The Painlevé paradox studied at a 3D slender rod

This paper aims at investigating the dynamical behaviors of a 3D rod moving on a rough surface with so-called Painlevé paradox. The condition for the occurrence of the Painlevé paradox in the rod is studied according to the theoretical results obtained from LCP's method for spatial multibody systems. Numerical results obtained by inserting a compliant contact model into the rigid body model present a support for the assumption that a tangential impact is related to the spatial paradoxical situations. Furthermore, the tangential impact is analyzed by using the Darboux-Keller's shock dynamics and are found with the same properties as the one in the planar rod: A tangential stick appears at the contact point during the impulsive process. With the help of the Stronge's coefficient, an impact rule is developed to describe the dynamical behaviors of the 3D rod with paradoxical situations. Comparisons between numerical results obtained from Darboux's model and the ones obtained from the compliant contact model are carried out and show well agreements.