## 3 Numerical Simulation
### Simulation Results and Discussion
#### 3.2.2 Results Analysis of Generalized Eigen Decomposition  
Table 3 shows a comparison between the identified results obtained using the GED method and the theoretical results. The comparison indicates that, without considering the impact of noise, the identified frequency, damping ratio, and mode shape closely matched the theoretical results, demonstrating excellent agreement between the two sets of data. Figure 8 presents the recovered modal responses in random vibration along with their corresponding power spectral density (PSD). It is important to note that the sequence of the recovered modal responses was not rearranged in order to display the original results obtained using the GED method. For example, the use of the term \"modal response 1\" in Figure 8 simply refers to the first response recovered by the GED method, and it does not necessarily correspond to mode 1 of the structure. This is a minor issue that can be easily resolved by rearranging the sequence of the recovered modal responses, as demonstrated in Figure 6 and explained in Section 2.5. It can be observed that the multicomponent structure responses were completely separated into monotone modal responses.  
In addition, as presented in Figure 9, the identified frequencies and damping ratios were almost identical with different sample (window) lengths of the time histories of the structure responses. The MAC values of the identification remained very high with different sample (window) lengths of the time histories of the structure responses; all were above 0.90. The results indicate that the accuracy of the GED method is insensitive to the sample length.  
Figure 7: The identification results by CP with different sample lengths of the system responses of the bridge system: (**a**) frequency; (**b**) damping ratio; (**c**) MAC.  
Compared with the CP method, it is clear to see that the performance of this method was better, especially in the identification accuracy of the close space modes. When the 1-5% RMS noise was added to the structure responses, the procedures of the GED method were performed on the noise-contaminated system responses, and the identification results are given in Table 3. It can be seen that the identification results were seriously affected by the noisy data (failing to identify the mixed mode 6, 8 and 9); that is, the GED method was not robust once the measured noise was added.  
\begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline \multirow{2}{*}{**Mode**} & \multicolumn{2}{c}{**Theoretical Value**} & \multicolumn{3}{c}{**Identified (Without Noise)**} & \multicolumn{3}{c}{**Identified (I–5\% RMS Noise)**} \\ \cline{2-9}  & \(f\) **(Hz)** & \(\xi\) & \(f\) **(Hz)** & \(\xi\) & \(f\) **(Hz)** & \(\xi\) & \(f\) **(Hz)** & **MAC** \\ \hline
1 (1 st B) & 4.69 & 0.01 & 4.67 & 0.011 & 1 & 4.71 & 0.009 & 1 \\
2 (2 nd B) & 6.51 & 0.01 & 6.51 & 0.01 & 1 & 6.55 & 0.011 & 1 \\
3 (1 st T) & 9.82 & 0.01 & 9.82 & 0.009 & 1 & 9.83 & 0.008 & 0.95 \\
4 (2 nd T) & 12.07 & 0.01 & 12.08 & 0.009 & 1 & 12.03 & 0.012 & 0.98 \\
5 (3 rd B) & 12.31 & 0.01 & 12.34 & 0.011 & 0.97 & 12.37 & 0.006 & 1 \\
6 & 14.47 & 0.01 & 14.51 & 0.012 & 0.93 & – & – & – \\
7 (4 th B) & 17.60 & 0.01 & 17.60 & 0.008 & 1 & 17.63 & 0.008 & 0.98 \\
8 & 19.07 & 0.01 & 19.05 & 0.01 & 1 & – & – & – \\
9 & 23.36 & 0.01 & 23.36 & 0.01 & 1 & – & – & – \\ \hline \hline \end{tabular}
\end{table}
Table 3: Identification results by generalized eigen decomposition in random vibration.  
Figure 8: Modal responses recovered by generalized eigen decomposition of the bridge subject to white noise excitation (the first five modal responses).