# Bearing Anomaly Recognition Using an Intelligent Digital Twin Integrated with Machine Learning
## 2 Proposed Methodology
### Digital Twin
#### 2.2.1 Signal Modeling: The First Step to Design the Digital Twin  
The first part of this stage is the vibration signal modeling in the normal state for the bearing. Therefore, first, the vibration signal in the normal state using the five degrees of freedom (DOF) mathematical Lagrangian technique is modeled. The mathematical vibration signal modeling for the bearing in the normal state is expressed by the following function [10; 41]. \[f_{M(q)}=M_{M(q)}\big{[}\ddot{q}\big{]}+B_{M}\big{(}q,\dot{q}\big{)}+\Delta_{M} \tag{1}\]  
Here, \(f_{M(q)}\), \(M_{M(q)}\), \(\ddot{q}\), \(B_{M}\big{(}q,\dot{q}\big{)}\), and \(\Delta_{M}\) are the force applied to the bearing, the matrices of mass for the bearing, the vibration signal acceleration for the bearing, the nonlinear term of the vibration signal that is one of the important challenges for mathematical modeling, and unknown conditions including faults and uncertainties, respectively. The nonlinear term of the vibration signal should be calculated using the following definition. \[B_{M}\big{(}q,\dot{q}\big{)}=V_{M}\big{[}\dot{q}\big{]}+D_{M}[q] \tag{2}\] \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Dataset Group** & **States** & **Load (hp)** & **Crack Sizes (inch)** \\ \hline \multirow{4}{*}{i} & NS & 0 & \\  & RPS & 0 & 0.007, 0.014, and 0.021 \\  & IFS & 0 & \\  & OFS & 0 & \\ \hline \multirow{4}{*}{ii} & NS & 1 & \\  & RFS & 1 & \\  & OFS & 1 & \\ \hline \multirow{4}{*}{iii} & NS & 2 & \\  & RFS & 2 & \\  & OFS & 2 & \\ \hline \multirow{4}{*}{iv} & NS & 3 & \\  & RFS & 3 & \\ \cline{1-1}  & IFS & 3 & \\ \cline{1-1}  & OFS & 3 & \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Summarized information of the CWRUBD [40]. Figure 2: Experimental testbed for the bearing data acquisition implemented by CWRUBD [10; 40]. Here, \(V_{M},\dot{q}\), and \(D_{M}\) are the stiffness of the bearing which is a nonlinear-based parameter, the velocity of the vibration signal of the bearing, and the damping of the bearing, respectively. Regarding (1), estimation of the parameter of the unknown condition is one of the main challenges of mathematical modeling of nonlinear systems such as the bearing. Therefore, based on [10; 41], the unknown conditions that are a combination of uncertainties and faults can be modeled using 5-DOF vibration signal modeling, and is defined by the following equation. \[\Delta_{M}=\alpha_{r}+\alpha_{i}+\alpha_{o} \tag{3}\]  
where \(\alpha_{r},\alpha_{i}\), and \(\alpha_{o}\) are the predicted effect of RFS, IFS, and OFS, respectively. The predicted RFS effect, IFS effect, and OFS effect are represented as the following definitions, respectively. \[\alpha_{r}=Max\Big{(}\alpha_{io}Cos\big{(}\alpha_{j}\big{)}+\alpha_{io}Sin \big{(}\alpha_{j}\big{)}-\epsilon_{\omega}-\alpha_{f},0\Big{)} \tag{4}\]  
\[\alpha_{i}=Max\Big{(}\alpha_{io}Cos\big{(}\alpha_{j}\big{)}-\alpha_{io}Sin \big{(}\alpha_{j}\big{)}-\Big{(}2\times\Big{(}\epsilon_{\omega}-\alpha_{f} \Big{)}\Big{)},0\Big{)} \tag{5}\]  
\[\alpha_{o}=Max\Big{(}\alpha_{io}Cos\big{(}\alpha_{j}\big{)}+1.5\times\alpha_{ io}Sin\big{(}\alpha_{j}\big{)}-\epsilon_{\omega}+\alpha_{f},0\Big{)} \tag{6}\]  
where the difference between the center of mass in IFS and OFS, (\(\alpha_{io}\)), is represented as follows:  
\[\alpha_{io}=\alpha_{xi}-\alpha_{xo} \tag{7}\]  
Here, \(\alpha_{xi}\) and \(\alpha_{xo}\) are the IFS center of mass and the OFS center of mass, respectively. The angular position in the RFS, IFS, and OFS, (\(\alpha_{j}\)), is computed as the following equation. \[\alpha_{j}=\frac{2\pi(j-1)}{k_{r}}+\epsilon_{\omega}+\alpha_{o} \tag{8}\]  
where \(k_{r}\) and \(\epsilon_{\omega}\) are the RFS numbers and the rotor velocity, respectively. Moreover, the deformation, (\(\alpha_{f}\)), is determined as follows:  
\[\alpha_{f}=\epsilon_{\omega_{a}}+\epsilon_{\omega_{b}} \tag{9}\]  
Here, (\(\epsilon_{\omega_{a}}\) and \(\epsilon_{\omega_{b}}\)) are the angular width at two positions, position \(a\) and \(b\), respectively.