# 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  
Due to the special geometry of the semi-conical shell workpiece and the use of the ball head endmill cutter, the cutter-workpiece engagement frequently varies in the whole milling process. In order to handle this issue, two approaches, namely, the Z-map method [33; 34] and the solid modeler method [35; 36], are commonly used to determine the area of engagement. Based on the improved Z-map method, in which the logical array is used in the cutter-workpiece interface determination, [PERSON] et al. [30] proposed a cutting force prediction method in 3-axis ball head end milling of a sculptured surface along an arbitrary tool path. The speed, accuracy, and performance of these two different approaches were compared and developed by [PERSON] et al. [37]. In their research, the solid modeler method was proved to be faster and more applicable for complex 5-axis machining, while the Z-map method was more flexible and simpler for 3-axis machining. Besides the conventional calculation of static cutting force, the calculation of the dynamic cutting force considering the tool run-out effect [38], tool geometry, orientation tilt and lead angles, modal parameters and cutting conditions [39] have been proposed. With these accurate methods, the cutting force and vibration in the process can be predicted, which is very useful. However, the static cutting force prediction approach proposed by [PERSON] et al. [30] is used in this paper because the cut segments of the dynamic cutting calculation need to be very small to guarantee the accuracy, which needs a large number of computations. More crucial is that the initial value plays a very important role in the calculation of the dynamic forces, which may lead to a big difference from the practical condition. Hence, the static cutting force modal is used in this paper, and each cut segment is set as feed per tooth with the maximum and minimum cutting force of each segmented cut are preserved. This can demonstrate the trend of force and possess high computational efficiency. The cutting force prediction approach proposed by [PERSON] et al. [30] is based on the traditional elemental mechanical cutting force model [40] and the chip thickness model for steady-state ball-end milling machining [41]. The key to this approach is the calculation of the feed turning angle and  
Figure 2: (**a**) Experimental setup of the semi-conical shell workpiece; (**b**) partial schematic of the contour line tool path; (**c**) schematic diagram of the workpiece and coordinate system \(\text{O}_{\text{w}}\text{-}\text{X}_{\text{w}}\text{Y}_{\text{w}}\text{Z}_{ \text{w}}\); (**d**) schematic diagram of the ball-endmill cutter and coordinate system \(\text{O}_{\text{c}}\text{-}\text{X}_{\text{c}}\text{Y}_{\text{c}}\text{Z}_{ \text{c}}\). cutter-workpiece contact area of each segmented position along the tool path because they are frequently changing. In order to define the feed turning angle, a dynamic rectangular coordinate system \(\rm{O_{c}\text{-}X_{c}Y_{c}Z_{c}}\) of the cutter is established as depicted in Figures 2d and 3a. The \(\rm{X_{c}}\)-direction designates the projected feed direction within the perpendicular plane of the cutter axis, the \(\rm{Z_{c}}\)-direction points upward along the cutter axis, and the feed vector of this cutting process is expressed as \(f_{\rm{AB}}\). The feed turning angle \(\gamma_{\rm{AB}}\) can then be defined as the angle between the \(\rm{X_{c}}\)-direction and the \(\rm{Y_{w}}\)-direction, which ranges from \(-\pi\) rad to \(\pi\) rad along each contour tool path. The sign of the \(\gamma_{\rm{AB}}\) is determined by the sign of the first component of \(f_{\rm{AB}}\), which means that it is negative when the cutter moves to the positive direction of the \(\rm{X_{w}}\)-axis and vice versa. In this finish milling process, the heights of each contour are ranging from 146 mm to 26 mm. The endmill cutter with a flat head will then be used for the milling of the lower position instead of the ball head, which will not be discussed in this paper.