Closed-form dynamic stiffness formulations for exact modal analysis of membranes in polar coordinates

Abstract

Exact dynamic stiffness (DS) formulations for free vibration of membranes and their assemblies in polar coordinates are presented in this paper. First, the exact general solutions of the governing differential equations of membranes in polar coordinates are derived, in the form of Bessel functions. Then, by substituting the general solutions into the displacement and force boundary conditions (BCs), the frequency-dependent elemental DS matrices of various shaped membranes are developed. Finally, assembling the elemental stiffness matrices and applying BCs leads to the final DS matrix. By utilizing the Wittrick–Williams (W–W) algorithm, the natural frequencies and mode shapes are computed. The difficult problem in applying the W–W algorithm, the so called j0 count, is resolved by determining the zeros of Bessel equation and applying an indirect method. The proposed method is applied to representative examples of individual membranes and their assemblies under arbitrary BCs. The efficiency and accuracy of the dynamic stiffness method (DSM) are demonstrated by comparing results by the DSM with those by the finite element method (FEM). The proposed method can serve as an exact but efficient tool for modal analysis of membranes in structural dynamics as well as fluid–structure interaction problems.