Solutions Stability of Initial Boundary Problem, Modeling of Dynamics of Some Discrete Continuum Mechanical System
https://doi.org/10.18255/1818-1015-2015-2-197-208
Abstract
The solution stability of an initial boundary problem for a linear hybrid system of differential equations, which models the rotation of a rigid body with two elastic rods located in the same plane is studied in the paper. To an axis passing through the mass center of the rigid body perpendicularly to the rods location plane is applied the stabilizing moment proportional to the angle of the system rotation, derivative of the angle, integral of the angle. The external moment provides a feedback. A method of studying the behavior of solutions of the initial boundary problem is proposed. This method allows to exclude from the hybrid system of differential equations partial differential equations, which describe the dynamics of distributed elements of a mechanical system. It allows us to build one equation for an angle of the system rotation. Its characteristic equation defines the stability of solutions of all the system. In the space of feedback-coefficients the areas that provide the asymptotic stability of solutions of the initial boundary problem are built up.
About the Authors
D. A. Eliseev
P.G. Demidov Yaroslavl State University
Russian Federation
аспирант, Sovetskaya str., 14, Yaroslavl, 150000, Russia
Russian Federation
аспирант, Sovetskaya str., 14, Yaroslavl, 150000, Russia
E. P. Kubyshkin
P.G. Demidov Yaroslavl State University
Russian Federation
д-р физ.-мат. наук, профессор кафедры математического моделирования, Sovetskaya str., 14, Yaroslavl, 150000, Russia
Russian Federation
д-р физ.-мат. наук, профессор кафедры математического моделирования, Sovetskaya str., 14, Yaroslavl, 150000, Russia
References
1. Елисеев Д. А., Кубышкин Е. П., “Уравнения движения твердого тела с двумя упругими стержнями”, Модел. и анализ информ. систем, 21:1 (2014), 66—72; English transl.: Eliseev D. A., Kubyshkin E. P., “Motion Equations of a Rigid Body with Two Elastic Rods”, Automatic Control and Computer Sciences, 48:7 (2014), 516–520.
2. Злочевский С. И., Кубышкин Е. П., “О стабилизации спутника с гибкими стержнями”, Космич. исслед., 27:5 (1989), 643; [Zlochevskiy S. I., Kubyshkin E. P., “O stabilizatsii sputnika s gibkimi sterzhnyami”, Kosmich. issled., 27:5 (1989), 643, (in Russian).]
3. Злочевский С. И., Кубышкин Е. П., “О стабилизации спутника с гибкими стержнями. II”, Космич. исслед., 29:6 (1991), 828; [Zlochevskiy S. I., Kubyshkin E. P., “O stabilizatsii sputnika s gibkimi sterzhnyami. II”, Kosmich. issled., 29:6 (1991), 828, (in Russian).]
4. Бербюк В. Е., Динамика и оптимизация робототехнических систем, Наукова думка, Киев, 1989; [Berbyuk V. E., Dinamika i optimizatsiya robototekhnicheskikh sistem, Naukova dumka, Kiev, 1989, (in Russian).]
5. Черноусько Ф. Л., Болотник Н. Н., Градецкий В. Г., Манипуляционные роботы: динамика, управление, оптимизация, Наука, М., 1989; English transl.: Chernousko F. L., Bolotnik N. N., Gradetsky V. G., Manipulation Robots: Dynamics, Control and Optimization, CRC Press, Boca Raton, 1994.
6. Кубышкин Е. П., “Оптимальное управление поворотом твердого тела с гибким стержнем”, ПММ, 56:2 (1992), 240–249; English transl.: Kubyshkin Ye. P., “Optimal control of the rotation of a solid with a flexible rod”, Journal of Applied Mathematics and Mechanics, 56:2 (1992), 205–214.
7. Krabs W., Chi-Long N., “On the Controllability of a Robot Arm”, Mathematical Methods in the Applied Sciences, 21 (1998), 25–42.
8. Krabs W., “Controllability of a Rotating Beam”, Lecture Notes in Control and Information Sciences, 185 (1993), 447–458.
9. Leugering G., “On control and stabilization of a rotating beam by applying moments at the base only”, Lecture Notes in Control and Information Sciences, 149 (1991), 182–191.
10. Gugat M., “Controllability of a slowly rotating Timoshenko beam”, ESAIM: Control, Optimisation and Calculus of Variations, 6 (2001), 333–360.
11. Krabs W., Sklyar G. M., “On the controllability of a slowly rotating Timoshenko beam”, Z. Anal Anwend., 18:2 (1999), 437–448.
12. Krabs W., Sklyar G. M., “On the stabilizability of a slowly rotating Timoshenko beam”, Z. Anal Anwend., 19:1 (2000), 131–145.
13. Krabs W., Sklyar G. M., Wozniak J., “On the set of reachable states in the problem of controllability of rotating Timoshenko beam.”, J. Anal Appl., 22:1 (2003), 215–228.
14. Taylor S., Yau S., “Boundary control of a rotating Timoshenko beam.”, E, 44, 2003, 143–184.
15. Кубышкин Е. П., Хребтюгова О. А., “Обобщенное решение одной начально-краевой задачи, возникающей в механике дискретно-континуальных систем”, Модел. и анализ информ. систем, 19:1 (2012), 84–96; English transl.: Kubyshkin E. P., Khrebtyugova O. A., “A Generalized Solution of an Initial Boundary Value Problem Arising in the Mechanics of Discrete–Continuous Systems”, Automatic Control and Computer Sciences, 47:7 (2013), 556–565.
16. Aoustin Y., Formalsky A., “On the Synthesis of a nominal Trajectory for Control Law of a one-Link flexible Arm”, The lnternational Joumal of Robotics Research, 16:1 (1997), 36–46.
17. Фролов К. В. и др., Вибрации в технике: Справочник в 6 томах, 1, Машиностроение, M., 1978; [Frolov K. V. i dr., Vibratsii v tekhnike: Spravochnik v 6 tomakh, 1, Mashinostroenie, M., 1978, (in Russian).]
18. Беллман Р., Кук К., Дифференциально-разностные уравнения, Мир, М., 1967; English transl.: Bellman R., Cooke K., Differential-Difference Equations, Academic Press, 1963.
Review
For citations:
Eliseev D.A., Kubyshkin E.P. Solutions Stability of Initial Boundary Problem, Modeling of Dynamics of Some Discrete Continuum Mechanical System. Modeling and Analysis of Information Systems. 2015;22(2):197-208. (In Russ.) https://doi.org/10.18255/1818-1015-2015-2-197-208
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