Scalar generalization of Newtonian restitution for simultaneous impact

•A single restitution condition based on a weighted linear combination of restitution inequality.•Guaranteed energy dissipation for all cases.•Model uses quadratic programming.•More physical realism than the popular linear complementarity based model for several impact configurations.

Simultaneous multiple impacts of solid bodies are modeled approximately within rigid body mechanics using impulse momentum relations, friction inequalities, and some kind of restitution model. The common restitution models are Newtonian, Poisson and energetic restitution. Of these, there is so far no satisfactory generalization of Newtonian restitution to simultaneous multiple impacts in three dimensions with friction. Here we propose a new generalization of Newtonian restitution which imposes a single scalar inequality regardless of the number of contacts. The inequality is a weighted sum of restitution inequalities for the different contacts, based on their respective coefficients of restitution. The weights used involve something we call local normal inertias, defined via the impulse momentum relations at each contact. The new generalized Newtonian restitution model, coupled with physical constraints of nonnegative normal impulses, non-interpenetration, and contact-wise friction inequalities, defines a feasible set of impulses. Our proposed impact model chooses the energy minimizing impulse within this feasible set, found using quadratic programming. Kinetic energy increases are never predicted by our model. Additionally, the model makes physically more realistic predictions than the popular linear complementarity approach in several cases, as we show using examples. For completeness, all relevant system matrices are described, and Matlab code is provided.