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Home » Issues » Archive of Issues » 2018 - 2 issue » pp. 240-251

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Russia Moscow
Year
2018
Volume
28
Issue
2
Pages
240-251
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Section Mechanics
Title On the dynamics of a pendulum mounted on a movable platform
Author(-s) Markeev A.P.abc, Sukhoruchkin D.A.a
Affiliations Ishlinsky Institute for Problems in Mechanics, Russian Academy of Sciencea, Moscow Aviation Instituteb, Moscow Institute of Physics and Technologyc
Abstract The motion of a mathematical pendulum mounted on a movable platform is considered. The platform rotates around a given vertical with a constant angular velocity $\omega$ and simultaneously executes harmonic oscillations with amplitude $A$ and frequency $\Omega$ along the vertical. The amplitude of oscillations is assumed to be small in comparison with the length $\ell$ of the pendulum $(A=\varepsilon \ell,\ 0<\varepsilon \ll 1) $. Three types of motions are considered. For the first two types, the pendulum is stationary relative to the platform and is located along its axis of rotation (hanging and inverted pendulum). For the third type of motions, the pendulum performs periodic oscillations with a period equal to the period of vertical oscillations of the platform. These oscillations have an amplitude of order $\varepsilon$ and at $\varepsilon = 0$ become relative equilibrium positions, in which the pendulum is a constant angle from the vertical. The motion of the third type exists if the angular velocity of rotation of the platform is large enough ($\omega^2 \ell>g$, $g$ is acceleration of gravity). In this paper, the problem of stability of these three types of pendulum motions for small values of $\varepsilon$ is solved. Both nonresonant cases and cases where resonances of the second, third and fourth orders occur in the system are considered. In the space of three dimensionless parameters of the problem, Lyapunov's stability and instability regions are singled out. The study is based on classical methods and algorithms due to Lyapunov, Poincaré and Birkhoff, as well as on modern methods of dynamical system analysis using Kolmogorov-Arnold-Moser (KAM) theory.
Keywords pendulum, resonance, Hamiltonian system, stability
UDC 531.36, 531.53
MSC 70E20, 70H14, 70K28
DOI 10.20537/vm180210
Received 17 May 2018
Language Russian
Citation Markeev A.P., Sukhoruchkin D.A. On the dynamics of a pendulum mounted on a movable platform, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2018, vol. 28, issue 2, pp. 240-251.
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