## 3 Illustration of a Plate-Isolator System as a 2-Stage Vibration Isolation System
### Horizontal (X and Z)  
For the excitation of the flywheel in the X and Z directions, the flywheel and structural plate can also be simplified into a 2-stage system ignoring the higher-order mode of the structural plate, as shown in Figure 4. The analytical solution of the force transmissibility is the same as Equation (7). However, due to the much higher in-plane stiffness of the structural plate compared with the out-of-plane stiffness, the response of the plate-isolator system in the X and Y directions within the frequency range of 0-100 Hz is similar to that of  
Figure 6: Influence of stiffness on the force transmissibility of the simplified 2-stage plate-isolator system: (**a**) variation with \(K_{\text{isolator}}\), \(K_{\text{plate}}\) = 3,378,239 N/m, \(C_{\text{plate}}\) = 10 Ns/m, \(C_{\text{isolator}}\) = 600 Ns/m; (**b**) variation with \(K_{\text{plate}}\), \(K_{\text{isolator}}\) = 56,321 N/m, \(C_{\text{plate}}\) = 10 Ns/m, \(C_{\text{isolator}}\) = 600 Ns/m. the isolator fixed at its bottom. It can be predicted from Figure 6b that reducing the in-plane stiffness of the structural plate to a relatively low value could improve the overall vibration isolation efficiency of the 2-stage system. However, achieving low in-plane stiffness for a honeycomb structural plate through mechanical optimization is difficult. Therefore, the next section will discuss how to enhance the lateral vibration isolation efficiency of the plate-isolator system by adopting the lateral elastic boundary on the structural plate. Illustration of a Plate-Isolator System with Elastic Boundary as a 3-Stage Vibration Isolation System  
Section 2 presents the findings of the simplified 2-stage model, which indicate that improving the performance of the plate-isolator system is challenging without reducing the stiffness of the structural plate to a relatively low value. Instead of optimizing the mechanical design of the structural plate, this section explores an alternative method of applying an elastic plate boundary. Figure 7 shows the plate-isolator system with the elastic boundary. The rigid constraint boundary of the structural plate is replaced with a spring constraint boundary, thereby simplifying the system into an elastic 3-stage system. The first stage is the elastic boundary  
### Horizontal (X and Z)  
For the horizontal (X and Z) directions, the stiffness \(K_{\rm plate}\) can be considered to be infinitely large. As a result, the node mass \(m_{\rm plate\text{-}central}=0\), and \(m_{\rm plate\text{-}boundary}=50\) kg. The finite element analysis results indicate that the fundamental frequencies of the flywheel vibration isolation platform in the X and Z directions are 13.354 Hz and 21.976 Hz, respectively. The mass of the flywheel is 8 kg and the equivalent stiffness of the isolator in the X and Z directions are \(K_{\rm isolator\text{-}x}=\) 56,321 N/m and \(K_{\rm isolator\text{-}z}=\)152,527 N/m. The influence of the lateral boundary stiffness of the structural plate on the force transmissibility of the plate-isolator is investigated using Equation (11), with the prediction results presented in Figure 9. When the lateral boundary stiffness of the structural plate is relatively small (specifically below 500,000 N/m in the X direction and 1,000,000 N/m in the  
Figure 8: Effect of the boundary elasticity and damping on force transmissibility of the 3-stage plate-isolator system in the Y direction: (**a**) variation with \(K_{\rm boundary}\) (\(C_{\rm plate}=10\) Ns/m, \(C_{\rm isolator}=600\) Ns/m, \(C_{\rm boundary}=100\) Ns/m); (**b**) variation with \(C_{\rm boundary}\) (\(C_{\rm plate}=10\) Ns/m, \(C_{\rm isolator}=600\) Ns/m, \(K_{\rm boundary}=0.1\)\(K_{\rm plate}\)). Z direction), the force transmissibility of the system has only one peak in 0-100 Hz. If the lateral boundary stiffness of the structural plate increases, the second peak will occur. The peak moves towards a higher frequency as the lateral boundary stiffness increases.