## 3 Numerical Analyses
### Displacement Transfer Ratio for Horizontal Excitation  
This section analyzes the characteristics of the vibration isolation system by calculating the displacement transfer ratio \(T_{r}\) during the steady-state response, while maintaining the contact state. As illustrated in Figure 7, unstable contact does not occur when the excitation acceleration or amplitude is small, so the analysis of \(T_{r}\) was performed with an excitation amplitude \(X_{0}=5\) mm. Figure 12 illustrates the \(T_{r}\) values of the horizontal and vertical motions for various friction coefficients. The results obtained were for a seismic isolator with \(f_{y}=5\) Hz, \(R=0.6\) m, and \(\alpha=0.1\), where \(f_{x}\) was 0.82 Hz for this device, according to Equation (13). In the case of the friction coefficient of \(\mu=0.005\) in Figure 12a, relative motion did not occur, as \(T_{r}=1\) until the excitation frequency reached 0.51 Hz, following which relative motion occurred. Moreover, resonance appeared at 0.82 Hz, and it is represented as a frequency response function of the one-degree-of-freedom system, where \(T_{r}\approx 1\) at \(\sqrt{2}\) times the resonance frequency. For frequency ranges greater than 1.16 Hz, \(T_{r}\) was less than 1, and it decreased as the excitation frequency increased. In Figure 12b, which presents the vertical transfer ratio  
Figure 11: Steady-state response and spring deflection under \(f_{y}=5\) Hz, \(\alpha=0.1\), \(\mu=0.02\), \(R=0.6\) m, and \(\zeta=0.3\): **(a)** Excitation conditions: \(X_{o}=20\) mm and \(f=0.55\) Hz. **(b)** Excitation conditions: \(X_{o}=20\) mm and \(f=0.82\) Hz. **(c)** Spring deflection: \(X_{o}=20\) mm and \(f=0.82\) Hz. with the same friction coefficient, \(T_{r}\) does not appear because no horizontal relative motion occurred until 0.51 Hz. However, as horizontal relative motion occurred in the frequency range greater than 0.51 Hz, \(T_{r}\) is displayed in this case. The vertical \(T_{r}\) value also reached its maximum at \(f_{x}\) of 0.82 Hz with the maximum horizontal \(T_{r}\). In Figure 12, where the coefficients of friction were 0.008 to 0.05, it can be observed that as the friction coefficient increased, the excitation frequency at which the relative motion began also increased. This means that as the frictional force increases, the excitation force must also increase to induce relative motion. The starting frequencies obtained from Equation (14) are displayed in Figure 12a. Furthermore, in the cases with a friction coefficient of 0.008 to 0.015, an unusual phenomenon was observed in which \(T_{r}\) was slightly less than 1 at an excitation frequency slightly above \(f_{s}\). This is owing to the small relative motion resulting from slight sliding of the superstructure, as well as the phase difference between the relative motion and excitation. In cases with a friction coefficient of approximately 0.008 to 0.012, a jump phenomenon could be observed, whereby \(T_{r}\) was suddenly increased. As the excitation frequency increased, \(T_{r}\) decreased to slightly less than 1, following  
Figure 12: Displacement transfer ratio of peak–peak value of the steady-state response to that of the excitation displacement for friction coefficient \(\mu\), when \(f_{y}=5\) Hz, \(R=0.6\) m, \(\alpha=0.1\), \(\zeta=0.3\), and \(X_{o}=5\) mm: (**a**) Horizontal direction (peak–peak value of \(U\) versus that of \(X\)). (**b**) Vertical direction (peak–peak value of \(y\) versus that of \(X\)). Figure 13 illustrates \(T_{r}\) when \(f_{y}\) was 3, 5, 6, and 7 Hz and \(f_{x}\) was 0.72, 0.82, 0.89, and 0.96 Hz from Equation (13) or Figure 5, respectively. In the case of \(f_{y}\) = 3 Hz, because \(f_{s}\) was 0.71 Hz, which is almost equal to the natural frequency 0.72 Hz, the resonance was suppressed and the maximum \(T_{r}\) was 1.1, which is slightly larger than 1. However, as indicated in Figure 13, in the other cases, the maximum \(T_{r}\) also increased because the difference between \(f_{s}\) and \(f_{y}\) increased.