Preamble

Simplified Model Analysis of the Interaction Between Waves and Flexible Piezoelectric Membrane Structures

Yang Wenyu, Liu Chunrong
(School of Civil Engineering and Architecture, Xiamen University of Technology)

Abstract

Piezoelectric materials can be utilized to capture ocean wave energy, converting wave energy directly into electrical energy. By integrating piezoelectric membranes with flexible membranes to form a submerged breakwater structure, a novel wave energy conversion device can be created that serves the dual functions of wave power generation and wave attenuation. This study derives the dimensionless equations of motion for a flexible piezoelectric membrane structure and investigates the interaction between waves and the structure using a numerical wave tank based on potential flow theory. A method for predicting the added mass of the flexible piezoelectric membrane structure is presented, and the primary control parameters influencing the structural motion response are discussed. Under wave action, the motion response of the flexible piezoelectric membrane structure satisfies a nonlinear vibrational differential equation, where the nonlinear term is governed by the dimensionless elastic modulus of the membrane. The resonance periods of the flexible piezoelectric membrane structure were obtained, and a fitted formula for calculating the dimensionless added mass was developed. Furthermore, for dimensionless initial tensions ranging from $1.31 \times 10^4$ to $6.33 \times 10^4$, the critical control parameters for the linear and nonlinear responses of the flexible piezoelectric membrane were determined.

1. Introduction

The utilization of piezoelectric materials to harvest ocean wave energy has emerged as a promising field, offering a direct mechanism for converting mechanical wave energy into electricity. When flexible piezoelectric membranes are integrated into submerged breakwater designs, they provide a multifunctional solution capable of both coastal protection through wave attenuation and sustainable energy production. Understanding the complex fluid-structure interaction (FSI) between ocean waves and these flexible membranes is critical for optimizing their design and efficiency.

2. Theoretical Formulation

2.1 Dimensionless Equations of Motion

To analyze the dynamic behavior of the flexible piezoelectric membrane, we establish a mathematical model based on the balance of forces acting on the structure. The dimensionless equation of motion for the flexible piezoelectric membrane can be derived as follows:

[FIGURE:1]

The governing equation accounts for the inertial forces, internal tension, and the external hydrodynamic pressure exerted by the fluid. The dimensionless form allows for a more generalized analysis of the structural response across various scales and environmental conditions.

2.2 Numerical Wave Tank Simulation

The interaction between the waves and the membrane is simulated using a numerical wave tank (NWT) based on potential flow theory. This approach assumes the fluid is inviscid, incompressible, and the flow is irrotational. The velocity potential $\Phi$ satisfies the Laplace equation:

$$\nabla^2 \Phi = 0$$

Boundary conditions are applied at the free surface, the seabed, and the fluid-structure interface to solve for the pressure distribution and the resulting displacement of the flexible membrane.

3. Results and Discussion

3.1 Added Mass Prediction

One of the critical factors in the dynamic analysis of submerged flexible structures is the added mass, which represents the inertia added to the system because the vibrating structure must move some volume of surrounding fluid. This study proposes a prediction method for the added mass of the flexible piezoelectric membrane. Based on the numerical results, a fitted formula for the dimensionless added mass $\bar{m}_a$ was obtained, facilitating easier engineering calculations for similar configurations.

3.2 Nonlinear Motion Response

The motion response of the flexible piezoelectric membrane under wave action is characterized by a nonlinear vibrational differential equation. The analysis reveals that the nonlinearity is primarily driven by the dimensionless elastic modulus of the membrane material.

[TABLE:1]

When the wave amplitude increases or the membrane stiffness varies, the geometric nonlinearity of the membrane becomes significant. The study identifies the resonance periods where the displacement of the membrane is maximized, which is crucial for maximizing energy harvesting efficiency.

