## 3 Mechanical Properties of Isolation Systems Equipped with SLDRFFB and SLDTFFB
### Motion Equations of Isolation Systems Equipped with SLDRFFB and SLDTFPB
By substituting Equation (4) into Equations (1) and (2), and ignoring higher order items, we have  
\[\frac{\sum\limits_{i=0}^{n}m_{i}r_{Ri}^{2}+l_{i}}{\sum\limits_{i=0}^{n}m_{i}r_ {Ri}}m_{0}\frac{\ddot{x}_{0}}{r_{R0}}+m_{0}\frac{\sum\limits_{i=1}^{n}m_{i}r_{ Ri}\ddot{u}_{i}}{\sum\limits_{i=0}^{n}m_{i}r_{Ri}}+m_{0}\frac{x_{0}}{r_{R0}}+m_{0} \frac{\sum\limits_{i=1}^{n}m_{i}gu_{i}}{\sum\limits_{i=0}^{n}m_{i}r_{Ri}}+ \frac{m_{0}r_{R}}{\sum\limits_{i=0}^{n}m_{i}r_{Ri}}\sum\limits_{i=0}f=m_{0} \ddot{x}_{g}(t), \tag{5}\]  
\[m_{i}\bigg{(}\ddot{u}_{i}+r_{R1}\frac{\ddot{x}_{0}}{r_{R0}}\bigg{)}+m_{i}g \frac{x_{0}}{r_{R0}}+k_{i}u_{i}-k_{i}u_{i-1}+k_{i+1}u_{i}-k_{i+1}u_{i+1}+c_{i} i_{i}-c_{i}\ddot{u}_{i-1}+c_{i+1}\dot{u}_{i}-c_{i+1}\dot{u}_{i+1}=m_{i}\ddot{x}_{g}(t) \tag{6}\]  
where \(m_{i}\) is mass of the \(i\)-th story, which is lumped at the surface of the floor, \(r_{Ri}\) is the distance between the circle center of the spherical orbit and the center of the \(i\)-th story, \(l_{i}\) is inertia moment of the \(i\)-th story, \(\sum f\) is the algebraic sum of the friction force, \(k_{i}\) is horizontal stiffness of the \(i\)-th story and \(c_{i}\) is horizontal damping of the \(i\)-th story. Specifically, \(i=0\) represents the isolation layer. \(k_{n+1}=0\), \(c_{n+1}=0\) and \(u_{0}=0\) should be adopted as zero in practical implementation since they represent unreal stiffness, damping over the top story and relative displacement of the isolation layer. Degrees of freedom in the isolation system equipped with SLDTFPB are outlined in Figure 5. The inclination angle of sliding frame \(\theta\) and horizontal displacements \(x_{i}\) are used in the analysis. The relative displacements from the center of the sliding frame to the top and bottom of the isolation layer are the same so that displacement responses of the isolation layer are expressed as Equation (7). \[\left\{ \begin{array}{l}x_{T0}=2\Big{(}r_{T}\cos\alpha-\frac{1}{2}h_{T}\Big{)} \sin\theta\\ y_{T0}=2\Big{(}r_{T}\cos\alpha-\frac{1}{2}h_{T}\Big{)}(1-\cos\theta)\end{array}  \right., \tag{7}\]  
where \(r_{T}\) is the radius of the spherical shell in SLDTFPB, \(\alpha\) is the angle of the line between the bottom of the side column in the sliding frame and the corresponding orbit center and \(h_{T}\) is height difference in the shaft of the side column in the sliding frame between the top and bottom of the isolation layer. Equation (7) shows that the movement orbit of the isolation layer in SLDTFPB is an arc whose center  
Figure 4: Isolation structure equipped with SLDRFPB. height is \(2r_{T}\cos\alpha-h_{T}\) and radius is \(r_{T0}=2r_{T}\cos\alpha-h_{T}\). The superstructure will move along the arc and keep upright when subjected to earthquakes. By substituting Equation (7) into Equations (1) and (3), the motion equations of the isolation layer and each story are further described by:  
\[ \begin{split}&\Big{(}\frac{m_{b}(r_{T}\cos\alpha)^{2}+h_{b}}{2r_{ T}-h_{T}}+m_{0}(2r_{T}-h_{T})\Big{)}\ddot{x}_{0}+\bigg{(}\frac{r_{T}}{2r_{T}-h_{T}}m_{ b}\cos\alpha+m_{0}+\sum\limits_{i=1}^{n}m_{i}\bigg{)}gx_{0}+k_{1}(2r_{T}-h_{T})(x_{0}- x_{1})\\ &=(m_{b}r_{T}\cos\alpha+m_{0}(2r_{T}-h_{T}))\ddot{x}_{g}(t)-r_{T} \text{sgn}\big{(}x_{0}\big{)}\sum f-c_{1}(2r_{T}-h_{T})\big{(}\dot{x}_{0}-\dot{ x}_{1}\big{)}\\ &\qquad\qquad m_{i}\ddot{x}_{i}+k_{i}x_{i}-k_{i}x_{i-1}+k_{i+1}x _{i}-k_{i+1}x_{i+1}+c_{i}\dot{x}_{i}-c_{i}\dot{x}_{i-1}+c_{i+1}\dot{x}_{i}-c_ {i+1}\dot{x}_{i+1}=m_{i}\ddot{x}_{g}(t),\end{split}  \tag{8}\]  
where \(m_{b}\) is total mass of basements which are sliding frames inside SLDTFFB, and \(l_{b}\) is total inertia moments of basements. By comparing Equations (5), (6), (8) and (9), it can be found that mechanical properties of SLDRFPB and SLDTFFB are complicated and different from those of classical FPB. Seismic loads will not only cause lateral acceleration but also angular acceleration. This is because the superstructure equipped with SLDRFPB and sliding basement in SLDTFFB rolls along their orbits of the spherical shell, respectively.