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B n =B b × B t = B b × (κ · B n) = −κ · B t; B t = κ · B n. 89) where κ denotes the curvature of our convex contour at point s, and the normal vector n is pointing inward. The binormal B b is constant for the plane case and thus independent of s. Now we imagine a point Σ moving along the contour with the velocity s. ˙ As a consequence the normal and the tangent vectors change their direction with respect to the body-fixed frame B. We describe this effect by differentiating the tangential and the normal vector with respect to time with ddt = dds · dd ts = dds · s˙ we come out with ˙ Bn = B n · s˙ = −κs˙ · B t, ˙ = B t · s˙ = +κs˙ · B n.
22 and differentiate it with respect to time resulting in ˙ IB · ABI + AIB · A ˙ BI = 0 and from that we get A ˙ IB · ABI = −AIB · A ˙ BI = −(A ˙ IB · ABI )T . 29 includes the following facts. Obviously the matrix expression ˙ IB · ABI is skew-symmetric. Additionally, it must represent angular rotaA tional velocities, which results from the rows of the matrix A being unitvectors in the coordinate system under consideration. 2 Kinematics 23 unit-vectors can only come out with a rotation, because the magnitudes of the vectors themselves do not change.
Some of these magnitudes we get by differentiation. In that case we should not forget the original state of the relative kinematic magnitudes for the special contact event under consideration, for example, in normal direction of a contact the event contact is indicated by the relative distance becoming then a constraint. In tangential direction the relative tangential velocity indicates the stick or slip situation. We shall come back to these properties later. 103 have not yet been fulfilled. The relative distance rD is then not perpendicular to the two surfaces Σ1 and Σ2 , but it represents some straight connection between the future contact points C1 and C2 .