Milling Stability Prediction Considering Axial Geometric Contact Effects

To overcome the limitations of existing three-degree-of-freedom milling stability models in representing axial cutting conditions, this study develops a stability prediction framework that accounts for both axial segmentation and the axial contact angle. A three-degree-of-freedom dynamic model of the milling system is first formulated by introducing the axial contact angle. The tool axis is then discretized, so that the cutting force coefficients can be evaluated in different axial sections and the non-uniform distribution of cutting forces along the tool can be captured more accurately. After incorporating the regenerative mechanism, the milling dynamics are expressed in the form of a linear time-delay differential equation. To enhance the numerical accuracy of the time-delay system solution, a full-discretization scheme using third-order Lagrange–Hermite interpolation is developed for constructing the state transition matrix. The stability boundary is subsequently determined based on Floquet theory, from which the stability lobe diagram is generated. The proposed model and solution procedure are validated by comparison with existing methods and by time-domain simulation. The results show that, when the spindle speed ranges from 5000 to 10,000 rpm and the axial depth of cut ranges from 0 to 8 mm, the overall variation rate of the predicted stability region is 11.19% after incorporating axial discretization and 59.88% after considering the axial contact angle. The stable and unstable cutting responses obtained from time-domain simulations are consistent with the regions predicted by the stability lobe diagram, which supports the validity of the proposed approach. Further investigation shows that, for the established three-degree-of-freedom milling model and the specified cutting parameters, the axial contact angle exerts a pronounced nonlinear effect on the stability boundary. Specifically, as the axial contact angle ε increases within the range 0°<ε≤45°, the stable region gradually shrinks; when ε increases from 45° to 90°, the stable region expands instead. These observations can provide useful guidance for selecting milling parameters and identifying stable machining conditions.