Vibroacoustic analysis of multi-layered micro-perforated plates coupled to an acoustic enclosure

Abstract

A vibroacoustic study of multi-layered micro-perforated plates (MPP) coupled to an enclosure cavity excited by internal and external acoustic excitations is presented. Most previous research centered on the multi-layered MPPs structure itself, the absorption and transmission performances were often calculated or tested in a simple acoustic environment. In a more realistic setting, the surrounding systems may affect the efficiency of MPPs. The proper modeling of the MPPs within acoustic enclosures requires careful study. The break-out sound from an enclosure via multi-layered MPP and the energy transmission of the structure into an acoustic enclosure undergoing exterior excitation are studied with an emphasis on the use of MPP in acoustic enclosures to suppress specific resonances of the enclosure. Therefore, an analytical formulation is proposed to model the behavior of the coupled system to address this issue. Using a modified Fourier series for the acoustic pressure, the continuity of the normal velocities at the interfaces of the plate or the MPP and the acoustic medium is assured accurately. A modified variational form for the acoustic and structure medium yields a completely coupled vibroacoustic system. Comparison with the finite element method (FEM) and available literature findings prove the efficacy and precision of the current approach. The effect of important parameters on the vibroacoustic behavior of multi-layered MPPs structure connected to an acoustic enclosure is examined. These parameters include micro-perforation of the plate, hole diameter, air gap thickness, acoustic enclosure dimensions, and the insertion of absorbent material in the gap.

Keywords

Micro-perforated plate sound transmission loss noise insulation coupled structural-acoustic sound absorption coefficient modal analysis

Introduction

The interaction between multi-layered structures and acoustic enclosures is a topic of significant concern that needs to be taken into consideration in a variety of applications, including but not limited to buildings, automobiles, aircraft, and other similar areas. It is common practice to use multilayered structures for sound absorption and vibration damping. It is the purpose of these structures to enhance acoustic performance without adding extra weight, which is necessary for such applications.

Double-plates with air cavities offer superior acoustic insulation compared to single-plate structures, they are frequently used in noise control engineering. It has gained a significant amount of interest.^3–9 For instance, Wang et al.³ used the statistical energy analysis method to compute the sound transmission loss (STL) of a double panel with an air gap, while the STL of air-filled cavities in simply supported finite double panels was computed analytically by Xin et al.⁴ Sandwich panels with sound-absorbing cores have also been the focus of a significant number of studies in addition to air cavities.^1,2,10–17 For example, Bolton et al.^12,15 applied the Biot theory to compute the STL of double-plate configurations filled with an elastic porous material. Doutres and Atalla¹¹ suggested an approach to compute the transmission loss of a double-plate structure with layered blanket cores. Specifically, it was discovered that at structural resonance frequencies, STL is enhanced with sound-absorbing cores. Santoni et al.¹⁸ published a review of the various methods to predict the sound transmission loss of building partitions.

On the other hand, MPPs are excellent sound absorbers. Perforated panels were placed in front of a solid wall to accelerate particle velocity through the perforations and dissipate acoustic energy. Without adding a layer of porous materials, sub-millimeter-sized holes may offer relatively wide-band absorbers due to the shear forces induced by the vibration of the air moving through them. That allows the construction of absorbent walls that are lightweight, fiber-free, and highly functional. Numerous researchers have investigated, both theoretically and experimentally, MPPs' effectiveness as sound absorbers. Maa^19,20 first presented a model approach for the sound absorption coefficient (SAC) of MPPs using an electroacoustic analogy. Using rigid-frame porous material models, the SAC of MPPs was calculated by Atalla and Sgard.²¹ Ng and Hao-MING²² investigated the effectiveness of perforated honeycomb sandwich panels as sound absorbers for low-frequency range. It has been demonstrated that the configuration is capable to absorb sound for frequencies as low as 63 Hz. The SAC of MPPs was estimated using an empirical impedance model by Rao and Munjal²³ and Lee and Kwon.²⁴ Yuvaraj and Jeyanthi²⁵ examined a countersunk MPP and two layers of porous materials in an impedance tube. The device improves sound absorption, and porous materials extend its bandwidth. Although MPPs perform well as sound absorbers, they are inadequate for STL. When compared to a plate of the same thickness, the STL of an MPP is lower as shown by Chen²⁶ and Dupont et al.²⁷

Currently, researchers are now considering MPP and sandwich panel combinations for STL and SAC. Perforated sandwich panel face plates can absorb sound, while back plates can function as noise barriers. The acoustic performance of an MPP-air cavity-plate coupled system was initially analyzed analytically and experimentally by Dupont et al.²⁷ It was discovered that the STL could be increased with a good SAC. Honeycomb structures were inserted into the air cavity of the system by Toyoda and Takahashi²⁸ to enhance the STL at medium frequencies. To compute the absorption and transmission properties of multi-layered MPPs separated from a thin plate by an air gap, Bravo et al.^29,30 suggested using a fully coupled analysis model for the calculation of the SAC and STL. Airframe and MPP-back panel relative velocities were shown to control the SAC and STL at acoustical resonant frequencies. Mu et al.³¹ found that adding an MPP to the source as well as the transmitting face of the double-panel improved the mass-air-mass resonance. Using the transfer matrix technique, Kim et al.³² described in good detail the effects of micro-perforations on multi-layered infinite MPPs’ STL over the whole frequency range. The equivalent impedance, which is the combination of the impedances for the inertia term and the micro-perforation, was used to characterize the effect of the perforation. Employing the transfer matrix approach and assuming a condition of plane waves in the low-frequency domain, Kim et al.³³ analyzed multi-layered flexible MPP’s STL, placed in a rectangular cross-sectioned impedance tube, with emphasis on its behavior at resonance frequencies.

