Partitioning the parameter space according to different behaviors during three-dimensional impacts

The equations of motion that define three-dimensional rigid-body impact with finite friction and restitution cannot be solved in a closed form. Previous work has shown that for general shapes and initial conditions, the direction of sliding velocity keeps changing continuously throughout the duration of impact. The flow patterns defined by the trace of the sliding velocity can be classified into a finite number of qualitatively distinct physical behavior. We identify three dimensionless parameters that completely specify the sliding behavior, and determine regions in this parameter space that correspond to each of the different flow patterns. The qualitative behavior during impact can now be determined based on the region which contains the parameters for a given impact configuration. The analysis is also used to study the sensitivity of the sliding behavior to changes in shape or configuration of the body and to rule out the occurrence of certain ambiguities in the post-sticking behavior during impact.