3.3 Critical Control Parameters

The transition from linear to nonlinear response is governed by specific control parameters. For a range of dimensionless initial tensions $T_0$ between $1.31 \times 10^4$ and $6.33 \times 10^4$, we have identified the critical thresholds. Below these thresholds, the system behaves linearly, and simplified linear models may suffice. However, beyond these critical values, nonlinear effects must be accounted for to accurately predict the structural displacement and the resulting voltage output of the piezoelectric components.

4. Conclusion

This paper presents a simplified model for analyzing the interaction between waves and a flexible piezoelectric membrane structure. By deriving dimensionless equations and employing a numerical wave tank, we have clarified the influence of added mass and elastic modulus on the system's dynamic response. The provided fitting formulas and critical control parameters offer valuable guidelines for the design and optimization of submerged piezoelectric breakwaters as dual-purpose wave energy converters and coastal protection structures.

关键词

Simplified analysis of interactions between water waves and flexible piezoelectric membrane structure

YANG Wenyu, LIU Chunrong
School of Civil Engineering and Architecture, Xiamen University of Technology

Keywords: flexible piezoelectric membrane; resonance period; added mass; wave attenuation effect; wave energy

Abstract

This paper investigates the interaction between water waves and a flexible piezoelectric membrane structure through a simplified analytical approach. By considering the coupling between the fluid domain and the structural deformation, we examine the resonance characteristics and the wave attenuation performance of the membrane. Key parameters such as the resonance period and the added mass effect are analyzed to evaluate the structure's efficiency in harvesting wave energy and its potential as a flexible breakwater. The results provide theoretical insights into the design and optimization of piezoelectric-based wave energy converters.

Introduction

With the increasing global demand for renewable energy, wave energy has emerged as a significant focus of marine engineering research due to its high energy density and vast availability. Traditional rigid wave energy converters often face challenges regarding structural fatigue and high maintenance costs in harsh marine environments. In recent years, flexible structures, particularly those utilizing piezoelectric materials, have gained attention for their ability to undergo large deformations while converting mechanical strain directly into electrical energy.

Flexible piezoelectric membranes offer a dual-functionality: they can act as wave attenuators (breakwaters) to protect coastal infrastructure while simultaneously harvesting energy from the ambient wave field. However, the fluid-structure interaction (FSI) involving flexible membranes is highly complex, as the membrane's motion is significantly influenced by the surrounding fluid's inertial and damping effects, often characterized as "added mass."

Theoretical Model

To analyze the interaction between water waves and the flexible piezoelectric membrane, we employ a simplified linear potential flow theory. The fluid is assumed to be inviscid, incompressible, and the flow is irrotational.

1.1 Governing Equations and Boundary Conditions

The velocity potential $\Phi(x, z, t)$ satisfies the Laplace equation within the fluid domain:
$$\nabla^2 \Phi = 0$$

At the free surface, the linearized kinematic and dynamic boundary conditions are applied. For the flexible membrane, the equation of motion can be expressed by incorporating the tension $T$, the mass per unit area $m_s$, and the fluid pressure difference $\Delta p$ across the membrane:
$$m_s \frac{\partial^2 \eta}{\partial t^2} - T \frac{\partial^2

361024 Xiamen

China

Abstract

Piezoelectric materials can be used to capture ocean wave energy converting it directly into e- lectricity. The combination of piezoelectric membrane and flexible membrane to form a submerged break- water structure can be used as a new type of wave energy conversion device which has the dual functions of wave energy generation and wave dissipation. In this paper the dimensionless motion equation of the pi- ezoelectric flexible membrane system was derived and the interaction between the wave and the flexible piezoelectric membrane structure was studied by the numerical wave flume based on the potential flow the- ory. A method for predicting the additional mass of the flexible piezoelectric membrane was presented the main control parameters which affect the motion response of the flexible piezoelectric membrane were discussed. The response of the flexible piezoelectric membrane under wave action satisfied the nonlinear vibration differential equation and the nonlinear term was controlled by the dimensionless elastic modulus of the membrane. The resonant period of the flexible piezoelectric membrane was obtained and the fit-

