Estimation of the Bistable Zone for Machining Operations for the Case of a Distributed Cutting Force Model

Abstract

Regenerative machine tool chatter is investigated for a singledegreeoffreedom model of turning processes. The cutting force is modeled as the resultant of a force system distributed along the rake face of the tool, whose magnitude is a nonlinear function of the chip thickness. Thus, the process is described by a nonlinear delaydifferential equation, where a short distributed delay is superimposed on the regenerative point delay. The corresponding stability lobe diagrams are computed and are shown numerically that a subcritical Hopf bifurcation occurs along the stability boundaries for realistic cuttingforce distributions. Therefore, a bistable region exists near the stability boundaries, where largeamplitude vibrations (chatter) may arise for large perturbations. Analytical formulas are obtained to estimate the size of the bistable region based on center manifold reduction and normal form calculations for the governing distributeddelay equation. The locally and globally stable parameter regions are computed numerically as well using the continuation algorithm implemented in ddebiftool. The results can be considered as an extension of the bifurcation analysis of machining operations with point delay.