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Abstract

A microperforated panel (MPP) often suffers from a limited absorption bandwidth and poor sound absorption in the low frequency range. The present study aimed to propose an MPP-panel-type resonator (PR) compound structure that can simultaneously widen the half-absorption bandwidth and improve the poor absorption at the low frequency range. A vibroacoustic model is developed and compared to finite element simulations to demonstrate its accuracy. The vibration characteristics of the compound structure and the surface impedance are analyzed to reveal the mechanism forming the absorption characteristics of an MPP-PR structure. It is found that backing an MPP with a panel-type resonator (PR) creates a low frequency absorption peak but followed by an absorption valley that decreases the absorption bandwidth. The analysis showed that the phase difference between velocity of air particles in perforation and the PR panel velocity dictates the absorption characteristics associated with the PR resonance. A multi-resonators panel-type resonator (MPR) is found that the absorption valley associated with the host panel resonance splits into multiple smaller amplitude absorption dips improving absorption performance. The proposed MPP-MPR compound structure demonstrated a relatively wider half-absorption bandwidth with improved low frequency absorption.

Keywords

microperforated panel panel-type resonator sound absorber coupled vibroacoustic system

1. Introduction

MPPs are effective sound absorbers in the mid and high frequency range proposed by Maa (Maa, 1998) but typically suffers from poor absorption at the low frequency range and a limited absorption bandwidth.

Researchers have therefore tried to widen the absorption bandwidth by using multiple layers of MPP arranged in series (Bucciarelli et al., 2019; Sakagami et al., 2006; Yang et al., 2019). They have also proposed more compact configuration by arranging different MPPs in parallel (Mosa et al., 2019, 2020; Prasetiyo et al., 2016; Qian and Zhang, 2022) and also using turned double layer MPP (Zhao and Lin, 2022). Previous studies have tried to modify the backing air cavity into L-shaped cavities (Gai et al., 2017); coiled cavities (Prasetiyo et al., 2021; Wu et al., 2022); sub-cavities with different dimensions arranged in parallel (Guo and Min, 2015). Dividing backing cavities into sub-cavities are effectively arranging an MPP-cavity system with different Helmholtz resonant frequencies in parallel.

The addition of local resonators is a common approach to improve the low frequency absorption of MPP and one of which is the Helmholtz resonator (HR) (Park, 2013; Sanada and Tanaka, 2013) which can be tuned to a low frequency by changing its geometric parameters but suffers from a narrow absorption bandwidth. Hence, researchers have proposed the use of HRs with different resonant frequencies arranged in parallel or series with the MPP (Gai et al., 2016; Mahesh and Mini, 2021). Regardless of the arrangement, an additional HR contributes to an absorption peak near its resonant frequency. Apart from HR, researchers have investigated panel-type and membrane-type resonator backing MPP to improve the low frequency absorption (Gai et al., 2018; Zhao et al., 2016, 2018; Zhao and Fan, 2015; Zhu et al., 2018). Panels/membranes were tuned to a relatively low fundamental resonant frequency to achieve a low frequency absorption peak but the absorption mechanism was not investigated.

The MPPs are commonly assumed to be rigid, however, an unexpected absorption peak has been observed in experiments (Lee and Swenson, 1991). A previous study has developed a theoretical model that accounted for the acoustic-structural coupling (Takahashi and Tanaka, 2002) but the structural resonance was not considered. The effect of MPP structural resonance on sound absorption was later investigated by Lee et al. (Lee et al., 2005; Lee and Lee, 2007). These studies have demonstrated that the structural resonance of MPP generates an additional absorption peak which was not predicted by a rigid MPP analytical model. However, when the MPP structural resonance coincided with the absorption peak due to perforation, an absorption dip was observed. Researchers have further investigated the effect of adding local resonators on MPP, to form a metamaterial panel (Ren et al., 2019; Zhao et al., 2021), on the absorption characteristics at the absorption dip region. When the resonator’s resonant frequency coincided with the resonant frequency of MPP, the absorption valley associated with MPP structural resonance splits into two absorption dips with smaller amplitudes.

In summary, developing a wideband MPP absorber with improved low frequency absorption remains a challenge. Works that revolved around improvements to absorption characteristics of MPP are often discussed but the mechanism forming the absorption characteristics is not yet fully explored, limiting the way that such structures can be optimized to extend the low frequency absorption bandwidth. Therefore, the novelty in this study is to propose an MPP-MPR compound structure to simultaneously improve the poor absorption of MPP at low frequencies and widen the continuous half-absorption bandwidth. This study also investigates the mechanism forming the absorption characteristics associated with the local resonance of MPR, which, to the best of authors’ knowledge, has not been previously focused upon. In the present work, a fully coupled vibroacoustic model is developed based on the modal coupling method to predict the absorption characteristics of an MPP-MPR structure and its accuracy is verified by comparing theoretical results to those from finite element model. The theoretical and numerical results will be analyzed to explain the mechanism forming the absorption characteristics of the compound sound absorbing structure. The paper is arranged as follows. In Section 2, the vibroacoustic model developed will be presented. Next, the results obtained from the theoretical and numerical models are discussed in detail in Section 3 in which the bandwidth widening effect of an MPP-MPR structure will be studied. Finally, the conclusions will be presented in Section 4.

