Skip to main content
SpringerLink
Log in

Vibrations of shallow shells rectangular in the horizontal projection with two freely supported opposite edges

  • Published:
Mechanics of Solids Aims and scope Submit manuscript

Abstract

The exact mode shapes of linear vibrations of a shallow shell rectangular in the horizontal projection with two freely supported opposite edges are obtained. These shapes are used to construct a discretemodel of vibrations of a shallow shell in geometrically nonlinear deformation. The harmonic balance method is used to study the free and forced nonlinear vibrations under internal resonance. The Lyapunov stability of the obtained periodic vibrations is analyzed.

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. E. I. Grigolyuk, “To Calculations of Stability of Shallow Arches,” Inzh. Sb. 9, 177–200 (1951).

    Google Scholar 

  2. A. W. Leissa and A. S. Kadi, “Curvature Effects on Shallow Shell Vibrations,” J. Sound Vibr., 16(2), 173–187 (1971).

    Article  ADS  MATH  Google Scholar 

  3. A. S. Vol’mir, A. A. Logvinskaya, and V. V. Rogalevich, “Nonlinear Free Vibrations of Rectangular Plates and Cylindrical Panels,” in Proc. 8th All-Union Conf. on Theory of Shells and Plates (Nauka, Moscow, 1973), pp. 426–431 [in Russian].

    Google Scholar 

  4. M. Amabili, “Nonlinear Vibrations of Doubly Curves Shallow Shells,” Int. J. Nonlin.Mech. 40(5), 683–710 (2005).

    Article  MATH  Google Scholar 

  5. A.W. Leissa, “The Free Vibration of Rectangular Plates,” J. Sound Vibr. 31(3), 257–293 (1973).

    Article  ADS  MATH  Google Scholar 

  6. L. Zhang and Y. Xiang, “Vibration of Open Circular Cylindrical Shells with Intermediate Ring Supports,” Int. J. Solids Struct. 43(13), 3705–3722 (2006).

    Article  MATH  Google Scholar 

  7. V. V. Bolotin (Editor), Vibrations in Technology. Handbook in 6 Volumes, Vol. 1: Vibrations of Linear Systems (Mashinostroenie, Moscow, 1978) [in Russian].

    Google Scholar 

  8. A. S. Vol’mir, Nonlinear Dynamics of Plates and Shells (Nauka, Moscow, 1972) [in Russian].

    Google Scholar 

  9. V. S. Gantkevich, Natural Vibrations of Plates and Shells (Naukova Dumka, Kiev, 1964) [in Russian].

    Google Scholar 

  10. M. Amabili, Nonlinear Vibrations and Stability of Shells and Plates (Cambridge Univ. Press, Cambridge, 2008) [in Russian].

    Book  MATH  Google Scholar 

  11. K. Avramov, Yu. Mikhlin, and E. Kurilov, “Asymptotical Analysis of Nonlinear Dynamics of Simply Supported Cylindrical Shells,” Nonlin. Dyn. 47(4), 331–352 (2007).

    Article  MathSciNet  MATH  Google Scholar 

  12. I. Breslavskii, K. V. Avramov, Yu. Mikhlin, and R. Kochurov, “Nonlinear Modes of Snap-Through Motions of a Shallow Arch,” J. Sound Vibr. 311(1–2), 297–313 (2008).

    Article  ADS  Google Scholar 

  13. V. S. Anishchenko, Complex Vibrations in Simple Systems (Nauka, Moscow, 1990) [in Russian].

    Google Scholar 

  14. V. A. Yakubovich and V. M. Starzhinskii, Linear Differential Equations with Periodic Coefficients and Their Applications (Nauka, Moscow, 1972) [in Russian].

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Podgorny Institute for Problems in Mechanical Engineering, National Academy of Sciences of Ukraine, Dm. Pozharskogo 2/10, Kharkiv, 61046, Ukraine

    K. V. Avramov

  2. National Technical University “Kharkiv Polytechnic Institute”, Frunze 21, Kharkiv, 61002, Ukraine

    I. D. Breslavskii

Authors
  1. K. V. Avramov
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. I. D. Breslavskii
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to K. V. Avramov.

Additional information

Original Russian Text © K.V. Avramov, I.D. Breslavskii, 2013, published in Izvestiya Akademii Nauk. Mekhanika Tverdogo Tela, 2013, No. 2, pp. 86–95.

About this article

Cite this article

Avramov, K.V., Breslavskii, I.D. Vibrations of shallow shells rectangular in the horizontal projection with two freely supported opposite edges. Mech. Solids 48, 186–193 (2013). https://doi.org/10.3103/S0025654413020106

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.3103/S0025654413020106

Keywords

Search

Navigation