Calculation for Normal Sound Absorption Coefficient of

Journal of Technology and Social Science (JTSS)

24

J. Tech. Soc. Sci., Vol.5, No.1, 2021

Calculation for Normal Sound Absorption Coefficient of Porous Media Using FE Model for Impedance Tube

Yoshio Kurosawa1,a 1 Faculty of Science and Technology, Teikyo University, 1-1 Toyosatodai, Utsunomiya City

320-8551, Japan

a<[email protected]>

Keywords: impedance tube, FEM, TMM, normal sound absorption, Biot-Allard model

Abstract. The impedance tube is used for measuring the normal sound absorption coefficient of a

sound absorbing material such as an automobile. Such as urethane foam and felt with high density,

the sound absorption coefficient measurement result may be different depending on the contact

situation between the cut sample and the impedance tube wall. In order to elucidate this phenomenon,

the impedance tube was modeled with finite elements and modeled the contact condition by placing a

spring between the cut sample and the impedance tube wall. By changing the spring coefficient, it

was tried to reproduce the way of vibration of the cut sample occurring at the time of actual sound

absorption, to elucidate the influence of improvement of prediction accuracy and the contact

condition of cut sample and impedance tube wall on sound absorption coefficient. It was introduced

calculation result by transfer matrix method and calculation result using finite element model by

Biot-Allard theory, comparison of experiment result. In addition, the results of comparing the results

of FEM and TMM by changing the Young's modulus and flow resistance of the sample are

introduced.

1. Introduction

In transportation equipment such as automobiles and buildings such as houses, the need for noise

reduction has been increasing. Sound insulation material using porous bodies that places of

utilization are also increasing. In designing and developing sound insulation material using porous

bodies, it has become common to predict and examine acoustic performance using commercial

acoustic analysis software to reduce prototype cost and shorten development period. However, in

order to predict acoustic performance using such analysis software, you have to need a accurate

material parameters (Biot parameters) to run that software. As the material parameter used at this

time, sometimes use measurement values using dedicated measuring instruments. However, it is

often the case that the opposite identification value is used from the normal incident sound

absorption coefficient of impedance tube. In fiber materials such as urethane foam and high density

felt, the sound absorption coefficient measurement result may be different depending on the contact

situation between the cut sample and the impedance tube wall. Of course, the value of the Biot

parameter identified from it also changes. In this study, the impedance tube was modeled with FEM

and modeled the contact situation by placing a spring between the cut sample and the impedance

tube wall. By changing the spring coefficient, how to vibrate the cut sample occurring during actual

sound absorption is reproduced and comparison with calculated results by transfer matrix method or

measurement results, tried to elucidate the influence of the contact situation between the cut sample

and the impedance tube wall on the sound absorption coefficient.


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2. About Biot-Allard Model

In this study, JCA model [1] and Biot-Allard model [1] were used as modeling of porous bodies.

The Biot-Allard model is a theoretical formula that airborne sound transmitted by the incident sound

passing through the gap of the poroelastic medium, predict the displacement of the solid borne sound

traveling through the inside (skeleton) of material. Displacement of skeleton part considering

interaction between solid borne sound and airborne sound : us and fluid displacement : u

f using its,

(1),(2)expression are represented.

(1)

(2)

：porosity，：Density of the porous frame， f ：fluid density（In this paper is air）， a ：

Equivalent density of the fluid taking into consideration viscous damping in interaction of the

skeleton and fluid． a Is shown in equation (3).

(3)

：Solid loss coefficient，：Flow resistance，：Tortuosity factor，：Viscosity characteristic head.

Next is elastic modulus P , Q , R , shown in equation (4)

(4)

bK ：Bulk modulus of skeleton(vacuum)， N ：Shear modulus of skeleton(vacuum), bK , N

equation (5), fK shown in equation (6).

−

+=

+

+=

NK

jEN

b)21(3

)1(2

)1(2

)1(

(5)

12

2

0

16

'1

'

81)1(

−

+

+−−

=

j

j

PK f

(6)

)()()()()())1((22

2

2

2

2

uut

GuNuQu sNPt

u

t

u fssffs

aas

−

−++−=

−

+− ･･

)()()()()(2

2

2

2

2fssf

s

a

f

afuuuu

tGuQuR

tt

−

++=

−

+ ・・

s

+=

)(1

G

j fa f

222

241)(,

+=fa

jG　

( )

( )

−

−++

f

f

fb

KR

KQ

KKNP

1

1

3

42


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：Ratio of specific heat， P0：Pressure of the equilibrium，：Thermal diffusivity，

：The thermal characteristic head，K f ：Equivalent rigidity of fluid considering thermal

damping in skeletal part and fluid transaction.

3. Analysis Result

3.1 FE Model and Biot Parameters

Fig.1 shows the FE model of the impedance tube and cut sample (GW: glass wool) used for the

analysis this time. Length 300 mm, diameter 100 mm, Lx = 100 mm, Dx = 50 mm and the thickness of

GW was 25 mm. The mesh pitch was 2 mm in the radial direction and 10 mm in the longer direction

and it was about 28000 elements, and a spring was attached to the circumference of the GW. P1 and P2

are the sound pressures of the microphones 1 and 2 attached to the each positions. The transfer function

was required from the sound pressures of the microphones 1 and 2, and the normal incident sound

absorption coefficient (hereinafter, sound absorption coefficient is the normal incident sound

absorption coefficient) was calculated. Table 1 shows the Biot parameters of the glass wool [2] and

urethane foam used in this calculation.

