## 3 Sensitivity Study on Vertical Seismic Isolation Performance with VIF
\begin{table}
\begin{tabular}{l c} \hline \hline
**Parameters** & **Values** \\ \hline Superstructure mass (kg) & 1000 \\ Stiffness of vertical spring, \(K_{\text{VS}}\) (N/m) & Variable depending on VIF \\ Damping of vertical spring, \(\text{CVS}\) (kg/s) & 62.8 \\ Elastic stiffness of steel damper, \(K_{\text{e}}\) (N/m) & 621,700 \\ Tangential stiffness of steel damper, \(K_{\text{t}}\) (N/m) & 36,460 \\ Characteristic strength, \(F_{\text{slide}}\) (N) & 990 \\ Struct. Damping of steel damper, \(C_{\text{SD}}\) (kg/s) & 0 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Preliminary design parameter values for sensitivity analysis. To investigate the sensitivity of the design VIF on the vertical seismic isolation performance, the design VIF was considered in the range of 1 Hz\(\sim\)5 Hz. The equivalent damping ratio and stiffness of the steel damper corresponding to the hysteretic bilinear model in Table 3 are 30.7% and 75,291 N/m, respectively. The equivalent stiffness of the steel damper has a corresponding natural frequency of about 1.38 HZ for a vertical design load of 10 kN. Therefore, the actual VIF will be determined by considering both spring and steel damper stiffness. Figure 5 shows the result of the calculated vertical response spectrum for the super structure according to the design VIF determined by the stiffness value of the vertical spring. As shown in the figure, it can be seen that the lower the design VIF, the higher the seismic isolation performance is due to the frequency shift effect. When the design VIF is 1.0 Hz, the spectral acceleration response of the superstructure is significantly reduced compared with the input response spectrum throughout frequencies. When the design VIF increases to 3 Hz, the superstructure exhibits the vertical seismic isolation effect in the range of about 6 Hz to 50 Hz, and the ZPA value is almost similar to that of the input earthquake without vertical seismic isolation effect. When the design VIF exceeds 3.0 Hz, the vertical seismic isolation effect is greatly reduced in the overall frequency range, and the ZPA value of the superstructure becomes larger than that of the input earthquake. Figure 4: Schematics of simple seismic analysis model used for sensitivity analysis: (**a**) Finite element model of vertical seismic isolation device; (**b**) bilinear model of steel damper. Figure 5: Spectral acceleration response of superstructure for various design VIF. Figure 6 shows the maximum vertical spring displacement response according to the design VIF. As shown in Figure 6a, when only a vertical spring is used without using a steel damper, the vertical spring displacement becomes very large (e.g., 101.3 mm when VIF = 3 Hz). Therefore, it is not possible to design a vertical seismic isolation device actually accommodating the LRB dimensions in Table 1. When the steel damper with the design characteristics of Table 3 is used, the vertical spring displacement can be significantly reduced to 30 mm or less, as shown in Figure 6b. In addition, it can be seen that the actual VIF is larger than the design value assumed for the vertical stiffness spring due to the stiffness effect of the steel damper, showing the hysteretic characteristic. The lower the design VIF, the larger the stiffness effect of the steel damper. When the design VIF is 1.0 Hz, the actual VIF increases by 80%. Table 4 summarizes the sensitivity analysis results of the vertical seismic isolation performance according to the design VIF. As shown in the table, it can be seen that as the design VIF increases, the effective frequency range in which the actual seismic isolation effect can be obtained is rapidly reduced. When the design VIF is equal to or higher than 3.0 Hz, the ZPA response becomes larger than that of the input floor response spectrum, and then the vertical seismic isolator has the opposite effect of amplifying the vertical seismic response in the high-frequency region.