## 2 Research Methodology
### Bearing Faults
First, it is not always possible to place sensors correctly on a real machine, which may lead to a decrease in the accuracy of the assessment. Another problem is the need to retrain the system for minor changes. Together with high computational costs, this is a problem for the practical application of the method. Along with contact bearings, magnet bearings are widely used in rotor machines. Their condition itself is an essential indicator of common machine conditions. Moreover, the parameters of such bearings depend on operational regimes in complex and implicit ways. Therefore, it is necessary to have a reliable way to estimate the condition of bearings. The research by [59] aims to establish the nonlinear coupling coefficients between the rotational speed and stiffness of magnetic bearings for a rotor through linear regression and a neural network (NN). Two elements and four degrees of freedom were utilized for rotor modeling using a finite element model:  
\[\left([C(\omega)]-\omega^{2}[M]\right)\{Y\}=\{F\}, \tag{37}\]  
Figure 25: Scatter defect classification diagrams for different ANN architectures [61]: (**a**) 2D CNN; (**b**) DCGAN; (**c**) TF-DLGAN. where \(\{F\}\) is the column vector of external nodal forces; \(\{Y\}\) is the column vector of nodal displacements; \(\omega\) is the angular speed; \([M]\) is the inertia matrix; \([C(\omega)]\) is the stiffness matrix that depends on the angular speed. Inertia and stiffness matrices for the finite elements were determined based on finite element analysis (FEA) theory. To normalize the parameters under study, their maximum values were experimentally determined. Experimental data were obtained from both a test stand and simulation software. The expected outcome for both approaches was the determination of influence coefficients describing the rotational speed vs. stiffness relationship, i.e., the coefficients of a quadratic polynomial:  
\[c(\omega)=c_{0}+a\omega+b\omega^{2}, \tag{38}\]  
where \(c_{0}\) is the initial bearing stiffness; \(a\) and \(b\) are the initial slope and curvature of the curve \({}^{\alpha}\)-\(\omega\)\". By regression analysis, the influence coefficients and critical frequencies for the quadratic dependence of the bearing stiffness upon rotational speed were determined through simulation. The weight coefficients were determined from the latter's linear regression matrix equation. The influence coefficients were then computed using experimentally measured values of the critical frequencies. Simulation data were used to train the network, and the critical frequencies used subsequently as input for the trained NN were actual values. As a comparison, experimental data were used to compute the actual influence coefficients and were compared with the values predicted using both methods. It was seen that the NN-based predictions were close to the actual values, while linear regression yielded a considerable relative error. This discrepancy is because linear regression inherently assumes linear parameter dependence, while the neural network does not rely on the provided relationships but constructs its model based on weight distributions. The weight distributions may represent complex dependencies that are difficult to write down analytically or identify explicitly. The high linear error validates that rotational speed and stiffness have a nonlinear relationship. In the study by [64], the authors consider a Chebyshev interval method accelerated by a Smolyak sparse grid to propagate four bounded uncertainties--the shaft modulus, bearing length, oil viscosity, and unbalance--through an oil-film-bearing rotor model. The third-order polynomial surrogate needs only about 70 finite element evaluations, reproducing a 100-point brute-force scan within 1% while cutting CPU costs by roughly 96%. The results provide clear upper and lower envelopes for the frequency-response peaks, such as in Figure 26, revealing how the bearing length shifts critical speeds and separates global from local parameter influence for a robust threshold setting.