# Fault Diagnosis of Rolling Bearings Based on Improved Kurtogram in Varying Speed Conditions
## 3 Simulations
### Simple Simulation of the Influence of Multiple Impact Sources  
To match the simulation closer to the real situation, a deterministic component and two random shocks are added to a fault impulse signal, and a considerable amount of Gaussian noise is added too. The simulation signal parameters are given in Table 3. The different components and their frequency spectra are shown in Figure 12.  
The FK, WPTK and Protrugram are applied to analyze the mixed signal, and the results are shown in Figures 13-15, respectively. It is worth noting that in the following paragraphs, a wavelet packet basis  
Figure 11: The results obtained by the FOSCK for processing the mixed signal with different resonant frequencies: (**a**) FOSCK, (**b**) the envelope of the band-pass filtered signal, (**c**) the resampling envelope signal of (**b**,**d**) the envelope order spectrum of (**c**).  
\begin{table}
\begin{tabular}{c c c c c c c c c c c} \hline \hline
**N (s)** & \(\mathbf{B_{n}}\) & \(\boldsymbol{\varphi_{m}}\) & \(\boldsymbol{f}\) (Hz) & \(\boldsymbol{f_{o}}\) & \(\boldsymbol{f_{s}}\) & \(\boldsymbol{f_{r1}}\) & \(\boldsymbol{f_{r2}}\) & \(\boldsymbol{\beta_{1}}\) & \(\boldsymbol{\beta_{2}}\) & \(\boldsymbol{\tau_{n}}\) & \(\boldsymbol{SNR}\) (dB) \\ \hline
1 & 1 & 0 & 10–15 & 4.5 & 20 & 5 & 7.3 & 1.2 & 3 & 0.01 & \(-5\) \\ \hline \hline \end{tabular}
\end{table}
Table 3: Parameters of the simulation model.  
Figure 10: Simple simulation: (**a**) the simulated signal, (**b**) the frequency spectra of (**a**), (**c**) the noise-added signal with SNR\(=-5\) dB and (**d**) the frequency spectra of (**c**).  
db10 is used in the WPTK as that given in [14], and a bandwidth (BW) that includes the 3 rd harmonic of the characteristic frequency is selected in the Protrugram according to the rules mentioned in [28]. The analysis results show that both the FK and the WPTK failed to detect the fault-sensitive resonance frequency, and their optimal frequency bands were located near 7300 Hz, which corresponded to the random shock. From the envelopes of the filtered signal shown in Figures 13 and 14b,c, the amplitudes of the fault impulses are relatively small, while the random shocks are obviously. The FCO and its harmonics are difficult to identify from the envelope order spectra, as shown in Figures 13 and 14d. Differing from the FK and WPTK, the Protrugram is shown in Figure 15a with BW equals to 300 Hz and the step is 50 Hz. The optimal frequency band relevant to the maximum kurtosis is the fault-sensitive frequency band. The corresponding envelope of the filtered signal and its resampling envelope signal are shown in Figure 15b,c, respectively. The FCO and its harmonics are clearly shown in Figure 15d. The FOSCK is shown in Figure 16a, the optimal frequency band corresponding to the maximum OSCK is calculated at the 5.5 th decomposition level, and its frequency band is (4792, 5000) Hz. The fault impulses are clear in the envelopes shown in Figure 16b,c. The FCO and its harmonics are clearly visible in Figure 16d, which means the FOSCK is robust to the varying speed and random shocks.  
Figure 12: Simulated signals: (**a**) a deterministic component signal, (**b**) the fault impulse signal, (**c**) the random shocks, (**d**) the synthetic signal without noise added, (**e**) the synthetic signal with noise-added and SNR \(=-5\) and (**f-j**) the frequency spectra of the different components.  
Figure 13: The results obtained by the FK for processing the mixed signal: (**a**) FK, (**b**) the envelope of the band-pass filtered signal, (**c**) the resampling envelope signal of (**b**,**d**) the envelope order spectrum of (**c**).