When a machine tool works on a surface that has already been machined (see Fig. 1), it is well known that, in many circumstances, the dynamic interaction of the tool with the wave left on the surface can cause the system to develop unstable vibrations. This type of self-excited tool oscillation is called regenerative chatter. Because chatter vibrations can cause poor surface finish on the workpiece and rapid tool wear, much work has been done on the modeling and analysis of the dynamics of regenerative chatter in machining; see, for example, the treatise of Tobias~1, the review article by Tlusty~2, or the more recent workshop proceedings edited by Moon~3. This work has produced a one degree-of-freedom lumped-parameter delay differential equation model, that Moon and Johnson~4 call the classical model for regenerative chatter, mx + c dx + kx = -K_tw(f + x(t) - x(t-tau)). (1) Here, for simplicity, we imagine a turning operation, so that the time delay tau is the reciprocal of the spindle speed OMEGA i.e. the time required for the workpiece to complete one revolution, f is the nominal feed rate, w is the axial depth of cut, and K_t is the relevant cutting coefficient (assumed constant). Analysis of this model (see, e.g., Tobias~1, Stepan~5) leads to the familiar depth of cut vs. spindle speed stability lobe picture of regenerative tool chatter (Fig. 2), where the chatter frequency is somewhat larger than the natural frequency of the most flexible mode of the machine-tool structural system. The analysis also supports the intuitive idea (see Tlusty~2) that the most stable cutting operation corresponds to the case when there is no change in the initial chip thickness. Thus, in the most stable situation, any surface waviness initially left by a perturbation in the motion of the tool is not amplified, because the tool returns to the same position after each revolution, i.e., x(t) = x(t-tau) This can be shown to occur when the spindle period is an integer multiple of the oscillation period of the tool.
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