## 5 Conclusions  
We studied a two-degrees-of-freedom isolation system, considering the geometrical nonlinearity and the coupling between the vertical and horizontal vibrations; the system was configured to slide on a dome or ball attached to a column damper under a concave plate. The following conclusions can be drawn:  
1. In the case of the RIS, vertical vibration caused by horizontal vibration occurs, and the control method for large vertical vibration is limited. However, the structure proposed in this study demonstrates effective control of the horizontal and vertical vibrations;
2. The starting frequency at which relative motion occurs was obtained as a function of the friction coefficient and excitation amplitude. This value increases as the friction coefficient increases and the excitation amplitude decreases. This is the primary design parameter for controlling the relative motion because the displacement transfer ratio can be controlled by the relationship between the natural frequency and the system;
3. The contact stability chart was obtained in the frequency and acceleration amplitude domains. As the unstable region of the stability chart appears near the resonance frequency where the vibration response is large, the region decreases as the friction coefficient and viscosity coefficient increase. The stability chart can be used to select design parameters to ensure stable contact in the isolator design;
4. The steady-state response to horizontal excitation indicated that harmonic components of n and 2n orders, which are nonlinear, appear in the vertical and horizontal responses;
5. The displacement transfer ratios were obtained to perceive the characteristics of the isolation system against the design parameters. The relative motion is locked until the excitation frequency is equal to \(f_{\mathrm{s}}\), and a jump occurs as the excitation frequency increases. Furthermore, as \(f_{\mathrm{s}}\) approaches the natural frequency, the transfer ratio decreases significantly, and when the \(f_{\mathrm{s}}\) frequency is larger than the natural frequency, it becomes 1 or less. Resonance can be avoided or controlled by setting the design factors of the friction and viscous damping coefficients, vertical natural frequency, and interval between dampers;
6. Considering the above, the design factors were set using actual earthquake data, and effective vertical and horizontal vibration reduction was achieved by the numerical analysis. Therefore, if the spring stiffness and the number of insulators are set properly, the application of the isolation system presented in this study can be applied from light to heavy equipment.  
Figure 18: Comparison of responses of RIS and the proposed system to horizontal seismic excitation.  
For the isolation proposed in this study, vibration control characteristics exhibited by the design parameters, such as the vertical design natural frequency, damping coefficient, friction coefficient, post spacing, and concave plate curvature, were presented. Setting and predicting the design factors was made possible in virtual space. Future work will investigate factors such the various concave plate curvatures and shapes, and will build a prototype with a detailed design. The numerical results of the isolator will be experimentally performed and verified using the prototype.  
All authors discussed and agreed on the idea, made equal scientific contributions, and participated in the writing. All authors have read and approved the final manuscript.  
This research was funded by Korea Institute of Energy Technology Evaluation and Planning (KETEP), the Ministry of Trade, Industry & Energy (MOTIE), the Korea Electric Power Corporation, and Korea Western Power Co. Ltd. under grant number 20172020108700.  
The authors declare