5.1. Case 1: Experimental Gearbox Data Analysis
The proposed method was adopted to analyze gear vibration data collected from the laboratory of testing technology and fault diagnosis, North China Electric Power University (NCEPU). The experimental gear fault device was mainly composed of a driving motor, bearing, gearbox, shaft, turntable and governor.
Figure 13a,b show photos of the experimental gear fault device and a gearbox structure drawing, respectively. In the experiment, gear vibration data were collected by using the accelerometer installed on the housing of the reduction gearbox with a sampling frequency of 5120 Hz. The experimental data collection system was mainly composed of an accelerometer, cable conductor, amphenol connector, signal conditioner, acquisition card and acquisition software, where the type of the data acquisition card was ADA16-8/2 (LPCI) with a single-terminal 8-channel input and 2-channel output. The motor speed could be adjusted by looking at the tachometer and turning the speed control knob. In addition, gear loading could be adjusted by switching on the brake and setting the level of braking torque. The specific steps can be found in the operating instructions of vibration analysis and the fault diagnosis test platform system for rotating machinery of QPZZ-II. The experimental gearbox was made up of two parts (i.e., the pinion and big gear). The pinion had 55 teeth, whereas the big gear had 75 teeth. In this experiment, the gearbox operated under five health conditions, including normal (condition 1), big gear pitting fault (condition 2), big gear fracture fault (condition 3), big gear pitting and pinion wear compound fault (condition 4), big gear fracture and pinion wear compound fault (condition 5). In addition, in this experiment, through speed adjustments, the motor operated at a rotating speed of about 800 rpm, but the actual speed and environmental interference under different health conditions differed somewhat, which indicates that the amplitude of the healthy condition may be greater than that of the unhealthy condition at a certain time point: the rotating frequencies of the small gear and big gear can be approximatively inferred as
fr1 = 13.3 Hz and
fr2 = 9.8 Hz, respectively. To verify the proposed method, 50 sets of gear vibration data under each health conditions were collected and each gear vibration signal consisted of 4096 data points. The training:testing data proportion was 1:1, i.e., the number of training samples and testing samples was the same, which was 125.
Table 2 details the gear health conditions and sample selection. The time domain waveforms and amplitude spectra of gear vibration signals under different health conditions are shown in
Figure 14. Notably, the plotted gear vibration signal belongs to the standardized results. The standardized formula is expressed as
x = (
x − mean(
x))/std(
x), where
x is the collected original gear vibration signal, mean(
x) is the mean value of
x and std(
x) is the standard deviation of
x. As can be seen from
Figure 14, the waveforms and spectra in different gear health conditions have certain similarities, especially for condition 3, condition 4 and condition 5, which implies that an effective method should be adopted to identify them. In order to facilitate the understanding, the identification performance of the proposed method was compared and analyzed from the following several aspects:
(1) The proposed method was utilized to analyze the collected gear vibration data. According to the flowchart of the proposed method, MHCO was first used to process different gear fault signals, where the optimal SE scale of MHCO was determined as 8 by using SCFNR. Notably, in this experimental data analysis, the search range of the flat SE scale
was set as 1 to
, where
fs is the sampling frequency,
fr is the rotating frequency of the input or output shaft and
denotes the down round operation. Due to the rotating frequency of the output shaft,
fr2 = 9.8 Hz is smaller than that of the input shaft, i.e., when the maximum SE scale
=
, fault signatures of different gear health condition can all be covered. Hence, in experimental case 1, the flat SE scale
.
Figure 15 plots the filtered results obtained by MHCO for different gear fault signals. Subsequently, the GCMLZC of all data samples was calculated for fault feature extraction. For analysis,
Figure 16a,b show the GCMLZC of one data sample for different gear health conditions before and after morphological convolution filtering, respectively. In the GCMLZC method, without a loss of generality, the largest scale factor
is set as 20. As can be seen from
Figure 16, GCMLZC with morphological convolution filtering has a better differentiation than GCMLZC without noise reduction. This proves the necessity of morphological convolution filtering in fault identification. Finally, the extracted GCMLZC was fed into the softmax classification model for automatically identifying different gear health conditions.
