Vibrations, damping and powerdissipation in Car Tyres

Martin Fraggstedt

Stockholm 2008

Doctoral ThesisTRITA-AVE 2008:24

ISSN 1651-7660ISBN 978-91-7178-996-9

Royal Institute of TechnologySchool of Engineering Sciences

Department of Aeronautical and Vehicle EngineeringThe Marcus Wallenberg Laboratory for Sound and Vibration Research

Postal address Visiting address Contact

Royal Institute of Technology Teknikringen 8 Tel: +46 8 790 9202MWL / AVE Stockholm Fax: +46 8 790 6122SE-100 44 Stockholm Email:mfragg@kth.seSweden

Abstract

Traffic is a major source of green house gases. The transport fieldstands for 32 % of the energy consumption and 28 % of the totalCO2 emissions, where road transports alone causes 84 % of these fig-ures. The energy consumed by a car travelling at constant speed, isdue to engine inefficiency, internal friction, and the energy needed toovercome resisting forces such as aerodynamic drag and rolling resis-tance. Rolling resistance plays a rather large role when it comes to fueleconomy. An improvement in rolling resistance of 10 % can yield fuelconsumption improvements ranging from 0.5 to 1.5 % for passengercars and light trucks and 1.5 to 3 % for heavy trucks.

The objective of this thesis is to estimate the power consumptionin the tyres. To do this a car tyre is modelled with waveguide finiteelements. A non-linear contact model is used to calculate the contactforces as the tyre is rolling on a rough road. The contact forces com-bined with the response of the tyre is used to estimate the input powerto the tyre structure, which determines a significant part of the rollingresistance. This is the first rolling resistance model based on physicalprinciples and design data.

The elements used in the waveguide finite elements tyre model arederived and validated. The motion of the tyre belt and side wall isdescribed with quadratic anisotropic curved deep shell elements thatincludes pre-stress and the motion of the tread on top of the belt bycurved quadratic, Lagrange type, homogenous, isotropic two dimen-sional solid elements. The tyre model accounts for: the curvature, thegeometry of the cross-section, the pre-stress due to inflation pressure,the anisotropic material properties and the rigid body properties ofthe rim and is based on data provided by Goodyear.

To validate the tyre model, mobility measurements and an exper-imental modal analysis have been made. The model agrees very wellwith point mobility measurements up to roughly 250 Hz. The eigen-frequency prediction is within five percent for most of the identifiedmodes. The estimated damping is a bit too low especially for the anti-symmetric modes. The non-proportional damping used in the model isbased on an ad hoc curve fitting procedure against measured mobilities.

The non-linear contact force prediction, made by the division ofapplied acoustics, Chalmers University of Technology takes the tyre,the road texture and the tread pattern into account.

The dissipated power is calculated through the injected power andthe power dissipated within each element. It is shown that a roughroad leads to more dissipation than a smooth road. A demonstrationon real existing motor ways, for which rolling resistance measurementsalso have been made, show the potential of the method.

The damping is very important for the rolling resistance predic-tion. The damping properties of the tyremodel are therefore updatedbased on measurement, equivalent structure modelling and viscoelasticmaterial models. This updated model is slightly better at the pointmobility prediction and is far better at predicting the damping level ofthe identified modes from the experimental modal analysis.

Doctoral Thesis

The thesis consists of an introduction and the following four papers:

Paper A

S. Finnveden, M. Fraggstedt, Waveguide finite elements for curved struc-tures, Journal of Sound and Vibration 312 (2008) 644-671.

Paper B

M. Fraggstedt, S. Finnveden, A Waveguide Finite Element Model Of APneumatic Tyre, Submitted August 2007 to Journal of Sound and Vibra-tion.

Paper C

M. Fraggstedt, S. Finnveden, Power dissipation in car tyres, 2008. To besubmitted to the Journal of Sound and Vibration .

Paper D

M. Fraggstedt, S. Finnveden, The influence of the road on rolling resistance,2008.

Paper E

M. Fraggstedt, S. Finnveden, Estimates of the visco-elastic properties of cartyres, 2008. To be submitted to the Journal of Sound and Vibration.

Contribution from the author of this thesis

Paper A

Developed the straight elements. Performed the validation study. Cowrotethe paper.

Paper B

Experimental modal analysis and the mobility measurements. Performedsimulations. Fine tuning of the model developed by the supervisor. Wrotethe paper.

Paper C

Performed the power calculations. Literature study on rolling resistance.Wrote the paper.

Paper D

Performed the power calculations. Wrote the report.

Paper E

Performed the measurements, simulation and the optimisation. Wrote thepaper.

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Material from this thesis has been presented at eight workshops in theITARI project plus at three conferences:

SVIB, Nordic Vibration Research Conference, Stockholm Sweden,M. Fraggstedt , Estimation of Damping in Car Tyres, 2004.

Novem conference 2005, S. Finnveden, C.-M. Nilsson and M. Fraggstedt,Waveguide FEA of the Vibration of Rolling Car Tyres.

Euronoise 2006, Tampere, Finland, M. Fraggstedt, Rolling Resistance OfCar Tyres, 2006.

