Effect of the Gravity and Magnetic Field to Find Regular Precessions ‎of a Satellite-gyrostat with Principal Axes on a Circular Orbit

Document Type : Research Paper


Department of Mathematics, Faculty of Science, Mansoura University, Mansoura 35516, Egypt


We consider the motion of a magnetized satellite-gyrostat in a circular orbit due to the combined influence of uniform gravity and magnetic fields. Based on the Lagrangian equations, the necessary conditions for the existence of regular precessions are determined in which the axis of precession i\s perpendicular to the orbital plane. All possible regular precessions and permanent rotations are determined and classified. We show the usage of Lagrange equations taking Eulerian angles as generalized coordinates for determining the regular precessions is more effective and accurate than utilization of Euler-Poisson equations.


Main Subjects

Publisher’s Note Shahid Chamran University of Ahvaz remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

[1] Beletsky, V.V., Khentov, A.A., Rotational motion of a magnetized satellite, Moscow, Nauka, 1985.
[2] Likins, P. W., Stability of a symmetrical satellite in attitude fixed in an orbiting reference frame, Journal of the Astronautical Sciences, 12, 1965, 18-24.
[3] Balli, R., Pucci, E., Regular Precessions of Gyrostatic Satellite in a Circular Orbit, Celestial Mechanics, 17, 1978, 317-324.
[4] Yehia, H., Nabih, H., Stability Analysis of Plane Vibrations of a Satellite in a Circular Orbit, MCSER-CEMAS-Sapienza University of Rome-Italy, 2013.
[5] Beletskii, V.V., Motion of an artificial satellite about its center of mass, Nauka, Moscow-Russian, 1965.
[6] Markeev, A.P., Stability of plane oscillations and rotations of a satellite in a circular orbit, Kosmich. Issled-russian, (13), 1975, 322-336.
[7] Markeev, A.P., Bardin, On the stability of planar oscillations and rotations of a satellite in a circular orbit, Celestial Mechanics, 85, 2003, 51-66.
[8] Mothe, R.C.M., Orbital Spin: A New Hypothesis to Explain Precession of Equinox The Third Motion of Earth, International Journal of Astronomy and Astrophysics, 4, 2014, 20-28.
[9] Grioli, G., Esistenzae Determinazione Delle Prezessioni Regolari Dinamicamente Possibili per un solido Pesante Asimmetrico, Annali di Matematica Pura ed Applicata, 26, 1947, 271-281.
[10] Kharlamova, E.I., On a motion of a rigid body having a fixed point, Mekh. Tverd. Tela, 2, 1970, 35-37.
[11] Yehia, H.M., On the regular precession of an asymmetric rigid body acted upon by uniform gravity and magnetic fields, Egyptian Journal of Basic and Applied Sciences, 2, 2015, 200-205.
[12] Yehia, H.M., Regular precession of a rigid body (gyrostat) acted upon by an irreducible combination of three classical fields, Journal of the Egyptian Mathematical Society, 25, 2017, 216-219.
[13] Ol’shanskii V.Yu., On the Regular Precessions of an Asymmetric Liquid-Filled Rigid Body, Prikladnaya Matematika i Mekhanika, 82, 2018, 559–571.
[14] Ol’shanskii, V.Yu., Analysis of regular precession conditions for asymmetrical liquid-filled rigid bodies, Celestial Mechanics and Dynamical Astronomy, 46, 2020, 1-20.
[15] Leimanis, E., The General Problem of the motion of Couple Rigid Bodies about a fixed Point, Springer-Verlag, Berlin, Heidelberg, New York, 1965.
[16] Markeev, A.P., Sokol’skii, A.G., Chekhovskaya, T.N., Stability of the Conical Precession of a Dynamically Symmetric Rigid Body, Pis’ma Astron. Zh., 3, 1977, 333-336.
[17] Ivan, I.Sh., Andrej, G.S., Hyperboloidal Precession of a Dynamically Symmetric Satellite Construction of normal forms of the Hamiltonian, Celestial Mechanics and Dynamical Astronomy, 62, 1995, 289-304.
[18] Peter, C.H., Spacecraft Attitude Dynamics, Dover Publications, Inc. Mineola, New York, 1986.
[19] Markeev, A.P., Stability of the Cylindrical Precession of a Satellite in an Elliptic Orbit, Izv. Akad. Nauk SSSR, Mekh. Tverd. Tela, 2, 2008, 3-12.
[20] Markeev A.P., On the Stability of Steady Rotation of a Satellite around the Normal to the Orbital Plane, Prikladnaya Matematika i Mekhanika, 83, 2019, 691-703
[21] Andrzej, J.M., Regular Precessions in the Restricted Problem of the Rotational Motion, Acta Astronautica, 44, 1994, 301-316.
[22] Bushenkov, V.A., Ovchinnikov, M.Yu., Smirnov, G.V., Attitude Stabilization of a Satellite by Magnetic Coils, Acta Astronautica, 50, 2002, 721–728.
[23] Makeyev, N.N., Stability of Regular Precession a Gyrostat-magnetic in a Magnetic Field, Bulletin of The Perm University, 4, 2014, 35-42.
[24] Beleskey, V.V., Motion of a Satellite about Its Center of Mass in the Gravitational Field, Moscow: Moscow State University Press, 1975.
[25] Sukhov, E.A., On orbital stability of resonant periodic motions originating from hyperboloidal precession of a dynamically symmetric satellite, IOP Conference Series: Materials Science and Engineering, 927(1), 2020, 012021.
[26] Yehia, H.M., On the motion of a rigid body acted up on by potential and gyroscopic force I. the equation of motion and their transformation, Journal of Theoretical and Applied Mechanics, 5, 1986, 747-754.
[27] Gorr, G.V., Maznev, A.V., Shchetinina, E.K., Precessional Motions in Rigid Dynamics and Systems of Coupled Rigid Bodies, DNU, Donetsk, 2009.