rigid rolling rotation to position

ω

Ωn

T X0

Xc

Figure 2.6 / Constant cornering motion in the ALE approach. Figure reproduced from(SIMULIA, 2009c).

2.5 STEADY-STATE TRANSPORT ANALYSIS 23

Y, described by

Y = Rs · (X−X0) + X0, (2.12)

where the spinning rotation matrix Rs is defined as

Rs = exp(ωt), (2.13)

with ω the skew-symmetric matrix associated with the rotation vector ω = ωT. This rigidrolling rotation is followed by a deformation to a point x, and a subsequent corneringrotation around n to position y so that

y = Rc · (x−Xc) + Xc, (2.14)

where Rc is the cornering rotation given by

Rc = exp(Ωt), (2.15)

and Ω is the skew-symmetric matrix associated with the rotation vector Ω = Ωn. Thevelocity of the particle then becomes

v = y = Rc · (x−Xc) + Rc · x. (2.16)

To describe the deformation of the body a map χ(Y, t), shown in Figure 2.7, is intro-duced, which gives the position of a point x at time t as a function of its location Y attime t so that

x = χ(Y, t). (2.17)

X

Rs

χ

Initial configuration

configurationCurrent

Reference configuration

Y

x

Figure 2.7 / ALE decomposition using a rigid rotation and deformation step.


The time derivative of (2.17) is given by

x =∂χ

∂Y· ∂Y

∂t+∂χ

∂t, (2.18)

where

∂Y

∂t= Rs · (X−X0) = ωT× (Y −X0). (2.19)

Noting that Rs = ω ·Rs = ωT ·Rs, and introducing the circumferential direction

S = T× Y −X0

R, (2.20)

where R = |Y − X0| is the radius of a point on the reference body, the velocity of thereference body can now be written as

∂Y

∂t= ωRS (2.21)

and (2.18) can be written as

x = ωR∂Y

∂t· S +

∂χ

∂t= ωR

∂χ

∂S+∂χ

∂t, (2.22)

where S is the distance-measuring coordinate along the streamline. The derivative of(2.15) is given by

Rc = Ω ·Rc = Ωn ·Rc. (2.23)

The velocity of the particle can be written as

v = Ωn× (x−Xc) + ωRRc ·∂χ

∂S+ Rc ·

∂χ

∂t. (2.24)

The acceleration is obtained by differentiation of (2.24)

a = Ω2(nn− I) · (x−Xc) + 2ωΩRn×Rc ·∂χ

∂S+ 2Ωn×Rc ·

∂χ

∂t(2.25)

+ω2R2Rc ·∂2χ

∂S2+ 2ωRRc ·

∂2χ

∂S∂t+ Rc ·

∂2χ

∂t2.

To obtain expressions for the velocity and acceleration in the reference frame tied to thebody, the following transformations are used

vr = RcT · v, ar = Rc

T · a, (2.26)

such that

vr = Ωn× (x−Xc) + ωR∂χ

∂S+∂χ

∂t(2.27)


and

ar = Ω2(nn− I) · (x−Xc) + 2ωΩRn× ∂χ

∂S+ 2Ωn× ∂χ

∂t(2.28)

+ω2R2∂2χ

∂S2+ 2ωR

∂2χ

∂S∂t+∂2χ

∂t2.

For steady-state conditions it holds that ∂χ∂t

= 0 and these expressions reduce to

vr = Ωn× (x−Xc) + ωR∂χ

∂S(2.29)

and

ar = Ω2(nn− I) · (x−Xc) + 2ωΩRn× ∂χ

∂S+ ω2R2∂

2χ

∂S2. (2.30)

The first term of (2.30) can be seen as the acceleration that gives rise to centrifugal forcesresulting from rotation about n. The second term can be identified as the accelerationthat gives rise to Coriolis forces. The last term combines the acceleration that give rise toCoriolis and centrifugal forces resulting from rotation about T. When the deformationis uniform along the circumferential direction, both Coriolis effects vanishes so that theacceleration gives rise to centrifugal forces only.The velocity of the center of the body X0 is given by

v0 = Ωn× (X0 −Xc) (2.31)

since the motions due to rolling and deformation vanish on the axis.In the case of straight line rolling Ω→ 0 (2.29) and (2.30) reduce to

v = v0 + vr = v0 + ωR∂χ

∂S(2.32)

and

a = ar = ω2R2∂2χ

∂S2. (2.33)

2.5.1 Contact conditions for steady-state rolling

Given two points on the surfaces of two bodies in contact, the relative velocity can beexpressed as

v = vD − vR, (2.34)

where vD is the velocity of a point on the deformable body, see (2.27), and vR the veloc-ity of a point on the rigid foundation. This can be split into the normal and tangentialcomponents. The rate of penetration is

h = −n · v = n · vR − ωRn · ∂χ∂S− n · ∂χ

∂t. (2.35)


For any point in contact it follows that n · ∂χ/∂S = 0, therefore

h = n · vR − n · ∂χ∂t, (2.36)

which in incremental form reduces to the standard contact condition

∆h = n · (∆xR −∆xD), (2.37)

with ∆xR and ∆xD the position increment of points on the rigid foundation and thedeformable body respectively. For steady-state conditions it follows that n ·∆xR = 0 and∆xD = 0.

The rate of slip is given by

γi = ti · v, (2.38)

where ti, i = 1, 2 are two orthogonal unit vectors tangent to the contact surface such thatn = t1 × t2. For steady-state conditions it holds that ∂χ/∂t = 0, which gives the finalexpression for the slip velocity

γi = Ωti · (n× (x−Xc)) + ωRti ·∂χ

∂S− ti · vR. (2.39)

A non-slip or stick condition is obtained when (2.39) is equal to zero. In a steady-staterolling situation no relative tangential motion indicates that locally the rotating object isrolling and not sliding on the surface. Sliding of the object will occur if the tangentialforces exceed a certain limit.

2.5.2 Frictional stress for steady-state rolling

Stick

The stick condition for the classical Coulomb law, with constant friction coefficient µ (in2D) is shown in Figure 2.8. This condition states that no relative motion occurs if

τ ≤ τcrit, (2.40)

where τcrit = µp, with p the (normal) contact pressure. This constraint can be handledin ABAQUS in two different ways, using a Lagrange multiplier or approximating this norelative motion constraint using a regularization step.

In case of the Lagrange multiplier method the constraint is obeyed exactly, however theadditional multipliers increase the cost of the analysis and convergence problems mayoccur in areas where the contact conditions change (SIMULIA, 2009c).When a regularization step is applied the non-differentiability of Coulomb’s law at zero


slip velocity is avoided. The regularization function is chosen such that the transitionfrom stick to slip is smooth. Some examples are shown in Figure 2.9. These functionshave the drawback that the transition from stick to slip is approximated and if the penaltyparameter of the regularization function is too large the model is not able to providerealistic predictions of stick-slip motion. The great advantage of the differentiability ishowever a simpler and more robust numerical algorithm, as shown by Wriggers (2006),and this is also the default implementation in ABAQUS.

γ

τcrit

Figure 2.8 / Classic Coulomb’s friction law.

γ

τcrit

Figure 2.9 / Examples of regularization functions for Coulomb’s friction law.

Slip

If the frictional stress is larger than τcrit, sliding occurs and the frictional stress is limitedby

τ = τcrit. (2.41)

For the 3D situation, it is assumed that rubber friction is isotropic, such that the frictionalresponse is the same in both directions. Sliding occurs if the equivalent frictional stress


is larger than τcrit, where τeqv is defined as

τeqv = ‖τ‖ =√τ 2

1 + τ 22 . (2.42)

Similarly the equivalent slip rate is defined as

γeqv = ‖γ‖ =√γ2

1 + γ22 . (2.43)

Furthermore for isotropic friction the direction of the slip and the frictional stress coin-cide, therefore it holds that

τiτeqv

=γiγeqv

(2.44)

and the frictional stress during sliding in direction i is given by

τi = µpγiγeqv

. (2.45)

2.6 Implementation of a friction law in steady-state rolling

For the implementation of friction laws other than the classical Coulomb model the sub-routine FRIC of ABAQUS can be used to prescribe the frictional stresses. The main inputsof this routine are the contact pressure and the slip velocity, given by (2.39). Besides thefrictional stress also the derivatives of the frictional stress with respect to the slip velocityand the contact pressure should be provided. This is different compared to a static analy-sis in ABAQUS, where the derivative of the frictional stress with respect to the slip distanceis required.The friction law is implemented locally on each node j in contact, so (2.11) becomes

µj(pj, γj) =

(pjp0


−h2 log2

(‖γ‖jVmax

)], (2.46)

where pj and ‖γ‖j are the normal contact pressure and equivalent slip velocity at nodej respectively. Furthermore, a threshold parameter plimit is used to prevent the pressureterm from going to infinity if the contact pressure approaches zero. If the nodal con-tact pressure is lower than this threshold, the contact pressure pj is substituted with thethreshold value plimit . The subscript j is omitted in the remainder of the text for read-ability, however all equations are evaluated for each node in contact.

To handle the non-differentiability at zero slip velocity a regularization step is applied. Apiecewise linear regularization function is chosen, with a user-defined critical slip velocityγcrit as penalty parameter. An illustration of the regularization function is shown inFigure 2.10. This parameter is used to calculate a slope ks, which is given by

ks =µ(p, γ)p

γcrit. (2.47)

2.6 IMPLEMENTATION OF A FRICTION LAW IN STEADY-STATE ROLLING 29

Now the frictional stress in stick can be formulated as

τi = ksγi, (2.48)

while the frictional stress for slip follows from (2.45)

τi = µ(p, γ)pγiγeqv

. (2.49)

γcrit γ

1

−1

Figure 2.10 / Piecewise linear regularization function with user-defined critical slip ve-locity γcrit.

In the subroutine additional outputs at every node, such as the friction coefficient µ, theequivalent slip velocity γeqv and the frictional power τeqvγeqv are written to the ABAQUS

output database. For assembly of the global stiffness matrix the following matrix is alsorequired

J =

∂∆τ1∂∆γ1

∂∆τ1∂∆γ2

∂∆τ1∂∆p

∂∆τ2∂∆γ1

∂∆τ2∂∆γ2

∂∆τ2∂∆p

, (2.50)

where the elements for both the stick and slip situation, using friction law (2.46), areprovided in Appendix A. A schematic overview of the in- and outputs of the subroutine isshown in Figure 2.11.

γi

p

τi

γeqv

τeqvγeqv

µ

J

fric

Figure 2.11 / Interface of the subroutine FRIC.


2.7 Validation of the implemented friction law

2.7.1 Cylindrical model

To validate the implementation of the friction law several tests are performed with a sim-ple cylindrical tire-like model. This model consists of a rubber cylinder, the relevantparameters are shown in Table 2.1, which is first pressed against a flat rigid surface andsecondly a constant forward velocity is prescribed, while the rotational velocity is varied.For this illustrative example, the rubber material model of the example tire of ABAQUS

(SIMULIA, 2009b) is used. The initial shear modulus of this model is given by 2C10 andthe parameter D1 governs the compressibility of the material, in this particular case therubber is modeled as fully incompressible.

Table 2.1 / Numerical values of the used parameters in the ABAQUS cylindrical test model.

Parameter Value UnitGeometry Rinner 400 [mm]

Router 500 [mm]Width 235 [mm]

Material model ρ 1 · 10−9 [tonnes/mm3]Hyperelastic, C10 1.0 [N/mm2]reduced polynomial D1 0.0 [N/mm2]

Load Fz −10000 [N]

Forward velocity Vx 13889 [mm/s]

Rotational velocity ω [1 50] [rad/s]

2.7.2 Comparison with standard Coulomb friction law

To make a comparison with the standard Coulomb friction model in ABAQUS two cylindersare used, see also Figure 2.12, one where the default Coulomb model of ABAQUS is usedand one where (2.46) is applied in the contact interface. By choosing the parameters kequal to zero and µs equal to µm, a constant friction coefficient is obtained. For bothmodels a constant friction coefficient of µ = 0.5 is used and γcrit is set to 50 mm/s.The forward velocity is kept constant at 50 km/h and the rotational velocity is increasedfrom 1 to 50 rad/s. This guarantees that a braking, free-rolling and traction situation is

2.7 VALIDATION OF THE IMPLEMENTED FRICTION LAW 31

obtained, since ωfree ≈ 27.5 rad/s. The computed frictional stresses of two nodes in thecontact area are shown in Figure 2.13 together with the error between these nodes and thecorresponding nodes on the other cylinder. It can be seen that the error is zero for bothnodes and this also holds for all other nodes, which shows that the standard availableCoulomb model can be reproduced using (2.46).

X Y

Z

Figure 2.12 / Cylindrical test model used for comparison with the standard Coulombfriction model of ABAQUS.

0 5 10 15 20 25 30 35 40 45 50−0.4

−0.2

0

0.2

0.4

Friction

al sh

ea

r str

ess [M

Pa

]

node 1

node 2

0 5 10 15 20 25 30 35 40 45 50−1

−0.5

0

0.5

1x 10

−6

ω [rad/s]

Err

or

[MP

a]

Figure 2.13 / Frictional shear stresses as a function of rotational velocity for two nodes onthe cylinder with friction model (2.46) and the error between the corresponding nodeson the cylinder with the default Coulomb model of ABAQUS.


2.7.3 Validation of friction law for varying friction coefficient

To validate the implemented friction law for varying friction coefficients the same simula-tion is done with only one cylindrical model. However now the parameters as presentedin Table 2.2 are used in the friction law. The effect of the threshold parameter plimit isshown in Figure 2.14. The contact pressures and slip velocities of this simulation are

Table 2.2 / Numerical values of parameters for (2.46).

Parameter p0 k µs µm h Vmax plimitValue 0.5 0.5 0.2 1.1 0.5 695 0.2

Unit [MPa] [-] [-] [-] [-] [mm/s] [MPa]

logged and used to off-line compute µ(p, γ), which can then be compared with loggedvalues of µ. In Figure 2.15 the logged values for µ are shown for two contact nodes to-gether with the error between these values and the recomputed values in MATLAB basedon the contact pressures and slip velocities of these nodes. These error signals confirmthat the implementation of the friction law is correct.

0 0.2 0.4 0.6 0.8 10.7

0.8

0.9

1

1.1

1.2

1.3

1.4

1.5

1.6

Pressure [MPa]

µp [−

]

Figure 2.14 / Visualization of the effect of parameter plimit at 0.2 MPa, which limits thepressure term in (2.46) to prevent numerical problems at low contact pressures.

2.8 STEADY-STATE FREE-ROLLING 33

0 5 10 15 20 25 30 35 40 45 500

0.5

1

1.5

µ [−

]

node 1

node 2

0 5 10 15 20 25 30 35 40 45 50−1

−0.5

0

0.5

1x 10

−5

ω [rad/s]

Err

or

[−]

Figure 2.15 / Friction coefficient as function of the rotational velocity for two nodes onthe contact surface and the error between ABAQUS and the off-line recomputed µ, basedon logged contact pressure and slip velocity.

2.8 Steady-state free-rolling

The rotational velocity of a free-rolling tire is related to the forward velocity Vx as follows

ωfree =Vxre, (2.51)

where re is the effective rolling radius defined at zero torque instead of zero longitudi-nal force (Pacejka, 2006). The unloaded radius, ru, of the tire can be used to make anestimation of the free-rolling rotational velocity. A braking or traction situation occurs ifthe rotational velocity is not equal to the free-rolling rotational velocity at the same for-ward velocity. Since the steady-state transport analysis requires that both the rotationalvelocity and the forward velocity are prescribed, the free-rolling solution is not known apriori. However the drive (or brake) torque T is defined as zero in case of free-rollingand this can be used to iteratively find the rotational velocity, which corresponds to free-rolling. While the forward velocity is kept constant during a steady-state rolling step, therotational velocity is incrementally updated based on previous values of the torque. Theupdate is based on the Newton-Raphson method

xn+1 = xn −f(xn)

f ′(xn). (2.52)

In this case the goal is to find zero torque as function of ω, so (2.52) can be rewritten as

ωn+1 =Tnωn−1 − Tn−1ωn

Tn − Tn−1

, (2.53)


where the derivative is approximated using the finite difference method and n is an in-crement in the steady-state rolling step.For the implementation a bound on the maximum update for ω is used to prevent conver-gence problems within an increment, which can occur if the difference in ω between twoincrements is too large. This method works well if ω is initialized close to the free-rollingrotational velocity. In Figure 2.16 a representative curve of the brake torque for increasing

T

ωfree

ω

Positive update ω

Figure 2.16 / Representative curve of the brake torque T as function of rotational velocity.In the dashed region a fixed update for ω is used and in the vicinity of ωfree the Newton-Raphson method is applied.

ω is shown. It can be seen that (2.53) will not work if ω is initialized in the dashed re-gion before the torque reaches its maximum. To overcome this, a positive update for ω ismade if the initial rotational velocity is far away from the free-rolling situation, e.g. a fullylocked wheel ω = 0. Once the solution is near the vicinity of free-rolling (2.53) is usedagain. This combination makes it possible to compute the entire braking characteristicof a tire.The implementation of the algorithm to find the free-rolling steady-state solution is vali-dated using the example tire of ABAQUS (SIMULIA, 2009b). In this example a 175 SR14

tire is used. A forward velocity of 10 km/h is prescribed and the obtained ωfree of 9.027

rad/s corresponds very well with the 9.026 rad/s given in the documentation.

2.9 Conclusions

In this chapter the choice of the numerical modeling framework for steady-state rollingtires with friction is motivated and described. Based on a literature overview to obtainthe handling characteristics of rolling tires, a choice is made for the steady-state transportanalysis of ABAQUS to compute the forces acting on the tire under different driving con-ditions. This is a computationally efficient method to obtain the global steady-state force

2.9 CONCLUSIONS 35

and moment characteristics of a tire under different driving conditions. Additionally,frictional effects, inertia effects, and history effects in the material can all be accountedfor. However, this method has also some limitations. The underlying surface must be(rigid) flat, convex or concave, which means that it is not possible to model a rough roadsurface. Therefore possible surface effects should be captured in the parameters of a fric-tion model. Furthermore, the current implementation in ABAQUS is only suitable for themechanical domain, a fully coupled thermo-mechanical analysis can not be performed.Most work found in literature still uses the Coulomb friction model for the tire-road in-teraction problem, while it is clear from experiments with elastomers that rubber frictiondepends on various parameters, such as sliding velocity, contact pressure, surface rough-ness and temperature. Because of these dependencies the Coulomb friction model is notsufficient to model the tire-road interaction. The goal of this thesis is to include some ofthese dependencies in an enhanced friction model for FE tire simulations.The choice of an improved friction model is based on the chosen numerical method toobtain the global steady-state force and moment characteristics and an overview of fric-tion models to describe the frictional response of rubber. The chosen friction model isdirectly dependent on contact pressure and sliding velocity, while the global effect of tem-perature and surface roughness is captured in the parameters of the friction model.The mathematical formulation of the steady-state transport method is reviewed and theimplementation of the friction model in ABAQUS is derived. Practical issues related to thenumerical implementation are identified and discussed. Furthermore, the implementa-tion of the friction model is validated using different test conditions. Finally, an algorithmis presented to obtain the steady-state rotational velocity of a free-rolling tire, which is notknown in advance. This algorithm has been adapted such that it can be used to computethe entire braking characteristic of a tire, even when the brake torque is not constant forincreasing rotational velocities.

CHAPTER THREE

Simulation procedure to computehandling characteristics with an FE model

Abstract / In this chapter the strategy for identification of the parameters of the friction model isexplained. The parameters are identified sequentially using a two step experimental / numericalapproach, which is based on lab and full scale experiments. Furthermore details of the used testtire and the corresponding FE model are provided. The different simulation steps required tocompute the three basic handling characteristics (pure braking, pure cornering and combinedslip) are discussed, while for each step the choice of boundary conditions is motivated.

3.1 Introduction

In this chapter the identification procedure to obtain the parameters of the friction modelis explained and the different simulation steps to compute the three basic handling char-acteristics with an FE tire model are discussed.The FE tire model has to fulfill several requirements. Information about the local frictioncoefficient, slip velocity and frictional power is made available as output of the frictionmodel. The forces and moments acting on the axle should also be available as output tomake a comparison with experiments possible. Additionally several inputs, such as ge-ometry and material properties, are required to construct the FE model. Different typesof simulations can be performed by prescribing specific boundary conditions on the FEmodel.Furthermore, the use of the steady-state transport algorithm introduces a constraint onthe underlying surface, which must be flat. In Figure 3.1 these requirements are incor-porated into the different blocks, as shown in Figure 1.3, and the rough road surface isreplaced with a flat surface.