The fitting formula for the dimensionless added mass of the flexible piezoelectric membrane was derived. When the dimensionless initial tension ranges from $1.31 \times 10^3$ to $6.33 \times 10^3$, the critical control parameters are provided. These results establish a theoretical foundation for the design and engineering application of flexible piezoelectric membrane structures. As a clean energy source, ocean wave energy harvesting is an effective alternative to fossil fuels. Common wave energy conversion principles involve capturing wave energy and converting it into mechanical energy, which is then transformed into electrical energy. Piezoelectric materials have emerged as novel materials capable of capturing ocean wave energy by directly converting it into electricity. Burns first integrated piezoelectric materials with wave energy development to design a wave-driven piezoelectric power generation device. Subsequently, researchers have proposed various structural configurations for piezoelectric wave energy converters. The most common designs involve cantilever beams equipped with piezoelectric patches attached to buoy structures or oscillating systems coupled with piezoelectric materials, typically installed near the free surface of the seawater. Nabavi conducted field tests on piezoelectric conversion devices installed on buoy structures in the offshore areas of Boston Harbor and San Francisco Bay. The output power reached $11.4 \text{ kW/m}^3$ per unit volume of piezoelectric material, demonstrating the promising development prospects of piezoelectric wave energy generation.

Combining piezoelectric membranes with flexible membranes to form a closed flexible piezoelectric membrane structure installed on the seabed can serve as both a novel wave energy conversion device and a submerged breakwater. This configuration fulfills the dual functions of power generation and wave attenuation. Consequently, the response of the flexible piezoelectric membrane under wave action plays a critical role in its power generation and wave dissipation performance. Scholars both domestically and internationally have conducted extensive theoretical, experimental, and numerical studies on flexible membrane breakwaters.

Ohyama experimentally investigated the wave transmission and reflection coefficients of bottom-mounted, water-filled flexible membrane breakwaters. Phadke utilized boundary element and finite element methods to study the resonant and nonlinear responses of bottom-mounted water-filled flexible membranes under wave action.

Stamos conducted a comparative experimental study on the wave attenuation effects of rigid and flexible bottom-mounted breakwaters, finding that semi-cylindrical flexible bottom-mounted breakwaters exhibit superior wave reflection performance.

Based on boundary element and finite element methods, the dynamic Lagrangian method was developed to simulate the nonlinear response of the interaction between three-dimensional flexible membrane breakwaters and waves. Zhao utilized numerical simulations and experiments to investigate the wave attenuation mechanism of the interaction between flexible water-bag submerged breakwaters and regular waves.

Potential flow theory was employed to study the hydroelastic interaction between waves and semi-circular flexible membrane breakwaters. Analytical solutions were developed, and the wave reflection and transmission coefficients, as well as the deformation of the flexible membrane, were discussed.

A viscous flow model was used to simulate the interaction between water-filled membrane breakwaters and waves. The study analyzed the effects of internal pressure and elastic modulus, as well as the vorticity field around the water-filled membrane and its structural response. The results suggest that wave reflection and energy dissipation effects are more significant under lower internal pressure conditions.

Experimental studies were conducted on the wave attenuation effects of horizontally and vertically placed flexible membrane breakwaters, analyzing the influence of parameters such as membrane porosity. By integrating piezoelectric materials with flexible membranes, the linear and nonlinear responses of flexible piezoelectric membranes under wave action were studied. The control parameters affecting the power generation efficiency and wave attenuation of the flexible piezoelectric membrane were discussed, revealing that increasing the operating frequency of the piezoelectric material can enhance power generation efficiency.

In this study, the dimensionless equations of motion for the flexible piezoelectric membrane structure are derived, and a numerical wave tank is utilized to simulate the interaction between waves and the flexible piezoelectric membrane structure. A prediction method for the added mass of the flexible piezoelectric membrane structure is presented, and the range of critical control parameters for its linear and nonlinear responses is obtained. These findings aim to provide a theoretical basis for the design and engineering application of flexible piezoelectric membrane structures.

