25 Model-Based Decision Support Methods Applied to the Conservation of Musical Instruments: Application to an Antique Cello 225 Fig. 25.2 Left: scheme of the cleats and cracks of the cello, based on radiographs provided by the laboratory of the Musée de la musique, Paris, right: bridge excitation and measurement point and direction on a Belgian cello Table 25.1 Orthotropic material properties for maple, spruce and ebony species, taken from [14, 15] Maple Spruce Ebony Specific gravity [−] 0.64 0.44 1.09 EL [MPa] 12,200 12,840 18,000 ER [MPa] 1820 1000 2450 ET [MPa] 1060 650 1520 νLR, νRL [−] 0.37, 0.05 0.37, 0.03 0.56, 0.07 νRT, νTR [−] 0.65, 0.37 0.48, 0.3 0.95, 0.7 νLT, νTL [−] 0.45, 0.03 0.4, 0.02 0.7, 0.06 GLR [MPa] 1375 810 1660 GRT [MPa] 430 46 540 GTL [MPa] 1010 790 1300 as shown in the Fig. 25.2, right part. The bridge admittances are often considered as a signature of the soundboard musical instrument dynamics and has been widely measured [17] but never modeled previously on a numerical model of a cello. The synthesis of the admittance of a cello can unlock some issues that are common in experiments, such as the reproducibility of the measure [18] and the interpretation of the results in the case of geometrical and material differences between instruments, since numerical models can change each parameter at once. 25.3 Results The computed modal bases are rich and only low frequency canonical modes will be considered. These modes, labeled as T1, C2, C3 and C4 according to usual nomenclature [19], are shown in the Fig. 25.3 and the evolution of the eigenfrequencies for the corresponding modes for each model V1, V2 and V2_2 are given in the Table 25.2. It is shown that changing the arch height can lead to a variation of the eigenfrequencies of up to 8% for low frequencies mode. In addition, for higher frequencies, the computed and experimental modal bases are not correlated above the 20th modes which states for a completely different behavior above 250 Hz. Thus, such geometrical parameters like arch heights and thickness are keys for the good correlation of a model and a real instrument and need to be characterized. The admittances at the bridge for the cases V2 (without repairs) and V2_2 (without repairs) are given in the Fig. 25.4. First, the usual A0 acoustic mode is not displayed on the admittance since the fluid-structure interaction was not implemented in the model. The admittance shape below 230 Hz is like the one given in [19] (with lower frequencies) and, under this point, no significative differences are shown between the repaired and not repaired cases. Above this frequency, differences occur in the admittances’ shapes, and increase with increasing frequencies, which highlights the facts that repairs affect the dynamical behavior of the cello, even in the low and mid-frequency domains.
RkJQdWJsaXNoZXIy MTMzNzEzMQ==