Liapunov exponents used to study chaotic vibration of beam and plates
by
Daniela Baran
Elie Carafoli National Institute for Aerospace Research, Bucharest, Romania

The great developing of numerical analysis of the dynamic systems emphasizes the existence of a strong dependence of the initial conditions, described in the phase plane by attractors with a complicated geometrical structure. The complexity of the geometrical structures of such attractors leads to the notion of strange attractor. Strange attractors are defined in many ways but we adopt here the definition of Holmes and Guckenheimer, an attractor is strange if it contains a homoclinical transversal orbit. The Liapunov exponents are used to determine if there is a real strong dependence on the initial conditions: there is at least a positive exponent if the system has a chaotic evolution and all the Liapunov exponents are negative if the system has not such an evolution. To draw these conclusions is sufficient to determine the largest Liapunov exponent, which is easier to calculate. In this paper we shall use the greatest Liapunov exponent to study two well-known problems who leads to chaotic motions: the problem of the buckled beam and the panel flutter problem. In the problem of the buckled beam we verify the results obtained with the Melnikov theorem with the maximum Liapunov exponent [1]. The flutter of a buckled plate is also a problem characterized by strong dependence of the initial conditions, existence of attractors with complicated structure existence of periodic unstable motions with very long periods (sometimes infinite periods). Several authors, including Holmes, Holmes and Marsden, Dowell have analyzed this system from different points of view. Using the largest Liapunov exponent method we obtain the results obtained by Dowell [2].

Date received: January 21, 2004

Copyright © 2004 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cakt-30.