Analytical Investigation on the Vibrations of Clamped Plates of Complicated Shape

Abstract

The problem of free transverse vibrations of plates of rectangular, circular, and elliptical shape is discussed in standard textbooks. In the case of plates of more complicated shape, it would be necessary to find first the natural system of coordinates. Even with the natural coordinate system known, it is very probable that the method of separation of variables would not be applicable for the solution of the partial differential equation. It is shown in this study that it may be advantageous to conformally transform the given shape onto a unit circle. Several numerical techniques are available for the determination of the mapping function. The boundary conditions become, for a clamped plate, u(r,θ,t)|r=1=0, (∂u/∂r) (r,θt)|r=1=0. A closed-form solution of the transformed differential equation seems unlikely. Galerkin's method is used for the determination of the fundamental frequency. The method is illustrated in the case of a clamped circular plate with two flat sides. [Research sponsored by the U. S. Office of Naval Research.]