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Automotive Innovation

, Volume 1, Issue 4, pp 374–380 | Cite as

Simplified Method of Simulating Double-Layer Micro-Perforated Panel Structure

  • Wan ChenEmail author
  • Chihua Lu
  • Zhien Liu
  • Songze Du
Article
  • 376 Downloads

Abstract

The micro-perforated panel (MPP) structure has been widely used in various noise control applications, and thus its acoustic performance prediction has been receiving increasing attention. The acoustic performance of simple MPP structures, such as a MPP sound absorber, has been predicted using an analytical calculation method. However, this is not a suitable approach toward predicting the acoustic performance of complicated MPP structures, owing to the structural complexity of these structures. Moreover, the many perforations of submillimeter scale diameter render the MPP structures very difficult to analyze using numerical simulation. Thus, this study focused on two different simplified MPP simulation methods: the transfer admittance method and the equivalent fluid method, and their application on double-layer MPP structures. Based on the two simplified MPP simulation methods, the transmission loss value of the double-layer MPP mufflers with two sets of different structural parameters was calculated, respectively. The predicted results were compared with the impedance tube measurements. The results revealed that the two simplified MPP simulation methods could effectively predict the acoustic performance of double-layer MPP structures. Moreover, the prediction based on the transfer admittance method can outperform the two simplified simulation methods.

Keywords

Double-layer MPP structure Acoustic simulation Transfer admittance Porous material Experimental test 

List of symbols

MPP

Micro-perforated panel

FEM

Finite element method

CFD

Computational fluid dynamics

t

Panel thickness

d

Perforation diameter

p

Perforation ratio

D1, D2

Cavity depth

b

Perforation spacing

TL

Transmission loss

vn1, vn2

Normal particle velocity

α1, α2, α3, α4, α5, α6

Transfer admittance coefficient

p1, p2

Sound pressure

β

Transfer admittance

K

Diameter ratio

Z

Specific acoustic impedance

R

Specific acoustic resistance

X

Specific acoustic reactance

η

Dynamic viscosity

ρ

Air density

ω

Angular frequency

f

Frequency

φ

Porosity

σ

Flow resistivity

α

Tortuosity

εe

Correction length

Av

Viscous characteristic length

At

Thermal characteristic length

1 Introduction

The micro-perforated panel (MPP) structure is very promising as the basis for the next generation of sound absorbing constructions. This structure has enough acoustic resistance and sufficiently low acoustic reactance to provide good absorption properties by reducing the perforation diameter to submillimeter scale. The MPP combined with a uniform air-back cavity and a rigid back wall can be used to construct conventional single-layer MPP sound absorbers, which were first proposed by Maa [1]. Based on Maa’s pioneering work, many theoretical methods have been proposed to predict the acoustic performance of MPP sound absorbers. The most common method is the acoustic-electric analogy method [2, 3, 4], which has been used to calculate the normal sound absorption coefficient of many different types of MPP sound absorbers. The transfer matrix method [5, 6] has also been used to predict the sound absorption performance of a single-layer MPP sound absorber or a multilayer MPP sound absorber. Additionally, the impedance transfer method [7, 8] has been used to calculate the surface acoustic impedance of MPP sound absorbers. In summary, the acoustic performance of simple MPP sound absorbers can be easily predicted based on the abovementioned analytical calculation methods. However, in practical applications, MPPs are more often applied to a complicated muffler, with a complex interior structural environment. Moreover, the incidence sound wave is also more frequently subject to an oblique incidence condition. Therefore, in these cases, it is not suitable to use any of the commonly used analytical calculation methods to predict the overall acoustic performance of the MPP structures.

To ensure the prediction accuracy, many scholars have used the acoustic finite element method (FEM) to simulate the acoustic performance of MPP structures. For example, Gerdes et al. [9] established a geometrical perforation model and adopted a computational fluid dynamics (CFD) approach to calculate the acoustic transfer impedance of a MPP. Subsequently, they used the acoustic transfer impedance results to describe the MPP layer in the finite element (FE) model of a MPP sound absorber. However, the large number of perforations in the MPP increased the difficulty of the geometrical modeling process, which in turn increased considerably the complexity of the entire simulation calculation.

