A Linear Approach for Multiple-point Impact in Multibody Systems

Abstract. The dynamical analysis of multibody systems undergoing simultaneous multiplepoint collisions is a relevant problem in various fields, such as robotics and biomechanics. Different approaches to study collisions can be found in literature, going from totally impulsive ones (dealing only with the preand postimpact states and assuming that the system configuration is unchanged) to totally continuous ones (where the time integration of the equations of motion is done and the system configuration changes throughout the collision interval). Simultaneity cannot be treated properly in impulsive approaches because the mathematical formulation shows indetermination. These approaches try to overcome this drawback by defining single-point collision sequences (not necessarily realistic) and the final results are often sequence-dependent. Continuous methods are better suited to deal with simultaneity, but they often result into complicated models. This article proposes a simple hybrid linear approach based on a vibrational dynamical model. The dynamics at the colliding points is simulated through linear stiff springs undergoing very small deformations and thus generating a vibratory behavior of the system. The overall system configuration is assumed to be constant as far as the inertia matrix is concerned. The collision end corresponds to a zero spring force. Only planar application cases will be presented, but the approach is suitable for 3D multiple-point smooth collisions in multibody systems with perfect constraints. Though only the case of perfectly elastic collisions will be shown, the methodology can be extended to collisions showing any degree of inelasticity.