## 4 Numerical Analysis
### Numerical Analysis of Nonlinear Building Structures  
Figure 7 shows the control signal and control force exerted on the structure by the AMD during the seismic loads, respectively. Figure 7: _Cont._Figures 8 and 9 show the responses (displacement and acceleration) to control and non-control of the nonlinear building structure by the proposed MENC algorithm in the event of the California earthquake and the Northridge earthquake, respectively. The California earthquake and the Northridge earthquake were not used in the training. Figure 8: _Cont._  
Figure 7: Control signal and control force of the AMD system under the seismic loads: (**a**) California earthquake; (**b**) Northridge earthquake; (**c**) Landers earthquake; (**d**) Cape earthquake; (**e**) San Simeon earthquake; (**f**) Loma Prieta earthquake. The decreasing rate of the maximum responses on the seismic loads is also shown in Table 3. The decreasing rates of the maximum displacement and acceleration on the third floor were 78.12% and 88.48%, respectively. Therefore, it was found that the proposed MENC algorithm could significantly reduce the structural response under unknown seismic load if applied to the AMD system. Figure 8: Comparison of control and non-control responses of the building structure under the California earthquake: (**a**) displacement; (**b**) acceleration. Figure 9: Comparison of control and non-control responses of the building structure under the Northridge earthquake: (**a**) displacement; (**b**) acceleration. The modal energies of the nonlinear building structures during the California and Northridge earthquakes are shown in Figures 10 and 11, respectively. The total energy, including the modal energy and the control energy of the building during both earthquakes, was remarkably reduced through the proposed MENC algorithm. Also, it was found that the first modal energy affected the whole structure's energy because the first mode is dominant in this structure's case. The total modal energy of the nonlinear building structure during the remaining four earthquakes is shown in Figure 12. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline  & \multicolumn{2}{c}{**Max. Displacement (mm)**} & \multicolumn{3}{c}{**Max. Acceleration (m/s\({}^{2}\))**} \\ \cline{2-7}  & **Non-Control** & **Control by** & **Decreasing** & **Non-Control** & **Control by** & **Decreasing** \\  & **MENC** & **Rate** & **MENC** & **Rate** \\ \hline California & & & & & & \\ earthquake & 40.10 & 11.17 & 72.14\% & 9.38 & 2.66 & 71.64\% \\ \hline Northridge & & & & & & \\ earthquake & 37.06 & 17.31 & 53.29\% & 10.71 & 2.29 & 78.62\% \\ \hline Landers earthquake & 28.11 & 6.73 & 76.06\% & 7.39 & 1.62 & 78.09\% \\ \hline Cape earthquake & 28.85 & 9.04 & 68.67\% & 8.23 & 1.85 & 77.52\% \\ \hline San Simeon & & & & & & \\ earthquake & 25.96 & 6.60 & 74.58\% & 7.12 & 0.82 & 88.48\% \\ \hline Loma Prieta & & & & & & \\ earthquake & 28.97 & 6.34 & 78.12\% & 7.12 & 1.08 & 84.83\% \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Comparison of the maximum responses in control and non-control of the building structure under the seismic loads. Figure 10: Comparison of modal energies in the control and non-control of the building structure under the California earthquake: (**a**) first modal energy; (**b**) second modal energy; (**c**) third modal energy; (**d**) total modal energy. Figure 11: Comparison of modal energies in the control and non-control of the building structure under the Northridge earthquake: (**a**) first modal energy; (**b**) second modal energy; (**c**) third modal energy; (**d**) total modal energy. Figure 12: _Cont._  
Figure 13 shows the relationships between the restoring force and the displacement on the first floor of the building structure caused by the seismic loads. Figure 13 also shows the non-controlled and controlled responses caused by the seismic loads, indicating that the target structure behaved nonlinearly.