ABSTRACT: In terms of safety and environment, reduction of the noise generated by tire vibrations on a road is very significant. In order to study the vibration properties of vehicle tires, various methods have been presented in literature. In most of these methods, the global structure of tires has been modelled as circular ring, orthotropic plate, periodic or full 3-dimensional models. A brief review of the characteristics of these models and comparison of their dynamic behaviour are the main purpose of the current study. The tire is supposed to be subjected to an excitation caused by contact of the tire and road. Study of vibrational responses demonstrates that the validity of each model is limited to a certain frequency range. To employ the circular ring and orthotropic plate models, first, we require to estimate some structural and material data associated to the nature of these models. To this end, the vibrational response of a 3D model is considered to determine some properties such as radial and tangential stiffnesses in circular ring model or bending, foundation stiffness, and tension in orthotropic plate model. Furthermore, the effect of inflation pressure on the dynamic behavior of the mentioned models is examined. For verification, the dynamical behaviour of a tire is studied experimentally. Application of the present study can be contemplated in the prediction of rolling noise and rolling resistance.

KEY WORDS: Tire models; Circular ring; Orthotropic plate; Periodic FEM.

1 GENERAL GUIDELINES

A lot of research on the vibration properties of vehicle tires has been done during the last decades. In this purpose, one of the simplest methods is the model of rotating ring on the elastic foundation. Due to the completeness and simplicity, since the 1960s, this method has drawn the attractions of numerous researchers. Development of the method is pioneered by Clark [1], Tielking [2], and Bohm [3] who presented a method for calculating the dynamic behavior of a loaded pneumatic tire modeled as an elastically supported cylindrical shell. In these works, the tire sidewall effects were modeled by the radial springs. Pacejka [4] modeled the tire as a circular ring under pressure. By considering the circumferential springs for the elastic foundation, he developed models for the lateral vibration. The effect of structural damping on the study of dynamic response of the classical ring on the foundation for the first time was considered by Padovan [5] in 1976. Later, Potts et al. [6] studied the vibration of a rotating ring on an elastic foundation in terms of the material and geometric properties of the tire. In order to study the free vibration of a circular ring tire located on an elastic foundation, a finite element method was presented by Kung et al. [7] in 1987. Huang [8,9] studied the response of a rotating ring subjected to the harmonic and periodic loadings. In 2008, Wei [10] proposed an analytical approach to analyze the forced transient response of the tires modeled based on the ring on the elastic foundation. The next method to model the tire structure is the model of Timoshenko beam. In the recent decade, Pinnington and Briscoe [11] developed a one-dimensional wave model to describe the tire dynamics. The tire belt is represented as a tensioned Timoshenko beam in order to derive arbitrary sidewall

impedance. The waves, which propagate along the tire, take shear and rotational effects into account. The validity of the circular ring models is limited to the frequency less than 400Hz, when the wavelength is large enough compared to the width of the tire.

In order to study the vibrational tire properties in higher frequencies in 1989 Kropp [12] proposed the orthotropic plate model on a winkler foundation, where the tire belt is modeled as a finite plate which has different tangential and lateral properties. The foundation represents the effect of sidewalls as well as the inflation pressure. Also, the external tension forces due to the inflation pressure are considered in this model. In 2001, Hamet [13] proposed an analytical approach to study the impulse response of a tire modeled by a thin orthotropic plate under tension supported by an elastic foundation. Later, Larson and Kropp [14] developed a double-layer tire model including the tangential motion and the local deformation of the tread. Their model is appropriate for the modeling of radial and tangential vibrations at the high-frequency range. However, the model of the orthotropic plate is completely dependent on the results of experiments. In order to estimate the structural properties of the orthotropic plate tire model, Perisse et al. [15] presented a procedure for the experimental modal testing of a smooth tire in low and medium frequency.

