Painlevé paradox and dynamic jam of a three-dimensional elastic rod

Painlevé paradox is a well-known problem in non-smooth dynamics. Till now, all relevant researches only focus on the Painlevé paradox of a rigid rod sliding on a rough surface with large coefficient of friction \(\mu \). However, if the compliance of the whole rod is considered, the Painlevé paradox problem will transform into a ‘bouncing phenomenon’ caused by jam (self-locking), which has not been studied. The aim of this paper was to analyze the relationship between the Painlevé paradox of the rigid rod and the dynamic jam behavior of the elastic rod. The full transient method is used to calculate the dynamic response of elastic rod. In comparison with the rigid rod, the compliance of the whole rod will bring a different critical value of coefficient of friction \(\mu _\mathrm{c}\). The stress waves are excited by the abrupt change of the contact forces, and they propagate in the rod. A ‘quasi-stick’ state of the contact surface is found, which supports the feasibility of the single-point compliant contact model. In addition, the peak value of normal contact force \(F_{3}\) and normal contact impulse \(P_{3}\) is constant when the system parameters \((\theta , \mu )\) are located in the region of Painlevé paradox.