Multi-scale continuum mechanics: From global bifurcations to noise induced high-dimensional chaos

Ira B. Schwartz, David S. Morgan, Lora Billings, Ying-Cheng Lai

Research output: Contribution to journalArticle

9 Citations (Scopus)

Abstract

Many mechanical systems consist of continuum mechanical structures, having either linear or nonlinear elasticity or geometry, coupled to nonlinear oscillators. In this paper, we consider the class of linear continua coupled to mechanical pendula. In such mechanical systems, there often exist several natural time scales determined by the physics of the problem. Using a time scale splitting, we analyze a prototypical structural-mechanical system consisting of a planar nonlinear pendulum coupled to a flexible rod made of linear viscoelastic material. In this system both low-dimensional and high-dimensional chaos is observed. The low-dimensional chaos appears in the limit of small coupling between the continua and oscillator, where the natural frequency of the primary mode of the rod is much greater than the natural frequency of the pendulum. In this case, the motion resides on a slow manifold. As the coupling is increased, global motion moves off of the slow manifold and high-dimensional chaos is observed. We present a numerical bifurcation analysis of the resulting system illustrating the mechanism for the onset of high-dimensional chaos. Constrained invariant sets are computed to reveal a process from low-dimensional to high-dimensional transitions. Applications will be to both deterministic and stochastic bifurcations. Practical implications of the bifurcation from low-dimensional to high-dimensional chaos for detection of damage as well as global effects of noise will also be discussed.

Original languageEnglish (US)
Pages (from-to)373-386
Number of pages14
JournalChaos
Volume14
Issue number2
DOIs
StatePublished - Jun 2004

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continuum mechanics
Continuum mechanics
Global Bifurcation
Continuum Mechanics
Chaos theory
chaos
Chaos
High-dimensional
Mechanical Systems
Slow Manifold
Continuum
pendulums
Pendulum
Pendulums
continuums
Natural Frequency
resonant frequencies
Natural frequencies
Time Scales
rods

ASJC Scopus subject areas

  • Applied Mathematics
  • Physics and Astronomy(all)
  • Statistical and Nonlinear Physics
  • Mathematical Physics

Cite this

Multi-scale continuum mechanics : From global bifurcations to noise induced high-dimensional chaos. / Schwartz, Ira B.; Morgan, David S.; Billings, Lora; Lai, Ying-Cheng.

In: Chaos, Vol. 14, No. 2, 06.2004, p. 373-386.

Research output: Contribution to journalArticle

Schwartz, Ira B. ; Morgan, David S. ; Billings, Lora ; Lai, Ying-Cheng. / Multi-scale continuum mechanics : From global bifurcations to noise induced high-dimensional chaos. In: Chaos. 2004 ; Vol. 14, No. 2. pp. 373-386.
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