# Weak Fault Detection of Tapered Rolling Bearing Based on Penalty Regularization Approach
## 4 Experimental Verification
Note that some other unrelated components in Figure 11d are greatly reduced compared with the envelope spectra of the raw signal and the de-noised signal generated by the LFLO method. Figure 11: _Cont._  
In addition, the distribution of sparse coefficients generated by LFLO and the proposed AHNPR method are given in Figure 12a,b, respectively. One can see that the sparse coefficients are underestimated by the LFLO method along with the sparse coefficients associated with additive noise that reduce sparsity. In comparison with the LFLO result, the sparse coefficients of the proposed AHNPR method are enhanced and amplified. Figure 11: The results of LFLO and the proposed AHNPR method. (**a**) The time-domain waveform generated by LFLO; (**b**) envelope spectrum of the results based on the LFLO method; (**c**) the time-domain waveform generated by the proposed AHNPR method; (**d**) envelope spectrum of the results based on the proposed AHNPR method. Figure 12: The optimized coefficients of the de-noised signal generated by LFLO and the proposed AHNPR method. (**a**) The result using the LFLO method; (**b**) the result using the proposed AHNPR method. As to the second experiment, Figure 13a,b present the raw vibration signal on the drive end (i.e., the left side of the input shaft, 4096 sampling points were selected, i.e., 0.8 s) and its envelope spectrum. In this case, the outer race fault frequency cannot be observed. This may be caused by the fact that the outer race fault is quite weak and the transfer path of the accelerometers located on the left side are longer than in the previous experiment. As such, the proposed AHNPR algorithm is applied to separate the fault feature from other unrelated components and the results are displayed in Figure 13c,d. The parameters of the proposed method can be obtained as follows: the standard deviation \(\sigma\) = 4.44, regularization term parameter \(\lambda\) = 7.5 \(\times\) 0.7 \(\times\) 4.4 = 23.31, and parameter \(\tau\) is set to \(\tau\) = 0.8. The outer race fault frequency and its harmonics are detected in the corresponding envelope spectrum. In addition, interestingly, the sidebands are distributed in both fault frequency and its harmonics, after calculating the difference between the fault frequencies and sidebands. It should be noted that the difference values are consistent with the rotational frequency (16.67 Hz). Therefore, the outer race fault of bearings can be more clearly diagnosed. The above result demonstrates that the proposed algorithm can detect the bearing fault and eliminate the background noise interference, regardless of whether the accelerometers are located on the right or left side of the input shaft. Figure 13: _Cont._  
In previous research [35], another non-convex penalty regularization approach, namely the asymmetric convex penalty regularization (ACPR) method, was proposed for weak fault detection for gearboxes. To explore the advantage of the proposed approach over the ACPR method, the diagnosis results under same operation environments are compared, the ACPR method is introduced to process the same data that collected from the right side and left side of the input shaft, respectively, and the rules of parameter settings are based on the reference [35]. The sparse component of right-side of input shaft generated by the ACPR method is shown in Figure 14a, and its envelop spectrum is shown in Figure 14b. Likewise, the sparse component of left-side of input shaft generated by the ACPR method is presented in Figure 14c, and its envelop spectrum is presented in Figure 14d. From the time-domain waveform of Figure 14a,b, the separating effect generated by the ACPR method is better than the results that generated by proposed method, and the noise components are decimated drastically, but the fault frequency of the bearing outer race and its harmonic cannot be extracted using the ACPR method, as shown in Figure 14b,d.