## 3 Numerical Analyses
### Steady-State Response for Horizontal Excitation  
To investigate the response characteristics of seismic and vibration isolators with spring-viscous dampers, we analyzed the steady-state response of the superstructure when a horizontal sine wave disturbance occurs in the base structure. This disturbed displacement of the base structure can be expressed as \(X_{o}\sin 2\pi ft\), where \(X_{o}\) and \(f\) are the excitation amplitude and frequency, respectively. Figure 10 illustrates the steady-state responses of the isolation device to the horizontal excitation at certain specific frequencies with an amplitude \(X_{o}=5\) mm, where the design parameters were \(f_{y}=5\) Hz, \(\alpha=0.1\), \(\mu=0.01\), \(R=0.6\) m, and \(\zeta=0.3\). Figure 10a indicates that when the excitation frequency was 0.55 Hz, the horizontal excitation and absolute displacement of the superstructure were the same. That is, horizontal relative displacement of the superstructure does not occur with respect to the horizontal vibration, which means that the superstructure does not slip against the dome. As the horizontal excitation force was less than the horizontal friction force, it can be observed that no sliding occurred, and the superstructure moved with the same displacement as the horizontal  
Figure 8: Unstable contact region change due to friction coefficient when \(R=0.6\) m, \(\mu=0.02\), and \(\alpha=0.1\). Figure 9: Unstable contact region change owing to damping ratio when \(R=0.6\) m, \(\zeta=0.3\), and \(\alpha=0.1\). excitation displacement, and because there was no relative movement, no vertical displacement occurred. The response at an excitation frequency of 0.82 Hz is illustrated in Figure 10b. As indicated in Equation (14) and Figure 7, when \(R=0.6\) m, \(\alpha=0.1\), and \(f_{y}=5\) Hz, the response in Figure 10b is resonance, because \(f_{x}\) is 0.82 Hz. The steady-state response was approximately 1.8-times larger than the excitation amplitude. Owing to the nonlinear effect, the horizontal response exhibited harmonics, which were thrice the excitation frequency, while the vertical response exhibited dominant harmonics, which were twice the excitation frequency. Figure 10c,d illustrate the steady-state response of the system to base excitation at 1 and 2 Hz, respectively. In the horizontal response of Figure 10c,d, harmonics that are three times the excitation frequency dominate, and in the horizontal response, harmonics that are twice the excitation frequency dominate. Furthermore, in the case of Figure 10, the sign of the spring deformation in Equation (8) is always observed to be positive, so that the spring remained compressed and the dome was always in contact with the superstructure. Figure 10: _Cont._  
Figure 11 presents a case in which only the amplitude of the excitation was changed to 20 mm under the conditions of Figure 10. Figure 11a indicates that no relative motion occurred until the excitation amplitude increased. As the excitation amplitude increased, relative movement is believed to have occurred because the excitation force became larger than the horizontal friction force. Figure 11b illustrates the response of the resonance state as indicated in Figure 10b, and a response that is 12.8-times larger than the excitation amplitude was exhibited. Figure 11c illustrates the deformation of the spring-viscous damper spring under the conditions of Figure 11b. As indicated in Figure 11c, the springs on the left and right were deformed alternately around the static deflection in Equation (11). Therefore, as indicated in Equation (8), it can be observed from Figure 11c that the spring was tensioned at approximately 12 mm (the negative sign of deflection). This means that the dome and superstructure were not in contact. In Figure 11b, the excitation amplitude is 20 mm and the excitation frequency is 0.82 Hz; thus, the maximum acceleration is 0.53 m/s\({}^{2}\). The projection of these values in Figure 7 reveals that they lie in the unstable contact area.