Three-Dimensional Rigid-Body Impact Mechanics for Automobile Collisions

Two-dimensional (planar) rigid-body impact mechanics for application in automobile collisions have been described by a number of researchers over the last several decades. Little has been discussed, however, regarding three-dimensional rigid-body impact mechanics in this regard. Two commercially available accident simulation programs, PC-Crash and Virtual CRASH, offer three-dimensional rigid-body impact mechanics as one of their collision models but documentation of the complete development of their three-dimensional equations, particularly with respect to necessary constraint strategies at the impulse center, are not readily available. In this paper, a three-dimensional rigid-body impact mechanics derivation is presented. In order to solve the set of impact mechanics equations of motion it is necessary to develop constraint relationships. The constraint strategy described in the literature pertaining to the PC-Crash Full-Impact/Sliding-Impact scenarios for two dimensions is extended to the three-dimensional case and the ramifications regarding post-impact relative velocity at the impulse center is discussed. A second strategy in which an impulse component is aligned with the contact plane component of the initial relative velocity at the impulse center is presented and compared to the PC-Crash scenarios. Lastly, while these two strategies incorporate a single impulse ratio/friction parameter for the contact plane, a strategy involving two independent impulse ratio/friction parameters is briefly discussed.