# Model Updating Using Frequency Response Functions Based on Sherman-Morrison Formula
## 4 Case Study
### Experimental Verification
#### 4.3.2 Parameter Updating  
Take the FRFs h\({}_{13}\) measured by Accelerometer 1 as an example. Combined with the rules in Section 2.3, the frequency of the model updating is selected as the frequency point near the resonance  
Figure 14: Accelerometers and excitation point: (a) Experimental arrangement; (b) FE model. Figure 13: Dimension of the aluminum solar panel frame. region, and to avoid the problem of amplitude nonlinearity, the frequency points of the third resonances are ignored as is shown in Figure 15.  
Where \(\text{Error}_{\text{b}}\) and \(\text{Error}_{\text{a}}\) is the relative error between the frequency values of the FE model before and after the updating and the experimental value, respectively. As can be seen from Table 7, after several iterative computations, the maximum relative error of the frequency is changed from 9.13% to 2.99%. Although the error between the second order frequency and the experimental value increased slightly after updating, the average error of the first four orders decreased from 7.03% to 1.72%, which verifies the effectiveness of the proposed method in this paper. The identification results of accelerometer mass are shown in Table 8. Since the accelerometer is attached with wires, the identification results will be affected to some extent, so it can be considered that the identification results are within the ideal range, which greatly improves the reliability of the FE model and verifies the feasibility of the model updating method proposed in this paper. As shown in Figure 16, due to the additional mass of the accelerometers between the real experimental model and the FE model, there is obviously a big gap between the FRFs of the FE model before updating and the experimental structure. While after several iterative computations, the error between the obtained FRFs and the real experimental value h\({}_{13}\) is quite close, which shows that the proposed method has a good model updating capability and engineering application. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Mode No.** & **Experimental/Hz** & \begin{tabular}{c} **Before** \\ **Updating/Hz** \\ \end{tabular} & \begin{tabular}{c} **Error\({}_{\text{b}}\)/\%** \\ \end{tabular} & \
\begin{tabular}{c} **After** \\ **Updating/Hz** \\ \end{tabular} &
\begin{tabular}{c} **Error\({}_{\text{a}}\)/\%** \\ \end{tabular} \\ \hline
1 & 2.15 & 2.34 & 8.84 & 2.15 & 0 \\
2 & 7.81 & 7.92 & 1.41 & 7.62 & \(-\)2.43 \\
3 & 12.71 & 13.87 & 9.13 & 13.09 & 2.99 \\
4 & 26.76 & 29.10 & 8.74 & 27.15 & 1.46 \\ \hline \hline \end{tabular}
\end{table} 
Table 7: Comparison of Frequency Values between FE Model and Experimental Model. Figure 15: Selected frequency range. At the same time, the experimental modal shapes are measured to compare with the FE modal shapes, to further verify the accuracy of the proposed method. Due to the limitation of test conditions, it is difficult to measure the displacement of all nodes. Therefore, 24 points as shown in Figure 17a are selected to measure the displacement in the \(Z\) direction by the hammer. The first 4 modal shapes of the experiment model are obtained by MaCras. Meanwhile, 24 nodes of the FE model corresponding to the measured points of the experimental model are selected, as shown in Figure 17b. The first 4 modal shapes of the FE model before and after updating are obtained through FE analysis. Using modal assurance criteria (MAC) to verify the mutual independence and consistency of the modal shapes between the first 4 order of the FE model before and after updating and the corresponding experimental model:  
\[MAC=\frac{\left|\phi_{u}{}^{T}\phi_{v}\right|^{2}}{\phi_{u}{}^{T}\phi_{u}\phi_ {v}{}^{T}\phi_{v}} \tag{33}\]  
The \(\phi_{u}\), \(\phi_{v}\) are the modal vector of the FE model and the experimental model, respectively, which are all the column vector, \(u\), \(v\) is the modal order.