37

38 3 SIMULATION PROCEDURE TO COMPUTE HANDLING CHARACTERISTICS WITH AN FE MODEL

INPUTS

INTERACTION MODEL

FE ModelOUTPUTS

GeometryMaterial dataBoundary conditions

3D Force & MomentFootprintFriction coefficientFrictional powerSlip velocity

Flat road

Figure 3.1 / Schematic overview of the interaction model in the steady-state numericalmodeling framework.

This chapter focuses on the description and the effect of the boundary conditions appliedat each simulation step. Obviously, the inputs to the FE tire model, such as geometryand material properties, determine the actual structural response of the tire. However,the simulation steps remain the same, regardless of the type of tire investigated. There-fore, the modeling approach can be discussed and evaluated without detailed informationabout the specific tire considered. The exact description of the geometry and the materialproperties of the tire under investigation is proprietary information of Vredestein and istherefore not discussed in detail.

The chapter is organized as follows. In Section 3.2 the strategy to identify the parametersof the friction model is discussed. The design of the test tire, which is used for experi-mental validation, is presented in Section 3.3. Next, the different steps and correspondingboundary conditions to create the 3D FE tire model, are given in Section 3.4. The simu-lation procedure to compute the braking characteristic is described in Section 3.5. Afterthat, the simulation steps to obtain the cornering characteristic and the combination ofcornering and braking are explained in Section 3.6. A summary of the followed steps isgiven in Section 3.7.

3.2 APPROACH TO IDENTIFY PARAMETERS OF THE FRICTION MODEL 39

3.2 Approach to identify parameters of the friction model

Due to the choice of a phenomenological friction model several experiments need to becarried out to gather enough data to extract the parameters of the model. Since the pro-posed friction law (2.11) depends on a contact pressure related part and a slip velocityrelated part, a two step experimental / numerical approach is developed and shown inFigure 3.2. In this approach the parameters of the friction model are identified sequen-tially using two different experimental setups.

LAT



TNO Tyre Test Trailer

Inputs

Frictionmodel

FE Tire

Virtual prototyping

Full scale

LAT wheelFEM

Lab scale



Figure 3.2 / Schematic overview of the two step experimental / numerical approach toobtain friction information using both lab and full scale experiments.

First, friction experiments are performed on a laboratory abrasion and skid tester (LAT) toinvestigate the influence of contact pressure on the frictional force. These measurements,which are conducted with very low rolling velocities and under free-rolling conditionsto minimize slip velocity, are used to identify the contact pressure dependent frictionparameters. A numerical model of this lab setup is used to validate the contact pressurepart of the friction model by comparing simulated forces with measured forces. This isdescribed in Chapter 4.Secondly, braking experiments at different velocities with the test tire are conducted toobtain slip velocity data for the friction model. The friction model is then coupled to theFE model of the test tire and the remaining parameters are identified and validated bycomparing simulated axle forces with measured forces. This is the subject of Chapter 5.Finally, the fully identified friction model is used to investigate the predictive capabilityof the friction model by comparing simulated axle forces due to cornering and combinedslip with experiments. The discussion of these results is presented in Chapter 6.


In this research only dry friction is considered, which means that all experiments areperformed on dry surfaces and it is assumed that no external lubricants are present. Thepresented two step experimental / numerical approach can also be applied under wet andicy conditions, although performing reproducible experiments under these conditionswill be challenging.

3.3 Design of the test tire used for experimental validationof the FE model

The passenger tire under investigation in this research has been specifically built for thisproject, which allowed for some freedom in the design of the tread. With the currentimplementation of the steady-state transport capability of ABAQUS it is nowadays possibleto include the full geometry of treaded tires. A limitation of this procedure is howeverthat solution accuracy can degrade if a treaded tire is used. Particularly the accuracy ofthe longitudinal stress, when the angular extent of the so-called base pitch sector is toolarge, is affected as stated by Qi et al. (2007). To avoid this undesirable numerical erroron the FE results, a tire with three longitudinal groves has been designed. This designguarantees continuous streamlines along circumferential direction.The longitudinal groves are placed slightly asymmetrically with respect to the center ofthe tire to create a tire with a narrow and a wide shoulder. Due to this asymmetry twodifferent footprint sizes will result during steering to the left and right side. This allowsto investigate the influence of contact area and contact pressure on the generated forcesduring steering both in experiments and in the FE model.The radial tire, which has been manufactured by Vredestein, is shown in Figure 3.3. Thegroves and the inner tread blocks have length L2, while the shoulders have length L1 andL3 respectively.

Figure 3.3 / The test tire with three longitudinal groves, placed asymmetric with respectto the center of the tire. The tire has been manufactured by Vredestein.

3.4 FE TIRE MODEL 41

3.4 FE tire model

The structural FE tire model is constructed in several steps, shown in Figure 3.4. Firsta 2D cross section of the tire is created. This section is mounted on the rim and theninflated to the desired inflation pressure. Secondly a 3D model is created by revolvingthis 2D cross section around the axle. The 3D tire is loaded to make contact with the roadand finally a steady-state rolling step is conducted.The structural response of the finite element model has been validated statically by Vre-destein. This has been done using measurements of the contact pressure distribution andcontact area. Furthermore the radial and axial stiffness of the tire has been validated withexperiments on a flat plank tire tester, which have been conducted by Vermond (2008).

2D AxisymmetricRim mounting and inflation

3DFootprint analysis

3DSteady-state rolling

Figure 3.4 / Modeling steps for 3D steady-state rolling tire simulations.

3.4.1 2D tire cross section

The 2D cross section contains all the geometric and material information of the tire. Aradial tire consists of different rubber compounds and several reinforcement materials,which are described in detail by Gent and Walter (2005) in chapter 1. These reinforce-ment layers are the dominant load carrying parts of the cord-rubber composite and areindicated in Figure 3.5 by the lines inside the contour of the cross section. The remain-ing part of the cross section consists of the different compounds, for example the tread,sidewalls and innerliner.The rubber compounds are modeled as slightly compressible hyperelastic materials. Gen-eralized 3- and 4-node linear, hybrid with constant pressure, axisymmetric elements withtwist are used to mesh the rubber compounds. These elements have three active degreesof freedom, two displacements and one rotation, which corresponds to the twist angle.Large deformations, which occur during steering, can be described more accurately withquadratic elements, however for problems involving contact, linear elements are recom-mended (SIMULIA, 2009c). The reinforcement belts and plies are modeled with so-called rebar elements. These elements are used to define layers of uniaxial reinforcementin membrane, shell or surface elements and can be used to model the cord properties ofthe reinforcements under a specific angular orientation. The advantage of the rebar ele-ments is that these layers are treated as a smeared layer with a constant thickness equal


to the area of each reinforcing bar divided by the reinforcing bar spacing, without theneed to model each cord separately (SIMULIA, 2009c, chapter 2.2). The rebar elementsused are generalized 2-node linear, with twist, axisymmetric surface elements. These el-ements have the same three active degrees of freedom as mentioned above. The finalmesh is shown in Figure 3.61, where it can be seen that the corners of the tread blocksare rounded to reduce large gradients and improve convergence.

Figure 3.5 / Geometrical shape of the cross section, including reinforcement layers. Theaspect ratio is distorted for confidentiality reasons.

Figure 3.6 / Final mesh of the cross section, with the reinforcements layers as embeddedrebar elements.

1Mesh layout provided by Vredestein.


3.4.2 Rim mounting and inflation

The rim has to be modeled as well to transmit road induced forces to the axle. Therim itself is modeled with two separate analytical rigid bodies as shown in Figure 3.7.These two bodies are connected to the fixed world with two reference nodes, located onthe z-axis. Frictionless contact surfaces are defined on the rim and part of the tire toestablish contact between the tire and the rim. As first step the tire is mounted ontothe rim. This is done by translating the rigid bodies along the z-axis using displacementboundary conditions, shown in Figure 3.8, such that the distance between the rim andthe tire complies with the standards specified by the European Tyre and Rim TechnicalOrganisation (2010). Finally the tire is inflated using a distributed surface load, whichis applied to the inner surface of the cross section. This pressure load is equal to thedesired inflation pressure and forces the tire into the final position, which can be seen inFigure 3.9.

Figure 3.7 / Initial positions of the tire and the rim.

3.4.3 3D tire model

The 3D tire model is created by revolving the axisymmetric 2D cross section of the tire,where a non-uniform discretization in circumferential direction is used to discretize the3D model into seventy two sectors. The used symmetric model generation procedure(SIMULIA, 2009c, chapter 10.4) converts the axisymmetric elements, used for the com-pounds, into general 3D, 6- and 8-node linear, hybrid, elements. The axisymmetric rebarelements are converted to 4-node quadrilateral, reduced integration elements. All ele-


Figure 3.8 / Mounting of the tire onto the rim, using displacement boundary conditionson the rim.

Figure 3.9 / Final positions of the tire and the rim after inflation of the tire.

ment types have three active translational degrees of freedom.An additional translational joint is added to connect the two separate rigid bodies togetherand prevent relative rotation of the two rigid bodies. The joint is linked to a new referencenode, which represents the axle of the wheel. The reaction forces acting on this node willbe used to compare with measurements.The final circumferential discretization is obtained by a number of mesh refinementsnear the contact zone until the reaction forces acting on the wheel converge to a steady


result. Each mesh refinement does however increase computation time, therefore thepossibility of using cylindrical elements has been investigated. These elements are avail-able for precise modeling of regions in a structure with circular geometry and can spanlarge angles, which could reduce the number of used sectors significantly. However thecylindrical elements in contact with the rim introduced convergence problems and aretherefore not used.Furthermore a flat rigid body, which represents the road, is added. This surface is linkedto a rigid body reference node. The motion of the entire road can now be prescribed byapplying boundary conditions at this rigid body reference node, which in this case is con-strained in all six degrees of freedom. The complete 3D model is shown in Figure 3.10.

Figure 3.10 / Non-uniform discretization of the FE tire including the rim and road sur-face.

3.4.4 Mesh effect on lateral force in 3D model

When the 3D model is generated the stresses and strains of the 2D model are transferredto the three-dimensional model and a static equilibrium step is performed. The elementformulations for the two-dimensional and three-dimensional elements are not identicaland as a result, there is a slight difference between the equilibrium solutions generatedby the two- and three-dimensional model (SIMULIA, 2009b, chapter 3.1).In this equilibrium step the contact conditions between the rim and tire are changedto prevent the tire from slipping along the rim in circumferential direction. When slipoccurs during the steady-state transport analysis, the solution obtained is no longer thecorrect steady-state solution, because convective effects are ignored. It is assumed that


no convective effects are present between surfaces during steady-state transport analysis,which means that two points on the contacting surfaces are still in the same contactposition after a rigid spinning motion. If however slip occurs, these points would notbe in the same position after a rigid spinning motion. This is not taken in account andleads therefore to an incorrect solution. To ensure that no slip takes place the frictionproperties are changed to ‘rough’ friction, which means that no slip is allowed. This isused in combination with a so-called no separation condition in normal direction, whichprevents separation of two surfaces once contact has been established. Both constraintsare enforced by Lagrange multipliers.Boundary conditions and loads are not transferred and are therefore redefined in the newanalysis to match the loads and boundary conditions, which were used in the 2D model,e.g. the inflation pressure. The axle reference node is constrained, using boundaryconditions, for all six degrees of freedom in this equilibrium step.

When there is no contact with the road surface, the force on the axle in all three directionsafter this static equilibrium step should be zero, since the reference point on the axle isnow the only connection to the fixed world. This is however not the case if the mesh ofthe 2D cross section is not chosen carefully. If for instance the mesh, shown in Figure3.11 is used, a lateral force of 14.8 N is obtained in the 3D model. This is due to theasymmetrical mesh in combination with default contact and solver tolerances, which leadto a small difference in the contact pressure distribution on both sides, when mountedonto the rim. This in turn leads to a small force difference on the two rigid bodies. Whenthe 3D model is generated this small force error is amplified along the circumferentialdirection and creates a significantly nonzero force in lateral direction.This can be prevented by using the mesh as shown in Figure 3.6, which is fully symmetricin the part that makes contact with the rim. The discretization error, which is still present,is now equal at both sides and therefore cancels out. As a result zero lateral force in the3D model is obtained after the equilibrium step.

Figure 3.11 / Close-up of a mesh, which leads to a nonzero lateral force in the 3D model.

3.5 SIMULATION PROCEDURE TO COMPUTE THE BRAKING CHARACTERISTIC OF A ROLLING TIRE 47

3.4.5 Static loading of the tire

After the equilibrium step the tire is loaded in two subsequent static analysis steps. Thefirst of these static steps establishes the initial contact between the road and the tire byprescribing a vertical displacement on the reference node of the wheel. Since this is astatic analysis, it is recommended that contact is established with a prescribed displace-ment, as opposed to a prescribed load, to avoid convergence difficulties that can arise dueto unbalanced forces (SIMULIA, 2009b). The prescribed boundary condition is removedin the second static step, and replaced by a desired vertical load on the reference node ofthe wheel.Additionally, the contact between the tire and the road has been made frictionless, sincethe frictional behavior in a steady-state transport step is different from the frictional mod-els used in a static analysis as already discussed in Section 2.6. This can create disconti-nuities between the solutions of a steady-state transport analysis and a static analysis. Ifa zero coefficient of friction is used in all analysis steps prior to a steady-state analysis asmooth transition is ensured (SIMULIA, 2009c, chapter 6.4).

3.5 Simulation procedure to compute the braking character-istic of a rolling tire

Before an actual braking characteristic of a rolling tire can be computed using the methodpresented in Section 2.8, friction between tire and road must be activated. Therefore anadditional steady-state step is introduced in which the friction model is activated. Inthis step the friction coefficient is increased linearly over the step to prevent convergenceproblems. To this end the local friction coefficient (2.46) is pre-multiplied with a param-eter R (see also Appendix A), which varies linearly between zero and one during the firststeady-state step.In this first steady-state transport step the tire is brought up to speed by setting the trans-lational velocity of the tire to the desired forward velocity Vx, also referred to as longitu-dinal velocity, while initializing the rotational velocity at 1 rad/s. This corresponds to analmost locked wheel. During the last simulation step the rotational velocity of the tire isincreased every increment and the corresponding steady-state solution is computed untilthe steady-state free-rolling solution is obtained. The complete simulation procedure issummarized in Figure 3.12.The obtained longitudinal force characteristic is referred to as the pure braking charac-teristic in case no (global) lateral slip is present (Pacejka, 2006).


2D model

3D model

equilibrium step

inflation

axle displacement

axle force

STATIC

ω = 1

ω = ωfree

Steady State Transport

Free-rolling

Friction model

Enable friction

Friction

Frictionless

and rim mounting

2D-3D

algorithm

Figure 3.12 / Schematic overview of the simulation procedure to obtain the braking char-acteristic of a rolling tire.

3.5.1 Mesh effect on the force equilibrium in vertical direction

When inertia effects are taken into account during a steady-state rolling step Coriolis andcentrifugal forces are generated. These forces are caused by deformation and spinningof the tire and the centrifugal force will increase with increasing rotational velocity. How-ever, when a non-uniform mesh in circumferential direction is used, a small numericallyinduced force in the direction of the denser mesh, see Appendix B, is generated when in-ertia effects are taken into account. This force is small with respect to the applied load onthe tire and as such does not affect the computed longitudinal and lateral forces. There-fore, inertia effects are included, since the generated internal forces lead to increasedstresses in circumferential direction and stiffening of the tire for increasing rotationalvelocity. Furthermore, the wheel is loaded (and not the road) in the FE model to keep thenormal force on the axle constant, because the load on the wheel is also kept constantduring experiments.

3.5 SIMULATION PROCEDURE TO COMPUTE THE BRAKING CHARACTERISTIC OF A ROLLING TIRE 49

3.5.2 Effect of penalty parameter in the friction model on the longitudi-nal force

For the implementation of the friction model, a user-defined critical slip velocity γcrithas been introduced in Section 2.6. Physically this penalty parameter can be interpretedas the value of the sliding velocity below which sticking occurs. Simulation results aremore accurate for smaller values of the penalty, since the approximation of the transitionbetween stick and slip improves for smaller values as indicated in Figure 2.10. Howeversolving the contact conditions of the FE model can become more difficult and as a resultcomputation time increases. Hence using a larger penalty makes convergence of thesolution more rapid at the expense of solution accuracy. So a trade-off between accuracyand computation time has to be made.To investigate the influence of the value of the penalty parameter on the accuracy of thesolution, several simulations, using different values for γcrit have been performed. Inthese simulations the braking characteristic of the tire is computed for a forward velocityof 100 km/h and a load of 5000 N is applied on the axle. Furthermore a constant frictioncoefficient of 0.8 is used.The built-up of the longitudinal force around free-rolling is dominated by the longitudinalslip stiffness of the tire. The value of the penalty parameter has however a direct influenceon the longitudinal force, since this force is generated during the transition from stickto full slip. If a too large penalty value is chosen the gradient of the longitudinal forcearound free-rolling decreases and an underestimation of the longitudinal slip stiffness isobserved in the numerical results.

The longitudinal force obtained with the default penalty value of ABAQUS is used as refer-ence solution. The default value of ABAQUS is a function of the rotational velocity and theradius of the rolling body, given by 0.01ωR, which provides an accurate solution (SIMU-LIA, 2009c). This value is compared to the penalty value 0.04ωR used in the exampletire of ABAQUS (SIMULIA, 2009b), which solves a comparable problem, and a critical slipvelocity as function of the maximum applied sliding velocity. Typically, values between1% to 5% give realistic results (van Breemen, 2009).In the subroutine FRIC, where the penalty is used, the current rotational velocity of thetire is not available. Therefore the critical slip velocity is specified directly as a percentageof the forward velocity of the tire, which is equal to the maximal possible sliding veloc-ity. Several simulations with percentages up to 5% of the maximum sliding velocity havebeen performed to find the value that matches the reference solution.

In Figure 3.13 the longitudinal forces around free-rolling for three different penalty val-ues are shown. It can be seen that the value taken from the example tire has a significantinfluence on the longitudinal force, while the difference between the other two is negligi-ble. The relative computation times are 1.0 for the default value, 0.88 for the value takenfrom the example tire and 0.99 for the sliding velocity respectively. Although it is numer-


ically slightly more expensive than the value of the example tire, which compares to 4%of the maximum sliding velocity, a more accurate representation of the longitudinal slipstiffness of the tire is obtained. Therefore a penalty value of 1% of the maximum slidingvelocity, as shown in Figure 3.13, is used in further simulations.

70 75 80 85

−4000

−3500

−3000

−2500

−2000

−1500

−1000

−500

0

ω [rad/s]

Lo

ngitu

nid

al fo

rce

[N

]

Sliding velocity

Example tire

Abaqus default

Figure 3.13 / Effect of penalty parameter in the friction model on the longitudinal forcearound free-rolling at 100 km/h.

3.6 Simulation procedure to compute the cornering andcombined slip characteristics

Once the rotational velocity, which corresponds to the free-rolling situation, has been ob-tained, for a given forward velocity, the cornering characteristic of a rolling tire can becomputed. This lateral force characteristic is referred to as the pure cornering character-istic in case no (global) longitudinal slip is present (Pacejka, 2006).The lateral force characteristic is a function of the side slip angle α, which is given by

α = arctan

(Vsy|Vx|

), (3.1)

with Vsy the lateral slip speed, which is equal to the lateral velocity Vy on a flat road.To induce a side slip angle on the tire in the FE model another steady-state rolling stepis performed. In this step the slip angle is gradually increased from zero degrees at thebeginning of the step to a desired slip angle at the end of the step. This can easily be

3.6 SIMULATION PROCEDURE TO COMPUTE THE CORNERING AND COMBINED SLIP CHARACTERISTICS 51

done by decomposing the velocity V into a lateral and longitudinal velocity component asshown in Figure 3.14. The decomposition is a function of the slip angle and is given by

Vy = sin(α)V (3.2)

Vx = cos(α)V (3.3)

These two components are prescribed in the translational reference frame of the steady-state step. The increment size within the steady-state step can be controlled with a limiton the maximum allowable increment update. In this way a series of steady-state solu-tions at each increment, which corresponds to slip angles between zero and the desiredslip angle, is obtained within one steady-state step.

