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Nelin. Dinam., 2010, Volume 6, Number 3, Pages 605–622 (Mi nd27)  

This article is cited in 10 scientific papers (total in 10 papers)

Nonlinear oscillations of sympathetic pendulums

A. P. Markeev

Institite for Problems in Mechanics, Russian Academy of Sciences

Abstract: Nonlinear problem of motion of two identical pendulums connected by an elastic spring in the neighborhood of their stable vertical equilibrium is investigated. Stiffness of the spring is supposed small, i.e. the case close to resonance $1:1$ is considered. The problem of existence and orbital stability of periodical motions of the pendulums arising from the equilibrium is solved. It is indicated existence of motions asymptotic to one of the periodical motions. An analysis of quasi-periodical motions of an approximate system is given in which members up to the forth order inclusively in the normalizing Hamiltonian of the problem are taken into account. Using KAM-theory the question is considered of preservation of these motions in the complete nonlinear system in which members of all orders in the series expansion of Hamiltonian in the sufficiently small neighborhood of the equilibrium are taken account.

Keywords: pendulum; nonlinear oscillation; resonance; stability

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Document Type: Article
UDC: 531.36,531.53
MSC: 70E55, 70H12, 70E50
Received: 23.08.2010

Citation: A. P. Markeev, “Nonlinear oscillations of sympathetic pendulums”, Nelin. Dinam., 6:3 (2010), 605–622

Citation in format AMSBIB
\by A.~P.~Markeev
\paper Nonlinear oscillations of sympathetic pendulums
\jour Nelin. Dinam.
\yr 2010
\vol 6
\issue 3
\pages 605--622

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    This publication is cited in the following articles:
    1. Markeyev A.P., “Non-Linear Oscillations of a 1:1 Resonance Hamiltonian System”, J. Appl. Math. Mech., 75:6 (2011), 631–646  crossref  mathscinet  mathscinet  zmath  isi  elib  elib  scopus
    2. A. P. Markeev, “O dvizhenii svyazannykh mayatnikov”, Nelineinaya dinam., 9:1 (2013), 27–38  mathnet
    3. Markeev A.P., “On the Stability of Nonlinear Vibrations of Coupled Pendulums”, Mech. Sol., 48:4 (2013), 370–379  crossref  mathscinet  isi  elib  scopus
    4. S. P. Bezglasnyi, N. I. Kutyreva, “Stabilizatsiya nestatsionarnykh dvizhenii mayatnika na vraschayuschemsya osnovanii”, Izvestiya Samarskogo nauchnogo tsentra Rossiiskoi akademii nauk, 16:4-1 (2014), 77–82  elib
    5. S. P. Bezglasnyi, E. S. Batina, E. E. Piyakina, “Parametricheskoe upravlenie s ogranicheniem dvizheniyami dvukhmassovogo mayatnika”, Trudy MAI, 2014, no. 72, 4  elib
    6. Evdokimenko A., Tkhai V., “On Sympathetic Pendulums Dynamics”, 2015 International Conference on Mechanics Seventh Polyakhovs Reading, ed. Tikhonov A., IEEE, 2015  isi
    7. Lev A. Smirnov, Alexey K. Kryukov, Grigory V. Osipov, Jürgen Kurths, “Bistability of Rotational Modes in a System of Coupled Pendulums”, Regul. Chaotic Dyn., 21:7-8 (2016), 849–861  mathnet  crossref
    8. M. A. Kovaleva, V. V. Smirnov, L. I. Manevich, “Statsionarnaya i nestatsionarnaya dinamika sistemy dvukh garmonicheski svyazannykh mayatnikov”, Nelineinaya dinam., 13:1 (2017), 105–115  mathnet  crossref  elib
    9. Evdokimenko A.P., “On Equilibrium Configurations and Their Stability For a System of Two Coupled Pendulums”, Mech. Sol., 52:3 (2017), 266–277  crossref  isi  scopus
    10. A. P. Markeev, “On Nonlinear Resonant Oscillations of a Rigid Body Generated by Its Conical Precession”, Nelineinaya dinam., 14:4 (2018), 503–518  mathnet  crossref
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