Quasi-zero frequency seismic vibration damping metamaterial based on quadruped coordination mechanism

Abstract

The application discloses a quasi-zero-frequency earthquake damping metamaterial based on a four-foot cooperative mechanism, comprising a plurality of periodically arranged unit cell structures, each unit cell structure comprising a soil matrix, a steel block and a four-foot structure arranged in sequence from bottom to top; the four-foot structure is connected with a connecting rod, and the connecting rod is connected with a resonant mass block through a negative Poisson's ratio material. The application can produce an ultra-wide band gap starting from quasi-zero frequency through the cooperative integration of local resonance, inertia amplification and a negative Poisson's ratio material, and is used for the effective attenuation of low-frequency surface waves caused by earthquakes.

Description

Technical Field

[0001] This invention belongs to the field of vibration reduction and isolation technology in earthquake engineering, and specifically relates to a quasi-zero frequency earthquake vibration reduction metamaterial based on a quadrupedal body cooperative mechanism. Background Technology

[0002] Rayleigh surface waves, exhibiting the slowest energy attenuation and greatest destructive power during propagation, are a primary cause of damage to buildings at medium to long distances. Therefore, effectively attenuating low-frequency seismic waves remains a significant challenge for structural protection. Seismic metamaterials (SMs), as novel artificially designed microstructures utilizing their bandgap properties to control elastic wave propagation, have become a research hotspot in earthquake engineering.

[0003] Currently, the bandgap generation mechanisms of seismic metamaterials mainly include three types: Bragg scattering, local resonance (LR), and inertial amplification (IA). The bandgap initiation frequency of Bragg scattering metamaterials is related to the lattice constant, resulting in excessively large structural sizes and making them difficult to apply in the low-frequency range. Local resonance metamaterials generate bandgap by embedding scatterers in the matrix, but often require a large resonant mass and have a narrow bandgap width. Inertial amplification metamaterials transmit and amplify the displacement of a large mass block through connecting rods, achieving a wide bandgap without significantly increasing the mass, but it is difficult to extend to quasi-zero frequencies.

[0004] In recent years, the application of negative Poisson's ratio (NPR) materials in seismic metamaterials has attracted attention. Ding et al. proposed an NPR-integrated metamaterial that can achieve a quasi-zero frequency initiation bandgap, Akintoye et al. studied periodic foundations containing NPR materials, and Huang et al. achieved Lamb wave attenuation below 10 Hz. However, existing NPR-based seismic metamaterials suffer from high parameter sensitivity, high cost, and complex manufacturing. Therefore, this paper proposes a novel seismic metamaterial that can synergistically integrate multiple bandgap mechanisms into a compact and practical structure, achieving continuous attenuation from quasi-zero frequency to ultra-wideband. Summary of the Invention

[0005] To overcome the above technical problems, the present invention aims to provide a quasi-zero frequency earthquake damping metamaterial based on a quadrupedal body cooperative mechanism. By synergistically integrating local resonance (LR), inertial amplification (IA), and negative Poisson's ratio (NPR) materials, an ultra-wideband bandgap starting from quasi-zero frequency can be generated for effective attenuation of low-frequency surface waves caused by earthquakes.

[0006] To achieve the above objectives, the technical solution adopted by the present invention is as follows: A quasi-zero frequency earthquake vibration reduction metamaterial based on a tetrapod cooperative mechanism includes multiple periodically arranged unit cell structures. Each unit cell structure includes a soil matrix 1, a steel block 2, and a tetrapod structure 3 arranged from bottom to top. The quadruped structure 3 is connected to the connecting rod 4, and the connecting rod 4 is connected to the resonant mass block 5 through the negative Poisson's ratio material 6.

[0007] By synergistically integrating local resonance (LR), inertial amplification (IA), and negative Poisson's ratio (NPR) materials, an ultrawideband bandgap starting from quasi-zero frequency can be generated for effective attenuation of low-frequency surface waves induced by earthquakes.

[0008] The quadruped structure 3 includes two sets of symmetrical quadruped units, each set consisting of four conical legs intersecting at the geometric center; the conical legs are divided into three upper conical legs and one lower embedded conical leg; The length of the three upper conical legs is 0.35 to 0.45 times the height of the embedded conical legs, and the ratio of the root diameter to the end diameter of the three upper conical legs is approximately 1.6 to 1.8:1.

