# 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
\begin{table}
\begin{tabular}{c c} \hline \hline
**Parameters** & **Value** \\ \hline Number of rollers & 9 \\ Stiffness of roller & \(5.96\times 10^{7}\) (\(\frac{\mathrm{N}}{\mathrm{m}}\)) \\ Mass of outer (kg) & 2.7 (Kg) \\ Stiffness of outer & \(1.31\times 10^{5}\) (\(\frac{\mathrm{N}}{\mathrm{m}}\)) \\ Mass of shaft (kg) & 1.36 (Kg) \\ Stiffness of shaft & \(23.3\times 10^{6}\) (\(\frac{\mathrm{N}}{\mathrm{m}}\)) \\ Damping & 654 (\(\frac{\mathrm{N}}{\mathrm{m}}\)) \\ Roller diameter & 7.940 (mm) \\ Pitch diameter & 39.04 (mm) \\ Defect size & 7 (mm) \\ Defect depth & 2 (mm) \\ Inside diameter & 25 (mm) \\ Outside diameter & 52 (mm) \\ Ball diameter & 7.94 (mm) \\ Thickness & 15 (mm) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Information of the bearing for mathematical modeling of the CWRUBD [10; 40]. Here, \(W_{AUT-LAG}(k),e_{AUT-LAG}(k),\Delta_{AUT-LAG}(k),\ Z_{AUT-LAG}(k)\), and \(\big{(}\gamma_{AUT-LAG_{i}},\ \gamma_{AUT-LAG_{i}},\ \beta_{AUT-LAG}\big{)}\) are the state of the signal modeling based on the AUT-LAG technique for the bearing's digital twin, the error of the bearing's digital twin modeling using the AUT-LAG technique, the unknown condition of the bearing's digital twin modeling that is computed using the AUT-LAG technique, the bearing's digital twin output modeling using the AUT-LAG technique, and the coefficients for the state-space bearing's digital twin tuning using the AUT-LAG method, respectively. Furthermore, the unknown condition and the error of the bearing's digital twin modeling that are computed using the AUT-LAG technique are represented as the following calculation. \[\left\{ \begin{array}{l}\Delta_{AUT-LAG}(k)=Z(k)-Z_{AUT-LAG}(k)\\ e_{AUT-LAG}(k)=Z_{AUT-LAG}(k+1)-Z_{AUT-LAG}(k)\end{array} \right. \tag{14}\]  
The AUT-LAG technique is more robust and stable than the AUT method, but this technique suffers from the nonlinear behavior of the signal. The machine-learning algorithm based on the support vector regression (SVR) is recommended to improve the effectiveness of the bearing's digital twin modeling. Hence, the combination of AUT-LAG and SVR (henceforth called SAL) to identify the signal of the bearing is represented as the following function. \[\left\{ \begin{array}{l}W_{SAL}(k+1)=\left[\begin{array}{cc}\gamma_{SAL_{1 }}W_{SAL}(k)+\gamma_{SAL_{2}}\{Z_{AUT-LAG}(k)+Z_{SVR}(k)\}\\ +\Delta_{SAL}(k)\end{array}\right]+\epsilon_{SAL}(k)\\ Z_{SAL}(k)=(\beta_{SAL})^{T}W_{SAL}(k)\end{array} \right. \tag{15}\]  
Here, \(W_{SAL}(k),\epsilon_{SAL}(k),\Delta_{SAL}(k)\), \(Z_{SAL}(k),Z_{SVR}(k)\), and \(\big{(}\gamma_{SAL_{1}},\gamma_{SAL_{2}},\beta_{SAL}\big{)}\) are the state of the signal modeling based on the SAL technique for the bearing, the error of the bearing modeling using the SAL technique, the unknown condition of the bearing modeling that is computed using the SAL technique, the bearing output modeling using the SAL technique, the nonlinearity compensator of the bearing's output compensation using the SVR technique, and the coefficients for the state-space bearing's performance tuning using the SAL method, respectively. Additionally, the unknown condition and the error of the bearing's modeling that are computed using the SAL procedure are represented as the following function. \[\left\{ \begin{array}{l}\Delta_{SAL}(k)=Z(k)-Z_{SAL}(k)\\ e_{SAL}(k)=Z_{SAL}(k+1)-Z_{SAL}(k)\end{array} \right.. \tag{16}\]  
The nonlinearity compensator of the bearing's output compensation using the SVR technique, \((Z_{SVR}(k))\), is introduced using the following definition. \[Z_{SVR}(k)=\sum_{i}\big{(}\delta_{i}^{+}-\delta_{i}^{-}\big{)}K(\theta_{i}, \theta)+\varnothing. \tag{17}\]  
Here, \((\delta_{i}^{+},\delta_{i}^{-}),K(\theta_{i},\theta)\), and \(\varnothing\) are the Lagrange constants, the nonlinear kernel functions that are selected in this work to improve the power of compensation, and the function's bias, respectively. The Gaussian kernel function is suggested in this work and this function is introduced based on variance, \((\sigma)\), using the following equation.