Power Dissipation in Car Tyres

Martin Fraggstedt

Stockholm 2006

Licentiate ThesisTRITA-AVE 2006:26

ISSN 1651-7660

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 ContactRoyal Institute of Technology Teknikringen 8 Tel: +46 8 790 8015MWL / 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 traveling at constant speed, isdue to engine ineffiency, 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 modeled 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.

The tyre model accounts for: the curvature, the geometry of thecross-section, the pre-stress due to inflation pressure, the anisotropicmaterial properties and the rigid body properties of the rim. The modelis based on design data. The motion of the tyre belt and side wall isdescribed with quadratic anisotropic, deep shell elements that includespre-stress and the motion of the tread on top of the tyre by quadratic,Lagrange type, homogenous, isotropic two dimensional elements.

To validate the tyre model, mobility measurements and an exper-imental modal analysis has 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. Above 500 Hz there is an error ranging from 1.5 dBup to 3.5 dB for the squared amplitude of the point mobility.

The non proportional damping used in the model is based on an adhoc curve fitting procedure against measured mobilities.

The contact force predictions, made by the division of appliedacoustics, Chalmers University of Technology, are based on a non-linearcontact model in which the tyre structure is described by its flexibilitymatrix. Topographies of the surface are scanned, the tread pattern isaccounted for, and then the tyre is ’rolled’ over it. The contact forcesare inserted into the tyre model and the response is calculated. Thedissipated power is then calculated through the injected power and thepower dissipated within each element. Results are promising comparedto literature and measurements.

Licentiate Thesis

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

Paper AFraggstedt M. and Finnveden S. A Waveguide Finite Element Model Of APneumatic Tyre, 2006. To be submitted.

Paper BFraggstedt M. and Finnveden S. Power dissipation in car tyres, 2006. To besubmitted.

Contribution from the author of this thesis

Paper AExperimental modal analysis and the mobility measurements. Performedsimulations. Fine tuning of the model developed by the supervisor. Writingthe paper.

Paper BPerformed the power calculations. Litterature study on rolling resistance.Writing the paper.

The material from this thesis has been presented at five workshops in theITARI project plus at three conferences:

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

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

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

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Contents

1 Introduction 11.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Car tyres . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.3 Waveguide finite elements . . . . . . . . . . . . . . . . . . . . 3

2 Summary of the papers 52.1 A waveguide finite element model of a pneumatic tyre . . . . 52.2 Power dissipation in car tyres . . . . . . . . . . . . . . . . . . 6

3 Future Work 7

4 Conclusion 8

5 Acknowledgements 8

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 traveling at constant speed, is due to en-gine ineffiency, internal friction, and the energy needed to overcome resistingforces such as aerodynamic drag and rolling resistance, which is the topic ofthis thesis.

The rolling resistance Fr is defined as the energy consumed per unitof distance traveled [2]. The unit is Nm/m = N which is equivalent toa drag 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].

Normally the rolling resistance is given as a dimensionless constant timesthe gravity force,

Fr = Cr m g, (1)

1

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.

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. The model was originally designed for tyre road noisepredictions.

The analytical investigations available in the literature are all based onrather simple equivalent structures. Stutts and Soedel [5] used a tensionband on a viscoelastic foundation. Kim and Savkoor [6] used an elastic ringsupported on a viscoelastic foundation. Yam et al [7] based there calculationon experimental modal parameters. Popov et al [8] modeled a truck tyre,based on the model developed by Kim and Savkoor [6]. The stiffness anddamping parameters needed, came from an experimental modal analysis.

The model used in this study has the correct geometry and stiffness pa-rameters as it is based on design data provided by the tyre manufacturerGoodyear. None of the models above are treating a rough road even thoughthe road texture and roughness have a significant effect on the rolling resis-tance [9].

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.The three major sub regions of the tyre are the upper side wall, the lowerside wall and the central area. The ply is a layer of embedded fabric inthe rubber. At the lower side walls the ply encloses a volume filled withboth steel wires and hard rubber materials, this makes the lower side wallareas relatively stiff. The upper side wall areas are on the other hand quiteflexible, since the ply layer there is simple and there is less steel in there.The central area consists of the belt and the tread. The belt consists of arubber embedded steel lining (breakers) in the circumferential direction togive support and rigidity. The tread is an about 13 mm thick rubber layerwhich is there to provide the grip. This makes the central area rigid with

2

respect to bending waves in the circumferential direction but fairly flexiblewhen it comes to motion within the cross-section. The high loss factor ofthe tread rubber makes the latter motion 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.

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. The model accounts for: thecurvature, the geometry of the cross-section, the pre-stress due to inflationpressure, the anisotropic material properties and the rigid body propertiesof the rim.

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 FE yield equations of motion for systems with wave-propagationalong a single direction in which the structure is uniform. It is then possible

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to separate the solution to the wave equation into one part depending onthe cross-section, one part depending on the coordinate along the waveguideand 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 decreases thecalculation time compared to ordinary finite elements since only the cross-section has to be discretised and the number of degrees of freedom is re-duced. Another advantage compared to conventional FE methods is that itis straight forward to identify and analyse different wave types, which allowsa physical understanding of the structure under investigation. The abilityto handle infinite waveguides is an additional good feature of this method.

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 [10].

’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 [11]. At the ends, the spec-tral elements can be coupled to other spectral elements or to regular finiteelements.

A modal solution is suited for a structure with rotational symmetry, suchas a car tyre. The response is assumed to be a sum of the waves (eigen-vectors) with real integer wave numbers, resulting from a twin parametereigenvalue problem. The amplitude of these waves are then treated as un-knowns in the strong form of the wave equation. The wave equation is thenmultiplied with one specific eigenvector and the result is integrated over thelength of the waveguide. The orthogonality between the eigenvectors, overthe length of the waveguide, filters out the coefficients corresponding to theeigenvector. The non-proportional damping used in the present analysis,however, leads to non-orthogonal eigenvectors and therefore this method isnot 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. The advantage with this direct methodology in the frequency domain

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is that is uncomplicated to handle fluid-structure interactions. The car tyreincluding the air cavity has been modeled successfully by Nilsson [10] with awaveguide FE approach similar to the one presented here. Also, frequencydependant materials are easily included. This is an especially good qualitywhen considering a structure such as a car tyre, which is built from rubber,whose material properties show a strong frequency dependency. This is theprocedure used in the present analysis.

Straight waveguide finite elements were first formulated by Alaami [12]and Lagasse [13] in 1973. Curved waveguides are used by Hladky-Hennion[14] and Nilsson [10]. The elements in the present tyre model are describedin [15]. In reference [10] there is a comprehensive review of the applicationsof waveguide FE for vibro-acoustic problems.

2 Summary of the papers

2.1 A waveguide finite element model of a pneumatic 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.

To validate the tyre model, mobility measurements and an experimen-tal modal analysis has been made. The calculations agrees very well withmeasurements up to roughly 250 Hz for the radial point mobilities, see Fig-ures 2 and 3 for excitation in the middle of the tread. The eigenfrequencyprediction are within five percent for the identified modes, except for theaxial semi rigid body mode (error 12 %), the anti-symmetric mode of ordertwo (error 10 %) and the anti-symmetric mode of order seven (error 7 %).The estimated damping, especially for the anti-symmetric modes, is a bittoo low.

The ’cut-on’ frequency, of the belt bending modes, is the lowest frequencyat which the corresponding waves are propagated. It comes earlier in theprediction than in the measurement. This is perhaps due to ageing of thetyre since comparable measurements performed in the spring of 2001 [16] isin agreement with the calculation. In the range 500 -1000 Hz there is anerror ranging from 1.5 dB up to 3.5 dB for the squared amplitude of thepoint mobility. For the transfer mobilities, the error is larger since they aremore sensitive to the exact position of the accelerometer, particularly so forthe anti-resonances, see Figure 4.

The non proportional damping is found with an ad hoc curve fittingprocedure based on the measured mobilities.

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2.2 Power dissipation in car tyres

The tyre model described in Paper A is used to estimate the power consumedby visco elastic losses. External forces resulting from a non-linear contactmodel, for three different roads are inserted and the responses are calculated.The dissipated power is then equated to the injected power as well as to thesum of the power dissipated within the elements.

The contact force predictions are made by Frederic Wullens of the di-vision of applied acoustics, Chalmers University of Technology (CTH) asdescribed in reference [17]. It is based on a non-linear contact model inwhich the response of the tyre is described with its flexibility matrix . To-pographies of the surface are scanned, the tread pattern is accounted for,and then the tyre is ’rolled’ over it in the time domain. The nonlinear con-ditions used are: i) the tyre cannot indent into the road, ii) if a point isnot in contact the force is zero and iii) the force cannot be negative (roadpulling tyre down). Only forces acting normal to the road is considered.

The contact forces are used to calculate the response of the tyre. Whenthe forces and the motion is known the injected power can be calculated. Thepredicted power dissipation compares favorably with those from literature[4] and with measurements. The power dissipation is larger on the roughroad than on the smooth road, this showing the great influence of the road onthe rolling resistance. To the best of the author’s knowledge, this influenceis neglected in all previous works.

The dissipated power for a test road managed and scanned by Renault,as a function of frequency and wave order can be seen in Figure 5 and 6respectively. The reason that the frequency spectrum looks so rough is thatonly two revolutions have been used for the calculation. If the contact forceswere truly periodic every other frequency component would cancel out. Byusing more non identical revolutions the result would probably look muchsmoother. A significant part of the dissipation occurs below 100 Hz and ata wave order around 3.

By studying the power dissipated within the elements it can be concludedthat there are nearly no losses occurring in the side wall, see Figures 7 and8, which is in conflict with [3] who says that roughly 30 % of the totaldissipated power appears in the upper and lower side wall. The overalldamping level in the model is estimated quite accurately (see Paper A), butthe distribution of the damping, in the different parts of the tyre, is probablywrong. Since the visco elastic data is very important for a rolling resistanceprediction, the damping should be established in a more scientific way, andthis development will be reported at a later stage.

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3 Future Work

Future work consists in fine tuning the tyre model with regards to dampingand to use longer contact forces in the time domain. Based on measure-ments of the dynamic shear modulus a frequency dependant tread will beintroduced. The damping of the belt and side wall will also be estimatedin a more scientific way based on an optimisation routine where the modaldamping ratios will be used as an error criterion.

Longer contact forces will lead to a finer frequency resolution, whichis needed for the accurate evaluation of the power consumed at the tyreresonances in the 100 Hz region. Also, more revolution would lead to abetter and perhaps smoother power spectrum.

An investigation of the influence of certain tyre parameters would alsobe interesting. It would be possible to change the speed, the load on thetyre and perhaps also to model wear of the tyre.

Preliminary tests with a frequency dependant tread have been made andwill briefly be explained. The tyre model presented in paper A is updated toinclude a frequency dependent tread resulting from a dynamic shear modulusmeasurement. The shear modulus data is fitted to a fractional Kelvin- VoigtModel, described in for example [18], which has the following appearance,

G = G0(1 + (iω

ω0)α). (3)

In equation (3) the parameters that are fitted to the measured data is G0,ω0 and α. G0 is equivalent to the static shear modulus parameter, ω0 hasdimension [rad/s] while α is dimensionless. Note that the Fourier transformof the fractional derivative of order α of x(t) is (iω)α times the Fouriertransform of x(t) [19]. The values of the fitted parameters are in Table 1

G0 [Pa] ω0 [rad/s] α

5.25 106 3.84 103 0.40

Table 1: Fitted parameters in fractional Kelvin - Voigt model.

The loss factor is defined as:

η =Im(G)Re(G)

. (4)

The frequency dependence of the tread is such that the loss factor is zero atzero frequency and then increases. The real part and the loss factor of thedynamic shear modulus is seen in Figures 9 and 10.

The damping of the belt is tampered a bit to get a similar agreement withthe point mobility measurement as the original model, see Figures 11 and 12.The rolling resistance calculation is then re-done with the new model but

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with the old contact forces. The contact forces calculation depends on theflexibility matrix of the tyre, so the result should be interpreted with care.For the Renault road the original model gave a total power loss of 805.7Watts whereas the model with frequency dependent tread gives a value of645.0 Watts.

The main part of the power loss occur around 50 Hz where the lossfactor, from the measurements of the tread is much smaller (η = 0.16) thanthe one used in the original model (η = 0.3) the losses are consequentlyreduced. See Figure 13 for the power loss versus frequency for the originaland the new model.

4 Conclusion

A car tyre is modeled with wave guide 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 roughnessis seen to have a significant effect on the dissipated power, which, to the bestof the authors’ knowledge, is neglected in all previous works

5 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 would also like to thank to Ulf Carlsson, Kent Lindgren and DaniloPrelevic for assisting me with the measurements, and Dr Jenny Jerrelind forher suggestions on the outline of this thesis. Special thanks to Carl-MagnusNilsson. Finally i would like to thank my family, my friend and the peopleat MWL.

References

[1] COM 370, White Paper, European transport policy for2010: time to decide, 2001.

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[2] ISO 18164 Passenger car, truck, bus and motorcycle tyres- Methods of measuring rolling resistance, 2005.

