## 2 Materials and Methods
### The Present RUL Prediction Method  
As shown in Figure 2, the addressed time-varying degradation model for bearing RUL prediction can be divided into the following eight steps. **Step 1:** The original non-stationary time domain signal is transformed into the angular domain stationary signal by equal angle resampling. **Step 2:** The bearing life data are transformed into the angular domain under variable speed conditions towards a complete angular dataset. The stable angular domain signal of the bearing is extracted at each time, and the degradation characteristic index (kurtosis, root mean square, etc.) of the rolling bearing can be obtained. **Step 3:** Kurtosis is used to monitor the occurrence of bearing faults. The starting point of bearing RUL prediction is determined. The specific process is as follows. Firstly, the mean value \(\mu\) and standard deviation \(\sigma_{k}\) of kurtosis under the normal working condition of the bearing are calculated. Then, the kurtosis interval \([\mu-3\sigma_{k},\mu+3\sigma_{k}]\) is defined as the normal working interval of the bearing. When the kurtosis of the bearing at a certain time exceeds the defined normal interval, it is judged that the bearing enters the degradation stage. This time is regarded as the starting point for the bearing RUL prediction is then started. **Step 4:** The envelope analysis of the vibration signal at the starting point of the prediction of the RUL of the bearing is carried out to obtain the fault characteristic. According to the results of envelope analysis, the fault frequency is obtained, and the bearing fault location is determined by calculation. Finally, the initial crack at the bearing fault location is set in the finite element model. **Step 5:** The relative root mean square value (RRMS) is employed as the input of the degradation model to predict the bearing RUL. The specific definition of RRMS is expressed by  
\[X_{RRMS}(t)=\frac{X_{RMS}(t)-X_{RMS}(T)}{X_{RMS}(T)} \tag{19}\]  
where \(X_{RRMS}(t)\) is the RRMS value of the signal at time \(t\), \(X_{RMS}(t)\) is the RMS value of the signal at time \(t\), and \(X_{RMS}(T)\) is the RMS value of the signal at RUL starting time \(T\). **Step 6:** The real-time finite element model of the bearing is used to obtain the numerical relationship between physical parameters \(\Delta K\) and the crack length \(x\). \[\Delta K=\Delta\sigma\sqrt{x} \tag{20}\]  
Through integrating \(L\) above both sides of Equation (16), the function can be obtained by  
\[ \begin{array}{l}\frac{x_{l}-x_{l-1}}{L_{l}-L_{t-1}}=\alpha x_{l-1}{}^{\beta} \\ x_{l}=x_{l-1}+\alpha x_{l-1}{}^{\beta}\Delta L\end{array}  \tag{21}\]  
where \(\alpha=C(\Delta\sigma)^{m}\), \(\beta=3m/2\), \(C\) and \(m\) are constants related to the material, \(\Delta\sigma\) is a constant, \(x\) is the crack length, and \(L\) is stress cycles. Figure 2: Bearing RUL prediction process using the present approach. The relationship between stress cycles \(L\) and the running time \(t\) of the bearing is linear, and the coefficient is simplified to 1. According to references [32; 33; 34], under ideal conditions, the characteristic index of vibration signal has a linear relationship with the crack length. In order to simplify, the coefficient is set to 1. Combining Equations (5), (6) and (21) leads to the state transition equation and observation equation in the bearing degradation process as  
\[ \begin{split} x_{t}&=x_{t-1}+\alpha x_{k-1}^{\beta} \Delta t+w_{t}\\ y_{t}&=x_{t}+v_{t}\end{split}  \tag{22}\]  
where \(w_{t},v_{t}\) are the system random noise, \(\Delta t\) is the time interval, \(x_{t}\) is state value, and \(y_{t}\) is the measured value. **Step 7:** The degradation index (RRMS) is brought into the degradation model by using the steps of the UPF algorithm, and the model parameters are updated by the measured values to estimate the degradation index value at the current time. The degradation model is used to transfer the existing distribution and predict the degradation index at the next time.