The vibration of a tyre is predicted using the wave finite element (WFE) method. The material
properties are considered to be frequency dependent. The WFE method starts from a
conventional finite element (FE) model of only a short section of a structure, which is uniform in
one direction. Existing element libraries can be fully utilised and the size of the FE model can be
very small. Free wave propagation characteristics are extracted from the dynamic stiffness
matrix of the FE model. The forced response is calculated using the wave approach. An
approach for determining the amplitudes of the directly excited wave is proposed to reduce
numerical difficulties. The method is applied to a tyre FE model formed using ANSYS. The
material properties of rubber are considered to be frequency dependent. The free wave
propagation is shown including nearfield waves. The predicted forced response is compared
with experimental data. Good agreement can be seen on the whole

Brennan, Michael

Mace, Brian

Waki, Yoshiyuki

English

Southampton (e-Prints Soton)

                                      19th INTERNATIONAL CONGRESS ON ACOUSTICS                          MADRID, 2-7 SEPTEMBER 2007                                                                              VIBRATION ANALYSIS OF A TYRE MODEL USING THE WAVE FINITE ELEMENT METHOD   PACS: 43.50.Lj  Waki, Yoshiyuki a; Mace, Brian b; Brennan, Michael c ISVR, University of Southampton, Highfield, Southampton SO17 1BJ, United Kingdom;  a yw@isvr.soton.ac.uk, b brm@isvr.soton.ac.uk, c mjb@isvr.soton.ac.uk   ABSTRACT The vibration of a tyre is predicted using the wave finite element (WFE) method. The material properties  are  considered  to  be  frequency  dependent.  The  WFE  method  starts  from  a conventional finite element (FE) model of only a short section of a structure, which is uniform in one direction. Existing element libraries can be fully utilised and the size of the FE model can be very  small.  Free  wave  propagation  characteristics  are  extracted  from  the  dynamic  stiffness matrix  of  the  FE  model.  The  forced  response  is  calculated  using  the  wave  approach.  An approach for determining the amplitudes of the directly excited  wave is proposed to reduce numerical difficulties. The method is applied to a tyre FE model formed using ANSYS. The material  properties  of  rubber  are  considered  to  be  frequency  dependent.  The  free  wave propagation is shown including nearfield waves. The predicted forced response is compared with experimental data. Good agreement can be seen on the whole.    1. INTRODUCTION  Tyre noise is becoming a significant source of traffic noise [1] and understanding the vibrational behaviour of a tyre is thus becoming more important. At high frequencies, where FE models become impractically large, knowledge of the wave properties of a structure is of great value. Wave approaches can then be used. In this paper, the WFE method [2,3] is applied to model the free and forced vibration of a tyre. The formulation is first reviewed. Element matrices of a short  section  of  a  uniform  structure  are  post-processed  to  yield  the  wave  behaviour.  The matrices can be obtained from a conventional FE method and a commercial package. This is in contrast  to  the  spectral  finite  element  method  [4].  A  short  section  of  an  unloaded  tyre  is modelled  in  ANSYS.  Complete  structural  details  can  be  included,  which  may  be  difficult  in analytical approaches (e.g. [5]).    2. FORMULATION OF THE WFE METHOD This  section  describes  the  formulation  of  the  eigenvalue  problem for  determining  free  wave propagation characteristics from a conventional FE model.  2.1 Dynamic stiffness matrix Consider a short section of length  ∆ of a uniform waveguide. The dynamic equation of the section can be written as [3]    = Dq f ;  2 jω ω = + − D K C M  (Eq. 1)  where  D  is the dynamic stiffness matrix,  q  is the displacement vector,  f  is the force vector and  , , K C M  are  the stiffness, damping and mass matrices, which may  be formed using a commercial FE package. Time harmonic motion j t eω  is implicit through this paper. If the section has interior nodes, the associated degrees of freedom (DOF) can be condensed by assuming that no external force is applied to the interior nodes [2]. (Eq. 1) can then be expressed in matrix form as     19th INTERNATIONAL CONGRESS ON ACOUSTICS – ICA2007MADRID 2    LL LR L LRL RR R R=            D D q fD D q f  (Eq. 2)  where the subscripts L and  R  represent the left and right hand side of the section.   