Vibration of Continuous Systems, 2nd Edition

13Vibration of Membranes

13.1 INTRODUCTION

A membrane is a perfectly flexible thin plate or lamina that is subjected to tension. It has negligible resistance to shear or bending forces, and the restoring forces arise exclusively from the in‐plane stretching or tensile forces. The drumhead and diaphragms of condenser microphones are examples of membranes.

13.2 EQUATION OF MOTION

13.2.1 Equilibrium Approach

Consider a homogeneous and perfectly flexible membrane bounded by a plane curve C in the xy plane in the undeformed state. It is subjected to a pressure loading of intensity f(x, y, t) per unit area in the transverse or z direction and tension of magnitude P per unit length along the edge as in the case of a drumhead. Each point of the membrane is assumed to move only in the z direction, and the displacement, w(x, y, t), is assumed to be very small compared to the dimensions of the membrane. Consider an elemental area of the membrane, dx dy, with tensile forces of magnitude P dx and P dy acting on the sides parallel to the x and y axes, respectively, as shown in Fig. 13.1. After deformation, the net forces acting on the element of the membrane along the z direction due to the forces P dx and P dy will be (see Fig. 13.1(d))

Figure 13.1 (a) Undeformed membrane in the