Free vibrations of rectangular plates simply supported at two opposite edges and elastically restrained at the two other edges. An analytical and a semi-analytical method

Abstract

The present study provides a novelty in treating rectangular plates simply supported at two opposite edges and elastically restrained at the two other edges using two approaches. On one hand, the analytical solution of the plate vibration equation satisfying the rotational and translational spring boundary conditions is obtained for different plate aspect ratios leading to the corresponding modes and frequencies. On the other hand, the Rayleigh-Ritz method (RRM) is applied using plate functions defined as products of beam functions satisfying the beam appropriate end conditions in each direction. i.e. simply supported in the direction of the two opposite edges, and elastic end conditions in the other direction. The results obtained by the two methods coincide for all frequencies and modes examined, showing the effectiveness of the RRM applied to this problem in a general and systematic way. This makes the RRM ready for use in further studies concerning similar plates with added point masses or springs, or plate vibrating in the nonlinear regime. Also, useful numerical results corresponding to both the frequencies and mode shapes are presented for various plate modes, stiffness values and plate aspect ratios.