Vibration Analysis of Toroidal Shells with Hollow Circular Cross-Section Having Variable Thickness
Publication: Journal of Engineering Mechanics
Volume 142, Issue 9
Abstract
A three-dimensional (3D) Ritz method of analysis is presented for determining the free vibration frequencies of completely free, toroidal shells of revolution with hollow circular cross-section having variable thickness. Displacement components , , and in the radial, circumferential, and axial directions, respectively, are taken to be sinusoidal in time, periodic in , and ordinary algebraic polynomials in the and directions. Strain and kinetic energies of the torus are formulated, and upper bound values of the frequencies are obtained by minimizing the frequencies. As the degree of the polynomials is increased, frequencies converge to the exact values. Convergence to four-digit exactitude is demonstrated for the first five frequencies of the torus. Comparisons are made between the frequencies from the present 3D Ritz method, a 3D finite-element method, experimental methods, and thin and thick ring theories.
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References
Baleres, T., and Armenakas, A. E. (1973). “Free vibrations of ring-stiffened toroidal shells.” Am. Inst. Aero. Astro. J., 11(12), 1637–1644.
Buchanan, G. R., and Liu, Y. J. (2005). “An analysis of the free vibration of thick-walled isotropic toroidal shells.” Int. J. Mech. Sci., 47(2), 277–292.
Cowper, G. R. (1966). “The shear coefficient in Timoshenko’s beam theory.” J. Appl. Mech., 33(2), 335–340.
Fang, Z. (1992). “Free vibration of fluid-filled toroidal shells.” J. Sound Vib., 155(2), 343–352.
Galletly, G. D. (1998). “Elastic buckling of complete toroidal shells of elliptical cross-section subjected to uniform internal pressure.” Thin-Walled Struct., 30(1–4), 23–34.
Hoppe, R. (1871). “Vibration Eines Ringes in seiner Ebene.” Journal für die reine und angewandte Mathematik, 73, 158–170.
Jha, A. K., Inman, D. J., and Plaut, R. H. (2002). “Free vibration analysis of an inflated toroidal shell.” J. Vib. Acoust., 124(3), 387–396.
Jiang, W., and Redekop, D. (2002). “Polar axisymmetric vibration of a hollow toroid using the differential quadrature method.” J. Sound Vib., 251(4), 761–765.
Kantorovich, L. V., and Krylov, V. I. (1958). Approximate methods in higher analysis, Noordhoff, Groningen, Netherlands, 266–268.
Kirkhope, J. (1976). “Out-of-plane vibration of thick circular ring.” J. Eng. Mech., 102, 239–247.
Kirkhope, J. (1977). “In-plane vibration of a thick circular ring.” J. Sound Vib., 50(2), 219–227.
Kosawada, T., Suzuki, K., and Takahashi, S. (1985). “Free vibrations of toroidal shells.” Trans. Jpn. Soc. Mech. Eng., 51(461), 8–16 (in Japanese).
Kosawada, T., Suzuki, K., and Takahashi, S. (1986). “Free vibrations of thick toroidal shells.” Trans. Jpn. Soc. Mech. Eng., 29(255), 3036–3042 (in Japanese).
Leissa, A. W. (1993). Vibration of shells, Acoustical Society of America, Melville, NY.
Leung, A. Y. T., and Kwok, N. C. T. (1994). “Free vibration analysis of a toroidal shell.” Thin-Walled Struct., 18(4), 317–332.
Lincoln, J. W., and Volterra, E. (1967). “Experimental and theoretical determination of frequencies of elastic toroids.” Exp. Mech., 7(5), 211–217.
Love, A. E. H. (1934). Treatise on the mathematical theory of elasticity, Cambridge University Press, Cambridge, U.K.
McGee, O. G., and Leissa, A. W. (1991). “Three-dimensional free vibrations of thick skewed cantilever plates.” J. Sound Vib., 144(2), 305–322.
Ming, R. S., Pan, J., and Norton, M. P. (2002). “Free vibrations of elastic circular toroidal shells.” Appl. Acoust., 63(5), 513–528.
Seely, F. B., and Smith, J. O. (1952). Advanced mechanics of materials, 2nd Ed., Wiley, New York.
Sokolnikoff, I. S. (1956). Mathematical theory of elasticity, 2nd Ed., McGraw-Hill, New York.
Tzou, H. S., and Wang, D. W. (2002). “Micro-sensing characteristics and modal voltages of linear/non-linear toroidal shells.” J. Sound Vib., 254(2), 203–218.
Volterra, E. (1967). “Vibrations of circular elastic rings.” Israel J. Tech., 5(4), 225–233.
Yamada, G., Kobayashi, Y., Ohta, Y., and Yokota, S. (1989). “Free vibration of a toroidal shell with elliptical cross-section.” J. Sound Vib., 135(3), 411–425.
Ye, T., Jin, G., and Zhang, Y. (2015). “Vibrations of composite laminated doubly-curved shells of revolution with elastic restraints including shear deformation, rotary inertia and initial curvature.” Comp. Struct., 133, 202–225.
Zhang, F., and Redekop, D. (1992). “Surface loading of a thin-walled toroidal shell.” Comp. Struct., 43(6), 1019–1104.
Zhou, D., Au, F. T. K., Lo, S. H., and Cheung, Y. K. (2002). “Three-dimensional vibration analysis of a torus with circular cross section.” J. Acoust. Soc. Am., 112(6), 2831–2839.
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© 2016 American Society of Civil Engineers.
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Received: May 21, 2015
Accepted: Mar 22, 2016
Published online: May 9, 2016
Published in print: Sep 1, 2016
Discussion open until: Oct 9, 2016
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