## 2 Research Methodology
### Rotor-Stator Interaction  
The primary reasons for rotor-stator friction are decreased working clearance between the rotor and stator caused by thermal expansion, shaft deflection, imbalance, misalignment, or bearing wear [78]. Additionally, in certain instances, friction may result from the entering material into the clearance or because of a deposit of wear debris. The effects of rotor-stator friction negatively affects the system in terms of increased component wear and unstable system conditions, such as self-excited vibration and fretting corrosion [79]. Impact contact scenarios are specifically hazardous, as they can lead to dynamic loading, material delamination, and failure of blades or disks [80]. In the study by [57], empirical wavelet transform (EWT) was considered as a new approach to the analysis of multi-component signals and is suggested based on the traditional wavelet transformation:  
\[z(t)=f(t)+iH(f)(t)=\alpha(t)e^{i\theta(t)}, \tag{32}\]  
where the actual values of functions \(\alpha(t)\) and \(\Omega(t)\) represent the real and imaginary parts of \(z(t)\). \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 ISOMAP [55] & Allows nonlinear data dimensionality reduction while preserving global geometry. & High computational costs for large samples. & \\ \cline{2-4}  & Effective for detecting hidden structures in complex multidimensional data. & Sensitive to the choice of the number of neighbors, which affects the results. & N/A \\ \hline Inverse eigensensitivity method (IES) & Effective for updating finite element models by adjusting parameters [53]. & Sensitive to initial assumptions about model parameters [26]. & N/A \\ \hline Principal component analysis (PCA) & Reduces data dimensionality while saving the most significant characteristics [28; 55]. & Loss of information with rejecting insignificant components [55]. & N/A \\ \cline{2-4}  & Reduces noise in data and improves the interpretability of results [28]. & Assumes linearity of data, which can be a limitation for complex nonlinear dependencies [28]. & N/A \\ \hline Natural frequency difference (NFD) & Improves the accuracy of updated finite element models [53]. & Sensitive to noise in measurements [55]. & \\ \cline{2-4}  & Suitable for identifying differences between experimental and numerical modal characteristics [55]. & It does not consider the shape of the vibration modes, which can lead to an incomplete picture of the discrepancy [53]. & FD \\ \hline Characteristic equation method [53] & Accurately determines the natural frequencies and oscillation forms of the system. & Sensitive to errors in system parameters, which can affect the accuracy of the results. & N/A \\ \hline Newton–Raphson method [55] & Converges quickly to a solution with a good initial approximation. & It can cause instability if the function’s derivative is close to zero. & N/A \\ \hline Runge–Kutta methods [29] & Provides high accuracy in solving a wide range of problems in numerical modeling. & Sensitive to the choice of the integration step: a too-significant step can reduce accuracy, and a too-small step can increase computational costs. & N/A \\ \hline \hline \end{tabular}  
* Signal processing domain (if applicable): FD—frequency domain; TFD—time-frequency domain.  
\end{table} 
Table 5: _Cont._This paper suggests an adaptive parameter-free empirical wavelet transform (APEWT) to implement adaptive partitioning of the Fourier spectrum in EWT. To alleviate the defects of Hilbert's transformation in instantaneous frequency and amplitude estimation, a quadrature-derivative-based normalized Hilbert transform (QDNHT) is suggested. In this study, the proposed novel time-frequency analysis method, consisting of APEWT and QDNHT, is compared with empirical mode decomposition (EMD), ensemble empirical mode decomposition (EEMD), and local characteristic-scale decomposition (LCD).