Archive of Issues
Russia Moscow
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 Science^{a}, Moscow Aviation Institute^{b}, Moscow Institute of Physics and Technology^{c} 
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 KolmogorovArnoldMoser (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. 240251. 
References 

Full text 