# Dynamic Behavior of a Simple Rolling Seismic Isolator with a Position Restoring Device
## 3 Numerical Analysis
### Steady-State Response and Frequency Response
In the frequency range higher than 0.93 Hz, the response magnitude of the device is smaller than \(U\) and decreases with increasing frequency. For frequencies below 0.93 Hz, \(T_{r}\) is greater  
Figure 7: The change of harmonic response according to the change of damping coefficient when \(U=0.03\) m, \(f=2\) Hz and \(k/M=5\). than or equal to 1, and a peak value is present. Further, in the region larger than 0.93 Hz, \(T_{r}\) increases as the \(k/M\) increases, which is equivalent to the increase of the damping value of the vibration system. As shown in Figure 9, the response of \(U=0.01\) m shows a different frequency response characteristic, which is that \(T_{r}\) is not 1 at 0.93 Hz, unlike the other \(U\) values. In this case, the maximum \(T_{r}\) is 2.577 at 0.81 Hz, the relative displacement is 15.7 mm (absolute displacement is 25.7 mm), and relative motion occurs only within the width of the wedge hole (\(\pm\)20 mm) at all excited frequencies. On the other hand, the relative movement where the spring unit moves out from the wedge hole easily occurs when the excitation amplitude or frequency are increased, as in the other cases. Therefore, in the case of \(U=0.01\) m in Figure 9, the frequency response characteristic seems to be exceptional, unlike in other cases. In addition, we can see in Figures 8 and 9 that the jumping phenomenon occurs, wherein the relative motion increases suddenly at some frequencies. This frequency is called the jump frequency \(f_{p}\). Relative motion between the superstructure and the base table occurs as the inertia force becomes larger than the frictional force. When the relative displacement becomes larger and the amplitude becomes larger than the width \(a\) of the wedge hole, the spring unit of the PRD moves away from the wedge hole. At this time, a jump phenomenon occurs, where the motion of the superstructure suddenly becomes large. In the case of \(k/M=5\) in Figure 8, relative motion starts at 0.44 Hz, and movement occurs in the wedge hole (a = 20 mm) up to 0.58 Hz, with \(T_{r}=1.592\) (absolute displacement 47.8 mm, relative displacement 17.8 mm). At 0.59 Hz, which is larger than the excitation frequency (that is, the jump frequency of 0.58 Hz), the \(T_{r}\) suddenly increases to 4.573 (absolute displacement 137.2 mm, relative displacement 107.2 mm), causing a jump in which the motion is greatly excited from the wedge hole. For \(k/M=7.5\), a jump occurs at 0.71 Hz. In this case, the \(T_{r}\) is 1.36 (absolute displacement: 40.8 mm, relative displacement: 10.8 mm) at the excitation frequency of 0.66 Hz, corresponding to the natural frequency of the system, and movement is limited only within the wedge hole. Therefore, large amplitude resonance does not occur. This can be applied to a design that avoids resonance. As \(U\) in Figure 9 increases, the \(f_{p}\) becomes smaller. If \(U\) is very large, the jump does not occur, and the transfer rate curve will be similar to that of a typical one-degree-of-freedom linear vibration system, such as \(k/M=0\) (without PRD). In other words, if the amplitude of the excitation is very large, the motion moves out of the wedge hole in all the frequency domains, which means that the PRD only plays the role of damping. Considering this jump phenomenon from the viewpoint of energy, the kinetic energy of the superstructure must be at least greater than the potential energy of the PRD spring. That is, the necessary condition of the frequency at which the jump will occur can be expressed by the following equation. \[\frac{1}{2}M(2\pi f_{p}\ U)^{2}>\ \frac{1}{2}k(b+y_{0})^{2}-\frac{1}{2}k\ y_{ \ o}^{2}\, \tag{16}\]  
\[f_{p}>\frac{1}{2\pi}\ \frac{\sqrt{b(b+2y_{o})}}{U}\sqrt{\frac{k}{M}}\ . \tag{17}\]  
From Equations (16) and (17), it can be seen that \(f_{p}\) is proportional to Figure 8: Overall response vs. excitation amplitude ratio according to the parameter \(k/M\) in case of \(U=0.03\) and \(\zeta_{r}=0.3\).