# Feed Rate Variation Strategy for Semi-Conical Shell Workpiece in Ball Head End Milling Process
## 2 Vibration Simulation of the Milling Process
### Cutting Force Prediction
\[\mathrm{d}F_{i}=K_{ic}\mathrm{d}A_{c}+K_{ic}\mathrm{d}s \tag{1}\]  
where \(\mathrm{d}F_{i}\) represents \(\mathrm{d}F_{t}\), \(\mathrm{d}F_{r}\) and \(\mathrm{d}F_{a}\) that are instantaneous forces in tangential, radial and axial directions as shown in Figure 2d. \(K_{ic}\) and \(K_{ic}\) stand for \(K_{tc}\), \(K_{cc}\), \(K_{ac}\), \(K_{te}\), \(K_{re}\) and \(K_{av}\) that are cutting force components and components of edge effects in tangential, radial and axial directions, respectively. \(\mathrm{d}A_{c}\) and \(\mathrm{d}s\) indicate instantaneous chip load and elemental length of the cutting edge in the axial direction. Different from the common flat head endmill cutter, the cutting coefficients are not constant along the cutting edge of the ball head endmill cutter because of the changes of local diameter and local helix  
Figure 4: Schematic diagrams: (**a**) cutter–workpiece engagements in the view of the \(X_{\rm w}\)-direction; (**b**) different contour tool paths in the view of the \(Z_{\rm w}\)-direction and distribution of the chosen positions; (**c**–**h**) simulation results of the cutter–workpiece engagements from the chosen positions. angle in the spherical part of the cutter. In order to conduct this issue, [PERSON] et al. [42] proposed a new identification method for ball-end milling and compared the different results obtained from the proposed model, common model and experimental results. It can be observed that the results from three methods share the same force trend, whereas a larger force is obtained by the common model. Considering the force trend is the same, and a larger force can be limited by a larger force threshold, cutting coefficients shown in Table 2 are chosen from reference book [43], which are obtained from calibration experiments with the same material of the workpiece and cutter. In order to improve the computational efficiency, only the maximum and minimum cutting force of each segmented cuts are preserved and shown since only the most critical situations matter. The simulated component cutting force from different directions and the resultant cutting force from three different contour paths are shown in Figure 5. Furthermore, the thickness direction of the semi-conical shell workpiece is the most flexible direction corresponding to the sphere radius direction of the cutter, as shown in Figure 2d. Hence, the radial cutting force of the cutter is given in Figure 5e, which will be useful in the next section. From Figure 5, it can be observed that the cutting force is stable at the beginning and the end of the contour path, whereas the large value and complex variation of the cutting force occurred at the middle of each contour path. This can be explained by the relationship between the adjacent contour paths, \begin{table}
\begin{tabular}{c c} \hline \hline
**Cutting Force Coefficients** & **Value** \\ \hline Cutting force component in the tangential direction, \(K_{tc}\) (N mm\({}^{-2}\)) & 895.5 \\ Cutting force component in the radial direction, \(K_{rc}\) (N mm\({}^{-2}\)) & 327.5 \\ Cutting force component in the axial direction, \(K_{ac}\) (N mm\({}^{-2}\)) & 64.2 \\ Component of edge effects in the tangential direction, \(K_{te}\) (N mm\({}^{-1}\)) & 22.5 \\ Component of edge effects in the radial direction, \(K_{te}\) (N mm\({}^{-1}\)) & 21.2 \\ Component of edge effects in the axial direction, \(K_{ac}\) (N mm\({}^{-1}\)) & 3.7 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Cutting force coefficients. Figure 5: Simulated cutting force along the contour from different heights: (**a**) cutting force in the \(X_{w}\)-direction; (**b**) cutting force in the \(Y_{w}\)-direction; (**c**) cutting force in the \(Z_{w}\)-direction; (**d**) resultant cutting force; (**e**) radial cutting force of the ball head endmill cutter. which are distributed sparsely at the middle of the contour path, as shown in Figure 2b. There will be a slot milling phenomenon at the same position, which can be seen in Figure 4e-f.