In order to validate the effectiveness of the proposed method in this paper for rolling bearing degradation stage prediction, three experimental cases were conducted consecutively.
5.1. Case 1
The accelerated life test was implemented at the Hangzhou Bearing Test and Research Center to acquire the whole-lifetime vibration data. The test rig consisted of a monitoring system, transmission system, loading system, lubrication system and computer control system as shown in
Figure 11. The bearing life enhancement test machine was ABLT-1A from Hangzhou Bearing Test and Research Center Co., Ltd., Hangzhou, China, the acceleration sensor was AI002 from Yangzhou Jingming Technology Co., Ltd., Yangzhou, China, and the dynamic signal test and analysis system was JM5937 from Yangzhou Jingming Technology Co., Ltd., Yangzhou, China. The bearings used in the test were tapered roller bearings 7008 AC from Luoyang Bearing Research Institute Co., Ltd., Luoyang, China.
Table 1 and
Table 2 show the test conditions and bearing parameters, respectively.
One datum was recorded every minute. In order to observe the complete degradation process of the bearing from normal state to minor failure and then to severe failure, the relative method was used in the experiment to determine the failure threshold of the bearing. When the maximum amplitude of the bearing vibration signal exceeded 3A (A is the maximum amplitude of the bearing during normal operation), the bearing was considered to have completely failed and the test was immediately terminated. The accelerated life test was carried out successively until the crest factor of vibration signal exceeded the set value; the total number of sampling vibration data was 8000.
The vibration data in this case are exactly the same as those of Experiment 2 in Reference [
39]. The waveform of the original vibration signal is shown in
Figure 12, and the vibration data did not undergo any preprocessing. From
Figure 12, it can be found that the vibration amplitude has an upward trend. Before 200 min, the vibration amplitude slightly decreases as time goes on and then enters a stable stage, indicating that the rolling bearing enters the normal operation period after a short run-in period. From around 3000 min, the vibration amplitude increases and then begins to decrease, being basically in a fluctuating state. At this stage, the rolling bearing may have undergone slight failure and started to enter the failure period. After 6000 min, the vibration amplitude sharply increases in fluctuation. At this stage, the rolling bearing already experienced serious failure.
A sequence with
N data is used to represent a degradation stage, so the vibration signal can be divided into 8000/
N degradation stages. The HGE of each degradation stage was calculated to obtain the degradation feature sequence of rolling bearing. In order to investigate the effect of data length
N on HGE, HFE and HSE calculations, these three statistics were all applied to analyze degradation stages with different lengths—
N = 200,
N = 400,
N = 800 and
N = 1000—and the results are shown in
Figure 13, where
q = 3,
e = 0.2 SD,
f = 2, node = 1. It can be seen from
Figure 13 that for four different lengths of degradation stages, the degradation feature sequences, which were extracted using HGE entropy, show a phased downward trend overall.
According to reference [
16], the entropy algorithm measures the complexity and regularity of nonlinear signals. When the rolling bearing is in normal operation, its motion is extremely irregular, which means that the complexity of the vibration signal of the rolling bearing is the highest. When local fault defects occur, the vibration signal of the rolling bearing exhibits obvious periodic signal characteristics, which means that the complexity of the vibration signal is reduced. When the defects of rolling bearings further expand, the periodic characteristics of their vibration signals becomes more obvious, which means that the complexity of the vibration signals is further reduced. The results calculated via HGE are consistent with the above phenomenon, which indicates that the results calculated via the proposed method are consistent with real nonlinear dynamic systems. It is worth mentioning that when the degradation stage length
N = 200, HGE characterizes the degradation trend of rolling bearings more accurately, which indicates that HGE has a lower requirement for data length and that reliable calculation results can be obtained with a shorter data length.
To demonstrate the superiority of HGE, the HSE and HFE of the degradation stage were calculated separately under the same parameters, as shown in
Figure 13. From
Figure 13, it can be seen that when
N = 200 and
N = 400, the degraded feature sequences based on HSE and HFE do not exhibit obvious regularity; when
N = 800 and
N = 1000, the degraded feature sequences based on HSE and HFE show a decreasing trend only when
t > 4000 min.
In order to convincingly establish the superiority of HGE, kurtosis, which is a commonly used time-domain feature extraction method, was employed to extract the rolling bearing degraded features. The kurtosis values of each degradation stage were calculated, as shown in
Figure 14. From
Figure 14, it can be found that when
N = 200,
N = 400 and
N = 800, the degraded feature sequences based on kurtosis do not exhibit obvious regularity. Although the degradation feature sequence based on kurtosis shows a downward trend when
N = 1000, this trend is not consistent with the evolution trend of vibration amplitude.
