Skip to main content
SpringerLink
Log in

Nonlinear vibrations of viscoelastic cylindrical shells taking into account shear deformation and rotatory inertia

  • Original Paper
  • Published:
Nonlinear Dynamics Aims and scope Submit manuscript

Abstract

The vibration problem of a viscoelastic cylindrical shell is studied in a geometrically nonlinear formulation using the refined Timoshenko theory. The problem is solved by the Bubnov–Galerkin procedure combined with a numerical method based on quadrature formulas. The choice of relaxation kernels is substantiated for solving dynamic problems of viscoelastic systems. The numerical convergence of the Bubnov–Galerkin procedure is examined. The effect of viscoelastic properties of the material on the response of the cylindrical shell is discussed. The results obtained by various theories are compared.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Timoshenko, S.P., Woinowsky-Krieger, S.: Theory of Plates and Shells, 2nd edn. McGraw-Hill, New York (1987)

    Google Scholar 

  2. Donnell, L.H.: Beams, Plates, and Shells. McGraw-Hill, New York (1976)

    MATH  Google Scholar 

  3. Volmir, A.S.: The Nonlinear Dynamics of Plates and Shells. Nauka, Moscow (1972) (in Russian)

    Google Scholar 

  4. Timoshenko, S.P.: Vibration Problems in Engineering. Van Nostrand, New York (1958)

    Google Scholar 

  5. Mindlin, R.D.: Influence of rotatory inertia and shear on flexural motions of isotropic elastic plates. J. Appl. Mech. 19(1), 31–38 (1951)

    Google Scholar 

  6. Uflyand, Ya.S.: Distribution of waves at transverse vibrations of cores and plates. Appl. Math. Mech. 12(3), 287–300 (1948)

    Google Scholar 

  7. Reissner, E.: The effect of transverse shear deformation on the bending of elastic plates. J. Appl. Mech. 12, A69–A88 (1945)

    Google Scholar 

  8. Ambartsumyan, S.A.: General Theory of Anisotropic Shells. Nauka, Moscow (1974) (in Russian)

    Google Scholar 

  9. Il’yushin, A.A., Pobedrya, B.E.: Fundamentals of the Mathematical Theory of Thermoviscoelasticity. Nauka, Moscow (1970) (in Russian)

    Google Scholar 

  10. Christensen, R.M.: Theory of Viscoelasticity. Academic, New York (1971)

    Google Scholar 

  11. Rzhanitsyn, A.R.: Theory of Creep in Russian. Stroyizdat, Moscow (1968)

    Google Scholar 

  12. Rabotnov, Yu.N.: Elements of the Hereditary Mechanics of Solids. Nauka, Moscow (1977) (in Russian)

    Google Scholar 

  13. Bogdanovich, A.E.: Nonlinear Dynamic Problems for Composite Cylindrical Shells. Elsevier Science, New York (1993)

    Google Scholar 

  14. Awrejcewicz, J., Krys'ko, V.A.: Nonclassical Thermoelastic Problems in Nonlinear Dynamics of Shells. Applications of the Bubnov—Galerkin and Finite Difference Numerical Methods. Springer-Verlag, Berlin (2003)

    MATH  Google Scholar 

  15. Kubenko, V.D., Koval'chuk, P.S.: Influence of initial geometric imperfections on the vibrations and dynamic stability of elastic shells. Int. Appl. Mech. 40(8), 847–877 (2004)

    Article  Google Scholar 

  16. Kayuk, Ya.F.: Geometrically Nonlinear Problems of the Theory of Plates and Shells. Naukova Dumka, Kiev, Ukraine (1987) (in Russian)

    Google Scholar 

  17. Amabili, M.: Nonlinear vibrations of circular cylindrical Panels. J. Sound Vib. 281, 509–535 (2005)

    Article  Google Scholar 

  18. Dogan, V., Vaicaitis, R.: Nonlinear response of cylindrical shells to random excitation. Nonlinear Dyn. 20, 33–53 (1999)

    Article  MATH  Google Scholar 

  19. Shirakawa, K.: Effects of shear deformation and rotatory inertia on vibration and buckling of cylindrical shells. J. Sound Vib. 91(3), 425–437 (1983)

    Article  MATH  Google Scholar 

  20. Krysko, V.A., Awrejcewicz, J., Bruk, V.: On the solution of a coupled thermo-mechanical problem for non-homogeneous Timoshenko-type shells. J. Math. Anal. Appl. 273(2), 409–416 (2002)

