# Model Updating Using Frequency Response Functions Based on Sherman-Morrison Formula
## 4 Case Study
### Truss Structure
#### 4.1.3 Parameter Updating  
Due to the inevitable interference of various noises in the real experiment, in order to simulate the real experiment state and test the anti-noise interference ability of the method proposed in this paper, 5% random white noise was added to the target value which was obtained in Section 4.1.1 in the form of \(\mathbf{H^{*}}(\omega)=\mathbf{H^{*}}(\omega)\) (\(1+0.05\)\(\mu\)), where the \(\mu\) is a random number of \(-1\)\(\sim\)\(1\), \(\mathbf{H^{*}}(\omega)\) is the target value obtained in Section 4.1.1, and \(\mathbf{H^{*}}(\omega)\) is the FRFs after adding random white noise. Then, the parameter updating formula shown in Equation (27) was used for 15 iterative computations. The frequency comparison between the FE model and the target model before and after modal updating is shown in Table 3, and the comparison results of the updated parameters are shown in Table 4. The convergence process of updated parameters is shown in Figure 5. The comparison of FRFs before and after updating is shown in Figures 6 and 7. \begin{table}
\begin{tabular}{c c} \hline
**Perturbation** & **Error\%** \\ \hline \(10^{-3}\) & 0.01 \\ \(10^{-2}\) & 0.24 \\ \(10^{-1}\) & 1.78 \\ \hline \end{tabular}
\end{table} 
Table 1: Errors between Proposed Method and Analytical Derivatives. \begin{table}
\begin{tabular}{c c} \hline
**The Analytic Derivatives Method** & **The Proposed Method** \\ \hline
**Time Cost/s** & 0.842 & 0.653 \\ \hline \end{tabular}
\end{table} 
Table 2: Comparison of the Time Cost. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Element ID**} & \multicolumn{2}{c}{**Target**} & \multicolumn{2}{c}{**Before**} & \multirow{2}{*}{**Error\({}_{b}\)/\%**} & \multicolumn{2}{c}{**After**} & \multirow{2}{*}{**Error\({}_{a}\)/\%**} \\  & **Value/Gpa** & **Updating/Gpa** & & & **Updating/Gpa** & \\ \hline
1 & 77.00 & 70.00 & 10.00 & 77.66 & 0.85 \\
6 & 84.00 & 70.00 & 20.00 & 84.62 & 0.74 \\
11 & 84.00 & 70.00 & 20.00 & 83.61 & \(-\)0.47 \\
14 & 84.00 & 70.00 & 20.00 & 83.29 & \(-\)0.85 \\
21 & 77.00 & 70.00 & 10.00 & 76.64 & \(-\)0.47 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Comparison of Parameters Before and After Model Updating Under 5% White Noise. Figure 5: Convergence process of designed parameters under 5% white noise. Figure 6: Comparison of FRFs data measured at Node 4 under 5% random white noise. The model updating process uses the frequencies of the first 5 order of the target value 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. Where \(\rm Error_{D}\) and \(\rm Error_{a}\) 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 3, under the excitation of 5% random white noise, there is a significant difference between the frequency of the FE model and the target model before the modal updating, and the maximum frequency error has reached \(-3.26\%\). This indicates that the reliability of the FE model is questionable, so it is necessary to update the initial FE model to improve the accuracy and reliability. After using the Sherman-Morrison formula proposed in this paper, the maximum error of the frequency of the first 7 order after modal updating is only \(-0.50\%\), which verifies that the model updating method using FRFs based on Sherman-Morrison formula has a good anti-noise performance. Where \(\rm Error_{D}\) and \(\rm Error_{a}\) is the relative error between the elastic modulus of the FE model before and after the updating and the target value, respectively. The model updating results of elastic modulus are shown in Table 4, under the excitation of 5% random white noise, the maximum relative error of the corrected elastic modulus is changed from 20.00% to \(-0.85\%\).