Journal of Mathematical Sciences

, Volume 220, Issue 4, pp 514–532 | Cite as

On the Construction of Coordinate Functions for the Ritz Method in the Numerical Analysis of Nonaxially Symmetric Eigenoscillations of a Dome-Shaped Shell of Revolution

  • Yu. V. Trotsenko

We propose systems of coordinate functions that can be used in the Ritz method aimed at finding the natural modes and eigenfrequencies of nonaxially symmetric oscillations of thin-walled dome-shaped shells of revolution. The basis functions are constructed with regard for the specific features of the spectral problem, which guarantees the uniform convergence of the process of calculations. As an example, we find the dynamic characteristics for a shell in the form of a spherical dome.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    A.W. Leissa, Vibration of Shells, Acoustical Society of America, New York (1993).Google Scholar
  2. 2.
    G. L. Anderson, “On Gegenbauer transforms and forced torsional vibrations of thin spherical shells,” J. Sound Vibrat., 12, 265–267 (1970).CrossRefMATHGoogle Scholar
  3. 3.
    B. Goller, “Dynamic deformations of thin spherical shells based on analytical solutions,” J. Sound Vibrat., 73, 585–596 (1980).CrossRefMATHGoogle Scholar
  4. 4.
    K. Mukherjee and S. K. Chakraborty, “Exact solution for larger amplitude free and forced oscillation of a thin spherical shell,” J. Sound Vibrat., 100, 339–342 (1985).CrossRefGoogle Scholar
  5. 5.
    A. M. Al-Jumaily and F. M. Najim, “An approximation to the vibrations of oblate spheroidal shells,” J. Sound Vibrat., 207, 561–574 (1997).CrossRefGoogle Scholar
  6. 6.
    I. Gavrilyuk, M. Hermann, V. Trotsenko, Yu. Trotsenko, and A. Timokha, “Axisymmetric oscillations of a cupola-shaped shell,” J. Eng. Math., 68, 165–178 (2010).MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    I. Gavrilyuk, M. Hermann, V. Trotsenko, Yu. Trotsenko, and A. Timokha, “Eigenoscillations of a thin-walled azimuthally closed, axially open shell of revolution,” J. Eng. Math., 85, 83–97 (2014).MathSciNetCrossRefGoogle Scholar
  8. 8.
    E. P. Doolan, J. J. H. Miller, and W. H. A. Schilders, Uniform Numerical Methods for Problems with Initial and Boundary Layers, Boole Press, Dublin (1980).MATHGoogle Scholar
  9. 9.
    A. G. Aslanyan and V. B. Lidskii, Distribution of Eigenfrequencies of Thin Elastic Shells [in Russian], Nauka, Moscow (1974).Google Scholar
  10. 10.
    V. V. Novozhilov, Theory of Thin Shells [in Russian], Sudostroenie, Leningrad (1962).Google Scholar
  11. 11.
    A. V. Karmishin, V. A. Lyaskovets, V. I. Myachenkov, and A. N. Frolov, Statics and Dynamics of Thin-Walled Shell Structures [in Russian], Mashinostroenie, Moscow (1975).Google Scholar
  12. 12.
    N. V. Valishvili, Methods for the Analysis of the Shells of Revolution on Computers [in Russian], Mashinostroenie, Moscow (1976).Google Scholar
  13. 13.
    W. Wasow, Asymptotic Expansions for Ordinary Differential Equations, Wiley, New York (1965).MATHGoogle Scholar
  14. 14.
    M. I. Vishik and L. A. Lyusternik, “Regular degeneration and boundary layer for linear differential equations with small parameter,” Usp. Mat. Nauk, 12, Issue 5(77), 3 – 122 (1957).Google Scholar
  15. 15.
    V. A. Trotsenko and Yu. V. Trotsenko, “Solution of the problem of free oscillations of a nonclosed shell of revolution under the conditions of its singular perturbations,” Nelin. Kolyv., 8, No. 3, 415–432 (2005); English translation: Nonlin. Oscillat., 8, No. 3, 414–429 (2005).Google Scholar
  16. 16.
    M. S. Zarghamee and H. R. Robinson, “A numerical method for analysis of free vibration of spherical shell,” AIAA Journal, 5, No. 7, 1256–1261 (1967).CrossRefMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  1. 1.Institute of Mathematics, Ukrainian National Academy of SciencesKievUkraine

Personalised recommendations