## 2 Research Methodology
### Misalignment
The vectors of global unbalanced force and misaligned AMB force are represented by \(f_{umb}\) and \(f_{UNB}^{mis}\), respectively [48]:  
\[f_{UNB}^{mis}=K_{s}^{mis}\eta_{UNB}^{mis}+K_{i}^{mis}i_{c}^{mis}+f_{c}^{mis}, \tag{24}\]  
where \(K_{s}^{mis}\), \(K_{i}^{mis}\) and \(f_{c}^{mis}\) are the modified AMB force-displacement stiffness matrix, force-current stiffness matrix, and constant force vector for the misaligned \(q\)-th AMB, respectively; \(\eta_{UNB}^{mis}\) and \(i_{c}^{mis}\) are the displacement and current vectors at the AMB location, respectively. For magnetic bearing simulation, the authors derived stiffness coefficients comparable to the resistance of the magnetic field in the bearing. Using the classical rotational motion equation and the forces operating in the magnetic field, they obtained the equation of motion for a magnetic bearing system with both the traditional and equivalent magnetic stiffness terms. In constructing such an equation set for the entire rotor model, the variables outnumber the equations and create an underdetermined system. To solve this problem, the authors introduced one step with a controlled value of axis displacement. Another system of equations was formulated for this step. The systems of equations were then combined, and the outcome was an overdetermined system of equations. The solution was obtained by finding optimal sets of parameters according to the least squares method. The parameters obtained from this system are natural and equivalent stiffness, eccentricity, phase of unbalance, and axis displacement. The second problem was eliminating the rotational degrees of freedom from the model. They accomplished this using dynamic degree-of-freedom reduction, a technique in which undesirable degrees of freedom are expressed in terms of more convenient ones. This substitute was then inserted into the equation of motion, creating a system of equations with only the degrees of freedom necessary. The method was verified using Simulink simulations with positive outcomes (Figure 13). Misalignment can cause other unwanted effects, such as decreased system efficiency and friction between system parts. The article by [49] discusses the impact of non-uniform contact of the rotor and disk surfaces and parallel misalignment of the rotor and motor axes on system stability and vibration response [74; 75]. The rotor system model is presented in the form of a finite element single-mass model with an added deliberate extra displacement of the shaft end:  
\[(M_{d}+M_{s})\ddot{U}+(C_{d}+C_{s}+C_{b}+D_{d}+D_{s})\dot{U}+(K_{s}+K_{b})U=F_ {c}+F_{e}+F_{g}, \tag{25}\]where \(U\) is the displacement vector of the rotor system; \(F_{c}\) is the additional excitation force caused by the parallel misalignment of the coupling; \(F_{g}\) is the gravity of the rotor system; \(K_{b}\), \(C_{b}\) are the supporting stiffness and supporting damping matrix, respectively; \(F_{e}\) is the unbalanced force of the rotor system; \(M\) is the mass matrix; \(K\) is the stiffness matrix; \(C\) is the damping matrix; \(D\) is the gyro matrix; indices \(s\), \(d\), \(b\) represent the shaft, disk, and bearings, respectively. The Heaviside function described the disk-rotor surface coupling in conditions of interference fit, while the contact or separation condition was defined by the ratio integral of normal and tangential forces at some connection points [49]:  
\[F_{n}=\int p_{(\beta)}l_{d}cos(\beta)ds; \tag{26}\] \[F_{\rm T}=\int sgn\Big{(}\Delta v_{(\beta)}\Big{)}\mu_{fd}p_{( \beta)}l_{d}cos(\beta)ds,\]  
where \(p_{(\beta)}\) is the normal contact stress; \(l_{d}\) is the axial length of disk-shaft contact; \(\Delta v_{(\beta)}\) is the difference between the tangential velocity of the shaft and the disk; \(\mu_{fd}\) is the friction coefficient between the disk and the shaft. The solution of the kinematic equations was reduced to the solution of the system of equations on the Jacobian matrix, which was computed by the combination of the Newton-Raphson and Newmark methods.