In most of these investigations, the absorption and transmission properties of multi-layered structures were evaluated without considering the influence of the surrounding system. Numerous studies on the cavity-plate systems excited by internal or external noise sources which may be airborne or structure-borne and their vibroacoustic behaviors have been published. By using modal coupling analysis, Pan³⁴ demonstrated how to calculate the acoustic and vibrational fields in a coupled plate cavity. Calculations were done to determine the frequency response as well as the sound field's transient decay in the cavity. The transmission of energy from a double plate connected by a mechanical link partition into an acoustic cavity was studied by Cheng et al.³⁵ The influence of the air gap and mechanical links on the soundproofing and insulation characteristics of these structures was explored using a fully coupled vibroacoustic formulation. Through the application of a secondary force, Pan et al.³⁶ and Pan and Hansen ³⁷ suggested active sound transmission control into a cavity through a panel. Bravo and Elliott³⁸ developed a coupled analytical method to examine cavity-panel-cavity and cavity-panel problems and the variability of the sound transmission loss in the low-frequency range was discussed. The impedance-mobility approach^39,40 was employed to investigate the break-out of sound from a rectangular enclosure with a plate(s). The pressure distribution as well as the radiation pattern of a structural-acoustic coupling system composed of a cavity and a semi-infinite space separated by a wall were predicted by Seo and Kim.⁴¹ Using the finite element analysis, Becot and Sgard⁴² investigated the capability of poroelastic materials to control the noise radiated outside a cavity in the low-frequency range. Attaching poroelastic to a panel, a considerable reduction in board frequency range can be achieved. The acoustic cavity delimited by a flexible and holed structure was the subject of research by Kim and Kim.^43,44 It was demonstrated that, based on the frequency range and wave number, the cavity can effectively communicate with the surrounding environment via the vibrating structural system and the hole. Clasen and Langer⁴⁵ used a two-dimensional FEM model to predict flanking transmissions in building acoustics. Through the use of a modified Fourier series approach, Du et al.^46,47 analyzed the cavity-panel system, reinforcing full velocity continuity at the cavity/panel interface. Using the patch-mobility technique, Chazot and Guyader⁶ predicted the transmission loss of a double panel coupled to a source room. To further investigate the transmission loss of a double panel filled with poro-granular material using the fluid-fluid Biot's model, the authors extend their previous paper.¹⁰ Wang et al.⁴⁸ presented a method for estimating the break-out sound from an acoustic cavity-backed plate. The model considers the interface’s strong structural-acoustic couplings. However, few studies focused on MPP coupled to an acoustic cavity. Sound absorption of MPPs supported by air volume or honeycomb structure was studied by Yang and Cheng⁴⁹ in small enclosures. Backing cavity and enclosure interaction were found to have a significant impact on the MPP’s capacity to absorb sound. However, the analytical model used in these studies consists of a typical MPP and a rigid back wall coupled to an acoustic enclosure. Thomazelli and Bertoli⁵⁰ investigated a potential alternative to optimize the production and development of a tunable MPP absorber capable of changing its geometrical configurations in response to the acoustic requirements of each room.

However, none of the prior research examined the vibroacoustic behavior of a multi-layered MPP structure coupled with an acoustic enclosure. In building acoustic problems, multi-layered MPPs might be utilized to minimize external noise transmitted to the acoustic enclosure. The structure might also be utilized to reduce reflections and transmissions produced by internal noise sources. Multi-layered MPP behavior must be evaluated in such applications by considering the interaction with acoustic enclosures. Existing research has significant limitations: (a) The sound transmission and absorption properties of multi-layered MPPs were evaluated without considering the influence of the surrounding system. Nonetheless, the surrounding systems and excitations have a significant impact on the multi-layered MPP's performance, and (b) most of the studies of cavity panel systems assume an external sound source or internal sound source, and researchers are focused on determining the cavity pressure and the sound transmission loss related with the configuration. However, the use of microperforated plates to introduce damping in a closed cavity and reduce reflections has not been examined. It is essential to model and predict the multilayered MPP's effectiveness in a complex vibro-acoustic environment to have a thorough understanding of vibro-acoustic behavior and how it responds to changes in the various system variables. Furthermore, the development of a strong and precise strategy capable of generically addressing numerous system aspects would be of tremendous value to both academics and engineers. To the author's knowledge, despite the progress achieved previously, the in-situ behavior of multilayered MPPs in a strongly coupled vibroacoustic system has never been systematically reported in the literature.

Intending to address the aforementioned shortcomings of previous models and clarify the vibroacoustic behavior of the multi-layered MPP and its sensitivity to changes in any of the system variables, a general structural-acoustic coupling model of a simply supported MPP separated from a plate by an air gap and coupled to an acoustic enclosure is presented. A vibro-acoustic formulation is proposed to model the behavior of the coupled system. A modified Fourier series is used to describe the sound pressure inside the cavity. The displacement and acoustic pressure solutions are obtained using the modified variational principle for the coupled system. Comparisons with the finite element technique (FEM) and previously published results provide evidence of the present approach's efficiency and accuracy. The effect of critical factors on the STL of the structure coupled to an enclosure is investigated, including micro-perforation of the plate, hole diameter, air gap thickness, acoustic enclosure dimensions, and the influence of absorbent material installed in the gap. It should be highlighted that the present approach is readily extensible to multi-layered MPPs coupled with an acoustic enclosure.

Theoretical formulations

Description of the coupled system

Figure 1 depicts the coupled system that is the subject of the current investigation. This system is an outer plate and an inner micro-perforated plate separated by an air gap Vg and coupled to an enclosure cavity. The MPP and the plate are rectangular, homogeneous, and isotropic. The simply supported boundary condition for the MPP and the plate is assumed. The following two cases are examined. A uniform, oblique plane sound wave with an incidence elevation angle $ϕ$ and an azimuth angle $θ$ is applied to the outer plate, whilst the inner MPP, is coupled to the enclosure with a volume V in the first case, and a point source is located inside the enclosure in the second. Both the air gap and the enclosure have acoustically rigid walls except for the surfaces occupied by the two plates. However, an arbitrary impedance surface condition can be easily incorporated.

Figure 1.

A schematic of an MPP-Cavity-Plate structure coupled to an enclosure.

The acoustic pressure within the acoustic enclosure should satisfy both the wave equation and the corresponding boundary conditions.

The wave equation is given as:

\nabla^{2} p_{e} + k^{2} p_{e} = - j ω ρ_{0} {\bar{Q}}_{0}

(1)

Where $p_{e}$ is the acoustic pressure within the enclosure, $k = ω c_{0}$ is the wavenumber, and $\nabla^{2} = \frac{\partial^{2}}{\partial x^{2}} + \frac{\partial^{2}}{\partial y^{2}} + \frac{\partial^{2}}{\partial z^{2}}$ is the Laplacian operator.

{\bar{Q}}_{0} = Q_{0} δ (x - x_{0}) (y - y_{0}) (z - z_{0})

(2)

Where $Q_{0}$ denotes the source strength and $δ$ is the Dirac delta function.

The boundary conditions on the enclosure boundaries:

\frac{\partial p_{e}}{\partial z} = j ω ρ_{0} \bar{v_{2}}, on the MPP

(3a)

\frac{\partial p_{e}}{\partial n} = 0, on the rigid wall

(3b)

where $ρ_{0}$ and $c_{0}$ are the density of air and speed of sound respectively, and ${\bar{v}}_{2}$ is the average velocity of the MPP.

Similarly, the acoustic pressure within the air gap satisfies the wave equation including the continuity conditions on the boundaries:

\nabla^{2} p_{g} + k^{2} p_{g} = 0

(4a)

\frac{\partial p_{g}}{\partial z} = j ω ρ_{0} v_{1}, on the plate

(4b)

\frac{\partial p_{g}}{\partial z} = - j ω ρ_{0} \bar{v_{2}}, on the MPP

(4c)

\frac{\partial p_{g}}{\partial z} = 0, on the rigid wall

(4d)

Where $p_{g}$ is the acoustic pressure inside the enclosure, $v_{1}$ is the velocity of the outer plate.

As can be seen in Figure 2, the average velocity $\bar{v_{2}}$ of the MPP is related directly to the velocity of the plate $v_{2}$ as well as the velocity of the fluid $v_{f}$ as it passes through the hole.

