The method of penalizing functions for elastic/anelastic impacts

The importance of the Erdmann-Weierstrass corner conditions in modelling systems constituted by bodies colliding during their motion is well known, as in many cases such equations are sufficient to determine the systems velocities after an impact. In this paper, the Erdmann-Weierstrass corner conditions are obtained through the method of the penalizing functions, which considers an auxiliary problem in which the impacts are rendered smooth, by allowing elastic deformations during the impact; the non-smooth path of motion is obtained as the limit of the smooth case, when the elastic constants involved become infinite. The case in which some kinetic energy is lost during impacts is also considered through the use of a suitable dissipation function in the smooth case, thus obtaining the classic rules for impacts with a coefficient of restitution smaller than one.