# Intelligent Diagnosis of Compound Faults of Gearboxes Based on Periodical Group Sparse Model
## 3 Simulated Signal Analysis  
The proposed method is compared and analyzed with the resonance decomposition approach based on the l1-parametric penalty term to verify the effectiveness of the proposed algorithm. In this section, two simulations are used to verify the analysis. First, a simulation signal containing two different fault components is constructed to show the intra-group sparsity of a single fault and the inter-group sparsity between different faults after decoupling the proposed model. The simulation signal with two fault components of 55 Hz and 30 Hz is shown in Figure 4. In Figure 4a, the apparent periodic component cannot be found. From Figure 4c, it can be seen that the 55 Hz fault feature and its multiplication are obvious, and the 30 Hz component has been drowned out by the noise. The decoupling separation is performed through the use of RSSD in reference [11] and the method proposed in this paper, and the processing results are shown in Figure 5. The resonance decomposition, as a typical sparse model based on the l1-parametric penalty term, can only keep good inter-group sparsity between single faults since there is no constraint effect of the l2 norm penalty term fails to produce good sparsity among different faults. From Figure 5a,c, it can be seen that the fault frequencies and their multiplicities are not obvious and are affected significantly from the disturbance components. Figure 5b,d show the performance difference between the sparse model based on the periodic group and the sparse model based on the l1 norm. Two different fault features are clearly extracted by the decomposition of the proposed method, and the two fault components maintain good inter-group sparsity, while the within-group sparsity between single fault components is good. In addition, the magnitude of the decoupled signal is not underestimated due to the weighting effect of the vanes in the penalty term, which gives the model good amplitude preservation and maintains a good reconstruction accuracy. Figure 4: Simulated signals of two different components: (a) time domain signals; (b) frequency domain signal; (c) envelope spectrum. Subsequently, a compound fault simulation signal containing a bearing fault \(x_{1}(t)\), a gear fault \(x_{2}(t)\) and a Gaussian white noise \(n(t)\) is constructed as follows:  
\[x(t)=x_{1}(t)+x_{2}(t)+n(t) \tag{25}\]  
\[x_{1}(t)=\sum_{k}a_{k}h_{1}(t-kT_{1}-\tau_{k})\ \ h_{1}(t)=\mathrm{e}^{-2\xi\pi f _{n1}t}\sin\left(2\pi f_{n1}\sqrt{1-\xi^{2}t}\right) \tag{26}\]  
\[x_{2}(t)=\sum_{m=0}^{M}A_{m}\left[1+\sum_{k=0}^{K}A_{m,k}\cos(2\pi kf_{r}t+ \alpha_{m,k})\right]\cos(2\pi mf_{z}t+\beta_{m}) \tag{27}\]  
where, the pulse amplitude of the bearing fault \(a_{k}=1\), the bearing fault period \(T_{1}=1/70\), resonance frequency \(f_{n1}=2000\), damping ratio \(\xi=0.02\); the amplitude of the gear fault signal \(A_{m}=0.6\), the rotation frequency of the shaft where the gear is located \(f_{r}=15\) Hz, the gear meshing frequency \(f_{z}=200\) Hz, and the phase of the order meshing frequency \(\beta_{m}\) is 0. The \(k\)th amplitude \(A_{m,k}\) and phase \(\alpha_{m,k}\) of the \(m\) order meshing frequency is 0; add a noise with a variance of 0.5. The simulation signals of the composite fault are shown in Figure 6. Figure 6a,b show the fault signals of the bearing and gear, respectively. Figure 5c shows the simulated signal of the composite fault containing noise, from which no significant periodic pulse is found. The envelope spectrum of Figure 6d reveals only the meshing frequency \(f_{m}\) and the rotational frequency \(f_{r}\), with no bearing-related fault information. The proposed method is used to decouple the composite fault signal. Where, the values of \(N_{0}\) and \(N_{1}\) are set to 4 as recommended in the reference [22]. The processing results are shown in Figure 7. As shown in the time domain plots in Figures 6c and 7a, the periodic fault pulse components of the bearings and gears are very obvious.