2.2 Free wave propagation in a uniform waveguide After applying the periodicity condition  R L λ = q q , the transfer matrix can be expressed in the form of an eigenvalue problem as [2]   q qf fλ   =       φ φTφ φ  (Eq. 3)  where the transfer matrix  T  is formed from the elements of the dynamic stiffness matrix. The eigenvalues of (Eq. 3) represent the wavenumbers k , i.e.   jkii e λ− ∆ =   (Eq. 4)  for the ith wave mode. The wavenumbers may be real, imaginary or complex. The associated ith wave mode is given by the eigenvector;    T T Ti qi fi   =   φ φ φ   (Eq. 5)  where T ⋅   is  the  transpose.  The  wave  mode  contains  information  about  both  the  nodal displacements and associated internal forces. For uniform waveguides, there exist positive- and negative-going  wave  pairs  of  the  form i jki e λ∆ ± =∓   and  the  eigenvalues  and  associated eigenvectors are ( ) , i i λ+ φ  and ( ) 1 , i i λ− φ . The superscripts ±  represent positive- and negative-going  waves.  Positive-going  waves  are  such  that  1 i λ <   or,  if  1 i λ = ,  the  energy  flow  is positive, i.e.  { }H Re 0 > f q ￿  where H ⋅  represents the complex conjugate transpose.  Numerical difficulties may occur when the eigenvalue problem, (Eq. 3), is formed and solved. Zhong’s  method  [6]  may  be  used  to  improve  the  matrix conditioning.  Numerical  issues  and implementations are discussed in detail in [7].   3. FORCED RESPONSE CALCULATION USING THE WAVE APPROACH The forced response can be obtained using the wave approach. The amplitude of the directly excited waves is first found and the total wave amplitudes are determined by considering the propagation  and  reflection  of  the  directly  excited  waves  in  general.  A  well-conditioned formulation to determine the amplitude of the excited waves is proposed.  3.1 Excited wave amplitude When an external force is applied to an infinite waveguide, continuity of displacement and force equilibrium give    , q q ext f f+ + − − + + − − = + − = Φ e Φ e f Φ e Φ e 0   (Eqs. 6)  where ± e  is a column vector of excited wave amplitudes,  ext f  is the external force vector and the matrices  q± Φ , f± Φ  contain the displacement or force eigenvectors, i.e.   1 q q qn± ± ± =     Φ φ φ ￿  and  n is the number of wave modes. The excited wave amplitudes, ± e , may be determined from (Eqs. 6). However, numerical problems are likely to occur because of ill-conditioning. A numerical implementation is therefore proposed which exploits the orthogonality of the left and right eigenvectors of the transfer matrix [2]. When all the eigenvalues are distinct, orthogonality of  eigenvectors  implies  ( )T1 n diag d d = Ψ Φ ￿ ,  where  Ψ   is  the  left  eigenvector  matrix    19th INTERNATIONAL CONGRESS ON ACOUSTICS – ICA2007MADRID 3 formed  in  the  same  manner  as  the  right  eigenvector  matrix.  The  eigenvectors  can  be normalised so that  1 i d =  and T = Ψ Φ I , where  I  is the identity matrix. With the normalised eigenvectors, premultiplying (Eqs. 6) by T Ψ  leads to   T T , . q ext q ext+ + − − = − = − e Ψ f e Ψ f   (Eqs. 7)  (Eqs. 7) are always well-conditioned. In practice, only the first  ( ) m n ≤  waves might be retained, these being waves for which  ( ) Im k  are small. The rapidly decaying waves often give a small contribution to the response.  3.2 Total wave amplitude When a waveguide forms a ring or a toroidal shape such as a tyre, the wave amplitudes ± a  at the excitation point are given by    ( ) { } ( ) { }1 1, L L− − + + − − − = − = − − a I τ e a I τ e e   (Eqs. 8)  where  ( ) L τ  is the wave propagation matrix whose diagonal elements are i jk L e− , and  L is the circumferential  length  of  the  waveguide.  The  displacement  is  then  given  by ( )1ni qi i qiia a+ + − −== + ∑ q φ φ .    4. APPLICATION TO TYRE VIBRATION ANALYSIS The WFE method is applied to a tyre model. The dispersion relationship is determined including nearfield waves. The point mobility is calculated and compared with experimental data.  4.1 Tyre model A test tyre with smooth tread (195/65R15) was provided by Bridgestone Corporation together with geometric and material property data. A section of the tyre was modelled using ANSYS 7.1 as shown in Figure 1. A tyre is a complicated structure which is composed of several different rubbers, steel and textile fibres. The solid element SOLID46, which generates equivalent single material properties for a layer structure, was used. A series of elements was concatenated to represent  the  curvature  in  the  circumferential  direction.  The  angle  of  1.8  degrees  in  the circumferential direction was modelled. An internal pressure of 200kPa was simulated by the application  of  surface  loads  on  the  elements.  