VIBRATION ANALYSIS OF ARBITRARILY SHAPED MEMBRANES USING NON-DIMENSIONAL DYNAMIC INFLUENCE FUNCTION

Abstract

In the present study, a theoretical formulation based on the collocation method is presented for the vibration analysis of arbitrarily shaped membranes. The mathematical relation between the two points of selected collocation points is given by a special function, the so-called non-dimensional dynamic influence function. Unlike the collocation methods in the literature, approximate functions used in this paper are simple, one-dimensional functions of which the only independent variable is the distance between the two points. The function is also a wave-type solution that satisfies exactly the given governing differential equation and physically describes the displacement reponse of a point in an infinite membrane due to a unit displacement excited at another point. The approximate solution is obtained by linear superposition of non-dimensional dynamic influence functions, and then boundary conditions are applied at the discrete points. The system matrix is always symmetric regardless of the boundary shape of the membrane, and the calculated eigenvalues rapidly converge to the exact values thanks to the special function employed in this study. Moreover, the method gives the associated mode shapes successfully without using interpolation functions between the boundary nodes. The validity and efficiency of the method proposed in this paper are illustrated through several numerical examples.