## 1 Introduction
A lot of research on the vibration mechanism of the lattice sandwich beam is basically based on several classical boundary support conditions, including clamped support, free support, simple support and cantilever support. However, there are few studies on general elastic boundary conditions. [PERSON] et al. [34; 35; 36; 37] first proposed an improved Fourier series method to reveal the vibration mechanisms of beam and plate structures. On this basis, a large number of scholars have put forward many original research results. For a [PERSON] beam with general elastic support conditions, [PERSON] and her team [38] developed a unified method to study their structure response by using virtual springs to simulate arbitrary boundary conditions. By further extending their research, [PERSON] et al. [39] proposed an improved Fourier-Ritz method to analyze the vibration characteristics of composite laminated beams under general boundary conditions. For a Mindlin rectangular plate, [PERSON] et al. [40; 41] studied its transverse vibration response by employing the improved Fourier series method with elastic point support boundary conditions. In addition, [PERSON] and [PERSON] [42] studied the bending behavior and local failure of glass fiber-reinforced composite (GFRC) and pultruded sandwich panels (PSPs). [PERSON] et al. [43] were devoted to investigating the sound insulation performance of a simply supported sandwich plate with an hourglass lattice core. [PERSON] et al. [44] experimentally and numerically investigated the free vibration, quasi-static compressive and split Hopkinson pressure bar impact responses of hybrid composite pyramidal truss sandwich panels. [PERSON] et al. [45] revealed elastoplastic deformation and a local indentation failure pattern of the truss-core sandwich panel and material ductility effects to the failure of the truss-core sandwich panel. [PERSON] et al. [46] used the concept of phononic crystals to improve the vibroacoustic properties of truss-core sandwich beams. However, most of the current research is confined to using classical boundary conditions, and there are few studies on the vibration characteristics of lattice sandwich beamsunder general elastic boundary conditions. In practical engineering, the structure boundary is not an ideal classical boundary condition, but a more general elastic boundary. Therefore, the theoretical analysis method under general elastic boundary support is more conducive to reveal the actual vibration characteristics of the structure. However, the introduction of elastic boundary conditions will lead to a very complicated theoretical modeling process, and the derivative of the vibration differential equation will be discontinuous or jumping. For this reason, this paper presents a unified solution for all boundary conditions, including classical boundary conditions and elastic boundary conditions. In the present method, the truss core of lattice sandwich beams is equivalent to a homogeneous layer, and the lattice sandwich beam is simplified to a laminated beam. By setting tension springs and torsion springs at both ends of the beam, the general elastic support boundary is simulated. Then, the displacement function of lattice sandwich beams is extended to the improved Fourier series expression, and our sine terms are introduced to overcome all the relevant discontinuities or jumps of elastic boundary support conditions. On this basis, the unknown series coefficients of the displacement function are treated as the generalized coordinates and solved using the Rayleigh-Ritz method. The correctness of the present method is verified by comparing with existing literature. The geometric parameters are studied and the vibration characteristics of the structure under elastic boundary conditions are analyzed. This study aims to accumulate data for analyzing the vibration of the lattice sandwich beam.