Modeling and Experimental Verification of Vibration and Noise

Modeling and Experimental Verification of Vibration and Noise Caused by the Cavity Modes of a Rolling Tire Under Static Loading

Z.C. Feng Department of Mechanical and Aerospace Engineering, 2413A Lafferre Hall, University of Missouri, Columbia, MO 65211 Perry Gu 43 Calypso, Irvine, CA 92618 ABSTRACT Tire cavity noise refers to the vehicle noise due to the excitation of the acoustic mode of a tire air cavity. Although two lowest acoustic modes are found to be sufficient to characterize the cavity dynamics, the dynamical response of these two modes is complicated by two major factors. First, the tire cavity geometry is affected by the static load applied to the tire due to vehicle weight. Second, the excitation force from the tire-road contact changes position as the tire rotates. In this paper, we first develop dynamic equations for the lowest cavity modes of a rotating tire under the static load. Based on the model, we obtain the forces transmitted to the wheel from the tire resulting from the random contact force between the tire and the road surface. The transmitted forces along the fore/aft direction and the vertical direction show two peaks at frequencies that are dependent both on the tire static load and on the vehicle speed. We also analyze the dynamic spectra of the cavity air pressure. Our results show the presence of dominant peaks in the noise spectra. We further report experimental data on spindle responses and the dynamic pressure recorded by a sensor inside a tire. The results are in satisfactory agreement with the model prediction. Our work thus provides a basic understanding for the interaction of tire cavity excitation and a tire/wheel assembly which is critical to develop strategies of mitigating the tire cavity noise in the early stage of tire/wheel design. Keywords: tire cavity noise, acoustic modes, rotational effect, vehicle noise. 1. INTRODUCTION With the steady advancement in engine and drive-train noise reduction, noises caused by the interactions of the tire and the road are major culprits that adversely affect riding comfort in passenger cars [1-2]. This is especially true for electric cars for which the tire/road noise and wind noise are predominant noise 1   

sources. The tire-road interaction noises can be attributed to two main causes: structural and acoustical. The structural modes refer to the vibration of the deformable solid constituting the tire structure. The acoustic modes refer to the vibration of the air molecules inside the tire cavity. There have been intensive studies on the tire structural noise [3-9]. In the meantime, it has been found that the noise associated with the first acoustic mode of the tire air cavity is especially annoying since this noise has sharp peaks with frequencies typically in the range of 190-250Hz [10-14] under normal 60-80 km/h cruising conditions. The tires of a vehicle traveling on road surfaces are subject to dynamic forces from the tire-road interaction. The dynamic forces excite the acoustic modes of the tire cavities. The dynamic pressure of the air inside the tire cavity acts on the wheel which is supported by the suspension mounts. Through the suspension mounts, the force causes body vibration to generate vehicle interior noise. Although the dynamic characteristics of the parts involved in the transmission path are well known, trying to prevent cavity noise at the design stage has proven to be very frustrating to automotive engineers. It has been observed that for the same tire, wheels made of different materials such as steel and aluminum produce different noise levels; same wheels with different tires produce very different cavity noise levels. The current design guideline to prevent tire cavity noise is to allow 20 to 30 Hz modal separation between non-deformed tire cavity frequency and wheel modal frequencies. This guideline is shown to be inadequate in practice. In some cases, even worse tire cavity noise results from meeting this general design guideline. Although the noise recorded in the cabin is most likely dependent on the coupling between the acoustic mode of the tire and the vibrational mode of the wheel [15-19], we separate the force transmission through the tire acoustic cavity from the rest of the tire-wheel combination different from the study in [15]. We specifically focus our attention on the effect of the static load on the tire and the tire rotation rate as determined by the vehicle speed. In [20], a phenomenological model has been proposed that takes into account the tire load and the tire rotation rate. The model prediction on the transmissibility is compared 2   