\bar{v_{2}} = v_{2} (1 - σ) + v_{f} σ

(5)

$σ$ denotes the perforation ratio, and it is defined as $σ = π d^{2} / 4 U^{2}$ , where $d$ denotes the hole’s diameter and $U$ represents the distance between the holes of the MPP.

Figure 2.

Average velocity on the surface of the MPP.

As was shown in prior research,⁵¹ the impedance of the MPP is related to the pressure difference.

Z_{r e s i s t} (v_{f} - v_{2}) + Z_{r e a c t} v_{f} = Δ p

(6)

According to Maa,²⁰ the impedance of the hole, $Z = Z_{r e s i s t} + Z_{r e a c t}$ , is composed of a resistive component and an imaginary component. The former corresponds to the resistive force between the fluid and the inner surface of the hole whereas the second one describes the inertia force of the fluid in the hole. They are given as:

Z_{r e s i s t} = \frac{8 η_{0} h_{2}}{{(d / 2)}^{2}} (\sqrt{1 + \frac{X^{2}}{32}} + \frac{\sqrt{2} dX}{32 h_{2}})

(7)

Z_{r e a c t} = j ρ ω h_{2} (1 + \frac{1}{\sqrt{9 + \frac{X^{2}}{2}}} + \frac{8 d}{3 π h_{2}})

(8)

where $η_{0}$ represents the air’s viscosity coefficient, $h_{2}$ denotes the thickness of the MPP, and $X = (\frac{d}{2}) \sqrt{ρ ω / η_{0}}$ .

The elimination of the term $v_{j, f}$ from equations (5) and (6) lead to the following relation:

\bar{v_{2}} = γ v_{2} + \frac{σ Δ p}{Z}

(9)

Where,

γ = 1 - σ (\frac{Z_{r e a c t}}{Z})

(10)

In the absence of perforation, $γ = 1$ and $σ = 1$ .

The governing equation of the outer plate is given as a function of the plate displacement $w_{1}$ (and $v_{1} = j ω w_{1}$ ):

D_{1} \nabla^{4} w_{1} - M_{1} ω^{2} w_{1} = p_{0} - p_{g}

(11)

Where $M_{1}$ is the mass per unit area and $D_{1}$ is the flexural rigidity of the plate, it can be written as:

D_{1} = E_{1} {h_{1}}^{3} / 12 (1 - {υ_{1}}^{2})

(12)

$E_{1}$ represents Young’s modulus, $υ_{1}$ Poisson’s ratio, $ρ_{1}$ density, and $h_{1}$ thickness of the plate.

In equation (12), damping of the structure is taken into consideration by inserting a complex Young’s modulus $E_{1} (1 + j η_{1})$ , where $η_{1}$ is the loss factor of the plate.

The sound pressure exerted on the incident panel is a combination of three different pressures: the incident pressure, the reflected pressure, and the radiated pressure which is extremely insignificant. Because the impedance of the incident plate is comparable to that of a rigid boundary when subjected to air loading, it is safe to assume that the magnitudes of the incident and reflected pressure waves are the same.^35,52,53 Therefore, by ignoring the pressure radiated toward the outside, we can determine that the blocked pressure, also known as the excitation pressure, is twice as great as the magnitude of the incident wave⁵³:

p_{0} = 2 p_{i n c} \exp (j ω t - j k z \cos (ϕ) - j k y \sin (ϕ) \sin (θ) - j k x \sin (ϕ) \cos (θ))

(13)

where $p_{i n c}$ is the incident pressure’s amplitude.

Similarly, the governing equation of the inner MPP is given as a function of MPP displacement $w_{2}$ (and $v_{2} = j ω w_{2}$ )

D_{2} \nabla^{4} w_{2} - M_{2} ω^{2} w_{2} = p_{g} - p_{e}

(14)

Where $M_{2}$ is the mass per unit area and $D_{2}$ is the flexural rigidity of the MPP.

Solution procedure of the coupled system

The velocity, $v_{j}$ , for the MPP and the plate may be expressed as a finite series of structural modes as follows:

v_{j} (x, y) = j ω \sum_{m = 1}^{\infty} \sum_{n = 1}^{\infty} q_{j, m n} φ_{m} (x) φ_{n} (y)

(15)

Where $ω$ is the excitation’s angular frequency and $q_{j, m n}$ are the corresponding modal coefficients.

For the case of simply supported boundary conditions, $φ_{m} (x) = \sin (m π x / L_{x})$ and $φ_{n} (x) = \sin (n π x / L_{y})$ .

To ensure differential continuity at the structural-acoustic coupling interface, the acoustic pressure inside the enclosure and the air gap may be represented as.^46,47

\begin{matrix} p_{g} (x, y, z) = \sum_{m_{x} = 0}^{M_{x}} \sum_{m_{y} = 0}^{M_{y}} \sum_{m_{z} = 0}^{M_{z}} A_{m_{x} m_{y} m_{z}}^{g} \cos (λ_{m_{x}} x) \cos (λ_{m_{y}} y) \cos (λ_{m_{z}} z) \\ + \sum_{m_{x} = 0}^{M_{x}} \sum_{m_{y} = 0}^{M_{y}} (ξ_{1 L_{z}} (z) a_{m_{x} m_{y}}^{g} + ξ_{2 L_{z}} (z) b_{m_{x} m_{y}}^{g}) \cos (λ_{m_{x}} x) \cos (λ_{m_{y}} y) \end{matrix}

(16)

\begin{matrix} p_{e} (x, y, z) = \sum_{m_{x} = 0}^{M_{x}} \sum_{m_{y} = 0}^{M_{y}} \sum_{m_{z} = 0}^{M_{z}} A_{m_{x} m_{y} m_{z}}^{e} \cos (λ_{m_{x}} x) \cos (λ_{m_{y}} y) \cos (λ_{m_{z}^{'}} z) \\ + \sum_{m_{x} = 0}^{M_{x}} \sum_{m_{y} = 0}^{M_{y}} ξ_{2 L_{z}} (z) b_{m_{x} m_{y}}^{e} \cos (λ_{m_{x}} x) \cos (λ_{m_{y}} y) \end{matrix}

(17)

Where $λ_{m_{s}} = m_{s} π / L_{s}, (s = x, y, z)$ . The supplemental functions $ξ_{1 L_{s}} (s)$ $ξ_{2 L_{s}} (s)$ can be found in.⁴⁷

Theoretically, an acoustic analysis of an enclosure may be expressed in a variational form. This can lead to a solution that is preferable to the one that can be obtained by just solving the Helmholtz equation. To accomplish this goal, a modified variational approach^54–56 is used to define the distribution of the acoustic pressure. This method involves searching for the minimum of the associated modified variational function.

