## 3 Hybrid Methodology  
In this paper, the authors propose a comprehensive approach for establishing a real-time imbalance and shaft bowing diagnosis system for a rotor system and use a Jeffcott rotor-bearing system as the application example. This approach combines mathematical modeling techniques with ML-based prediction methods to enable the accurate and timely detection of multiple faults in the system, as shown in Figure 2. The ability to assess the onset of faults is a crucial aspect of both physics-based and machine learning approaches when analyzing the overall system. ML techniques are increasingly being used to diagnose faults in rotating machinery by analyzing data from sensors on the machinery. These models can learn to detect patterns that indicate faults and classify the type and severity of the problem, which can help maintenance teams diagnose issues earlier and prevent costly downtime. To overcome this challenge, a hybrid approach will be developed. This approach will combine machine learning techniques with expert knowledge and physical models to augment the limited fault data. Data-driven methodologies leverage statistical theories and artificial intelligence (AI) algorithms to analyze gathered data--particularly fault data--in order to facilitate fault diagnosis. However, the efficacy of data-driven techniques is contingent upon a substantial amount of accurately labeled data being available for network training. Notably, researchers frequently encounter challenges in acquiring sufficient labelled datasets for industrial rotor systems. To address this data constraint issue, the physical model employs a knowledge-based approach, presenting a viable way to overcome the lack of relevant data. Nevertheless, it is imperative to acknowledge the limitations of this data-driven approach, particularly in scenarios involving multiple faults, where its reliability may be compromised. In such instances, utilizing a physical model to generate fault data for ANN training emerges as a more dependable solution, enhancing the robustness of the overall diagnostic process. Two steps are involved in this process, namely model construction and real-time diagnosis, as shown in Figure 2. The model construction phase involves a combination of physical modeling and ML techniques. By using a physical model, one can generate sufficient sets of simulated data for ANN-supervised training, which helps to produce a more reliable model for diagnosing imbalances and shaft bowing in a Jeffcott rotor-bearing system. The simulated dataset is randomly generated using a systematic approach, with the imbalance parameters ranging from 0 to 1000 (g\(\cdot\)mm) and the bowing parameters ranging from 0 to 500 (mm). Next, the derived model is implemented into a real rotor system for real-time diagnosis, as displayed in phase 2. In this phase, cases with two  
Figure 2: A novel hybrid methodology for real-time imbalance and shaft bow diagnoses. different residual bowing faults plus various imbalance combinations are tested to acquire the real response features from an experimental setup involving a rotor rig. Note that in the designed scenarios the bowing only occurs under two conditions, \(s=0.5\) mm run at 1600 rpm and 3200 rpm and \(s=4\) mm run at 680 rpm, due to the difficulty of bending the shaft to achieve a small, permanent bow. Moreover, in the \(s=4\) mm bowing case, the rotor can be run only at very low speed due to safety concerns. The real response features (\(f_{1},f_{2},f_{3},\) and \(f_{4}\)) are then calculated from the measured response of the rotor rig and fed into the trained FNN for instant fault component identification, i.e., \(\mathbf{m}^{\mathsf{T}}\)= {\(U,\,\alpha,\,s,\,\theta\)}. By combining phase 1 and phase 2, this hybrid approach can be used to overcome the data availability limitations and provide an accurate diagnosis model for imbalances and bow faults in rotating machinery. To achieve this goal, we employed the most common ML-based approaches used for rotor fault diagnosis, which are ANNs.