## 3 Illustration of a Plate-Isolator System as a 2-Stage Vibration Isolation System  
To gain insight into the basic dynamic characteristics of the plate-isolator system, the plate-isolator system is simplified into a 2-stage isolation system, as illustrated in Figure 4. In the simplified 2-stage isolation system, the structural plate is represented as a linear spring with a mass node on its upper end. The node mass \(m_{\text{plate}}\) equals the first-order modal mass of the plate. The spring stiffness \(K_{\text{plate}}\) is calculated according to the fundamental modal frequency and the corresponding modal mass of the plate. The vibration isolation platform is also represented as a linear spring, and the flywheel is modeled as a mass node. The spring stiffness \(K_{\text{isolator}}\) is calculated according to the fundamental modal frequency of the vibration isolation platform and the mass of the flywheel. The coefficients \(\text{C}_{\text{plate}}\) and  
Figure 3: Comparison of force transmissibility between the signal vibration isolation platform and the plate-isolator system: (**a**) force loading mode; (**b**) load modes of structural plate; (**c**) force transmissibility. Assuming a harmonic excitation \(F[t]=F\text{Sin}[\omega t]\) is applied on the flywheel, the kinetic equations of the 2-stage system can be derived:  
\[\left\{ \begin{array}{l}K_{1}x_{1}(t)+K_{2}x_{1}(t)-K_{2}x_{2}(t)+C_{1}x_{1}^{ \prime}(t)+C_{2}x_{1}^{\prime}(t)-C_{2}x_{2}^{\prime}(t)+M_{1}x_{1}^{\prime \prime}(t)=0\\ -F(t)-K_{2}x_{1}(t)+K_{2}x_{2}(t)-C_{2}x_{1}^{\prime}(t)+C_{2}x_{2}^{\prime}(t )+M_{2}x_{2}^{\prime}(t)=0\end{array} \right. \tag{1}\]  
where \(m_{plate}=M_{1}\), \(m_{flywheel}=M_{2}\), \(K_{plate}=K_{1}\), \(C_{plate}=C_{1}\), \(K_{isolator}=K_{2}\), \(C_{isolator}=C_{2}\), and \(x_{1}(t)\) and \(x_{2}(t)\) denote the displacement of mass nodes \(m_{\text{plate}}\) and \(m_{\text{flywheel}}\), respectively. Note that these stiffness and damping parameters are different for the vertical and horizontal directions. Respectively assume:  
\[x_{1}(t)=A_{1}\sin(\omega t)+A_{2}\cos(\omega t) \tag{2}\]  
\[x_{2}(t)=B_{1}\sin(\omega t)+B_{2}\cos(\omega t) \tag{3}\]  
Substituting Equations (2) and (3) into Equation (1), we can derive the following:  
\[\left\{ \begin{array}{l}K_{1}(A_{1}\sin(\omega t)+A_{2}\cos(\omega t))+K_{2} (A_{1}\sin(\omega t)+A_{2}\cos(\omega t))-K_{2}(B_{1}\sin(\omega t)+B_{2}\cos (\omega t))\\ +C_{1}(A_{1}\omega\cos(\omega t)-A_{2}\omega\sin(\omega t))+C_{2}(A_{1}\omega \cos(\omega t)-A_{2}\omega\sin(\omega t))\\ -C_{2}(B_{1}\omega\cos(\omega t)-B_{2}\omega\sin(\omega t))+M_{1}\big{(}-A_{1 }\omega^{2}\sin(\omega t)-A_{2}\omega^{2}\cos(\omega t)\big{)}=0\\ -F(t)-K_{2}(A_{1}\sin(\omega t)+A_{2}\cos(\omega t))+K_{2}(B_{1}\sin(\omega t) +B_{2}\cos(\omega t))\\ -C_{2}(A_{1}\omega\cos(\omega t)-A_{2}\omega\sin(\omega t))+C_{2}(B_{1}\omega \cos(\omega t)-B_{2}\omega\sin(\omega t))\\ +M_{2}\big{(}-B_{1}\omega^{2}\sin(\omega t)-B_{2}\omega^{2}\cos(\omega t) \big{)}=0\end{array} \right. \tag{4}\]  
By equaling the coefficients of \(\text{Sin}[\omega t]\) and \(\text{Cos}[\omega t]\) of the two kinetic equations in Equation (4) to 0 respectively, four linear equations of \(A_{1}\), \(A_{2}\), \(B_{1}\), and \(B_{2}\) can be obtained. By solving the linear equations, the vibration amplitudes \(\left(A_{1}^{2}+A_{2}^{2}\right)^{\frac{1}{2}}\) and \(\left(B_{1}^{2}+B_{2}^{2}\right)^{\frac{1}{2}}\) take the form:  
\[\left(A_{1}^{2}+A_{2}^{2}\right)^{\frac{1}{2}}=\frac{\sqrt{F^{2}(K_{2}^{2}+C_ {2}^{2}\omega^{2})}}{\sqrt{ \begin{array}{l}\left(K_{1}(K_{2}-M_{2}\omega^{2 })+\omega^{2}(-K_{2}(M_{1}+M_{2})+M_{1}M_{2}\omega^{2})\right)^{2}\\ +\omega^{2}(2C_{1}C_{2}M_{2}\omega^{4}+C_{1}^{2}(K_{2}-M_{2}\omega^{2})^{2}+C_{ 2}^{2}(K_{1}^{2}-2K_{1}(M_{1}+M_{2})\omega^{2}+\omega^{2}(C_{1}^{2}+(M_{1}+M_{ 2})^{2}\omega^{2}))\right)\end{array} } \tag{5}\]  
Figure 4: Plate-isolator system and its 2-stage simplified model.