Chaotic Vibrations of Beams: Numerical Solution of Partial Differential Equations

[+] Author and Article Information
N. S. Abhyankar

Department of Aerospace and Mechanical Engineering, Syracuse University, Syracuse, NY 13244

E. K. Hall, S. V. Hanagud

School of Aerospace Engineering, Georgia Institute of Technology, Atlanta, GA 30332

J. Appl. Mech 60(1), 167-174 (Mar 01, 1993) (8 pages) doi:10.1115/1.2900741 History: Received December 13, 1988; Revised March 07, 1991; Online March 31, 2008


The objective of this paper is to examine the utility of direct, numerical solution procedures, such as finite difference or finite element methods, for partial differential equations in chaotic dynamics. In the past, procedures for solving such equations to detect chaos have utilized Galerkin approximations which reduce the partial differential equations to a set of truncated, nonlinear ordinary differential equations. This paper will demonstrate that a finite difference solution is equivalent to a Galerkin solution, and that the finite difference method is more powerful in that it may be applied to problems for which the Galerkin approximations would be difficult, if not impossible to use. In particular, a nonlinear partial differential equation which models a slender, Euler-Bernoulli beam in compression issolvedto investigate chaotic motions under periodic transverse forcing. The equation, cast as a system of firstorder partial differential equations is directly solved by an explicit finite difference scheme. The numerical solutions are shown to be the same as the solutions of an ordinary differential equation approximating the first mode vibration of the buckled beam. Then rigid stops of finite length are incorporated into the model to demonstrate a problem in which the Galerkin procedure is not applicable. The finite difference method, however, can be used to study the stop problem with prescribed restrictions over a selected subdomain of the beam. Results obtained are briefly discussed. The end result is a more general solution technique applicable to problems in chaotic dynamics.

Copyright © 1993 by The American Society of Mechanical Engineers
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