The above analysis shows that HGE presents an inherently more effective measure for characterizing the degradation feature of rolling bearings than HSE, HFE and kurtosis, and HGE also has the advantage of lower data length requirements. Therefore, HGE can be used to extract the degradation feature of rolling bearings.
The large time span caused by calculating the HGE of each degradation stage when the sample length is too long is not conducive to the timely mining of degradation information during the operation of rolling bearings. In this article, we set 200 data as a degradation stage (N = 200), and the vibration signal was divided into 40 degradation stages. The HGE of each degradation stage was calculated to obtain the degradation feature sequence of rolling bearings.
Due to the fact that a rolling bearing is in the optimal operation period for the first 4000 min, there is no practical significance to predicting its degradation during this stage. Therefore, a dynamic prediction was made for the degradation stage sequence corresponding to the last 4000 min (stage 20 to 40, a total of 21 degradation stages). The large sample sequence of 21 degradation stages predicted via the GBMC model is shown in
Figure 15. We sorted the data in the large sample sequence of 21 degradation stages from small to large and then grouped them, taking
Z = 10. We drew a histogram, shown in
Figure 16.
According to Equation (44), we calculated the estimated true values of 21 degradation stages to obtain the degradation feature prediction sequence. According to Equations (45)–(47), the estimated interval of the degradation stage sequence can be obtained where the significance level is set as 0.05, as shown in
Figure 17a. From
Figure 17a, it can be seen that the prediction sequence of the degradation stage generally shows a downward trend. Between 4000 and 6000 min, the prediction sequence of the degradation stage is almost consistent with the degradation stage sequence. Between 6000 and 8000 min, due to the continuous degradation of the bearing vibration performance, the extracted degradation stage values show a disorderly fluctuation state, but the prediction sequence still captures the continuous downward trend of the degradation stage. From
Figure 17a, it can also be found that the estimated interval of the degradation stage sequence almost includes all the degradation stage values, with only two degradation stage values falling outside the estimated interval. According to Equation (48), the prediction reliability
HB is 90.91%, indicating that the estimated interval can effectively track the evolution trend of the degradation stage sequence.
The dynamic uncertainty of the degradation stage sequence can be obtained according to Equation (49), as shown in
Figure 17b. From
Figure 17b, it is clear that the uncertainty shows a decreasing trend between 4000 and 5000 min, indicating that after slight degradation of the rolling bearing, it begins to enter a “self-healing” period and the uncertainty begins to decrease. Between 5000 and 5800 min, the uncertainty shows a stable fluctuation trend, indicating that the rolling bearing enters a brief period of stable operation after “self-healing”, and the uncertainty stabilizes at a small value stage; after 5800 min, the uncertainty shows a continuous upward trend, indicating further deterioration of the rolling bearing.
From the above analysis, it is clear that the evolution process of the degradation stage of rolling bearings is characterized from different perspectives by the estimated true value X0 (prediction value), estimated interval [XL, XU], prediction reliability HB, and dynamic uncertainty U. The estimated true value reflects the evolution trend of the degradation stage of rolling bearings, the estimated interval reflects the upper and lower boundaries of the prediction results, the reliability reflects the credibility of the prediction results and the uncertainty reflects the dynamic fluctuation range of the prediction results.
In order to verify the superiority of the proposed model, the grey bootstrap (GB) model [
43], which combines grey prediction theory and the bootstrap method, was used to predict the degradation stages, as shown in
Figure 18. From
Figure 18, it can be seen that although the prediction value of the degradation stage obtained via the GB model shows an overall downward trend, which partly reflects the evolution trend of the degradation stage, there are more true values of the degradation stage that fall outside the estimated interval than GBMC, and the prediction reliability
HB is only 86.36%, indicating that the estimated interval cannot effectively capture the range of the rolling bearing degradation stage.
For comparison, the AR method was also used to predict the degradation stage of the bearing, and the results are shown in
Figure 19. The performance of GBMC and AR was evaluated based on average absolute error and correlation coefficient, and the comparison results are listed in
Table 3. Consulting
Table 3, although the prediction errors of bearing degradation stages obtained via the AR method are slightly smaller than the GBMC, the correlation coefficient obtained via the GBMC is larger than that obtained via the AR method. It should also be pointed out that compared to the AR method, the GBMC has the advantage of dynamic evaluation of prediction results.
Through the above analysis, it is clear that compared with the GB model and AR method, the proposed GBMC method of this paper can effectively predict and evaluate the degradation stage of rolling bearing. This provides a new prediction and evaluation method for the online monitoring and operational performance evaluation of rolling bearings.