with the experimental data and a good agreement was found. The present paper has two objectives. First, we derive the dynamical model in a rigorous manner using Lagrange’s equations. Although the dynamical model is essentially the same as the phenomenological model in [20], the new approach represents a nontrivial progress in modeling the cavity modes in a loaded and rotating tire. Second, we analyze the spectra of the sound pressure inside the tire and compare the results with the experimental measurements. The agreement between the model prediction and the experiments thus serves to demonstrate the relevance of the dynamic model in understanding the tire noise generation mechanism. In the following section, we present the derivation of the dynamic model using Lagrange’s equations. In Section 3, we present the dynamic responses of the cavity modes using frequency response functions. In Section 4, we derive the spectral density of the acoustic pressure inside the tire. The tire rotation introduces a time dependent relationship between the modal dynamics and the sound pressure. In Section 5, the model prediction is compared with the experimental data. 2. EQUATIONS GOVERNING THE CAVITY ACOUSTIC MODES The acoustic wave inside the tire cavity is governed by the wave equation. The complicated geometry of the tire cavity makes it impossible to obtain analytical solution of the wave equations. Since only the lowest acoustic modes have been identified as the major contributors to the cavity noise, we derive the equations governing the lowest two acoustic cavity modes using Lagrange’s equations. Consider the toroidal tire acoustic cavity. In Figure 1, the coordinate system o is attached to the rotating tire. The angle of rotation measured from a fixed horizontal axis is denoted by  . We assume that air density fluctuation in the cavity be:

 '   0 [ x(t ) cos   z (t ) sin  ]

3   

(1)

where x(t ) and z (t ) are the dimensionless amplitudes of the two modes and  0 is the gas density inside the tire at equilibrium. Air is assumed to be stationary relative to the tire. The angle  is measured from the  -axis which is attached to the rotating tire. The two acoustic modes correspond to density gradients along the  -axis and the  -axis respectively. Therefore x(t ) and z (t ) can be interpreted as the coordinates of the “center of mass” of the air particles inside the tire. To satisfy the continuity equation of the linear acoustic problem [21]

 '   0   v'  0 , t

(2)

we let the velocity fluctuation be the following:

v '  r[ x (t ) sin   z (t ) cos  ]e θ ,

(3)

where the unit vector e θ is defined in Figure 1; r and  specify the position of the air particle in the coordinate attached to the rotating tire. The kinetic energy in the case of linear acoustics approximation is given by [13, 21]

1 1 T    0 | v ' | 2 dV    0 [ x (t ) sin   z (t ) cos  ] 2 r 2 dV 2 2



1 x 2 z 2  0 [  (1  cos 2 ) r 2 dV  xz  sin 2 r 2 dV   (1  cos 2 ) r 2 dV ] 2 2 2

(4)

In this work, we only consider the cavity noise under constant-speed driving conditions. In calculating the kinetic energy, we have ignored the contribution to the kinetic energy from the tire rotation. In effect, we are ignoring the Coriolis force and the centrifugal force in the acoustic problem. The potential energy is given by 4   

0c 2 c2 ' 2 U  (  ) dV   [ x(t ) cos   z (t ) sin  ] 2 dV 2 0 2 

0c 2 x 2 2

(

2

 (1  cos 2 ) dV  xz  sin 2 dV 

z2 2



(1  cos 2 ) dV )

(5)

In the absence of any static load, we assume that the tire cavity is axi-symmetric and carry out the integration over  easily. When a static load is applied, the integration volume is defined by the deformed tire geometry. For a loaded and rotating tire, the tire deformation depends on time as well. However, it is reasonable to assume that the tire cavity geometry does not change other than translating at



the vehicle speed. Therefore the two integrals r 2 dV and

 dV are only affected by the tire static load but

not affected by the tire rotation. For convenience, we introduce the following notations

r

2

dV  I 0

(6)

and

 dV  V .

