Analytical coupled vibroacoustic modeling of membrane-coupled Helmholtz resonator

Abstract

The membrane-coupled Helmholtz resonator (MCHR) offers distinct advantages for ultralow-frequency sound absorption. In this study, an analytical model of the MCHR is developed to investigate its vibroacoustic behavior under perpendicular acoustic incidence. The MCHR comprises a membrane backed by a shallow cavity and enclosed within a rigid frame containing multiple apertures. Governing equations are first derived based on membrane deformation and incompressible air motion in the aperture. Subsequently, a coupled membrane–Helmholtz model is then formulated by incorporating cavity-pressure variations induced by the relative motion between the membrane and the aperture air. The accuracy and applicability of the theoretical model are validated through comparison with finite element simulations. Furthermore, the influence of key parameters—including aperture radius and depth, as well as membrane elastic modulus and loss factor—on sound absorption performance is examined quantitatively. Results show that the first absorption peak arises primarily from Helmholtz resonance between the aperture and the cavity, whereas the second and third peaks are mainly due to energy dissipation in the aperture and the membrane, both resulting from membrane resonance. The established analytical model of the MCHR can thus serve as an effective tool for designing this class of membrane-type acoustic metamaterials.

Keywords

membrane-coupled Helmholtz resonator vibroacoustic behavior analytical model ultralow-frequency sound absorption absorption mechanism

1. Introduction

The attenuation and absorption of low-frequency sound have attracted significant attention owing to the growing severity of noise pollution. Conventional sound-absorbing materials, primarily porous materials (Cao et al., 2018; Tang et al., 2024; Zhao et al., 2023) and composite materials (Baek and Kim 2020; Gwon et al., 2016; Li et al., 2023) achieve excellent absorption performance through frictional and viscous dissipation within their structures. However, these solutions typically require a thickness comparable to the working wavelength, making it difficult to simultaneously obtain low-frequency absorption and a thin design.

To address this limitation, researchers have developed various Helmholtz-type low-frequency sound absorbers by introducing local resonance mechanisms (Guo et al., 2021; Omoteso et al., 2025; Xue et al., 2025). Helmholtz resonators can achieve high sound absorption near their resonant frequency. For example, Ingard (1953) investigated the influence of neck geometry on the resonance frequency of a Helmholtz resonator. Shi and Mark (2015) designed a Helmholtz structure with a spiral neck tube, significantly extending the effective neck length and enabling efficient low-frequency noise reduction. Huang et al. (2019) notably reduced the overall thickness by embedding the neck into the cavity. Cai et al. (2014) achieved an even thinner design by spirally embedding the Helmholtz resonator into a rectangular plate. Subsequently, Li et al. (2012) designed a “folded-space” back cavity, attaining near-perfect low-frequency sound absorption with a thickness of only 1/223 of the wavelength. Later, Cui and Harne (2019) replaced the rigid cavity wall of a Helmholtz resonator with a flexible wall, inducing mixed resonances through multiphysics coupling and thereby achieving multiple absorption peaks.

To pursue even lighter and thinner sound-absorbing designs, researchers have developed membrane-type acoustic metamaterials for low-frequency absorption (Chen et al., 2014; Li et al., 2024; Mei et al., 2012; Nie et al., 2026). Yang et al. (2008) first proposed a membrane-type acoustic metamaterial capable of breaking the mass-density law, where vibrational eigenfrequencies could be tuned via a small central mass. Ma et al. (2014) obtained hybrid resonance by tuning membrane and cavity parameters, achieving perfect sound absorption through impedance matching at the peak frequency. Later, Liu et al. (2019) proposed a membrane metamaterial composed of multiple resonant cells, realizing perfect absorption via impedance matching based on the acoustic siphon effect. Furthermore, Lee (2016) and Mak et al. (2021) observed that introducing apertures into the cavity significantly alters the acoustic characteristics, especially in the low-frequency range—a finding of considerable research interest. More recently, Wang et al. (2025) established a membrane-coupled Helmholtz resonator (MCHR) and systematically studied the effect of apertures on sound absorption using numerical simulation. Their results indicate that introducing an aperture into a sealed cavity generates an ultralow frequency absorption peak, with energy dissipation primarily originating from the aperture region. Nevertheless, to optimize the key structural parameters of the MCHR and enhance the characterization of its sound absorption properties, it is imperative to establish a reliable analytical model that accurately captures its vibroacoustic behavior. This approach is also highly valuable for initial MCHR design in practical engineering applications. Moreover, analytical modeling provides advantages in computational efficiency and flexibility, enabling a deeper investigation into the underlying physical mechanisms of the MCHR system.