5.2. Case 2
To illustrate the universality of the proposed method, another accelerated life test was also conducted at the Hangzhou Bearing Test and Research Center, where the basic layout of the test rig was the same as in case 1.
Table 4 lists the detailed test conditions. The total number of sampling vibration data is 5600. The vibration data in this case are exactly the same as those of Experiment 3 in Reference [
39]. The vibration data did not undergo any preprocessing. The waveform of the vibration signal is shown in
Figure 20. From
Figure 20, it can be seen that the vibration amplitude of the bearing remains at a small value between 0 min and 400 min, indicating that the bearing was in the initial running in period; during 401–5000 min, the vibration amplitude of the bearing is in a stable fluctuation state, indicating that the bearing was in a normal operating period; and after 5000 min, the vibration amplitude of the bearing rapidly increases, indicating that the bearing entered a deterioration period.
HGE, HFE and HSE were all applied to analyze degradation stage with three different lengths:
N = 200,
N = 400,
N = 800, where
q = 3,
e = 0.2 SD,
f = 2, node = 3; the results are shown in
Figure 21. From
Figure 21, it is clear that for three different data lengths, the degradation feature sequences extracted based on HGE can accurately depict the degradation trend of bearing. In contrast, the degradation features extracted based on HFE and HSE cannot effectively reflect the degradation trend of bearings, especially when the data length of each degradation stage is short (
N = 200); the extracted degradation feature sequence shows an irregular state. The kurtosis values of each degradation stage were calculated, as shown in
Figure 22. From
Figure 22, it can be seen that for these three data lengths, the degraded feature sequences based on kurtosis did not exhibit significant regularity.
As in case 1, 200 data were taken as a degradation stage (
N = 200), and the vibration signal was divided into 28 degradation stages. We calculated the HGE of each degradation stage to obtain the degradation feature sequence of the rolling bearings. The dynamic prediction was made for the degradation stage sequence corresponding to the last 1600 min (stages 20 to 28, a total of 9 degradation stages). The large sample sequence of nine degradation stages predicted by the GBMC model is shown in
Figure 23. We sorted the data in the large sample sequence of nine degradation stages from small to large, and then grouped them, taking
Z = 10. We drew a histogram, shown in
Figure 24.
According to Equation (44), we calculated the estimated true values of nine degradation stages to obtain the degradation feature prediction sequence. According to Equations (45)–(47), the estimated interval of the degradation stage sequence was obtained, where the significance level was set as 0.05, as shown in
Figure 25a. From
Figure 25a, it can be seen that between 4000 and 5000 min, the prediction value of the degradation stage is almost consistent with the degradation stage value. Between 5000 and 5600 min, the predicted sequence captured the downward trend of the degradation stage. From
Figure 25a, it can also be found that the estimated interval almost includes all the degradation feature values, with only one degradation feature value falling outside the estimated interval. According to Equation (48), the prediction reliability
HB is 90%, indicating that the estimated interval can effectively track the evolution trend of the degradation stage. The dynamic uncertainty of the degradation stage sequence was calculated as shown in
Figure 25b. From
Figure 25b, it is clear that between 4000 and 5000 min, the uncertainty fluctuates steadily, indicating that the bearing was in a stable operating period and the uncertainty stabilized at a small value stage; after 5000 min, the uncertainty rapidly increased, indicating that the rolling bearing underwent serious deterioration.
The prediction results based on the GB model are shown in
Figure 26, from which it can be found that although the prediction value of the degradation stage obtained via the GB model shows an overall downward trend, reflecting to some extent the evolution trend of the rolling bearing degradation stage; the prediction reliability
HB is only 80%. The prediction results based on the AR method are shown in
Figure 27, and the comparison results are listed in
Table 5. To see
Table 5, although the prediction errors and the correlation coefficient of the bearing degradation stages obtained via the AR method are almost the same as that of the GBMC; the GBMC has the advantage of dynamic evaluation of prediction results.
Therefore, the above analysis indicates that the proposed GBMC-based degradation stage prediction method is superior to the GB-based and AR-based degradation stage prediction methods.
