Simulation of a non-smooth dynamical model of the piano key

The grand piano action has been developped empirically and provides a remarkably accurate control of the hammer velocity and the impact time [1]. The numerical simulation of the action is necessary to improve numerical keyboards and to better understand haptic controllability. Following previous efforts [2, 3], we propose a simulation based on an explicit model and a solver of non-smooth dynamical problems. We also discuss the physical quantity – force or displacement – with which the model is to be controlled. A scheme of the grand piano key is presented in Figure 1, with the proposed corresponding kinematical diagram in Figure 2. The 2D model [3] includes six rigid bodies in rotation with dry and viscous articular friction, thirteen contact zones with nonlinear springs (Equation 1), three of them (hammer-jack, jack-escapement button, hammer-check) being also subject to Coulomb friction. The reaction force Ffelts of a felt to a displacement δ is given by: Ffelts(δ) = k δ r + p δ̇ δ (1) damper hammer lever whippen jack key friction zone felt material ground check