Orbital steady motions of a system of two bodies with an elastic connection

Steady motions of a pair of mass points connected by a rigid rod were studied by Abrarova and Karapetyan 1], the motions of a pair of mass points connected by an elastic spring were considered by Burov and Stepanov 2]. The present article is devoted to the existence, stability and bifurcation of the steady motions for a system composed of two bodies connected by an elastic torsional spring. Each of these bodies is a massless bar with mass points at the ends. One supposes that these bars are elastically connected and the system moves in the plane of the orbit of its mass centre. The trivial steady motions are studied (the principal axes of inertia of the system are directed as the radius-vector of the centre of inertia and as the tangent to the orbit). The non-trivial steady motions (the principal axes of inertia of the system are not directed as the radius vector or as the tangent to the orbit and the bars are not orthogonal each other) are also studied. We expose the complete study of the stability and of the bifurcation diagrams.