\int \int \int_{V_{g}} \frac{1}{2} [p_{g} \frac{jk}{ρ_{0} c_{0}} p_{g} - {(\nabla p_{g})}^{T} \frac{j}{ρ_{0} ω} (\nabla p_{g})] dV + \int \int_{S_{MPP 1}} p_{g} v_{1} dS - \int \int_{S_{MPP 2}} p_{g} {\bar{v}}_{2} dS = 0

(18)

Using equation (7) we get

\begin{matrix} \int \int \int_{V_{g}} \frac{1}{2} [p_{g} \frac{jk}{ρ_{0} c_{0}} p_{g} - {(\nabla p_{g})}^{T} \frac{j}{ρ_{0} ω} (\nabla p_{g})] dV + \\ \int \int_{S_{MPP 1}} p_{g} v_{1} dS - \int \int_{S_{MPP 2}} p_{g} (γ_{2} v_{2} + \frac{σ_{2} Δ p_{2}}{Z_{2}}) dS = 0 \end{matrix}

(19)

Where $Δ p_{2} = p_{g} - p_{e}$

The characteristic equation may be found by inserting the admissible function specified in equations (15)–(17) into equation (18), and then carrying out the variational operation in terms of the generalized coordinate vector.

The resultant matrix equation for the air gap is as follows:

[K_{g} + j ω Z_{g} + ω^{2} M_{g}] P_{g} + ω^{2} C_{p_{g}, w_{1}} W_{1} - ω^{2} C_{p_{g}, w_{1}} W_{2} + j ω C_{p_{g}, p_{e}} P_{e} = {0_{g}}

(20)

In the same manner, for the acoustic enclosure, we get:

[K_{e} + j ω Z_{e} + ω^{2} M_{e}] P_{e} + ω^{2} C_{p_{e}, w_{1}} W_{2} + j ω C_{p_{e}, p_{g}} P_{g} = {Q_{e}}

(21)

Using equation (15), the left-hand side of equation (11) can be rewritten as

D_{1} \nabla^{4} w_{1} - M_{1} ω^{2} w_{1} = M_{1} \sum_{m = 1}^{\infty} \sum_{n = 1}^{\infty} q_{1, m n} (ω_{1, m n}^{2} - ω^{2}) φ_{m} (x) φ_{n} (y)

(22)

Where $ω_{1, m n}$ is the natural frequency that corresponds to the $(m, n)$ mode of the first MPP, given as

ω_{1, m n} = π^{2} h_{1} {\frac{m^{2}}{L_{1}^{2}} + \frac{n^{2}}{L_{2}^{2}}} \sqrt{\frac{E_{1}}{12 ρ_{1} (1 - {v_{1}}^{2})}}

(23)

When both sides of equation (11) are multiplied by $φ_{m^{'}} (x) φ_{n^{'}} (y)$ , and then the equation is integrated across the surface of the plate while using the advantage of the orthogonal feature of the modes, the result is:

M_{1, m n} q_{1, m n} (ω_{1, m n}^{2} - ω^{2}) = \int \int_{S} p_{0} φ_{m} (x) φ_{n} (y) - \int \int_{S} p_{g} φ_{m} (x) φ_{n} (y)

(24)

Following the same steps for the MPP, we get:

M_{2, m n} q_{2, m n} (ω_{2, m n}^{2} - ω^{2}) = \int \int_{S} p_{g} φ_{m} (x) φ_{n} (y) - \int \int_{S} p_{e} φ_{m} (x) φ_{n} (y)

(25)

The enclosure and air gap equations (20) and (21) and the MPPs equations (24) and (25) form a set $(N_{e} + N_{g} + M_{plate} + M_{MPP})$ of coupled equations that may be put in a matrix form as follows:

[K + j ω C + ω^{2} M] X = Q

(26)

where $M$ and $K$ are diagonal matrices representing the mass and stiffness of the whole system, respectively, $C$ is the modal coupling matrice for cavity-hole-cavity, $X$ is the vector of the unknown complex modal amplitudes of the different parts of the coupled system(plate, air cavity, MPP, and enclosure), Q represents the generalized excitation vector. In the first case, when uniform pressure is applied to the outer plate, the point source inside the enclosure is set to zero, However, in the second case, the outside excitation pressure is suppressed.

It should be noted that the formulated equations are restricted to the case of plate-cavity-MPP coupled to an acoustic enclosure. However, it could be easily extended to multilayered MPPs-Cavity-Plate coupled with an acoustic enclosure.

Calculation of the transmission loss

Transmission loss in the case of an enclosure as receiving room

The transmission loss or sound reduction index of an MPP-Cavity-Plate partition is defined by the following relation:

T L = 10 \log_{10} (\frac{〈 P_{0}^{2} 〉}{〈 P_{e}^{2} 〉})

(27)

Where $〈 P_{0}^{2} 〉$ and $〈 P_{e}^{2} 〉$ are given by:

〈 P_{0}^{2} 〉 = \frac{1}{2 S} \int_{S} p_{0} p_{0}^{*} dS

(28)

〈 P_{e}^{2} 〉 = \frac{1}{2 V_{e}} \int_{V_{e}} p_{e} p_{e}^{*} dV

(29)

where $p_{0}$ is the complex amplitude of the blocked sound pressure and $p_{e}$ is the inner enclosure pressure.

Transmission loss in the case of an enclosure as a source room

The transmission loss of the partition is defined by the relation:

T L = 10 \log_{10} (\frac{Π_{i n c}}{Π_{rad}}) (dB)

(30)

where $Π_{i n c}$ and $Π_{rad}$ are the sound powers that impinge on and radiate from the MPP-cavity-plate, respectively, at a given frequency. The impinging sound power is typically estimated by treating the interior enclosure as a reverberant room, and it is approximately computed using

Π_{i n c} (ω) = \frac{1}{N} \sum_{k = 1}^{N} \frac{{| p_{e}^{k} |}^{2} S}{4 ρ_{0} c_{0}}

(31)

Where $p_{e}^{k}$ is the acoustic pressure computed at the $k^{th}$ position of the randomly chosen points in the enclosure and $S$ is the surface of the plate.

The radiation power can be determined from the relation below:

Π_{t r a n s} = \frac{1}{2} Re {\int \int_{S} p_{tr} v_{1}^{*} dA}

(32)

Where $p_{tr}$ is the acoustic pressure radiated by the outer plate on its surface expressed by Rayleigh’s integral of the radiated pressure. It is computed numerically.

Numerical results and discussions

In this section, the proposed model is validated and a parametric study is performed following the theoretical modeling and solution approach presented previously. The equations derived in earlier sections have been incorporated into a computer program. The current method for estimating the dynamical responses and the STL of the MPP-Cavity-Plate structure connected to an enclosure in both cases, internal acoustic point source and external plane wave, will be validated first to verify its reliability and accuracy. Then, an investigation into the effects of different significant parameters on the vibroacoustic behavior of the coupled system is carried out, including micro-perforation of the plate, hole diameter, air gap thickness, acoustic enclosure dimensions, and the influence of absorbent material introduced in the gap.

