## 5 Background Noise Elimination Method
### Background Elimination and Comparative Analysis  
According to the conclusion given in Section 4, the double-teeth spalling fault feature are the most clear when the two teeth are adjacent. For the sake of discussion, the experimental signal when the two spalling faulty teeth are adjacent is processed, and the experimental conditions are listed in Table 2.  
Due to the background noise, the periodic impulse cannot be recognized in the time-domain signal, and the faulty feature frequency is also not clear, as shown in Figure 12. In other words, the background noise in the experimental signal leads to the recognition effect of the double-teeth spalling fault feature, and then affects the accuracy of dynamic model validation. Therefore, the identification of double-teeth spalling fault feature from the noisy signal is of great significant. First, the proposed ALO-VMD method is applied to process the experimental signal. To verify the optimization performance of ALO, the standard particle swarm optimization (PSO) is tested and compared. Here, the population of the ants is set to 30, and the maximum iterations is set to 10, the range of decomposition layer \(K\) = [2,11], and the range of penalty factor \(\hat{a}\) = [50,8000]. Figure 19 and Table 3 show the comparison results of ALO and PSO. The results show that the optimum speed and optimization accuracy of the ALO algorithm is better than that of the PSO algorithm. Then, the optimal parameters K = 3 and \(\hat{a}\) = 1626 are obtained by ALO. As the result, three components and their corresponding EWK are obtained, as shown in Table 4. Since the EWK values of u1 and u2 are larger than zero, they are determined as sensitive components and selected as a reconstructed signal, while other modes are noisy modal components. Subsequently, the reconstructed signal is demodulated by the Hilbert envelope and the envelope spectrum are shown in Figure 20. Compared with Figure 12c, the periodic impulse vibration, which was excited by the spalling fault can be clearly observed in Figure 20a. The period of the impulse vibration is 0.04 s, which is equal to the rotating period of the pinion. In addition, in a rotation period of the pinion, the impulse vibration excited twice, and the interval time of the two impulse vibrations is close to the single tooth meshing period. From the envelope spectrum, as shown in Figure 20b, the faulty feature frequency _fs_ and its multiplications (_2 fs_, _3 fs_, _4 fs_, and _5 fs_) can be clearly observed in the spectrum of the reconstructed signal. The results are consistent with the previous simulated results mentioned in Section 4, and it is demonstrated that the fault feature is extracted effectively by the proposed ALO-VMD method. For comparisons, the PSO-VMD, the fixed-parameter VMD, the EMD, and the LMD are employed to process the same signal, and the results are displayed in Figure 21. The optimal \(K\) and \(\hat{a}\) in PSO-VMD are 3 and 3580, respectively, and the \(K\) and \(\hat{a}\) in the fixed-parameter VMD are set as 4 and 2000, respectively. Similar to the ALO-VMD method, the reconstructed signal of other methods is also selected according to the value of the EWK index. For clarity, the visible abscissa range of the spectrum is set to 0-300 Hz. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Component** & **u1** & **u2** & **u3** \\ \hline EWK & 0.286 & 0.605 & \(-\)0.891 \\ Selection & \(\surd\) & \(\surd\) & \\ \hline \hline \end{tabular}
\end{table} 
Table 4: The efficient weighted kurtosis (EWK) value of each component. Figure 20: The results of an experimental signal by the proposed ALO-VMD method: (**a**) The time-domain signal and (**b**) the envelope spectrum. As shown in Figure 21, only the faulty feature frequency _fs_ and its multiplication 2_fs_ can be identified in the spectrums of the LMD, and three fault feature frequencies fs, 2_fs_, and 4_fs_ are displayed by the EMD. In contrast, five fault frequencies are existing in the envelope spectrums of fixed-parameter VMD, PSO-VMD, and the proposed ALO-VMD.