## 3 Numerical Analyses
### Response Characteristics for Vertical Excitation  
The vibration isolator in this study had the structural characteristic of the superstructure moving in the horizontal and vertical directions; therefore, it could exhibit vibration isolation performance in both directions. Thus, we considered the response characteristics for the vertical excitation. As can be observed in Figure 2, the device structure was symmetrical, so no horizontal movement occurred when only vertical excitation was applied. Therefore, from Equation (9), the motion equation for vertical excitation can be expressed as follows:  
\[\ddot{y}+\omega_{y}^{2}(y-Y)+2\zeta\omega_{y}(\dot{y}-\dot{Y})=0\;\;\text{or} \;\;\ddot{y}+\omega_{y}^{2}y+2\zeta\omega_{y}\dot{y}\;=\omega_{y}^{2}Y+2\zeta \omega_{y}\dot{Y}, \tag{16}\]  
where Equation (16) is the typical motion equation of the base excitation model [39]. Therefore, if the excitation is vertical in the sinusoidal function, the displacement transfer ratio is:  
\[T_{r}=\frac{\sqrt{1+\left(2\dot{c}r\right)^{2}}}{\sqrt{\left(1-r^{2}\right)^{ 2}+\left(2\dot{c}r\right)^{2}}}, \tag{17}\]  
Figure 15: Displacement transfer ratio of horizontal direction when \(f_{y}\) = 5, \(R\) = 0.6 m, \(\alpha\) = 0.1, \(\mu\) = 0.01, and \(X_{o}\) = 5 mm. where \(r=\frac{f}{f_{y}}\) and \(f\) is the excitation frequency. Thus, the principle of Equation (17) can be applied to protect the superstructure from vertical excitation. **4. System Response for El Centro Earthquake**  
This section evaluates the proposed isolation system using data of the El Centro earthquake. Firstly, for the design parameter settings, the two seismic signals and superimposed power spectral densities of the stable region were verified, as illustrated in Figure 16. As indicated in the figure, it was recommended to select a design parameter below \(f_{y}=5\) Hz, because the excitation force at \(f_{y}=10\) Hz overlapped with the unstable region, while at \(f_{y}=7\) Hz, it was in the stable region but close to the boundary. Therefore, in this study, \(f_{y}=3\) Hz was selected to ensure sufficient stability. Figure 17 illustrates the horizontal and vertical responses of a device with the design parameters in Figure 7a, when the NS (North-South components) signal and UP (vertical components) signal of the El Centro earthquake ([[http://www.vibrationdata.com/elcentro.htm](http://www.vibrationdata.com/elcentro.htm)]([http://www.vibrationdata.com/elcentro.htm](http://www.vibrationdata.com/elcentro.htm))) were excited horizontally and vertically, respectively [40]. Moreover, Table 2 compares the maximum response values for the device of Figure 7b. The maximum acceleration response of the isolator (I) was reduced by 45% (\(\blacktriangledown\)0.15 g) and 40% (\(\blacktriangledown\)0.091 g) in the horizontal and vertical directions, respectively, compared to the maximum acceleration input. In isolator (II), it was decreased by 40% (\(\blacktriangledown\)0.131 g) and 17% (\(\blacktriangledown\)0.039 g), respectively. These results can be further improved by changing the damping ratio or friction coefficient. Therefore, it can be concluded that the proposed device is effective for excitation in both the horizontal and vertical directions. Figure 16: Overlaps of contact stability charts of Figure 9 and power spectral density of two El Centro earthquake waveforms (NS signal and UP signal): (**a**) \(f_{y}=3\) Hz. (**b**) \(f_{y}=5\) Hz. (**c**) \(f_{y}=7\) Hz. (**d**) \(f_{y}=10\) Hz. Also, it was compared with a rolling bearing isolation system (RIS) without a vertical spring damper. Figure 18 compares the horizontal response of the RIS and the isolator (I) due to horizontal excitation of an El Centro horizontal earthquake signal. The design parameters of RIS were \(R=0.6\) m, a natural frequency of 0.66 Hz, and rolling resistance coefficient 0.004, and the design parameters of the isolator (I) were \(f_{y}=3\) Hz, \(\alpha=0.1\), \(\mu=0.02\), \(R=0.6\) m, and \(\zeta=0.3\).