## 4 Dynamic Model Updating Examples
### Engineering Example: Dynamic Model Updating of an Intermediate Case  
This section describes the updating of an intermediate case (IMC) of a gas turbine to validate the potential of the proposed CMPES method for complex engineering structures. IMC is a vitally important transition channel between a low-pressure compressor and a high-pressure compressor, and it is the main load-bearing component because the thrust generated by the engine is transmitted through it. The IMC is welded by titanium alloy consisting of the outer casing, strut, splitter, and inner casing. The cross-sectional shapes of the twelve hollow struts (abbreviated as 1#-12#) are not the same due to the different functions. Holes with different apertures, cross-sectional shapes, and locations are processed on the multi-layer thin-walled cases. In addition, many installation accessories are assembled on it, including mount, rectifier rotating rocker arms, and lubricating oil ducts. Therefore, it is a great challenge to establish an accurate FE model. The initial FE model of the IMC, ignoring local details such as holes, chamfers, and accessories, was built by 18,936 three-dimensional elements, as illustrated in Figure 7. According to the cross-sectional shapes, the struts can be divided into three kinds: wide strut (1#), middle struts (3#, 7#, and 11#), and narrow struts (2#, 4#, 5#, 6#, 8#, 9#, 10#, and 12#). To simplify the dynamic FE model, the irregular cross-sectional shapes of these struts were all equivalent to a hollow thin-walled rectangle with constant bending stiffness of the narrow struts, shown in Figure 7b. All the elements were assigned to the same material attribute of titanium alloy: density was 4500 kg/m\({}^{3}\), [PERSON]'s modulus was 109.8 GPa, and Poisson's ratio was 0.3. \begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline \multirow{2}{*}{\
\begin{tabular}{c} **Mode** \\ **No.** \\ \end{tabular} } & \multirow{2}{*}{
\begin{tabular}{c} **“Experimental”Frequency** \\ **(Hz)** \\ \end{tabular} } & \multicolumn{4}{c}{**MUM**} & \multicolumn{4}{c}{**CMPES**} \\ \cline{3-10}  & & & **Frequency** & **FRE (\%)** & **MAC** & **1-MAC** & **Frequency** & **FRE (\%)** & **MAC** & **1-MAC** \\ \hline
1 & 12.01 & 12.17 & 1.34 & 1.00 & 0.00 & 12.01 & 0.00 & 1.00 & 0.00 \\
2 & 54.73 & 54.86 & 0.24 & 1.00 & 0.00 & 54.73 & 0.00 & 1.00 & 0.00 \\
3 & 62.38 & 62.52 & 0.21 & 1.00 & 0.00 & 62.38 & 0.00 & 1.00 & 0.00 \\
4 & 260.71 & 260.85 & 0.05 & 1.00 & 0.00 & 260.71 & 0.00 & 1.00 & 0.00 \\
5 & 1110.40 & 1110.46 & 0.01 & 1.00 & 0.00 & 1110.41 & 0.00 & 1.00 & 0.00 \\
6 & 1150.50 & 1150.57 & 0.01 & 1.00 & 0.00 & 1150.52 & 0.00 & 1.00 & 0.00 \\
7 & 1208.40 & 1208.67 & 0.02 & 1.00 & 0.00 & 1208.37 & 0.00 & 1.00 & 0.00 \\
8 & 2130.40 & 2130.63 & 0.01 & 1.00 & 0.00 & 2130.46 & 0.00 & 1.00 & 0.00 \\
9 & 2245.80 & 2245.80 & 0.00 & 1.00 & 0.00 & 2245.77 & 0.00 & 1.00 & 0.00 \\ RMSE & & & & 0.46 & & & 0.00 & & 0.00 \\ \(CV\) & & & & 0.00 & & & & 0.00 & \\ \hline \hline \end{tabular}
\end{table} 
Table 10: Comparison of the updated results of the F-shaped structure by the MUUM and CMPES methods based on the simulated “experimental” data. Figure 7: The initial dynamic FE model of the IMC. **(a)** Isometric view. **(b)** Left view and partial view of the strut section. **(c)** Front view. In order to update the FE model, the multiple reference impact technique was employed to conduct the quasi-free-free experimental modal test in the radial plane (the _xoy_ plane in Figure 7). As shown in Figure 8, the testing IMC was suspended horizontally by a soft rope reeved through bolt holes on the rear flange of the outer ring to approximately represent the free-free boundary in the radial plane. Considering the distribution characteristics of the struts and the measurable location of response and excitation points, the test model of 372 response points for the testing IMC was established.