## 5 Conclusions  
This study proposed a novel hybrid approach that combines model-based and ML approaches to investigate imbalance and shaft bow monitoring for Jeffcott rotor-bearing systems in online settings. Initially, the physical model of the Jeffcott rotor system involved the development of a mathematical model to determine the rotor parameters, generating a substantial amount of simulated fault data for the ANN model. The validated physical model served as the basis for generating a significant volume of simulated data (10,000 datasets), which was subsequently employed to train ML models such as our FNN. This dataset was used in training, validating, and testing the models, ensuring their effectiveness in diagnosing multiple faults. However, the FNN with 40 nodes exhibited the lowest RMSE, indicating better performance. The FNN was tested in diverse conditions, including varying fault and frequency ratios. Under these diverse conditions, using the simulated datasets, the FNN \begin{table}
\begin{tabular}{l l c c c c c} \hline \hline
**Number** & **Faults** & **Real** & **Diagnosed** & **Error** & **Comp** & **Error\%** \\ \hline \multirow{4}{*}{1 (same quadrant)} & \(U\) & 6 & 5.12 & 14.7\% & \(\Delta U_{\mathit{f}}\) & \(-\)15.9 \\  & \(\alpha\) & 45\({}^{\circ}\) & 54.71\({}^{\circ}\) & 9.71\({}^{\circ}\) & \(\Delta U_{\bot}\) & 14.4 \\  & \(s\) & 4.0 & 4.17 & 4.2\% & \(\Delta S_{\mathit{f}}\) & 0.8 \\  & \(\theta\) & 60\({}^{\circ}\) & 74.84\({}^{\circ}\) & 14.84\({}^{\circ}\) & \(\Delta S_{\bot}\) & 26.7 \\ \hline \multirow{4}{*}{2 (in-phase)} & \(U\) & 12 & 14.86 & 23.8\% & \(\Delta U_{\mathit{f}}\) & 11.7 \\  & \(\alpha\) & 90\({}^{\circ}\) & 64.39\({}^{\circ}\) & 25.61\({}^{\circ}\) & \(\Delta U_{\bot}\) & 53.5 \\  & \(s\) & 4.0 & 4.27 & 6.8\% & \(\Delta S_{\mathit{f}}\) & \(-\)4.7 \\  & \(\theta\) & 90\({}^{\circ}\) & 116.83\({}^{\circ}\) & 26.83\({}^{\circ}\) & \(\Delta S_{\bot}\) & 48.2 \\ \hline \multirow{4}{*}{3 (anti-phase)} & \(U\) & 24 & 30.28 & 26.0\% & \(\Delta U_{\mathit{f}}\) & 24.3 \\  & \(\alpha\) & 225\({}^{\circ}\) & 215.06\({}^{\circ}\) & 9.94\({}^{\circ}\) & \(\Delta U_{\bot}\) & 21.8 \\  & \(s\) & 4.0 & 3.72 & 7.0\% & \(\Delta S_{\mathit{f}}\) & \(-\)7.1 \\  & \(\theta\) & 30\({}^{\circ}\) & 32.71\({}^{\circ}\) & 2.71\({}^{\circ}\) & \(\Delta S_{\bot}\) & 4.4 \\ \hline \multirow{4}{*}{4 (perpendicular)} & \(U\) & 30 & 32.16 & 7.2\% & \(\Delta U_{\mathit{f}}\) & \(-\)5.2 \\  & \(\alpha\) & 67.5\({}^{\circ}\) & 39.63\({}^{\circ}\) & 27.87\({}^{\circ}\) & \(\Delta U_{\bot}\) & 50.1 \\  & \(s\) & 4.0 & 3.0 & 25.0\% & \(\Delta S_{\mathit{f}}\) & \(-\)29.7 \\  & \(\theta\) & 150\({}^{\circ}\) & 169.19\({}^{\circ}\) & 19.9\({}^{\circ}\) & \(\Delta S_{\bot}\) & 25.4 \\ \hline \hline \end{tabular}
\end{table} 
Table 7: Diagnosis error percentage at 650 rpm (\(\tau\) = 0.28) for initial bowing of 4 mm. consistently demonstrated exceptional performance, particularly when the imbalance and shaft bowing did not dominate each other. Moreover, to achieve higher diagnosis accuracy for shaft bowing, the rotor should be run at low speeds, while higher speeds are better for imbalance diagnosis. The combinations of fault components (_U_, \(a\), \(s\), _th_) were applied to the Jeffcott rotor experiment, and the experimental data were obtained and fed into the trained FNN for instant diagnosis. The FNN effectively identified and quantified faults within the system. The results showed that both the imbalance and bowing faults exhibited their lowest diagnosis error values, as these two faults were in-phase at sub-critical running speeds. At transcritical speeds, the imbalance error was lowest; however, the bowing error significantly increased because the bowing response rapidly diminished. These findings highlight the superiority of the hybrid approach and its potential for enhancing the precision and reliability of imbalance and shaft bowing diagnosis in rotor systems.