Free vibration analysis of arbitrarily shaped membranes using the trigonometric p-version of the finite-element method

The trigonometric p-version of the finite-element method is formulated and applied to free-vibration analysis of arbitrarily shaped membranes. This is accomplished using a trigonometrically enriched curved quadrilateral element. New shape functions are expressed in terms of trigonometric sine functions. These functions are numerically more stable than orthogonal polynomials as the order is increased. In this method, large elements are used and curved segments are represented accurately by using the blending function method. The accuracy of the solution is sought by increasing the trigonometric order while fixing the mesh. Frequency values are first found for a closed sectorial membrane and comparisons are made with exact values. Frequency values are also found for an open sectorial membrane and comparisons are made with exact values and with values from linear and isoparametric quadratic quadrilateral finite-element solutions. Furthermore, highly accurate frequency values are found for a square membrane with a central circular opening. It is first shown that the method converges rapidly downwards as the trigonometric order is increased. It is also shown that, for equivalent numbers of d.o.f.s, the method gives a much higher accuracy than linear and isoparametric quadrilateral finite-element solutions.