## 2 Research Methodology
### Cracks  
Rotor cracks [69] are one of the essential risks, as they might occur after an incredibly long time with no symptomatic effects but will eventually increase the vibration levels and disrupt structural stability. They result from material fatigue wear, heat stress, shock \begin{table}
\begin{tabular}{p{113.8 pt} p{113.8 pt} p{113.8 pt} p{113.8 pt}} \hline \hline
**Method** & **Advantages** & **Disadvantages** & **Domain \({}^{*}\)** \\ \hline Joint input-state estimation algorithm (JISEA) [31] & Provides simultaneous estimation of both system states and unknown excitatory forces. & High sensitivity to noise and errors in input data. & N/A \\ \cline{2-4}  & It is more efficient than the classical Kalman filter for structural dynamics problems. & Requires significant computing power, especially for complex systems. & \\ \hline Modal balancing (MB) method [22] & Allows effective compensation for a distributed unbalanced load. & Requires accurate modeling and experimental verification. & TD/FD \\ \hline  & High accuracy in estimating or identifying [25; 26]. & Depends on reliable FEM input or experimental data [25; 26]. & \\ \cline{2-4} Artificial neural network (ANN) & It supports virtual balancing without trial runs [25]. & ANN performance is sensitive to architecture [25]. & N/A \\ \cline{2-4}  & Suitable for nonlinear and implicit process modeling [25; 26]. & Less effective under noise or with linear-only models [26]. & \\ \hline Modal balancing (MB) method [22] & Allows adequate compensation for a distributed unbalanced load. & Requires accurate modeling and experimental verification. & TD/FD \\ \hline Differential evolution (DE) method [28] & Good results with nonlinear and multidimensional problems. & It may require many iterations to achieve an accurate solution. & N/A \\ \hline Singular value decomposition (SVD) [22] & It allows the isolation of the main components and reduces low and medium noise levels. & It is sensitive to high noise levels in the input data, which can distort the results. & N/A \\ \hline Newton–Raphson method & Fast convergence to roots with a good initial approximation [25; 36]. & It may not converge, or it gives incorrect results with a poor initial approximation [25]. & N/A \\ \hline Runge–Kutta methods & High accuracy in solving differential equations without calculating higher-order derivatives [28; 37]. & Sensitive to the choice of integration step, which can affect the accuracy of the results [29]. & N/A \\ \hline Least squares method & It is suitable for regularizing problems with many variables [27]. & Requires the correct choice of regularization parameters to avoid over-computation [31]. & N/A \\ \hline  & Provides complete statistics (mean, variance, skewness, kurtosis) of critical speeds thresholds. & Quickly becomes computationally expensive & \\ \cline{2-4}  & Scales naturally when additional random variables or nonlinearities are unreduced. & Supplies no derivatives or analytical expressions & N/A \\ \hline \hline \end{tabular} \({}^{*}\)Signal processing domain (if applicable): TD—time domain; FD—frequency domain; TFD—time-frequency domain.  
\end{table} 
Table 2: Cont. loads, or manufacturing defects [70]. A hidden or unnoticed crack could lead to equipment failure and massive financial loss. The study by [40] considers methods for diagnosing a rotor with a breathing crack under unbalanced forces by applying strain energy release rate (SERR) factors and the neutral axis method. The system model applied is developed based on a finite element model (FEM) of a Timoshenko beam, whose element flexibility is a function of stress distribution. The rotor motion has been modeled based on classical equations of motion, which were solved using the Newmark method.