(7)

0

In addition to time independence assumption of the tire cavity geometry, we further assume that the deformed cavity geometry has a reflection symmetry about a vertical line as shown in Figure 1 (b). With these two assumptions, we may simplify the integrals in (4) and (5) by using the angle  ' defined in Figure 1(b). That is:

   '  

 2

(8)

The variable  ' is introduced such that all integrals are defined over a symmetric domain for

 '  [ ,  ] . Consequently, the volume integrations of odd functions of  ' are zero:

 sin 2 dV  0 '

5   

(9)

 sin 2 r

' 2

dV  0 .

(10)

Applying these properties, we obtain

 cos(2 )r

2

 sin(2 )r

dV   cos(2 '  2   )r 2 dV   cos(2 )  cos(2 ' )r 2 dV

(11)

dV   sin(2 '  2   )r 2 dV  sin(2 )  cos(2 ' )r 2 dV

(12)

2

Similarly, substituting (8) into the integrals in (5), we obtain

 cos 2 dV   cos(2 ) cos 2 dV

(13)

 sin 2 dV  sin(2 ) cos 2 dV

(14)

'

'

Note that the integrals in (11)-(14) are all zero for an unloaded axi-symmetric tire. Introduce the following notations:

 cos(2 )r '

2

dV   1 I 0

 cos 2 dV   V '

2

0

(15) (16)

where I 0 and V0 are defined in (6) and (7);  1 and  2 are dimensionless. The latter two are zero for an unloaded symmetric tire. They are small under the assumption that the tire deformation is small. With the above simplification, the kinetic and potential energies are:

1 x 2 z 2 T   0 I 0 [ (1   1 cos 2 )  xz 1 sin 2  (1   1 cos 2 )] 2 2 2

 

 

(17)

c2 x2 z2      U   0V0 [ (1   2 cos 2 )  xz 2 sin 2  (1   2 cos 2 )]                        (18)  2 2 2

Using Lagrange’s equations, we obtain the equations of motion in the following:

 2 sin 2   x  1   1 cos 2   1 sin 2   x c 2V0 1   2 cos 2 0    sin 2 1   cos 2   z  I   sin 2 1   2 cos 2   y  1 1 2    0  6   

(19)

Since  1 and  2 are small compared with one, applying the following approximation for arbitrary matrices

A and B : (I  A) 1 (I  B)  (I  A)(I  B)  I   ( A  B) ,

(20)

where I is the identity matrix, we obtain from (19) the following equations of motion for the two lowest modes of the acoustic cavity,

( 2   1 ) sin 2   x   0  x c 2V0 1  ( 2   1 ) cos 2    z  I  (   ) sin 2 1  ( 2   1 ) cos 2   z    2 1 0 

(21) 

For a tire rotating at a constant speed  =  , we may assume   t without loss of generality. The two equations in (21) are coupled by terms with periodic coefficients in the form of cos 2t and sin 2t . Solutions of ordinary differential equations with periodic coefficients are cumbersome in general. Instead of solving these equations, we proceed to calculate physical quantities we are interested, pretending the solutions of the (21) were already found. Focusing on the noise transmission path from the tire cavity to the vehicle suspension, we analyze the net forces acting on the wheel by the acoustic cavity. The acoustic pressure is given by [21]

p '  c 2  '  c 2  0 [ x(t ) cos   z (t ) sin  ] .

(22)

The net force in the fore/aft and vertical directions are thus given by

FX    p ' (t ,  ) cos(   )d A

(23a)

FZ    p ' (t ,  ) sin(   )d A

(23b)

where the integration is over the inner wall of the tire cavity or the surface of the wheel. Since the deformation of the wheel (mostly made of steel or aluminum) can be ignored, by substituting (23) into the above, we can carry out the integration to obtain: 7   

FX  we rw c 2  0 ( x cos t  z sin t )

(24a)

FZ  we rw c 2  0 ( x sin t  z cos t ) .

