32 Modelling of Sympathetic String Vibrations in the Clavichord Using a Modal Udwadia-Kalaba Formulation 279 Fig. 32.1 Photo from above of the Hubert clavichord, with indications as to the substructures being modeled in our numerical simulations 0 500 1000 1500 2000 2500 3000 3500 4000 Frequency (Hz) -120 -100 -80 -60 -40 -20 0 20 40 60 Spectral amplitude Experience Simulation Fig. 32.2 Spectral comparison between the experimental signal measured with the vibrometer and the simulation of the C5 string having excited the F3 string of the Hubert clavichord note the presence of the fundamental frequency peak of the C5 RS which is at 491 Hz and its harmonics, being present because of the impulse response given to all substructures by the tangent excitation. Figure 32.2 shows a good agreement between the numerical simulation and measurement. So with this simplified model, we can take account of much of the physics being involved despite of the complexity of this instrument. For example, the coupling of the string with the bridge is quite simplified in the model. However, some spectral components do not have the same spectral amplitudes. In particular, we see that the partial at 200 Hz is absent in the simulation. We conjecture that this frequency peak comes from a soundboard mode of the clavichord which was not taken into account in the model. As for the other partials, their lack of spectral energy is probably due to a lack of precision in the estimation of the damping of the strings, and/or from some inaccuracy of the simulated string excitation. To further improve the model, we should consider all the 74 sympathetic strings of the Hubert clavichord in our simulation, which implies much longer computations. However, repeating the same measurement with all strings being free, the vibratory response of the RS of the C5 string remains quite unchanged. So we may not need to consider all the strings in the model to obtain a better result. Also, to improve our results, we should proceed to a more precise study of the damping of the strings and of the excitation features, to have a better estimation of the spectral amplitude of each partial of the computed response.
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