## 3 Mechanical Properties of Isolation Systems Equipped with SLDRFFB and SLDTFFB
### Motion Equations of Isolation Systems Equipped with SLDRFFB and SLDTFPB  
Since motion of isolation systems equipped with the developed SLDRFFB or SLDTFPB constitutes a rotational component, and bearing reaction cannot be determined straightforward, D'Alembert principle is not suitable for establishing motion equations of the isolation system with SLDRFFB or SLDTFPB. The Lagrange principle is more convenient to establish the motion equations. Assume that:  
1. Mass of each story is evenly distributed on the surface. 2. Superstructure satisfies rigid floor assumption. 3. Columns have infinite vertical stiffness, and superstructure is \"shear-type\". 4. Considering one-dimensional horizontal seismic action, the vertical acceleration of the superstructure is caused by the inclination of the isolation layer. 5. Each slider is in point contact with the sliding surface so that the bearing reaction points to the center of the spherical surface. 6. Seismic loads are expressed as the product of mass and ground acceleration. According to the Lagrange principle, the equations of motion are described by Equations (1)-(3),  
\[\frac{d}{dt}\bigg{(}\frac{\partial T}{\partial\dot{\theta}}\bigg{)}-\frac{ \partial T}{\partial\theta}+\frac{\partial V}{\partial\theta}=\frac{\partial \Big{(}W_{f}+W_{gm}+W_{c}\Big{)}}{\partial\theta}, \tag{1}\]  
\[\frac{d}{dt}\bigg{(}\frac{\partial T}{\partial\dot{u}_{i}}\bigg{)}-\frac{ \partial T}{\partial u_{i}}+\frac{\partial V}{\partial u_{i}}=\frac{\partial \Big{(}W_{f}+W_{gm}+W_{c}\Big{)}}{\partial u_{i}}, \tag{2}\]  
\[\frac{d}{dt}\bigg{(}\frac{\partial T}{\partial\dot{x}_{i}}\bigg{)}-\frac{ \partial T}{\partial x_{i}}+\frac{\partial V}{\partial x_{i}}=\frac{\partial \Big{(}W_{f}+W_{gm}+W_{c}\Big{)}}{\partial x_{i}}, \tag{3}\]  
where variables include \(\theta\), \(u_{i}\), or \(u_{i}\) represent the inclination angle of isolation layer, the relative displacements between each story of the superstructure and isolation layer and the horizontal displacements of each story between each story of the superstructure and ground, \(T\) is kinetic energy of the system, \(V\) is potential energy of the system, \(W_{gm}\) is work of ground motion, which is determined by horizontal response of each degree of freedom, \(W_{f}\) represents work of friction force, based on the relative displacement response of two closed surfaces, and \(W_{c}\) is work of the damping force in the superstructure. Figure 4 shows an isolation structure equipped with SLDRFFB. As shown in Figure 4, degrees of freedom of the isolation system equipped with SLDRFFB includes the inclination angle of isolation layer \(\theta\) and relative displacements \(u_{i}\) between each story of the superstructure and isolation layer. When the bottom of the spherical shell is located at the origin of coordinates, displacement responses of the isolation layer are described by Equation (4). \[\left\{ \begin{array}{l}x_{R0}=r_{R0}\sin\theta\\ y_{R0}=r_{R}-r_{R0}\cos\theta\end{array} \right., \tag{4}\]  
where \(r_{R}\) is the radius of the spherical shell in SLDRFPB, and \(r_{R0}=r_{R}-h_{R0}\) is the radius of the movement orbit of the center of the bottom of the superstructure, which is equal to the difference between the radius of the spherical shell \(r_{R}\) and the depth of the isolation layer \(h_{R0}\). We assume \(\sin\theta\approx\theta\), \(\cos\theta\approx 1\) due to the small inclination angle, which will be validated in Section 4.