[3] Hall D.E. and Moreland J.C. Fundamentals of rolling re-sistance, Rubber Chemistry and Technology 74 (3): 525-539 JUL-AUG, 2001.

[4] Wennerstrm E. Fordonsteknik, 8th edition, in swedish,KTH, 2004.

[5] Stutts D.S. and Soedel W. A Simplified Dynamic Modelof the Effect of Internal Damping on the rolling resistancein pneumatic tires, Journal of Sound and Vibrarion 155(1), 153-164, 1992.

[6] Kim S.-J., and Savkoor A.R. The Contact Problem of In-Plane Rolling of Tires on a Flat Road, Vehicle SystemDynamics Supplement 27, pp. 189-206, 1997.

[7] Yam L.H., Guan D.H., Shang J. and Zhang A.Q. Study ontyre rolling resistance using experimental modal analysis,Int. J. Vehicle Design, Vol. 30, No. 3, pp. 251-262, 2002.

[8] Popov A.A., Cole D.J., Cebon D. and Winkler C.B. En-ergy Loss in Truck Tyres and Suspensions. Vehicle SystemDynamics Supplement 33 , pp. 516-527, 1999.

[9] Hoogvelt R.B.J., Hogt R.M.M., Meyer M.T.M. andKuiper E. Rolling resistance of passenger car andheavy vehicle tyres a literature survey, TNO report01.OR.VD.036.1/RH, December 11th 2001.

[10] Nilsson C.-M. Waveguide finite elements applied on a cartyre. Doctorial thesis. Aeronautical and Vehicle Engineer-ing, KTH 2004.

[11] Birgersson F., Finnveden S. and C.-M. Nilsson. A spectralsuper element for modelling of plate vibration. Part 1:general theory, Journal of Sound and Vibration 287 (2005)297314.

[12] Alaami B. Waves in prismatic guides of arbitrary crosssection. Journal of Applied Mechanics, (December):1067-1071, 1973.

[13] Lagasse P.E. Higher-order finite-element analysis of to-pographic guides supporting elastic surface waves. TheJournal of the Acoustical Society of America Volume53(4):1116-1122, 1973.

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[14] Hladky-Hennion A.-C. Finite element analysis of thepropagation of acoustic waves in waveguides. Journal ofSound and Vibration, 194(2), 119-136, 1996.

[15] Finnveden S., Fraggstedt M. Waveguide finite elementsfor curved structures, TRITA-AVE 2006:38.

[16] Andersson P. High Frequency tyre vibration. Lic. thesis,Chalmers University of Technology, 2002.

[17] Wullens F. Excitation of tyre vibrations due to tyre/roadinteraction, PhD thesis, Applied Acoustics, ChalmersUniversity of Technology, 2004.

[18] Koeller R. C. Applications of fractional calculus to thetheory of viscoelasticity Transactions of the American So-ciety of Mechanical Engineers Journal of Applied Mechan-ics 51, 299-307, 1984.

[19] Bagley R. L. and Torvik P. J. Fractional calculus-A dif-ferent approach to the analysis of viscoelastically dampedstructures, American Institute of Aeronautics and Astro-nautics Journal21, 741-748, 1983.

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Figure 2: Magnitude of point mobility for excitation in the middle position.Measured (solid) and calculated (dashed).

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Figure 4: 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).

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0 50 100 150 200 250 300 350 400 450 5000

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Figure 5: Dissipated power as a function of frequency. The bandwidth is5.6 Hz. Most of the dissipation occurs below 100 Hz.

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Figure 6: Dissipated power as a function of wave order. A substantial partof the dissipated power occur at a wave order of around 3.

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Figure 7: Power dissipation in the different elements. Belt elements (solid),Tread elements (dashed)

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Figure 8: The arrow indicates elements where a lot of power is consumed

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Figure 9: Real part of dynamic shear modulus. Measured (solid) and cal-culated with equation (3) (dashed)

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Figure 10: Measured (solid) and calculated with equation (3) (dashed) loss-factor.

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Figure 11: Magnitude of point mobility for excitation in the middle position.Measured (solid), original model (dashed) and new model with frequencydependent tread (dotted).

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Figure 13: Dissipated power as a function of frequency. The bandwidth is5.6 Hz. Original model (solid) and model with frequency dependent tread(dashed).

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A Waveguide Finite Element Model of a

Pneumatic Tyre

Martin Fraggstedt and Svante Finnveden

The Marcus Wallenberg Laboratory for Sound and Vibration Research (MWL),Royal Institute of Technology, 100 44 Stockholm, Sweden

E-mail: mfragg@kth.se

Abstract

A car tyre is modeled with waveguide finite elements. The model, which is basedon design data, includes: the curvature, the geometry of the cross-section, the pre-stress due to inflation pressure, the anisotropic material properties and the rigidbody properties of the rim. The motion of the tyre belt and side wall is describedwith quadratic anisotropic, deep shell elements that includes pre-stress and themotion of the tread on top of the tyre by quadratic, Lagrange type, isotropic twodimensional elements.

The model agrees very well with measurements up to roughly 250 Hz for thepoint mobilities and above this frequency, the error is in the range 1.5 dB up to 3.5dB for the squared amplitude. The error is larger for the transfer mobilities sincethey are more sensitive, to the exact location of the accelerometer, in particularso for the anti-resonances. The eigenfrequency prediction is within five percent formost of the identified modes resulting from an experimental modal analysis. Theestimated damping is a bit too low especially for the anti-symmetric modes. Thenon proportional damping used in the model is based on an ad hoc curve fittingprocedure using measured mobilities. In an companion paper the model is usedto estimate the rolling resistance with promising results. However, since the viscoelastic data is extremely important for rolling resistance estimations further workconcerning the damping is needed.

1 Introduction

Traffic has been a major noise polluter for over half a century. For higherspeeds it has been found that tyres are the major contributor to traffic noise.Also the vehicle interior noise due to tyres are becoming more important asother noise sources such as engines, exhaust systems and gear boxes are bettermanaged.

The negative effect on the environment has also been highlighted for a numberof years, since traffic is a major source of green house gases. The transport fieldis representing 32 % of the energy consumption and 28 % of the total CO2emissions, where road transports alone stands for 84 % of these figures [1].Furthermore, when tyres are worn small rubber particles are emitted. Theseparticles are highly toxic and adds to the environmental problems.

The energy consumed by a car traveling at constant speed, is due to engine in-effiency, internal friction, and the energy required to overcome resisting forcessuch as aerodynamic drag and rolling resistance. Rolling resistance is ratherimportant when it comes to fuel economy. A rolling resistance reduction of10 % can yield fuel consumption improvements ranging from 0.5 to 1.5 % forpassenger cars and light trucks and 1.5 to 3 % for heavy trucks [2].

In a companion paper [3] the tyre model presented here is used to calculatethe power dissipation in a car tyre, as it is rolling on a rough road, whichdetermines a significant part of the rolling resistance.

Tyres are very important for the dynamics of the car as they provide thegrip needed for cornering, braking and acceleration. They are also to a greatextent involved in the cars handling abilities. Finally it is the tyres and thesuspension system that assures a comfortable ride.

The motivation for an accurate tyre model is the need to understand moreabout the dynamics, tyre noise emissions and power dissipation of car tyres.

1.1 Car tyres

A tyre is made up of many different materials including steel, fabric and ofcourse a number of rubber compounds, see Figure 1. The materials are usedin a variety of ways to achieve different dynamic properties in the tyre subregions. The three major sub regions are the upper side wall, the lower sidewall and the central area. The ply is a layer of fabric that is embedded inrubber. At the lower side walls the ply encloses a volume filled with both steelwires and hard rubber materials, this makes the lower side wall areas quitestiff. On the contrary, the upper side wall areas are quite flexible, since theply layer there is plain and there is less steel in there. The central area ismade up of the belt and the tread. The belt consists of a rubber embeddedsteel lining (breakers) in the circumferential direction to assure support andrigidity. The tread is an about 13 mm thick rubber layer which provides thegrip. This makes the central area rigid with respect to flexural waves in thecircumferential direction but fairly flexible when it comes to motion withinthe cross-section. Since the rubber in the tread is associated with a high lossfactor the latter motion is highly damped.

2

The tyre studied here is a Goodyear, radial, passenger car tyre, with thedimensions 205/55ZR16, mounted on an Argos rim. The tyre is ’slick’, i.e. itdoes not have a tread pattern or groves, but in all other aspects has propertiestypical of a production tyre.

The tyre model used in the present study is based on design data kindlyprovided by the tyre manufacturer Goodyear. The rotational symmetry ofthe tyre is utilized by a waveguide finite element approach, where only thecross-section is discretised, and hence the calculation time is reduced.

The model presented here accounts for: the curvature, the geometry of thecross-section, the pre-stress due to inflation pressure, the anisotropic materialproperties and the rigid body properties of the rim. The motion of the tyrebelt and side wall is described with quadratic anisotropic, deep shell elementsthat includes pre-stress and the motion of the tread on top of the tyre byquadratic, Lagrange type, isotropic two dimensional elements.

Fig. 1. The tyre consists of three major sub regions. Upper side wall, lower side walland the central area. The central area is in turn divided into the belt and the tread.

1.2 Existing dynamic tyre models

There are numerous earlier attempts to describe the dynamic behaviour of thecar tyre. There are a number of different approaches based on equivalent struc-

3

ture modeling. Andersson [4] and Larsson et al. [5] used simple orthotropicplate strips and Muggleton et al. [6] used an assembly of ortotropic plate strips.Bohm [7], Kropp [8], Kung et al. [9] and Dohrmann [10] used a circular ringmodel to capture the dynamic behaviour whearas Molisani et al. [11] and Kimet al. [12] used circular thin shells. Pinnington et al. [13], chose to use straightbeams, Pinnington [14] curved beams and Larsson et al. [15] coupled elasticlayers. A finite element model was used by Kung et al. [9] and Richards [16].Nilsson [17] used a waveguide finite element model built up from pre-stressedconical thin shell elements. This approach is similar to the one presented here.References [11], [16], [17] and [18] also take the air cavity into account which isof main importance for structure borne noise into the passenger compartmentbut is disregarded here.

1.3 Waveguide finite elements

A waveguide is a device which, constrains or guides the propagation of mechan-ical waves along a path defined by the physical construction of the waveguide.Here a waveguide is also assumed to have constant geometrical and materialproperties along one direction.

Waveguide finite elements (FE) yield equations of motion for systems withwave-propagation along a single dimension in which the structure is uniform,the solution to the wave equation is then separable into one part dependingon the cross-section, one part depending on the coordinate along the structureand one part depending on time.

One example of a waveguide is a generalised beam in which longitudinal,torsional, shearing and flexural waves can travel. The essence of a waveguideapproach is to look at waves propagating in the structure.

The main advantage of waveguide FE is that it reduces the calculation burdencompared to standard finite elements since only the cross-section has to bediscretised and the number of degrees of freedom is reduced. Another advan-tage compared to conventional FE methods is that different wave types arereadily identified and can be analysed, allowing for a physical understandingof the investigated structure. The ability to handle infinite waveguides is anadditional good feature of this method.

Forced response solutions for waveguide FE models can be handled in severaldifferent ways. In the present analysis an assumed modes procedure is em-ployed. The response is given by an exponential Fourier series in the spatialdomain. This approach is suitable, since the tyre is a circular structure andthe solutions to the wave equation will be periodic with respect to the circum-ferential angle. The sum is inserted into the the variational statement, and

4

upon variation follows the equations of motion. This will be explained furtherin subsequent sections.

Straight waveguide finite elements were first formulated by Alaami [19] andLagasse [20] in 1973. Curved waveguides are used by Hladky-Hennion [21] andNilsson [17]. The elements in the tyre model presented here are described in[22]. A review of the applications of waveguide FE for vibro acoustic problemsis given in [17].

This paper starts with a presentation of the tyre model including data defini-tion and the equations of motion. The free response of the tyre is then validatedagainst an experimental modal analysis and the forced response against pointand transfer mobility measurements.

2 Waveguide finite element model

2.1 Variational statement

For a conservative system Hamilton’s principle states that the true motion ofthe system is the one that minimizes the difference between the time integralsof the strain and kinetic energy minus the work by external forces. The fol-lowing functional is therefore stationary, subject to the boundary conditions,

H =∫V

∫(ep − ek − B)dtdV , (1)

where V is the domain of the element, t is the time, ep, ek are the strain andkinetic energy densities and B is the work by external forces.

By considering a harmonic time dependence of the form e−iωt, where ω is theangular frequency, one can let time stretch from minus infinity to plus infinityand apply Parseval’s identity. This leads to the following expression for thestrain energy in the frequency domain

epot =1

2

1

2π

∫V

∫ε∗TCεdωdV , (2)

where,∗ denotes complex conjugate, T denotes transpose, C is the rigidity ma-trix, which is real valued for a conservative system. The vector ε contains theengineering components of strain, which are linear functionals of the displace-ment u and it’s spatial derivatives.

5

The kinetic energy is given on a similar form,

ekin =1

2

1

2π

∫V

∫ω2ρu∗TudωdV , (3)

where ρ is the density. The work done by external forces is given by,

eext force =1

2π

∫V

∫f∗TudωdV , (4)

where f is the generalized force vector.

