# Sparse Low-Rank Based Signal Analysis Method for Bearing Fault Feature Extraction
## 4 Simulation Analysis  
In the simulation part, two simulated signals with outer race defect are employed to test the utility of the proposed FISLRA. To further demonstrate the advantage of FISLRA, comparative analysis of FISLRA and two other methods (TSLRM and SK) is performed. As mentioned above, TSLRM is susceptible to HCs. Therefore, two simulated signals \(y_{1}\) and \(y_{2}\) are analyzed to test the superiority of FISLRA relative to TSLRM. The results obtained are compared. The vibration signal \(y_{1}\) contains Harmonic Components \(h(t)\), Repetitive Transients \(d(t)\), Random Shocks \(r(t)\) and white noise \(n(t)\). The vibration signal \(y_{2}\) contains the fault-induced impulse \(d(t)\), random shocks \(r(t)\), white noise \(n(t)\) but no HCs. The simulated signal \(y_{1}\) is modeled as follows:  
\[ \begin{array}{l}y_{1}(t)=d(t)+h(t)+r(t)+n(t)\\ =\sum\limits_{i}D_{i}S_{d}(t-iT_{d}-\tau_{i})+\sum\limits_{n}A_{n}\cos(2\pi f _{n}t+\alpha_{n})\\ +\sum\limits_{i}R_{i}S_{i}(t-T_{i})+n(t)\\ \end{array}  \tag{14}\]  
\[S(t)=e^{-\alpha_{n}t}\sin(2\pi ft) \tag{15}\]  
where the first part denotes the repetitive transients \(d(t)\). \(D_{i}\) represents the amplitude of the \(i\)th impulse; \(T_{d}\) is the fault period of the repetitive transients; \(\tau_{i}\) denotes the random slip, and usually \(\tau_{i}\) is set to 1%-2% of the fault period; \(\alpha_{r}\) is damping ratio of the impulse response and \(f_{r}\) is the resonance frequency. The second part denotes the harmonic components \(h(t)\) generated by gearbox or rotor. \(\alpha_{n}\) is the initial phase and \(A_{n}\) is the amplitude; \(f_{n}\) stands for the rotating frequency. The third item represents the random shocks \(r(t)\), where \(T_{i}\) and \(R_{i}\) are occurrence time and amplitude of the \(i\)th shock, respectively. The white noise is represented by \(n(t)\). The corresponding sub-components of simulated signal \(y_{1}\) are drawn in Figure 4a-d. Time length of simulated data is 1 s and the sampling frequency is 20,000 Hz. Table 1 lists the settings of other parameters. The composition and parameter settings of the vibration signal \(y_{2}\) are exactly the same as those of the vibration signal \(y_{1}\), except that HCs are not included. After adding the white noise with SNR of \(-3\) dB, the waveform of the simulated signal \(y_{2}\) is demonstrated in Figure 5, of which fault \begin{table}
\begin{tabular}{c c c c c c c c c c c c} \hline \multicolumn{6}{c}{**Repetitive Transients**} & \multicolumn{6}{c}{**Harmonic Components**} & \multicolumn{6}{c}{**Random Shocks**} \\ \hline \(D_{1}\) & \(T_{d}\) & \(f_{r}\) & \(\alpha_{r}\) & \(f_{1}\) & \(f_{2}\) & \(A_{1}\) & \(A_{2}\) & \(\alpha_{1}\) & \(\alpha_{2}\) & \(f_{r}\) & \(\alpha_{r}\) \\
1.3 & 1/37 & 2500 & 300 & 10 & 20 & 1 & 0.4 & \(\pi\)/3 & \(\pi\)/6 & 6000 & 800 \\ \hline \end{tabular}
\end{table} 
Table 1: Parameter setting of signals in simulation analysis. Figure 4: Sub-component of simulated signal \(y_{1}\). (**a**) Repetitive transients \(d(t)\); (**b**) rotor vibration \(h(t)\); (**c**) random shocks \(r(t)\); (**d**) white noise \(n(t)\). features are disturbed by noise. Thus, TSLRM and FISLRA are used to extract fault features from the simulated signal \(y_{2}\). The results obtained by applying TSLRM and FISLRA are demonstrated in Figure 6. The repetitive transients are successfully extracted by both methods, so the ability of the method based on sparse low rank model for fault detection can be proved. The difference is that the extracted fault feature components obtained by TSLRM are disturbed by random shocks (shown in Figure 4c) that are marked with red dots, but the results obtained by the proposed FISLRA are not disturbed by random shocks. This shows that the robustness of the proposed method FISLRA is better against random impact interference than TSLRM. To further certify the superiority of FISLRA over TSLRM, the simulated signal \(y_{1}\) is analyzed.