Discrete-Time Prediction of Chatter Stability, Cutting Forces, and Surface Location Errors in Flexible Milling Systems

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Abstract
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.

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