For dissipative motion Hamilton’s principle does not apply and instead a mod-ified version is used [23],[24]. Dissipative forces are introduced through the useof a complex, possibly frequency dependent, rigidity matrix. By replacing thecomplex conjugates of the strain and displacement with the complex conju-gate of strain and displacement in a mathematically designed adjoint system,which has negative damping, the functional that is minimized for the truemotion is found. For linear vibration the different frequency component donot couple, it is therefore posible to consider one frequency at a time. Theresulting functional is named the Lagrangian L,

L =∫V

εaTCε−ω2ρuaTu − fHu − uaTfdV, (5)

where H denotes complex conjugate and transpose and a denotes complexconjugate in the adjoint system.

2.2 Tyre model

A waveguide finite element model is suitable since the tyre has constant cross-sectional material and geometrical properties. This is because the consideredtyre does not have a tread pattern.

The idea is to discretise the cross-section using standard FE technique but tokeep the analytical approach along the direction of the waveguide. For thesestructures the response can be described by

u(x, r, φ) = Ψ(x, r)T v(φ), (6)

where u is the response variable, x, r are the coordinates of the cross-section,φ measures the angle around the structure, Ψ is a vector of polynominal FEshape functions, each non zero within one finite element only, and the entriesof the vector v are the ’nodal’ displacement.

6

The engineering strain vector is given by spatial derivatives of the displace-ments and can be written in the following form involving the nodal displace-ments.

ε (x, r, φ) = E0 (x, r) v (φ) + E1 (x, r)∂v (φ)

∂φ, (7)

where E0 and E1 depend on the strain-displacement relations and the FE basisfunctions [22].

The variant of Hamilton’s principle mentioned above is implemented, displace-ments of the form (6) are assumed, then standard FE procedures for elementformulation and assembling leads to the following Lagrangian [22].

L+ =∫ π

−π

(1∑

n=0

1∑m=0

∂nvaT

∂φnAnm

∂mv

∂φm− ω2vaTMv − vaTf − fHv

)dφ, (8)

where f is the corresponding generalised consistent nodal force vector. Thus,the product of the excitation pressure, per unit length, and the FE shapefunctions is integrated over the excited elements area, which produces thenodal force vector. The matrices Anm and M describes the elastic and theinertia forces of the structure.

By performing a variation of the adjoint systems displacements the equationof motion will follow, this procedure will be explained more in detail in asubsequent section when the tyre is connected to a rigid rim.

2.3 Input data definition

The geometry and most of the elastic data that defines the tyre model arebased on an Abaqus finite element model given by Goodyear. The big dis-advantage with this procedure is the need to rely on a tyre manufacturer forinput data. The advantage is that the model is based on design data, thereforecommunication with the design engineers is possible. Also, it is of very goodquality.

The input data file for the Abaqus model is converted to fit the syntax usedby the waveguide FE program (written in Matlab). In the Abaqus model theelastic data for the tread is simply given by a single value for the shear modulusG = 5.81MPa. Since rubber is nearly incompressible the value of Poisson’sratio was set to ν = 0.4995. The data for the deep shell elements given byGoodyear was adapted slightly. A discussion of these adaption follows in thesubsequent sections.

7

2.3.1 Pre-load

The amount of pre-load force (per unit length) attached to each element isgiven as 3 membrane forces and 3 bending forces per element. The waveguidemodel only considers the membrane forces [22],[25], hence the bending forcesare ignored, which could be one of the reasons that slight adaptation is needed.When studying the pre-load forces it is seen that the shear forces and thecircumferential forces at the side wall is very small. In fact, they have a randomappearance. For stability reasons these elements in the static force vectors areset to zero. It is also noted that the force in the direction along the cross-section (ξ -direction) is zero at the rim. This cannot be correct since staticequilibrium for a strip going around the tyre requires the force in the ξ -direction to be inversely proportional to the radius as discussed in [22]. Thepre-load used here is defined by

Nξ(n) = Nξ,mRm

R(n), (9)

where n is the element number, R(n) is the radius at the middle of the element,Nξ(n) is the pre load at element n, Rm is the radius at the middle of the tyreand Nξ,m is the pre-load at the middle of the tyre. The pre-load defined byequation (9) gives static equilibrium and is just a slight adjustement of thepre-load used in the Abaqus model.

The peaks were shifted a bit in frequency between the simulation and themeasurements. Since the pre-load is in a first approximation proportional tothe inflation pressure, which is measured with a rather simple pressure gauge,it is multiplied with a factor 0.95 to have better agreement with measurements.

2.3.2 Transverse shear rigidity

The Abaqus model defines the 6 x 6 matrix in the upper left corner of thematrix D in [22] equation (60). The transverse shear rigidity is not explicitlydefined, therefore the transverse shear rigidities are set to be equal to theinplane shear rigidity, i.e.

C[7, 7] = C[8, 8] = C[3, 3] (10)

2.3.3 Visco-elastic data

The elastic data, as mentioned above, is defined by Goodyear’s Abaqus model.The dissipative properties on the other hand are defined by an ad hoc curvefitting procedure based on measured mobilities. The procedure to calculatemobilities is explained in a following section where also the numerical resultsare shown.

8

Solid rubber is highly damped and based on numerical experiments a lossfactor η = 0.3 was used for the tread. To find a baseline for the dampingof the deep shell elements, the 3-dB band widths of the resonances in thelow frequency region (100-200 Hz) were estimated. At these frequencies thetyre behaves like a pre-stressed curved beam on an elastic foundation. Thedamping for the higher order beam modes became too high and thereforesome of the baseline damping was distributed to damping proportional to thepre-load. Following this idea the rigidity matrix of the deep shell elements Dis multiplied with a factor 1− iηb, where ηb = 0.01 and the pre-load by a factor1 − iηp, where ηp = 0.01.

To get the right frequency trend concerning damping, the diagonal elements ofD, for the belt which is beneath the tread elements, which describes the rigid-ity of inplane shear, bending across the belt and bending in the circumferentialdirection is multiplied by factors 1− iη3 , 1− iη4 and 1− iη5 respectively, whereη3 = 0.1, η4 = 0.15 and η5 = 0.05. The increase of the apparent damping athigher frequencies is modeled through a factor 1 + iηv multiplying the massmatrix, where

ηv (f) =

⎧⎪⎨⎪⎩

0, f < 260 Hz

0.05 + 0.05 · f/3000, f ≥ 260 Hz(11)

This is the only frequency dependent damping even though it is well knownthat the visco-elastic properties of rubber varies largely with frequency. Forthe considered tyre, however, the large increase of apparent damping at the’cut-on’ of the higher order cross-sectional modes, around 350 Hz, is mostprobably due to that a large portion of the strain energy is in the rubber, andless in the steel wires within the belt.

2.3.4 Rim data

The rim is described as a rigid body having properties defined by the carmanufacturer Renault: mass M = 8.87 kg, mass moment of inertia about theaxis of rotation Jx = 0.2422 kg m2, and mass moment of inertia about a radialaxis Jr = 0.1487 kg m2. The rim’s centre of gravity is displaced 4.5 cm, in theaxial-direction towards the spokes, from the middle of the cross-section.

A rim resonance has been observed at 290 Hz [17], but this does not appearto have any significant effect on the mobilty measurements.

9

2.3.5 Mesh definition

Two different tyre cross-section meshes have been used. The medium meshwith 42 elements, 113 nodes and 516 degrees of freedom (DOF) is seen inFigure 2 and the enriched Abaqus full mesh with 214 elements, 569 nodesand 2562 DOF in Figure 3. Assuming that the same nodal density is usedin the 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 was sufficient up to at least 1000 Hz.

3 Equations of motion for the tyre-rim assembly

The tyre-rim assembly is modeled to be free since this is the easiest boundarycondition to achieve in the lab. In the lab the tyre is hung in chains and rubberstraps attached to the ceiling. This section describes the coupling betweenthe rim and the tyre leading to the equations of motion, upon which thecalculations of a free and a forced response is discussed.

3.1 Rim modeling

The rim’s motion is described in the Cartesian coordinate system seen tothe left in Figure 4. The tyre motion, on the other hand, is described in thecylindrical coordinate system seen to the right in Figure 4. The x-axes coincide.At 12 o’clock, φ = 0 and er = ey, eφ = ez . The cylindrical coordinate axesare related to the Cartesian axes by

ex = ex,

er = cos φey + sin φez,

eφ = − sin φey + cos φez.

(12)

The rigid rim’s motion has six degrees of freedom: three displacements andthree rotation. In a Cartesian coordinate system, with its origin at the rim’smass centre, the displacement, U, at location [x, y, z] is given by

UTex = Ux + zΦy − yΦz,

UTey = Uy − zΦx + xΦz,

UTez = Uz + yΦx − xΦy,

(13)

where Ux, Uy and Uz are the displacements at the mass centre and Φx, Φy andΦz are the rotations.

10

The rim’s height is 2h and it’s width is 2b. It’s mass centre is off-set a distanced in the x -direction from the centre of the tyre carcass towards the spokes.The rim is thus connected to the tyre at the locations given by.

[x, y, z] = [bi, h cos φ, h sin φ] = ri (14)

where b1 = b − d at node 1, bN = − (b + d) at node N and φ takes any valuein the interval [0, 2π]. The rim’s displacement at the connection to the tyre isgiven by equations (13) and (14). It is equally given in cylindrical coordinatesby

vTi ex = Ux + h Φy sin φ − h Φz cos φ ,

vTi er = Uy cos φ + Uz sin φ + b Φz cos φ − b Φy sin φ , (15)

vTi eφ = h Φx − Uy sin φ + Uz cos φ − b Φy cos φ − b Φz sin φ ,

where vi contains the tyre’s three displacement components at node 1 or N.The rotation about the φ -axis is equally given at both the tyre nodes as

φφ = cos φ Φz − sin φ Φy. (16)

3.2 Dynamic model

3.2.1 Variational statement

The kinetic energy of the rim is given by

Ek,rim = −1

2

1

2πω2uaTMrimu, u = [ Ux Uy Uz Φx Φy Φz ]T, (17)

where the entries to the diagonal matrix Mrim are given by the rim’s massand mass moments of inertia.

At the connections, the rim and tyre motions are equal. This restraint isincluded into the functional describing the tyre and rim motion with Lagrangemultipliers. Thus the functional is given by

L = L+ − ω2uaTMrimu +∫ π

−πλaT

⎛⎜⎝⎡⎢⎣ v1

vN

⎤⎥⎦−

⎡⎢⎣ B1

BN

⎤⎥⎦u

⎞⎟⎠

+ λT

⎛⎜⎝⎡⎢⎣ va

1

vaN

⎤⎥⎦−

⎡⎢⎣ B1

BN

⎤⎥⎦ua

⎞⎟⎠ dφ

(18)

where the vectors λa and λ contains Lagrange multipliers and L+ is definedin equation (8). The matrices Bi are, from equation (15) and (16), given by

11

Bi =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎣

1 0 0 0 h sin φ −h cos φ

0 cos φ sin φ 0 −bi sin φ bi cos φ

0 − sin φ cos φ h −bi cos φ −bi sin φ

0 0 0 0 − sin φ cos φ

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦

. (19)

3.2.2 Assumed modes procedure

Periodic solutions for v are assumed on the form of

v(φ) =∞∑

n=−∞vn(ω) einφ, (20)

where

vn =1

2π

∫ π

−πv(φ) e−inφ. (21)

In equations (20) and (21) n can be interpreted as the number of wavelengthsgoing around the tyre in the circumferential direction and will here be referredto as the wave order. The same approach is used for va, λ, λa, f and Bi. Theseexpressions are inserted into the Lagrangian

L =∫ π

−π

∑n

∑m

⎛⎝ 1∑

p=0

1∑q=0

(in)pvaTn Apqvm(im)q

⎞⎠ei(n+m)φ

− ω2vaTn Mvmei(n+m)φ

+ λaTn

⎛⎜⎝⎡⎢⎣ v1

vN

⎤⎥⎦

m

−⎡⎢⎣ B1

BN

⎤⎥⎦

m

u

⎞⎟⎠ ei(n+m)φ

+ λTm

⎛⎜⎝⎡⎢⎣ va

1

vaN

⎤⎥⎦

n

−⎡⎢⎣ B1

BN

⎤⎥⎦

n

ua

⎞⎟⎠ ei(n+m)φdφ

−∫ π

−π

∑n

∑m

(vaT

n fm + f∗nvm

)ei(n+m)φdφ − ω2uaTMrimu

(22)

12

and it follows that

L = 2π∑n

⎡⎣⎛⎝ 1∑

p=0

1∑q=0

(−1)pvaT−nApqvn(in)p+q − ω2vaT

−nMvn

⎞⎠

+ λaT−n

⎛⎜⎝⎡⎢⎣ v1

vN

⎤⎥⎦

n

−⎡⎢⎣ [B1]n

[BN ]n

⎤⎥⎦u

⎞⎟⎠

+ λTn

⎛⎜⎝⎡⎢⎣ va

1

vaN

⎤⎥⎦−n

−⎡⎢⎣ [B1]−n

[BN ]−n

⎤⎥⎦ua

⎞⎟⎠

−(vaT−nfn + fH

−nvn

)⎤⎦− ω2uaTMrimu .

