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Use of Fractal Dimension in the Characterization of Chaotic Structural Dynamic Sytems

[+] Author and Article Information
E. Hall, S. Kessler, S. Hanagud

School of Aerospace Engineering, Georgia Institute of Technology, Atlanta GA 30332-0150

Appl. Mech. Rev 44(11S), S107-S113 (Nov 01, 1991) doi:10.1115/1.3121342 History: Online April 30, 2009

Abstract

The purpose of this paper is to investigate the use of fractal dimensions in the characterization of chaotic systems in structural dynamics. The investigation focuses on the example of a simply-supported, Euler-Bernoulli beam which when subjected to a transverse forcing function of a particular amplitude responds chaotically. Three different nonlinear models of the system are studied: a complex partial differential equation (PDE) model, a simplified PDE model, and a Galerkin approximation to the simpler PDE model. The responses of each model are examined through zero velocity Poincaré sections. To characterize and compare the chaotic trajectories, the box counting fractal dimension of the Poincaré sections are computed. The results demonstrate that the fractal dimension is a spatial invariant along the length of the beam for the specific class of forcing function studied, and thus it can be used to characterize chaotic motions. In addition, the three models yield different fractal dimensions for the same forcing which indicates that fractal dimensions can also be used to quantify whether a simplification of a chaotic model accurately predicts the chaotic behavior of the full-blown model. Thus the conclusion of the paper is that fractal dimensions may play an important role in the characterization of chaotic structural dynamic systems.

Copyright © 1991 by American Society of Mechanical Engineers
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