## 3 Mechanical Properties of Isolation Systems Equipped with SLDRFFB and SLDTFFB
### Bearing Reaction and Friction Force of SLDRFPB and SLDTFPB  
According to the Coulomb friction principle, the friction coefficient under static and dynamic conditions is constant. Friction force in Equations (5) and (8) is determined by the state of motion and the reaction force. When the friction coefficient and curvature radius of the spherical shell surface remain unchanged, the term of friction force in the equation of motion yields the product of the symbolic function of the velocity of the isolation layer and the algebraic sum of the bearing reaction force, i.e.,  
\[\sum f=-\text{sgn}\big{(}\dot{x}_{0}\big{)}\sum\mu N, \tag{10}\]  
where \(\mu\) is friction coefficient between the sliding block and sliding surface, and \(N\) is the bearing reaction corresponding to friction force. In particular, the friction force in SLDTFFB is divided into the top and the bottom parts. The values of two parts are different, but the results of motion state judgement are the same. Figure 5: Isolation structure equipped with SLDTFFB. Bearing reaction cannot be solved based on the Lagrange principle. In that case, we establish the equilibrium equation according to D'Alembert's principle. Lateral and vertical resultant forces of statically determinate frame structures with single span and multi-story are equal to zero. Friction force is difficult to describe straightforward because it changes with the motion state. The friction coefficient is usually less than 0.1, or even less than 0.05, so that the friction force has little effect on the reaction force. It is more convenient to consider the friction coefficient as 0 when solving the force. A force analysis sketch of the isolation system equipped with SLDRFPB is shown in Figure 6.  
When the friction force is zero, the equilibrium equation of the isolation system equipped with SLDRFPB can be written as:  
\[\left[ \begin{array}{cc}\cos(\theta-\alpha_{R})&\cos(\theta+\alpha_{R})\\ \sin(\theta-\alpha_{R})&\sin(\theta+\alpha_{R})\end{array} \right]\left[ \begin{array}{c}N_{RI}\\ N_{Rr}\end{array} \right]=\left[ \begin{array}{c}\sum m_{i}g+\sum m_{i}r_{Ri} \sin(\theta+\alpha_{i})\ddot{\theta}\\ \sum m_{i}\ddot{x}_{g}-\sum m_{i}r_{Ri}\cos(\theta+\alpha_{i})\ddot{\theta} \end{array} \right] \tag{11}\]  
In this case, each bearing reaction can be simplified as:  
\[\left[ \begin{array}{c}N_{RI}\\ N_{Rr}\end{array} \right]=\left(\frac{m}{\sin 2\alpha_{R}}\right)\left[ \begin{array}{c}\sin(\theta+\alpha_{R})g-\cos(\theta+\alpha_{R})^{\prime} \ddot{x}_{g}+\sum\frac{r_{Ri}\cos(\alpha_{i}-\alpha_{R})}{n}\ddot{\theta}\\ -\sin(\theta-\alpha_{R})g+\cos(\theta-\alpha_{R})\ddot{x}_{g}-\sum\frac{r_{ Ri}\cos(\alpha_{i}+\alpha_{R})}{n}\ddot{\theta}\end{array} \right], \tag{12}\]  
where \(\alpha_{R}\) represents the angle between the vertical direction and the direction from the sliding block to the center of the spherical orbit, \(N_{RI}\) and \(N_{Rr}\) represent the left and right bearing reactions of columns respectively, and \(\alpha_{i}\) represents the angle between the vertical direction and the direction from the edge of the \(i\)-th story to the center of the spherical shell. It is noted that friction force has little influence on redistribution of the bearing reaction force, especially the algebraic sum of bearing reaction force when the friction coefficient is small. The algebraic sum of bearing reaction is expressed in terms of the result of Equation (12) under static condition,  
\[N_{RI}+N_{Rr}\approx\frac{r_{R0}}{r_{R}}\sum_{i=0}^{n}m_{i}g \tag{13}\]  
It can be seen that seismic load, location of sliding frame and state of motion have less influence on the result of Equation (13), so that the value of bearing reaction in the equation of motion can be adopted as the bearing reaction force of the original position. The conclusion can be further extended to statically indeterminate structures such as multi-span frame structures since the bearing reaction is distributed more uniformly. Figure 6: Force sketch of the isolation system equipped with SLDRFPB.