Free vibrations of rectangular cantilever plates. Part 1: out-of-plane motion

An analysis of the free transverse vibrations of a cantilevered plate is performed by means of a variational approximation procedure. Since the variational equation with all natural conditions is available for the three-dimensional elastic equations but not for the plate equation, the required variational equation for the plate is obtained by means of a low order expansion in the three-dimensional equation. The uncoupled flexural portion of the resulting system is used in the problem treated in this work. The problem is treated by first obtaining the exact solution for flexural waves in the thin plate by satisfying the flexural equation for the plate with two opposite edges free. The solution results in a set of dispersion curves. A number of the resulting waves are used in what remains of the variational equation, in which all conditions occur as natural conditions. Roots of the resulting transcendental equation are calculated, which yield the eigensolutions and associated eigenfrequencies. The results compare very well with those from finite element method treatment, which shows that this procedure is very accurate. It also provides an understanding in terms of the waves that make up the vibration, which is not provided by any of the other methods.