## 2 Equations of Motion
### Derivation of Equations of Motion  
As illustrated in Figure 4, the forces are the normal reactions \(N_{r(l)}\) acting on the concave surface of the dome, the frictional forces caused by the relative motion of the dome and concave surface, and the gravitational load \(Mg\) of the superstructure. Moreover, \(F_{r(l)}\) and \(g\) are the forces exerted by the spring-viscous damper and gravitational acceleration, respectively. To derive the motion equation, assume that the superstructure with mass \(M\) is a rigid body, the dome is always in contact, and its mass is ignored or included in the mass of the superstructure. The superstructure moves within the range of curvature.  
Figure 3: Deflection of spring owing to relative displacement \(x\) of superstructure.  
According to Figure 4, the equilibrium equations of force for the horizontal and vertical directions can be expressed as follows:  
\[\begin{array}{l}\sum F_{x}=N_{r}\sin\theta_{r}-\mu N_{r}\cos\theta_{r}\text{ sgn}(\dot{x})-N_{l}\sin\theta_{l}-\mu N_{l}\cos\theta_{l}\text{sgn}(\dot{x})=M( \ddot{X}+\ddot{x}),\\ \sum F_{y}=-N_{r}\cos\theta_{r}-\mu N_{r}\sin\theta_{r}\text{sgn}(\dot{x})-N_{l }\cos\theta_{l}+\mu N_{l}\sin\theta_{l}\text{sgn}(\dot{x})+Mg=M\ddot{y},\end{array} \tag{4}\]  
where '\(sgn\)' denotes the sign of the parameter in parentheses, and the vertical component of the normal forces \(N_{r(l)}\) acting on the contact surface serves as the vertical force \(F_{r(l)}\) of the spring-viscous damper. The relationship between these forces is expressed as follows:  
\[N_{r}=F_{r}/\cos\theta_{r}\text{ and }N_{l}=F_{l}/\cos\theta_{l}. \tag{5}\]  
The forces \(F_{r}\) and \(F_{l}\) acting on the spring-viscous damper are expressed as the sum of the spring and damping forces of each damper, as follows:  
\[F_{r(l)}=k(y_{st}+y+y_{r(l)}-Y)+c(\dot{y}+\dot{y}_{r(l)}-\dot{Y}), \tag{6}\]  
where \(y\) is the dynamic vertical displacement and \(y_{st}\) is the static deflection, which refers to the preload amount of the spring, and is expressed as:  
\[y_{st}=\frac{Mg}{2k}. \tag{7}\]  
The contact between the dome and superstructure can be confirmed by observing the sign of the following spring deflection equation, as expressed in Equation (6):  
\[\delta=y_{st}+y+y_{r(l)}-Y. \tag{8}\]  
A positive sign in Equation (8) indicates that the spring is compressed, while a negative signifies tension. If the sign of Equation (8) is negative, as in the example in Figure 4, the dome and superstructure exist in the unstable contact state, because the spring cannot actually be tensioned. Thus, the contact between the dome and superstructure can be confirmed by observing the sign of Equation (8) during the numerical analysis.  
Substituting Equation (5) into Equation (4) and dividing the result by the mass \(M\), the motion equations in the horizontal and vertical directions can, respectively, be obtained as follows:  
Figure 4: Free-body diagram of isolation mounting system.  
\[\begin{array}{l}\ddot{x}-D_{r}\tan\theta_{r}+\mu D_{r}\text{sgn}(\dot{x})+D_{l} \tan\theta_{l}+\mu D_{l}\text{sgn}(\dot{x})=-\ddot{X},\\ \ddot{y}+D_{r}+\mu D_{r}\tan\theta_{r}\text{sgn}(\dot{x})+D_{l}-\mu D_{l}\tan \theta_{l}\text{sgn}(\dot{x})-g=0.\end{array} \tag{9}\]  
In Equation (9), \(D_{r}\) and \(D_{l}\) are expressed as follows, respectively:  
\[\begin{array}{l}D_{r}=\frac{1}{2}\Big{\{}(2\pi f_{y})^{2}(y_{st}+y+y_{r}-Y)+2 \zeta(2\pi f_{y})(\dot{y}-\dot{x}\tan\theta_{r}-\dot{Y})\Big{\}},\\ D_{l}=\frac{1}{2}\Big{\{}(2\pi f_{y})^{2}(y_{st}+y+y_{l}-Y)+2\zeta(2\pi f_{y})( \dot{y}+\dot{x}\tan\theta_{l}-\dot{Y})\Big{\}},\end{array} \tag{10}\]  
where \(f_{y}\) is the linearized natural frequency of the vertical motion and \(\zeta\) is the damping ratio, which are respectively defined as follows:  
\[f_{y}=\frac{1}{2\pi}\sqrt{\frac{2k}{M}},\ \ \zeta=\frac{c}{2\sqrt{2 kM}}\ \ \text{ and }\ \ y_{st}=\frac{g}{(2\pi f_{y})^{2}}. \tag{11}\]