## 2 Theoretical Formulations
### Energy Functions of Lattice Sandwich Beams
 \tag{3}\]  
where \(E_{x}\) is the elastic modulus of the upper and lower panels, and \(G_{\mathrm{zze}}\) represents the equivalent shear modulus of the core layer. \(\sigma_{t}\) and \(\sigma_{b}\) represent the normal stress of the upper panel and that of lower panel in the \(x\)-axis direction, respectively; \(\tau_{c}\) represents the shear stress of the core layer. The next step is to start deriving kinetic and potential energy expressions of the whole structure, obtain the Lagrange energy functional expression and use the Rayleigh-Ritz method to calculate the natural frequency of the structure. The total potential energy \(U\) of a lattice sandwich beam under arbitrary boundary conditions consists of two parts: one is the strain energy \(U_{B}\) of lattice sandwich beams, and the other is the elastic potential energy \(U_{S}\) stored in the restraint springs at both ends of lattice sandwich beams. Their specific expressions can be given as follows:  
\[U=U_{B}+U_{S} \tag{4}\]  
\[ \begin{array}{l}U_{B}=\frac{1}{2}\int_{0}^{L}\int_{A}\left(\sigma\cdot \varepsilon+\tau\cdot\gamma\right)dAdx\\ =\frac{\eta}{2}\int_{0}^{L}\left(\int_{-h/2}^{-c/2}\sigma_{t}\varepsilon_{t} dz+\int_{-c/2}^{c/2}\tau_{c}\gamma_{c}dz+\int_{c/2}^{h/2}\sigma_{b}\varepsilon_{b}dz \right)dx\\ \\ U_{S}=\frac{1}{2}k_{0}w^{2}(x)\big{|}_{x=0}+\frac{1}{2}K_{0}\left(\frac{ \partial w(x)}{\partial x}\right)^{2}\Big{|}_{x=0}+\frac{1}{2}K_{0}\theta^{2} (x)\big{|}_{x=0}\\ +\frac{1}{2}k_{L}\left.w^{2}(x)\right|_{x=L}+\frac{1}{2}K_{L}\left(\frac{ \partial w(x)}{\partial x}\right)^{2}\Big{|}_{x=L}+\frac{1}{2}K_{L}\theta^{2} (x)\big{|}_{x=L}\end{array}  \tag{6}\]  
where \(k_{0}\) and \(k_{L}\) represent the rigidity constants of the translational springs at the left (where \(x=0\)) and right (where \(x=L\)) ends of lattice sandwich beams, respectively. Similarly, \(K_{0}\) and \(K_{\mathrm{L}}\) represent the rigidity constants of the torsion springs at the left (where \(x=0\)) and right (where \(x=L\)) ends of the lattice sandwich beams, respectively. For vibration systems, the units of rigidity for the translational and torsional springs are N/m and Nm/rad, respectively. For restraint springs at both ends of the lattice sandwich beam, it is generally assumed that rigidity is provided without considering the effect of their own weight. Additionally, for a lattice sandwich beam under arbitrary boundary conditions, its entire kinetic energy \(T\) is determined only by the kinetic energy of the lattice sandwich beam and has nothing to do with the elastic constraints at either end of the beam. At this time, the overall kinetic energy of the system \(T\) is expressed as follows:\[ \begin{array}{l}T=\frac{1}{2}\int_{0}^{L}\int_{A}\rho\bigg{[}\left(\frac{\partial u _{t}}{\partial t}\right)^{2}+\left(\frac{\partial u_{t}}{\partial t}\right)^{2} \bigg{]}dAdx\\ =\frac{B}{2}\int_{0}^{L}\int_{-h/2}^{-c/2}\rho_{f}\bigg{[}\left(\frac{\partial u _{t}}{\partial t}\right)^{2}+\left(\frac{\partial u_{t}}{\partial t}\right)^{2 }\bigg{]}dz+\int_{-c/2}^{c/2}\rho_{c}\bigg{[}\left(\frac{\partial u_{t}}{ \partial t}\right)^{2}\\ +\left(\frac{\partial u_{t}}{\partial t}\right)^{2}\bigg{]}dz+\int_{c/2}^{h/2 }\rho_{f}\bigg{[}\left(\frac{\partial u_{t}}{\partial t}\right)^{2}+\left( \frac{\partial u_{t}}{\partial t}\right)^{2}\bigg{]}dz\bigg{\}}dx\end{array}  \tag{7}\]  
where \(\rho_{f}\) is the density of the upper panel and lower panel, and \(\rho_{c}\) is the equivalent density of the core layer.