1 数学模型

The flexible piezoelectric membrane structure, its external circuit, and the numerical wave tank are illustrated in [FIGURE:1]. The structure consists of a central piezoelectric layer sandwiched between two insulating flexible membranes, with electrodes at the ends connecting it to an external circuit. This flexible piezoelectric assembly is installed at the bottom of the wave tank and is filled with fluid, effectively dividing the entire flow field into an external domain and an internal domain.

Given that the thickness of the flexible piezoelectric membrane is typically on the millimeter scale—negligible compared to the overall dimensions of the structure—it is treated as a single arc segment within the model. Based on this simplification, a global coordinate system $(x, z)$ is established to describe the coupled motion of the fluid flow and the flexible piezoelectric structure. As the specific deployment site for this device has not yet been designated, this study focuses on fundamental mechanism research; consequently, the incident wave conditions in the numerical wave tank are set as standard sinusoidal waves.

Key words flexible piezoelectric membrane resonance period added mass wave energy dissipation

1176 应用力学学报

Governing equations for the flow field of the membrane and numerical wave flume: Based on potential flow theory, the flow fields both inside and outside the flexible piezoelectric membrane structure satisfy the Laplace equation:
$$\nabla^2 \Phi_i = 0$$
where $\Phi$ represents the velocity potential function, the subscript $e$ denotes the external flow field, and the subscript $i$ denotes the internal flow field.

On the free surface ($z = 0$), the boundary conditions satisfied by the velocity potential $\Phi_e$ of the external flow field are:
$$\frac{\partial^2 \Phi_e}{\partial t^2} + g \frac{\partial \Phi_e}{\partial z} = 0$$
At the bottom of the flow field, the boundary condition satisfied by the velocity potential is $\frac{\partial \Phi}{\partial z} = 0$. Furthermore, the normal velocities of the internal and external fluids in contact with the flexible piezoelectric membrane boundary must be consistent with the velocity of the membrane itself.

∂ φ i ∂ n = ∂ φ e ∂ n （ 4 ）

Governing Equations for Flexible Piezoelectric Membrane Structures

Due to the extremely small thickness of flexible piezoelectric membrane structures, the inertial force per unit area of the structure is negligible compared to the magnitude of the pressure difference between the internal and external fluids. Therefore, we consider only the equilibrium relationship between the membrane tension and the fluid pressure differential across the membrane, expressed as:

$$\Delta p = p_{in} - p_{out} = T \kappa$$

where $p_{in}$ and $p_{out}$ represent the internal and external pressures of the membrane structure, respectively; $T$ denotes the membrane tension per unit width; and $\kappa$ represents the curvature of the membrane.

The relationship between the internal and external pressures of the flexible piezoelectric membrane structure and their corresponding velocity potentials satisfies the unsteady Bernoulli equation, namely:

$$p + \rho \frac{\partial \phi}{\partial t} + \frac{1}{2} \rho |\nabla \phi|^2 = C(t)$$

∂ φ e ∂ t + 1 2 ∇ φ e · ∇ φ e + p e ρ + gz = 0 （ 6 ）

$g z = T$; where $\rho$ is the density of water; $T_{m0}$ is the initial tension of the membrane under static equilibrium conditions; and $r_I$ is the initial radius of the membrane. Substituting these into Equation (

$$\frac{\partial \phi_i}{\partial t} - \frac{\partial \phi_e}{\partial t} = \frac{1}{2} \left( \nabla \phi_e \cdot \nabla \phi_e - \nabla \phi_i \cdot \nabla \phi_i \right) + \frac{T_{m0}}{\rho r_I} - \frac{T_m \kappa}{\rho} \quad (8)$$

The membrane tension $T_m$ can be expressed as:

In the equation, the terms within the brackets on the right side represent the tension provided by the piezoelectric layer; the subsequent term represents the tension provided by the flexible membrane; $E_p$ is the elastic modulus of the piezoelectric layer; $h_p$ is the thickness of the piezoelectric layer; $d_{31}$ is the piezoelectric constant of the piezoelectric layer; $E_s$ is the elastic modulus of the flexible membrane; $h_s$ is the thickness of the flexible membrane; and $\epsilon$ is the strain of the membrane structure.

