# A Novel Rolling Bearing Fault Diagnosis Method Based on MFO-Optimized VMD and DE-OSELM
## 4 VMD Optimization Based on the Improved MFO
### Simulated Analysis  
A mathematical model was utilized to simulate the signal of rolling bearing inner ring faults for the purpose of evaluating the effectiveness of the MFO-optimized VMD method and the \(EWCS\) index:  
\[x(t)=s(t)+n(t)=\sum_{i}A_{i}h(t-iT-\tau_{i})+n(t) \tag{26}\]  
\[h(t)=\exp(-Ct)\cos(2\pi f_{n}t) \tag{27}\]  
\[A_{i}=1+A_{0}\cos(2\pi f_{r}t) \tag{28}\]  
The simulation signal was synthesized by adding a periodic impact component \(s(t)\) with Gaussian white noise \(n(t)\). The sampling frequency (\(f_{s}\)) was set to 12,000 Hz, while the rotation frequency (\(f_{r}\)), the structural resonance frequency (\(f_{n}\)), the fault frequency (\(f_{i}\)), and the damping ratio (C) were set to 33 Hz, 3000 Hz, 79 Hz, and 500, respectively. Additionally, the signal-to-noise ratio (SNR) of the Gaussian white noise was set as \(-5\) dB. The resulting simulation signal was plotted in both the time domain and the envelope spectrum, as presented in Figure 3.  
Figure 3 shows that the periodic impact characteristics that resulted from the structure failure are enveloped by noise, which makes it difficult to extract the impact characteristics precisely and directly. As such, signal processing techniques are necessary to extract the impact characteristics. The optimal parameter combination for VMD was determined using the improved MFO. Initially, we set the population size to 20 and a maximum of 10 iterations, with \(m\) and \(p\) values both set to 10. The outcome of the parameter search is illustrated in Figure 4.  
Figure 4 demonstrates that the MAEE value of 9.2493800, achieved with parameter combination <3, 773>, was obtained on the third iteration. To demonstrate the proposed improved MFO algorithm's efficiency, we compared its performance to three other popular optimization algorithms: the original MFO, GWO, and PSO. Each algorithm had an initial population size of 20 and was restricted to a maximum of 10 iterations. Each algorithm ran for 10 iterations, and Table 2 presents the results obtained. Table 2 reveals that GWO exhibits the highest MAEE value amongst the four algorithms across the ten operations. Despite the fact that the MAEE values obtained by the PSO, MFO, and improved MFO3 algorithms are equivalent, the improved MFO algorithm has the smallest mean square error and standard deviation across all results, demonstrating its efficacy. The optimal parameter combination of <3, 773> was used to decompose the simulation signal by VMD. To validate the proposed EWCS index, we computed the Peak Signal-to \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Algorithm** & **Optimum Value** & **MSE** & **STD** \\ \hline GWO & 9.249382 & 7.2257 \(\times\) 10\({}^{-12}\) & 1.8724 \(\times\) 10\({}^{-15}\) \\ PSO & 9.249380 & 1.1715 \(\times\) 10\({}^{-6}\) & 7.2087 \(\times\) 10\({}^{-4}\) \\ MFO & 9.249380 & 1.9533 \(\times\) 10\({}^{-7}\) & 4.4167 \(\times\) 10\({}^{-4}\) \\ Improved MFO & 9.249380 & 5.9336 \(\times\) 10\({}^{-12}\) & 1.5860 \(\times\) 10\({}^{-6}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Results of four optimization algorithms. Figure 4: The improved MFO convergence curve of the simulation signal. Figure 3: (**a**) Time–domain of the simulation signal. (**b**) Frequency–domain diagram of the simulation signal. Noise Ratio (PSNR), Mean Squared Error (MSE), and EWCS value of the three resulting IMFs obtained from the decomposition. The computed results are presented in Table 3, where it can be observed that IMF2 exhibits the highest PSNR and the smallest MSE, thus implying that this IMF contains abundant fault information. This observation is consistent with the outcome of the proposed method, which screens effective IMFs based on EWCS value, and it substantiates the validity of the proposed method.