Modeling and simulating the dynamics of the grand piano action

The grand piano action has been developed empirically over two centuries. In its modern version, it provides a remarkably accurate control of the hammer velocity and its impact time. By means of an elementary 1-DOF model, we show why it is mostly preferable to consider and simulate the dynamics by computing the reaction force in response to a motion imposed by the finger rather than the opposite. We present a complete dynamical model based on that proposed earlier by Lozada: 6 rotating bodies (key, damper, whippen, jack, escapement lever, hammer), pivots subject to dry and viscous friction, 13 contact zones with reacting and dissipative forces, 3 of them (hammer-jack, jack-escapement button, 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 is proposed. The results of the numerical simulation of the model are in very good agreement with the measurements for levels of playing ranging from piano to forte.