Stability Analysis of a Rigid Body with Attached Geometrically Nonlinear Rod by the Energy-Momentum Method

This paper applies the energy-momentum method to the problem of nonlinear stability of relative equilibria of a rigid body with attached flexible appendage in a uniformly rotating state. The appendage is modeled as a geometrically exact rod which allows for finite bending, shearing and twist in three dimensions. Application of the energy-momentum method to this example depends crucially on a special choice of variables in terms of which the second variation block diagonalizes into blocks associated with rigid body modes and internal vibration modes respectively. The analysis yields a nonlinear stability result which states that relative equilibria are nonlinearly stable provided that; (i) the angular velocity is bounded above by the square root of the minimum eigenvalue of an associated linear operator and, (ii) the whole assemblage is rotating about the minimum axis of inertia

Stability Analysis of a Rigid Body with a Flexible Attachment Using the Energy-Casimir Method

We consider a system consisting of a rigid body to which a linear extensible shear beam is attached. For such a system the Energy-Casimir method can be used to investigate the stability of the equilibria. In the case we consider, it can be shown that a test for (formal) stability reduces to checking the positive definiteness of two matrices which depend on the parameters of the system and the particular equilibrium about which the stability is to be ascertained

A block diagonalization theorem in the energy-momentum method

We prove a geometric generalization of a block diagonalization theorem first found by the authors for rotating elastic rods. The result here is given in the general context of simple mechanical systems with a symmetry group acting by isometries on a configuration manifold. The result provides a choice of variables for linearized dynamics at a relative equilibrium which block diagonalizes the second variation of an augmented energy these variables effectively separate the rotational and internal vibrational modes. The second variation of the effective Hamiltonian is block diagonal. separating the modes completely. while the symplectic form has an off diagonal term which represents the dynamic interaction between these modes. Otherwise, the symplectic form is in a type of normal form. The result sets the stage for the development of useful criteria for bifurcation as well as the stability criteria found here. In addition, the techniques should apply to other systems as well, such as rotating fluid masses

Stability of Relative Equilibria. Part II: Application to Nonlinear Elasticity

No Abstrac

Stability of coupled rigid body and geometrically exact rods, block diagonalization and the energy, momentum method

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 analysis of uniformly 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 accommodating 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 bloch diagonalization procedure, in which the second variation of the energy augmented with the linear and angular momentum block diagonalizes, separating the rotational from the internal vibration modes