## 2 Research Methodology
### Residual Rotor Bending
The second correlation check was performed between the updated FEM model and the experimental model. The experiments indicated a considerable discrepancy between the initial simulation results and experimental data (Figure 18). Figure 17: Responses of the rotor with different magnitudes of residual bow. Adopted from [55]. However, after updating the inverse model, the results were the same. In addition, the maximum depth of tool cutting was tried so as to establish how far the cutting can extend without vibration. Similar to misalignment, residual shaft bending can induce unwanted contact between the rotor system parts. The study by [29] focused on the impact of imbalance on rotor-to-housing contact. The model considered is a Jeffcott rotor with a significant imbalance deflection, creating disk-stator friction. As a result of an elastic impact, a contact force will resolve into normal and tangential forces whose magnitudes are clearance-to-eccentricity ratio dependent. Through kinetic energy and deformation energy equations of the shaft and disk, non-conservative assumptions of the contact force, and the extended Hamiltonian principle, the authors have achieved an extended governing equation of motion for the rotating system with the contact interaction. With the introduction of dimensionless velocity and geometrical parameters substituting values, employing the Galerkin procedure, discretization of the governing equation yielded a set of equations describing frictional forces. The reduced motion equation considers nonlinear relationships between displacement and deformation, strain and stress, mechanical friction, and excitation due to mass imbalance. These equations were then used to study frictional impact, stability, and dynamic chaos by bifurcation diagrams, which were validated by the results simulated in a time domain and phase space. The research results include bifurcation diagrams (Figure 19), which reveal instability regions as functions of rotational frequency, imbalance amplitude, and friction coefficient between the rotor and stator. Figure 19: Bifurcation diagram with the effect of the stiffness of the rubbing surface. Adopted from [29]. Figure 18: MAC plot for updated FE and experimental modes. Adopted from [53]. Using Fourier transform analysis and phase mapping, the authors identified subharmonic responses that indicate a rotor-stator friction interaction (Figure 20). It was ascertained that the system is susceptible to the rotational speed of the rotor; minor changes will destabilize the entire system. A comparative analysis of the residual rotor bending detection methods is summarized in Table 5.  
\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 \multirow{4}{*}{ \begin{tabular}{} \end{tabular}  } & Ability to consider complex geometries and material properties [29]. & \multirow{4}{*}{ \begin{tabular}{} \end{tabular}  } \\  & Suitable to analyze resistance to self-oscillations and the effect of defects [53]. & Sensitivity to boundary conditions and grid-partitioning parameters [29]. & N/A \\ \hline \multirow{4}{*}{ \begin{tabular}{} \end{tabular}  } & Improves the correspondence of the numerical model to real experimental data. & The dependence on the accuracy of experimental data to adjust the model. & N/A \\ \cline{2-4}  & & It loses temporal resolution and is not suitable for analyzing nonstationary processes. & \\ \cline{2-4}  & & Sensitive to the choice of window length and interpolation methods. & \\ \hline \multirow{4}{*}{\ \begin{tabular}{} \end{tabular}  } & Allows analyzing nonstationary signals with high time-frequency resolution [55]. & Higher computational costs compared to FFT [28]. & \multirow{4}{*}{ \begin{tabular}{} \end{tabular}  } \\  & Effective for detecting local signal features and short-term anomalies [28]. & The choice of the wave basis significantly affects the accuracy of the results [55].