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Contents

1 Introduction 1

1.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Car tyres . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.3 Waveguide finite elements . . . . . . . . . . . . . . . . . . . . 31.4 Tyre model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.4.1 Existing tyre models. . . . . . . . . . . . . . . . . . . . 51.4.2 Waveguide FE tyre model. . . . . . . . . . . . . . . . 5

2 Analytical modelling of rolling resistance 6

2.1 Rolling resistance estimation based on dissipated power . . . 62.2 Rolling resistance - dynamic method . . . . . . . . . . . . . . 7

3 Tyre model updating 8

3.1 Damping - Viscoelastic materials. . . . . . . . . . . . . . . . . 83.2 Equivalent structure modelling . . . . . . . . . . . . . . . . . 9

4 Summary of the papers 10

4.1 Paper A. Waveguide finite elements for curved structures . . 104.2 Paper B. A waveguide finite element model of a pneumatic tyre 114.3 Paper C. Power dissipation in car tyres . . . . . . . . . . . . . 124.4 Paper D. The influence of the road on rolling resistance . . . 134.5 Paper E. Estimates of the viscoelastic properties of car tyres. 13

5 Future Work 15

6 Conclusion 15

7 Acknowledgements 15

1 Introduction

1.1 Background

For over fifty years traffic has been an irritating noise polluter. For higherspeeds tyres have been found to be the major contributor for traffic noise.Also the interior noise in the vehicle due to the tyres are becoming moreimportant as other noise sources such as engines, exhaust systems and gearboxes are better managed.

The negative effect on the environment has been highlighted for a numberof years, given that traffic is a major source of green house gases. Thetransport field is representing 32% of the energy consumption and 28% ofthe total CO2 emissions, where road transports alone stands for 84 % ofthese figures [1].

When it comes to the dynamics of the car the tyres are crucial, as theyprovide the grip required for cornering, braking and acceleration. In addi-tion, tyres are also highly involved in the cars handling abilities. As a finalpoint it is the tyres and the suspension system that assures a comfortableride.

The energy consumed by a car travelling at constant speed, is due toengine inefficiency, internal friction, and the energy needed to overcomeresisting forces such as aerodynamic drag and rolling resistance, which isone of the main topics of this thesis.

The rolling resistance Fr is defined as the energy consumed per unit ofdistance travelled [2]. The unit is Nm/m = N which is equivalent to adrag force in Newtons. Tyres are made of reinforced rubber, which is aviscoelastic material. As it deforms a part of the energy is stored elasticallybut the remainder is dissipated as heat. These hysteretic losses, as well asaerodynamic drag and friction in the contact patch and with the rim arelosses that contribute to the total drag force on a moving vehicle. Rollingresistance has a rather large impact when it comes to fuel economy. A 10% improvement in rolling resistance can give fuel consumption reductionsranging from 0.5 to 1.5 % for passenger cars and light trucks and 1.5 to 3% for heavy trucks [3].

In undergraduate texts, the rolling resistance is given as a dimensionlessconstant times the gravity force,

Fr = Cr m g, (1)

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where m is the mass, g is the constant of gravity and Cr is the rolling resis-tance coefficient. Cr is normally in the range 0.01-0.02 with a typical valueof 0.012 for a passenger car tyre on dry asphalt [4]. The power consumedby this force is

P = V Fr = V Cr m g (2)

where V is the speed of the vehicle. In equation (1) the only explicit pa-rameter is the load. The variation with other parameters are concealed inCr. Studies has shown that the rolling resistance coefficient is influenced bya number of parameters such as speed, driving torque, acceleration, rubbercompound, internal and ambient temperature, road texture, road roughness,and wear. The model is however usually sufficient for some applications. Inthis thesis the influence of the road texture and tyre characteristics is stud-ied.

The aim of this thesis is to model a radial car tyre with waveguide finiteelements and to use this model to estimate the power dissipation as the tyreis rolling on a rough road. These losses determine a significant part of therolling resistance. As a demonstration the procedure is used on real existingmotor ways, for which rolling resistance measurement data are available,making a comparison possible. This is the first model based on physicalprinciples to estimate the rolling resistance.

1.2 Car tyres

Car tyres are made of several different materials including steel, fabric and ofcourse numerous rubber compounds, see Figure 1. To get different dynamicproperties in the tyre sub regions the materials are used in many ways. Thethree major sub regions of the tyre are the upper side wall, the lower side walland the central area. The ply is a layer of embedded fabric in the rubber.At the lower side walls the ply encloses a volume filled with both steel wiresand hard rubber materials, this makes the lower side walls relatively stiff.The upper side walls are on the other hand quite flexible, since the ply layerthere is simple and there is less steel in there. The central area consistsof the belt and the tread. The belt consists of a rubber embedded steellining in the circumferential direction to give support and rigidity. Thetread is an about 13 mm thick rubber layer which is there to provide thegrip. This makes the central area rigid with respect to bending waves in thecircumferential direction but fairly flexible when it comes to motion withinthe cross-section. The high loss factor of the tread rubber makes the lattermotion highly damped.

The tyre studied here is a Goodyear, radial, passenger car tyre, withthe dimensions 205/55ZR16, mounted on an Argos rim. The tyre is ’slick’,i.e. it does not have a tread pattern or groves, but in all other aspects hasproperties typical of a production tyre.

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Figure 1: The tyre consists of three major sub regions. Upper side wall,lower side wall and the central area.

To make use of the rotational symmetry of the tyre a waveguide finiteelement approach is employed, where only the cross-section is discretised,and hence the calculation time is reduced.

1.3 Waveguide finite elements

A waveguide is a wide-ranging term for a device, which constrains or guidesthe propagation of mechanical waves along the waveguide. Here it is alsoassumed that a waveguide has constant geometrical and material propertiesalong one direction.

Waveguide Finite Elements (FE) yield equations of motion for systemswith wave-propagation along a single direction, in which the structure isuniform. It is then possible to separate the solution to the wave equationinto one part depending on the cross-section, one part depending on thecoordinate along the waveguide and one part depending on time.

As an example of a waveguide, a generalised beam, in which longitudinal,torsional, shearing and flexural waves can travel, can be considered. Themain idea with a waveguide approach is to study waves propagating in thestructure.

The most important benefit with waveguide FE, is that it decreasesthe calculation time compared to ordinary finite elements since only thecross-section has to be discretised and the number of degrees of freedom isreduced. Another advantage compared to conventional FE methods, is thatit is straight forward to identify and analyse different wave types, whichallows a physical understanding of the structure under investigation. The

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ability to handle infinite waveguides, is an additional good feature of thismethod.