2. Analytical modelling of a coupled MPP-MPR vibroacoustic system

An analytical model of a coupled MPP-MPR vibroacoustic system is developed by extending the modal coupling method through the inclusion of the effect of MPP perforations and the sound radiation associated with MPP flexibility. The configuration used in this study is shown in Figure 1.

Figure 1.

The configuration of MPP-MPR structure investigated in this study consists of an MPP backed by cavity I,MPR and cavity II that is rigidly terminated.

2.1. Modelling of a coupled MPP-MPR vibroacoustic system by the modal coupling method

The incident acoustic pressure wave is taken to be normal to the MPP. The pressure radiated “upward” due to structural sound radiation of MPP is neglected as it is commonly assumed to be negligible as compared to the incident pressure. Therefore, the backed pressure acting on the MPP, $P_{b} = 2 . P_{i n c} . e^{i ω t}$ where $ω$ is the excitation frequency. The equations of motion for MPP and MPR can be described as (Halim et al., 2011a, 2011b; Fahy et al., 2007):i) MPP

D_{M P P} \nabla^{4} w_{M P P} (x, y, t) + m_{M P P} {\ddot{w}}_{M P P} (x, y, t) = P_{b} - P_{I} (z = 0)

(1)

ii) MPR

D_{M P R} \nabla^{4} w_{M P R} (x, y, t) + m_{M P R} {\ddot{w}}_{M P R} (x, y, t)

= P_{I} (z = L_{z I}) - P_{I I} (z = 0) - \sum_{g = 1}^{G} m_{g} {\ddot{θ}}_{g} δ (x - x_{g}, y - y_{g})

(2)

iii) Resonators

m_{g} {\ddot{θ}}_{g} = - k_{g} (θ_{g} - w_{M P R} (x_{g}, y_{g})) - c_{g} ({\dot{θ}}_{g} - {\dot{w}}_{M P R} (x_{g}, y_{g}))

(3)

where

\nabla^{4} = {(\frac{\partial^{2}}{{\partial x}^{2}} + \frac{\partial^{2}}{{\partial y}^{2}})}^{2}

and

D_{M P P / M P R} = \frac{E {h_{M P P / M P R}}^{3}}{12 {(1 - ν)}^{2}}

The parameters of the plates and resonators are as follows: $D_{M P P / M P R}$ is the flexural rigidity of MPP/MPR; E is the Young’s modulus; $h_{M P P / M P R}$ is the thickness of MPP/MPR; $υ$ is the Poisson’s ratio; $m_{M P P / M P R}$ is the surface mass density of MPP/MPR; $m_{g}$ is the mass of g-th resonator and G is the number of resonators used. The function of $δ$ (.) denotes the Dirac delta function. The symbol $w_{M P P}$ , $w_{M P R}$ , and $θ_{g}$ denotes the vertical displacement of MPP, MPR, and resonators, respectively. By assuming the steady-state harmonic solution, the amplitude of resonator displacement can be derived from equation (3) as

{θ_{g}}^{*} = \frac{c_{g} ω i + k_{g}}{- m_{g} ω^{2} + c_{g} ω i + k_{g}} . W_{M P R}

(4)

where

k_{g}

and

c_{g}

represent the stiffness and damping coefficient of g-th resonator, respectively. The cavities are considered to be rigid-walled rectangular enclosures and acoustic wave equations are expressed as (Halim et al., 2011a, 2011b; Fahy et al., 2007):i) Cavity I

\nabla^{2} P^{I} (x, y, z, t) - \frac{1}{c^{2}} {\ddot{P}}^{I} (x, y, z, t) = ρ_{a} {\ddot{U}}_{M P P} \partial (z - 0) - ρ_{a} {\ddot{w}}_{M P R} \partial (z - L_{z I})

(5)

ii) Cavity II

\nabla^{2} P^{I I} (x, y, z, t) - \frac{1}{c^{2}} {\ddot{P}}^{I I} (x, y, z, t) = ρ_{a} {\ddot{w}}_{M P R} \partial (z - 0)

(6)

where c denotes speed of sound and

ρ_{a}

is the density of air. Here,

{\dot{U}}_{m p p}

is the spatially averaged velocity adjacent to the MPP in cavity I

{\dot{U}}_{M P P} = (1 - σ) {\dot{W}}_{M P P} + σ {\dot{U}}_{f}

(7)

where

σ

is the perforation ratio of MPP and

{\dot{U}}_{f}

is the velocity of air particles in perforation. The impedance of perforation by considering the structural vibration of MPP can be described as

Z_{M P P}^{R} ({\dot{U}}_{f} (x, y) - {\dot{W}}_{M P P} (x, y)) + Z_{M P P}^{I} ({\dot{U}}_{f} (x, y)) = P_{b} - P^{I} (x, y, z = 0)

(8)

Rearranging equation (8) and substituting into equation (7)

{\dot{U}}_{M P P} = (1 - σ + \frac{σ Z_{M P P}^{R}}{Z_{m p p}}) {\dot{W}}_{M P P} + \frac{σ}{Z_{m p p}} [P_{b} - P^{I} (x, y, z = 0)]

(9)

Here $Z_{m p p} = Z_{M P P}^{R} + Z_{M P P}^{I}$ and the terms $Z_{M P P}^{R}$ and $Z_{M P P}^{I}$ can be expressed as (Maa, 1998)

