## 4 SDOF Systems with Nonlinear Eddy Current Dampers
### Response to Seismic Excitations
#### 4.3.2 Displacement and Acceleration Responses to Real Earthqu  
The system parameters are the same as before, and the responses to different ground motions are calculated using the Runge-Kutta algorithm. The peak displacement responses (PDRs) and the peak acceleration responses (PARs) are shown in Tables 1 and 2, respectively. Here, we define the response reduction ratio \(\rho\) as follows  
\[\rho=\frac{A_{o}\ -\ A_{d}}{A_{o}} \tag{9}\]  
where \(A_{o}\) is the peak displacement or acceleration response without a damper and \(A_{d}\) is the peak displacement or acceleration response with a damper. The larger the reduction ratio, the better the response control performance of supplemental devices. The reduction ratios of displacement and acceleration are compared for ECDs and FVDs in Figures 9 and 10, respectively. The points above the zero line indicate that the control performance of ECD is better than that of FVD, and vice versa. From Figures 9 and 10, it is found that the displacement and acceleration reduction ratios of ECDs are larger than those of FVDs in most cases, and the displacement reduction ratios are generally greater than the acceleration reduction ratios for both ECDs and FVDs. It should also be pointed out that there are still some cases where ECDs do not work better than FVDs for Kobe and Loma Prieta seismic waves. In the previous Section 4.2, it is found that although the control performance of the ECDs are better than that of FVDs in the resonance frequency band, in the frequency band of a certain distance from the resonance or far from the resonance, the control performance of the ECDs are slightly worse in some cases. That is the reason why ECDs do not work better than FVDs when the dominant frequency band of the seismic excitation is far from the resonance. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{\
\begin{tabular}{c} Seismic \\ Wave \\ \end{tabular} } & \multirow{2}{*}{
\begin{tabular}{c} **Without** \\ **Damper (m)** \\ \end{tabular} } & \multicolumn{2}{c}{**FVD-SDOF**} & \multicolumn{2}{c}{**ECD-SDOF**} \\ \cline{3-6}  & & & **\(\alpha\)** & **PDR (m)** & \(\rho\)** & **PDR (m)** & \(\rho\)** \\ \hline \multirow{4}{*}{Electronro} & & 0.2 & 0.0097 & 93.75\% & 0.0158 & 89.81\% \\  & & 0.4 & 0.0200 & 87.12\% & 0.0186 & 88.04\% \\  & 0.1553 & 0.6 & 0.0324 & 79.12\% & 0.0273 & 82.43\% \\  & & 0.8 & 0.0404 & 73.96\% & 0.0333 & 78.56\% \\  & & 1 & 0.0498 & 67.91\% & 0.0441 & 71.63\% \\ \hline \multirow{4}{*}{Irpinia} & & 0.2 & 0.0399 & 79.82\% & 0.0416 & 78.97\% \\  & & 0.4 & 0.0634 & 67.92\% & 0.0579 & 70.69\% \\  & 0.1977 & 0.6 & 0.0846 & 57.22\% & 0.0724 & 63.39\% \\  & & 0.8 & 0.1033 & 47.75\% & 0.0890 & 54.95\% \\  & & 1 & 0.1155 & 41.54\% & 0.1049 & 46.94\% \\ \hline \multirow{4}{*}{ChiChi} & & 0.2 & 0.0225 & 97.86\% & 0.0225 & 97.86\% \\  & & 0.4 & 0.0321 & 96.95\% & 0.0316 & 97.00\% \\  & 1.0532 & 0.6 & 0.0460 & 95.63\% & 0.0417 & 96.04\% \\  & & 0.8 & 0.0967 & 90.81\% & 0.0561 & 94.67\% \\  & & 1 & 0.1640 & 84.43\% & 0.0787 & 92.53\% \\ \hline \multirow{4}{*}{Kobe} & & 0.2 & 0.0817 & 35.67\% & 0.0778 & 38.80\% \\  & & 0.4 & 0.0883 & 30.48\% & 0.0802 & 36.89\% \\  & & 0.6 & 0.0950 & 25.24\% & 0.0863 & 32.11\% \\  & & 0.8 & 0.0963 & 24.19\% & 0.0936 & 26.37\% \\  & & 1 & 0.0969 & 23.76\% & 0.0981 & 22.80\% \\ \hline \multirow{4}{*}{Northridge} & & 0.2 & 0.0178 & 87.68\% & 0.0161 & 88.82\% \\  & & 0.4 & 0.0300 & 79.18\% & 0.0248 & 82.76\% \\  & & 0.6 & 0.0373 & 74.13\% & 0.0329 & 77.15\% \\  & & 0.8 & 0.0458 & 68.23\% & 0.0396 & 72.50\% \\  & & 1 & 0.0578 & 59.92\% & 0.0447 & 68.97\% \\ \hline \multirow{4}{*}{Niigat} & & 0.2 & 0.0220 & 95.14\% & 0.0252 & 94.43\% \\  & & 0.4 & 0.0426 & 90.59\% & 0.0384 & 91.52\% \\ \cline{1-1}  & & 0.6 & 0.0721 & 84.10\% & 0.0554 & 87.77\% \\ \cline{1-1}  & & 0.8 & 0.1208 & 73.34\% & 0.0795 & 82.47\% \\ \cline{1-1}  & & 1 & 0.1733 & 61.77\% & 0.1023 & 77.44\% \\ \hline \end{tabular}
\end{table} 
Table 1: Peak displacement responses (PDRs) of SDOF-ECDs and SDOF-FVDs.