Free and forced in-plane vibration of rectangular plates with non-uniform elastic boundary conditions

In this paper, the free and forced in-plane vibration analysis of rectangular plates are performed for the first time using an improved Fourier series method, in which the boundary restraining spring stiffness can vary in any functional pattern along each edge. Two-dimensional improved Fourier series displacement forms are constructed with four supplementary polynomials introduced into the standard 2-D Fourier series to make the field functions sufficiently smooth in the whole solving domain. Energy formulations are employed to describe the in-plane dynamics of plate system, in which the in-plane concentrated point force is taken into account in the form of work term. All the unknown Fourier series coefficients are then solved through the Rayleigh-Ritz procedure. Several numerical examples are given to demonstrate the correctness and effectiveness of the proposed model through the comparison with those calculated via finite element analysis (FEA). The results show that these two results can agree very well with each other for various non-uniform boundary conditions. Based on the established model, the in-plane vibration response is also studied. Some curves and contours are obtained to illustrate how the boundary restraining stiffnesses affect the in-plane point and transfer mobility of rectangular plate structure.