## 4 Results and Analysis  
### Vibration Response Analysis of Shafting under Different Blade Wear Degrees  
Equation (17) is solved by the Runge-Kutta method; the values of the basic parameters of the shafting of hydropower units are shown in Table 5.  
Figure 10: Time-domain diagram of vibration response of the turbine runner system in x direction: (**a**) unworn, (**b**) wear 22.461 mm. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Parameter** & **Value** & **Parameter** & **Value** \\ \hline \(m_{1}\) & \(1.4\times 10^{5}\) kg & \(a\) & 2.810 m \\ \(m_{2}\) & \(1.8\times 10^{4}\) kg & \(b\) & 1.925 m \\ \(k_{1}\) & \(2.9\times 10^{9}\) N/m & \(c\) & 8.705 m \\ \(k_{2}\) & \(3.2\times 10^{9}\) N/m & \(d\) & 2.010 m \\ \(k_{3}\) & \(1.2\times 10^{9}\) N/m & \(J_{p1}\) & \(1.35\times 10^{6}\) kg\(\cdot\)m\({}^{2}\) \\ \(k_{4}\) & \(9.0\times 10^{7}\) N/m & \(J_{p2}\) & \(1.2\times 10^{5}\) kg\(\cdot\)m\({}^{2}\) \\ \(k_{5}\) & \(6.0\times 10^{8}\) N/m & \(J_{d1}\) & \(1.5\times 10^{5}\) kg\(\cdot\)m\({}^{2}\) \\ \(c_{1}\) & \(5.1\times 10^{6}\) N/s/m & \(J_{d2}\) & \(6.0\times 10^{4}\) kg\(\cdot\)m\({}^{2}\) \\ \(c_{2}\) & \(3.2\times 10^{6}\) N/s/m & \(R\) & 3.9 m \\ \(c_{3}\) & \(4.0\times 10^{5}\) N/s/m & \(L\) & 1.6 m \\ \(c_{4}\) & \(4.0\times 10^{5}\) N/s/m & \(k_{j}\) & 1.2 \\ \(e_{1}\) & \(0.5\times 10^{-3}\) m & \(l_{j}\) & 1000 A \\ \(e_{2}\) & \(0.3\times 10^{-3}\) m & \(\mu_{0}\) & \(4\pi\times 10^{-7}\) H\(\cdot\)m \\ \(\omega\) & 31.4 rad/s & \(\delta_{0}\) & \(7.0\times 10^{-3}\) m \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Basic parameters of shafing of hydropower units. Figure 9: Time-domain comparison of displacement in the x direction of the generator system with unworn blades and worn blades of 22.461 mm. It can be seen from the above figure that the vibration response time-domain diagram of the generator rotor changes little with the wear degree of the runner blade, and its trough only changes from \(-1.33\times 10^{-5}\) m to \(-1.38\times 10^{-5}\) m. The changing trend of the vibration response of the turbine runner system is very obvious. Figure 11 shows the time-domain comparison of the turbine runner system under five different wear degrees of the blades. The figure shows that the wear degree of runner blades significantly affects the lateral vibration response of the runner system. The peak-peak value of the amplitudes under different wear degrees in Figure 11 are extracted, and the results are shown in Table 6. The peak-peak value in the x direction increased by 1.01 times, while the peak-to-peak value in the y direction increased by 0.86 times during the transition from no wear to wear of 22.461 mm. Figures 12 and 13 are frequency-domain diagrams of the vibration response in the x direction when the rotor blades are in two wear states at a rated speed of the spindle system. It can be seen from the figure that the main vibration frequency of the generator rotor and the turbine runner is equal to the rotation frequency. The main frequency amplitude of the generator rotor increases by only 2.43%, while that of the turbine runner increases by 95.14%. The frequency component of the generator rotor appears at half frequency, which indicates that the spindle bending and tilting, considering the gyroscopic effect, causes a nonlinear coupled self-excited vibration. There is only a small peak value at half frequency of the response of the turbine runner system, which indicates that the gyro effect has no obvious influence on the turbine runner system. Figure 14 shows the frequency-domain comparison of the vibration response of the turbine runner system at rated speed. The amplitude of the vibration response corresponding to the rotation frequency increases with an increase in the wear degree of the runner blade, and because the mean value of the radial excitation force is not zero, the DC component appears at 0 Hz on the spectrum diagram.