Natural frequencies of membranes having mixed boundary conditions

Abstract

A number of approximate methods have been given for determining the natural frequencies of membranes of arbitrary shape and uniform boundary conditions. In this paper, a method is presented whereby natural frequencies can be determined for membranes of arbitrary shapes with a portion of the boundary clamped (no displacement) and a portion free (vanishing slope). The method is developed through expanding the displacements in a series of functions, each of which satisfies the governing differential equation, but not the mixed boundary condition. A variational principle is used to find those expansion coefficients which give the best approximation to the desired boundary condition. The required variational principle is developed from Hamilton's principle through the same relaxation of the class of admissible functions used to develop Reissner's principle from the Theorem of Minimum Potential Energy. Results for a rectangular membrane, clamped on three edges, with the fourth edge partially clamped and partially free, are given, and are compared with results obtained for the same geometry through a collocation procedure and through a finite element analysis.