Application of a Shooting Method to Regenerative Chatter During Turning

A two-degree-of-freedom (2-DOF) model comprising nonlinear delay differential equations (DDEs) is analyzed for self-excited oscillations during orthogonal turning. The model includes multiple time delays, possibility of tool leaving cut, additional process damping (due to flank interference), ploughing force, and shear-angle/friction-angle variation. An algorithm, based on an existing shooting method for DDEs, is developed to simulate tool dynamics and seek periodic solutions. The multiple-regenerative and tool-leaving-cut effects are simulated via an equivalent 1-DOF system by introducing a time shift. While the limit cycle amplitude and minimum-period obtained via shooting and via direct numerical integration compare well, the latter method converges very slowly, thus establishing the efficiency of the former. Numerical studies involving the machining parameters are presented. Only period-1 motion was observed for the range of cutting parameters considered here. Features of a subcritical Hopf bifurcation appear in the amplitude versus width-of-cut plane. This implies the possibility of subcritical instability characterized by sudden onset of finite-amplitude chatter. Additional process damping causes a reduction in chatter amplitudes as well as the subcritical instability to occur at a larger width of cut. An increase in width of cut causes frequent tool-leaving-cut events and increased chatter amplitudes. The frequency of tool disengagement increases with cutting velocity, despite cutting force in the shank direction remaining constant over a certain velocity range. The chatter amplitude at first increases and then decreases when the cutting velocity or the uncut chip thickness is increased. The present plant model and dynamics could be useful for real time active control of tool chatter.