Ensuring supercritical Hopf bifurcation in tool chatter using a nonresonant limit cycle absorber

Abstract

Regenerative tool chatter in turning is generally subcritical, resulting in rapid vibration escalation and tool separation upon stability loss. In contrast, a supercritical bifurcation would yield smaller vibrations, enabling a gradual return to stability. This study demonstrates that a light, nonresonant, secondary, limit cycle vibration absorber (LCVA) can alter the subcritical Hopf bifurcation in regenerative machine-tool chatter to supercritical. The absorber, being nonresonant, does not require precise tuning. We first obtain asymptotic approximations for different segments of the linear stability boundary to be used subsequently in the multiple scales analysis. Amplitude evolution equations thus obtained are recast into a canonical form, which reveals that three nondimensional parameters govern the coupled system’s qualitative behaviours. We find a criterion to ensure a supercritical Hopf bifurcation, and demonstrate supercriticality along the complete second stability lobe. We also show that the resulting tool chatter is completely quenched in the initial stages of instability and further incursions into the linearly unstable regime results in conditional appearance of chatter with finite amplitude tool chatter without separation. Tool chatter robustly appears at even larger incursions. Thus, bistability is shifted from the linearly stable regime to the unstable regime, allowing machining close to or beyond the linear stability boundary which results in better productivity. Numerical results from DDE-BIFTOOL substantiate these results.