Modeling and numerical simulation of a piano.
Juliette Chabassier, Antoine J. Chaigne · The Journal of the Acoustical Society of America · 2011
A complete model of a piano is built, in order to account for the acoustical behavior of the instrument from excitation to sound. A nonlinear hammer strikes the strings. The precursor at the bridge can be explained by the presence of a longitudinal vibration in the string, which is nonlinearly coupled to the transversal vibration using the geometrically exact model. A nonstandard condition must be written for the bridge model, in order to transmit transversal and longitudinal vibrations of the strings to the soundboard. A Reissner Mindlin plate model is used for the soundboard, which radiates in the air. The coupling must be reciprocal so that a global energy is preserved. A numerical discretization is proposed for the whole system. A first difficulty is due to the nonlinearity of both strings and hammer. Another one arises from the different couplings of the system: hammer/string, string/soundboard, and soundboard/air. An innovating, energy preserving scheme is built for the nonlinear system of string equations, and an energy technique is adopted for the whole problem to ensure numerical stability. The resulting complete numerical scheme conserves a discrete and consistent global energy. Numerical results are presented and compared to measurements.