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This paper reviews the dynamics of machining and chatter stability research since the first stability laws were introduced by Tlusty and Tobias in the 1950s. The paper aims to introduce the fundamentals of dynamic machining and chatter stability, as well as the state of the art and research challenges, to readers who are new to the area. First, the unified dynamic models of mode coupling and regenerative chatter are introduced. The chatter stability laws in both the frequency and time domains are presented. The dynamic models of intermittent cutting, such as milling, are presented and their stability solutions are derived by considering the time-periodic behavior. The complexities contributed by highly intermittent cutting, which leads to additional stability pockets, and the contribution of the tool's flank face to process damping are explained. The stability of parallel machining operations is explained. The design of variable pitch and serrated cutting tools to suppress chatter is presented. The paper concludes with current challenges in chatter stability of machining which remains to be the main obstacle in increasing the productivity and quality of manufactured parts.

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This paper presents a discrete-time modeling of dynamic milling systems. End mills with arbitrary geometry are divided into differential elements along the cutter axis. Variable pitch and helix angles, as well as run-outs can be assigned to cutting edges. The structural dynamics of the slender end mills and thin-walled parts are also considered at each differential element at the tool-part contact zone. The cutting forces include static chip removal, ploughing, regenerative vibrations, and process damping components. The dynamic milling system is modeled by a matrix of delay differential equations with periodic coefficients, and solved with an improved semidiscrete-time domain method in modal space. The chatter stability of the system is predicted by checking the eigenvalues of the time-dependent transition matrix which covers the tooth period for regular or spindle periods for variable pitch cutters, respectively. The same equation is also used to predict the process states such as cutting forces, vibrations, and dimensional surface errors at discrete-time domain intervals analytically. The proposed model is experimentally validated in down milling of a workpiece with 5% radial immersion and 30 mm axial depth of cut with a four fluted helical end mill.