Forced vertical vibrations of an elastic elliptic plate on an elastic half space – a direct approach using orthogonal polynomials

The forced vibration of an elastic half space produced by a rigid elliptic indenter oscillating about an axis perpendicular to the plane face of the half space is considered. The boundary conditions lead to a two-dimensional dual integral equation in terms of the unknown normal stress. By appropriate substitution, the dual integral equation is ®rst reduced to a two-dimensional Fredholm integral equation. This is transformed to an in®nite set of equations using Abelian transformations. Next, the Abel-transformed variable of the unknown normal stress is expanded in terms of orthogonal Jacobi polynomials, and by solving the system of linear equations, orthogonal polynomial solutions are obtained. The method used to obtain the orthogonal polynomial solutions of this problem is new and the major advantage of this expansion technique is that it is valid for all frequencies. Detailed numerical work has been given for the total load on the disc for di€erent values of frequencies