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


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Article Info

DATE

2019-01-03

AUTHORS

S. N. Gavrilov, E. V. Shishkina, Yu. A. Mochalova

ABSTRACT

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. More... »

PAGES

1-10

References to SciGraph publications

  • 2004-07. Localization of nonlinear waves in elastic bodies with inclusions in ACOUSTICAL PHYSICS
  • 2010-06. Sufficient conditions on the existence of trapped modes in problems of the linear theory of surface waves in JOURNAL OF MATHEMATICAL SCIENCES
  • 2016-12. Evolution of a trapped mode of oscillation in a continuous system with a concentrated inclusion of variable mass in DOKLADY PHYSICS
  • 2013. Trapped Modes and Edge Resonances in Acoustics and Elasticity in DYNAMIC LOCALIZATION PHENOMENA IN ELASTICITY, ACOUSTICS AND ELECTROMAGNETISM
  • 2017-04. Oscillations of a string on an elastic foundation with space and time-varying rigidity in NONLINEAR DYNAMICS
  • 2015-01. Localization in a Bernoulli-Euler beam on an inhomogeneous elastic foundation in VESTNIK ST. PETERSBURG UNIVERSITY, MATHEMATICS
  • 2000-08. Resonance vibrations of elastic waveguides with inertial inclusions in TECHNICAL PHYSICS
  • 2011-01. Oscillations of a beam with a time-varying mass in NONLINEAR DYNAMICS
  • 2014-12. Vibration of a membrane strip with a segment of higher density: analysis of trapped modes in MECCANICA
  • 2001-07. Mode Localization in Dynamics and Buckling of Linear Imperfect Continuous Structures in NONLINEAR DYNAMICS
  • 2002-05. Vibration of a flexible plate in contact with the free surface of a heavy liquid in TECHNICAL PHYSICS
  • 2014-10. On oscillations of a beam with a small rigidity and a time-varying mass in NONLINEAR DYNAMICS
  • 2012-04. Motion of the exfoliation boundary during localization of wave processes in DOKLADY PHYSICS
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    http://scigraph.springernature.com/pub.10.1007/s11071-018-04735-3

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    http://dx.doi.org/10.1007/s11071-018-04735-3

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