Vx = Vx

y

Top view tire

V

αVx

Vy

Figure 3.14 / Introduction of a side slip angle α from straight line rolling (left side) bydecomposition of the translational velocity vector into a longitudinal and lateral velocitycomponent (right side).

When a nonzero side slip angle is present, the previously computed free-rolling rotationalvelocity is not valid anymore. It follows from (3.3) that the longitudinal velocity reducesfor increasing slip angle. This indicates that the rotational velocity of the tire should alsodecrease to prevent longitudinal slip.It is possible to use the free-rolling algorithm to find the corresponding free-rolling so-lution for every possible slip angle, but this is cumbersome to do. This would requirethat after every increment several additional increments are necessary to update the ro-tational velocity. Therefore it is chosen to approximate the free-rolling rotational velocityfor nonzero slip angles using

ωα = cos(α)ωfree, (3.4)


which takes the effect of nonzero slip angles on the rotational velocity into account,thereby reducing longitudinal slip.

Finally the combined slip characteristic can be computed, which is a combination of purelongitudinal and pure lateral slip. For this last handling characteristic, the tire is rollingfreely at a nonzero slip angle and is then braked until wheel lock.In the simulation model this can be achieved with yet another steady-state step, which isdone after the steady-state cornering step. In this step the translational velocities given by(3.2) and (3.3) are kept constant, while the rotational velocity ωα is reduced to zero.

Table 3.1 / Settings of the rotational and translational velocities in the two referenceframes during a steady-state rolling step to compute the three basic steady-state handlingcharacteristics of a rolling tire.

Pure Braking Pure Cornering Combined slipRotational Translational Rotational Translational Rotational Translationalvelocity velocity velocity velocity velocity velocity

Start SSR step Start SSR step Start SSR stepω = 1 Vx = V ω = ωfree Vx = V ω = ωα Vx = cos(α)V

Vy = 0 Vy = 0 Vy = sin(α)V

End SSR step End SSR step End SSR stepω = ωfree Vx = V ω = ωα Vx = cos(α)V ω = 0 Vx = cos(α)V

Vy = 0 Vy = sin(α)V Vy = sin(α)V

In Table 3.1 an overview of the settings in the two reference frames during the steady-staterolling steps is presented, which are used to obtain the three basic handling characteris-tics of a rolling tire.

It is of course also possible to start directly with a steady-state cornering (or a combinedslip) step after the static steps, as shown in figure 3.12, if the free-rolling rotational ve-locity is already known. In that case one should initialize the rotational and translationalvelocities to the values presented in Table 3.1 in the initial steady-state rolling step, wherethe friction model is enabled.

3.7 CONCLUSIONS 53

3.7 Conclusions

In this chapter the simulation procedure to compute the steady-state handling character-istics with a given FE tire model has been described. Additionally, the strategy for theidentification of the parameters of the friction model has been introduced. The parame-ters will be identified sequentially using a two step experimental / numerical approach,which is subject of the following chapters.The design of the test tire, which is used for both experimental validation of the FE modeland identification of the friction model, has been presented together with a short descrip-tion of the FE model. In order to develop an accurate tire-road interaction model a goodFE tire model is a necessity. The different components required to construct the FE tiremodel contribute significantly to the overall accuracy of the computed steady-state han-dling characteristics. The development of the FE tire model itself is however beyond thescope of this thesis and a tire model provided by Vredestein is used. Therefore only theeffect of boundary conditions on the FE tire model has been discussed for each simula-tion step.Before an actual braking characteristic of a rolling tire, using a steady-state transport step,can be computed, friction between the tire and the road must be activated. This has beenaccomplished by the introduction of an additional steady-state transport step in which thefriction model is activated. In this step the friction coefficient is increased linearly overthe step to ensure a smooth transition between the frictionless and frictional situation.Finally, the approach to compute the pure cornering and combined slip characteristics,once the rotational velocity for free-rolling is known, has been been presented.

CHAPTER FOUR

Friction parameter identification using aLaboratory Abrasion and skid Tester1

Abstract / In this chapter the influence of contact pressure on the frictional force between a smallrubber wheel and a rotating disk is investigated and parameters of the tire friction model areidentified. Friction data is obtained using a laboratory abrasion and skid tester. From this datathe parameters of the tire friction model are identified and implemented in an FE model for thetire/disk contact. The different steps followed to build this FE model are also described. Sub-sequently, results from simulations are compared with experiments for a variety of operatingconditions. The chapter is ended with a discussion of the results.

4.1 Introduction

Indoor tire testing can provide a good alternative for several outdoor tire tests. Themain advantage of indoor testing are the more controllable environmental conditions.Information about the tire frictional behavior for model parameterization could beacquired by extensively testing tires under different conditions, but these results are stillinfluenced by the structure of the tire itself. It is therefore advantageous to use small-scale testing under controlled conditions. However, the experimental characterizationof frictional properties of rubber compounds is cumbersome due to the necessity ofcomplex measurement systems as shown by Huemer et al. (2001a); Gäbel and Kröger(2006); Blume et al. (2003) and Garro et al. (1999).The Laboratory Abrasion and skid Tester 100 (LAT 100), manufactured by VMI HollandBV (2009), is one of the few commercially available experimental setups. This machine

1Parts of this chapter have been presented in van der Steen et al. (2010b).

55

56 4 FRICTION PARAMETER IDENTIFICATION USING A LABORATORY ABRASION AND SKID TESTER

was specially developed for compound testing in the tire industry. It can measure theskid, traction and wear of small rolling tires, diameter 80 mm, with a special disk,simulating road conditions. In this way the complexity of a complete tire is reduced to asolid tire made out of one compound.

In this chapter the influence of contact pressure on the frictional force between a smallrubber wheel and a disk is investigated and parameters of the tire friction model areidentified. Several different loading and velocity conditions are used in the LAT 100 toobtain the frictional and stiffness characteristics of one compound. An FE model of theLAT 100 is constructed in ABAQUS, where the framework derived in Chapter 2 is used toefficiently compute steady-state free-rolling solutions. The obtained numerical resultsare compared with measured data for a variety of operating conditions.

This chapter is organized as follows. In Section 4.2 the followed strategy, using lab scaleexperiments, is presented. In Section 4.3, a description of the experimental setup is given.Next, the different steps in the modeling approach are motivated in Section 4.4. Theidentification of the velocity independent parameter set of the friction model is presentedin Section 4.5. After that, the results of both the model and the experiments are shownin Section 4.6, while in Section 4.7 a discussion of the results is presented. Finally,conclusions are drawn in Section 4.8.

4.2 Identification using lab scale experiments

The chosen friction model (Section 2.4.2) can be decomposed into the product of a contactpressure dependent part and slip velocity dependent part as

µ(p, v) = µp(p)µv(v). (4.1)

This chapter focuses on the identification of the contact pressure part, which is given by

µp(p) =

(p

p0

)−k. (4.2)

An overview of the modeling approach using the LAT 100 setup is shown in Figure 4.1.Several measurements, under very low rolling velocities and free-rolling conditions, areused as input for the friction model. Once the parameters of (4.2) are identified thefriction model is coupled to an FE model of this small tire and the FE model is validatedby comparing the computed and experimentally found hub forces. In the next chapterthe friction model is extended with the slip velocity term using measurements of the testtire.

4.3 EXPERIMENTAL SETUP 57

LAT

LAT wheelFEM

Frictionmodel

Lab scale






Inputs

FE Tire

Virtual prototyping

Full scale

Figure 4.1 / Schematic overview of the modeling approach to obtain friction informationfrom lab scale experiments.

4.3 Experimental setup

In the experimental setup, shown in Figure 4.2, a small solid tire is pressed on a drivenabrasive corundum disk. The normal force, Fz, on the tire can be preset between 40

and 140 N and a servo-controlled load cylinder is used to keep the normal force constantduring measurements. The disk can only rotate in a clockwise direction and the velocitycan be set between 0.002 and 100 km/h. It is possible to place the wheel under differentside slip angles, α, which is also shown in Figure 4.2, by moving the complete module,with the force sensor and the wheel, over a guidance rail. The sample tire, which is madefrom the tread rubber compound, is solid and without a profile. Furthermore the wheelis fixed in all directions except for the rolling direction. The fixation of the tire is forcebased using two metal plates, which prevent the tire from slipping between the plates.Besides these two metal clamping plates, a spacer ring is inserted into the center hole ofthe wheel, before the wheel is placed onto the measurement hub.The friction present between the abrasive disk and small tire drives the tire and the result-ing forces are measured with a triaxial force sensor. Besides abrasion experiments alsothe slip characteristics, such as pure cornering, of a compound on different surfaces canbe measured at the lower speed range. For small side slip angles the lateral force, Fy, isdominated by the dynamic stiffness of the compound, while for large side slip angles thefrictional force dominates (VMI Holland BV, 2009). An extensive overview of the setupand the different measurement capabilities can be found in the work of Broeze (2009).


(a)

a

Triaxial force sensor

Abrasion disk

W

Side slip angleFz

Fy

Fx

(b)

Figure 4.2 / a) The Laboratory Abrasion and skid Tester 100. b) Schematic top and sideview of the setup.

4.4 FE model of the setup

In this section the FE model, including the abrasion disk, is described. The derivation ofthe used material model is discussed first. After that the structural model is described;the modeling procedure is simplified by exploiting the axisymmetric properties of thesolid tire. A 2D cross section of the wheel is constructed first and later used to generate afull 3D model by revolving this cross section.

4.4.1 Material model

One of the components of every FE model is a description of the materials under study.To characterize a specific material in ABAQUS several standard material models are avail-able, the parameters of these models can be specified directly or fitted on experimentaldata. The rubber compounds in tires can be defined as the formulation of a mixture ofrubber and additives, which meets the needs of the tire component application. Usuallya compound consists of one or more polymers, vulcanizing agents, accelerators, rein-forcing fillers, antidegradents, plasticizers, softeners and tackifiers. All these additives inthe compound contribute to a more complex stress-strain response compared to a pureelastomer.

4.4 FE MODEL OF THE SETUP 59

Experimental data

The compound of the used sample tire has been tested at the Deutsches Institut fürKautschuktechnologie e.V. to characterize the material response. The result of one ofthe quasi-static uniaxial compression and tension experiments is shown in Figure 4.3.In this experiment a dumbbell shaped sample is compressed and stretched to −30 and+50% strain respectively in three consecutive cycles. Two softening effects occur duringthis experiment, the so-called Payne effect (Payne, 1963) and Mullins effect (Mullins andTobin, 1957). The Payne effect is a softening effect for small strains (0.1%), attributed tobreaking apart aggregates of filler particles. The Mullins effect is a substantial softeningeffect at higher strains, attributed to progressive detachment of rubber molecules fromfiller particles. Both these softening effects can be seen in Figure 4.3. Furthermore itcan be seen that for the third curve the stress converges towards the stress of the secondcycle, which indicates that three cycles are sufficient to induce damage in the test sample.Additionally, hysteresis is present in all of the cycles.

−30 −20 −10 0 10 20 30 40 50−2

−1.5

−1

−0.5

0

0.5

1

Strain percentage [%]

Norm

aliz

ed e

ngin

eering s

tress [

−]

cycle 1

cycle 2

cycle 3

Figure 4.3 / Normalized engineering stress-strain response of a quasi-static uniaxial com-pression and tension test.

Choice of material models

Only the available material models in ABAQUS are considered, since developing new nu-merical material models is beyond the scope of this research. Therefore several assump-tions are made to choose a material model. It is assumed that:

• The expected strains on the sample tire range from −10% to 20%.


• After several rotations of the tire, the material is in a converged damaged cycle andPayne and Mullins effects can be ignored.

• The material is nearly incompressible.

• Hysteresis is also present in the rolling tire.

Following the guidelines of the ABAQUS manual (SIMULIA, 2009a) a hyperelastic modelis chosen for the long-term elastic response and a viscoelastic model is chosen for therate-dependent material behavior. For the determination of hyperelastic material param-eters a fit procedure is available. Experimental test data can be provided directly and theappropriate values of the coefficients for all available hyperelastic models are determined,after which the best model can be selected.To provide stress-strain data for the determination of the long-term coefficients it is as-sumed that the long-term hyperelastic response can be approximated with the averagedstress of the loading and unloading stresses of the third cycle. Furthermore emphasisis placed on the strains ranging from −10% to +20% by providing more data points inthis region. The fit error between all strain energy potentials and the provided data isthe smallest for the Van der Waals strain energy potential (SIMULIA, 2009a, chapter18.5) and is therefore chosen. The provided data points and the Van der Waals model areshown in Figure 4.4.

−30 −20 −10 0 10 20 30 40 50−1.5

−1

−0.5

0

0.5

1


Norm

aliz

ed e

ngin

eering s

tress [

−]

Data points

Van der Waals hyperelastic model

Figure 4.4 / Approximated long-term normalized stress-strain response data points forthe fitting procedure and the evaluated Van der Waals strain energy potential.

The hysteresis is included by adding a linear viscoelastic part to the long-term hyperelasticpart. The implementation in ABAQUS is done by means of a Prony series (SIMULIA,2009a, chapter 18.7). This series is fitted on experimental frequency data of the shear


storage, G′, and loss modulus, G′′, measured in-house by Vredestein2.

Validation of the material models

As a validation step for the material models the dumbbell experiment is reproduced withan FE model of the dumbbell. In the numerical model one clamp is fixed and on the otherclamp a time dependent force is prescribed. This force is calculated using the nominalstress σe and the undeformed cross sectional area A0,

F = A0σe. (4.3)

The nominal strain

εe =l − LL

, (4.4)

in the experiment is obtained by tracking the current length l of the dumbbell bar and theinitial length L. In the simulation model the current distance between two nodes on thedumbbell surface and their initial distance are used to compare with the measured nomi-nal strain, shown in Figure 4.5. The axisymmetric section is modeled with linear, hybrid,4-node, axisymmetric elements3. In Figure 4.6 the calculated (normalized) stress-strainresponse is compared to the experimental response. It can be seen that the response ofthe model is slightly softer than the actual material, since the dumbbell in the numericalmodel is compressed and stretched slightly further than in the experiment. Furthermorethere is a small deviation in the hysteresis curve, which indicates that the energy dissipa-tion in the experiment is larger. However the overall shape of the stress-strain response iscaptured qualitatively well considering the made assumptions and the available materialmodels.

l

Figure 4.5 / Dumbbell axisymmetric cross section and location of nodes for determina-tion of the nominal strain.

2These parameters are company confidential.

3Mesh layout provided by Vredestein.


−30 −20 −10 0 10 20 30 40 50−2

−1.5

−1

−0.5

0

0.5

1


Norm

aliz

ed

eng

inee

ring

str

ess [

−]

Experiment

Simulation

Figure 4.6 / Comparison between measured and calculated stress-strain response withthe hyperelastic and viscoelastic material models.

4.4.2 2D model

The 2D axisymmetric model in ABAQUS is defined in a cylindrical coordinate system,where the 2D cross section is placed at a reference plane θ = 0. A close look at Table4.1 shows that the outer radius of the spacer ring is slightly larger than the inner radiusof the tire. This, together with the metal plates, already prestresses the tire. The ax-ial deformation of the wheel due to clamping of the side disks has been experimentallydetermined as 0.6 mm. Furthermore the tire has been designed with rounded corners,which have a radius of 1 mm. This small curvature improved the overall reproducibilityof measurements on the setup as shown by Broeze (2009) and it also ensures that theoutward normal on the surface is continuous in the numerical model.

Table 4.1 / Geometric data of the parts for the 2D models.

Part Inner radius [mm] Outer radius [mm] Height [mm]Tire 16.5 39 18

Ring 15 17.5 15

Side disk 8 30 4

A numerical efficient approach is to exclude all the details of the fixation and only modelthe cross section of the tire. This is only valid if no slip occurs as contact between tireand metal plates is established. In that case all resultant forces due to contact with theabrasion disk are transferred to the measurement hub. Slip has not been observed in anyof the experiments, so it is assumed that there is no slip between the tire and the fixation


parts. The no slip constraint can be achieved by first using displacement boundary con-ditions to prestress the wheel and afterwards replacing these boundary conditions with akinematic coupling constraint for all degrees of freedom to a reference node. This cou-pling constraint is chosen such that the part of the tire that is normally in contact withthe ring and side disks is now fully constrained and the no slip condition is obeyed.To show the validity of this approach the stress distribution is compared to a model withthe three fixation components included. In this model the spacer ring and side disks aremodeled as rigid bodies, since these are much stiffer than the rubber compound. Thetire section, for both models, consists of linear, hybrid, 4-node, axisymmetric elements.All parts are positioned such that there is an initial penetration of the spacer ring and theside disks in the rubber tire. A special feature of ABAQUS (SIMULIA, 2009a) is availableto solve these ‘overclosures’ before other simulation steps are carried out. Once the over-closure has been solved, the friction properties are changed such that slip is preventedand the two models can be compared with each other.Figure 4.7 shows the 2D mesh layout and the final stress distribution of the two mod-els. It can be seen that the simplified model is a good approximation, since the stressdistribution outside the fixation area is equal. Therefore the simplified model is chosen,because it saves computation time in the 3D model, since all 3D contact of the fixation isavoided.

Viewport: 2 ODB: /home/rsteen/tmp/Groschwheel_572elem.odb

Step: Shrinkfitstep, Shrinkfit to push wheel inside the ring and side−disksIncrement 6: Step Time = 1.000

ODB: Groschwheel_572elem.odb Abaqus/Standard Version 6.8−2 Mon Jun 08 11:59:04 CEST 2009

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X

Y

Z

(b)

Viewport: 2 ODB: /home/rsteen/tmp/Groschwh...erm_kinematic_restart.odb

(Avg: 75%)S, Mises

+6.500e−02+8.208e−02+9.917e−02+1.162e−01+1.333e−01+1.504e−01+1.675e−01+1.846e−01+2.017e−01+2.187e−01+2.358e−01+2.529e−01+2.700e−01

Step: Step−1Increment 1: Step Time = 1.000Primary Var: S, MisesDeformed Var: U Deformation Scale Factor: +1.000e+00

ODB: Groschwheel_VDW_longterm_kinematic_restart.odb Abaqus/Standard Version 6.8−2 Mon Jun 08 11:59:52 CEST 2009

X

Y

Z

X

Y

Z

(c)

Figure 4.7 / a) The mesh layout of the simplified model and difference in stress distribu-tions of the b) simplified model and c) detailed model.

4.4.3 3D model

A non-uniform discretization in circumferential direction is used to discretize the 3Dmodel. Seventy sectors of general 3D, 8-node linear, hybrid, brick elements are used,each node having three active translational degrees of freedom. The sector angle differsfrom 0.75 in the contact area up to 10.5 at the top of the tire, shown in Figure 4.8a. The


computed stresses and strains of the 2D model are transferred to all these sectors andequilibrium for this situation is calculated.The abrasion disk of the experimental setup is included as a flat analytical rigid body.The origin of the global coordinate system is set at the center of the wheel, such that theX-axis corresponds to the longitudinal direction and the Y -axis to the lateral direction onthe tire. The center of the abrasion disk as a function of the side slip angle can now becalculated as

xdisk = − sin(α)Rdisk, (4.5)

ydisk = cos(α)Rdisk, (4.6)

where Rdisk is the constant (2D) distance of 165 mm between the center of the disk andthe center of the tire, which is shown in Figure 4.8b. The coordinates of the disk are usedto prescribe the rotational velocity of the disk.

(a) Side view tire model.

Abrasion disk

Ω

Y XRdisk

αV

(b) Top view entire model.

Figure 4.8 / a) Non-uniform discretization of the small tire. b) Position of the origin ofthe coordinate system.

Contact between abrasion disk and tire

As a first validation step the size of the contact patch is compared with ink measurementsunder a static load. For this purpose the abrasion disk on the setup is replaced with a flatand smooth glass disk on which paper is taped. For the numerical model a time periodof one second is used to apply the load, the results are shown in Figure 4.9. The relativeerror of the contact area is determined by

errorrel =

√(∑ni=1(AModi − AExpi)

2∑ni=1(AExpi)

2

), (4.7)


where AExp and AMod are the experimentally obtained and computed contact areas respec-tively, and n is equal to the number of experiments. The relative error in this case is 6.1%,from which it can be concluded that there is a good correlation with the measurements.

40 60 80 100 120120

140

160

180

200

220

240

260

Fz [N]

Con

tact a

rea [

mm

2]

Experiment

Model

Figure 4.9 / Measured and simulated area of the contact patch under static load.

