# Dynamic Behavior of a Simple Rolling Seismic Isolator with a Position Restoring Device
## 3 Numerical Analysis
### Steady-State Response and Frequency Response  
To investigate the response characteristics of the seismic isolation system equipped with PRD, the steady-state response of the superstructure was analyzed when the base table had a sinusoidal disturbance. This disturbed displacement of the base table can be expressed as \(u=U\sin 2\pi ft\), where \(f\) is the excitation frequency. As shown in Equations (9)-(13), the seismic response is affected by \(k/M\), \(\xi_{r}\), and \(U\). Therefore, the characteristics of the response were analyzed according to the changes of \(k/M\), \(\xi_{r}\), and \(U\). Figure 6 compares the response of device without and with PRD for the excitations of a specific frequency; the design parameters of the isolation systems are shown in Table 1 and that of PDR are \(k/M=5\) and \(\xi_{r}=0.2\). The natural frequency of the system without PRD was calculated to be 0.66 Hz by Equation (15). From the numerical results shown in Figure 6a, it can be seen that the resonance frequency at 0.66 Hz corresponds to the natural frequency. The amplitude at the resonance frequency tends to become very small attenuation despite the presence of PRD. This is because as the \begin{table}
\begin{tabular}{c c c c} \hline \hline
**The Ratio of \(k/M\)** & **0 (without PRD)** & **5** & **10** \\ \hline
**Time Taken to Return (s)** & 16.6 & 12.55 (\(\mathbf{\Psi}\)23.4\%) & 10.05 (\(\blacktriangledown\)39.3\%) \\
**Offset (mm)** & 1.2 & - & - \\
**Damping Ratio \(\zeta_{r}\)** & 0 & 0.3 & 0.5 \\
**Time Taken to Return (s)** & 8.14 & 5.06 (\(\blacktriangledown\)37.8\%) & 3.71 (\(\blacktriangledown\)54.4\%) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Summary of transient analysis result (\(\blacktriangledown\): Decease). Figure 4: Transient responses of the superstructure according to change of \(\zeta_{r}\) in case of \(k/M=5\). Figure 5: Transient response and envelope for the seismic isolation system (RIS) with PRD (initial displacement = 0.05 mm, \(k/M=0,\,5,\,10\) and \(\zeta_{r}=0.3\)). amplitude increases due to resonance, the ratio of the width of the wedge hole to the range of motion becomes smaller, thereby reducing the effect of the PRD. Figure 6b,c show the steady-state response of the system to base excitation at \(u=0.03\sin 4\pi t\), corresponding to 2 Hz and at \(u=0.03\sin 6\pi t\), corresponding to 3 Hz. In Figure 6b, which shows the response curve at 2 Hz, which is three times the natural frequency, the harmonic is dominant at the resonance frequency. On the other hand, it can be confirmed that the harmonic component does not appear when the excitation frequency is 3 Hz, which is not an integral multiple of the resonance frequency as shown Figure 6c. To investigate the harmonic phenomena according to the design parameters, the influence of damping, which is controllable among parameters, is examined, and the results are shown in Figure 7. As shown in Figure 7, the harmonic phenomenon is negligible in the damping value above 0.3. Therefore, it is possible to control the harmonic phenomenon by setting an appropriate damping coefficient. Figure 6: Steady-state response to harmonic excitation under \(k/M=5\) and \(\zeta_{r}=0.2\). The effect of design parameters on the steady-state response for the absolute displacement of the system was investigated through base excitation at constant amplitude and frequency. As mentioned above, the displacement transmissibility ratio \(T_{r}\) is defined as the ratio of the overall peak value of the steady-state response to \(U\), because the steady-state response contains harmonic components. It was obtained for various values of \(k/M\) and \(U\), when \(\zeta_{r}\) = 0.3. The curves in Figures 8 and 9 represent \(T_{r}\) for \(k/M\) and \(U\), respectively. In Figures 8 and 9, the resonant frequency appears at 0.66 Hz, and represents a typical frequency response function of a one degree of freedom system with \(T_{r}\) = 1 at 0.93 Hz, which is \(\sqrt{2}\) times the resonant frequency.