(23)

The scalar products required for the evaluation of expression [Bi]n are givenby

∫ π

−πe±iφ sin φ dφ = ±iπ,

∫ π

−πe±iφ cos φ dφ = π,

∫ π

−πe0dφ = 2π, (24)

with all other possible combinations of trigonometric functions and exponen-tials evaluating to zero. It follows that

[Bi]0 =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎣

1 0 0 0 0 0

0 0 0 0 0 0

0 0 0 h 0 0

0 0 0 0 0 0

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦

, (25)

[Bi]1 =1

2

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎣

0 0 0 0 −ih −h

0 1 −i 0 ibi bi

0 i 1 0 −bi ibi

0 0 0 0 i 1

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦

, (26)

and

[Bi]−1 = [Bi]∗1 , [Bi]n = 0, |n| ≥ 2. (27)

3.2.3 Equations of motion

The equations of motion for the structure follow upon the variation of theadjoint system displacements and the Lagrange multipliers λa. It is only the

13

wave orders n = −1, 0 and 1 that are coupled to the rim’s motion. Conse-quently, for wave orders |n| ≥ 2, the equations of motion and the boundaryconditions are simply (

K (n) − ω2 M)

vn = fn (28)

⎡⎢⎣ v1

vN

⎤⎥⎦

n

= 0 (29)

u = 0 (30)

where

K(n) =1∑

p=0

1∑q=0

(−1)pApq(in)p+q (31)

For wave orders n = −1, 0 and 1 , the equations of motion are similarly givenby (

K(n) − ω2M)vn + λn = fn (32)

⎡⎢⎣ v1

vN

⎤⎥⎦

n

−⎡⎢⎣ B1

BN

⎤⎥⎦

n

u = 0 (33)

− 1

2πω2Mrim u −

⎡⎢⎣ B1

BN

⎤⎥⎦

T

1

λ−1 −⎡⎢⎣ B1

BN

⎤⎥⎦

T

0

λ0 −⎡⎢⎣ B1

BN

⎤⎥⎦

T

−1

λ1 = 0. (34)

The equations (32)-(34) are collected in matrix form

⎡⎢⎢⎢⎢⎢⎣D 0 ET

0 −ω2Mrim/2π −BH

E −B 0

⎤⎥⎥⎥⎥⎥⎦

⎡⎢⎢⎢⎢⎢⎣V

u

Λ

⎤⎥⎥⎥⎥⎥⎦ =

⎡⎢⎢⎢⎢⎢⎣

[fT−1 fT

0 fT1

]T0

0

⎤⎥⎥⎥⎥⎥⎦ , (35)

where E is an incidence matrix, being zero except for one entry in each of itstwelve rows that is unity, and

D =

⎡⎢⎢⎢⎢⎢⎣K (−1) − ω2 M 0 0

0 K (0) − ω2 M 0

0 0 K (1) − ω2 M

⎤⎥⎥⎥⎥⎥⎦ (36)

14

B =

⎡⎢⎢⎣⎡⎢⎣ B1

BN

⎤⎥⎦

T

−1

⎡⎢⎣ B1

BN

⎤⎥⎦

T

0

⎡⎢⎣ B1

BN

⎤⎥⎦

T

1

⎤⎥⎥⎦

T

(37)

V =

⎡⎢⎢⎢⎢⎢⎣

[v]−1

[v]0

[v]1

⎤⎥⎥⎥⎥⎥⎦ , Λ =

⎡⎢⎢⎢⎢⎢⎣

λ−1

λ0

λ1

⎤⎥⎥⎥⎥⎥⎦ (38)

3.3 Free response

When the force term in equations (28) and (35) is zero, a twin parametergeneralised eigenvalue problem for the waveorder, n and the angular frequency,ω follows. This can on a general form be written as,

[DGen(ω, n) − ω2MGen(ω)]V = 0, (39)

where DGen and MGen are generalised possibly frequency dependent stiffnessand mass matrices, respectively and V is the eigenvector. The relation betweenthe frequency and the wave order is sometimes called the dispersion relation.

By solving this eigen value problem for integer values of n, thus solving for thefrequency, the result is the dispersion curve, see Figure 5 for the undampedtyre. To produce Figure 5 the medium mesh was used. In the dispersion curveeach dot corresponds to a natural frequency.

By inspecting the mode shapes corresponding to these solutions the dispersioncurves can be divided into different branches each having their own charac-teristic waveform. The first two branches from the left are the symmetric andanti-symmetric belt modes illustrated in Figures 6(a) and 6(b). The thirdbranch from the left is the symmetric belt bending family (see Figure 6(c)).This family is not propagated below the ’cut-on’ frequency at approximately350 Hz. The waveforms corresponding to the fourth branch are anti-symmetricbending modes. Due to crossings and general complexity of the higher orderbranches they are harder to describe in a simple way, but they will of coursecontribute to the response.

The rigid body motion of the tyre is not modeled correctly. This is not becauseof the FE formulation but has with the virtual work from the pre-load to do.Even when an exact rigid body motion is inserted into the strain-displacementrelation a small strain is induced [22]. This can be seen in the dispersion curve

15

where there are rigid body modes around 20 Hz at wave orders zero and one,see Figure 5. The rigid body modes will however not influence the forcedresponse calculation in the interesting frequency region, also, when the rollingtyre is considered, the rim will be blocked and the rigid body mode will notbe present.

3.4 Forced response

To simulate a mobility measurement the applied force is modeled as a constantpressure over a quadratic area. This is because; a) in the experiments the forceis distributed and b) a point force would mainly lead to local deformation ofthe tread and not to a structural response. The extension in the direction ofthe crosssection is handled through FE procedures for consistent loads andleads to a generalized force. For a generalised force that is constant in anangular interval,

f (φ) = F/2Δ, φ ∈ [φ0 − Δ, φ0 + Δ] (40)

and is zero outside this interval,

fn =1

2π

∫ π

−π

F

2Δ

(H(φ0 − Δ) − H(φ0 + Δ)

)e−inφdφ

=1

2π

F

2Δ

∫ φ0+Δ

φ0−Δe−inφdφ =

−1

2π

F

2Δ

e−inΔ − einΔ

ine−inφ0

=Fe−inφ0

2π

sin nΔ

nΔ

(41)

This is the expression for the force that is inserted into equation (28) for waveorders |n| ≥ 2 and equation (35) for wave orders n = −1, n = 0 and n = 1.

4 Measurements

In order to validate the tyre model a modal analysis and a point mobilitymeasurement have been taken. The visco elastic properties of the tyre werebased on the mobility measurements.

4.1 Modal analysis

The rim and tyre was elastically suspended with steel chains and rubber strapsattached to the ceiling, see Figure 7. The suspension modes are far below the

16

first elastic mode in terms of frequency.

The excitation was performed using two electrodynamic shakers mounted inthe radial direction of the tire, i.e. perpendicular to the belt, see Figure 8.Two force transducers were glued to the tread in the excitation points. Sincetwo shakers is used, two columns of the accelerance matrix is generated anda polyreference method can be used to extract the modal parameters.

The response of the belt were measured using ten accelerometers, which weremounted to the response points using bee’s wax (Figure 9). The response wasmeasured both in the radial and axial directions. The measured signals is thenled through a VXI hardware analyser connected to an IDEAS software.

The shakers were fed with a burst random source signal, hence the shakersonly vibrates a fraction of the time window. In this case 80 percent. Table 1lists the measurement set-up parameters.

Number of spectral lines 1601

fmax[Hz] 1000

fmin[Hz] 0

Sampling frequency [Hz] 2560

Frame size 4096

Δt[ms] 0.390625

Δf [Hz] 0.625

Frame length [s] 1.6

Pre trigger [s] 0.16

Number of avergaes 30Table 1Set-up parameters in the Ideas software.

A poly reference method supported by the software was applied to the mea-sured frequency response functions. The modes that have been identified areclassified either as symmetric belt modes (S) or anti-symmetric belt modes(AS). A list of the identified modes is found in Table 2 in which the brack-eted values comes from a medium mesh calculation. The damping, which wasbased on point mobility measurements, underestimates the lossfactors of theanti-symmetric modes, also the trend of increased damping with frequency isstronger in the measurements than in the calculations.

The calculated eigenfrequencies agree well with the measured ones between100 and 200 Hz. For wave orders zero and one the rim and tyre is interactingin the motion, these modes are called semi rigid body modes and are depicted

17

in Figure 10, and are not to be confused with the rigid body modes mentionedearlier. The first one is the axial semi rigid body mode of order zero whichhas a ’telescopic’ mode shape where the rim and the tyre is moving out ofphase in the axial direction. The model overestimates this eigenfrequency byslightly more than ten percent. This mode, however. is not considered to bethat important for a rolling tyre. There should be a rotational zero order modewhere the rim and the tyre is rotating, out of phase, in the circumferentialdirection but this mode has not been found.

The modes which have one wavelength going around the tyre are coupled dueto the fact that the mass center of the rim is shifted 4.5 cm, in the axialdirection, from the geometric centre of the tyre. If this wasn’t so one of themodes would be classified as symmetric (purely translational) and the otherone anti-symmetric (only rocking motion). For wave orders larger than one,the rim does not move in the model.

There are nine modes overlapping in the 100 Hz region (of which 5 respond),because of this the parameter extraction has not been fully successfull for thesemodes. The modes of order two are not resolved: they do not have orthogonalmode shapes. Examples of mode shapes can be seen in Figures 11 and 12.

No mode from the belt bending family, that occurs at 350 Hz and above, wasresolved, probably due to the high damping of these modes and a high modaloverlap. According to the calculation some 15 modes overlaps in this frequencyregion, see Figure 5.

The acoustic mode, which occurs at approximately 225 Hz, when there isexactly one wavelength around the tyre in the air cavity, is not present in themodel since the tyre cavity is not modeled, however, it has previously beenmodeled with good accuracy [17].

4.2 Mobility measurements

The tyre was hung in the same way as for the modal analysis measurement(freely suspended). An impedance head was used to take the measurement.The impedance head was placed on the belt on three different positions, inthe middle, translated 5 cm closer to the spokes in the axial direction andtranslated 5 cm further away from the spokes in the axial direction, see Figure13.

An additional accelerometer was used to take transfer mobilities, see Figure 14.A swept sine technique was used, giving good coherence and a fine frequencyresolution. Measurements were made for the frequency range 30-1000 Hz. Alist of instrumentation used can be found in Table 3.

18

Description [Wave order] Frequency [Hz] Lossfactor [% of critical damping]

0 Axial semi rigid body mode 56.19 (62.9) 1.91 (0.7)

1 As + S 93.20 (89.9) 3.27 (1.24)

(2 As + S)* 98.36 (101.4) 3.73 (3.03)

(2 As + S)* 102.41(112.5) 3.63 (2.46)

1 As + S 111.21 (106.2) 4.86 (2.6)

3 S 121.35 (122.5) 3.17 (2.84)

4 S 143.83 (145.6) 2.83 (2.71)

3 As 153.67 (158.1) 4.53 (2.95)

5 S 168.25 (170) 3.12 (2.72)

4 As 188.9 (185.6) 4.82 (3.08)

6 S 194.73 (195) 3.25 (2.87)

5 As 215.64 (205.6) 5.12 (3.23)

7 S 224.1 (220.3) 3.65 (3.15)

Acoustic mode 225.44 (-) 0.32 (-)

6 As 240.2 (223.8) 5.55 (3.43)

8 S 255.4 (245.6) 4.20 (3.54)Table 2Identified modes. The bracketed values comes from a medium mesh calculations.*The modes of order two are not fully resolved.

Item Type Serial number

Shaker LDS V203 513R4/25

Impedance head B&K 8001 10109

Accelerometer B&K 4393V 1929299

Amplifier ( x 3) B&K 2635 1447225

1571336

669794

Analyser Siglab -Table 3Instrumentation

The weight of the impedance head below the force transducer was less than5 g and the excitation area was a 15 mm diameter disk. For the calculation asquare excitation area of 20 mm times 20 mm was used since this agreed best

19

with the measurements. Convergence for the considered frequency region wasreached when 60 wave orders was used for each frequency, thus for 60 waveorders and 361 frequencies the calculations takes approximately 25 minuteson a modern (2006) 2 Ghz laptop.

Figures 15 and 16 show the magnitude and the phase of measured and calcu-lated point mobilities for excitation in the middle position. It should be notedthat this measurement is the main basis for the visco elastic data estimation.

The small peak at rougly 90 Hz, which is only visible in the calculation, isa semi rigid body mode response that in the measured curve probably iscontained within the wide peak at 100 Hz. According to the modal analysisthere are many modes present in this peak and the small shift in frequencyof the semi rigid body mode makes the peak divide into two peaks in thecalculation. The peak at 225 Hz is the acoustic mode mentioned earlier andis not present in the calculation since the air cavity is not modeled.

