## 2 Theoretical Formulations
### Displacement Function and Unified Solution  
In order to find the vibration responses of lattice sandwich beams with arbitrary boundary conditions, the improved Fourier series is used to describe the displacement function of the structure in this paper, thereby overcoming the potential discontinuities or jumps of arbitrary boundary conditions during derivation. Hence, the transverse displacement \(w\) and the rotation angle \(\theta\) of a lattice sandwich beam under arbitrary boundary conditions are expressed as follows [28]:  
\[w(x,t)=\Phi(x)\mathrm{e}^{i\omega t} \tag{8}\]  
\[\theta(x,t)=\Psi(x)\mathrm{e}^{i\omega t} \tag{9}\]  
\[\Phi(x)=\sum_{m=0}^{\infty}A_{m}\cos\lambda_{m}x+\sum_{n=1}^{4}A_{n}^{a}\sin \lambda_{n}x \tag{10}\]  
\[\Psi(x)=\sum_{m=0}^{\infty}B_{m}\cos\lambda_{m}x+\sum_{n=1}^{4}B_{n}^{a}\sin \lambda_{n}x \tag{11}\]  
where \(\lambda_{m}=m\pi/L\), \(\lambda_{n}=n\pi/L\) and \(\Phi(x)\) and \(\Psi(x)\) are the mode shape functions of the lattice sandwich beam under arbitrary boundary conditions, which are described by the improved Fourier series. The improved Fourier series is given as a Fourier cosine series along with four sine terms in this paper. These four sine terms are considered auxiliary functions. \(A_{m}\) and \(B_{m}\) are the coefficients of the cosine Fourier series, and \(A_{n}^{a}\) and \(B_{n}^{a}\) are those of the four sine terms, where \(n=1\), \(2\), \(3\), \(4\). As mentioned above, the discontinuity of the original displacement and its derivative can be solved by adding auxiliary functions, and the rapid convergence of the calculation process can also be promoted. Once the structural displacement function is given an explicit expression, the next task is to substitute these displacement function expressions into the kinetic and potential energy expressions. Then, these coefficients in the displacement function are solved by the Rayleigh-Ritz method. The kinetic and potential energy expressions contain the same time term, \(\mathrm{e}^{i\omega t}\), which will also exist after complex derivative and integration operations. For the sake of convenience and simplicity, the time term \(\mathrm{e}^{i\omega t}\) will be reduced in the following derivation process. Substituting Equations (2), (3), (8) and (9) into Equation (5) and performing the operation of merging the similar items, the strain energy UB of lattice sandwich beams will be given as:  
\[ \begin{array}{l}U_{B}=\frac{B}{2}\int_{0}^{L}\bigg{[}\left(A_{\mathrm{t}}+A _{b}\right)\left(\frac{\partial\Psi(x)}{\partial x}\right)^{2}+2(B_{t}-B_{b}) \left(\frac{\partial\Psi(x)}{\partial x}\right)\\ \cdot\left(\frac{\partial^{2}\Phi(x)}{\partial x^{2}}\right)+\left(D_{\mathrm{t }}+D_{b}\right)\left(\frac{\partial^{2}\Phi(x)}{\partial x^{2}}\right)^{2}+A _{\mathrm{c}}\Psi^{2}(x)\bigg{]}dx\end{array}  \tag{12}\]  
Here  
\[A_{\mathrm{t}}=\left(c^{2}/4\right)\int_{-h/2}^{-c/2}E_{x}dz,\ B_{\mathrm{t}}= \left(c/2\right)\int_{-h/2}^{-c/2}E_{x}zdz,\ D_{\mathrm{t}}=\int_{-h/2}^{-c/2}E _{x}z^{2}dz\]  
\[A_{\mathrm{c}}=\int_{-c/2}^{c/2}E_{x}dz,\ A_{\mathrm{b}}=\left(c^{2}/4\right) \int_{c/2}^{h/2}E_{x}dz,\ B_{\mathrm{b}}=\left(c/2\right)\int_{c/2}^{h/2}E_{x} zdz,\ D_{\mathrm{b}}=\int_{c/2}^{h/2}E_{x}z^{2}dz\]Substituting Equations (2), (3), (8) and (9) into Equation (6) and performing the operation of merging the similar items, the potential energy US stored in restraint springs at the two ends of the lattice sandwich beams will be given as:  
\[ \begin{array}{l}U_{s}=\frac{B}{2}\left[k_{0}\Phi^{2}(x)\big{|}_{x=0}+\left.K_{0 }\Big{(}\frac{\partial\Phi(x)}{\partial x}\Big{)}^{2}\right|_{x=0}+\left.K_{0} \Psi^{2}(x)\right|_{x=0}\\ +k_{1}\Phi^{2}(x)\big{|}_{x=L}+\left.K_{L}\Big{(}\frac{\partial\Phi(x)}{ \partial x}\Big{)}^{2}\right|_{x=L}+\left.K_{L}\Psi^{2}(x)\right|_{x=L}\end{array}  \tag{13}\]  
Substituting Equations (2), (3), (8) and (9) into Equation (7) and performing the operation of merging the similar items, the kinetic energy \(T\) of the structure will be given as:  
\[