# Model Updating Using Frequency Response Functions Based on Sherman-Morrison Formula
## 2 Theoretical Analysis
### Frequency Response Function (FRF)'s Model Updating Based on Sensitivity Analysis  
The purpose of the model updating is to minimize the difference of FRFs between the experimental and the FE model. The difference between the FRFs of experimental model \(\mathbf{H}^{*}(\omega)\) and the FRFs of FE model \(\mathbf{H}(\omega)\) are used to construct the objective function as follows:  
\[min\|\delta\|^{2} \tag{17}\]  
\[\delta=\mathbf{H}^{*}(\omega)-\mathbf{H}(\omega) \tag{18}\]  
where \(\delta\) is the difference between the experimental FRFs and the corresponding analysis values, which is called the residual vector. Equation (4) is rewritten as:  
\[\mathbf{S}=\frac{\partial\mathbf{H}}{\partial\mathbf{P}_{m}}=\underset{\Delta\mathbf{P}\to 0}{ lim}\frac{\Delta\mathbf{H}(\omega)}{\Delta\mathbf{P}}=\underset{\Delta\mathbf{P}\to 0}{ lim}\left[-\mathbf{H}^{*}(\omega)\,\frac{\Delta\mathbf{Z}}{\Delta\mathbf{P}_{m}}\mathbf{H}( \omega)\right] \tag{19}\]\[\delta=\Delta\textbf{H}(\omega)=-\textbf{H}^{*}(\omega)\Big{(}\Delta\textbf{K}+j \omega_{r}\Delta\textbf{C}-\omega_{r}{}^{2}\Delta\textbf{M}\Big{)}\textbf{H}(\omega) \tag{20}\]  
where the changes of the total mass matrix \(\Delta\textbf{M}\) and the total stiffness matrix \(\Delta\textbf{K}\) of the structure can be expressed by the sum of partial derivatives of each updating element:  
\[\Delta\textbf{M}=\sum_{i=1}^{nm}\frac{\partial\textbf{M}}{\partial\textbf{P}_{ i}^{M}}\Delta\textbf{P}_{i}^{M} \tag{21}\]  
\[\Delta\textbf{K}=\sum_{j=1}^{ns}\frac{\partial\textbf{K}}{\partial\textbf{P}_{ j}^{S}}\Delta\textbf{P}_{j}^{S} \tag{22}\]  
where \(nm\) and \(ns\) are the total number of mass and stiffness updating units, and \(\Delta\textbf{P}_{i}^{M}\) represents the change of the \(i\)th mass updating parameter, \(\Delta\textbf{P}_{j}^{S}\) represents the change of the \(j\)th stiffness updating parameter. By substituting Equations (21) and (22) into Equation (20), and setting \(\Delta\textbf{C}=0\), which means ignoring the updating of damping parameter  
\[\delta=\sum_{i=1}^{nm}\left\{-\textbf{H}^{*}(\omega)\left(-\omega_{r}^{2} \frac{\partial\textbf{M}}{\partial\textbf{P}_{i}^{M}}\right)\textbf{H}(\omega )\right\}\Delta\textbf{P}_{i}^{M}+\sum_{j=1}^{ns}\left\{-\textbf{H}^{*}( \omega)\left(\frac{\partial\textbf{K}}{\partial\textbf{P}_{j}^{S}}\right) \textbf{H}(\omega)\right\}\Delta\textbf{P}_{j}^{S} \tag{23}\]  
Equation (23) is rewritten as follows  
\[\delta=\left[ \begin{array}{cc}S^{M}&S^{S}\\ \end{array} \right]\left[ \begin{array}{c}\Delta\textbf{P}^{M}\\ \Delta\textbf{P}^{S}\\ \end{array} \right]=S\Delta\textbf{P} \tag{24}\]  
where \(S^{M}\) and \(S^{S}\) are the sensitivity matrix of mass and stiffness, respectively, \(S\) is the sensitivity matrix. \(\Delta\textbf{P}^{M}\) and \(\Delta\textbf{P}^{S}\) are the variation of mass and stiffness parameters, and \(\Delta\textbf{P}\) is the vector of all the variation of stiffness and mass parameters. The model updating of parameters is an iterative process of convergence, and the regularization process is required to solve Equation (24). Generally, the following estimation criteria are obtained by adding constraint information for the variation of updating parameters:  
\[min\left\{\|S\Delta\textbf{P}-\delta\|_{2}^{2}+\lambda^{2}\|\Delta\textbf{P} \|_{2}^{2}\right\} \tag{25}\]  
where \(\lambda^{2}\) is the regularization parameter whose value can be determined by the \(L\)-curve method. It can be obtained that the estimated variation of the updating parameters according to Equation (25):  
\[\Delta\textbf{P}=\left[\textbf{S}^{T}\textbf{S}+\lambda^{2}\textbf{I}\right] ^{-1}\textbf{S}^{T}\delta \tag{26}\]  
Therefore, the iterative formula of the updating parameters is:  
\[\textbf{P}_{n+1}=\textbf{P}_{n}+\left[\textbf{S}_{n}{}^{T}\textbf{S}_{n}+ \lambda^{2}\textbf{I}\right]^{-1}\textbf{S}_{n}{}^{T}\delta_{n} \tag{27}\]  
where \(n\) and \(n+1\), respectively, represents the values corresponding to the analysis model in the iteration of step \(n\) and \(n+1\).