# The Vibration Isolation Design of a Re-Entrant Negative Poisson's Ratio Metamaterial
## 1 Introduction  
Metamaterials are artificially designed materials with periodic lattice arrangements, exhibiting advanced mechanical and physical properties that surpass those found in natural materials, such as mechanics [1; 2], acoustics [3; 4; 5; 6; 2], and electromagnetics [7; 8]. Notably, negative Poisson's ratio (NPR) metamaterials have garnered widespread attention in fields like aerospace due to their remarkable properties, including a negative Poisson's ratio [9; 10; 11; 12; 13; 14; 15], negative stiffness [16; 17; 18; 19; 20], and negative thermal expansion coefficient [21; 22; 23]. In the mechanical application of metamaterials, the deformation behavior, bearing capacity, and the ability to resist vibration and impact have always been the research focuses. [PERSON] et al. [24] investigated the influence of hierarchical structures on bandgap distribution and energy absorption efficiency, revealing significant potential in widening bandgaps and enhancing energy absorption efficiency. [PERSON] et al. [25] proposed novel face-centered cubic plate and truss micro-lattices as new types of metamaterials and found that the absorption coefficient's characteristics are fundamentally shaped by the geometric morphology of voids and cavities. The exceptional mechanical performance stems from the near-membrane stress state of the plate structures and the superior ductility and strength of the base material. [PERSON] et al. [26] theoretically and numerically studied the dynamic behavior of triangular and hexagonal lattice cylindrical structures under constant-velocity impact. The results showed that for lattice cylindrical shells, introducing a positive density gradient can enhance early-stage energy absorption at high velocities. For lattice sandwichcylindrical shells, introducing a density gradient can effectively reduce the peak fracture force but has minimal impact on energy absorption. [PERSON] et al. [27] demonstrated that the deformation behavior of expandable lattice cylindrical shells can be represented by NPR modes, while the different deformation behaviors of lattice sandwich cylindrical shells (e.g., \"Z\" mode, circular mode, diamond mode, and mixed mode) are primarily determined by the proportions of the core material. [PERSON] et al. [28] rigorously validated and confirmed the numerical and analytical homogenization of honeycomb structures with high relative densities and varying representative unit quantities. [PERSON] et al. [29] highlighted the existence of bandgaps at relatively low frequencies in periodic locally resonant materials (LRM), which can be utilized for vibration attenuation or shock absorption. [PERSON] et al. [30] analyzed the equivalent mechanical behavior and bandgap characteristics of star-shaped honeycomb structures, and the numerical results showed that a negative Poisson's ratio provides an enhanced effective Young's modulus for the considered honeycomb structure. When the Poisson's ratio is negative, the bandgap exists at much lower frequencies, and the cumulative bandgap width remains constant. [PERSON] et al. [31] proposed a novel honeycomb lattice geometric shape that combines seamless arrangements of a traditional core and an expandable core, offering significant advantages in tuning high-frequency bandgap behavior. All of these works show that metamaterials have good application potential in aerospace, and the rational design of artificial micro-structured metamaterials with advanced mechanical and physical properties that are not accessible in nature is challenging and important. [PERSON] et al. [32] investigated the existence of Minnaert resonance in three-dimensional systems and found that weak resonance always occurs with appropriately chosen parameters, and enhanced or even strong resonance can be achieved through geometric shape selection. [PERSON] et al. [33] proposed a single-phase acoustic metamaterial composed of periodically arranged divergent star-shaped units that exhibit low-frequency and wide bandgaps and validated it by experimental research.