## 3 Mechanical Properties of Isolation Systems Equipped with SLDRFFB and SLDTFFB
### Comparison of Mechanical Properties of Isolation Systems Equipped with SLDRFPB and SLDTFPB  
Mechanical properties of isolation systems equipped with SLDRFPB and SLDTFPB can be compared according to Equations (20)-(27). It can be seen in Equations (21) and (25) that vertical distribution of mass in the superstructure has an influence on the equivalent radius of the isolation layer. Figure 11 compares the relations between the equivalent radii of the two developed SLDFFBs and story number. A group of frame structures with different story numbers are equipped with the same SLDRFPB or SLDTFPB. Height of each story is adopted as h = 3 m and frame spans are adopted as L = 35.3 m. Story number varies from 1 to 20. As shown in Figure 11, the equivalent radius of SLDRFPB decreases with increasing of number of stories. This is because the increase of distance between the center of gravity and the center of the circle leads to the decrease of the average rotation radius of the whole superstructure when it rolls around the center of the circle. On the other hand, the equivalent radius of SLDTFPB increases a little with the number of stories because the track of the superstructure equipped with SLDTFPB does not change with the number of stories. Furthermore, with the increase of mass of the superstructure, a smaller mass ratio between the rotational sliding frame and the translational superstructure brings less influence on the equivalent radius of SLDTFPB. Figure 10: Equivalent radius of a frame structure equipped with SLDTFPB. Figure 9: Equivalent radius of a frame structure equipped with SLDTFPB. In order to further investigate the isolation effect of the two developed SLDFFPs with corresponding parameters, equivalent radii of SLDRFFB and SLDTFPB with subequal track radius of superstructure are compared in Figure 12. The horizontal axis about curves of SLDRFPB in Figure 12 represents the depth of SLDRFFB, \(h_{R0}\), and the horizontal axis about curves of SLDRTPB in Figure 12 represents a 9/2 the depth of SLDTFPB, \(h_{T0}\), which cannot be readily seen in this figure. Span of the sliding frame in SLDTFPB is 1/3 span of the superstructure, and the mass of each sliding frame is 1/3 the mass of each story. According to the geometric relationship in Equations (28) and (29), \(h_{R0}/h_{T0}\) is selected as 9/2 so that \(r_{0}\) in SLDRFFB and SLDTFPB is the same. From Figure 12, it can be seen that these curves which represent \(r_{R0}\), \(r_{T0}\) and \(r_{Te}\) basically coincide. The equivalent radius of SLDRFFB is always less than that of SLDTFPB. \[r_{R0}=\frac{L^{2}-4h_{R0}^{2}}{8h_{R0}}\approx\frac{L^{2}}{8h_{R0}}, \tag{28}\]  
\[r_{T0}=\frac{\left(\frac{k}{3}\right)^{2}-4h_{T0}^{2}}{4h_{T0}}-h\approx\frac {L^{2}}{36h_{T0}}. \tag{29}\]  
In general, the equivalent radius of SLDRFFB is smaller than the radius of the spherical shell, and the equivalent friction coefficient and equivalent mass of SLDRFFB are larger than the friction coefficient of the sliding surface and mass of the isolation layer. Thus, restoring force, friction force and seismic load are magnified in different scales. The equivalent radius of SLDTFPB is almost  
Figure 11: Equivalent radii of different stories’ frame structures equipped with SLDTFPB and SLDRFFB. Figure 12: Equivalent radii of isolation systems equipped with SLDRFFB and SLDTFPB, respectively. twice as much as the spherical shell and is close to the orbit radius of the superstructure's motion. The equivalent friction coefficient of SLDTFPB is similar to the friction coefficient of the sliding surface, and the equivalent mass of SLDTFPB is a little larger than the mass of the isolation layer. Generally, isolation frequency of the isolation layer with large equivalent radius is small, and damping capacity of the isolation layer with large equivalent friction coefficient is large.