1

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

Tesla Motors, Inc., 12259 Crenshaw Blvd., Hawthorne, California, CA 90250

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

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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 40 to 50 mph 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 [15-18], we focus on the acoustic mode alone in

this paper. 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 [19], 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 with the experimental data and a good agreement was found. The present paper has two

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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 [19], the new

approach represents a non-trivial progress in the understanding of the dynamics of 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 ξηo is fixed to the rotating tire. The angle

of rotation measured from a fixed horizontal axis is denoted by ϕ . We assume that air density fluctuation

in the cavity be:

]sin)(cos)([0' θθρρ tztx += (1)

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where )(tx and )(ty are the amplitudes of the two modes and 0ρ is the gas density inside the tire at

equilibrium. The angle θ is measured from the ξ -axis which is fixed to the tire. To satisfy the continuity

equation of the linear acoustic problem [20]

0'0

'

=⋅∇+∂

∂ vρρt

, (2)

we let the velocity fluctuation be the following:

θev ]cos)(sin)([' ϕθθ &&& ++−= tztxr , (3)

where the unit vector θe is defined in Figure 1. These assumptions agree with those given in [13].

reθe

ϕ

ξη

Xo

θ

Figure 1. The coordinate system for the tire.

The kinetic energy in the case of linear acoustics approximation is given by [13, 20]

∫∫ ++−== dVrtztxdVT 220

20 ]cos)(sin)([

21||

21 ϕθθρρ &&&'v

∫∫∫ +−= dVrzdVrzxdVrx 22222220 coscossin2sin(

21 θθθθρ &&&&

5

)cos2sin2 2222 ∫∫∫ ++− dVrdVrzdVrx ϕθϕθϕ &&&&& (4)

Note that terms in the last row of the above do not affect the Lagrange’s equation for ϕ& =constant,

corresponding to the constant speed driving condition under which the cavity noise is typically a major

dominate noise source. We therefore will ignore them from now on, without loss of generality.

The potential energy is given by

∫∫ +== dVtztxc

dVcU 22

02'

0

2

]sin)(cos)([2

)(2

θθρ

ρρ

)sinsincos2cos(2

2222

0

2

dVzdVxzdVxc θθθθρ ∫∫ ∫ ++= (5)

The volume integrals above must be evaluated over the entire tire cavity. In the absence of any static load

on the tire, we may assume that the tire cavity is axi-symmetric and carry out the integration over θ

easily. However, when a static load is applied, the tire deforms to a complicated shape and these volume

integrals are very difficult to calculate. We introduce approximations in order to obtain results that

capture the essential effect of the tire deformation. Moreover, we want to include the effect of the static

load on the acoustic mode when the tire rotates by an arbitrary angleϕ .

The deformation of a statically loaded tire can be represented by its changed mean radius and cross

section area. In [13], these are expressed as Fourier series in the angle θ . Since we consider a tire which

has rotated from the horizontal direction, we expand the series in the following:

...'3cos'2cos'cos(1[ 32100 +++++= θθθε ccc rrrrRr

...)]'3sin'2sin'sin 321 ++++ θθθ sss rrr (6)

6

...'3cos'2cos'cos(1[ 32100 +++++= θθθε ccc aaaaAA

...)]'3sin'2sin'sin 321 ++++ θθθ sss aaa (7)

where

ϕθθ +=' (8)

and ε is a small parameter representing small deflection of the tire. With further simplification of the tire

cross section geometry, Yamauchi and Akiyoshi [13] have obtained the natural frequencies by ignoring

high order terms in ε . Because of the orthogonality of the circular functions, only terms with subscripts 0

and c2 contribute to the final results. Note that for small parameter 1ε and 2ε , we have the following

approximations:

11 1)1( εε nn +=+ . (9)

and

)(1)1)(1( 2121 εεεε ++=++ . (10)

Without loss of generality, we can therefore account for the tire deformation by multiplying the integrand

in the kinetic energy integrals by

'2cos21 211 θεε ++=r (11)

and the integrand in the potential energy integrals by

'2cos21 432 θεε ++=r . (12)

The small parameters 1ε , 2ε , 3ε and 4ε account for the combined effect of the mean radius and the

cross sectional area in the integrals.

