# Hybrid Empirical and Variational Mode Decomposition of Vibratory Signals
## 5 Conclusions  
A hybrid algorithm for the estimation of uncertain arbitrary frequency harmonic vibration components in mechanical systems is described. This hybrid estimation approach can be implemented for nonstationary vibration system signals. By integrating the Empirical Mode Decomposition and Variational Mode Decomposition algorithms, the hybrid approach exhibits superior performance in both accuracy and computational efficiency compared to traditional methods in specific cases, particularly when analyzing harmonic vibratory signals with nonlinear and nonstationary behavior. While the hybrid estimation strategy shows improvements over EMD and VMD when used independently, and even over the Fast Fourier Transform in certain variable-frequency operational scenarios, it is not suggested that the hybrid method universally outperforms the FFT. Instead, this study highlights its ability to address specific challenges, such as mode mixing and closely spaced frequencies, where traditional methods may fall short. The results across the case studies, quantified using the mean squared error, consistently approached zero, indicating the method's ability to provide highly accurate frequency and mode estimations in complex vibratory systems. In the first case study, involving a two-degrees-of-freedom mechanical system, the hybrid estimation algorithm demonstrated exceptional precision in computing natural frequencies and harmonic vibration modes. This precision, while highly effective for these scenarios, depends on the careful calibration of parameters and the inherent characteristics of the analyzed signals, which can limit its generalizability to other signal types without additional adjustments. The second case study, which involved a signal composed of harmonic vibratory components with closely spaced frequencies, highlighted the hybrid algorithm's ability to effectively address the challenge of mode mixing. By leveraging EMD for detecting mode mixing and VMD for refining the estimation of individual components, the hybrid approach enhanced both the accuracy and efficiency of the analysis. This case underscores its potential for applications in vibration monitoring and fault detection, where precise mode separation is critical. However, its dependence on accurate initial conditions and computational resources for parameter optimization presents limitations for real-time applications. The hybrid framework successfully leverages the strengths of EMD in breaking down signals into adaptive mode functions and the robustness of VMD in maintaining decomposition precision in the presence of noise and mode mixing. Nevertheless, this method shares some limitations with EMD and VMD, such as its sensitivity to noise levels and difficulties in handling signals with extremely similar frequencies in possibly perturbed, uncertain nonlinear vibration systems. These limitations point to opportunities for further optimization and enhancement, particularly in developing adaptive parameter selection mechanisms and improving computational efficiency. Future work should focus on addressing these limitations by exploring automated methods for parameter tuning and extending the hybrid approach to handle more diverse signal characteristics in uncertain multi-degree-of-freedom vibrating systems. Additionally, the development of computational efficiency confirmation indices and comparisons across a broader range of scenarios, including highly noisy environments, will further validate the method's utility and expand its applicability. Finally, the hybrid EMD-VMD algorithm establishes itself as a powerful and efficient tool for the detailed analysis of arbitrary frequency harmonic vibration signals, overcoming the challenges of nonlinearity and nonstationarity in specific contexts. While not universally superior to the FFT or standalone EMD and VMD, its ability to integrate the strengths of these methods demonstrates its relevance for tackling complex vibration system signal analysis challenges.