A plane wave basis method for the vibration analysis of membranes and plates

Abstract

A new boundary method for modelling structural vibrations, called the Plane Wave Basis Method, is developed to estimate the natural frequencies and mode shapes of membranes and plates with various boundary conditions. Since its formulation may be derived from the Indirect Boundary Element Method, this method is studied and applied to the vibration of arbitrary shaped membranes and clamped plates. Furthermore, a new boundary element technique that deals with equations of the type JCU = b{x, y) is presented. Based on the spatial Fourier transform, it may be used with any type of fundamental solution and does not need any domain integration. This approach has been applied to determine the forced response of membranes to surface waves. The alternative formulation using the plane wave basis method is based on use of the Trefftz functions or T-function. Thus, the Trefftz methodology is introduced and one of its application, called the Exterior Boundary Element Method or Modified Trefftz Method, is applied to the vibration of clamped membranes. In both cases, the plane wave basis formulation expresses the transverse displacement as a superposition of propagating waves and evanescent waves. This method is highly effective in simplifying the programming and reducing the computational expense. The vibration of clamped membranes and of square, triangular, trapezoidal, rhombic and elliptical plates with different boundary conditions such as clamped, simply supported, sliding clamped, point supported, free and combinations of the aforementioned, are analysed. In most of the cases, the results agree well with the exact values or the values which have been found so far by various other approximate methods. However, problems are encountered when dealing with free polygonal plates; it is thought that the reason for this is attributable to the corner points. Although several different models of corners were studied, none of them was found to be satisfactory.

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