# Intelligent Diagnosis of Compound Faults of Gearboxes Based on Periodical Group Sparse Model
## 4 Experimental Verification and Analysis  
### PHM Data Challenge Fault Dataset  
Gearbox failure data from the 2009 PHM Data Challenge dataset. These vibration data represent the operating condition of a typical industrial gearbox. The gears on the input and output shafts have 32 and 80 teeth, respectively. The two gears on the intermediate shaft have 96 and 48 teeth, respectively. The primary and secondary ratios are 3 and 1.67, respectively, and one sensor is installed on the input and one on the output side of the gearbox for data acquisition. The gearbox structure used for data collection is shown in Figure 8. In this paper, the performance of the proposed algorithm is tested by using the data set of spur gears with the output side sensor with a significant coupling modulation of the output side signal. Figure 6: Constructed simulation signal of bearing-gear: (**a**) bearing fault signal; (**b**) gear fault signal; (**c**) simulation signal of compound fault of bearing-gear including noise; (**d**) the envelope spectrum of the bearing-gear compound fault. Figure 7: Decoupled two fault signals: (**a**) time domain of bearing fault signal; (**b**) envelope spectrum of bearing fault signal; (**c**) time domain of gear fault signal; (**d**) envelope spectrum of gear fault. The fault data set contains two groups as shown in the green area (group A) and red area (group B) in Figure 8. Group A contains three fault types: input shaft unbalance, intermediate shaft rolling body fault (input side), and output shaft outer ring fault (input side). Group B contains six fault types: input shaft unbalance, input shaft inner ring fault (input side), intermediate shaft rolling body fault (input side), output shaft outer ring fault (input side) and output shaft 80 gear broken teeth. The bearing parameters are the following: the rolling body diameter is 7.94 mm, the bearing pitch diameter is 33.5 mm, and the rolling body number is 8. The sampling frequency is 66.67 KHz. The input shaft speed of 1800 rpm/min in the data set of group A was selected and the fault frequencies were calculated to be 30 Hz, 19.92 Hz and 18.31 Hz. The composite fault signal is shown in Figure 9. The fault features are submerged except for the rotational frequency component. The proposed algorithm is used to decouple the compound fault signals. The root mean square error (RMSE) is shown in Equation (8). The RMSE indicates the ability to maintain the amplitude of the features. \(\lambda\) and \(\rho\) balance the inter-group sparsity and intra-group sparsity, respectively. The RMSE of the two parameters at different ranges are shown in Figure 10, and \(\lambda=\rho=1\times 10^{-4}\) is chosen. Since the noise estimation intensity is generally consistent for both data sets, the same parameters are set for the following categories of operating conditions. \[\text{RMSE}=\sqrt{\frac{1}{\text{N}}\Big{(}\|\text{x}-\text{f}(\text{x})\|_{2 }^{2}\Big{)}} \tag{28}\]  
The decoupled vibration signals of different states are cut off into segments of 950 points each in length. Each segment is used as a sample. A total of 800 sets of data are selected from the group A fault data set for analysis, where 400 sets are randomly selected as the training data and the remaining 400 sets are used as the test data. The wavelet packet energy features of different states are extracted, and the dimension of sample data is changed from 950\"400 to 32\"400. A total of 1440 sets of data are selected from the failure data set B for analysis, where 720 sets are randomly selected as the training data and the remaining 720 sets are used as the test data. After extracting wavelet packet energy features for different health states, the sample data dimension is transformed from 950\"720 to 32\"720. When using the proposed algorithm to identify and classify different health states of group A, each parameter is set: the number of atoms of the sub-dictionary corresponding to each category is 80 and the sparsity is 1.