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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<ykurosawa@mps.teikyo-u.ac.jp>
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.
Journal of Technology and Social Science (JTSS)
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J. Tech. Soc. Sci., Vol.5, No.1, 2021
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
Journal of Technology and Social Science (JTSS)
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J. Tech. Soc. Sci., Vol.5, No.1, 2021
: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
Journal of Technology and Social Science (JTSS)
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J. Tech. Soc. Sci., Vol.5, No.1, 2021
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
Journal of Technology and Social Science (JTSS)
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J. Tech. Soc. Sci., Vol.5, No.1, 2021
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]
Journal of Technology and Social Science (JTSS)
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J. Tech. Soc. Sci., Vol.5, No.1, 2021
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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J. Tech. Soc. Sci., Vol.5, No.1, 2021
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
Journal of Technology and Social Science (JTSS)
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J. Tech. Soc. Sci., Vol.5, No.1, 2021
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)
Journal of Technology and Social Science (JTSS)
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J. Tech. Soc. Sci., Vol.5, No.1, 2021
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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J. Tech. Soc. Sci., Vol.5, No.1, 2021
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.
Journal of Technology and Social Science (JTSS)
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J. Tech. Soc. Sci., Vol.5, No.1, 2021
·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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