Vibrations of Complete Hollow Spheres with Variable Thickness
Publication: Journal of Engineering Mechanics
Volume 141, Issue 9
Abstract
A three-dimensional (3D) method of analysis is presented for determining the free-vibration frequencies of complete hollow spherical shells of revolution with variable thickness. Unlike conventional shell theories, which are mathematically two-dimensional (2D), the present method is based on the 3D dynamic equations of elasticity. Displacement components , , and in the radial, circumferential, and axial directions, respectively, are taken to be periodic in and in time, and algebraic polynomials in the - and -directions. Potential (strain) and kinetic energies of the complete hollow spheres are formulated, and the Ritz method is used to solve the eigenvalue problem, thus yielding upper-bound values of the frequencies 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 complete hollow spheres. Comparisons are also made between the frequencies from the present 3D method, a 2D thin-shell theory, and two other 3D analyses.
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References
Buchanan, G. R., and Ramirez, G. R. (2002). “A note on the vibration of transversely isotropic solid spheres.” J. Sound Vib., 253(3), 724–732.
Buchanan, G. R., and Rich, B. S. (2002). “Effect of boundary conditions on free vibration of thick isotropic spherical shells.” J. Vib. Cont., 8(3), 389–403.
Chang, Y.-C., and Demkowicz, L. (1995). “Vibrations of a spherical shell, comparison of 3-D elasticity and Kirchhoff shell theory results.” Comput. Assist. Mech. Eng. Sci., 2(3), 187–206.
Chen, W. Q., and Ding, H. J. (2001). “Free vibration of multi-layered spherically isotropic hollow spheres.” Int. J. Mech. Sci., 43(3), 667–680.
Chen, W. Q., Ding, H. J., and Xu, R. Q. (2001). “Three-dimensional free vibration analysis of a fluid-filled piezoceramic hollow sphere.” Compos. Struct., 79(6), 653–663.
Chen, W. Q., Wang, L. Z., and Lu, Y. (2002). “Free vibrations of functionally graded piezoceramic hollow spheres with radial polarization.” J. Sound Vib., 251(1), 103–114.
Chiroiu, V., and Munteanu, L. (2007). “On the free vibrations of a piezoceramic hollow sphere.” Mech. Res. Commun., 34(2), 123–129.
Chree, C. (1889). “The equations of an isotropic elastic solid in polar and cylindrical coordinates, their solutions and application.” Trans. Cambridge Philos. Soc. Math. Phys. Sci., 14, 250–269.
Cohen, H., and Shah, A. H. (1972). “Free vibrations of a spherically isotropic hollow sphere.” Acustica, 26(6), 329–340.
Ding, H., and Chen, W. (1996). “Nonaxisymmetric free vibrations of a spherically isotropic spherical shell embedded in an elastic medium.” Int. J. Solids. Struct., 33(18), 2575–2590.
Grigorenko, Y. M., and Kilina, T. N. (1990). “Analysis of the frequencies and modes of natural vibration of laminated hollow sphere in two- and three-dimensional formulations.” Sov. Appl. Mech., 25(12), 1165–1171.
Hasheminejad, S. M., and Mirzaei, Y. (2011). “Exact 3d elasticity solution for free vibrations of an eccentric hollow sphere.” J. Sound Vib., 330(2), 229–244.
Jiang, H., Young, P. G., and Dickinson, S. M. (1996). “Natural frequencies of vibration of layered hollow spheres using exact three-dimensional elasticity equations.” J. Sound Vib., 195(1), 155–162.
Kang, J.-H., and Leissa, A. W. (2001). “Three-dimensional field equations of motion, and energy functionals for thick shells of revolution with arbitrary curvature and variable thickness.” J. Appl. Mech., 68(6), 953–954.
Kang, J.-H., and Leissa, A. W. (2004). “Three-dimensional vibration analysis of solid and hollow hemispheres having varying thickness with and without axial conical holes.” J. Vib. Control, 10(2), 199–214.
Kantorovich, L. V., and Krylov, V. I. (1958). Approximate methods in higher analysis, Noordhoff, Groningen, Netherlands.
Lamb, H. (1882). Proc. London Math. Soc., 13, 189–212.
Leissa, A. W. (2005). “The historical bases of the Rayleigh and Ritz method.” J. Sound Vib., 287(4–5), 961–978.
McGee, O. G., and Leissa, A. W. (1991). “Three-dimensional free vibrations of thick skewed cantilever plates.” J. Sound Vib., 144(2), 305–322. (errata (1991) 149(3), 539-542).
McGee, O. G., and Spry, S. C. (1997). “A three-dimensional analysis of the spheroidal and toroidal elastic vibrations of thick-walled spherical bodies of revolution.” Int. J. Numer. Methods Eng., 40(8), 1359–1382.
Poisson, S. D. (1829). “Mémoire sur l’équilibre et le mouvement des corps élastiques.” Mém. Acad. Sci., 8(5).
Qatu, M. S. (2002). “Recent research advances in the dynamic behavior of shells: 1989-2000, part 2: Homogeneous shells.” Appl. Mech. Rev., 55(5), 415–434.
Ritz, W. (1909). “Über eine neue methode zur lösung gewisser variationsprobleme der mathematischen physik.” J. Reine Angew. Math., 135, 1–61.
Sâto, Y., and Usami, T. (1962). “Basic study on the oscillation of a homogeneous elastic sphere, part II. Distribution of displacement.” Geophys. Mag., 31(3), 25–47.
Seyyed, M., Hasheminejad, S. M., and Mirzaei, Y. (2011). “Exact 3d elasticity solution for free vibrations of an eccentric hollow sphere.” J. Sound Vib., 330(2), 229–244.
Shah, A. H., Ramkrishnan, C. V., and Datta, S. K. (1969a). “Three-dimensional and shell-theory analysis of elastic waves in a hollow sphere, part I: Analytical foundation.” J. Appl. Mech., 36(3), 431–439.
Shah, A. H., Ramkrishnan, C. V., and Datta, S. K. (1969b). “Three-dimensional and shell-theory analysis of elastic waves in a hollow sphere, part 2: Numerical results.” J. Appl. Mech., 36(3), 440–444.
Shatalov, M. Y., Joubert, S. V., Coetzee, C. E., and Fedotov, I. A. (2009). “Free vibration of rotating hollow spheres containing acoustic media.” J. Sound Vib., 322(4–5), 1038–1047.
Singh, B. M., Rokne, J., and Dhaliwal, R. S. (2008). “Vibrations of a solid sphere or shell of functionally graded materials.” Eur. J. Mech. A. Solids, 27(3), 460–468.
Sokolnikoff, I. S. (1956). Mathematical theory of elasticity, 2nd Ed., McGraw-Hill, New York.
Wang, H. M. and Ding, H. J. (2007). “Radial vibration of piezoelectric/magnetostrictive composite hollow sphere.” J. Sound Vib., 307(1–2), 330–348.
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Published In
Journal of Engineering Mechanics
Volume 141 • Issue 9 • September 2015
Copyright
© 2015 American Society of Civil Engineers.
History
Received: May 15, 2014
Accepted: Jan 20, 2015
Published online: May 4, 2015
Published in print: Sep 1, 2015
Discussion open until: Oct 4, 2015
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