Chapter 32 Modelling of Sympathetic String Vibrations in the Clavichord Using a Modal Udwadia-Kalaba Formulation J.-T. Jiolat, J.-L. Le Carrou, J. Antunes, and C. d’Alessandro Abstract The vibratory and acoustic modeling of musical instruments is important for several purposes in cultural heritage preservation, performance studies and musical creation. On the one hand, building a model helps understanding the key features of an instrument, and then is useful for evaluation, documentation and preservation of historical models. On the other hand, modeling and simulation can help for improving existing instruments, or even designing new instruments by extension of the model. The clavichord is an early keyboard instrument equipped with a very simple mechanics. The strings are excited by small metal wedges or blades (the tangents) placed at the end of the keys. The tangent remains in contact with the strings for the duration of the note, defining the vibrating length of the string. All strings are coupled at a same bridge. A string is divided into three sections: a damped section (DS) between the hitch-pin and the tangent; the played section (PS), excited by the tangents, between the tangent and the bridge; and the resting section (RS) between the bridge and the tuning pin. Because of the coupling through the bridge of the PS and RS, the RS is set into vibration, acting as sympathetic strings. The vibratory responses of the RS is modelled using a modal approach based on the Udwadia-Kalaba formulation. Firstly, a review of the method is presented, accompanied with measurements performed on an instrument (copy of a Hubert 1784 fretted clavichord), which include an experimental modal analysis at the instrument bridge and measurements of string motions. Then, simulation results are reported and compared with experimental measurements. Keywords Sympathetic vibration · Clavichord · Udwadia-Kalaba formulation · String coupling · Modal analysis 32.1 Introduction The sound of string instruments results of the vibratory behavior of coupled mechanical subsystems. These couplings can be studied by using physical modeling of several kinds. For instance, in the case of the concert harp, the coupling of the strings and the soundboard has been modeled by means of transfer matrices [5]. Also, it could be modeled by using finite element methods or experimental modal analysis, in particular using substructure techniques. In the case of the guitar, the couplings have been modeled by extracting the modal parameters of the soundboard at the bridge locations where the strings and the structure motions are coupled [1]. In the clavichord, a string is divided into three functional sections: a damped section (DS) between the hitch-pin and the tangent; the played section (PS), excited by the tangents, between the tangent and the bridge; and the resting section (RS) between the bridge and the tuning pin (see Fig. 32.1). The RS of the string is not directly excited by the tangent but is subjected to the motion constraint at the bridge. Then it is set into vibration, acting as sympathetic strings. Our objective is to predict the vibratory response of the RS of strings, set indirectly into vibration as a consequence of the excitation of one PS. To proceed accordingly, we first present the Udwadia-Kalaba (U-K) formulation and its modal extension, in order to compute the vibratory responses of a set of coupled mechanical substructures. Then, having extracted the necessary experimental modal parameters from our studied clavichord, we present some results from our numerical simulation. J.-T. Jiolat ( ) · J.-L. Le Carrou · C. d’Alessandro Sorbonne Université, CNRS, Institut Jean Le Rond d’Alembert, Equipe LAM, Paris, France J. Antunes Centro de Ciencias e Tecnologias Nucleares, Instituto Superior Técnico, Universidade de Lisboa, Bobadela LRS, Portugal © Society for Experimental Mechanics, Inc. 2020 R. Barthorpe (ed.), Model Validation and Uncertainty Quantification, Volume 3, Conference Proceedings of the Society for Experimental Mechanics Series, https://doi.org/10.1007/978-3-030-12075-7_32 277
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