## Materials and Methods
### Modling of Digital Twin for Magnetic Bearing (MB)
Six degrees of freedom (DOFs) are at play in the shaft's motion, including translation and rotation in the radial horizontal, radial vertical, and axial directions, respectively. Two axial-radial magnetic bearings bias provide suspension for the four-degrees-of-freedom radial motion. The radial dynamics (4-DOF) rotor system can be characterized as follows:  
\[ \begin{array}{l}\mathrm{I}_{y}\ddot{\beta}-l_{z}\Omega\dot{\alpha}=l_{hA}f_{ xA}-l_{bB}f_{xB}\\ \mathrm{m}\dot{x}=f_{xA}+f_{xB}\\ \mathrm{I}_{x}\ddot{\alpha}+\mathrm{I}_{z}\Omega\dot{\beta}=l_{bB}f_{yB}-l_{bA }f_{yA}\\ \mathrm{m}\dot{y}=\mathrm{f}_{yA}+\mathrm{f}_{yb}\end{array}  \tag{9}\]  
The mass of the rotor is denoted by \(m\). \(\mathrm{I}_{x}\), \(\mathrm{I}_{y}\), and \(\mathrm{I}_{z}\) indicate moments of inertia around the x, y, and z axes. Equation (10) presents \(\mathrm{F}_{\mathrm{b}}=[f_{xA}f_{xB},f_{yA}f_{yB}]^{\mathrm{T}}\). The bearing coordinate system represents the radial suspension forces acting on the four degrees of freedom. \[\mathrm{Fb}=\mathrm{K}_{\mathrm{x}}\mathrm{q}_{\mathrm{b}}+\mathrm{K}_{ \mathrm{i}}\mathrm{I} \tag{10}\]  
\(I=diag(i_{xA}\), \(i_{xB}\), \(i_{yA}\), \(i_{yB})\) denoted the current coil matrix of magnetic bearing. \(K_{x}=diag(K_{xA}\), \(K_{xB}\), \(K_{yA}\), \(K_{yB})\) is the displacement stiffness matrix. \(K_{z}=diag(K_{xA1}\), \(K_{zB1}K_{iA2}\), \(K_{82})\) is the current stiffness matrix, present in Equations (16)-(19). The limitations for a magnetically suspended shaft can be summed up as 4-DOFs radial suspension forces \(f_{xA}f_{yB}\), \(f_{yA}\), and \(f_{yB}\), according to the mathematical model of rotor dynamics described by Equation (10). The dynamic equation of the x plane is represented by the upper part of [M], [K], and [G], while the lower part represents the dynamic equation of the y plane. They are all 4N x4N-order sparse symmetric matrices, as shown in Equation (11). The dynamic equations present below at the non-rotor ends and non-supported:  
\[\mathrm{M}_{\mathrm{i}}\ddot{U}_{\mathrm{i}}+\mathrm{G}_{\mathrm{i}}\dot{U} _{\mathrm{i}}+K_{\mathrm{i}}^{1}\mathrm{U}_{\mathrm{i}-1}+K_{\mathrm{i}}^{2} \ \mathrm{U}_{\mathrm{i}}+K_{\mathrm{i}}^{3}\ \mathrm{U}_{\mathrm{i}}+K_{\mathrm{i}}^{4}\ \mathrm{U}_{\mathrm{i}+1}=\mathrm{F} \tag{11}\]  
Figure 8: Geometric model of the magnetic bearing rigid shaft. Table 2 displays the dimensions and parameters of a rigid rotor model supported by radial active magnetic bearings, whereas Table 3 displays the control parameters of a local active magnetic bearing actuator. The electromagnet is driven by a constant voltage from the dc-link, maintained by a simple proportional current controller acting as a current amplifier model, as seen in Figure 10. We also take into account the voltage's saturation level. The dynamics of the coil were taken into account. The current amplifier has a fixed output voltage of Vdc, where \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Symbol** & **Meaning** & **Value** & **Units** \\ \hline \(m\) & Mass of rotor & 24.26 & kg \\ \hline \(l_{z}\) & Inertia of rotor in z axis & 0.04525 & kgm\({}^{2}\) \\ \hline \(l_{x}=l_{y}\) & Inertia of the rotor in x, and y axis & 0.67388 & kgm\({}^{2}\) \\ \hline \(a\) & Position of Bearing A & \(-0.2607\) & m \\ \hline \(c\) & Sensor position of bearing A & \(-0.1107\) & m \\ \hline \(k\)s\(A\) & Position stiffness of bearing A & \(-3.4\times 10^{6}\) & N/m \\ \hline \(K\)i\(A\) & Current stiffness of bearing A & 765 & N/A \\ \hline \(b\) & Position of Bearing B & 0.2883 & m \\ \hline \(d\) & Sensor position of bearing B & 0.1373 & m \\ \hline \(k\)s\(B\) & Position stiffness of bearing B & \(-3.4\times 10^{6}\) & N/m \\ \hline \(K\)i\(B\) & Current stiffness of bearing B & 765 & N/A \\ \hline \hline \end{tabular}
\end{table} 
Table 2: The dimensions and parameters of a rigid rotor model.