## 3 Numerical Simulation
### Simulation Results and Discussion  
#### 3.2.1 Results Analysis of Complexity Pursuit  
Figure 5 illustrates the measured structure responses, while Figure 6 displays the reconstructed modal responses, indicating that the coupled random structure responses were effectively separated into distinct single-component modal responses. For conciseness, only the first five modes are given in Figure 6. Table 2 presents a comparison between the identified results obtained via the CP method and the theoretical results. If the noise effect is disregarded, the identified values of frequency (_f_), damping ratio (_g_), and mode shapes exhibited excellent agreement with the theoretical results, except for the mode shape of mode 4. As depicted in Figure 5, the fourth and fifth modes were closely spaced and challenging to differentiate on the basis of the PSD of the structural response. However, the CP method successfully separated these modes (as shown in Figure 6). Despite this, the CP method was unable to precisely determine the mode shape of these close modes. The sixth mode, eighth mode, and ninth mode were mixed modes, which could be accurately separated using the CP method.  
As depicted in Figure 7, the identified frequencies and damping ratios remained consistent despite variations in the length of the time histories used to analyze the structure responses. The MAC values obtained from analyzing the structure responses with different sample lengths revealed high consistency, with all values above 0.90 (except mode 4).  
Figure 5: Measured structure responses of the bridge subject to white noise excitation (the five responses from positions 1 to 5).  
These results suggest that the accuracy of the CP method was not significantly affected by variations in sample length, indicating the method's robustness.  
It is worth mentioning that the identification ability of CP method hardly declined with 1-5% RMS noise added to the output data. As shown in Table 2, except for the damping ratio and mode shape of mode 4, other modal parameters were completely identified. The findings indicate that the CP method is highly robust to the effects of measured noise, as it was able to accurately recover the modal responses of the structure despite the presence of noise.  
\begin{table}
\begin{tabular}{c c c c c c c c c} \hline \multirow{2}{*}{**Mode**} & \multicolumn{2}{c}{**Theoretical Value**} & \multicolumn{2}{c}{**Identified (Without Noise)**} & \multicolumn{2}{c}{**Identified (1–5\% RMS Noise)**} \\ \cline{2-9}  & \(f\) **(Hz)** & \(\xi\) & \(f\) **(Hz)** & \(\xi\) & **MAC** & \(f\) **(Hz)** & \(\xi\) & **MAC** \\ \hline
1 (1 st B) & 4.69 & 0.01 & 4.67 & 0.009 & \(1\) & 4.66 & 0.012 & \(1\) \\
2 (2 nd B) & 6.51 & 0.01 & 6.51 & 0.011 & \(1\) & 6.54 & 0.012 & \(1\) \\
3 (1 st T) & 9.82 & 0.01 & 9.84 & 0.011 & \(1\) & 9.76 & 0.008 & 0.92 \\
4 (2 nd T) & 12.07 & 0.01 & 12.04 & 0.008 & 0.74 & 12.11 & 0.006 & 0.69 \\
5 (3 rd B) & 12.31 & 0.01 & 12.32 & 0.009 & 0.99 & 12.34 & 0.009 & 0.94 \\
6 & 14.47 & 0.01 & 14.46 & 0.009 & 0.96 & 14.56 & 0.009 & 0.90 \\
7 (4 th B) & 17.60 & 0.01 & 17.64 & 0.009 & 0.99 & 17.57 & 0.012 & 0.90 \\
8 & 19.07 & 0.01 & 19.07 & 0.009 & \(1\) & 18.90 & 0.008 & 0.92 \\
9 & 23.36 & 0.01 & 23.35 & 0.008 & 0.99 & 23.34 & 0.007 & 0.96 \\ \hline \end{tabular}
\end{table}
Table 2: Identification results by complexity pursuit in random vibration.  
Figure 6: Modal responses recovered by complexity pursuit of the bridge subject to white noise excitation (the first five modal responses).