Considering that the piezoelectric constants of materials are typically very low, the terms on the right side of the equation containing the piezoelectric constant can be neglected. Under the assumption of small deformations, in polar coordinates, the radial displacement of the flexible piezoelectric membrane from its equilibrium position is denoted by $u_r$.

The boundary conditions for the membrane are defined at the connection points between the membrane and the bottom boundary.

κ = 1 r I - 1 r 2 I ∂ 2 ζ ∂ θ

ζ θ ， ( ) t = 0 （ 12 ）

2 无量纲化的柔性压电膜结构的运动

To describe the motion of the flexible piezoelectric membrane structure and analyze the primary parameters influencing its behavior, the governing equations of motion were non-dimensionalized. The wave period $T$, the density of water $\rho_w$, and the gravitational acceleration $g$ were employed as the fundamental scales to non-dimensionalize the relevant physical quantities.

The non-dimensional radial displacement of the membrane, $\bar{w}$, can be expressed in the form of a Fourier series as $\bar{w} = \sum_{n=0}^{\infty} \bar{w}_n \cos(n\theta)$. Given that the wave amplitude is relatively small, the higher-order terms in Eq. (13) can be neglected. Following non-dimensionalization, the expression becomes:

ζ * （ θ ， t * ） = ∑

The dimensionless curvature $\kappa^$ can be expressed as $\kappa^ = \kappa g(T)^2$.

The dimensionless membrane tension can be expressed as follows:
[TABLE:1]
In the formula, the dimensionless elastic modulus of the membrane structure is defined as $\Pi$, and the dynamic strain of the membrane is denoted by $\epsilon$.

κ * = 1 r * I - 1 （ r * I ）

In equation (17), $\Delta \theta = \theta_1 - \theta_0$, where $\theta_0$ and $\theta_1$ represent the angles corresponding to the two ends of the membrane.

k = 1 A k r * ( ) I

ε ′ = 1 4 π Δ ( ) θ

is the number of modes. Substituting into Eq. (1), we obtain:

∂ φ * i ∂ t * - ∂ φ * e ∂ t * = - π Δ ( ) θ

2 T

k = 1 A k （ t ） k 2 sin k π Δ θ （ θ -

2 E

k = 1 A k （ t ） k 2 ·

Integrating the expression yields:

2 A

Based on the linear wave hypothesis, the velocity potential of the fluid $\Phi(x, y, z, t)$ satisfies the Laplace equation within the fluid domain:

$$\nabla^2 \Phi = 0$$

Assuming the fluid is incompressible, inviscid, and the flow is irrotational, the velocity potential can be decomposed into a time-independent spatial potential $\phi(x, y, z)$ and a harmonic time component $e^{-i\omega t}$, where $\omega$ is the angular frequency of the incident wave.

Boundary Conditions

To solve for the velocity potential, the following boundary conditions must be satisfied:

1. Free Surface Condition: On the mean free surface ($z=0$), the combined kinematic and dynamic boundary condition is expressed as:
   $$\frac{\partial \phi}{\partial z} - \frac{\omega^2}{g} \phi = 0$$
   where $g$ is the acceleration due to gravity.

2. Seabed Condition: At the sea floor ($z=-h$), the vertical velocity must be zero:
   $$\frac{\partial \phi}{\partial z} = 0$$

3. Body Surface Condition: On the wetted surface of the structure $S_B$, the normal velocity of the fluid must match the normal velocity of the body:
   $$\frac{\partial \phi}{\partial n} = V_n$$
   where $n$ is the unit normal vector pointing into the fluid and $V_n$ is the velocity component of the structure in the normal direction.

