Hybrid State Chart Modelling For Nonlinear Bouncing Ball Dynamics

Abstract

This contribution highlights two aspects of the classical bouncing ball modeling and simulation. On modelling level, the ball characterises more a big bubble than a ball, so that drag forces cannot be neglected and impacts must be modelled by distortion, and additionally the implication of the big-sized ball for different atmospheres (comparing Earth and Mars). What looks like fun - may be used for education in modelling and simulation, and may become serious science once. On implementation level, the presented MATLAB/Stateflow version is a purely discrete approach: fly and distortion are modelled by state charts, updated by triggered stepsize events which drive an ODE solver. Switching between impact, distortion, and fly is triggered by state events following a predictive event finding strategy: as the event function is the state ball height, also first and second derivative of event function are known. Thus allows approximating the event function by a polynomial of first order or second order near the impact, the known zero of which may be used for adjusting a smaller step size, and in last consequence, to give the impact time. The idea may be generalized for contact problems in mechanical systems.