[0009] The embedded conical leg below is embedded in the soil matrix 1. The embedding depth of the embedded conical leg in the soil matrix 1 is 0.2 to 0.25 times the unit cell lattice constant, and the ratio of root diameter to end diameter is about 1.8 to 2.2:1.

[0010] The tapered leg changes axially from the root diameter to the end diameter, forming a cone.

[0011] The three upper conical legs intersect at symmetrical angles, and the upper conical legs are perpendicular to the horizontal direction. θ 1, θ 1 is an angle of 15° to 25° between the tapered legs. θ 2, θ 2 represents an angle of 10° to 20°.

[0012] The three upper tapered legs and the embedded tapered legs are directly and rigidly connected at their geometric centers via a connecting base. Four planar structures matching the tapered legs are arranged on the surface of the connecting base. This configuration achieves efficient coordination of the local resonance (LR) and inertial amplification (IA) mechanisms by increasing the stiffness of the embedded ends, reducing the effective mass of the upper portion, and optimizing multi-directional vibration coupling. This increases the low-frequency bandgap's onset frequency, bringing it closer to quasi-zero frequency, and widens the overall bandgap width.

[0013] The lattice constant a of the unit cell structure is 2m, the height H of the soil matrix 1 is 40m, and the height h of the steel block 2 is 0.6m.

[0014] The connecting rod 4 is a rectangular thin plate structure. Both ends of the rectangular thin plate structure are inclined along the X direction, forming two branches with a connection angle θ3 = 18°. θ3 is the angle between the connecting rod 4 and the plane containing the axis of the tapered leg. The height h1 = 0.6 m, thickness b = 0.01 m, and width h2 = 0.1 m are all present. This design realizes an inertial amplification (IA) mechanism, effectively amplifying and transmitting the displacement of the resonant mass block. The resonant mass block 5 has dimensions h3 = 2 m and a side length a2 = 0.1 m. It is connected to the connecting rod 4 via a negative Poisson's ratio material 6, with dimensions h2 × a1, where a1 = 0.02 m.

[0015] The resonant mass block 5 is a rectangular prism with a vertical height h3 = 2 m. h 3 = 2m, the horizontal cross-section has a side length of a2 = 0.1 m. a A square mass block with a diameter of 2 = 0.1m is positioned above the negative Poisson's ratio material 6 and flexibly connected to the connecting rod 4 via the negative Poisson's ratio material 6. The negative Poisson's ratio material 6 has a Poisson's ratio of -0.1 to -0.3 and a Young's modulus of (2.0 to 3.0) × 10⁻⁶. 4 Pa, with a density of 120–150 kg / m³.

[0016] Each branch of the connecting rod 4 is wrapped around / connected to two embedded conical legs (two conical supports buried in the soil), and then the two branches converge in the middle plate of the rectangular thin plate structure, which is connected to the resonant mass block 5 in the vertical direction through the negative Poisson's ratio material 6.

[0017] Both the quadruped structure 3 and the connecting rod 4 are made of steel, with a Young's modulus of (1.9~2.1)×10¹¹ Pa, a Poisson's ratio of 0.25~0.35, and a density of 7800~7900 kg / m³.

[0018] The Young's modulus of the soil matrix 1 is (1.8~2.5)×10 7 Pa, Poisson's ratio is 0.25 to 0.35, and density is 1700 to 1900 kg / m³.

[0019] The band gap generated by the unit cell structure includes 0.03–2.12 Hz, 2.75–8.79 Hz, 9.22–12.16 Hz and 12.97–16.33 Hz, with a total bandwidth of 14.43 Hz, covering 91% of the surface wave region below the acoustic cone.

[0020] The single-cell structure generates multiple band gaps through a coordinated LR-IA-NPR mechanism: the first band gap (0.03 ~ 2.12 Hz) is mainly formed by the inertial amplification (IA) mechanism induced by the negative Poisson's ratio material, combined with the low-frequency translational and torsional modes of the resonant mass; the second band gap (2.75 ~ 8.79 Hz) and the third band gap (9.22 ~ 12.16 Hz) are generated by the local resonance (LR) mechanism of the tetrapod structure and the LR-IA coupling mode, in which the three upper conical legs act as vibrators and the embedded conical legs act as springs. The fourth band gap (12.97 ~ 16.33 Hz) is dominated by local vibrations at the upper end of the tetrapod. The total bandwidth is 14.43 Hz, covering 91% of the surface wave region below the acoustic cone.