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Evaluating the integral

)coscossin2sin(21

1222

12

1222

0 ∫∫∫ +−= dVrrzdVrrzxdVrrxT θθθθρ &&&& (13)

we obtain:

)]2cos1(2sin2)2cos1([21

212

2212

00 ϕεεφεϕεεπρ ++++−+= zzxxIT &&&& (14)

Similarly, evaluating the integral

)sincossin2cos(2 2

2222

22

0

2

dVrzdVrxzdVrxcU θθθθρ ∫∫ ∫ ++= (15)

we obtain:

)]2cos1(2sin2)2cos1([2 43

2443

20

0

2

ϕεεϕεϕεεπρ

−++−++= zxzxAcU (16)

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

+⎥⎦

⎤⎢⎣

⎡⎥⎦

⎤⎢⎣

⎡++

−+zx&&

&&

ϕεεϕεϕεϕεε

2cos12sin2sin2cos1

212

221

02cos12sin

2sin2cos1

434

443

0

02

=⎥⎦

⎤⎢⎣

⎡⎥⎦

⎤⎢⎣

⎡−+−

−++zx

IAc

ϕεεϕεϕεϕεε

(17)

Since we regard the tire deformation caused by the static load to be small, 1ε , 2ε , 3ε and 4ε are small

compared with one. Applying the following approximations for arbitrary matrices A and B :

AIAI εε +=− −1)( , (18)

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and

)())(( BAIBIAI ++=++ εεε , (19)

where I is the identity matrix, we obtain the following equations of motion for the two lowest modes of

the acoustic cavity,

+⎥⎦

⎤⎢⎣

⎡+⎥

⎦

⎤⎢⎣

⎡zx

zx

&

&

&&

&&02ςω ⎥

⎦

⎤⎢⎣

⎡⎥⎦

⎤⎢⎣

⎡+−−++−

+−++−+zx

ϕεεεεϕεεϕεεϕεεεε

ω2cos)(12sin)(

2sin)(2cos)(1

421342

42421320

0= (20)

where ς represents small damping that we added to account for the dissipation of the acoustic energy.

These equations can be written in more compact form:

0)2(sin]2cos~[2 200 =−+++ zxxx ϕεϕεωςω αβαβ&&& (21a)

0)2(sin]2cos~[2 200 =−−++ xzzz ϕεϕεωςω αβαβ&&& (21b)

where

)1(~13

200 εεωω −+= (22)

and

)( 4220 εεωεαβ += . (23)

For a tire rotating at a constant speed Ω =ϕ& , equations (21a) and (21b) contain terms with periodic

coefficients in the form of tΩ2cos and tΩ2sin . Tire rotation rate is much slower compared with the

natural frequencies of the lowest acoustic mode. If these terms with periodic coefficients are ignored,

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equations (21a) and (21b) represent two uncoupled oscillators whose natural frequencies are 0~ω [19].

Even though the static load splits a symmetrical cavity mode into two cavity modes, the cavity may be

regarded as having two modes whose frequencies are slowly changing with time, varying around 0~ω .

These two slowly changing frequencies have the same average frequency at 0~ω . We also observe that the

static load on the tire causes the two acoustic modes to couple to each other [19].

Instead of solving equations (21a) and (21b) with time periodic coefficients, we realize that the effect of

the acoustic modes on the vehicle dynamics is our real interest. Considering the fact that the tire cavity

response and its interaction with the tire/wheel assembly is a major noise transfer path, we analyze the net

forces acting on the wheel by the acoustic cavity. The acoustic pressure is given by [20]

]sin)(cos)([02'2' θθρρ tztxccp +== . (24)

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

AdtpFX ∫ +−= )cos(),( θϕθ (25a)

AdtpFZ ∫ +−= )sin(),( θϕθ (25b)

where the integration is over the inner wall of the tire cavity or the surface of the wheel. Since the

deformation of the wheel can be ignored, by substituting (24) into the above, we can carry out the

integration to obtain:

)sincos(02 tztxcrwF weX Ω−Ω−= ρπ (26a)

)cossin(02 tztxcrwF weZ Ω+Ω−= ρπ . (26b)

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where ew is the effective width of the wheel and wr 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,

tztxX Ω−Ω= sincos (27a)

tztxZ Ω+Ω= cossin , (27b)

the forces on the wheel are now simply:

XcrwF weX 02 ρπ−= (28a)

ZcrwF weZ 02 ρπ−= . (28b)

Therefore, the forces on the wheel in the horizontal and vertical directions are proportional to X and Z .

