Multiple Impacts: A State Transition Diagram Approach

Abstract

Impact happens when two or more bodies collide, generating very large impulsive forces in a very short period of time during which kinetic energy is first absorbed and then released after some loss. This paper introduces a state transition diagram to model a frictionless multibody collision. Each state describes a different topology of the collision characterized by the set of instantaneously active contacts. A change of state happens when a contact disappears at the end of restitution, or when a disappeared contact reappears as the relative motion of two bodies goes from separation into penetration. Within a state, (normal) impulses are coupled differentially subject to relative stiffnesses at the active contact points and the strain energies stored there. Such coupling may cause restart of compression from restitution during a single impact. Impulses grow along a bounded curve with first-order continuity, and converge during the state transitions. To solve a multibody collision problem with friction and tangential compliance, the above impact model is integrated with a compliant impact model. The paper compares model predictions to a physical experiment for the massé shot, which is a difficult trick in billiards, with a good result.