## 2 Theoretical Formulations  
### Description of the Model  
As shown in Figure 1a,b, a lattice sandwich beam under arbitrary boundary conditions is taken as the research object. In this paper, three different lattice cores are considered, namely, a 3D-Kagome core, a tetrahedral core and a pyramid core, as shown in Figure 2. The coordinate system involved in this paper is shown in Figure 1a,b, where the x-y plane is fixed at the mid-plane of the lattice sandwich beam, the \(z\)-axis is perpendicular to the x-y plane and its direction points downward. The length, width and height of the lattice sandwich beam are defined by \(L\), \(B\) and \(h\), respectively. The lattice sandwich beam contains a different number of elements in the direction of length and width. Different lengths and widths of sandwich beams can be obtained by selecting different numbers of units. Figure 1: _Cont._  
To explore the vibration mechanism of lattice sandwich beams with arbitrary boundary conditions, the artificial virtual spring technology is first used in this paper. The specific process is to set transverse translational springs and torsion springs at both ends of the beam, based on which the shear force and bending moment at the boundary support can be simulated. Then, by arbitrarily adjusting the rigidity values of the transverse translational springs and torsion springs, arbitrary boundary conditions can be easily simulated. For example, a boundary condition with clamped ends supported can be obtained by setting the rigidity values of the transverse translational springs and the torsion springs at the boundary support to infinity. In practical computations, a maximum number is often used instead of an infinite value. Similarly, setting the rigidity values of these springs at the boundary support to an extremely small value can be considered a free supported boundary. In theoretical modeling, the upper and lower panels of sandwich beams are thin and hard. Generally, only transverse bending deformation and axial deformation are considered, while shear deformation is ignored. In contrast, the influence of the core layer on the overall flexural rigidity of the lattice sandwich beam is largely ignored because the core layer is generally thicker and lighter. Hence, for a thicker core layer, only the shear deformation is generally taken into account, and the shear stress is considered to be constant in the direction of height. Therefore, based on the above deformation assumptions, the lattice sandwich beam is generally simplified to an equivalent laminated beam composed of the upper and lower panels and an equivalent homogeneous material in the middle layer. Three different lattice cores are involved in this paper, namely, a 3D-Kagome core, a tetrahedral core and a pyramidal core, and are shown in Figure 2. The diameter, length and inclination of the circular truss will be represented by \(2r\), \(l\) and \(\alpha\), respectively. According  
Figure 1: Schematic diagram of lattice sandwich beams with arbitrary boundary conditions: (**a**) The structural parameters of the structural system. (**b**) The cross-sectional view of the structural system. (**c**) The kinematical parameters of the system. Figure 2: Schematic diagram of three different lattice core layers. (**a**) Pyramidal truss core. (**b**) Tetrahedral truss core. (**c**) 3D-Kagome truss core.