On the Vibration of a Circular Membrane with Added Mass

Abstract

The characteristic frequencies of a circular membrane carrying a rigidly attached concentric circular mass of finite area and vibrating symmetrically are studied. The main problem is that of determining the ranges of the parameters for which the frequencies will always be above or always be below those of the unloaded case. One of the chief tools is the minimum principle appropriate to the problem. The first term in the asymptotic expansion of the higher frequencies is obtained.

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.…”

A differential quadrature procedure for free vibration of circular membranes backed by a cylindrical fluid-filled cavity

“…In one of the earliest work, Ingard [3] derived an analytical solution in the form of an integral for the vibration problem of an ideal stretched circular membrane under a plane sound wave. The characteristic frequencies of a circular membrane carrying a rigidly attached concentric circular mass of finite area and vibrating symmetrically were studied by Cohen and Handelman [4]. Romily [5] proposed an exact solution, in the form of an infinite series, for the problem of transmission of an axially symmetric sound wave through an ideal stretched membrane in a rigid circular tube.…”