Modeling and simulating the dynamics of the grand piano action

International audienceThe grand piano action has been developed empirically over two centuries. In its modern version, it provides a remarkablyaccurate control of the hammer velocity and its impact time. By means of an elementary 1-DOF model, we showwhy it is mostly preferable to consider and simulate the dynamics by computing the reaction force in response to amotion imposed by the finger rather than the opposite. We present a complete dynamical model based on that proposedearlier by Lozada: 6 rotating bodies (key, damper, whippen, jack, escapement lever, hammer), pivots subject to dryand viscous friction, 13 contact zones with reacting and dissipative forces, 3 of them (hammer-jack, jack-escapementbutton, hammer-check) being also subject to Coulomb friction. This model introduces discontinuities on the velocities.The problems raised by the usual regular-dynamics formulation are discussed and a non-smooth dynamics approach isproposed. The results of the numerical simulation of the model are in very good agreement with the measurements forlevels of playing ranging from piano to forte