Global Dynamics of Low Immersion High-speed Milling. Global Dynamics of Low Immersion High-speed Milling

In the case of low immersion high-speed milling, the ratio of time spent cutting to not cutting can be considered as a small parameter. In this case the classical regenerative vibration model of machine tool vibrations reduces to a simpli ed discrete mathematical model. The corresponding stability charts contain stability boundaries related to period doubling and Neimark-Sacker bifurcations. The subcriticality of both types of bifurcations is proved in this paper. Further, global period-2 orbits are found and analyzed. In connection with these orbits, the existence of chaotic motion is demonstrated for realistic high-speed milling parameters. High-speed milling is one of the most e cient cutting processes used in industry. In the process of optimizing this technology, it is a challenging task to understand its special dynamical properties. Although this eld has a vast literature (e.g. [2, 1, 15, 4, 3, 10]), very little is known about the nonlinear dynamics of high-speed milling processes. In this paper we investigate a nonlinear discrete time model, whose linear counterpart was constructed rst by Davies et al. [4]. This nonlinear model is simple enough to have closed form results, which qualitatively describe complicated phenomena found by simulations in a delay equation model of the process [3, 11]. Particularly, the stability analysis in Davies et al.[4] shows that the xed point of the model can lose its stability in two ways: either by a Neimark-Sacker bifurcation or by a period doubling bifurcation. Here we prove that both bifurcations are subcritical. We also investigate another period2 motion. This second period-2 motion corresponds to the tool cutting only every second period. Similarly, this motion can also bifurcate in the two ways mentioned above. In the case when the unstable ∗PACS: 05.45.-a †Department of Applied Mechanics, Budapest University of Technology and Economics, P.O. Box 91, H-1521, Budapest, Hungary, e-mail: [email protected] ‡Department of Applied Mechanics, Budapest University of Technology and Economics, P.O. Box 91, H-1521, Budapest, Hungary, e-mail: [email protected] §Department of Engineering Mathematics, University of Bristol, BS8 1TR, Bristol, United Kingdom, e-mail: [email protected]