Figure 17 shows the identification results of the proposed method for the first trial. Seen from
Figure 17, the identification accuracy rate of the proposed method reached 98.4%, which indicates that only two data samples were misidentified; therefore, the proposed method is preliminarily proven to be effective in identifying gear fault types.
(2) To further verify the validity of the proposed method, comparisons among the proposed method and four representative complexity indexes (i.e., MLZC, multiscale dispersion entropy (MDE) [
31], multiscale permutation entropy (MPE) [
32] and multiscale sample entropy (MSE) [
33]) were performed. To avoid randomness in the identification results of different methods and to ensure a fair comparison, all methods were preprocessed by the same morphological filtering (MHCO), the largest scale factor
of all methods (i.e., GCMLZC, MLZC, MDE, MPE and MSE) were set as 20 and 10 trials were conducted. In addition, in the MDE method, the embedded dimension
m = 3, time delay
d = 1, the number of classes
c = 5. In the MPE method, the embedded dimension
m = 3 and time delay
d = 1. In the MSE method, the embedded dimension
m = 3 and the similarity tolerance
, where
SD is the standard deviation of the analyzed signal.
Figure 18 shows the identification accuracies of different methods in 10 trials. In addition,
Table 3 gives the detailed identification results of different methods, including the maximum, minimum and mean identification accuracy. Seen from
Figure 18 and
Table 3, the average identification accuracy (98.24%) of the proposed method was bigger than that of other methods (i.e., MLZC, MDE, MPE and MSE), whereas the standard deviation (0.3373) of the proposed method was lower than that of other methods, which means that the identification ability and stability of the proposed method are superior to other methods mentioned in this paper. Therefore, the effectiveness of the proposed method in gear fault identification is further validated by the above comparison.
(3) To consolidate the fault identification results, the fivefold cross-validation method was also applied to analyze the same gear vibration signal. Concretely, the data sample was first divided into five parts (each part had 50 samples), where four parts (i.e., 200 samples) were alternately regarded as the training samples and the remaining part (i.e., 50 samples) served as the testing sample. Next, five trials of different methods were performed, and the average identification accuracy values of five results were regarded as the ultimate identification accuracy of different methods.
Table 4 gives the detailed diagnosis results obtained by different methods. As shown in
Table 4, the proposed method achieved an average identification accuracy of 98.80%, whereas other complexity methods (i.e., MLZC, MDE, MPE and MSE) obtained 96.40%, 97.60%, 94.40% and 86.40% accuracy, respectively. The identification accuracy of the proposed method is clearly higher than that of other comparison methods. Consequently, the effectiveness and superiority of the proposed method is demonstrated once again.
5.2. Case 2: Engineering Data Analysis for Wind Turbine Gearbox
In this section, the proposed method was adopted to analyze the practical vibration data from a 1.5 MW wind turbine gearbox, which is located on a wind farm in northern China.
Figure 19 shows a structural diagram of the wind turbine transmission system, which mainly consisted of a vane, spindle, rotor, gearbox and generator. The analyzed wind turbine gearbox adopted three-stage transmission (i.e., planetary stage, middle stage and high-speed stage), and was an FD1660 type. The rated power of the wind turbine gearbox was 1660 KW, and the weight of the gearbox was approximately 16,800 kg. In addition, the generator speed could be adjusted by using the electrical control system of the wind turbine.
Table 5 lists the teeth numbers of each stage gear of the wind turbine gearbox, where Z
0 denotes the teeth number of the planet gear, Z
1 denotes the teeth number of the sun gear, Z
2 represents the teeth number of the inner ring gear, Z
3 and Z
5 are the teeth numbers of the big gear and small gear in the middle stage, respectively, and Z
4 and Z
6 are the teeth numbers of the big gear and small gear in the high-speed stage, respectively. In engineering data analysis, gear vibration data were collected by an accelerometer (see
Figure 19) glued onto the casing of the gearbox with a sampling frequency of 32,768 Hz. The wind turbine gearbox operated under four gear health conditions (i.e., normal, pitting fault of small gear in middle stage, spalling fault of big gear in high-speed stage, fracture and wear compound fault small gear in high-speed stage). When gear vibration data collection was conducted for each health condition, the wind speed was stable at about 12 m/s (corresponding to an input shaft speed of about 17 rpm and a power of about 1500 kW), and the speed of the high-speed shaft was stable at about 1400 rpm. Thus, the rotating frequencies of the high-speed shaft and middle shaft can be approximatively calculated as
fh = 23.33 Hz and
fm = 6.27 Hz, respectively.
