Second-order total freedom analysis of 3D objects in a single point contact

Abstract

This paper studies the instantaneous second-order total freedom of two smooth objects initially in contact at a single point. The analytical determination of the set of physically allowed second-order motions is formulated in the Euclidean space using the screw theoretic concepts of twist and twist-derivative. Mathematical expression that characterizes the nature of second-order motion for an arbitrary contact geometry is derived in terms of twist coordinates, twist-derivative coordinates and the principle normal curvatures of the two surfaces at the point of contact. It is shown that the characteristic of this expression is equivalent to that of the derivative of reciprocal product of the instantaneous twist and contact normal line. The efficacy of the theory developed is demonstrated through two illustrative examples.