However, applying finite element (FE) approaches, one can model more accurately the structural features of a tire. Thus, especially during the past decade, a considerable number of studies have been focused to investigate the wave propagation in a symmetrical periodic element of a tire based on FE

Tire/ Road Dynamic Contact; Study of Different Approaches to Modelling of a Tire

L. Pahlevani, D. Duhamel, H. Yin, G. Cumunel

Université Paris-Est, Ecole des Ponts ParisTech, Laboratoire Navier UMR 8205, 77455 Marne la Vallee, France email: ladan.pahlevani@enpc.fr, denis.duhamel@enpc.fr, yin@enpc.fr, gwandal cumunel@enpc.fr

Proceedings of the 9th International Conference on Structural Dynamics, EURODYN 2014Porto, Portugal, 30 June - 2 July 2014

A. Cunha, E. Caetano, P. Ribeiro, G. Müller (eds.)ISSN: 2311-9020; ISBN: 978-972-752-165-4

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techniques. Brillouin [16] and Mead [17] applied the Flo-quet’s principle or the transfer matrix to study the wavepropagation in a 3-dimensional(3D) periodic structure. In2005, Houillon et al. [18] and Mace et al. [19] developeda FE method to determine the propagation constants andwave modes. Their works were focused to obtain disper-sion relations and their application in energetic methods.Recently, Duhamel et al. [20] employed a similar methodto calculate dispersion relations but for point force re-sponses. Their approach was called the Waveguide FiniteElement (WFE). Furthermore, this technique was appliedby Waki et al. [21] to predict the free wave propagationand the forced response of a tire. The results obtainedby using WFE method is similar to those obtained withthe classical FE approach. But, the computational costof the WFE method is very low compared to the usualFE. To this end, one can easily use this model to analyzestructures with complex geometries and material distri-butions. In addition, applying a reduction technique, thenumber of degree of freedom (DOF) in a periodic elementcan be greatly reduced so that it can significantly shortenthe computation time.

In the current work, we mainly review the above men-tioned models of tire with a focus on their vibrationalresponse. For this purpose, we calculate the dynamicbehavior of these models, discuss their assumptions andlimitations, and compare their results with the experi-mental results. In case studies, the effects of inflationpressure in tire is studied. In order to verify the resultsgiven by periodic 3D model, a full 3D tire model isanalyzed numerically via a FE technique.

2 BRIEF REVIEW ON TIRE MODELS

In this section, we summarize the existing models of tireproposed in literature and discuss the assumptions, limi-tations, and drawbacks of each model. The models can beclassified into five categories; rotating ring, Timoshenkocircular beam, orthotropic plate, periodic 3D and full 3Dmodels. The properties of these models are demonstratedin detail at the following.

2.1 Rotating Ring Model

The rotating ring model is one of the simplest methodsused to model a tire. In this model, it is supposed thatthe automobile tire is composed of two main parts; thebelt band (tread) and the sidewalls. Based on the infla-tion pressure of the tire, the sidewall may provide three-way (radial, tangential, and lateral) elastic foundation forthe belt. Here, the tread is modeled as a rotating ring andthe elastic properties of sidewall are modeled by the dis-tributed springs, kr and kθ in radial and circumferentialdirections, respectively, i.e. the lateral stiffness is ignored(Fig. 1). It is assumed that there is a punctual contact be-tween the ring and the road. In addition, the slip betweenthe ring and the road surface is ignored. Considering ur

and uθ as the displacements in radial and tangential di-rections, respectively, for the rotating tire with no contact,

the equations of motion are written as [9]

EI

R4(u

′′′′

r − u′′′

θ ) +EA

R2(ur + u

′

θ) +pb

R(ur + 2u

′

θ − u′′

r )

+ krur + ρAΩ2(2u′θ − u′′

r ) + ρA(ur − 2Ωuθ) = qr

EI

R4(u′′′

r − u′′θ )−

EA

R2(u′

r + u′′θ ) +

pb

R(uθ − 2u′

r − u′′θ )

+ kθuθ − ρAΩ2(2u′r + u′′

θ ) + ρA(uθ + 2Ωur) = qθ (1)

where, R, b, h, and ρ are mean radius, tread with, aver-aged thickness, and density of the tire, respectively. EIand EA indicate the bending and the membrane stiffness,respectively. I and A is moment inertia and surface of ringcross-section and E corresponds to the Young’s modulus oftire. p denotes the internal pressure and Ω indicates therotational velocity. q(qr, qθ) applys the external load ofthe system. In these equations, primes and dotes indicatethe differentiations with respect to theta and t, respec-tively. The displacements, ur and uθ, can be obtained byapplying modal analysis as the following

(ur, uθ) =n=+∞∑n=−∞

An(1, iCnei(nθ+ωnt)),

Cn =n3EI

R4 + nEAR2 + 2npb

R + 2ρAΩ(nΩ− ωn)

n2(EIR4 + EA

R2 ) +pb(n2+1)

R + kθ + ρA(n2Ω2 − ω2n)