In this paper, a theoretical model for a membrane-coupled Helmholtz resonator (MCHR) is developed to investigate its sound absorption performance and underlying mechanisms. The resonator consists of a membrane backed by a shallow cavity and enclosed by a rigid frame with apertures. The governing equations are first derived from the membrane deformation and the incompressible motion of air within the aperture. A coupled model of the membrane and Helmholtz structure is then formulated by incorporating cavity pressure changes resulting from the relative motion between the membrane and the aperture air. The theoretical predictions show close agreement with numerical simulations, validating the model’s accuracy. The results indicate that the model effectively predicts how key parameters—including aperture radius, depth, and membrane elastic modulus—affect both the peak absorption frequency and overall sound absorption performance. The first absorption peak originates primarily from Helmholtz resonance between the aperture and the cavity, while the second and third peaks are attributed primarily to energy dissipation from the aperture and the membrane, both of which are induced by membrane resonance. These findings provide valuable insights for the design of future membrane-coupled Helmholtz resonator, and the established model can thus serve as an effective tool for designing this class of membrane-type acoustic metamaterials.

2. Analytical model of the MCHR

2.1. Vibroacoustic modeling of the MCHR

2.1.1.Case of the MCHR with one aperture

Figure 1 shows the schematic of the proposed membrane-coupled Helmholtz resonator (MCHR, hereinafter), which consists of a circular rubber membrane, enclosed with a shallow cavity and a rigid frame with apertures in lower surface of the cavity. The radius of both the membrane and the cavity is r₁, while their thicknesses are h_p and d, respectively. A cylindrical aperture of radius r₂ and depth d₁ is integrated into the frame’s lower surface. The membrane is pre-stretched through the application of an initial stress denoted by τ.

Figure 1.

Schematic illustration of an MCHR with an aperture in lower surface of the cavity.

Consider a plane sound wave is normally incident on the membrane of the MCHR, the circular membrane—assumed to be fully flexible—experiences a transverse displacement w that varies with the radial coordinate r. As illustrated in Figure 1(b), the air inside the apertures is considered to move as a rigid body of mass per unit area M_h = ρd_h, which was suspended on a spring acted by the pressure change in the cavity due to the volume change ΔV, and where ρ is the air density and d_h = d₁ + 0.85r₂ is the equilibrium length of aperture (Du et al., 2012; Kinsler et al., 2000). Accordingly, two equations of motion are established to describe the membrane displacement, w and displacement of equivalent air mass in aperture (Huang et al., 2016; Park et al., 2024), ξ

D \nabla^{4} w + m_{p} \frac{\partial^{2} w}{\partial t^{2}} - T \nabla^{2} w = P_{0} - Δ p

(1)

M_{h} \frac{d^{2} ξ}{d t^{2}} + R_{h} \frac{d ξ}{d t} = Δ p

(2)

Where $D = \frac{E_{p} {h_{p}}^{3}}{12 (1 - ν^{2})} + \frac{σ {h_{p}}^{3}}{12}$ is the effective bending stiffness, E_p, v, ρ_p and h_p are the Young’s modulus, Poisson’s ratio, mass density, and thickness of the membrane, respectively. m_p = ρ_ph_p is the density per unit area, T = σh_p is the uniform tension per unit length. P₀ is the external acoustic pressure, incident sound pressure, acting on the membrane. $R_{h} = \sqrt{2 ρ μ ω} (\frac{d_{1}}{r_{1}} + 1)$ is the acoustic resistance of the aperture, Δp represents the pressure change in the cavity due to the volume change ΔV, which is produced by the relative motion of the membrane and the air within aperture. (Note that ξ is assumed to be positive when the mass is pushed into the cavity)

