Modal approach to the dynamics of string instruments over the audible range

Abstract

The present theoretical analysis addresses the dynamics of the bow instruments in the whole audible range of frequencies. Several scenarios are considered for the string-bridge-soundbox system: natural vibrations and frequency spectrum, external periodic excitation on some bridge point, and vibration induced by the bow rubbing. The single parts of the musical artifact are conveniently modelled and the bridge interfacing between the strings and the top plate of the soundbox is studied considering two possible vibrating modes of the bridge: a translational mode along its symmetry axis and a rotational mode around the waist centre. The bridge mechanical model is thus definable from both the kinematical and the dynamical aspects, and the relations between the forces and displacements of the strings and the sound box are formulated.The coupled natural modes of vibration are obtained in non-dissipative conditions, finding that the whole frequency spectrum is roughly given by the union of the uncoupled spectra, of the strings and the harmonic box. Then, introducing some level of dissipation, the frequency response to a transverse harmonic excitation on an arbitrary bridge point and the admittance on another arbitrary point of the instrument are studied, obtaining a useful tool to identify or verify the mechanical characteristics of the system. At last, the response to the bow excitation is analysed introducing an appropriate model for the nonlinear stick-slip contact between the bow and the strings and solving the motion equations by modal decomposition.