# Hybrid Empirical and Variational Mode Decomposition of Vibratory Signals
## 4 Cases Studies
### Case 1: Estimation of Vibratory Components Using EMD, VMD, and the Hybrid EMD-VMD Method  
For the first case, a set of vibratory components with close frequencies was selected to present a challenge for the technique and highlight the advantages over conventional methods. A signal composed of six vibratory components with different amplitudes and frequencies was defined. The parameters of the signal are presented in Table 1:The signal was generated by summing these vibratory components, as shown in:  
\[\mathcal{F}(t)=\sum_{i=1}^{6}F_{i}\cos(\Omega_{i}t)\]  
The generated signal consists of the sum of six vibratory components, each defined by its respective amplitude and frequency. The sampling interval is defined as a 10 s window consisting of 1000 samples, ensuring adequate temporal resolution for the representation of the signal's frequency components. Figure 10 illustrates the generated signal. The generated signal was decomposed using three different decomposition methods: Empirical Mode Decomposition, Variational Mode Decomposition, and a hybrid method combining both techniques. First, the EMD algorithm was applied to decompose the signal into intrinsic mode functions. This technique allows the decomposition of the signal into intrinsic vibratory components without assuming any a priori basis. Subsequently, the VMD method was applied to perform mode decomposition. VMD addresses the mode mixing problem by decomposing the signal into a specific number of modal components, providing a more robust solution in the presence of noise and closely spaced frequency components. The hybrid method combines EMD and VMD to improve the accuracy of vibratory component estimation. First, EMD is applied to identify potential mode mixing by comparing the number of maxima and minima with the number of zero crossings in the signal. Then, VMD, guided by the information obtained from EMD, is used to determine the appropriate number of components to estimate. The results of the decompositions are presented in Figure 11. Each of the six estimated vibratory components is compared with the original components of the signal. The graphs show comparisons between the original vibratory components and those obtained by each decomposition method. The estimated frequencies \(\widehat{\Omega_{i}}\) and amplitudes \(\widehat{F_{i}}\) of the components are evaluated in Figure 12 and Figure 13, respectively, demonstrating that the hybrid \begin{table}
\begin{tabular}{c c c c} \hline \hline \(i\) & \(\mathbf{F_{i}[u]}\) & \(\mathbf{\Omega_{i}}\) **[rad/s]** & \(\mathbf{f_{i}}\) **[Hz]** \\ \hline
1 & 6 & 157.08 & 25 \\
2 & 2 & 94.25 & 15 \\
3 & 5 & 62.83 & 10 \\
4 & 1.5 & 31.42 & 5 \\
5 & 4 & 18.85 & 3 \\
6 & 2 & 6.28 & 1 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Amplitudes \(F_{i}\) and frequencies \(\Omega_{i}\) of the vibratory components. Figure 10: Generated signal composed of six vibratory components. method offers greater accuracy in the estimation of vibratory components, especially in signals with closely spaced frequencies and nonlinear components. Additionally, the instantaneous amplitudes \(\widehat{F}_{i}\) and frequencies \(\widehat{\Omega_{i}}\) of the estimated components were analyzed. The following figures present the estimated amplitudes and frequencies for each of the six vibratory components. To effectively illustrate the outcomes attained through the various decomposition and analysis methods introduced in this study, a comprehensive table is constructed, as shown in Table 2. This table consolidates the results, displaying the average values derived from each analytical strategy, allowing for a clear comparative overview. The techniques covered include EMD, VMD, and the hybrid EMD-VMD method. Each of these methods offers a unique approach to signal analysis, tailored to extract specific  
Figure 11: Tracking of estimated vibratory component references: comparison between EMD, VMD, and the hybrid EMD–VMD method. features such as frequency and amplitude from the data.