## 2 Ground Motions Considered
### Numerical Simulations and Results  
A to-be-protected object above the SRB models was assumed to be important equipment with a mass of 1000 kg located on the first floor of a building structure, e.g., a storage system for wafers used in high-tech industries. For simplicity, the SDOF assumption still held in the numerical simulation, i.e., the isolation mode predominated the overall dynamic behavior of the isolated equipment. At each time step of ground motions of Groups 1, 2, and 3, the Newmark-\(\beta\) method with linear acceleration was used for solving Equations (4) and (5) correspondingly when at the slope and arc rolling stages. During two or more consecutive time steps in numerical integration, there could be no relative motion between the upper and lower bearing plates, as shown in Figure 1b. Therefore, the sticking phase [40], which will significantly affect the post-earthquake re-centering performance, was taken into consideration. By setting \(\ddot{u}\) in Equations (4) and (5) equal to zero, the critical characteristic strengths at the slope and arc rolling stages, which are the criteria to identify if the slipping or sticking phase occurs [40], could be defined and calculated by Equations (6) and (7), respectively. More precisely, if \(F_{SRB}^{C}\) is calculated to be smaller than the designed damping force \(F_{SRB}\) at a specific time step (e.g., at \(t_{i}\)), the sticking phase will occur. In this case, at the time step \(t_{i}\), \(\ddot{u}\) and \(\dot{u}\) should be set to zero, and \(u\) should remain the same as that at the previous time step \(t_{i-1}\). Note that \(F_{SRB}^{C}\) at the slope rolling stage is only related to \(\ddot{u}_{g}\) once \(\theta\) is determined, while \(F_{SRB}^{C}\) at the arc rolling stage is also related to \(u\), in addition to \(\ddot{u}_{g}\), once \(R_{SRB}\) is determined. \[F_{SRB}^{C}=\Big{|}M\ddot{u}_{g}+\frac{Mg\theta_{SRB}}{2}\text{sgn}(u)\Big{|}, \tag{6}\]  
\[F_{SRB}^{C}=\Big{|}M\ddot{u}_{g}+\frac{Mg}{4R_{SRB}}u\Big{|}. \tag{7}\]  
With sloping angles of 3, 4, 6, and 8 degrees at the slope rolling stage, the magnitudes of \(F_{SRB}^{C}\) after excitations, i.e., \(\ddot{u}_{g}=0\), were correspondingly 256.83 N, 342.43 N, 513.65 N, and 684.87 N, of which the first two and the last two are smaller than and larger than their designed damping forces, respectively. It also means that for the designs with sloping angles of 3 and 4 degrees, if the magnitudes of \(\ddot{u}_{g}\) during external disturbance are correspondingly larger than 0.030 \(g\) and 0.012 \(g\) with the same direction as \(u\) or larger than 0.082 \(g\) with the opposite direction to \(u\), the motion between the superior and inferior bearing plates can still be in the slipping phase. The horizontal maximum and residual displacement responses of SRB-3, SRB-4, SRB-6, and SRB-8 subjected to ground motions of Groups 1, 2, and 3 are presented in Figures 5-7, respectively. As for the maximum horizontal displacement responses subjected to ground motions of Group 1, another presentation, i.e., the maximum horizontal displacement responses versus PGV in a logarithm scale, is also provided for better observation. The maximum horizontal displacement responses of SRBs have been numerically demonstrated to be almost linearly proportional to PGV on a logarithm scale [41]. Figure 4: Hysteresis loops of four different SRB models with the same horizontal acceleration control performance. As observed from these figures, based on retaining the same horizontal acceleration control performance, adopting a constant sloping angle and damping force for SRBs might not entirely control their horizontal maximum and residual displacement responses simultaneously.