Non-smooth dynamics for an efficient simulation of the grand piano action

Models with impact or dry friction, yielding discontinuous velocities or accelerations, have motivated research for appropriate numerical methods in the community of non-smooth dynamics. In this work, we apply such methods on the grand piano action. This multibody system has two properties of interest in terms of modelling and simulation: it is extremely sensitive to small misadjustments, and its functioning strongly relies on dry friction and stick-slip transitions-known to be crucial for the touch of the pianist. Using numerical methods of non-smooth contact dynamics, the non-smooth character of dry friction was conserved, in contrast to classical approaches based on regularization which additionally impose the somewhat arbitrary choice of a regularizing parameter. The use of such numerical method resulted in computations about a few hundred times faster than those reported in recent literature. For the first time, the presented predictions of the piano action's simulations are forces (in particular, the reaction force of the key on the pianist's finger), instead of displacements which filter out most of the dynamical subtleties of the mechanism. The comparisons between measured and simulated forces in response to a given motion are successful, which constitutes an excellent validation of the model, from the dynamical and the haptic points of view. Altogether, numerical methods for non-smooth contact dynamics applied to a non-smooth model of the piano action proved to be both accurate and efficient, opening doors to industrial and haptic applications of sensitive multibody systems for which dry friction is essential.