The stability of a family of non-trivial equilibrium orientations to the attracting centre of a gyrostat with an elastic rod

Abstract

The motion of a system consisting of a gyrostat and an elastic rod in a circular Kepler orbit in a central Newtonian force field is considered in a restricted formulation. The gyrostat is treated as a rigid body in which there is a dynamically rotating flywheel and a statically counterbalanced flywheel. The uniform elastic rod, which is rectilinear in the undeformed state, is rigidly fixed to the gyrostat housing at one end. The axis of the undeformed rod is arbitrarily located in the principal plane of inertia of the gyrostat. The relative displacements of the points of the system as a result of a small deformation of its elastic link are represented in the form of an infinite series of its expansion (without its a priori truncation) in a specified system of functions, which depend on the spatial coordinates, with unknown time-dependent coefficients. The orientation of the system for an attracting centre is defined by indicating the position with respect to the associated system of coordinates of the unit vectors of the normal to the plane of the orbit and the radius vector or transversal of the orbit at the centre of mass of the system. Here, these two unit vectors are located in the principal central plane of inertia of the gyrostat, containing the axis of the undeformed rod. The deformations of the rod, which naturally depend on the orientation and the gyrostatic moment which ensures equilibrium of the chosen orientation (non-trival equilibrium since, in this case, generally speaking, the rod is deformed), and its stability in the Lyapunov sense are determined for the two single parameter families of uniaxial orientations of the system to an attracting centre which have been separated out in this way.