Russian Journal of Nonlinear Dynamics
RUS  ENG    ЖУРНАЛЫ   ПЕРСОНАЛИИ   ОРГАНИЗАЦИИ   КОНФЕРЕНЦИИ   СЕМИНАРЫ   ВИДЕОТЕКА   ПАКЕТ AMSBIB  
Общая информация
Последний выпуск
Архив
Импакт-фактор

Поиск публикаций
Поиск ссылок

RSS
Последний выпуск
Текущие выпуски
Архивные выпуски
Что такое RSS



Rus. J. Nonlin. Dyn.:
Год:
Том:
Выпуск:
Страница:
Найти






Персональный вход:
Логин:
Пароль:
Запомнить пароль
Войти
Забыли пароль?
Регистрация


Rus. J. Nonlin. Dyn., 2020, том 16, номер 4, страницы 607–623 (Mi nd732)  

Mathematical problems of nonlinearity

On Nonlinear Oscillations and Stability of Coupled Pendulums in the Case of a Multiple Resonance

A. P. Markeevab, T. N. Chekhovskayab

a Ishlinsky Institute for Problems in Mechanics RAS, prosp. Vernadskogo 101-1, Moscow, 119526 Russia
b Moscow Aviation Institute (National Research University), Volokolamskoe shosse 4, Moscow, 125080 Russia

Аннотация: The points of suspension of two identical pendulums moving in a homogeneous gravitational field are located on a horizontal beam performing harmonic oscillations of small amplitude along a fixed horizontal straight line passing through the points of suspension of the pendulums. The pendulums are connected to each other by a spring of low stiffness. It is assumed that the partial frequency of small oscillations of each pendulum is exactly equal to the frequency of horizontal oscillations of the beam. This implies that a multiple resonance occurs in this problem, when the frequency of external periodic action on the system is equal simultaneously to two its frequencies of small (linear) natural oscillations. This paper solves the nonlinear problem of the existence and stability of periodic motions of pendulums with a period equal to the period of oscillations of the beam. The study uses the classical methods due to Lyapunov and Poincaré, KAM (Kolmogorov, Arnold and Moser) theory, and algorithms of computer algebra.
The existence and uniqueness of the periodic motion of pendulums are shown, its analytic representation as a series is obtained, and its stability is investigated. For sufficiently small oscillation amplitudes of the beam, depending on the value of the dimensionless parameter which characterizes the stiffness of the spring connecting the pendulums, the found periodic motion is either Lyapunov unstable or stable for most (in the sense of Lebesgue measure) initial conditions or formally stable (stable in an arbitrarily large, but finite, nonlinear approximation).

Ключевые слова: nonlinear oscillations, resonance, stability, canonical transformations

Финансовая поддержка Номер гранта
Российский научный фонд 19-11-00116
This work was carried out under the grant of the Russian Science Foundation (project No. 19-11-00116) at the Moscow Aviation Institute (National Research University) and at the Ishlinsky Institute for Problems in Mechanics of the Russian Academy of Science.


DOI: https://doi.org/10.20537/nd200406

Полный текст: PDF файл (373 kB)
Список литературы: PDF файл   HTML файл

Реферативные базы данных:

Тип публикации: Статья
MSC: 70E55, 70H09, 70H14
Поступила в редакцию: 31.10.2020
Принята в печать:16.11.2020

Образец цитирования: A. P. Markeev, T. N. Chekhovskaya, “On Nonlinear Oscillations and Stability of Coupled Pendulums in the Case of a Multiple Resonance”, Rus. J. Nonlin. Dyn., 16:4 (2020), 607–623

Цитирование в формате AMSBIB
\RBibitem{MarChe20}
\by A. P. Markeev, T. N. Chekhovskaya
\paper On Nonlinear Oscillations and Stability of Coupled Pendulums in the Case of a Multiple Resonance
\jour Rus. J. Nonlin. Dyn.
\yr 2020
\vol 16
\issue 4
\pages 607--623
\mathnet{http://mi.mathnet.ru/nd732}
\crossref{https://doi.org/10.20537/nd200406}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=4198783}


Образцы ссылок на эту страницу:
  • http://mi.mathnet.ru/nd732
  • http://mi.mathnet.ru/rus/nd/v16/i4/p607

    ОТПРАВИТЬ: VKontakte.ru FaceBook Twitter Mail.ru Livejournal Memori.ru


    Citing articles on Google Scholar: Russian citations, English citations
    Related articles on Google Scholar: Russian articles, English articles
  • Russian Journal of Nonlinear Dynamics
    Просмотров:
    Эта страница:41
    Полный текст:29
    Литература:2
     
    Обратная связь:
     Пользовательское соглашение  Регистрация посетителей портала  Логотипы © Математический институт им. В. А. Стеклова РАН, 2021