Exact stable regions in the parameter space of machine tool chatter

One of the most common problems that occur during machining is Machine tool chatter, which adversely affects surface finish, dimensional accuracy, tool life and machine life. Machine tool chatter can be modeled as a linear time invariant differential equation with time delay or delay differential equation. Infinite dimensional nature of delay differential equations is apparent in the study of time delay systems. The analytical stability methods are thus more difficult for these differential equations and approximate methods do not give accurate results. In this paper, a new method is developed to determine the exact stable region(s) in the parameter space of machine tool chatter. In this method, first, the bifurcation points are determined. Then, the Lambert function is used to decide on the stability characteristics of each particular region. The advantages of this method are simple implementation and applicability to high order linear time delay systems. By resulting stability regions from this method, we can choose an optimal spindle speed to suppress the chatter. The new approach is the most acceptable method with comparison to traditional graphical, computational and approximate methods due to excellent accuracy and other advantages.