By William Elwood Byerly
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Extra resources for An introduction to generalized coordinates in mechanics and physics
5. A Geometric Impact Law We give now a geometric interpretation of Moreau's impact law (Moreau, 1988) which is a unilateral version of Newton's classical kinematic impact law, see Glocker (2001) for a detailed discussion. For brevity, explicit dependence on time t is suppressed. l satisfy (11) 50 Figure 3. The geometry of impacts with global dissipation index. This decomposition is unique. The tenu v plays the role of the tangential component of u- which remains unchanged by the impact. The normal component vol is "inverted" by the impact rule V X := -EVol, where E is the global coefficient of restitution, a magnitude expressing the global dissipation behaviour of the impact when chosen between 0 and 1.
In this case u+ == u-, thus no impact occurs. 6. Extension to Re-entrant Corners Finally we extend the impact law (10)-(12) to re-entrant corner points. Note that such a situation can not be generated by the intersection of smooth simple unilateral constraints, but occurs in practice when, for example, two rectangular blocks hit each other at their corners. 2 and equations (4), (5a) we first leave the symmetric formulation (11) and replace it by the equivalent conditions v E Cq'D, vi. E C;-Cq'D c civ.
Nevertheless, this is a simple enough assumption, which leads to a number of definite conclusions: first, we obtain a kinetic coefficient of restitution when there is only one contact interval; this coefficient is qualitatively correct for the eight steel bars which have been tested by Stoianovici and Hurmuzlu, and quantitatively correct for the five longer ones. In section 4, we go beyond the first microcontact time interval and we produce an approximation of the solution, which enables us to find the smallest initial angle for which there is a second micro contact time interval soon after the first.