Could Chalk Hopping Be Caused by Reverse Chatter?

Anyone who has ever used a chalkboard is probably familiar with the phenomenon of “chalk hopping,” where the chalk unexpectedly skips across the chalkboard, leaving a dotted line in its wake. Such behavior is ubiquitous to mechanical systems with moving parts in contact, where it is almost always undesirable. It is widely believed that hopping behavior is a physical manifestation of either the classical Painlevé paradox or a related phenomenon called dynamical jam. The present paper poses the question of whether chalk hopping might be caused by a different, and much more recently discovered, instability called “reverse chatter,” in which two bodies initially in sustained contact can lose contact through a sequence of impacts with increasing amplitude. Previous simulations of reverse chatter have considered only constant external loads, which do not adequately model the forces exerted on a piece of chalk. The current work presents simulation results for a model system in the presence of a control algorithm that mimics the human hand by attempting to keep the chalk in contact with the chalkboard. The simulations reveal that there exist physically realistic parameter values for which a loss of contact occurs that cannot be attributed to either the classical Painlevé paradox or dynamical jam, but which can only be attributed to reverse chatter. Furthermore, the subsequent motion of the system after losing contact is found to be strikingly similar to that of chalk hopping on a chalkboard, to a hitherto unparalleled degree. These results show that neither the classical Painlevé paradox nor dynamical jam is necessary for hopping behavior, and suggest that reverse chatter may be the most plausible explanation for chalk hopping.