A General Formulation of the Milling Process Equation: Contribution to Machine Tool Chatter Research—5

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Abstract
In the cutting process, forces are induced at each of the cutter teeth in contact with the workpiece, and these forces in turn excite the machine tool structure. Due to the inherent feedback which exists between the cutting forces and the structure deflection, there are conditions under which this system becomes unstable. When this occurs, a condition of self-excited chatter exists. In the theory of self-excited chatter for single point tools wherein the tool is continuously in contact with the workpiece, the system can be described by a time-invariant equation, and has been rather fully developed. However, the milling process cannot be described in this manner. It is characterized by a multitooth cutter, and the cutting process itself is interrupted. Also, the direction of the cutting forces generated by each tooth does not remain constant with respect to the machine tool structure as for turning operations, but changes direction as a function of cutter position. In this paper, more complete description of the milling process is formulated. The resulting equation is a general nth order vector-matrix linear equation with periodic coefficients and a transport lag. Or equally, it is a linear differential-difference equation with time-varying coefficients. This equation is then expressed as n-first order equations (state variable form), consistent with current literature and in a form compatible for digital computer analysis.

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