Motivated by our previous work [19], we introduce the change of variables in (27a) and (27b) to obtain

the following equations:

02])1([)(2 2200 =Ω+Ω−++Ω++ ZXZXX a

&&&& εωςω (29a)

02])1([)(2 2200 =Ω−Ω−++Ω−+ XZXZZ &&&&

βεωςω (29b)

where

4231 εεεεεα +++−= (30a)

4231 εεεεε β −−+−= , (30b)

Equations (29a) and (29b) agree with the phenomenological model in [19]. Notice that, associated with

each acoustic mode, the air molecules inside the tire move from one place to another. The dynamic

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variables X and Z may be visualized as the center of mass that characterizes the group motion of the air

molecules. In the above equations, 0ω is the natural frequency of the acoustic cavity of the unloaded tire.

It has the degeneracy of two for a perfectly symmetric tire. The coefficient ς is the damping ratio. The

tire is assumed to be subjected to a static load acting on the outer surface along the Z direction with fixed

center position. The coefficients αε and βε represent the effects on the natural frequencies of the two

acoustic modes by the tire load. Consequently, they can be determined based on the changes of the

resonance frequencies.

For a non-rotating tire, 0=Ω , equations (29) show that the static load causes the frequencies of the two

cavity modes to split into

)1(20 αεωω +=H (31a)

and

)1(20 βεωω +=V . (31b)

The frequencies Hω and Vω refer to the resonance frequencies of a stationary loaded tire along the

fore/aft and vertical directions respectively. Once Hω and Vω are known, from (31) we obtain:

12

0

−⎟⎟⎠

⎞⎜⎜⎝

⎛=

ωω

εαH (32a)

and

12

0

−⎟⎟⎠

⎞⎜⎜⎝

⎛=

ωω

ε βV . (32b)

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Therefore our present model requires the knowledge of frequencies Hω and Vω corresponding to the

application of the static load. They can be determined experimentally or numerically. The experimental

determination is more straightforward. The numerical determination would require two steps. The tire

cavity geometry must first be determined. Following that, the cavity resonance frequency can be

calculated. Since the static load causes the tire overall profile to shrink vertically and to elongate

horizontally, we intuitively expect αε to be negative and βε positive. This is borne out by the test results

[19]. In the phenomenonlogical model presented in [19], these two parameters are assumed to be

proportional to the static load. Such an assumption is not needed if they are determined for each applied

static load.

The rotation of the tire causes the two modes to couple. By dropping the damping terms in (29), we can

obtain the eigenvalues which give the new resonance frequencies [19] denoted as 1ω and 2ω where

222222220

21 )(2)(

41~ Ω++−+Ω+= VHHV ωωωωωω (33a)

222222220

22 )(2)(

41~ Ω++−−Ω+= VHHV ωωωωωω . (33b)

and

)](211[~ 2

00 βα εεωω ++= )(21 22

HV ωω += . (34)

Note that in the absence of symmetry-breaking static load, 00~ ωωωω === HV . Equations (33a) and

(33b) simplifies to Ω+= 01 ωω and Ω−= 02 ωω .

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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:

02))1([)(2 2200 =Ω+Ω−++Ω++ ZXZXX a

&&&& εωςω (35a)

rcfXZXZZ =Ω−Ω−++Ω−+ &&&& 2))1([)(2 2200 βεωςω (35b)

where rf represents the random force and c is a proportionality coefficient. Thus the stochastic response

can be obtained following the standard techniques for linear systems subject to a stochastic input [21, 22].

