## 2 Research Methodology
### Unbalance  
Unbalance [65] is among the most common issues in rotating equipment due to irregular mass distribution around the rotation axis. The primary causes are errors during manufacturing, component wear, non-homogeneous material buildup, and mechanical damage [66]. The consequences of unbalance include increased vibrations, bearing wear, support structure loads, and potential failure hazards. Timely detection and balance correction are essential for maintaining rotating equipment's stable and secure operation. The study by [14] presents three varying methods of obtaining imbalance. The methods apply to fine-tuning the model parameters so that unbalanced computed values agree with the measured values under equivalent conditions in the physical model. Equivalent loads were adjusted until the eccentricity of the individual unbalance matched the measured value in the first method. The minimization of the difference between the actual and approximate eccentricity values was achieved using the least squares method:  
\[\int\left|\left\{\Delta\widetilde{F}_{c}(\beta,t)\right\}-\left\{\Delta \widetilde{F}(t)\right\}\right|^{2}dt\to min, \tag{1}\]  
where \(\Delta\widetilde{F}_{c}\) and \(\Delta\widetilde{F}\) are the theoretical and experimental values of the equivalent loads, respectively. Since there were substantial errors in the determined parameters, the method was refined by reducing the effect of modal expansion, thereby reducing the error due to this effect. The refined method showed a considerable reduction in error compared to the original process. The third approach, however, seeks to minimize the amplitudes of vibration rather than the loads [14]:  
\[\int\left|\left\{\Delta r_{ij}\right\}-\left\{\Delta r_{M}\right\}\right|^{2} dt\to min, \tag{2}\]  
where \(j\) is the location of measured vibrations; \(r_{ij}\) is the vibrations calculated at location \(j\) with the fault at location \(i\); \(\left\{r_{M}\right\}_{j}\) is the vibrations measured at location \(j\). In this case, the varied parameters are the values of equivalent loads applied in the model at damage locations. This method yielded the best outcomes; nevertheless, error levels were intolerably high for some cases. The inherent linearity of the equivalent loads in the model can explain the significant errors obtained. One improvement to this approach \begin{table}
\begin{tabular}{c c c} \hline
**Issue** & **Identification Methods** & **References** \\ \hline \multirow{8}{*}{Bearing faults} & Artificial neural network (ANN) & [59; 61; 62] \\ \cline{2-3}  & Convolutional neural networks (CNN) & [16; 38; 44; 61; 63] \\ \cline{2-3}  & Continuous wavelet transform (CWT) & [61] \\ \cline{2-3}  & Long short-term memory (LSTM) & [61] \\ \cline{2-3}  & Generative adversarial networks (GAN) & [61] \\ \cline{2-3}  & Least squares method & [51; 59] \\ \hline \end{tabular}
\end{table} 
Table 1: _Cont._can be the application of nonlinear laws in modeling the equivalent loads. Further, the application of nonlinear relationships between the equivalent loads and damping, as well as stiffness, can also lead to producing improved results by taking into consideration the inherent effects due to imbalance. Figure 1 presents the equivalent vertical loads estimated in the system and the FFT of the vertical forces at a specific node. Diagnostic models of unbalance tend to place it as a point defect, whereas in nature, depending on the underlying cause, the unbalance can be distributed. The study by [22] examines a diagnostic and balancing method for distributing unbalance under the influence of rotor shaft bending. The authors believe that discrete value modeling of rotor bending-induced imbalance is not precise enough and cannot adequately characterize the phenomenon's nature.