Treatments of over-restrained boundaries for doubly connected plates of arbitrary shape in vibration analysis

The problem of the free flexural vibrations of plates of doubly connected arbitrary shape is investigated. A hybrid energy approach which combines both the advantages of the Rayleigh-Ritz method and the Lagrange multiplier method is developed for analysing the aforementioned plate problems. The Rayleigh-Ritz method with a set of pb-2 shape functions is used to formulate the plates with classical boundary conditions as a single continuum element, while the Lagrange multipliers are imposed at discrete points in the inner plate boundaries, for instance, when the overrestrained boundary has occurred. The admissible pb-2 shape function consists of the product of a two-dimensional polynomial and a basic function. The basic function is defined by the product of the equations of the prescribed piecewise continuous boundary shape each raised to the power of 0, 1 or 2, corresponding respectively to a free, simply-supported or clamped edge. This set of functions automatically satisfies all the kinematic boundary conditions of the plate at the outset. Numerical results for several examples of doubly connected plates of arbitrary shape are presented to demonstrate the applicability and accuracy of the present method. © 1992.