# A Kriging Model Based Finite Element Model Updating Method for Damage Detection
## 4 Experimental Study
Unlike the numerical simulation mentioned in Section 3, in this case study, a larger parameter range is beneficial in getting the right updating parameter values. However, this may lead to selected frequencies to become close to a theoretical resonant frequency, which would result in a sharply increased objective function value, as illustrated in Figure 9. \begin{table}
\begin{tabular}{c c c} \hline \hline
**State** & **Condition** & **Description** \\ \hline State 1 & Baseline condition & - \\ State 2 & 19.1\% base mass increment & 1.2 kg additional mass on the basd \\ State 3 & 19.1\% 1 st-story mass increment & 1.2 kg additional mass on the 1 st story \\ State 4 & 21.8\% 1 st-story stiffness reduction & Replace one column (1 st story) \\ State 5 & 43.7\% 1 st-story stiffness reduction & Replace one column (1 st story) \\ State 6 & 21.8\% 2 nd-story stiffness reduction & Replace two column (2 nd story) \\ State 7 & 43.7\% 2 nd-story stiffness reduction & Replace one column (2 nd story) \\ State 8 & 21.8\% 3 rd-story stiffness reduction & Replace two column (3 nd story) \\ State 9 & 43.7\% 3 rd-story stiffness reduction & Replace one column (3 nd story) \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Summary of structural state conditions. Figure 7: Theoretical model for the test structure. Since the experimental data has considerable noise, the number of selected frequency points for model updating should be much higher than the number of the updating parameters to reduce the effects of noise and damping. If such an increase appears in too many frequency points, there will be severe fluctuations in the response surface of the final objective function formed at these frequency points, which are undesirable and useless in identification. Figure 8: The frequency response function (FRF) of the first floor of the real structure and the updated model: (**a**) Absolute value of \(\mathrm{H}(1,1)\); (**b**) Absolute value of the real part of \(\mathrm{H}(1,1)\); (**c**) Absolute value of the imaginary part of \(\mathrm{H}(1,1)\). Figure 9: The objective function of 50 Hz when only m1 and m2 are variable using experimental data of state 1. It is not difficult to relate this increase to the sharp decline of the real parts of the FRFs near the natural frequency as shown in Figure 8. In contrast, the imaginary parts of the FRFs show better correlation near the natural frequency. Thus if the selected frequency is near the natural frequency of experimental data, 3 Hz on either side of a peak of the real part of the FRFs is used as a criterion in this case, the imaginary part of the FRFs only is chosen to calculate the FDAC to get a smooth response surface. It turns out that this measure reduces the complexity of the frequency response surface. This improvement is shown in Figure 10. It is clear that the response surface is much smoother than before after such a special treatment which saves a great deal of computation time in constructing the Kriging model, because the fluctuating portion of the response surface is largely reduced. It should be noted that both the Kriging model based method and the GA method benefit from the improvement, namely that their computing time decreases significantly. Finally, the updating results of each state by the two methods are shown in Figure 11.  
It is clear that the updating results of both methods match the real values quite well. The parameter that has experienced the main change can be clearly identified with small false alarms in other parameters, due to the measurement noise and the modeling error in the theoretical model. The successful application in these states is sufficient to demonstrate the superiority of the method using FDAC value to build the objective function. Furthermore, the Kriging method shows similar updating results to those by the GA method, which means the proposed method has the advantage of a typical good surrogate model and ensures the updating accuracy at the same time.