# Feed Rate Variation Strategy for Semi-Conical Shell Workpiece in Ball Head End Milling Process
## 2 Vibration Simulation of the Milling Process  
This paper focuses on a practical machining process in which the outer surface of the semi-conical shell workpiece is milled with a ball head endmill cutter in a 3-axis machining center, as shown in Figure 2a. The geometric parameters of the workpiece and cutter are shown in Table 1, where \(R_{1}\) and \(R_{2}\) represent the radius of the right and left end of the workpiece, respectively. \(h\) represents the thickness of the workpiece. The material of the workpiece is the Al-Cu cast aluminum alloy with alloy code ZL205A in GB/T1173 or 3522 in ISO. It is regarded as the strongest aluminum alloy, which possesses good plastic and corrosion resistance, excellent machining and welding performance. The partial schematic diagram of the contour tool path is shown in Figure 2b, and the height difference between adjacent contour lines is set as 0.3 mm. A stationary rectangular coordinate system O\({}_{\text{w}}\)-X\({}_{\text{w}}\)Y\({}_{\text{w}}\)Z\({}_{\text{w}}\) of the workpiece is built to help describe the condition as shown in Figure 2c, and the geometric parameters of the workpiece are also shown in the same figure. The schematic diagram of the ball head endmill cutter is given in Figure 2d, and it should be noted that there is a difference between adjacent contour lines. When the cutter moves along the negative direction of the Y\({}_{\text{w}}\)-axis, it is an up milling process, which possesses a larger vibration and vice versa. In the machining process, the depth of cut is set as 0.2 mm constantly in the sphere radius direction of the ball head cutter, and the spindle speed is set as 3600 rpm. Because of the weak rigidity property of the area with a large radius, a conservative constant feed rate of 800 mm min\({}^{-1}\) is chosen to avoid large workpiece vibration caused by the cutting force and flexibility.  
In order to find the optimal feed rate variation, the vibration analyses containing machining stability and forced vibration response are calculated with the theory of mechanical vibration where the cutting force and dynamic performance are needed. The cutting force is obtained by [PERSON]'s method [30], which takes the special geometry of the workpiece and cutter, variable cutter-workpiece interface and  
\begin{table}
\begin{tabular}{c c c} \hline \hline  & **Parameter Name** & **Parameter Value** \\ \hline \multirow{6}{*}{Workpiece} & \(R_{1}\) (mm) & 26 \\  & \(R_{2}\) (mm) & 147 \\  & \(L\) (mm) & 842 \\  & \(h\) (mm) & 10 \\  & \(\Phi\) (rad) & \(\pi\) \\  & Material & Al-Cu cast aluminum alloy \\ \hline \multirow{6}{*}{Endmill cutter} & Number of teeth & 2 \\  & Helix angle (deg) & 25 \\ \cline{1-1}  & Diameter (mm) & 20 \\ \cline{1-1}  & Manufacturer and model & X-BTC2002 of DHF \\ \cline{1-1}  & Material & tungsten steel \\ \hline \hline \end{tabular}
\end{table}
Table 1: Geometric parameters of the workpiece and cutter.  
contour tool path into consideration. The SGM [31] is used to calculate the dynamic performance of the semi-conical shell workpiece, which is very feasible and accurate.