Steady-state free-rolling solution

After the equilibrium and the initial loading step a steady-state solution for the rollingsituation has to be found. Only the rotational velocities of the disk and tire need to bespecified to compute a steady-state solution. On the experimental setup it is not possibleto accelerate or decelerate the tire. The tire is only able to rotate freely, which means thatthe moment My (or driving torque) is zero. Furthermore, a constant velocity profile ofthe abrasion disk has to be applied on the setup, which corresponds to the velocity at thecenter contact point of the tire. The rotational velocity of the disk is straightforwardlycalculated by

Ωdisk =VdiskRdisk

, (4.8)

where Vdisk is the prescribed forward velocity. The rotational velocity of the tire whichresults in free-rolling is not known a priori. Therefore an initial rotational velocity isgiven and the corresponding steady-state solution is computed. Based on the value of thedriving torque the rotational velocity of the tire is adjusted, by means of the algorithmdescribed in Section 2.8, and a new steady-state solution is computed until the torque is


within a specified tolerance of 1 Nmm. The initial rotational velocity can be estimated us-ing the fact that at free-rolling the velocity of the wheel approaches the prescribed forwardvelocity of the disk.

ωwheel =Vdisk cos(α)

r, (4.9)

where r is the radius of the tire and cos(α) takes the orientation of the wheel into account.

4.5 Parameter identification

To acquire frictional information, experiments on rolling tires with large side slip anglesof ±40 are conducted under different normal forces at 0.1 km/h. At this low velocityit is assumed that the sliding velocity does not contribute significantly to the friction.Furthermore wear at lower velocities is greatly reduced by rolling measurements insteadof measurements with blocked wheels. For these large side slip angles the limit of themaximal achievable lateral force is reached. If the lateral force is divided by the normalforce, a decreasing global friction coefficient for increasing load is found. This agrees withresults found in literature, see e.g. Dorsch et al. (2002); Huemer et al. (2001a); Blumeet al. (2003) and Lindner (2005). The friction coefficient is approximated by regressionthrough the data points, using the model

µ(Fz) = p1Fp2z , (4.10)

where Fz is the applied normal force and p1 and p2 are empirically determined coeffi-cients. The obtained coefficient p2 of −0.254 in this power law is in good agreement with−0.33, which is found for the load dependence of rubber friction based on hemispheri-cal surface asperities with elastic Hertzian contact as shown by Schallamach (1952). Themeasurements and the approximation using (4.10) are shown in Figure 4.10.

For the implementation of (4.2) the friction coefficient must be expressed as a functionof the contact pressure. This can be accomplished by substitution of the applied load bythe average contact pressure, as presented by Dorsch et al. (2002). The measured contactarea, as shown in Figure 4.9, is approximated with a linear least square error fit, anddividing the applied load by the measured contact area the average contact pressure iscalculated. As expected, it follows that the average contact pressure rises for increasingload, see Figure 4.11.The calculated average contact pressure is substituted for the applied load in the experi-ments to obtain a pressure dependent relation for the measurements. Now the parame-ters p0 and k can be identified. The evaluation of (4.2) together with the experiments canbe seen in Figure 4.12.

4.6 COMPARISON OF NUMERICAL AND EXPERIMENTAL RESULTS 67

40 50 60 70 80 90 100 110 120 1300.8

0.9

1

1.1

1.2

1.3

1.4

1.5

Fz [N]

µp [−

]

Experiments

p1F

z

p2

Figure 4.10 / Measured friction coefficient as function of normal force at 0.1 km/h andthe approximation using equation (4.10).

40 60 80 100 1200.2

0.25

0.3

0.35

0.4

0.45

0.5

0.55

0.6

Fz [N]

Pre

ssure

[M

Pa]

Figure 4.11 / Average contact pressure as function of normal force based on the mea-sured contact area.

4.6 Comparison of numerical and experimental results

Simulations, using (4.2) to calculate the friction coefficient for the tire/disk contact,are carried out for a forward velocity of the disk of 0.1 km/h. The side slip angle isvaried from −40 to +40 and a load of 40, 75 or 120 N is applied. Furthermore, fourmeasurements are made for each side slip angle and load.


0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.650.8

0.9

1

1.1

1.2

1.3

1.4

1.5

Pressure [MPa]

µp [−

]

Experiments

(p/p0)−k

Figure 4.12 / Measured friction coefficient as function of average contact pressure andthe approximation using equation (4.2).

In Figure 4.13 the mean longitudinal forces, and the highest and lowest measured value,of four measurements are shown together with the results of the numerical model forthe entire range of side slip angles. It can be seen that for zero degree slip angle thecomputed longitudinal forces are nonzero due to the viscoelastic material model. Ifviscoelastic material is included, the point where the resultant normal force acts, movestowards the leading edge. This creates an additional torque, which results in a rollingresistance force at zero degree slip angle in the free-rolling situation (Gent and Walter,2005). If these numerical results are compared with measurements, it can be seen thatthere is a good qualitative match. For both the model and the experiments the rollingresistance force increases for higher loads. Furthermore it can be seen that overallthe numerical model matches the experiments qualitatively and the non-symmetricaltriangular shape around the zero degree slip angle is correctly predicted. It should benoted that the force sensor is calibrated in the range −5 to −100 N. Therefore careshould be taken when interpreting the actual values of these measurements, since theseare based on extrapolation of the sensor calibration data.The lateral forces are shown in Figure 4.14, where it can be seen that there is a goodqualitative match for all loads and side slip angles. In the region |α| > 10 the lateralforce is dominated by the generated frictional force, which achieves its maximum around|α| ∼= 20. Quantitatively, the model predicts however a lower lateral force than observedin experiments for all normal loads. This is probably due to discrepancies in the size ofthe contact area, as will be discussed further in Section 4.7.


In the region for |α| < 5 the change in lateral force is governed by the cornering stiffness

CFα =∂Fy∂α

∣∣∣∣α=0

, (4.11)

which can be approximated by a linear least square error fit through data points for Fy at−5, 0 and 5. The results of this approximation are shown in Figure 4.15, where it canbe seen that the cornering stiffness increases for increasing load. Although quantitativelythe cornering stiffness of the model is higher than observed in the experiments, theincrease of the cornering stiffness with increasing load is correctly predicted by themodel.

The nonzero lateral forces at zero degree side slip angle are due to turn slip (Pacejka,2006), which occurs due to the path curvature. Turn slip is present since the radius ofthe abrasion disk is too small to consider it as driving on a straight road. This effect iswell captured in the model and can be shown when the slip velocity field is visualized.In Figure 4.16 the slip velocity field for the free-rolling tire at zero degree side slip angleis shown, with an observer placed on the wheel for the direction of the slip vectors. Thex-y nodal position is given in the reference ISO system, the derivation of the slip velocityfield is given in Appendix C. As expected at free-rolling slip develops at the trailing edge ofthe contact patch. The orientation of the slip vectors clearly indicates driving on a curvedpath. The rotational and translational velocities of the wheel are constant. The rotationalvelocity of the disk is also constant, but the tangential velocity of the disk increases withthe radius of the disk. As a result the direction of the slip vectors changes over the widthof the tire.


−50 −40 −30 −20 −10 0 10 20 30 40 50−1

−0.8

−0.6

−0.4

−0.2

0

α [°]

Fx [N

]

40 [N] Experiment

40 [N] Model

(a)

−50 −40 −30 −20 −10 0 10 20 30 40 50−2

−1.5

−1

−0.5

0

α [°]

Fx [

N]

75 [N] Experiment

75 [N] Model

(b)

−50 −40 −30 −20 −10 0 10 20 30 40 50−4

−3

−2

−1

0

α [°]

Fx [N

]

120 [N] Experiment

120 [N] Model

(c)

Figure 4.13 / Longitudinal force versus side slip angle for experiments and numericalmodel at 0.1 km/h under a) 40 N, b) 75 N and c) 120 N normal load. The bar around theexperimental data points indicates the highest and lowest measured value.


−50 −40 −30 −20 −10 0 10 20 30 40 50−50

0

50

α [°]

Fy [N

]

40 [N] Experiment

40 [N] Model

(a)

−50 −40 −30 −20 −10 0 10 20 30 40 50−100

−50

0

50

100

α [°]

Fy [

N]

75 [N] Experiment

75 [N] Model

(b)

−50 −40 −30 −20 −10 0 10 20 30 40 50−150

−100

−50

0

50

100

150

α [°]

Fy [N

]

120 [N] Experiment

120 [N] Model

(c)

Figure 4.14 / Lateral force versus side slip angle for experiments and numerical modelat 0.1 km/h under a) 40 N, b) 75 N and c) 120 N normal load. The bar around theexperimental data points indicates the highest and lowest measured value.


30 40 50 60 70 80 90 100 110 120 1302

3

4

5

6

7

8

9

Fz [N]

Corn

ering

stiff

ness [N

/deg

]

Experiment

Model

Figure 4.15 / Cornering stiffness at different loads.

−8−6−4−202468

−8

−6

−4

−2

0

2

4

6

8

y

x

Figure 4.16 / Slip velocity field for the free-rolling tire at α = 0 and Fz = 120 N.

4.7 Discussion

The presented model is used to investigate the mismatch with the observed overall lowercornering stiffness of the experiments. A sensitivity analysis with respect to side slip an-gle, geometry and velocity is performed for a load of 40 N around zero degree side slipangle. The results of this analysis are presented in Table 4.2. It can be seen that the sideslip angle has a large influence, while variations of the radius of the disk have little effect

4.7 DISCUSSION 73

Table 4.2 / Lateral force for varying parameters.

α Rdisk [mm] velocity [km/h]−1 0 1 148.5 165 181.5 0.09 0.10 1.10

Fy [N] −1.52 −6.00 −11.62 −6.29 −6.00 −5.55 −5.68 −6.00 −6.04

on the lateral force. Furthermore, a 10% velocity difference has also limited influence.This again confirms that the lateral force at small slip angles is dominated by the corner-ing stiffness. As shown in Table 4.2, a misalignment of 1 in the test setup would causean error of 5 N in Fy. After close examination of the setup a misalignment of about−0.5

of the side slip angle has been discovered, which partly explains the discrepancy betweenexperiments and simulations at α = 0. However, this difference is not enough to ex-plain the lower cornering stiffness and the presented experimental results are thereforenot corrected for this misalignment.

Another discrepancy could be due to uncertainties and inaccuracies in the materialmodel. The overall lower cornering stiffness found in the experiments indicates thatthe material behavior is too stiff. After several simulations it appeared that the materialresponse in the ALE steps is stiffer than in quasi-static steps. If for instance the foot-print size at zero degree side slip angle for a rolling wheel at a load of 120 N is comparedwith the contact area shown in Figure 4.9, a reduction of twelve percent is found. Ifthe viscoelastic model is turned off and only the hyperelastic model is used, there is nosignificant difference between the rolling and non rolling footprint size. This might bedue to discrepancies between the viscoelastic material model and the actual viscoelasticbehavior of the rubber wheel.

The difference in contact area also causes the lower computed lateral forces in the region|α| > 20 in Figure 4.14. If the reduction of the contact areas is calculated between zeroand forty degree side slip angle, which is shown in Table 4.3, it follows that there is a largedifference between zero and forty degree. This means that the average contact pressureat ±40 degree side slip angle is much higher than shown in Figure 4.12.

Table 4.3 / Contact areas for zero and forty degree side slip angle and the correspondingreduction for viscoelastic and only hyperelastic material model.

Material α = 0 [mm2] α = 40 [mm2] Reduction [%]Viscoelastic 202 155 23

Hyperelastic 291 186 36


As presented in Table 4.3, the reduction in the contact area at ±40 degree side slip anglevaries between 23% and 36%. Therefore, an average reduction of 30% with respect tozero degree side slip angle is assumed and a correction of 30% is applied to the measuredcontact area at zero degree side slip angle. If this corrected smaller contact area is usedfor substitution of the applied load, the average contact pressure is increased as shown inFigure 4.17. It can be seen that the coefficient p0 in (4.2) increases, while the exponent kremains the same.In Figure 4.18 the lateral force for a load of 120 N is shown when this correction is appliedand only the hyperelastic material model is used. Now the lateral forces are much closerto the experimental values, but there is still a difference with the experiments. This couldbe due the sensitivity of the force sensor to moments, which introduces an additionaloffset in the lateral force. This in turn could explain the substantial difference betweenthe absolute values of the measured lateral forces at 40 and −40 as observed in Figures4.14 and 4.18. These nonequal limits are not expected for an isotropic compound andthe relatively low velocity used, since the lateral force reaches a maximum if the frictionlimit is reached. This has also been confirmed by Grosch (2009), who developed theexperimental setup. Therefore the absolute values of the lateral forces should be equaland constant for |α| ≥ 30, which is the case for the numerical model and therefore showthe validity of the applied modeling approach.

0.4 0.5 0.6 0.7 0.80.8

0.9

1

1.1

1.2

1.3

1.4

1.5

Pressure [MPa]

µp [−

]

Experiments

(p/p0)−k

Figure 4.17 / Measured friction coefficient as function of corrected average contact pres-sure and the approximation using equation (4.2).

From an experimental point of view it would be interesting to further investigate theinfluence of different surface roughness profiles of the abrasion disk and temperature

4.8 CONCLUSIONS 75

−50 −40 −30 −20 −10 0 10 20 30 40 50−150

−100

−50

0

50

100

150

α [°]

Fy [N

]

120 [N] Experiment

120 [N] Corrected model

Figure 4.18 / Lateral force versus side slip angle for experiments and corrected numericalmodel at 0.1 km/h under 120 N normal load.

effects of the rubber material, which become important when higher velocities than 0.1

km/h are applied. However as already indicated by Broeze (2009) the influence of themechanical design on the results should be clarified first.

4.8 Conclusions

In this chapter the pressure dependent part of the tire friction model has been identified.To this end a numerical and experimental analysis of the Laboratory Abrasion and skidTester 100 has been carried out. The numerical model is implemented in ABAQUS, wherea 2D cross section of the tire is created to generate the 3D model. This model is staticallyvalidated with footprint measurements. A pressure dependent friction model has beenderived from the identified friction characteristics. This friction model has been imple-mented in the FE model for the tire/disk contact. The steady-state numerical modelingframework as discussed in chapter 2 has been used to efficiently compute the steady-statefree-rolling solution under different loads and side slip angles. It has been shown thatthe rolling resistance in the longitudinal direction is captured as observed in the experi-ments. The computed steady-state forces show a good qualitative match with experimentsfor all side slip angles and loads. Moreover the present turn slip is also captured well withthe model. Furthermore, the influence on the lateral force of the friction model is onlyvisible for large side slip angles, while for small slip angles the friction coefficient hasa very limited influence and the dynamic stiffness of the rubber compound dominatesthe response. An even better quantitative result is obtained when the calculated averagecontact pressure is corrected using the results of the numerical model.

CHAPTER FIVE

Friction parameter identification usinglongitudinal slip characteristics1

Abstract / The proposed friction model is decomposed into the product of a contact pressure de-pendent part, which has been identified in Chapter 4, and a slip velocity dependent part. In thischapter the identification of the slip velocity dependent part, using measured axle forces, is pre-sented. The complete identified friction model is coupled to the FE model of the tire under testing.The steady-state transport approach is used to efficiently compute the steady-state longitudinal slipcharacteristics, which show a good quantitative agreement with the experiments.

5.1 Introduction

The strategy developed to capture observed effects of dry friction on the characteristicsof rolling tires is presented in this chapter. Braking experiments at different velocities,with the tire as discussed in Chapter 3, have been conducted to obtain a slip velocitydependent parameter set for the tire friction model. The parameters are identified usingmeasured axle forces, but to exclude small variations in vertical load, which occur duringexperiments, the semi-empirical Magic Formula (MF) model (Pacejka, 2006) is used toevaluate the longitudinal slip characteristics at a constant vertical load.For the identification procedure measurements at 0.8 times the load index (L.I.) are used,while measurements at 0.4 and 1.2 times the load index are used for validation of thetire friction model. With the complete identified parameter set of the friction model thesteady-state longitudinal slip characteristics are computed using the procedure describedin Chapter 3. The obtained numerical results are compared with the Magic Formula.

1Parts of this chapter have been presented in van der Steen et al. (2010a).

77

78 5 FRICTION PARAMETER IDENTIFICATION USING LONGITUDINAL SLIP CHARACTERISTICS

The chapter is organized as follows. In Section 5.2 a description of the followed strat-egy, using full scale experiments, is given. The conducted tire measurements used forthe Magic Formula are provided in Section 5.3. After that, the identification process isexplained in Section 5.4. With the identified tire friction model the steady-state longitudi-nal slip characteristics are computed. The free-rolling rotational velocities are comparedwith measurement data in Section 5.5. Next the results of both the FE model and the MFare compared in Sections 5.6 and 5.7, while in Section 5.8 a discussion of the results ispresented. Finally, conclusions are drawn in Section 5.9.

5.2 Identification using full scale experiments

The approach followed to capture observed effects of dry friction on the handling charac-teristics of rolling tires is described in this section. The friction model (Section 2.4.2) hasbeen decomposed into the product of a contact pressure dependent part and slip velocitydependent part as

µ(p, v) = µp(p)µv(v). (5.1)

A method to obtain the two parameters of the pressure related part (p0 and k) of thefriction model has been presented in Chapter 4 using the Laboratory Abrasion and skidTester. This chapter focuses on the identification of the remaining velocity related part,which is given by

µv(v) =

[µs + (µm − µs) exp

−h2 log2

(v

Vmax

)]. (5.2)

Most experimentally determined friction laws in literature do not provide information ofthe sliding velocities that occur during real operating conditions. Experimental data isusually available in the range of low sliding velocities only, e.g. Huemer et al. (2001a);Dorsch et al. (2002); Blume et al. (2003), while in terms of handling performance muchhigher sliding velocities are experienced. Therefore braking experiments at differentvelocities with the test tire will be used to obtain the velocity dependent data set foridentification of the friction model. The friction model is validated by comparingsimulated axle forces with measured forces, which is shown in Figure 5.1.

The drawback of using high velocity tire testing is that no measurements can be madein the contact area and only axle measurements are thus available for identification pur-poses. Therefore this ‘global’ axle data is used to identify the parameters of the frictionmodel, which is implemented locally in the contact area of the FE model. Furthermorethe parameters will be made dependent on the forward velocity, such that the FE model

5.3 TIRE FORCE AND MOMENT MEASUREMENTS 79

can also be used for other speeds than used in experiments, which broadens the applica-bility of this method.

LAT




Inputs

Frictionmodel

FE Tire

Virtual prototyping

Full scale

LAT wheelFEM

Lab scale



Figure 5.1 / Schematic overview of the modeling approach to obtain friction informationfrom full scale experiments.

5.3 Tire force and moment measurements

Figure 5.2 / TNO Tyre Test Trailer.

To assess the real tire behavior, experiments have been carried out with the TNO TyreTest Trailer, see Figure 5.2, on a proving ground. This trailer is typically used to measurethe steady-state force and moment slip characteristics of a tire on a real road. For thisresearch, experiments have been carried out to measure the longitudinal force, Fx, asfunction of the longitudinal slip, κ, during straight line braking up to wheel lock. Vari-


ous operating conditions have been considered by changing the vertical load, Fz, trailerforward velocity, Vx, and tire inflation pressure. The tire behavior is measured for threevertical loads (0.4, 0.8 and 1.2 times the load index), three inflation pressures (nomi-nal pressure and nominal pressure ±0.5 bar) and five forward velocities (from 20 to 100

km/h).Although the vertical load is controlled to the set value, small variations in vertical loadoccur in the experiments. These variations are caused by road unevenness, trailer motionand the relatively fast tire deformation that occurs during braking. For a fair comparisonwith the FE model, it is desirable to have the longitudinal slip characteristics available at aprecise vertical load. To achieve this, the semi-empirical Magic Formula model has beenused. For each velocity a Magic Formula dataset has been identified on the brake mea-surements at the three loads with the parameter identification software MF-Tool (TNO,2010). Then the interpolation capabilities of the Magic Formula are used to obtain themeasured slip characteristics at the exact vertical load.

