## 4 Dynamic Model Updating Examples
### Engineering Example: Dynamic Model Updating of an Intermediate Case
Only twelve translational acceleration sensors were bolted to the flanges, marked with arrows in the upper left corner of Figure 8. All the response points were excited five times by an impact hammer one by one to measure the response of each sensor. The analytical bandwidth ranged from 100 to 880 Hz, about four times the operating speed. Then, the first fourteen incomplete modes were identified in the range based on the poly-reference least squares complex exponential method. The modal analysis with free-free boundary condition was performed on the initial FE model. The first twenty natural frequencies and mode shapes were determined. After accomplishing the coordinate pairing in the response direction between the FE nodes and the experimental response points (see Figure 8), the MAC values in the radial plane between the initial analytical and incomplete experimental modes (realization method is the same as Section 4.1.2) were calculated. Figure 9 shows the correlation analysis results of the initial analytical and experimental modes. It can be seen that the maximal MAC value was 0.75, which was much smaller than that in Table 6. When performing correlation pairing with a larger MAC threshold, such as 0.5, the case 0\(\leftrightarrow\)n occurs frequently. If a smaller MAC threshold of 0.25 is used, though the number of pairing modes increases, the case 1\(\leftrightarrow\)n increases sharply. Therefore, the correlation pairing results change significantly due to different MAC thresholds. Figure 8: Physical photo of free–free experimental modal test of the IMC. (**a**) Experimental model and the location of the sensors. (**b**) Impact modal test instrument. (**c**) Suspension of the IMC. When the MAC threshold of 0.5 was used, the initial analytical results could only be paired with six experimental modes. The order of the third and fourth modes was swapped. The majority of FRE values of these six modes were also more than 20%. Moreover, the analytical natural frequencies were higher than the corresponding experimental results. The \(RMSE_{FRE}\) and \(RMSE_{MAC}\) of the initial analysis results were 22.68% and 0.3709, respectively. The discrepancies between them were apparent. Therefore, the initial FE model needed to be updated. Considering the structural characteristics of the IMC and the simplifications of the dynamic FE model, the density and Young's modulus of the four groups of components were used as updating parameters distinguished by colors in Figure 7: the outer casing, splitter, and inner casing in green, the wide strut in purple, the middle struts in red, and the narrow struts in blue. Similar to Section 4.1, the non-dimensional ratios of material parameters to the corresponding titanium alloy values were used. The lower and upper boundaries of all updating parameters were 0.50 and 1.50. The optimized parameters used in this example are the same listed in Table 4. Then, the initial FE model was updated by the proposed CMPES method. More details of the varied correlation pairing results due to individual differences in the same generation and evolution were given to show the advantage of the correlated mode auto-pairing strategy and population evolution screening mechanism. The objective function values \(\mathbf{J}^{(0)}\) of all individuals in the initial population were calculated by Equation (18), shown in Figure 10a. Three typical individuals in the initial population were marked: the worst individual (individual no. 32, box), the individual closest to the mean (individual no. 34, diamond), and the best individual (individual no. 8, circle). The MAC matrices and corresponding FRE bar graphs of these three individuals determined by the correlated mode auto-pairing strategy are shown in Figure 10b-g. The numbers of correlation pairing modes and the swapped mode orders differed. The corresponding FRE values changed significantly among individuals. It is not possible  
Figure 9: Correlation analysis results of the initial analytical and experimental modes of the IMC. (**a**) MAC matrix.