# Modélisation et simulation numérique d'un piano par modèles physiques

1 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 : The purpose of this study is the time domain modeling and numerical simulation of a piano. We aim at explaining the vibratory and acoustical behavior of the piano, by taking into account the main elements that contribute to sound production. The soundboard is modeled as a bidimensional thick, orthotropic, heterogeneous, frequency dependant damped plate, using Reissner Mindlin equations. The vibroacoustics equations allow the soundboard to radiate into the surrounding air, in which we wish to compute the complete acoustical field around the perfectly rigid rim. The soundboard is also coupled to the strings at the bridge, where they form a slight angle from horizontal. Each string is modeled by a one dimensional damped system of equations, taking into account not only the transversal waves excited by the hammer, but also the stiffness thanks to shear waves, as well as the longitudinal waves arising from geometric nonlinearities. The hammer is given an initial velocity that projects it towards a choir of strings, before being repelled. The interacting force is a nonlinear function of the hammer compression. The final piano model that will be discretized is a coupled system of partial differential equations, each of them exhibiting specific difficulties (nonlinear nature of the string system of equations, frequency dependant damping of the soundboard, great number of unknowns required for the acoustic propagation), in addition to couplings' inherent difficulties. On the one hand, numerical stability of the discrete scheme can be compromised by nonlinear and coupling terms. A very efficient way to guarantee this stability is to construct a numerical scheme which ensures the conservation (or dissipation) of a discrete equivalent of the continuous energy, across time steps. A major contribution of this work has been to develop energy preserving schemes for a class of nonlinear systems of equations, in which enters the string model. On the other hand, numerical efficiency and computation time reduction require that the unknowns of each problem's part, for which time discretization is specific, hence different, be updated separately. To achieve this artificial decoupling, adapted Schur complements are performed after Lagrange multipliers are introduced. The potential of this time domain piano modeling is emphasized by realistic numerical simulations. Beyond greatly replicating the measurements, the program allows us to investigate the influence of physical phenomena (string stiffness or nonlinearity), geometry or materials on the general vibratory behavior of the piano, sound included. Spectral enrichment, " phantom partials " and nonlinear precursors are clearly revealed when large playing amplitudes are involved, highlighting how this approach can help better understand how a piano works.
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https://pastel.archives-ouvertes.fr/pastel-00690351
Contributor : Juliette Chabassier <>
Submitted on : Monday, April 23, 2012 - 11:53:18 AM
Last modification on : Thursday, January 14, 2021 - 11:56:04 AM
Long-term archiving on: : Monday, November 26, 2012 - 3:30:13 PM

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• HAL Id : pastel-00690351, version 1

### Citation

Juliette Chabassier. Modélisation et simulation numérique d'un piano par modèles physiques. Analyse numérique [math.NA]. Ecole Polytechnique X, 2012. Français. ⟨pastel-00690351⟩

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