Stabilityand HOPF Bifurcation Analysis In Amachine Model Of Regenerative Vibration With Time Delay

In the metal cutting process, tool and workpiece often arise relative vibration. According to the causes of the relative vibration can be divided into two: one is caused by external excitation of forced vibrations; the other is self-excited vibration caused by the cutting process itself. Relative vibration will reduce the quality of cutting machined surface and affect the life of the machine tool as well. The harm to the precision of modern numerical control machines is particularly serious, because the kind of workpiece precision machine which guarantee the high precision index will becomes pointless. Therefore, we must find out the vibrational factors of cutting vibration model. The analysis of the experimental and theoretical show that diverse causes of chatter, chatter factors are too numerous.In this thesis, we study a machine model of regenerative vibration with time delay, the model describes the process of the tool cutting the workpiece. This model and the traditional model is different, because it considers the rotation of the workpiece, the vibration of the tool itself is also considered. Studying the dynamic properties of the model will help us understand the processes and causes of the vibration. In this paper, the main task is to research the model of regenerative vibration with time delay from the perspective of stability and bifurcation, and we can obtain the sufficient conditions for stability of the machine tool.First of all, we proved that the existence of equilibrium of the system, then the sufficient conditions of the stability and the existence of Hopf bifurcations at the equilibrium are obtained by analyzing the distribution of the characteristic values. Furthermore, an explicit algorithm for determining the direction of the Hopf bifurcation and the stability of the bifurcating periodic solutions are derived by using the normal form and the center manifold theory. At last, several numerical simulations to support our theoretically analytical conclusions are carried out using Matlab soft.