Stability and long-time behaviour of a rigid body containing a damper

This paper provides a comprehensive analysis of stability and long-time behaviour of a coupled system constituted by two rigid bodies separated by a thin layer of lubricant. We show that permanent rotations of the whole system, with the solids at relative rest, are exponentially stable if and only if the axis of rotation is the principal axis of inertia corresponding to the largest moment of inertia of the outer body. All other equilibria are normally hyperbolic, and hence unstable. In addition, we show that all solutions to the governing equations converge to an equilibrium configuration, no matter which initial conditions are chosen. Numerical evidence of the above results as well as conditions ensuring attainability of the stable configurations are also presented. For the stability analysis we use a linearization principle for dynamical systems possessing a center manifold. The characterization of the long-time behaviour is obtained by a careful analysis of the partially dissipative system of equations governing the motion of the coupled system.