## 4 Conclusions  
In this paper, a unified method is proposed to study the free vibration characteristics of lattice sandwich beams under various boundary conditions. In the existing methods, two different types of constraint springs are used to simulate the general elastic support boundary at both ends of lattice sandwich beams. By changing the rigidity of the boundary restraint spring, various boundary conditions can be obtained easily without modifying the solving algorithm and solving process. On this basis, the influence of the structural parameters on the free vibration characteristics of lattice sandwich beams with three different truss cores under different boundary conditions are investigated, including beam length, panel thickness, core height, truss radius, inclination angle and structural damping. From the numerical results, the following conclusions can be drawn:  
1. The analysis result shows that, when the rigidity of the restraint springs is greater than \(10^{10}\), it can be regarded as a clamped boundary condition. When the rigidity of the restraint springs is less than \(10^{4}\), the support effect of the restraint spring is very weak. It can be regarded as a free boundary condition. When the rigidity of the restraint springs is between \(10^{4}\) and \(10^{10}\), it can be regarded as an elastic boundary condition. 2. It can be found that the influence of boundary conditions on structural vibration is very important. As the rigidity of the restraint spring at both ends of the lattice sandwich beam continues to increase, its frequency parameters (\(\Omega\)) also increase. Among them, the frequency parameters (\(\Omega\)) are the largest under the lumped ends. The elastic boundary support reduces the node effect of the boundary support. 3. It can be seen that under different boundary conditions, the free vibration frequency of the structure decreases with the increase of beam length but increases with the increase of core height; the free vibration frequencies of the structure will become larger due to the increase of panel thickness under C-C and SS-SS boundary conditions, but it will become smaller as the panel thickness increases under E3-E3, E3-SS and E3-C boundary conditions. 4. It can be known that with the increase of the truss radius, the free vibration frequency of the structure first increases and then decreases. The main reason is that a smaller radius aids rigidity control, and a larger radius aids quality control. With the increase of the inclination angle, the free vibration frequency of the structure also first increases and then decreases. 5. It can be learned that when the influence of damping is considered, the displacement amplitude of the lattice sandwich beam will continue to decrease with time and that the greater the damping, the more the attenuation. Figure 23: Time-response history at the middle point of the lattice sandwich beam, when \(\zeta=0\) and \(\zeta=0.01\). **Author Contributions:** Conceptualization, Y.J. and R.Y. ; methodology, Y.J. and R.Y. ; software, H.L. and H.X. ; formal analysis, Y.J., R.Y. and H.X. ; investigation, H.C. and H.X. ; resources, H.L. and H.C.; date curation, Y.J. and R.Y. ; writing-original draft preparation, H.L. and H.X. ; writing-review and editing, Y.J., R.Y. and H.X. ; visualization, H.C. and H.X. ; supervision, Y.J. and H.C.; project administration, H.C.; funding acquisition, H.L. All authors have read and agreed to the published version of the manuscript. **Funding:** This paper is financially supported by the National Natural Science Foundation of China (Grant No.52071095), the basic research and cutting-edge technology projects of the State Administration for Science Technology and Industry for National Defense (Grant No.JCKY2019604C003) and funds of the State Key Laboratory of Vibration, shock and Noise (6142204190207). **Institutional Review Board Statement:** Not applicable. **Informed Consent Statement:** Not applicable.