## 3 Results
### Model Updating and Validation  
The 10 groups of updating parameters and characteristic quantities shown in Table 3 are taken as training samples of the RBF neural network. The function newrbe in the MATLAB R2019a very quickly creates an exact RBF network with zero errors on the design vectors. In order to compare with the updated results which only use resonance frequencies as the characteristic quantities, the sample points are divided into two groups:  
Group I: the first three-order resonance frequencies and the first-order antiresonant frequency;  
Group II: only the first four-order resonance frequencies. They are inputted into the RBF neural network, respectively. The output vector is the parameters to be updated, and the mapping relationship between the frequencies and the updating parameters is established. Finally, the target values (i.e., measurement values or truth values) are inputted into the trained network, and results from I and II are obtained, respectively, as shown in Table 4.  
By comparing result I and result II, the error of each updated value in result I is significantly smaller than that in result II, and the maximum error is less than 7%, while \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \multirow{2}{*}{**Updating Parameter**} & \multirow{2}{*}{**Target Value**} & \multirow{2}{*}{**Initial Value**} & \multirow{2}{*}{**Initial Error**} & \multicolumn{3}{c}{**Result I**} & \multicolumn{3}{c}{**Result II**} \\ \cline{5-8}  & & & & **(\%)** & & **Updated** & **Error** & **Updated** & **Error** \\ \cline{5-8}  & & & & **Value** & **(\%)** & **Value** & **(\%)** & **Value** & **(\%)** \\ \hline \(E\) (GPa) & 77 & 70 & \(-9.09\) & 77.31 & 0.4 & 73.84 & \(-4.1\) \\ \(m\) (kg) & 6.5 & 5 & \(-23.08\) & 6.15 & \(-5.3\) & 5.56 & \(-14.46\) \\ \(k_{t}\) (\(10^{7}\) N/m) & 1 & 2 & 100 & 0.93 & \(-7\) & 2.19 & 119 \\ \(k_{r}\) (\(10^{4}\) N\(\cdot\)m/rad) & 2 & 5 & 150 & 1.89 & \(-5.5\) & 2.28 & 14 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Updated parameters obtained from the two groups. \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline \multirow{2}{*}{**Test Number**} & \multirow{2}{*}{\(E\)**} & \multirow{2}{*}{\(m\)**} & \multirow{2}{*}{\(k_{t}\)**} & \multirow{2}{*}{\(k_{r}\)**} & \multirow{2}{*}{**Resonance Frequency**} & \multicolumn{3}{c}{**Antiresonant Frequency**} \\  & & & & & & **1 st** & **2 st** & **3 st** & **4 st** & **1 st** \\ \hline
1 & 69 & 5.25 & 1.25 & 3.5 & 11.533 & 54.869 & 224.99 & 405.5 & 30.5 \\
2 & 70 & 6 & 2.25 & 6 & 12.484 & 55.647 & 229.97 & 434.8 & 31.25 \\
3 & 71 & 6.75 & 0.5 & 3 & 10.864 & 51.213 & 220.62 & 357.5 & 31.25 \\
4 & 72 & 4.75 & 1.5 & 5.5 & 12.778 & 58.985 & 232.91 & 428 & 31 \\
5 & 73 & 5.5 & 2.5 & 2.5 & 10.823 & 54.669 & 230.84 & 425.8 & 31.25 \\
6 & 74 & 6.25 & 0.75 & 5 & 12.289 & 55.114 & 230.52 & 392.5 & 31.75 \\
7 & 75 & 7 & 1.75 & 2 & 9.9153 & 51.66 & 231.08 & 415.5 & 32.75 \\
8 & 76 & 5 & 2.75 & 4.5 & 12.537 & 59.144 & 239.01 & 449 & 32.25 \\
9 & 77 & 5.75 & 1 & 1.5 & 9.414 & 53.385 & 231.68 & 402.5 & 32.5 \\
10 & 78 & 6.5 & 2 & 4 & 11.962 & 55.769 & 239.26 & 439.8 & 33.5 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Resonance and antiresonant frequencies of the beam with 10 levels. the error of tensile stiffness \(k_{t}\) in result II even reaches 119%, further deviating from the target value. In order to verify the influence of the updating results on the FRF curves, the two groups of updated parameters obtained in Table 4 were, respectively, substituted into the ABAQUS 2016, and the characteristic quantities of the objective function shown in Table 5 and the FRF curves shown in Figure 6 were obtained. As can be seen from Table 5, the relative error between the characteristic quantities of the objective function and the updated value by using the updated parameters of result I in Table 4 is less than 3%, and smaller than that of result II. Compared with the relative error of 0.5% of the finite element model updating by the first four-order modal frequencies and the first-order effective modal mass in Ref.