Exact and approximate dynamic analysis of circular arches using DQM

A modified version of the differential quadrature method is applied to two versions of the sixth-order differential equation of motion governing free in-plane inextensional vibrations of circular arches (see Henrych, 1981). All the boundary conditions can be imposed exactly, without introducing delta points (see e.g. Bert and Malik, 1996). Consequently. the results are calculated with high precision, and a comparison between exact and approximate frequencies becomes possible. The convergence rate of the discretization method is shown to be very fast, even for the higher eigenvalues, so that a small number of Lagrangian coordinates permits a good approximation to the true results. It is shown that the approximate formulation leads to noticeable errors for the first frequencies of deep arches, whereas shallow arches and higher-order frequencies can be safely calculated with the simplified approach. The paper ends with some tables in which the first ten free vibrations frequencies for clamped arches, two-hinged arches and cantilever arches are compared with some known results from the literature. (C) 1999 Elsevier Science Ltd. All rights reserved.