Δ p = - \frac{ρ c^{2}}{V_{0}} Δ V

(3)

Δ V = S_{h} ξ - S_{p} 〈 w 〉

(4)

Here, $〈 w 〉$ denotes the net displacement averaged over the membrane surface. Using Equations (3) and (4), equations (1) and (2) can be rewritten as:

D \nabla^{4} w - m_{p} ω^{2} w - T \nabla^{2} w = P_{0} - \frac{ρ c^{2}}{V_{0}} (S_{p} 〈 w 〉 - S_{h} ξ)

(5)

\frac{d^{2} ξ}{d t^{2}} + \frac{R_{h}}{M_{h}} \frac{d ξ}{d t} = {ω_{h}}^{2} (\frac{S_{p}}{S_{h}} 〈 w 〉 - ζ)

(6)

ω_{h} = c \sqrt{\frac{S_{h}}{V_{0} d_{h}}}

(7)

Here, ωh is the resonance frequency of the Helmholtz resonator under the assumption of a rigidly enclosed cavity, c is the speed of sound in air, and S_p and S_h are the surface areas of the membrane and the aperture, respectively. Equation (6) can be rearranged to express a direct relationship between ξ and $〈 w 〉$ .

ζ = \frac{{ω_{h}}^{2}}{\frac{R_{h}}{M_{h}} ω i + {ω_{h}}^{2} - ω^{2}} \frac{S_{p}}{S_{h}} 〈 w 〉

(8)

Thus, equation (5) can be decoupled from equation (6) and expressed solely in terms of the variable w.

D \nabla^{4} w - m_{p} ω^{2} w - T \nabla^{2} w = P_{0} - \frac{ρ c^{2} S_{p}}{V_{0}} \frac{\frac{R_{h}}{M_{h}} ω i - ω^{2}}{\frac{R_{h}}{M_{h}} ω i + {ω_{h}}^{2} - ω^{2}} 〈 w 〉

(9)

2.1.2. Case of the MCHR with multiple apertures

In contrast to the single-aperture configuration of the MCHR in Figure 1, multiple apertures are introduced on the lower surface of the cavity, as shown in Figure 2, where three identical cylindrical apertures spaced 120° apart.

Figure 2.

Schematic illustration of an MCHR with multiple apertures in lower surface of the cavity.

The vibrational membrane under acoustic excitation, depicted in Figure 2, is governed by the same equation as in equation (1). The rigid frame with multiple apertures is modeled as a perforated plate, where the air within each aperture moves as a rigid body. Consequently, the equation of motion for the air mass inside an aperture can be expressed as equation (2) (Huang et al., 2016; Park et al., 2024).

The volume change ΔV caused by the relative motion of the membrane and the aperture will be re-expressed as:

Δ V = N S_{h} ξ - S_{p} 〈 w 〉

(10)

The N represents the number of apertures, then by using Equations (3) and (11), the equation (2) can be further specified as follows:

\frac{d^{2} ξ}{d t^{2}} + \frac{R_{m}}{M_{m}} \frac{d ξ}{d t} = {ω_{h}}^{2} (\frac{S_{p}}{N S_{h}} 〈 w 〉 - ζ)

(11)

And the equation (11) can be rearranged, directly relating ξ and $〈 w 〉$

ζ = \frac{{ω_{h}}^{2}}{\frac{R_{m}}{M_{m}} ω i + {ω_{h}}^{2} N - ω^{2}} \frac{S_{p}}{S_{h}} 〈 w 〉

(12)

Then, equation (1) can be decoupled from equation (12), represented only of the single variable w

D \nabla^{4} w - m_{p} ω^{2} w - T \nabla^{2} w = P_{0} - \frac{ρ c^{2} S_{p}}{V_{0}} \frac{\frac{R_{m}}{M_{m}} ω i - ω^{2}}{\frac{R_{m}}{M_{m}} ω i + {ω_{h}}^{2} N - ω^{2}} 〈 w 〉

(13)

Equation (13) shows that when the number of apertures N equals 1, equation (9) becomes a special case of equation (13).

