On the 6 × 6 cartesian stiffness matrix for three-dimensional motions

Abstract

It has been recently observed by Pigoski et al. [1, 2], and by Ciblak and Lipkin [3] that the Cartesian stiffness matrix associated with a linear elastic coupling between two rigid bodies is, in general, asymmetric if the resulting forces and moments do not sum to zero. 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 motions in SE(3), the special Euclidean group of rigid body motions in three dimensions, are considered, the resulting 6 × 6 Cartesian stiffness matrix is asymmetric in any inertial (fixed) or body-fixed reference frame. We prove the general result that in any subgroup of SE(3), if the system is subject to external forces and moments, the Cartesian stiffness matrix is symmetric if finite displacements along the basis twists are used to generate the stiffness matrix commute. Further, we derive several useful properties of stiffness matrices using ideas from Lie theory. In particular, we offer a simple proof to show that the stiffness matrix in the body-fixed reference frame is the transpose of the stiffness matrix in the inertial reference frame, a result also derived in [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.