Linearly elastic annular and circular membranes under radial, transverse, and torsional loading. Part II: Vibrations about deformed equilibria

Abstract

Equilibrium configurations of annular and circular membranes involving large displacements were determined in Part I of this study. Now small vibrations about such equilibrium states are analyzed for annular membranes. The membrane is linearly elastic, initially flat, and taut. Foppl-von Karman theory is used to obtain the linearized vibration equations. The problems considered include inward and outward radial stretching, transverse displacement at the inner edge, transverse pressure, a vertical distributed load with a vertically sliding outer membrane edge, torsion at the inner edge along with outward stretching, and in-plane vibrations for cases having a flat equilibrium shape. In many of the cases the transverse motion is coupled with radial and circumferential motions. A shooting method is used to obtain vibration frequencies and corresponding vibration modes with different numbers of nodal diameters and nodal circles. The effects of Poisson’s ratio, in-plane radial and circumferential inertias, the ratio of the radii of the inner and outer edges, and the loading magnitude on the vibration frequencies are investigated.