## 1 Introduction
### Calculation of the Transfered Power Flow of the Coupling System  
Let the external excitation \(p_{\rm E}(\sigma,t)=P_{\rm E}\cdot{\rm e}^{i\omega t}\) acting on the upper plate be a homogeneous simple harmonic force (pressure). For the upper plate, its vibration differential equation:  
\[\
abla^{4}w^{\prime}{}_{n}(\sigma,t)+\rho h\frac{\partial^{2}w^{\prime}{}_{n}( \sigma,t)}{\partial t^{2}}=\sum\limits_{l=1}^{N_{\rm I}}F_{\rm EI}(t)\delta( \sigma-\sigma_{l})-P_{\rm E}{\rm e}^{i\omega t} \tag{35}\]  
where \(w^{\prime}{}_{n}(\sigma,t)\) is the vibration displacement response of the upper plate; \(F_{\rm EI}(t)\) is the force of the \(l\)th isolation unit on the upper plate. The oscillation superposition method is used to make \(w^{\prime}{}_{n}(\sigma,t)\) expressed as a series solution \(w^{\prime}{}_{n}(\sigma,t)=\sum\limits_{j=1}^{\infty}W_{\rm Ej}(\sigma)q_{\rm Ej }(t)\), and its normal vibration velocity \(u^{\prime}{}_{n}(\sigma,t)=\bar{w}^{\prime}{}_{n}(\sigma,t)\), so it is obtained:  
\[\mathbf{U}^{\prime}_{n}=\big{[}u^{\prime}_{n}(\sigma_{1}),u^{\prime}_{n}(\sigma_{ 2}),\ldots,u^{\prime}_{n}(\sigma_{N_{\rm I}})\big{]}^{\rm T}=-{\rm i}\omega \bigg{(}\widetilde{\mathbf{W}}_{\rm E}^{\rm T}H_{\rm E}Q_{\rm E}+\widetilde{\mathbf{W }}_{\rm E}^{\rm T}H_{\rm E}\widetilde{\mathbf{W}}_{\rm E}F_{\rm Et}\bigg{)}=\mathbf{A} ^{(1)}\mathbf{Q}_{\rm E}+\mathbf{A}^{(2)}F_{\rm Et} \tag{36}\]  
In the equation, \(\mathbf{H}_{\rm E}=\big{(}-\omega^{2}\mathbf{M}_{\rm E}+\mathbf{\Omega}_{\rm E}\mathbf{M}_{ \rm E}\big{)}^{-1}\) is a transfer matrix; \(\mathbf{M}_{\rm E}={\rm diag}[M_{\rm EI},M_{\rm EZ},M_{\rm EZ3},\ldots]\) and \(\mathbf{\Omega}_{\rm E}={\rm diag}[\omega_{\rm E1},\omega_{\rm E2},\omega_{\rm E3 },\ldots]\) are the diagonal matrices of the modal mass \(M_{\rm Ej}\) and the natural frequency \(\omega_{Ej}\) of the upper plate, respectively; \(\widetilde{\mathbf{W}}_{\rm E}=[\mathbf{W}_{\rm E}(\sigma_{1}),\mathbf{W}_{\rm E}(\sigma _{2}),\ldots,\mathbf{W}_{\rm E}(\sigma_{l})]\) is a matrix consisting of \(\mathbf{W}_{\rm E}(\sigma)\), and \(\mathbf{W}_{\rm E}(\sigma)=[\mathbf{W}_{\rm E1}(\sigma),\mathbf{W}_{\rm E2}(\sigma),\mathbf{W} _{\rm E3}(\sigma),\ldots]^{\rm T}\) is the column vector of the form function \(W_{\rm Ej}(\sigma)\) (\(j=1,2,3,\ldots\)) of the upper plate; \(\mathbf{Q}_{\rm E}\) is the column vector of generalized forces acting on the upper plate. For the lower plate and chamber coupling unit,  
\[\mathbf{U}_{n}=\mathbf{C}\cdot\mathbf{F}_{\rm Eb}=[\mathbf{C}_{rs}]\mathbf{F}_{\rm Eb} \tag{37}\]  
where \(\mathbf{C}_{rs}={\rm i}\omega[\mathbf{W}_{\rm B}(\sigma_{r})]^{\rm T}\big{(}\mathbf{ \Omega}_{\rm B}\sqrt{\mathbf{M}_{\rm B}}\big{)}^{-1}\mathbf{X}_{\rm B}\mathbf{H}\mathbf{W}_{ \rm B}(\sigma_{s})\), \((\tau,s=1,2,\ldots,N_{\rm I})\). The admittance matrix \(\mathbf{C}\) can in turn be expressed as:  
\[\mathbf{C}={\rm i}\omega\widetilde{\mathbf{W}}_{\rm B}^{\rm T}\Big{(}\mathbf{\Omega}_{\rm B }\sqrt{\mathbf{M}_{\rm B}}\Big{)}^{-1}\mathbf{X}_{\rm B}\mathbf{H}\widetilde{\mathbf{W}}_{\rm B} \tag{38}\]  
where \(\widetilde{\mathbf{W}}_{\rm B}^{\rm-}=[\mathbf{W}_{\rm B}(\sigma_{1}),\mathbf{W}_{\rm B}( \sigma_{2}),\ldots,\mathbf{W}_{\rm B}(\sigma_{l})]\) is a matrix consisting of \(\mathbf{W}_{\rm B}(\sigma)\).