## 5 Experiment Results
\[f_{bpfr}=\frac{D}{2d}\Bigg{(}1-(\frac{d}{D})^{2}\cos^{2}\alpha\Bigg{)}f_{r} \tag{23}\]7 (**a**) The ACFHNR value for the full life cycle of bearing 1; (**b**) The sensitivity value of ACFHNR. \[f_{bpfr}=\frac{D}{2d}\Bigg{(}1-(\frac{d}{D})^{2}\cos^{2}\alpha\Bigg{)}f_{r} \tag{23}\]8 (**a**) The kurtosis value for the full life cycle of bearing 1; (**b**) The sensitivity value of kurtosis. From Figures 17-20, at least two conclusions can be drawn: (1) Both the RMS and the ACFHNR index are able to detect faults earlier than the kurtosis factor, as they can detect the fault information at group 532 (at a time of about 88 h 40 min) while the kurtosis factor detected the fault information at group 647 (at time of about 107 h 50 min); (2) Compared with the kurtosis and RMS indexes, the sensitivity value of the ACFHNR index is the largest in the incipient fault stage, where the sensitivity value of ACFHNR index is 0.0261, while the sensitivity values of kurtosis and the RMS are 0.0033 and 1.55714 \(\times\) 10\({}^{-4}\), respectively. According to the above conclusions, it can be found that the ACFHNR index is superior to the traditional RMS and kurtosis indexes for detecting the early faults of rolling bearings. Moreover, the early fault vibration signal of group 533 is analyzed. The sample lengths are set as 10,000. The original vibration signal and its corresponding envelope demodulation spectrum are shown in \[f_{bpfr}=\frac{D}{2d}\Bigg{(}1-(\frac{d}{D})^{2}\cos^{2}\alpha\Bigg{)}f_{r} \tag{23}\]9 Theouter race fault frequency \(f_{bpfo}=~{}236\) HZ can be detected, which further indicates the effectiveness of the ACFHNR index for detecting the early weak fault vibration signal directly. To further prove the usefulness of the proposed ACFHNR-SVD-TEO method, the vibration signal of group 533 is processed. \begin{table}
\begin{tabular}{c c c c} \hline
**The Index** & **ACFHNR** & **Kurtosis** & **RMS** \\ \hline \hline Threshold value & 0.15615 & 3.76849 & 0.0812 \\ \hline \end{tabular}
\end{table} 
Table 2: The threshold values of the three indexes. Figure 21: **(a)**The original vibration signal of group 533; **(b)** The envelope demodulation spectrum of group 533. Figure 20: The discrete probability distribution for parameters at the stationary stage: (**a**) ACFHNR; (**b**) root-mean-square (RMS); (**c**) kurtosis. \begin{table}
\begin{tabular}{c c c c} \hline
**The Index** & **ACFHNR** & **Kurtosis** & **RMS** \\ \hline \hline Sensitivity & 0.0261 & 0.0033 & \(1.55714\times 10^{-4}\) \\ \hline \end{tabular}
\end{table} 
Table 3: The sensitivity value of the three indexes in the incipient fault stage. The delay step is set as \(m=15\). Also, both the kurtosis values and the ACFHNR index values corresponding to these 15 singular components are calculated, as shown in Figure 22. It can be seen that the ACFHNR value of the first singular component is the largest, which means that the first component is the effective signal feature. Thus the signal of the first singular component is processed using the proposed method, as shown in Figure 23. The fault frequencies \(f_{bpfo}\), \(2f_{bpfo}\), \(3f_{bpfo}\), and \(4f_{bpfo}\) can be clearly detected. Once again, this processed result proves the usefulness of the proposed method to detect the early faults of the rolling element bearings.