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IssuesArchive of Issues2012-3pp.285-297

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V.S. Aslanov and S.P. Bezglasnyi, "Stability and Instability of Controlled Motions of a Two-Mass Pendulum of Variable Length," Mech. Solids. 47 (3), 285-297 (2012)
Year 2012 Volume 47 Number 3 Pages 285-297
DOI 10.3103/S002565441203003X
Title Stability and Instability of Controlled Motions of a Two-Mass Pendulum of Variable Length
Author(s) V.S. Aslanov (Korolyov Samara State Aerospace University, Moskovskoe sh. 34, Samara, 443086 Russia, aslanov_vs@mail.ru)
S.P. Bezglasnyi (Korolyov Samara State Aerospace University, Moskovskoe sh. 34, Samara, 443086 Russia, bezglasnsp@rambler.ru)
Abstract The problem of parametric control of plane motions of a two-mass pendulum (swing) is considered. The swing model is a weightless rod with two lumped masses one of which is fixed on the rod and the other slides along it within bounded limits. The control is the distance from the suspension point to the moving point. The proposed control law of swing excitation and damping consists in continuously varying the pendulum suspension length depending on the phase state. The stability of various controlled motions, including the motions near the upper and lower equilibria, is studied. The Lyapunov functions that prove the asymptotic stability and instability of the pendulum lower position in the respective cases of the pendulum damping and excitation are constructed for the proposed control law. The influence of the viscous friction forces on the pendulum stable motions and the onset of stagnation regions in the case of its excitation is analyzed. The theoretical results are confirmed by graphical representation of the numerical results.
Keywords two-mass pendulum, controlled system, asymptotic stability, Lyapunov function
References
1.  T. G. Strizhak, Methods for Studying `Pendulum'-Type Dynamical Systems (Nauka, Alma-Ata, 1981) [in Russian].
2.  K. Magnus, Vibrations. Introduction to the Study of Oscillatory Systems (Mir, Moscow, 1982) [in Russian].
3.  S. L. Chechurin, Parametric Vibrations and Stability of Periodic Motion (Izd-vo LGU, Leningrad, 1983) [in Russian].
4.  L. D. Akulenko, Asymptotic Methods of Optimal Control (Nauka, Moscow, 1987) [in Russian].
5.  A. P. Markeev, Theoretical Mechanics (CheRo, Moscow, 1999) [in Russian].
6.  Yu. F. Golubev, Foundations of Theoretical Mechanics (Izd-vo MGU, Moscow, 2000) [in Russian].
7.  A. P. Seyranian, "The Swing: Parametric Resonance," Prikl. Mat. Mekh. 68 (5), 847-856 (2004) [J. Appl. Math. Mech. (Engl. Transl.) 68 (5), 757-764 (2004)].
8.  A. A. Zevin and L. A. Filonenko, "A Qualitative Investigation of the Vibrations of a Pendulum with a Periodically Varying Length and a Mathematical Model of a Swing," Prikl. Mat. Mekh. 71 (6), 989-1003 (2007) [J. Appl. Math. Mech. (Engl. Transl.) 71 (6), 892-904 (2007)].
9.  L. D. Akulenko and S. V. Nesterov, "The Stability of the Equilibrium of a Pendulum of Variable Length," Prikl. Mat. Mekh. 73 (6), 893-901 (2009) [J. Appl. Math. Mech. (Engl. Transl.) 73 (6), 642-647 (2009)].
10.  L. D. Akulenko, "Parametric Control of Vibrations and Rotations of a Compound Pendulum (a Swing)," Prikl. Mat. Mekh. 57 (2), 82-91 (1993) [J. Appl. Math. Mech. (Engl. Transl.) 57 (2), 301-310 (1993)].
11.  E. K. Lavrovskii and A. M. Formal'skii, "Optimal Control of the Pumping and Damping of a Swing," Prikl. Mat. Mekh. 57 (2), 92-101 (1993) [J. Appl. Math. Mech. (Engl. Transl.) 57 (2), 311-320 (1993)].
12.  V. M. Volosov and B. I. Morgunov, Averaging Method in the Theory of Nonlinear Oscillatory Systems (Izd-vo MGU, Moscow, 1971) [in Russian].
13.  L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamlrelidze, and E. F. Mishchenko, Mathematical Theory of Optimal Processes (Nauka, Moscow, 1969) [in Russian].
14.  I. G. Malkin, Theory of Motion Stability (Nauka, Moscow, 1966) [in Russian].
Received 04 June 2010
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