## Guest talk: Prof. Alan Champneys (10.01.2018)

On January 10th, 2018, Prof. Alan Champneys, University of Bristol, is visiting the research training group (invited by Miriam Steinherr).
We cordially invite you to his talk titled:

*Unfolding the Painlevé paradox in contact mechanics
*

Wednesday 10th January 2018, 16:00, room MZH 2340, MZH building, Bibliothekstr. 5.

** Abstract **

* Unfolding the Painlevé paradox in contact mechanics*

A review is given of the so-called Painlevé paradox in contact mechanics of rigid bodies with friction, where there is sufficient coupling between normal and tangential degrees of freedom. Here it is shown how so-called impact without collision may occur and both non-uniqueness and non-existence paradoxes can occur. The canonical example is that of the instability that happens when chalk is pushed rather than dragged across a blackboard in which repeated 'impacts without collision' can occur due to an effective negative stiffness. Taking rigid bodies with a unique point contact, an attempt is made to explain and unfold various singularities by taking the limit of a compliant formulation in the infinite-stiffness limit. It is shown how all open sets of initial conditions can be uniquely resolved in this way. Two isolated singularities persist. One is the onset of reverse chatter (an accumulation of impacts in reverse time) and the other is the so-called Génot and Brogliato point (a G-spot) in which lift-off and onset of negative stiffness coincide. Both are shown to attract sets of initial conditions. For the latter we propose a new asymptotic analysis in the presence of a small parameter that regularises the rigid contact into an elastic one. The dynamics in the inner region is found to feature a canard-like trajectory in some cases, perturbations from which are found to be represented by a certain generalised hypergeometic function. The asymptotic behaviour of this function for large argument is found to determine whether continuation from the G-spot is via direct lift-off or via a so-called impact without collision in which there is an O(1) change in contact velocity over an infinitesimal time interval. The results are backed up by numerical computations on both a simple toy example and on a frictional impacting mechanism.