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Chaotic Motion in Mechanical and Engineering Systems

  • A. Steindl
  • H. Troger
Part of the International Centre for Mechanical Sciences book series (CISM, volume 319)

Abstract

We start with the definition of a chaotic process by relating the time evolution of a deterministic mechanical system which is governed by Newton’s laws to the stochastic sequence of heads and eagles following from the process of repeatedly tossing a fair coin. Then we give conditions which must be fulfilled by a mechanical system to meet this definition which basically requires the occurence of transversal homoclinic points and consequently results in the existence of the horseshoe and shift map.

In order to apply this approach to a mechanical system with finitely many or infinitely many degrees of freedom reductions of the dimension of the phase space in which the evolution of the system will be represented must be performed. Here the concepts of center manifolds and inertial manifolds are introduced. Further an extensive treatment of the Melnikov method and the explanation of a numerical method which both allow to establish the existence of transversal homoclinic points is given.

Applications are presented for simple mechanical systems like the pendulum with oscillating support, a satellite on an elliptic orbit, a two bar robot performing a prescribed endpoint motion and the clattering oscillations in a gear box. In addition also the planar, chaotic oscillations of a fluid conveying pipe are considered which are governed by an infinite dimensional dynamical system.

Keywords

Unstable Manifold Homoclinic Orbit Chaotic Motion Heteroclinic Orbit Galerkin Approximation 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag Wien 1991

Authors and Affiliations

  • A. Steindl
    • 1
  • H. Troger
    • 1
  1. 1.Technische Universität WienWienAustria

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