Dynamics Beyond Dynamic Jam; Unfolding the Painlevé Paradox Singularity

Abstract

This paper analyzes in detail the dynamics in a neighborhood of a Génot--Brogliato point, colloquially termed the G-spot, which physically represents so-called dynamic jam in rigid body mechanics with unilateral contact and Coulomb friction. Such singular points arise in planar rigid body problems with slipping point contacts at the intersection between the conditions for onset of lift-off and for the Painlevé paradox. The G-spot can be approached in finite time by an open set of initial conditions in a general class of problems. The key question addressed is what happens next. In principle, trajectories could, at least instantaneously, lift off, continue in slip, or undergo a so-called impact without collision. Such impacts are nonlocal in momentum space and depend on properties evaluated away from the G-spot. The answer is obtained via an analysis that involves a consistent contact regularization with a stiffness proportional to $1/\varepsilon^2$ for some $\varepsilon$. Taking a singular limit as $\varepsilon \to 0$, one finds an inner and an outer asymptotic zone in the neighborhood of the G-spot. Matched asymptotic analysis then enables continuation from the G-spot in the limit $\varepsilon \to 0$ and also reveals the sensitivity of trajectories to $\varepsilon$. The solution involves large-time asymptotics of certain generalized hypergeometric functions, which leads to conditions for the existence of a distinguished smoothest trajectory that remains uniformly bounded in $t$ and $\varepsilon$. Such a solution corresponds to a canard that connects stable slipping motion to unstable slipping motion through the G-spot. Perturbations to the distinguished trajectory are then studied asymptotically. Two distinct cases are distinguished according to whether the contact force becomes infinite or remains finite as the G-spot is approached. In the former case it is argued that there can be no such canards and so an impact without collision must occur. In the latter case, the canard trajectory acts as a dividing surface between trajectories that momentarily lift off and those that do not before taking the impact. The orientation of the initial condition set leading to each eventuality is shown to change each time a certain positive parameter $\beta$ passes through an integer. Finally, the results are illustrated in a particular physical example, namely the frictional impact oscillator first studied by Leine, Brogliato, and Nijmeijer.