Elsevier

Thin-Walled Structures

Volume 49, Issue 9, September 2011, Pages 1191-1194
Thin-Walled Structures

Technical Note
Approximate elliptical integral solution for the large amplitude free vibration of a rectangular single mode plate backed by a multi-acoustic mode cavity

https://doi.org/10.1016/j.tws.2011.03.002Get rights and content

Abstract

The nonlinear structural–acoustic problem considered in this study is the large amplitude free vibration of a rectangular elastic plate backed by a cavity. Very few classical solutions for this nonlinear structural–acoustic problem have been developed, although there are many for nonlinear plate or linear structural–acoustic problems. Thus, the main contributions of this study paper include (1) a concise multi-acoustic single structural modal formulation that is derived from two coupled partial differential equations representing the nonlinear structural free vibration and the acoustic pressure induced and (2) the approximate elliptical integral solution that is obtained by solving one residual equation only, and well agrees with that obtained from a harmonic balance finite element analysis. It is found that the natural frequency convergences with the increase in the numbers of acoustic modes and harmonic terms, and the effects of vibration amplitude, air cavity depth, and aspect ratio on the nonlinear natural frequency are also examined.

Introduction

Many studies on structural–acoustic interaction have been presented over recent decades (such as van Hal et al. [1], Dey et al. [2], and Everstine [3]). The problem of a panel backed by a cavity is one of those focused by many researchers. In practice, double panels and panel absorbers are usually made of thin metal or plastic sheet designed for light weight purposes. Therefore, a nonlinear approach must be adopted to analyze vibrations of such panels. Although the problem of a panel backed by a cavity has been of considerable interest to many researchers, e.g. Lee et al. [4], [5]; Lyon [6]; Pretlove [7]; Jackson [8], most of them have adopted linear approaches. In the study by Lee [9], the harmonic balance finite element analysis was used to determine the solution to the problem. However, this approach requires a significant amount of computational effort, pre-processing input, and the setting up of the harmonic balance equations. In finite element approach for solving the problems of nonlinear plate vibrations (e.g. Lee et al. [10]; Reddy and Huang [11]; Decha-Umphai and Mei [12]; Benamar et al. [13]), it has been common to setup a set of residual equations or global matrix equations, and then solve them for the eigenvalue solutions. All these approaches require a significant effort as an eigenvalue problem has to be solved. The present study uses the multi-acoustic and single structural mode approach to develop a concise elliptical integral solution for the large amplitude free vibrations of a rectangular elastic plate backed by a cavity.

Section snippets

Theoretical formulation

Fig. 1 shows the theoretical model, which contains a flexible and simply supported rectangular plate backed by a cavity. The boundary at z=c is flexible, and the other walls are acoustically reflective and structurally rigid. According to single structural mode approach from Chu and Herrmann [14], the governing equation for the large amplitude vibration of a rectangular plate is given byρd2Adt2+ρω02Α+βΑ3=0

Consider the acoustic pressure within the cavity acting on the plate. Eq. (1a) is modified

Results

Using Eqs. (13), (14a), the fundamental natural frequency of a simply supported square aluminum plate of 0.3048 m×0.3048 m×1.2192 mm backed by a 0.0508 m cavity at various vibration amplitude ratios is obtained. The material properties are: Young's modulus E=7×1010 N/m2, Poisson's ratio ν=0.3, and mass density ρ=2700 kg/m3. The first 25 symmetrical acoustic modes and the first four harmonic terms are used for this convergence study. The frequency ratio is defined as ωn/ωo. Table 1 shows that the nine

Conclusions

A multi-acoustic mode and single structural mode formulation, which is based on the classical nonlinear plate equation and homogeneous wave equation, has been presented for the large amplitude free vibrations of a rectangular plate backed by a cavity. The nonlinear natural frequency is obtained by solving one equation with one unknown. The present elliptic integral solution agrees reasonably well with those obtained with the harmonic balance finite element approach. The convergence study has

Acknowledgments

The research reported in this paper was fully supported by a grant from the Research Grants Council of the Hong Kong Special Administrative Region, China [9041356 (CityU 116408)].

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