(2)

where, ωn indicates the natural frequencies of the systemand An and Cn are constants. During the rotation, thecircular ring is subjected to the two external loadings; thefirst one is corresponding to the weight of vehicle and theother is the excitation due to the contact between the ringand the road. In the current study, the weight of vehicleis ignored. Generally, the equation of motion in the timedomain can be expressed as

Mu(t) + Cu(t) +Ku(t) = q(t) (3)

where M , C, and K are the mass, damping and stiffnessmatrices, respectively, while u and q denote the displace-ment and force vectors. Substituting Eq.(3) into Eq.(2)and considering the harmonic function einθ, one can ob-tain matrix kn as[

−i(d1 − 2Ωωn) d2 − ω2n

d3 − ω2n i(d1 − 2Ωωn)

](4)

where,

d1 =1

ρA[n3EI

R4+ n

EA

R2+ 2n

pb

R+ 2nρAΩ2]

d2 =1

ρA[n2EI

R4+

EA

R2+

pb(n2 + 1)

R+ ρAn2Ω2 + kθ]

d3 =1

ρA[n4EI

R4+

EA

R2+

pb(n2 + 1)

R+ ρAn2Ω2 + kr]

If the determinant of kn equals zero, the natural frequen-cies of the system are easily computed as the roots of this

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equation

ω4n − (4Ω2 + d2 + d3)ω

2n + 4d1Ωωn + d2d3 − d21 = 0 (5)

In the frequency domain, we have

u(ω) = G(ω)Q(ω), (6)

where G is the Green’s function which can be approxi-mated by a linear combination of Nm modes as

G(ω) = [−ω2M + iωC +K]−1

=

Nm∑n=1

φnφtn

(−ω2 + 2iξnωnω + ω2n)

, (7)

where, ωn and ξn indicate the natural frequency anddamping of each mode and φn are the mass-normalizedmode shape for mode n.

Fig. 1- Circular ring model

2.2 Timoshenko Circular Beam Model

In Timoshenko circular beam model, the belt is modeledas a Timoshenko beam to accommodate bending, shear,and the rotary inertia effects that are significant at highfrequencies. Similar to the circular ring model presented inthe previous section, the sidewall of the tire is replaced bythe radial and tangential springs, kr and kθ. ConsideringR as the mean radius of the tire, in the linear case, one canobtain the equations of motion of the Timoshenko circularbeam as the form of

(GA

R+

GI

R3)(u

′′

r − u′

θ

R+ α

′)− EA

R(ur + u

′

θ

R)

− EI

R3(ur + u

′

θ

R− α

′) + p− krur + ρAΩ2

(−R+ u′′

r − 2u′

θ − ur) + ρA(ur − 2Ωuθ) = qr

EA

R(u

′

r + u′′

θ

R) +

EI

R3(u

′

r + u′′

θ

R− α

′) + (

GA

R+

GI

R3)

(u

′

r − uθ

R+ α)− kθuθ − ρAΩ2(u

′′

θ + 2u′

r − uθ)

+ ρA(uθ − 2Ωuθ) = qθ

EI

R2(−u

′

r + u′′

θ

R+ α

′′)−GA(

u′

r − uθ

R+ α)− ρIΩ2α

′′

= 0 (8)

where, G and ν indicate the shear modulus and Poisson’scoefficient, respectively, and α denotes rotation along thez axis. The general format of the equation of motion intime domain is referred by Eq.(3). For the current model,in order to obtain matrix K, a numerical approach basedon the finite difference approximation technique accom-panying by the Newton-Raphson method is applied. Thefinite difference operators are defined as

u′(θi) =

u(θi+1)− u(θi−1)

2h

u′′(θi) =

u(θi+1)− 2u(θi) + u(θi−1)

h2(9)

where, θ = (θ1 = 0, θ2, . . . , θN = 2π(N−1)N ), N is the num-

ber of points considered on the circular beam. Note thatu(0) = u(2π).After calculating K matrix, we can determine the fre-quency response function of the system by applying themodal analysis.

