TY - JOUR

T1 - On the three-dimensional vibrations of a hollow elastic torus of annular cross-section

AU - Zhou, Ding

AU - Liu, Weiqing

AU - McGee, O. G.

N1 - Copyright:
Copyright 2011 Elsevier B.V., All rights reserved.

PY - 2011/4

Y1 - 2011/4

N2 - 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.

AB - 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.

KW - Elastic hollow torus (tori)

KW - Elastic ring

KW - Elastic toroid

KW - Horn, or spindle tori

KW - Ritz method

KW - Three-dimensional vibrations

KW - Toroidal coordinates

UR - http://www.scopus.com/inward/record.url?scp=79954437082&partnerID=8YFLogxK

U2 - 10.1007/s00419-010-0420-0

DO - 10.1007/s00419-010-0420-0

M3 - Article

AN - SCOPUS:79954437082

VL - 81

SP - 473

EP - 487

JO - Archive of Applied Mechanics

JF - Archive of Applied Mechanics

SN - 0939-1533

IS - 4

ER -