## 4 Numerical Analysis and Experimental Verification
### Numerical Analysis Using Simulated Data
The FNN performance results under three different operational ranges, namely sub-critical speed (\(\tau<1\)), near-critical speed (\(\tau\approx 1\)), and trans-critical speed (\(\tau>1\)), were also investigated. The RMSE was used as the metric for the performance evaluation in each scenario, and lower RMSE values indicate better performance or closer alignment with the FNN predictions. The data illustrated in Table 2 demonstrate the performance of the FNN across various fault dominations at different speeds. Let us first look at the third column for the imbalance-dominant case, where it can be observed that the estimated error sum of the FNN is the lowest (green face) when running at near-critical speed \(\tau\approx 1\), then \(\tau>1\) and \(\tau<1\). The identification error sum of the shaft bowing case, however, exhibits no significant differences for the different speed ranges. For the cases that are shaft-bowing-dominant, there are no significant differences in imbalance and bowing estimation errors under various speeds. When these two faults are of equal weight, the lowest estimation errors for imbalances and shaft bowing happen in different speed regions. Imbalances are best identified at \(\tau>1\), while the bowing identification process shows the lowest error with the rotor running at \(\tau<1\). \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Hidden** & **Elapsed** & \(U_{x}\) & \(U_{y}\) & \(s_{x}\) & \(s_{y}\) & \(\sum\)**RMSE** \\ \hline
10 & 9 & \(1.88\times 10^{-6}\) & \(1.89\times 10^{-6}\) & \(2.05\times 10^{-7}\) & \(1.37\times 10^{-7}\) & \(4.11\times 10^{-6}\) \\
20 & 16 & \(2.63\times 10^{-7}\) & \(2.14\times 10^{-7}\) & \(2.43\times 10^{-7}\) & \(2.07\times 10^{-7}\) & \(9.26\times 10^{-7}\) \\
30 & 24 & \(3.13\times 10^{-7}\) & \(2.85\times 10^{-7}\) & \(1.37\times 10^{-7}\) & \(1.37\times 10^{-7}\) & \(8.72\times 10^{-7}\) \\
40 & 38 & \(1.59\times 10^{-7}\) & \(1.88\times 10^{-7}\) & \(9.35\times 10^{-8}\) & \(7.67\times 10^{-8}\) & \(5.17\times 10^{-7}\) \\
50 & 45 & \(2.18\times 10^{-7}\) & \(2.73\times 10^{-7}\) & \(1.57\times 10^{-7}\) & \(1.87\times 10^{-7}\) & \(8.34\times 10^{-7}\) \\
60 & 58 & \(4.59\times 10^{-7}\) & \(4.34\times 10^{-7}\) & \(1.17\times 10^{-7}\) & \(1.18\times 10^{-7}\) & \(1.13\times 10^{-6}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: RMSE values for the imbalance and shaft bowing components of different node setups. Figure 4: Comparison of different hidden node setups. From the results shown above, it can be concluded that imbalances are better diagnosed at higher speeds (\(\tau\approx 1\) or \(\tau>1\)), while the shaft bowing is better diagnosed at lower speeds (\(\tau<1\)). This conclusion can be explained and verified using the frequency responses of these two types of faults, as shown in Figure 5. In this figure, it can clearly be seen that the imbalance response is significantly magnified at higher speeds, while conversely the shaft bowing response diminishes with increasing speeds. In other words, shaft bowing dominates at sub-critical speeds and imbalances take over after the critical speed is reached. Both reach their maximum response at near-critical speeds. Moreover, in Table 2, it is noteworthy that the equal scenario case exhibits much smaller RMSE values than the others for both the imbalance and shaft bowing faults. This means that the FNN diagnosis of the simultaneous imbalance and shaft bowing faults would be the most accurate when the shaft bowing and imbalance faults are not unduly dominant. Damping is another factor that affects the response amplitude and phase. In most rotor-bearing systems, the damping ratio is relatively low. Nevertheless, to look into the damping effect in depth, cases with equal fault-weight ratios but with different damping ratios (underdamped, critically damped, and overdamped) were tested at sub-critical speeds (\(\tau<1\)). The diagnosis accuracy results for the fault components are shown in Table 3 as RMSE values.