2.2. Sound absorption coefficient of the MCHR

Consider a plane acoustic wave incident perpendicularly on the MCHR. The objective is to determine the specific acoustic impedance, defined as the ratio of the pressure to the normal particle velocity on the membrane surface. A trial solution to the governing equation for the membrane, equation (1), is therefore considered as follow:

w (r) = A [J_{0} (k_{1} r) - J_{0} (k_{1} R)] + B [J_{0} (k_{2} r) - J_{0} (k_{2} R)]

(14)

where

{k_{1}}^{2} = \frac{b}{2} + \sqrt{a - \frac{b^{2}}{4}} i {k_{2}}^{2} = \frac{b}{2} - \sqrt{a - \frac{b^{2}}{4}} i

(15)

a = - \frac{m_{p} ω^{2}}{D} b = - \frac{T}{D}

(16)

Details of the derivation of the trial solution is presented in Appendix A, and the J₀ denotes the first kinds of the zeroth- order Bessel function, and the net displacement $〈 w 〉$ can then be evaluated as follows:

\begin{array}{l} 〈 w 〉 = \int_{0}^{R} \int_{0}^{2 π} w (r) r d r d θ \\ = A J_{2} (k_{1} R) + B J_{2} (k_{2} R) \end{array}

(17)

Accordingly, the general solution to the governing equation for the membrane can be written in the following form.

w (r) = A J_{0} (k_{1} r) + B J_{0} (k_{2} r) + \frac{Q}{a}

(18)

Q = \frac{P_{0}}{D} - \frac{ρ c^{2} S_{p}}{V_{0} D} \frac{\frac{R_{m}}{M_{m}} ω i - ω^{2}}{\frac{R_{m}}{M_{m}} ω i + {ω_{h}}^{2} N - ω^{2}} [A J_{2} (k_{1} R) + B J_{2} (k_{2} R)]

(19)

The displacement and its slope are zero at the boundary r = R, leading to

w (R) = 0

(20)

\frac{d w (R)}{d r} = 0

(21)

By substituting displacement expression (18) into the interface conditions (20) and (21), the unknown coefficients A and B can be expressed in terms of the incident pressure amplitude P0. The acoustic impedance of the membrane can then be obtained as follows:

Z = \frac{P_{0}}{i ω \bar{w}}

(22)

where

\bar{w}

is the average displacement of the membrane, which is computed by integrating the displacement w over the corresponding region

\bar{w} = \frac{\iint_{S_{p}} w d s}{S_{p}}

(23)

The normalized acoustic impedance of the MCHR respect to the air as

Z_{t} = \frac{P_{0}}{i ω \bar{w} ρ c} = X_{t} + Y_{t} i

(24)

Here, X_t and Y_t denote the normalized acoustic resistance and reactance, respectively, representing the real and imaginary parts of the specific acoustic impedance Z_t. Finally, the reflection coefficients R and absorption coefficients α of the MCHR can be consequently given by:

R = \frac{Z_{t} - 1}{Z_{t} + 1}

(25)

α = 1 - {| \frac{Z_{t} - 1}{Z_{t} + 1} |}^{2} = \frac{4 X_{t}}{{(1 + X_{t})}^{2} + {Y_{t}}^{2}}

(26)

3. Validation of the theoretical modeling

To validate the developed vibroacoustic model, the acoustic and vibrational properties of the MCHR predicted by the proposed theoretical model are compared with results from finite element analysis performed in COMSOL Multiphysics software. The COMSOL Multiphysics software incorporates acoustic-structure interaction with geometric nonlinearities. The MCHR configuration, shown in Figure 2, comprises a membrane and a cavity with multiple apertures. The corresponding geometric and material parameters are listed in Table 1.

