Abstract:
We consider the motion of an asymmetric gyrostat under the attraction of a uniform Newtonian field. It is supposed that the center of mass lies along one of the principal axes of inertia, while a rotor spins around a different axis of inertia. For this problem, we obtain the possible permanent rotations, that is, the equilibria of the system. The Lyapunov stability of these permanent rotations is analyzed by means of the Energy–Casimir method and necessary and sufficient conditions are derived, proving that there exist permanent stable rotations when the gyrostat is oriented in any direction of the space. The geometry of the gyrostat and the value of the gyrostatic momentum are relevant in order to get stable permanent rotations. Moreover, it seems that the necessary conditions are also sufficient, but this fact can only be proved partially.
This work has been partially supported by the Spanish Ministry of Economy, projects MTM2014-59433-C2-2-P and ESP2013-44217-R. A. E. also acknowledges support from the group E-48 of the Aragon Government and FEDER funds.
Citation:
Manuel Iñarrea, Víctor Lanchares, Ana I. Pascual, Antonio Elipe, “On the Stability of a Class of Permanent Rotations of a Heavy Asymmetric Gyrostat”, Regul. Chaotic Dyn., 22:7 (2017), 824–839
\Bibitem{InaLanPas17}
\by Manuel I\~narrea, V{\'\i}ctor Lanchares, Ana I. Pascual, Antonio Elipe
\paper On the Stability of a Class of Permanent Rotations of a Heavy Asymmetric Gyrostat
\jour Regul. Chaotic Dyn.
\yr 2017
\vol 22
\issue 7
\pages 824--839
\mathnet{http://mi.mathnet.ru/rcd293}
\crossref{https://doi.org/10.1134/S156035471707005X}
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Xue Zhong, Jie Zhao, Kaiping Yu, Minqiang Xu, “Stability Analysis of Resonant Rotation of a Gyrostat in an
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A. A. Burov, “Motion of a Variable Body with a Fixed Point in a Time-Dependent Force Field”, Mech. Solids, 58:8 (2023), 2750
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A. A. Burov, “Motion of a Variable Body with a Fixed Point in a Time-dependent Force Field”, Prikladnaya matematika i mekhanika, 87:6 (2023), 984
Anton V. Doroshin, Mikhail M. Krikunov, “The generalized method of phase trajectory curvature synthesis in spacecraft attitude dynamics tasks”, International Journal of Non-Linear Mechanics, 147 (2022), 104246
Víctor Lanchares, Manuel Iñarrea, Ana Isabel Pascual, Antonio Elipe, “Stability Conditions for Permanent Rotations of a Heavy Gyrostat with Two Constant Rotors”, Mathematics, 10:11 (2022), 1882
Vladimir S. Aslanov, Dmitry A. Sizov, “Chaotic pitch motion of an aerodynamically stabilized magnetic satellite in polar orbits”, Chaos, Solitons & Fractals, 164 (2022), 112718
A. A. Elmandouh, F. H. Alsaad, “The stability of certain motion of a charged gyrostat in Newtonian force field”, Adv. Astron., 2021 (2021), 6660028
A. A. Elmandouh, A. G. Ibrahim, “Hamiltonian structure, equilibria, and stability for an axisymmetric gyrostat motion in the presence of gravity and magnetic fields”, Acta Mech., 230:7 (2019), 2539–2548
A. V. Doroshin, “Regimes of regular and chaotic motion of gyrostats in the central gravity field”, Commun. Nonlinear Sci. Numer. Simul., 69 (2019), 416–431
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