Model validation

Numerical validation

This section gives analytical and numerical comparisons to verify the accuracy and precision of the current method. For this purpose, an MPP separated from a flexible plate by an air gap and connected to an enclosure is simulated, and the results obtained using the proposed method are compared to those predicted using finite element analysis software (Comsol). The coupled system considered consists of simply supported MPP and plate ( $L_{x} \times L_{y} = 0.2 m \times 0.3 m$ ), separated by a 0.1m air cavity and coupled to an acoustic enclosure ( $L_{z} = 0.5 m$ ). The properties of air are: $ρ_{0} = 1.225 kg / m^{3}$ and $c_{0} = 343 m / s$ , and the viscosity is $η_{0} = 17.9 \times 10^{- 6} kg / ms$ . In this study, we consider aluminum plates with the following material properties: Young’s modulus $E = 7.2 \times 10^{10} Pa$ , Poisson’s ratio $v = 0.34$ , density $ρ_{p} = 2700 kg / m^{3}$ , and the loss factor is assumed to be $η = 0.01$ . The sidewalls around the air gap cavity and the enclosure are perfectly rigid. The thicknesses of the MPP and the plate are, respectively, 0.001 and 0.002 m. The MPP parameters are $σ = 0.1 %$ and $d = 0.3 mm$ .

To verify the efficiency of the proposed model for acoustic source inside the enclosure. A point source of strength $Q_{0} = 10^{- 4} m^{3} / s$ is placed inside the acoustic enclosure at (0.05, 0.08, 0.1 m). For the case when the coupled system is excited by an external acoustic source, a plane wave is applied to the incidence part. COMSOL Multiphysics was used to create the FE model. shown in Figure 3. To imitate an infinite and non-reflective acoustic domain, a Perfectly Match Layer (PML) is placed at the end of the incident domain. The air in the acoustic enclosure, air gap cavity, and incident field are compressible without considering thermal conductivity and viscosity. As a result, the COMSOL “Pressure Acoustics” module is used. During the simulation, the MPP is modeled as linear elastic isotropic material using COMOL’s “Solid Mechanics” module. During the simulation, thermal conduction and viscosity of the air within the holes must be considered. As a consequence, the “Thermal-Acoustics” module is used. The normal accelerations of the air and plate at the interface of the pressure acoustic field and plate are the same in the FE model. The FE model applies continuous normal stress, acceleration, and adiabatic conditions at the interface of the thermal acoustic field and pressure acoustic field. In addition, the velocity of the air at the interface of the thermal acoustic field and plate coupling boundary is identical to that of the plate, with isothermal temperature variation. The FE model incorporated the boundary conditions of the MPPs, enclosure, and air gap. Tetrahedral elements mesh the FE model, as shown in Figure 3. The dimensions and physics of each part affect the size of the elements. In the current method, the number of modes used in both structural displacement and acoustic pressure decomposition is the most important factor impacting the solution’s accuracy. In general, precision may be attained by increasing the number of modes until convergence within the desired frequency spectrum. For internal excitation, the proposed approach and the numerical method are compared in predicting the dynamic responses of the coupled system in Figure 4. The space-averaged acoustic pressure level inside the enclosure and the air gap cavity is plotted in Figure 4(a). The velocity responses at (0.1, 0.1 m) on the radiating plate and (0.1, 0.1 m) on the inner MPP are shown in Figure 4(b). The proposed model's calculated vibrational and acoustic responses agree very well with those computed using the numerical method, as shown in Figure 4. The current model is validated by comparing the computed STL with the results obtained from the FEM. Figure 5 depicts the STL findings of the structure under sound excitation with elevation angle $ϕ = 0$ and azimuth angle $θ = 0$ derived from the present approach and fem simulation. As shown, the predicted STL achieved by the present technique agrees well with the numerical method’s findings.

Figure 3.

Finite element model of an MPP-Cavity-Plate coupled to an acoustic enclosure.

Figure 4.

Comparison of the current method and FEM numerical method on predicting the velocity responses on the MPP (0.1, 0.1, 0) and the panel (0.1, 0.1, 0) and the space-averaged sound pressure inside the acoustic cavity and air gap when the coupled system is excited from inside by an acoustic point source.

Figure 5.

Comparison of calculated STLs using the present approach and FEM results for MPP-Cavity-Plate coupled to an acoustic enclosure when excited from outside by a plane wave.

Comparison with finite-sized partitions in free field

Comparisons are made between the predicted TL when multi-layered MPP partitions are coupled to an acoustic enclosure with those previously published in the literature.²⁹ In this selected study, Bravo et al investigated the propagation of two-dimensional plane waves over an insulating partition set in an acoustically rigid stiffened baffle with finite size $0.42 m \times 0.61 m$ . The configuration consists of an MPP separated from a flexible plate by an air cavity of 0.048m with clamped boundaries. Simulations were conducted using the same physical parameters as in the reference.²⁹ The plate and the MPP are made from aluminum with the following properties: Young’s modulus of $6.9 \times 10^{10} Pa$ , the density of $2700 kg / m^{3}$ , Poisson’s ratio of 0.33, and structural damping ratio of 0.01. The plate is 2.5 mm thick and the MPP is 0.5 mm thick with 0.5 mm diameter holes, and a perforation ratio of 0.78%. The depth of the chosen coupled enclosure in the current study is 0.55 m with a source of strength $Q_{0} = 10^{- 4} m^{3} / s$ placed at (0.05, 0.08, 0.1 m) when the coupled system is excited from the inside.

Figures 6 and 7 display the findings for the TL assuming internal source (acoustic point source) and external source (incident plane wave), respectively. Comparing these findings to those of an MPP-Cavity-Plate partition set in an acoustically rigid stiffened baffle reveals that the agreement is excellent. The proposed approach, when the partition is connected to an acoustic enclosure, can accurately anticipate the overall trend for the TL across a wide frequency spectrum. It provides further details on the modal effects of an acoustic enclosure on power transmission due to the presence of acoustic resonances. The STL dips corresponding to structural resonances in the simply supported case are shifted to lower frequencies in comparison with those of the clamped case,²⁹ this can be explained by the fact that the natural frequencies of the simply supported system are lower than their counterparts of the clamped system.

Figure 6.

STL of MPP-Cavity-Plate coupled to an acoustic enclosure when excited from the inside (acoustic point source) and configuration of Bravo et al.²⁹

Figure 7.

STL for MPP-Cavity-Plate coupled to an acoustic enclosure when excited from the outside by plane wave and configuration of Bravo et al.²⁹

Parametric study

Several major system parameters that impact the vibroacoustic behavior/acoustic performance of the MPP-Cavity-Plate structure coupled to an acoustic enclosure are investigated by employing numerical modeling. In numerical simulations, the connected enclosure’s dimensions and properties are specified as follows. The dimension of the enclosure is $L_{x} \times L_{y} \times L_{z} = 0.5 m \times 0.35 m \times 0.55 m$ . The depth of the air gap (hg) considered is 0.07 m. The properties of air are: $ρ = 1.225 kg / m^{3}$ , and $c = 343 / s$ , and the viscosity is $η_{0} = 17.9 \times 10^{- 6} kg / ms$ . The plate and the MPP ( $L_{x} \times L_{y} = 0.5 m \times 0.35 m$ ) are made of aluminum plates whose material properties are: Young’s modulus $E = 7.2 \times 10^{10} Pa$ , Poisson’s ratio $v = 0.34$ , density $ρ_{p} = 2700 kg / m^{3}$ , and the loss factor $η = 0.01$ . The plate is 2 mm thick and the MPP is 1mm thick with holes of 0.3 mm in diameter. In the first case, the outer plate is under acoustic excitation with an elevation angle $ϕ = 0$ and azimuth angle $θ = 0$ and in the second case, the source of strength $Q_{0} = 10^{- 4} m^{3} / s$ is located inside the enclosure at (0.12, 0.1, 0.1). A perfectly rigid impedance boundary is assumed for the acoustic enclosure and air gap cavity. Several relevant system parameters, such as micro-perforation of the plate, hole diameter, air gap thickness, acoustic enclosure dimensions, and the insertion of the absorbent material in the gap are evaluated via numerical analysis to study the impacts of these variables on the vibroacoustic of the coupled system.

