## 2 Calculation of Runner Fluid Exciting Force Considering Blade Wear
### Numerical Simulation of Radial Fluid Exciting Force of Runner  
Figure 5 shows the time-domain comparison of the radial fluid exciting force on the runner at different wear thicknesses under the conditions of rated speed and rated inlet flow. As can be seen from the figure, no matter what kind of wear state the runner is in, the radial exciting force it is subjected to shows significant periodic characteristics in general.  
Figure 6 shows the frequency-domain comparison of the radial exciting force of the runner under different wear degrees (the influence of the DC component has been removed). When the blade is not worn, the amplitude of the exciting force in the x direction and y direction is about 320 N and 360 N, respectively, and the amplitude of the blade frequency is slightly higher than that at the rotation frequency. With an increase in the blade wear degree, the amplitude of the exciting force at the rotation frequency and the blade frequency shows an increasing trend. The reason is that the blade wear leads to a decrease in the guiding performance of the runner system, the energy conversion efficiency becomes low, and part of the fluid energy is converted into radial force. However, the excitation force at high frequency has a strong correlation with the inherent characteristics of the blade and has a great influence on the key vortex structures such as the blade passage vortex, the Karman vortex, and the draft tube vortex zone.  
\begin{table}
\begin{tabular}{c c} \hline
**Different Operating Time/Year** & **Wear Thickness/mm** \\ \hline
1 & 3.209 \\
3 & 9.626 \\
5 & 16.043 \\
7 & 22.461 \\ \hline \end{tabular}
\end{table}
Table 2: Blade wear at different time stages.  
Figure 5: Time-domain comparison of radial exciting force of runner system: (**a**) x direction, (**b**) y direction.  
Figure 7 shows the trend diagram of the radial exciting force amplitude changing with the blade wear degree at different frequencies. Taking the radial exciting force amplitude of the runner in the X-axis direction as an example, when the blade wear thickness gradually increases, the amplitude of the rotating frequency exciting force increases from 321.53 N to 981.82 N, increasing by about 2.05 times. The amplitude of the blade frequency increased from 472.44 N to 1263.59 N, increasing by about 1.67 times. The amplitude of the exciting force at the frequency of the guide vanes fluctuated at around 500 N, and there is no obvious increasing trend because the generation area of the static and static interference frequency of the guide vanes is in front of the runner inlet, and the wear of the blade has little influence on the upstream flow field. In addition, by observing the changing trend of the amplitude of the exciting force at the rotation frequency and blade frequency, it can be seen that the sudden change in the amplitude mainly occurs when the blade is just worn. Subsequently, with the continuous deepening of the blade wear degree, the amplitude has a gradual increasing trend.  
Figure 6: Frequency-domain comparison of radial excitation force of runner system: (**a**) x direction frequency-domain diagram, (**b**) y direction frequency-domain diagram.  
Figure 7: Radial excitation force amplitude variation trend diagram: (**a**) X-direction, (**b**) Y-direction.