## 4 Numerical Validation  
To study the effectiveness of the proposed method, a simulated signal, denoted by \(x(t)\), was applied. The signal is a mathematical model, used to describe the displacement response of a damped system under free vibration [27]. It can be written as follows:  
\[x(t)=5e^{-2 zt}\sin(78.5t+0.46)+20e^{-3 At}\sin 500.5t+20e^{-2.2 At}\sin 1401.1t+ 5e^{-3.84t}\sin(2745.6t+0.1) \tag{11}\]  
where \(A_{n}\) is the amplitude, \(\xi_{n}\) is the damping ratio, \(f_{n}\) is the natural frequency, and \(\theta_{n}\) is the phase angle of the \(n\)th mode. \(N\) is the total number of modes. Here, the number of modes is 4. The parameters used for generating \(x(t)\) are as follows: \(f_{1}=12.5\) Hz, \(f_{2}=79.7\) Hz, \(f_{3}=223.1\) Hz and \(f_{4}=437.2\) Hz, Amplitudes are \(A_{1}=A_{4}=5\), \(A_{2}=A_{3}=20\). Phase angles are \(\theta_{1}=0.46,\ \theta_{4}=0.1\), and \(\theta_{2}=\theta_{3}=0\), and the correlated damping ratios are \(\xi_{1}=2.8\%\), \(\xi_{2}=0.68\%\), \(\xi_{3}=0.16\%\), and \(\xi_{4}=0.14\%\). Here, the sampling frequency is 4096 Hz, and the sampling points total 8192. Classical EMD was performed on the simulated signals of the displacement response, and eight IMFs and one residual term were obtained. The first four order IMFs and their corresponding spectrum are shown in Figure 3. It can be seen from Figure 3 that the displacement simulation signal was not decomposed into four single IMF components, but into eight IMFs, and the decomposed low frequency components were not what we had expected and should not have appeared, this was evidence of false modes. In addition, it can also be seen from Figure 3 that the first generated IMFs covered a wide frequency range and obviously contained three frequency components. The second decomposed IMF component was a single frequency component; however, mode mixing appeared again in the third IMF component obtained by EMD; namely, two frequency values appeared in the single component of the third IMF. The displacement simulation signal of Equation (11) was also analyzed using the EEMD method. The first six orders of IMFs decomposed using the EEMD method are shown in Figure 4. In the process of EEMD, the added white noise level was set to 0.1, and the overall average mean number was 50. As can be seen from Figure 4, EEMD could almost decompose signals into single components, except the first IMF, and there was very obviously two frequency components (around 100 Hz); that is, modal mixing occurred. It can also be seen from Figure 4 that energy leakage existed in the decomposition process of the EEMD method, and it was obvious that the two different IMFs generated in the low-frequency domain contained the same frequency component (labeled by the red dotted rectangle). Meanwhile, one positive aspect was that EEMD successfully identified low order frequency components, which is of great significance for modal analysis and structural damage detection of large structures. The frequency range is theoretically the difference between two adjacent frequencies of the Fourier spectrum, but the smaller the frequency range is, the better the band-pass filtering effect will be. Therefore, in this paper, the frequency range is no more than 5%. Then, the simulated signal was processed by filtering through the band-pass filters with four specified frequency bands and IMFs, after the processing of each filter were obtained accordingly. For instance, there were 10 IMFs, and a residual could be obtained from the first frequency-band after the processing of EMD, as shown in Figure 6. Finally, the required IMF was automatically identified using clustering analysis shown in Figure 7. Here, Euclidean distance was used to cluster the decomposition results, with no need to adjust any indexes or thresholds, and only the nearest distance default in MATLAB was used to efficiently determine the real IMF. Figure 5: IMFs obtained by IEMD and the corresponding FFT frequency\(-\)spectrum. Figure 6: IMFs obtained by the first band\(-\)filtering.