## 5 Experiment Results
In the test bed, two PCB 353B33 High Sensitivity Quartz ICP (Integrated Circuits Piezoelectric) accelerometers (PCB Piezotronics Inc., Depew, NY, USA) are mounted for data acquisition (one vertical and one horizontal) for each bearing, and a radial load of 2721.6 kg is applied to the bearings. Meanwhile, the vibration signals are measured at an interval of every 10 min. The sampling frequency is set to 20 KHz, and the data length is 20,480 points [38]. The shaft rotating speed of the motor is 2000 rpm. Four bearings, named Bearings 1 to 4, are installed on one shaft, as shown in \[f_{bpfr}=\frac{D}{2d}\Bigg{(}1-(\frac{d}{D})^{2}\cos^{2}\alpha\Bigg{)}f_{r} \tag{23}\]0 At the end of the accelerated experiment, an outer race fault is discovered in test bearing 1. The fault frequency of the outer race fault can be computed as follows:  
\[f_{bpfr}=\frac{D}{2d}\Bigg{(}1-(\frac{d}{D})^{2}\cos^{2}\alpha\Bigg{)}f_{r} \tag{23}\]1  
where \(f_{r}\) is the rotating frequency, D is the pitch diameter, d is the balling diameter, Z is the number of the balls, and \(\alpha\) is the contact angle. Also, according to Equation (23), the fault frequency of the out racer of the bearing can be computed, which is about 236 HZ [29]. Based on this prior fault frequency, the ACFHNR index value of all the fault vibration signals in the whole life of the testing are calculated, as shown in Figure 17a. For comparison purposes, the kurtosis and RMS (Root Mean Square) values are computed. The results are shown in Figures 18a and 19a, in which the sub graphs in the figures are the local enlargement of dashed line boxes. In the run-to-failure experiment, it is assumed that the bearing worked at the stationary stage from group 10 to group 500. The discrete probability distribution for the ACFHNR, kurtosis, and RMS index at the stationary stage is Gaussian distribution, as presented in \[f_{bpfr}=\frac{D}{2d}\Bigg{(}1-(\frac{d}{D})^{2}\cos^{2}\alpha\Bigg{)}f_{r} \tag{23}\]2 The alarm thresholds for each parameter are set as the mean value plus four times the stand deviation [23]. Then the threshold of the three parameters can be obtained, as shown in \[f_{bpfr}=\frac{D}{2d}\Bigg{(}1-(\frac{d}{D})^{2}\cos^{2}\alpha\Bigg{)}f_{r} \tag{23}\]3 Also, the alarm time for RMS, kurtosis and the ACFHNR index are derived. The alarm time of RMS is 88 h 40 min, while that of kurtosis is 107 h 50 min, and the ACFHNR warn time is 88 h 40 min, as shown in Figures 17a, 18a and 19a. Meanwhile, compared with the envelope harmonic-to-noise ratio (EHNR) index proposed in the literature [23], the ACFHNR index can detect faults 1 h and 50 min earlier, where the alarm time of the EHNR index is 90 h 30 min [23]. Moreover, we proposed the concept of sensitivity to quantify the three indexes, which can be calculated by Equation (25), where \(\Delta d\) is the difference value of the three indexes in the time \(\Delta t\). The sensitivity can be deemed the gradient of the fitted curve in the incipient fault stage. \[f_{bpfr}=\frac{D}{2d}\Bigg{(}1-(\frac{d}{D})^{2}\cos^{2}\alpha\Bigg{)}f_{r} \tag{23}\]4 The bearing fault test bed from the Intelligent Maintenance System, University of Cincinnati, (**a**) The physical map; (**b**) The sketch map. \[f_{bpfr}=\frac{D}{2d}\Bigg{(}1-(\frac{d}{D})^{2}\cos^{2}\alpha\Bigg{)}f_{r} \tag{23}\]5  
The sensitivity values in the incipient stages of the three indexes are calculated. For the ACFHNR and RMS indexes, the data from group 532 to group 700 are calculated, as shown in Figures 17b and 18b. For the kurtosis index, the data from group 647 to group 700 are calculated, as shown in Figure 19b. The sensitivity values for the three indexes are presented in Table 3, and it can be found that the sensitivity value of ACFHNR is much larger than that of kurtosis or RMS. \[f_{bpfr}=\frac{D}{2d}\Bigg{(}1-(\frac{d}{D})^{2}\cos^{2}\alpha\Bigg{)}f_{r} \tag{23}\]6 (**a**) The RMS value for the full life cycle of bearing 1; (**b**) The sensitivity value of RMS.