A Nonlinear Model of a Regenerative Vibrating Machine Tool

We study the existence of Hopf bifurcations in a model of a regenerative vibrating machine tool, by using the same methods as in the Bifurcation Theorem and in the theory of central varieties. In a paper published in 1984, H. M. Shi and S. A. Tobias [6] have studied the theory of finite amplitude machine tool instability. They considered there some experimental results and have shown that there exist unstable finite amplitudes that are periodically unstable moments of the machine tool in the neighborhood of its asymptotically stable states. Following [2], we apply the subcritical Hopf bifurcation in the delay equation model for machine tool vibrations. We have the mechanical model of a machine tool with regenerative vibrations in Figure 1, which presents the orthogonal cuts of such a model taken from [2]. Here f is the thickness of the splinter. Consider the splinter given by: ∆l = l − l0 + x(t), (1) where l is the length of the initial cut, l0 is the length of the cut made in equilibrium state, x(t) characterizes the position of the knife and is depending on the component Fx of the cutting force.