## 3 Results
### Run-to-Failure Bearing Experiments
#### 3.2.2 Experimental Set-Up  
The variable speed experimental data are transformed in the angular domain. The spectrum of the original non-stationary signal is shown in Figure 9a, and the stationary signal after angular domain transformation is shown in Figure 9b. As shown in Figure 10a, the characteristic indexes (kurtosis, root mean square) are extracted from the angle domain signal. The degradation start time of the bearing is 1150 min, and the failure time is 1450 min. The RRMS of each moment was calculated, and the degradation characteristics of rolling bearings were predicted by the method presented in this paper. It was further mapped to the bearing RUL, as shown in Figure 10b. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Type**} & **Pitch** & **Ball** & **Contact** & **Rated** & **Numbers of** \\  & **Diameter** & **Diameter** & **Angle** & **Load** & **Rollers** \\ \hline ERK16 & \(D\) (mm) & \(d\) (mm) & (rad) & \(C_{r}\) (kN) & \(Z\) \\  & 39.04 & 7.94 & 0 & 14 & 9 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Main geometric parameters of rolling bearing ERK16. Figure 10: (**a**) Degradation characteristics; (**b**) real RRMS and predicted RRMS. Figure 9: (**a**) The spectrum of the original non-stationary signal; (**b**) the stationary signal after angular domain transformation. To explain the influence of different fault locations on the stress intensity factor of time-varying model parameters, a plane open crack with length of 1 mm and depth of 1 mm was set in the inner race and outer race of the bearing in FEM. According to the experimental conditions, the inner race of the bearing was fixed, and a radial load of 8500 N was applied to the bearing. The relationship between the stress intensity factor and the square root of the crack length is shown in Figure 11. It can be seen that the crack location has little effect on the stress intensity factor. \(\Delta K=\Delta\sigma_{1}\sqrt{x}=5.675\times 10^{5}\times\sqrt{x}\) for inner race, and \(\Delta K=\Delta\sigma_{2}\sqrt{x}=5.753\times 10^{5}\times\sqrt{x}\) for outer race were set in the experiments. To illustrate the superiority of the bearing degradation model based on time-varying parameters proposed in this paper, RRMS extracted directly from the original time domain was introduced into the bearing degradation model based on the traditional empirical formula. RMSE of state estimation error is obtained as shown in Figure 12. RMSE obtained by directly extracting features from time domain is 0.8327, while the error obtained by extracting features after angular domain transformation becomes 0.2596. Therefore, it can be seen that angular domain transformation has better effect in Processing non-stationary time domain signals with variable rotational speed. RMSE of the degradation model based on the traditional empirical formula is 0.4708, while RMSE of the degradation model based on the time-varying parameters is reduced to 0.2596. Therefore, the degradation model based on the time-varying parameters proposed in this paper is closer to the actual degradation state of the bearing with high prediction accuracy. Figure 11: Relationship between \(\Delta K\) and \(\sqrt{x}\) at different positions. Figure 12: RMSE predicted by four methods. We further verify the effectiveness of the developed method in bearing RUL prediction. The predicted degradation characteristic RRMS is mapped to the RUL of the bearing. The time when the bearing reaches the failure threshold is calculated, as shown in Table 5. The prediction error of RUL based on time domain analysis and the traditional fixed model is 36 min, while the prediction error of RUL is reduced to 27 min after angular domain transformation. The prediction error of RUL based on angular domain transformation and the empirical fixed degradation model is 25 min, while the prediction error of RUL based on angular domain transformation and time-varying parameter degradation model is 12 min.