## 2 Research Methodology
### Unbalance
Regarding manifestations of unbalance as a statistical class characterized by a particular set of parameters in a given hyperspace, the task of identifying unbalance can be reduced to clustering for the input parameter set, such as identification of membership  
Figure 2: Estimated unbalance eccentricity amplitudes in different planes. Adopted from [22]. for this set in the given class or not. The research by [24] evaluates the performance of imbalance diagnosis through two techniques: the k-nearest neighbors (KNN) technique and the decision tree (DT) technique. The test rig consists of a rotor with four disks, with openings for the placement and the intensity variations in unbalances. Vibration is measured by accelerometers fitted onto the shaft supports. The FFT is performed on the data acquired from the accelerometers to convert it to a continuous spectrum. This is used as the input data for both methods. For the KNN algorithm, the distances were computed with Euclidean, Manhattan, and Minkowski metrics, respectively:  
\[\sqrt{\sum_{i=1}^{k}(x_{i}-y_{i})^{2}},\ \sqrt{\sum_{i=1}^{k}|x_{i}-y_{i}|}, \text{and}\ \sqrt[\sum_{i=1}^{k}(|x_{i}-y_{i}|)^{q}. \tag{7}\]  
In the decision tree algorithm, three elements were built according to the entropy values of the parameters of classified objects. As a result of the study, the authors provided charts for comparison of expected and experimentally derived data on the position and mass of the imbalances. According to them, the KNN technique yielded more precise results in all three parameters studied (Figure 4). Figure 4: Prediction accuracy, %. Adopted from [24]. Figure 3: Defects’ locations on the decomposition diagrams for the first and second decomposition levels. Adopted from [23]. The least accurate was the estimation of the imbalance mass, which can have an error of 25%. The lower accuracy associated with the decision tree technique is due to the complicated formalization of data necessary for building the tree structure. Moreover, the suggested methodology does not calculate the angular coordinate of the corresponding imbalance. Another limitation of the proposed approach is that classification by the k-nearest neighbors' technique is based on discrete values. This implies that the precise location of the imbalance along the shaft can be established only within predefined ranges. To this extent, therefore, this limitation hinders the method's precision in pinpointing imbalance locations exactly. Another study [37] uses deep learning methods for imbalance identification and proposes a fuzzy logic approach to analyze the instability of unbalanced nonlinear rotor systems. The authors note that actual rotor-bearing systems are subject to uncertainties due to manufacturing tolerances, assembly misalignment, material degradation, and variations in operating conditions. Classical deterministic models cannot capture these uncertainties, and therefore, these models make erroneous predictions regarding the system's behavior. To overcome this challenge, the study introduces a fuzzy mathematical formalism, allowing the representation of parameter uncertainties by fuzzy sets instead of fixed quantities or probabilistic distributions. This formalism is particularly suitable for modeling nonlinear phenomena in rotor dynamics, such as bearing stiffness fluctuations, damping properties, and unbalanced forces. The authors developed equations of motion for a nonlinear Jeffcott rotor with fuzzy system parameters in the stiffness and damping coefficients of the system:  
\[X(z)=\sum_{i=0}^{m}A_{i}z^{i};\;Y(z)=\sum_{i=0}^{m}B_{i}z^{i}, \tag{3}\]0  
\[X(z)=\sum_{i=0}^{m}A_{i}z^{i};\;Y(z)=\sum_{i=0}^{m}B_{i}z^{i}, \tag{3}\]1  
\[X(z)=\sum_{i=0}^{m}A_{i}z^{i};\;Y(z)=\sum_{i=0}^{m}B_{i}z^{i}, \tag{3}\]2  
where \(k_{T}\) indicates the translational stiffness coefficient; \(f^{e}\) and \(f^{b}\) are the mass unbalance and bearing forces, respectively; \(K_{1}\) and \(K_{3}\) are the linear and nonlinear stiffness coefficients; \(C\) is the linear damping coefficient.