## 5 Result Analysis
### FRF Analysis
After obtaining the frequency response function matrix of the machining, depending on the application of extended Sherman-Morrison-Woodbury formula presented in previous, we calculated the original frequency response function at the midpoint of each cutting step in the process and extracted the modal parameters. Thereby, to verify the accuracy of the formula, the finite element model of different cutting steps was established, the finite element analysis was conducted, and the analysis results were post-processed to extract the dynamic characteristic parameters. The dynamic characteristic parameters of each cutting step in the process obtained by both methods were plotted for each dynamic parameter changing with the cutting path, and the final results are shown in Figure 10. In the figure, the red solid circles represent the results of the finite element analysis, the blue five-pointed stars represent the results predicted by the extended Sherman-Morrison-Woodbury formula, and the blue curves are the results that were fitted by the least square method according to the blue five-pointed star. According to the prediction by the extended Sherman-Morrison-Woodbury formula, the results are fitted by the least square method, and the relationship of the dynamic characteristic parameters of the cutting process with the tool path is shown in Table 4.  
Figure 9(a) indicates that, initially, the amount of material removal was less, and the natural frequency of the workpiece along the machining path increased slowly, with the increase in the amount of material removal, the natural frequency of the workpiece increased significantly. The more material is removed, the greater the natural frequency of the workpiece is changed. Namely, in the thin-walled parts of the multi-axis milling process with a large amount of cutting, the influence of material removal on the dynamic characteristics of the workpiece cannot be ignored. From Figure 9(b), it can be seen that in the process of machining, the modal mass, the modal stiffness, and the modal damping variation trends are consistent, they first increase and then decrease, and the whole trend is a parabolic shape. From the graph, we can find that the symmetry axis of the three parabolas is not in the midpoint position of the cutting tool path, which is mainly caused by material removal. In addition, comparing the predicted results by finite element analysis and the application of extended Sherman-Morrison-Woodbury formula, we find that the results of the finite element analysis simulation and the results predicted by applying the Sherman-Morrison-Woodbury formula have a \begin{table}
\begin{tabular}{l l l l} \hline \hline
**Model Parameters** & **Expressions** & \(\mathbf{R^{2}}\) & **Diagrams** \\ \hline Natural frequency, Hz & \(\omega(\gamma)=0.02924\gamma^{2}-0.2687\gamma+1249\) & 0.9999 & Figure 9(a) \\ Modal mass, kg & \(m(\gamma)=-1.261\times 10^{-5}\gamma^{2}+7.307\times 10^{-4}\gamma+2.136\times 10^{-2}\) & 0.9985 & Figure 9(b) \\ Modal stiffness, N/m & \(k(\gamma)=-19.11\gamma^{2}+1195\gamma+32780\) & 0.9998 & Figure 9(c) \\ Modal damping, N-s/m & \(c(\gamma)=-5.776\times 10^{-4}\gamma^{2}+3.428\times 10^{-2}\gamma+1.15\) & 0.9921 & Figure 9(d) \\ \hline \hline \end{tabular}
\end{table} 
Table 4: The relationship of dynamic characteristics with the change of the processing path. high consistency; both the size and change tendency are consistent. In other words, the method of using the extended Sherman-Morrison-Woodbury formula to obtain the dynamic characteristic of the workpiece during the machining process is accurate. When the variation of the dynamic characteristic of the cutting process with the tool path has been obtained, we can predict the machining process stability, optimize the cutting parameters, and improve the processing efficiency.