# Hybrid Empirical and Variational Mode Decomposition of Vibratory Signals
## 4 Cases Studies
### Case 3: Vibratory Component Analysis in a Six Degrees of Freedom System  
The objective of this study is the precise decomposition of vibratory modes using different methodologies, which are then compared with the analytical response of the system. This vibratory system is described by the coupled second-order differential equations:  
\[m_{1}\ddot{x_{1}}+c_{1}\dot{x_{1}}+(k_{1}+k_{2})x_{1}-k_{2}x_{2} = 0\] \[m_{2}\ddot{x_{2}}+c_{2}\dot{x_{2}}+k_{2}x_{2}-k_{2}x_{1}+k_{3}x_{ 2}-k_{3}x_{3} = 0\] \[m_{3}\ddot{x_{3}}+c_{3}\dot{x_{3}}-k_{3}x_{2}+(k_{3}+k_{4})x_{3}-k _{4}x_{4} = 0\] \[m_{4}\ddot{x_{4}}+c_{4}\dot{x_{4}}-k_{4}x_{3}+(k_{4}+k_{5})x_{4} -k_{5}x_{5} = 0\] \[m_{5}\ddot{x_{5}}+c_{5}\dot{x_{5}}-k_{5}x_{4}+(k_{5}+k_{6})x_{5} -k_{6}x_{6} = 0\] \[m_{6}\ddot{x_{6}}+c_{6}\dot{x_{6}}-k_{6}x_{5}+(k_{6}+k_{7})x_{6} = 0\]  
The characteristic equation of the system is determined by  
\[\det(M\textsl{s}^{2}+K)=0\]  
The parameter values are presented in Table 7. \begin{table}
\begin{tabular}{c c c c c} \hline  & **EMD** & **VMD** &
\begin{tabular}{c} **Hybrid** \\ **EMD–VMD** \\ \end{tabular} & **FFT** \\ \hline  & \multicolumn{4}{c}{**Position 1**} \\ \hline \(\Omega_{11}\) **[rad/s]** & 0.0009 & 0.8546 & 0.0000 & 0.0016 \\ \(\Omega_{12}\) **[rad/s]** & 0.0001 & 0.0004 & 0.0000 & 0.0001 \\ \(X_{1}^{(1)}\) **[m]** & 0.0000 & 0.0002 & 0.0000 & 0.0001 \\ \(X_{1}^{(2)}\) **[m]** & 0.0000 & 0.0000 & 0.0000 & 0.0001 \\ \hline  & \multicolumn{4}{c}{**Position 2**} \\ \hline \(\Omega_{21}\) **[rad/s]** & 0.0004 & 0.8784 & 0.0000 & 0.0016 \\ \(\Omega_{22}\) **[rad/s]** & 0.0016 & 0.0004 & 0.0000 & 0.0001 \\ \(X_{2}^{(1)}\) **[m]** & 0.0000 & 0.0003 & 0.0000 & 0.0001 \\ \(X_{2}^{(2)}\) **[m]** & 0.0000 & 0.0002 & 0.0000 & 0.0001 \\ \hline \end{tabular}
\end{table} 
Table 6: Mean squared error for frequency and amplitude estimates. \begin{table}
\begin{tabular}{c c} \hline
**Parameter** & **Value** \\ \hline  & **Masses [kg]** & \\ \hline \(m_{1}\) & 0.1 \\ \(m_{2}\) & 0.25 \\ \(m_{3}\) & 0.15 \\ \(m_{4}\) & 0.2 \\ \(m_{5}\) & 0.3 \\ \(m_{6}\) & 0.25 \\ \hline  & **Stiffness Coefficients [N/m]** & \\ \hline \(k_{1}\) & 1000 \\ \(k_{2}\) & 300 \\ \(k_{3}\) & 350 \\ \(k_{4}\) & 450 \\ \(k_{5}\) & 500 \\ \(k_{6}\) & 750 \\ \(k_{7}\) & 400 \\ \hline \end{tabular}
\end{table} 
Table 7: Parameter values of the six-degrees-of-freedom system. The natural frequencies are derived from the roots of the characteristic equation:  
\[ \begin{array}{ll}115.54i,&-115.54i,\\ 93.120i,&-93.120i,\\ 82.682i,&-82.682i,\\ 60.299i,&-60.299i,\\ 38.941i,&-38.941i,\\ 20.982i,&-20.982i\end{array} \]  
Thus, the natural frequencies for this set of parameters are  
\[ \begin{array}{llll}\omega_{n1}&=&20.982\;[\text{rad}/\text{s}]\\ \omega_{n2}&=&38.941\;[\text{rad}/\text{s}]\\ \omega_{n3}&=&60.299\;[\text{rad}/\text{s}]\\ \omega_{n4}&=&82.682\;[\text{rad}/\text{s}]\\ \omega_{n5}&=&93.120\;[\text{rad}/\text{s}]\\ \omega_{n6}&=&115.54\;[\text{rad}/\text{s}]\end{array} \]  
In this analysis, the Empirical Mode Decomposition, Variational Mode Decomposition, and hybrid EMD-VMD methods were used to estimate the natural frequencies. These methods enable the decomposition of system signals into their natural frequency components, providing insights into the vibrational dynamics of the system. The accuracy of the frequency estimates obtained by each method is evaluated in Table 8, showing the performance and ability of each technique to identify the vibrational characteristics of the system. The accuracy of each method was further evaluated by calculating the percentage error, shown in Table 9, and the mean squared error (MSE) of the frequency estimates, presented in Table 10. The results demonstrate the superiority of the hybrid EMD-VMD method for complex signal decomposition, achieving lower average errors and higher accuracy compared to the EMD and VMD methods.