On the three-dimensional vibrations of a hollow elastic torus of annular cross-section

Ding Zhou, Weiqing Liu, O. G. McGee

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This paper offers a three-dimensional elasticity-based variational Ritz procedure to examine the natural vibrations of an elastic hollow torus with annular cross-section. The associated energy functional minimized in the Ritz procedure is formulated using toroidal coordinates r,θ comprised of the usual polar coordinates (r, θ) originating at each circular cross-sectional center and a circumferential coordinate φ around the torus originating at the torus center. As an enhancement to conventional use of algebraic-trigonometric polynomials trial series in related solid body vibration studies in the associated literature, the assumed torus displacement, u, v and w in the r, θ and φ toroidal directions, respectively, are approximated in this work as a triplicate product of Chebyshev polynomials in r and the periodic trigonometric functions in the θ and φ directions along with a set of generalized coefficients. Upon invoking the stationary condition of the Lagrangian energy functional for the elastic torus with respected to these generalized coefficients, the usual characteristic frequency equations of natural vibrations of the elastic torus are derived. Upper bound convergence of the first seven non-dimensional frequency parameters accurate to at least five significant figures is achieved by using only ten terms of the trial torus displacement functions. Non-dimensional frequencies of elastic hollow tori are examined showing the effects of varying torus radius ratio and cross-sectional radius ratio.

Original languageEnglish
Pages (from-to)473-487
Number of pages15
JournalArchive of Applied Mechanics
Issue number4
StatePublished - Apr 2011


  • Elastic hollow torus (tori)
  • Elastic ring
  • Elastic toroid
  • Horn, or spindle tori
  • Ritz method
  • Three-dimensional vibrations
  • Toroidal coordinates


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