(24b)

where we is the effective width of the wheel and rw is the radius of the wheel. Therefore, once the dynamics of the acoustic modes are determined, we can obtain the forces acting on the wheel. On the other hand, if we introduce the following coordinate transformation,

U  x cos t  z sin t

(25a)

W  x sin t  z cos t ,

(25b)

the forces on the wheel are now simply:

FX  we rw c 2  0U

(26a)

FZ  we rw c 2  0W .

(26b)

Therefore, the forces on the wheel in the horizontal and vertical directions are proportional to U and

W respectively. Motivated by our previous work [20], we introduce the change of variables in (25). Corresponding to these changes of variables, we have the following relationship:

2 U   2 U   0           W   2 0  W   0

0  U  cos t      2  W   sin t

 sin t   x cos t   z

(27)

Substituting (21) into the above , we finally have:

c 2V0 U  [ (1   1   2 )   2 ]U  2W  0 I0

(28a)

c 2V0 W  [ (1   1   2 )   2 ]W  2U  0 . I0

(28b)

If we let

H  c

V0 (1   2   1 ) I0

8   

(29)

V  c

V0 (1   2   1 ) . I0

(30)

we obtain the following two equations:       U  ( H2   2 )U  2W  0  

  

 

 

(31a)

  (   )W  2U  0         W

 

 

 

(31b) 

2 V

2

Note that the special case of   0 corresponds to a statically loaded tire when x and z coincide with the horizontal and vertical directions respective. For this special case, this is no coupling between the two modes and the corresponding natural frequencies are given by (29) and (30), which can be calculated if the tire geometric deformation has been obtained. Since obtaining the tire deformation can be a computationally cumbersome process, it is often easier to obtain these two natural frequencies experimentally. When tire angular speed is constant but non-zero, the two equations in (31) are coupled. However, the following characteristic equation for the two couple modes is easily obtained:

( H2   2   2 )(V2   2   2 )  4( ) 2  0 , which gives the two resonance frequencies denoted as

12   22 

V2   H2 2

V2   H2 2

(32)

1 and 2 as follows:

 2 

1 2 (V   H2 ) 2  2( H2  V2 ) 2 4

(33)

 2 

1 2 (V   H2 ) 2  2( H2  V2 ) 2   4

(34) 

Note that the two resonance frequencies are dependent on the tire rotation speed as well as the natural frequencies of a loaded tire. In the absence of the static load, V   H   0 , where  0 the resonance frequency of the an un-deformed tire. Equations (33) and (34) simplifies to 1   0   and

 2   0   . In order to illustrate the dependence of the frequencies on the tire speed, we use the following data from tests:  0  2 (217) rad/sec;  H  2 (207) rad/sec; V  2 (222) rad/sec. 9   

The dependence of the resonance frequencies on the tire speed is shown in Figure 2 with all quantities non-dimensionalized by  0 . The two curves are nearly identical to the numerical solutions of the convective one-dimensional wave equation obtained by Gunda and Gau of Goodyear Technical Center [21]. They are also similar to the experimental curves given in Figure 4 of [13]. 3. DYNAMIC RESPONSES TO STOCHASTIC TIRE-ROAD INTERACTIONS With the cavity model in hand, it is easy to study the dynamic responses to stochastic tire-road interactions by adding the stochastic forces to the model. In the following we represent the tire-road interaction as a random force in the vertical direction:

U  2 0 (U  W )  ( H2   2 )U  2W  0

(35a)

W  2 0 (W  U )  (V2   2 )W  2U  f r

(35b)

where f r represents the random force and  is a proportionality coefficient;  represents the viscous damping of the acoustic modes. Thus the stochastic response can be obtained following the standard techniques for linear systems subject to a stochastic input [23, 24]. The response is constructed based on the system’s response to sinusoidal inputs. To obtain the system response to sinusoidal inputs, we let

f r  exp(i i t ) ,

(36)

U  G H ( i ) exp(i i t )

(37a)

W  GV ( i ) exp(i i t ) ,

(37b)

and substitute them into equations (35) to obtain the following algebraic equations:

[ i2  2i 0 i  ( H2   2 )]G H  2( 0  i i )GV  0

(38a)

 2( 0  i i )G H  [ i2  2i 0 i  (V2   2 )]GV   .

