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.
@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},
volume = {48},
year = {2014},
pages = {1241-1278},
doi = {10.1051/m2an/2013136},
mrnumber = {3264353},
zbl = {1301.00044},
language = {en},
url = {http://dml.mathdoc.fr/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, Tome 48 (2014) pp. 1241-1278. doi : 10.1051/m2an/2013136. http://gdmltest.u-ga.fr/item/M2AN_2014__48_5_1241_0/
[1] Recherche/Crèation 1. Modalys (2007). http://forumnet.ircam.fr/701.html.
[2] and , From touch to string vibrations. I: Timing in the grand piano action. J. Acoust. Soc. Amer. 88 (1990) 52.
[3] , Observations on the transient components of the piano tone. KTH (1993).
[4] , , , , and , Physically informed signal processing methods for piano sound synthesis: a research overview. EURASIP J. Appl. Signal Process. 2003 (2003) 941-952.
[5] , S, Bilbao and R. Kronland-Martinet, The simulation of piano string vibration: from physical models to finite difference schemes and digital waveguides. J. Acoust. Soc. Amer. 114 (2003) 1095-1107.
[6] , and , Optimal shear correction factors in hierarchical plate modelling. Math. Modell. Sci. Comput. 1 (1993) 1-30. | MR 1235486 | Zbl 0788.73039
[7] S Bilbao, Conservative numerical methods for nonlinear strings. J. Acoust. Soc. Amer. 118 (2005) 3316-3327.
[8] X Boutillon, Model for piano hammers: Experimental determination and digital simulation. J. Acoust. Soc. Amer. 83 (1988) 746-754.
[9] and , Generation of longitudinal vibrations in piano strings: From physics to sound synthesis. J. Acoust. Soc. Amer. 117 (2005) 2268-2278.
[10] and , Numerical simulation of piano strings. I. A physical model for a struck string using finite-difference methods. J. Acoust. Soc. Amer. 95 (1994) 1112-1118.
[11] , and , Transitoires de piano et non linéarités des cordes: mesures et simulations. Proc. of the 10th French Acoustical Society Meeting (in french) (2012).
[12] and , Energy based simulation of a Timoshenko beam in non-forced rotation. Application to the flexible piano hammer shank. Wave Motion, submitted in (2013).
[13] and , Stability and dispersion analysis of improved time discretization for simply supported prestressed Timoshenko systems. Application to the stiff piano string. Wave Motion 50 (2012) 456-480. | MR 3039536
[14] and , Energy preserving schemes for nonlinear hamiltonian systems of wave equations. application to the vibrating piano string. Comput. Methods Appl. Mech. Engrg. 199 (2010) 2779-2795. | MR 2740757 | Zbl 1231.74146
[15] , Design and tone in the mechanoacoustic piano. Part II. Piano structure. J. Acoust. Soc. Amer. 100 (1996) 695-708.
[16] , Piano strings and “phantom” partials. J. Acoust. Soc. Amer. 102 (1997) 659.
[17] . The shear coefficient in timoshenko's beam theory. ASME, J. Appl. Math. 33 (1966) 335-340. | Zbl 0151.37901
[18] , Modélisation du couplage corde - chevalet - table d'harmonie dans le registre aigu du piano. JJCAAS 2006 (2006) 1-1.
[19] , , and , Time-domain simulation of a guitar: Model and method. J. Acoust. Soc. Amer. 114 (2003) 3368-3383.
[20] , La table d'harmonie du piano - études modales en basses et moyennes fréquences. Thèse de Doctorat (2010) 1-190.
[21] and , Physical modeling of the piano. EURASIP J. Appl. Signal Process. 2004 (2004) 926-933.
[22] , Pianoteq. Available at http://www.pianoteq.com.
[23] , and , Dynamic modeling and experimental testing of a piano action mechanism with a flexible hammer shank. J. Comput. Nonlinear Dyn. 3 (2008) 1-10.
[24] and , Theoretical Acoustics. Princeton University Press (1968).
[25] and , Piano tone synthesis using digital filters by computer simulation. Acoustics, Speech, and Signal Processing, IEEE International Conference on ICASSP'86 11 (1986) 1293-1296.
[26] and , Dispersion of waves in piano strings. J. Acoust. Soc. Amer. 83 (1988) 305-317.
[27] , and , Time-domain modeling and numerical simulation of a kettledrum. J. Acoust. Soc. Amer. 105 (1999) 3545-3562.
[28] , The effect of transverse shear deformation on the bending of elastic plates. J. Appl. Mech. 12 (1945) 69-77. | MR 12579 | Zbl 0063.06470
[29] , Hysteretic model of the grand piano hammer felt. J. Acoust. Soc. Amer. 97 (1995) 2577.
[30] L.T.-Tsien, Global classical solutions for quasilinear hyperbolic systems. Wiley (1994). | MR 1291392 | Zbl 0841.35064
[31] , and , Modeling the dynamics of a compliant piano action mechanism impacting an elastic stiff string. J. Acoust. Soc. Amer. 125 (2009) 4034-4042.
[32] , Coupled piano strings. J. Acoust. Soc. Amer. 62 (1977) 1474.