# 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
It can be observed that the contour path at the HP is mainly affected by the first modal at the beginning and the end of the path. The contour path at the MP is mainly affected by the second and third modal at the beginning and the end of the path. The contour path at the LP is mainly affected by the third modal near the \(-200\) mm of \(X_{\text{m}}\)-direction along the contour path. With the use of the modeling method proposed by [PERSON] et al. [44], the frequency response function (FRF) of the semi-conical workpiece at any point (\(x\), \(y\)) in \(Z_{\text{m}}\)-direction with the dynamic force-excited position (\(x_{F},y_{F}\)) can be expressed as \(H(\omega,x,y)\):  
\[H(\omega,x,y)=\sum_{i=1}^{mnum}\frac{w_{\text{nor},i}(x,y)w_{\text{nor},i}(x_{F },y_{F})}{\omega_{i}^{2}-\omega^{2}+2\xi_{i}\omega\omega_{i}} \tag{11}\]  
where \(mnum\) denotes the number of models been considered, \(\xi_{i}\) and \(\omega_{i}\) are the \(i\) th modal damping ratio and natural frequency, respectively. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Methods**} & \multicolumn{5}{c}{**First Five Natural Frequencies (Hz)**} \\ \cline{2-6}  & **1** & **2** & **3** & **4** & **5** \\ \hline SGM with M = N = 15 & 600.54 & 985.30 & 1602.4 & 1913.6 & 2147.0 \\ FEM & 583.94 & 957.09 & 1533.1 & 1722.5 & 1995.5 \\ EMA & 586.44 & 966.23 & 1572.4 & 1703.2 & 1989.3 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Comparison of the first five natural frequencies for a semi-conical shell with different methods. Figure 7: First three modal shapes of the semi-conical shell workpiece with three chosen contour tool paths: **(a)** first modal shape; **(b)** second modal shape; **(c)** third modal shape. With the use of Equation (11), the magnitude and real part of the simulated FRF are shown in Figure 8, in which the force excited position and response position are at the same position (0, \(\pi\)/2). The first three modes are considered, and the modal damping ratio of each mode is set as 0.01. The results of the impact tests are also shown in the same figure, and they are very close to the results of the simulation. The experimental FRF possesses a higher magnitude and a lower natural frequency, but the difference is very small. The position (0, \(\pi\)/2) is the node of the first mode and only the second and third modes can be observed from the FRF. The machining instability is the severe vibration in the machining process due to the weak rigidity of the tool or workpiece, which is also known as the chatter phenomenon. The regenerative effect is the principal factor of the chatter, and the stability lobe diagram (SLD) is an important and effective tool to show the relationship between machining stability and machining parameters. Based on the Fourier series approach, the method proposed by [PERSON] [45; 46] has been widely used in practical conditions and embedded into much commercial software. However, a more simple and convenient method based on the average tooth angle is used in this paper because only one direction of the workpiece is considered as weak rigidity, and the complex shapes of the cutting edge can be ignored. The instantaneous chip thickness \(a_{\mathrm{w}}\) and cutting depth \(a_{\mathrm{p}}\) are demonstrated in Figure 9, and the obtained SLD can guide the selection of the spindle speed \(\Omega\) and cutting depth \(a_{\mathrm{p}}\) to achieve stable machining.