# Model Updating Using Frequency Response Functions Based on Sherman-Morrison Formula
## 2 Theoretical Analysis
### Frequency Range Selection  
The FRFs of the real experimental model and the FE model around the \(n\)th natural frequency is shown in Figure 1. The modeling errors will cause the FRFs of the same mode order to move along the coordinate axis to a certain extent, but the basic shape does not change significantly.  
The approximate estimation of the FRFs \(\mathbf{H}_{d}(\omega)\) near the resonance region can be expressed as  
\[\mathbf{H}_{d}(\omega)\cong\mathbf{H}(\omega+\delta\omega_{n}),n=1,\cdots,m \tag{28}\]  
where  
\[\delta\omega_{n}=\omega_{n}-\omega_{dn},n=1,\cdots,m \tag{29}\]  
where, \(\omega_{n}\) and \(\omega_{dn}\) are the \(n\)th natural frequency of the real experimental model and the FE model, respectively, \(\delta\omega_{n}\) is the frequency variation of the real experimental model and the FE model, and \(m\) is the order of the natural frequency taken by simulation.  
The frequency range of model updating is expressed as \(\overleftarrow{\omega}\). According to Equations (28) and (29), \(\mathbf{H}_{d}(\omega)\) on \(\overleftarrow{\omega}\) can be approximately expressed as  
\[\mathbf{H}_{d}(\overline{\omega})\cong\mathop{\cup}\limits_{n=1}^{m}\mathbf{H}( \overline{\omega}+\delta\omega_{n})=\mathop{\cup}\limits_{n=1}^{m}\left(-( \overline{\omega}+\delta\omega_{n})^{2}(\mathbf{M}+\delta\mathbf{M})+j(\overline{ \omega}+\delta\omega_{n})(\mathbf{C}+\delta\mathbf{C})+\mathbf{K}+\delta\mathbf{K}\right)^{-1} \tag{30}\]  
After repeated iterative calculations, the structural parameters of the FE model will be constantly updated and gradually approach to the parameters of the real experimental model. Therefore, in Equations (28) and (29), the initial frequency \(\omega_{n}\) and FRFs \(\mathbf{H}(\omega)\) will be constantly updated iteratively and converge to the natural frequency and FRFs of the real experimental model.  
At present, most of the damping models cannot accurately simulate the actual damping effect but will increase the model errors and affect the result of model updating. Therefore, the damping parameter is not updated in this paper, so the influence of structural damping on the result of model updating should be reduced. From the analysis in Section 2.2, the damping has little effect on the frequency response amplitude when far away from the resonance peak. Therefore, the selection of appropriate frequency range can alleviate the influence of damping on the results of model updating, and the influence of damping can be ignored when the model is updated based on FRFs. The frequency  
Figure 1: The frequency response function (FRF)’s of the real experimental model and the finite element (FE) model around the \(n\)th natural frequency.  
close to resonance would be affected by damping, so the small region of about 4 Hz around resonance was not considered in the model updating process [14].  
If the two adjacent resonant peaks are too close to each other, the variation of FRFs in the frequency band between them cannot be guaranteed to decrease monotonically in the updating process, so this region is excluded to avoid the amplitude nonlinear problem. In each iteration, the frequency range also needs to be adjusted according to the change of the resonance point.