Forced Vibration of Damped Circular and Annular Membranes

Abstract

The response of circular and annular membranes to sinusoidal vibration is described theoretically. In contrast to prior investigations, emphasis is placed here on the forced vibration of membranes with damping, which is accounted for by writing the membrane tension as a complex quantity. The membranes, of outer radius a, are driven symmetrically as follows: by a ring force of arbitrary radius; by a force that is uniformly distributed within a concentric circle, also of arbitrary radius; or by a central “point” force. The point force can be visualized as the limiting case of the ring- or distributed-force excitation when the radius of application μa—μ being some numerical multiplier ⩽1.0—approaches zero. As this limit is approached, both the force transmitted to the membrane boundary and the transfer impedance become independent of the value of μ; however, the expressions for driving-point impedance approach a common value that is proportional to [loge μ]−1. Since, in practice, the impressed force cannot be applied at a point of zero radius, μ can realistically be assigned a small value (μ≠0) for which the logarithmic term remains finite and for which meaningful calculations of driving-point impedance can be made. In the example considered, the area over which the point force is applied is 10−6 times smaller than the membrane area (μ=0.001). The physical significance of graphical results that show the frequency dependence of impedance and transmissibility is discussed, and the effect of mass loading the membrane is described. Simple expressions are given from which the membrane damping factor can be deduced if the transmissibility across the membrane is measured at resonance.