Forced response solutions for waveguide FE models can be handled inseveral different ways. Four of these methods for forced responses will bebriefly explained.

For infinite waveguides an approach based on Fourier transforms maybe used. The equations of motion are transformed to the wave number do-main through a spatial Fourier transform. The solution in the wave numberdomain then has to be transformed back to the spatial domain through aninverse Fourier transform which generally involves residue calculus [[5], [6]].

’Super Spectral Elements’, (SSE), are derived by using wave solutions,given from a generalised eigenvalue problem, as test and shape functionsin the variational form of the wave equation [7]. At the ends, the spec-tral elements can be coupled to other spectral elements or to regular finiteelements.

Certain convenient boundary conditions are fullled by a wave solutiontogether with a companion wave travelling in the opposite direction. Forsuch boundary conditions, the eigenmodes of a finite length structure aredirectly identified by a two-dimensional waveguide FE analysis, which hasa much lower computational cost than a conventional three-dimensional FEanalysis. Once the eigenmodes are identified, a modal analysis gives theforced response of the structure. In reference [6] the convenient boundarycondition for a straight waveguide was the shear diaphragm condition, inwhich all motion within the cross section is blocked while all motion alongthe waveguide is free. Another convenient boundary condition is that for acircular structure, such as a car tyre. In this case the response is a periodicfunction of the angular coordinate. The non-proportional damping usedin the present analysis, however, leads to non-orthogonal eigenmodes andtherefore this method is not used.

In an assumed modes procedure the response is assumed to be an expo-nential Fourier series in the spatial domain. This approach is suitable, sincethe tyre is a circular structure and the solutions to the wave equation will beperiodic with respect to the circumferential angle. The sum is inserted intothe variational statement, and upon variation follows the equations of mo-tion. One advantage with this direct methodology in the frequency domainis that it is uncomplicated to handle fluid-structure interactions [8]. Thecar tyre including the air cavity has been modeled successfully by Nilsson[5] with a waveguide FE approach similar to the one presented here. Also,frequency dependant materials are easily included. This is an especiallygood quality when considering a structure such as a car tyre, which is builtfrom rubber, whose material properties show a strong frequency dependency.This is the procedure used in the present analysis.

Straight waveguide finite elements were first formulated by Alaami [9]and Lagasse [10] in 1973. Curved waveguides are used by Hladky-Hennion

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[11] and Nilsson [5]. In reference [5] there is a comprehensive review of theapplications of waveguide FE for vibro-acoustic problems.

The waveguide finite elements used in the tyre model are developed andvalidated in Paper A.

1.4 Tyre model

1.4.1 Existing tyre models.

There are several earlier attempts to describe the dynamic behaviour of thecar tyre. There are many different approaches based on equivalent structuremodelling. Andersson [12] and Larsson et al. [13] used simple orthotropicplate strips and Muggleton et al. [14] used an assembly of ortotropic platestrips. Bohm [15], Kropp [16], Kung et al. [17] and Dohrmann [18] useda circular ring model to capture the dynamic behaviour whearas Molisaniet al. [19] and Kim et al. [20] used circular thin shells. Pinnington andBriscoe [21], chose to use straight beams, Pinnington [22] curved beams andLarsson et al. [23] coupled elastic layers. A finite element model was usedby Kung et al. [17] , Richards [24] and Pietrzyk [25]. Nilsson [5] used awaveguide finite element model built up from pre-stressed conical thin shellelements. This approach is similar to the one presented here. Kim et al. [26]divided the tyre response into different frequency regimes in which diffrentwave types are dominating. They found that below 500 Hz most waves areinefficient sound radiators due to the cancelation effect. In references [19],[24], [5], [27] and [28] the air cavity is also taken into account which is ofmain importance for structure borne noise into the passenger compartment,but is disregarded here.

More recent attempts include the work by Lopez et al. [29] on thevibrations in a deformed rolling tyre using a FE model in a fixed (eulerian)frame, and the work by Brinkmeier et al. [30], where FE methods are usedto simulate tyre noise.

1.4.2 Waveguide FE tyre model.

The Waveguide FE tyre model is based on an Abaqus finite element modelgiven by Goodyear [25], specifying the geometry and most of the elasticdata. The big disadvantage with this procedure is the need to rely on a tyremanufacturer for input data. The advantage is that the model is based ondesign data, therefore communication with the design engineers is possible.It is of very good quality. The model was originally designed for tyre roadnoise predictions.

The motion of the tyre belt and side wall is described with quadraticanisotropic, deep shell elements that include pre-stress and the motion ofthe tread on top of the tyre by quadratic, Lagrange type, isotropic twodimensional elements.

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Two different tyre cross-section meshes have been used. The mediummesh with 42 elements, 113 nodes and 516 degrees of freedom (DOF) is seenin Fig. 2 and an enriched Abaqus full mesh with 214 elements, 569 nodesand 2562 DOF in Fig. 3. Assuming that the same nodal density is used inthe circumferential direction, the full mesh is equivalent to a conventionalFE model with a total of 2562 × 1300 ≈ 3.3 106 DOF. Based on numericalexperiments involving forced response and dispersion curves calculations, itwas concluded that the medium mesh is sufficient up to at least 1000 Hz.

The Abaqus model does not include any damping information and arather simplistic damping definition is used in most of the work presentedhere. It is based on an ad hoc curve fitting procedure against measuredmobilities.

The model accounts for: the curvature, the geometry of the cross-section,the pre-stress due to inflation pressure, the anisotropic material propertiesand the rigid body properties of the rim.

The tyre model is described and validated in Paper B.