The response is constructed based on the system’s response to sinusoidal inputs.

To obtain the system response to sinusoidal inputs, we let

)exp( tif ir ω= , (36)

)exp()( tiGX iiH ωω= (37a)

)exp()( tiGZ iiV ωω= , (37b)

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

0)(2)])1((2[ 022

002 =+Ω+Ω−+++− ViHaii GiGi ωςωεωωςωω (38a)

cGiGi VbiiHi =Ω−+++−++Ω− )])1((2[)(2 2200

20 εωωςωωωςω . (38b)

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Solving these two equations for )( iHG ω and )( iVG ω , we then obtain plots of |)(| iHG ω and

|)(| iVG ω versus the input frequency iω . 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 [19]. From the

measurements, we have obtained

secrad/)217(20 πω = , )207(2πω =H rad/sec, and )222(2πω =V rad/sec.

From equations (32), we obtain

0900.0−=αε , and 0466.0=βε .

Figure 2 is the transmissibility for 03.0/ 0 =Ω ω and 05.0/ 0 =Ω ω . The two peaks occur at the

frequencies close to the two resonance frequencies given by (33). 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 1ω

( 1ω > 2ω ). As the vehicle speed increases, the peak at 2ω becomes more significant. In equation (33), the

frequencies Hω and Vω 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.

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0

0.1

0.2

0.3

0.4

0.5

0.7 0.9 1.1 1.3

X Transmissibility

ωι/ω0

Ω=0.03

Ω=0.05

0

0.1

0.2

0.3

0.4

0.5

0.7 0.9 1.1 1.3

Z Transmissibility

ωι/ω0

Ω=0.03Ω=0.05

Figure 2 Transmissibility of a loaded tire. 03.0/ 0 =Ω ω , and 05.0/ 0 =Ω ω 01.0=ς , 01.0=c , and

10 =ω .

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4. SPECTRAL DENSITY OF ACOUSTIC PRESSURE INSIDE THE TIRE CAVITY

Once the frequency response functions )( iHG ω and )( iVG ω are determined, it is straightforward to

obtain the spectral density function of )(tX and )(tZ corresponding to white noise tire-road interaction

force rf . Following the notations in [21, 22], we have

)()()( * ωωω HHX GGS = (39a)

)()()( * ωωω VVZ GGS = , (39b)

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: )(tx and )(tz ,

using equations (27), we obtain

)sin()()cos()()( 00 ϕϕ +Ω++Ω= ttZttXtx (40)

where the angle 0ϕ indicates the location of the pressure sensor at time t=0. Knowing the spectral density

function of )(tX and )(tZ , we derive the spectral density of )(tx in the following.

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

)]()([)( ττ += txtxERx

)cos()]()([)]()([21 τττ Ω+++= tZtZEtXtXE

)sin()]()([)]()([21 τττ Ω+−+− tXtZEtZtXE . (41)

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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

)]()()()([41)( Ω++Ω−+Ω++Ω−= ωωωωω ZZXXx SSSSS

)]()()()([41

Ω++Ω+−Ω−−Ω−+ ωωωω ZXXZZXXZ SSSSi

(42)

where the functions )(ωXS and )(ωZS are given in (39), and

)()()(21)( * ωωττπ

ω ωτXZ

iXZXZ GGdeRS == −

∞

∞−∫ (43a)

)()()(21)( * ωωττπ

ω ωτZX

iZXZX GGdeRS == −

∞

∞−∫ . (43b)

Figure 3 shows the spectral density calculated from equation (42) for parameters given in Figure

2. Since the frequency response function )( iHG ω and )( iVG ω are dominated by two peaks at 1ω and

2ω ( 1ω > 2ω ), we expect four peaks at Ω±1ω and Ω±2ω . These four peaks are seen in Figure 3. Note

that the center two peaks at Ω−1ω and Ω+2ω are very close to each other. They may be

indistinguishable in experimental data with large uncertainties.

