# Model Updating Using Frequency Response Functions Based on Sherman-Morrison Formula
## 2 Theoretical Analysis
### Sensitivity Analysis Based on SMW  
The dynamic equation of the multi-degrees-of-freedom system in the frequency domain can be expressed as  
\[\Big{(}-\omega^{2}\mathbf{M}+j\omega\mathbf{C}+\mathbf{K}\Big{)}\mathbf{x}(\omega)=\mathbf{F}(\omega) \tag{1}\]  
where \(\mathbf{M}\) is the mass matrix, \(\mathbf{K}\) is the stiffness matrix, \(\mathbf{C}\) is the damping matrix, \(\mathbf{x}(\omega)  
Equation (5) can be expressed in differential form as  
\[\delta\mathbf{Z}=\left(\delta\mathbf{K}+j\omega_{r}\delta\mathbf{C}-\omega_{r}^{2}\delta\mathbf{M}\right) \tag{6}\]  
where \(\delta\mathbf{Z}\), \(\delta\mathbf{M}\), \(\delta\mathbf{C}\), and \(\delta\mathbf{K}\) represent the variations of dynamic stiffness matrix, mass matrix, damping matrix, and stiffness matrix, respectively. However, the damping of the structure is related to the material performance of individual members and the difference of FE modeling. The influence of damping on the FRF amplitude is significant in resonance and decreases rapidly as the damping moves away from resonance. Therefore, the selection of appropriate frequency range can mitigate the impact of damping on the model updating results. We assume that a system of \(n\)-degrees of freedom is subject to mass perturbation \(\delta m_{i}\) at the \(i\)th-degree of freedom, and stiffness perturbation \(\delta k_{i}\) at the \(j\)th-degree of freedom [32], the perturbations of the mass matrix \(\delta\mathbf{M}\) and the stiffness matrix \(\delta\mathbf{K}\) can be expressed as  
\[\delta\mathbf{M}=\left[ \begin{array}{ccccc}0&\cdots&0&\cdots&0\\ \vdots&\ddots&\vdots&\vdots&\vdots\\ 0&\vdots&\delta m_{i}&\vdots&0\\ \vdots&\vdots&\vdots&\ddots&\vdots\\ 0&\cdots&0&\cdots&0\end{array} \right] \tag{7}\]  
\[\delta\mathbf{K}=\left[ \begin{array}{ccccc}0&\cdots&0&\cdots&0\\ \vdots&\ddots&\vdots&\vdots&\vdots\\ 0&\vdots&\delta k_{j}&\vdots&0\\ \vdots&\vdots&\vdots&\ddots&\vdots\\ 0&\cdots&0&\cdots&0\end{array} \right] \tag{8}\]  
We can rewrite the perturbations of the mass matrix \(\delta\mathbf{M}\) and the stiffness matrix \(\delta\mathbf{K}\) as  
\[\left\{ \begin{array}{l}\delta\mathbf{M}=\mathbf{G}_{uv}\mathbf{G}_{uv}{}^{T}\\ \delta\mathbf{K}=\mathbf{X}_{uv}\mathbf{X}_{uv}{}^{T}\end{array} \right. \tag{9}\]  
Where \(u\) and \(v\) represent the nodes on each side of the unit. \(\mathbf{G}_{uv}\) and \(\mathbf{X}_{uv}\) are expressed as  
\[\mathbf{G}_{uv}=\sqrt{\delta m_{i}}\left[ \begin{array}{ccccc}0&\cdots&0&\cdots&0 \\ \vdots&\ddots&\vdots&\vdots&\vdots\\ 0&\vdots&1&\vdots&0\\ \vdots&\vdots&\vdots&\ddots&\vdots\\ 0&\cdots&0&\cdots&0\end{array} \right] \tag{10}\]  
\[\mathbf{X}_{uv}=\sqrt{\delta k_{j}}\left[ \begin{array}{ccccc}0&\cdots&0&\cdots&0 \\ \vdots&\ddots&\vdots&\vdots&\vdots\\ 0&\vdots&1&\vdots&0\\ \vdots&\vdots&\vdots&\ddots&\vdots\\ 0&\cdots&0&\cdots&0\end{array} \right] \tag{11}\]Therefore, the variation of the system dynamic stiffness matrix \(\delta\mathbf{Z}\) can be expressed as  
\[\delta\mathbf{Z}=-\omega_{r}{}^{2}\mathbf{G}_{uv}\mathbf{G}_{uv}{}^{T}+\mathbf{X}_{uv}\mathbf{X}_{uv}{ }^{T} \tag{12}\]  
The inverse for the variation of dynamic stiffness matrix \(\delta\mathbf{Z}\) which is the variation of FRFs \(\delta\mathbf{H}\) is obtained by applying the Sherman-Morrison formula  
\[\delta\mathbf{H}=\delta\mathbf{Z}^{-1}=\mathbf{H}^{\prime}(\omega_{r})-\mathbf{H}(\omega_{r}) \tag{13}\]  
where \(\mathbf{L}\) is a \(2N\times 2N\) matrix given by  
\[\mathbf{L}=\left[ \begin{array}{cc}\mathbf{I}_{N\times N}-{\omega_{r}}^{2}\mathbf{G}_{uv} ^{T}\mathbf{H}\mathbf{G}_{uv}&\omega_{r}\mathbf{G}_{uv}{}^{T}\mathbf{H}\mathbf{X}_{uv}\\ -\omega_{r}\mathbf{X}_{uv}{}^{T}\mathbf{H}\mathbf{G}_{uv}{}^{T}&\mathbf{I}_{N\times N}+\mathbf{X}_ {uv}{}^{T}\mathbf{H}\mathbf{X}_{uv}\end{array} \right] \tag{14}\]  
By substituting Equation (13) into Equation (4), the sensitivity of the FRFs with respect to perturbation parameters can be obtained.