5.4 Friction parameter identification

The parameter identification consists of several steps, which are explained in this section.A schematic overview of these steps is shown in Figure 5.3. First the longitudinal braking

Magic Formulaκ− Fx µMF (κ) = Fx

FzµMFv(κ) = µMF (κ)

µp

Fit parameters

Vmax = κmaxVx

Implement local:µlocal(p, v) = µp(p)µv(v)|κ|

µv

Figure 5.3 / Schematic overview of the identification procedure.

force at 0.8 times the load index, using the Magic Formula, is divided by the appliedvertical load to obtain a global friction coefficient µMF as function of κ. By introducingthe global longitudinal slip as

κ = − v

Vx, (5.3)

with v the sliding velocity given by

v = Vx − ωre, (5.4)

5.4 FRICTION PARAMETER IDENTIFICATION 81

where ω is the rotational velocity and re the effective rolling radius, it follows that κ variesfrom −1 at wheel lock to zero at free-rolling. Since both κ and Fx are negative duringbraking, the absolute values are taken to obtain a positive friction coefficient µMF for |κ|in the interval [0 1].Secondly the global friction coefficient is divided by µp. This value is computed as Fx/Fzat wheel lock, obtained by an FE simulation. In this simulation the friction model derivedin Chapter 4 is implemented, which depends on contact pressure only. This leads to avelocity dependent part of the global friction coefficient as function of κ

µv(κ) =


−h2 log2

(κ

κmax

)], (5.5)

with

κmax =VmaxVx

. (5.6)

The result of these steps is shown in Figure 5.4 for all tested velocities. From this figure itcan be seen that the overall response of the tested tire is similar for all velocities. Around10−15% slip the maximum of the friction coefficient is reached and for higher slip ratiosthe friction coefficient drops, this behavior is well-known from experiments and also doc-umented in literature (Yamazaki et al., 2000; Garro et al., 1999; Gent and Walter, 2005;Pacejka, 2006).Furthermore, it can be seen that there is however a difference in the peak values of thefriction coefficient. This can be explained by the hysteretic friction component, whichresults from internal friction of the rubber. During sliding asperities of a rough surfaceexert oscillating forces on the rubber surface. This leads to cyclic deformations of therubber and to energy dissipation caused by the internal damping of the rubber (Pers-son, 2001). Since all experiments have been performed on the same proving ground,the wavelengths (λi) remain equal but the excitation frequencies change with differentforward velocity Vx (fi ∼ Vx/λi). This causes different losses in the rubber material forvarying forward velocity (Moore, 1972).Additionally, the frictional energy increases with increasing velocity, which leads to a tem-perature increase in the contact area. The temperature also affects the viscoelastic prop-erties of the rubber, which in turn change the frictional behavior.Both the effects on the rubber material of the surface roughness and the temperatureincrease are not explicitly incorporated in the proposed friction model, therefore theseeffects are captured in the velocity part of the friction model. To identify the parametersin (5.5) the following assumptions are made to simplify the identification procedure:

• The point where the peak in the friction model is located (κmax) can be chosen thesame for all velocities.

• The value for µMFv for a fully locked tire (|κ| = 1) can be chosen the same for allvelocities.


0 0.2 0.4 0.6 0.8 1−0.2

0

0.2

0.4

0.6

0.8

1

1.2

|κ| [−]

µM

Fv [

−]

20 km/h

40 km/h

60 km/h

80 km/h

100 kmh

Figure 5.4 / Velocity part of the global friction coefficient as function of |κ| for five forwardvelocities.

The global longitudinal force equals the integral of the shear stress over the contact sur-face. The slip velocity increases however over the contact length from the leading edgetowards the trailing edge, when the tire starts to slip. Therefore the shear stresses nearthe leading edge are higher than the shear stresses at the trailing edge. As a result themaximum local peak friction coefficient must be higher than the global friction coeffi-cient. Therefore the value of κmax is chosen at κ = 0.05, such that the maximum frictioncoefficient µm will be higher than the friction coefficient shown in Figure 5.4. In thismanner the influence of the velocity part of the friction model on the longitudinal slipstiffness is small. In other words, the structural response of the tire dominates the longi-tudinal slip stiffness.If κ = 1 is substituted in (5.5) the following expression for µlock is obtained

µlock =


−h2 log2

(1

κmax

)]. (5.7)

By fixing the value of µlock, as the average value of µMFv at κ = 1, an explicit expressionfor the parameter h2 as function of µs and µm is given by

h2 =ln(

µm−µs

µlock−µs

)log2 (κmax)

. (5.8)

5.4 FRICTION PARAMETER IDENTIFICATION 83

The remaining parameters µs and µm of (5.5) are determined separately for all five veloc-ities by minimizing a least-square error

min[µs µm]

12

m∑i=1

(µv([µs µm], κi)− µMFv(κi))2 , [µs µm] ∈ R2. (5.9)

subject to:

−µs < 0

µs − µlock < 0

µlock − µm < 0

The interval for κ is chosen from the maximum of µMFv to κ = 1 with an equidistantdistribution for κ to put an equal weight on the used data points, this is illustrated inFigure 5.5 for one velocity. In this way the longitudinal slip stiffness of the tire is notincluded in the identification process. The obtained parameters for µm as function of

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.5

0.6

0.7

0.8

0.9

1

|κ| [−]

µM

Fv [−

]

Figure 5.5 / Data points of one velocity used for the identification of µs and µm.

forward velocity are shown in Figure 5.6. These five parameters values are approximatedusing a linear least-square error fit as function of the forward velocity

µm = a1Vx + a2. (5.10)

Using (5.10) for the value of µm a second optimization is done for every velocity for µs

minµs

12

m∑i=1

(µv(µs, κi)− µMFv(κi))2 , µs ∈ R. (5.11)

subject to:

−µs < 0

µs − µlock < 0


The obtained parameters for µs as function of forward velocity are shown in Figure 5.7,

20 30 40 50 60 70 80 90 1000.9

0.95

1

1.05

1.1

Vx [km/h]

µm

[−

]

Figure 5.6 / Obtained parameter values of µm as function of forward velocity.

the five parameters values are approximated using a second order linear least-square errorfit as function of the forward velocity

µs = a3V2x + a4Vx + a5. (5.12)

20 30 40 50 60 70 80 90 1000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Vx [km/h]

µs [−

]

Figure 5.7 / Obtained parameter values of µs as function of forward velocity.

Now (5.5) can be evaluated in the entire interval of forward velocities between 20 and 100

km/h using (5.8), (5.10) and (5.12) for the values of h2, µm and µs. The last step in theprocedure, see Figure 5.3, consists of switching from (5.5) back to (5.2) by

Vmax = κmaxVx. (5.13)

5.5 FREE-ROLLING ROTATIONAL VELOCITY: COMPARISON OF FEM PREDICTION AND EXPERIMENTS 85

5.5 Free-rolling rotational velocity: Comparison of FEM pre-diction and experiments

The procedure described in Section 3.5 is followed to compute the entire braking char-acteristic of the tire. The tire is brought up to speed at the desired forward velocity inthe first steady-state transport step, while initializing the rotational velocity at 1 rad/s. Inthe second steady-state transport step the rotational velocity of the tire is increased everyincrement until the free-rolling situation is reached. Simulations are carried out for allfive forward velocities and the three load conditions. The computed free-rolling rotationalvelocity is compared with the, filtered and averaged, measured rotational velocity just be-fore a brake test starts. The results are shown in Table 5.1, where it can be seen that thereis a good match for all loads and velocities.

Table 5.1 / Free-rolling rotational velocity (in rad/s) for five forward velocities and threeload conditions of experiments and FE model.

Velocity 0.4 L.I. [N] 0.8 L.I. [N] 1.2 L.I. [N]20 Experiment 18.1 18.7 18.0

[km/h] FE Model 17.6 17.7 17.7

40 Experiment 36.3 36.2 35.1

[km/h] FE Model 35.2 35.4 35.4

60 Experiment 54.1 53.7 54.2

[km/h] FE Model 52.7 53.0 53.1

80 Experiment 71.1 71.5 72.3

[km/h] FE Model 70.3 70.7 70.8

100 Experiment 88.4 89.4 90.1

[km/h] FE Model 87.8 88.3 88.5

Furthermore it can be seen that for the FE model the free-rolling rotational velocity in-creases with increasing load. This is expected, since the rotational velocity of a free-rollingtire is directly related to the forward velocity as

ωfree =Vxre, (5.14)

where re is the effective rolling radius, which decreases when the load increases. The


value of the free-rolling rotational velocity can be used to determine the longitudinal slipin a more practical way as follows

κ =ω − ωfreeωfree

. (5.15)

This formulation is used to compare the simulated steady-state braking characteristicswith the Magic Formula.

5.6 Comparison between FEM and MF predictions

The longitudinal forces as function of the longitudinal slip are shown in Figures 5.8,where the three load conditions are grouped for each velocity. In these figures the MagicFormula is used to describe the measured slip characteristics at the desired vertical loads.In the MF-Tool software the parameters of the MF are chosen such that the error betweenthe MF and experiments is minimized. As a result, it can happen that the longitudinalforce at κ = 0 is positive. However, the measured longitudinal force at free-rolling isalways negative due to a rolling resistance force.The longitudinal forces for the lowest and highest load conditions are computed usingthe same parameters in the friction model as for the middle loads, which have been usedin the identification process. Therefore these results illustrate the predictive capability ofthe friction model.It can be seen that the axle force of the FEM curves at 0.8 times the load index, whichare used in the identification process, match the Magic Formula both qualitatively andquantitatively. This shows that it is possible to fit the parameters of the velocity part ofthe friction model using measured axle forces and implement one parameter set of thefriction model locally at each node.Although some deviations can be seen for the higher load index, especially at 20 km/h, itcan be concluded that there is also a good quantitative agreement between the FE predic-tions and the MF for 0.4 and 1.2 times the load index. Since the only difference in thesesimulations is the applied load, it follows that the decoupling of the friction model into avelocity and a contact pressure part is a valid assumption.The deviation with the MF is most obvious at the lowest velocity of 20 km/h. This is dueto the choice of one value for µlock for all velocities. It can be seen in Figure 5.4 that µlockat 20 km/h is significantly higher than at the other velocities, which leads to an underes-timation of the friction coefficient for this velocity at increasing slip ratios. This effect isamplified by increasing the load on the tire, which can be seen in Figure 5.8a, since thevelocity part of the friction model is proportional to the applied load.

5.6 COMPARISON BETWEEN FEM AND MF PREDICTIONS 87

−1 −0.9 −0.8 −0.7 −0.6 −0.5 −0.4 −0.3 −0.2 −0.1 0

−10000

−8000

−6000

−4000

−2000

0

κ [−]

Fx [N

]

MF

FEM

(a) 20 km/h.

−1 −0.9 −0.8 −0.7 −0.6 −0.5 −0.4 −0.3 −0.2 −0.1 0−10000

−8000

−6000

−4000

−2000

0

κ [−]

Fx [

N]

MF

FEM

(b) 40 km/h.

−1 −0.9 −0.8 −0.7 −0.6 −0.5 −0.4 −0.3 −0.2 −0.1 0−10000

−8000

−6000

−4000

−2000

0

κ [−]

Fx [N

]

MF

FEM

(c) 60 km/h.


−1 −0.9 −0.8 −0.7 −0.6 −0.5 −0.4 −0.3 −0.2 −0.1 0−10000

−8000

−6000

−4000

−2000

0

κ [−]

Fx [N

]

MF

FEM

(d) 80 km/h.

−1 −0.9 −0.8 −0.7 −0.6 −0.5 −0.4 −0.3 −0.2 −0.1 0−10000

−8000

−6000

−4000

−2000

0

κ [−]

Fx [

N]

MF

FEM

(e) 100 km/h.

Figure 5.8 / Longitudinal force as function of longitudinal slip for both Magic Formula(MF) and FE Model at 0.4, 0.8 and 1.2 times the load index.

5.7 Effect of inflation pressure on the longitudinal force

Besides the load and forward velocity also the inflation pressure in the experiments hasbeen varied. When the inflation pressure is decreased, at a fixed load, the contact areaincreases, which leads to a lower contact pressure. This normally results in higher lon-gitudinal forces, which is confirmed by the Magic Formula as shown in Figure 5.9. Theeffect of a lower contact pressure in the contact area on the friction model is that the fric-tion coefficient increases and a higher longitudinal force can be generated. It can be seenin Figure 5.9 that the increase in longitudinal force is overestimated. Furthermore thedeformations for the simulation at the highest load are so large, due to the under-inflatedand overloaded tire, that convergence is not obtained for the lower range of longitudinalslip.

5.7 EFFECT OF INFLATION PRESSURE ON THE LONGITUDINAL FORCE 89

In contrast to an under-inflated tire, an over-inflated tire has a higher contact pressureand the tire is riding more on the center of the tread rather than on the shoulders of thetire. The higher contact pressure leads to a lower friction coefficient and this results ina lower longitudinal force, shown in Figure 5.10. Now it can be seen that the FE pre-dictions slightly underestimate the longitudinal force, compared to the Magic Formularesults. An explanation for deviations of the longitudinal force with respect to the MagicFormula could be due the change in contact pressure distribution. If the contact pressuredistribution in the contact area is inspected, shown in Figure 5.11, it follows that the con-tact pressure is outside the measured range on the LAT 100 for a large part of the contactarea (Figure 4.17). This indicates that an accurate prediction of the friction coefficient cannot be guaranteed, since (4.2) is extrapolated outside the measurement range.

−1 −0.9 −0.8 −0.7 −0.6 −0.5 −0.4 −0.3 −0.2 −0.1 0−10000

−8000

−6000

−4000

−2000

0

κ [−]

Fx [

N]

MF

FEM

Figure 5.9 / Longitudinal force as function of longitudinal slip for MF and FE Model at0.4, 0.8 and 1.2 times the load index at 0.5 bar under nominal pressure at 60 km/h.

−1 −0.9 −0.8 −0.7 −0.6 −0.5 −0.4 −0.3 −0.2 −0.1 0−10000

−8000

−6000

−4000

−2000

0

κ [−]

Fx [N

]

MF

FEM

Figure 5.10 / Longitudinal force as function of longitudinal slip for MF and FE Model at0.4, 0.8 and 1.2 times the load index at 0.5 bar above nominal pressure at 60 km/h.


0 0.2 0.4 0.6 0.8 10

50

100

150

200

250

Num

be

r o

f nod

es in

co

nta

ct

0 0.2 0.4 0.6 0.8 10

50

100

150

200

250

Num

ber

of

nodes in c

onta

ct

0 0.2 0.4 0.6 0.8 10

50

100

150

200

250

Contact pressure [MPa]

Num

ber

of nodes in

conta

ct

Figure 5.11 / Contact pressure distribution for the under-inflated (top), nominal inflated(middle) and over-inflated (bottom) tire at 0.8 L.I. for peak longitudinal force.

5.8 DISCUSSION 91

5.8 Discussion

The approach proposed in this thesis allows to compute steady-state braking character-istics under different loads and velocities with a good accuracy compared to the MagicFormula. The interpolation of the parameters µm and µs, using (5.10) and (5.12), allowsto use the friction model at other velocities than the five velocities used in experiments.This broadens the applicability of the friction model, but at the five test velocities, thedeviation with the MF increases. It is shown in Figure 5.6 that the largest deviation withrespect to the fit occurs at 40 km/h and this is directly reflected in the results, shownin Figure 5.8b, where the peak force is overestimated. If only the tested velocities areof interest, the match with the MF at these five velocities can be enhanced by using theoptimized parameters µs and µm for each separate velocity.The derived parameter set is not unique, but depends on the choices of µlock and κmax.The effect of the choice for one value of µlock has already been discussed, but it was alsoassumed that the location of κmax can be chosen the same for all velocities. Additionalsimulations are performed, which confirm that κmax at 0.05 is indeed reasonable. In onemodel the peak value is shifted to κmax = 0.01 and a second model the peak is shiftedto κmax = 0.10, which is around the point where the maximum of the longitudinal forceis located. The results of these simulations are shown in Figure 5.12. It can be seen that

−1 −0.8 −0.6 −0.4 −0.2 0−7000

−6000

−5000

−4000

−3000

−2000

−1000

0

κ [−]

Fx [N

]

MF

FEM κmax

0.05

FEM κmax

0.10

(a) 60 km/h.

−1 −0.8 −0.6 −0.4 −0.2 0−7000

−6000

−5000

−4000

−3000

−2000

−1000

0

κ [−]

Fx [N

]

MF

FEM κmax

0.05

FEM κmax

0.01

(b) 100 km/h.

Figure 5.12 / Longitudinal force as function of longitudinal slip for both Magic Formula(MF) and FE Model for two values of κmax at 0.8 times the load index.

if the peak location is shifted to 0.10 the tire starts to slide slightly earlier and an under-estimation of the longitudinal force is observed. Furthermore it turns out, also for othervelocities, that the peak longitudinal force can not be reached when κmax = 0.10. How-ever if a value of 0.01 is used, the results are more or less the same. This indicates that


if the value of κmax is chosen before the point where the peak longitudinal force occurs,the presented approach leads to accurate predictions of the tire response.

5.9 Conclusions

In this chapter the parameter identification of the sliding velocity part of the tire frictionmodel has been presented. The described identification procedure is based on measure-ments of the longitudinal force as function of the longitudinal slip during straight linebraking under different driving velocities. With the obtained parameters of the sliding ve-locity part, the entire phenomenological friction model has been identified. The derivedmodel captures observed effects of dry friction on the longitudinal slip characteristics ofa rolling tire.The steady-state numerical modeling framework as discussed in chapter 2 has been usedto efficiently compute the steady-state braking solution under different loads and veloc-ities. It has been shown that the computed steady-state forces are in good quantitativeagreement with experiments for all loads. This supports the validity of fitting the param-eters of the velocity part of the friction model using measured axle forces and implement-ing one parameter set of the friction model locally at each node in contact. Furthermore,it has been shown that the decoupling of the friction model into a velocity and a pressurepart is a valid assumption. However the results at different inflation pressures suggestthat extrapolation of the contact pressure dependent part of the friction model leads todeviations with the Magic Formula.In the next chapter the predictive capability of the derived friction model is investigatedby computing the forces and moments that occur during steady-state cornering and com-bined slip conditions.

CHAPTER SIX

Predictive capability of the FE tire model

Abstract / The fully identified friction model is used to compute the pure cornering and combinedslip characteristics. The obtained results are discussed and the FE model is used to demonstratepossible extensions in future applications. Inevitable variations on the forces and moments, whichoccur during measurements, are provided to put the predicted forces and moments into the rightperspective. Finally, the predicted characteristics are compared with the Magic Formula and it isshown that the handling performance of the tire can be adequately predicted with the identifiedfriction model.

6.1 Introduction

The final goal of the FE model is to use it in an early design stage and predict the tirebehavior, such as handling, rolling resistance and wear, by creating virtual prototypesinstead of real prototypes. In this chapter the fully identified friction model is used toinvestigate the predictive capability of the friction model by comparing simulated axleforces of cornering and combined slip characteristics with Magic Formula evaluations.Besides the validation of the steady-state handling characteristics under dry conditions,the FE model is also used to illustrate possible future extensions, such as wear.Furthermore details about the tire experiments are provided to show the inevitable vari-ations during measurements on supposedly identical tires. These uncertainties in thecharacteristics arise from changing chemical and viscoelastic properties of tire materialscombined with effects of wear and environmental conditions. These variations can beused to put the predicted cornering and combined slip characteristics of the FE modelinto a better perspective.

This chapter is organized as follows. In Section 6.2, the predicted handling characteristicsfor pure cornering and combined slip conditions are given and discussed. After that,

93

94 6 PREDICTIVE CAPABILITY OF THE FE TIRE MODEL

observations about the experimental data are presented in Section 6.3, while in Section6.4 a comparison of the FE model with the Magic Formula is made and the results arediscussed. Finally, conclusions are summarized in Section 6.5.

6.2 Prediction of the handling characteristics

6.2.1 Pure cornering characteristic

The pure cornering characteristics have been measured for side slip angles ranging from−12 to 12. This range is also used in the simulations, where the procedure presentedin Section 3.6 is applied. As starting point for the rotational velocities in the steady-staterolling step, the free-rolling rotational velocities for a zero degree slip angle as given inTable 5.1 are used. The calculation of the cornering characteristic is therefore split intotwo separate simulations, one for positive and one for negative slip angles, which bothstart at zero degree slip angle.The forward velocity in the simulations is 60 km/h, which is equal to the standard testvelocity during experiments. During a cornering manoeuvre not only a lateral force isgenerated, but also a self-aligning moment Mz and an overturning moment Mx.

