Extension of Rayleigh–Ritz method for eigenvalue problems with discontinuous boundary conditions applied to vibration of rectangular plates

Abstract

The Rayleigh–Ritz (R–R) method is extended to eigenvalue problems of rectangular plates with discontinuous boundary conditions (DBC). Coordinate functions are defined as sums of products of orthogonal polynomials and consist of two parts, each satisfying the BC in its respective region. These parts are matched by minimizing the mean square error of functions and their x-derivatives at the interface between regions. Matching defines a positive definite 2N^2 × 2N^2 matrix Q whose eigenvectors form the orthogonal coordinate functions. The corresponding eigenvalues measure the matching error of the two parts at the interface. When applying the R–R method, the total error is the sum of the matching error and that arising from the finite number of coordinate functions. Although most of the coordinate functions correspond to the zero eigenvalue, these do not suffice and additional functions corresponding to small but finite eigenvalues must be included. In three examples with discontinuous BC of the clamped, simply supported and free kind, the calculated frequencies match closely those from a finely discretized finite element method.