[0021] Multiple unit cell structures are arranged periodically along the horizontal direction to form a seismic surface wave barrier. Among them, the embedded conical legs of the four-legged structure 3 are buried in the soil matrix 1, and the seismic acceleration is attenuated by more than 95%.

[0022] The beneficial effects of this invention are: This invention achieves deep low-frequency wave isolation through intelligent mechanical design by synergistically integrating local resonance (LR), inertial amplification (IA), and negative Poisson's ratio (NPR) materials, without relying on heavy or bulky NPR materials. LR-IA coupling significantly broadens the attenuation bandwidth, while the NPR material promotes quasi-zero frequency initiation. The total bandgap width generated by this invention reaches 14.43 Hz, covering 91% of the surface wave region below the acoustic cone, with seismic acceleration attenuation exceeding 95%. Complex vibration modes, including torsional and coupled LR-IA motions, promote robust Rayleigh wave attenuation. Experimental results and time-history analysis based on real seismic records verify the superior performance of the LRIA-NPR-SM of this invention.

[0023] This invention achieves efficient integration of LR, IA, and NPR mechanisms through the synergistic combination of a quadruped structure, connecting rods, resonant mass, and negative Poisson's ratio material. Specifically, the embedded legs of the quadruped structure act as springs, while the upper legs act as oscillators to generate local resonance (LR). The connecting rods and resonant mass constitute an inertial amplification (IA) system, amplifying and transmitting the displacement of the mass. The negative Poisson's ratio material, through its tensile expansion properties, alters the overall stress field and deformation mode, promoting quasi-zero frequency crack initiation and inducing torsional coupling motion. This intelligent mechanical design achieves deep low-frequency wave isolation without relying on large-mass or large-volume negative Poisson's ratio materials. LR-IA coupling significantly broadens the attenuation bandwidth, while the NPR material promotes quasi-zero frequency initiation.

[0024] The invention generates a total bandgap width of 14.43 Hz, covering 91% of the surface wave region below the acoustic cone, with seismic acceleration attenuation exceeding 95%. Complex vibration modes, including torsional and LR-IA coupled motions, promote robust Rayleigh wave attenuation. Experimental results and time history analysis based on real seismic records verify the superior performance of the LRIA-NPR-SM invention. Attached Figure Description

[0025] Figure 1 This is a diagram of the unit cell structure model of the tetrapod seismic metamaterial of this invention.

[0026] Figure 2 This is a comparison diagram of the three structural configurations of the present invention.

[0027] Figure 3 This is a diagram of the band structure of the present invention.

[0028] Figure 4 This is the transmission spectrum of the present invention.

[0029] Figure 5 This is the acceleration response diagram of the present invention. Detailed Implementation

[0030] The present invention will now be described in further detail with reference to the accompanying drawings.

[0031] like Figure 1 As shown, this invention discloses a quasi-zero frequency earthquake vibration reduction metamaterial based on a tetrapod cooperative mechanism. Inspired by the tetrapod wave-damping structure of a coastline, the unit cell structure consists of a soil matrix 1, a steel block 2, a tetrapod structure 3, connecting rods 4, a resonant mass block 5, and a negative Poisson's ratio material 6. The soil matrix 1 is a soil block with a height H = 40m and a side length a = 2m. The steel block 2 is a rigid base with a height h = 0.6m, fixedly connected to the soil matrix 1. The tetrapod structure 3 includes two sets of symmetrical tetrapod units, each set consisting of four cylindrical legs intersecting at the geometric center, fixedly connected to the steel block 2.

[0032] The three upper conical legs have a length l2 = 1m and diameters R = 0.25m and d1 = 0.15m, respectively. The embedded conical leg of the soil body 1 is buried at a depth of h = 0.6m, with heights of l = 2.6m and l1 = 2.3m, and diameters of d = 0.30m and d2 = 0.15m, respectively. The connecting rod 4 is connected to the embedded conical leg of the quadruped structure 3 along the X-direction at a connection angle θ3 = 18°, with a height of h1 = 0.6m and a thickness of b = 0.01m. The resonant mass block 5 has dimensions of h3 = 2m and a side length of a2 = 0.1m. It is connected to the connecting rod 4 via a negative Poisson's ratio material 6, with dimensions of h2 × a1, where a1 = 0.02m.

