Physics-based modeling techniques of a twelve-string Portuguese guitar: A non-linear time-domain computational approach for the multiple-strings/bridge/soundboard coupled dynamics

The Portuguese guitar is a pear-shaped instrument with twelve metal strings which is widely used in Portuguese traditional music. Unlike most common guitars, it has a curved top-plate and a specific violin-like bridge which is not rigidly fixed to the soundboard of the instrument. From the dynamical point of view, if the bridge transmits the strings vibrations to the instrument body in order to maximize the radiated energy, it also couples all the component parts of the instrument which therefore interact by structural coupling. This can originate various audible effects such as beating behavior and the excitation of numerous sympathetic resonances enhanced by the large number of strings of the instrument, and this is certainly why the Portuguese guitar has such distinct sound compared to other guitars. In this paper, a fully coupled time-domain model of the Portuguese guitar is developed and a series of simulations are presented to emphasize the various coupling phenomena involved in sound production. To reproduce the main musical features, the model includes the coupled dynamics of the twelve strings supported by a bridge which interact with the body of the instrument, described through Finite-Element modeling of the soundboard of a typical Portuguese guitar. Further simple models have been devised for the string/fret interaction and the pluck excitation. Since nonlinear effects are quite apparent in the behavior of string musical instruments, the string dynamics is modeled by the Kirchhoff-Carrier equations which describe large-amplitude string vibrations, and includes the coupling between both polarizations of string motion. The coupling between the strings and the soundboard at the bridge is provided by a model of the bridge kinematics, built on the basis of simple geometrical rationale, so that the two perpendicular string motions can exchange energy back and forth. By a close examination of the energy transfers between the various subsystems of the model, we first assess the correct behavior of the physical model and then examine the respective influence of the string nonlinearity and the bridge on the nonplanar motion of the string. The fully coupled model which pertains to the restricted group of studies which deals with the complete physical-based modeling of a multi-stringed instrument, captures many important phenomena observed in practice, among which the pitch glide effect and the mutual excitation of sympathetic vibrations.