# Model Updating Using Frequency Response Functions Based on Sherman-Morrison Formula
## 4 Case Study
### Solar Wing
#### 4.2.2 Parameter Updating  
The elastic modulus of the connecting beams and the solar wing, and the density and the cross-sectional area of the connecting beams are updated in turn with the strategy of step by step. Meanwhile, 5% random white noise was added to the target value of the FRF data obtained in Section 4.2.1 to simulate the real experimental situation in the form of \(\mathbf{H^{\ast\ast}}(\omega)=\mathbf{H^{\ast}}(\omega)\left(1+0.05\mu\right)\), where the \(\mu\) is a random number of \(-1\)-\(1\), \(\mathbf{H^{\ast}}(\omega)\) is the target value obtained in Section 4.2.1, and \(\mathbf{H^{\ast\ast}}(\omega)\) is the FRFs after adding random white noise. Then, the parameter updating formula shown in Equation (27) was used for 20 iterative computations. The model updating process uses the frequencies of the first 5 order of the target model and their corresponding FRFs, while the fifth and sixth frequencies and their corresponding FRFs do not directly participate in the updating, but are used to evaluate the accuracy of the model updating method which is proposed in this paper. Combined with the rules in Section 2.3, the frequency of the model updating is selected as the frequency point near the resonance region that is shown in Figure 10. Remove the low frequency band containing the first two resonance regions, and to avoid the problem of amplitude nonlinearity, the frequency points between the fourth and fifth resonances are ignored. 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 target value, respectively. As can be seen from Table 5, under the excitation of 10% random white noise, the maximum relative error of the frequency is changed from \(-5.49\%\) to \(-1.10\%\), which verifies the effectiveness and anti-noise performance of the proposed method in this paper. Where \(\text{Error}_{\text{b}}\) and \(\text{Error}_{\text{a}}\) is the relative error between the parameters of the FE model before and after the updating and the target value, respectively. The model updating results of parameters are shown in Table 6, under the excitation of 10% random white noise, the maximum relative error of the updated parameters is changed from 30.00% to \(-0.24\%\), which greatly improves the reliability of the FE model and verifies the feasibility of the model updating method in this paper. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Parameters** & **Target Value** & \begin{tabular}{c} **Before** \\ **Updating** \\ \end{tabular} & \begin{tabular}{c} **Error\({}_{\text{b}}\)/\%** \\ **Updating** \\ \end{tabular} & \
\begin{tabular}{c} **After** \\ **Updating** \\ \end{tabular} &
\begin{tabular}{c} **Error\({}_{\text{a}}\)/\%** \\ **Updating** \\ \end{tabular} \\ \hline E\({}_{1}\)/Cpa & 273.00 & 210.00 & 30.00 & 272.35 & \(-0.24\) \\ E\({}_{2}\)/Cpa & 84.00 & 70.00 & 20.00 & 84.23 & 0.27 \\ \(\rho_{1}\)/kg/m\({}^{3}\) & 9027.50 & 7850.00 & 15.00 & 9002.22 & \(-0.28\) \\ \(A_{1}\)/m\({}^{2}\) & \(4.80\times 10^{-4}\) & \(4.00\times 10^{-4}\) & 20.00 & \(4.82\times 10^{-4}\) & 0.42 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Comparison of Parameters Before and After Model Updating Under 5% White noise. Figure 10: Selected frequency range. As shown in Figures 11 and 12, due to modeling errors between the target model and the FE model, there is obviously a big gap between the FRFs of the FE model before updating and the target model under 5% random white noise. While after 20 iterative computations, the error between the obtained FRFs and the target value is quite close, which shows that the model updating method using FRFs based on Sherman-Morrison formula can effectively reduce noise interference. At the same time, the influence of damping on the amplitude of FRFs is effectively reduced by choosing the appropriate frequency range. Therefore, the results verify that the model updating method using FRFs based on Sherman-Morrison formula has a good anti-noise performance.