DOI 10.15507/2079-6900.24.202201.66-75
Original article
ISSN 2079-6900 (Print)
ISSN 2587-7496 (Online)
MSC2020 34A05, 70E17, 70E40
On the Movement of Gyrostat under the Action of Potential and Gyroscopic Forces
A. A. Kosov, E. I. Semenov
Matrosov Institute for System Dynamics and Control Theory of Siberian Branch of Russian Academy of Sciences (Irkutsk, Russian Federation)
Abstract. A system of differential equations is considered that describes the motion of a gyrostat under the action of the moment of potential, gyroscopic and circular-gyroscopic forces. The form of the moment of forces is indicated for which the system has the three first integrals of a given form. An analog of V.I. Zubov’s theorem for representing solutions of gyrostat equations by power series is given, and the possibility of using this approach to predict motions is shown. For an analogue of the Lagrange case, integration in quadratures is performed. Analogues of the case of full dynamical symmetry and the Hess case are also indicated. Based on the principle of optimal damping developed by V.I. Zubov, a design of the control moment created by circular-gyroscopic forces is proposed, which ensures that one of the coordinates reaches a constant (albeit unknown in advance) value or the transition of the state vector to the level surface of the particular Hess integral. A numerical example is given, for which a two-parameter family of exact almost periodic solutions, represented by trigonometric functions, is found.
Key Words: gyrostat, moment of potential and gyroscopic forces, first integrals, integrability, exact solutions, analogues of classical cases, control
For citation: A. A. Kosov, E. I. Semenov. On the Movement of Gyrostat under the Action of Potential and Gyroscopic Forces. Zhurnal Srednevolzhskogo matematicheskogo obshchestva. 24:1(2022), 66–75. DOI: https://doi.org/10.15507/2079-6900.24.202201.66-75
Submitted: 10.11.2021; Revised: 19.02.2022; Accepted: 24.02.2022
Information about the authors:
Alexander A. Kosov, Leading researcher, Matrosov Institute for System Dynamics and Control Theory of Siberian Branch of Russian Academy of Sciences (134, Lermontov Str., Irkutsk, 664033, Russia), Ph. D. (Physics and Mathematics), ORCID: https://orcid.org/0000-0003-1352-1828, kosov_idstu@mail.ru
Eduard I. Semenov, Senior researcher, Matrosov Institute for System Dynamics and Control Theory of Siberian Branch of Russian Academy of Sciences (134, Lermontov Str., Irkutsk, 664033, Russia), Ph. D. (Physics and Mathematics), ORCID: https://orcid.org/0000-0002-9768-9945, edwseiz@gmail.com
All authors have read and approved the final manuscript.
Conflict of interest: The authors declare no conflict of interest.
This is an open access article distributed under the terms of the Creative Commons Attribution 4.0 International License. 