Approximate Solution for Longitudinal Vibration of Non-Uniform Beams by Differential Transform Method (DTM)

Abstract

In this paper, differential transform method is applied for longitudinal vibration analysis of beams with non-uniform cross section. Natural frequencies and corresponding normalized mode shapes are calculated for different cases in cross section and boundary conditions. Comparison of results with the previous solutions proves the accuracy and versatility of the presented paper.     2 0 sin A xA a bx  . Li (4) combined the transfer matrix method and closed form solution of one step non-uniform rod to obtain a single frequency equation for a multi-step non-uniform rod. In another papers, he derived an exact solution for free longitudinal vibrations of one-step non-uniform rods with classical and non-classical boundary conditions (5) and an analytical solution for determine the longitudinal natural frequencies and mode shapes for a one step non-uniform bar with several boundary conditions (6). Li, et al. (7) found closed form solution for the longitudinal vibration of non-uniform structures with lumped masses and spring supports whose cross-section varies as   1 n