## 5 Result Analysis
### Case Study: Multi-Axis Milling of Aero-Engine Blade
The stability of the multi-axis milling can be predicted when the dynamic characteristic variation of the workpiece is obtained. Under the premise of a stable machining process, increasing the cutting parameters is beneficial to shorten the production cycle and improve the production efficiency. According to the corrected dynamic characteristics (see Figure 10) of the workpiece system with respect to cutting positions along machining direction, the 3D-SLD of the thin-walled workpiece milling was evaluated for different tool positions and spindle speeds by the semi-discrete method, as shown in Figure 14. Figure 14a is a 3D-SLD. The three coordinate axes represent the spindle speed, tool position, and axial depth, respectively. The combinations of the cutting parameters in the zones of the curved lines below are stable cases, and those in the zones of the curved lines above are unstable cases. At different locations of the workpiece in the machining process, the processing parameters can, according to Figure 14a, select the best combination of spindle speed and axial depth of the cut, making the whole cutting process remain in a stable condition. Figure 14b presents a new graphic chart only determined by the \"tool position\" axis and \"axial cutting depth\" axis. Which is easy to apply to optimize the cutting parameters such as the spindle speed and axial cutting depth. Without a loss of generality, six cutting conditions, as shown in Table 2, are implemented, in which three different spindle speeds (5000 rpm, 10,000 rpm and 16,000 rpm) are selected under two different axial cutting depths (\(a_{p}\) = 8 mm: pink dotted line; \(a_{p}\) = 12 mm: purple dotted line). The photographs of machined workpiece under six cutting conditions and its partial enlargement diagram are included in Figure 11.0  
Figure 11.1 Stability limit diagram. \begin{table}
\begin{tabular}{l c c c} \hline \hline
**Model Parameters** & **Expressions** & \(\mathbf{R^{2}}\) & **Diagrams** \\ \hline Natural frequency, Hz & \(\omega(\gamma)=-8.611\times 10^{-6}\gamma^{3}+1.752\times 10^{-3}\gamma^{2}\) & & \\  & \(-1.261\times 10^{-2}\gamma+1022\) & 0.9835 & Figure 13a \\ Modal mass, kg & \(m(\gamma)=-2.434\times 10^{-6}\gamma^{2}+3.184\times 10^{-4}\gamma\) & 0.9996 & Figure 13b \\ Modal stiffness, N/m & \(k(\gamma)=-2.548\gamma^{2}+338.2\gamma+22960\) & 0.9999 & Figure 13c \\ Modal damping, N-s/m & \(c(\gamma)=-7.803\times 10^{-4}\gamma^{2}+9.828\times 10^{-2}\gamma\) & 0.9856 & Figure 13d \\  & \(+7.326\times 10^{-1}\) & & \\ \hline \hline \end{tabular}
\end{table} 
Table 5: The relationship of dynamic characteristics with the change of the processing path. From Figure 14, it has been observed that the variation of the dynamic characteristics of the thin-walled part with a complex curved surface and variable thickness introduces a change of the axial cutting depth SLD with respect to the spindle speed. If it changes a lot, it would be possible that a few stable cases exist. For example, under condition 2, the speed is 10,000 rpm (green solid line), \(a_{p}=8\) mm (pink dotted line). Similar to the illustrations presented above, the part of the green solid line below the pink dotted line are the unstable zones, and the part of the green solid line above the pink dotted line are the stable zones. In other words, in condition 2, the workpiece experienced unstable cutting, stable cutting, and unstable cutting in a total of three processes. It should be underlined that, Figure 14 shows the stability limit diagram results in the cutting conditions with \(a_{p}=8\) mm and \(12\) mm, because the dynamic characteristics used are assessed in these cutting conditions, as described in the previous part. To verify the accuracy of the dynamic stability limit diagram, as above, the photographs of the workpiece machined surface under six cutting conditions were obtained, as shown in Table 6. In Table 6, the regions denoted by the labels a-n are in accordance with those in Figure 14b.