## 5 Result Analysis
### Case Study: Multi-Axis Milling of Aero-Engine Blade  
The experiments have proven that the accuracy of the dynamic characteristics of the workpiece is obtained by using the extended Sherman-Morrison-Woodbury formula. Next, the method is applied to the typical multi-axis milling process. We used the FEM to obtain the initial dynamic characteristics of the workpiece, and then used the proposed method to obtain the dynamic characteristics of the workpiece during the machining process. The model of the multi-axis milling time-varying parameter system of the blade is shown in Figure 11. The dynamic characteristic variation of the workpiece when the top of the blade material is removed during the semi-finishing process was simulated. The blade is a typical multi-axis milling molding part, the material is Al7050, the height is 213.3 mm, the blank margin is 3 mm, the blank thickness is 8-20 mm, the middle part of the blade is thicker, the thickness is 20 mm, the two sides are thinner, the thickness is 8 mm. In the machining process, the tool used is a carbide ball-end cutter with a diameter of 20 mm and an 82 mm overhang. Due to the workpiece overhang being longer, the rigidity is small, so the dynamic characteristics of the workpiece have a significant impact on the processing stability of the multi-axis milling system. The multi-axis milling semi-finishing at the top of the blade of the tool path can be obtained with CAM software. The total length of the tool path was 125.48 mm. Along the path and the whole process was divided into five cutting steps, the five cutting steps at the midpoint of the position coordinates were P1: 10 mm, P2: 37.65 mm, P3: 62.74 mm, P4: 87.84 mm, P5: 60.3 mm, as shown in Figure 11.  
In the machining, the relationship between the amount of material removal and the tool path is shown in Figure 12. As can be seen from the figure, the amount of material removal along the tool path is a nonlinear change. Because of the complex spatial characteristics of the tool movement, so in the traditional analytic method it is difficult to obtain the variation of material removal. The variation of the amount of material removal \(m(\gamma)\) with the cutting path \(\gamma\) in this picture can be expressed as  
\[m(\gamma)=2\times 10^{-7}\gamma^{2}+2.26\times 10^{-5}\gamma,\ R^{2}=0.9997. \tag{18}\]  
The original and the cross-point frequency response function of the midpoint of five cutting steps before and after machining were obtained using the FEM. We calculated the variation amount of the modal mass, modal stiffness, and modal damping in each structural modification, and then took the  
Figure 11: Model of multi-axis milling of blade. obtained calculation results as the initial condition, using the proposed method to extract the modal parameters, the results are shown in Figure 13.  
The blue stars represent the dynamic characteristics of the different positions predicted by using the extended Sherman-Morrison-Woodbury formula. On the basis of the results, the relationship of the dynamic characteristic parameters of the cutting process with the tool path was obtained by the least square method, as shown in Table 5.  
Figure 12: The variation amount of the material removal along the cutting path. Figure 13: The relationship between the dynamic characteristics of the workpiece and the cutting path. Figure 13 indicates that in the complicated blade multi-axis milling process, various forms of the tool movement and the material removal effect directly lead to the change of the dynamic characteristics of the blade (such as natural frequency, modal mass, modal stiffness, modal damping) when the cutting path is more complicated. Meanwhile, the natural frequency increased with the cutting path, the other three first increased and then decreased. The variation trend was consistent with the previous experimental results. Therefore, the existing cutting strategies, cutting process and cutting parameters are conservative, and the processing efficiency is low.