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 Nelin. Dinam., 2014, Volume 10, Number 3, Pages 245–263 (Mi nd441)

Weakly coupled oscillators in the presence of elactic support in the conditions of acoustic vacuum

Irina P. Kikot, Leonid I. Manevich

N. N. Semenov Institute of Chemical Physics, Russian Academy of Sciences, Kosygina st. 4, Moscow, 117977, Russia

Abstract: A weightless string without preliminary tension with two symmetric discrete masses, which are influenced by elastic supports with cubic characteristics, is investigated both by numerical and analytical methods. The most important limit case corresponding to domination of resonance low-energy transversal oscillations is considered. Since such oscillations are described by approximate equations only with cubic terms (without linear ones), the transversal dynamics occurs n the conditions of acoustic vakuum. If there is no elastic supports nonlinear normal modes of the system under investigation coincide with (or are close to) those of corresponding linear oscillator system. However within the presence of elastic supports one of NNM can be unstable, that causes formation of two another assymmetric modes and a separatrix which divides them. Such dynamical transition which is observed under certain relation between elastic constants of the string and of the support, relates to stationary resonance dynamics. This transition determines also a possibility of the second dynamical transition which occurs when the supports contribution grows. It relates already to non-stationary resonance dynamics when the modal approach turns out to be inadequate. Effective description of both dynamical transitions can be attained in terms of weakly interacting oscillators and limiting phase trajectories, corresponding to complete energy echange between the oscillators.

Keywords: string with discrete masses, elastic support, nonlinear dynamics, asymptotical method, complete energy exchange, limiting phase trajectory, energy localization.

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UDC: 534.015.1; 534-6
MSC: 70K30, 70K50, 70K75
Revised: 11.09.2014

Citation: Irina P. Kikot, Leonid I. Manevich, “Weakly coupled oscillators in the presence of elactic support in the conditions of acoustic vacuum”, Nelin. Dinam., 10:3 (2014), 245–263

Citation in format AMSBIB
\Bibitem{KikMan14} \by Irina~P.~Kikot, Leonid~I.~Manevich \paper Weakly coupled oscillators in the presence of elactic support in the conditions of acoustic vacuum \jour Nelin. Dinam. \yr 2014 \vol 10 \issue 3 \pages 245--263 \mathnet{http://mi.mathnet.ru/nd441} 

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This publication is cited in the following articles:
1. I. P. Koroleva (Kikot), L. I. Manevich, “Ostsillyatornaya tsep na uprugoi podlozhke v usloviyakh akusticheskogo vakuuma”, Nelineinaya dinam., 11:3 (2015), 487–502
2. Kseniya G. Silina, Irina P. Kikot, Leonid I. Manevitch, “Energy Exchange and Localization in the Planar Motion of a Weightless Beam Carrying Two Discrete Masses”, Regul. Chaotic Dyn., 20:2 (2015), 109–122
3. Ryl'nikova M.V., Manevitch L.I., Eremenko V.A., Smirnov V.V., “Utilization of Elastic Energy of Rock Mass as a Source of Renewable Energy”, J. Min. Sci., 51:6 (2015), 1180–1190
4. I. P. Koroleva (Kikot), L. I. Manevich, “Ostsillyatornaya tsep c izgibnoi zhestkostyu na uprugoi podlozhke v usloviyakh, blizkikh k akusticheskomu vakuumu”, Nelineinaya dinam., 12:3 (2016), 311–325
5. L. I. Manevich, I. P. Koroleva (Kikot'), “Limiting phase trajectories as an alternative to nonlinear normal modes”, IUTAM Symposium Analytical Methods in Nonlinear Dynamics, Procedia IUTAM, 19, eds. P. Hagedorn, E. Clerkin, Elsevier Science BV, 2016, 144–151
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