The ’cut-on’ of the belt bending modes is roughly 10 % (35 Hz) lower in thecalculation than in the measurement. This is perhaps due to the ageing of therubber compound making it stiffer and thus increasing the eigenfrequenciesassociated with the ’cut-on’. Measurements taken two years earlier by Nilsson[17] on the same tyre, and measurements taken four years earlier by Andersson[4], on a tyre from the same batch, show a ’cut-on’ at 350 Hz which is inagreement with the calculation. The phase show the same level of accuracy asthe magnitude.

Figure 17 show the magnitude of a transfer mobility with excitation in themiddle and a response point located 23.5 cm away in the circumferential di-rection and 4.3 cm above the geometric centre of the tyre. The discrepency islarger than for the point moblity since the result is very sensitive to the exactlocation of the measurement equipment. In particular the level and exact fre-quency of the anti-resonances are sensitive depending on the exact speed bywhich the various waves travel leading to positive and/or negative interference.

For excitation in the upper position (5 cm above the geometric centre towardsthe spokes) the ’cut-on’ of the bending mode is not visible (See Figure 18 and19). This is because the excitation is in a nodal point point of these modes.The magnitude of a transfer mobilty where the response is measured 12.3 cmaway in the circumferential direction and 4.7 cm above the geometric centreof the tyre is in Figure 20. The calculation overestimates the response for thehigher order modes (140-250 Hz) by between 1.5 and 5 dBs for the squaredamplitude of the mobilty.

When the excitation is in the lower position, the anti-symmetric modes shouldbe excited since the mass centre of the rim is located 4.5 cm above geometriccentre and the excitaion is 5 cm below the geometric centre. This is seen in the

20

simulation in Figure 21 between the third and fourth large peak at roughly153 Hz but is just barely noticeable in the measurements. One explanation forthis is that the damping of the anti-symmetric modes is to low in the model(see Table 2). The phase is seen in Figure 22. A transfer mobility, wherethe response is measured 23.5 cm away in the circumferential direction and4.3 cm above the geometric centre of the tyre is in Figure 23, the transfermobility prediction above 300 Hz is less accurate than for excitation in theupper position.

5 Conclusions

A car tyre is modeled with waveguide finite elements. The model includes: thecurvature, the geometry of the cross-section, the pre-stress due to inflationpressure, the anisotropic material properties and the rigid body propertiesof the rim. The model is based on design data. The motion of the tyre beltand side wall is described with quadratic anisotropic, deep shell elements thatinclude pre-stress and the motion of the tread on top of the tyre by quadratic,Lagrange type, isotropic two dimensional elements.

The model agrees very well with measurements up to roughly 250 Hz for thepoint mobilities. The eigenfrequency prediction are within five percent for theidentified modes, except for the axial semi rigid body mode ( error 12 %), theanti-symmetric mode of order two (error 10 %) and the anti-symmetric modeof order seven (error 7 %). The estimated damping is a bit too low, especially sofor the anti-symmetric modes. The ’cut-on’ of the belt bending modes comesearlier in the prediction than in the measurement. This is probably due toageing of the tyre since similar measurements performed in the spring of 2001[4] is in agreement with the prediction. Above 500 Hz there is an error rangingfrom 1.5 dB up to 3.5 dB for the squared amplitude of the point mobility. Theerror is larger for the transfer mobilities since they are more sensitive, to theexact location of the accelerometer, in particular so for the anti-resonances. Arim resonance has been observed at 290 Hz [17], but this does not appear tohave any significant effect on the mobilty measurements.

The non proportional damping used in the model is based on an ad hoc curvefitting procedure against measured mobilities. Since the visco elastic data isextremely important for rolling resistance estimations, the damping should beestablished in a more scientific way, and on going work considers this devel-opment.

21

6 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, and Wolfgang Kropp, Chalmers, forhelpful discussion and to Wolfgang Gnorich and Andrzej Pietrzyk, Goodyear,for advice and for sharing data for tyres. The final tyre model and the workpresented in this thesis were funded by the European Commission, ITARI,FP6-PL-0506437. Special thanks to Carl-Magnus Nilsson.

References

[1] COM 370, White Paper, European transport policy for 2010: timeto decide 2001.

[2] Hall D.E. and Moreland J.C. Fundamentals of rolling resistance, Rubber Chemistry and Technology 74 (3): 525-539 JUL-AUG2001.

[3] Fraggstedt M. and Finnveden S. Power dissipation in car tyres.Paper B in this thesis, 2006.

[4] Andersson P. High Frequency tyre vibration. Lic. thesis, ChalmersUniversity of Technology, 2002.

[5] Larsson K. ,Barrelet S. and Kropp W. The modeling of thedynamic behaviour of tyre tread blocks. Applied Acoustics, 63:659-677, 2002.

[6] Muggleton J.M., Mace B.R. and Brennan M.J. Vibrationalresponse prediction of a pneumatic tyre using an orthotropic two-plate wave model. Journal of Sound and Vibration, 264:929-950,2003.

[7] Bohm F. Mechanik des gurtelreifens. Ingenieur Archiv, 35:82-102,1966.

[8] Kropp W. Structure-borne sound on a smooth tyre. AppliedAcoustics, 26:181-192, 1989.

[9] Kung L.E, Soedel W. and Yang T.Y. Free vibration of a pneumatictyre tire-wheel unit using a ring on a an elastic foundation anda finite element model.Journal of Sound and Vibration, 107(2),1986.

[10] Dohrmann C.R. Dynamics of a tire-wheel suspension assembly.Journal of Sound and Vibration, 210(5):627-642, 1998.

22

[11] Molisani L.R., Burdisso R.A. and Tsihlas D. A coupled tirestructure/acoustic cavity model. International Journal of Solidsand Structures, 40:5125-5138, 2002.

[12] Kim Y.-J and Bolton J.S Effects of rotation on the dynamicsof a circular cylindrical shell with application to tire vibrations.Journal of Sound and Vibration, 275: 605-621, 2004.

[13] Pinnington R.J. and Briscoe A.R. A wave model for a pneumatictyre belt. Journal of Sound and Vibration, 253(5):941-959, 2002.

[14] Pinnington R.J. A wave model of a circular tyre. Part 1: beltmodeling. Journal of Sound and Vibration, 290 101132 2006.

[15] Larsson K. and Kropp W. A high-frequency three-dimensionaltyre model based on two coupled elastic layers. Journal of Soundand Vibration, 253(4):889-908, 2002.

[16] Richards T.L. Finite elements analysis of structural-acousticcoupling in tyres. Journal of Sound and Vibration, 149(2):235-243, 1991.

[17] Nilsson C.-M. Waveguide finite elements applied on a car tyre.Doctorial thesis. Aeronautical and Vehicle Engineering, KTH2004.

[18] Pinnington R.J. Radial force transmission to the hub froman unloaded stationary tyre. Journal of Sound and Vibration,253(5):941-959, 2002.

[19] Alaami B. Waves in prismatic guides of arbitrary cross section.Journal of Applied Mechanics, (December):1067-1071, 1973.

[20] Lagasse P.E. Higher-order finite-element analysis of topographicguides supporting elastic surface waves. The Journal of theAcoustical Society of America Volume 53(4):1116-1122, 1973.

[21] Hladky-Hennion A.-C. Finite element analysis of the propagationof acoustic waves in waveguides. Journal of Sound and Vibration,194(2), 119-136, 1996.

[22] Finnveden S., Fraggstedt M. Waveguide finite elements for curvedstructures, TRITA-AVE 2006:38.

[23] Morse P.M. and Feshbach H. Methods of theoretical physics,Chapter 3. 1953.

[24] Finnveden S. Exact spectral finite element analysis of a railwaycar structure, Acta Acustica 2 (1994) 461-482.

[25] Washizu k. Variational methods in elasticity and plasticity,Pergamon Press, 1975.

23

−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

]

Fig. 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

]

Fig. 3. Enriched Abaqus mesh.

24

x

y

z

r

ϕ

y z

x

Fig. 4. Sketch of tyre and rim. Left: Cartesian coordinate system for describing therim’s motion. Right: cylindrical coordinate system for describing the tyre motion.

0 100 200 300 400 500 600 700 800 900 10000

5

10

15

20

25

30

Frequency [Hz]

Wav

e O

rder

[Dim

ensi

onle

ss]

Fig. 5. Dispersion relation for the undamped tyre. Each dot represents a resonancewith a corresponding wave form.

25

(a) Symmetric Belt mode.

(b) Anti-symmetric Belt mode.

(c) Belt bending mode.

Fig. 6. The three low-frequency mode families.

26

Fig. 7. The tyre is suspended through chains and rubber straps attached to theceiling.

27

Fig. 8. Two shakers were used to excite the tyre.

Fig. 9. The accelerometers were attached with bee’s wax. The pieces of tape servesas measurement point indicators.

28

Fig. 10. The semi rigid body modes. Top left: axial mode of order zero (telescopicmode). Top right: rotational mode of order zero. Bottom left: symmetric mode oforder one (translational). Bottom right: anti-symmetric mode of order one (rocking).

29

Fig. 11. The symmetric bending mode of order four located at around 143 Hz.

Fig. 12. The anti-symmetric belt mode of order three located at around 153 Hz.

30

Fig. 13. Impedance head in the upper position.

Fig. 14. The accelerometer was attached with bee’s wax.

31

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

Fig. 15. Magnitude of point mobility for excitation in the middle position. Measured(solid) and calculated (dashed).

101

102

103

−2

−1.5

−1

−0.5

0

0.5

Frequency (Hz)

Pha

se o

f poi

nt m

obili

ty (

rad)

Fig. 16. Phase of point mobility for excitation in the middle position. Measured(solid) and calculated (dashed).

32

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

Fig. 17. Magnitude of transfer mobility for excitation in the middle position. Theresponse is measured 23.5 cm avay in the circumferential direction and 4.3 cm abovethe geometric centre. Measured (solid) and calculated (dashed).

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

Fig. 18. Magnitude of point mobility for excitation in the upper position. Measured(solid) and calculated (dashed).

33

101

102

103

−2

−1.5

−1

−0.5

0

0.5

Frequency (Hz)

Pha

se o

f poi

nt m

obili

ty (

rad)

Fig. 19. Phase of point mobility for excitation in the upper 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

Fig. 20. Magnitude of transfer mobility for excitation in the upper position. Theresponse is measured 12.3 cm avay in the circumferential direction and 4.7 cm abovethe geometric centre. Measured (solid) and calculated (dashed).

34

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

Fig. 21. Magnitude of point mobility for excitation in the lower position. Measured(solid) and calculated (dashed).

101

102

103

−2

−1.5

−1

−0.5

0

0.5

Frequency (Hz)

Pha

se o

f poi

nt m

obili

ty (

rad)

Fig. 22. Phase of point mobility for excitation in the lower position. Measured (solid)and calculated (dashed).

35

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

Fig. 23. Magnitude of transfer mobility for excitation in the lower position. Theresponse is measured 23.5 cm avay in the circumferential direction nd 4.3 cm abovethe geometric centre. Measured (solid) and calculated (dashed).

36

Power Dissipation in Car Tyres

Martin Fraggstedt and Svante Finnveden

The Marcus Wallenberg Laboratory for Sound and Vibration Research (MWL),Royal Institute of Technology, 100 44 Stockholm, Sweden

E-mail: mfragg@kth.se

Abstract

A car tyre is modeled with waveguide finite elements. A non-linear contact model isused to calculate the contact forces as the tyre is rolling on a rough road. The contactforces combined with the response of the tyre is used to estimate the input powerto the tyre structure, which determines a significant part of the rolling resistance.The input power must, based on energy conservation, equal the dissipated powerwithin the structure. By looking at the dissipated power within each element it ispossible to predict where the power loss occurs.

The power calculation has been made for three different roads and the results arepromising compared to literature and experiments.

1 Introduction

Traffic has a negative effect on the environment, for over fifty years it has beenan irritating noise polluter. For higher speeds tyres have been found to be themajor contributor for traffic noise. Also the interior noise in the vehicle due tothe tyres are becoming more important as other noise sources such as engines,exhaust systems and gear boxes are better managed.

Traffic is also a major source of green house gases. In the United States thecontribution from the road transport sector to CO2 emissions from fossil fuelconsumption was 24 % in 2002 [1]. The energy consumed by a car, travelingat constant speed, is due to engine ineffiency, internal friction, and the en-ergy needed to overcome resisting forces such as aerodynamic drag and rollingresistance. Tyre rolling resistance accounts for approximately 20 % of CO2emissions from cars and 30 to 40 % from trucks. This means that approx-imately 5 % of the total CO2 emissions from fossil fuel is generated in thetyres. Here we are studying the effect of rolling resistance, by looking at thepower dissipated, caused by visco-elastic forces as a tyre is rolling on a roughroad.

1.1 Car tyres

Car tyres are made of several different materials including steel, fabric andof course numerous rubber compounds, see Figure 1. The three major subregions of the tyre are: the lower side wall, the upper side wall and the centralarea. The ply is a layer of fabric that is embedded in rubber.At the lower sidewalls the ply encloses a volume filled with both steel wires and hard rubbermaterials, this makes the lower side wall areas relatively stiff. The upper sidewall areas are on the other hand quite flexible, since the ply layer there isplain and there is less steel in there. The central area is made up of the beltand the tread. The belt consists of a rubber embedded steel lining (breakers)in the circumferential direction to assure support and rigidity. The tread is anabout 13 mm thick rubber layer which provides the grip.

