## 4 Conclusions  
Bull-nose end milling is an excellent finishing operation since it can provide greater material removal rates and smoother surface finishes. However, the regenerative chatter often occurs in the bull-nose end milling of thin-wall cylindrical parts, which have a ring-shaped and closed geometry structure. The dynamics of the cutter subsystem and the workpiece subsystem, the CWE, and the cutting force coefficients have a significant influence on the chatter stability. The objective of this article is to establish an accurate and applicable model to predict chatter stability during bull-nose end milling of thin-walled cylindrical parts. The main conclusions are summarized as follows:  
1. A 3-DOF chatter stability model is developed, which is applicable to forecast the regenerative chatter in the bull-nose end finish milling of thin-walled cylindrical parts. It reflects the dynamics of the cutter subsystem in the \(x\)-axis and \(y\)-axis directions and the dynamic of the workpiece subsystem in the surface normal (namely, \(z\)-axis) direction. The SDM is employed to solve the dynamic equation describing the 3-DOF milling system;
2. A slice-intersection-based method for calculating the engagement boundary curve is established. The method is applicable to determining the CWE geometries with different milling conditions, such as slotting and following cuts. Furthermore, a lead angle of 0 degrees is recommended to maximize the machined strip width;
3. The approach to layered cutting force coefficient identification is presented considering the effect of varying cutter diameter. The specific coefficients for each cutting disk element can be determined by substituting the corresponding height into the cubic fitting polynomial;
4. The validation tests of the proposed model are performed on a four-axis CNC machine tool. The predicted SLD agrees well with the experimental data. The spindle speed and the axial depth of the cut can be optimally chosen using the SLD to effectively avoid regenerative chatter and achieve smooth surfaces.  
Conceptualization, X.Z. and D.Z.; methodology, X.Z. and B.W.; software, X.Z.; validation, B.W.; formal analysis, C.Z. and D.Z.; data curation, X.Z. and M.X.; writing--original draft preparation, X.Z.; writing--review and editing, C.Z. and M.X.; funding acquisition, X.Z. and C.Z. All authors have read and agreed to the published version of the manuscript.  
This research was supported by the National Key Research and Development Program of China, grant number 2020 YFF0304905; and the research fund of the Key Laboratory of High Performance Manufacturing for Aero Engine (Northwestern Polytechnical University), Ministry of Industry and Information Technology, grant number HPM-2020-05.  
Not applicable.  
Figure 12: Finished surfaces of the thin-walled cylinder. **(a)** Finished surface at point A, Ra = 0.836 \(\mu\)m; **(b)** Finished surface at point B, Ra = 13.273 \(\mu\)m; **(c)** Finished surface at point C, Ra = 5.924 \(\mu\)m.  
**Acknowledgments:** The authors would like to thank [PERSON] for calculating the CWEs and programming the toolpaths in Siemens NX 8.0 software.  
**Conflicts of Interest:** The authors declare no conflict of interest.