Z_{M P P}^{R} = \frac{32 μ h_{M P P}}{σ d^{2}} K_{a}, K_{a} = {[1 + \frac{q^{2}}{32}]}^{0.5} + \frac{\sqrt{2}}{32} (\frac{q d}{h_{M P P}})

(10a)

Z_{M P P}^{I} = \frac{{i ρ}_{a} ω h_{M P P}}{σ} K_{b}, K_{b} = 1 + {[1 + \frac{q^{2}}{2}]}^{- 0.5} + 0.85 \frac{d}{h_{M P P}}

(10b)

q = d \sqrt{\frac{ω ρ_{a}}{4 μ}}

(11)

Here $i = \sqrt{- 1}$ ; $μ$ is the viscosity of air; d is the perforation diameter. The MPP and MPR are simply supported and the acoustic cavities are assumed to be a rectangular rigid-walled cavity. The displacement of MPP, MPR, and acoustic pressure response are the summation of all modal modes based on modal decomposition method and can be expressed as

w_{M P P / M P R} = \sum ϕ_{j}^{M P P / M P R} (x, y) . Q_{j}^{M P P / M P R} . e^{i ω t}

(12)

P^{I / I I} = \sum ψ_{f}^{I / I I} (x, y, z) . Q_{f}^{I / I I} . e^{i ω t}

(13)

where

Q

denotes the complex modal amplitude with superscript indicating the different structural/acoustic components and subscript denoting the mode index of structural/acoustic components. The symbol

ϕ_{j}

denotes the j-th structural mode, and

ψ_{f}

denotes the f-th acoustic mode, which are, respectively, expressed as

ϕ_{j}^{M P P / M P R} (x, y) = \sin (\frac{r π x}{L_{x}}) \sin (\frac{s π y}{L_{y}})

(14)

ψ_{f}^{I / I I} (x, y, z) = \cos (\frac{l π x}{L_{x}}) \cos (\frac{m π y}{L_{y}}) \cos (\frac{n π z}{L_{z I / z I I}})

(15)

Here r = s = [1,2,3,….] and l = m = n = [0,1,2,….]. Substituting equation (12) into equations (1, 2) and equation (13) into equations (5,6) without an external structural/acoustic excitation, the equations can be expressed as

D_{M P P / M P R} [\sum_{j}^{J} {(\nabla}^{4} ϕ_{j}^{M P P / M P R}) . Q_{j}^{M P P / M P R} . e^{i ω t}] = {(β_{j}^{M P P / M P R})}^{2} m_{M P P / M P R} . [\sum_{j}^{J} . Q_{j}^{M P P / M P R} . e^{i ω t}]

(16)

\sum_{f}^{F} \nabla^{2} ψ_{f}^{I / I I} . Q_{f}^{I / I I} . e^{i ω t} = \frac{{β_{f}^{I / I I}}^{2}}{c^{2}} \sum_{f}^{F} ψ_{f}^{I / I I} . Q_{f}^{I / I I} . e^{i ω t}

(17)

where

β

denotes the natural frequency, the superscript denotes the components and subscript denotes the mode index. Equations (9), equations (12, 13) and equations (16, 17) are then substituted back into equations (1, 2) and equations (5, 6). Due to the steady-state harmonic response assumption, the temporal terms on both sides of equations are cancelled out and the governing equations are analyzed in frequency domain only. Structural (acoustic) mode shapes are multiplied on both sides of the governing equations and integrated over surface area (volume) to exploit the orthogonality of mode shapes. Thus, the governing equations are decomposed into a set of uncoupled equations in modal space expressed as:i) MPP

m_{M P P} ({β_{j}^{M P P}}^{2} - ω^{2}) M_{j}^{M P P} Q_{j}^{M P P} + \sum_{f}^{F} C_{j, f}^{M P P - I} Q_{f}^{I} = P_{b} N_{j}^{M P P}

(18)

ii) MPR

m_{M P R} ({β_{j}^{M P R}}^{2} - ω^{2}) M_{j}^{M P R} Q_{j}^{M P R} - ω^{2} \sum_{g}^{G} {m_{g} . [\frac{c_{g} ω i + k_{g}}{- m_{g} ω^{2} + c_{g} ω i + k_{g}}] . ϕ_{j}^{M P R} (x_{g}, y_{g}) . [\sum_{j}^{J} ϕ_{j}^{M P R} (x_{g}, y_{g}) . Q_{j}^{M P R}]} - \sum_{f}^{F} C_{j, f}^{M P R - I} Q_{f}^{I} + \sum_{f}^{F} C_{j, f}^{M P R - I I} Q_{f}^{I I} = 0

(19)

iii) Cavity I

[\frac{({β_{f}^{I}}^{2} - ω^{2})}{{ρ_{a} c}^{2}} . M_{f}^{I} + (\frac{i ω σ χ_{f}}{Z_{m p p}})] . Q_{f}^{I} + ω^{2} (1 - σ + \frac{σ Z_{M P P}^{R}}{Z_{m p p}}) \sum_{j}^{J} C_{j, f}^{M P P - I} Q_{j}^{M P P} - ω^{2} \sum_{j}^{J} C_{j, f}^{M P R - I} Q_{j}^{M P R} = \frac{i ω {σ P}_{b}}{Z_{m p p}} . N_{f}^{I}

