Numerical simulation of a piano

Numerical simulation of a piano

2012

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 the main elements that contribute to sound production into account. The soundboard is modeled as a bidimensional thick, orthotropic, heterogeneous, frequency dependent damped plate, using Reissner-Mindlin equations. The structural acoustics equations allow the soundboard to radiate into the surrounding air, in which we 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 with the horizontal plane. Each string is modeled as a one dimensional damped system, taking not only the transversal waves excited by the hammer into account, but also the stiffness through a shear wave equation, as well as the longitudinal waves due to geometric nonlinearities and variation of tension during motion. The function of the hammer is represented by 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 has to 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 dependent damping of the soundboard, large number of unknowns required for the acoustic propagation), in addition to difficulties inherent to coupled systems. 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 the present work thus consisted in developing energy preserving schemes for a class of nonlinear systems of equations, as the string model. On the other hand, numerical efficiency and computation time reduction require that the unknowns of each step of the problem, requiring specific, hence different, discrete time formulation, be updated separately. To achieve this artificial decoupling, adapted Schur complements are performed after introduction of Lagrange multipliers.

The capacity of this time domain piano modeling is illustrated by realistic numerical simulations. Not only does the program greatly replicate the measurements, it also allows us to investigate the influence of physical phenomena (string stiffness or nonlinearity), geometry and material properties on the general vibratory and acoustical behavior of the piano. Spectral enrichment, << phantom partials >> and nonlinear precursors are clearly reproduced when large playing amplitudes are involved, highlighting how this approach can help to gain a better understanding how piano works.

Modeling a piano, from hammer excitation to sound propagation...

Juliette CHABASSIER

PhD Ecole Polytechnique (France)

Field Applied Mathematics

ADVISOR 1 Patrick JOLY

ADVISOR 2 Antoine CHAIGNE

Copyright © Benoit Fabre 2010

Copyright © Benoit Fabre 2010

Copyright © Benoit Fabre 2010

Copyright © Juliette Chabassier 2012

Copyright © Juliette Chabassier 2012