Geodesic
On the Movement of Gyrostat under the Action of Potential and Gyroscopic Forces
Žurnal Srednevolžskogo matematičeskogo obŝestva, Tome 24 (2022) no. 1, pp. 66-75.

Voir la notice de l'article provenant de la source Math-Net.Ru

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.
Keywords: gyrostat, moment of potential and gyroscopic forces, first integrals, integrability, analogues of classical cases, control.
Mots-clés : exact solutions 
@article{SVMO_2022_24_1_a5,
     author = {A. A. Kosov and \`E. I. Semenov},
     title = {On the {Movement} of {Gyrostat} under the {Action} of {Potential} and {Gyroscopic} {Forces}},
     journal = {\v{Z}urnal Srednevol\v{z}skogo matemati\v{c}eskogo ob\^{s}estva},
     pages = {66--75},
     publisher = {mathdoc},
     volume = {24},
     number = {1},
     year = {2022},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/SVMO_2022_24_1_a5/}
}
TY  - JOUR
AU  - A. A. Kosov
AU  - È. I. Semenov
TI  - On the Movement of Gyrostat under the Action of Potential and Gyroscopic Forces
JO  - Žurnal Srednevolžskogo matematičeskogo obŝestva
PY  - 2022
SP  - 66
EP  - 75
VL  - 24
IS  - 1
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/SVMO_2022_24_1_a5/
LA  - ru
ID  - SVMO_2022_24_1_a5
ER  - 
%0 Journal Article
%A A. A. Kosov
%A È. I. Semenov
%T On the Movement of Gyrostat under the Action of Potential and Gyroscopic Forces
%J Žurnal Srednevolžskogo matematičeskogo obŝestva
%D 2022
%P 66-75
%V 24
%N 1
%I mathdoc
%U http://geodesic.mathdoc.fr/item/SVMO_2022_24_1_a5/
%G ru
%F SVMO_2022_24_1_a5
A. A. Kosov; È. I. Semenov. On the Movement of Gyrostat under the Action of Potential and Gyroscopic Forces. Žurnal Srednevolžskogo matematičeskogo obŝestva, Tome 24 (2022) no. 1, pp. 66-75. http://geodesic.mathdoc.fr/item/SVMO_2022_24_1_a5/

[1] V. V. Golubev, Lectures on Integration of the Equations of Motion of a Rigid Body about a Fixed Point, Israeli Program for Scientific Translations, Israeli, 1960, 287 pp. | MR | Zbl

[2] I. N. Gashenenko, G. V. Gorr, A. M. Kovalev, Classical problems in the dynamics of rigid body, Naukova Dumka Publ., Kiev, 2012, 441 pp. | MR

[3] V. I. Zubov, Analytical dynamics of the system of bodies, LSU publishing house, Leningrad, 1983, 344 pp.

[4] S. Nikolov S., N Nedkova, “Dynamical Behavior of a Rigid Body with One Fixed Point (Gyroscope). Basic Concepts and Results. Open Problems: a Review”, Journal of Applied and Computational Mechanics, 1:4 (2015), 187–206

[5] A. V. Belyaev, “On the general solution of the problem of the motion of a heavy rigid body in the Hess case”, Sbornik: Mathematics, 206:5 (2015), 621-649 | MR | Zbl

[6] M. Romano, “Exact analytic solution for the rotation of a rigid body having spherical ellipsoid of inertia and subjected to a constant torque”, Celestial Mech. Dyn. Astr., 100:3 (2008), 181–189 | MR | Zbl

[7] G. V. Gorr, A. V. Maznev, “On solutions of the equations of motion of a rigid body in the potential force field in the case of constant modulus of the kinetic moment”, Rigid Body Mechanics, 47 (2017), 12–24 | MR

[8] G. V. Gorr, A. V. Maznev, “Precession and isoconic motions of a rigid body under the potential and gyroscopic forces”, Rigid Body Mechanics, 45 (2015), 26–39 | MR

[9] H. M. Yehia, A. A. Elmandouh, “Regular Precession of a Rigid Body (Gyrostat) Acted upon by an Irreducible Combination of Three Classical Fields”, Journal of Physics A. Mathematical and Theoretical, 46:14 (2013), 142001 | MR | Zbl

[10] H. M. Yehia, A. A. Elmandouh, “A new conditional integrable case in the dynamics of a rigid body-gyrostat”, Mech. Res. Commun, 78 (2016), 25–27 | MR

[11] A. A. Kosov, E. I. Semenov, “On first integrals and stability of stationary motions of gyrostat”, Physica D: Nonlinear Phenomena, 430 (2022), 133103 | MR | Zbl

[12] V. I. Zubov, Problem of stability of control processes, Publishing house Sudostroenie, Leningrad, 1980, 253 pp. | MR