(20)

iv) Cavity II

\frac{({β_{f}^{I I}}^{2} - ω^{2})}{{ρ_{a} c}^{2}} . M_{f}^{I I} . Q_{f}^{I I} + ω^{2} \sum_{j}^{J} C_{j, f}^{M P R - I I} Q_{j}^{M P R} = 0

(21)

where

M_{j}^{M P P / M P R} = \int_{0}^{L y} \int_{0}^{L x} {ϕ_{j}^{M P P / M P R}}^{2} d x d y

M_{f}^{I / I I I} = \int_{0}^{L_{z I / z I I}} \int_{0}^{L y} \int_{0}^{L x} {ψ_{f}^{I / I I}}^{2} d x d y d z

χ_{f} = \int_{0}^{L y} \int_{0}^{L x} {ψ_{f}^{I} (z = 0)}^{2} d x d y

N_{j}^{M P P} = \int_{0}^{L y} \int_{0}^{L x} ϕ_{j}^{M P P} d x d y

C_{j, f}^{M P P - I} = \int_{0}^{L y} \int_{0}^{L x} ψ_{f}^{I} (z = 0) ϕ_{j}^{M P P} d x d y

C_{j, f}^{M P R - I} = \int_{0}^{L y} \int_{0}^{L x} ψ_{f}^{I} (z = L_{z I}) ϕ_{j}^{M P R} d x d y

C_{j, f}^{M P R - I I} = \int_{0}^{L y} \int_{0}^{L x} ψ_{f}^{I I} (z = 0) ϕ_{j}^{M P R} d x d y

Equations (18–21) are solved for the complex modal amplitudes in structural and acoustic domains.

2.2. Sound absorption coefficient of the MPP-MPR compound structure

The acoustic impedance of MPP and MPR attributed to their structural flexibility are denoted as $Z_{M P P . s}$ and $Z_{M P R}$ , respectively. The respective specific impedance is calculated as

Z_{M P P . s} = \frac{P_{b} - P_{I . a v g} (z = 0)}{{\dot{W}}_{M P P . a v g}}

(22)

Z_{M P R} = \frac{P_{I . a v g} (z = L_{z 1}) - P_{I I . a v g} (z = 0)}{{\dot{W}}_{M P R . a v g}}

(23)

where

p_{I / I I . a v g} = \frac{\int_{0}^{L x} \int_{0}^{L y} P_{I / I I} (x, y, t) d x d y}{L_{x} . L_{y}}

, and

{\dot{w}}_{M P P / M P R . a v g} = \frac{\int_{0}^{L x} \int_{0}^{L y} {\dot{W}}_{M P P / M P R} (x, y, t) d x d y}{L_{x} . L_{y}}

The impedance of cavities by assuming the cavity is rigidly terminated which is expressed as

Z_{c a v I} = - i ρ c \cot (k L_{z I})

(24)

Z_{c a v I I} = - i ρ c \cot (k L_{z I I})

(25)

where k denotes the acoustic wavenumber; and

L_{z I}

L_{z I I}

are the cavity depth of cavity I and cavity II, respectively.

The specific surface impedance, $Z_{s u r f}$ of the MPP-MPR compound structure with the connection of impedance as shown in Figure 2 is described as

Z_{s u r f} = {[\frac{1}{Z_{M P P}} + \frac{1}{Z_{M P P . s}}]}^{- 1} + {[\frac{1}{Z_{c a v I}} + \frac{1}{Z_{c a v I I} + Z_{M P R}}]}^{- 1}

(26)

Figure 2.

The acoustic impedance connection of an MPP-MPR structure based on an electro-acoustic analogy.

The absorption coefficient can then be calculated from equation (26)

α = 1 - {| \frac{Z_{s u r f} - ρ_{a i r} c}{Z_{s u r f} + ρ_{a i r} c} |}^{2}

(27)

The accuracy of this analytical model is verified by comparing the results obtained with peer reviewed papers which investigated the effect of MPP structural resonance on sound absorption (Lee and Lee, 2007) and backing a rigid MPP with a panel-type resonator to create a low frequency absorption peak (Zhao and Fan, 2015). Although the structure studied are not entirely the same, the current model predicted sound absorption characteristics which are highly in agreement with results in literatures.

2.3. Numerical model of the MPP-MPR compound structure

The analytical model is verified against the numerical model simulated in finite element software, ANSYS. The numerical model is configured by placing the MPP-MPR compound sound absorber in an impedance tube as depicted in Figure 3. The cavities are defined as acoustic regions while MPP and MPR are defined as structural regions. Radiation boundary is implemented on the left end of the impedance tube to simulate a non-reflecting boundary. A plane acoustic wave excitation is initiated from the left end of impedance tube. The thermo-viscous losses in MPP perforations are accounted by assigning “Thermo-Viscous BLI Boundary” boundary condition on acoustic domain in perforations. The fluid-structure interaction between the acoustic domain and MPP and MPR are accounted by assigning “Fluid Solid Interface” on surface of acoustic domain contacting with the structural domain. The MPP and MPR are set to have simply supported boundary conditions in this investigation. As for resonators unit, the mass element is simply rigid aluminum cubes and attached to MPR via spring similar to a lumped mass-spring-dashpot model. The resonators are only allowed to vibrate in transverse direction by restraining the vibration in other directions using “Frictionless” boundary on mass element surface. In practice, the mass can be prepared from any rigid material with the designated mass and the stiffness element can be made from elastomer with the designed cross-sectional area and length to achieve the desired resonant frequency. The absorption coefficient can be extracted and plotted directly in the post-processing step. The solving range is set to be from 100 Hz to 800 Hz with resolution of 10 Hz. In post-processing, the averaged pressure and averaged velocity on the surface of MPP is also extracted to calculate the surface impedance of the overall MPP compound sound absorber.

