## 2 Research Methodology
### Bearing Faults  
The leading causes of bearing defects include fatigue failure, poor or contaminated lubrication, overloading, incorrect installation procedures, electrical erosion, and corrosion [81, 82]. Since the earliest signs of defects can be inconspicuous, it is essential to apply modern diagnostic methods to detect damage early and prevent severe consequences [83, 84]. The paper by [60] introduces a novel diagnostic method for bearing faults using the Teager energy operator. The proposed method uses both a simple energy operator and a Fourier transform step on the equation of motion:  
\[m\ddot{x}(t)+kx(t)=0, \tag{34}\] \begin{table}
\begin{tabular}{l l l l} \hline \hline
**Method** & **Advantages** & **Disadvantages** & **Domain * \\ \hline Fast Fourier transform (FFT) & A fast method of spectral analysis that allows estimating the frequency composition of signals in rotary systems [49]. & Not suitable for analyzing rapidly changing signals [57]. & FD \\ \cline{2-4}  & Effective for detecting fundamental harmonics and resonant frequencies in vibration analysis [57]. & & \\ \hline Hilbert transform & An effective demodulation method for mono-component signals. & & \\ (HT) [57] & Suitable for estimation of the instantaneous frequency and the instantaneous amplitude of signals. & Violates the conditions of [PERSON]’s theorem, which may affect the accuracy of the analysis. & TD/FD \\ \hline Wavelet transform (WT) [57] & High time-frequency resolution for analyzing nonstationary signals. & & \\ (ML) [57] & An adaptive method for decomposing a signal into internal modal functions (IMFs) allows for analyzing complex nonstationary processes. & & \\ (ML) [57] & Allows analytical evaluation of the influence of minor nonlinear effects on the behavior of rotor systems. & It does not always accurately describe transient transitions, as it is based on approximations. & N/A \\ \hline \multirow{6}{*}{Newton-Raphson method} & Used for the iterative solution of kinetic equations in rotary systems, which allows considering nonlinear contact forces between the disk and the shaft [49]. & For complex mechanical systems, additional refinement of the Jacobi matrix is required to achieve stable convergence [55]. & \\ \cline{2-4}  & Allow to improve the accuracy of numerical modeling by combining with the Newmark-\(\beta\) method [49]. & Sensitive to the choice of the initial approximation [57]. & \\ \cline{2-4}  & Provides fast convergence in solving nonlinear equations [57]. & & \\ \hline \multirow{6}{*}{Newmark-\(\beta\) method} & It allows effectively modeling the bearing parameters’ impact on the rotor vibration characteristics [49]. & Sensitive to the choice of initial conditions, which can affect the accuracy of the numerical solution [57]. & N/A \\ \cline{1-1} \cline{2-4}  & & \\ \hline \hline \end{tabular}  
* * Signal processing domain (if applicable): TD—time domain; FD—frequency domain; TFD—time-frequency domain.  
\end{table} 
Table 6: _Cont._and the detection of impulses begins by using the Teager energy operator to process signals:  
\[\Psi[x(t)]=\dot{x}^{2}(t)-x(t)\cdot\ddot{x}(t), \tag{35}\]  
where \(m\) is the mass; \(k\) is the stiffness; \(x\) is the displacement that varies over time \(t\). Fourier transform analysis was conducted on the instantaneous Teager energy series to determine the impulses' repeating frequency and generate the Teager energy spectrum [60]:  
\[\Psi[x(n)]=x^{2}(n)-x(n+1)\cdot x(n-1), \tag{36}\]  
where \(n\) is the discrete time index. The bearing fault can be determined by examining the characteristic frequency of the bearing element faults and their prominent frequencies. The experimental results demonstrate that the proposed method successfully extracts the characteristic frequencies of the bearing element faults while effectively detecting weaker inner race and rolling element fault symptoms. Despite the undeniable advantages, the method has several limitations. The technique uses the Teager energy operator (TEO), which shows sensitivity toward short-term signal impulses.