Fig. 1. The tyre consists of three major sub regions. Upper side wall, lower side walland the central area. The central area is in turn divided into the belt and the tread.

The objective of this work is to estimate the hysteretic losses due to the de-formation of the tyre as it is rolling on a rough road. To do that a waveguideFinite Elements (FE) model for the tyre, described in reference [2], is used.The model includes: the curvature, the geometry of the cross-section, the pre-stress due to inflation pressure, the anisotropic material properties and therigid body properties of the rim. The model is based on design data. The mo-tion of the tyre belt and side wall is described with quadratic anisotropic, deepshell elements that include pre-stress and the motion of the tread on top of

2

the tyre by quadratic, Lagrange type, homogenous, isotropic two dimensionalelements. The non proportional damping used in the model is based on mea-sured mobilities. The external forces acting on the tyre as it is rolling comesfrom a non-linear contact model [3]. Only normal forces are considered.

The tyre in the present study 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 has prop-erties typical of a production tyre. The tread pattern has an effect on the tyrevibration [4]. When the tread pattern is cut out of the tread, mass is removedand the bending stiffness is reduced. This generally leads to higher eigenfre-quencies and higher levels for the point mobility. Circumferential groves canbe included in the model since this would not alter the rotational symmetry.Commercial tread patterns however would have to be modeled as an equiva-lent tread pattern. It should me noted though, that the tread pattern is ac-counted for in the contact force calculation, so, as long as the model is slightlyadapted to fit mobility measurements the methodology presented here shouldwork. Local deformation in the tread and tread pattern becomes increasinglyimportant above roughly 800 Hz [5] which is well above the frequencies wherepower dissipation occurs for a rolling tyre.

The remaining of this section discusses rolling resistance. In the upcomingsection the waveguide FE model is described. After that a brief description ofthe contact forces follows. The contact forces are then in Section 4 inserted intothe tyre model and the displacement response is calculated. The calculationof the power consumed by the tyre is described and is followed by results andconcluding remarks.

1.2 Influence of tyre parameters on rolling resistance

The rolling resistance Fr is defined as the energy consumed per unit of distancetravelled [6]. The unit is Nm/m = N which is equivalent to a drag force inNewtons. Rubber is a visco elastic material, as it deforms a part of the energyis stored elastically but the remainder is dissipated as heat. These hystereticlosses, as well as aerodynamic drag and friction in the contact patch and withthe rim are losses that contribute to the total drag force on a moving vehicle.Rolling resistance plays a rather large role when it comes to fuel economy. Animprovement in rolling resistance of 10 % can yield fuel consumption improve-ments ranging from 0.5 to 1.5 % for passenger cars and light trucks and 1.5to 3 % for heavy trucks [7].

The rolling resistance is usually given as a dimensionless constant times the

3

gravity force,

Fr = Cr m g, (1)

where m is the mass, g is the constant of gravity and Cr is the rolling resistancecoefficient. Cr is normally in the range 0.01-0.02, with a typical value of 0.012for a passenger car tyre on dry asphalt [8].

The power consumed by this force is

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

where V is the speed of the vehicle. In this uncomplicated semi empiricalmodel for the power dissipation the only explicit parameters are the load andthe speed. Any non-linear dependence on these parameters and all the otherparameters are hidden in Cr. This model is usually good enough for someapplications but studies have shown that the rolling resistance coefficient isinfluenced by a number of parameters which will be discussed in the remainingof this section.

There are two different kinds of passenger car tyres, radial and diagonal, whichdiffers in construction. The radial tyre has been completely dominant the lastdecade. The rolling resistance of a radial tyre is allways lower than for diagonaltyres. Although radial tyres have a larger deflection, the internal deformationof a diagonal tyre is much bigger because of movement within the differentlayers, which does not occur for radial tyres [9].

The rubber compound has a large effect on the rolling resistance. By changingto a tread compound with a smaller loss factor, for example silica, the rollingresistance can be reduced with as much as 20 % [9]. Traditionally, loweringthe rolling resistance has led to a reduction of the wet grip performance, butthis problem seems to be manageable today.

The relation of the rolling resistance of passenger car tyres to tyre load isalmost linear. Over the practical load range, the rolling resistance coefficientis nearly constant, only slightly decreasing with increasing load [9].

Another important parameter is the inflation pressure. When the inflationpressure is increased the deflection and deformation of the tyre will be smallerleading to less hysteretic losses and a lower rolling resistance. The effect ismore pronounced for higher loads [9].

Rolling resistance will increase with deflection. Deflection is defined as thedifference between the loaded radius and the unloaded radius. However, usingdeflection in rolling loss studies is often avoided because of the difficulty ofmeasuring it accurately. For increasing vertical load, rolling resistance willincrease at constant deflection [10]

4

The rolling resistance can be considered to be constant until a certain speed.When exceeding this speed, standing waves will occur in the periphery of thetyre, which will increase the rolling resistance. The speed where this effect willoccur depends on the construction of the tyre [9].

The rolling loss will increase with increasing braking and driving torque. Yet,almost all rolling loss tests are done under free-rolling conditions. Tests withbraking and driving torques are more difficult to execute. An increase of 500Nm, can double the rolling loss [10].

The material characteristics of rubber show a strong temperature dependence[11], see Figure 2. In the later end of the transition region, and in the rubberregion a temperature increase leads to a softer rubber with smaller losses. Thisis the background to the discussion that will now follow.

Fig. 2. Schematic characteristics of rubber material temperature dependence [11].

With the increase of the internal tyre temperature, the hysteresis will de-crease and thus the rolling resistance will decrease. If the ambient tempera-ture and speed are constant, the internal tyre temperature will change untilit reaches the equilibrium condition. When a cold tyre starts to roll, the in-ternal temperature is increasing (approximately 40◦C for passenger car tyresuntil equilibrium temperature is reached) and the rolling resistance drops by30 % since the hysteretic losses are smaller at higher temperature for rubbercompounds. This means that driving short distances, during which the tyredoes not reach its equilibrium temperature, has a negative impact on rollingresistance. Tyres optimised for rolling resistance, will not develop their fullpotential during short urban trips [9].

The vehicle speed and the tyre temperature are directly related. In the non-steady state temperature condition, the speed change will have a large in-fluence on the rolling resistance. At time zero, the temperature of the tyreis independent of speed and is equal to the ambient temperature. As timeprogresses, rolling loss values decreases and temperature rises until no furtherchanges occur, and the equilibrium temperature has been reached. A sudden

5

speed increase of 20 km/h, momentarily increases the rolling loss by 9 %.In practice the tyre will warm up much faster, than in a laboratory environ-ment, because of the fact that driving a car on the road includes acceleration,deceleration and cornering [12].

Increasing the ambient temperature will increase the equilibrium tyre temper-ature by the same amount. Experimental data suggests that for each degreeincrease in ambient temperature, rolling loss decreases by 0.4 % for heavyvehicle tyres to 0.8 % for passenger car tyres [12].

By reducing tread thickness e.g. because of tyre wear, the rolling resistancewill decrease. The difference in rolling resistance between a new and worn tyrecan reach 25 % [9].

The relation between tyre dimensions and rolling resistance is inconsistent.The reason is that it is very difficult to compare different sizes of tyres, becauseby increasing the tyre size, it’s construction (e.g. tread, belt, and cords) willalso change [9].

Another central parameter is the road and its shape. Depending on the wave-length of the unevenness, the unevenness is divided into road texture and roadroughness [12]. The road texture concerns wavelengths up to the length of thecontact patch (typically 100 up to 150 mm) and is in turn divided into: i)micro texture (wavelength smaller than 1 mm), ii) macro texture (wavelengthbetween 1 mm and 10 mm), and iii) mega texture (wavelength between 10mm and 100 mm) [9]. The full texture range from micro to mega texture hasa significant influence on the rolling resistance [9]. An increase of the macrotexture will increase the rolling resistance [9].

For wavelengths in the same range as the contact path or larger, the un-evenness is called road roughness. The road roughness causes a deflection ofthe contact path as a whole and also excites the suspension system [12] and istherefore best studied in outdoor tests. Since the suspension is involved in thispower loss mechanism it is sometimes called driving resistance to separate itfrom the rolling resistance [9].

The road condition has a significant influence on the rolling resistance. Therolling resistance is almost doubled at a velocity of 120 km/h when comparinga dry and a wet road with 0.5 mm water depth [10].

Hall [7] claims, based on temperature distribution in the tyre, that roughly 40% of the rolling resistance is generated in the shoulder (the part of the centralarea which is closest to the upper side wall, see Figure 1), 30 % in the crown(the middle part of the central area), 15 % in the upper side wall and 15 % inthe lower side wall.

6

1.3 Measuring rolling resistance

The rolling resistance of tyres can be measured in different ways: 1) In a lab-oratory environment such as a test drum, a twin drum or a flat-belt, or 2) Bymeasurements on the road such as traction measurements, coast-down tests,tow force measurements of a trailer and fue1 consumption measurements.

Rolling resistance is linked directly to energy consumption and because ofthat another way to determine rolling resistance is through measuring thetyre temperature in relation to road and ambient temperature.

In the ISO standard [6] the test drum method is proclaimed. The drum shouldbe at least 1.5 m in diameter. There are four alternative methods to do therolling resistance measurement: a) the force method where the reaction forceat the spindle is measured. b) the tourque method where the tourque inputis measured at the test drum. c) the power method where the input power ismeasured at the test drum. d) the deceleration method where the decelerationof the test drum and tyre assembly is measured.

Each way of measuring the rolling resistance has advantages and/or disadvan-tages such as accuracy, correlation to real use, necessary hardware and timespan. For that reason it is difficult to compare the results from different testsand sources. The knowledge of the environment, such as ambient tempera-ture, wind and road type is also important. The rolling resistance of a tyre istherefore not an absolute quantity or property.

A round robin test was performed, in the autumn of 2004, to investigate theeffect of different roads for the rolling resistance and the energy consumption[13]. For the rolling resistance measurements two different measurement sys-tems based on trailers were used. Two kinds of tyres, slicks and profiled, wereused on 8 roads for the slick tyre and 11 roads for the profiled tyre. For the pro-filed tyre the measurement systems have a correlation coefficient of 85 %. Forthe slick tyre the correlation coefficient is only 34 %. The conclusion drawn in[13] is that for profiled tyres differences in rolling resistance for different roadscan be quantified even though the absolute values cannot.

For the energy consumption an electric car with the ability to measure theconsumed energy and a diesel car where the fuel consumption can be closelymonitored were used. The correlation coefficient is in this case 28 % (46 % ifone measurement is excluded). The conclusion is that external circumstancesinfluence the measurements so much that differences in energy consumptionfor different roads are poorly estimated.

7

1.4 Modeling of rolling resistance

The rolling resistance models can be categorised as empirical, thermal, viscoelastic or thermo-visco elastic [12]. The only models discussed here will be thevisco elastic models.

Stutts and Soedel [14] modeled the pneumatic tyre as a tension band on avisco elastic foundation. The ground contact region of the rolling tyre in steadystate was found by solving a two-point boundary value problem with specifiedradial deflections, treating the ground contact end points as unknowns. Therolling resistance was determined by integrating the forces acting throughthe elastic foundation on the wheel axle due to the steady state deformationof the tread band. Bending stiffness of the tension band is neglected anddissipative forces are introduced through one lumped viscous force term for thefoundation. They revealed the presence of a critical angular velocity beyondwhich the response changes from a damped exponential to a damped harmonicform. The physical parameters, of the tyre were calculated from experimentaldata. Without damping the contact region is symmetric and the net rollingresistance force is zero. When damping is introduced there is a forward shiftof the contact region and the rolling resistance force appears.

Kim and Savkoor [15] used an elastic ring supported on a visco elastic foun-dation. Additional elastic spring elements on the outer surface of the ring areincluded to model the compliance of the tread rubber. The contact problemof a free-rolling tyre is formulated for prescribed normal deflection and sub-jected to constraints of both normal contact and friction. Coulomb friction isassumed to apply locally, Coriolis effects are accounted for and bending of thetreadband is allowed according to the Bernoulli-Euler assumption. The equa-tions of motion are solved with a modal expansion method. Three differentdamping models are applied, viscous, structural and a Maxwell four-elementmodel. The structural damping leads to a speed independent rolling resistancein accordance with litterature.

Yam et al [16] based their calculation on experimental modal parameters andthe equilibrium and kinematics of a rolling tyre. The tyre carcass deformationis calculated using the transfer matrix resulting from an experimental modalanalysis. The calculation of tyre rolling properties starts from the static statusof the tyre, and the results after the tyre rotates through one discretized angleare calculated successively, until the rolling of the tyre is steady. The distri-butions of vertical reaction and horizontal tractional forces on the footprintof a rolling tyre are calculated. The rolling resistance, under different verticalloads, inflation pressures and rolling speeds are investigated numerically andis verified against experiments and litterature.

