Analytical coupled vibro-acoustic modeling of tensioned membrane backed by the rectangular cavity

Highlights

• A unified novelty procedure is proposed to study the sound radiation and sound insulation from the membrane backed by a rectangular cavity for the first time.

• The analytical vibro-acoustic coupled model consisting of the tensioned membrane and the closed cavity is established, which is suitable for different boundary conditions.

• The influences of various parameters on sound radiation power and sound transmission loss of the coupled system are studied in detail.

Abstract

This paper proposes a unified approach to establish the analytical vibro-acoustic coupled model consisting of the tensioned membrane and the closed cavity. A 2-D Fourier cosine series with auxiliary functions is used to derive the displacement of the membrane structure. The sound pressure inside the cavity is expressed as a 3-D Fourier series with additional terms by the method of wave acoustics. Then, the vibro-acoustic coupled model between the membrane and the closed cavity is established based on the Hamilton's principle, which is suitable for different boundary conditions. The program developed by this method can be used to study the sound radiation power and sound transmission loss of coupled systems with various parameters. To demonstrate the accuracy of the proposed method, the free and forced responses of the coupled model are compared with results obtained by finite element method. The membrane with general boundary conditions is well investigated. Besides, the acoustic characteristics of the coupled model are analyzed by adjusting membrane thickness, membrane tension, cavity depth, membrane mounting position and source position, including sound radiation power and sound transmission loss. And the findings have been interpreted appropriately.

Introduction

Membrane structures have been widely used in engineering for its advantages of lightweight, easy folding and high specific strength. For example, inflatable membrane structures are increasingly used in airship, aerocraft and UAV wing structure, which deal with the engineering application problems that traditional structural materials cannot solve [1], [2], [3], [4]. Due to the high flexibility of the membrane structure, inflatable membrane structures are sensitive to acoustic excitations. If the acoustic excitation frequency is consistent with the natural frequency of the structure, the structure will be destroyed by resonance [5,6]. Besides, the internal and external fluid medium have a significant influence on the vibration characteristics of the membrane structure. Therefore, in order to better observe and control the vibration of the inflatable membrane structure, it is of great significance to conduct the vibro-acoustic coupling analysis of membrane-cavity coupled model.

Many studies have been conducted to analyze the vibro-acoustic characteristics of flexible membrane structures. They cover homogenous and non-homogenous models, general and arbitrary shaped geometric configurations, in which membranes of general shape, such as rectangular and circular membranes. In the case of simple structures, the exact solutions for their vibrations can usually be derived [7], [8], [9]. However, for complex structures, it is generally difficult to obtain exact solutions and can only be solved by numerical method or semi-analytical method, mainly including Rayleigh-Ritz method [10,11], finite difference method [12], direct, indirect and multipole Trefftz method [13,14], discrete singular convolution method [15], dimensionless dynamic influence function method [16,17], differential quadratics method [18], wave propagation method [19] and boundary element method (BEM) [20], finite element method (FEM) [21], [22], [23], spectral element method (SEM) [24], [25], [26], [27], etc. Based on Gorman's superposition method (GSM) [28], many researchers also obtain the general solutions of Kirchhoff plates [29,30], Mindlin plates [31], and composite plates [32] in the frequency domain.

In the last century, Takahashi et al. [33] proposed the theory of sound absorption and transmission of single permeable membrane, and carried out theoretical studies on the structure composed of air layer, sound absorption layer and permeable membrane surface, providing an effective method for predicting the acoustic characteristics of membrane structure. Wang [34] first studied the vibration characteristics of a composite membrane combined with two circular membranes by using the internal matching method. Kang and Lee [16] proposed a multi-domain method to subdivide the membrane into several parts for the free vibration analysis of arbitrarily shaped membranes. This method is particularly effective for highly concave membranes and multi-connected membranes with holes, providing an effective method for the study of complex composite membranes. Liu et al. [35] proposed a method for computing the dynamic stiffness of membranes under arbitrary boundary conditions, which is applicable to the single membrane structure and its components. Liu et al. [36] took large deployable space antenna (LDSA) as the object, and first studied the Andronov-Hopf bifurcations, Pomeau-Manneville intermittent chaos and nonlinear vibrations of the structure in the case of 1:3 internal resonance. Then, the reduced scale model was designed to analyze the LDSA structure, and it was found that when the excitation frequency was consistent with the low-order frequency, the energy of the antenna would transfer from the low frequency vibration mode to the high frequency vibration mode [37]. Besides, the eccentric rotating ring truss antenna under parametric and external excitation was also studied by the perturbation analysis [38]. However, nonlinear vibration analysis of structures seems to be inadequate. Therefore, Delapierre et al. [39] used Galerkin method to derive the reduced order model of weak nonlinear vibration of membrane structures and studied the transverse nonlinear vibration of annular membranes at constant angular velocity. Zhang et al. [40] studied the nonlinear vibration of composite laminated cylindrical shells and established the nonlinear partial differential governing equation of eccentric rotating composite laminated cylindrical shell by using Donnell thin shear deformation theory, von Karman-type nonlinear relation and Hamilton's principle [41].