[0033] Upper tapered leg length l 2=1 m Embedded tapered leg height l =2.6 m ), the total length of the embedded conical leg itself (i.e., the distance from the intersection of the geometric centers of the four-legged body to the tip of the embedded leg, corresponding to the attached... Figure 1 Parameters in l ).

[0034] The three upper tapered legs and the embedded tapered legs are directly and rigidly connected at their geometric centers via a connecting base. Four planar structures matching the tapered legs are arranged on the surface of the connecting base. This configuration achieves efficient coordination of the local resonance (LR) and inertial amplification (IA) mechanisms by increasing the stiffness of the embedded ends, reducing the effective mass of the upper portion, and optimizing multi-directional vibration coupling. This increases the low-frequency bandgap's onset frequency, bringing it closer to quasi-zero frequency, and widens the overall bandgap width.

[0035] The connecting rod 4 is a rectangular thin plate structure. The two ends of the rectangular thin plate structure are inclined along the X direction to form two branches with a connection angle θ3 = 18°. θ3 is the angle between the connecting rod 4 and the plane containing the axis of the tapered leg. The height h1 = 0.6 m, the thickness b = 0.01 m, and the width h2 = 0.1 m.

[0036] This design implements an inertial amplification (IA) mechanism, which effectively amplifies and transmits the displacement of the resonant mass block.

[0037] The resonant mass block 5 is a rectangular prism with a vertical height h3 = 2 m. h 3 = 2m, the horizontal cross-section has a side length of a2 = 0.1 m. a A square mass block with a diameter of 2 = 0.1m is positioned above the negative Poisson's ratio material 6 and flexibly connected to the connecting rod 4 via the negative Poisson's ratio material 6. The negative Poisson's ratio material 6 has a Poisson's ratio of -0.1 to -0.3 and a Young's modulus of (2.0 to 3.0) × 10⁻⁶. 4 Pa, with a density of 120–150 kg / m³.

[0038] Both the quadruped structure 3 and the connecting rod 4 are made of steel, with a Young's modulus of (1.9~2.1)×10¹¹ Pa, a Poisson's ratio of 0.25~0.35, and a density of 7800~7900 kg / m³.

[0039] The Young's modulus of the soil matrix 1 is (1.8~2.5)×10 7 Pa, Poisson's ratio is 0.25 to 0.35, and density is 1700 to 1900 kg / m³.

[0040] The band gap generated by the unit cell structure includes 0.03–2.12 Hz, 2.75–8.79 Hz, 9.22–12.16 Hz and 12.97–16.33 Hz, with a total bandwidth of 14.43 Hz, covering 91% of the surface wave region below the acoustic cone.

[0041] The single-cell structure generates multiple band gaps through a coordinated LR-IA-NPR mechanism: the first band gap (0.03 ~ 2.12 Hz) is mainly formed by the inertial amplification (IA) mechanism induced by the negative Poisson's ratio material, combined with the low-frequency translational and torsional modes of the resonant mass; the second band gap (2.75 ~ 8.79 Hz) and the third band gap (9.22 ~ 12.16 Hz) are generated by the local resonance (LR) mechanism of the tetrapod structure and the LR-IA coupling mode, in which the three upper conical legs act as vibrators and the embedded conical legs act as springs. The fourth band gap (12.97 ~ 16.33 Hz) is dominated by local vibrations at the upper end of the tetrapod. The total bandwidth is 14.43 Hz, covering 91% of the surface wave region below the acoustic cone.

[0042] like Figure 2 As shown, to verify the function of each component, this invention designed three structural configurations for comparative study: Structure I (LRIA-NPR-SM) is a complete structure containing a resonant mass block and NPR material; Structure II removes the NPR material; Structure III removes both the resonant mass block and the NPR material.

[0043] The material parameters used in this invention are shown in Table 1: Table 1 Material Parameters The working principle of this invention is as follows: Based on the phononic crystal bandgap theory, the band structure of the unit cell was calculated using Bloch's theorem and the finite element method (COMSOL Multiphysics). Bloch periodic boundary conditions were applied to all four sides of the soil matrix 1, with the upper surface free and the lower surface fixed. The wave vector was scanned along the Brillouin zone path Γ→X→M→T→Y.