8

Popov et al [17] modelled a truck tyre, based on the model developed by Kimand Savkoor [15]. The stiffness and damping parameters needed, came from anexperimental modal analyis. Structural damping is used. Sophisticated rollingresistance measurements using a large test drum were also performed. Themodel is used to obtain the rolling resistance due to tread compression, whichamounts to 56 % of the measured rolling resistance force. All contributionsare however not calculated.

The models above are all based on rather simple equivalent structures, suchas elastic rings, whereas the model used in the present study has the correctgeometry and stiffness parameters as it is based on design data. None of themodels above are treating a rough road even though the road texture androughness have a significant effect on the rolling resistance.

2 Waveguide finite element model

2.1 Variational statement

Without external forces and with dissipation neglected, Hamilton’s principlestates that the true motion of the system is the one that minimizes the differ-ence between the time integrals of the strain and kinetic energy.

∫ t2

t1δ(U − T )dt =

∫ t2

t1δ(L)dt =0 (3)

where t is the time and U , T are the strain and kinetic energy. The functionalU −T will be referred to as the Lagrangian L. By considering a harmonic timedependence of the form e−iωt, where ω is the angular frequency, one can lettime stretch from minus infinity to plus infinity and apply Parseval’s identity.For linear vibration the different frequency components do not couple, it istherefore possible to consider one frequency at a time. The resulting bilinearfunctional is,

L =1

2

1

2π

∫V

(ε∗)T C ε − ρω2(u∗)T udV (4)

where ∗ denotes complex conjugate, T denotes transpose, C is the rigiditymatrix, which is real valued for a conservative system and ρ is the density.The vector ε contains the engineering components of strain, which are linearfunctionals of the displacement u and it’s spatial derivatives.

When dissipation is included the above functional (equation (4)) does not havea stationary minimum for the true displacements. The losses may, however, behandled by using a variational principle similar to that of Hamilton [18],[19].

9

Dissipative forces are introduced through the use of a complex, possibly fre-quency dependent, rigidity matrix. By replacing the complex conjugates ofthe strain and displacement with the complex conjugate of strain and dis-placement in an adjoint negatively damped system the new functional willbe stationary for true motion. This is conceptually more complex but doesnot add to the calculation burden. The work done by external forces f is alsoadded as in reference [2]. The resulting functional is the new lagrangian L,

L =∫V

(εaT C ε − ω2ρuaT u − fH u − uaT f

)dV, (5)

where H denotes complex conjugate and transpose and a denotes complexconjugate of the response in the adjoint system.

2.2 Tyre model

A waveguide FE model is a good tool when the structure has constant cross-sectional material and geometrical properties along one coordinate direction,e.g. smooth car tyres. The idea is to discretise the cross-section using standardFE technique but to keep the analytical approach along the direction of thewaveguide. For these structures the response can be described by

u(x, r, φ) = Ψ(x, r)Tv(φ), (6)

where u is the response variable, x, r are the coordinates of the cross-section,φ measures the angle around the structure, Ψ is a vector of polynomial FEshape functions, each non zero within one finite element only, and the entriesof the vector v are the ’nodal’ displacement.

The tyre cross-section mesh with 42 elements, 113 nodes and 516 degrees offreedom (DOF) is seen in Figure 3. The motion of the tyre belt is describedwith quadratic anisotropic, deep shell elements that include pre stress and themotion of the tread on top of the tyre by quadratic, Lagrange type, isotropictwo dimensional elements.

In this investigation the rigid rim is assumed to be blocked. The response atthe connection to the rim is therfore zero,

vc1 = vc2 = 0, (7)

where vc1 and vc2 contains the DOFs associated with the two nodes connectedto the rim.

The modified version of Hamilton’s principle mentioned above is employed,displacements of the form (6) are assumed, standard FE procedures for ele-

10

−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

]

Fig. 3. Tyre mesh. The belt is modelled with quadratic anisotropic, deep shell el-ements that includes prestress and the tread on top of the tyre by quadratic ,Lagrange type, isotropic two dimensional elements.

ment formulation and assembling are used, and finally variation of the adjointsystems displacements, leads to the following Euler-Lagrange equation [20].

[−A11

∂2

∂φ2+ (A01 − A10)

∂

∂φ+ A00 − ω2M

]v(φ, ω) = f(φ, ω), (8)

where f is the corresponding generalised consistent force vector, the matricesAij and M describe the elastic and inertia forces of the structure. Dissipativeforces are introduced through the imaginary parts of the matrices Aij and M.The damping is based on measured point mobilities and is described in [2].

The solutions to equation (8) will be periodic with respect to the circumferen-tial angle φ, since the tyre is a circular structure. This means that the solutioncan be expressed as an exponential Fourier series

v(φ, ω) =∑n

vn(ω)einφ, (9)

and similarly for the force vector. In equation (9) n is equivalent to the numberof wavelengths going around the tyre in the circumferential direction and willhere be referred to as the wave order.

To arrive at the desired equation the assumed solution is inserted into equa-tion (8), which is then multiplied by e−imφ, finally the reulting expression is

11

integrated around the tyre. This procedure filters out the coeficients due tothe orthogonality of the complex exponential basis function over the intervall[0 2π]. The result is

[n2A11 + in(A01 − A10) + A00 − ω2M

]vn(ω) = fn(ω), (10)

which also can be written

Dn(ω)vn(ω) = fn(ω). (11)

3 Contact Forces

The contact forces are calculated by Chalmers University of Technology (CTH),as described in reference [3]. The contact force prediction is based on a non-linear contact model in which the tyre structure is described by its flexibilitymatrix. Topographies of the surface are scanned, the tread pattern is accountedfor, and then the tyre is ’rolled’ over it. The procedure is done in the timedomain and uses a Lagrange multipliers approach. The nonlinear conditionsused are: i) the tyre cannot indent into the road, ii) if a point is not in contactthe force is zero and iii) that the force cannot be negative (road pulling tyredown) ([21] chapter 6). Only forces acting normal to the road is considered.

The tyre in this investigation has, as mentioned earlier, no tread pattern.The tread pattern has an effect on the tyre vibration [4]. The tread pattern is,however, taken into account in the contact force computation, so as long as thetyre model is somewhat tailored to fit mobility measurements the procedurepresented here should work. The power dissipation is mainly at low frequencies(below 100 Hz), so in a first approximation the tread does not have to bemodeled in great detail, but further investigation on the influence of the treadis needed.

3.1 Evaluation Of Generalised Contact Forces

The contact forces were determined for a speed of 80 km/h and consist of twofull revolutions. The total load on the tyre varies with time but the mean loadis within 10 % of 300 kg for all roads. The pressure, p (force per unit lengthand unit circumferential angle) resulting from the contact model, is describedby a three-dimensional force matrix, F, specifying the values of the contact

12

pressure [3].

p(x, φ, t) = F(p, q, k)Δt

ΔφΔxδ(t − tk)

· (H(φ − (φq − Δφ/2)) − H(φ − (φq + Δφ/2)))

· (H(x − (xp − Δx/2)) − H(x − (xp + Δx/2))) ,

(12)

φq = 2π(q − 1)

Q; q = 1, 2, ..., Q; Q = 512

xp = [−0.07 0.06 ... 0.07] ; p = 1, 2, ..., 15

tk = (k − 1)Δt; k = 1, 2, ... , K; K = 1024

Δx = 0.01 m, Δφ =2π

Qrad, Δt = 0.1734 ms

where H is the Heaviside function, p is the pressure and Δx Δφ is the areaupon which the force acts. The coordinate φ is around the tyre, x is in thetransverse direction along the tyre contour and t is the time.

The nodal force vector, f = f(φ, t) (force per unit angle), is produced byweighting of the pressure with the FE shape functions and integrating over x.

f(φ, t) =∫

Ψ(x, r)p(x, φ, t)dx (13)

The part of p which depends on x is defined as

F(s, q, k) =∫

Ψ(x, r)F(p, q, k)1

Δx· (H(x − (xp − Δx/2)) − H(x − (xp + Δx/2))) dx

(14)

where the new index s represents the nodes in the tyre road contact zone.With this formulation the nodal force vector can be written:

f(φ, t) = F(s, q, k)Δt

Δφδ(t − tk)

· (H(φ − (φq − Δφ/2)) − H(φ − (φq + Δφ/2)))(15)

The next task is to make a Fourier series expansion in time and in the cir-cumferential direction.In the circumferential direction the series is given by

f(φ, t) =N∑

n=−N

fn(t)einφ. (16)

The Fourier coefficients in (16) are given by

fn(t) =1

2π

∫ π

−πf(φ, t)e−inφdφ =

1

2π

∫ 2π

0f(φ, t)e−inφdφ. (17)

13

Using equation (15) and that f (φ, t) is a periodic function in φ :

fn(t) =Δt

2πδ(t − tk)

∫ 2π

0

F(s, q, k)

Δφe−inφ

· (H(φ − (φq − Δφ/2)) − H(φ − (φq + Δφ/2))) dφ

=Δt

2πδ(t − tk)

sin(nΔφ/2)

nΔφ/2

Q∑q=1

F(s, q, k)e−inφq

(18)

The contact forces are mostly zero but when the investigated location is incontact, there is a rather wild outburst. It would be unfortunate if such anoutburst occurred at the ends of the summation in (18) and if this is the case,the entries are shifted. Thus, the entries to f are shifted so that the first entryis for φ = φs:

fn(t) =1

2π

∫ 2π

0f(φ, t)e−inφdφ =

e−inφs

2π

∫ 2π

0f(θ + φs, t)e

−inθdθ (19)

which follows from that f is a periodic function of φ. This procedure determinesfn for n = 0, 1, ... , Q−1 ,, which equally gives fn for n = −Q/2, ... , Q/2−1 ,since f−n = fQ−n.

Finally, the generalised nodal force vector is transformed to the frequencydomain. Thus, it is assumed that the forces are periodic, described by thefollowing Fourier series:

fn(t) =N∑

n=−N

fn(ωm)eiωmt (20)

where

fn(ωm) =1

T

∫ T

0fn(t)e−iωmtdt (21)

ωm = 2πm/T, T = KΔt, M = K/2 (22)

fn(ωm) =1

T

∫ T

0fn(t)e−iωmtdt

=1

T

∫ T

0

K∑k=1

fn(tk) Δtδ(t − tk)e−iωmtdt

=Δt

T

K∑k=1

fn(tk) e−iωmtk

=1

K

K∑k=1

fn(tk) e−i2πm(k−1)

K

(23)

The summation in (23) is made with the Matlab procedure fft. This producesthe Fourier coefficients for ωm, m = 0, . . . , (K − 1) , which equally deter-mines, after reordering, the coefficients for m = −M, . . . , (M − 1), sincefn (ω−m) = fn (ωK−m). The contact forces fn (t) are not truly periodic and, to

14

avoid Gibb’s phenomena, a Hanning window is applied before the FFT.The resulting relation is:

f (φ, t) =N∑

n=−N

M−1∑m=−(M−1)

fn (ωm) ei n φ+iωmt (24)

3.1.1 On Negative Frequencies

The nodal force vector f (φ, t) is a real valued quantity and it follows that:

fn (t) = f∗−n (t) (25)

The spatially Fourier transformed force fn (t), on the other hand, is not real anda similar result for the generalised force vector fn (ω) is perhaps not obvious.However, by analysing the real and imaginary parts of fn (t), which both arereal by definition, and using the relation (25), it follows that

fn (ω) = f∗−n (−ω) (26)

Moreover, the system matrix in equation (11) has this property and it followsthat the response vector vn (ω) is also such that

vn (ω) = v∗−n (−ω) (27)

The response at positive frequencies thus gives the one at negative frequen-cies. Consequently, the calculation burden and the memory requirements arereduced if the response for all wave orders and positive frequencies are con-sidered only.

4 Vibration Response Of A Rolling Tyre

The tyre is rotating with a fixed angular speed, Ω , which is given by

Ω = 4π/T , (28)

since the contact forces supplied by CTH consists of two revolutions. Now,Newton’s law applies for a fixed piece of matter: a particle. The equations ofmotion are therefore solved in a Lagrangian coordinate system that is fixedto the rotating tyre. In doing so, the Corioli forces are neglected. Also, thecentrifugal force is neglected and it is assumed that the increased static tensionand radial expansion that it induces are already included in the definitionof the tyre’s steady state. Upon this basis, the tyre vibration is predicted.Thus, the generalised force vectors fn (ωm) are inserted into equation (10).

15

The solution of these equations produces the generalised tyre displacementvectors un (ωm). The tyre displacement as a function of location and time isthen given by

v (φ, t) =N∑

n=−N

M−1∑m=−(M−1)

vn (ωm) ei n φ+iωmt (29)

5 Dissipated power

Energy is always conserved, it is just transformed from one form to another.The energy going into the system must therefore equal the dissipated energyin the system. The same argument is valid for the power, consequently, theinjected power must equal the dissipated power.

5.1 Dissipated power based on external forces

The direct approach to calculate the time average of the input power is basedon the following relation:

P =1

T

∫ 2π

0

∫ T

0fT(φ, t)

∂v

∂tdtdφ, (30)

where P is the time averaged input power, f is the nodal force vector and∂v(φ, t)/∂t the nodal velocity vector.

