## 3 Case Study
### Shaking Table Modal Test  
To examine the robustness of the proposed MEMD-based identification methods in seismic excitations, the acceleration response of a 12-storey reinforced frame in shaking  
Figure 5: Comparison of the mode shapes for a numerical model of the structure.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Mode**} & \multicolumn{2}{c}{**MAC**} & \multicolumn{2}{c}{**NMD**} \\ \cline{2-5}  & **MFBFFT** & **MSSI** & **MFBFFT** & **MSSI** \\ \hline
1X & 1.0000 & 1.0000 & 0.11\% & 0.15\% \\
2X & 0.9964 & 0.9991 & 6.02\% & 2.97\% \\
1Y & 0.9977 & 0.9975 & 4.83\% & 5.03\% \\
2Y & 0.9998 & 0.9945 & 1.26\% & 7.42\% \\ \hline \hline \end{tabular}
\end{table}
Table 3: Correlation index of the identified and FEM mode shapes.  
Figure 6: Singular spectrum from SVD of Toeplitz matrix. (**a**) Original data; (**b**) IMFs decomposed by MEMD.  
table model tests [39] is utilized as a case study. The length scale of the model is 1/10 with an overall height of 3.6 m, beams of 30 mm \(\times\) 30 mm, and columns of 50 mm \(\times\) 60 mm, as shown in Figure 7. To measure the dynamic response of the model, more than 61 operating conditions were carried out and 25 channels of accelerometers were instrumented in the model. More information about the shaking table modal was presented by [PERSON] [40]. In this study, the acceleration response of seven channels in X-direction with the sampling frequency of 255 Hz during El Centro excitation is adopted to identify the modal parameters. The original responses and corresponding IMFs of A6 and A7 points are shown in Figure 8. Absolutely, the signals to be decomposed (i.e., X6 and X7 in Figure 8) contain nonstationary features.  
The identified frequencies and damping ratios of the first four modes are summarized in Tables 4 and 5, together listing the frequencies of FEM and other results identified by ERA methods [12]. It can be observed that the frequencies identified from the seismic response are still comparable with the FEM despite the original data being a typical nonstationary process and the length of data being only 2688. Similarly, the MARD is used to evaluate the accuracy of the results on the basis of the FEM. The MARD of frequencies is 4.7% by the MFBFFT method and 3.7% by the MSSI method, indicating the proposed MEMD-based methods are capable to identify the natural frequencies from the seismic response. For the damping ratio, the variation between different methods is relatively large and the identified COVs of the damping ratio remain at a high level, implying greater posterior uncertainties. It can be seen from Table 5 that there is a tendency for the modal damping ratio to decrease with the increase in modes. Due to the exact value of the damping ratio being unavailable, the mean values of the results identified by the different methods are regarded as the benchmark instead. The MARDs of damping ratio are larger than 10% except that, by the HHT method, it is only 3.7%. By employing the EMD method, the MARD of the FBFFT method decreases to 10.2% from 15.6% and by the SSI method, it falls to 21.6% from 23.3%, respectively.  
Figure 7: View of frame model and accelerometers of the shaking table test.  
Figure 8: First five IMFs of the shaking table test by MEMD decomposition.  
The stabilization diagrams during the SSI method are displayed in Figure 9. Owing to the limited length of the response, the frequency resolution of the power spectrum density function is not high enough, making it difficult to find out the exact system orders as shown in Figure 9a. As for the selected IMFs decomposed by the MEMD method, extra vibration modes are filtered out, leaving only one dominant frequency component as shown in Figure 9b, which indicates a system order of two or four (a result from the remaining noise).