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Soundboards are key parts of pianos, since they transfer the energy from the strings to the acoustic field. For the present simulations, the soundboards are modeled as orthotropic plates with varying thickness, where the additional substructures (bridges, ribs, apron) are considered as local variations of mass and rigidity. The soundboard is assumed to be fixed along its edges, except along the apron where it is supposed to be free.
Modes of vibration are essential to characterize the properties of a soundboard. More than thousand modes are involved in the radiation of a piano tone. Low frequency modes (first row) are uniformly distributed over the complete surface, but they do not radiate sound very efficiently. In contrast, medium (second row) and high (third row) modes are localized over a small portion of the soundboard, and are efficient radiators. The tone color of a piano greatly depend on the distribution of modal frequencies of the soundboard: this is a global quality of the instrument. Contribution of the soundboard modes are clearly audible in the initial transients of the tones in the medium and high range of notes.
Another important properties of the soundboard is the modal density, which is the number of modes per Hz. This parameter critically depends on the soundboard's thickness and on its substructures (ribs, bridge, apron). The modal density is linked to the acoustical efficiency of the soundboard: thick and rigid soundboards usually have a smaller modal density than the thin ones, and radiate globally more sound power. However, such a global result should be considered with care, since the modal density varies with frequency. Here, the figure shows a comparison of calculated modal densities for three different historic pianos: GH05 (G. Hecher piano, 1805), JBS51 (J.B. Streicher, 1851) and JBS73 (J. B. Streicher and Son, 1873).
One attractive feature of the simulation program used in this project is its capability of computing the total energy of one piano component. The figure here shows the time evolution of the Soundboard Energy Ratio (SER) for the note D#3 played on two instruments (J.B. Streicher and Son 1873 (JBSS73) and N. Streicher 1819 (NS19)). The SER is, at each time, the total energy of the soundboard divided by the initial energy imparted to the piano by the hammer blow. This, again, is a global quantity, which characterizes the instrument irrespective of the observation point. It can be seen that the SER is higher for the JBSS73, compared to the NS19, but that the energy decreases much faster. Similar differences are observed on the acoustic energy for these two pianos.
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