Time domain simulation of a piano. Part 1: model description
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 48 (2014) no. 5, p. 1241-1278
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The purpose of this study is the time domain modeling 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 dependent 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 the horizontal plane. 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 is a coupled system of partial differential equations, each of them exhibiting specific difficulties (nonlinear nature of the string system of equations, frequency dependent damping of the soundboard, great number of unknowns required for the acoustic propagation), in addition to couplings' inherent difficulties.

DOI : https://doi.org/10.1051/m2an/2013136
Classification:  00A71,  00A65,  65P05,  65N25,  35Q72,  35L05
Keywords: piano, modeling, energy, precursor and phantom partials, damping mechanisms
@article{M2AN_2014__48_5_1241_0,
     author = {Chabassier, J. and Chaigne, A. and Joly, P.},
     title = {Time domain simulation of a piano. Part 1: model description},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
     publisher = {EDP-Sciences},
     volume = {48},
     number = {5},
     year = {2014},
     pages = {1241-1278},
     doi = {10.1051/m2an/2013136},
     zbl = {1301.00044},
     mrnumber = {3264353},
     language = {en},
     url = {http://www.numdam.org/item/M2AN_2014__48_5_1241_0}
}
Chabassier, J.; Chaigne, A.; Joly, P. Time domain simulation of a piano. Part 1: model description. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 48 (2014) no. 5, pp. 1241-1278. doi : 10.1051/m2an/2013136. http://www.numdam.org/item/M2AN_2014__48_5_1241_0/

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