To avoid the geometry modeling of massive perforations, Zuo et al. [10] first obtained the acoustic impedance of a MPP according to Maa’s formula and then simulated the MPP by defining the transfer admittance relationships between both of its sides such that the transmission loss of a single-layer MPP muffler could be predicted. From a different perspective, Atalla [11] proposed that a rigid perforated plate could be modeled as an equivalent fluid by following the Johnson–Allard approach, given that the equivalent tortuosity is appropriately defined. Thus, it becomes possible to use the existing finite element models of rigid porous materials to model rigid MPPs. On the basis of Atalla’s idea, Hou [12, 13] further expanded the modeling method and demonstrated that a flexible MPP can be modeled in a similar manner by using poro-elastic finite elements, which allow the prediction of the panel’s structural resonances. The two abovementioned simplified MPP simulation methods have been applied to the acoustic prediction of single-layer MPP structures. However, this application has not yet been investigated for complicated MPP structures, such as a double-layer MPP muffler. Moreover, many studies have focused only on one of these two simplified MPP simulation methods, and little effort has been put into investigating them simultaneously and comparing their accuracy. To fill this research gap, the two abovementioned simplified MPP simulation methods were used, respectively, to predict the acoustic performance of double-layer MPP mufflers with a wide silencing bandwidth.

In this study, first, the double-layer MPP structures were introduced. Then, the transmission loss of one double-layered MPP muffler was predicted using the transfer admittance method and the equivalent fluid method, respectively. The muffler was tested with an impedance tube to validate the predicted results. Additionally, the other muffler with different structural parameters was treated in a similar manner to further validate the results.

2 Double-Layer MPP Structures

2.1 Double-Layer MPP Sound Absorber

To better understand the double-layer MPP muffler under investigation, a brief introduction to the double-layer MPP sound absorber will be given first. A conventional single-layer MPP sound absorber consists of a MPP, rigid backing wall, and air cavity between them. This type of MPP sound absorber offers an outstanding alternative to traditional porous materials, but has an obvious disadvantage: it is effective only within a narrow band around its resonance frequency, owing to its resonator nature, which typically renders its sound absorption capacity inadequate for a general purpose absorber. In an effort to widen the sound absorption frequency range, many studies have reported that the introduction of extra absorption peaks may be the most effective and promising approach. Thus, compound MPP sound absorbers have been repeatedly proposed. A double-layer MPP sound absorber is the most common compound MPP sound absorber and was first proposed by Maa [1, 14]. Thereafter, several studies have followed with similar proposals [15, 16]. A double-layered MPP sound absorber has two MPP layers, two air cavities, and a rigid backing wall. The distance between the two MPP layers is D1 and the distance between the inner layer MPP and the rigid wall is D2, as illustrated in Fig. 1. Note that, in this study, the two MPP layers had the same properties. Therefore, the acoustic performance of the double-layer MPP sound absorber depended on the panel thickness t, perforation diameter d, perforation ratio p (ratio of the perforation surface area to the total surface area of the panel), and cavity depth D1 and D2.
Fig. 1

Schematic diagram of double-layer MPP sound absorber

2.2 Double-Layer MPP Muffler

Two double-layer MPP mufflers with different structural parameters were selected as the research object. To clarify the two simplified MPP simulation methods, one of the two mufflers was considered for a detailed illustration. The muffler was made out of stainless steel. The two MPP layers of the muffler were both straight cylindrical tubes and had the same structural parameters. Similarly, the spacing between the inner MPP and the outer MPP was D1 and the spacing between the outer MPP and the rigid wall was D2, as shown in Fig. 2. The detailed values of the MPP’s structural parameters and the muffler’s cavity depths are listed in Table 1. Moreover, the MPP perforations were circular holes arranged in a square manner. Thus, the MPP perforation ratio was p = πd2/(4b2), where b is the spacing between the centers of two adjacent perforations.
Fig. 2

Schematic diagram of double-layer MPP muffler

Table 1

Muffler’s MPP structural parameters and cavity depths

t (mm)

d (mm)

p (%)

D1 (mm)

D2 (mm)

0.5

0.58

8.26

9

1.8

The transmission loss is determined by the structure of the muffler and is often used to evaluate its acoustic performance [17, 18, 19]. Thus, we focused on the transmission loss. The transmission loss value of the double-layer MPP muffler was calculated and measured, as will be discussed below.