(38b)

10   

Solving these two equations for G H ( i ) and GV ( i ) , we then obtain plots of | G H ( i ) | and

| GV ( i ) | versus the input frequency  i . They are referred as X transmissibility and Z transmissibility in the following respectively. Parameters used in the following plots are based on test results of a particular tire [20]. Figure 3 shows the transmissibility for  /  0  0.03 and  /  0  0.05 . The two peaks occur at the frequencies close to the two resonance frequencies given by (33) and (34). Note that the Z transmissibility has a larger peak at the higher resonance frequency. This is because the higher resonance frequency corresponds to the vertical mode and the forcing from road is acting along the vertical direction. If the tire does not rotate, the horizontal mode is not excited. For slow tire rotation, coupling effect to the horizontal mode is weak. At slow vehicle speed, the transmissibility is dominated by a single peak close to 1 ( 1 >  2 ). As the vehicle speed increases, the peak at  2 becomes more significant. In equations (33) and (34), the frequencies

H and V refer to the resonance frequencies of a stationary loaded tire along the fore/aft and

vertical directions respectively. They are dependent on the load acting on the tire. In other words, the two peak frequencies in the transmissibility are dependent on the tire static load. In addition, the separation of the two peaks increases with the vehicle speed. 4. SPECTRAL DENSITY OF THE ACOUSTIC PRESSURE INSIDE THE TIRE CAVITY Once the frequency response functions G H ( i ) and GV ( i ) are determined, it is straightforward to obtain the spectral density function of U (t ) and W (t ) corresponding to white noise tire-road interaction force f r . Following the notations in [23, 24], we have

SU ( )  G H* ( )G H ( )

(39a)

SW ( )  GV* ( )GV ( ) ,

(39b)

11   

where the asterisk stands for complex conjugate of the function. If a pressure sensor is installed inside the tire cavity to measure the acoustic response: x(t ) and z (t ) , using equations (25), we obtain

x(t )  U (t ) cos(t   0 )  W (t ) sin(t   0 )

(40)

where the angle  0 indicates the location of the pressure sensor at time t=0. Knowing the spectral density function of U (t ) and W (t ) , we derive the spectral density of x(t ) in the following.

Using (40), we first obtain the autocorrelation function of x(t )

R x ( )  E[ x(t ) x(t   )] 1  {E[U (t )U (t   )]  E[W (t )W (t   )]} cos( ) 2 1  {E[U (t )W (t   )]  E[W (t )U (t   )]}sin( ) . 2

(41)

The above result is obtained after discarding explicit time dependent terms by assuming that the random processes are stationary. Take the Fourier transform of the autocorrelation function. Express the sinusoidal functions as exponential functions. Finally, we have

1 S x ( )  [ SU (  )  SU (  )  SW (  )  SW (  )] 4 

1 [ SUW (  )  SWU (  )  SUW (  )  SWU (  )] 4i

(42)

where the functions SU ( ) and SW ( ) are given in (39), and

SUW ( ) 

1 2

1 SWU ( )  2



R

UW

( )e i d  GV ( )G H* ( )

(43a)

WU

( )e i d  G H ( )GV* ( ) .