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0.7 0.9 1.1 1.3

Pressure Spe

ctrum

ωι /ω0

10 dB

ω1−Ω

ω1+Ωω2+Ω

ω2−Ω

0.7 0.9 1.1 1.3

Pressure Spe

ctrum

ωι/ω0

10 dB

ω1−Ω

ω1+Ωω2−Ω

ω2+Ω

Figure 3. Power spectrum in log scale. Top panel: 03.0/ 0 =Ω ω ; bottom panel 05.0/ 0 =Ω ω .

Other parameters are 01.0=ς , 01.0=c , and 10 =ω . Arrows indicate the four peaks.

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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.

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

30mph and 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 Ω =0.030 and

Ω =0.050 respectively.

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155 180 205 230 255 280

Acceleration (m

/s^2)

Frequency (Hz)

Spindle Fore/Aft Response: 30 mph

Spindle Fore/Aft Response: 50 mph

1.0 m/s^2

155 180 205 230 255 280

Acceleration (m

/s^2)

Frequency (Hz)

Spindle Vertical Response: 30 mph

Spindle Vertical Response: 50 mph

1.0 m/s^2

Figure 4. Wheel acceleration response at vehicle speed 30 mph and 50 mph respectively. Top panel:

fore/aft acceleration; bottom panel: vertical acceleration.

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Figure 4 shows the acceleration spectra of wheel spindle vibration for the driving conditions at 30 mph

and 50 mph. The spectra are plotted over the frequency range approximately the same as the

dimensionless range in Figure 2. Since Figure 4 shows the response, not the transmissibility as shown in

Figure 2, the response level increases as the vehicle speed increases. Nevertheless we observe from

Figure 4 the following. 1)The experimental data shows 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 2.

155 165 175 185 195 205 215 225 235 245 255 265 275 285

Inside Tire Microphone Pressure

Frequency (Hz)

10 dB

10 dB

Vehicle Speed: 50 mph

Vehicle Speed: 30 mph

215 Hz227 Hz210 Hz

199 Hz

216 Hz)

237 Hz

213 Hz

191 Hz

Figure 5. Spectrum of the pressure sensor measurements. The pressure sensors are mounted inside the

tire.

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Figure 5 shows the spectra of the pressure response of the cavity microphone for the driving conditions at

30 mph and 50 mph respectively. The spectra are plotted over the frequency range approximately the

same as the dimensionless range in Figure 3. 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 3. Moreover, the spreading of the side peaks from the main peak increases with the vehicle speed.

CONCLUSIONS AND DISCUSSIONS

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 [19] 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

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 6 for two different tire

rotational speeds. According to equations (33a) and (33b), the two peaks are at Ω+= 01 ωω and

Ω−= 02 ωω . Therefore, the two peaks at Ω−1ω and Ω+2ω in the pressure spectrum coalesce

at 0ω for arbitrary Ω . In Figure 7, we have plotted the pressure spectrum for this special case for

03.0/ 0 =Ω ω and 05.0/ 0 =Ω ω . We note that the other side peaks near Ω+1ω and Ω−2ω

23

disappears. 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.

0

0.1

0.2

0.3

0.4

0.5

0.7 0.9 1.1 1.3

X Transmissibility

ωι/ω0

Ω=0.03

Ω=0.05

0

0.1

0.2

0.3

0.4

0.5

0.7 0.9 1.1 1.3

Z Transmissibility

ωι/ω0

Ω=0.03

Ω=0.05

24

Figure 6 Transmissibility of an un loaded tire. 03.0/ 0 =Ω ω , and 05.0/ 0 =Ω ω 01.0=ς , 01.0=c ,

and 10 =ω .

0.7 0.9 1.1 1.3

Pressure Spe

ctrum

ωι/ω0

10 dB

ω0

0.7 0.9 1.1 1.3

Pressure Spe

ctrum

ωι/ω0

ω0

10 dB

Figure 7. Power spectrum in log scale. Top panel: 03.0/ 0 =Ω ω ; bottom panel 05.0/ 0 =Ω ω .

Other parameters are 01.0=ς , 01.0=c , and 10 =ω . An arrow indicates only one peak.

25

For the special case of no asymmetry of an unloaded tire, the results of developed model are in

perfect agreement with the published research results.

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