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Fractal Control Boundaries of Driven Oscillators and Their Relevance to Safe Engineering Design

  • J. M. T. Thompson
  • M. S. Soliman
Part of the International Centre for Mechanical Sciences book series (CISM, volume 319)

Abstract

When a metastable, damped oscillator is driven by strong periodic forcing, the catchment basin of constrained motions in the space of the starting conditions {x(0), (0)} develops a fractal boundary associated with a homoclinic tangling of the governing invariant manifolds. The four-dimensional basin in the phase-control space spanned by {x(0), (0), F,ω}, where F is the magnitude and ω the frequency of the excitation, will likewise acquire a fractal boundary, and we here explore the engineering significance of the control cross section corresponding, for example, to x(0) = (0) = 0. The fractal boundary in this section is a failure locus for a mechanical or electrical system subjected, while resting in its ambient equilibrium state, to a sudden pulse of excitation. We assess here the relative magnitude of the uncertainties implied by this fractal structure for the optimal escape from a universal cubic potential well. Absolute and transient basins are examined, giving control-space maps analogous to familiar pictures of the Mandelbrot set. Generalizing from this prototype study, it is argued that in engineering design, against boat capsize or earthquake damage, for example, a study of safe basins should augment, and perhaps entirely replace, conventional analysis of the steady-state attracting solutions.

Keywords

Fractal Dimension Fractal Boundary Control Space Basin Boundary Homoclinic Tangency 
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

  • J. M. T. Thompson
    • 1
  • M. S. Soliman
    • 1
  1. 1.University College LondonLondonUK

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