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 Nelin. Dinam., 2017, Volume 13, Number 1, Pages 105–115 (Mi nd553)

Original papers

Stationary and nonstationary dynamics of the system of two harmonically coupled pendulums

M. A. Kovaleva, V. V. Smirnov, L. I. Manevich

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

Abstract: An analysis is presented of the nonlinear dynamics of harmonically coupled pendulums without restrictions to oscillation amplitudes. This is a basic model in many areas of mechanics and physics (paraffin crystals, DNA molecules etc.). Stationary solutions of equations of motion corresponding to nonlinear normal modes (NNMs) are obtained. The inversion of the NNM frequencies with increasing oscillation amplitude is found. An essentially nonstationary process of the resonant energy exchange is described in terms of limiting phase trajectories (LPTs), for which an effective analytic representation is obtained in slow time-scale. Explicit expressions of threshold values of dimensionless parameters are found which correspond to the instability of NNMs and to the transition (in parametric space) from the full energy exchange between the pendulums to the localization of energy. The analytic results obtained are verified by analysis of the Poincaré sections describing evolution of the initial system.

Keywords: essentially nonlinear systems, coupled pendulums, nonlinear normal modes, limiting phase trajectories

 Funding Agency Grant Number Russian Foundation for Basic Research 16-33-60186 ìîë_à_äê

DOI: https://doi.org/10.20537/nd1701007

Full text: PDF file (5885 kB)
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Document Type: Article
UDC: 531.539.2
MSC: 70K43, 37E99
Accepted:28.01.2017

Citation: M. A. Kovaleva, V. V. Smirnov, L. I. Manevich, “Stationary and nonstationary dynamics of the system of two harmonically coupled pendulums”, Nelin. Dinam., 13:1 (2017), 105–115

Citation in format AMSBIB
\Bibitem{KovSmiMan17} \by M.~A.~Kovaleva, V.~V.~Smirnov, L.~I.~Manevich \paper Stationary and nonstationary dynamics of the system of two harmonically coupled pendulums \jour Nelin. Dinam. \yr 2017 \vol 13 \issue 1 \pages 105--115 \mathnet{http://mi.mathnet.ru/nd553} \crossref{https://doi.org/10.20537/nd1701007} \elib{http://elibrary.ru/item.asp?id=28841003}