A State Transition Diagram for Simultaneous Collisions with Application in Billiard Shooting

Abstract

This paper models a multibody collision in the impulse space as a state transition diagram, where each state represents a phase during which impacts are “active” at only a subset of the contact points. A state transition happens whenever an active impact finishes restitution, or an inactive impact gets reactivated, depending on whether the two involved bodies are instantaneously penetrating into each other or not. The elastic energy due to an impact is not only affected by the impulse at the corresponding contact point, but also by other impulses exerted on the two involved bodies during the impact. Consequently, Poisson’s impulse-based law of restitution could result in negative energy. A new law governing the loss of elastic energy during restitution is introduced. Convergence of the impulse sequence generated by the state transition diagram is established. The collision outcome depends on the ratios of the contact stiffnesses rather than on their individual values. The collision model is then applied in an analysis of billiard shooting in which the cue stick impacts the cue ball, which in turn impacts the pool table. The system is driven by the normal impulses at the two contacts with the tangential impulses determined via a contact mode analysis.