An Infinite-Length System Possessing a Unique Trapped Mode Versus a Single Degree of Freedom System: A Comparative Study in the Case of Time-Varying Parameters

Гаврилов, С. Н.; Shishkina, Ekaterina V.; Mochalova, Yu. A.

doi:10.1007/978-3-030-11665-1_13

Cited by 4 publications

(19 citation statements)

References 38 publications

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“…Since the frequency equation is defined only for the trapped mode frequency (and it is not valid in the neighbourhood of this frequency), this operation is senseless. We also carefully analyzed our subsequent studies [32,31,33], where the approach based on the method of multiple scales was applied to a number of problems, and now we are sure that the same error was never repeated.…”

Section: Comparison Between Analytic and Numerical Resultsmentioning

confidence: 99%

“…Formula (4.20), which describes the evolution for the amplitude of the trapped mode in the case M = 0, K < 0, coincides with the corresponding formula for a linear oscillator with spring of slowly time-varying stiffness [50] (the Liouville-Green approximation). The particular case M = 0, K < 0 of the problem under consideration is the only one known for us system [33,28,32,31,26] with trapped mode for which the corresponding formula has this simple classical form.…”

Section: 18)mentioning

confidence: 99%

“…It is a general approach, which allows us to investigate non-stationary free localized oscillation in infinite systems with time-varying parameters in the case if the corresponding system with constant parameters possesses a single trapped mode. The method was applied to several model problems concerning an infinite string on elastic foundation with a discrete inclusion, namely a string with time-varying tension [28], string with time-varying mass [31], string with oscillator of time-varying stiffness [32]. Also, the method was successfully applied to the problem concerning a beam with discrete inclusion of a time-varying stiffness [33], and a resonant solution of a model problem concerning a string [34] was obtained.…”

Section: Introductionmentioning

confidence: 99%

“…Thus, all necessary auxiliary results are presented in Appendix. The material of Appendix A-Appendix E mostly involves some known results or their generalizations (see [51,26,28,32,27,52,53]). The final result (Appendix F) concerning the large-time asymptotics of non-stationary free and forced oscillation in the system with constant parameters is the generalization of the results obtained in [40], though we use a bit different technique for the asymptotic evaluation of integrals.…”

Section: Introductionmentioning

confidence: 99%

“…We have introduced and used the normalized amplitudes investigating other problem considered in[32].…”

mentioning

confidence: 99%

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Non-stationary oscillation of a string on the Winkler foundation subjected to a discrete mass-spring system non-uniformly moving at a sub-critical speed

Gavrilov,

Shishkina,

Poroshin

2021

Preprint

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We consider non-stationary free and forced transverse oscillation of an infinite taut string on the Winkler foundation subjected to a discrete mass-spring system non-uniformly moving at a given sub-critical speed. The speed of the mass-spring system is assumed to be a slowly time-varying function less than the critical speed. To describe a non-vanishing free oscillation we use an analytic approach based on the method of stationary phase and the method of multiple scales. The moving oscillator is characterized by a partial frequency, which can be greater or less than the cut-off frequency. Accordingly, a sub-critical uniformly accelerated motion generally has two stages. At the first stage there exists a trapped mode, and, therefore, a part of the wave energy is localized near the moving oscillator and does not propagate away. For this stage we obtain the analytic solution in a simple form describing non-vanishing free oscillation and verify it numerically. For the second stage there is no trapped mode, and all the wave energy propagate away. This stage is investigated numerically, and some unexpected results are obtained. Additionally, we consider the case of the oscillator with a destabilizing spring. The dynamics of the system in the latter case is quite different from the case of commonly used stabilizing spring, since the system loses the stability during an accelerated motion. We also take into consideration the forced oscillation caused by an external load being a superposition of harmonics with time-varying parameters (the amplitude and the frequency).

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Section: Comparison Between Analytic and Numerical Resultsmentioning

confidence: 99%