Necessary  constraints  including  the  fixed boundary conditions at the bottom of the section were imposed. The number of DOFs was 324 after the condensation of interior DOFs.                Figure 1.-Tyre model; (a) in the section, (b) in the circumferential direction.   4.2 Frequency dependent material property The material properties of a rubber depend on frequency. The stiffness matrix was decomposed as  ( ) ( ) fibre rubber tension f f = + + K K K K   where  2 f ω π =   is  frequency,  the  stiffness  matrices (a)  (b)    19th INTERNATIONAL CONGRESS ON ACOUSTICS – ICA2007MADRID 4 fibre K  and  tension K  represent the frequency independent contributions of the fibres and the in-plane tension due to internal pressure. The latter was derived from the difference between two stiffness matrices associated with FE models with and without internal pressure. The frequency dependent properties of the rubber were included into  ( ) rubber f K . Since the stiffness matrix is proportional to the Young’s modulus  E  if the Poisson ratio is assumed constant,  ( ) rubber f K  at a frequency  f  is given by    ( ) ( )( ) ( ) ( ) { } 001 rubber rubberE ff f j fE fη = + K K   (Eq. 9)  where  ( ) f η  is the frequency dependent loss factor and  0 f  is a reference frequency at which the stiffness matrix  ( ) 0 rubber f K  is evaluated.  In  this  numerical  example,  rubbers  are  assumed  to  behave  like  the  American  National Standards Institute (ANSI) standard polymer for which data is available in the literature [8]. In the frequency range of interest ( ) 0 2kHz f < ≤ ,  E  and η  may be approximated by     ( ) ( ) ( ) log log E E E f f α β = ⋅ + ,  (Eq. 10)   ( ) ( )2 log f f η η η α β = ⋅ + .  (Eq. 11)  The coefficients  0.1, 0.01, 0.1 E η η α α β = = =  were estimated from the literature [8] and  E β  was determined from given material data. For the tyre model, the effect of the stiffness change is small in the frequency range analysed because the magnitude of  fibre K  (and  tension K ) is much larger than that of the change in  ( ) rubber f K .  4.3 Free wave propagation The dispersion relationship is determined for an undamped case for the sake of clarity. Figure 2 shows  the  polar  wavenumber  (rad/rad)  below  600Hz  for  modes  where  the  tread  motion  is predominantly  in the plane of the tyre cross-section and not in the circumferential direction. Results  for  real,  imaginary  and  complex  (conjugates)  wavenumbers  are  shown.  Complex conjugate wavenumbers are only plotted close to their cut-off frequencies for clarity. Natural frequencies occur if Re(k)=1,2,··· and also if Re(k)=0 for the purely breathing modes [9] although they do not occur in the frequency range shown for the tyre model.  The anti-symmetric A1 mode cuts-on first. This comprises the side-to-side motion of the tread as illustrated in the figure. In the illustrations of wave modes, the solid line is the original shape and the dashed line is the deformed shape. The A1 and symmetric S1 wave modes are bouncing modes where the belt mass is vibrating on the sidewall stiffness. The A2, S2 wave modes are typical cross-sectional modes. Curve veering between two wave modes implies that the two wave modes are not orthogonal, and the deformed shape in general changes with frequency.   Negative  group  velocity  can  be  observed  especially  for  lower  order  wave  modes.  As  the imaginary part of the complex conjugate wavenumbers becomes zero, two waves cut-on with a nonzero  real  wavenumber.  One  wavenumber  increases  with  frequency  while  the  other decreases implying a negative group velocity. The wave associated with the negative group velocity becomes a pure nearfield (with imaginary wavenumber) as frequency increases, and cuts-on again as a predominantly shear wave. The wave mode shapes change relatively rapidly with  frequency,  as  the  wave  modes  for  TS1  at  different  frequencies  illustrate.  Above  about 400Hz, the existence of branches with negative group velocity cannot be clearly seen and the dispersion relationship becomes similar to that of a flat structure. The ‘apparent’ ring frequency is therefore estimated to be around 400 Hz, which agrees with previous literature [9].    19th INTERNATIONAL CONGRESS ON ACOUSTICS – ICA2007MADRID 5    -0.1 0 0.10.20.250.30.35       -0.1 0 0.10.20.250.30.35               0 100 200 300 400 500 600-4-20246810Frequency [Hz]-Im |k|                         Re |k|        [rad/rad] -0.1 0 0.1         -0.1 0 0.10.20.250.30.35         -0.1 0 0.1                           Figure 2.-Relationship between frequency and the polar wavenumber: ―: real, ····: imaginary, −·−: complex conjugate wavenumbers. Anti-symmetric (Ai) and symmetric (Si) wave modes, i=1,2,3. Ti denotes shear wave modes.   