Limit cycle oscillations of a violin string

Abstract

In this work, we write and solve a first principles model for the motion of a bowed string. We find limit cycle oscillations driven by stick-slip friction. The shape of these oscillations is in accordance with the Helmholtz-Rayleigh motion. We observe that when bow force, bow speed, and other parameters are varied, the stable limit cycle occurs in a narrow region of parameter space. This explains why it is difficult for amateurs to produce musically acceptable sounds from the instrument.