Inflatable membrane structure is a typical thin-walled structure, which is usually regarded as a structure-cavity coupled system in practical analysis. Therefore, compared with the analysis of simple structure, the researches on coupled system are more valuable. Dowell and Voss [42] first studied in this field and analyzed the modal problems of the elastic plate and sound cavity coupled system. Subsequently, Pretlove [43] proposed the theoretical solution of free vibration response of the simply supported rectangular plate with back cavity structure, and further studied the theoretical solution of forced vibration response of the elastic plate [44]. In addition, in order to obtain the natural frequencies of the coupled model, researchers have also adopted many methods, such as the variational iteration method, harmonic balance method, direct integration method and matrix iteration method [45,46]. Jain and Sonti [47] proposed an analytical method for deriving the coupling matrix of structure-acoustic system based on impedance method [48]. In recent years, an improved Fourier series method (IFSM) has been widely used in the study of structural vibration and vibro-acoustic coupling [49], [50], [51], [52], [53], [54], [55]. Based on this method, Zhang et al. [56,57] analyzed the vibro-acoustic coupling characteristics of the laminated rectangular plate-cavity coupled system. Kong et al. [55] also studied the rectangular plate-cavity coupled model with various parameters and analyzed its sound radiation power. Rajalingham et al. [58] divided the coupled system into a dynamical system composed of two subsystems by using the multi-modal method and studied the free vibration response of a circular membrane supported by a cylindrical cavity. In order to better understand the coupling relationship between the structure and the surrounding fluid medium, the cavity-structure-field theoretical model was established to consider the interaction between structure, internal sound field and external sound field, which indicates that the coupling effect resulted in additional peaks and troughs of the average pressure inside the cavity [59]. Muffler is a relatively common membrane structure-sound field coupled model in practice. Du et al. [60], [61], [62] established 1-D and 2-D acoustic coupled models of pipeline membrane muffler by using the energy principle [63], [64], [65], [66], and carried out in-depth analysis on the performance of membrane with different elastic constraints. Mi and Yu [67] proposed a new membrane resonator based on the Helmholtz resonator, which combined with air cavity and tensioned membrane for the low-frequency noise control in the pipeline. The resonator is connected to the main waveguide through two necks and can be used to evaluate the acoustic performance of U-shaped pipeline as the main structure. Except for thin plates and membranes with regular shapes, researchers have also carried out corresponding studies on irregularly shaped structures. Li and Cheng [68] adopted the integrated mode method to study the sound pressure inside the irregular cavities, and established a completely coupled vibro-acoustic model of flexible plate supported by the sound cavity, which is suitable for cases with tilted walls inside the cavity. Shi et al. [69] studied the influences of the shapes, geometric parameters and impedance values of trapezoidal cavity on the acoustic properties of the trapezoidal cavity based on the IFSM. Tarazaga et al. [70] described the acoustic-structure coupling dynamic characteristics of pressurized optical membranes based on impedance, mainly aiming to control the vibration of the membranes by acoustic excitations.

Until now, many investigations have been conducted on vibro-acoustic properties of membrane structures. However, there is no unified analysis method to study the sound radiation and sound insulation performance of membrane structures with back cavity. This paper proposes a unified approach to establish the analytical vibro-acoustic coupled model consisting of the tensioned membrane and the closed cavity, in which the membrane size can be different from the cavity wall. Then, the sound radiation power and sound transmission loss of the membrane-cavity coupled system under the internal excitation have been investigated. Compared with the FEM, this method only needs a few elements with very few degrees of freedom to obtain the calculation results of the coupled model between the membrane structure and the closed cavity, and the time required is greatly reduced. And this method could be adopted to deal with different boundary conditions, including elastic ones, just by adjusting the spring values at the edges. In addition, the program derived from this method can be applied to coupled systems with various parameters, which reduces the repetitive modeling work in FEM. To demonstrate the accuracy of the proposed method, the free and forced response of the coupled model are compared with results obtained by FEM. The membrane with general boundary conditions is well investigated. Besides, the acoustic characteristics of the coupled model are analyzed by adjusting the membrane size, membrane tension, cavity depth, membrane mounting position and source position, which mainly included sound radiation power and sound transmission loss.