On planar oscillations of a body with a variable mass distribution in an elliptic orbit

Abstract

Planar motion of an orbiting body having a variable mass distribution in a central field of gravity is under analysis. Within the so-called ‘satellite approximation’ planar attitude dynamics is reduced to the 3/2-degrees of freedom description by one ODE of second order. The law of the mass distribution variations implying an existence of the special relative equilibria, such that the body is oriented pointing to the attracting centre by the same axis for any value of the orbit eccentricity is indicated. For particular example of an orbiting dumb-bell equipped by a massive cabin, wandering between the ends of the dumb-bell. For this example stability of the equilibria such that the dumb-bell ‘points to’ the attracting centre by one of its ends is studied. The chaoticity of global dynamics is investigated. Two important examples of a vibrating dumb-bell and of a dumb-bell equipped by a cabin wandering between its endpoints are considered. The dynamics of space objects, including moving elements, has been investigated by many authors. These studies usually have been connected with the necessity to estimate the influence of relative motions of moving parts, for example, crew motions [ 1 , 2 ], circulation of liquids [ 3 ], etc. on the attitude dynamics of a spacecraft. The development of projects of large-scale space systems with mobile elements, in particular, of satellite systems with tethered elements and space elevators, has posed problems related to their dynamics. Various aspects of the role of mass distribution even for the simplest orbiting systems, like dumb-bell systems are known since the publications [ 4 – 7 ], etc. The possibility of the sudden loss of stability because of the mass redistribution has been pointed out in reference [ 8 ] (see also references [ 9 – 13 ]). The considered system belongs to the mentioned class of systems and represents by itself one of the simplest systems allowing both analytical and numerical treatment, without supplementary simplifying assumptions such as smallness of the orbital eccentricity. Another set of applied problems is related to orientation keeping of the system for deployment and retrieval of tethered subsatellites as well as for relative cabin motions of space elevators. In particular, the problem of the stabilization/destabilization possibility for the given state of motion due to rapid oscillations of the cabin exists. This could be the subject of another additional investigation.