## 3 Hybrid Methodology
ANNs can learn to classify different types of faults based on vibrations, currents, or acoustic data from sensors on the machinery. In particular, ANNs can be utilized to diagnose imbalances and shaft bowing in rotating machinery, which are common problems that can lead to excessive vibrations, reduced machine performance, and even catastrophic failure. Among the various types of ANNs, we had tested various types such as feed-forward neural networks (FNNs), RNNs, and LSTMs and found that their differences in accuracy are almost nondifferentiable. Hence, an FNN was selected for its simple structure and computational efficiency. In the FNN, the information flows in one direction, from the input layer through one or more hidden layers to the output layer. By training the FNN with historical data from the rotor system, it can learn the underlying patterns and relationships between the various system parameters and multiple faults. This enables the FNN to make predictions about future fault events based on real-time sensor data. Figure 3 demonstrates the applications of the machine learning framework for diagnosing multiple faults in a Jeffcott rotor-bearing machine, showcasing its potential for multifaceted fault detection and analyses. 1. Acquiring data: The physical model is used to generate the datasets randomly. Nevertheless, the training set can be generated from measured data as much as one need. It is notable that these imbalance and shaft bowing components, \(\mathbf{m}^{\mathsf{T}}\)= {\(U,\,\alpha,\,s,\,\theta\)}, are random inputs to the physical model, and the response features components at the disk centre after the forwarding calculation are the output, i.e., Equation (24). 2. Data preparation: The generated datasets will be considered raw data for further processing via supervised training. The simulated inputs and outputs are reversed during the network training. The response components \(\mathbf{f}^{\mathsf{T}}\), i.e., the 4 parameters, are used as the input to the ANN, and the 4 parameters \(\mathbf{m}^{\mathsf{T}}\) are used for the target. 3. ANN architecture: An appropriate model needs to be selected and configured for the problem at hand. Typically, a feed-forward neural network (FNN) with one or more hidden layers is used. There are several parameters in each model that can be varied to arrive at the best model parameter for each case. The proposed framework is modeled using MATLAB version: 9.15.0 (R2023b) software. 4. Model training: The FNN is trained with a single hidden layer architecture with a varying number of nodes using 10,000 datasets, which are randomly generated from a physical model. The first 70 percent of the datasets are utilized for training, the second 15 percent are used for validation, and the final 15 percent are used for testing. 5. Model testing: To rigorously evaluate the FNN model's performance, the root mean squared error (RMSE) is employed to evaluate the alignment between the randomly generated values (so-called real data) and estimated values. A lower RMSE value signifies better accuracy and model performance. The formula for the RMSE is: \[RMSE=\sqrt{\frac{1}{n}\sum_{i=1}^{n}\left(y_{i}-\hat{y_{i}}\right)^{2}}\] (26)where \(y_{i}\) denotes the real value for the \(i\)th point, \(\hat{y}_{i}\) denotes the estimated value, and n denotes the number of datapoints. 6. Diagnosis: The trained FNN is tested using simulated and real data acquired from the experimental Jeffcott rotor setup to diagnose the multi-fault components, i.e., \(U,\alpha,s\), and \(\theta\). Based on the output of the FNN, the machine operator can monitor the growth of imbalances and shaft bowing faults to take necessary actions.