## Mechanical Properties of Ring-Friction Pendulum Isolation Bearings
### Vertical Seismic Isolation Performance of the Ring Spring-Friction Pendulum Bearing  
According to the study of [PERSON] [21], shear damage occurs in the soil above the Dakai station during a strong earthquake, and the earthquake peaks at about 6 s. The plastic shear strain runs through the soil above. The soil loses shear resistance. The gravity of the overlying soil acts on the top of the structure under the vertical earthquake. Due to the sudden increase in vertical pressure on the structure, the structure's ductility in the horizontal direction reduces and the shear resistance weakens, which causes compression-shear damage in the weak part of the structure. Therefore, by setting vertical isolation devices, the vertical stiffness of the isolated structure can be reduced, thus reducing the self-oscillation frequency of the structure, which ultimately serves the purpose of reducing the transmission of vertical seismic effects. Figure 1: Structure diagram of the ring spring–friction pendulum isolation bearing. During an earthquake, the overlying soil of the structure loses its shear capacity, and eventually, the overlying soil becomes an additional structure mass. As a result, the stiffness of the soil and structure is much greater than that of the vibration isolation bearing. Therefore, this paper considers the soil and structure as one part. The seismic isolation structure is considered as a single-degree-of-freedom viscoelastic system [22], and the force diagram of the system is shown in Figure 2.  
When the single-degree-of-freedom viscoelastic system is subjected to dynamic load, the transfer rate equation is as follows:  
\[\text{TR}=\Biggr{\{}\frac{1+\left[2\xi(\omega/\omega_{\text{n}})\right]^{2}}{ \left[1-\left(\omega/\omega_{\text{n}}\right)^{2}\right]^{2}+\left[2\xi(\omega /\omega_{\text{n}})\right]^{2}}\Biggr{\}}^{1/2} \tag{1}\]  
where \(\xi\) is the damping ratio of the seismic isolation structural system, \(\omega\) is the frequency of dynamic loads, and \(\omega_{\text{n}}\) is the self-vibration frequency of seismically isolated structures. It can be seen from the above equation that transmissibility TR is a function of the frequency ratio \(\omega/\omega_{\text{n}}\). Figure 3 shows the function curve of transmissibility TR and the frequency ratio \(\omega/\omega_{\text{n}}\) for different damping ratios. It can be seen from the figure that damping is the main factor affecting transmissibility when the frequency ratio is in the range of \([1,\sqrt{2}]\). When the damping ratio is larger, the transmissibility is larger. However, when the frequency ratio is less than 1, the transmissibility of the resonant force is always less than 1, no matter how much the damping is. Therefore, in order to make the transmissibility of dynamic load smaller, the stiffness \(K\) of the system should be adjusted to as large a value as possible so that the self-oscillation frequency of the system reduces to below \(\omega\sqrt{2}/2\). Because in this frequency range, although the greater the damping, the higher the transmissibility, the impact is small and the increase is not large; in this case, the frequency ratio is the key factor affecting the transmissibility TR [23; 24]. Figure 3: Transmissibility of harmonic oscillation. Figure 2: Mass-spring-damper system. The vertical seismic isolation performance of the ring spring-friction pendulum bearing is mainly provided by the ring spring.