Synchronization of impacting mechanical systems with a single constraint

The property of maximal monotonicity, which is fulfilled by most commonly used impact laws, is used to significantly reduce the complexity of the distance function. Based on the simplified distance function, a Lyapunov function is constructed to investigate the synchronization problem for two identical one-dimensional mechanical systems. Sufficient conditions for the uncoupled individual systems are provided under which local synchronization is guaranteed. Furthermore, we present an interaction law which ensures global synchronization, also in the presence of grazing trajectories and accumulation points (Zeno behavior). The results are illustrated using numerical examples of a 1-DOF mechanical impact oscillator which serves as stepping stone in the direction of more general systems.