Exact solution for vibration analysis of an eccentric elliptical membrane

Abstract

The classical method of separation of variables in elliptical coordinates in conjunction with the translational addition theorems for Mathieu functions are used to investigate free transverse vibrations of an elastic membrane of elliptical planform with an arbitrarily located elliptical perforation. Subsequently, the elaborated method of eigenfunction expansion is employed to obtain an exact time‐domain series solution, in terms of products of angular and radial Mathieu functions, for the forced transverse oscillations of the eccentric membrane. The analytical solution is illustrated through numerical examples including circular/elliptical membranes with a circular perforation or with an elliptical perforation of selected geometric, orientation, and location parameters. The first five natural frequencies are tabulated, and selected vibration mode shapes are presented in graphical form. Also, the displacement responses of representative membranes in a practical loading configuration (i.e., a uniformly distributed step load) are calculated. The accuracy of solutions is ensured through proper convergence studies, and the validity of results is demonstrated with the aid of a commercial finite element package as well as by comparison with the existing data. The set of data reported herein is believed to be the first rigorous attempt on the free/forced vibrational characteristics of eccentric elliptical membranes for a wide range of geometric parameters.