# Enhancing Friction Pendulum Isolation Systems Using Passive and Semi-Active Dampers
## 3 Isolator Design
### Sensitivity Analysis  
Previously to set the design process, a sensitivity analysis is carried out. Firstly, the single FPS (Figure 4a) is analyzed under the serviceability earthquake in order to study how the friction coefficient affects the FPS under low-intensity earthquakes. Thus, a value of \(\mu\) will be chosen in order to avoid breakaway effects. Secondly, the FPS+VD and the phase-controlled FPS+SD are studied using the value of \(\mu\) obtained in the first analysis. For the passive one, the viscous damping \(c_{a}\) with respect to the frequency of the pendulum \(f_{a}\) is studied. For the smart version, the unblocking damping (normal functioning) \(c_{min}\) with respect to \(f_{a}\) is analyzed (see Equation (9)). Table 1 summaries the parameters adopted for the sensitivity analysis. \(J_{1,P}\) and \(J_{2,P}\) are shown in Figure 9 for the single FPS under the serviceability earthquake. It can be observed that a low friction coefficient between 0.03-0.06 provides better performance under the low-intensity earthquake. In Figure 9a, the peak acceleration \(J_{1,P}\) is shown, and values of \(J_{1,P}\) between 0.32-0.53 can be achieved for all \(f_{a}\) values if \(\mu\in(0.03,0.06)\). If one pays attention to \(J_{2,P}\) (Figure 9b) the peak isolator displacement, for \(\mu<0.03\), much higher isolator displacement is demanded. From Figure 9a, it can be observed that higher values of \(\mu\) produces a stepwise degradation of the performance and will get the isolator involved in undesirable residual displacements leading to breakaway effects. Now, the sensitivity analysis for the FPS+VD and for the phase-controlled FPS+SD are carried out using \(\mu=0.05\). Figures 10 and 11 show the contour plots of \(J_{1}\) and \(J_{3}\) for both cases. The values of \(J_{1}\) are significantly smaller for the semi-active version. In both cases, from smaller plate sizes than a particular one, the isolated structure performance (Figure 10a,b) may be worse than the non-isolated one. However, the semi-active controlled version improves the performance for a broad range of plate sizes. From Figure 10b, it can be observed that the performance of the FPS+SD is almost constant from a minimum size of the FPS (maximum value of \(f_{a}\)). Hence, any \(f_{a}\) smaller than 0.85 Hz will provide the best performance. From the RMS values (Figure 10d), the isolator shows a performance that is almost independent of \(c_{min}\). However, this fact does not happen with the passive version (Figure 10c). Additionally, the influence of \(f_{a}\) (the concave plate size) is quite low for a \begin{table}
\begin{tabular}{c c c c} \hline \hline \multicolumn{4}{c}{**Modal Parameters**} \\ \hline  & **Structure** & **Value** & \\ \hline \(m_{p}\) & Mass & 24,000 & kg \\ \(f_{p}\) & Frequency & 2.0 & Hz \\ \(\zeta_{p}\) & Damping ratio & 1.0 & \% \\ \(c_{p}\) & Damping & 6031.9 & kg/s \\ \(k_{p}\) & Stiffness & \(3789.9\times 10^{3}\) & N/m \\ \hline \multicolumn{4}{c}{**FPS+VD/FPS+SD**} & **Value** & \\ \hline \(m_{a}\) & Mass & 240 & kg \\ \(f_{a}\) & Frequency & \([0.50\) to \(1.10]\) & Hz \\ \(\mu\) & Friction coefficient & \([0.01\) to \(0.20]\) & \% \\ \(c_{a}\) & Viscous damper & \([0.01c_{p}\) to \(0.50c_{p}]\) & kg/s \\ \(c_{min}\) & Semi-active damper & \([0.01c_{p}\) to \(0.50c_{p}]\) & kg/s \\ \(c_{max}\) & Semi-active damper & \(10c_{p}\) & kg/s \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Parameters for the sensitivity analysis. Figure 9: Sensitivity analyses. Response of the single FPS under the serviceability earthquake. semi-active case. Interestingly, the contour plots for \(J_{3}\), the inter-story drift, show similar qualitative behavior to \(J_{1}\). Again, the isolator will work from a particular size of the plate. Finally, it is worthy to mention that the contour plots obtained for phase-controlled and energy-controlled FPS+SD are quite similar. As an example and for comparison with Figure 11d, Figure 12 shows \(J_{3,RMS}\) for the energy-controlled FPS+SD.