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Free vibration analysis of shallow and deep ellipsoidal shells having variable thickness with and without a top opening

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Abstract

A three-dimensional (3-D) method of analysis is presented for determining the natural frequencies and the mode shapes of hemi-ellipsoidal domes having non-uniform thickness with and without a top opening by the Ritz method. Instead of mathematically two-dimensional (2-D) conventional thin shell theories or higher-order shell theories, the present method is based upon the 3-D dynamic equations of elasticity by the Ritz method. Mathematically minimal or orthonormal Legendre polynomials are used as admissible functions in place of ordinary simple algebraic polynomials which are usually applied in the Ritz method. The analysis is based upon the circular cylindrical coordinates instead of the shell coordinates which are normal and tangential to the shell mid-surface. Potential (strain) and kinetic energies of the hemi-ellipsoidal dome having variable thickness with and without a top opening 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 Legendre polynomials is increased, the frequencies converge to the exact values. Convergence to four-digit exactitude is demonstrated for the first five frequencies. Numerical results are presented for a variety of shallow and deep hemi-ellipsoidal domes having variable thickness of five values of aspect ratios with and without a top opening, which are completely free and fixed at the bottom. The frequencies from the present 3-D analysis are compared with those from other 3-D analysis and a 2-D thin shell theory.

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References

  1. Leissa, A.W.: Vibration of shells. Washington D.C.: U.S. Government Printing Office 1973 (reprinted by The Acoustical Society of America 1993)

  2. DiMaggio, F.L., Silbiger, A.: Free extensional torsional vibrations of a prolate ellipsoidal shell. Contract Nonr 266(67), Proj 385–414, Tech Rept No 2, Office of Naval Research (1960)

  3. Silbiger, A., DiMaggio, F.L.: Extensional axisymmetric second class vibrations of a prolate ellipsoidal shell. Contract Nonr 266(67), Proj 385-414, Tech Rept No 3, Office of Naval Research (1961)

  4. DiMaggio, F.L., Silbiger, A.: Free extensional torsional vibrations of a prolate ellipsoidal shell. J. Acoust. Soc. Am. 33(1), 56–58 (1961)

    Article  Google Scholar 

  5. Shiraishi, N., DiMaggio, F.L.: Perturbation solution for the axisymmetric vibrations of prolate ellipsoidal shells. J. Acoust. Soc. Am. 34(11), 1725–1731 (1962)

    Article  Google Scholar 

  6. Gontkevich, V.S.: Natural Vibrations of Shells in a Liquid. Naukova Dumka, Kiev (1964). (In Russian)

    Google Scholar 

  7. Hwang, C.: Extensional vibration of axisymmetrical shells. AIAA J. 3(1), 23–26 (1965)

    Article  MathSciNet  MATH  Google Scholar 

  8. Penzes, L.E., Burgin, G.: Free vibrations of thin isotropic oblate ellipsoidal shells. Rept No CD/CBTD, Convair Div, General Dynamics, pp. 65–113 (1965)

  9. Nemergut, P.J., Brand, R.S.: Axisymmetric vibrations of prolate ellipsoidal shells. J. Acoust. Soc. Am. 38(2), 262–265 (1965)

    Article  Google Scholar 

  10. Hayek, S., DiMaggio, F.L.: Axisymmetric vibrations of submerged ellipsoidal shells. Contract Nonr. 266(67) Proj., 355-414, Tech. Rept. No. 4, Office of Naval Research (1965)

  11. Ross, E.W., Matthews, Jr. W.T.: Frequencies and mode shapes for axisymmetric vibration of shells. Tech. Rept. AMRA TR 66-04, U.S. Army Materials Research Agency (1966)

  12. DiMaggio, F.L., Rand, R.: Axisymmetric vibrations of prolate ellipsoidal shells. J. Acoust. Soc. Am. 40(1), 179–186 (1966)

    Article  Google Scholar 

  13. Penzes, L.E., Burgin, G.: Free vibrations of thin isotropic oblate-ellipsoidal shells. J. Acoust. Soc. Am. 39(1), 8–13 (1966)

