14 / 15
Е.А. Сухов, Б.С. Бардин
14
Инженерный журнал: наука и инновации
# 11·2017
Numerical and analytical plotting of periodic motion
and investigating motion stability in the case of a symmetric
satellite
©
E.A. Sukhov
1
, B.S. Bardin
1,2
1
Moscow Aviation Institute (MAI), Moscow, 125993, Russia
2
Blagonravov Mechanical Engineering Research Institute of the Russian Academy of
Sciences, Moscow, 101990, Russia
A specific case of motion of a solid dynamically symmetric satellite along a circular orbit
in reference to the centre of mass is its hyperboloid precession. If the hyperboloid preces-
sion is stable, the equations of satellite motion allow for existence of periodic motion
families that describe the oscillations of the satellite's dynamical symmetry axis in the
vicinity of the hyperboloid precession. It is possible to derive these families in the form of
convergent series in powers of the small parameter, i.e. the oscillation amplitude. There
exist two types of these motions: short-term and long-term. If the amplitude is not small,
it is necessary to employ a numerical method in order to plot the motions. In the three-
dimensional space of the problem parameters, the authors plotted the region where long-
term motions exist that stem from the hyperboloid precession of a symmetric satellite. We
deal with the cases of resonance being present and third order resonance being absent.
We conducted a first-order investigation of the orbital stability problem for long-term
motions. We provide the problem statement and the results of plotting the periodic mo-
tions analytically in the absence of resonances. We describe in brief the method for plot-
ting the periodic solution families numerically. We present the results of numerical and
analytical plotting of long-term solution families stemming from the hyperboloid preces-
sion in the vicinity of the resonance. We draw conclusions on the first-order orbital sta-
bility of said solutions for small amplitudes.
Keywords:
Hamiltonian system, periodic motions, dynamically symmetric satellite, nu-
merical continuation of solution families, regular precession, orbital stability, resonance
REFERENCES
[1]
Duboshin G.N.
Byul. ITA AN SSSR — Bulletin of the Institute of Theoretical
Astronomy of the USSR Academy of Sciences
, 1960, vol. 7, no. 7, pp. 511–520.
[2]
Kondurar V.T.
Astronomicheskiy zhurnal — Astronomy Reports
, 1959, vol. 36,
no. 5, pp. 890–901.
[3]
Beletskiy V.V.
Dvizhenie sputnika otnositelno tsentra mass v gravitatsionnom
pole
[Satellite motion in a gravitational field in reference to the centre of mass].
Moscow, Lomonosov Moscow State University Publ., 1975, 308 p.
[4]
Sokolskiy A.G., Khovanskiy S.A.
Kosmicheskie issledovaniya — Cosmic
Research
, 1979, vol. 17, no. 2, pp. 208–217.
[5]
Markeev A.P.
Lineynye gamiltonovy sistemy i nekotorye zadachi ob
ustoychivosti dvizheniya sputnika otnositelno tsentra mass
[Linear Hamiltonian
systems and certain stability problems for motion of a satellite in reference to its
centre of mass]. Moscow, Izhevsk, Regular and Chaotic Dynamics Publ.,
Computer Research Institute Publ., 2009, 369 p.
[6]
Chernousko F.L.
Prikladnaya matematika i mekhanika — Journal of Applied
Mathematics and Mechanics
, 1964, vol. 28, no. 1, pp. 155–157.