2 Analytical modelling of rolling resistance

The analytical rolling resistance investigations available in the literature areall based on rather simple equivalent structures. Stutts and Soedel [31] useda tension band on a viscoelastic foundation. Kim and Savkoor [32] used anelastic ring supported on a viscoelastic foundation. Yam et al [33] basedtheir calculation on experimentally deduced modal parameters. Popov etal [34] modeled a truck tyre, based on the model developed by Kim andSavkoor [32]. The stiffness and damping parameters needed, came from anexperimental modal analysis.

The model used in this study, has the correct tyre geometry and stiffness,and it includes the road texture effect. None of the models above are treatinga rough road even though the road texture and roughness have a significanteffect on the rolling resistance [35].

2.1 Rolling resistance estimation based on dissipated power

The approach used throughout this work is to estimate the rolling resistancefrom the power dissipation in the tyre due to internal viscoelastic forces.

The contact forces are calculated by Chalmers University of Technology(CTH), based on a non-linear contact model in which the tyre structureis described by its flexibility matrix [36]. Topographies of the surface arescanned, the tread pattern is accounted for, and then the tyre is ’rolled’over it. The calculations are made in the time domain and is based on aLagrange multipliers approach. The non-linear conditions used are: i) thetyre cannot indent into the road, ii) if a point is not in contact the force

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is zero and iii) that the force cannot be negative (road pulling tyre down)([37] chapter 6). Only forces acting normal to the road are considered.

Newton’s law applies for a fixed piece of matter: a particle. The equa-tions of motion are therefore solved in a Lagrangian coordinate system fixedto the rotating tyre. In doing so, the Corioli forces are neglected. The cen-trifugal force is also neglected and it is assumed that the increased statictension and radial expansion that it induces are already included in thedefinition of the tyre’s steady state. Upon this basis, the tyre vibration ispredicted.

Energy is always conserved, it is just transformed from one form to an-other. The energy going into the system must therefore equal the dissipatedenergy in the system. The same argument is valid for the power, conse-quently, the injected power must equal the dissipated power.

Since only normal forces are considered, the dissipated power stems fromdamping alone.

In Paper C the dissipated power procedure is developed and the resultsare shown to be in the right range compared to measurements. It is alsoshown that a rough road dissipated more power than a smooth road. PaperD contains a demonstration of the procedure on real existing motor waysfor which measured rolling resistance data are available. The results arepromising and if the roads are different enough a ranking is possible.

To clarify the rolling resistance - power dissipation equivalence a simpleexample can be considered. To keep a car running at a constant speed, inthe presence of rolling resistance forces, requires that energy is added. Thisadded energy must be equal or greater than the energy dissipated in thetyres. The equal sign applies if there are no other dissipation mechanisms,such as friction in the tyre rim interface or losses in the suspension system,than the internal dissipation in the tyre.

2.2 Rolling resistance - dynamic method

One could argue that once the contact forces are known the rolling resistancecan be estimated from those alone. The power needed to keep the tyre rollingat a given angular velocity Ω subject to a resisting moment M is given by

P = MΩ = Fd Ω (3)

, where F is the contact force resultant and d is the centre of contact forces,see Figure 4. To produce the typical power 800 W, at a speed of 80 km/h anda vertical load corresponding to 300 kg, d should be around 3 mm, but thespatial force resolution is only 4 mm meaning that the dynamic method isunreliable, and numerical experiments shows that it often fails. The energyapproach used in this work is more robust.

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3 Tyre model updating

The tyre model from Paper B, is in Paper E updated to include viscoelasticproperties making it more realistic. This is made in a modular way withinverse methods. The reason that this work is undertaken is because therolling resistance is depending solely on the damping which therefore hasto be modelled as good as possible. The necessary building blocks in theupdating procedure are: i) Measurements on tread samples, a beam sampleand the whole tyre, ii) equivalent structure modelling, iii) the initial tyremodel and iv) a viscoelastic material model. The tyre is divided into threesubstructures, tread, belt and sidewall. Each measurement result providesinput for each substructure.

The updating procedure concerns the stiffness and damping of the wholetyre. A good article on model updating is reference [44]. Relevant articlesregarding inverse methods includes McIntyre and Woodhouse’s [45] work onthe experimental determination of material properties of orthotropic sheetmaterial and the study by Geng et al. [46], on damping in truck tyres, wherea generally distributed viscous damping model is fitted to observations froman experimental modal analysis. The modal techniques, used in this work,can only provide estimations of the loss factor at resonant frequencies. Thisfact can, at least for simple structures, be circumvented by instead making,a more challenging, spatially distributed measurement to estimate the lossfactor from the complex wave order [47], while this is not pursued here.

3.1 Damping - Viscoelastic materials.

Since the rubber in the tyre is highly damped, the elastic properties alone,are not sufficient to accurately model tyre dynamics. The viscoelastic prop-erties are needed too. A good damping description is also crucial for rollingresistance estimations. The tyre model used in Paper B, C and D has a verysimplistic damping model which is updated in Paper E.

Damping can be introduced in many ways. For structure borne soundapplications the linear structural damping, with a frequency independentlossfactor, is widely used. This model is, however, physically unrealisticsince it leads to non-casual behavior [38]. The standard linear model, wherethe response depends on a convolution between the response history andrelaxation functions, first introduced by Boltzman [39], does not have thisdrawback. However, it turns out that many relaxation functions, and there-for many unknown parameters, are needed to accurately describe the fre-quency dependence of rubber materials over a wide frequency range [40].Another approach is the fractional derivative model. A five parameter frac-tional derivative model in the frequency domain is suggested by [40]. It isargued that the reason that only five parameters are needed is that theseparameters are more consistent with the physical principles involved [40].

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Rubber is a visco-elastic material exhibiting frequency dependant mate-rial characteristics. A rather simple mathematical model describing thevisco-elastic behaviour of rubber, valid for a harmonic time dependencee−iωt, is the fractional Kelvin-Voigt model described in for example [41].This model uses three fractional derivative material model (FDMM) pa-rameters to describe the frequency dependence.

