Skip to main content
SpringerLink
Log in

Oscillations of a string on an elastic foundation with space and time-varying rigidity

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

Abstract

The dynamics of a string on an elastic foundation with time- and coordinate-dependent coefficients have been studied. Asymptotic solutions have been constructed for the following cases: for an arbitrary value of the elastic foundation coefficient at small and large time values, and for small and large coefficients of the elastic foundation at arbitrary times. Also a special case originated from an ageing process has been studied. The ageing process is described by an expression approximating some well-known experimental data. The existence of localized modes along the x coordinate is shown. The existence of these localized modes can lead to a spatial resonance phenomenon under certain conditions. For the case of an arbitrary elastic foundation coefficient value at small and at large times, the spatial resonance phenomenon is observed at small, special frequencies. This effect depends also on a special phase and mode number. For large mode numbers, this special resonance seems to be not possible.

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

Fig. 1
Fig. 2
Fig. 3
Fig. 4

Similar content being viewed by others

References

  1. Lu, N., Wang, X., Vlassak, J.: Failure by simultaneous grain growth, strain localization, and interface de-bonding in metal films on polymer substrates. J. Mater. Res. 24, 375–385 (2009)

    Google Scholar 

  2. Floris, C., Lamacchiaa, F.P.: Analytic solution for the interaction between a viscoelastic Bernoulli–Navier beam and a winkler medium. Struct. Eng. Mech. 38(5), 593–618 (2011)

    Article  Google Scholar 

  3. Golecki, J., Jeffrey, A.: Two-dimensional dynamical problems for incompressible isotropic linear elastic solids with time dependent moduli and variable density. Acta Mech. 5, 118–180 (1968)

  4. Hsieh, J.H., Wang, C.M., Li, C.: Deposition and characterization of TaN-Cu nanocomposite thin films. Surf. Coat. Technol. 200(10), 3179–3183 (2006)

    Article  Google Scholar 

  5. Abramian, A., Vakulenko, S.: Oscillations of a beam with a time varying mass. Nonlinear Dyn. 63, 135–147 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  6. Abramyan, A.K., Vakulenko, S.A., Indeitsev, D.A., Semenov, B.N.: Influence of dynamic processes in a film on damage development in an adhesive base. Mech. Solid 47(5), 498–504 (2012)

    Article  Google Scholar 

  7. Abramyan, A., Vakulenko, S., Indeitsev, D., Bessonov, N.: Destruction of thin films with damaged substrate as a result of waves localization. Acta Mech. 226(2), 295–309 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  8. Lebedev, D.V., Chuklanov, A.P., Bukharaev, A.A., Druzhinina, O.S.: Measuring young’s modulus of biological objects in a liquid medium using an atomic force microscope with a soecial probe. Tech. Phys. Lett. 35(4), 371–374 (2009)

    Article  Google Scholar 

  9. Chen, K., Schweizer, K.S.: Theory of aging, rejuvenation, and the nonequilibrium steady state in deformed polymer glasses. Phys. Rev. E 82, 041804 (2010)

    Article  Google Scholar 

  10. Tomlins, P.E.: Comparison of different functions for modelling the creep and physical ageing effects in plastics. Polymer 37(17), 3907–3913 (1996)

    Article  Google Scholar 

  11. Kovalenko, E.V.: The solution of contact problems of creep theory for combined ageing foundations. PMM USSR 48(6), 739–745 (1984)

    MathSciNet  MATH  Google Scholar 

  12. Arya, J.C., Bojadziev, G.N.: Damping oscillating systems modeled by hyperbolic differential equations with slowly varying coefficients. Acta Mech. 35(3), 215–221 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  13. Verhulst, F.: Nonlinear Differential Equations and Dynamical Systems, 2nd edn. Springer, Berlin (1996)

    Book  MATH  Google Scholar 

  14. Birman, V., Byrd, L.W.: Modeling and analysis of functionally graded materials and structures. Appl. Mech. Rev. 60(5), 195–216 (2007)

    Article  Google Scholar 

  15. Fermi, E.: Notes on Quantum Mechanics. The University of Chicago Press, Chicago (1961)

    Google Scholar 

  16. de Bruijn, N.G.: Asymptotic Methods in Analysis. Dover Books on Mathematics, North-Holland (1958)

    MATH  Google Scholar 

Download references

Acknowledgements

This work is supported by a grant of the Government of Russian Federation, Grant 074-U01.

Author information

Authors and Affiliations

  1. Institute of Problems in Mechanical Engineering, Russian Academy of Sciences, V.O., Bol’shoy Pr., 61, St. Petersburg, Russia, 199178

    A. K. Abramian

  2. Delft Institute of Applied Mathematics, Faculty of EEMCS, Delft University of Technology, Mekelweg 4, 2628 CD, Delft, Netherlands

    W. T. van Horssen

  3. ITMO University, Saint Petersburg Russia, 49 Kronverksky Pr., St. Petersburg, Russia, 197101

    S. A. Vakulenko

Authors
  1. A. K. Abramian
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. W. T. van Horssen
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. S. A. Vakulenko
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to A. K. Abramian.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Abramian, A.K., van Horssen, W.T. & Vakulenko, S.A. Oscillations of a string on an elastic foundation with space and time-varying rigidity. Nonlinear Dyn 88, 567–580 (2017). https://doi.org/10.1007/s11071-016-3261-8

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11071-016-3261-8

Keywords

Search

Navigation