Table 1.

The geometric parameters of the MCHR with multiple apertures.

Geometric	Size	Material	Size
d	21.7 mm	E _p	2.0 × 10⁶ Pa
d ₁	10 mm	ρ _p	1300 kg/m³
h _p	0.2 mm	v	0.469
r ₁	45 mm	η	0.07
r ₂	1.6 mm	σ	4.0 × 10⁵ Pa

Figure 3 shows the absorption coefficients for the MCHR with multiple apertures, as calculated from equation (26), comparing results from the analytical model and the finite element simulation. The analytical results (solid lines) show good agreement with the finite element simulation (circled marks) for different aperture numbers N, thereby validating the accuracy of the analytical model.

Figure 3.

Comparison of absorption coefficients of the MCHR under different apertures between analytical, finite element method.

As shown in Figure 3, the frequency of the first absorption peak increases gradually with the number of apertures N. Specifically, as N increases from 1 to 12, this peak frequency rises from 90.2 Hz to 270.4 Hz. Concurrently, the magnitude of the absorption coefficient at this peak first increases and then decreases. The second absorption peak frequency of the MCHR also gradually increases with the aperture number N, while the absorption performance of the third absorption peak shows much less sensitivity to the variation in aperture number.

To evaluate the energy absorption performance predicted by the vibroacoustic model, the MCHR with three apertures (N = 3) serves as an example. The membrane mode shapes at the three absorption peak frequencies, as predicted by the analytical and finite element methods, are compared in Figure 4(a). The analytical model identifies peak frequencies at 150 Hz, 800 Hz, and 1160 Hz, while the finite element method yields peaks at 147 Hz, 779 Hz, and 1144 Hz. It can be seen that the prediction results of the analytical model and the finite element method are in high agreement, confirming the accuracy of the model. Notably, the model successfully captures the geometric deformation effects of the membrane. The first mode shape features significant vibrational displacement distributed over a large portion of the membrane surface. The second and third modes are characterized by the significant resonant vibration of the membrane region.

Figure 4.

(a) Mode shapes of the membrane in the MCHR. First column: Analytical solutions; Second column: FE solutions. (b) Airflow velocity field at the absorption peak frequency plotted in grayscale.

Figure 4(b) illustrates the distribution of the airflow velocity v in the cavity plotted in grayscale, with overlaid streamlines. The velocity magnitude is normalized by the source particle velocity v₀ = P₀/Z₀, where Z₀ and P₀ denote the air impedance and the pressure amplitude of the incident plane wave, respectively. Notably, for the first absorption peaks of the MCHR (N = 3) identified in Figure 3, the streamlines are mainly concentrated in the apertures, and the darkest regions correspond to the apertures, and the airflow velocity inside the holes is more than 2 orders of the magnitude higher than in other regions. However, for the second and third absorption peaks, the velocity is mainly concentrated in the large vibrational displacement of membrane and the aperture region, and is more than an order of magnitude higher than in other regions. Correlating these results with the mode shapes in Figure 4(a), the first absorption peak originates primarily from the resonant motion of the air in the apertures (the Helmholtz resonance). The second and third peaks are predominantly due to structural resonance of the membrane, and the significant membrane vibration at these peak frequencies induces substantial airflow through the apertures, ultimately leading to energy dissipation from both the vibrating membrane and the apertures.

4. Results and discussions

Based on the developed analytical model, the effects of key parameters on the MCHR’s sound absorption performance are investigated. The structural parameters include the radius and depth of the apertures, as well as the pretension and elastic modulus of the membrane. Furthermore, the influence of damping parameters—specifically, the aerodynamic viscosity of the apertures and the loss factor of the membrane—is investigated.