3 Numerical Simulation Using Transfer Admittance Method

3.1 Transfer Admittance of MPP

The particle velocity and sound pressure relationships between both sides of the MPP are expressed by its transfer admittance, as follows:
$$\left[ \begin{aligned} v_{{{\text{n}}1}} \hfill \\ v_{{{\text{n}}2}} \hfill \\ \end{aligned} \right] = \left[ {\begin{array}{*{20}c} {\alpha_{1} } & {\alpha_{2} } \\ {\alpha_{4} } & {\alpha_{5} } \\ \end{array} } \right]\left[ {\begin{array}{*{20}c} {p_{1} } \\ {p_{2} } \\ \end{array} } \right] + \left[ {\begin{array}{*{20}c} {\alpha_{3} } \\ {\alpha_{6} } \\ \end{array} } \right]$$
(1)
where vn1 and vn2 are the normal particle velocity on the inner and outer surfaces of the MPP, respectively; p1 and p2 are the sound pressure on the inner and outer surfaces of the MPP, respectively; α1, α2, α4, and α5 are the transfer admittance coefficients; α1 = β, α2 = − β, α4 = − , α5 = , β is the transfer admittance of the MPP, and K is the ratio of the inner diameter to the outer diameter of the MPP tube; α3 and α6 are the sound source coefficients; in the acoustic calculation of the muffler α3 = α6 = 0 [20].
Thus, to define the particle velocity and the sound pressure relationships between both sides of the MPP, the transfer admittance of the MPP must first be calculated. According to Maa [1], whose basic idea is that the MPP is a parallel connection of the perforations, the acoustic transfer impedance of the MPP is expressed as follows:
$$Z{ = }R + jX$$
(2)
with
$$R = \frac{32\eta t}{{pd^{2} }}\left[ {\sqrt {1 + \frac{{k^{2} }}{32}} + \frac{\sqrt 2 k}{8}\frac{d}{t}} \right]$$
(3)
$$X = \frac{\omega \rho t}{p}\left[ {1 + \frac{1}{{\sqrt {9 + \frac{{k^{2} }}{2}} }} + 0.85\frac{d}{t}} \right]$$
(4)
and
$$k = \frac{d}{2}\sqrt {\frac{\omega \rho }{\eta }}$$
(5)
where R is the specific acoustic resistance of the MPP; X is the specific acoustic reactance of the MPP; η is the dynamic viscosity, η = 1.82 × 10−5 kg/(ms); ρ is the air density, ρ = 1.225 kg/m3; and ω = 2πf represents the angular frequency (f is the frequency of the incident acoustic wave).
Then, the transfer admittance of the MPP is expressed as follows:
$$\beta = \frac{1}{Z}$$
(6)

3.2 Transmission Loss Calculation

The MPPs of the double-layer MPP muffler can be simulated by creating the corresponding transfer admittance properties. Because the acoustic transfer impedance of a MPP is different at different frequencies, an acoustic transfer impedance table must be prepared before the simulation calculation. The acoustic transfer impedance table is obtained based on complicated calculations with Eqs. 35 and is presented in Table 2. The frequency range of the impedance table was set to approximately 10–7000 Hz, and the frequency step was set to 10 Hz. As mentioned in Sect. 2.2, the structural parameters of the inner layer MPP and outer layer MPP had the same values; thus, the MPPs could be simulated using the same impedance table.
Table 2

Acoustic impedance of MPP

Frequency (Hz)

Acoustic resistance

Acoustic reactance

10

11.8196

1.0791

20

12.4082

2.1553

30

12.8730

3.2286

6990

64.9178

674.9958

7000

64.9616

675.9426

Considering that the MPPs of this muffler were metal panels with large rigidity, the panel vibration effect under any acoustic loading could be ignored. The geometry model of the MPPs was not established, but the air cavity geometry model of the muffler was established, and the schematic diagram of its cross section is shown in Fig. 3. Surfaces 1 and 2 are the inner and outer surfaces, respectively, of the inner layer gap representing the original inner layer MPP. Surfaces 3 and 4 are the inner and outer surfaces, respectively, of the outer layer gap representing the original outer layer MPP.
Fig. 3

