## Materials and Methods
### Modling of Digital Twin for Magnetic Bearing (MB)
By transforming Equation (21) into complex form, it becomes  
\[\eta_{\Delta}(\text{t})=\xi e^{j(wt+\theta)}v_{\Delta}(\text{t})=\sigma e^{j(wt +\gamma)} \tag{27}\]  
Substituting Equation (27) into Equation (26) obtains  
\[\xi_{\rho}{}^{j(wt+\theta)}=-\frac{mw^{2}+2k_{x}}{mw^{2}}A_{\eta}\ e^{j(wt+ \gamma)}\sigma e^{j(wt+\gamma)}=-\frac{I_{\text{r}}w^{2}+2 kxl_{b}^{2}}{I_{ \text{r}}w^{2}}A_{\
u}\ e^{j(wt+\beta_{\gamma})} \tag{28}\]According to Equation (28), the amplitude and phase of the rotor mass imbalance can be obtained online by  
\[\xi=A_{\eta}\frac{mw^{2}+2k_{x}}{mw^{2}}\sigma=A_{\psi}\frac{I_{r}w^{2}+2 kxl_{b}^ {2}}{I_{r}w^{2}}\theta=\varnothing_{\theta}+\pi\gamma=\varnothing_{\gamma}+\pi \tag{29}\]  
Furthermore, to model 3D vibration dynamics with an unbalanced rotor mass, the amplitudes and phases solved by Equation (29) are the same as the A-end and B-end counter discs. Use m\({}_{\rm A}\) and \(\varnothing_{\rm A}\) for the mass and phase at the A-end of the counter disc. m\({}_{\rm B}\) and \(\varnothing_{\rm B}\) are used to represent the mass and phase at the B end of the counter disc. This can be said about the counterweight matrix:  
\[\rm{m}\Delta=\left\{ \begin{array}{ccc}m_{A}r_{a}&\cos(\varphi_{A})\\ m_{B}r_{B}&\cos(\varphi_{B})\\ m_{A}r_{a}&\sin(\varphi_{A})\\ m_{B}r_{B}&\sin(\varphi_{B})\\ \end{array} \right\} \tag{30}\]  
The generalized coordinate system's force moment must be zero for the principle of equivalence to apply and there are  
\[ \begin{array}{ccc}\rm{m}_{bx}w^{2}r_{b}l_{cB}-\rm{m}_{ax}w^{2}r_{a}l_{cA}=I _{r}w^{2}\sigma&\cos&\gamma\\ \rm{m}_{ax}w^{2}r_{a}-\rm{m}_{bx}w^{2}r_{b}=mw^{2}\xi\cos\theta\\ \rm{m}_{by}w^{2}r_{b}l_{cB}-\rm{m}_{ay}w^{2}r_{a}l_{cA}=I_{r}w^{2}\sigma&\cos &\gamma\\ \rm{m}_{ay}w^{2}r_{a}-\rm{m}_{by}w^{2}r_{b}=mw^{2}\xi\cos&\theta\\ \end{array}  \tag{31}\]  
where the mass components are divided into radial and orthogonal directions by the letters m\({}_{ax}\), m\({}_{ay}\), m\({}_{bx}\), and m\({}_{by}\). The counterweight radius is given by r\({}_{a}\) and r\({}_{b}\) and \(\omega\) is the rotor's rotating angular velocity. Finally, the discovered rotor mass imbalance's equivalent-counterweight amplitudes and phases can be solved as  
\[\rm{m}_{A}=\sqrt{{m_{ax}}^{2}+{m_{ay}}^{2}} \tag{32}\]  
\[\rm{m}_{B}=\sqrt{{m_{bx}}^{2}+{m_{by}}^{2}} \tag{33}\]  
\[\varphi_{A}=\arctan(\frac{m_{ay}}{m_{ax}}) \tag{34}\]  
\[\varphi_{B}=\arctan(\frac{m_{by}}{m_{bx}}) \tag{35}\]  
The logical model is mainly used to establish dynamics consistent with the physical model. The logical model needs to be simultaneous simulation and remote control of digital twin models. In order to make the virtual department and accurately simulate the system's running process, it is necessary to convert various codes of the entity model, such as MATLAB apps program, etc. We can use information from the connected asset to define a model in MATLAB. Using its multi-domain modeling capabilities, Simulink may also be used to develop a model based on physical principles. In order to create a digital twin, data-driven or physics-based models can be fine-tuned with information from the live asset. These digital duplicates have many applications, including but not limited to prediction, what-if analysis, anomaly detection, fault isolation, and more. MATLAB facilitates data-driven approaches such as machine learning, deep learning, neural networks, and system identification (45:55). A model is often trained or extracted from one set of data and then validated or tested on another. For identification and analysis, in this paper, a MATLAB app neural network is used for a data-driven approach. The logical model deals with the virtual reality interaction mechanism, code analysis, simulation, and remote monitoring. According to the digital twin data transmission reference architecture, the primary input data transmission sources are the physical model. Corresponding to the data interface technology, the input-output flow is mainly designed with the RESTful architecture.