EXACT PERTURBATION FOR THE VIBRATION OF ALMOST ANNULAR OR CIRCULAR PLATES

A perturbation solution is presented to analytically determine the eig ensolutions for self-adjoint plate vibration problems on nearly annula r or circular domains, The irregular domain eigensolutions are calcula ted as perturbations of the corresponding annular or circular domain e igensolutions. These perturbations are determined exactly. The simplic ity of these exact solutions allows the perturbation to be carried thr ough third order for distinct unperturbed eigenvalues and through seco nd order for degenerate unperturbed eigenvalues. Furthermore, this sim plicity allows the resulting orthonormalized eigenfunctions to be read ily incorporated into response, system identification, and control ana lyses. The clamped, nearly circular plate is studied in detail, and th e exact eigensolution perturbations are derived for an arbitrary bound ary shape deviation. Rules governing the splitting of degenerate unper turbed eigenvalues at both first and second orders of perturbation are presented. These rules, which apply for arbitrary shape deviation, ge neralize those obtained in previous works where specific, discrete asy mmetries and first order splitting are examined The eigensolution pert urbations and splitting rules reduce to simple, algebraic formulae in the Fourier coefficients of the boundary shape asymmetry. Elliptical p late eigensolutions are calculated and compared to finite element anal ysis and for the fundamental eigenvalue, to the exact solution given b y Shibaoka (1956).