D(ω) = D0(1 + (−iω

ω0

)α). (4)

,where ω is the angular frequency, D0 is a characteristic static stiffness (stiff-ness at zero frequency), and ω0 and α describes the frequency dependance.ω0 has dimension [rad/s] and α is dimensionless. Note that the Fouriertransform of the fractional derivative of order α of x(t) is (−iω)α times theFourier transform of x(t) [40], which is the reason for using the term ’frac-tal’. When ω0 tends to infinity the model becomes purely elastic and whenα equals unity the model reduces to the classic Kelvin-Voigt material model.

The lossfactor is defined by:

η = −

Im(D)

Re(D). (5)

The lossfactor at zero frequency is zero as it should be, but this simplemodel does not lead to a vanishing loss factor when the frequency goes toinfinity. This is, however, not a problem since the focus of this investigationis up to 1000 Hz where this simple model is deemed sufficient.

The fractional Kelvin - Voigt model is a simplification compared to thefive parameter model. The reason that the three parameter model is used,is exclusively to reduce the number of parameters and thereby simplifyingthe analysis and making the damping estimation more robust. It should beemphasized that any fitting function is possible, as long as it depicts theobserved material behaviour sufficiently accurately within the consideredfrequency domain, [42].

The tread of the tyre is made of homogenous, isotropic, almost incom-pressible rubber and its stiffness properties are thus defined by the shearmodulus alone. Results from a dynamic shear modulus measurement ontread samples is in Paper E used to determine the FDMM parameters ofthe tread.

3.2 Equivalent structure modelling

The elastic properties of the deep shell elements used in the waveguide FEmodel are given by Goodyear. There are in total 29 shell elements in thetyre model (see Figure 2) of which 15 are unique The local stiffness matrixof each shell element is an 8 by 8 matrix , D, with in total 21 independent

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elements (23 if the the transverse shear stiffnesses are different to the inplaneshear stiffness).

Damping is added by making the stiffness matrices complex . A completedescription of the damping thus requires 21 · 15 = 315 lossfactors, which,furthermore, are frequency dependent. This is not feasible and therefor somekind of parametrisation has to be made.

The idea is to model each shell element by an equivalent structure madeup of rubber embedded steel wires. It is inspired by reference [43], wherePinnington studies wave transmission in flexible tubes reinforced by steelwires.

The equivalent structure depends on a number of geometrical parameterssuch as layer thicknesses, angle of wires, steel to rubber ratios. It alsodepends on two elastic parameters: the shear modulus of rubber and theYoung’s modulus of steel, which facilitates the modelling.

The equivalent structure is first determined for the static case. The shellelements can be divided into two groups based on the angle of the wires:belt and sidewall elements.

Viscoelastic properties can be included by instead of using static stiffnessvalues making use of Equation (4), where the static stiffness is known andα and ω0 are to be determined.

Measurements of the eigenfrequencies and corresponding loss factors ofa beam sample cut out from the the tyre belt with tread on top of it areused to determine the FDMM parameters of the belt. Modal parametersfrom the experimental modal analysis on the complete tyre made earlier,are used to determine the FDMM parameters of the side wall.

This work was very successful and the updated tyre model has a morerealistic damping behavior, which also agrees better with measurements.

4 Summary of the papers

4.1 Paper A. Waveguide finite elements for curved struc-

tures

A Waveguide finite elements formulation for curved structures is derived.The formulation is valid for structures having constant properties along oneaxis, such as infinite straight beams, cylinders or car tyres.

The formulation is based on a modified Hamilton’s principle valid forgeneral viscoelastic motion. This is proved rigorously mathematically. Thismeans that frequency dependent losses in the structure can be easily takeninto account also at the theoretical level.

Curved and straight isoparametric solid and deep shell elements are de-veloped.

To validate the developed elements a finite length cylinder with sheardiaphragm end conditions is considered. This example structure is chosen

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because all four elements can be used to model the vibration field. Anothernice property of this structure is that the exact solution is known fromliterature. The calculated results compares well with those from literature.

For very thin shells, the low order modes can have eigenfrequencies be-low higher order modes. This means that to accurately resolve them, asubstantially higher nodal density is needed than one would initially expect.

4.2 Paper B. A waveguide finite element model of a pneu-

matic tyre

A waveguide finite elements model based on design data is used to describethe dynamic properties of a passenger car tyre. The response of the tyrebelt and side wall is described with quadratic anisotropic, deep shell elementsthat include pre-stress and the motion of the tread on top of the tyre byquadratic, Lagrange type, isotropic two dimensional elements. These arethe curved elements derived in paper A.

To validate the tyre model, mobility measurements and an experimentalmodal analysis have been made. The point mobility prediction agrees verywell with measurements up to roughly 250 Hz for the radial point mobilities,see Figures 5 and 6 for excitation in the middle of the tread. The eigenfre-quency prediction are within five percent for the identified modes, exceptfor the axial semi rigid body mode (error 12 %), the anti-symmetric mode oforder two (error 10 %) and the anti-symmetric mode of order seven (error 7%). The predicted damping is overestimated for the symmetric mode familyand underestimated for the anti-symmetric mode family.

The ’cut-on’ frequency, of the belt bending modes, is the lowest fre-quency at which the corresponding waves are propagated. It appears atlower frequencies in the prediction than in the measurement. This is per-haps due to aging of the tyre since comparable measurements performed inthe spring of 2001 is in agreement with the calculation [12]. In the range500 - 1000 Hz there is an error ranging from 1.5 dB up to 3.5 dB for thesquared amplitude of the point mobility. For the transfer mobilities, theerror is larger since they are more sensitive to the exact position of theaccelerometer, particularly so for the anti-resonances, see Figure 7.