Figure 3.

(a) The configuration of numerical model set up in ANSYS to simulate the sound absorption performance of such a structure. (b) The side view of simulation configuration.

3. Absorption characteristics with the absorption mechanism analysis

3.1. The absorption characteristics of backing MPP with a panel-type resonator

In this section, an additional panel-type resonator (PR) is added to the back of the MPP to form an MPP-PR sound absorber in order to improve the poor sound absorption of MPP at the low frequency range. The use of an MPP-PR demonstrates the ability to create a low frequency absorption peak (Zhao et al., 2018). In this previous work, however, the mechanism forming the absorption characteristics were not studied. Therefore, the PR vibration and surface impedance of MPP-PR will be analyzed to explain the absorption mechanism associated with the PR vibration. Understanding the mechanism forming the absorption trend of an MPP-PR will enable to determine the weakness of this structure and propose modifications for improvement in the later section. The absorption peak associated with the Helmholtz resonance of MPP is called the perforation peak for clarity of this discussion.

The absorption characteristics of an MPP-PR structure is shown in Figure 4. The first absorption peak is attributed to the resonance of PR while the second and third absorption peaks are attributed to Helmholtz resonance of MPP and the structural resonance of MPP, respectively. The use of PR exhibits an additional low frequency absorption peak at around 280 Hz. In terms of half absorption bandwidth, application of PR demonstrates a bandwidth of 319 Hz as compared to 268 Hz in the case of rigid MPP without PR.

Figure 4.

The absorption coefficient of MPP with the application of PR calculated by the theoretical and finite element models. The MPP-PR structure has the following dimensions: $L_{x} = L_{y} = 0.095 m$ , $L_{z I} = L_{z I I} = 0.08 m$ , $σ = 0.012, d = 1 m m,$ $h_{1} = 1.2 m m$ , $h_{2} = 0.5 m m$ .

The air-frame relative velocity, $Δ V$ and the phase angle between PR spatially averaged velocity, $V_{PR}$ and air particle velocity in perforation, $V_{a i r}$ , $\arg (\frac{V_{PR}}{V_{a i r}})$ are shown in Figure 5. At the absorption peak frequency, the $V_{PR}$ is in phase with $V_{a i r}$ as shown in Figure 5(b) causing lower pressure in the cavity between MPP and PR hence the pressure difference across MPP perforation is greater. This results in greater relative air-frame velocity in perforation as shown in Figure 5(a), thus, enhancing viscous losses in the perforations.

Figure 5.

(a) The magnitude of relative velocity between air particles and MPP spatially averaged velocity, $Δ V = | V_{a i r} - V_{M P P} |$ and (b) the phase angle between velocity of air particles, $V_{a i r}$ and the spatially averaged PR velocity, $V_{PR}$ . The first square marker corresponds to frequency of the absorption peak while the second marker corresponds to the frequency of the absorption dip.

Although the application of PR can create a low frequency absorption peak, however, a significant absorption dip is still present. At the frequency corresponding to the absorption dip, the resonance of PR causes an approximately 180 $°$ phase change so a high phase difference between $V_{PR}$ and $V_{a i r}$ is observed. The PR vibrating in the opposite direction of air particles causes greater pressure in the cavity thus the pressure difference across perforation is reduced. This resulted in the lower air-frame relative velocity as shown in Figure 5(a). This demonstrates a major difference between the effect of PR resonance and structural resonance of MPP on the absorption characteristics of the overall sound absorber. The in-phase relationship between $V_{PR}$ and $V_{a i r}$ causes absorption peak while the in-phase relationship between MPP velocity and $V_{a i r}$ causes an absorption dip.

The surface impedance is shown in Figure 6. The first peak in the resistance and reactance is associated with the resonance of PR, while the second peaks are attributed to the structural resonance of MPP. From the theoretical result obtained, the resistance is not greatly affected outside the resonant frequency range but the presence of PR increases the acoustic reactance significantly at frequencies below the PR resonance. The increase in reactance due to PR resonance fulfils the impedance matching condition in which the acoustic reactance crosses 0 at 280 Hz hence an absorption peak is observed. At a frequency that corresponds to the absorption dip, both the resistance and reactance are at the peak magnitude causing a severe impedance mismatch hence most of the acoustic wave is being reflected. Therefore, further modification of the surface impedance due to PR resonance is important to minimize the absorption dip, which will be proposed in the present study.

Figure 6.

The influence of PR structural resonance on the normalized acoustic surface (a) resistance and (b) reactance of the MPP-PR compound structure predicted by theoretical and finite element models. The peaks in resistance and reactance occur near the PR resonant frequency. The square marker corresponds to the frequency of absorption peak associated with structural resonance of PR.