Schematic diagram of muffler’s air cavity cross section

The finite element (FE) model of the muffler should be established after obtaining the geometrical model of the air cavity. Generally, the length of the element is less than 1/6 of the wavelength corresponding to the highest calculation frequency. The investigated double-layer MPP muffler was mainly used for high-frequency noise control. Thus, to ensure the calculation accuracy, the maximum element size of the FE model was set to 2 mm. Figure 4 shows the entire FE muffler model. As can be seen, both ends of the muffler’s mesh model represent the mesh model of the end connectors used to fix the muffler for experimental measurement. Then, the FE model of the muffler was imported into the LMS acoustics module of the Virtual Lab software. The inner layer MPP was simulated by defining the transfer admittance properties between surfaces 1 and 2; that is, by inputting the above impedance table and converting it to the transfer admittance coefficients α1, α2, α4, and α5 at each frequency. Similarly, the outer layer MPP could also be simulated by defining the transfer admittance properties between surfaces 3 and 4. The fluid material properties of the air domains were the mass density of 1.225 kg/m3 and the corresponding sound velocity of 340 m/s. A unit velocity boundary condition was defined at the inlet of the FE model, while a full absorption boundary condition was defined at its outlet. After performing a harmonic response, the transmission loss value of the double-layer MPP muffler was derived. The plot of the transmission loss curve is shown in Fig. 5.
Fig. 4

Finite element (FE) muffler model

Fig. 5

Transmission loss calculated by numerical simulation based on transfer admittance method

4 Numerical Simulation by Equivalent Fluid Method

4.1 Porous Material Model

A porous material consists of a frame and a fluid medium. Johnson-Champoux-Allard porous material models are categorized into three types according to frame stiffness; namely the rigid model, limp model, and elastic model [21].

The rigid porous material model ignores the frame deformation, and its acoustic performance depends on the fluid parameters (sound velocity, mass density, specific heat ratio, Prandtl number, and dynamic viscosity) and the pore parameters (porosity, flow resistivity, tortuosity, and viscous and thermal characteristic lengths). In this case, sound absorption was produced by the air’s viscous resistance, which is similar to the sound absorption principle of the MPP. Based on the rigid porous material model, the limp porous material model considers the influence of the inertial load on the acoustic wave. Thus, it has an additional characteristic parameter (frame density), which refers to that of a rigid porous material model. Among the three models, the elastic porous material model is the most complicated. Considering the frame vibration effect under acoustic loading, the elastic model has three additional characteristic parameters (frame density, Young’s modulus, and Poisson’s ratio), which refer to those of the rigid porous material model.

4.2 MPP Equivalent Porous Material Model

A thin porous material structure may become equivalent to a MPP by transforming the MPP structural parameters to the relevant parameters of the corresponding porous material model. Typically, the fluid medium of the MPP structures is air. Thus, only the pore parameters need to be calculated when establishing the equivalent porous material model for a rigid MPP. According to Atalla [11], the parameters are transformed as follows:
$$\varphi = p$$
(7)
$$\sigma = \frac{32\eta }{{\varphi d^{2} }}$$
(8)
$$\alpha_{\infty } = 1 + \frac{{2\varepsilon_{e} }}{t}$$
(9)
when the perforations are circular holes and arranged in a square manner, the following relationships hold:
$$\varepsilon_{\text{e}} = 0.24\sqrt {\pi d^{2} } \left( {1 - 1.14\sqrt \varphi } \right),\sqrt \varphi < 0.4$$
(10)
$$A_{\text{v}} = A_{\text{t}} = \frac{d}{2}$$
(11)
where φ is the porosity, σ is the flow resistivity, α is the tortuosity, εe is the correction length, Av is the viscous characteristic length, and At is the thermal characteristic length.

The three porous material models correspond to three MPP equivalent porous material models, respectively [12, 13]. An appropriate MPP equivalent porous material model was selected based on the material properties of the MPP. During the parameter conversion, the panel density was considered in the limp MPP equivalent porous material model, and the panel density, Young’s modulus, and Poisson’s ratio were considered in the elastic MPP equivalent porous material model.

4.3 Transmission Loss Calculation

The MPPs of the double-layer MPP muffler have large rigidity. Thus, a rigid MPP equivalent porous material model was chosen for the muffler’s simulation calculation. Based on Eqs. 711, the relevant parameters of the MPP equivalent porous material model were calculated and are presented in Table 3. Figure 6 shows the FE model of the muffler’s outlet. The equivalent porous material structure was modeled on the basis of the MPP geometry model. The FE model was imported into the acoustics module of the LMS Virtual Lab software. The two layers of the MPPs were simulated by defining the corresponding porous material properties. The fluid material properties of the air domains and the boundary conditions were the same as those mentioned in Sect. 3.2. The muffler’s transmission loss results obtained by the simulation calculation are shown in Fig. 7.
Table 3

Parameters of MPP equivalent porous material model

φ (%)

σ (Pa·s· m−2)

α

Av (mm)

At (mm)