The non-proportional damping is found with an ad hoc curve fitting pro-cedure based on the measured mobilities. The applied damping is structuraland frequency independent in the modal frequency region (below 250 Hz).For higher frequencies losses proportional to the mass matrix, increasinglinearly with frequency, are added to the mix, since the apparent dampinglevel is higher in this frequency region.

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4.3 Paper C. Power dissipation in car tyres

The tyre model described in Paper B is used to estimate the power consumedby viscoelastic losses. External forces resulting from a non-linear contactmodel, for two different roads are inserted and the responses are calculated.The dissipated power is then equated to the injected power. The dissipatedpower can also be studied element wise, making it possible to pinpoint wherethe dissipation occurs.

The contact force predictions are made by the division of applied acous-tics, Chalmers University of Technology (CTH) as described in reference[36]. It is based on a non-linear contact model in which the response of thetyre is described with its flexibility matrix. Topographies of the surface arescanned, the tread pattern is accounted for, and then the tyre is ’rolled’ overit in the time domain. The nonlinear conditions used are: i) the tyre cannotindent into the road, ii) if a point is not in contact the force is zero and iii)the force cannot be negative (road pulling tyre down). Only forces actingnormal to the road is considered.

The contact forces are used to calculate the response of the tyre. Apower spectral density comparison of the measured and predicted accelera-tion can be seen in Figure 8. The measured acceleration signal comes froman accelerometer placed in a groove of a tyre rotating on a tyre test drum.The low frequency peak is underestimated by around 3 dB and at higherfrequencies the curves diverge even more. The curves do bear similaritiesand the result is promising. One possible explanations for the discrepanciesis that the measurements are taken for a ’warm’ tyre since the tyre heatsup during the measurements and the calculation is made for a ’cold’ tyremodel. This temperature increase could change the viscoelastic propertiesof the rubber compounds.

When the force and motion are known the injected power can be calcu-lated. The predicted power dissipation is underestimated but has the sameorder of magnitude compared to literature [4] and measurements. The powerdissipation is larger on the rough road than on the smooth road, this show-ing the influence of the road on the rolling resistance. To the best of theauthor’s knowledge, this influence is neglected in all previous calculations ofrolling resistance.

The dissipated power for a slick tyre running on a smooth road (ISOroad), as a function of frequency and wave order can be seen in Figure 9and 10 respectively. The reason that the frequency spectrum looks so roughis that two revolutions have been used for the calculation. If the contactforces were truly periodic every other frequency component would cancelout. A significant part of the dissipation occurs below 100 Hz and at a waveorder around 3. The fact that the dissipation is low frequency and that theresponse is underestimated for those frequencies, is probably the reason whythe power dissipation is underestimated. Another explanation could be that

12

friction forces are omitted in the contact force calculation.By studying the power dissipated within the elements it can be con-

cluded that most of the power dissipation occurs in the tread and in thebelt elements below the tread, see Figures 11 and 12. Roughly. 15 % ofthe losses occur in the side wall which is in conflict with reference [3], whosays that roughly 30 % of the total dissipated power appears in the upperand lower side wall. The overall damping level in the model is estimatedquite accurately (see Paper B), but the distribution of the damping, in thedifferent parts of the tyre, is probably wrong. Since the viscoelastic dataare very important for rolling resistance predictions, the damping shouldbe established in a more scientific way. Furthermore, rolling resistance is alow frequency phenomena meaning that the use of a constant structural lossdefinition cannot be justified.

4.4 Paper D. The influence of the road on rolling resistance

The calculation procedure developed in Paper C is used to predict the rollingresistance of real existing roads. The investigated roads were used in around robin test for rolling resistance measurements in the autumn of 2004.Six of the roads were re-visited in 2007 when texture measurements weremade. The measured rolling resistance results spreads a lot, depending onmeasurements system and the tyres used, but the ranking of the roads isconsistent. The predicted values by the procedure developed in Paper C arelow when compared to the measured values. Also the predicted differencesbetween the roads are smaller in the calculation. Nevertheless, if the roadsare different enough, a ranking based on calculations is possible.

4.5 Paper E. Estimates of the viscoelastic properties of car

tyres.

The tyre model introduced in Paper B is updated. As mentioned in PaperC, the viscoelastic properties of the tyre are very important when rollingresistance is of interest. In this work the fractional Kelvin - Voigt viscoelasticmaterial model is put to use. The updating procedure is made in a modularway.

The tread of the tyre, for which solid elements are used, is consideredisotropic meaning that only two stiffness parameters are needed. Poisson’sratio and the dynamic shear modulus. Rubber is near incompressible, mean-ing that Poisson’s ratio is close to 0.5. A measurements of the dynamic shearmodulus have been made in the interesting frequency region. The parame-ters in the fractional derivative model are then fitted to the measurements ina least square sense. Comparisons of the measured and simulated dynamicshear modulus and lossfactor are in Figures 13 and 14, showing a very goodagreement.

13

The belt and sidewall, for which shell elements are used, are more com-plex and needs many stiffness parameters to be described. Each shell elementis replaced by an equivalent structure defined by two stiffness parametersalong with a number of geometry parameters. The reduction of stiffnessparameters by a factor 10 is the motivation for this work.

The equivalent structure consists of two rubber layers sandwiching alayer with rubber embedded steel wires. The geometry parameters are; theangle of the steel wires, the thickness of the layer with steel wires, the totalthickness of the element, the distance from the origin to the edge of theelement and the position of the layer with wires within the element.

As a first step the equivalent structure is fitted for the static case. Itturns out that the shell elements can be arranged in two subgroups basedon the angle of the steel wire: belt elements and sidewall elements. For thebelt elements the angle is around 21 degrees and for the sidewall elementsthe angle is 90 degrees.