5.3. Case 3
The experimental data were the full life cycle vibration data of rolling bearings from the Center for Intelligent Maintenance Systems (IMS), University of Cincinnati [
44]. The test rig [
11] consisted of a spindle, four test bearings, an AC motor and a rub belt, as shown in
Figure 28. The test bearing was a Rexnord ZA-2115 double row bearing. The shaft was driven by an AC motor and coupled by rub belts. A radial load of 6000 lb was added to the shaft and bearing by a spring mechanism. The rotation speed was 2000 rpm. The accelerometer was installed on each bearing housing. The sampling frequency was 20 kHz, and each file contained 20 480 data. The collection interval between each two files was 10 min. The data collection started at 10:32:39 on 12 February 2004 and ended at 6:22:39 on 19 February 2004. A total of 984 files were collected, and the experiment lasted for 163 8 h. At the end, the outer ring of bearing 1 was seriously deteriorated. Due to the large amount of data in each package and the fluctuation range of magnitude being almost consistent, 10 representative data were selected from each package to form the bearing life data, the data length was 984 × 10 and the waveform of the vibration data is shown in
Figure 29.
It can be seen from
Figure 29 that the vibration amplitude of the rolling bearing is relatively stable before 7000 min and the fluctuation range is not large, indicating that the rolling bearing is in the normal operation period. From 7000 min, the vibration amplitude of rolling bearings begins to gradually increase, indicating that rolling bearing performance begins to deteriorate until it fails.
HGE, HFE and HSE were all applied to analyze degradation stage with three different lengths:
N = 200,
N = 400,
N = 800 and
N = 1000, where
q = 3,
e = 0.2 SD,
f = 2, node = 3; the results are shown in
Figure 30. From
Figure 30, it is clear that for four different data lengths, the degradation feature sequences extracted based on HGE can accurately depict the degradation trend of the bearing. Before 7000 min, the entropy value of HGE fluctuates smoothly with little change, while after 7000 min, the entropy value of HGE begins to gradually decline. The trend in HGE is consistent with the trend in rolling bearing vibration signal. In contrast, the degradation features extracted based on HFE and HSE cannot effectively reflect the degradation trend of bearings, especially when the data length of each degradation stage is short (
N = 200 and
N = 400); the extracted degradation feature sequence shows an irregular state. The kurtosis values of each degradation stage were calculated, as shown in
Figure 31. From
Figure 31, it can be found that when
N = 200 and
N = 400, the degraded feature sequences based on kurtosis do not exhibit obvious regularity. Although the degradation feature sequence based on kurtosis shows an upward trend when
N = 800 and
N = 1000, this trend is opposite to the degradation trend of rolling bearings.
As in case 1 and case 2, 200 data were taken as a degradation stage (
N = 200), and the vibration signal was divided into 49 degradation stages. (The last degradation stage contains 240 data.) We calculated the HGE of each degradation stage to obtain the degradation feature sequence of the rolling bearings. The dynamic prediction was made for the degradation stage sequence corresponding to the last 6000 min (stage 20 to 49, a total of 30 degradation stages). The large sample sequence of 30 degradation stages predicted via the GBMC model is shown in
Figure 32. We sorted the data in the large sample sequence of 30 degradation stages from small to large and then grouped them, taking Z = 10. We drew a histogram, shown in
Figure 33.
According to Equations (44)–(47), the estimated true value and the estimated interval of the degradation stage sequence can be obtained, where the significance level was set as 0.05, as shown in
Figure 34a. From
Figure 34a, it can be seen that between 4000 and 7000 min, the predicted sequence can capture the smooth fluctuation trend of the degradation stage. Between 7000 and 10,000 min, the predicted sequence can capture the downward trend of the degradation stage. From
Figure 34a, it can also be found that the estimated interval almost includes all the degradation feature values, with four degradation feature values falling outside the estimated interval. According to Equation (48), the prediction reliability
HB is 83.87%, indicating that the estimated interval can effectively track the evolution trend of the degradation stage. The dynamic uncertainty of the degradation stage sequence was calculated as shown in
Figure 34b. From
Figure 34b, it is clear that between 4000 and 7000 min, the uncertainty stabilizes at a small-value stage, indicating that the bearing was in a stable operating period; after 7000 min, the uncertainty rapidly increased, indicating that the rolling bearing underwent serious deterioration.
The prediction results based on the GB model are shown in
Figure 35, from which it can be found that although the prediction value of the degradation stage obtained via the GB model shows an overall downward trend, reflecting to some extent the evolution trend of rolling bearing degradation stage, the prediction reliability
HB is only 77.42%. The prediction results based on the AR method are shown in
Figure 36, and the comparison results are listed in
Table 6. Consulting
Table 6, although the prediction errors and the correlation coefficient of bearing degradation stages obtained via the AR method are almost the same as that of the GBMC, the GBMC has the advantage of dynamic evaluation of prediction results.
Therefore, the above analysis indicates that the proposed GBMC-based degradation stage prediction method is superior to the GB-based and AR-based degradation stage prediction methods.