## 3 Case Study  
### Numerical 5-Storey Frame Structure  
The finite element model of a 5-storey, 3-bay by 5-bay frame structure as shown in Figure 1 is established by SAP 2000 to validate the MEMD-based modal identification method. The overall height of the modal is 15 m with a story height of 3 m and a bay of 2.5 m. The sections of the beam and column elements are the size: 0.3 m \(\times\) 0.5 m and 0.4 m \(\times\) 0.4 m, respectively. All of them are made of C30 concrete. The natural frequencies of the first two sway modes are 3.5374 Hz and 3.7821 Hz, respectively. The modal damping ratios for calculating the dynamic response are assumed uniformly to be 3%. The input excitation of the model is 1 dB white Gaussian noise with a sampling frequency of 50 Hz. The structural responses of 5 gauge points as shown in Figure 1 are picked up from the SAP 2000. To validate the accuracy and robustness of the MEMD-based method, a sinusoidal noise is added to the extracted response and the amplitude is a quarter of the standard deviation of the associated response as Equation (11). \[\widetilde{x}_{i}(t)=x_{i}(t)+\frac{\sigma_{i}}{4}[\sin(10\pi t)+\sin(30\pi t)] \tag{11}\]  
where \(x_{i}(t)\) is the response of \(i\)th point, \(\sigma_{i}\) is the associated standard derivation and \(\widetilde{x}_{i}(t)\) is the noise-added response. Figure 2 displays the noise-added structural responses of point A5 along the direction of the \(X\) and \(Y\) axis. The associated noise-added signal at each point is decomposed by the MEMD method and the X-direction results of the first five IMFs are displayed in  
Figure 1: A 5-storey frame structure. The identified frequencies and damping ratios of sway modes are summarized in Tables 1 and 2, together listing the results identified by the finite element method (FEM), the peak-pick, and the half-power bandwidth methods. Meanwhile, the mean absolute relative deviation (MARD) [36] is calculated to evaluate the accuracy of modal identification, as following equation. \[\text{MARD}=\frac{1}{n}\sum_{j=1}^{n}\left|\left(\theta_{j}-\theta_{jr}\right)/ \theta_{jr}\right| \tag{12}\]  
where \(\theta_{j}\) and \(\theta_{jr}\) are the identified and exact modal parameters at the \(j\)th mode, respectively; \(n\) is the number of the identified modes. Figure 4: Hilbert spectrum of the first five IMFs. (**a**) IMF1; (**b**) IMF2; (**c**) IMF3; (**d**) IMF4; (**e**) IMF5. It can be noted that the frequencies identified from the noise-added response by several methods are satisfactory and the relative error, MARD, is almost less than 1%, revealing the high precision of the identified frequencies. However, compared with the frequency, more deviations in damping ratio can be seen, especially by the HPBW and the HHT methods which may be strongly affected by the limited length and nonstationary characteristics of the acceleration response. By employing the MEMD, it can be observed that by the FBFFT method the MARD of damping ratio falls to 3.92% from 6.42%, and by the SSI method, it reduces from 6.17% to 1.50%, indicating that with the employment of the MEMD the errors of damping ratio caused by sinusoidal noise effect in the original signal can be significantly decreased. The mode shapes of sway vibration modes are shown in Figure 5. The mode shapes of the frame structure are normalized by making the modal contribution coefficient of the top measuring point A5 the unit. The results in Figure 5 demonstrate that the mode shapes identified by MSSI and MFBFFT methods are almost consistent with that given by the FEM.