## 2 Equations of Motion  
### Description of Vibration Isolator Model  
The practical system is a complex structure in which multiple isolation devices support the superstructure, as shown in Figure 1. The one device is a symmetrical structure in which multiple spring dampers, to each of which a ball-type dome is attached, are vertically fixed to the lower plate to support an upper plate made of a concave surface. To consider the theoretical vibration characteristics of this system, structural simplification is needed. If we consider only two degrees of freedom in the vertical and horizontal directions without rotational motion, the behavior of multiple isolators is the same motion, so the system can be simplified to the distributed mass at one isolator and isolator, as shown in Figure 2a. The rotational motion is ignored because several mounts on the horizontal plane are used symmetrically on the vertical plane and the vertical displacement is small in the seismic excitation. The base structure also receives ground excitation, while the superstructure contains the equipment to be protected.  
When relative motion occurs between the base structure and superstructure, energy loss takes place, owing to the changes in the vertical potential energy, friction force, and viscous damping force. In Figure 2, point \(O\) is the origin of the \(x\)-\(y\) coordinate system set in static equilibrium, in which the system sets the \(x\)-axis as the horizontal line through the contact points of the dome and superstructure, and the \(y\)-axis as the vertical line passing through the center of the base structure.  
Figure 1: Configuration and installation diagram of the proposed vibration isolation mounting system.  
Figure 2: Model of isolator system with double spring-dampers: (**a**) Geometry configuration. (**b**) Displacement relationship of motion.  
According to the geometric relationship in Figure 2b, the deflection of the right and left dampers is, respectively, expressed as follows:  
\[y_{r}=R(\cos\theta_{o}-\cos\theta_{r})\text{ and }y_{l}=R(\cos\theta_{o}-\cos \theta_{l}) \tag{1}\]  
where,  
\[\theta_{o}=\sin^{-1}\left(\frac{a}{R}\right),\ \ \ \ \ \ \theta_{r}=\sin^{-1} \left(\frac{a-x}{R}\right),\ \ \ \ \ \ \theta_{l}=\sin^{-1}\left(\frac{a+x}{R}\right),\]  
\[\cos\theta_{o}=\sqrt{1-\left(\frac{a}{R}\right)^{2}},\ \ \ \ \ \cos\theta_{r}=\sqrt{1-\left(\frac{a-x}{R}\right)^{2}},\ \ \ \ \ \cos\theta_{l}=\sqrt{1-\left(\frac{a+x}{R}\right)^{2}}. \tag{2}\]  
When the superstructure only moves horizontally, the vertical traveling velocity of two domes or contact points that are used to predict the damping force of the spring-viscous damper can be obtained by differentiating Equation (1) with respect to time, and the results are as follows:  
\[\dot{y}_{r}=-x\tan\theta_{r}\text{ and }\dot{y}_{l}=\dot{x}\tan\theta_{l}. \tag{3}\]  
In the design, \(\frac{a}{R}\) is possible in the range of 0.5 or less. On the other hand, this value must be properly selected to ensure the maximum permissible moving displacement of the superstructure. So, the deflection of the spring with the displacement \(x\) varies in the isolation system with \(a=0.2R\), owing to the fact that the range of \(-0.3R\leq x\leq 0.3R\) was numerically interpreted and is shown in Figure 3. Figure 4 illustrates that if the spring is not preloaded, the dome and superstructure may not make contact. Therefore, the spring must be properly preloaded, because the dome and superstructure must always be in contact for the vertical spring-viscous damper to function effectively.