# A Novel Ring Spring Vibration Isolator for Metro Superstructure
## 4 Analysis of Vibration Isolation Structure  
### Model Parameters  
Our research takes a typical six-story frame structure as an example (Figure 11), and uses software SAP2000 to model the common structure and vibration isolation structure. The building plane is distributed in a rectangular shape. A single secondary beam is set between the main beams in the Y direction. The basic layout of the building is five spans, which are six meters in the X direction, and three spans, which are four meters in the Y direction. The structure has six floors, and each floor is 3.6 m high. The concrete strength grade is C30. The steel bars are HRB400 and HRB335. The section size of the frame column, the frame beam and the secondary beam is 600 mm \(\times\) 600 mm, 300 mm \(\times\) 600 mm and 300 mm \(\times\) 500 mm, respectively. The thickness of the slab is 110 mm. The self-weight of the structure and the uniformly arranged constant load of the floor are set as 3 kN/m\({}^{2}\). The live load of the floor is set as 2 kN/m\({}^{2}\). For the metro vibration simulation of buildings, the most commonly used direct method is to input the vertical acceleration at the bottom of the building. [PERSON] et al. [22] at South China University established a building model and compared simulation data with the measured data. The parameter analysis verified the correctness of the model, so it is desirable to input the measured vibration acceleration excitation in the base. Since the  
Figure 11: Schematic diagram of the three-dimensional structure model. focus of this paper is to establish a comparison between ordinary structures and vibration isolation structures to explore the vibration isolation effect of vibration isolators instead of accurately assessing the actual vibration level of the building, the interactions between soil and structure are not considered. The ten-second measured vibration acceleration of the free field is directly input in the building base (Figure 12). From the spectra curve of the metro vibration, the energy of metro vibration is shown to be mainly concentrated in the range of 20 Hz to 50 Hz, which is consistent with the research results [23]. In order to cover the main frequency band 30 Hz to 50 Hz in the analysis and to consider the low-order vibration mode of the building, the two frequency points of 1 Hz and 50 Hz are used to determine these two coefficients. Therefore, the damping coefficient is: \(\alpha\) = 0.3696 Hz, \(\beta\) = 1.872 \(\times\) 10\({}^{-4}\) s.  
Considering that vibration caused by the metro is microvibration and the structure is in the elastic state, the elastic material model has been used in the entire finite element analysis. The beam-column element is simulated by the frame element. The floor element is the shell element. Under slight vibration, the damping ratio of concrete can be taken as 0.03 [24]. In addition to its simplicity, the Rayleigh damping structure also satisfies orthogonal conditions. SAP2000 has developed the \"Damping-Friction Spring\" unit for the widespread application of friction springs at this stage. Its damping model is based on the viscoelastic model of [PERSON], and its nonlinear force-deformation relationship is [25]:  
\[f(t)=c(t)\Delta t \tag{6}\]  
where \(f(t)\) is the unit force, \(c\) (\(t\)) is the controllable damping constant (positive) and \(\Delta t\) is the damper deformation. As is shown in Formula (6), the unit dissipates energy based on displacement (independent of speed) [26], which is specifically manifested as internal friction related to the deformation during loading and unloading. The nonlinear force-deformation relationship is shown in Figure 13.  
Figure 12: Vertical acceleration curve: (**a**) time history; (**b**) spectra. The stiffness K\({}_{1}\) of loading is greater than the stiffness K\({}_{2}\) of unloading. Due to the change in the direction of the frictional force, the transition between K\({}_{1}\) and K\({}_{2}\) is made by a great elastic stiffness K\({}_{0}\).