(43b)

 

R



12   

Figure 4 shows the spectral density calculated from equation (42) for parameters given in Figure 3. Since the frequency response function G H ( i ) and GV ( i ) are dominated by two peaks at 1 and

 2 ( 1 >  2 ), we expect four peaks at 1   and  2   . These four peaks are seen in Figure 4. Note that the center two peaks at 1   and  2   are very close to each other. They may be indistinguishable in experimental data with large uncertainties.   4. COMPARISON WITH THE EXPERIMENTAL DATA We have conducted vibrational and acoustic measurements on vehicles and tires. Accelerometers are fixed to the spindle of the suspension to measure the accelerations of the spindle along the fore/aft and vertical directions respectively. Although we cannot directly measure the forces transmitted to the spindle, the acceleration measurements provide resonance peaks which are expected to correspond to the peaks of the transmissibility studied in section 3. The acoustic measurement is on the interior pressure of the tire. The experimental setup includes a wireless microphone inside the tire cavity to measure cavity dynamic pressure. The microphone and its signal transmitter are built into a case which is attached to a wheel using hose clamps. An antenna is mounted on the fender of a testing vehicle to receive the dynamic pressure signal from the transmitter inside the tire cavity. The antenna is connected to a receiver whose output is connected to the frontend of a data acquisition system. The vehicle was tested for stationary and driving conditions. For the stationary condition, the modal frequencies of the tire cavity are obtained by impacting tire treads using an impact hammer. For the driving condition, the pressure response of the tire cavity microphone and the accelerometer responses of vehicle spindle vibrations are recorded simultaneously. When the static load on the tire is relieved, the cavity resonance mode has a frequency of 217 Hz. When the static load is applied, the slightly asymmetric acoustic cavity now has two resonance frequencies, 207 Hz and 222 Hz, in the interested frequency range. 13   

The tire used in our test is P225/50R17. The tire outer diameter is 656.8mm. Based on the diameter,

we calculated the tire rotation rate to be 6.500 Hz and 10.833 Hz when the vehicle travels at 48.28 km/h (30mph) and 80.47 km/h (50 mph) respectively. Based on the symmetric cavity resonance frequency of 217 Hz, the dimensionless tire rotation rates corresponding to these two vehicle speeds are  =0.030 and  =0.050 respectively.   Figure 5 shows the acceleration spectra of wheel spindle vibration for the driving conditions at 30 mph and 50 mph. Since Figure 5 shows the response, not the transmissibility as shown in Figure 3, the response level increases as the vehicle speed increases. Nevertheless we observe from Figure 5 the following. 1) The experimental data show two nearly equal peaks for the fore/aft direction and a significantly larger high frequency peak than lower frequency peak for the vertical direction; 2) The separation of the two peaks increases with vehicle speed. Both observations are in agreement with the model predictions shown in Figure 3.   Figure 6 shows the spectra of the pressure response of the cavity microphone for the driving conditions at 30 mph and 50 mph respectively. The presence of side peaks is clear despite the uncertainties of the data. The side peaks to the right of the main peak are more distinguishable, in agreement with Figure 4. Moreover, the spreading of the side peaks from the main peak increases with the vehicle speed. 5. CONCLUSIONS We have reported our work aimed at an understanding of the effect of tire rotation and static load on the dynamics of the tire cavity. The work reported in this paper extends the results obtained in [20] in which a phenomenological model was presented. Here the mathematical model for the tire cavity mode is derived from energy principles. The model is then directly used to derive the transmissibility of the tire when the cavity modes are included. Based on the model, the cavity pressure response to random road noise is

14   

obtained. It was found that acoustic pressure inside the tire has side peaks in addition to the main peak. The model predicted frequencies of the side peaks are found to agree with the experimental data.

Our model and its predictions can be considered as generalization of existing research on symmetric tires. In the absence of the static load, the tire symmetry is preserved. However, according to equations (38a) and (38b), coupling between the fore/aft and vertical degrees of freedom is still present. The transmissibilities are shown in Figure 7 for two different tire rotational speeds. According to equations (33) and (34), the two peaks are at 1   0   and

 2   0   . Therefore, the two peaks at 1   and  2   in the pressure spectrum coalesce at  0 for arbitrary  . In Figure 8, we have plotted the pressure spectrum for this special case for  /  0  0.03 and  /  0  0.05 . We note that the side peaks near 1   and  2   disappear. Although this is to be expected for a symmetric tire, proving this based on the expression in (42) is beyond the scope of the present work. ACKNOWLEDGEMENT

We are grateful to the three reviewers for their very detailed and constructive comments. We thank Dr. Andrzej Pietrzyk for beneficial discussions and for bringing to our attention numerical work done at Goodyear.