H(𝜔) =𝑃2(𝜔)

𝑃1(𝜔) (7)

Z0 =P0

u0= jρc

−H(ω) sin kLx+sin k(Lx+Dx)

H(ω) cos kLx−cos k(Lx+Dx) (8)

𝛼0 = 1 − |𝑍0−𝜌𝑐

𝑍0+𝜌𝑐|

2 (9)

H()：Transfer function，Z0：Acoustic impedance，0：Normal incident sound absorption

coefficient，Lx：The distance of microphone 1 and cut sample, Dx：The distance of microphone 1

and 2，：Air density，c：Sound velocity，𝑢0：Particle velocity of air，k：Wave number (＝𝜔

𝑐)

Table1 Biot parameters for glass wool and urethane foam

Unit Glass wool （GW）

Urethane foam (VO)

Thickness mm 25 20

Porosity - 0.99 0.968

Flow resistivity Ns/m4 12000 46000

Tortuosity - 1.01 3.8

Viscous characteristics length

μm 130 60

Thermal characteristics length

μm 264 10

Density kg/m3 32 35

Young’s modulus Pa 20020 2000000

Poisson’s ratio - 0.001 0.3

Loss factor - 0.01 0.09


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Fig. 1. Fe model of impedance tube (φ100mm)

3.2 Calculation Result (glass wool)

Here, the calculation result of the sound absorption coefficient of glass wool at 100 to 1600 Hz is

shown. Fig. 2 shows the comparison between the sound absorption coefficient when changing the

spring coefficient in the FEM using the JCA model and the sound absorption coefficient calculated by

the transfer matrix method (TMM in figure) using the JCA model. The free in a figure show the

calculation result of the springless model, 1, 10, 100, 1000 show the value of the spring constant N/m.

It is understood that the calculation results of FEM and TMM are congruence regardless of the value of

the spring constant. This is because the JCA model does not take skeletal vibration into consideration

and it is not affected by boundary conditions.

Fig. 2. Calculation results of normal incidence sound absorption coefficient of JCA model

Fig. 3 is a graph compare FEM and TMM of the Biot-Allard model. With the Biot parameter used

this time, the absorption coefficient calculation results of the JCA model and the Biot-Allard model by

the transfer matrix method almost accords with each other except for the 250 Hz band and almost show

the tendency of the straight line. In FEM, peaks appear in 2~3, and when the spring constant raise, the

peak frequency increases too. Since the experimental result and the calculation result of the transfer

matrix method almost accords with each other, it can be said that the Biot-Allard model is not

inadequate for FE modeling glass wool.

25mm

100mm

300mm


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Frequency (Hz)

Ab

so

rptio

n c

oe

ffic

ien

t

Fig. 3. Calculation results of normal incidence sound absorption coefficient of Biot-Allard model

It was confirmed in figure, how the skeleton vibration affects the sound absorption rate peak

generated. Fig. 4 shows the Biot (1000) of Fig. 5 taken out for easy viewing. Fig. 5 shows the

displacement of the sound source side center of the glass wool, and fig. 6 shows the sound pressure of

the microphones 1 and 2. The peak of the sound absorption coefficient completely doesn't accord with

the peak of the skeletal vibration and there is no correlation. On the other hand, the peak frequency of

skeletal vibration and the peak frequency of microphone sound pressure are in accord, indicating that

the correlation is high. Fig. 7 shows the vibration mode of glass wool at 270 Hz. It is the primary mode

out of the plane, and the peaks of the other frequencies were almost similar vibration modes.

Fig. 4. Calculation results of absorption coefficient of Biot-Allard model (spring 1000)

Fig. 5. Calculation results of GW displacement

1.E-07

1.E-06

1.E-05

1.E-04

1.E-03

1.E-02

1.E-01

0 500 1000 1500

Dis

pla

ce

me

nt (m

)

Frequency [Hz]


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Frequency (Hz)

So

und

pre

ssu

re (

dB

)

Fig. 6. Calculation results of sound pressure (Mic.1 & Mic.2)

Fig. 7. Vibration mode (270[Hz]). The red part has a large displacement.

3.3 Calculation Result (urethane foam)

Here, calculation results of urethane foam (VO) at 100 to 1600 Hz are shown. Fig. 8 shows the

comparison with the sound absorption coefficient when changing the spring coefficient in Biot-allard

model of FEM and the sound absorption coefficient calculated by the JCA model (TMM (Rigid) in the

figure) of the transfer matrix method and the Biot-Allard model (TMM (Biot) in the figure). In the Biot

parameter used this time, the sound absorption coefficient calculation results of the JCA model used

transfer matrix and the Biot-Allard model were completely different. This is because the Young's

modulus is larger than that in the case of glass wool and the influence of skeleton vibration appears. In

the calculation result of FEM, increasing the spring constant increased the peak frequency too like a

Fig. 3, but the sound absorption rate did not change when the spring constant was 105 or more. The

reason why the spring constant was increased, it was closer to the calculation result of the transfer

matrix method, but it did not match, it is thought that the influence of skeleton vibration differs

between infinite plate and determinate case.