Lateral force

The computed lateral forces for 0.4, 0.8 and 1.2 times the load index are shown in Figure6.11. In the region for |α| < 2.5 the change in lateral force is governed by the corneringstiffness CFα, and the built-up of the lateral force is linear. It can be seen that the cor-nering stiffness depends on the applied load. The cornering stiffness typically increasesup to a fraction of the load index and then falls off in magnitude as the load increasesfurther. It is desirable that the location of this peak is above the tire load index for goodhandling (Gent and Walter, 2005, chapter 8).As the slip angle becomes larger, more and more of the available contact area starts toslide and a maximum amount of lateral force will be generated at a certain slip angle,in this case around ±10.5. Although not visible in Figure 6.1, beyond this peak value,increasing the slip angle decreases the lateral force due to increasing slip velocity. Thisis analogous to the response of the tire subjected to pure braking, where the longitudinalforce for increasing longitudinal slip decreases.It can also be seen that the magnitude of the lateral force at positive slip angles is slightlylarger than for negative slip angles. This can be explained by the non-symmetric tire tread

1The deformations at the highest load and positive slip angle are so large, that for slip angles above 9.2

no solution is obtained with the current model.

6.2 PREDICTION OF THE HANDLING CHARACTERISTICS 95

−15 −10 −5 0 5 10 15−8000

−6000

−4000

−2000

0

2000

4000

6000

8000

α [°]

Fy [

N]

0.4 L.I.

0.8 L.I.

1.2 L.I.

Figure 6.1 / Simulated lateral force as function of side slip angle at 0.4, 0.8 and 1.2 timesthe load index, at a forward velocity of 60 km/h.

design, see Section 3.3, in combination with the used friction model. When a slip angleis present, the footprint is reshaped into a trapezoid. A right turn (positive slip angle)leads to a long left shoulder and a short right shoulder, whereas a left turn leads to a longright shoulder and a short left shoulder. The long shoulder is also exposed to a highermagnitude of normal stress than the more lightly loaded short shoulder, which meansthat the lateral force is mainly due to the integral of the lateral shear stresses on the longshoulder. With the asymmetric test tire, two different footprint sizes are obtained duringsteering to the left and right. The larger surface area on the wide left shoulder, as shownin Figure 6.2b for a positive slip angle, leads to a lower average contact pressure, whichin turn leads to a higher lateral force. In contrast, the smaller surface area on the rightshoulder, as shown in 6.2a leads to a higher average contact pressure, which in turn leadsto a lower lateral force with respect to the positive slip angle. The black areas in both fig-ures, which indicate the highest local friction coefficients, are due to the low magnitudeof the normal stress and their contribution to the overall lateral force is small.

The calculated sliding velocities during steering are more or less the same for steeringto the right and left, which means that the slip velocity dependent friction coefficientdistribution is similar for left and right turns as is shown in Figure 6.3. Here the highestlocal friction coefficients are found at the leading edge of the contact patch, where theslip velocity is the lowest. As a result the lateral force acting on the wheel is higher forpositive slip angles than for negative slip angles due to the difference in footprint size.


(a) Negative slip angle of 12. (b) Positive slip angle of 12.

Figure 6.2 / Distribution of local pressure dependent friction coefficient, where the scalevaries from 0 to 2 and darker colors indicate higher coefficients. The bottom of the figurecorresponds to the leading edge of the contact patch.

(a) Negative slip angle of 12. (b) Positive slip angle of 12.

Figure 6.3 / Distribution of local slip velocity dependent friction coefficient, where thescale varies from 0 to 2 and darker colors indicates higher coefficients. The bottom of thefigure corresponds to the leading edge of the contact patch.


Self-aligning moment

The location of the lateral force resultant is a function of the slip angle and gives rise toa moment around the z-axis, which is known as the self-aligning moment and can bedescribed by

Mz = −tFy +Mzr, (6.1)

with t the pneumatic trail and Mzr a small residual torque (Pacejka, 2006). The self-aligning moments are shown in Figure 6.4, where it can be seen that the moments firstincrease and after a few degrees slip angle decrease again. The location of the lateral forceresultant moves towards the origin of the reference coordinate system for increasing slipangle (Pacejka, 2006) and the pneumatic trail thus decays with increasing slip angle. Thepneumatic trail can even move ahead of the origin at very large slip angles, such that theself-aligning moment changes sign.

−15 −10 −5 0 5 10 15−200

−150

−100

−50

0

50

100

150

200

α [°]

Mz [N

m]

0.4 L.I.

0.8 L.I.

1.2 L.I.

Figure 6.4 / Simulated self-aligning moment as function of side slip angle at 0.4, 0.8 and1.2 times the load index, at a forward velocity of 60 km/h.

Plysteer and conicity

In Figure 6.5 a close-up of Figures 6.1 and 6.4 is shown, where it can be seen that boththe lateral forces and self-aligning moments at zero degree slip angle are nonzero. This iscaused by two effects known as plysteer and conicity. Plysteer effects are due to structuraltire design, such as the distance between the belts, tread pattern, as well as the frictional


−1 −0.5 0 0.5 1

−2000

−1000

0

1000

2000

α [°]

Fy [

N]

0.4 L.I.

0.8 L.I.

1.2 L.I.

(a) Lateral force.

−1 −0.5 0 0.5 1−100

−50

0

50

100

α [°]

Mz [N

m]

0.4 L.I.

0.8 L.I.

1.2 L.I.

(b) Self-aligning moment.

Figure 6.5 / Close-up of the lateral forces and self-aligning moments around zero degreeslip angle.

dynamics of rolling tires, which is discussed in detail by Ohishi et al. (2002). The gen-erated lateral force due to plysteer does change sign, if the rolling direction of the tire isreversed and plysteer is therefore also known as pseudo side slip.Conicity can occur if the tire belt is located slightly off-center, which is the result of manu-facturing variances (Gent and Walter, 2005, chapter 8) and is not present in the FE model.Conicity has the effect that the rolling radius varies from one side to the other and the tireacts as if it has a conical cross section. This also results in the development of a lateralforce and aligning torque. The lateral force does not change sign if the rolling directionis reversed, which is similar to a cambered wheel and this effect is also known as pseudocamber.The residual cornering force is the cornering force at zero self-aligning torque, similarlyresidual aligning torque, Mzr, is the aligning torque at zero cornering force. Both aredirectly related to vehicle handling, since under free control (zero aligning torque) thenonzero residual lateral force pushes the vehicle to the side. To maintain a straight lineon the road, a driver needs to counterbalance the residual aligning torque to have zerolateral force.Automobile manufactures often demand a specific range of plysteer residual aligningtorque, which is taken into account in the overall vehicle design to reduce tire-inducedpull effects. This can lead to different tire designs for a nominally identical tire and asa result pull problems can occur if the original equipment tires are replaced (Gent andWalter, 2005, chapter 8).


Overturning moment and loaded radius

The overturning moment Mx at the wheel axis is a function of the lateral force and theloaded radius of the tire and is shown in Figure 6.6.

−15 −10 −5 0 5 10 15−2500

−2000

−1500

−1000

−500

0

500

1000

1500

2000

2500

α [°]

Mx [

Nm

]

0.4 L.I.

0.8 L.I.

1.2 L.I.

Figure 6.6 / Simulated overturning moment as function of side slip angle at 0.4, 0.8 and1.2 times the load index, at a forward velocity of 60 km/h.

−15 −10 −5 0 5 10 150.88

0.89

0.9

0.91

0.92

0.93

0.94

0.95

0.96

0.97

α [°]

r l / r

u [−

]

0.4 L.I.

0.8 L.I.

1.2 L.I.

Figure 6.7 / Normalized loaded radius as function of side slip angle at 0.4, 0.8 and 1.2times the load index, at a forward velocity of 60 km/h.


The loaded radius rl of the FE model is approximated as the distance from the wheelcenter to a single node in the middle of the contact area. This node has also a lateraland longitudinal displacement during steering, so only the vertical displacement is takeninto account to calculate the loaded radius of the tire. The loaded radius, normalizedwith the unloaded radius ru, as function of applied load is shown in Figure 6.7, whereit can be seen that indeed a bell-shape, as discussed by Pottinger in (Gent and Walter,2005, chapter 8), is obtained. Furthermore the decrease of the loaded radius at large slipangles increases for increasing load. These results are in good agreement with similardata presented in Pacejka (2006).

6.2.2 Combined slip characteristic

Combined slip is an important characteristic, since during normal vehicle operation cor-nering is often combined with a torque, driving or braking, action. With the FE modelcombined slip characteristics, under several slip angles, have been computed at a con-stant forward velocity of 60 km/h. These simulations are started at the free-rolling sit-uation and then the rotational velocity is reduced to zero. In Figures 6.8 and 6.9 thelongitudinal forces as function of longitudinal slip are shown for 0.4 and 0.8 times theload index respectively.

It can be seen that for increasing slip angle the peak longitudinal force decreases. Therate at which the force decreases for increasing α is directly related to the isotropic frictionmodel, where the friction coefficient is assumed to be independent of the direction of the

−1 −0.8 −0.6 −0.4 −0.2 0−3500

−3000

−2500

−2000

−1500

−1000

−500

0

κ [−]

Fx [N

]

0°

+3°

+5°

+8°

−3°

−5°

−8°

Figure 6.8 / Longitudinal force as function of longitudinal slip for 7 slip angles at 0.4times the load index, at a forward velocity of 60 km/h.


−1 −0.8 −0.6 −0.4 −0.2 0−7000

−6000

−5000

−4000

−3000

−2000

−1000

0

1000

κ [−]

Fx [

N]

0°

+3°

+5°

+8°

−3°

−5°

−8°


γ1

γ2

τ1τ2γeqv τeqv

Figure 6.10 / Representation of the slip circle with slip velocity vector and correspondingfrictional shear stress vector.

slip velocity vector. This can be visualized with a so-called slip circle as shown in Figure6.10. Since the generated frictional shear stress is oriented with the slip velocity vector,the shear stress in longitudinal direction (τ1) becomes smaller if side slip is present.For a locked wheel, under a slip angle α, all local slip velocity vectors are aligned with theforward velocity, V , and the resulting slip velocity equals the forward velocity. Thereforethe lateral and longitudinal slip vectors are the projections of the forward velocity vectoron the lateral and longitudinal directions, given by sin(α)V and cos(α)V , respectively.Due to the relatively small slip angles it follows that the longitudinal slip velocity is muchhigher than the lateral slip velocity and the longitudinal force approaches the solutionfor zero slip angle at wheel lock. This also implies that the lateral force for increasinglongitudinal slip should decrease, which is confirmed in Figures 6.11 and 6.12, where the


lateral force as function of longitudinal slip is shown.2

−1 −0.8 −0.6 −0.4 −0.2 0−3000

−2000

−1000

0

1000

2000

3000

κ [−]

Fy [N

]

0°

+3°

+5°

+8°

−3°

−5°

−8°

Figure 6.11 / Lateral force as function of longitudinal slip for 7 slip angles at 0.4 timesthe load index, at a forward velocity of 60 km/h.

−1 −0.8 −0.6 −0.4 −0.2 0−6000

−4000

−2000

0

2000

4000

6000

κ [−]

Fy [N

]

0°

+3°

+5°

+8°

−3°

−5°

−8°


2For 3 slip angle at 0.8 L.I. no solution is obtained for longitudinal slip ratios above 71% with thecurrent model.


The shape of the self-aligning moment curve at combined slip is a complex combinationof several effects. At free-rolling, the self-aligning moment is given by the pure corneringcharacteristic as shown in Figure 6.4. When a brake torque is applied, a longitudinalforce is generated and the tire deforms in longitudinal direction due to flexibility of thecarcass. This results in additional contributions to the self-aligning moment. In Figure6.13 a brush model with flexible carcass is shown, where it can be seen that the deflectionof the carcass results in a deflection uc in longitudinal direction and a deflection vc inlateral direction. Furthermore the location of the resultant of the longitudinal force hasalso an offset vo with respect to the x-axis due to the asymmetric tread of the tire. There-fore the total self-aligning moment originates from both the magnitude and the resultantforce location of the lateral and longitudinal force.In Figures 6.14 and 6.15 the self-aligning moment as function of longitudinal slip isshown. It can be seen that for small values of κ the additional deflection uc causes anincrease of the self-aligning moment. Once the lateral force starts to decrease, the lon-gitudinal force increases and the location of the resultant of the longitudinal force gen-erates an additional moment. At κ = −1, Mz is the sum of the lateral offset caused bythe asymmetric tread of the tire times the longitudinal force, the lateral carcass deflectionis reduced if the tire slips completely, and a small lateral force times the longitudinal de-flection of the carcass. This last term changes sign for positive and negative slip angles,because the direction of the lateral force changes for positive and negative slip angles.

Wheel spin axis

Carcass

Wheel plane

Vy

xα

Trailing edge Leading edge

Fy

Fx

F

vo

uc

vc

t

Figure 6.13 / Extended brush model with flexible carcass under combined slip situation.


−1 −0.8 −0.6 −0.4 −0.2 0−60

−40

−20

0

20

40

60

κ [−]

Mz [

Nm

]

0°

+3°

+5°

+8°

−3°

−5°

−8°

Figure 6.14 / Self-aligning moment as function of longitudinal slip for 7 slip angles at0.4 times the load index, at a forward velocity of 60 km/h.

−1 −0.8 −0.6 −0.4 −0.2 0−100

−50

0

50

100

150

κ [−]

Mz [N

m]

0°

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−3°

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6.2.3 Friction power distribution in the footprint

The frictional power per unit area, τeqvγeqv, is calculated in order to illustrate other poten-tial applications of the model. The integral over time is equal to the energy per unit areaand known as the shear energy intensity (Gent and Walter, 2005, chapter 7). This shearenergy intensity can be used as a predictor of tire profile wear as shown by Pottinger andMcIntyre (1999). As an illustration the shear power intensity is shown in Figure 6.16for four distinct driving conditions. Note that, under straight line driving with constantvelocity (free-rolling), the shear power intensity is not visible on the used scale.

(a) Free-rolling, α = 0, κ = 0. (b) Pure braking, α = 0, κ = −1.

(c) Pure cornering, α = 8, κ = 0. (d) Combined slip, α = 8, κ = −1.

Figure 6.16 / Distribution of frictional power for different driving conditions at 0.8 timesthe load index, at a forward velocity of 60 km/h. Darker colors indicate higher intensity.The bottom of the figure corresponds to the leading edge of the contact patch.


This is expected, since under free-rolling the sliding velocity is very low. At wheel lock,the sliding velocity is maximal and the frictional power intensity increases. For the purecornering situation the distribution originates from the lateral shear stresses at a rela-tively low slip velocity. Finally the frictional power intensity for combined slip at wheellock is shown, where the highest shear stresses and slip velocities are present.A large part of the frictional power is converted into heat, which results in increased sur-face temperature. In Figure 6.17 snapshots, corresponding to the four driving conditions,of the surface temperature are shown. The surface temperature has been measured withan infrared camera mounted onto the fixed frame of Tyre Test Trailer, behind the rotatingmeasurement tower.

(a) Free-rolling, α = 0, κ = 0. (b) Pure braking, α = 0, κ = −1.

(c) Pure cornering, α = 8, κ = 0. (d) Combined slip, α = 8, κ = −1.

Figure 6.17 / Measured surface temperature distribution for different driving conditionsat 0.8 times the load index, at a forward velocity of 60 km/h. Darker colors indicatehigher temperature. The bottom of the figure corresponds to the leading edge of thecontact patch.

6.3 FORCE AND MOMENT MEASUREMENTS AND MAGIC FORMULA 107

This means that the history of points traveling through the entire contact zone is capturedand these images thus show a ‘smeared’ temperature distribution. Nevertheless, there isa clear resemblance with the calculated shear power intensity.Therefore an useful extension could be to incorporate temperature effects in the FEmodel based on the shear power intensity. Different strategies can be followed here.Giessler et al. (2010) used the friction heat to predict tread temperature increase and theeffect on tire traction under ice and snow conditions. The friction model of Huemer etal. (2001a) incorporated temperature effects in the material properties using the WLFtransformation. Hofstetter et al. (2006) used a fully coupled thermo-mechanical FEmodel to convert a part of frictional power into heat, which increases the tread temper-ature. This temperature increase also results in changing viscoelastic material properties.

As mentioned, the shear energy intensity is used as a predictor of wear. After pure brak-ing experiments on the proving ground, flat spots on the tires have been observed. Ad-ditionally, for the combined slip experiments uneven wear of the inner tread blocks hasbeen noticed, which indicate that shear energy intensity is a contributing factor in thecomplex wear process. Another possible future extension of the FE model could there-fore be to use this frictional power per unit area to predict wear of the tread profile. Thisis done by e.g. Smith et al. (2008), where the steady-state approach of ABAQUS is used topredict wear of a tread profile under different driving conditions. It is even possible toadapt the mesh to the worn configuration with the current version of ABAQUS (SIMULIA,2009b).

6.3 Force and moment measurements and Magic Formula

In this section some observations about the experimental data are discussed before acomparison of the FE model with the Magic Formula is made. As presented by Pottinger(Gent and Walter, 2005, chapter 8), an individual tire does not have a single well-definedset of force and moment characteristics. This uncertainty in the characteristics arisesfrom the changing chemical and viscoelastic properties of the tire materials combinedwith effects of wear.As already mentioned in Chapter 5, experiments have been carried out with the TNO TyreTest Trailer. Besides the straight line braking experiments at different velocities, the stan-dard measurement program to obtain a Magic Formula dataset has been executed (TNO,2008). This dataset is constructed from pure braking, pure cornering and combined slipmeasurements under different operation conditions and using several tires. All thesemeasurements are combined in the parameter identification software MF-Tool to identifyone single Magic Formula parameter set. With this single so-called tire property file allsteady-state handling characteristics can be evaluated. Since the MF is a semi-empirical


approach, with a high fitting accuracy, this tire file provides a good description of mea-sured handling characteristics at that moment in time.To investigate external influences on the tire behavior, this complete dataset has beenmeasured four times (see Table 6.1). The first set has been measured at proving groundone, while the second set has been measured several months later at proving ground two.This proving ground has a different road surface, but tires from the same batch have beenused for these measurements. The third and fourth dataset have been measured againat proving ground one, but one year later than the first measurements. For the third seta newly manufactured batch of tires has been used, while the fourth set has also beenmeasured with tires from the first batch.The four different Magic Formula datasets represent all the same tire, which therefore canprovide information about the sensitivity to external influences on the measured forcesand moments, such as environmental conditions, road surface or aging.

Table 6.1 / Specifications of the different sets used to obtain the Magic Formula datasets.Tires from batch 1 have been stored at room temperate in between measurements.

Set nr. Proving ground Date Batch1 1 Nov. 2007 1

2 2 Sep. 2008 1

3 1 Nov. 2008 2

4 1 Nov. 2008 1

In the following figures only the results for 0.8 times the load index are discussed. Theresults for the other two loads tested show similar trends.In Figure 6.18 and Figure 6.19 the pure cornering characteristics are shown. It canbe seen that the lateral force is similar for all sets under all slip angles. The presentdifferences are however reflected in the self-aligning moment, where the moment inset 2 declines faster than the other sets. This can be the result of a different roadsurface. Furthermore the peak value at negative slip angle of set 3 is 15% higher than theother three, which could be due to the manufacturing variations between the two batches.

The combined slip characteristics are only shown for three different slip angles for im-proved visibility. When the longitudinal force as function of longitudinal slip is inspected,some remarkable differences can be seen in Figure 6.20. The longitudinal force at wheellock is significantly lower for sets 2 and 4, which is probably an aging effect. As indicatedby Pottinger (Gent and Walter, 2005, chapter 8), additional crosslinking of the rubbertakes place over time, which affects the viscoelastic properties. It can be seen that at set

6.3 FORCE AND MOMENT MEASUREMENTS AND MAGIC FORMULA 109

2 the lowest longitudinal forces have been measured, which suggest a road surface ef-fect as well. These figures indicate that both the longitudinal and lateral force can showdeviations up to 15% between the different sets.

−15 −10 −5 0 5 10 15−6000

−4000

−2000

0

2000

4000

6000

α [°]

Fy [N

]

Set 1

Set 2

Set 3

Set 4

Figure 6.18 / Lateral force as function of side slip angle at 0.8 times the load index, at aforward velocity of 60 km/h.

−15 −10 −5 0 5 10 15

−100

−80

−60

−40

−20

0

20

40

60

80

100

α [°]

Mz [N

m]

Set 1

Set 2

Set 3

Set 4

Figure 6.19 / Self-aligning moment as function of side slip angle at 0.8 times the loadindex, at a forward velocity of 60 km/h.


