## 1 Introduction
### Formulas and Considerations  
The simplified and linearized equation of motion of FPBs for a single-degree-of-freedom (SDOF) system shown in Figure 1a is given in Equation (1) [3; 4; 5; 6].  
\[F=-M\Big{(}\ddot{u}+\ddot{u}_{g}\Big{)}=\frac{Mg}{R_{FPB}}u+F_{FPB}\text{sgn} \Big{(}\dot{u}\Big{)}, \tag{1}\]  
where \(F\) is the inertia force; \(M\) is the mass of the protected object above the isolation bearing; \(g\) is the acceleration of gravity; \(R_{FPB}\) is the curvature radius of the spherical sliding surface; \(F_{FPB}\) is the friction damping force, which is usually approximated by the product of \(Mg\) and the friction coefficient \(\mu_{FPB}\) between two mutual-contact sliding surfaces; \(u\), \(u\), and \(\ddot{u}\) are the displacement, velocity, and acceleration of the upper bearing plate (or the protected object) relative to the lower bearing plate (the ground usually), respectively; \(\ddot{u}_{g}\) is the horizontal acceleration excitation.  
As observed from Figure 2 and Equation (1), once the design displacement \(D_{D}\), \(R_{FPB}\), and \(F_{FPB}\) are determined, the effective horizontal stiffness \(K_{\textit{eff}}\) and the effective horizontal period \(T_{\textit{eff}}\) at \(D_{D}\) can be mathematically calculated by Equations (2) and (3), respectively [3; 4; 5; 6].  
\[K_{\textit{eff}}=\frac{Mg}{R_{FPB}}+\frac{F_{FPB}}{D_{D}}=\Big{(}\frac{1}{R_{ FPB}}+\frac{\mu_{FPB}}{D_{D}}\Big{)}Mg, \tag{2}\]  
\[T_{\textit{eff}}=2\pi\sqrt{\frac{Mg}{\big{(}\frac{Mg}{R_{FPB}}+\frac{F_{FPB}}{ D_{D}}\big{)}g}}=2\pi\sqrt{\frac{1}{\big{(}\frac{1}{R_{FPB}}+\frac{\mu_{FPB}}{ D_{D}}\big{)}g}}. \tag{3}\]  
The unique features of FPBs can be directly observed from Equations (2) and (3) [3; 4; 5; 6]. The effective horizontal stiffness is proportional to the mass of the protected object above, thus easily overcoming the inherent and accidental eccentricity problems, especially when the protected object is a highly irregular architectural design. In addition, the effective horizontal period is independent of the mass of the protected object above. In other words, the effective horizontal period, which is the most important design parameter when performing the linear static or dynamic analysis [37], will not be affected if the mass of the protected object after construction varies in use or is different from the initial assumed mass.  
The simplified and linearized equations of motion of SRBs at the slope and arc rolling stages for the SDOF system shown in Figure 1b are given in Equations (4) and (5), respectively [32; 33; 34].  
\[F=-M\Big{(}\ddot{u}+\ddot{u}_{g}\Big{)}=\frac{Mg\theta_{SRB}}{2}\text{sgn}(u )+F_{SRB}\text{sgn}\Big{(}\dot{u}\Big{)}, \tag{4}\]\[F=-M\Big{(}\ddot{u}+\ddot{u}_{g}\Big{)}=\frac{Mg}{4R_{SRB}}u+F_{SRB}\text{sgn} \Big{(}u\Big{)}, \tag{5}\]  
where \(\theta_{SRB}\) is the sloping angle of the sloped rolling surface; \(F_{SRB}\) is the friction damping force; \(R_{SRB}\) is the curvature radius of the arc rolling surface.  
The unique feature of SRBs can be directly observed from Equation (4) and Figure 2[29; 30; 31; 32; 33]. With a fixed sloping angle and damping force, the horizontal acceleration transmitted to the protected object above can remain constant regardless of any intensity and frequency content of external disturbance. Accordingly, SRBs do not have a fixed natural period and can offer maximum horizontal decoupling between the protected object and external disturbance.