## 3 Illustration of a Plate-Isolator System as a 2-Stage Vibration Isolation System
### Vertical (\(Y\) Direction)  
When the flywheel vibrates vertically, as illustrated in Figure 4, the fundamental frequency of the edge fixed structural plate (1000 mm \(\times\) 1000 mm \(\times\) 30 mm, the total mass is 50 kg) is calculated to be \(f=51.956\) Hz via FEA. The corresponding fundamental modal mass is 31.7 kg. In the 2-stage system, the equivalent mass of the plate equals the fundamental modal mass of the plate, i.e., \(m_{\text{plate}}=31.7\) kg. The equivalent stiffness of the plate is \(K_{\text{plate}}=3,378,239\) N/m and is calculated based on the fundamental frequency and modal mass. The mass of the flywheel is 8 kg and the fundamental frequency of the vibration isolation platform is \(f=13.354\) Hz; thus, the equivalent stiffness of the vibration isolation platform is \(K_{\text{isolator}}=56,321\) N/m. Based on the disturbance force expression of Equation (7), the force transmissibility of the simplified 2-stage vibration isolation system under harmonic excitation is studied (Figures 5 and 6). The response curves exhibit two peaks: the first peak corresponds to the isolator mode, while the second peak corresponds to the structural plate mode. The analytical solutions indicate that the increased stage (i.e., the structural plate) introduces an additional peak on each frequency response curve, resulting in a higher isolation starting frequency. To optimize a vibration isolation system, it is common to optimize the damping and stiffness of the system. However, the parameter analysis results presented in Figures 5 and 6 demonstrate that optimizing the damping or stiffness is ineffective for this two-stage plate-isolator system. The effects of plate damping \(C_{\text{plate}}\) and isolator damping \(C_{\text{isolator}}\) on the force transmissibility are shown in Figure 5. Figure 5a demonstrates that a large damping coefficient of \(C_{\text{plate}}\) = 10,000 Ns/m is required to effectively suppress the second peak caused by the plate elasticity. If the damping of the structural plate is further increased, the response of the system approaches that of the vibration isolation platform itself. Similarly, Figure 5b illustrates that increasing the damping of the vibration isolation platform (\(C_{\text{isolator}}\) = 10,000 Ns/m) effectively reduces the peak value of the system response. If the damping of the vibration isolation platform is further increased, the overall response of the system approaches that of the structural plate itself. Therefore, the results in Figure 5 indicate that the force transmissibility of the plate-isolator system cannot be significantly improved by increasing the damping of the structural plate or the vibration isolator. Infinitely increasing the structural plate damping can eliminate the influence of the structural plate elasticity and ensure the vibration isolation effect of the isolator. However, infinite or quite large damping is usually difficult to achieve in practice for a structural plate. The influences of the vibration isolator stiffness \(K_{\text{isolator}}\) and the structural plate stiffness \(K_{\text{plate}}\) are presented in Figure 6. The analysis results in Figure 6a indicate reducing the isolator stiffness \(K_{\text{isolator}}\) has little effect on the second peak and high-frequency response. Therefore, optimizing the stiffness of the vibration isolator alone cannot significantly improve the vibration isolation effect of the plate-isolator system. On the other hand, Figure 6b suggests that by reducing the stiffness of the structural plate, the frequency corresponding to the second peak of the force transmissibility shifts forward. Meanwhile, the force transmissibility at high frequency decreases. When the stiffness of the structural plate is reduced to 1/10 of the original value, the force transmissibility of the plate-isolator system exhibits one obvious peak. The overall response is similar to that of a soft 1-DOF vibration isolation system.