On the 6 ́6 stiffness matrix for three dimensional motions

In this paper, methods of differential geometry and properties of Lie groups are used to show that in any conservative system subjected to a non zero external load, if one examines motions in SE(3), the special Euclidean group of rigid body motions in three dimensions, the resulting 6 ́6 stiffness matrix is asymmetric in any inertial (fixed) or body-fixed reference frame. We prove that in any subgroup of SE(3), when external forces or moments act on the system, the stiffness matrix, in general, is asymmetric if the Lie bracket of any two basis vectors is non zero. Further, we derive several useful properties of stiffness matrices using the ideas of covariant differentiation and the adjoint action of SE(3) on its Lie algebra, se(3). Finally, we outline a method to construct a symmetric stiffness matrix by choosing an appropriate moving reference frame that is not fixed to any rigid body. Introduction This paper considers the static analysis of conservative systems in which the associated potential energy, V, is a function of position only. We consider two rigid bodies, one fixed and the other free, connected by any coupling such that the system is conservative. The system is in equilibrium when the partial derivative of V with respect to each generalized coordinate, qi, vanishes. The stiffness matrix, K, consists of the second partial derivatives of the potential energy with respect to the generalized coordinates. Since