Stability of coupled rigid body and geometrically exact rods: Block diagonalization and the energy-momentum method

Abstract

This paper develops and applies the energy-momentum method to the problem of nonlinear stability of relative equilibria. The method is applied in detail to the stability of uniformily rotating states of geometrically exact rod models, and a rigid body with an attached flexible appendage. Here, the flexible appendage is modeled as a geometrically exact rod capable of accomodating arbitrarily large deformations in three dimensions; including extension, shear, flexure and twist. The model is said to be ‘geometrically exact’ because of the lack of restrictions of the allowable deformations, and the full invariance properties of the model under superposed rigid body motions. We show that a (sharp) necessary condition for nonlinear stability is that the whole assemblage be in uniform (stationary) rotation about the shortest axis of a precisely defined ‘locked’ inertia dyadic. Sufficient conditions are obtained by appending the restriction that the angular velocity of the stationary motion be bounded from above by the square root of the minimum eigenvalue of an associated linear operator. Specific examples are worked out, including the case of a rod attached to a rigid body in uniform rotation. Our technique depends crucially on a special choice of variables, introduced in this paper and referred to as the block diagonalization procedure, in which the second variation of the energu augmented with the linear and angular momentum block diagonalizes, separating the rotational from the internal vibration modes.