2.3 Orthotropic Plate Model

In this model, the tire is simulated by the three dimen-sional plate as the tire belt, which has different tangentialand lateral properties, and the sidewalls modeled by a thinplate under tension on an elastic foundation (due to theinflation in the tire), Fig. 2. Based on the Kirchhoff hy-pothesis for thin plates, the equation of motion can bewritten as

[−T0x∂2

∂x2− T0y

∂2

∂y2+Bx

∂4

∂x4+ 2

√Bxy

∂2

∂x2

∂2

∂y2

+By∂4

∂y4+ s+m

′′ ∂2

∂t2]u(x, y, t) = F

′′(x, y, t) (10)

where, T0x and T0y are membrane tensions caused bythe inflation air in the tire. Bx, By, and Bxy are thelongitudinal bending, transversal bending, and the crossstiffness of the belt, respectively. For the orthotropicplates Bxy ≈

√BxBy. s, indicates the stiffness of the

elastic support of the belt. m′′and F

′′are the mass of

plate and the acting force per unit area, respectively.In the analysis, only harmonic motion is considered andthe common factor eiωt is omitted. Material losses areintroduced by adding an imaginary part to the bendingstiffness, tension and the stiffness of the foundation. Notethat only, the radial motion of tire (the vertical motionof the plate) is considered. The corresponding boundaryconditions are defined asu(x+ lx, y, t) = u(x, y, t), i.e. the belt is circularu(x, y, t) = 0 at y = 0, y = ly, i.e. the plate is simplysupported at the sides,

and, ∂2

∂y2 (x, y, t) = 0 at y = 0, y = ly.

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Fig. 2- Orthotropic plate model

Using Green’s function technique the solution of the equa-tion can be expressed as following

u(x, y, t) =

∫ ∫ ∫F

′′(x0, y0, τ)G(x, y, t)dx0dy0dτ (11)

where G(x,y,t) is solution of

[−T0x∂2

∂x2− T0y

∂2

∂y2+Bx

∂4

∂x4+ 2

√Bxy

∂2

∂x2

∂2

∂y2

+By∂4

∂y4+ s+m

′′ ∂2

∂t2]G(x− x0, y − y0, t− τ)

= δ(x− x0)δ(y − y0)δ(t− τ) (12)

Modeling the circular tire by an infinitely long strip andcalculating the corresponding Green’s function in termsof the superposition of normal modes one can obtain[13]

G(x, y, t) =4

lxly

1

m′′

∞∑n=1

sin(kyny0)sin(kyny)

∞∑m=0

ϵm

× cos[kxm(x− x0)]sin[Ωnm(t− τ)]

Ωnme−ηnmΩnm(t−τ)H(t− τ)

(13)

where Ωnm are the eigenfrequencies associated to thewavenumbers kxm = 2πm

lxand kyn = πn

ly. Ωnm is defined

as

Ωnm =Re([k4xmBx + 2k2xmk2yn√BxBy + k4ynBy

+ (k2xm + k2yn)T0 + s]12 /m

′′) (14)

The constant ϵ has the values ϵ0 = 12 and ϵm =0 = 1. η is

damping for each mode which it is calculated as

η =1

ΩnmIm([k4xmBx + 2k2xmk2yn

√BxBy + k4ynBy

+ (k2xm + k2yn)T0 + s]12 /m

′′) (15)

2.4 Periodic 3D Model

A symmetrical periodic element of the tire, as shown inFig. 3, is considered. The equation for time harmonicmotion of a periodic section can be written as

Du = q (16)

where, D = K + iωC − ω2M , is the dynamic stiffnessmatrix, u and q denote nodal DoFs and force vector, re-spectively. K, C, and M are stiffness, viscous dampingand mass matrices which are obtained from conventionalFE methods. If D is decomposed into left (L) and right(R) boundaries, interior degrees of freedom are indicatedby I, and also it is assumed that there are no externalforces on the interior nodes, the equation of motion maybe expressed asDLL DLR DLI

DRL DRR DRI

DIL DIR DII

×

uL

uR

uI

=

qLqR0

(17)

By eliminating the interior degrees of freedom this equa-tion can be condensed, also in order to decrease the num-ber of degrees of freedom, a reduced basis can be used[20]. After calculation of equivalent mass and stiffness ofthe system, one can apply modal analysis to obtain thefrequency response function.

Fig. 3- Symmetrical periodic element of a tire

2.5 Full 3D Model

The last model considered is full 3-dimensional tire model.This model is devoted solely to the verification of theresults obtained by periodic 3D model. For this purpose,the tire is analyzed numerically by the finite elementtechniques. Abaqus software is applied to model andanalyze the tire.

