## 2 Research Methodology
### Residual Rotor Bending  
Residual shaft bending is a serious fault, leading to a higher radial runout, excessive bearing loads, system imbalances, and extra bending moments. These can produce uneven bearing assembly load distributions, lower the system's operating efficiency of the mechanism, and even have dangerous resonance effects [76,77]. The theoretical and experimental identification of the failures produced by unbalance and residual shaft bending, two of the more common faults found in rotating machines, is treated in [28]. The identification procedure is one followed through a mathematical modeling system. With finite element modeling of the dynamic system, the faults were identified through a correlational analysis of the rotor responses in the time domain. The identification equation is derived from the Lyapunov matrix equation of a reduced model of the system containing only the measured degrees of freedom, and the fault parameters are thereafter identified by the least squares method:  
\[M_{\xi}^{\pi}(t)+P\dot{\xi}(t)+K_{\xi}^{\pi}(t)=H\cdot n_{un}(t)+Bn_{b}(t) \tag{28}\]  
where \(M\) is the mass matrix; \(P=C+\Omega\cdot G\) is the force proportional to the velocity matrix, which contains the damping \(C\) and gyroscopic \(G\) matrices; \(\Omega\) is the rotor running speed; \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 Phase measurement & It allows the evaluation of the dynamic characteristics of a rotor system in real time [27]. & Requires accurate calibration of measuring equipment [27]. & \\ \cline{2-4}  & High sensitivity to changes in the system state [48]. & Difficult to use for complex rotor systems with multiple disturbance sources [27]. & TD/FD \\ \hline Time-frequency analysis (TFA) [51] & An effective method for analyzing transients in rotary systems. & High computational complexity, especially for large datasets. & TFD \\ \cline{2-4}  & Allows for the detection of frequency components that are not visible in conventional spectral analysis. & High computational costs for large matrices. & N/A \\ \hline Singular value decomposition (SVD) [27] & Reduces data dimensionality and highlights the most important system characteristics. & High computational costs for large matrices. & N/A \\ \hline Newmark-\(\beta\) method & Allows controlling accuracy and stability by selecting the parameters \(\beta\) and \(\gamma\)[50]. & Sensitive to choosing initial parameters [50]. & N/A \\ \cline{2-4}  & Suitable for analyzing both linear and nonlinear systems [49]. & & \\ \hline Newton–Raphson method [49] & Provides fast convergence for finding the roots of nonlinear equations. & It does not work if the derivative of the function is close to zero, which can cause instability. & N/A \\ \hline Runge–Kutta methods & High accuracy in solving differential equations without calculating derivatives at each point [48]. & Adaptive step selection may be required, complicating implementation [49]. & N/A \\ \cline{2-4}  & Stability in solving equations [49]. & & implementation [49]. \\ \hline Least squares method [51] & Effective for solving overdetermined equation systems. & It may not work well for very sparse data. & N/A \\ \hline \hline \end{tabular}  
* * Signal processing domain (if applicable): TD—time domain; FD—frequency domain; TFD—time-frequency domain.  
\end{table} 
Table 4: _Cont._is the force proportional to the displacement matrix; \(H\) is the imbalance matrix; \(B\) is the shaft bow matrix; \(n_{un}\) and \(n_{b}\) are the input vectors that correspond to unbalance and the shaft bow, respectively. The differential evolution optimization method allowed for estimating the physical properties of the bearing, coupling, and rotor damping. These theoretically identified faults were considered for seven unbalance cases and an arc location, followed by experimental measures that imposed four different unbalance situations on the rotor to identify both the unbalance and the unknown shaft arc.