Equilibrium positions of a weight on a cable fixed to a dumbbell-shaped space station moving along a circular geocentric orbit

The motion of a space object in the gravitational field of the Earth is considered. The object consists of an extended space station and a weight, which is free to move along the cable fixed to the ends of the station. It is assumed that the station is composed of two masses coupled by a weightless rod, while the cable is weightless and non-stretched. The equations of motion of such a system are derived for the case when the motion proceeds in a single plane, while the center of mass of the system moves along a circular geocentric orbit. The conditions of the cable tension (conditions of being on tie) are derived. The phase portrait of the weight motion along the cable is constructed when the station is oriented to the attracting center or is perpendicular to this position. The possibility to leave the tie in this case is analyzed. Equilibrium configurations of the system are found, i.e., such motions of the object under consideration at which the weight does not change its position relative to the station. Lyapunov stability of such configurations is analyzed for two situations: when the station is composed of equal masses and when masses at the ends of the station are different. In particular, for the case of different masses it is established that there exist such positions of equilibrium at which the dumbbell is located at an angle to the direction to the attracting center. In some cases these positions can be stabilized (if the weight is fixed on the cable).