Fig. 9 shows the results of comparing displacement of urethane foam with spring constant. It can be

seen that the peak frequency increases as the spring constant increases.


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Fig. 8. Calculation results of normal incidence sound absorption coefficient of Biot-Allard

model

Fig. 9. Calculation results of VO displacement

3.4 Comparison of Measurement Results and Calculation Results (urethane foam)

Here, it was compared the measurement result with the calculation result. The impedance tube and

the cut sample used for the measurement are shown in Fig.10. The inside diameter is 64 mm, and a

pipe length is 400 mm. Acoustic excitation (volume acceleration) was performed with a hose

speaker, and displacement of urethane foam was measured with a laser doppler vibrometer. The FE

model used in the calculation is shown in Fig.11. The yellow spheroidal object in Fig.11 is a point

sound source simulating a speaker, and the black part is a urethane model.

Fig. 12 shows the calculation results of the sound absorption coefficient obtained from impedance

tube measurement result and transmission matrix method using Biot parameters identified from

sound absorption coefficient. It can be said that the Biot parameter was identified without any

problem.

Fig. 13 is a graph comparing the measurement result of the sound absorption coefficient of

urethane and the calculation result of FEM. Comparing the measured value and Biot (Free), the

frequencies where the peaks appear are roughly in concert, but the sound absorbing rate is not much.

Compared with the one with the spring, it is understood that it almost in concert with the measured

value. In addition, there is a tendency to approach the measured value by increasing the hardness of

the spring.

Fig. 14 and Fig. 15 are comparing the sound pressure levels of each of the microphones 1 and 2. In


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both microphones cases, you can find that it is close to the measurement result by increasing the

spring constant.

Fig. 16 show the vibration displacement of urethane. The upper figure shows the mode diagram at

270 Hz in the case of no spring and lower figure shows mode diagram at 430 Hz when the spring

constant is 100,000 N/m. The displacement from blue to red is larger. It turns out that both are

vibrating in the primary mode out of the plane.

Fig. 10. Experimental setup of impedance tube and laser doppler vibrometer

F

Fig. 12. Compare experimental results and calculation results

Fig. 13. Calculation results of absorption coefficient of VO

Fig. 11. FE model for impedance tube (φ100mm) (φ64mm)


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Fig. 16. Displacement of VO. The red part has a large displacement.

Fig. 15. Sound pressure level of Mic2

a.270 [Hz] (Free)

b. 430 [Hz] (100000)

Fig. 14. Sound pressure level of Mic1


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3.5 Comparison between TMM and FEM (parameter study)

Next, in order to find the coincide condition the FEM of Biot-Allard model and the sound

absorption coefficient calculation result of TMM, a parameter study of Young's modulus of

urethane foam was carried out. Fig. 17 shows the comparison of the sound absorption coefficient

calculation results of FEM and TMM in the case of Young's modulus 103, 105, 107 from the left. As

the Young's modulus was increased, it became closer to the calculation result of TMM regardless of

the spring constant, and it was coincidence could confirme that when the Young's modulus was set

to about 107. Skeletal vibration decreases when the Young's modulus is increased, and it is thought

that it is because it approached the JCA model.

a. Young's modulus = 1,000 b. Young's modulus=100,000

c. Young's modulus =10,000,000

Fig. 17. Comparison of sound absorption coefficient when Young's modulus is changed by

TMM and FEM.

4.Conclusion

The impedance tube was modeled with a finite element and analyzed it and found the following.


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·The sound absorption coefficient calculation result of FEM using glass wool JCA model (without

skeletal vibration) gained good congruence with the sound absorption coefficient measurement

result by impedance tube and calculation result of TMM sound absorption coefficient using JCA

model. The result of FEM's absorption coefficient calculation by the Biot-Allard model was

completely different from the measurement result by TMM.

·The peak frequency of the sound absorption coefficient and the peak frequency of the

displacement of the skeletal vibration do not coincide. Skeletal vibration correlates strongly with

microphone sound pressure.

·The result of calculation of the sound absorption coefficient of VO (urethane foam),when the

boundary condition of the tube wall harden it approaches the measurement result or the calculation

result by TMM.

· The reason why the sound absorption coefficients of TMM and FEM are in coincidence is when

the Young's modulus is very large.

References

[1] K. Hirosawa, H. Suzuki, H. Nakagawa, and K. Takahashi, " Parameter study on Biot model of

porous elastic material at normal incident sound absorption coefficient", Japan Acoustical

Society Building Acoustics Study Group (Tokyo, Japan), April 2015.

[2] J. F. Allard, and N. Atalla, “Propagation of sound in porous media”, John Wiley & Sons, Inc.

2009.

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Calculation for Normal Sound Absorption Coefficient of