# 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
Three contour paths from different heights are chosen to be studied in detail, which is situated at the heights of 120 mm, 80 mm, and 40 mm. They are abbreviated as HP (high position), MP (middle position) and LP (low position), respectively. The variations of the feed turning angle from different heights are shown in Figure 3b. It can be observed that the feed turning angle starts from \(\pi\)/2 at the beginning of the contour path, and it is increasing gradually until the cutter reaches the middle of each contour path. Near the left part of the workpiece, where the radius is small, the maximum feed turning angle \(\pi\) is reached and then its sign changes to negative. It continues to increase until the cutter reaches the end of the contour path, where it is \(-\pi\)/2. The length of each contour path can also be seen from Figure 3b, and a longer distance is passed for machining a whole contour path at the lower height. It must be pointed out that each contour tool path is established based on the principle that the depth of cut is 0.2 mm constantly in the sphere radius direction of the ball head cutter on the \(\rm{Y_{w}}\)-\(\rm{Z_{w}}\) plane of the cutter-workpiece contact point. On the \(\rm{Y_{w}}\)-\(\rm{Z_{w}}\) plane, the area of intersection between cutter and workpiece is constant no matter the height of the contour position, as shown in Figure 4a. This can be verified by a simple mathematical differential calculation. Furthermore, this can indicate that the resultant force from the contour at lower height should be larger than that at the high position because the actual cutting linear velocity is different due to the same spindle speed and different actual radius. However, it is not accurate to determine the cutting force within the two-dimensional (2D) cross-section, especially for a machining process with complex geometry contact areas between the cutter and workpiece. It is necessary to determine the contact area at each cut segment, and the distance between two adjacent segments is feed per tooth along the tool path in the calculation process. With the use of the improved Z-map method, the contact area is expressed by a 2D logical square array where the contacting elements are expressed as \"1\" and others are expressed as \"0\". The logical calculation is used to simplify the calculation process and improve computational efficiency. Six different positions from the contour tool paths are chosen to show the simulated contact area, as shown in Figure 4b, and the results are shown in Figure 4c-h. For the sake of comparison, the obtained contact areas shown in  
Figure 3: (a) Schematic diagram of the workpiece, contour tool path and coordinate system \(\rm{O_{c}\text{-}X_{c}Y_{c}Z_{c}}\); (b) simulation results of the feed turning angle along the contours from different heights. Figure 4c-h are located at the same position \(-100\) mm of the \(X_{\rm w}\)-axis. The black circle is regarded as the projection of the cutter in the \(Z_{\rm w}\)-direction, and the green-filled area is the contact area between the cutter and workpiece. It can be observed that the area of the contact area is decreasing gradually with the height reduction of the contour position. It can also be observed that the contact area is situated close to the tool center at the beginning of the whole milling process, and it is moving away from the tool center with the height reduction of the contour position. The symmetry of the workpiece and contour tool path can also be reflected in the results of the contact areas. With the obtained results of the feed turning angle and cutter-workpiece contact area, the cutting force can be calculated with the elemental mechanical cutting force model [40]. In that model, the cutting force is a linear function of the uncut chip thickness area, which can be represented with Equation (1).