4.1. The MCHR’s damping parameters

First, keeping other parameters constant, Figure 5 shows the sound absorption curves for different membrane loss factors η and aerodynamic viscosity coefficients µ = µ₁ *µ₀ in the open region, where µ0 = 17.9 × 10⁶ Pa · s.

Figure 5.

Sound absorption spectrum of the MCHR under different damping parameters: (a) of the membrane loss factors η; (b) aerodynamic viscosity coefficients μ inside the aperture.

Figure 5 presents the sound absorption spectra for varying membrane loss factors η. As η increases from 0.0001 to 0.7, the first absorption peak remains largely unaffected. In contrast, the second and third absorption peaks show high sensitivity to the η, with the third peak being the most significantly influenced. However, all the three absorption peaks in the MCHR are highly sensitive to changes in the aerodynamic viscosity coefficient µ within the apertures. These observations indicate that the first absorption peak is primarily caused by the apertures, while the absorption mechanisms of the second and third absorption peaks are attributed to the combined effects of the apertures and membrane.

4.2. The MCHR’s structural parameters

Based on the preceding analysis, the first absorption peak of the MCHR is mainly attributed to the resonance effect of the air in the apertures (the Helmholtz resonance), and the aperture radius r₂ and aperture depth d₁ are key design parameters for the aperture. Therefore, the influence of the aperture radius r₂ and aperture depth d₁ on the sound absorption characteristics of the MCHR was analyzed, as shown in Figure 6.

Figure 6.

Sound absorption spectrum of the MCHR under different structural parameters: (a) aperture depth d₁; (b) aperture radius r₂.

The absorption peak frequencies of the MCHR, particularly the first peak, shift significantly with changes in the aperture radius r₂ and depth d₁. Specifically, with other parameters held constant, the peak frequency decreases with increasing depth d₁ but increases with increasing radius r₂. Furthermore, with the increase of depth d1 and radius r₂, the sound absorption effect of the MCHR exhibits a clear trend of first increasing and then decreasing.

Furthermore, the ratio of the aperture radius r₂ and depth d₁ in the MCHR to acoustic wavelength at the three absorption peak frequencies shown in the Figure 6, denoted as Q = d₁/λ or Q = r₂/λ, is defined and calculated. It is found that the value of Q remains consistently below 0.03, regardless of the specific values chosen for the aperture radius and depth. This quantitatively confirms the dominance of local resonance and the sub-wavelength operating principle of the MCHR. Within such a deep sub-wavelength regime, the oscillating air inside the aperture behaves as a lumped mass element rather than a propagating wave. Its motion is predominantly governed by inertia and viscous damping, with wave effects being negligible. This validates the core assumption in the analytical model of treating the aperture as a lumped acoustic mass and resistance.

When multiple identical apertures are integrated into the lower cavity surface (Figure 2), the structure can be modeled as a perforated plate comprising numerous acoustically parallel Helmholtz resonators. The resonant frequency f₀ can be expressed as follows:

f_{0} = \frac{c}{2 π} \sqrt{\frac{ψ}{(d_{1} + 1.7 r_{2}) d}}

(27)

Where $ψ = N r_{2}^{2} / r_{1}^{2}$ represents the perforation rate, defined as the ratio of the total aperture area to the total plate area. Equation (27) indicates that the resonance frequency f₀ increases with both the aperture number N and radius r₂, but decreases with aperture depth d₁. This is consistent with the trend of the influence of radius r₂ and depth d₁ on the first absorption peak frequency of MCHR in Figure 6, and also aligns with the trend of the influence of the aperture number N on the first peak frequency in Figure 3.

Substituting the structural parameters of the MCHR into equation (27) yields a resonant frequency f₀ = 246.2 Hz for the perforated-plate model. This value is considerably higher than the first absorption peak frequency (150 Hz) of the three-aperture MCHR shown in Figure 3. The discrepancy arises because equation (27) assumes perfectly rigid cavity walls, whereas the MCHR incorporates a flexible membrane. The resulting reduction in the effective stiffness of the system lowers the first absorption peak frequency well below the ideal Helmholtz resonance f₀. Although a fully quantitative prediction of this peak remains challenging, the analytical model still enables a qualitative design of the structural parameters.

