## 3 Results  
### Identification of Resonance and Antiresonant Frequencies Based on FEM  
According to Figure 1, the finite element model of the cantilever beam was constructed by using ABAQUS 2016. The beam was divided into 10 equal beam elements and one concentrated mass unit, among which the spring element is used for elastic support. Since only the validity of the proposed method is verified and the study object is a finite element model with the same number of grids, the influence of mesh number on the accuracy of the finite element analysis results is ignored here. In addition, with large-scale structures such as offshore platforms and offshore wind turbines, it is impossible to arrange too many vibration measuring points due to the limitations of test conditions such as underwater structures or high-rise structures. So, node 4 under the mass unit is selected to carry out the FEM updating considering actual conditions.  
In the finite element analysis of the model, the measured impulse force signal of the force hammer with a sampling frequency of 2048 Hz and sampling time of 4 s is taken as the excitation signal at node 4 (i.e., the driving point) in the finite element model in a vertically downward direction. Because the excitation signal is the measured hammering impulse signal, the signal contains noise. The acceleration response signals of node 4 under two conditions of initial value and target value are calculated by using ABAQUS 2016. The point FRFs of node 4 are calculated by using \"ftestimate\" in MATLAB R2019a, respectively, as shown in Figures 4 and 5. There are three antiresonant frequencies and four resonance frequencies within the considered frequency range according to the characteristics of point FRF at node 4. Since it is difficult for large-scale civil engineering structures to excite high-order modal information, only low-order modal can be obtained. In addition, we want to adopt the minimum characteristic quantities to modify the updating parameters. Therefore, the first four-order resonance frequencies and the first-order antiresonant frequency are identified in the paper as characteristic quantities of the objective function, as shown in Table 2. It is worth noting that there is a peak value between 57.25 Hz and 230 Hz shown in Figure 4, but according to the above, the phase corresponding to the peak does not change \(-180^{\circ}\), so the frequency is not resonance frequency. This is because measurement noise can result in spurious non-physical \"identified\" modes. Figure 5 is the same as Figure 4. The ordinate of both figure is consistent with Figure 6 in Section 3.3.  
preliminarily selected according to the number of updating parameters (i.e., \(s\)) and the number of each factor level (i.e., \(q\)). By comparing the discrepancy of four factors in the corresponding use table, \(U_{10}\)*(\(10^{8}\)) with the minimum value of 0.2236 is selected as the uniform design table. The upper bound (green) and lower bound (yellow) limits for each updating parameter should include the initial value and the target value, which are divided into 10 equal parts (i.e., 10 levels), and then they are evenly dispersed into 10 groups of test samples with the usage table, as shown in the updating parameters of Table 3. The corresponding point FRFs of the 10 groups of node 4 are calculated, respectively. The first four-order resonance frequencies and the first-order antiresonant frequency are identified, as shown in the characteristic quantities of Table 3.