Each shell element is now described with two stiffness parameters, cor-responding to steel and rubber respectively. The stiffness parameters aremade frequency dependent according to the fractional Kelvin - Voigt ma-terial model. The static stiffness’ are considered known. The FDMM pa-rameters of the steel are set to give a lossfactor of around 0.1 %, while onlymarginally effecting the real part and are therafter considered to be known.The rubber FDMM parameters are determined based on results from mea-surements. Two sets of FDMM parameters for the belt and sidewall areused leading to a total of four unknown parameters.

To determine the FDMM parameters of the belt, measurements on abeam specimen cut out from the tyre were made. The beam consisted ofthe tyre belt with the tread on top of it. Modal parameters for the three firstbending modes were extracted. The tread properties are known from thetread measurements. The beam is modelled with regular 3D finite elements.The geometry is specified by a submesh of the tyremodel mesh. In anoptimisation routine the modal damping of the measured and calculatedmodes are compared in a least square sense.

Result from the experimental modal analysis described in Paper B pro-vided information to find the FDMM parameters for the sidewall. An opimi-sation routine compares the calculated and measured eigenfrequencies andlossfactors.

The updated tyre model remedies some of the problem with the origi-nal tyremodel. The damping estimation is better, see Figures 15 and 16.Furthermore, the damping at low frequencies is modelled in a more realisticway.

The measured and simulated point mobility show excellent agreement inthe modal frequency region, see Figures 17 and 18.

14

5 Future Work

The most obvious next step is to use the updated tyre model for rollingresistance predictions. Unfortunately there was no time to finalise this study.

An investigation of the influence of certain tyre parameters would beinteresting. It would be possible to change the speed, the load on the tyreand perhaps also to model wear of the tyre.

The contact modelling can be improved to include friction forces andwith faster and more memory empowered computers the spatial resolutioncan be increased. Material non-linearities effects may also be included.

A statistical study, similar to the demonstration in Paper D, on thecorrelation between road texture and rolling resistance is possible and wouldbe very informative.

6 Conclusion

A car tyre is modelled with waveguide finite elements. The model is em-ployed to calculate the power dissipation as the tyre is rolling on a roughroad showing promising agreement with measurements. The road rough-ness is seen to have a significant effect on the dissipated power, which isneglected in all previous works. It is concluded that rolling resistance isa low frequency phenomena and the damping properties of the tyre modelhave therefore been updated. This is, to the best of the authors knowledge,the first rolling resistance model, based on physical principles and designdata.

7 Acknowledgements

The early development of the tyre model was funded by the Swedish Re-search Council (621-2002-5661) and the European Commission (G3RD-CT-2000-00097). Many thanks to the members of the Ratin consortium and inparticular to Roger Pinnington, ISVR, for helpful discussion, to WolfgangGnorich and Andrzej Pietrzyk, Goodyear, for advice and for sharing datafor tyres and to Wolfgang Kropp, Patrik Andersson and Frederic Wullens,Applied Acoustics, Chalmers, for advise and calculation of contact forces.The final tyre model and the work presented in this thesis were funded bythe European Commission, ITARI, FP6-PL-0506437. The measurement ofrolling resistance was made by Gdansk University of Technology.

I am very thankful to my supervisor Svante Finnveden, for his construc-tive ideas, support and encouragement. He also has a decent taste in music.

Many thanks to the EDSVS and especially Professor Bjorn Petersson for

15

making my stay at Technische Universitat Berlin possible. I would also liketo thank to Ulf Carlsson, Kent Lindgren and Danilo Prelevic for assistingme with measurements. Special thanks to Carl-Magnus Nilsson.

Finally I would like to thank my family, my friend and the people at MWL.

References

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[2] Passenger car, truck, bus and motorcycle tyres - methods of measuringrolling resistance. ISO 18164, 2005.

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16

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[21] R.J. Pinnington and A.R. Briscoe. A wave model for a pneumatic tyrebelt. Journal of Sound and Vibration, 253:941–959, 2002.

[22] R.J. Pinnington. A wave model of a circular tyre. part 1: belt modeling.Journal of Sound and Vibration, 290:101–132, 2006.

[23] K. Larsson and W. Kropp. A high-frequency three-dimensional tyremodel based on two coupled elastic layers. Journal of Sound and Vi-bration, 253:889–908, 2002.

[24] T.L. Richards. Finite elements analysis of structural-acoustic couplingin tyres. Journal of Sound and Vibration, 149:235–243, 1991.

[25] A. Pietrzyk. Prediction of the dynamic response of a tire. In Proc ofInterNoise, The Hague, pages 2547–2550, 2001.

[26] B.S. Kim, G.J. Kim, and T.K. Lee. The identification of sound gener-ating mechanisms of tyres. Applied Acoustics, 68:114–133, 2007.

[27] R.J. Pinnington. Radial force transmission to the hub from an unloadedstationary tyre. Journal of Sound and Vibration, 253:941–959, 2002.

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[28] S. Finnveden, C-M. Nilsson, and M. Fraggstedt. Waveguide fea of thevibration of rolling car tyres. In Proc Novem, Saint Raphae, 2005.

[29] I. Lopez, R.E.A. Blom, N.B. Roozen, and H. Nijmeijer. Modellingvibrations on deformed rolling tyres - a modal approach. Journal ofSound and Vibration, 307:481–494, 2007.

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[31] D.S. Stutts and W. Soedel. A simplified dynamic model of the effect ofinternal damping on the rolling resistance in pneumatic tires. Journalof Sound and Vibration, 155(1):153–164, 1992.

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[35] R.B.J. Hoogvelt, R.M.M. Hogt, M.T.M. Meyer, and E. Kuiper. Rollingresistance of passenger car and heavy vehicle tyres a literature survey.TNO report, 2001.

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[41] M. Sjoberg. Rubber isolators - measurements and modelling using frac-tional derivatives and friction. SAE paper No 2000-01-3514, 2000.