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REFERENCES [1] W.A. Leasure and E.K. Bender, Tire-road interaction noise. Journal of Acoustical Society of America Vol. 58 (1), 39-50. 1975. [2] U. Sandberg, J.A. Ejsmont, Tyre/Road Noise Reference Book, Informex, Kisa, Sweden, 2002. [3] Y.B. Chang, T.Y. Yang, and W. Soedel, Dynamic analysis of a radial tire by finite elements and modal expansion, Journal of Sound and Vibration, 96 (1), 1984, 1-11. [4] Y.J. Kim, J.S. Bolton. Effects of rotation on the dynamics of a circular cylindrical shell with applications to tire vibration. Journal of Sound and Vibration, 275 (3-5), 2003. 605-621. [5] R.J. Pinnington. A wave model of a circular tyre. Part 1: belt modeling. Journal of Sound and Vibration, 290 (1-2), 2006, 101-132. [6] I. Lopez, R.E.A. Blom, N.B. Roozen, and H. Nijmeijer, Modelling vibrations on deformed rolling tyres-a modal approach. Journal of Sound and Vibration, 307, 2007, 481-494. [7] I. Lopez, R.R.J.J. van Doorn, R. van der Steen, N.B. Roozen, H. Nijmeijer, Frequency loci veering due to deformation in rotating tyres. Journal of Sound and Vibration, 324, 622-639, 2009. [8] P. Kindt, P. Sas, W. Desmet, Development and validation of a three-dimensional ring-based structural tyre model. Journal of Sound and Vibration, 326, 852-869, 2009. [9] J. Cesbron, F. Anfosso-Ledee, D. Duhamel, H.P. Yin, D. Le Houedec. Experimental study of tyre/road contact forces in rolling conditions for noise prediction. Journal of Sound and Vibration, 320, 125-144, 2009. [10] T. Sakata, H. Morimura, and H. Ide, 1990. Effect of tire cavity resonance on vehicle road noise. Tire Science and Technology, 18, 68-79. 16   

[11] J. Thompson, 1995. Plane wave resonance in the air cavity as a vehicle interior noise source. Tire Science and Technology, 23, 2-10. [12] R. Gunda, S. Gau, and C. Dohrmann, 2000. Analytical model of tire cavity resonance and coupled tire/cavity modal model. Tire Science and Technology, 28, 33-49. [13] H. Yamauchi and Y. Akiyoshi, 2002. Theoretical analysis of tire acoustic cavity noise and proposal of improvement technique. JSAE Review, 23, 89-94. [14] P. Gu, A. Ni, J. Park, and D. Schaffer. Study of tire acoustic cavity resonance. Ford Global Noise and Vibration Conference, October 21-22, 2002. [15] E. Rustighi, S.J. Elliott, S. Finnveden, K. Gulyas, T. Mocsai and M. Danti, 2008, Linear stochastic evaluation of tyre vibration due to tyre/road excitation, Journal of Sound and Vibration 310, pp. 11121127. [16] R. W. Scavuzzo, L.T. Charek, P.M. Sandy, and G.D. Shteinhauz. Influence of wheel resonance on tire acoustic cavity noise. SAE International Congress & Exposition, Detroit, Michigan, February 28March 3, 1994, Paper Number 940533, 1-6. [17] P. Gu, Y. Chen, Z, Li, H. Kim, S. Bi, and B. Schandevel, “Development of Vehicle and Systems Design Specifications for Mitigating Tire Cavity Noise”, Ford Global Noise and Vibration Conference, June 11-12, 2008. [18] L.R. Molisani, R.A. Burdisso, and D. Tsihlas. A coupled tire structure/acoustic cavity model. International Journal of Solids and Structures, 40, 2003, 5125-5138. [19] T. Hayashi. Experimental analysis of acoustic coupling vibration of wheel and suspension vibration on tire cavity resonance. SAE 2007-01-2345.

