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