Estimates of the bistable region in metal cutting

The classical model of regenerative vibration is investigated with new kinds of nonlinear cutting force characteristics. The standard nonlinear characteristics are subjected to a critical review from the nonlinear dynamics viewpoint based on experimental results available in the literature. The proposed nonlinear model includes finite derivatives at zero chip thickness and has an essential inflexion point. In the case of the one degree of freedom model of orthogonal cutting, the existence of unstable self-excited vibrations is proven along the stability limits, which is strongly related to the force characteristic at its inflexion point. An analytical estimate is given for a certain area below the stability limit where stable stationary cutting and a chaotic attractor coexist. It is shown how this domain of bi-stability depends on the theoretical chip thickness. The comparison of these results to experimental observations and also to the subcritical Hopf bifurcation results obtained for standard nonlinear cutting force characteristics provides relevant information on the nature of the cutting force nonlinearity.