caa a22 4500 445812095 CHVBK 20180317145215.0 cr unu---uuuuu 170323e20110401xx s 000 0 eng 10.1007/s00419-010-0420-0 doi (NATIONALLICENCE)springer-10.1007/s00419-010-0420-0 On the three-dimensional vibrations of a hollow elastic torus of annular cross-section [Elektronische Daten] [Ding Zhou, Weiqing Liu, O. McGee III] 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 $$({r,\theta , \varphi})$$ comprised of the usual polar coordinates (r, θ) originating at each circular cross-sectional center and a circumferential coordinate $${\varphi}$$ 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 $${\varphi }$$ 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 $${\varphi}$$ 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. Springer-Verlag, 2010 Three-dimensional vibrations nationallicence Ritz method nationallicence Elastic hollow torus (tori) nationallicence Elastic ring nationallicence Horn, or spindle tori nationallicence Elastic toroid nationallicence Toroidal coordinates nationallicence Zhou Ding College of Civil Engineering, Nanjing University of Technology, 210009, Nanjing, People's Republic of China aut Liu Weiqing College of Civil Engineering, Nanjing University of Technology, 210009, Nanjing, People's Republic of China aut McGee III O. Department of Mechanical Engineering, Howard University, Washington, DC, USA aut Archive of Applied Mechanics Springer-Verlag 81/4(2011-04-01), 473-487 0939-1533 81:4<473 2011 81 419 https://doi.org/10.1007/s00419-010-0420-0 text/html Onlinezugriff via DOI 1 research-article jats NATIONALLICENCE 856 40 https://doi.org/10.1007/s00419-010-0420-0 text/html Onlinezugriff via DOI NATIONALLICENCE 700 1- Zhou Ding College of Civil Engineering, Nanjing University of Technology, 210009, Nanjing, People's Republic of China aut NATIONALLICENCE 700 1- Liu Weiqing College of Civil Engineering, Nanjing University of Technology, 210009, Nanjing, People's Republic of China aut NATIONALLICENCE 700 1- McGee III O. Department of Mechanical Engineering, Howard University, Washington, DC, USA aut NATIONALLICENCE 773 0- Archive of Applied Mechanics Springer-Verlag 81/4(2011-04-01), 473-487 0939-1533 81:4<473 2011 81 419 Metadata rights reserved Springer special CC-BY-NC licence nationallicence BK010053 XK010053 XK010000 NATIONALLICENCE NATIONALLICENCE NL-springer