## 4 Numerical Analysis and Experimental Verification
### Experimental Verification and Real-Time Diagnosis
This variation is caused by various practical engineering issues that can impact the results, adding complexities beyond the reduced theoretical models. For \(s=4\) mm at 650 rpm in this state, the rotor's frequency ratio is lower than the critical speed. Figure 5 shows that when the frequency ratio is smaller than one, the \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Number** & **Faults** & **Real** & **Diagnosed** & **Error** & **Comp** & **Error\%** \\ \hline \multirow{4}{*}{1 (same quadrant)} & \(U\) & 6 & 4.4 & 26.7\% & \(\Delta U_{//}\) & \(-\)27.7 \\  & \(\alpha\) & 45\({}^{\circ}\) & 35.24\({}^{\circ}\) & 9.76\({}^{\circ}\) & \(\Delta U_{\perp}\) & 12.4 \\  & \(s\) & 0.5 & 0.69 & 38.0\({}^{\circ}\) & \(\Delta s_{//}\) & 36.5 \\  & \(\theta\) & 60\({}^{\circ}\) & 54.41\({}^{\circ}\) & 8.59\({}^{\circ}\) & \(\Delta s_{\perp}\) & 20.6 \\ \hline \multirow{4}{*}{2 (in-phase)} & \(U\) & 12 & 13.34 & 11.2\% & \(\Delta U_{//}\) & 10.3 \\  & \(\alpha\) & 90\({}^{\circ}\) & 97.24\({}^{\circ}\) & 7.24\({}^{\circ}\) & \(\Delta U_{\perp}\) & 14.0 \\  & \(s\) & 0.5 & 0.31 & 38.0\% & \(\Delta s_{//}\) & \(-\)38.4 \\  & \(\theta\) & 90\({}^{\circ}\) & 96.18\({}^{\circ}\) & 6.18\({}^{\circ}\) & \(\Delta s_{\perp}\) & 6.7 \\ \hline \multirow{4}{*}{3 (anti-phase)} & \(U\) & 24 & 26.72 & 11.3\% & \(\Delta U_{//}\) & 2.8 \\  & \(\alpha\) & 225\({}^{\circ}\) & 202.39\({}^{\circ}\) & 22.61\({}^{\circ}\) & \(\Delta U_{\perp}\) & 42.8 \\ \cline{1-1}  & \(s\) & 0.5 & 0.54 & 8.0\% & \(\Delta s_{//}\) & 5.9 \\ \cline{1-1}  & \(\theta\) & 30\({}^{\circ}\) & 18.68\({}^{\circ}\) & 11.32\({}^{\circ}\) & \(\Delta s_{\perp}\) & 21.2 \\ \hline \multirow{4}{*}{4 (perpendicular)} & \(U\) & 30 & 28.41 & 5.3\% & \(\Delta U_{//}\) & \(-\)7.0 \\  & \(\alpha\) & 67.5\({}^{\circ}\) & 56.53\({}^{\circ}\) & 10.97\({}^{\circ}\) & \(\Delta U_{\perp}\) & 18.0 \\ \cline{1-1}  & \(s\) & 0.5 & 0.31 & 38.0\% & \(\Delta s_{//}\) & \(-\)39.0 \\ \cline{1-1}  & \(\theta\) & 150\({}^{\circ}\) & 139.88\({}^{\circ}\) & 10.12\({}^{\circ}\) & \(\Delta s_{\perp}\) & 10.9 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Diagnosis error percentage at 3200 rpm (\(\tau=1.39\)) for initial bowing of 0.5 mm. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Case** & **Faults** & **Real** & **Diagnosed** & **Error** & **Comp** & **Error\%** \\ \hline \multirow{4}{*}{1 (same quadrant)} & \(U\) & 6 & 7.03 & 17.2\% & \(\Delta U_{//}\) & 17.0 \\  & \(\alpha\) & 45\({}^{\circ}\) & 48.80\({}^{\circ}\) & 3.08\({}^{\circ}\) & \(\Delta U_{\perp}\) & 6.3 \\  & \(s\) & 0.5 & 0.36 & 28.0\% & \(\Delta s_{//}\) & \(-\)28.4 \\  & \(\theta\) &system's behavior indicates that the response is mostly influenced by the shaft bowing amplitude. This crucial finding suggests that in specific circumstances, the influence of the shaft bowing on the system's behavior is greater than that of the imbalance. The small discrepancy in the error margins indicates that both types of faults have a substantial role in the observed vibrations, despite their theoretical differences. The consistent error gaps highlight the complex relationship between imbalance and shaft bow effects on the system's dynamic behavior. This shows a situation where both causes have similar effects, and their combined impact is seen in the total inaccuracy in the system's response. The more significant disparities between the actual and diagnosed faults can be prominently observed for all experiments, unlike the scenarios using simulated data, in which very high accuracy rates were achieved. These larger errors may be attributed to several causes. For instance, there may have been an existing imbalance and bowing before we intentionally added the trial weight or because the notable clearance between the shaft and bearing was not a good fit for the simple support assumption. Both of these issues would generate vibration noises and reduce the diagnosis accuracy in the experiments.