## 2 Research Methodology
### Rotor-Stator Interaction
The comparative results demonstrate the effectiveness of the proposed new technique (Figure 21). Finally, the new method was applied to fault diagnosis of a rotor system with local friction, and analysis of the experimental data indicates that the new method can diagnose the friction faults of a rotor well and is better than the EMD and EEMD methods. The APEWT has some drawbacks, including the end effect and decomposition dependence on spectral partitioning. As a rule, the impact of friction has a nonlinear nature, which must be carefully considered in investigations, as it can significantly increase the accuracy of results. The study by [56] explores a modeling approach for nonlinear friction between a rotor and a stator, considering the possible clearance in the bearing assembly. The model is based on a single-mass Jeffcott rotor, where the stator is represented as a highly elastic element connected to the rotor through a clearance with an elastic-damping link. The contact phase between the rotor and the stator is modeled as a mixed viscous-dry friction phenomenon, which causes instantaneous changes in angular velocity. To describe friction interaction, the authors introduce the instantaneous friction force and a coefficient of restitution, which characterizes an elastic impact. The coefficient of restitution is assumed to depend on the damping coefficient of the stator. Simulations revealed various contact patterns per rotor revolution, depending on the rotational speed, while keeping the geometric conditions constant:  
\[M\ddot{z}+\Big{(}D+\delta D_{f}\Big{)}\dot{z}+Kz+\delta K_{f}(z-ce)+\Delta K(x- jy)+j\delta F_{\tau}\bigg{(}1+\frac{a}{|z|}\bigg{)}e^{j\psi}=mr\omega^{2}e^{j( \omega t+\alpha)}+Se^{j\gamma}, \tag{33}\]  
where \(x\) and \(y\) are the rotor lateral displacements in two orthogonal directions; \(M\), \(D\), and \(K\) are the modal mass, damping, and rotor modal stiffness matrices; \(K_{f}\) is the local radial stiffness of the stator; \(m\), \(r\), and \(\alpha\) are the unbalanced mass, radius, and angular orientation, respectively; \(S\) and \(\gamma\) are the amplitude and angular orientation of the radial force applied to the rotor; \(K_{f}\) and \(D_{f}\) are the stiffness and damping during the local rotor-stator contact interaction; \(F_{\tau}\) is a mixed viscous/dry friction force during contact with the stator; \(a\) is the the rotor radius at the contact area; \(c\) is the rotor-to-stator radial clearance; \(\delta\) is a contact indicator function (\(\delta=0\) if \(|z|<c\); \(\delta=1\) if \(|z|\geq c\)). Additionally, bifurcation diagrams were constructed (Figure 22). It was found that stator damping plays a crucial role in system stability, influencing the size of chaotic regime zones. The experiments also demonstrated that a higher coefficient of restitution causes the rotor to \"bounce\" after impacts, which increases instability. In contrast, a larger clearance reduces the overall system's stiffness, lowering the rotor's natural frequency and shifting resonance zones. The study was validated through experiments on different test rigs, simulating realistic clearances in bearing assemblies and rotor-stator friction. The experimental results showed good agreement with the simulation findings, confirming the accuracy of the proposed model. A comparative analysis of the methods for rotor-stator friction detection is summarized in Table 6.