## 4 Simulation Verification
### Results
#### 4.4.3 A Mass Slow-Time-Varying 3-DOF Structure with 10% White Gaussian Noise  
Ten percent white Gaussian noise is added to the vibration displacement response signal of the slow-time-varying 3-DOF structure, and the modal parameters of the structure are identified.  
In the mass slow-time-varying 3-DOF structure with 10% white Gaussian noise, we also select four moments including \(t\) = 100 s, \(t\) = 650 s, \(t\) = 1500 s and \(t\) = 1974.375 s. Table 11 shows the MAC value at the instantaneous moment. Figure 19 shows the modal shape of the instantaneous moment. Figure 20 shows the time-varying natural frequencies identified by the MWLPP. Figure 21 shows the time-varying MAC values of modal shapes identified by the MWLPP. Table 12 shows the average error \(\delta_{i-avg}\) of MWLPP in identifying natural frequencies. In the identification process of operational modal parameters of structures, modal parameters of some windows (moments) are not identified, i.e., modal parameters are missing. Those windows that modal parameters are not identified are described as un-identified windows. Table 13 shows the number of un-identified windows by MWLPP. Table 14 shows the MAC\({}_{i-avg}\) of modal shapes identified by MWLPP.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Order** & \(t\) = 100 s & \(t\) = 650 s & \(t\) = 1500 s & \(t\) = 1974.375 s \\ \hline
1 & 0.9995 & 1 & 0.9938 & 1 \\
2 & 0.9988 & 0.9997 & 0.9707 & 0.8243 \\
3 & 0.9970 & 1 & 0.9818 & 0.8731 \\ \hline \hline \end{tabular}
\end{table}
Table 11: The MAC value of four moments including \(t\) = 100 s, \(t\) = 650 s, \(t\) = 1500 s and \(t\) = 1974.375 s in the mass slow-time-varying 3-DOF structure with 10% white Gaussian noise.  
Figure 18: The MAC (model assurance criterion) value of modal shapes identified by the MWLPP in the density slow-time-varying cantilever beam structure.  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline  & \begin{tabular}{c} **First Order** \\ **Un-Identified** \\ **Number** \\ \end{tabular} & \begin{tabular}{c} **Rate** \\ **Run-Identified** \\ **Number** \\ \end{tabular} & \begin{tabular}{c} **Second Order** \\ **Run-Identified** \\ **Number** \\ \end{tabular} & \begin{tabular}{c} **Rate** \\ **Run-Identified** \\ **Number** \\ \end{tabular} & \begin{tabular}{c} **Third Order** \\ **Run-Identified** \\ **Number** \\ \end{tabular} & \begin{tabular}{c} **Rate** \\ **Number** \\ \end{tabular} \\ \hline \
\begin{tabular}{c} MWLPP \\ (with 10\% white Gaussian noise) \\ \end{tabular} & 0 & 0 & 674 & 0.876\% & 1062 & 1.380\% \\
\begin{tabular}{c} MWLPP \\ (without white Gaussian noise) \\ \end{tabular} & 0 & 0 & 808 & 1.049\% & 747 & 0.970\% \\ \hline \hline \end{tabular}
\end{table}
Table 13: The number of un-identified windows by MWLPP in the mass slow-time-varying 3-DOF structure with 10% white Gaussian noise.  
Figure 19: The modal shapes identified by MWLPP of four moments including (**a**) \(t=100\) s, (**b**) \(t=650\) s, (**c**) \(t=1500\) s and (**d**) \(t=1974.375\) s in the mass slow-time-varying 3-DOF structure with 10% white Gaussian noise.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Method** & \(\textbf{MAC}_{1-avg}\) & \(\textbf{MAC}_{2-avg}\) & \(\textbf{MAC}_{3-avg}\) \\ \hline \
\begin{tabular}{c} MWLPP \\ (with 10\% white Gaussian noise) \\ \end{tabular} & 0.9981 & 0.9260 & 0.9219 \\
\begin{tabular}{c} MWLPP \\ (without white Gaussian noise) \\ \end{tabular} & 0.9979 & 0.9344 & 0.9254 \\ \hline \hline \end{tabular}
\end{table}
Table 14: The MAC\({}_{i-avg}\) of modal shapes identified by MWLPP in the mass slow-time-varying 3-DOF structure with 10% white Gaussian noise.