Geodesic
On control of motion of a parametric pendulum
Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, Tome 15 (2015) no. 1, pp. 67-73.

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The paper is devoted to a passive control problem. The problem of control of plane motions of a two-mass parametric pendulum in a uniform gravitational field is considered. The problem is important for and necessary in software design of automated systems for control of mechanisms. In particular, it can be applied to various modeling problems of pendulum motions of mechanical systems. The pendulum is modeled by two equivalent weightless rods with two equivalent point masses moving along the circle centered at the pivot. The control is carried out by varying continuously the angle between two rods. It is a function that depends on the representative point of the gravity center of pendulum in the phase plane. Two control processes of excitation and damping pendulum near the lower equilibrium position by swing principle are constructed. The problem is resolved by the method of Lyapunov's functions known from the classical theory of stability. The control is obtained in the form of closed form solution in the class of continuous functions. The obtained results are an important contribution to development of control mechanisms in engineering. 
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S. P. Bezglasnyi. On control of motion of a parametric pendulum. Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, Tome 15 (2015) no. 1, pp. 67-73. http://geodesic.mathdoc.fr/item/ISU_2015_15_1_a9/

[1] Krasil'nikov P. S., “The Non-linear oscillations of a pendulum of variable length on a vibrating base”, J. Prikl. Mat. Mekh., 76:1 (2012), 25–35

[2] Andreev A. S., “Lyapunov Functions Method in the Problems of Control”, J. Srednevolzhskogo Math. Obschestva, 12:4 (2010), 64–73 (in Russian)

[3] Andreev A. S., “On the stability of equilibrium position of the non-autonomous mechanical system”, J. Prikl. Mat. Mekh., 60:3 (1996), 388–396

[4] Bezglasnyi S. P., Mysina O. A., “Stabilization of Program Motions of a Rigid Body on a Moving Platform”, Izv. Saratov Univ. (N. S.), Ser. Math. Mech. Inform., 8:4 (2008), 44–52 (in Russian)

[5] Bezglasnyi S. P., Batina E. S., Vorobyov A. S., “Synthesis of Asymptotically Stable Motion of a Robot Arm Manipulator”, Izv. Saratov Univ. (N. S.), Ser. Math. Mech. Inform., 13:4(1) (2013), 36–42 (in Russian)

[6] Strizhak T. G., Methods for Studying «Pendulum»-Type Dynamical Systems, Nauka, Alma-Ata, 1981, 253 pp. (in Russian)

[7] Seyranian A. P., “The swing: Parametric resonance”, J. Prikl. Mat. Mekh., 68:5 (2004), 757–764

[8] Akulenko L. D., Nesterov S. V., “The stability of the equilibrium of a pendulum of variable length”, J. Prikl. Mat. Mekh., 73:6 (2009), 642–647

[9] Akulenko L. D., “Parametric control of vibrations and rotations of a compound pendulum (a swing)”, J. Prikl. Mat. Mekh., 57:2 (1993), 301–310

[10] Lavrovskii E. K., Formal'skii A. M., “Optimal control of the pumping and damping of a swing”, J. Prikl. Mat. Mekh., 57:2 (1993), 311–320

[11] Aslanov V. S., Bezglasnyi S. P., “Stability and instability of controlled motions of a two-mass pendulum of variable length”, Mechanics of Solids, 47:3 (2012), 285–297 | DOI

[12] Aslanov V. S., Bezglasnyi S. P., “Gravitational stabilization of a satellite using a movable mass”, J. Prikl. Mat. Mekh., 76:4 (2012), 405–412

[13] Bezglasnyi S. P., Pijakina E. E., Talipova A. A., “Limited Control of Dual-mass Pendulum Motion”, Avtomatizacija processov upravlenija, 34:4 (2013), 35–41 (in Russian)

[14] Malkin I. G., Theory of Motion Stability, Nauka, M., 1966, 530 pp. (in Russian)