Nonlinear dynamics and playability of bowed instruments: From the bowed string to the bowed bar

摘要

The numerical simulation of bowed-string instruments has been traditionally implemented using wave propagation methods. In this paper we use a time-domain modal approach coupled with an explicit model for the frictional forces in order to compare the non-linear dynamical behaviour of two characteristic bowed musical systems: bowed strings and bowed bars. In order to understand the influence of the tuning undercut in the dynamical response of the bowed bar, we select three different bar profiles: a constant cross-section bar with inharmonic relationships of the modal frequencies; a traditional vibraphone bar with the first three bending modes into an approximately harmonic relationship of f_2 = 4f_1 and f_3 = 10f_1 (1-4-10) and a shape optimized bar with a 1-2-3 natural frequency relationship (f_n/f_1). The optimisation procedure used in the latter case can lead one to obtain unusual shapes for the instrument bars, which can originate new instruments with specific timbre qualities and playability characteristics. The results obtained show that the uniform and vibraphone bowed bar dynamics are mainly dominated by the first bending mode in contrast with the higher modal activity of bowed string motions. Although the vibraphone bar exhibits a higher level of playability (meaning how quickly a "musical" self-excited regime arises and how stable it is) in relation to the uniform bar, due to the higher harmonicity of the modal frequencies, the unorthodox tuning used for the harmonic (1-2-3) bar enables the emergence of similar waveforms as that of the bowed string as well as similar characteristic times (within an oscillation period), during which the bar is sticking to the bow.