A New Basis Set For Computing The Resonant Frequencies Of Vibrating Membranes

Abstract

The collocation method for computing the resonant frequencies of membranes is investigated. The membrane displacement is written as a series expansion of solutions of Helmholtz's equation in cylindrical co-ordinates. The solution is approximated by truncating the series and the result is sampled at a finite number of points along the perimeter. The resonant frequencies are then found from the zeros of the determinant of the resulting system of equations. For geometries with minimal or near minimal boundaries, the series expansion converges in just a few terms and no serious numerical problems are encountered. However, for elongated or irregular geometries, a large number of terms in the expansion are required and the system determinant becomes unbalanced, resulting in numerical underflow. The problem is traced to the large order behavior of Bessel's function and is rectified by defining a set of subdomain basis functions with suitably normalized Bessel functions that are well behaved over their domain of definition.The proposed method is demonstrated by applying it to the computation of the resonant frequencies of a vibrating elliptical membrane clamped over its perimeter. It is, however, not restricted to this particular problem. The approach is completely general and can be applied to a wide variety of two-dimensional eigenvalue problems.