## 2 Research Methodology
### Unbalance
To counter this issue, a model is proposed in which unbalance is treated as a continuously distributed value along the rotor shaft:  
\[X(z)=\sum_{i=0}^{m}A_{i}z^{i};\;Y(z)=\sum_{i=0}^{m}B_{i}z^{i}, \tag{3}\]  
where \(X(z)\) is the projection of the global eccentricity curve on the x-z plane; \(Y(z)\) is the projection of the global eccentricity curve on the y-z plane; \(A_{i}\), \(B_{i}\) are the polynomial coefficients. The magnitude of the unbalance depends on the value of the local deflection; therefore, cubic interpolation of rotor deflection was used to calculate the distribution function of unbalance. Sensor measurement data and a finite element model (FEM) of the rotor were used in computing the values of deflections. Forces caused by unbalance were calculated via so-called eccentricity coefficients, which are dependent on the deflection function. Following the distributed unbalanced force function, compensation point masses were calculated. An unbalanced assessing criterion was gained, i.e., the sum of the two Euclidean norms of these functions in planes [22]:  
\[\|X(z)\|=\sqrt{\int_{0}^{l}\left[A_{3}z^{3}+A_{2}z^{2}+A_{1}z+A_{0}\right]^{2} dz}; \tag{4}\]  
\[\|R(z)\|=\sqrt{\|X(z)\|^{2}+\|Y(x)\|^{2}}. \tag{5}\]  
The proposed technique was verified experimentally. The results were consistent with those of the simulations. Moreover, balancing masses derived from the proposed technique decreased the system's response to figures near the values calculated. Among the research, the outcomes increased modeling accuracy over discrete unbalanced models and offered a  
Figure 1: Amplitude–frequency response of the system. Adopted from [14]. criterion-based method for rotor suitability assessment. However, for the approach to be applied, some conditions in measurement need to be met: sensors are at equal intervals, which is not invariably possible in actual rotors. Also, the model does not account for the masses and corresponding unbalances of the disks. Figure 2 presents the estimated distributed unbalanced eccentricity values for both the X and Y axes. Mechanical damage to rotor sections can lead to material chipping and imbalance. Therefore, it can be assumed that localizing such damage determines the position of the imbalance. The study by [23] does not connect directly to rotor dynamics but considers a technique for the localization of damage using deep learning methods. The study consists of two parts. The first part deals with a defective localization technique based on the mode-shape analysis of the blade. The equation of movement (6) is presented in complex form:  
\[\Big{(}K+i\omega C-\omega^{2}M\Big{)}xe^{i\omega t}=Fe^{i\omega t}, \tag{6}\]  
where \(F\) is the force vector; \(i\) is the imaginary operator; \(\omega\) is the frequency of the harmonic excitation; \(C=\alpha\cdot M+\beta\cdot K\) is the hysteretic damping matrix; \(\alpha\) and \(\beta\) are the constants independent of the natural frequencies. A weighted summation of all the modal shapes of the structure is used in computation, and a different weight is assigned to each mode. The data are then applied to the Debes wavelet transform (Db6) to produce a 2D map of the signals and establish the locations of the blade damage where singularities are present. To estimate the degree of the damage, the author employs the random forest (RF) technique that chooses the optimal solution from an ensemble of decision trees with varying settings. The training dataset was obtained by simulating the blade damage at various locations with varying degrees. With the provided training data, the natural frequencies of the blade are calculated, and the RF algorithm is applied to establish a relationship between the damage locations, the natural frequency data, and the respective damage measure (Figure 3). The approach was validated against a reduced-order blade model that had been derived from frequency data that had been extracted from a full-scale blade. The relative difference between the experimentally derived results and simulated data was determined to be 6-8%.