An impedance model for thin microperforated panels∗

Xianhui LI(1) , Tuo XING, Liying ZHU, Congshuang JIANG, Wenjiang WANG, Bin ZHANG
Beijing Municipal Institute of Labor Protection, Taoranting Road 55, Beijing 100054, PR China

Abstract
The classical model for the sound absorption of microperforated panels (MPPs) is well developed. For a thin
MPP, however, interaction between the air flow on the two sides of the panel may invalidate the impedance end
correction in the model. In this paper, a correction length model is proposed to predict the transfer impedance ofa thin MPP. Numerical analysis shows that the impedance jumps take place within a distance less than one halfof the perforation radius. Provided that the panel thickness is larger than the perforation radius, impedance end
correction can be well approximated by adding 1.2 times of the radius to the panel thickness. Sound inducedvibration also significantly affects the sound absorption. Herein it is modeled by parallelly connecting theaverage impedance of a flexible panel to the transfer impedance of a rigid MPP. The experimental results fortwo different MPP configurations validate the model.
Keywords: Microperforated panels, Impedance model, Sound absorption

1INTRODUCTION

MPPs were proposed by Maa for efficient sound absorption more than thirty years ago (1-3). The classical
MPP model combined the oscillating viscous flow model in small diameter tubes (4, 5) and Ingard’s resistiveend correction for perforations (6). Recently, many revised models have been proposed to improve the prediction accuracy in the sound absorption. For examples, Allam and Åbom associated the coefficients with edge
geometry and used 4 for sharp edge holes and 2 for round edge ones (7). Bolton and Kim found that the coefficient could be better represented as a function of frequency (8). Herdtle et. al. developed a 2D axisymmetricnumerical model to simulate a viscous, incompressible, oscillating flow in the time domain (9). They derived anend correction model represented by an additional hole length. Temiz et al. investigated the effect of the edge
shape geometry on the end correction using non-dimensional parameters (10). They concluded that the end correction coefficients are functions of the shear wave number and the edge geometry. Based on the viscothermal
wave theory (11), Li derived an impedance end correction model for sharp-edged circular perforations (12). It
is revealed that a static flow resistance term should be introduced to correctly model the energy dissipation due
to the acoustic flow distortion outside the perforation.
The above models have been successfully used in predicting the sound absorption of the MPPs with moderate
thicknesses. For a thin MPP, however, interaction between the air flow on the two sides of the panel is not
negligible, which may invalidate the impedance end corrections in these models. In this paper, a correction
length model is proposed to predict the transfer impedance of a thin MPP. It shows that the impedance jumps
take place within the distance less than one half of the perforation radius. Provided that the panel thickness
is larger than the perforation radius, contribution from the end effect can be well approximated by adding 1.2
times of the radius to the thickness. Sound induced vibration also significantly affects the transfer impedance.
Herein the average impedance of a flexible panel is derived, which is parallelly connected to transfer impedanceof a rigid MPP to obtain the overall impedance. Numerical and experimental results validate that the proposedmodel offers good prediction for the sound absorption of the MPPs.
∗(1) Email: lixianh@vip.sina.com

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2MODELING

2.1 Classical MPP model
The classical model was derived from the wave propagation in narrow tubes in which the oscillatory viscousboundary layer spans the hole diameter (1-3). For circular holes, the normal acoustic transfer impedance anMPP with constant-diameter holes can be expressed as
ZMPP = ZPerf + ZEnd ,(1)where the impedance due to the perforations isZPerf =−1√2 J1 (k − j)
jρωL
√
1− √,
σk − j J0 (k − j)(2)

in which ρ is the density of air, ω is the angular frequency, L is the panel thickness, and σ is the porosity ofthe MPP. In addition, k is the perforation constant defined asp(3)k = r ρω/η,where η is the dynamic viscosity, r is the hole radius, and J0 and J1 are the Bessel functions of the first kindof the zeroth and first order, respectively.
The impedance end correction term ZEnd involves both resistive and reactive parts. Maa (1) used the surface
resistance (6), defined bypRs = 2ηρω/2(4)to add resistive end effects and included a reactance correction term derived from the radiation impedance of apiston. The total end correction can written as
r