4. Radiation Condition: At an infinite distance from the structure, the scattered waves must propagate outward toward infinity:
   $$\lim_{r \to \infty} \sqrt{r} \left( \frac{\partial \phi_s}{\partial r} - ik\phi_s \right) = 0$$
   where $k$ is the wavenumber and $\phi_s$ represents the scattering potential.

[FIGURE:1]

Wave Forces and Hydrodynamic Coefficients

The total velocity potential $\phi$ can be decomposed into the incident wave potential $\phi_I$, the diffraction potential $\phi_D$, and the radiation potential $\phi_R$ resulting from the six-degree-of-freedom motion of the structure:

$$\phi = \phi_I + \phi_D + i\omega \sum_{j=1

2 A

= - π 2

φ * i = ∑

k = 1

φ * e = ∑

k = 1

the velocity potentials of the incident and scattered waves. By substituting Eq. (1) into Eq. (2), we obtain the dimensionless modal amplitudes for the vibration of the flexible piezoelectric membrane structure.

K A = π 2

K B = π 4

3 E

M a = ∫

μ = - ∫

where $\bar{K}$ is the dimensionless stiffness of the membrane structure; $\bar{M}$ is the dimensionless added mass; $\bar{C}$ is the damping coefficient; $\bar{F}$ is the external excitation; and $\bar{G}$ represents the coupled excitation of the states. The numerical solution methods for this system of equations can be found in \cite{Ref_Source}.

F ex = ∫

3 分析与讨论

Regarding the linear strain response of the flexible piezoelectric membrane structure, it can be observed that the response of the structure satisfies a nonlinear differential vibration equation. The coefficient of the nonlinear term in this equation is determined by the dimensionless elastic modulus of the membrane structure. When the value of the dimensionless elastic modulus is sufficiently small, the nonlinear term becomes negligible. In such cases, the governing equation (21) simplifies to a linear vibration equation, and the flexible piezoelectric membrane structure exhibits only a linear response.

Conversely, when the value of the dimensionless elastic modulus exceeds a certain critical threshold, the nonlinear term can no longer be ignored, causing the flexible piezoelectric membrane structure to exhibit nonlinear response characteristics. For instance, when the dimensionless parameter is set to $0.0001$, the nonlinear term in the governing equation (21) is considered negligible. Under these conditions, the flexible piezoelectric membrane structure demonstrates a linear response, which is further corroborated by the dynamic strain-time variation patterns illustrated in [FIGURE:1].

（ E * = 0. 000 1 ， T 0 = 70 N/ m ）

under different wave periods （ E * = 0. 000 1 ， T 0 = 70 N/ m ）

1178 应用力学学报

When the initial tension is set to $70 \text{ N/m}$, the strain response of the membrane structure under different incident wave periods consistently follows a sine or cosine curve, indicating that the structure exhibits purely linear behavior. The amplitude of the strain response initially increases and then decreases as the wave period varies. Specifically, the strain response amplitude reaches its maximum value at a period of $T = 0.88 \text{ s}$. When the initial tension falls within the range of...

100 N/ m

In all cases, the flexible piezoelectric membrane structure exhibits a linear strain response consistent with Figure \ref{fig:linear_response}. When the initial tension is fixed and a period scan is performed, a maximum strain response of the flexible piezoelectric membrane structure is consistently observed at a specific incident wave period. According to (22), the dynamic strain of the membrane structure can be expressed as the sum of the amplitudes of each vibration mode. Based on previous research regarding flexible membrane motion, the first-order vibration mode of the membrane structure cannot exist independently; its amplitude $A_1$ is a function of the subsequent modal amplitudes $A_n$ ($n > 1$).

[FIGURE:1] provides the time-series curves for the amplitudes of the $n$-th order vibration modes of the membrane structure. The amplitudes of vibration modes above the fourth order are extremely small, and their contribution to the dynamic strain can be neglected. When $T = 0.0001$, the amplitudes of the second and third vibration modes provide the primary contribution to the strain response of the membrane structure, with the second-order mode amplitude contributing the most. In studies concerning the linear response of piezoelectric flexible membranes, when $T$ ranges from $0.01$ to $0.4$, the amplitude of the first-order vibration mode contributes most significantly to the strain response of the membrane structure.

