## 4 Experimental Setup and Procedure  
To verify the accuracy of the method aimed at obtaining the instantaneous dynamic characteristics in the multi-axis milling process using the extended Sherman-Morrison-Woodbury formula, an experimental study is conducted. The experiment mainly consists of two parts: the modal experiment and the cutting experiment. The former mainly aims at obtaining the original and the cross-point frequency response function at different positions of the workpiece before and after machining, while the latter mainly aims at getting the workpiece after machining and conducting  
Figure 7: The flowchart for the calculation of frequency response functions (FRFs) (black solid arrow: cancellation of material removing effects; red dotted arrow: dummy inverse process). the modal experiment. It worth noting that, in the cutting experiment, whether one measures the cutting force of the workpiece during machining, or one measures the vibration of the workpiece with the laser displacement sensor during machining, neither is relevant to the instantaneous dynamic characteristics of the workpiece under discussion in this section, and the measurement is mainly used for other analysis. The devices used in the cutting and modal experiments are shown in Figure 8. The machine used in the process of the cutting experiments is a high-speed milling machining center VMC0540d with a maximum speed of 30,000 rpm, maximum power of 42 kW, and maximum torque of 22 N\(\bullet\)m. The cutter used in the experiment is a cemented carbide end milling tool with a diameter of 20 mm, helix angle of 30\({}^{\circ}\), overhang length of 67 mm, and tool tooth number 3. Compared with the workpiece to be machined, the tool stiffness is much larger than that of the workpiece, so that the milling cutter can be assumed to have a rigid structure, and the workpiece is considered to be flexible. In the experiment, the workpiece material is aluminum alloy 7050, and physical parameters of the workpiece are shown in Table 1. Fixed with clamps, the overhang is 50 mm and works along the longitudinal direction of the workpiece, as shown in Figure 8a. In the experiment, the feed per tooth is 0.05 mm, and the radial cutting depth linearly varies from 0 mm to 2 mm. When cutting along the length direction of the completed workpiece, the radial cutting depth is changed to 2 mm. The workpiece is divided into five (\(n=5\)) cutting steps along the machining path. The length of each cutting step is 13.4 mm, and the midpoint of each cutting step is taken as the excitation point and the response point of the modal experiment. The dynamic characteristics of the whole length of the cutting step are substituted by the original frequency response function at the midpoint of each cutting step. The position coordinates of the midpoint of the five cutting steps are P1: 6.7 mm, P2: 20.1 mm, P3: 33.5 mm, P4: 46.9 mm, and P5: 60.3 mm. The original and cross-point frequency response function at the midpoint of five steps before and after the cutting process was measured, respectively. The modal parameters are extracted from the original frequency response function. To verify the accuracy of the semi-discrete method and to obtain the stability limit diagram of delay differential equations with time-varying parameters of thin-walled parts with a complex curved surface and variable thickness during the multi-axis milling process, an experimental study was conducted. In the experiment, different machining conditions were set up, see Table 2, and the surface \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Workpiece**} & \multicolumn{3}{c}{**Geometric Dimensions (mm)**} & \multirow{2}{*}{**Density**} & \multirow{2}{*}{**Elastic**} & \multirow{2}{*}{**Poisson**} \\ \cline{2-2} \cline{5-6}  & & & & & **(kg/m\({}^{3}\))** & \\ \hline Al 7075 & 67 & 64 & 4 & \(2.82\times 10^{3}\) & 70.3 & 0.33 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Geometric and physical parameters of the workpiece. Figure 8: Experimental device.