[42] L. Kari. On the waveguide modelling of dynamic stiffness of cylindri-cal vibration isolators. part 1: The model, solution and experimentalcomparison. Journal of Sound and Vibration, 244(2):211–233, 2001.

[43] R.J. Pinnington. The axisymmetric wave transmission properties ofpressurized flexible tubes. Journal of Sound and Vibration, 204:271–289, 1997.

[44] S. Zivanovic, A. Pavic, and P. Reynolds. Finite element modelling andupdating of a lively footbridge: The complete process. Journal of Soundand Vibration, 301:126–145, 2007.

[45] M.E. McIntyre and J. Woodhouse. On measuring the elastic and damp-ing constants of orthotropic sheet materials. Acta metall., 36:1397–1416, 1988.

[46] Z. Geng, A.A. Popov, and D.J. Cole. Measurement, identification andmodelling of damping in pneumatic tyres. International Journal ofMechanical Sciences, 49:1077–1094, 2007.

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19

−0.1 −0.05 0 0.05 0.1

0.18

0.2

0.22

0.24

0.26

0.28

0.3

0.32

0.34

x [m]

r [m

]

Figure 2: Medium mesh.

−0.1 −0.05 0 0.05 0.1

0.18

0.2

0.22

0.24

0.26

0.28

0.3

0.32

0.34

x [m]

r [m

]

Figure 3: Enriched mesh.

20

Figure 4: The dissipated power can also be estimated from the forces di-rectly. This approach is not as stable as the energy approach and requiresa much finer spatial resolution.

101

102

103

−70

−65

−60

−55

−50

−45

−40

−35

Frequency [Hz]

Mag

nitu

de o

f poi

nt m

obili

ty d

B r

el 1

(m

/Ns)

2

Figure 5: Magnitude of point mobility for excitation in the middle position.Measured (solid) and calculated (dashed).

21

101

102

103

−2

−1.5

−1

−0.5

0

0.5

Frequency (Hz)

Pha

se o

f poi

nt m

obili

ty (

rad)

Figure 6: Phase of point mobility for excitation in the middle position.Measured (solid) and calculated (dashed).

101

102

103

−100

−90

−80

−70

−60

−50

−40

Frequency [Hz]

Mag

nitu

de o

f tra

nsfe

r m

obili

ty d

B r

el 1

(m

/Ns)

2

Figure 7: Magnitude of transfer mobility for excitation in the middle posi-tion. The response is measured 23.5 cm avay in the circumferewntial direc-tion and 4.3 cm above the geometric centre. Measured (solid) and calculated(dashed).

22

0 200 400 600 800 1000 1200 1400 1600 1800 2000−5

0

5

10

15

20

25

30

35

40

Frequency [Hz]

Pow

er s

pect

ral d

ensi

ty o

f acc

eler

atio

n dB

[rel

1 (

m/s

2 )2 /Hz]

Figure 8: Power spectral density of the Acceleration signal for the rotatingaccelerometer, measured; solid, simulated; dashed. The measured curve isan average over 320 revolutions and the calculated curve is a spatial averagearound the tyre.

0 50 100 150 200 250 300 350 400 450 5000

5

10

15

20

25

30

35

40

Frequency [Hz]

Pow

er [W

]

Figure 9: Dissipated power as a function of frequency. The bandwidth is5.6 Hz. Most of the dissipation occurs below 100 Hz.

23

0 5 10 15 20 25 300

10

20

30

40

50

60

Waveorder [−]

Pow

er [W

]

Figure 10: Dissipated power as a function of wave order. A substantial partof the dissipated power occur at a wave order of around 3.

0 5 10 15 20 25 300

5

10

15

20

25

30

Element number

Pow

er [W

]

Figure 11: Power dissipation in the different elements. Belt elements (solid),Tread elements (dashed). Most of the dissipation occurs in the tread.

24

−0.1 −0.05 0 0.05 0.1

0.2

0.22

0.24

0.26

0.28

0.3

0.32

X (m)

Y (

m)

Figure 12: The ellipses and circles indicates elements where a lot of poweris consumed.

0 200 400 600 800 1000 12007

7.5

8

8.5

9

9.5

10

10.5

11

11.5

12x 10

6

Frequency [Hz]

Re(

G)

[Pa]

Figure 13: Real part of dynamic shear modulus. Measured (solid) andcalculated with equation (4) (dashed)

25

0 200 400 600 800 1000 12000.18

0.2

0.22

0.24

0.26

0.28

0.3

0.32

0.34

0.36

0.38

Frequency [Hz]

Loss

fact

or

Figure 14: Measured (solid) and calculated with equation (4) (dashed) loss-factor.

3 4 50

0.5

1

1.5

2

2.5

3

3.5

4

Wave order [−]

Dam

ping

[Per

cent

of c

ritic

al d

ampi

ng]

Figure 15: Measured (X), calculated (Paper B) (Rings) and calculated (up-dated model, Paper E) (Squares) lossfactors for the symmetric mode family.

26

3 4 50

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

5.5

Wave order [−]

Dam

ping

[Per

cent

of c

ritic

al d

ampi

ng]

Figure 16: Measured (X), calculated (Paper B) (Rings) and calculated (up-dated model, Paper E) (Squares) lossfactors for the anti symmetric modefamily.

102

103−65

−60

−55

−50

−45

−40

−35

−30

Frequency [Hz]

Mag

nitu

de o

f poi

nt m

obili

ty d

B r

el 1

(m

/Ns)

2

Figure 17: Magnitude of point mobility for excitation in the middle position.Measured (solid), model in Paper B (dashed) and updated model in PaperE (dash-dotted).

27

101

102

103−2

−1.5

−1

−0.5

0

0.5

Frequency (Hz)

Pha

se o

f poi

nt m

obili

ty (

rad)

Figure 18: Phase of point mobility for excitation in the middle position.Measured (solid), model in Paper B (dashed) and updated model in PaperE (dash-dotted).

28