In summary, the absorption characteristics of an MPP-PR associated with the resonance of PR is affected by the phase difference between $V_{PR}$ and $V_{a i r}$ that affects the air-frame relative velocity, which eventually affects the absorption characteristics. The PR is able to create an absorption peak below the perforation peak and slightly widen the half-absorption bandwidth. However, the absorption dip is a major setback as it reduces the improvement in the half-absorption bandwidth significantly and causes discontinuity in the half absorption bandwidth.

3.2. Extending half-absorption bandwidth with the use of a multi-resonator panel-type resonator

In the previous section, backing a flexible MPP with PR exhibits better absorption characteristics in frequencies below 308 Hz; however, the absorption bandwidth is compromised. Hence, this section proposed the installation of resonator units on the PR to create a MPR as shown in Figure 1, to try to eliminate the large amplitude absorption valley. The absorption characteristics of MPP-MPR will be analyzed in this section to explain the effect of backing an MPP with an MPR. The symbol $m_{r}$ denotes the total resonator mass where $m_{r} = \sum_{1}^{G} m_{g}$ .

For this initial study, 4 similar resonators are equally spaced and installed on the PR. The resonator mass, stiffness, and its corresponding resonant frequency are given in Table 1. In order to address the absorption dip associated with the resonance of PR, the resonant frequency of resonator unit is set to be 340 Hz which coincides with the absorption dip frequency.

Table 1.

The total resonator mass to panel mass ratio, resonator mass, stiffness, and its corresponding normalized resonant frequency of each resonator.

Mass ratio, $\frac{m_{r}}{m_{PR}}$	Resonator mass, $m_{g}$ , g	Stiffness, N/m	Resonant frequency, Hz
0.10	0.3046	1390	340
0.15	0.4569	2085	340
0.20	0.6092	2780	340

The absorption characteristics of MPP-MPR with different mass ratios calculated by the theoretical and finite element models are shown in Figure 7. An additional absorption peak is observed at frequencies around 360 Hz, which is attributed to the local resonance of resonator units. The presence of resonators split the original deep absorption valley at 340 Hz into two smaller amplitude absorption dips at frequencies around 300 Hz and 400 Hz, respectively. Different ratios of $\frac{m_{r}}{m_{PR}}$ generally have a similar trend of absorption with certain differences in the absorption peak frequency at around 253Hz and the absorption dip amplitude at the frequency around 400 Hz. It was observed that the greater mass ratio shifted the first absorption peak to slightly to lower frequencies.

Figure 7.

The sound absorption coefficient of an MPP backed by a single frequency MPR with different ratios of the total resonator mass to the panel mass.

The difference between the $V_{PR}$ at the absorption peak (360 Hz) and the absorption dip (400 Hz) is not substantial for all three cases as shown in Figure 8(a). The phase angle between $V_{PR}$ and $V_{a i r}$ , $\arg (\frac{V_{PR}}{V_{a i r}})$ for all three cases at the absorption peak frequency (square marker) is relatively low as shown in Figure 8(b) indicating that the host panel is vibrating relatively in phase with air particles in perforation. At the absorption dip, however, the $\arg (\frac{V_{PR}}{V_{a i r}})$ is relatively high for all three cases indicating the host panel is vibrating in opposite direction of the air particles. With the relatively close magnitude of $V_{PR}$ at the absorption peak and dip, it is found out that the modification to phase due to local resonance of resonators has resulted in different absorption characteristics. This observation agreed with the conclusion drawn in Section 3.1 that reveals the relationship between absorption characteristics and the phase angle between $V_{PR}$ and $V_{a i r}$ .

Figure 8.

(a) The magnitude of spatially averaged velocity of the host panel, $| V_{PR} |$ and (b) the phase difference between spatially averaged velocity of the host panel, $V_{PR}$ and velocity of air particle in perforation, $V_{a i r}$ . The square markers correspond to the absorption peak while the diamond-shaped markers correspond to the absorption dip associated with the local resonance of the resonator units.

The surface impedance calculated from theoretical and finite element models is shown in Figure 9. The local resonance of resonators reduces $V_{PR}$ at 340 Hz substantially due to restoring forces generated by resonators on the host plate as shown in Figure 8(a) and it also modifies the phase of the $V_{PR}$ as shown in Figure 8(b). As a consequence, the dominant resistance and reactance peak at 340 Hz is split into two peaks with greatly reduced amplitudes. Thus, the better impedance matching condition results in the MPP-MPR structure having two absorption dips with much higher absorption coefficient. The width of peak in $\arg (\frac{V_{PR}}{V_{a i r}})$ associated with local resonance of resonators increases with an increasing mass ratio. The wider peak resulted in a more gradual drop in the phase difference between the absorption peak at around 360 Hz and the absorption dip at around 400 Hz. Thus, the magnitude of fluctuation in the resistance and reactance in this frequency range decreases with an increasing mass ratio causing the fluctuation in the absorption coefficient at frequencies around $340 H z < \frac{ω}{2 π} < 420 H z$ to be more gradual.

Figure 9.

The normalized acoustic surface (a) resistance and (b) the reactance of the different mass ratio under the influence of local resonance of MPR and the structural resonance of MPP.

