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Research Papers

# Free In-Plane Vibration Analysis of Rectangular Plates With Elastically Point-Supported Edges

[+] Author and Article Information
Jingtao Du, Wanyou Li

College of Power and Energy Engineering, Harbin Engineering University, Harbin 150001, P. R. China

Zhigang Liu1

College of Power and Energy Engineering, Harbin Engineering University, Harbin 150001, P. R. Chinazgliu56@yahoo.com.cn

Wen L. Li, Xuefeng Zhang

Department of Mechanical Engineering, Wayne State University, 5050 Anthony Wayne Drive, Detroit, MI 48202-3902

1

Corresponding author.

J. Vib. Acoust 132(3), 031002 (Apr 14, 2010) (11 pages) doi:10.1115/1.4000777 History: Received January 20, 2009; Revised June 25, 2009; Published April 14, 2010; Online April 14, 2010

## Abstract

In comparison with the transverse vibrations of rectangular plates, far less attention has been paid to the in-plane vibrations even though they may play an equally important role in affecting the vibrations and power flows in a built-up structure. In this paper, a generalized Fourier method is presented for the in-plane vibration analysis of rectangular plates with any number of elastic point supports along the edges. Displacement constraints or rigid point supports can be considered as the special case when the stiffnesses of the supporting springs tend to infinity. In the current solution, each of the in-plane displacement components is expressed as a 2D Fourier series plus four auxiliary functions in the form of the product of a polynomial times a Fourier cosine series. These auxiliary functions are introduced to ensure and improve the convergence of the Fourier series solution by eliminating all the discontinuities potentially associated with the original displacements and their partial derivatives along the edges when they are periodically extended onto the entire $x-y$ plane as mathematically implied by the Fourier series representation. This analytical solution is exact in the sense that it explicitly satisfies, to any specified accuracy, both the governing equations and the boundary conditions. Numerical examples are given about the in-plane modes of rectangular plates with different edge supports. It appears that these modal data are presented for the first time in literature, and may be used as a benchmark to evaluate other solution methodologies. Some subtleties are discussed about corner support arrangements.

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Copyright © 2010 by American Society of Mechanical Engineers
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## Figures

Figure 2

Two cases of elastic point support arrangement considered in the calculation

Figure 3

The first six in-plane modes for the square plate of case 1-a with the restraining coefficients kp=kn=109 for each elastic point support

Figure 4

The first six in-plane modes for the rectangular plate of case 1-b with the restraining coefficients kp=kn=109 for each elastic point support

Figure 5

Three corner support arrangements for square plate

Figure 1

A rectangular plate with arbitrary in-plane elastic point edge supports

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