Simulation of the Oscillation Regimes of Bowed Bars: A Nonlinear Modal Approach

It is still a challenge to properly simulate the complex stick-slip behavior of multi-degree-of-freedom systems. In the present paper we investigate the self-excited nonlinear responses of bowed bars, using a time-domain modal approach, coupled with an explicit model for the frictional forces, which is able to emulate stick-slip behavior. This computational approach can provide very detailed simulations and is well suited to deal with systems presenting a dispersive behavior. The effects of the bar supporting fixture are included in the model, as well as a velocity-dependent friction coefficient. We present the results of numerical simulations, for representative ranges of the bowing velocity and normal force. Computations have been performed for constant-section aluminum bars, as well as for real vibraphone bars, which display a central undercutting, intended to help tuning the first modes. Our results show limiting values for the normal force F_N and bowing velocity y˙_bow, for which the “musical” self-sustained solutions exist. Beyond this “playability space”, double period and even chaotic regimes were found for specific ranges of the input parameters F_N and y˙_bow. As also displayed by bowed strings, the vibration amplitudes of bowed bars also increase with the bow velocity. However, in contrast to string instruments, bowed bars “slip” during most of the motion cycle. Another important difference is that, in bowed bars, the self-excited motions are dominated by the system first mode. Our numerical results are qualitatively supported by preliminary experimental results.