Despite improvements in the absorption dip region due to the presence of resonators, a notable absorption dip still exists at around 300 Hz. However, the frequency range where $α < 0.5$ that causes discontinuity in the half-absorption bandwidth is now narrower. The width of frequency range of $α < 0.5$ is 80 Hz for the MPP-PR while it is only 37 Hz, 46 Hz, and 54 Hz for mass ratio, $\frac{m_{r}}{m_{PR}}$ equivalent to 0.1, 0.15, and 0.2, respectively. It is observed that the greater the mass ratio, the further the distance between two absorption peaks. This is due to heavier resonators installed on MPR panel slightly reduces the panel resonant frequency as shown in Figure 8(b). The near 180 $°$ phase change associated with the MPR panel resonance is observed to occur at lower frequencies with an increasing mass ratio. Thus, it is found out that the greater mass ratio provides better sound absorption in the frequency range of $340 H z < \frac{ω}{2 π} < 420 H z$ but separated the two absorption peaks further apart. In terms of half absorption bandwidth, MPP-MPR demonstrates a bandwidth of 354 Hz, 362 Hz, and 359 Hz for mass ratio, $\frac{m_{r}}{m_{PR}}$ equivalent to 0.1, 0.15, and 0.2, respectively, as compared to 319 Hz in the case of MPP-PR.

In summary, MPP-MPR creates an additional absorption peak attributed to the local resonance of resonator units and splits the absorption valley into two smaller absorption dips. The local resonance of resonators modified the vibration characteristics and the phase of $V_{PR}$ that helps to reshape the acoustic impedance to improve the low frequency sound absorption. Despite a notable absorption dip still exists, the presence of resonators has generally widened the half-absorption bandwidth.

3.3. Extending the half-absorption bandwidth with an MPR with multi-frequency resonators

Despite the improvement in the half-absorption bandwidth by MPP-MPR structure, an absorption dip of $α < 0.5$ still exists at frequencies around 300 Hz. Therefore, the MPP backed by MPR with multi-frequency resonators (mMPR) is proposed in this section to target the existing absorption dip, and the effect of using multi-frequency resonators will be analyzed in detail. The mMPR consists of four resonators, of which two resonators with a resonant frequency of 300 Hz, close to the frequency of the absorption dip, and two resonators remain 340 Hz as described in Section 3.2. The ratio of $\frac{m_{r}}{m_{PR}}$ is set to be 0.1.

The absorption characteristics of MPP-mMPR is shown in Figure 10. The theoretical result has shown that an additional absorption peak is observed at around 300 Hz. The initial absorption valley at 300 Hz is further split into two smaller amplitude absorption dips. The $V_{PR}$ at absorption peak has slightly greater magnitude than dip as shown in Figure 11(a). At the absorption peak frequency, the local resonance of the resonators induces two peaks in $\arg (\frac{V_{PR}}{V_{a i r}})$ as shown in Figure 11(b), causing a relatively low phase difference. When the $V_{PR}$ is in phase with $V_{a i r}$ , two absorption peaks at around 300 Hz and 360 Hz are observed while the out of phase vibration between the $V_{PR}$ and $V_{a i r}$ exhibits two absorption dips at around 280 Hz and 320 Hz.

Figure 10.

The absorption characteristics of an MPP backed by single frequency or multi-frequency MPR with the ratio of total resonator mass to panel mass, $\frac{m_{r}}{m_{PR}} = 0.1$ .

Figure 11.

(a) The magnitude of spatially averaged velocity of the host panel, $| V_{PR} |$ and (b) the phase difference between spatially averaged velocity of mMPR and velocity of air particle in perforation. The square markers correspond to the absorption peak while the diamond-shaped markers correspond to the absorption dip associated with the local resonance of the resonator units.

Every set of additional resonators is observed to split the original resistance and reactance peaks into two lower amplitude peaks as shown in Figure 12, bringing the peak amplitudes closer to the magnitudes of 1 and 0, respectively. Hence, the presence of resonators with different resonant frequencies is crucial in frequency-reshaping the surface impedance to improve the impedance matching condition in the low frequency range. The initial absorption dip at 300 Hz with $α < 0.4$ has been split into two lower amplitude absorption dips with $α$ >0.5. This observation demonstrates a general trend that by adding resonators with different resonant frequencies, the absorption dip associated with mMPR panel resonance can be divided into smaller amplitude absorption dips offsetting the major weakness of an MPP-PR structure. In this study, the half-absorption bandwidth of a flexible MPP backed with a multi-frequency MPR is 406 Hz as compared to 268 Hz of a rigid MPP (51% increment) under similar MPP parameters. It is also worth mentioning that the average absorption coefficient in the low frequency range (calculated from 200 Hz to 500 Hz) of MPP-PR, MPP-MPR, and MPP-mMPR are 0.548, 0.686, and 0.733, respectively, demonstrating a significant improvement in low frequency sound absorption.

Figure 12.

The normalized surface acoustic (a) resistance and (b) reactance of the single frequency MPR or multi-frequency MPR under the influence of local resonance of MPR and structural resonance of MPP.

In summary, the major drawback of backing the MPP with a PR is the absorption valley associated with the panel resonance of PR, which has been addressed in the present study by proposing the mMPR. The absorption valley can thus be split into two smaller amplitude absorption dips with every set of resonators with different resonant frequency added. The results obtained from the discussions above contributes to a guideline in designing a compact broadband MPP-mMPR sound absorber with the improved low frequency sound absorption.