3 PREDICTION OF STRUCTURAL PROPERTIES RE-QUIRED IN CIRCULAR RING AND ORTHOTROPICPLATE MODELS

As described in the previous sections, in order to study thedynamic behavior of tires, two models of circular ring andorthotropic plate models, beside of 3D models, are em-ployed. In order to apply these models, first, we requireto estimate structural and material data associated to thenature of these models. For this purpose, a homogeneousgrooved tire is modeled by Abaqus software. We considerthat the inflation pressure in the tire is equal to 2 bars.The mechanical and structural properties of the tire are

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Table 1: The mechanical and structural properties of thehomogeneous tire

Internal radious 165.1 mmWidth of tread 165 mmHeight of sidewall 115.5 mmYoung modulus 80 MPaPoisson coefficient 0.49

summerized in Table 1. In the present study, a propor-tional damping system is considered, so that the viscousdamping matrix C is defined as

C = αM + βK (18)

where, M and K are mass and stiffness matrices of themodel respectively. α and β are damping factors whosevalues are computed based on an experimental measure-ment on the tire of Michelin R13/165/65 in which, α =14.4 and β = 8×10−5. The computed radial driving pointmobilitiy is shown in Fig. 4.

102

103

−45

−35

−25

Frequency [Hz]

20lo

g 10(Y

/Y0|)[

dB r

ef. Y

0=1

m/N

s]

Fig. 4- The computed radial driving point mobility

Substituting the first and second resonance frequencies ofthe given results into Eq.(5), one can determine the stiff-ness of the radial and tangential springs kr and kθ asdefined in circular ring model. Table 2 displays the re-sults.

In case of orthotropic plate model, based on the methoddescribed by Andersson et al. [22], foundation stiff-ness, s, tension, T0x and T0y, and bending stiffness,Bx, By, and Bxy, and their corresponding damping val-ues can be estimated. The results are presented in Table 3.

4 RESULTS AND DISCUSSIONS

To compare the various models of tire, several numericalstudies will be addressed in this section. Furthermore, theeffect of inflation pressure is addressed. In all the studies

Table 2: Structural and material properties required incircular ring model

kr 1.273e6 N/m2

kθ 9.643e5 N/m2

Table 3: Structural and material properties required inorthotropic plate model

T0x 4.8e4(1 + 0.09i) N/mT0y 8.3e4(1 + 0.09i) N/mbx 2.3(1 + 0.3i) Nmby 1.2(1 + 0.3i) Nmbxy 1.6(1 + 0.3i) Nms 3e4(1 + 0.01i) N/m3

presented in this section, the models are subjected to apunctual excitation force and the reported results arecorresponding to the excitation point.

In the first study, the radial point mobility calculatedby the models of circular ring, orthotropic plate andperiodic 3D are compared in Fig. 5. As it is expected,the models of rotating ring and Timoshenko circularbeam are in a good correspondence with the periodic 3Dmodel at low frequencies whereas, at high frequencies, thepoint mobility given by orthotropic plate model is verysimilar to the corresponding results of periodic 3D model.In order to verify the results given by tire models, anexperimental measurement is done. The results displayedin Fig. 5 approve that there is a good agreements betweenthe calculated and measured mobilities.

Second, the results of a full 3D and a periodic 3D modelsare compared. Fig. 6 demonstrates the point mobilitiespertinent to each model. As it is seen, the results oftwo models have a good agreement with each other,whereas, the computation time for the periodic 3D modelis significantly less than the full 3D model.

Finally, the effect of inflation pressure is examined. Thepoint mobilities given by the periodic 3D model at pres-sures 0, 1, and 2 bars are plotted at Fig. 7. It is observedthat, at the low frequencies enhancement of the pressurecauses the mobilities decline while the resonance frequen-cies increase. But, at the high frequencies the

102

103

−55

−45

−35

−25

Frequency [Hz]

20lo

g 10(Y

/Y0|)[

dB r

ef. Y

0=1

m/N

s]

Timoshenko Circular Beam

Rotating Ring

Periodic 3D Orthotropic PlateExperiment

Fig. 5- Comparison of the radial driving point mobilityobtained by the numerical and experimental models oftire

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102

103

−50

−40

−30

Frequency [Hz]

20lo

g 10(Y

/Y0|)[

dB r

ef. Y

0=1

m/N

s]

Periodic 3D

Full 3D

Fig. 6- Comparison of the radial driving point mobilityobtained by the full and periodic 3D models of tire

variation of the inflation pressure has no essential influ-ence on the mobility.

