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RESEARCH PAPERS

Exact Perturbation for the Vibration of Almost Annular or Circular Plates

[+] Author and Article Information
R. G. Parker

Department of Mechanical Engineering, Ohio State University, Columbus, OH

C. D. Mote

Department of Mechanical Engineering, University of California, Berkeley, Berkeley, CA 94720

J. Vib. Acoust 118(3), 436-445 (Jul 01, 1996) (10 pages) doi:10.1115/1.2888203 History: Received September 01, 1994; Online February 26, 2008

Abstract

A perturbation solution is presented to analytically determine the eigensolutions for self-adjoint plate vibration problems on nearly annular or circular domains. The irregular domain eigensolutions are calculated as perturbations of the corresponding annular or circular domain eigensolutions. These perturbations are determined exactly. The simplicity of these exact solutions allows the perturbation to be carried through third order for distinct unperturbed eigenvalues and through second order for degenerate unperturbed eigenvalues. Furthermore, this simplicity allows the resulting orthonormalized eigenfunctions to be readily incorporated into response, system identification, and control analyses. The clamped, nearly circular plate is studied in detail, and the exact eigensolution perturbations are derived for an arbitrary boundary shape deviation. Rules governing the splitting of degenerate unperturbed eigenvalues at both first and second orders of perturbation are presented. These rules, which apply for arbitrary shape deviation, generalize those obtained in previous works where specific, discrete asymmetries and first order splitting are examined. The eigensolution perturbations and splitting rules reduce to simple, algebraic formulae in the Fourier coefficients of the boundary shape asymmetry. Elliptical plate eigensolutions are calculated and compared to finite element analysis and, for the fundamental eigenvalue, to the exact solution given by Shibaoka (1956).

Copyright © 1996 by The American Society of Mechanical Engineers
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