[0044] like Figure 3As shown, band structure calculations indicate that structure I (LRIA-NPR-SM) generates four band gaps: the first band gap is 0.03–2.12 Hz, the second band gap is 2.75–8.79 Hz, the third band gap is 9.22–12.16 Hz, and the fourth band gap is 12.97–16.33 Hz, with a total bandwidth of 14.43 Hz. In comparison, structure II (without NPR) has a total bandwidth of 12.96 Hz, and structure III (without mass block, without NPR) has a total bandwidth of only 6.47 Hz. This suggests that adding NPR material creates the first band gap starting at 0.03 Hz and fills the gap between the second and third band gaps, while adding the resonant mass block creates two new low-frequency band gaps.

[0045] Vibration modal analysis revealed that in the first bandgap, the resonant mass 5 vibrates along the X direction (IA mechanism), while simultaneously generating torsional motion in the X and Y directions (NPR-induced IA mechanism). In the second bandgap, the upper part of the quadruped structure 3 vibrates (LR mechanism: embedded conical legs act as springs, and the three upper conical legs act as oscillators). In the third bandgap, the resonant mass generates maximum vibration (LR mechanism), while the top of the quadruped also vibrates (coupled LR-IA mode). In the fourth bandgap, vibrational energy is concentrated at the upper end of the quadruped.

[0046] Parametric analysis shows that the side length a2 of the resonant mass block 5 is the key parameter determining the bandgap frequency: when a2=0 (no mass block), the first and second bandgap completely disappear, confirming that the LR mechanism strictly depends on the resonant mass; when a2=10~60 mm, the two bandgap expands significantly; when a2=60~200 mm, the expansion rate is stable, exhibiting a mass-coupling saturation effect. The Poisson's ratio of the NPR material has a significant impact on the bandgap characteristics: the tensile expansion characteristics of the NPR material change the global stress field and deformation mode, reshaping the bandgap through the redistribution of vibrational energy and the transfer of dominant modes, rather than translating the bandgap.

[0047] like Figure 4 As shown, the transmission spectrum analysis employed a finite periodic model, with 100 unit cells arranged along the X-direction to form a seismic barrier. Perfectly matched layers (PMLs) of 3 a thickness were placed on the sides and bottom, with a soil depth of 20 a. The excitation point was 25 a from the barrier, and the receiver point was located at the right end of the barrier at a distance a. The results show that the transmission attenuation ranged from -0.05 dB to -55 dB, with minimal attenuation in the 0–2 Hz range (≤ -25 dB) and significant attenuation reaching -50 dB in the 2–20 Hz range. Displacement field analysis indicated that the Rayleigh wave amplitude significantly attenuated at 1 Hz, and at 9 Hz, the surface wave was converted into a downward-propagating body wave.

[0048] like Figure 5As shown, time history analysis was performed using El-Centro seismic waves. The results show that the peak ground acceleration (PGA) in the X direction attenuates by more than 95%, and the PGA in the Z direction attenuates by more than 90%. Fourier spectroscopy confirms the significant isolation effect within the bandgap region and also demonstrates excellent wave suppression capability in the low-frequency range of 0–1 Hz. The combined results of wave seismic records and Taft seismic waves further validate the effectiveness of this invention.

[0049] To verify the practical effect of this invention, an experiment was conducted using a 1:50 scale model. 24 experimental samples were fabricated using SLA 3D printing technology, with the main structure being epoxy resin (E=4.35×10⁻⁶). 9 (Pa, ν=0.368, ρ=1180 kg / m³), the NPR component material parameters are consistent with the simulation. Sand was used as the soil matrix in the experiment, with a glass water bath width of 0.48 m and a 2 cm thick sponge layer at the bottom as the absorbing boundary. Samples were arranged in 6×4 arrays along the X and Y directions, partially buried in the sand. An ECON CL-YD-303 vibratory hammer was used for excitation, and an accelerometer and M+P dynamic signal analyzer were used to collect data. The experimental results and numerical simulation results matched well in the 0–1200 Hz frequency range, and the bandgap frequency range was highly consistent with the FEM prediction, verifying the isolation effectiveness of the present invention.

