## 6 Results and Discussion  
The proposed MVO-optimized VMD method combined with Wavelet Threshold Denoising (WTD) demonstrates superior performance in both simulated and real-world denoising tasks regarding vibration signals. The results are discussed in detail below:  
(1) Verification of Noise Reduction Performance  
In the simulation experiments, the proposed method significantly enhances the signal-to-noise ratio (SNR) and reduces the mean square error (MSE). Specifically, the SNR of the noisy signal increases from 5.7327 to 16.3866, while the MSE decreases to 0.58293. In contrast, the standalone Wavelet Threshold Denoising method yields an SNR of 8.113 and an MSE of 1.5107, indicating that the MVO-VMD-MPE-WTD approach outperformed traditional methods. Additionally, a comparison of different decomposition levels (e.g., SNR of 13.9798 for K = 5 vs. 16.3866 for K = 12) shows that the MVO-optimized decomposition level of 12 provided the best noise reduction results, further validating the efficacy of the proposed method.  
(2) Effectiveness of Adaptive Parameter Optimization  
The selection of the number of modes (K) and penalty factor (\(\alpha\)) is critical in traditional VMD. If the choice is inappropriate, it can easily lead to excessive or insufficient signal decomposition. By utilizing the MVO algorithm to adaptively optimize these parameters, the proposed method avoids the subjectivity of manual selection and enhances the precision and efficiency of signal decomposition. The experimental results indicate that, for the case study in this paper, the optimal number of decomposition levels is 12 and the penalty factor \(\alpha\) is 2000, demonstrating that this parameter combination can minimize noise while preserving the key characteristics of the signal.  
(3) Comparison with Other Optimization Algorithms  
To assess the superiority of the MVO algorithm, it is compared with other optimization methods such as NGO and AOA. Although all algorithms successfully optimized VMD parameters, MVO demonstrates greater computational efficiency and noise reduction performance. For instance, MVO required 1113.3 s of computation time, which is significantly lower than NGO (1180.2 s) and AOA (1266.6 s). Furthermore, the noise reduction error ratio of the MVO-VMD-MPE-WTD method is 1.14, considerably better than the standalone Wavelet Threshold Denoising method (6.6242) and the MVO-VMD method that discards noisy IMFs (3.9179). This highlights the superior efficiency and effectiveness of the proposed method.  
(4) Performance in Real Engineering Applications  
In the denoising processing of flow-induced vibration measurement signals of an arcuate spillway radial gate, the proposed approach exhibited strong adaptability and robustness. The noisy signal, characterized by numerous spikes, is significantly smoothed after denoising. The power spectral density analysis shows that the main frequency peaks after denoising (8.20 Hz and 18.31 Hz) closely match the natural frequencies obtained from numerical simulation (8.27 Hz and 18.20 Hz), with an error margin of less than 1%. This result confirms that the proposed method effectively retains the essential frequency components of the signal while substantially improving signal quality.  
(5) Retention of Signal Characteristics and Accuracy Validation  
The analysis of power spectral density further demonstrates that the key frequency components of the signal remained stable after denoising, confirming that the proposed method not only reduces noise but also preserves critical signal features. Comparison with numerical simulation results reinforces the method's applicability in real engineering contexts, as the denoised signal accurately reflects the actual vibration characteristics, illustrating the reliability and practicality of the proposed method in handling complex signals.