Title: Dynamics beyond dynamic jam; unfolding the Painlevé paradox singularity,
Abstract: This paper analyses in detail the dynamics in a neighbourhood 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
non-local in momentum space and depend on properties evaluated away from the
G-spot. The results are illustrated on a particular physical example, namely
the a frictional impact oscillator first studied by Leine et al.
The answer is obtained via an analysis that involves a consistent contact
regularisation with a stiffness proportional to $1/\varepsilon^2$. Taking a
singular limit as $\varepsilon \to 0$, one finds an inner and an outer
asymptotic zone in the neighbourhood of the G-spot. Two distinct cases are
found 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.