[0050] It is evident that the quasi-zero frequency earthquake-damping metamaterial based on the tetrapod cooperative mechanism of this invention can generate an ultra-wideband bandgap in the low-frequency range, effectively reducing surface waves generated by earthquakes. This invention achieves deep low-frequency isolation through intelligent mechanical design rather than bulky structural components, and can be applied to fields such as earthquake engineering vibration reduction. Multiple unit cell structures can be periodically arranged in the soil surrounding a building to reduce the impact of elastic waves generated by earthquakes on ground structures.

Claims

1. A quasi-zero frequency seismic vibration reduction metamaterial based on a tetrapod cooperative mechanism, characterized in that, It includes multiple periodically arranged unit cell structures, each unit cell structure including a soil matrix (1), a steel block (2), and a tetrapod structure (3) arranged from bottom to top. The quadruped structure (3) is connected to the connecting rod (4), and the connecting rod (4) is connected to the resonant mass block (5) through the negative Poisson's ratio material (6).

2. The quasi-zero frequency earthquake vibration reduction metamaterial based on a tetrapod cooperative mechanism according to claim 1, characterized in that, The quadruped structure (3) includes two sets of symmetrical quadruped units, each set consisting of four conical legs intersecting at the geometric center; the conical legs are divided into three upper conical legs and one lower embedded conical leg; The length of the three upper conical legs is 0.35 to 0.45 times the height of the embedded conical legs, and the ratio of the root diameter to the end diameter of the three upper conical legs is 1.6 to 1.8:1; The lower embedded conical leg is embedded in the soil matrix (1), and the ratio of root diameter to end diameter is approximately 1.8 ~ 2.2:

1.

3. The quasi-zero frequency earthquake vibration reduction metamaterial based on a tetrapod cooperative mechanism according to claim 2, characterized in that, The tapered leg changes axially from the root diameter to the end diameter, forming a cone.

4. The quasi-zero frequency earthquake vibration reduction metamaterial based on a tetrapod cooperative mechanism according to claim 2, characterized in that, The three upper conical legs intersect at symmetrical angles. The three upper conical legs intersect at symmetrical angles, and the upper conical legs are perpendicular to the horizontal direction. θ 1, θ 1 is an angle of 15° to 25° between the tapered legs. θ 2, θ 2 represents an angle of 10° to 20°.

5. The quasi-zero frequency earthquake vibration reduction metamaterial based on a tetrapod cooperative mechanism according to claim 2, characterized in that, The three upper tapered legs and the embedded tapered legs are directly and rigidly connected at their geometric center via a connecting base; The surface of the connecting base is provided with four planar structures that match the ends of the tapered legs.

6. The quasi-zero frequency earthquake vibration reduction metamaterial based on a tetrapod cooperative mechanism according to claim 2, characterized in that, The connecting rod (4) is connected to the embedded conical leg of the quadruped structure (3) along the X direction.

7. The quasi-zero frequency earthquake vibration reduction metamaterial based on a tetrapod cooperative mechanism according to claim 2, characterized in that, The resonant mass block (5) is a rectangular prism. The resonant mass block (5) is located above the negative Poisson's ratio material (6) and is flexibly connected to the connecting rod (4) through the negative Poisson's ratio material (6).

8. The quasi-zero frequency earthquake vibration reduction metamaterial based on a tetrapod cooperative mechanism according to claim 7, characterized in that, The connecting rod (4) is a rectangular thin plate structure. The two ends of the rectangular thin plate structure are inclined along the X direction to form two branches. The connection angle θ3 = 18°. θ3 is the angle between the connecting rod (4) and the plane containing the axis of the tapered leg. Each branch of the connecting rod (4) is wrapped around / connected to two embedded conical legs (two conical supports buried in the soil), and then the two branches meet in the middle plate of the rectangular thin plate structure, the middle plate being connected to the resonant mass block (5) in the vertical direction through the negative Poisson's ratio material (6).

9. The quasi-zero frequency earthquake vibration reduction metamaterial based on a tetrapod cooperative mechanism according to claim 1, characterized in that, Multiple unit cell structures are arranged periodically along the horizontal direction to form a seismic surface wave barrier, wherein the embedded conical legs of the four-legged structure (3) are buried in the soil matrix (1).

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