## 2 Theoretical Formulation
### Governing Equation
\begin{array}{l}\sigma_{\text{x,c}}=\frac{E_{\text{c}}}{1-h_{\text{c}}^{2 }}\big{[}u_{\text{c},x}+\mu_{\text{c}}v_{\text{c},y}-z\big{(}w_{,xx}+\mu_{ \text{c}}w_{,yy}\big{)}\big{]}\\ \sigma_{\text{y,c}}=\frac{E_{\text{c}}}{1-h_{\text{c}}^{2}}\big{[}u_{\text{c },y}+\mu_{\text{c}}v_{\text{c},x}-z\big{(}w_{,yy}+\mu_{\text{c}}w_{,xx}\big{)} \big{]}\\ \sigma_{\text{x,b}}=\frac{E_{\text{b}}}{1-h_{\text{b}}^{2}}\big{[}u_{\text{b}, x}+\mu_{\text{b}}v_{\text{b},y}-z\big{(}w_{,xx}+\mu_{\text{b}}w_{,yy}\big{)} \big{]}\\ \sigma_{\text{y,b}}=\frac{E_{\text{b}}}{1-h_{\text{b}}^{2}}\big{[}u_{\text{b}, y}+\mu_{\text{b}}v_{\text{b},x}-z\big{(}w_{,yy}+\mu_{\text{b}}w_{,xx}\big{)} \big{]}\\ \tau_{\text{xy,c}}=\frac{E_{\text{c}}}{2(1+\mu_{\text{c}})}\big{(}u_{\text{c}, y}+v_{\text{c},x}-2 zw_{,xy}\big{)}\\ \tau_{\text{xy,b}}=\frac{E_{\text{b}}}{2(1+\mu_{\text{b}})}\big{(}u_{\text{b}, y}+v_{\text{b},x}-2 zw_{,xy}\big{)}\\ \tau_{\text{xz,d}}=\frac{E_{\text{d}}}{2(1+\mu_{\text{d}})}\gamma_{\text{xz,d}},\tau_{\text{yz,d}}=\frac{E_{\text{d}}}{2(1+\mu_{\text{d}})}\gamma_{\text{yz,d}}\end{array}  \tag{4}\]  
where \(\mu\) and \(E\) represent the Poisson ratio and elasticity modulus, respectively. For the constrained layer, its strain energy \(U_{\text{c}}\) is:  
\[U_{\text{c}}=\frac{1}{2}\int_{a}^{b}\int_{c}^{d}\int_{-h_{\text{d}}/2}^{h_{ \text{d}}/2}\big{(}\sigma_{\text{x,c}}\varepsilon_{\text{x,c}}+\sigma_{\text{y, c}}\varepsilon_{\text{y,c}}+\tau_{\text{xy,c}}\gamma_{\text{xy,c}}\big{)}dxdydz \tag{5}\]  
For the damping layer, based on assumption e, its strain energy \(U_{\text{d}}\) is:  
Figure 2: Displacement of the sandwich plate in \(x\)-\(z\) plane. \[U_{\rm d}=\frac{1}{2}\int_{a}^{b}\int_{c}^{d}\int_{-h_{\rm d}/2}^{h_{\rm d}/2} \Big{(}\tau_{\rm xz,d}\gamma_{\rm xz,d}+\tau_{\rm yz,d}\gamma_{\rm yz,d}\Big{)} dxdydz \tag{6}\]  
Substituting Equations (3) and (4) into Equations (5) and (6), \(U_{\rm c}\) and \(U_{\rm d}\) can be expanded as:  
\[U_{\rm c}=\frac{1}{2}\int_{a}^{b}\int_{c}^{d}\left\{ \begin{array}{c}\frac{E_ {\rm d}h_{\rm c}}{1-\mu_{\rm c}^{2}}\Big{[}u_{\rm c,x}^{2}+v_{\rm c,y}^{2}+2\mu _{\rm c}u_{\rm c,x}v_{\rm c,y}+\frac{1-\mu_{\rm c}}{2}\big{(}u_{\rm c,y}+v_{\rm c,x}\big{)}^{2}\Big{]}\\ +\frac{E_{\rm c}h_{\rm c}^{2}}{12\big{(}1-\mu_{\rm c}^{2}\big{)}}\Big{[}w_{xx }^{2}+w_{\rm 3y}^{2}+2\mu_{\rm c}w_{\rm,xx}w_{\rm,yy}+2(1-\mu_{\rm c})w_{\rm,xy}^{ 2}\Big{]}\end{array} \right\}dxdy \tag{7}\]  
\[U_{\rm d}=\frac{G_{\rm d}}{2\mu_{\rm d}^{2}}\int_{a}^{b}\int_{c}^{d}\Big{[}(u _{\rm c}-u_{\rm b}+w_{x}h)^{2}+\big{(}v_{\rm c}-v_{\rm b}+w_{y}h\big{)}^{2} \Big{]}dxdy \tag{8}\]  
The strain energy of the base plate, \(U_{\rm b}\), can be expressed as:  
\[U_{\rm b}=\frac{1}{2}\int_{0}^{L}\int_{0}^{W}\left\{ \begin{array}{c}\frac{E_ {\rm b}h_{\rm b}}{1-\mu_{\rm c}^{2}}\Big{[}u_{\rm b,x}^{2}+v_{\rm b,y}^{2}+2 \mu_{\rm b}u_{\rm b,x}v_{\rm b,y}+\frac{1-\mu_{\rm b}}{2}\big{(}u_{\rm b,y}+v_ {\rm b,x}\big{)}^{2}\Big{]}\\ +\frac{E_{\rm b}h_{\rm b}^{2}}{12\big{(}1-\mu_{\rm b}^{2}\big{)}}\Big{[}w_{xx }^{2}+w_{\rm,yy}^{2}+2\mu_{\rm b}w_{xx}w_{\rm,yy}+2(1-\mu_{\rm b})w_{\rm,xy}^{ 2}\Big{]}\end{array} \right\}dxdy \tag{9}\]  
The total strain energy of the sandwich plate is:  
\[U=U_{\rm c}+U_{\rm d}+U_{\rm b} \tag{10}\]  
For the kinetic energy of the sandwich plate, the transverse inertia is only considered and the total kinetic energy is:  
\[T=\frac{1}{2}\bigg{[}\int_{a}^{b}\int_{c}^{d}\big{(}\rho_{\rm c}h_{\rm c}+\rho _{\rm d}h_{\rm d}\big{)}\tilde{w}^{2}dxdy+\int_{0}^{L}\int_{0}^{W}\rho_{\rm b} h_{\rm b}\tilde{w}^{2}dxdy\bigg{]} \tag{11}\]  
where \(\rho_{i}\) (\(i=\rm c\), \(\rm d\) and \(\rm b\)) is the density for different layers. The potential energy of the moving load can be expressed as:  
\[W=W_{F}+W_{M} \tag{12}\]  
For moving mass, the force due to weight and inertia can be expressed as:  
\[F_{M}=\big{[}Mg+M\tilde{w}(x_{M},y_{M},t)\big{]}\delta(x-x_{M})\delta(y-y_{M}) \tag{13}\]  
where \(x_{F}=v_{F}\times t\), \(y_{F}\) is a constant value and \(\delta(. )\) denotes the Dirac delta-function.