Arbitrary precision frequencies of a free rectangular thin plate

Abstract

A variational method developed by the authors and named whole element method (WEM) is used to find the arbitrary precision frequencies of a rectangular thin plate (within the Germain}Lagrange theory) having its four borders free of constraint. WEM consists is proposing an adequate functional and a sequence representing the plate transversal displacement w(x, y). Such a sequence is made of a linear combination of functions belonging to a complete set in L2. The sequence, and not each co-ordinate function, is required to satisfy the essential or geometric conditions. The sequence generation is systematic and no analysis of the classical natural modes of the plate is needed. In particular, trigonometric functions which a priori belong to a complete set in the domain are used in the present analysis. The solving equations involving very simple sums arise from the minimization of the functional. WEM is based on theorems which show the ultimate exactness of the eigenvalues and the uniform convergence of the essential functions of the problem. To the authors knowledge this problem has no classical solution.