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Journal of Elasticity

, Volume 115, Issue 2, pp 193–224 | Cite as

Effect of the Boundary Conditions and Influence of the Rotational Inertia on the Vibrational Modes of an Elastic Ring

  • Nicolas Clauvelin
  • Wilma K. Olson
  • Irwin Tobias
Article

Abstract

We present the small-amplitude vibrations of a circular elastic ring with periodic and clamped boundary conditions. We model the rod as an inextensible, isotropic, naturally straight Kirchhoff elastic rod and obtain the vibrational modes of the ring analytically for periodic boundary conditions and numerically for clamped boundary conditions. Of particular interest are the dependence of the vibrational modes on the torsional stress in the ring and the influence of the rotational inertia of the rod on the mode frequencies and amplitudes. In rescaling the Kirchhoff equations, we introduce a parameter inversely proportional to the aspect ratio of the rod. This parameter makes it possible to capture the influence of the rotational inertia of the rod. We find that the rotational inertia has a minor influence on the vibrational modes with the exception of a specific category of modes corresponding to high-frequency twisting deformations in the ring. Moreover, some of the vibrational modes over or undertwist the elastic rod depending on the imposed torsional stress in the ring.

Keywords

Elastic rods Vibrations Topology Boundary-value problem Numerical continuation 

Mathematics Subject Classification (2010)

74K10 34B15 65L10 

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Copyright information

© Springer Science+Business Media Dordrecht 2013

Authors and Affiliations

  • Nicolas Clauvelin
    • 1
  • Wilma K. Olson
    • 2
  • Irwin Tobias
    • 3
  1. 1.BioMaPS Institute for Quantitative BiologyRutgers, the State University of New JerseyPiscatawayUSA
  2. 2.BioMaPS Institute for Quantitative Biology and Department of Chemistry and Chemical BiologyRutgers, the State University of New JerseyPiscatawayUSA
  3. 3.Department of Chemistry and Chemical BiologyRutgers, the State University of New JerseyPiscatawayUSA

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