## 6 Conclusions  
In this paper, a dynamic model of a gear system with double-teeth spalling fault is presented with considering the influence of TVMS, transmission error, and bearing support stiffness. The effectiveness  
Figure 21: The envelope spectrum of five decomposition methods for the experimental signal.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Method** & **ALO-VMD** & **PSO-VMD** & **Fixed-Parameter VMD** & **EMD** & **LMD** \\ \hline SNR & 10.464 & 9.916 & 9.854 & 9.217 & 9.101 \\ RMSE & 1.577 & 1.669 & 1.772 & 1.964 & 2.004 \\ \hline \hline \end{tabular}
\end{table}
Table 5: The efficient weighted kurtosis (EWK) value of each component.  
of the proposed dynamic model is validated by analyzing experimental results. The effect of the number of spalling teeth, and the relative position of the two faulty teeth are investigated. Both of the experimental tests and dynamic simulation results have shown that the double-teeth spalling fault will generate impulse vibration twice in one rotating period of the fault gear, and the interval of the two impulses is related to the number of healthy teeth, which can be expressed as \((H+1)t_{m}\). Moreover, the features of double-teeth spalling fault are significant when the two spalling teeth are adjacent.  
In addition, a novel parameter-adaptive VMD method based on the ALO is proposed to eliminate the background noise from the experimental signal for ensuring the accuracy of dynamic model verification results and reducing the difficulty of fault feature analysis. The optimal parameters \(K\) and \(\hat{a}\) is obtained by the ALO, and the raw signal obtained from experiments is decomposed by using the ALO-VMD. According to the EWK index, several signal components whose value is greater than zero are summed as a reconstructed signal. The reconstructed signal is then demodulated by the Hilbert envelope, and the fault features are analyzed by the envelope spectrum. The results confirm that, (1) the ALO has a good optimization ability and can provide the VMD with the optimal parameters \(K\) and \(\hat{a}\), (2) by comparing with the PSO-VMD, fixed-parameter VMD, EMD, and LMD, it can be found that the proposed ALO-VMD method is more effective in eliminating the noise than other methods.  
Compared with References [19; 22; 23], the effect of tooth fillet foundation deflection is not considered in the proposed gear model. As reported in Reference [54], an equation for the fillet foundation deflection is valid only when the normal stress at the tooth root is linearly distributed and confined usable for large gears. However, due to the limitation of experimental conditions, the width of the fillet foundation is greater than that of teeth of the spalling pinion in this paper. This may be a limitation of the proposed model that needs to be further studied. Moreover, the spalling effects of random distribution are also worthy of further investigation.  
Methodology, Validation, Writing--original draft, L.S. Conceptualization, J.W. Funding acquisition, B.P. Writing--review and editing, J.W. and Y.X. Resources, Q.Z. Software, C.L. All authors have read and agreed to the published version of the manuscript. This research was funded by the National Natural Science Foundation of China (Grant 51475425).  
The authors sincerely thank Shuanghuan Driveline Co., Ltd. for providing the necessary facilities and machinery to build the prototype of the gear system.  
The authors declare no conflict of interest.