Modelling of Sympathetic String Vibrations in the Clavichord Using a Modal Udwadia-Kalaba Formulation

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.

Interactions between nonlinear resonators represented in Volterra series

“…This approach has also been applied to the case of nonlinear resonators [20]. Another recent approach to work with constraints is based on the Udwadia-Kalaba method [21,22] which has been applied to a beam and string instruments [23] [24] [25]. Lately non-smooth dynamics methods have been used in the field of musical acoustics to work with such interaction problems [26].…”