title: ÐÑÑÐ»ÐµÐ´Ð¾Ð²Ð°Ð½Ð¸Ðµ ÑÐ»Ð°Ð±Ð¾-Ð½ÐµÐ»Ð¸Ð½ÐµÐ¹Ð½Ð¾Ð¹ Ð¼Ð¾Ð´ÐµÐ»Ð¸ ÐºÐ¾Ð»ÐµÐ±Ð°Ð½Ð¸Ð¹ Ð´Ð²Ð¾Ð¹Ð½Ð¾Ð³Ð¾ Ð¼Ð°ÑÑÐ½Ð¸ÐºÐ°
title: Ð²ÑÐ¿ÑÑÐºÐ½Ð°Ñ ÐºÐ²Ð°Ð»Ð¸ÑÐ¸ÐºÐ°ÑÐ¸Ð¾Ð½Ð½Ð°Ñ ÑÐ°Ð±Ð¾ÑÐ° Ð±Ð°ÐºÐ°Ð»Ð°Ð²ÑÐ°
titleType: Subtitle
title: Research of a weakly nonlinear model of oscilations of a double pendulum
Creators: name: ÐÑÐ°Ð²ÑÐ¸Ð½ÑÐºÐ¸Ð¹, Ð¡ÐµÑÐ³ÐµÐ¹
Publisher: name: Ð¡Ð°Ð½ÐºÑ-ÐÐµÑÐµÑÐ±ÑÑÐ³ÑÐºÐ¸Ð¹ Ð¿Ð¾Ð»Ð¸ÑÐµÑÐ½Ð¸ÑÐµÑÐºÐ¸Ð¹ ÑÐ½Ð¸Ð²ÐµÑÑÐ¸ÑÐµÑ ÐÐµÑÑÐ° ÐÐµÐ»Ð¸ÐºÐ¾Ð³Ð¾
Year: 2023
Subjects: Ð´Ð²Ð¾Ð¹Ð½Ð¾Ð¹ Ð¼Ð°ÑÐµÐ¼Ð°ÑÐ¸ÑÐµÑÐºÐ¸Ð¹ Ð¼Ð°ÑÑÐ½Ð¸Ðº
ÑÐ»Ð°Ð±Ð°Ñ Ð½ÐµÐ»Ð¸Ð½ÐµÐ¹Ð½Ð¾ÑÑÑ
Ð°ÑÐ¸Ð¼Ð¿ÑÐ¾ÑÐ¸ÑÐµÑÐºÐ¸Ðµ Ð¼ÐµÑÐ¾Ð´Ñ
ÑÐ¾ÑÐ¼Ñ Ð¸ ÑÐ°ÑÑÐ¾ÑÑ Ð½ÐµÐ»Ð¸Ð½ÐµÐ¹Ð½ÑÑ ÐºÐ¾Ð»ÐµÐ±Ð°Ð½Ð¸Ð¹
double mathematical pendulum
weak nonlinearity
asymptotic methods
modes and frequencies of nonlinear oscillations

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description: This paper considers oscillations of a double mathematical pendulum with identical links and end weights. Exact nonlinear equations of motion of the system are derived, from which a classical linear model of small oscillations is obtained, as well as a weakly nonlinear oscillation model that considers cubic nonlinearity. A well-known solution to the problem of small oscillations of a double pendulum is given, which serves as a basis for further research. With the help of asymptotic methods of nonlinear mechanics, an approximate solution is constructed for a weakly nonlinear model, first when the system moves along each of the nonlinear vibration modes, and then a complete solution that is valid under arbitrary initial conditions. It is shown that the found complete solution has a nontrivial structure, representing polyharmonic oscillations at eight different frequencies. The resulting approximate formulas are accompanied by graphical illustrations that compare the behavior of the angles of deviation of the pendulum links from the vertical when using linear and weakly nonlinear models, as well as the original nonlinear model, which is calculated using numerical integration. The conclusions drawn are of interest for analytical mechanics and the theory of oscillations and can also be used in practice in problems of robotics and biomechatronics.