Spectral analysis of vibrating plates with general shape

Abstract

Lightweight plate structures are widely used in many engineering and practical applications. The analysis and design of such structures call for efficient computational tools, since exact analytical solutions for vibrating plates are currently known only for some standard shapes in conjunction with a few basic boundary conditions. The present paper deals with the adoption of a set of eigenfunctions evaluated from a simple structure as a basis for the analysis of plates with both general shape and general boundary conditions in the Rayleigh-Ritz condensation method. General boundary conditions are introduced in the functional of the potential energy by additional terms, and both trigonometric and polynomial interpolation functions are implemented for mapping the shape of the plate in Cartesian coordinates into natural coordinates. Flexural free vibration analysis of different shaped plates is then performed: skew, trapezoid and triangular plates, plates with parabolic edges, elliptic sector and annular plates. The proposed method can also be directly applied to variable thickness plates and non-homogeneous plates, with variable density and stiffness. Purely elastic plates are considered; however the method may also be applied to the analysis of viscoelastic plates. The results are compared to those available in the literature and using standard finite element analysis.