−1 −0.8 −0.6 −0.4 −0.2 0

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Fx [

N]

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]

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6.4 COMPARISON OF THE FE MODEL AND THE MAGIC FORMULA 111

6.4 Comparison of the FE model and the Magic Formula

6.4.1 Pure cornering

The comparison of the lateral force for pure side slip at three load conditions is shownin Figure 6.22, where a comparison is made with set 1. It can be seen that the overallprediction of the FE model, for all three loads, is good.Three observations can be made from this figure. First, the built-up of the lateral force forthe 0.8 and 1.2 load conditions in the FE model is faster than for the MF. This indicatesthat the cornering stiffness of the FE model at these loads is higher than the MF. Second,for high positive slip angles the predicted force of the FE model is at most 3% lower thanthe MF. Third, the predicted force at high negative slip angles is at most 10% lower thanthe MF, while the difference between positive and negative slip angles is less obviousfor the MF. This is again a strong indication that the influence of contact pressure inthe friction model is slightly overestimated and the friction coefficient drops too fast forincreasing contact pressure, see Section 5.7. An overestimation of the influence of contactpressure results in an overall lower lateral force than the MF and at negative slip anglesthis effect is amplified further by the smaller size of the footprint with respect to a positiveslip angle. Nevertheless, the correlation is good for the entire range of slip angles and allloads.

It can be seen in Figure 6.23 that also the self-aligning moment is accurately predicted, es-pecially considering the variations between different datasets. The self-aligning moment,

−15 −10 −5 0 5 10 15−8000

−6000

−4000

−2000

0

2000

4000

6000

8000

α [°]

Fy [N

]

0.4

0.8

1.2

MF

FEM

Figure 6.22 / Lateral force as function of side slip angle at 0.4, 0.8 and 1.2 times the loadindex, at a forward velocity of 60 km/h.


−15 −10 −5 0 5 10 15−200

−150

−100

−50

0

50

100

150

200

α [°]

Mz [N

m]

0.4

0.8

1.2

MF

FEM

Figure 6.23 / Self-aligning moment as function of side slip angle at 0.4, 0.8 and 1.2 timesthe load index, at a forward velocity of 60 km/h.

as given by (6.1), depends on the length of the pneumatic trial and the magnitude of thelateral force. In the FE model the lateral force for negative slip angles is already lowerthan the MF, which explains the faster decay at negative slip angles. The pneumatic trailas function of slip angle is shown in Figure 6.24, where it can be seen that the pneumatictrail of the FE model is just a few millimeters shorter than the smooth pneumatic trailgiven by the MF, which lead to a smaller peak value of Mz in the FE model. Additionally,when the lateral force is close to zero and changes sign in the FE model, the location ofthe lateral force resultant is not well-defined. This causes the observed jump at−0.1 slipangle.The small differences in the pneumatic trail could be a result of the local friction model.However conicity, which is not present in the FE model, has an effect on the measure-ments. Furthermore, in the experiments a rolling resistance force is always present dueto the viscoelastic materials in the tire. Rolling resistance is not present in the currentmodel, which also leads to a different footprint with respect to footprints during actualexperiments.

Cornering and aligning stiffness

In the region for |α| < 2.5 the change in lateral force is governed by the corneringstiffness

CFα =∂Fy∂α

∣∣∣∣α=0

, (6.2)


−15 −10 −5 0 5 10 15−30

−20

−10

0

10

20

30

40

50

60

α [°]

Pn

eum

atic tra

il [m

m]

MF

FEM

Figure 6.24 / Pneumatic trail as function of side slip angle at 0.8 times the load index, ata forward velocity of 60 km/h.

which is approximated by a linear least square error fit through data points for Fy between−2.5 and 2.5 slip angle. Similarly, the change in self-aligning moment is governed bythe aligning stiffness

CMz =∂Mz

∂α

∣∣∣∣α=0

, (6.3)

which is approximated by a linear least square error fit through data points for Mz be-tween−2.5 and 2.5 slip angle. These stiffnesses are important parameters in determin-ing the linear range behavior of vehicles. The computed cornering and aligning stiffnesstogether with the MF are shown in Figure 6.25. It can be seen that the peak of the cor-nering stiffness for increasing load is not reached and therefore lies well above the tiresload index, which is good for handling. Furthermore the computed stiffness at 0.4 and0.8 are in good agreement with the MF, the stiffness at 1.2 is 15% higher than the MF.This indicates that the structural response of the FE model for this high load deviatesmore with respect to the Magic Formula. Additionally the cornering stiffness of set 2

deviates from the other three, which could originate from different environmental condi-tions, since this set has been measured at proving ground 2. The aligning stiffnesses ofthe FE model are slightly underestimated with respect to the MF, which is to be expectedsince the pneumatic trail in the FE model is shorter.


0 0.5 1 1.50

500

1000

1500

2000

Fz/L.I. [−]

Corn

erin

g s

tiffn

ess [

N/d

eg

]

Set 1

Set 2

Set 3

Set 4

FE

(a)

0 0.5 1 1.50

20

40

60

80

100

120

Fz/L.I. [−]

Alig

nin

g s

tiffne

ss [N

m/d

eg]

Set 1

Set 2

Set 3

Set 4

FE

(b)

Figure 6.25 / a) Cornering stiffness and b) aligning stiffness as function of normalizedload.

Forward velocity

The Magic Formula does not consider the forward velocity as input parameter, whichmeans that velocity induced effects are not taken into account. As already mentioned, themagnitude of the lateral force eventually peaks. Beyond this peak value, increasing theslip angle decreases the lateral force due to increasing slip velocity. This indicates thatthere should be a velocity effect present and the lateral force at high slip angles shouldbe higher for lower velocities. The sliding velocities during cornering are however muchlower compared to braking, which suggests that the effect might not be as clear as in abraking experiment.To investigate this, a simulation and a measurement at proving ground 2, have beenperformed with a forward velocity of 40 km/h. For this single measurement and alsoa single measurement at 60 km/h, two MF datasets have been generated. These setsshould be used with care, since possible disturbances are also fitted, and are thereforeonly used for illustrative purposes. It can be seen in Figure 6.26 that there is a smalleffect visible both for the FE model and the MF. However, as expected, it is a very smalleffect, which is only visible for very high slip angles. It is however captured in the FEmodel. Furthermore this shows that, for pure cornering, tire testing at different velocitiesmight not be necessary, since the velocity influence is negligible compared to the forcedeviations observed for the different datasets.


0 5 10 15−6000

−5500

−5000

−4500

−4000

−3500

−3000

α [°]

Fy [

N]

FE 60 km/h

FE 40 km/h

(a)

0 5 10 15−6000

−5500

−5000

−4500

−4000

−3500

−3000

α [°]

Fy [

N]

MF 60 km/h

MF 40 km/h

(b)

Figure 6.26 / Close-up of lateral force as function of slip angle at two forward velocitiesfor a) the FE model and b) the MF.

6.4.2 Combined slip

The validation for combined slip characteristics is presented for 4 slip angles for the sakeof visibility. Furthermore it is chosen to compare the FE model with sets 1 and 3, sincethe braking experiments used for identification in Chapter 5 have also been performedwith new tires. The validation of the longitudinal forces under combined slip for two loadconditions are shown in Figures 6.27 and 6.28 respectively. It follows that the trend ofthe FE model is a bit closer to set 1, but for both load conditions and all slip angles theFE predictions represent the tire behavior really well. If the lateral forces are inspected,see Figures 6.29 and 6.30, all FE predictions are within the bounds given by the two MFsets. This shows that the FE model adequately predicts the forces under the complexcombined slip situation.

Self-aligning moment

The measurements of the self-aligning moment during combined slip fluctuate severelyand are less reliable. This is a well-known issue in tire testing and as a result it is hardto fit the Magic Formula coefficients. Therefore the MF under combined slip is adaptedbased on physical insights (Pacejka, 2006), but the self-aligning moment can still deviatewith respect to the different datasets due to these fluctuations in the measurements. Asa result the obtained self-aligning moments with the MF might not be representative ofthe tested tire.In Figure 6.31 and 6.32 the self-aligning moments for two of the MF datasets are com-pared to the FE prediction. There are large differences between the two MF datasets,which makes model validation, based on these results, difficult. Further investigation of


the exact behavior of the self-aligning moment under combined slip is therefore neces-sary to judge to predicted moments of the FE model.

−1 −0.8 −0.6 −0.4 −0.2 0−3500

−3000

−2500

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0

κ [−]

Fx [

N]

0°

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+8°

Set 1

Set 2

FE


−1 −0.8 −0.6 −0.4 −0.2 0

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]

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FE



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]

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FE


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]

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FE



−1 −0.8 −0.6 −0.4 −0.2 0−60

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20

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κ [−]

Mz [

Nm

]

0°

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+3°

+8°

Set 1

Set 2

FE


−1 −0.8 −0.6 −0.4 −0.2 0−100

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50

100

150

κ [−]

Mz [N

m]

0°

−5°

+3°

+8°

Set 1

Set 2

FE


6.5 CONCLUSIONS 119

6.5 Conclusions

In this chapter the predictive capability of the FE model has been presented and a com-parison with the Magic Formula has been made.It has been shown that, for a pure cornering simulation, the magnitude of the lateral forceat positive slip angles is slightly larger than for negative slip angles. This is the result ofthe non-symmetric tire tread in combination with the used friction model.The effect of plysteer, which leads to offsets in the computed lateral force and self-aligningmoment characteristics, has been discussed. The overturning moment and approximatedloaded radius as function of slip angle are in good agreement with similar data presentedin literature. The combined slip characteristics have been presented for several slip anglesand the observed shape of the longitudinal and lateral forces as function of longitudinalslip has been explained. It has been shown that the self-aligning moment under com-bined slip is a combination of the location of both the longitudinal and the lateral forceresultant. These locations depend not only on the local friction model, but also on theflexibility of the carcass.The resemblance between computed shear power distribution, under different drivingconditions, and measured surface temperature clearly illustrate the potential of the FEmodel. The shear power distribution can be used in future developments of the model toincorporate temperature or wear effects.It has been shown that an individual tire does not have a single well-defined set of forceand moment characteristics. Therefore four datasets have been measured and the devia-tions between the four Magic Formula evaluations have been discussed.A comparison has been made between predicted handling characteristics of the FE modeland the Magic Formula. For both the pure cornering and the complex combined slip situ-ation, the forces and moments are accurately predicted with the current model. Observeddeviations with respect to the cornering characteristics, for negative slip angles, suggestthat the influence of contact pressure in the friction model is slightly overestimated.Finally, it can be concluded that all three basic handling characteristics are adequatelypredicted by using the identified friction model on a full scale tire.

CHAPTER SEVEN

Conclusions and recommendations

Abstract / In this chapter, the main conclusions of this thesis are summarized and recommenda-tions for future work are presented.

7.1 Conclusions

In this thesis the finite element (FE) computation of steady-state handling characteristicsof rolling tires, under driving situations varying from normal driving to extreme manoeu-vres, has been presented. The steady-state handling characteristics considered are purebraking, pure cornering and combined slip and have been defined as follows. In case ofpure braking, the tire is braked from free-rolling up to wheel lock. For pure cornering,the free-rolling tire is steered up to ±12 side slip angle and in case of combined slip, thetire is braked up to wheel lock when rolling at a constant slip angle.To compute these handling characteristics the Coulomb friction model is often used tomodel the tire-road interaction problem, while it is clear from experiments that tire-roadinteraction can not be captured accurately with this friction model. Additionally Coulombfriction is not a feasible choice for the simulation of extreme manoeuvres. To overcomethese limitations an enhanced friction model for the tire-road interaction problem is pro-posed. This leads to the main objective of this thesis:

To develop a robust and numerically efficient friction model for finite element tire sim-ulations and to create a framework for the identification and implementation of frictionrelated parameters. The friction model should capture observed effects of dry friction andit should be compatible with commercial FE codes.

The proposed friction model is directly dependent on contact pressure and sliding ve-locity, while the global effect of temperature and surface roughness is captured in theparameters of the friction model.

121

122 7 CONCLUSIONS AND RECOMMENDATIONS

A framework for the identification of the unknown parameters of this friction model hasbeen developed. This framework consists of a two step experimental / numerical ap-proach in which the parameters of the contact pressure and the sliding velocity part aretreated separately and identified sequentially by combining small scale lab experimentswith full scale outdoor experiments.Validation results of computed steady-state handling characteristics, which cover the en-tire operating range, confirm the effectiveness of the developed friction model.

For the FE analysis a currently available state-of-the-art numerical method of the commer-cial finite element package ABAQUS is used. The steady-state transport analysis of ABAQUS

is a computationally efficient method to obtain the global steady-state force and momentcharacteristics of a tire under different driving conditions. However, this method has alsosome limitations. The underlying road must be flat and fully coupled thermo-mechanicalsimulations are not possible in the current implementation. The choice for the improvedfriction model is based on an overview of friction models to describe the frictional re-sponse of rubber, while taking the limitations of the numerical method into account.

A numerical and experimental analysis of the commercially available Laboratory Abra-sion and skid Tester 100 has been presented, where measured hub forces have been usedto identify the local contact pressure dependent parameters of the tire friction model.A specially developed test tire has been used for the identification of the slip velocity de-pendent parameters. This tire has an asymmetric tread profile in order to study the effectof the contact pressure distribution on the friction force. Furthermore, the correspondingFE tire model has been used for the experimental validation.The slip velocity dependent parameters have been identified using measurements of thelongitudinal force at the axle under different driving velocities. To exclude small vari-ations in vertical load during experiments, the semi-empirical Magic Formula model,fitted on the experimental data, is used to evaluate the longitudinal slip characteristics ata constant vertical load. It has been shown that the computed steady-state longitudinalforces are in good quantitative agreement with experiments for all loads.Based on these results, it is concluded that the proposed identification framework, wherethe parameters are identified using measured axle forces and implemented locally at eachnode in contact, is a suitable approach to obtain the parameters of the friction model.

The predictive capability of the tire model in combination with the fully identified fric-tion model has been assessed. A comparison has been made between predicted handlingcharacteristics of the finite element model and experimental data for the pure corneringand combined slip range. For both the pure cornering and the complex combined slipsituation, the forces and moments are accurately predicted with the current model. Fornegative slip angles, observed deviations with respect to the cornering characteristics sug-gest that the influence of contact pressure in the friction model is slightly overestimated.Nevertheless, the presented comparison shows that all three basic handling characteris-tics are well predicted by using the identified friction model on a full scale tire.

7.2 RECOMMENDATIONS 123

7.2 Recommendations

In this final section, directions that can be pursued in future work are discussed.

The observed deviations with respect to the cornering characteristics for negative slip an-gles and the simulations with different inflation pressure, are most likely caused by anoverestimation of the influence of contact pressure in the friction model, where the con-tact pressure is extrapolated outside the measured range. Therefore, it is recommendedthat the followed identification method using the Laboratory Abrasion and skid Tester100 is investigated further. Alternatively, specially designed lab scale setups, such asused by Huemer et al. (2001b) to obtain a broader range of contact pressures could alsobe developed.

Within the presented numerical modeling framework, several extensions can be made.The current friction model does not depend on temperature. The temperature duringfrictional sliding of rubber is not constant. Therefore, it is recommended to incorporatetemperature effects in the FE model based on the shear power intensity. Different strate-gies can be followed here. Giessler et al. (2010) uses the friction heat only to predict treadtemperature increase and the effect on tire traction. The friction model of Huemer et al.(2001a) incorporates temperature effects in the material properties using the WLF trans-formation. Hofstetter et al. (2006) uses a fully coupled thermo-mechanical FE model toconvert a part of frictional power into heat, which increases the tread temperature. Thistemperature increase also results in changing viscoelastic material properties. This lastmethod seems the most promising, although both methods require an accurate descrip-tion of the viscoelastic material properties of the tread compound.

The handling characteristics considered in this thesis are based on dry roads, but it isnot clear what the exact contribution of surface texture on the frictional force is. A firststep in this direction in the context of the current project has been made by Hunnekes(2008), but further research in this area is necessary. Furthermore, the application of thepresented two step experimental / numerical approach under wet conditions should beassessed.

In this thesis, the friction model is combined with an existing FE tire model. Othercomponents of the FE model, such as material behavior and mesh design, contribute tothe structural response of the model as well. Hence, these components should also beimproved in order to further enhance the prediction capability of the entire model.

Real virtual prototyping has not been achieved yet. A full scale tire has been manufac-tured to identify the parameters of the friction model. A first step towards actual virtualprototyping should be to use the fully identified friction model on a different type of tire,e.g. with other size, aspect ratio or structural properties, with the same tread compound.In this manner the predictive capability of the friction model can be further assessed. Ina second step, a more complicated tread pattern can be used, although the accuracy of the

124 7 CONCLUSIONS AND RECOMMENDATIONS

solution of the ALE method should be carefully checked (Qi et al., 2007).Once these steps have been successfully completed, the possibility to use FE tire mod-els in vehicle dynamic simulations should be investigated. Although the computationaleffort is too large for vehicle dynamic simulations, it might be possible to fit a Magic For-mula model on a combination of predicted handling characteristics and less experimentaldata. In this way the required low computational cost for a full vehicle simulation is stillmaintained.

As discussed by Pottinger (Gent and Walter, 2005, chapter 8), an individual tire does nothave a single well-defined set of force and moment characteristics. This uncertainty in thecharacteristics arises from the changing chemical and viscoelastic properties of the tirematerials combined with effects of wear. Furthermore, due to manufacturing variationsother effects, such as conicity, are introduced in real tires. This is one of the reasons whyin this thesis the different Magic Formula sets obtained on tires from one batch deviatewith respect to each other, even when the environmental conditions are similar. A well-defined upper and lower bound around a MF dataset should be developed. On the FEmodel side, parametric studies with respect to manufacturing variations should be carriedout to obtain an upper and lower bound around the predicted handling characteristics.Furthermore, in the MF-Tool software the parameters of the MF are chosen such that theerror between the MF and experiments is minimized. As a result, it can happen that thelongitudinal force at κ = 0 is positive. This makes a quantitative comparison with the FEmodel prediction very difficult, since a small difference in longitudinal slip around zeroresults in a large longitudinal force difference. A well-defined error criterium should bedeveloped.

Currently, measurements for validation purposes of FE models are still necessary. Ob-taining good measurements of the forces and moments or other system quantities actingon rolling tires is a difficult task. Direct measurements of the stress distribution in thecontact area, with a high spatial resolution, during driving are preferred for validation ofthe local friction model. However, this is not yet possible. Therefore, the forces and mo-ments acting on the hub are used for validation in this thesis, although the measurementsof moments, during combined slip situations, fluctuate severely and are less reliable. Forvalidation of a local friction model, accurate measurements of the moments are neces-sary. Even if the force magnitude is correctly predicted, the local stress distribution in thecontact area determines the resultant force location and hence the magnitude of the mo-ment. Therefore, further research into more advanced measuring techniques, preferablydirectly in the contact area, is still required.

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APPENDIX A

Friction model implementation

In this appendix the full set of equations of the friction model is summarized.The friction law is described by

µj(pj, γj) =

(pjp0


−h2 log2

(‖γ‖jVmax

)], (A.1)

where pj and ‖γ‖j are the normal contact pressure and equivalent slip velocity at nodej respectively. The parameters p0 and k are related to the contact pressure and theparameters µs, µm, h and Vmax are related to the sliding velocity.

The parameter R, in the interval [0 1], is used to model frictionless contact (R = 0)during the static steps and frictional contact (R = 1) during the steady-state rolling steps.An intermediate steady-state rolling step is used to activate the friction model. In thisstep the parameter varies linearly from zero to one, which prevent convergence problemsduring the transition from frictionless to frictional contact.

With the user-defined critical slip velocity γcrit the slope

ks =µ(p, γ)p

γcrit. (A.2)

is calculated.

