# Fault Diagnosis of Rolling Bearings Based on Acoustic Signals in Strong Noise Environments
## 4 Simulation Verification Analysis
The other methods include empirical mode decomposition (EMD) and complete ensemble empirical mode decomposition (CEEMD). Each method decomposes the simulated signal and calculates its RFN, kurtosis, and ISEI as illustrated in Figure 9.  
The noise tolerance \(\tau\) and convergence error limit \(\varepsilon\) for VMD are set to the values reported in the literature [16], specifically \(\tau=0\) and \(\varepsilon=1\times 10^{-7}\). The convergence curve of the CPO algorithm, derived from the minimum envelope entropy value, is shown in Figure 10. After 18 iterations, the envelope entropy value reaches its minimum, yielding the parameter combination \(K=12\) and \(\alpha=1217\). The decomposed modal spectrum is shown in Figure 11, demonstrating that the VMD optimized by CPO effectively mitigates modal aliasing and endpoint effects. Each frequency band is concentrated around the center frequency, indicating a more thorough decomposition. The optimal component following C-VMD decomposition was selected based on the ISEI value, identifying V-IMF4 for envelope analysis. As shown in Figure 12, only \(f_{i}\) is visible, with its harmonic component being subtle. Under conditions of strong noise, the ability to extract fault information from the bearing remains limited. Figure 9: RFN, kurtosis, and ISEI of each component after decomposing the signal using each method (simulation with inner raceway fault). (**a**) The characteristic values for C-VMD’s IMF components; (**b**) the characteristic values for CEEMD’s IMF components; (**c**) the characteristic values for fixed-parameter FMD’s IMF components; (**d**) the characteristic values for EMD’s IMF components. The noise tolerance \(\tau\) and convergence error limit \(\varepsilon\) for VMD are set to the values reported in the literature [16], specifically, \(\tau=0\) and \(\varepsilon=1\times 10^{-7}\). The convergence curve of the CPO algorithm, derived from the minimum envelope entropy value, is shown in Figure 5.0 After 18 iterations, the envelope entropy value reaches its minimum, yielding the parameter combination \(K=12\) and \(\alpha=1217\). Figure 5.1 Spectrum of each IMF decomposed via parameter-optimized VMD (simulation with inner raceway fault). Figure 5.2 Convergence curve of parameters optimized by the CPO algorithm (simulation with inner raceway fault). The decomposed modal spectrum is shown in Figure 11, demonstrating that VMD optimized by CPO effectively mitigates modal aliasing and endpoint effects. Each frequency band is concentrated around the center frequency, indicating a more thorough decomposition. The optimal component following C-VMD was selected based on the ISEI value, identifying V-IMF4 for envelope analysis. As shown in Figure 12, only \(f_{i}\) is visible, with its harmonic component being subtle. Under conditions of strong noise, the ability to extract fault information from the bearing remains limited. The fixed-parameter FMD, EMD, and CEEMD methods denoise the simulated signal to obtain the best components according to the ISEI value, subsequently performing envelope spectrum analysis. As shown in Figures 13-15, the envelope spectra of the best components were selected after fixed-parameter FMD, EMD, and CEEMD denoised the simulated signal, respectively. No obvious fault characteristics were detected; however, the effective information was obscured by strong noise. The fault information is buried within this noise, and fixed-parameter FMD methods demonstrate limited capabilities in fault extraction when compared to S-FMD and C-VMD, which yield more significant results. While the envelope spectrum of the optimal component is derived from the EMD and CEEMD methods, following the denoising of the simulated signal, can indicate the fault characteristic frequency of \(f_{i}\), there is no discernible frequency doubling. This frequency doubling is entirely obscured by strong noise, which can easily lead to the misclassification of fault types.