# Fault Diagnosis of Rolling Bearings Based on Acoustic Signals in Strong Noise Environments
## 4 Simulation Verification Analysis  
By constructing a simulation signal of a fault in a rolling bearing [20], the feasibility of the SFMD method is verified. The simulation signal expression is:  
\[\left\{ \begin{array}{l}x(t)=s(t)+n(t)=\sum\limits_{i}A_{i}h(t-iT)+n(t)\\ A_{i}=1+A_{0}\text{cos}(2\pi f_{r}t)\\ h(t)=\text{exp}(-Ct)\text{cos}(2\pi f_{\pi}t)\end{array} \right. \tag{13}\]  
where \(s(t)\) is the periodic impact component;  
The amplitude \(A_{0}\) is 0.3;  
The rotation frequency \(f_{r}\) is 30 Hz;  
Figure 3: Flowchart of sound signal fault diagnosis. The attenuation coefficient \(C\) is 700;  
The resonance frequency \(f_{n}\) is 4 khz;  
The inner ring fault characteristic frequency is \(f_{i}=1/T=120\) Hz;  
\(n(t)\) is Gaussian white noise; the signal-to-noise ratio of the noise-containing signal is \(-13\) dB (calculation formula \(\mathrm{SNR}=20\mathrm{log}_{10}(v_{s}/v_{n})\); \(v_{s}\) and \(v_{n}\) are the effective values of the impact component and noise component, respectively); the sampling frequency \(f_{s}\) is 16 kHz; and the number of analysis points is 4096 points. As shown in Figure 4, fault information cannot be identified in the time domain diagram, spectrogram, or envelope spectrum of the simulated signal after adding noise. Using the proposed method for signal denoising processing, the convergence curve of the SBOA obtained based on the minimum envelope entropy value is shown in Figure 5. The iterations converged after 13 times, and the optimal parameter combination was [250; 5; 5; 10]. FMD optimized by the SBOA is S-FMD, and its decomposed components are F-IMF. VMD optimized by CPO is C-VMD, and its decomposed components are V-IMF. Figure 6 shows the decomposed time domain diagram after optimizing FMD. Figure 7 shows the kurtosis, harmonic noise ratio, and ISEI of each F-IMF. Figure 4: Simulated signals: (**a**) shock signal waveform; (**b**) time domain waveform; (**c**) spectrogram; (**d**) envelope spectrum. Figure 5: Convergence curve of parameter optimized by the SBOA (simulation with inner raceway fault). Figure 6: Parameter-optimized FMD results of the signal (simulation inner raceway fault). **(a)** Time domain diagram of F-IMF1; **(b)** time domain diagram of F-IMF2; **(c)** time domain diagram of F-IMF3; **(d)** time domain diagram of F-IMF4; **(e)** time domain diagram of F-IMF5. As shown in Figure 7, F-IMF4 has the largest kurtosis value. As shown in Figure 6, F-IMF4 (Figure 6d) has a large kurtosis because it contains three abnormal shocks, while RFN and ISEI indicate that F-IMF3 contains the most faults. An envelope spectrum analysis is performed on the F-IMF decomposed by each S-FMD. The envelope spectrum of F-IMF3 contains the most fault information. The envelope spectrum of F-IMF3 in Figure 8 clearly shows the first eight harmonic components of the inner raceway fault frequency \(f_{i}\) 120 Hz and the rotation frequency \(f_{r}\) (30 Hz). This signifies a substantial reduction in the strong noise component added to the simulated signal at \(-13\) dB, effectively mitigating noise interference and enhancing the extraction of fault information. This shows that using the kurtosis value as the selection criterion for the best mode is not applicable, but the ISEI comprehensive index is effective as the selection criterion. In addition, the RFN value of F-IMF3 is 3.31 dB, which is 16.31 dB higher than the \(-13\) dB signal-to-noise ratio of the original simulation signal. Figure 8: Envelope spectrum of F-IMF3 decomposed via parameter-optimized FMD (simulation with inner raceway fault). Figure 7: Feature values of the decomposed modes via S-FMD (simulation with inner raceway fault). The simulation signal is denoised separately using four methods including adaptive C-VMD. Based on the parameter settings in the literature [19], the fixed FMD parameters are set as follows: the filter length \(L\) is 40, the frequency band cutting number \(K\) is 5, the mode number \(n\) is 4, and the cycle period \(m\) is 15.