8.26

20,959.71

1.66

0.29

0.29

Fig. 6

FE model of muffler’s outlet

Fig. 7

Transmission loss calculated by numerical simulation based on equivalent fluid method

5 Experimental Verification

To verify and compare the accuracy of the two simplified MPP simulation methods with regard to the acoustic simulation of the double-layer MPP muffler, the transmission loss value of the muffler was measured with an impedance tube using the two-load method [22, 23]. Figure 8 shows the experimental setup in the laboratory. The impedance tube used in this study was a B&K Type 4206 impedance tube with an inner diameter of 29 mm and an effective measurement frequency range of 500–6400 Hz. Because the muffler’s inlet and outlet diameters were not the same as the diameter of the impedance tube, two end connectors were made to connect them seamlessly for the experiment (the end connectors were also considered in the numerical calculation). Four flush-mounted 1/2 B&K free-field microphones were used to measure the sound pressure inside the duct. A loudspeaker driven by a power amplifier was connected at one end of the impedance tube as the excitation source, and a removable cap was placed at the other end of the tube to provide two different termination conditions. The muffler sample to be tested was located between the two tubes. A four-channel B&K type 3560-C signal analyzer platform and a personal computer equipped with the PULSE 8.0 software were used for data acquisition and signal processing. Finally, the transmission loss value of the muffler was obtained by testing.
Fig. 8

Experimental setup

The comparison between the numerical simulation results and the experimental results of transmission loss for the double-layer MPP muffler is presented in Fig. 9. As can be seen, the numerical results based on the two simplified MPP simulation methods are in relatively good agreement with the experimental results, which indicates that the proposed transfer admittance method and the equivalent fluid method can both accurately predict the acoustic performance of a double-layer MPP structure. It can also be seen that the transmission loss curves based on the two simplified MPP simulation methods are almost identical below 4000 Hz. Moreover, in frequencies above 4000 Hz, the transmission loss results based on the transfer admittance method are closer to the experimental results at the peaks and dips in comparison with the transmission loss results based on the equivalent fluid method.
Fig. 9

Comparison of numerical simulations and experimental results of transmission loss for the double-layer MPP muffler: + indicates experimental measurement; – indicates prediction by transfer admittance method; · indicates prediction by equivalent fluid method

To further validate the above results, another double-layer MPP muffler with different structural parameters was treated in a similar manner. The detailed values of this muffler’s MPP structural parameters and cavity depths are listed in Table 4. Additionally, the comparison between the numerical simulation results and the experimental results of transmission loss is presented in Fig. 10. The comparison results are consistent with those mentioned above, which demonstrates that the above discussion is reasonable.
Table 4

Second muffler’s MPP structural parameters and cavity depths

t′ (mm)

d′ (mm)

p′ (%)

D1′ (mm)

D2 (mm)

0.5

0.5

4.91

9

1.8

Fig. 10

Comparison of numerical simulations and experimental results of transmission loss for second double-layer MPP muffler: + indicates experimental measurement; − indicates prediction by transfer admittance method; · indicates prediction by equivalent fluid method

6 Conclusions

This study adopted two different simplified MPP simulation methods to calculate the transmission loss of double-layer MPP mufflers. One method combined classical MPP sound absorber theory with MPP transfer admittance and simulated a MPP by defining a set of transfer admittance coefficients between its inner and outer surfaces. The other method considered the effect of the MPP material properties and replaced the MPP with an equivalent porous material model. To verify and compare their accuracy, the transmission loss value of the double-layer MPP mufflers was measured experimentally with an impedance tube. According to the comparison results between the two numerical simulations and the experimental data, conclusions may be illustrated as follows:
  1. (1)

    The transfer admittance method and the equivalent fluid method both provide an easy and efficient approach toward predicting the acoustic performance of double-layer MPP structures.

     
  2. (2)

    Acoustic prediction based on the transfer admittance method may be more accurate than that based on the equivalent fluid method at the peaks and dips of TL curve.

     

Notes

Acknowledgements

This study was supported by the National Key Laboratory Open Foundation of Tractor Power System (Grant No. SKT2017012) and the National Natural Science Foundation of China (Grant No. 51575410).

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Copyright information

© China Society of Automotive Engineers (China SAE) 2018

Authors and Affiliations

  1. 1.Hubei Key Laboratory of Advanced Technology for Automotive ComponentsWuhan University of TechnologyWuhanChina
  2. 2.Hubei Collaborative Innovation Center for Automotive Components TechnologyWuhan University of TechnologyWuhanChina

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