Analysis of sound radiation from a vibrating elastically supported annular plate using compatibility layer and radial polynomials

https://doi.org/10.1016/j.jsv.2021.116593Get rights and content

Highlights

  • Asymmetric vibrations of an elastically supported annular plate.

  • The acoustic power in the in the form of rapidly convergent expansion series.

  • Solving the problem using compatibility layer.

  • Improving numerical efficiency using the radial polynomials.

  • Numerical results valid within the low frequency range.

  • The effect of the plate’s boundary conditions.

  • The effect of fluid loading.

Abstract

This study investigates the problem of sound radiation from a thin vibrating annular plate. The plate is supported elastically at its circumference and embedded into the bottom of a circular cavity in a rigid plane. The selected model provides results covering the occurrence of sound radiation from the thin annular plates with any arbitrary boundary configurations. The cavity and the half-space are filled with air, and the air in the cavity plays the role of the compatibility layer. Therefore, the vibrations of the plate are coupled with that of air. Application of the compatibility layer allows the decomposition of the coupled problem of sound radiation to two problems: one for the vibration of the plate coupled with the cavity and the other for the sound radiation from the cavity to the half-space. Consequently, the plate is not coupled to the half-space directly. Therefore, the modal impedance coefficient of the plate is not required to solve the problem. The coefficients of the cavity are sufficient for the calculations, and they can be calculated significantly faster than in the case of performing a numerical integration while simultaneously satisfying the assumed accuracy by using radial polynomials. This also makes it possible to simplify numerical calculations. Finally, the total acoustic power, the acoustic pressure, and the vibration velocity have been obtained using the proposed approach and analyzed numerically. In the specific case when the cavity depth is reduced to zero, the nondimensionalized added virtual mass incremental as well as the modal impedance coefficients of the plate are calculated. Using the obtained results, the resonant frequencies can be accurately determined within a broad frequency band. The described method can be used for calculating the sound radiation for any arbitrary boundary configuration of the thin annular plates. In addition, the proposed approximation reduces the time required for the numerical calculations by roughly 2000 times compared to the time required for the numerical integration.

Introduction

Sound radiation from a thin vibrating annular plate has attracted the attention of many researchers in recent years. It has thus become a classical problem now. Leppington et al. [1] investigated the reaction from a fluid to the vibrations of a rectangular plate. Streng [2] analyzed the acoustic pressure on the surface of a vibrating stretched circular membrane in free space. Levine and Leppington [3] obtained the effective damping coefficient due to acoustic damping for a vibrating circular plate. Laulagnet [4] studied the sound radiation from an unbaffled rectangular plate. Lee and Singh [5] analyzed the sound radiation from a spinning annular disk using trial functions. Geveci and Walker [6] studied the nonlinear response of the dominant modes of rectangular plates, including the fluid loading effect. Lee and Singh [7], [8] examined the vibroacoustic behavior of a thick annular disk using analytical methods. Arenas [9] described the sound radiation from an annular plate using the resistance matrix. Mellow and Kärkkäinen [10] analyzed the sound radiation from a vibrating circular membrane in a baffle and in free space. Mellow [11] considered the axisymmetric sound field of a circular disk in an infinite baffle and presented efficient formulae in terms of the hypergeometric function. Mellow and Kärkkäinen [12] continued their investigations on the sound radiation from a vibrating circular membrane in a baffle and in free space. Mellow and Kärkkäinen [13] examined the sound field of a shallow spherical shell in an infinite baffle using similar methods. Mellow [14] continued studies on the sound radiation from a vibrating circular disk. Rdzanek and Rdzanek [15] presented asymptotic formulae for the acoustic power radiated via a single dominant axisymmetric mode of an elastically supported annular plate. Lee [16] tested the sound radiation from the in-plate vibrations of an annular plate with narrow slots. Vishwakarma et al. [17] studied the behavior of a microresonator with a clamped–free annular plate. Hasheminejad and Afsharmanesh [18] analyzed the  vibroacoustic response of annular sandwich disks. Rdzanek et al. [19] presented a low-frequency approximation for the radiation efficiency of a single dominant axisymmetric mode of an elastically supported annular plate. Huang and Chiang [20] performed an investigation on the sound radiation from electrostatic circular and annular speakers. Shakeri and Younesian [21] described the noise mitigation of vibrating annular plates. Chatterjee et al. [22] modeled the acoustic radiation from tapered clamped–free annular plates. On the other hand, the problem of sound radiation of  circular radiators has been efficiently treated using the radial polynomials. Aarts and Janssen [23] examined the axisymmetric sound radiation from a resilient circular radiator using the circle Zernike polynomials. Rdzanek [24] investigated the acoustic power of an asymmetrically vibrating thin circular plate using a similar method. The problem of plane wave transmission through a circular aperture in a thick wall was studied by Jun and Eom [25]. This problem was later on revisited by Rdzanek [26] using the radial polynomials. Sgard et al. [27] examined the boxed niche effect on the sound transmission through a single panel. Wrona et al. [28] shaped the vibroacoustic radiation response of a plate. They used added ribs and masses, achieving a significant reduction in the sound transmission. Wrona et al. [29] applied an active structural acoustic control to reduce the transmitted sound. They found that the arrangement of the actuators and sensors is crucial for achieving the desired results. Wyrwal et al. [30] presented a thorough theoretical study of a double-panel casing of a noisy device by developing a mathematical model for such a dynamical structure. The device was represented using a system of coupled partial differential equations subjected to appropriate boundary conditions, including all the fluid–structure interactions of the vibroacoustic and acoustic vibration character and the imperfect fastening of the wall panel edges to a rigid frame. Zou et al. [31] applied a three-dimensional sono-elastic method to solve the vibroacoustic problem of ships in the water of finite depth and validated their results experimentally. Zou et al. [32] studied the underwater acoustic radiation using structures that were arbitrarily covered with acoustic coatings.

