On the Vibration of Mass-Loaded Membranes

Abstract

The conventional treatment of a vibrating membrane with a small mass load, originated by Rayleigh, leads to perturbed eigenfrequencies slightly lower than the unperturbed values and independent of the area of the load so long as it is small compared to the area of the membrane. It is shown that this result is incorrect because of an invalid initial assumption about the amplitude of vibration of the load. Actually the perturbed frequencies are slightly higher than the unperturbed values, depend upon the area of the load, and approach the unperturbed values as the load approaches point mass. In addition a very low frequency mode exists whose frequency approaches zero as the load becomes smaller in area; this mode represents a perturbation of the trivial zero-frequency solution in the unperturbed case. The special case of a circular membrane loaded at the center is discussed in some detail.

Recent work on musical acoustics

“…Morse 1948). Kornhauser and Mintzer (1953) showed that a mass added to the membrane raised the frequencies of the normal modes of vibration and not the converse, as suggested by Rayleigh.…”

Acoustic Insulation Mechanism of Membrane-Type Acoustic Metamaterials Loaded with Arbitrarily Shaped Mass Blocks of Variable Surface Density

“…It is easy to see from Figure 11a that, when the mass block was placed eccentrically, its STL curve was shifted, but the magnitude of the main peaks and valleys (the peaks and valleys produced when the mass is coincident with the film center of mass) and the frequency bands generated did not change much; when the mass block was placed too close to the film boundary, its STL was significantly shifted. Whereas this differs from the behavior of the cylindrical mass block, which produces more peaks and valleys when eccentric [7][8][9][12][13][14][15][16][17][18][19][20], these additional sound insulation summits affected the frequency band and size of the main peaks. It was shown that the hexagonal cone mass block had a more stable sound insulation effect in the lower frequency band.…”

“…As a result, researchers have begun to investigate the STL theoretical aspects from membrane-type metamaterials. Firstly, Kornhauser and Mintzer [13] and Cohen and Handelman [14] analyzed the STL effect of a circular mass block when it is concentric with a circular film by studying the intrinsic frequency and vibration pattern of the circular film. Zhang et al [15] proposed an analytical method in which the solid mass is located in the center of the film through the Galerkin method.…”

Archives of Acoustics

“…Efficient theoretical models to estimate the transmission loss of MAMs are demanded on account of the high potential of MAMs in low frequency sound insulation. For membranes loaded with a rigid mass, some of the first theoretical studies were carried out (Kornhauser et al, 1953;Cohen et al, 1957), in which the influence of a concentric circular mass on the natural frequencies of a circular membrane has been investigated. Further theoretically analyses for this case was done, the natural frequencies and mode shapes are given for the concentrically loaded circular membrane (Wang, 2003) analytical model of a rectangular mass loaded a rectangular membrane specifically tailored for the application to MAMs, the forced vibration behaviour of membrane loaded with a mass was obtained by employing Galerkin method.…”