Chaotic non-planar vibrations of the thin elastica, Part II: Derivation and analysis of a low-dimensional model

J. P. Cusumano, F. C. Moon

Research output: Contribution to journalArticle

32 Citations (Scopus)

Abstract

Here, we develop a simple model for the bending-torsion vibrations of the thin elastica; the experimental observations were described in Part I. A geometrically exact rod theory is developed: Dimensional analysis demonstrates that a curvature constraint not used in previous analyses is appropriate in our case, and this is used to develop a coupled set of non-linear integro-differential equations for the problem. Using an additional simplifying assumption on the spatial derivative of the torsional field variable, a simplified set of partial differential equations is derived. It is shown that a two-mode projection of these model partial differential equations can be related to an intuitively appealing two-degree-of-freedom mechanical system. Numerical experiments on the two-mode model show that it captures much of the behavior observed in the physical experiments on the thin elastica. In particular, the model possesses a family of bending-torsion non-linear modes with a frequency-amplitude characteristic much like that found experimentally, and the driven problem loses planar stability in a fashion analogous to that observed with the elastica.

Original languageEnglish (US)
Pages (from-to)209-226
Number of pages18
JournalJournal of Sound and Vibration
Volume179
Issue number2
DOIs
StatePublished - Jan 12 1995

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derivation
vibration
partial differential equations
Torsional stress
Partial differential equations
torsion
Integrodifferential equations
dimensional analysis
differential equations
rods
degrees of freedom
projection
Experiments
curvature
Derivatives

All Science Journal Classification (ASJC) codes

  • Condensed Matter Physics
  • Mechanics of Materials
  • Acoustics and Ultrasonics
  • Mechanical Engineering

Cite this

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abstract = "Here, we develop a simple model for the bending-torsion vibrations of the thin elastica; the experimental observations were described in Part I. A geometrically exact rod theory is developed: Dimensional analysis demonstrates that a curvature constraint not used in previous analyses is appropriate in our case, and this is used to develop a coupled set of non-linear integro-differential equations for the problem. Using an additional simplifying assumption on the spatial derivative of the torsional field variable, a simplified set of partial differential equations is derived. It is shown that a two-mode projection of these model partial differential equations can be related to an intuitively appealing two-degree-of-freedom mechanical system. Numerical experiments on the two-mode model show that it captures much of the behavior observed in the physical experiments on the thin elastica. In particular, the model possesses a family of bending-torsion non-linear modes with a frequency-amplitude characteristic much like that found experimentally, and the driven problem loses planar stability in a fashion analogous to that observed with the elastica.",
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Chaotic non-planar vibrations of the thin elastica, Part II : Derivation and analysis of a low-dimensional model. / Cusumano, J. P.; Moon, F. C.

In: Journal of Sound and Vibration, Vol. 179, No. 2, 12.01.1995, p. 209-226.

Research output: Contribution to journalArticle

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