3.4. Effect of resonator mass ratio and damping ratio on overall sound absorption characteristics

The previous sections have discussed the effect of backing MPP with MPR and the absorption mechanism is revealed. Hence, a parametric study will be discussed in this section to investigate the effect of different resonator mass ratio and damping ratio of resonators on sound absorption characteristics of the overall MPP compound structure.

The effect of resonator mass ratio on absorption characteristics of MPP-MPR compound structure is depicted in Figure 13(a). The absorption peak at around 360 Hz is observed to be slightly decreasing with increasing mass ratio of resonators. The absorption peak at around 300 Hz is observed to have a notable decrease in amplitude and the absorption dip at around 280 Hz is significantly deteriorated with the increasing mass ratio of resonator. This observation is due to the fact that the peak associated with resonance of MPR is shifted to lower frequency with the increasing mass ratio of resonator. Thus, the resonators resonant frequency does not perfectly coincide with the absorption dip frequency causing a notable decrease of the absorption peak at around 300 Hz and the deterioration of absorption dip at around 280 Hz.

Figure 13.

The effect of (a) different total resonator mass to MPR panel mass ratio and (b) resonator damping ratio on the sound absorption characteristics of MPP-mMPR compound sound absorber.

The effect of resonator damping ratio, $ζ_{g}$ on sound absorption characteristics is shown in Figure 13(b). The increase in damping of resonators has significantly improved the absorption dip and flatten the sound absorption curve. This is due to the reduction in magnitude of perturbation to surface impedance when resonators resonate which eventually reduces the impedance mismatch. The only exception is the absorption peak associated with resonators of 300 Hz which experiences a slight decrease in amplitude with increasing damping ratio. The acoustic reactance in this frequency range is in the negative domain as depicted in Figure 12(b). Thus, reducing the magnitude of perturbation in acoustic impedance further increases the impedance mismatch which explains the slight drop of the absorption peak amplitude. Although there is no significant effect on the half-absorption bandwidth, however, the improvement in average absorption coefficient in the frequency range from 235 Hz to 435 Hz (i.e., the functional frequency range of the resonators) is evident as it increases from 0.766 to 0.854 (11.5% increment) when the damping ratio of resonators, $ζ_{g}$ increases from 0.03 to 0.07. However, it is worth mentioning that the damping of resonator unit can be difficult to be precisely controlled in practical application. Nevertheless, in general, the greater damping ratio gives a flatter sound absorption curve with a greater averaged sound absorption coefficient over the working frequency range of resonators.

In summary, the absorption peak associated with MPR panel resonance is shifted with varying resonator mass ratio. The resonant frequency of the resonators needs to be designed accordingly to avoid a relatively deep absorption valley. The damping ratio of resonators can be appropriately set to significantly improve the sound absorption performance in the frequency range targeted by MPR by controlling the depth of absorption valley. This demonstrates that the low frequency sound absorption can be further improved if the damping of resonators can be precisely controlled.

4. Conclusions

In this study, an MPP-MPR compound sound absorbing structure has been proposed to widen the half-absorption bandwidth and improve the low frequency sound absorption which can be potentially used in applications requiring minimal acoustic reflection. A fully coupled vibroacoustic model based on the modal coupling method has been developed and its accuracy in predicting the sound absorption characteristics is verified by comparing the analytical results with results obtained from finite element modelling. The key findings from this study are summarized as follows:

• Backing the MPP with a PR creates an absorption peak in the low frequency range followed by a wide and deep absorption valley, which greatly compromised the half-absorption bandwidth. The analysis showed that the absorption peak (dip) is attributed to the in phase (out of phase) relation between velocity of air particle and PR panel velocity, which can be regulated by adding resonators on PR.

• It has been demonstrated that the absorption valley associated with MPR panel resonance can be split into two smaller amplitude absorption dips. The improvement is attributed to the local resonance of resonators that modifies the phase of the host panel velocity and reshapes the surface impedance in the MPR resonance region, which improves the overall absorption performance in that particular frequency range.

• Resonators with different resonant frequencies can be used to further split the absorption dips to frequency-reshape the surface impedance, and significantly raise the absorption coefficient at the original absorption dip to a greater value to obtain a desired sound absorption performance.

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) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: The authors acknowledge the support received from Ningbo Science and Technology Bureau - Ningbo Natural Science Foundation Programme (Project code 202003N4183) China.

ORCID iD

Dunant Halim

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Bandwidth widening for sound absorption of a flexible microperforated panel backed by a multi-frequency panel-type resonator

Abstract

Keywords

1. Introduction

2. Analytical modelling of a coupled MPP-MPR vibroacoustic system

2.1. Modelling of a coupled MPP-MPR vibroacoustic system by the modal coupling method

2.2. Sound absorption coefficient of the MPP-MPR compound structure

2.3. Numerical model of the MPP-MPR compound structure

3. Absorption characteristics with the absorption mechanism analysis

3.1. The absorption characteristics of backing MPP with a panel-type resonator

3.2. Extending half-absorption bandwidth with the use of a multi-resonator panel-type resonator

3.3. Extending the half-absorption bandwidth with an MPR with multi-frequency resonators

3.4. Effect of resonator mass ratio and damping ratio on overall sound absorption characteristics

4. Conclusions

Footnotes

Declaration of conflicting interests

Funding

ORCID iD

References