# Model Updating Using Frequency Response Functions Based on Sherman-Morrison Formula
## 4 Case Study
### Truss Structure
#### 4.1.3 Parameter Updating
Compared with the result in noiseless environment, the maximum error of the elastic modulus is within 1%, which greatly improves the reliability of the FE model and verifies the feasibility of the model updating method in this paper. It can also be seen from the iterative convergence diagram of elastic modulus \(\mathbf{E}\) of the truss in Figure 5, under the excitation of 5% random white noise, after 15 iterative computations, each parameter tends to converge, and the maximum error of each parameter is within \(\pm 1.00\%\) after convergence. Therefore, the results verify that the model updating method using FRFs based on Sherman-Morrison formula is still applicable in noisy environment. As shown in Figures 6 and 7, the FRFs before and after updating are compared with the target value under 5% random white noise. There is obviously a big gap between the FRFs of the FE model before updating and the target model under 5% random white noise, and the FRFs data of the FE model almost completely coincides with the target value after updating, which shows that the model updating method using FRFs based on Sherman-Morrison formula can effectively reduce noise interference. Each iteration is based on the sensitivity of the response at all the frequency points involved in the model updating. However, too many frequency points involved in the updating will greatly increase the computation and affect the efficiency of model updating. The change of FRFs is often discontinuous at the resonance peak, which especially happens in the low-damping structure. Therefore, the iterative process will not converge when the response of the frequency points close to the resonance peak  
Figure 7: Comparison of FRFs data measured at Node 10 under 5% random white noise. are used for updating. So, it is generally necessary to select the frequency point carefully before model updating. Taking the FRFs of Node 10 under 5% random white noise as an example, three frequency points selection cases are presented are given as follows, and the influence of them on the results of model updating is compared by observing the convergence of the elastic modulus of each element. * Case 1: Select one frequency point every 10 Hz in the frequency band from 0 to 400 Hz and select 40 points in total. * Case 2: Select 40 points randomly in the frequency band from 0 to 400 Hz. * Case 3: Select 8 frequency points at the interval of 0.5 Hz from 10 Hz to the left of each resonance peak in the frequency band from 0 to 400 Hz and select 40 points in total. Figure 8 shows the comparison of elastic modulus updating results of three cases. Since the frequency points selected by Case 1 and Case 2 do not avoid the resonance peak, so the FRFs are discontinuous which causes the elastic modulus of the two cases is not convergent. While only after 7 iterations, the elastic modulus of Case 3 is convergent. However, due to the noise pollution of 5% in the FRFs data, the result does not converge to 0%, but the maximum error is within \(\pm\)1.5%. By comparing the results of model updating using all the FRFs data, it can be seen that only a small amount of FRFs data is needed to complete the model updating, which indicates that the correct selection of frequency points can effectively improve the computational efficiency of model updating, and further proves that the proposed method has high computational efficiency and accuracy. Furthermore, only a small amount of FRFs data can be selected to complete the model updating quickly, the result is not very perfect compared with the result of using all frequency points, which indicates that increasing the frequency point can correspondingly increase the information contained in the FRFs data, which is an effective means to improve the quality of model updating.