## 8 Future Research Directions  
Based on the effectiveness of the proposed method, future research can be extended in the following aspects.  
The first is optimization algorithm improvement. It would be valuable to explore the integration of the MVO algorithm with other global optimization algorithms, such as a Genetic Algorithm (GA) and Particle Swarm Optimization (PSO), to further enhance the parameter optimization of VMD and avoid the issue of local optima.  
The second is adaptive wavelet threshold selection. In this study, a fixed wavelet threshold is used. Future research could focus on developing adaptive threshold selection methods that dynamically adjust the threshold based on signal characteristics, thereby improving the overall denoising performance.  
The third is expanding application areas. While the current study primarily focuses on vibration signal processing for spillway radial gates, future work could extend the method to other complex, non-stationary signal processing scenarios, such as a rotating machinery fault diagnosis and seismic signal analysis, to validate its applicability and effectiveness in other fields.  
**Author Contributions:** Conceptualization, X.L. and Y.L.; methodology, X.L. and X.Z.; software, X.L. and C.W.; validation, X.L. and X.Z.; formal analysis, D.Z. and X.Z.; investigation, D.Z.; resources, Y.L., D.Z. and S.T.; data curation, X.L.; writing--original draft preparation, X.L.; writing--review and editing, Y.L.; visualization, Y.L.; supervision, S.T. and Y.L.; project administration, S.T., C.W. and Y.L.; funding acquisition, S.T., Y.L. and D.Z. All authors have read and agreed to the published version of the manuscript.  
**Funding:** This research was funded by Science and Technology Projects of CHINA HUANENG GROUP (grant No. HNKJ22-H108), Science and Technology Projects of China Power Construction Corporation Limited (grant No. DJ-ZDXM-2021-03), and the National Natural Science Foundation of China (grant No. 52179060, 52479060).  
**Institutional Review Board Statement:** Not applicable.  
**Informed Consent Statement:** Not applicable.  
**Data Availability Statement:** The raw data supporting the conclusions of this article will be made available by the authors on request.  
**Conflicts of Interest:** Authors [PERSON], [PERSON], and [PERSON] were employed by the company Power China Guiyang Engineering Corporation Limited. The remaining authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.  
**Nomenclature**  
\begin{tabular}{l l} \(A_{k}(t)\) & Instantaneous amplitude \\ \(\varphi_{k}(t)\) & Instantaneous phase \\ \([u_{k}]\) & K decomposed Intrinsic Mode Functions (IMFs) \\ \([w_{k}]\) & Central frequency components of each IMF \\ \(f\) & Input signal \\ \(S\) & Maximum scale factor \\ \(X_{1}\) & Artificial pure signal1 \\ \(X_{2}\) & Artificial pure signal2 \\ \(X_{3}\) & Artificial pure signal3 \\ \(X_{noise}\) & Random white noise signal \\ \(Y\) & Noise-contaminated simulated signal \\ \([M]\) & Mass matrix \\ \([\)C\(]\) & Damping matrix \\ \([K]\) & Stiffness matrix \\ \(X[k]\) & FFT result of the signal on the k-th frequency component \\ \(N[k]\) & FFT result of the noise on the k-th frequency component \\ \(s[n]\) & Signal value at the n-th time instant \\ \(n[n]\) & Noise value at the n-th time instant \\ \(P_{s}\) & Power of a noise-contaminated signal \\ \(P_{g}\) & Power of the denoised signal \\ \(SNR_{time}\) & Time-domain signal-to-noise ratio \\ \(SNR_{freq}\) & Frequency-domain signal-to-noise ratio \\ \(MSE\) & Mean squared error \\ \(K_{ij}\) & Variance contribution rate \\ \(k_{ij}\) & Distribution coefficient \\ NGO & Northern Goshawk Optimization \\ AOA & Archimedes optimization algorithm \\ RIME & Rime optimization algorithm \\ MVO & Multiverse Optimizer Algorithm \\ EMD & Empirical Mode Decomposition \\ VMD & Variational Mode Decomposition \\ _MPE_ & Multiscale Permutation Entropy \\ WTD & Wavelet Threshold Denoising \\ \end{tabular}