# Model Updating Using Frequency Response Functions Based on Sherman-Morrison Formula
## 4 Case Study
### Solar Wing
#### 4.2.1 FE Modeling  
A complete FE model of the solar wing was modeled by the FE analysis software 'Nastran' based on the size diagram of the solar wing model as shown in Figure 9. The solar wing is an up-down axisymmetric structure, consisting of two plates with identical materials and geometric shapes. Each plate is divided into 12 shell elements numbered 1-24, and the two plates are connected by beam elements numbered 25-34. The solar wing model consists of 34 elements and 47 nodes, each node contains translational and rotational DOFs in the direction of \(X\), \(Y\), and \(Z\). Since Node 44, 45, and 46 are  
Figure 8: Comparison of elastic modulus updating results of three cases: (**a**) Case 1; (**b**) Case 2; (**c**) Case 3.  
fixed, and the rest are hinged, the translational and rotational DOFs in the \(X\), \(Y\), and \(Z\) directions of these three nodes are all locked. Therefore, the total DOFs of the solar wing structure is 264. The material parameters of the finite element model are as follows: the elastic modulus of the beam is \(E_{1}=200\) Gpa, the density is \(\rho_{1}=7850\) kg/m\({}^{3}\), the cross-sectional area of the beam is \(A_{1}=4\times 10^{-4}\) m\({}^{2}\), and the poisson ratio is \(\mu_{1}=0.32\). The elastic modulus of the solar wing is \(E_{2}=70\) Gpa, the density is \(\rho_{2}=2700\) kg/m\({}^{3}\), the thickness is \(t=4\times 10^{-3}\) m, and the poisson ratio is \(\mu_{2}=0.34\). To simulate the actual damping of the model, the Rayleigh proportional damping model is adopted, as is shown below  
\[C=a\mathbf{M}+b\mathbf{K} \tag{32}\]  
where \(C\) represents the damping matrix of the system, \(\mathbf{M}\) represents the mass matrix of the system, \(\mathbf{K}\) represents the stiffness matrix of the system, \(a\) and \(b\) represent the damping coefficients. In this case, the actual damping coefficients are assumed to be \(a=0.2\) and \(b=0.0005\).  
The comparison of frequency values between the FE model and the target model under 5% white noise is shown in Table 5. The modeling errors of the target model that needs to be updated and the FE model are shown in Table 6. There are initial errors of 30.00%, 20.00%, 15.00%, and 20.00% in \(E_{1}\), \(E_{2}\), \(\rho_{1}\), and \(A_{1}\), respectively. It is assumed that the simulation experiment is a single point excitation, and the excitation point and response point in the target model is the same one. The frequency of the first 7 order and FRFs of Node 13 and Node 26 are obtained as the target value to conduct the model updating progress.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Mode No.**} & **Target** & **Before** & \multirow{2}{*}{**Error\({}_{b}\)/\%**} & **After** & \multirow{2}{*}{**Error\({}_{a}\)/\%**} \\  & **Value/Hz** & **Updating/Hz** & & & **Updating/Hz** & \\ \hline
1 & 0.25 & 0.24 & \(-\)4.00 & 0.25 & 0 \\
2 & 0.91 & 0.86 & \(-\)5.49 & 0.90 & \(-\)1.10 \\
3 & 2.09 & 1.99 & \(-\)4.78 & 2.08 & \(-\)0.48 \\
4 & 4.44 & 4.24 & \(-\)4.50 & 4.38 & \(-\)1.35 \\
5 & 5.69 & 5.43 & \(-\)4.57 & 5.68 & \(-\)0.18 \\
6 & 8.63 & 8.23 & \(-\)4.63 & 8.59 & \(-\)0.46 \\
7 & 9.35 & 8.93 & \(-\)4.49 & 9.29 & \(-\)0.64 \\ \hline \hline \end{tabular}
\end{table}
Table 5: Comparison of Frequency Values between FE Model and Target model Under 5% White noise.  
Figure 9: Dimensions of solar wing.