This study deals with the problem of asymmetric sound radiation from an elastically supported annular plate. The plate is embedded in the middle of a circular cylindrical cavity and lightly loaded by the fluid on both of its sides. Therefore, this is a coupled fluid–structure problem. The cavity has two outlets, one toward the upper half-space and the other one toward the lower half-space. To the best of our knowledge, such a general problem has not been solved so far. As can be understood from the literature, a few specific problems mentioned above have already been solved, including the axisymmetric radiation from such a plate. However, the results presented herein cover the sound radiation for any arbitrary boundary configuration of thin annular plates. This problem can be solved efficiently using the radial polynomials in a way similar to that presented by Rdzanek in their previous work [26], where the plate was connected to the half-space directly. However, in this study, the fluid inside the cavity plays the role of the compatibility layer (see Fig. 1(a)). As a result, the benefits of the radial polynomials can be used for virtually any arbitrary vibration velocity profile at the bottom of the cavity. This includes the annular plate, which has been used in this study as an example to demonstrate the use of the compatibility layer method. The benefits of this method are the accuracy and time efficiency of the results and lead to highly efficient numerical calculations. Consequently, the sound radiation from an asymmetrically excited annular plate can be examined comprehensively (see Fig. 1(b)). For this purpose, the problem of the eigenfunctions and eigenfrequencies of an elastically supported plate have been described in detail (see Fig. 1(c)). Finally, the light fluid loading effect has been analyzed using the added virtual mass incremental (AVMI) factor and the nondimensionalized added virtual mass incremental (NAVMI) factor in the case where the cavity depth is reduced to zero (cf. Amabili et al. [33] Eqs. (8)–(10)).

The results presented herein can be potentially applied to analytically solve the problems of sound radiation from thin annular plates. The problems of sound transmission can be dealt with likewise. The plates can be suspended elastically on one or both edges in a circular cylindrical cavity. One specific example of such problems is the analysis of the micromachined resonator presented by Vishwakarma et al. [17]. The resonator has the form of a thin annular plate clamped on its external circumference to a circular rigid cavity, whereas its internal edge is completely free. Such a system is then applied as the acoustic sensor. This is a special case of elastic mounting on both the edges and can be modeled by selecting specific values of the boundary stiffnesses. Another sample application has been proposed by Huang and Chiang [20] who investigated electrostatic circular and annular speakers with clamped edges. The results presented herein can potentially help analyze the acoustic field generated by such speakers. It should be noted that the clamped annular plate is a specific case of an elastically supported plate. The vibrations and sound radiation of such a plate can be modeled by selecting the appropriate boundary stiffness values.

Section snippets

Statement of the problem

The problem of sound radiation from an annular plate embedded in the middle of a rigid circular cylindrical cavity has been considered in this study. The outlet of the cavity is embedded into a flat rigid screen (cf. Fig. 1, Fig. 1). The radius of the cavity is a  (m) and is equal to the external radius of the plate, whereas the internal radius of the plate is b (0<b<a). The z coordinate of the entire plate is equal to zero. The plate is mounted elastically at both its edges (see Fig. 1(c) and

Numerical analysis

The numerical analysis has been performed by assuming arbitrary parameter values as follows (unless stated otherwise): the speed of sound c=344ms1, the density of air ϱ0=1.21kgm3, the radius of the plate a=150  mm, the thickness of the plate h=2  mm, the Young’s modulus E=210109  Pa, the damping coefficient η=104,,102, the density of steel ϱ=7850kgm3, the Poisson’s ratio ν=0.3, the obscuration ratio ε=0.5 (for the results corresponding to the selected different values, see Rdzanek and

Concluding remarks

In this study, the sound radiation from a vibrating elastically supported annular plate was considered. The main aim of this study was to examine whether the use of the compatibility layer method combined with the use of the radial polynomials can effectively examine the considered problem numerically. From the results obtained, it has been shown that this goal has been achieved. The major findings of this study are as follows: The proposed method of the compatibility layer enables efficient

CRediT authorship contribution statement

Wojciech P. Rdzanek: Conceptualization, Methodology, Software, Visualization, Writing – original draft. Jerzy Wiciak: Conceptualization, Methodology, Writing – review & editing. Marek Pawelczyk: Conceptualization, Methodology, Writing – review & editing.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgments

The research presented in this paper was partially supported under The Centre for Innovation and Transfer of Natural Sciences and Engineering Knowledge Project at The University of Rzeszów, Poland. The work described in this paper has been executed within statutory activities of the Faculty of Mechanical Engineering and Robotics of AGH—University of Science and Technology, Poland, No. 16.16.130.942. The research reported in this paper has been supported by the National Science Centre, Poland ,

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