4.4 Forced response The input mobility at the centre of the tread is calculated using the formulation of (Eqs. 7). For a point force excitation, only symmetric modes are then excited. The frequency dependent loss factor (Eq. 11) is now included in  rubber K . All wavenumbers become complex because of the damping.  Figure  3  shows  the  magnitude  of  the  predicted  and  measured  mobility.  The experiment was performed in a freely supported condition using rubber strings such that the response below 30Hz is dominated by the boundary conditions. Good agreement can be seen on the whole. It should be noted that only about 70 pairs of positive- and negative-going wave modes are used to extract the numerical result. This implies that only 140 DOFs can express the dynamic behaviour of the tyre model below 2kHz.  The response may be characterised by four frequency regions, i.e. (I) below the first resonance (at which the S1 mode cuts-on) where the sidewall stiffness dominates the system motion, (II) between  the  first  resonance  and  the  frequency  where  the  S2  mode  cuts  on,  in  which  the response is like that of a beam on a flexible foundation, (III) at middle frequencies where the response is somewhat constant and plate-like and (IV) at high frequencies where the response is similar to that of an elastic half space. However, the experiment shows more or less a flat response even at high frequencies. Possible reasons for this are as follows. In the experiment, the excitation area is finite (a circular area of diameter 23.5mm is excited) so that higher order A1 S1 A2 S2 A3 S3 Ts1  Ts2  TA2 Ts1    19th INTERNATIONAL CONGRESS ON ACOUSTICS – ICA2007MADRID 6 wave modes cannot be properly excited. In addition, there is a resonance of the experimental rig around 4kHz, which could contaminate the result. Agreement at low and middle frequencies could be improved by updating the FE model parameters.    10210310-310-2Frequency [Hz]Magnitude [m/Ns] Figure 3.-Magnitude of the Input mobility at the centre of the tread;  ―: the WFE result, ····: experiment.   5. CONCLUSIONS The wave finite element (WFE) method has been applied to a tyre, including the frequency dependent material properties. The free and forced vibration of the tyre was determined. The FE model of a short section of the tyre was formed using the commercial package ANSYS and subsequently  post-processed.  The  model  had  only  324  DOFs.  Geometrical  and  structural details were included. Only 70 wave mode pairs (140 DOFs) were retained to predict the forced response. The dispersion relationship was found including nearfield waves and complex wave behaviour  was  observed.  The  forced  response  was  determined  by  the  wave  approach  in conjunction  with  a  numerical  implementation,  which  is  always  well-conditioned.  Good agreement with experimental data was seen. It can be concluded that the WFE method is a powerful tool to investigate the dynamic behaviour of a complex structure with small calculation cost.    ACKNOWLEDGEMENT The  authors  gratefully  acknowledge  Bridgestone  Corporation  for  providing  the  tyres  and material data.    References: [1] U. Sandberg, J.A. Ejsmont: Tyre/Road Noise Reference Book. Informex (2002). [2] D. Duhamel, B.R. Mace, M.J. Brennan: Finite element analysis of the vibrations of waveguides and periodic structures. Journal of Sound and Vibration 294 (2006)  205-220.  [3] B.R. Mace, D. Duhamel, M.J. Brennan, L. Hinke: Finite element prediction of wave motion in structural waveguides. Journal of the Acoustical Society of America 117,5 (2005) 2835-2843.  [4] S. Finnveden: Evaluation of modal density and group velocity by a finite element method. Journal of Sound and Vibration 273 (2004) 51-75.  [5] W. Kropp, K. Larsson, F. Wullens, P. Andersson: High frequency dynamic behaviour of smooth and patterned passenger car tyres. Proceedings of the Institute of Acoustics 26,2 (2004) 1-12.  [6] W.X. Zhong, F.W. Williams: On the Direct Solution of Wave Propagation for Repetitive Structures. Journal of Sound and Vibration 181,3 (1995) 485-501. [7] Y. Waki, B.R. Mace, M.J. Brennan: On Numerical Issues for the Wave/Finite Element Method. ISVR Technical Memorandum No:964 (2006).  [8] W.M. Madigosky, G.F. Lee, J.M. Niemiec: A method for modeling polymer viscoelastic data and the temperature shift function. Journal of the Acoustical Society of America 119,6 (2006) 3760-3765.  [9] Y.J. Kim, J.S. Bolton: Modeling of Tire Treadband Vibration. internoise 2001, Hague (2001) 2605-2610.  I  II  III  IV 

Vibration analysis of a tyre model using the wave finite element method

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