## 4 Results and Analysis
### Prediction of Runner Wear Life Based on Amplitude Characteristics  
During the long-term service of hydropower units, the main shaft-system components face multiple degradation factors such as wear and fatigue, resulting in progressive deterioration of vibration performance. The amplitude degradation trajectory of shafting is not uniform growth; it is usually \"slow first and then fast\", that is, the amplitude growth is gentle in the early stage and significantly accelerates in the later stage until it approaches the critical state. The causes include the cumulative amplification of local stress concentration, gap increase, stiffness reduction, and other effects caused by various failure forms, as  
Figure 19: Vibration response phase locus and Poincare mapping diagram of turbine runner system: (**a**) unworn, (**b**) wear 22.461 mm. Figure 20: Comparison diagram of phase trajectory of turbine runner in x direction with different wear degrees. well as the enhancement and amplification of external exciting forces (such as hydraulic instability, flow path defects, mechanical disturbance, power system disturbance, etc.) to the vibration response under the condition of internal degradation of shafting. Therefore, the amplitude degradation trajectory is approximately regarded as an exponential function in this paper, whose approximate expression is as follows:  
\[y=a\exp(bt), \tag{19}\]  
where \(a\) and \(b\) are undetermined coefficients. \(y\) is the amplitude of the turbine runner, expressed in \(\upmu\)m. \(t\) is the wear time, expressed in h.  
According to Equation (3) and the previous research results, the relationship between the time-wear-thickness-fluid-excitation-force vibration response can be established. Plot the degradation trajectory of the turbine runner amplitude response of the main shaft system over time, where the time corresponding to different degrees of blade wear is represented on the x-axis, and the amplitude response of the turbine system is depicted on the y-axis, as shown in Figure 21. The parameters of the expression are obtained by fitting the curve with the least square method; the value of \(a\) is 12.08 and the value of \(b\) is 1.128 \(\times\) 10\({}^{-5}\). Table 8 shows the evaluation indicators fitted by the expression, and the results show that the fitting effect is good. Under standard operating conditions, the core requirement to ensure the safe and reliable operation of the hydro-generator set is to control the vibration amplitude within the limited range, or the relative change in the vibration amplitude should not exceed the set threshold. When the vibration of key systems such as the generator rotor and turbine runner is too large, it may affect the overall safety, stability, and reliability of the unit, and the allowable amplitude of the vibration response of the spindle system should be determined according to the standards to determine whether the spindle is in the vibration failure state. When the spindle vibration amplitude \(F_{t}\) reaches the failure threshold \(F_{p}\), it is considered that the unit has failed, and t is the unit failure time. The mathematical expression can be simply expressed as:  
\[t=\inf\bigl{\{}t:F_{t}\geq F_{p}\bigr{\}}, \tag{20}\]  
According to standards GB/T 8564-2023, GB/T 11348.5-2008, and ISO20816-5:2018 [29; 30; 31], the allowable transverse vibration amplitude of the spindle system studied in this paper is  
Figure 21: Amplitude degradation trajectory over time. \begin{table}
\begin{tabular}{c c c} \hline \hline
**SSE** & **R\({}^{2}\)** & **RMSE** \\ \hline
3.465 & 0.9444 & 1.075 \\ \hline \hline \end{tabular}
\end{table} 
Table 8: Goodness of fit statistic. 75 \(\upmu\)m. However, since the allowable value is the reference value for monitoring vibration data of units in engineering, and this paper only studies the amplitude change caused by the hydraulic excitation force under the condition of blade wear, the proportional coefficient is used to determine the allowable value of amplitude change caused by wear.