# Simulation-Based Model-Updating Method for Linear Dynamic Structural Systems
## 4 Results and Discussion  
The analytical model of the structural system obtained from eigenvalue solutions is compared to and correlated with the data obtained from the shaking table tests. The mathematical model of the structural system has five DOFs, and the experimental tests were performed on this building model. The simulation-based model-updating technique is used to compare and correct the analytical data with the test data. To deal with the problem that the analytical model does not represent the actual structural behavior of the building model, the parameter estimation method was applied. In this method, the mass matrix was assumed to be exact since the small-scale building model used as a numerical example is not a complicated system. The updated stiffness matrix has smaller coefficients than the first analytically calculated matrix. This is because of the assumptions in the stiffness formula used in the analytical model. The corrected damping matrix has higher coefficients than the first generated analytical matrix. The damping ratio obtained from free vibration tests by using the Rayleigh damping model was applied in the first analytically determined damping matrix. The natural frequencies of structural systems are accurately estimated from measurements. On the contrary, the modal damping ratios have a wide scatter depending on the selected analysis method [22]. The first five natural frequencies of the structural system obtained with the updated model parameters calculated in the previous section are presented comparatively in Table 2.  
As shown in Table 2, the calculated analytical, experimental, and updated natural frequencies of the structural system are compatible. The mode shapes of the building model with the updated system parameters are shown in Figure 9, analytically and experimentally.  
As can be seen in Figure 9, results closer to the experimental method were obtained in all modes with the model update method according to the mode shapes calculated by the theoretical approach. It is noteworthy that the mode shapes calculated with the updated system parameters, especially at high frequencies, are in good agreement with the experimental results.  
The simulation of the dynamic behavior of the building model with the updated system parameters and comparison with the measurement results is shown in Figure 10. Figure 10a shows that the simulation and measurement amplitude values of the building model under the influence of the El Centro earthquake converge. Updating the damping matrix of the building model has produced results that simulate the actual dynamic behavior. On the other hand, it is seen in Figure 10b that the free vibration data, both measurement and simulation results in the first 40 s, proceed in a harmonious manner. In addition, it was observed that the phase difference between the measurement and simulation results was eliminated after updating the stiffness coefficients of the building model.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline  & \multicolumn{3}{c}{**Natural Frequencies (Hz)**} \\ \hline
**Mode** & **Analytical** & **Experimental** & **Updated** \\ \hline
1 & 1.67 & 1.61 & 1.60 \\
2 & 4.88 & 4.83 & 4.68 \\
3 & 7.69 & 8.00 & 7.38 \\
4 & 9.87 & 9.95 & 9.48 \\
5 & 11.27 & 11.38 & 10.81 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Analytical, experimental, and updated natural frequencies of the structural system.  
The results show that the proposed model-updating method successfully represents the actual free vibration and seismic behavior of the building.