## Mechanical Properties of Ring-Friction Pendulum Isolation Bearings
### Horizontal Seismic Isolation Performance of the Ring Spring-Friction Pendulum  
The horizontal seismic isolation performance of the ring spring-friction pendulum bearing is mainly provided by the friction pendulum, which is slightly different from the traditional friction pendulum. Figure 7 shows the force diagram of the modified friction pendulum. In this paper, the friction of the slider in the friction pendulum was regarded as the classical Coulomb friction. The following assumptions were proposed:  
1. The value of the frictional force is proportional to the upper normal load and irrelevant to the contact area. 2. The value of the frictional force is irrelevant to the speed of the object's movement
3. The coefficient of friction is constant. Each friction pendulum contains two friction pairs, and the friction surface of each friction pair is at opposite angles. This design ensures that no matter if the two sliders move to the left or to the right during the sliding process, each slider can be in contact with the friction surface, making the structure more stable and less likely to tip over when subjected to horizontal forces. This paper assumes that the axis of symmetry of the friction pendulum intersects the vertical line of the friction surface at O, P is the center of gravity of slider A and slider B, the length of OP is \(R\), and the inclination of the friction surface is \(\theta\). So, the horizontal sliding displacement of the slider can be expressed as \(D=R\)sin\(\theta\), \(W\) is the structural gravity of the friction pendulum, the two sliders are subjected to positive pressures \(T_{1}\) and \(T_{2}\) and ring spring support forces \(N_{1}\) and \(N_{2}\), \(F\) is the horizontal restoring force of the two friction pendulum bearings' combined force,\(f_{1}\) and \(f_{2}\) are the friction forces on the slider, and \(K_{\
u}\) is the stiffness of the ring spring [27; 28]. According to the principle of bending moment balance, the following formula can be obtained:  
\[FR\cos\theta+\frac{2T_{1}D}{\cos\theta}=(f_{1}+f_{2})R+(N_{1}+N_{2})R \tag{4}\]  
and  
\[\left\{ \begin{array}{l}T_{1}=N_{1}\cos\theta\\ N_{1}=\frac{W}{2}-KD(t)\sin\theta\\ N_{1}+N_{2}=W\\ f_{1}+f_{2}=\mu W\cos\theta\text{sgn}\end{array} \right. \tag{5}\]  
Substituting Equation (5) into Equation (4), the horizontal restoring force can be expressed as the following equation:  
\[F=2K_{\
u}D\tan\theta+\mu\text{Wsgn}(D) \tag{6}\]  
and  
\[\text{sgn}=\left\{ \begin{array}{l}1\ \theta>0\\ 0\ \theta=0\\ -1\theta<0\end{array} \right\} \tag{7}\]  
Figure 7: Diagram of the friction pendulum bearing: (**a**) bearing deformation diagram and (**b**) force analysis of the friction pendulum bearing. The equivalent stiffness of the friction pendulum can be expressed as the following equation:  
\[K_{\rm h}=\frac{F}{D}=2K_{\rm v}\tan\theta+\frac{\mu\mathrm{Wsgn}(D)}{D} \tag{8}\]  
The hysteresis performance of the modified friction pendulum under cyclic loading in Abaqus software was simulated, and the correctness of the derivation of the above equation was verified. During the simulation, four models were created. A comparison was made by varying the friction surface's inclination angle \(\theta\), the friction coefficient of the friction surface \(\mu\), and the stiffness of the lower ring spring \(K_{\rm v}\). Next, the simulation results were compared with the formula. The critical parameters of the model are shown in Table 1. Figure 8 shows the dimensional drawing and the friction pendulum's 3D solid finite element model. The size of the four friction pendulums was the same, the material was Q345 steel, the modulus of elasticity was 200 GPa, Poisson's ratio was 0.33, the yield strength of steel was 345 MPa, and the density 7850 was kg/m\({}^{3}\). In order to ensure that the two upper support plates moved up and down at the same time, a rigid shell surface was placed on the upper part of the support plate. All loads were transferred to the lower friction pendulum through the shell surface.