## 2 Research Methodology
### Misalignment
This significantly improved convergence and reduced the number of iterations to obtain a solution. The analysis found that if the interference fit is insufficient or surface wear is induced by unfavorable operating conditions, the system becomes unstable at some rotational speed limits. This is so because, at high rotational speeds, the frictional forces generated by contact stress are insufficient to ensure that the rotor and disk rotate together as a single body. As a result, the disk begins to slip with impacts against the rotor, generating cyclic contact-separation phenomena. In the frequency  
Figure 13: FFT processed frequency domain responses (**a**) and phases (**b,c**) in case of unbalance and residual misalignment, respectively. Adopted from [48]. spectrum, this effect is revealed as a \"splitting\" of the rotor's fundamental frequency by an additional component. In addition, parallel misalignment between the rotor and motor axes also generates superharmonics in the spectrum, the most prominent of which are the \(\times 3\) and \(\times 4\) components. In addition, parallel misalignment influences slipping, which is reflected in the spectrum by a more significant separation between the fundamental frequency and its split harmonics (Figure 14). Experimental trials were utilized to confirm the modeling results, and the experimental results agreed with the simulation findings. Other parts that can suffer from misalignment are the bearings. Misalignment induces wear of the bearing elements, thus decreasing their durability. The article by [50] studied the impact of rotor shaft misalignment on the system over time. The overall goal of the approach was to form a numerical approach for the analysis of the nonlinear dynamic response of the rotor-bearing system, considering the time-dependent misalignment. In contrast to most of the research, where misalignment is regarded as a static factor, the authors, in their model, account for the dynamic misalignment resulting from the rotor deformation during rotation. This is calculated through the finite element method, from which rotor deformations can be obtained to estimate the angular deviation and lubricant layer thickness in the bearing:  
\[[M]\big{\{}\tilde{q}\big{\}}+[D]\big{\{}\hat{q}\big{\}}+[K]\big{\{}q\big{\}}= \{F_{u}\}-\big{\{}F_{g}\big{\}}+\{F_{b}\}+\{M_{b}\} \tag{27}\]  
where \([M]\), \([D]\), \([K]\) are the mass, damping, and stiffness matrix of the rotor, respectively; \(\{q\}\) is the displacement vector; \(\{F_{u}\}\), \(\{F_{g}\}\), \(\{F_{b}\}\), and \(\{M_{b}\}\) are the vectors of unbalanced force, gravity, nonlinear bearing force, and moments generated by misalignment, respectively. The damping matrix \([D]\) consists of a part proportional to the stiffness matrix, i.e., \([D]=\beta\cdot[K]+\omega\cdot[G]\), where \(\beta\) is the proportional coefficient related to stiffness; \(\omega\) is the angular velocity [50]. Then, the pressure distribution in the lubricating film is calculated using the finite difference method (FDM); the bearing load can be defined. These parameters are used to analyze the system's transient lubrication process and vibration response. The approach makes it possible to research the influence of misalignment on the lubricant layer instability (oil whip) and its interaction with the resonant oscillation frequencies of the rotor. Numerical simulations demonstrate that variations in misalignment influence the stability of the rotor system (Figure 15), as well as the amplitude of vibrations and the lubricant film behavior for various rotation speeds. However, the method has certain drawbacks that limit its application in practice. Firstly, numerical computations performed with the assistance of the finite element method (FEM) and the finite difference method (FDM) are  
Figure 14: Vibration response of the rotor system with parallel misalignment 1 mm. Adopted from [49]. expensive from a computational perspective, and it is challenging to apply the technique in real-time applications.