The displacement and the force are given by (29) and (24) respectively. Thevelocity is easily derived from the displacement.

∂v(φ, t)

∂t=

N∑n=−N

M−1∑m=−(M−1)

iωmvn(ωm)einφ+iωmt (31)

Since only the coefficients for positive frequencies has been calculated this isre-written using the relations (26) and (27),

∂v(φ, t)

∂t=

N∑n=−N

einφ

⎡⎣0 · ivn(0) +

M−1∑m=1

iωmvn(ωm)eiωmt

− iωmvn(−ωm)e - iωmt

⎤⎦

(32)

16

Which can be simplified to:

∂v(φ, t)

∂t=

N∑n=−N

einφ

[M−1∑m=0

iωmvn(ωm)eiωmt − iωmv∗−n(ωm)e−iωmt.

](33)

f(φ, t) =N∑

n=−N

einφ

[fn(0) +

M−1∑m=1

fn(ωm)eiωmt+f∗−n(ωm)e−iωmt

](34)

These formulas are now inserted in equation (30)

P =1

T

∫ 2π

0

∫ T

0

N∑q=−N

eiqφ

⎡⎣fT

q (0) +M−1∑p=1

fTq

(ωp)eiωpt + fH

−q(ωp)e

- iωpt

⎤⎦

·N∑

n=−N

[M−1∑m=0

iωmvn(ωm)eiωmt − iωmv∗−n(ωm)e - iωmt

]einφdtdφ

(35)

Since, ∫ 2π

0einφeiqφdφ =

⎧⎨⎩0 if n �= −q,

2π if n = −q,

many terms cancel leading to

P =2π

T

N∑n=−N

∫ T

0

⎡⎣fT

−n(0) +

M−1∑p=1

fT−n

(ωp)eiωpt + fH

n (ωp)e- iωpt

⎤⎦

·[

M−1∑m=0

iωmvn(ωm)eiωmt − iωmv∗−n(ωm)e - iωmt

]dt

(36)

Once again the integral vanishes for many terms since

∫ T

0eiωmteiωptdt =

⎧⎨⎩0 if p �= −m,

T if p = −m.

Finally the injected power is given by

P = 2πN∑

n=−N

M−1∑m=1

iωmfHn (ωm)vn(ωm) − iωmfT

−n(ωm)v∗−n(ωm) . (37)

5.2 Dissipated Power Using Internal Energy Considerations

Another way to form the time average of the input power is through the powerconsumed by dissipation. Starting from the power expression (equation 37),the nodal force can be expressed in terms of the nodal displacement via theequation of motion

Dn(ω)vn(ω) = fn(ω). (38)

17

This formal expression can be expanded to

(A00 + inA01 − inA10 + n2A11 − ω2M

)vn = fn, (39)

where the matrices Aij describe the stiffness of the structure and M is themass matrix. Inserting this expression for the force into the power expression(equation 37) and making use of symmetry of the stiffness matrices and themass matrix, leads to the following expression for the dissipated power

P = 4πN∑

n=−N

M−1∑m=1

ωm(vHn (ωm) Im(A00)vn(ωm) + invH

n (ωm) Im(A01)vn(ωm)

− invHn (ωm) Im(A10)vn(ωm) + n2vH

n (ωm) Im(A11)vn(ωm)

− ω2mvH

n (ωm) Im(M)vn(ωm))

(40)

Observe that since losses proportional to the mass matrix are included in themodel the mass matrix is also present in the expression.

The advantage with this procedure is that it can be done element vise, whichmeans that the elements where a substantial part of the power is consumedcan be identified.

6 Results

The above mentioned procedures to calculate the dissipated power has beenemployed for three different surfaces: a test road managed by Renault, a testroad replica with a surface according to ISO standards for pass by noise testingwhich can be classified as smooth, and a road replica with a rough surface.The two latter ”roads” are mounted on a test drum managed by Goodyear.The tyre referred to as airplane tyre differs only in tread pattern to the tyremodeled in this investigation. It has three longitudinal grooves. The nct5 tyreis a commercial tyre with properties very similar to the modeled tyre. It doeshowever have a tread pattern. The tread patterns are, as mentioned before,accounted for in the contact model. The calculations take a stiff two hours ona modern (2006) laptop for 512 frequencies, 200 wave order and 516 degreesof freedom. The speed was 80 km/h and the load was close to 300 kg. Theresults are given in Table 1.

For comparison Table 1 also presents values coming from a measurement bythe Gdansk University of Technology (TUG). These were made on the airplanetyre with three longitudinal grooves mentioned above. The results from these

18

Case Dissipated Power [W]

Airplane tyre measured by TUG on safety walk (sandpaper) 601.68

Airplane tyre measured by TUG on rough road () 866.55

Airplane tyre on ISO road (mean load 309.9 kg) 859.89

nct5 on Renault Road (mean load 319.3 kg) 805.79

nct5 on Goodyears ISO road (mean load 297.0 kg) 702.73

nct5 on Goodyears Rough road (mean load 312.4 kg) 787.72

Table 1Measured and calculated dissipated power for different roads.

measurements were given as a rolling resistance coefficient which has beenscaled in accordance with the semi-empirical model (load 300kg) to producevalues of the dissipated power, see equation (2). As can be seen in Table 1, thecalculated power gives reasonable values. The rough drum replica gives largerlosses than the smooth drum, which also is in accordance with literature.

The dissipated power for the Renault road as a function of frequency and waveorder can be seen in Figure 4 and 5 respectively. The reason that the frequencyspectrum looks so rough is that only two revolutions have been used for thecalculation. If the contact forces were truly periodic every other frequencycomponent would cancel out. By using more non identical revolutions the re-sult would probably look much smoother. A substantial part of the dissipationoccurs below 100 Hz and at a wave order around 3. Most of the dissipatedpower occurs in and below the tread. This is illustrated, for the Renault roadin Figure 6 and 7. The lack of symmetry is once again attributed to the factthat too few revolutions were used. There are almost no losses occurring inthe side wall which is in contradiction with [7] who claims that roughly 30 %of the total power loss appears in the upper and lower side wall. The totaldamping level in the model is estimated quite accurately [2], but the distri-bution of the damping, in the different parts of the tyre, is probably wrong.which possibly explains the discrepancy. No results at other speeds loads orinflation pressure are available.

7 Conclusions

A car tyre is modeled with waveguide finite elements. The model is based ondesign data and accounts for: the curvature, the geometry of the cross-section,the pre-stress due to inflation pressure, the anisotropic material properties andthe rigid body properties of the rim. The motion of the tyre belt and side wallis described with quadratic anisotropic, deep shell elements that include pre-

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stress and the motion of the tread on top of the tyre by quadratic, Lagrangetype, isotropic two dimensional elements.

The non proportional damping used in the model is based on measured mo-bilities.

External forces resulting from a non-linear contact model, for three differentroads are inserted and the responses are calculated. The dissipated power isthen calculated through the injected power and the power dissipated withinthe elements.

The results obtained with the two methods agrees perfectly with each other.Even if the power calculation should be the same for the different methodsit is still nice to see that they agree since there is an ocean of FE jugglingand administration between them. The calculation result compares favorablywith those from literature [8] and measurements. The rough road dissipatesmore power then the smooth road. There are almost no losses occurring inthe side wall which is in contradiction with [7] who claims that roughly 30 %of the total power loss appears in the upper and lower side wall. The totaldamping level in the model is fine [2], but the distribution of the damping, inthe different parts of the tyre, is probably wrong, which possibly explains thediscrepancies. Since the visco elastic data is extremely important for a rollingresistance estimation, the damping should be established in a more scientificway, concluding that the results for for rolling resistance are promising butthat further work is needed.

8 Future work

Future work consists in fine tuning the tyre model with regards to dampingand to use longer contact forces in the time domain. Based on measurementsof the dynamic shear modulus a frequency dependant tread will be introduced.The damping of the belt and side wall will also be estimated in a more scientificway based on an optimisation routine where the modal damping ratios will beused as an error criterion.

Longer contact forces will lead to a finer frequency resolution and more rev-olution would be used to form the time averaged input power leading to asmoother power spectrum.

An investigation of the influence of certain tyre parameters would also beinteresting. It would be possible to change the speed, the external load, andperhaps the wear of the tyre (by reducing the tread height). Finally, severaldifferent roads can be considered.

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9 Acknowledgements

This work was funded by the European Commission, ITARI, FP6-PL-0506437.Many thanks to Wolfgang Gnorich and. Andrzej Pietrzyk, Goodyear, for ad-vice and for sharing data for tyres and to Wolfgang Kropp, Patrik Anderssonand Frederic Wullens, Applied Acoustics, Chalmers, for advise and calculationof contact forces. The measurement of rolling resistance was made by GdanskUniversity of Technology.

References

[1] Environmental protection agency, EPA 430-R-02-003, Inventoryof U.S. Greenhouse gas emissions and sinks 2002.

[2] Fraggstedt M. and Finnveden S. A Waveguide Finite ElementModel of a Pneumatic Tyre. Paper A in this thesis, 2006.

[3] Wullens F. Excitation of tyre vibrations due to tyre/roadinteraction, PhD thesis, Applied Acoustics, Chalmers Universityof Technology, 2004.

[4] Andersson P., Larsson K., Wullens F. and Kropp W. HighFrequency Dynamic Behaviour of Smooth and PatternedPassenger Car Tyres, Acta Acustica United With Acustica Vol.90 (2004) 445 -456.

[5] Larsson K. and Kropp W. A high-frequency three-dimensionaltyre model based on two coupled elastic layers. Journal of Soundand Vibration, 253(4):889-908, 2002.

[6] ISO 18164 Passenger car, truck, bus and motorcycle tyres -Methods of measuring rolling resistance, 2005.

[7] Hall D.E. and Moreland J.C. Fundamentals of rolling resistance,Rubber Chemistry and Technology 74 (3): 525-539 JUL-AUG,2001.

[8] Wennerstrm E. Fordonsteknik, 8th edition, In swedish, KTH,2004.

[9] Hoogvelt R.B.J., Hogt R.M.M., Meyer M.T.M. and Kuiper E.Rolling resistance of passenger car and heavy vehicle tyres aliterature survey, TNO report 01.OR.VD.036.1/RH, December11th 2001.

[10] Schuring D. (Firestone Tire & Rubber Compagny) Rolling loss ofpneumatic tires, Rubber chemistry and technology, volume 53 p.600-727, 1980.

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[11] Sjoberg M. On Dynamic properties of rubber isolators, PhDthesis, Department of Vehicle Engineering, KTH 2002, ISSN 1103-470x.

[12] Schuring D. and Futamura S. (Central Research, BridgestoneFirestone Inc.) Rolling loss of pneumatic highway tires in theeighties, Rubber chemistry and technology, Vol. 63, pp. 3 15-367,1990.

[13] Round Robin Test Rolling Resistance / Energy ConsumptionDWW-2005-046, 2005.

[14] Stutts D.S. and Soedel W. A Simplified Dynamic Model of theEffect of Internal Damping on the rolling resistance in pneumatictires, Journal of Sound and Vibrarion 155 (1), 153-164, 1992.

[15] Kim S.-J., and Savkoor A.R. The Contact Problem of In-PlaneRolling of Tires on a Flat Road, Vehicle System DynamicsSupplement 27, pp. 189-206, 1997.

[16] Yam L.H., Guan D.H., Shang J. and Zhang A.Q. Study ontyre rolling resistance using experimental modal analysis, Int. J.Vehicle Design, Vol. 30, No. 3, pp. 251-262, 2002.

[17] Popov A.A., Cole D.J., Cebon D. and Winkler C.B. EnergyLoss in Truck Tyres and Suspensions. Vehicle System DynamicsSupplement 33 , pp. 516-527, 1999.

[18] Morse P.M. and Feshbach H. Methods of theoretical physics,Chapter 3. 1953.

[19] Finnveden S. Exact spectral finite element analysis of a railwaycar structure, Acta Acustica, 2, 461-482, 1994.

[20] Finnveden S. and Fraggstedt M. Waveguide finite elements forcurved structures, TRITA-AVE 2006:38.

[21] Andersson P. Modelling interfacial details in tyre/road contact-Adhesion forces and non-linear contact stiffness, PhD thesis,Applied Acoustics, Chalmers University of Technology, 2005.

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0 50 100 150 200 250 300 350 400 450 5000

10

20

30

40

50

60

70

80

Frequency [Hz]

Pow

er [W

]

Fig. 4. Dissipated power as a function of frequency. The bandwidth is 5.6 Hz. Mostof the dissipation occurs below 100 Hz.

0 5 10 15 20 25 300

20

40

60

80

100

120

Waveorder

Pow

er [W

]

Fig. 5. Dissipated power as a function of wave order. A substantial part of thedissipated power occur at a wave order of around 3.

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0 5 10 15 20 25 300

10

20

30

40

50

60

70

80

90

Element number

Pow

er [W

]

Fig. 6. Power dissipation in the different elements. Belt elements (solid), Treadelements (dashed)

−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

]

Fig. 7. The arrow indicates elements where a lot of power is consumed.

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