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[20] Z.C. Feng, Perry Gu, Yongjian Chen, and Zongbao Li, Modeling and Experimental Investigation of Tire Cavity Noise Generation Mechanisms for a Rolling Tire, SAE Int. J. Passeng. Cars - Mech. Syst. 2(1): 1414-1423, 2009. [21] Rajendra Gunda and Steve Gau, 2000, Effect of rotation/loading on acoustic modes of the tyre cavity, Tire Society Conference, Akron, 2000. [22] A.D. Pierce. Acoustics: An Introduction to Its Physical Principles and Applications. Acoustical Society of America, Woodbury, New York. p. 38, 1991. [23] P.H. Wirsching, T.L. Paez, and H. Ortiz. Random Vibration: Theory and Practice. John Wiley & Sons, Inc., New York, 1995. [24] D.E. Newland. An Introduction to Random Vibrations, Spectral & Wavelet Analysis. 3rd Ed. Longman Scientific & Technical, Essex England, 1993.

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Figure Captions Figure 1. (a) The coordinate system for the tire. (b) Definition of the variable  ' . Figure 2. Dependence of the two resonance frequencies on the tire speed  . The parameters are

 0  2 (217) rad/sec;  H  2 (207) rad/sec; V  2 (222) rad/sec. Figure 3. Transmissibility of a loaded tire.  /  0  0.03 , and  /  0  0.05   0.01 ,   0.01 , and

 0  1 . (a) Transmissibility in fore/aft direction; (b) Transmissibility in vertical direction. Figure 4. Power spectrum of the acoustic pressure inside the tire cavity plotted in log scale. (a):  /  0  0.03 ; (b)  /  0  0.05 . Other parameters are   0.01 ,   0.01 , and  0  1 . Arrows indicate the four peaks.  

Figure 5. Wheel acceleration response at vehicle speed 48.3 km/h and 80.5 km/h respectively: (a) fore/aft acceleration; (b) vertical acceleration.

Figure 6. Spectrum of the pressure sensor measurements at vehicle speed of 48.3 km/h and 80.5 km/h respectively. The pressure sensors are mounted inside the tire.

Figure 7. Transmissibility of an un-loaded tire at vehicle speeds corresponding to  /  0  0.03 and  /  0  0.05 respectively: (a) fore/aft transmissibility; (b) vertical transmissibility. Other parameters used are:   0.01 ,   0.01 , and  0  1 .

Figure 8. Power spectrum in log scale for an un-loaded tire: (a)  /  0  0.03 ; (b)  /  0  0.05 . Other parameters used are   0.01 ,   0.01 , and  0  1 . An arrow indicates the only peak.

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(a)

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(b)

Z

er





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

 



 

 

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X

  Figure 1   

20   

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er

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    Figure 2.  

21   

               (a) 

0.5 

X   Transmissibility

0.4



0.3 0.2 0.1 0 0.7

0.9

1.1

1.3

  

 

              (b) 

Z   Transmissibility

0.5  

0.4 0.3 0.2 0.1 0 0.7

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1.1   

22   

1.3  

Figure 3      

 

 

 

 

               (a) 

   

   

 

   

 

               (b) 

23   

 

 

Figure 4    

 

 

 

 

 

(a) 

 

  24   

 

 

 

 

 

 

(b) 

 

  Figure 5       

25   

  Figure 6              

 

 

  26 

 

 

 

 

 

          (a) 

   

 

 

 

          (b) 

  Figure 7

27   

(a)

  (b)

 

Figure 8

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