Effect of micro-perforation of the MPP

To examine the impact of the perforation ratio on the coupled system, the STLs for both cases, external acoustic excitation (plane wave) and internal acoustic excitation (acoustic point source), are plotted considering different perforation ratio values. As seen in Figures 8 and 9, as the micro-perforation ratios grow, the STLs deteriorate. Comparison between the case of the double plate ( $σ = 0.0 %$ ) and the current configuration ( $σ = 1 %$ ) coupled with an acoustic enclosure shows that the STL deteriorates when the micro-perforation is present. However, A significant gain is obtained at the resonant frequencies associated with the acoustic enclosure (592, 774 Hz, …) in Figure 8.

Figure 8.

STL of the partition when the coupled system is excited from outside for various perforation ratios.

Figure 9.

STL of the partition when the coupled system is excited from inside for various perforation ratios.

From the plot of the space-averaged sound pressure level for different perforation ratios (Figure 10), It can be seen that almost all of the resonances of the acoustic enclosure are damped when the micro-perforation is increased. This is because the MPP introduces modal damping into the system. Additionally, frequency shifts are seen at some peaks that are attributed to the reactance term of the impedance of the MPP. It should be noted that when the inner plate is not perforated, No damping or shifting has occurred at these peaks. This can be explained by the fact that the increase in perforation improves both absorption bandwidth and absorption amplitude as shown in the plot of the computed absorption coefficients of MPP-Cavity-Plate in free field conditions using the proposed method in Bravo et al.²⁹ in Figure 11.

Figure 10.

Space averaged SPL inside the enclosure when the coupled system is excited from inside for various perforation ratios.

Figure 11.

Comparison of the absorption coefficients of MPP-Cavity-Plate for three different perforation ratios of the MPP.

Effect of the hole diameter

In this section, the pore diameter is varied while the perforation ratio on the MPP is kept constant and equal $σ_{1} = 1 %$ . From Figures 12 and 13, it is found that the STLs are slightly degraded, However, a large gain is obtained at the resonant frequencies as the hole diameter decreases. however, when we lower the hole diameter to 0.2 mm, the STL for the case of outside excitation demonstrates identical behavior to larger diameters. Moreover, as shown in Figure 14, by decreasing the micro-pore diameter, and keeping the perforation rate constant, the peaks are more damped and shifted. However, when the hole diameter is lowered from 0.3 to 0.2 mm, the resonant frequencies are less damped. This is explained by the fact the absorption coefficient amplitude decreases when the diameter decrease from 0.3 to 0.2 mm while the absorption bandwidth remains almost the same.

Figure 12.

STL of the partition when the coupled system is excited from outside for varying micro-pore diameter.

Figure 13.

STL of the partition when the coupled system is excited from inside for varying micro-pore diameter.

Figure 14.

SPL inside the enclosure when the coupled system is excited from inside for varying micro-pore diameter.

Effect of the air gap thickness

To determine the impacts of air gap depth on STL, a series of numerical computations with variable air gap thickness are performed. The influence of air gap depth ( $h_{g}$ = 30 mm, 70 mm, and 100 m) on the STL of the current configuration under inside point source acoustic excitation and outside pressure excitation is shown in Figures 15 and 16. The perforation ratio of the MPP is $σ_{1} = 1 %$ and the hole diameter is 0.3 mm. It can be seen that the STL values rise as the air gap thickness increases since a reduction in air gap thickness increases system coupling. However, the locations of coupled system resonances (dips) remain largely unchanged. In sum, the thickness of the air gap has a substantial effect on the sound insulation capacities of the structure. From Figure 17, it is found that the peaks of the space-averaged sound pressure are more damped when the air gap thickness increases. This could be explained by the fact that the increase in the air gap thickness shifts the absorption bandwidth toward lower frequencies.

Figure 15.

Influence of air gap on STL of the partition when the coupled system is excited from outside.

Figure 16.

Influence of air gap on STL of the partition when the coupled system is excited from inside for micro-pore diameter.

Figure 17.

Influence of air gap on SPL inside the enclosure when the coupled system is excited from inside.

Effect of acoustic enclosure dimensions on STL

To determine the modal effects of the enclosure on STL, a series of numerical simulations with varied $L_{z}$ are performed. The effects of air enclosure depth ( $L_{z}$ = 0.55 m, 0.65 m, and 0.75 m) on the sound transmission loss of the structure connected to an acoustic enclosure in two cases are shown in Figures 6 and 7 when the perforation ratio of the MPP is equal to $σ_{1} = 0.1 %$ and $σ_{1} = 1 %$ . The hole diameter is 0.3 mm.

Figures 18 and 19 depict the associated transmission loss for both cases for two different MPP’s perforation ratios ( $σ_{1} = 0.1 %$ and $σ_{1} = 1 %$ ). The installation of an acoustic enclosure has significantly altered the prior findings since it is affected by its modal properties. Below the first enclosure-controlled mode, the energy transmission is mainly governed by structural resonances. For an external acoustic excitation (Figure 18), The size of the discrepancy between the curves is because the coupled enclosure’s modes amplify the mean square pressure in this enclosure for a given panel excitation. Moreover, a change in enclosure size modifies the resonance frequencies and results in a shift of the dips corresponding to excited enclosure modes in the case where the perforation ratio is $σ_{1} = 0.1 %$ . This is mainly induced by the acoustic mode parallel to the plate length. We can observe that when the cavity depth lowers, the corresponding associated frequencies of the acoustical cavity increase. As shown in Figure 18(b), the dip induced by the acoustic mode at 609 Hz is shifted to the value of 567 Hz when $L_{z}$ is varied from 0.55 to 0.65 m. However, when the perforation ratio is equal $σ_{1} = 0.1 %$ , the dips of the STL corresponding to the resonant frequencies corresponding to acoustic modes are suppressed. On the other hand, for internal acoustic excitation, Figure 19 displays the obvious disparity between the TL curves. The calculated TL for the partition can vary depending on the size of the source room considered, assuming all other conditions are held constant.

Figure 18.

STL of the partition when the coupled system is excited from outside for different dimensions of the enclosure: (a) a = 1% and (b) a = 0.1%

Figure 19.

STL of the partition when the coupled system is excited from inside for different dimensions of the enclosure.

Enclosures have a significant impact on the STL of multi-layered MPPs structures. Therefore, the transmission loss of multi-layered MPP coupled to an acoustic enclosure must be thoroughly examined. In general, changes to the depth of the enclosure have the largest effect on the acoustical natural frequencies, although the effect on the plates is negligible.