ρ
ηω
ZEnd =+ 1.7 jωr .
(5)
σ2ρ
2.2 Correction length model
When the thickness of an MPP is thin enough, the air flow on one side of the panel may interact with that on
the other side, which will affect the transfer impedance of the MPP according to (11). In order to investigate
the interaction between the air flows, a 2D axisymmetric finite element model is built with the thermoviscous
acoustics module of COMSOL Multiphysics R . The setup of the finite element model is shown in Figure 1.
The radius of the hole is 0.25 mm, the thickness of the panel is 0.3 mm, and the height of the computational
domain is 0.94 mm, which corresponds to a porosity of 7%. A plane wave is incident from the left side anda perfect matched layer is used to reduced of influence of the reflecting wave. On the right side, a plane wave
radiation boundary condition is used to model the propagation in a semi-infinite medium.
The normalized impedances of the MPP are computed along the axis of the hole at 250 Hz, 500 Hz, 1000 Hz,
2000 Hz, and 4000 Hz, which are represented by the curves with different colors in Figure 2. One can observe
the unit normalized impedances (blue circles) on the right side, which result from a plane wave propagating
in an unbounded medium. In the middle of the perforation delimited by two vertical red lines, these curves
follow exactly the impedances of the viscous wave in the circular hole given by Equation 2. Near the ends ofthe hole, there are obvious impedance jumps which take place within a distance much less than one half of thehole radius. Extending the curves in the middle section linearly to the right side, all the lines intersect Z = 1 atalmost the same position. Its distance from the right end of the hole ∆L is approximately 0.6r. It is deduced
from this observation that the impedance jumps at one end can be approximated by adding the length of thehole by 0.6r. The impedance jumps at the left end, although slightly different from those at the right end, can2143

Figure 1. Finite element model setup
be approximately by the same correction length too. Finally, the transfer impedance of the MPP including theend correction can be written as
ZMPP = (1 + 1.2r/L)ZPerf .
(6)
The above result is almost identical to the empirical model of Herdtle et. al. (9), where it is obtained from a
CFD simulation of an incompressible isothermal flow through a sharp edged straight hole.

Figure 2. Normalized impedance of an MPP with 0.25 mm radius and 0.3 mm thickness: (a) real part; (b)imaginary part
When the panel thickness is further reduced to 0.15 mm, the normalized impedances are shown in Figure 3.
Even though the panel thickness is smaller than the hole radius, the correction length derived above is still valid.
Further reducing the panel thickness to 0.1 mm, the impedance jumping regions on the left and right sides begin
to merge and different correction lengths are observed for the real and imaginary parts of the impedance, asshown in Figure 4. In such case, the end correction model given by Equation 6 is no longer valid. Fortunately,
the thickness of an MPP is seldom less than its perforation radius in practice; hence the air flows on the twosides of the panel can be considered as independent and their influence on the impedance can be approximatedby a correction length of 0.6r each. This result is in good agreement with that obtained from the time domain
simulation of a viscous, incompressible, oscillating flow (9).

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Figure 3. Normalized impedance of an MPP with 0.25 mm radius and 0.15 mm thickness: (a) real part; (b)imaginary part2.3 Effect of the sound-induced vibration
In this paper, a thin flexible MPP installed in an impedance tube is considered to demonstrate the sound induced vibration effect on the total transfer impedance. Since the size of the MPP is much smaller than theshortest wavelength involved, the incident plane wave approximately exerts a uniform pressure distribution overits surface. Suppose the nth mode shape of the vibrating MPP is φn (x), the average velocity due to the vibration
isR∆piω( φn (x) dS)2v̄Panel = ∑,(7)2 − ω 2 + 2iζ ωω ) Sm(ωnnnnn
where mn , ωn , and ζn are the modal mass, natural frequency, and modal damping ratio of the nth mode,
respectively, ∆p is the pressure drop across the MPP, and S is its area.
Due to its small size as compared with the wavelength, the velocity component δ v(x) = v(x) − v̄Panel of theMPP couples only to the evanescent wave along the normal direction. It follows that the piston-like motion
represented by v̄Panel is sufficient to describe to sound absorption characteristics of the MPP, and an effective
impedance is defined asR
−1∆piω( φn (x) dS)2ZPanel ==S ∑.
(8)22v̄Paneln mn (ωn − ω + 2iζn ωn ω)
Taking into account the contribution from the air flow through the perforations, the total average velocity isgiven according to (13) asv̄ = γ v̄Panel + ∆p/ZMPP ,(9)where γ = 1 − σ Im(ZMPP )/ZMPP .
Thus, combining Equations (8) and (9) yields the overall impedanceZall ≡ ∆p/v̄ = (γ/ZPanel + 1/ZMPP )−1 .

(10)When the porosity of the MPP is very small, γ ≈ 1 and Zall is actually the parallel connection of ZPanel andZMPP . Then the normal incident sound absorption coefficient of the absorber is
α=4ρcRe {Zall }(Re {Zall } + ρc)2 + (Im {Zall })2,

(11)

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Figure 4. Normalized impedance of an MPP with 0.25 mm radius and 0.1 mm thickness: (a) real part; (b)imaginary part
where c is the air sound speed.