The time-series curves for the amplitudes (where $n = 2$ to $5$ and $T = 0.0001$) are as follows:

A k （ k = 2 ～5 ， E * = 0. 000 1 ）

Natural Frequencies of Flexible Piezoelectric Membrane Structures: When a flexible piezoelectric membrane structure is subjected to varying initial tensions, the variation of the wave transmission coefficient relative to the incident wave period can be determined. The resulting relationship between the transmission coefficient and the wave period is illustrated in [FIGURE:1].

Variation curves of $C_T$ with respect to the wave period ($E^* = 0.0001$).

different wave periods （ E * = 0. 000 1 ）

As can be observed, when the initial tension is constant, the transmission coefficient generally increases first with the incident wave period, then decreases to a minimum value, and subsequently increases again. Taking the initial tension as...

70 N/ m

For example, when $T = 0.88$ s, the wave transmission coefficient reaches its minimum value. Simultaneously, as shown in [FIGURE:N], the dynamic strain of the membrane structure reaches its maximum value, indicating that the membrane structure undergoes resonance under the action of the waves. The resonant response of membrane structures plays a critical role in their design and practical application. Under the assumption of linearity, the frequency corresponding to this wave period is considered the natural frequency of the membrane structure. [TABLE:1] presents the resonance periods of the membrane structure under different initial tension conditions. As the initial tension increases, the resonance period of the membrane structure decreases.

[TABLE:1] Resonance period of flexible piezoelectric membrane structure under different initial tension conditions: $1.31 \times 10^1$, $2.43 \times 10^1$, $4.50 \times 10^1$, $7.83 \times 10^1$, $1.24 \times 10^2$, $2.08 \times 10^2$, $3.98 \times 10^2$, $6.33 \times 10^2$. Regarding the added mass of the flexible piezoelectric membrane structure: in the response control equations of the flexible piezoelectric membrane structure, the dimensionless parameters are primarily calculated using $\mathcal{K}$ and $\mathcal{A}$. The added mass is determined by the velocity potential of the internal and external flow of the piezoelectric flexible membrane structure and cannot be calculated directly. However, if the influence of nonlinear stiffness on the added mass is neglected, the added mass can be determined through linear theory.

According to linear vibration theory, the dimensionless resonance of the membrane structure can be expressed as a function of the dimensionless structural stiffness $K$ and the dimensionless added mass $A$. By determining the value of $K$ under resonance conditions according to Eq. (23), the following relationship is established.

To determine the value of $K_A$, the parameters used in the calculation are $\Delta \theta = 0.64674$ and $r^*_I = \dots$

Taking the resonance period at $L = 0.35$ m, the value of the dimensionless added mass $A$ of the structure can be back-calculated using Eq. (13). Based on dimensional analysis, the dimensionless added mass $A$ represents the added mass per unit width.

[FIGURE:N] illustrates the variation curve of the dimensionless added mass relative to $B/L$, where $B$ is the width of the membrane structure. In this numerical example, the results show that...

M a = M ρ g 2 T

The incident wavelength is $L = 0.6$ m, which can be determined according to the dispersion relation, namely $\omega^2 = gk \tanh(kh)$, where $T$ is the wave period, $g$ is the gravitational acceleration, and $h$ is the water depth. At a depth of $h = 0.3$ m, it can be observed that the dimensionless added mass $B/L$ follows a logarithmic relationship.

As $B/L$ increases, the value also increases; the corresponding fitting curve satisfies the relationship:

L = gT 2

If the natural frequency of the membrane structure is known, the dimensionless added mass of the membrane structure can be determined according to Eq. (13). Conversely, if the added mass of the membrane structure is known, its natural frequency can be predicted. This relationship provides a theoretical foundation for the design of flexible piezoelectric membrane structures.

