# Enhancing Friction Pendulum Isolation Systems Using Passive and Semi-Active Dampers
## 4 Example of Design  
The optimization process for the design of the isolated-structure is now applied to the structure described in Table 1. The isolator parameters considered for the optimization are also presented in the table. The optimization process has been run for three cases: (i) FPS+VD in which the optimization parameters are, \(f_{a}\) and \(c_{a}\), (ii) FPS+SD with phase control and (iii) FPS+SD with energy-predictive-based control. The optimization parameters for the last two cases are \(f_{a}\) and \(c_{min}\) (which corresponds to a damping ratio \(\zeta_{min}\)). From Step 1 of the optimization problem, a friction coefficient of \(\mu=0.05\) has been obtained. Note that this value is the same for three cases. Thus, Table 2 collects the configuration parameters once Step 2 is run and the minimum values of the performance function (\(\phi_{2}\)) (Equation (17)) together with its reduction with respect to the non-isolated structure. Furthermore, for comparison reasons, two more cases are included in this table, the optimum single FPS using \(\mu=0.05\) (low-friction FPS) and \(\mu=0.20\) (high-friction FPS), which correspond to the optimum friction coefficients for the serviceability and design earthquake, respectively. Note that, when a single FPS is subjected to the design earthquake, the optimum performance function is achieved for the maximum possible friction coefficient \(\mu_{max}\) (\(\mu=0.20\)) used within the optimization problem (see Table 1). Table 3 summarizes the results obtained for all the PIs. The PI based on the mechanical energy, \(J_{4,S}\) and \(J_{4,T}\) are also presented in the table. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**System** & \(f_{a}\) & \(\mu\) & \(\zeta_{(c,e,e_{min})}\) & \(\phi_{2}\) & **Reduction (\%)** \\ \hline Non-isolated structure & - & - & - & 0.833 & 0.00 \\ FPS\({}_{\mu=0.05}\) & 0.671 & 0.05 & - & 0.834 & 0.00 \\ FPS+VD & 0.647 & 0.05 & 0.500 & 0.791 & 5.04 \\ FPS+SD (Phase control) & 0.684 & 0.05 & 0.480 & 0.564 & 32.29 \\ FPS+SD (Energy control) & 0.745 & 0.05 & 0.240 & 0.571 & 31.45 \\ FPS\({}_{\mu=0.20}\) & 0.671 & 0.20 & - & 0.529 & 36.49 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Optimum parameters for the isolator configurations studied. From Tables 2 and 3, it can be observed that both semi-active designs behave better than the FPS+VD and much better than the FPS with \(\mu=0.05\). To achieve similar results as to the semi-active version with the single FPS, much higher values of \(\mu\) are needed, and consequently, the behavior under lower intensity earthquakes will be drastically affected. Paying attention to \(J_{3,P}\), which is the main PI considered within the performance function, the phase-controlled FPS+SD shows the best performance, achieving a reduction of 76%. The cumulative distribution function (CDF) curves for the bearing plate displacement and for the inter-story drift using the optimum configurations of Table 2 under the design earthquake (Figure 13) and the serviceability earthquake (Figure 14) are computed and plotted. The CDFs indicate the percentage of time of non-exceedance of a value of the considered magnitude. Figure 13a,b show that the FPS with high friction exhibits much minor bearing plate displacements in comparison with the FPS+VD, both semi-active versions, and the FPS with low friction. The FPS+VD and FPS with low friction show more displacement. FPS+SD with phase control and energy-predictive-based control have similar behavior and present minor displacement than the FPS+VD. From Figure 13c,d, it can be observed that the FPS+VD presents more capacity to control the drift than the single FPS and similarly, the semi-active versions show better performance than the FPS+VD. Finally, the phase-controlled FPS+SD always behaves slightly better than the energy-predictive-based one. Figure 14 shows the same plots as those of Figure 13 but for the serviceability earthquake. Now, it is clearly appreciated that all the low-friction FPSs show similar performance in terms of the drift.