A perturbation approach for the eigenfrequency analysis of separable Kirchhoff thin-plate problems on a rectangle

We introduce a method for the estimation of the eigenfrequencies for separable Kirchhoff thin-plate problems on a rectangle, particularly those involving the free boundary condition. We believe that this paper is the first to give such a treatment for the low end of the spectrum. The method is an adaptation of an asymptotic/perturbation method used for the treatment of various beam problems and, in the setting of the plate, is shown to be a generalization and refinement of the asymptotic methods of Bolotin and Keller and Rubinow (the wave propagation method). We compare our results with those of our own Legendre-tau approximation and, where available, with numerical results extant in the literature. Excellent agreement is found in each case.