3VALIDATION

To validate the proposed model, an absorber is made up of a microperforated steel disk of radius 49.8 mm andthickness 0.4 mm, backed by a rigid cylindrical cavity of depth 50 mm. The disk is uniformly perforated bycircular holes separated from each other by a distance of 5 mm and with a diameter 0.49 mm. The material hasa density of 7800 kg/m3 , Young’s modulus 210 GPa, Poisson’s ratio 0.3, and modal damping ratio 0.008. The
air has a density of 1.21 kg/m3 , sound speed 340 m/s, and dynamic viscosity 1.8×10−5 Pa·s. The absorber isinstalled in a impedance tube and tested according to ISO 10534-2 (14). The frequency range of the measurements is from 200 to 1600 Hz. Figure 5 shows the absorption coefficients of the absorber. It is observed thatfor the relatively thick MPP the flexural vibration has little effect on the absorber, except a narrow absorption
peak at its first axial symmetric mode. In such case, both Maa’s model and the proposed model provide good
estimations for the sound absorption coefficients.
To highlight the influence of the sound induced vibration on the absorber’s performance, another absorber ismade up of a microperforated aluminum disk of radius 49.8 mm and thickness 0.2 mm. The disk is uniformly
perforated by circular holes separated from each other by a distance of 5 mm and with a diameter 0.48 mm.
The material has a density of 2800 kg/m3 , Young’s modulus 69 GPa, Poisson’s ratio 0.3, and modal damping
ratio 0.02. All other parameters remain the same as the first example. Figure 6 shows the absorption coefficientsof the absorber. It is observed that the absorption peak is slightly lowered and shifted to a higher frequency
due to the sound induced vibration. The proposed model provides better estimations than Maa’s model aroundthe sound absorption peaks.

4CONCLUSIONSThis paper proposed a correction length model to predict the transfer impedance of a thin MPP. It is foundfrom the numerical simulation that the impedance jumps take place within a distance less than one half ofthe perforation radius. As long as the panel thickness is larger than the perforation radius, contribution fromthe two ends can be considered independently. Their effects can be well approximated by adding 1.2 timesof the perforation radius to the panel thickness. Sound induced vibration also significantly affects the sound2146

Figure 5. Normal incident sound absorption coefficient of the steel MPP; Experimental data (blue), the proposedmodel (black), Maa’s model (red)absorption. It can be modeled by parallelly connecting the average impedance of a flexible panel to the transfer
impedance of a rigid MPP. Impedance tube test shows that the proposed model offers a good estimation of thesound absorption coefficients of a thin flexible MPP.

ACKNOWLEDGEMENTS
This work is supported by Beijing Natural Science Foundation (1172007), Natural Science Foundation of China(11174041), Xicheng District Excellent Talent Project (20180029, 20180022), Beike Scholar Project (BS201901)
and Innovation Project (PXM2019-178304-000003) of BJAST.

REFERENCES[1] Maa, D. Y. Theory and design of microperforated panel sound-absorbing constructions, Sci. Sin. Vol 18 (1),1975, pp 55-71.
[2] Maa, D. Y. Microperforated-panel wideband absorbers, Noise Control Eng. J. Vol 29 (3), 1987, pp 77-84.
[3] Maa, D. Y. Potential of microperforated panel absorber, J. Acoust. Soc. Am. Vol 104 (5), 1998, pp 28612866.
[4] Rayleigh, J. W. S. The Theory of Sound, Dover Publications, NY (USA), 2nd edition, 1945.
[5] Crandall, I. B. Theory of Vibrating Systems and Sound. Van Nostrand, NY (USA), 1926.
[6] Ingard, U. On the theory and design of acoustic resonators, J. Acoust. Soc. Am. Vol 25 (6), 1953, pp1037-1061.
[7] Allam, S.; Åbom, M. A new type of muffler based on microperforated tubes, Trans. ASEM, J. Vib. Acoust.
Vol 133 (3), 2011, pp 1-8.

2147Figure 6. Normal incident sound absorption coefficient of the aluminum MPP; Experimental data (blue), theproposed model (black), Maa’s model (red)[8] Bolton, J. S.; Kim, N. N. Use of CFD to calculate the dynamic resistive end correction for micro-perforated
materials, Acoust. Aust. Vol 38, 2010, pp 134-139.
[9] Herdtle, T.; Bolton, J. S.; Kim, N. N.; Alexander, J. H.; Gerdes, R. W. Transfer impedance of microperforated materials with tapered holes, J. Acoust. Soc. Am. Vol 134 (6), 2013, pp 4752-4762.
[10] Temiz, M. A.; Lopez-Arteaga, I.; Efraimsson, G.; Åbom, M.; Hirschberg, A. The influence of edge geometry on end-correction coefficients in micro perforated plates, J. Acoust. Soc. Am. Vol 138 (6), 2015, pp3668-3678.
[11] Beltman, W. M. Viscothermal wave propagation including acousto-elastic interaction, Part I: Theory, J.
Sound Vib. Vol 227 (3), 1999, pp 555-586.
[12] Li, X. End correction model for the transfer impedance of microperforated panels using viscothermal wave
theory, J. Acoust. Soc. Am., Vol 141 (3), 2017, pp 1426-1436.
[13] Takahashi, D.; Tanaka, M. Flexural vibration of perforated plates and porous elastic materials under acoustic loading, J. Acoust. Soc. Am., Vol 112 (4), 2002, pp 1456-1464.
[14] International Organization for Standardization, ISO-10534-2: Determination of Sound Absorption Coefficient and Impedance in Impedance Tubes, Geneva, 1998.

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