Suppression of Regenerative Instabilities by Means of Targeted Energy Transfers

Abstract

Regenerative effects in machining arise from the fact that the cutting force exerted on a tool is influenced not only by the current position but also by that in the previous revolution. Hence, the equation of motion for the tool appears as a delay differential equation, and the regenerative instability results in steady-state periodic motions, called limit cycle oscillations. We study targeted energy transfers for suppressing regenerative instabilities by applying a nonlinear energy sink (NES) to a single-degree-of-freedom machine tool model. A series of bifurcation analysis by means of numerical continuation techniques demonstrate that there are three distinct suppression mechanisms; that is, recurrent burstouts and suppressions, and partial and complete suppressions of regenerative instabilities. We characterize each suppression mechanism numerically by means of wavelet and Hilbert transforms and analytically by means of the complexification-averaging (CX-A) technique. Furthermore, we extend the CX-A analysis to perform asymptotic analysis by introducing a reduced-order model and partitioning slow-fast dynamics. The resulting singular perturbation analysis yields parameter conditions and regions for the three suppression mechanisms, which exhibit good agreement with the bifurcations sets obtained from numerical continuation methods. The results will help design NESs for passively controlling regenerative instabilities in machine tools.