# Feed Rate Variation Strategy for Semi-Conical Shell Workpiece in Ball Head End Milling Process
## 2 Vibration Simulation of the Milling Process
### Dynamic Performance of Workpiece and Stability Prediction of Cutting System
In addition, the required equations to get the SLD are given as follows:  
\[a_{\mathrm{plim}}=\frac{-1}{2K_{s}\mathrm{Re}[H]N_{t}{}^{*}} \tag{12}\]  
\[\frac{f_{c}}{\Omega N_{t}}=N_{s}+\frac{\varepsilon}{2\pi} \tag{13}\]  
\[\varepsilon=2\pi-2\tan^{-1}\left(\frac{\mathrm{Re}[H]}{\mathrm{Im}[H]}\right) \tag{14}\]  
where \(a_{\mathrm{plim}}\) is the critical cutting depth to avoid chatter, \(f_{c}\) is the chatter frequency, \(N_{s}\) is the integer number of waves of vibration imprinted on the workpiece surface in one revolution, \(\varepsilon\) is the phase between current and previous tool vibration, \(K_{s}\) is the cutting force coefficient in the direction of weak rigidity, which is \(K_{rc}\) in this paper, \(N_{t}\) is the number of teeth on the cutter and \(N_{t}{}^{*}\) is the average number of teeth in the cut which can be denoted as:  
\[N_{t}{}^{*}=\frac{(\phi_{ex}-\phi_{st})N_{t}}{360} \tag{15}\]  
where \(\phi_{ex}\) and \(\phi_{st}\) are the start and exit angle in the cut, which can be obtained from the results of the cutter-workpiece engagements. Figure 8: (**a**) Magnitude of the frequency response function (FRF) from the simulation and experiment; (**b**) real part of the FRF from the simulation and experiment; (**c**) stability lobe diagram obtained with the FRF from the simulation and experiment. Combining the given equations and obtained FRF, the SLD from each position of the semi-conical workpiece can be obtained. As an example, the SLD for the position (0, \(\pi\)/2) is shown in Figure 7c. In the SLD, the curve stands for the critical depth of cut, and the area below this curve is the stable area, and it can be observed that the critical depth of cuts is different with the variation of the spindle speed. Therefore, in order to show the stability condition for each position of the workpiece, two three-dimensional (3D) SLDs are shown in Figure 10. The \(x\)-axis and \(y\)-axis of the figure are the positions in the \(\mathrm{X_{m}}\)-direction and \(\mathrm{Y_{m}}\)-direction of the workpiece, and the different colors show the value of the critical cutting depth. The critical cutting depth can be divided into absolute critical cutting depth and relative critical cutting depth. The former means the maximum of the critical cutting depth of all spindle speeds, as shown in Figure 10a. The latter means the critical cutting depth of one certain spindle speed, and the critical cutting depth of 3600 rpm is shown in Figure 10b. From Figure 10, it can be found that the distributions of the absolute critical cutting depth and relative critical cutting depth are almost the same, which are also in the same as stiffness distributions of the workpiece. The critical cutting depth of the area near the left and fixed boundary positions of the workpiece are very high. The minimum of the absolute critical cutting depth is 3.216 mm, which is obtained at the position (0, \(\pi\)/2). Therefore, the selection of the cutting depth for an area with a large radius should be more careful. The main difference between Figure 10a,b is that the distribution of area with lower critical cutting depth in Figure 10a is more than that in Figure 10b. With the same calculation as shown in Figure 4a, the maximum of the cutting depth \(a_{\mathrm{P}}\) used in the practical condition is 1.93 mm, which is lower than the minimum of the absolute critical cutting depth of all positions. Figure 10: 3D stability lobe diagrams (SLDs) of the semi-conical workpiece considering the first three modes in the \(\mathrm{Z_{m}}\)-direction: (**a**) absolute critical cutting depth for each position of the workpiece at all spindle speed; (**b**) relative critical cutting depth for each position of the workpiece with the spindle speed at 3600 rpm. Figure 9: Geometries diagram of the cutting and vibration in the milling process (**a**) in the view of the \(\mathrm{Z_{w}}\)-direction; (**b**) in the view of the \(\mathrm{X_{w}}\)-direction.