## 4 Conclusions  
A time-varying degradation model has been reported in this paper for predicting RUL of bearings under variable rotational speed. For non-stationary time-domain signals, it remains challenging to extract the degradation features accurately. Bearing degradation models established by theoretical knowledge have great error with the real degradation processing For this reason, angular domain transformation and time-varying parameters were integrated as the time-varying degradation model. The non-stationary temporal  
Figure 13: (**a**) Life prediction comparison of two models in the time domain; (**b**) life prediction comparison of two models in the angular domain.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline  & **Angle Domain** & **Developed** & **Time Domain** & **Time Domain** \\  & **Fixed Model** & **Model** & **Fixed Model** & **Time-Varying Model** \\ \hline Actual RUL & 1450 min & 1450 min & 1450 min & 1450 min \\ Predicted RUL & 1425 min & 1462 min & 1414 min & 1423 min \\ Error & 25 min & 12 min & 36 min & 27 min \\ CRA & 0.9824 & 0.9917 & 0.9751 & 0.9813 \\ \hline \hline \end{tabular}
\end{table}
Table 5: Comparison of RUL prediction results of four methods.  
signal of bearing lifecycle under variable speed was transformed as stationary angular domain representation. Real-time FEM of the bearing was established to track the bearing performance degradation state. The parameters of the bearing performance degradation model were updated according to the time-varying physical commitment calculated by FEM. The degradation characteristic index 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 characteristic value at the current time. The degradation model is used to transfer the existing distribution and predict the degradation characteristic at the next time. Finally, the remaining useful life of the bearing is predicted by calculating the time when the predicted degradation eigenvalue reaches the failure threshold. For the XJTU-SY experiment, compared with state-of-the-art prognostic method, the accuracy of the present model is improved by 5.3%. For the BPS experiment, compared with the traditional fixed model, the prediction accuracy of the present model is improved by 1.7%. Therefore, the present model can predict the bearing remaining useful life (RUL) more accurately under variable rotational speed.  
However, the proposed method still has some limitations. The developed method is only suitable for one-dimensional cracks. Regarding multi-dimensional cracks, which is a crucial and important issue for RUL prognosis, this will be studied in the future.  
Conceptualization, W.D. and X.H.; methodology, W.D. and X.H.; software, W.D. and X.H.; validation, W.D., X.H. and H.W.; formal analysis, X.H.; investigation, X.H.; resources, X.H., W.D. and H.W.; data curation, X.H. and W.D.; writing--original draft preparation, X.H.; writing--review and editing, X.H., W.D. and H.W.; visualization, X.H.; supervision, W.D., X.H. and H.W.; project administration, W.D. and X.H.; funding acquisition, W.D. All authors have read and agreed to the published version of the manuscript.  
The research is supported by the Foundation of Henan Key Laboratory of Underwater Intelligent Equipment (approved grant: KL03C2104), the Key Science and Technology Research Project of the Henan Province (approved grant: 192102210105), the Foundation for University Key Teacher of Henan Educational Department (approved grant: 2018 CGJS091), the Key Scientific Research Projects of Colleges and Universities in Henan Province (approved grant: 21A460033), and the Program for Innovative Research Team in University of Henan Province (approved grant: 20 IRTSTHN015).  
Publicly available datasets [35] were analyzed in this study; other datasets are proprietary and cannot be shared at this time.  
The authors declare no conflict of interest.