Non-stationary localized oscillations of an infinite string, with time-varying tension, lying on the Winkler foundation with a point elastic inhomogeneity

  • S. N. GavrilovEmail author
  • E. V. Shishkina
  • Yu. A. Mochalova
Original Paper


We consider non-stationary oscillations of an infinite string with time-varying tension. The string lies on the Winkler foundation with a point inhomogeneity (a concentrated spring of negative stiffness). In such a system with constant parameters (the string tension), under certain conditions a trapped mode of oscillation exists and is unique. Therefore, applying a non-stationary external excitation to this system can lead to the emergence of the string oscillations localized near the inhomogeneity. We provide an analytical description of non-stationary localized oscillations of the string with slowly time-varying tension using the asymptotic procedure based on successive application of two asymptotic methods, namely the method of stationary phase and the method of multiple scales. The obtained analytical results were verified by independent numerical calculations based on the finite difference method. The applicability of the analytical formulas was demonstrated for various types of external excitation and laws governing the varying tension. In particular, we have shown that in the case when the trapped mode frequency approaches zero, localized low-frequency oscillations with increasing amplitude precede the localized string buckling. The dependence of the amplitude of such oscillations on its frequency is more complicated in comparison with the case of a one-degree-of-freedom system with time-varying stiffness.


PDE with time-varying coefficients Method of multiple scales Trapped modes Localization 



The authors are grateful to Prof. D.A. Indeitsev for useful and stimulating discussions.

Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflict of interest.


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© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.Institute for Problems in Mechanical Engineering RASSt. PetersburgRussia
  2. 2.Peter the Great St. Petersburg Polytechnic University (SPbPU)St. PetersburgRussia

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