5 CONCLUSION

In summary, we outlined the characteristics of variousmodels of tire as circular ring, orthotropic plate and 3Dmodels. When the tire is subjected to an excitation, wecompared the vibrational responses of the models. Gen-erally, the results approve that the circular ring modelsare valid for low frequencies while the orthotropic platemodel provides reasonable result in high frequencies. Inaddition, the effect of the inflation pressure was studied.It is observed that there is a significant discrepancybetween the calculated point mobility and also resonancefrequency for different pressures at low frequency butby increasing the frequency the discrepancy fades away.Moreover, the results given by the models of tire wereverified by the experimental results. The presented studymay find potential applications in study of rolling noiseand rolling resistance.

ACKNOWLEDGMENTS

We acknowledge funding from ADEME under theDEUFRAKO ODsurf project.

REFERENCES

[1] S.K. Clark, The rolling tire under load. SAE Paper650493, 1965.[2] J.T. Tielking, Plane vibration charactrisrtics of apneumatic tire model. SAE Paper 650492, 1965.[3] F. Bohm, Mechanik des gurtelreifens. Ingenicar-Archiv35, 82-101, 1966.[4] H.B. Pacejka, Tire in-plane dynamics in: S.K. Clark,Mech. pneu. Tires 122, National Bureau of standsMonograph, Washington, DC, 1971.[5] J. Padovan, On viscoelasticity and standing wave intires. Tire Sci. Tech. 4, 233-246, 1976.[6] G.R. Potts, C.A. Bell, L.T. Charek, and T.K. Roy,Tire vibrations. Tire Sci. Tech., 202-225, 1977.[7] L.E. Kung, Radial vibration of pneumatic radial tires.General Tire Inc, 1987.

102

103

−50

−40

−30

Frequency [Hz]

20lo

g 10(Y

/Y0|)[

dB r

ef. Y

0=1

m/N

s]

P=0

P=2

P=1

Fig. 7- The effect of inflation pressure on the radialdriving point mobility obtained by periodic 3D model oftire

[8] S.C. Huang and W. Soedel, Response of rotating ringsto harmonic and periodic loading and comparison withthe inverted problem. J. sound vibr. 118(2), 253-270,1987.[9] S.C. Huang, The vibration of rolling tires in groundcontact. J. vehicle design 13, 78-95, 1996.[10] Y.T. Wei, L. Nasdala, and H. Rothert, Analysis offorced transient response for rotating tires using REFmodels. J. Sound Vibr. 320, 145-162, 2008.[11] Pinnington R.J., Briscoe A.R., A wave model for apneumatic tire belt. J. Sound Vibr. 253, 969-987, 2001.[12] W. Kropp, Structure born sound on smooth tire.Appl. Acoustics 26, 181-192, 1989.[13] J.F. Hamet, Tire/road noise: time domain Green’sfunction for the orthotropic plate model. Acustica 87,470-474, 2001.[14] K. Larsson and W. Kropp, A high-frequency three-dimensional tire model based on two coupled elasticlayers. J. Sound Vibr. 253, 889-908, 2002.[15] J. Prisse, J.M. Clairet, and J.F. Hamet J.F., Modaltesting of a smooth tire in low and medium frequency-estimation of structural parameters. IMAC XVIII,960-967, 2000.[16] L. Brillouin, Wave propagation in periodic structures,second edition, Dover, New York, 1953.[17] D.J. Mead, Wave propagation in continuous periodicstructures: research contributions from Southampton, J.Sound Vibr. 190, 495-524, 1996.[18] L. Houillon, M.N. Ichchouh, and L. Jezequel, Wavemotion in thin-walled structures, J. Sound Vibr. 281,483-507, 2005.[19] B.R. Mace, D. Duhamel, M.J. Brennan, and L. Hinke,Finite element prediction of wave motion in structuralwaveguides, J. Acous. Soc. Ame. 117, 2835-2843, 2005.[20] D. Duhamel, B.R. Mace, and M.J. Brennan, Finiteelement analysis of the vibrations of waveguides andperiodic structures. J. Sound Vibr. 294, 205-220, 2006.[21] Y. Waki, B.R. Mace, and M.J. Brennan, Free andforced vibrations of a tire using a wave/finite elementapproach. J. Sound Vibr. 323, 737-756, 2009.[22] P. Andersson, K. Larsson, and W. Kropp, A methodfor experimental collection of global material data fortires, Nordic Vibration Research, 2001.

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