Influence of porous material

To improve the transmission loss at resonant frequencies, sound-absorbing material with flow resistivity $20000 Ns / m^{4}$ is inserted in the gap between the MPP and the plate. The absorbent material is modeled as an equivalent fluid, and its material properties are computed using the empirical model of Delany and Bazley.⁵⁷ The associated weak form (18) may account for airborne propagation through low frame stiffness insulating materials filling the cavity, by replacing $c_{0}$ and $ρ_{0}$ with $c_{a}$ and $ρ_{a}$ which are complex and frequency-dependent. A robust description should consider poroelastic modeling but as demonstrated by Beranek,⁵⁸ when propagating through soundproofing materials with relatively low frame stiffness, longitudinal elastic vibrations attenuate much more than acoustic compression waves.

Figures 20 and 21 compare the computational results of a gap filled with fiberglass vs air. It is observed that the transmission loss increases above 500 Hz. At the resonances, however, a large gain is gained in both transmission loss in the two cases and sound pressure level peaks inside the acoustic enclosure plotted in Figure 22. The damping introduced by the absorbent material governs the response of the system. The damping is mostly attributable to the acoustic dissipation generated by viscous drag forces and thermal exchanges between the air and the material.

Figure 20.

STL of the partition when the coupled system is excited from outside for gap filled with fiberglass and gap filled with Air. The perforation ratio of the MPP is $σ_{1} = 1 %$ and the hole diameter is 0.3mm.

Figure 21.

STL of the partition when the coupled system is excited from inside for gap filled with fiberglass and gap filled with Air.

Figure 22.

SPL inside the enclosure when the coupled system is excited from inside for gap filled with fiberglass and gap filled with Air.

Conclusion

This study presents the results of a vibro-acoustic investigation of a multi-layered MPP structure coupled to an enclosure. The transmission loss of the structure was predicted using an analytical method. Using a modified variational model, the equations of the coupled system were derived. The 3D-enhanced Fourier series describes the acoustic pressure inside the air gap and the enclosure. MPPs are assumed to have the simply supported boundary condition. In addition, finite element computation demonstrates the validity and accuracy of the present model, with a great agreement being achieved. Then, a parametric investigation was conducted to determine the influence of micro-perforation of the plate, hole diameter, air gap thickness, acoustic enclosure dimensions, and the insertion of absorbent material in the gap on the vibroacoustic behavior of the coupled system

The results summarized here are the most significant: As anticipated, the STLs deteriorate with the increase of the micro-perforation ratio of the MPP. However, the dips in the STL corresponding to the resonances of the acoustic enclosure are damped. Regarding the sound pressure level inside the enclosure, the acoustic resonances are damped due to the modal damping introduced by the MPP. It is found that the STLs are improved at the resonant frequencies as the hole diameter decreases and the resonant frequencies are damped until an optimal value for the given parameters of the coupled system. In addition, by increasing the micro-pore diameter the peaks of the pressure inside the enclosure are damped and shifted due to the reactance term of the MPP. It was also discovered that the STL values rise as the thickness of the air gap grows, but the locations of coupled system resonances (dips) stay generally unaltered. In addition, the air gap thickness influences the damping of the resonance peaks of the SPL of the enclosure by shifting the absorption bandwidth. It has been shown that the introduction of an acoustic cavity may have a considerable influence on the vibro-acoustic behavior of the structure, resulting in additional dips corresponding to the excited acoustic modes only when the perforation ratio is small. Moreover, when absorbent material is inserted into the gap, transmission loss increases, particularly at the resonances, and the peaks of the sound pressure are damped.

The presence of an acoustic enclosure can result in a significant impact on the vibro-acoustic behavior of multi-layered MPPs structures, which might provide essential information for the design process and noise reduction of these structures. In comparison to prior techniques, it is easier to establish different structural and acoustic boundary conditions. In addition, the suggested approach may be immediately applied to other, more realistically complex geometries. Currently, additional research is being conducted to assess the applicability of heterogeneous MPPs for the same objective. For example, it has been demonstrated that the concept of inhomogeneous perforation can significantly increase bandwidth absorption at a low-frequency range when the effect is evaluated in an impedance tube. It is still necessary to quantify their performance after it has been incorporated into a more complex system. Future studies will also look into their transmission efficiency when coupled to an elastic structure and the factors underlying in this process.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

ORCID iD

Brahim El kharras

References

Liu

Daudin

Analytical modelling of sound transmission through finite clamped double-wall sandwich panels lined with poroelastic materials. Compos Struct 2017; 172: 359–373.

Liu

Diffuse field sound transmission through sandwich composite cylindrical shells with poroelastic core and external mean flow. Compos Struct 2016; 135: 383–396.

Wang

Rajaram

, et al. Predicting the sound transmission loss of sandwich panels by statistical energy analysis approach. J Vib Acoust 2010; 132(1). 011004.

Xin

Chen

CQ.

Sound transmission through simply supported finite double-panel partitions with enclosed air cavity. J Vib Acoust 2010; 132(1): 011008.

Price

Crocker

MJ.

Sound transmission through double panels using statistical energy analysis. J Acoust Soc Am 1970; 47: 683–693.

Chazot

Guyader

JL.

Prediction of transmission loss of double panels with a patch-mobility method. J Acoust Soc Am 2007; 121(1): 267–278.

Craik

RJM

. Non-resonant sound transmission through double walls using statistical energy analysis. Appl Acoust 2003; 64(3): 325–341.

Xin

TJ.

Analytical and experimental investigation on transmission loss of clamped double panels: implication of boundary effects. J Acoust Soc Am 2009; 125(3): 1506–1517.

Sun

Liu

Vibration analysis of hard-coated composite beam considering the strain dependent characteristic of coating material. Acta Mech Sinica 2016; 32(4): 731–742.

10.

Chazot

Guyader

JL.

Transmission loss of double panels filled with porogranular materials. J Acoust Soc Am 2009; 126(6): 3040–3048.

11.

Doutres

Atalla

Acoustic contributions of a sound absorbing blanket placed in a double panel structure: absorption versus transmission. J Acoust Soc Am 2010; 128(2): 664–671.

12.

Bolton

Green

Normal incidence sound transmission through double panel systems lined with relatively stiff, partially reticulated polyurethane foam. Appl Acoust 1993; 39: 23–51.

13.

Panneton

Atalla

Numerical prediction of sound transmission through finite multilayer systems with poroelastic materials. J Acoust Soc Am 1996; 100(1): 346–354.

14.

Kang

Bolton

JS.

A finite element model for sound transmission through foam-lined double-panel structures. J Acoust Soc Am 1996; 99(5): 2755–2765.

15.

Bolton

Shiau

Kang

YJ.

Sound transmission through multi-panel structures lined with elastic porous materials. J Sound Vib 1996; 191(3): 317–347.

16.

Ghanbari

Hossainpour

Rezazadeh

Studying thin film damping in a micro-beam resonator based on non-classical theories. Acta Mech Sinica 2016; 32(3): 369–379.

