Rotational motion of a free body induced by mass redistribution

Abstract

The effects of mass redistribution on the rotational motion of a free body, initially rotating uniformly about its axis of maximum moment of inertia, are studied using a simple three-body model composed of two points masses and a triaxial rigid body. An exact analytical solution for the motion of the original spin axis after redistribution of the masses is developed from the classical solution for the rotational motion of a free triaxial rigid body. Yhis motion in interpreted geometrically using the reciprocal inertia ellipsoid of the redistributed system and the angular momentum integral. Application of simple mass redistribution to the problem of spacecraft attitude control is discussed and results for a particular system are presented. T is a well-known fact1 that for a free, triaxial, rigid body, the condition of uniform rotation about its axis of in- termediate, principal moment of inertia is unstable. This in- stability manifests itself when small disturbances in the angular velocity components about the other principal axes occur; for then, the once-stationary axis of intermediate moment of inertia performs large-angle motions with respect to a nonrotating reference frame. Recently, Beachley and Dicker2 and Beachley,3 have suggested that this characteristic of rigid body rotational motion might be used to advantage to a spinning spacecraft in the sense that predictable large-angle motions of an axis fixed in the spacecraft might be induced by changing the mass distribution of the spacecraft.% A control system based on this idea might be more economical than a mass expulsion system which would rotate the angular momentum vector of the spacecraft. Changes in the mass distribution of the spacecraft may be accomplished by internal motion of control masses. The ef- fects of such internal mass motion on the rotational motion of a free system of bodies have been studied by several authors and a review of some of the work in this area is given by Lorell and Lange,5 who propose an automatic mass-trim system for spinning spacecraft which utilizes movable control masses. The primary purpose of this paper is not to study the effects of internal mass motion on the rotational motion of a free system of bodies, although these effects cannot be ignored en- tirely and will be considered. Our main goal is to obtain an analytical description of the spacecraft's rotational motion af- ter a redistribution of mass has occurred. Such a description has previously been obtained only in a limited way by numerical integration.3 We shall also provide geometrical in- terpretation of the analytical results and consider the use of simple mass redistribution as a means for spacecraft attitude control. In this paper we shall consider a system similar to those discussed in Ref. 3 and employ some results from the classical theory for the rotational motion of a free, triaxial, rigid body.