# Simulation-Based Model-Updating Method for Linear Dynamic Structural Systems
## 2 Modelling of the Structural System
### Dynamic Behavior of the Structural System  
The dynamic behavior of the structural systems can be obtained by using analytical and experimental methods. The analytical approach uses a mathematical model that represents the dynamic properties of the actual structure to solve the problem. The process of attaining the experimental model involves exciting the system and measuring the resulting structural responses. Supplementary to the dynamic properties of the building model presented in the preceding section, it is essential to analytically model the dynamic structural behavior and conduct a comparative analysis with experimental measurement results. The dynamic responses of the building model were measured using the shaking table through two experimental tests: free vibration and the El Centro earthquake simulation. A simulation model was created to validate the experimentally obtained data (Figure 4). In this dynamic model, the El Centro earthquake acceleration data serve as the  
\begin{table}
\begin{tabular}{c c c c} \hline \hline \multirow{2}{*}{**Mode (x Direction)**} & \multicolumn{2}{c}{**Natural Frequencies (Hz)**} & \multirow{2}{*}{**Error (\%)**} \\ \cline{2-2} \cline{4-4}  & **Analytical** & & **Experimental** \\ \hline
1 & 1.67 & 1.61 & 3.6 \\
2 & 4.88 & 4.83 & 1.0 \\
3 & 7.69 & 8.00 & 4.0 \\
4 & 9.87 & 9.95 & 0.8 \\
5 & 11.27 & 11.38 & 0.9 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Natural frequencies of the structural system.  
Figure 3: Mode shapes of the building model.  
input signal for the system, while the structural responses constitute the output signals. The building model is simulated by employing the analytical structural matrices outlined in the previous section, utilizing the state-space representation. The following section will provide a comprehensive explanation of the simulation block diagram designed to verify the measured free vibration responses of the building model.  
As a result of the simulation, the displacement responses of the five-story building model were obtained numerically. The comparative presentation of experimental and simulation results is shown in Figure 5.  
Shaking table tests were performed to investigate the experimental dynamic responses of the building model, while numerical results were computed using simulation models. Figure 5a shows that the simulation results exhibit higher amplitude values than the measured results during the earthquake effect. The higher amplitude values in the simulation data are related to the analytically calculated damping matrix. Damping decreases oscillation amplitudes by dissipating mechanical energy in a dynamic system. Observing Figure 5b, an accurate concordance between the results obtained through both methods is apparent within 30 s of the graph; however, differences become evident at lower amplitude levels. The phase difference between the results of the free vibration behavior of the building model obtained using experimental and numerical methods is another result observed from the time-displacement graph. Through the repeating of simulations with  
Figure 4: Block diagram representing the seismic behavior of the building model.  
Figure 5: Displacement responses of first floor: (**a**) El Centro earthquake, (**b**) Free vibration.  
varying damping and stiffness coefficients, it was concluded that the stiffness matrix affects the phase difference. In selecting the stiffness and damping coefficients for updating, the dynamic behavior of the building model under the effect of the El Centro earthquake and the results of the free vibration test were considered.