    Article  Google Scholar 

  21. Okazaki, A., Tatemichi, A.: Damping properties of two-layered cylindrical shells with an unconstrained viscoelastic layer. J. Sound Vib. 176(2), 145–161 (1994)

    Article  MATH  Google Scholar 

  22. Cheng, C.-J., Zhang, N.-H.: Dynamical behavior of viscoelastic cylindrical shells under axial pressures. Appl. Math. Mech. 22(1), 1–9 (2001)

    Article  MATH  Google Scholar 

  23. Kozlov, V.I., Karnaukhova, T.V.: Basic equations for viscoelastic laminated shells with distributed piezoelectric inclusions intended to control nonstationary vibrations. Int. Appl. Mech. 38(10), 1253–1260 (2002)

    Article  Google Scholar 

  24. Potapov, V.D.: Stability of Stochastic Elastic and Viscoelastic Systems. Wiley, Chichester, UK (1999)

    Google Scholar 

  25. Koltunov, M.A.: Creep and Relaxation. Visshaya Shkola, Moscow (1976) (in Russian)

    Google Scholar 

  26. Badalov, F.B., Eshmatov, Kh.,Yusupov, M.:Oncertain methods of solving of systems of integrodifferential equations encountered in viscoelasticity problems. Appl. Math. Mech. [Transl. Prikl. Mat. Mekh.] 51(5), 683–686 (1987)

    Google Scholar 

  27. Badalov, F.B., Eshmatov, Kh.: The brief review and comparison of integrated methods of mathematical modeling in problems of hereditary mechanics of rigid bodies. Int. J. Electron. Model 11(2), 81–90 (1989)

    Google Scholar 

  28. Badalov, F.B., Eshmatov, Kh.: Investigation of nonlinear vibrations of viscoelastic plates with initial imperfections. Sov. Appl. Mech. [Transl. Prikl. Mekh.] 26(8), 99–105 (1990)

    Google Scholar 

  29. Badalov, F.B., Eshmatov, Kh., Akbarov, U.I.: Stability of a viscoelastic plate under dynamic loading. Sov. Appl. Mech. [Transl. Prikl. Mekh.] 27(9), 892–899 (1991)

    Article  Google Scholar 

  30. Verlan, A.F., Eshmatov, B.Kh.: Mathematical simulation of oscillations of orthotropic viscoelastic plates with regards to geometric nonlinearity. Int. J. Electron. Model. 27(4), 3–17 (2005)

    Google Scholar 

  31. Eshmatov, B.Kh.: Mathematical modeling of problems of nonlinear vibrations viscoelastic orthotropic plates. In: Materials of International Conference on Computational and Informational Technologies for Research, Engineering, and Education, pp. 359–366, Alma-Ata, Kazakhstan, 7–9 October (2004)

  32. Eshmatov, B.Kh.: Nonlinear vibrations of viscoelastic orthotropic cylindrical shells in view of propagation of elastic waves. In: Materials of the XVII Session of the International School of Models of the Mechanics of the Continuous Environment, pp. 186–191, Kazan, Russia, 4–10 July (2004)

  33. Eshmatov, B.Kh.: Nonlinear vibrations of viscoelastic orthotropic plates from composite materials. In: Third M.I.T. Conference on Computational Fluid and Solid Mechanics, Boston, MA, USA, 14–17 June 2005

  34. Eshmatov, B.Kh.: Dynamic stability of viscoelastic plates at growing compressing loadings. Int. J. Appl. Mech. Tech. Phys. 47(2), 289–297 (2006)

    Article  Google Scholar 

  35. Eshmatov, B.Kh.: Mathematical model of problem about nonlinear vibrations and dynamical stability of viscoelastic orthotropic shells taking into account shear deformation and rotation inertia. Republican Sci. J. Proc. Acad. Sci. Republic Uzbekistan 1, 27–30 (2005)

    Google Scholar 

  36. Eshmatov, B.Kh.: Nonlinear vibrations and dynamic stability of viscoelastic orthotropic rectangular plates. J. Sound Vib. (to be published in 2007)

Download references

Author information

Authors and Affiliations

  1. Department of Informational Technology, Tashkent Institute of Irrigation and Amelioration, Tashkent, 700000, Uzbekistan

    B. Kh. Eshmatov

Authors
  1. B. Kh. Eshmatov
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to B. Kh. Eshmatov.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Eshmatov, B.K. Nonlinear vibrations of viscoelastic cylindrical shells taking into account shear deformation and rotatory inertia. Nonlinear Dyn 50, 353–361 (2007). https://doi.org/10.1007/s11071-006-9163-4

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11071-006-9163-4

Keywords

Search

Navigation