Use of trigonometric terms in the finite element method with application to vibrating membranes

Abstract

This paper deals with finite element solutions to the membrane eigenvalue problem. The transverse displacement of an arbitrary quadrilateral element is approximated by a four term polynomial plus arbitrary numbers of trigonometric terms that provide additional freedom to both the boundary and the interior of the element. The four corner point displacements, as well as the amplitudes of the trigonometric functions, are used as generalized coordinates, thus making it a simple matter to satisfy inter-element compatibility requirements. Results of natural frequency calculations made by using this element are given for membranes of various shapes, and comparisons are made with known solutions and with finite element solutions in which simple polynomial elements are used. It is found that the accuracy of the solutions obtained by using the trigonometric element is very good when compared to solutions from the polynomial elements. An advantage of the trigonometric elements is the way that elements of arbitrary order are easily formulated. With such elements it is usually necessary to subdivide a problem into elements only once. The number of system degrees of freedom may then be varied over a wide range simply by changing the number of trigonometric terms in the elements.