Сonstrained control of motions of a two-mass pendulum
Mathematics. Physics. Mechanics
Аuthors1*, 2**, 1***
1. Samara National Research University named after Academician S.P. Korolev, 34, Moskovskoye shosse, Samara, 443086, Russia
2. Space Rocket Centrе Progress, 18, Zemets str., Samara, 443009, Russia
In this note the problem of control of plane motions of a two-mass parametric pendulum is considered. The pendulum is modeled by two equivalent weightless rods with two equivalent point masses. They are fixed on the rods and moving along the circle with the center at the pivot. Pendulum can perform rotational or oscillatory motions in a vertical plane around the point of attachment. Pendulum motion occur in a uniform gravitational field. Two-mass pendulum has two equilibrium positions. Lower position is stable and upper position is instable. Friction forces is neglected.
Possibility of parametric control of pendulum excitation and swing damping in the vicinity of the lower equilibrium position are discussed. The control is realized by varying continuously the angle between two rods. It is a function that depends on the representative point of motion of the gravity center of pendulum. We assume about the restrictions on move of the gravity center of pendulum along the bisectrix of angle between rods.
The aim of this paper is to build new control laws with specified properties that implement processes of excitation and damping pendulum near the lower equilibrium position.
The problem is solved by the method of Lyapunov’s functions of the classical theory of stability.
In this paper the two control law processes of excitation and damping pendulum with the assumption of restrictions on the movement of the gravity center are constructed. The Lyapunov’s 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. It is shown that under controlled movements according to the first law occurs the asymptotic damping oscillation amplitude of pendulum for any initial conditions movements. When control according to the second law is the growth of the amplitude and the transition from oscillatory to rotational motions. The theoretical results are illustrated by graphical representation of the numerical results.
Results of paper can be used in modeling and control of plane pendulum motions of various mechanical systems.
Keywords:two-mass pendulum , management, Lyapunov's function, asymptotical stability
- Krasil’nikov P.S. Prikladnaya matematika i mekhanika, 2012, vol. 76, no. 1, pp. 36–51.
- Markeev A.P. Nelineinaya dinamika, 2010, vol. 6, no. 3, pp. 605–622.
- Andreev A.S. Zhurnal Srednevolzhskogo matematicheskogo obshchestva. 2010, vol. 12, no. 4, pp. 64–73.
- Bezglasnyi S.P., Mysina O.A. Izvestiya Saratovskogo universiteta. Matematika. Mekhanika. Informatika, 2008, vol. 8, no. 4, pp. 44-52.
- Bezglasnyi S.P., Batina E.S., Vorob’ev A.S. Izvestiya Saratovskogo universiteta. Matematika. Mekhanika. Informatika, 2013, vol. 13, no. 4 (1), pp. 36-42.
- Strizhak T.G. Metody issledovaniya dinamicheskikh sistem tipa «mayatnik» (Methods for Studying «Pendulum»-Type Dynamical Systems), Alma-Ata, Nauka, 1981, 253 p.
- Seiranyan A.P. Prikladnaya matematika i mekhanika, 2004, vol. 68, no. 5, pp. 847–856.
- Akulenko L.D., Nesterov S.V. Prikladnaya matematika i mekhanika, 2009, vol. 73, no. 6. pp. 893-901.
- Akulenko L.D. Prikladnaya matematika i mekhanika,1993, vol.57. no 2. pp. 82-91.
- Lavrovskii E.K., Formal’skii A.M. Prikladnaya matematika i mekhanika, 1993, vol. 57, no. 2, pp. 92-101.
- Aslanov V.S., Bezglasnyi S.P. Izvestiya RAN. Mekhanika tverdogo tela, 2012, no. 3. pp. 32-46.
- Aslanov V.S., Bezglasnyi S.P. Prikladnaya matematika i mekhanika, 2012, vol. 76, no. 4, pp. 565-575.
- Bezglasnyi S.P., Piyakina E.E., Talipova A.A. Avtomatizatsiya protsessov upravleniya, 2013, vol. 34, no. 4, pp. 35-41.
- Bezglasnyi S.P., Batina E.S., Piyakina E. E., Elektronnyi zhurnal «Trudy MAI», 2014, № 72, available at: http://www.mai.ru/science/trudy/published.php?ID=47314 (accessed 27.01.2014).
- Malkin I.G. Teoriya ustoichivosti dvizheniya (Theory of Motion Stability), Moscow, Nauka, 1966, 530 p.