# Weak Fault Detection of Tapered Rolling Bearing Based on Penalty Regularization Approach
## 4 Experimental Verification  
To demonstrate the validity of the proposed approach in engineering applications, the tapered rolling bearing with a faulty outer race is implemented. Experimental vibration data were collected from several accelerometers instrumented on a bearing end bracket from a large reducer, as shown in Figure 9. The geometrical parameters of the tested tapered bearing are listed in Table 1, and the fault frequency of the bearing outer race is 118.8 Hz. The sampling frequency is 51.2 KHz (right side of input shaft) and 5.12 KHz (left side of input shaft), the rotation frequency is 16.67 Hz and sampling length is 6.4 s. The tapered rolling bearing (FAG-32212-A) fault was located on the right side of the input shaft. In this experiment, the vibration data collected from the right side of input shaft (horizontal direction, i.e., non-drive end) were used in the first experiment. To create a more challenging condition, the vibration data collected from the left side of the input shaft (horizontal direction, drive end) were also analyzed in the second experiment. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Bearing Type** & **Fault Type** & \begin{tabular}{c} **Number of** \\ **Balls** \\ \end{tabular} & \begin{tabular}{c} **Inner** \\ **Diameter** \\ \end{tabular} & \
\begin{tabular}{c} **Outer** \\ **Diameter** \\ \end{tabular} &
\begin{tabular}{c} **Fault** \\ **Frequency** \\ \end{tabular} \\ \hline FAG-32212-A & Outer race & 9 & 60 mm & 110 mm & **118.8 Hz** \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The geometrical parameters of the tested tapered rolling bearing. Figure 8: The results of sparse Fourier coefficients using the LFLO and the proposed AHNPR method. (**a**) The result of the LFLO method; (**b**) the result of the proposed AHNPR method. The first experiment had two tasks, i.e., to estimate the weak fault signal from the original vibration signal and to calculate the sparsity of the sparse coefficient. The raw vibration signal (right side of input shaft, 4096 sampling points were selected, i.e., 0.08 s) and its envelop spectrum are displayed in Figure 10a,b, respectively. As shown in Figure 10b, although the spectrum peak at 237.5 Hz, which is consistent with the twice outer-race fault frequency, can be detected; however, the spectrum peak is masked by heavy background noise and frequency features are not clear enough to detect practical fault. Figure 10: Original vibration signal and its envelope spectrum. (**a**) Original vibration signal; (**b**) the envelope spectrum of the original vibration signal. Figure 9: Experimental set up for the tapered bearing test. The proposed AHNPR approach and the LFLO method are respectively employed to process the measured vibration signal and the obtained results are displayed in Figure 11. The parameter \(\tau\) was set to \(\tau\) = 0.8 and matrix \(A\) is also defined as \(\mathbf{A}_{m,n}=(1/\sqrt{N})\exp(j(2\pi/N)mn),m=0,1,\cdots,M-1;\ n=0,1,\cdots,N-1\). Thus, matrix \(B\) is \(\mathbf{B}=\sqrt{0.8/\lambda A}\). It can be calculated that \(\mathbf{B}^{\top}\mathbf{B}\) is a non-diagonal matrix, thus the augmented Huber function (AHF) \(\phi_{B}(x)\) is non-separable. Here, it should be noted that the healthy data are not available or are unknown, according to Equations (23) and (24). Taking the constant parameters to be \(\beta_{0}\) = 0.7 and \(\gamma\) = 7.5, the parameters of the proposed method can be obtained as follows: the standard deviation \(\sigma\) = 4.234, and regularization term parameters \(\lambda\) = 7.5 \(\times\) 0.7 \(\times\) 4.234 = 22.23. Comparing results with the time-domain waveform and envelope spectrum, one can observe improvements at two levels: the first is the significant enhancement in the contribution of periodical characteristic. The periodical characteristic with LFLO is weaker than that related to the proposed method. The second improvement is that the amplitudes of fault frequency and its harmonic are worthy of further enhancement.