# Hybrid Empirical and Variational Mode Decomposition of Vibratory Signals
## 4 Cases Studies
### Case 2: Vibratory Component Analysis in a Two Degrees of Freedom System  
The focus of this investigation is the accurate decomposition of vibratory modes using various methodologies, which are then compared against the analytical response of the system. The vibratory system is described by  
\[M\dot{\underline{X}}+KX=0\]  
\begin{table}
\begin{tabular}{c c c c c} \hline  & **Reference** & **EMD** & **VMD** & **Hybrid EMD–VMD** \\ \hline  & & \multicolumn{3}{c}{**Frequency [rad/s]**} \\ \hline \(\Omega_{1}\) & 157.08 & 163.73 & 157.14 & 157.08 \\ \(\Omega_{2}\) & 94.25 & 62.71 & 94.37 & 94.25 \\ \(\Omega_{3}\  
\[\omega_{n1} = \sqrt{\left|\frac{-a_{2}+\sqrt{a_{2}^{2}-4a_{0}a_{4}}}{2a_{4}}\right|}\] \[\omega_{n2} = \sqrt{\left|\frac{-a_{2}-\sqrt{a_{2}^{2}-4a_{0}a_{4}}}{2a_{4}} \right|}\]  
The two-degrees-of-freedom vibratory mechanical system is characterized by the following parameter values in Table 4.  
The amplitude ratios are given by  
\[r_{1} = \frac{-m_{1}\omega_{n1}^{2}+k_{1}+k_{2}}{k_{2}}=1\] \[r_{2} = \frac{-m_{1}\omega_{n2}^{2}+k_{  
Using the Fourier Transform, the frequency and amplitude components of \(x_{1}(t)\) and \(x_{2}(t)\) are quantified. Figure 15 shows the Fourier spectra for both signals. The signals \(x_{1}(t)\) and \(x_{2}(t)\) are decomposed into their vibratory components \(\widehat{x}_{11}\), \(\widehat{x}_{12}\), \(\widehat{x}_{21}\), and \(\widehat{x}_{22}\) using documented methods, as shown in Figure 16, which highlights the intrinsic modes extracted. The frequency parameters of \(x_{1}(t)\) and \(x_{2}(t)\) are shown in Figure 17, providing a detailed understanding of their oscillatory dynamics. Figure 16: Vibratory components of \(x_{1}(t)\) and \(x_{2}(t)\): analytical method versus EMD, VMD, and the hybrid method. Figure 15: Fourier analysis of \(x_{1}(t)\) and \(x_{2}(t)\): frequency and amplitude spectra. The amplitude parameters of \(x_{1}(t)\) and \(x_{2}(t)\) are shown in Figure 18, providing a clearer view of their vibratory behaviors. These amplitudes are estimated from the extracted modes. Finally, Table 5 consolidates the average values derived from the decomposition and analysis methods, offering a clear comparative overview. This structured presentation highlights the effectiveness of each method in estimating position, velocity, and acceleration features, facilitating informed decisions for future applications. The techniques covered include Empirical Mode Decomposition, Variable Mode Decomposition, Fourier Transform and the hybrid EMD-VMD method. Each of these methods offers a unique approach to signal analysis, tailored to estimating specific features such as frequency and amplitude from the data. The inclusion of average values in the table facilitates an intuitive understanding of the general performance and effectiveness of each method across different metrics such as position, velocity, and acceleration. This structured presentation not only aids in assessing the relative strengths and limitations of each technique but also provides a solid foundation for further discussion and analysis. By comparing the aggregated results, readers can discern patterns and the efficacy levels of each strategy, contributing to more informed decisions in future applications of these methods in similar contexts. The following table presents the MSE values for each method (EMD, VMD, EMD-VMD, and FFT) compared to the reference values for the frequencies and amplitudes of both position and velocity signals. As shown in Table 6, the MSE values indicate the accuracy of each method. Lower MSE values correspond to higher accuracy and better performance. The hybrid EMD-VMD method consistently shows lower MSE values across most metrics, demonstrating its effectiveness in accurately capturing the signal characteristics compared to the other methods. This validates the superiority of the hybrid method in handling complex signals with closely spaced frequencies and varying amplitudes.