On the Stability of Steady Rotation of a Satellite around the Normal to the Orbital Plane
Tóm tắt
We study the rotational motion of a satellite (a rigid body) around the center of mass in a central Newtonian gravitational field in a circular orbit. The stability problem of a steady motion is solved for the case when the symmetry axis of the satellite is perpendicular to the orbital plane, and the satellite itself rotates about the symmetry axis with a constant angular velocity (cylindrical precession). The problem depends on two parameters, the dimensionless value of the absolute angular velocity of rotation of the satellite and the ratio of its axial and equatorial moments of inertia. The rigorous stability and instability conclusions were obtained for the parameter values that were not previously studied. Together with the known results of domestic and foreign authors, these conclusions give a rigorous and complete solution to the stability problem of the cylindrical precession of the satellite in a circular orbit for all values of the problem parameters.
Tài liệu tham khảo
G. N. Duboshin, “On the rotational motion of artificial celestial bodies,” Byull. Inst Teor. Astron. Akad. Nauk SSSR 7 (7), 511–520 (1960).
V. T. Kondurar’, “Particular solutions to the general problem of translational-rotational motion of a spheroid under the action of ball attraction,” Astron. Zh. 36 (5), 890–901 (1959).
W. T. Thomson, “Spin stabilitzation of attitude against gravity torque,” J. Astronaut. Sci. 9 (1), 31–33 (1962).
T. R. Kane, E. L. March, and W. G. Wilson, “Discussion on the paper: “Spin stabilization of attitude against gravity torque”, by W.T. Thomson,” J. Astronaut. Sci. 9 (4), 108–109 (1962).
F. L. Chernous’ko, “On the stability of regular precession of a satellite,” J. Appl. Math. Mech. 28 (1), 181–184 (1964).
P. W. Likins, “Stability of symmetrical satellite in attitude fixed in an orbiting reference frame,” J. Astronaut. Sci. 12 (1), 18–24 (1965).
T. R. Kane, “Attitude stability of Earth-pointing satellites,” AIAA J. 3 (4), 726–731 (1965).
V. V. Beletskii, Satellite’s Motion around Center of Mass in a Gravitational Field (MSU, Moscow, 1975) [in Russian].
V. A. Sarychev, “Problems of artificial satellite orientation,” in Science and Engineering Results (VINITI, Moscow, 1978), Vol. 11 [in Russian].
A. P. Markeev, Linear Hamiltonian Systems and Some Problems on Satellite Stability Motion around Its Center of Mass (Research Centre “Regular and chaotic dynamics”, Institute of Computer Science, Moscow, Izhevsk, 2009) [in Russian].
I. G. Malkin, Theory of Stability of Motion (Office of Technical Information, Washington, 1952).
V. I. Arnol’d, V. V. Kozlov, and A. I. Neishtadt, Mathematical Aspects of Classical and Celestial Mechanics, 3rd ed. (Springer, Berlin, 2006).
A. P. Markeev, Libration Points in Celestial Mechanics and Cosmodynamics (Nauka, Moscow, 1978) [in Russian].
A. P. Markeev, “Stability of a canonical system with two degrees of freedom in the presence of resonance,” J. Appl. Math. Mech. 32 (4), 766–772 (1968).
A. P. Markeev, “On one case of regular precession stability for a rigid body in the gravitational field,” Temat. Sb. Nauchn. Tr. Mosk. Aviats. Inst., No. 460, 13–17 (1978).
A. G. Sokol’skii, “Regular precessions stability for symmetrical satellite,” Kosm. Issled. 18 (5), 698–706 (1980).
A. P. Markeev, “The problem of the stability of the equilibrium position of a Hamiltonian system at 3:1 resonance,” J. Appl. Math. Mech. 65 (4), 639–645 (2001).
B. S. Bardin and A. J. Maciejewski, “Transcendental case in stability problem of Hamiltonian system with two degrees of freedom in presence of first order resonance,” Qual. Theory Dyn. Syst. 12 (1), 207–216 (2013).
F. L. Gantmacher, Lectures in Analytical Mechanics (Mir, Moscow, 1970).
A. G. Sokol’skii, “On stability of an autonomous Hamiltonian system with two degrees of freedom under first-order resonance,” J. Appl. Math. Mech. 41 (1), 20–28 (1977).
A. Kurosh, Higher Algebra (Mir, Moscow, 1970).