Frequency analysis of composite annular membranes

Abstract

In this paper, the free vibration of an annular membrane consisting of three con- centric segments is considered. The frequency equation and mode shapes are obtained by the use of the Green's function method. A numerical example to vibration problem of non- homogeneous annular membrane is presented. solution of the problem and vibration analysis of membranes with discontinuously varying thickness is presented. In reference (2) the finite element method in the analysis was also used. The free vibration problem of annular membrane with many discontinuous variation of the density is the subject of paper (3). Although the formulation of the problem deals with the vibration of membrane which consist of m segments (each of constant density), the numerical examples concern the antisymmetric modes of composite membrane consisting of two segments. In this case the frequency equation is obtained by setting the determinant of a 4x4 matrix of coefficients to a derived system of equation, which equals zero. In many papers various methods are applied to solution of the eigenvalue problems. The authors of paper (4) in solving the eigenproblem for annular membrane propose the method of fundamental solution. In this method the free space Green's function are ap- plied. In this paper the free vibration problem of a composite annular membrane con- sisting of three segments of constant densities is presented. The solution of the problem (frequency equation, mode shapes) is derived by using the properties of Green's functions corresponding to the Helmholtz operator in an annular domain. An example of numerical frequency analysis is given.