    Article  MATH  Google Scholar 

  14. Ross, E.W., Matthews Jr., W.T.: Frequencies and mode shapes for axisymmetric vibration of shells. J. Appl. Mech. 34(1), 73–80 (1967)

    Article  Google Scholar 

  15. Rand, R., DiMaggio, F.L.: Vibrations of fluid-filled spherical and ellipsoidal shells. J. Acoust. Soc. Am. 42(6), 1278–1286 (1967)

    Article  MATH  Google Scholar 

  16. Yen, T., DiMaggio, F.L.: Forced vibrations of submerged ellipsoidal shells. J. Acoust. Soc. Am. 41(3), 618–626 (1967)

    Article  MATH  Google Scholar 

  17. Rand, R.H.: Torsional vibrations of elastic prolate spheroids. J. Acoust. Soc. Am. 44(3), 749–751 (1968)

    Article  MATH  Google Scholar 

  18. Penzes, L.E.: Free vibrations of thin orthortropic oblate ellipsoidal shells. J. Acoust. Soc. Am. 45(2), 500–505 (1969)

    Article  Google Scholar 

  19. Hayek, S.: Complex natural frequencies of vibration submerged ellipsoidal shells. Int. J. Solids Struct. 6(3), 333–351 (1970)

    Article  MATH  Google Scholar 

  20. Hwang, C.: Extensional vibration of axisymmetrical shells. In: AIAA Annals of Structures and Materials Conference, 5th, pp. 227–234. Palm Springs, California (1964)

  21. Bedrosian, B., DiMaggio, F.L.: Transient response of submerged ellipsoidal shells. Int. J. Solids Struct. 8(1), 111–129 (1972)

    Article  MATH  Google Scholar 

  22. Berger, B.S.: The dynamic response of a prolate ellipsoidal shell submerged in an acoustical medium. J. Appl. Mech. 41, 925–929 (1974)

    Article  Google Scholar 

  23. Cambou, J.P.: Vibrations libre et axisymetrique des reservoirs quasi-spherique. Acustica 34, 72–76 (1975)

    MATH  Google Scholar 

  24. Burroughs, C.B., Magrab, E.B.: Natural frequencies of prolate ellipsoidal shells of constant thickness. J. Sound Vib. 57(4), 571–581 (1978)

    Article  MATH  Google Scholar 

  25. Ross, C.T.F., Johns, T.: Vibration of submerged hemi-ellipsoidal domes. J. Sound Vib. 91(3), 363–373 (1983)

    Article  Google Scholar 

  26. Popov, A.L., Chernyshev, G.N.: Transition surfaces for short-wave vibrations of an ellipsoidal shell in a fluid. J. Appl. Math. Mech. 49(5), 620–624 (1985)

    Article  MATH  Google Scholar 

  27. Jones-Oliveira, J.B.: Fluid–solid interaction of a prolate ellipsoidal shell structures loaded by an acoustic shock wave. Ph.D. Dissertation, M.I.T. (1990)

  28. Chen, P.-T., Ginsberg, J.H.: Modal properties and eigenvalue veering phenomena in the axisymmetric vibration of ellipsoidal shells. J. Acoust. Soc. Am. 92(3), 1499–1508 (1992)

    Article  Google Scholar 

  29. Ross, C.T.F.: Vibration and elastic instability of thin-walled domes under uniform external pressure. Thin Walled Struct. 26(3), 159–177 (1996)

    Article  MathSciNet  Google Scholar 

  30. Al-Jumaily, A.M., Najim, F.M.: An approximation to the vibrations of oblate ellipsoidal shells. J. Sound Vib. 204(4), 561–574 (1997)

    Article  Google Scholar 

  31. Zhang, S.H., Danckert, J., Yuan, S.J., Wang, Z.R., Han, W.J.: Spherical and ellipsoidal steel structure products made by using integral hydro-bulge forming technology. J. Constr. Steel Res. 46(1–3), 338–339 (1998). (paper number 225, full paper on enclosed CD-ROM)

    Article  Google Scholar 

  32. Chen, P.-T.: Acoustic radiations for submerged elastic structures using natural mode expansions in conjunction with radiation modes approach. J. Sound Vib. 246(2), 245–263 (2001)

