On periodic motions of a satellite-gyrostat with a large inner angular momentum

We consider an axially symmetric satellite with a large inner angular momentum (i.e. the satellite is a gyrostat) directed along the symmetry axis. The satellite center of mass moves along a circular orbit around an attractive center, the satellite attitude motion being affected by the gravitational torque. We investigate periodic motions of the satellite symmetry axis relative to the orbital coordinate system. Such motions are described by the autonomous system of differential equations of the fourth order that contains a large parameter. The generating solutions appear as a regular precession in the orbital coordinate system and as a rest in absolute space, the symmetry axis being formed a nonzero angle with the orbital plane. Earlier we investigated the limiting case of such periodic motions, when the symmetry axis lied in the orbital plane in generating solutions. Earlier we investigated the limiting case of such periodic motions, when the symmetry axis lied in the orbital plane in generating solutions. The period of the limiting solution is equal to a half of orbital one. The periods of new solutions depend on the angle mentioned above. To prove the existence of the new periodic solution, we reduce the boundary value problem that specifies them to the system of integral equations. The later is solved by the method of successive approximations.