## 4 Numerical Analysis and Experimental Verification  
### Numerical Analysis Using Simulated Data  
The detection of imbalances and shaft bowing in Jeffcott rotor systems is an immensely significant task within numerous industrial domains. To tackle this challenge, the use of ANN models has proven to be highly effective. Our investigation commenced by employing an FNN that comprised multiple hidden layers, an input layer, and an output layer, all lacking feedback connections or loops [46]. In the context of diagnosing multiple faults, the input layer adeptly receives the vibration signals or response components originating from the horizontal and vertical directions of the Jeffcott rotor system. Subsequently, these signals are skillfully processed by the neurons residing within the concealed layers, which judiciously employ activation functions such as the tan-sigmoid function or the rectified linear unit (ReLU) function. The outcome obtained from the hidden layer is then skillfully mapped to the ultimate output through the neurons residing within the output layer. It is worth noting that the output layer typically employs a linear transfer function, such as the purelin transfer function, to carry out this mapping process. To train the FNN, 10,000 datasets are randomly generated, and 70% of them are used for training, 15% for validation, and the last 15% for testing. The Levenberg-Marquardt backpropagation (LM) training algorithm, specifically the _trainlm_ training function in MATLAB, was employed to train the network. To determine the most suitable number of hidden nodes, a trial-and-error approach was employed, as there is no precise method that can be used to select this parameter based solely on the number of inputs and outputs. However, in many ANN applications, including for FNNs applied to rotor systems, a single hidden layer has been proven sufficient [37; 47]. Once the FNN is trained, it can be used to accurately detect and diagnose imbalances and shaft bowing in Jeffcott rotor systems based on the input vibration signals. To examine the performance of the models, networks were trained with 10, 20, 30, 40, 50, and 60 hidden nodes. The results of the study revealed that a single hidden layer with 40 nodes had the lowest RMSE, as shown in Figure 4. The number of errors drops rapidly and reaches the minimum for the configuration involving 40 nodes. The elapsed time periods vary with the node number, as also shown in Figure 4, and it can be seen that the time increases almost linearly with the number of nodes. The errors of the testing set were reported in Table 1 in terms of \((U_{x},U_{y},s_{x},s_{y})\) rather than \((U,s,\alpha,\theta)\), because the phase angle values (\(\alpha\) and \(\theta\)) did not align accurately because of their periodic nature. This change was made to mitigate concerns regarding potentially misleading outcomes. In Table 1, the lowest numbers of errors for the four components all occur with 40 nodes. Figure 3: ML approach to diagnosing multiple faults in a Jeffcott rotor-bearing system. During our investigation of the performance of the FNN using simulated data, we proposed a variety of fault conditions so that the diagnosis accuracy rates could be observed and compared. Different fault combinations can be classified as imbalance-dominant, shaft-bowing-dominant, and equal. When the imbalance dominates, the \(U\) values range from 0.6 to 0.9 kg\(\cdot\)m and the \(s\) values from 0.5 to 0.1 mm. This means that the imbalance component is 100 times the shaft bowing component. In contrast, shaft bowing dominance involves \(U\) values ranging from 0.00001 to 0.00002 kg\(\cdot\)m and \(s\) values ranging from 2 to 3 mm, opposing the imbalance scenario. The \(U\) and \(s\) parameters vary from 0.002 to 0.003 kg\(\cdot\)m and 2 to 3 mm, respectively, in the equal case. In addition, the effects of imbalances and shaft bowing vary with the rotational speed.