Nonlinear Dynamics

, Volume 88, Issue 1, pp 567–580 | Cite as

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

  • A. K. AbramianEmail author
  • W. T. van Horssen
  • S. A. Vakulenko
Original Paper


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.


Time-varying rigidity Ageing Elastic foundation Localized modes Slender structure Thin film 



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


  1. 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. 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)CrossRefGoogle Scholar
  3. 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)Google Scholar
  4. 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)CrossRefGoogle Scholar
  5. 5.
    Abramian, A., Vakulenko, S.: Oscillations of a beam with a time varying mass. Nonlinear Dyn. 63, 135–147 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 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)CrossRefGoogle Scholar
  7. 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)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 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)CrossRefGoogle Scholar
  9. 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)CrossRefGoogle Scholar
  10. 10.
    Tomlins, P.E.: Comparison of different functions for modelling the creep and physical ageing effects in plastics. Polymer 37(17), 3907–3913 (1996)CrossRefGoogle Scholar
  11. 11.
    Kovalenko, E.V.: The solution of contact problems of creep theory for combined ageing foundations. PMM USSR 48(6), 739–745 (1984)MathSciNetzbMATHGoogle Scholar
  12. 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)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Verhulst, F.: Nonlinear Differential Equations and Dynamical Systems, 2nd edn. Springer, Berlin (1996)CrossRefzbMATHGoogle Scholar
  14. 14.
    Birman, V., Byrd, L.W.: Modeling and analysis of functionally graded materials and structures. Appl. Mech. Rev. 60(5), 195–216 (2007)CrossRefGoogle Scholar
  15. 15.
    Fermi, E.: Notes on Quantum Mechanics. The University of Chicago Press, Chicago (1961)Google Scholar
  16. 16.
    de Bruijn, N.G.: Asymptotic Methods in Analysis. Dover Books on Mathematics, North-Holland (1958)zbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2016

Authors and Affiliations

  • A. K. Abramian
    • 1
    Email author
  • W. T. van Horssen
    • 2
  • S. A. Vakulenko
    • 3
  1. 1.Institute of Problems in Mechanical EngineeringRussian Academy of SciencesSt. PetersburgRussia
  2. 2.Delft Institute of Applied Mathematics, Faculty of EEMCSDelft University of TechnologyDelftNetherlands
  3. 3.ITMO UniversitySaint Petersburg RussiaSt. PetersburgRussia

Personalised recommendations