17.

Meng

Xin

TJ.

External mean flow effects on sound transmission through acoustic absorptive sandwich structure. AIAA J 2012; 50(10): 2268–2276.

18.

Santoni

Davy

Fausti

, et al. A review of the different approaches to predict the sound transmission loss of building partitions. J Build Acoust 2020; 27(3): 1–27.

19.

Maa

D-Y.

Theory and design of microperforated panel sound-absorbing constructions. Sci China 1975; 18(1): 38–50.

20.

Maa

D-Y.

Potential of microperforated panel absorber. J Acoust Soc Am 1998; 104(5): 2861–2866.

21.

Atalla

Sgard

Modeling of perforated plates and screens using rigid frame porous models. J Sound Vib 2007; 303(1): 195–208.

22.

Hao-Ming

Double perforated honeycomb panel as a low frequency acoustic absorber. J Build Acoust 1997; 4(1): 21–37.

23.

Rao

Munjal

Experimental evaluation of impedance of perforates with grazing flow. J Sound Vib 1986; 108(2): 283–295.

24.

Lee

Kwon

YP.

Estimation of the absorption performance of multiple layer perforated panel systems by transfer matrix method. J Sound Vib 2004; 278: 847–860.

25.

Yuvaraj

Jeyanthi

Acoustic performance of countersunk micro-perforated panel in multilayer porous material. J Build Acoust 2019; 27(1): 1-18.

26.

Chen

K-T.

Study on the acoustic transmission loss of a rigid perforated screen. Appl Acoust 1996; 47(4): 303–318.

27.

Dupont

Pavic

Laulagnet

Acoustic properties of lightweight microperforated plate systems. Acta Acust United Ac 2003; 89(2): 201–212.

28.

Toyoda

Takahashi

Sound transmission through a microperforated-panel structure with subdivided air cavities. J Acoust Soc Am 2008; 124(6): 3594–3603.

29.

Bravo

Maury

Pinhède

Sound absorption and transmission through flexible micro-perforated panels backed by an air layer and a thin plate. J Acoust Soc Am 2012; 131(5): 3853–3863.

30.

Bravo

Maury

Pinhède

Enhancing sound absorption and transmission through flexible multi-layer micro-perforated structures. J Acoust Soc Am 2013; 134(5): 3663–3673.

31.

Toyoda

Takahashi

Sound insulation characteristics of multi-layer structures with a microperforated panel. Appl Acoust 2011; 72(11): 849–855.

32.

Kim

H-S

Kim

S-R

Kim

B-K

, et al. Sound transmission loss of multilayered infinite micro-perforated plates. J Acoust Soc Am 2020; 147(1): 508–515.

33.

Kim

H-S

P-S

Kim

B-KN

, et al. Sound transmission loss of multi-layered elastic micro-perforated plates in an impedance tube. Appl Acoust 2020, 166: 107348.

34.

Pan

The forced response of an acoustic-structural coupled system. J Acoust Soc Am 1992; 91: 949–956.

35.

Cheng

Gao

JX.

Energy transmission in a mechanically-linked double-wall structure coupled to an acoustic enclosure. J Acoust Soc Am 2005; 117: 2742–2751.

36.

Pan

Hansen

Bies

. Active control of noise transmission through a panel into a cavity: I. Analytical study. J Acoust Soc Am 1990; 87: 2098–2108.

37.

Pan

Hansen

CH.

Active control of noise transmission through a panel into a cavity. II: experimental study. J Acoust Soc Am 1991; 90: 1488–1492.

38.

Bravo

Elliott

SJ.

Variability of low frequency sound transmission measurements. J Acoust Soc Am 2004; 115: 2986–2997.

39.

Venkatesham

Tiwari

Munjal

ML.

Analytical prediction of the breakout noise from a rectangular cavity with one compliant wall. J Acoust Soc Am 2008; 124: 2952–2962.

40.

Venkatesham

Tiwari

Munjal

ML.

Analytical prediction of break-out noise from a reactive rectangular plenum with four flexible walls. J Acoust Soc Am 2010; 128: 1789–1799.

41.

Seo

Kim

YH.

Directional radiation pattern in structuralacoustic coupled system. J Acoust Soc Am 2005; 118: 92–103.

42.

Becot

Sgard

On the use of poroelastic materials for the control of the sound radiated by a cavity backed plate. J Acoust Soc Am 2006; 120: 2055–2006.

43.

Kim

YH.

Structural-acoustic coupling in a partially opened plate-cavity system: experimental observation by using nearfield acoustic holography. J Acoust Soc Am 2001; 109: 65–74.

44.

Kim

SM.

Solution of coupled acoustic problems: A partially opened cavity coupled with a membrane and a semi-infinite exterior field. J Sound Vib 2002; 254: 231–244.

45.

Clasen

Langer

Finite element approach for flanking transmission in building acoustics. J Build Acoust 2007; 14(1): 1–14.

46.

, et al. Vibro-acoustic analysis of a rectangular cavity bounded by a flexible panel with elastically restrained edges. J Acoust Soc Am 2012; 131: 2799–2810.

47.

Liu

, et al. Acoustic analysis of a rectangular cavity with general impedance boundary conditions. J Acoust Soc Am 2011130: 807–817.

48.

Wang

, et al. Prediction of break-out sound from a rectangular cavity via an elastically mounted panel. J Acoust Soc Am 2016; 139(2): 684–692.

49.

Yang

Cheng

Sound absorption of microperforated panels inside compact acoustic enclosures, J Sound Vib 2016; 360: 140–155.

50.

Thomazelli

Bertoli

An experimental study on the optimization of the production and efficiency of tunable Helmholtz absorbers for the modal control of small rooms. J Build Acoust 2019; 26(2): 1–23.

51.

Takahashi

Tanaka

Flexural vibration of perforated plates and porous elastic materials under acoustic loading. J Acoust Soc Am 2002; 112(4): 1456–1464.

52.

Xin

Chen

CQ.

Vibroacoustic behavior of clamp mounted double-panel 26 partition with enclosure air cavity. J Acoust Soc Am 2008; 124: 3604–3612.

53.

Carneal

Fuller

CR.

An analytical and experimental investigation of active structural acoustic control of noise transmission through double panel system. J Sound Vib 2004; 272: 749–771.

54.

Hua

Meng

A domain decomposition approach for vibration analysis of isotropic and composite cylindrical shells with arbitrary boundaries. Compos Struct 2013; 95: 307–321.

55.

Chen

Long

, et al. A modified variational approach for vibration analysis of ring-stiffened conical–cylindrical shell combinations. Eur J Mech A Solids 2013; 37: 200–215.

56.

Chien

. Method of high-order Lagrange multiplier and generalized variational principles of elasticity with more general forms of functional. Appl Math Mech 1983; 4: 143–157.

57.

Beranek

LL.

Acoustical properties of homogeneous, isotropic rigid tiles and flexible blankets. J Acoust Soc Am 1947; 19(4): 556–568.

58.

Delany

Bazley

EN.

Acoustical properties of fibrous materials. Appl Acoust 1970; 3(2): 105–116.