Modeling and simulation of a grand piano

Juliette Chabassier 1 Patrick Joly 2 Antoine Chaigne 3
1 Magique 3D - Advanced 3D Numerical Modeling in Geophysics
LMAP - Laboratoire de Mathématiques et de leurs Applications [Pau], Inria Bordeaux - Sud-Ouest
2 POEMS - Propagation des Ondes : Étude Mathématique et Simulation
Inria Saclay - Ile de France, UMA - Unité de Mathématiques Appliquées, CNRS - Centre National de la Recherche Scientifique : UMR7231
Abstract : A time-domain global modeling of a grand piano is presented. The string model includes internal losses, stiffness and geometrical nonlinea- rity. The hammer-string interaction is governed by a nonlinear dissi- pative compression force. The soundboard is modeled as a dissipative bidimensional orthotropic Reissner-Mindlin plate where the presence of ribs and bridges is treated as local heterogeneities. The coupling between strings and soundboard at the bridge allows the transmission of both transverse and longitudinal waves to the soundboard. The soundboard is coupled to the acoustic field, whereas all other parts of the structure are supposed to be perfectly rigid. The acoustic field is bounded artificially using perfectly matched layers (PML). The discrete form of the equations is based on original energy preserving schemes. Artificial decoupling is achieved, through the use of Schur complements and Lagrange multipliers, so that each variable of the problem can be updated separately at each time step. The capability of the model is highlighted by series of simulations in the low, medium and high regis- ter, and through comparisons with waveforms recorded on a Steinway D piano. Its ability to account for phantom partials and precursors, consecutive to string nonlinearity and inharmonicity, is particularly emphasized.
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Juliette Chabassier, Patrick Joly, Antoine Chaigne. Modeling and simulation of a grand piano. Journal of the Acoustical Society of America, Acoustical Society of America, 2013, 134, pp.648. ⟨10.1121/1.4809649⟩. ⟨hal-00873089⟩

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