224 R. Viala et al. lack of knowledge. In the same manner, the fine knowledge of the geometry can be either impossible or costly, involving experiments and devices [6, 7]. So, a major concern must be pointed out about the geometrical and material uncertainties, and the way they can affect the results of a model. Moreover, the climatic conditions that an instrument can undergo have to be considered for such problematics, as musical instruments constantly undergo relative humidity and temperature changes when outside the showcase [8]. In this study, a cello is modeled using the finite element method, and a geometrical characteristic is changed, the height of the top and back plates, while the material properties are unchanged. Secondly, the impact of repairs and defaults on the dynamical response of the model is evaluated, as a potential starting point for a decision support tool for restoration of musical instruments. The object of study of this work is an antique cello which exhibits many previously repaired cracks, and galleries produced by xylophagous insects, whose activity can be experimentally detected by acoustic emission [9]. The cello is made of different species of wood: spruce, maple, ebony and sometimes rosewood that are assembled together. Different experimental studies in the dynamical fields have been focused on several parts of the cello, such as the tailpiece dynamics [10, 11] and studied wolf notes [12]. The body of the instrument is here the main object of concern and is made of carved maple back and bent sides, and a carved spruce soundboard. The aim of the study is to evaluate the impact of an incorrect geometrical modeling on the dynamical response. For this purpose, the geometry of the cello is finely modeled using computer aided design and the finite element method, described below. 25.2 Analysis The studied structure is a cello, made by Pietro Guarneri in the eighteenth century and kept at the musée de la musique, Paris, under the label E.1555. The nominal model is made using the template of a model given in [13] and the data collected by curators and instrument makers about some properties of the E1555 cello. The Computer aided design (CAD) of the cello model has been made using the software SOLIDWORKS and is shown in the Fig. 25.1. The repair cleats and galleries are schematized in the Fig. 25.2. The geometry has been meshed with tetrahedral elements with quadratic interpolation, the number of nodes is approximately equal to 350,000, which leads to a value close to one million degrees of freedom. Most of the parts are made of wood (maple, spruce an ebony species). The wooden parts are modelled under the linear elastic hypothesis, with an orthotropic definition of the elastic parameters. The material properties identified are taken from [14, 15] and given in the Table 25.1. For the latter, three models will be considered: the first model made with nominal values obtained in [13], labelled as V1 and the one with the correct arch height of both back and top, which corresponds to an increase of 10 mm and a corresponding global shape change, labelled as V2. The photogrammetry method has been used for the measurements of the arch dimensions. In addition, a model is created based on the model V2 without cleats and gallery, labelled as V2_2, to evaluate the impact of repairs or defaults on the behavior of the cello. The bent parts orientation has been considered by changing the orientation of the local coordinate frames of the corresponding elements. The modal basis computation of 100 modes lasts one hour. Once the modes are computed, the comparison between each case is made with a modal assurance criterion, proposed in [16]. To highlight the capability of the finite element models, the bridge admittances of each case are computed. The admittance is computed with the application of an input force and an observation of either the displacement velocity and acceleration at the same point in the same direction, Fig. 25.1 Computer aided design, (CAD) of the cello model; left: front view, right: back view
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