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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. Abramian
  • W. T. van Horssen
  • S. A. Vakulenko
Original Paper
  • 155 Downloads

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.

Keywords

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

Notes

Acknowledgements

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

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Copyright information

© Springer Science+Business Media Dordrecht 2016

Authors and Affiliations

  • A. K. Abramian
    • 1
  • 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

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