Finite Duration Impacts With External Forces

This work considers the effect of external forces during finite duration collisions using an incremental model of impact. The deformation of the “rigid” body is modeled through an elastic element and the time interval over which contact occurs is of finite duration. Moreover, the work done by the external forces is nonzero during the collision. This model allows for an equivalent coefficient of restitution $e$ to be identified. In the presence of a constant external force the coefficient of restitution depends not only on the system parameters, but the initial relative velocity at the point of impact. For external forces which tend to bring the colliding bodies together, the colliding bodies remain in contact for sufficiently small impact velocities $(e=0)$ while for larger incoming speeds, the coefficient of restitution is positive. This state dependent restitution arises from the coupling of external forces to the collision model, and is not seen in more familiar models of impact. Finally, based on the results of the experimental system and the incremental model, the standard algebraic model of restitution is modified to include these finite duration effects.