## Materials and Methods
### Modling of Digital Twin for Magnetic Bearing (MB)
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Symbol** & **Meaning** & **Value** & **Units** \\ \hline \(k_{p}\) & Proportional control gain & 0.4 & A/V \\ \hline \(k_{s}\) & Sensor gain & 0.02 & V/\(\mu\)m \\ \hline \(k_{l}\) & Integral control gain & 0.1 & A/V \\ \hline \(k\)pi & Proportional control gain & 45 & V/A \\ \hline \(Vdc\) & DC link voltage & 124 & V \\ \hline \(imax\) & Bias current of a magnet coil & 1.4 & A \\ \hline \(\omega_{D}\) & Derivative control gain & 243 & rad/s \\ \hline \(L\) & Inductance of a magnet coil & 124 & mH \\ \hline \(R\) & Resistance of a magnet coil & 0.453 & ohm \\ \hline \end{tabular}
\end{table} 
Table 3: The control parameters of a local active magnetic bearing actuator. Figure 9: Block diagram for PID controller for each axis of the rotor. Vdc is the amplifier's D.C. input voltage. Each electromagnet's behavior follows a linear relationship with control current and position variations. Now, the important layer of digital twin modeling is explained briefly, which is named the behavior layer. Figure 11 shows that the behavior layer contains three modes. The first mode is the geometric analysis model, the second is the logical model, and the third is the data model to monitor the work of the vibration rotor with mass imbalance. Generally, in the behavior layer, we identify the rotor mass imbalance and dynamic vibration modeling online based on the three modes of this layer. The geometry analysis model includes the independent variables of rotational speed into the dynamics model and extends it to three-dimensional (3D) space obtained by the Ansys Twin Builder program. Furthermore, the parameter of the physical model for the rotors system (A and B) are obtained from sensor A and sensor B. The sensors' parameters are transferred to the behavior layer using MATLAB Simulink's closed-loop suspension control design with Ansys Twin Builder 3D dynamics modeling. First, the assembled 3D geometry from Figure 12 is imported into Ansys Twin Builder and the material parameters for each shaft section are configured. Second, the 4-DOFs radial suspension forces \(f_{xA},f_{zB},f_{yA}\), and \(f_{yB}\) are added at the bearing coordinate system's origin and a revolute motion around the z-axis at the generalized coordinate system's origin. Third, 4-DOFs are taken to control current \(i_{xA}\), \(i_{yA}\), \(i_{zB}\), and \(i_{yB}\) as input and 4-DOFs displacement \(x_{sA}\), \(y_{sA}\), \(x_{sB}\), and \(y_{sB}\) as output. The Ansys Twin Builder plant module is then developed to characterize the rotor system. Forth, the closed-loop rotor system is created, seen in Figure 13, by importing the Ansys Twin Builder plant module into Simulink and adding the suspension controller Gc(s), power amplifier Gp(s), and displacement sensor Gs(s). Figure 11: Behavior layer. Figure 10: Current amplifier model. In the MATLAB/Simulink software package, the control system and electromagnetic actuator models were created. Figure 11 illustrates the overall design. The input to the Ansys Twin Builder model is the four electromagnetic forces in the radial active magnetic bearings, and the output of the Ansys Twin Builder plant model is the displacements of the rotor at the sensors. The actuator model uses the rotor deviations on each radial bearing to compute the position stiffness forces, and the controller uses the rotor deviations on each sensor to create the control current. A flexible rotor suspended by radial AMBs is modeled in Ansys Twin Builder /View and imported into MATLAB/Simulink as a Ansys Twin Builder sub-module, which includes a function module. Figure 14 illustrates the Ansys Twin Builder sub-module contents. The mass imbalance of the rotor is an essential result of factors such as unequal shaft material and mistakes made during machining and assembly, which leads to the geometric axis not aligning with the inertial axis. On the one hand, the static imbalance and vibration  
Figure 14: Mechanical system model of rotor. Figure 12: The physical model for the rotor’s closed-loop system.