An exact series solution for the transverse vibration of rectangular plates with general elastic boundary supports

An analytical method is developed for the vibration analysis of rectangular plates with elastically restrained edges. The displacement solution is expressed as a two-dimensional Fourier series supplemented with several one-dimensional Fourier series. Mathematically, such a series expansion is capable of representing any function (including the exact displacement solution) whose third-order partial derivatives are (required to be) continuous over the area of the plate. Since the discontinuities (or jumps) potentially related to the partial derivatives at the edges (when they are periodically extended onto the entire x–y plane as implied by a two-dimensional Fourier series expansion) have been explicitly “absorbed” by the supplementary terms, all the series expansions for up to the fourth-order derivatives can be directly obtained through term-by-term differentiations of the displacement series. Thus, an exact solution can be obtained by letting the series simultaneously satisfy the governing differential equation and the boundary conditions on a point-wise basis. Because the series solution has to be truncated numerically, the “exact solution” should be understood as a solution with arbitrary precision. Several numerical examples are presented to illustrate the excellent accuracy of the current solution. The proposed method can be directly extended to other more complicated boundary conditions involving non-uniform elastic restraints, point supports, partial supports, and their combinations.