## 4 Numerical Comparisons between FPBs and SRBs
### Comparison Results and Discussions  
The protected object above the FPB models and the associated conditions were the same as those introduced in Section 3.3. For numerically simulating the FPB models, at each time step of ground motions of Groups 2 and 3, the Newmark-\(\beta\) method with linear acceleration was used for solving Equation (1). The same method as that described in Section 3.3 but a different formula, as given in Equation (10), was employed herein to consider the sticking phase [40] for the FPB models. \[F_{FPB}^{C}=\left|M\bar{u}_{g}+\frac{Mg}{RFB}u\right|, \tag{10}\]  
Figure 11: Illustrations of curved sliding and sloped rolling surfaces correspondingly designed for FPB and SRB models. Figure 10: Illustrations of design criteria for FPB models. The maximum horizontal acceleration and displacement responses, as well as the residual displacement responses of SRB-3, FPB-3-2M, FPB-3-3M, FPB-3-2, FPB-3-3, SRB-6, FPB-6-2M, FPB-6-3M, FPB-6-2, and FPB-6-3 subjected to ground motions of Groups 2 and 3, are presented in Figures 12-14, respectively. For better comparison, the response ratios of FPB-3-2M to SRB-3, FPB-3-3M to SRB-3, FPB-3-2 to SRB-3, FPB-3-3 to SRB-3, FPB-6-2M to SRB-6, FPB-6-3M to SRB-6, FPB-6-2 to SRB-6, and FPB-6-3 to SRB-6 are also presented. The numbers indicated in these figures represent the amounts of ground motions under which the control performances of the FPB models were better than those of the corresponding SRB models. In addition to mean responses, the 25 th and 75 th percentile responses are also provided for further statistical comparison. The numerical results of SRB-3 and SRB-6 at the arc rolling stage, i.e., the maximum horizontal acceleration responses less than 0.08-\(g\) shown in Figure 12, are covered in the figures and the following discussions. As observed from Figure 13, evidently, since all the maximum horizontal displacement responses were larger than zero, the situation that the SRB or FPB models were not activated (i.e., were not functional) under the ground motions considered did not occur. Figures 15 and 16 present the displacement and acceleration response histories as well as the hysteresis loops of SRB-3, FPB-3-2, SRB-6, and FPB-6-2 under 1999 Taiwan Chi-Chi, TCU074 (EW) (one of the ground motions of Group 2) and those of SRB-3, FPB-3-3, SRB-6, and FPB-6-3 under 1999 Taiwan Chi-Chi, TCU052 (EW) (one of the ground motions of Group 3), respectively. Note again that this numerical study aimed to discuss the potential merits of FPBs and SRBs quantitatively based on the design criteria described in Section 4.2, rather than to design practically acceptable horizontal isolation displacement as small (or satisfactory) as possible for FPBs and SRBs against the ground motion considered. As observed from Figures 12, 15 and 16, the distribution of maximum horizontal acceleration responses of the FPB models was much more scattered than that of the SRB models, even considering the responses of the SRB models at the arc rolling stage (only SRB-6 under one far-field ground motion). It is mainly because of the unique feature of SRBs--constant acceleration control. Under 15 (13) and 20 (21) sets (i.e., the majority) of far-field ground motions, FPB-3-2M (FPB-3-2) and FPB-6-2M (FPB-6-2) had better horizontal acceleration control performances than SRB-3 and SRB-6, respectively, in particular, the latter. Under 28 (28) sets and 37 (36) sets (i.e., the majority) of pulse-like near-fault ground motions, FPB-3-3M (FPB-3-3) and FPB-6-3M (FPB-6-3) had better horizontal acceleration control performances than SRB-3 and SRB-6, respectively, in particular, the latter. Without considering the extremes, the response ratios on average, regardless of being subjected to far-field or pulse-like near-fault ground motions, were less than unity. To sum up, under the ground motions considered, if simply observing the amount or the average of response ratios, the FPB models might potentially have an advantage over the SRB models in terms of horizontal acceleration control.