## 4 Dynamic Model Updating Examples
### Engineering Example: Dynamic Model Updating of an Intermediate Case
(**b**) Correlation analysis results described by MAC matrix with the MAC threshold of 0.25. (**c**) Correlation analysis results described by MAC matrix and FRE bar graph with the MAC threshold of 0.5.  
to capture these differences by using a constant correlation pairing method based on pre-analysis. The objective function values of the best individual in each population are shown in Figure 11. The value decreasing with evolution indicated that the correlation pairing results were improved. Convergence was achieved in the 83 rd generation, for the average change in objective function value in the last-fifteen-generation population was less than the given function tolerance in Table 4.  
The objective function values and correlation pairing differences among the typical individuals in the last-generation population are shown in Figure 7:0 Compared with Figure 10, the discrete degree of objective function values was significantly reduced. The numbers of correlation pairing modes of the typical individuals were increased. The mode swapping was improved. The FRE values were also significantly reduced. Figure 7:1 Objective function value of the best individual changes with the evolution. Figure 7:2 Objective function values and correlation pairing differences in the initial population. (**a**) Objective function values of \(\textbf{J}^{(0)}\); (**b**) MAC matrix of \(\textbf{MAC}^{(0,32)}\); (**c**) MAC matrix of \(\textbf{MAC}^{(0,34)}\); (**d**) MAC matrix of \(\textbf{MAC}^{(0,8)}\); (**e**) FRE bar graphs of \(\textbf{FRE}^{(0,32)}\); (**f**) FRE bar graphs of \(\textbf{FRE}^{(0,34)}\); (**g**) FRE bar graphs of \(\textbf{FRE}^{(0,8)}\). The correction pairing results of the best individual in the last-generation population are listed in Figure 7:3 Compared with the best individual in the initial population (see Figure 10d,g), the number of correlation pairing modes was increased from six to nine after updating, which improved the exploitation of experimental modal data. The MAC values of the corresponding modes were increased. The \(RMSE_{MAC}\) was decreased to 0.2591. Besides, the FRE values of the other modes were all within 10% except the eighth mode. The \(RMSE_{FRE}\) was reduced to 6.50%. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Mode** & **Mode** & **Experimental** & **Updated Analytical** & **FRE (\%)** & **MAC** & **1-MAC** \\
**No.** & **Description** & **Frequency (Hz)** & **Frequency (Hz)** & **Frequency (Hz)** & **FRE (\%)** & **MAC** & **1-MAC** \\ \hline
1 & Radial & 248.89 & 270.87 & 8.83 & 0.8752 & 0.1248 \\
2 & Radial & 297.38 & 272.17 & \(-\)8.48 & 0.8752 & 0.1248 \\
3 & Radial & 327.13 & 339.00 & 3.63 & 0.9420 & 0.0580 \\
4 & Radial & 345.59 & 341.18 & \(-\)1.28 & 0.8616 & 0.1384 \\
5 & Torsional & 389.10 & & & & \\
6 & Radial & 587.44 & 644.32 & 9.68 & 0.7409 & 0.2591 \\
7 & Radial & 647.03 & 649.33 & 0.36 & 0.8444 & 0.1556 \\
8 & Radial & 689.50 & 761.62 & 10.46 & 0.5987 & 0.4013 \\
9 & Axial & 739.64 & & & & \\
10 & Radial & 746.37 & 760.85 & 1.94 & 0.6187 & 0.3813 \\
11 & Axial & 774.78 & & & & \\
12 & Axial & 821.28 & & & & \\
13 & Local & 831.38 & & & & \\
14 & Radial & 843.40 & 867.74 & 2.89 & 0.6091 & 0.3909 \\ \(RMSE\) & & & & 6.50 & & 0.2591 \\ \(CV\) & & & & & & \\ \hline \hline \end{tabular}
\end{table} 
Table 11: The final updated modal results for the IMC. Figure 12: Objective function values and correlation pairing differences in the last-generation population. (**a**) Objective function values of \(\mathbf{j}^{(83)}\); (**b**) MAC matrix of \(\mathbf{MAC}^{(83,35)}\); (**c**) MAC matrix of \(\mathbf{MAC}^{(83,15)}\); (**d**) MAC matrix of \(\mathbf{MAC}^{(83,3)}\); (**e**) FRE bar graphs of \(\mathbf{FRE}^{(83,35)}\); (**f**) FRE bar graphs of \(\mathbf{FRE}^{(83,15)}\); (**g**) FRE bar graphs of \(\mathbf{FRE}^{(83,3)}\). By further comparison with the initial analytical results in Figure 9, the discrepancies of the final updated results were noticeably decreased.