A.1 Stick

The frictional stress in stick is given by

τi = ksγi. (A.3)

133

134 A FRICTION MODEL IMPLEMENTATION

The partial derivatives for the linearized incremental friction shear stress and contactpressure are derived using Maple, version 12.0 and are given by

∂∆τ1

∂∆γ1=

−2h2Rpγ21(

γ21 + γ2

2

)γcrit

(p

p0

)−k(µm − µs) log

(√γ2

1 + γ22

Vmax

)exp

−h2 log2

(√γ2

1 + γ22

Vmax

)

+Rp

γcrit

(p

p0


−h2 log2

(√γ2

1 + γ22

Vmax

)](A.4)

∂∆τ1

∂∆γ2=

−2h2Rpγ1γ2(γ2

1 + γ22

)γcrit

(p

p0


(√γ2

1 + γ22

Vmax

)exp

−h2 log2

(√γ2

1 + γ22

Vmax

)(A.5)

∂∆τ1

∂∆p=

(1− k)R(p

p0


−h2 log2

(√γ2

1 + γ22

Vmax

)]γ1

γcrit(A.6)

∂∆τ2

∂∆γ1=

−2h2Rpγ1γ2(γ2

1 + γ22

)γcrit

(p

p0


(√γ2

1 + γ22

Vmax

)exp

−h2 log2

(√γ2

1 + γ22

Vmax

)(A.7)

∂∆τ2

∂∆γ2=

−2h2Rpγ22(

γ21 + γ2

2

)γcrit

(p

p0


(√γ2

1 + γ22

Vmax

)exp

−h2 log2

(√γ2

1 + γ22

Vmax

)

+Rp

γcrit

(p

p0


−h2 log2

(√γ2

1 + γ22

Vmax

)](A.8)

∂∆τ2

∂∆p=

(1− k)R(p

p0


−h2 log2

(√γ2

1 + γ22

Vmax

)]γ2

γcrit(A.9)

A.2 SLIP 135

A.2 Slip

The frictional stress for slip is given by

τi = µ(p, γ)pγiγeqv

. (A.10)

The partial derivatives for the linearized incremental friction shear stress and contactpressure in slip are given by

∂∆τ1

∂∆γ1=

Rp√γ2

1 + γ22

(p

p0


−h2 log2

(√γ2

1 + γ22

Vmax

)]

− Rpγ21(

γ21 + γ2

2

)3/2 ( p

p0


−h2 log2

(√γ2

1 + γ22

Vmax

)](A.11)

− 2h2Rpγ21(

γ21 + γ2

2

)3/2 ( p

p0


(√γ2

1 + γ22

Vmax

)exp

−h2 log2

(√γ2

1 + γ22

Vmax

)

∂∆τ1

∂∆γ2=

−Rpγ1γ2(γ2

1 + γ22

)3/2 ( p

p0


−h2 log2

(√γ2

1 + γ22

Vmax

)](A.12)

− 2h2Rpγ1γ2(γ2

1 + γ22

)3/2 ( p

p0


(√γ2

1 + γ22

Vmax

)exp

−h2 log2

(√γ2

1 + γ22

Vmax

)

∂∆τ1

∂∆p=

(1− k)R(p

p0


−h2 log2

(√γ2

1 + γ22

Vmax

)]γ1√γ2

1 + γ22

(A.13)

∂∆τ2

∂∆γ1=

−Rpγ1γ2(γ2

1 + γ22

)3/2 ( p

p0


−h2 log2

(√γ2

1 + γ22

Vmax

)](A.14)

− 2h2Rpγ1γ2(γ2

1 + γ22

)3/2 ( p

p0


(√γ2

1 + γ22

Vmax

)exp

−h2 log2

(√γ2

1 + γ22

Vmax

)

136 A FRICTION MODEL IMPLEMENTATION

∂∆τ2

∂∆γ2=

Rp√γ2

1 + γ22

(p

p0


−h2 log2

(√γ2

1 + γ22

Vmax

)]

− Rpγ22(

γ21 + γ2

2

)3/2 ( p

p0


−h2 log2

(√γ2

1 + γ22

Vmax

)](A.15)

− 2h2Rpγ22(

γ21 + γ2

2

)3/2 ( p

p0


(√γ2

1 + γ22

Vmax

)exp

−h2 log2

(√γ2

1 + γ22

Vmax

)

∂∆τ2

∂∆p=

(1− k)R(p

p0


−h2 log2

(√γ2

1 + γ22

Vmax

)]γ2√γ2

1 + γ22

(A.16)

APPENDIX B

Mesh effect on the force equilibrium invertical direction

In a steady-state rolling step, the centrifugal force creates an additional, radially oriented,force when inertia effects are taken into account. This is an internal force and the nettoeffect should be zero. However, when a non-uniform mesh in circumferential directionis used the center of mass does not coincide with the rotation axis of the tire. Because themesh is fixed in space in the ALE method, this offset creates a small numerically inducedforce in the direction of the denser mesh when the tire spins with rotational velocity ω.

To illustrate this effect two simulations are carried out. In these simulations only a rota-tional velocity profile is prescribed, while the axle of the tire is fixed, as shown in FigureB.1. In the first simulation a uniform mesh is used and in the second simulation the

ω

z

x

Figure B.1 / Exaggerated view of the influence of centrifugal force on the tire, when arotational velocity profile is prescribed and the axle of the tire is fixed.

mesh as shown in Figure 3.10 is used. The reaction forces in vertical direction of bothsimulations are shown in Figure B.2. The reaction force in vertical direction is zero for

137

138 B MESH EFFECT ON THE FORCE EQUILIBRIUM IN VERTICAL DIRECTION

the uniform mesh, while for the nonuniform mesh a force proportional to ω2 in the neg-ative z direction is observed. This demonstrates that a pure rotational velocity on the tire,without a forward velocity, generates a small numerically induced force in the directionof the denser mesh.

0 10 20 30 40 50 60 70 80 90−1

−0.5

0

0.5

1

Rea

ction

fo

rce

axle

[N

]

0 10 20 30 40 50 60 70 80 900

50

100

150

ω [rad/s]

Reaction forc

e a

xle

[N

]

Figure B.2 / Reaction forces on the axle for uniform mesh (top) and reaction forces onthe axle for non-uniform mesh (bottom) as function of rotational velocity.

When the road is added and contact is present between the road and the tire, the gener-ated numerically induced force contributes to the overall force balance between the wheeland road reference node. In an FE analysis it is equivalent to prescribe the vertical force inupward direction on the reference node of the road and fixate the axle reference node or toprescribe the vertical force in downward direction on the reference node of the wheel andfixate the reference node of the road. Two additional simulations with the non-uniformmesh are carried out, in which the road is added and the braking characteristic of the tireis computed for a forward velocity of 100 km/h. In the first simulation the road is loadedwith Fz = 5000 N and in the second simulation the wheel is loaded on the axle with aforce Fz of −5000 N. In Figure B.3 the reaction forces of both simulations are shown,which confirms the decrease in reaction force on the axle and the increase of reactionforce on the road.

To avoid this force a uniform mesh should be used. However, then the ALE method losesits main advantage. Furthermore, to obtain a sufficient dense mesh in the contact area, avery large number of sectors is required which is computationally not feasible. It can beseen in Figure B.3 that this force is small with respect to the applied load on the tire andas such hardly affects the computed longitudinal and lateral forces. Therefore, inertia

139

0 10 20 30 40 50 60 70 80 90−5000

−4950

−4900

−4850

Re

actio

n f

orc

e a

xle

[N

]

0 10 20 30 40 50 60 70 80 905000

5050

5100

5150

ω [rad/s]

Re

actio

n f

orc

e r

oad

[N

]

Figure B.3 / Close-up of the longitudinal force on the axle as function of rotational velocityfor both the road load and the axle load at a constant forward velocity of 100 km/h.

effects are included and since the load on the wheel is kept constant during experiments,for all forward velocities, the wheel is loaded in the FE model to keep the force on the axleconstant.

APPENDIX C

Computation of slip velocity field for theLAT 100 setup

The slip velocity field is not always oriented with the global axes system in ABAQUS. Thisdepends on the used description for the rigid surface. In this case an analytical rigidsurface (SIMULIA, 2009a) is used to model the disk. When an analytical rigid surfaceis defined in ABAQUS the orientation of the slip vectors is determined by the sequenceof the line segments to construct the surface. The first slip direction is always orientedalong the direction of the line segments forming the surface. The second slip directionis defined such that the outward surface normal and the two surface tangents form aright handed orthogonal system. In the case of a rigid circular disk this gives a radialand a tangential oriented slip vector. Since the global coordinate system is Cartesianbased, with its center in the middle of the tire, it is not straightforward to interpret thedefault slip velocity output. For every point on the tire, that is in contact, a different localcoordinate system exists. By post-processing the slip velocity data it is possible to alignall the local orientations with the global axes system. The alignment of the slip vectorswith the global axes system is done in two subsequent steps, see also Figure C.1. Theorigin of the global axes system is located at the center of the wheel and the center of thedisk is positioned such that a specified slip angle is achieved. As a result of this, a fixedreference line, constructed through the center of the disk and the origin projected on theglobal X − Y plane, can be created. This is the line, with length b, in Figure C.1. Now itis possible to project vslip of every arbitrary node to a reference system C by rotating overan angle β. This angle is calculated using

βj = arccos

(a2j − b2 − c2

j

−2bcj

), (C.1)

141

142 C COMPUTATION OF SLIP VELOCITY FIELD FOR THE LAT 100 SETUP

where the lengths of the line segments aj and cj follows from the current nodal positionof node j. The rotation matrix is given by

Rβ =

[cos(β) sin(β)

− sin(β) cos(β)

]. (C.2)

Secondly an additional rotation over the side slip angle α,

Rα =

[ − sin(α) − cos(α)

cos(α) − sin(α)

], (C.3)

is required to project the reference system onto the global axes system. The componentsof the slip velocity vector in the global ABAQUS axes system can now be calculated as[

vslipXvslipY

]= Rα Rβ

[vslip1vslip2

]. (C.4)

vslip11

vslip21

X

Y

C1

C2

β1 αa1

b

c1

−β2

vslip22

vslip12

c2a2

Figure C.1 / Overview of rotation procedure to project local slip vectors, vslipj, onto the

global ABAQUS coordinate system, X-Y, for a positive side slip angle α.

Samenvatting

Bandmodellering is tegenwoordig een noodzakelijk onderdeel van het ontwerpproces vannieuwe banden. De automobielindustrie, overheden en consumenten eisen betere griponder alle weersomstandigheden, minder slijtage en meer recent ook minder geluid eneen lagere rolweerstand. Eindige elementen analyses worden gebruikt in het ontwerppro-ces van nieuwe banden, zodat rekening gehouden kan worden met deze, conflicterende,eisen. Het modelleren van banden met behulp van de eindige elementen methode kanhet inzicht in specifieke eigenschappen van een band verhogen, de ontwerptijd verkortenen de ontwerpkosten van nieuwe banden verlagen. Met de meeste eindige elementenmodellen is het nog steeds niet mogelijk experimenten op de weg goed te reproduce-ren. Naast de statische deformatie moet ook de dynamische responsie van een rollendeband op de weg nauwkeurig voorspeld worden. Het sturen, remmen en optrekken hangtaf van de gegenereerde wrijvingskrachten. Wrijving hangt niet alleen af van de (ma-teriaal)eigenschappen van het loopvlak van een band, maar ook van het wegdek en deweersomstandigheden. Het hoofddoel van dit proefschrift is het ontwikkelen van eenrobuust en numeriek efficient wrijvingsmodel geschikt voor banden simulaties met be-hulp van eindige elementen en het creëeren van een raamwerk voor de identificatie enimplementatie van wrijving gerelateerde parameters.

Het modelleren van een band in combinatie met de omgeving is een uitdaging, omdatverschillende fysische verschijnselen een rol spelen. In het algemeen zullen zowel demechanische, thermische en fluïdische effecten bijdragen aan de responsie van de band.Dit onderzoek spitst zich toe op het mechanische domein, waarvoor een numeriek raam-werk is gedefinieerd om simulaties met rollende banden met constante snelheid uit tevoeren. Dit raamwerk kan als uitgangspunt worden gebruikt in de verdere ontwikkelingvan het totale simulatiemodel. Eén van de doelstellingen van dit proefschrift is het ont-wikkelen en valideren van een wrijvingsmodel dat geschikt is voor eindige elementenanalyses en effecten van droge wrijving op de wegligging van rollende banden kan be-schrijven. Wrijving is een complexe interactie tussen twee materialen die met elkaar incontact zijn. Wrijving kan gemodelleerd worden op verschillende lengteschalen en er kan

143

144 SAMENVATTING

gebruik gemaakt worden van verschillende numerieke methoden. Dit kan echter leidentot lange rekentijden, wat niet praktisch is voor gebruik in een industriële applicatie. Omeen numeriek haalbare en relatief snelle oplossing te bieden, is er gekozen voor een fe-nomenologisch wrijvingsmodel, waar de parameters geïdentificeerd worden door middelvan een twee-staps experimenteel / numerieke aanpak.

Allereerst zijn er wrijvingsexperimenten uitgevoerd op een laboratoriumopstelling, ge-schikt voor slijtage- en wrijvingsmetingen, om de invloed van de contactdruk op de wrij-vingskracht te onderzoeken. In deze experimentele opstelling wordt een klein rubberwiel, met een variërende drifthoek, op een schijf gedrukt. De aanwezige wrijving tussende schijf en het wiel zorgt ervoor dat het wiel wordt aangedreven door de roterende schijf.De resulterende krachten op het wiel worden gemeten met een krachtsensor. Meerderemetingen onder verschillende belastingen en drifthoeken zijn uitgevoerd. Deze metin-gen, met lage rotatiesnelheid, zijn gebruikt om contactdrukgerelateerde parameters vanhet wrijvingsmodel te identificeren. De relevante onderdelen van deze opstelling zijn ge-modelleerd in het commerciële eindige elementen pakket ABAQUS. De prestaties onderconstante snelheid zijn geëvalueerd voor verschillende drifthoeken en vergeleken met demetingen. Simulaties tonen aan dat de aanwezige ‘turn slip’, die grote invloed heeft ophet slipsnelheidsveld aan de achterkant van het contactvlak, goed wordt beschreven methet model. Daarnaast komt de berekende spoorstijfheid goed overeen met de metingen.

In de tweede plaats zijn er met een autoband remmetingen op de weg verricht waarbijverschillende voorwaartse snelheden zijn gebruikt om een snelheidsafhankelijke para-meterset voor het wrijvingsmodel te verkrijgen. Het zo verkregen wrijvingsmodel is ver-volgens gekoppeld aan een eindige elementen model van de band. Dit model is eveneensin het eindige elementen pakket ABAQUS geconstrueerd. Het eindige elementen modelis statisch gevalideerd door middel van metingen van de contactdrukverdeling, de con-tactoppervlakte en de radiale en axiale stijfheid van de band. De ‘steady-state transport’methode van ABAQUS is vervolgens gebruikt om efficient de evenwichtsoplossingen on-der verschillende voorwaartse snelheden, zoals gebruikt in experimenten op de weg, teberekenen.

Tenslotte is de voorspellende waarde van het eindige elementen bandmodel in combi-natie met het voorgestelde wrijvingsmodel geëvalueerd. Het weggedrag tijdens rechtuitremmen, enkel sturen en een combinatie van remmen en sturen onder verschillendebelastingen, bandenspanningen en rijsnelheden is geëvalueerd en gevalideerd met expe-rimenten. Gebaseerd op deze vergelijking is geconcludeerd dat het weggedrag adequaatvoorspeld wordt.

Dankwoord

Onderzoek is eigenlijk nooit klaar en een proefschrift kan altijd nog net iets beter. Tijdenshet schrijven van deze laatste pagina’s begin ik langzaamaan te beseffen dat dit toch echthet einde inluidt van vier jaar onderzoek en dit proefschrift. Vanzelfsprekend heeft deafgelegde weg tijdens dit promotietraject de nodige hobbels gekend. Gelukkig heb ikdeze weg niet alleen gevolgd, maar heb ik dit pad mogen bewandelen met meerderepersonen zonder wie dit proefschrift er in de huidige vorm niet zou zijn. Daarvoor wil ikiedereen bedanken.

Henk, als eerste wil ik jou bedanken voor de mogelijkheid dit onderzoek uit te voeren.Daarnaast waardeer ik je directe communicatie, vertrouwen en steun enorm. Je hebt me,al sinds mijn afstuderen, altijd de vrijheid gegeven om mezelf steeds verder te verdiepenin de materie. Tegelijkertijd stuurde je tijdig bij als ik de grote lijnen uit het oog dreigdete verliezen. Ook daarvoor wil ik je bedanken.

Ines, zonder jouw enthousiasme en betrokkenheid was dit project heel anders verlopen!Ik wil je ontzettend bedanken voor de fijne samenwerking, onze inhoudelijke discussiesen misschien nog wel meer voor alle andere niet werkgerelateerde gesprekken.

I would like to thank the members of the reading committee, Matthias Kröger, DanielRixen and Marc Geers, for their careful reading and valuable feedback. Furthermore,I would like to thank Wim Desmet and Bart de Bruijn for taking part in the defensecommittee.

Bart, jouw enthousiasme tijdens ons eerste gesprek was een van de redenen om dit pro-ject uit te voeren. De afstand Eindhoven-Enschede is groot, desondanks was onze sa-menwerking goed. Ik wil je bedanken voor alle hulp en de kennis over banden en FEMdie je in korte tijd hebt overgedragen. Daarnaast was het fijn om van tijd tot tijd ABAQUS

frustraties te kunnen delen! Antoine, vanuit TNO was jij direct betrokken bij dit project.Bedankt voor alle inhoudelijke discussies en het nauwgezet commentaar op de papersen mijn proefschrift. Het uitvoeren van experimenten met ‘mijn’ banden was voor mijeen van de hoogtepunten in dit project. Willem, Bauke en Ton, bedankt voor jullie in-

145

146 DANKWOORD

zet en zeker ook voor de gezellige tijd na de lange werkdagen. Verder wil ik iedereenbij Vredestein en TNO bedanken die direct of indirect een bijdrage aan dit project heeftgeleverd.

Ik wil Jarno bedanken voor de experimentele bijdrage aan hoofdstuk vier. Al ben je mijnenige afstudeerder geweest, jouw inspanningen om de LAT 100 te doorgronden hebbengeweldig geholpen. Daarnaast wil ik Igo, Erwin, Peter, Ruud, Martijn, Michel, Roel, Je-roen en Bram bedanken voor hun bijdragen. Leo en Patrick wil ik bedanken voor deondersteuning bij alle hard- and software gerelateerde problemen tijdens dit project.

Verder wil ik alle (oud)collega’s van de DCT groep bedanken voor de uitzonderlijk goedesfeer tijdens en vooral ook na het werk. Petra en Lia, bedankt voor alle gezelligheid op hetsecretariaat en alle andere zaken die het leven van een promovendus een stuk prettigermaken. In het bijzonder wil ik mijn roomies Gerrit, Francois en Benjamin noemen.Bedankt voor de gezelligheid, het systeem, de vele discussies en de ontspanning in devorm van koffiepauzes doordeweeks en het af en toe brouwen (en drinken) van bier inhet weekend. Buiten de werkkring wil ik familie en vrienden bedanken voor de gebodenafleiding en de getoonde interesse in mijn werk.

Erik en Frank, bedankt dat jullie me terzijde willen staan. Het voelt goed om ditpromotietraject samen af te sluiten. Tenslotte zijn er twee mensen die, meer dan ze zelfbeseffen, hebben bijgedragen aan dit proefschrift. Pap en mam. Bedankt voor jullieonvoorwaardelijke steun en het vertrouwen tijdens alle goede en slechte momenten.Jullie hebben niet alleen mij, maar ons alle drie altijd gestimuleerd onze eigen weg tegaan. Dank daarvoor.

René van der Steenoktober 2010

Curriculum Vitae

René van der Steen was born on May 1st, 1981 in Borsele, the Netherlands. After finishinghis secondary education at the Rijksscholengemeenschap Goese Lyceum in Goes in 1999,he started his study Mechanical Engineering at the Eindhoven University of Technologyin Eindhoven, the Netherlands. He carried out his international internship at Queen’sUniversity, Kingston, Ontario, in Canada, where he worked on ‘Adaptive extremum-seeking control applied to bioreactors’. He received his Master’s degree cum laude inFebruary 2006 on the thesis entitled ‘Numerical and experimental analysis of multipleChua circuits’. This work was performed in the Dynamics and Control group at thedepartment of Mechanical Engineering. After completion of his Master’s thesis, Renéworked for four months as a visiting scientist in the Dynamics and Control group.In July 2006, he started as a Ph.D. student in the same group on the topic of tire modelingusing finite elements, with special attention to robust friction models. The project wasperformed in cooperation with Apollo Vredestein B.V., Enschede, the Netherlands andTNO Automotive, Helmond, the Netherlands. The main results of his Ph.D. research arepresented in this thesis.

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Documents

kautschuk

global axes

basic handling

van der steen

r pp0ks

vmi holland

commercial

shear energy