The control of linear and nonlinear responses in flexible piezoelectric membrane structures is illustrated by the variation curves of mass. According to the governing equations of the flexible piezoelectric membrane structure, the dimensionless elastic modulus of the structure is the primary control parameter determining its linear and nonlinear responses. Clearly, there exists a critical value for this parameter. When the value is below this critical threshold, the structure exhibits a linear response; when it exceeds the threshold, the structure exhibits a nonlinear response. Therefore, by fixing the value—effectively fixing the linear stiffness of the structure—one can analyze the dynamic response of the membrane structure following its interaction with waves to determine the conditions corresponding to a linear response.

1180 应用力学学报

Different scale scans were performed using the values provided in the study. [FIGURE:1] illustrates the time-series curves of the dynamic strain response of the membrane structure as $\beta$ varies within the range of $0.0005$ to $0.5$, with $\gamma = 1.31 \times 10^{-4}$. As observed in the figure, as the value of $\beta$ gradually increases, the dynamic strain response curves progressively deviate from a standard sinusoidal form, exhibiting an increasing trend of deformation.

When $\beta \le 0.003$, the membrane structure exhibits a linear response. However, when $\beta > 0.003$, the membrane structure begins to demonstrate nonlinear response characteristics. Therefore, $\beta = 0.003$ serves as the critical threshold for determining the linear response of the membrane structure. [FIGURE:2] presents the control parameter ranges for the linear and nonlinear responses of the flexible piezoelectric membrane structure, where the region to the left of the curve corresponds to the linear response zone, and the region to the right corresponds to the nonlinear response zone.

（ T * 0 = 1. 31 × 10 - 5 ）

（ T * 0 = 1. 31 × 10 - 5 ）

Regarding the wave attenuation performance of the flexible piezoelectric membrane structure, the transmission coefficient of the incident wave is presented. Within the range of $1.31 \times 10$ to $1.24 \times 10$, the transmission coefficient initially increases gradually, then decreases abruptly, and subsequently increases again (corresponding to an incident wave period smaller than the specified range). When the value lies between $2.08 \times 10$ and $6.30 \times 10$, the transmission coefficient increases steadily (corresponding to an incident wave period greater than the specified range). At $= 1.31 \times 10$ and $= 0.01$, the transmission coefficient approaches a specific value, indicating that the flexible membrane structure exhibits a near-linear response. If a transmission coefficient near or below a certain threshold is considered indicative of effective wave attenuation, an appropriate parameter control range can be identified, specifically $7.83 \times 10$. In the design of flexible piezoelectric membrane structures, if wave attenuation is the primary objective, the control parameters can be selected to minimize the transmission coefficient based on the aforementioned conditions. However, if the primary goal is wave energy harvesting, or a combination of power generation and wave attenuation, the control ranges for the design parameters require further discussion.

Variation law of the coefficient.

4 结

This study derives the dimensionless equations of motion for flexible piezoelectric membrane structures and identifies the primary dimensionless control parameters. Based on linear assumptions, the resonance period and added mass of the flexible piezoelectric membrane structure are discussed. Furthermore, the control parameters defining the linear response range are established, and the wave attenuation effectiveness of the structure is analyzed. The main conclusions are as follows:

1) The motion response of the flexible piezoelectric membrane structure under wave action satisfies a nonlinear differential equation of vibration. The nonlinear terms are governed by the dimensionless membrane elasticity. The resonance period of the structure is determined, showing that the resonance period decreases as the initial tension increases.

2) The dimensionless added mass of the flexible piezoelectric membrane structure follows a logarithmic law. For values of $B/L$ ranging from $1.31 \times 10^{-2}$ to $6.33 \times 10^{-1}$, the critical control parameter ranges for the linear and nonlinear responses of the structure are provided. these results provide a theoretical basis for the design of flexible piezoelectric membrane structures.

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