Mechanics of Solids (about journal) Mechanics of Solids
A Journal of Russian Academy of Sciences
in January 1966
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Online ISSN 1934-7936

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M. K. Nabiullin, "Stability of equilibrium states of a tether orbital space system," Mech. Solids. 39 (4), 5-14 (2004)
Year 2004 Volume 39 Number 4 Pages 5-14
Title Stability of equilibrium states of a tether orbital space system
Author(s) M. K. Nabiullin (Irkutsk)
Abstract The problem of stabilization of equilibrium of a tether space orbital system is considered. The system consists of a carrier body with flywheels and a probe satellite attached to the carrier body by means of a long heavy tether. The satellite is treated as a point mass and the tether as a flexible thread that does not resist bending and torsion. The center of mass of the carrier body with flywheels is assumed to move in a prescribed circular Keplerian orbit in the Newtonian central field. Particular solutions of nonlinear differential equations with ordinary and partial derivatives that correspond to the equilibriums of the tether orbital system in the orbital reference frame are found. The principal central axes of the system in the equilibriums are collinear to the axes of the orbital reference frame. The tether with the satellite can be arranged along the radius of the orbit and point toward or away from the attraction center or along the tangent to the orbit. Sufficient conditions for stability of these equilibriums are established and analyzed on the basis of the Lyapunov functional method.

Issues of nonlinear dynamics, stability, libration oscillations, and control of various tether systems in a circular orbit have been studied by numerous authors [1-8]. These studies take into account the aerodynamic forces, magnetic forces, and thrust forces applied to the carrier body. The stability of the tether system equilibriums were studied in [9-11]. The results of the present paper are compared with the familiar ones.
1.  V. A. Sarychev, Space Exploration. Volume 2 [in Russian], VINITI, Moscow, 1978.
2.  V. V. Beletskii and E. M. Levin, Dynamics of Tether Space Systems [in Russian], Nauka, Moscow, 1990.
3.  A. K. Misra and V. J. Modi, "A survey on the dynamics and control of tether satellite systems," in Tethers in Space: Proc. Int. Conf., Arlington, Va, Sept. 17-19, 1986, pp. 667-719, San Diego (Calif.), 1987.
4.  V. A. Sarychev, Yu. A. Sadov, A. L. Petrov, and O. O. Galaktionova, Some Specific Features of Nonlinear Oscillations of an Inextensible Thread in the Plane of a Circular Orbit. Preprint No. 35, In-t Prikl. Matem., Moscow, 1989.
5.  N. E. Bolotina and V. G. Vil'ke, "Stability of equilibrium states of a flexible heavy thread tied to a satellite in a circular orbit," Kosmicheskie Issledovaniya [Cosmic Research], Vol. 16, No. 4, pp. 621-626, 1978.
6.  S. I. Klimov, M. L. Pivovarov, E. N. Alekseeva, A. V. Prudkoglyad, and V. G. Rodin, "Utilization of tether systems for basic research of the magnetosphere. Dynamics of unfolding and operation of the tether system," Kosmicheskie Issledovaniya [Cosmic Research], Vol. 34, No. 1, pp. 106-109, 1996.
7.  V. V. Sidorenko, "Evolution of fast rotations of a tether orbital system," Kosmicheskie Issledovaniya [Cosmic Research], Vol. 33, No. 1, pp. 36-39, 1995.
8.  Yu. A. Sadov, "Equilibrium configurations of a tether orbital system with the atmosphere resistance being taken into account," Kosmicheskie Issledovaniya [Cosmic Research], Vol. 34, No. 1, pp. 73-80, 1996.
9.  M. K. Nabiullin, Steady-state motion and Stability of Flexible Satellites [in Russian], Nauka, Novosibirsk, 1990.
10.  M. K. Nabiullin, "Rotation and stability of an equilibrium of a tether orbital system," Izv. AN. MTT [Mechanics of Solids], No. 6, pp. 3-12, 1992.
11.  M. K. Nabiullin, Stationary Motions and Stability of Flexible Satellites, Advanced Series in Mathematical Science and Engineering, World Federation Publishing Company, Atlanta, 1995, 1996.
12.  A. I. Lur'e, Analytical Mechanics [in Russian], Fizmatgiz, Moscow, 1961.
13.  E. Janke, F. Emde, and F. S. Lösch, Special Functions [Russian translation], Nauka, Moscow, 1977.
14.  A. A. Movchan, "Stability of processes with respect to two metrics," PMM [Applied Mathematics and Mechanics], Vol. 24, No. 6, pp. 988-1001, 1960.
15.  V. V. Rumyantsev, "On the motion and stability of an elastic body with a cavity containing fluid," PMM [Applied Mathematics and Mechanics], Vol. 33, No. 6, pp. 346-357, 1969.
16.  T. K. Sirazetdinov, Stability of Distributed Systems [in Russian], Nauka, Novosibirsk, 1987.
17.  V. V. Rumyantsev, Stability of Steady-state Motions of Satellites [in Russian], Izd-vo VTs AN SSSR, Moscow, 1967.
18.  V. V. Beletskii, Motion of a Satellite about its Center of Mass [in Russian], Nauka, Moscow, 1965.
Received 04 July 2001
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