Figure 20 shows photographs of three gearbox faults. In the process of method validation, we obtained 100 data samples of each health condition. For each health condition, 50 samples were randomly selected as the training data, and the remainder was regarded as testing data. A total of 200 training and 200 testing samples were obtained, and each sample had 16,384 points. Apparently, it is a four-classification issue to be solved in essence.
Table 6 presents detailed information of wind turbine gearbox data.
Figure 21 shows the time domain waveform and amplitude spectrum of the gear vibration signal under different health conditions. Seen from
Figure 21, gear fault conditions are difficult to be identified directly through observing the time domain waveform and amplitude spectrum, because different gear vibration data have certain self-similarity. Therefore, it is necessary to adopt an effective method to process the practical gearbox data.
According to the flowchart in
Figure 12, the proposed method was adopted to analyze the practical gearbox data. In the proposed method, based on the SCFNR indicator, the optimal SE scale of MHCO was selected as 10. Similar to case 1, the search range of the flat SE scale
was set as 1 to
, where
fs is the sampling frequency,
fr is the rotating frequency of the high-speed shaft or middle shaft and
denotes the down round operation. The rotating frequency of the middle shaft
fm = 6.27 Hz was smaller than that of the high-speed shaft, i.e., when the maximum SE scale
=
, fault signatures of different gear health condition can all be covered. Hence, in experimental case 2, the flat SE scale
. Due to space limitations, the corresponding parameter optimization diagram is not included here.
Figure 22 shows the filtered signals of three gear faults. For fault feature extraction, we calculated the GCMLZC of the filtered signal of all samples. Similar to case 1, in the GCMLZC method, the largest scale factor
was selected as 20.
Figure 23a,b show the GCMLZC of gear vibration signals before and after applying the MHCO method, respectively. As shown in
Figure 23, after morphological convolution filtering, the degree of distinction of four gear health conditions is greater than that without filtering processing. This verifies the importance of morphological convolution filtering for signal preprocessing. Finally, the extracted GCMLZC was input into the softmax model for fault pattern identification.
Figure 24 shows the identification results of the proposed method in the first trial. As shown in
Figure 24, only one sample was misclassified, which indicates that the proposed method can obtain an identification accuracy of 99.5% (199/200). Thus, the proposed method exhibits good recognition performance for wind turbine gearbox faults.
Similarly to case 1, to further prove the effectiveness of the proposed method, the identification abilities of five methods (i.e., GCMLZC, MLZC, MDE, MPE and MSE) were compared. Similarly, 10 trials of different methods were conducted to ensure the fairness of the comparison results. In addition, in all comparison methods (i.e., GCMLZC, MLZC, MDE, MPE and MSE), without a loss of generality, the largest scale factor
was set as 20. In the MDE method, the embedded dimension
m = 3, time delay
d = 1 and the number of classes
c = 5. In the MPE method, the embedded dimension
m = 3 and time delay
d = 1. In the MSE method, the embedded dimension
m = 3 and the similarity tolerance
, where
SD is the standard deviation of the analyzed gear vibration signal.
Figure 25 shows the fault identification accuracy of different methods in 10 trials, and
Table 7 gives the detailed comparison results of different methods. Seen from
Figure 25 and
Table 7, the average identification accuracy (99.35%) of the proposed method was higher than that of other four methods (i.e., MLZC, MDE, MPE and MSE). In addition, standard deviation (0.2415) of the proposed method was less than that of other methods. This again indicates that the superiority of the proposed method in identifying wind turbine gearbox faults is verified.
To further consolidate the fault diagnosis results of wind turbine gearbox, we also used the fivefold cross-validation method to analyze the practical wind turbine gearbox data.
Table 8 shows the detailed fault identification results obtained by different methods in the fivefold cross-validation. As can be seen from
Table 8, the proposed method obtained an average identification accuracy of 99.50%, which is greater than that of the other comparison methods (i.e., MLZC, MDE, MPE and MSE). In other words, the proposed method has a stronger fault discriminant ability. This further proves that the proposed method is effective in extracting fault features from wind turbine gearboxes and identifying different gear fault categories.