The influence of membrane elastic modulus E_p on the sound absorption performance of the MCHR, as analyzed using the model, is shown in Figure 7. The results indicate that the frequency of the first absorption peak increases with the elastic modulus E_p, rising from 150.3 Hz to 240.4 Hz as varies from 0.1 GPa to 1000 GPa. Concurrently, the corresponding peak absorption coefficient decreases. Similarly, the second absorption peak frequency exhibits a significant increase due to its primary dependence on membrane resonance, which is directly governed by the membrane material’s stiffness.

Figure 7.

Sound absorption spectrum of the MCHR under different elastic moduli E_p.

5. Conclusion

Building on our previous numerical investigation of the acoustic absorption properties of a membrane-coupled Helmholtz resonator (MCHR) with lower-surface apertures, this study develops an analytical model for its coupled vibroacoustic behavior. The earlier work demonstrated that introducing apertures into a sealed cavity produces a lower-frequency absorption peak, with energy dissipation localized primarily in the aperture regions. Here, we present a high-precision analytical solution for the MCHR system, validated against finite element simulations. This model accurately assesses the influence of key parameters—including aperture radius and depth, as well as membrane elastic modulus and loss factor—on both the peak frequencies and the overall absorption performance. Our analysis indicates that the first absorption peak originates predominantly from Helmholtz resonance within the apertured cavity, while the second and third peaks result mainly from energy dissipation at the apertures and the membrane, both driven by resonant motion of the membrane itself. The established analytical modeling offers an advantage in computational efficiency and flexibility, facilitating a deeper investigation into the underlying physical mechanisms of the MCHR system.

Footnotes

ORCID iD

Wang Mingfei

Funding

The authors received no financial support for the research, authorship, and/or publication of this article.

Declaration of conflicting interests

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

Appendix

Considering the membrane of the MCHR that is excited by the pressure field, the governing equation describing its transverse deflection w is expressed as: (A1)

D \nabla^{4} w + m_{p} ω^{2} w - T \nabla^{2} w = P_{0} - Δ p

The homogeneous equation of equation (A1) can be expressed as: (A2)

D \nabla^{4} w + m_{p} ω^{2} w - T \nabla^{2} w = 0

(A3)

(\nabla^{4} + a + b \nabla^{2}) w = 0

where (A4)

a = - \frac{m_{p} ω^{2}}{D}, b = - \frac{T}{D}

Through algebraic factorization, the operator $(\nabla^{4} + a + b \nabla^{2})$ can be decomposed, allowing equation (A3) to be rewritten as (Du et al., 2012; Kinsler et al., 2000): (A5)

[\nabla^{2} + {k_{1}}^{2}] [\nabla^{2} + {k_{2}}^{2}] w = 0

where (A6)

{k_{1}}^{2} = \frac{b}{2} + \sqrt{a - \frac{b^{2}}{4}} i, {k_{2}}^{2} = \frac{b}{2} - \sqrt{a - \frac{b^{2}}{4}} i

From equation. (A3), the solution for w can be derived as follows: (A7)

[\nabla^{2} + {k_{1}}^{2}] w_{I} = 0

(A8)

[\nabla^{2} + {k_{2}}^{2}] w_{II} = 0

Considering the boundary condition that w = 0 at r = r₁, the trial solutions of equations (A7) and (A8) can be considered as: (A9)

w_{I} (r) = A [J_{0} (k_{1} r) - J_{0} (k_{1} R)]

(A10)

w_{II} (r) = B [J_{0} (k_{2} r) - J_{0} (k_{2} R)]

The solution to equation (A3) takes the form of a linear combination of wI and wII. Accordingly, the trial solution for the membrane governing equation is formulated as follows: (A11)

w (r) = A [J_{0} (k_{1} r) - J_{0} (k_{1} R)] + B [J_{0} (k_{2} r) - J_{0} (k_{2} R)]

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