    Article  Google Scholar 

  33. Kang, J.-H.: Field equations, equations of motion, and energy functionals for thick shells of revolution with arbitrary curvature and variable thickness from a three-dimensional theory. Acta Mech. 188, 21–37 (2007)

    Article  MATH  Google Scholar 

  34. Kang, J.-H., Leissa, A.W.: Vibration analysis of solid ellipsoids and hollow ellipsoidal shells of revolution with variable thickness from a three-dimensional theory. Acta Mech. 197, 97–117 (2008)

    Article  MATH  Google Scholar 

  35. Jin, F., Qian, Z.H., Wang, Z.K., Kishimoto, K.: Propagation behavior of Love waves in a piezoelectric layered structure with inhomogeneous initial stress. Smart Mater. Struct. 14, 515–523 (2005)

    Article  Google Scholar 

  36. Li, X.Y., Wang, Z.K., Huang, S.H.: Love waves in functionally graded piezoelectric materials. Int. J. Solids Struct. 41, 7309–7328 (2004)

    Article  MATH  Google Scholar 

  37. Cao, X., Jia, J., Ru, Y., Shi, J.: Asymptotic analytical solution for horizontal shear waves in a piezoelectric elliptic cylinder shell. Acta Mech. 226, 3387–3400 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  38. Thomas, B., Roy, T.: Vibration analysis of functionally graded carbon nanotube-reinforced composite shell structures. Acta Mech. 227, 581–599 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  39. Shim, H.-J., Kang, J.-H.: Free vibrations of solid and hollow hemi-ellipsoids of revolution from a three-dimensional theory. Int. J. Eng. Sci. 42, 1793–1815 (2004)

    Article  MATH  Google Scholar 

  40. Sokolnikoff, I.S.: Mathematical Theory of Elasticity, 2nd edn. McGraw-Hill Book Co., New York (1956)

    MATH  Google Scholar 

  41. Kang, J.-H., Leissa, A.W.: Three-dimensional field equations of motion, and energy functionals for thick shells of revolution with arbitrary curvature and variable thickness. J. Appl. Mech. 68, 953–954 (2001)

    Article  MATH  Google Scholar 

  42. Kang, J.-H., Leissa, A.W.: 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 (2004)

    Article  MATH  Google Scholar 

  43. Mikhlin, S.G.: Variational Methods in Mathematical Physics. Pergamon, Oxford (1964)

    MATH  Google Scholar 

  44. Mikhlin, S.G., Smolitskiy, K.L.: Approximate Methods for Solution of Differential and Integral Equations. American Elsevier Publishing Company Inc, New York (1967)

    Google Scholar 

  45. Beckmann, P.: Orthogonal Polynomials for Engineers and Physicists. Chap. 3. The Golem Press, Boulder (1973)

    MATH  Google Scholar 

  46. Courant, R., Hilbert, D.: Methods of Mathematical Physics, vol. I. Interscience Publishers, New York (1953)

    MATH  Google Scholar 

  47. Kantorovich, L.V., Krylov, V.I.: Approximate Methods of Higher Analysis. Noordhoff, Groningen (1958)

    MATH  Google Scholar 

  48. Leissa, A.W.: The historical bases of the Rayleigh and Ritz methods. J. Sound Vib. 287, 961–978 (2005)

    Article  Google Scholar 

  49. McGee, O.G., Leissa, A.W.: Three-dimensional free vibrations of thick skewed cantilever plates. J. Sound Vib. 144, 305–322 (1991). (Errata 149, 539–542)

    Article  Google Scholar 

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  1. Department of Architectural Engineering, Chung-Ang University, 221 Heuksuk-Dong, Dongjak-Ku, Seoul, 156-756, South Korea

    Soo-Min Ko & Jae-Hoon Kang

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Ko, SM., Kang, JH. Free vibration analysis of shallow and deep ellipsoidal shells having variable thickness with and without a top opening. Acta Mech 228, 4391–4409 (2017). https://doi.org/10.1007/s00707-017-1932-2

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  • DOI: https://doi.org/10.1007/s00707-017-1932-2

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