## 2 3-DOF Chatter Stability Model
### Calculation of CWE  
CWE signifies the instantaneous contact interface while a bull-nose end mill is cutting a thin-walled cylinder. It is one of the foundations for modeling the dynamics of the milling system. During the semi-finish milling or finish milling of thin-walled cylinders, the cutting edges on the toroidal surface of a bull-nose end cutter are applied to end milling, giving rise to a smaller axial depth of cut and a larger machined strip width [39; 40]. As shown in Figure 3, the CWE for the bull-nose end milling of thin-walled cylindrical parts consists of three boundaries. Boundary 1 is the cutter profile curve in the current feed direction view. Boundary 2 is the intersection curve between the uncut surface of the workpiece and the toroidal surface of the tool. Boundary 3 is the intersection curve between the workpiece transition surface left by the previous cutting path (CP) and the toroidal surface of the tool. Furthermore, the workpiece transition surface can be generated by a series of boundaries 1, which correspond to each cutting contact point along the previous cutting path. Therefore, the calculation of boundary 1 is the key to determining the CWE at an arbitrary cutting contact point. As shown in Figure 4, a slice \(S_{i}\) can be obtained by offsetting the workpiece's machined surface \(S_{m}\) outward. Furthermore, the intersection curve \(C_{i}\) between the slice \(S_{i}\) and the cutter's toroidal surface \(S_{c}\) can also be calculated. The boundary points \(P_{i}\) and \(Q_{i}\) are determined by employing the closest lines from the two auxiliary planes \(P_{r1}\) and \(P_{r2}\) to the intersection curve \(C_{i}\). The normal directions of the two reference planes \(P_{r1}\) and \(P_{r2}\) are simultaneously orthogonal to the workpiece feed direction and the cutter axis direction. The two boundary points are located on boundary 1 and represent the maximum view width of the intersection curve \(C_{i}\) in the workpiece feed direction view. Then, the CWE boundaries can be identified using the intersection curve \(C_{a}\) and a spline obtained by fitting a series of boundary points. The surface trimming tool can be used to extract the engagement geometry for the following cut condition. Figure 4: Calculation of CWE boundary 1. (**a**) Calculation of the intersection curve between each slice and the cutter toroidal surface; (**b**) Calculation of the two boundary points on the intersection curve. Figure 3: CWE at a cutting contact point. The flowchart of the CWE calculation algorithm is shown in Figure 5. And the detailed steps of the CWE calculation algorithm are described in the following:  
1. Input the machined surface of the workpiece \(S_{m}\) and the toroidal surface of the tool \(S_{c}\), and determine the cutting contact point \(P_{c}\) according to the cutter axis orientation and the milling line position. The machined surface of the workpiece \(S_{m}\) and the toroidal surface of the tool \(S_{c}\) are tangent to each other at the cutting contact point \(P_{c}\);
2. Input the total number of slices \(N_{s}\) and the machining allowance \(h\), and set the initial value of an index variable \(i\) to 1;
3. Offset the workpiece's machined surface \(S_{m}\) towards the outside by the distance \(\Delta h=i\cdot h/N_{s}\) to obtain a slice \(S_{i}\);
4. Calculate the intersection curve \(C_{i}\) between the slice \(S_{i}\) and the toroidal surface of the tool \(S_{c}\). The intersection curve \(C_{i}\) is essentially boundary 2 with the corresponding machining allowance;
5. Calculate the minimum distance points \(P_{i}\) and \(Q_{i}\) from the intersection curve \(C_{i}\) to the reference planes \(P_{r1}\) and \(P_{r2}\), and store the minimum distance points into the set of boundary points \(\{P\}\);
6. Go to Step (3) and repeat the process until the index variable \(i\) is greater than the total number of slices \(N_{s}\);
7. Fit the set of boundary points \(\{P\}\) of the CWE as a spline curve \(B\);
8.