# Development of Vibration Control Structure on Suspended Ceiling Using Pulley Mechanism
## 5 Simulation Analysis
### Simulation Results  
In Figure 23, the experimental results of maximum response values and time history data in displacement and acceleration at the SCS panel are compared with the analytical results for all the experimental cases under simulated earthquake waves. During the comparison of the maximum response value, while the experimental results are relatively larger than the analytical results, the experimental and numerical models' responses are in good agreement.  
Figure 21: Analytical model in the STERA 3D.  
Figure 20: Moment-rotation relationship for bending spring.  
It is considered that the repeated earthquake excitation to the specimen decreased the pre-tension force of the stay cable, and the joint rigidness caused the differences between the experimental and analytical results. The shake table test of Case CW60 was carried out after the Case CW0 in 2020. Therefore, due to the cumulative of the previous test case damage, the error in Case CW60 was made to become larger than Case CW0.  
The coefficient of determination, \(R^{2}\), is selected to quantitatively measure the accuracy of the simulation model in this study, and it is defined as Equation (12). In addition, Table 6 shows the \(R^{2}\) for the validation of each simulation model. The \(R^{2}\) value for all cases of the maximum acceleration and displacement response are 0.83 and 0.91, respectively. In particular, the minimum \(R^{2}\) value was 0.69 in the Case CW60. Besides, the simulation model accurately reproduced the time histories responses of the accelerations and displacements, including the frequency and the location of the wave crest.  
\[R^{2}=1-\frac{\sum\limits_{i=1}^{n}{{{(Exp_{i}-Ana_{i})}^{2}}}}{\sum\limits_{i =1}^{n}{{{(Exp_{i}-\overline{Exp})}^{2}}}} \tag{12}\]  
where \(Exp_{i}\) is the output by the shake table test, \(Ana_{i}\) is the output by the simulation analysis, and \(\overline{Exp}\) is the mean of \(Exp_{i}\).  
The results of the force-velocity relationship of the damper device in the shake table tests and simulation analyses are contrasted in Figure 24. The gaps in the hysteresis loop shape were confirmed; however, the error between the experimental and analytical results of the time histories are reproduced well. Thereby, the effect of displacement amplification was successfully simulated in the analytical model.  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Damper Type** & **Parameter** & \\ \hline \multirow{2}{*}{Bilinear model} & Initial stiffness \(K_{0}\) [N/mm] & 110 \\ \cline{2-3}  & Stiffness ratio \(K_{1}\) / \(K_{0}\) & 0.0001 \\ \cline{2-3}  & Yield force \(F_{fy}\) [N] & 6 \\ \hline \multirow{2}{*}{Nonlinear viscous dashpot model} & Damping coefficient \(C_{d}\) [N-(s/mm)\({}^{\alpha}\)] * & 0.06 \\ \cline{2-3}  & Velocity exponent \(\alpha\) & 0.9 \\ \hline \hline \end{tabular}  
* the unit of the damping coefficient \(C_{d}\) changes with velocity exponent \(\alpha\) (i.e., when \(\alpha=1\) is N-s/mm, and when \(\alpha=0.9\) is N-(s/mm)\({}^{0.5}\)).  
\end{table}
Table 5: Damper parameter in the analytical model.  
Figure 22: Restoring force characteristics of damper in the simulation model.  
Figure 23: Results comparison of the SCS response between shake table test and simulation analysis.  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Case Name** & **Acceleration Response** & **Displacement Response** \\ \hline Case C & 0.86 & 0.93 \\ \hline Case CW0 & 0.98 & 0.97 \\ \hline Case CW60 & 0.69 & 0.79 \\ \hline Case CWD0 & 0.70 & 0.99 \\ \hline Case CWD60 & 0.94 & 0.82 \\ \hline All cases & 0.83 & 0.91 \\ \hline \hline \end{tabular}
\end{table}
Table 6: Coefficient of correlation for the validation of simulation model.  
Figure 24: Results comparison of the damper response between shake table test and simulation analysis.