Bounds on natural frequencies of composite circular plates: Integral equation methods

The problem of finding upper and lower bounds to the natural frequencies of free vibration of a circular plate with stepped radial density has been investigated. Such problems, involving discontinuous coefficients in the governing differential equation, are formulated with an integral equation by using a Green function and the basic theory of linear integral equations. It is shown that eigenvalue estimation techniques based on the integral equation formulation are more effective than those based on the traditional differential equation formulation. By using the Galerkin method, the integral equation formulation is shown to provide more accurate upper bounds than the differential equation formulation for both the clamped and simply supported plates. Additionally, it is shown that the corresponding Green functions can be used to supply improvable lower bounds.