## 3 Numerical Analysis
### Verification
#### 3.2.1 Validation of Literature and FEM
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Method**} & \multicolumn{8}{c}{**Dimensionless Natural Frequency Order**} \\  & **1** & **2** & **3** & **4** & **5** & **6** \\ \hline Present & 163.44 & 231.14 & 1109.97 & 1323.37 & 2744.02 & 3692.69 \\ [PERSON] et al. (2020) & 164.17 & 232.24 & 1125.86 & 1331.65 & - & - \\ [PERSON] et al. (2020) & 164.27 & 231.57 & 1124.41 & 1332.04 & 2796.71 & 3729.08 \\ FEM & 163.51 & 230.80 & 1107.20 & 1324.50 & 2727.20 & 3700.10 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Comparative validation of the natural frequency (Hz) of the shaft–disk system. the same as those of the corresponding single plate, as follows: \(R_{i}=0.02\) m, \(R_{o}=0.2\) m, \(h_{d}=0.008\) m, and \(N=4\), of which the round hole \(R_{h}=0.11\) m, \(r_{h}=0.05\) m, the sector hole \(R_{1}=0.07\) m, \(R_{2}=0.15\) m, and \(d_{h}=0.02\) m. The rotors' shafts are the same, and the disks are located in the middle of the 2 nd section of the shaft. The geometric parameters of shaft are: \(L_{1}=0.3\) m, \(L_{2}=0.1\) m, \(L_{3}=0.2\) m, \(r_{1}=0.015\) m, \(r_{2}=0.02\) m, and \(r_{3}=0.015\) m.  
In the experiment, we used the hammering method to test the natural frequency and modal vibration pattern of the structure, and the required equipment came from the DongHua DHDAS dynamic signal test and analysis system. The experimental schematic and field diagrams are shown in Figures 9 and 10. The overall experimental process is divided into three major parts:  
1. Experimental pre-processing. The experimental parts are processed, and according to the size of the experimental parts in the modal analysis software, the model is drawn and the measurement point numbering is completed. Then, the corresponding measurement points are drawn. The experimental parts are set as the free boundary, and the elastic rope suspension is used to simulate the boundary conditions. The simulation of the experimental parts is completed and the nodes with large deformation in each order of the modal state are selected as the response test points in order to paste the sensor during the experiment. Figure 10: Modal test experiment of shaft–disk rotors. Figure 9: Schematic diagram of modal test experiment. 2. Experimental measurement. The experiment of disk structure adopts the method of single-point vibration pickup; the experiment of shaft-disk structure adopts the method of single-point excitation and multi-point response, so the modal test of the disk only needs to place a single acceleration sensor on the response test point, while the shaft-disk system needs to select a measurement point on the shaft and the disk and paste acceleration sensors. Afterwards, the device connection is completed and the device parameters are debugged. Finally, the force hammer is used to tap each measurement point in turn to obtain and store the response data. 3. Post-experiment processing. After completing the experiment, the obtained data are processed through the modal analysis software, the intrinsic frequency is calculated, and the modal vibration pattern is plotted. Table 5 give the experimental results and their comparisons with the model, and Difference 1 and Difference 2 in the tables represent the deviations of this method from the experimental and finite element results, respectively. The natural frequencies of the two experimental parts are in good agreement, the comparison deviation of the shaft-disk rotor is not more than 2.5%. Considering the processing error of the experimental parts, the measurement error, and the imperfect simulation of the boundary conditions, in addition to many other factors, it can be assumed that the above deviations are in the acceptable range. Taking into account that the number of measurement points is much smaller than the number of nodes in the model, the comparison of the experimental modal results is not good, but in general, the two modal patterns are basically the same.