A double-layer sandwich topology metamaterial plate and application thereof

Abstract

The application belongs to the technical field of mechanical topological insulator, and relates to a double-layer sandwich topological metamaterial plate and application thereof. The double-layer sandwich metamaterial plate is composed of a double-layer sheet and a middle sandwiched torsional-coupled resonator; the torsional-coupled resonator comprises a cylindrical mass block and a plurality of inclined rods for fixed connection. The application contains two kinds of crystal cell units, i.e. a rhombic crystal cell unit and a hexagonal crystal cell unit, wherein the rhombic crystal cell unit contains two torsional-coupled resonators, and the hexagonal crystal cell unit contains six torsional-coupled resonators. The quantum valley Hall effect is simulated by changing the size of the two resonator mass blocks of the rhombic crystal cell unit, and the topological interface state with vibration localized at the interface is realized. The quantum spin Hall effect is simulated by changing the distance from the resonator to the center of the hexagonal crystal cell unit, and the topological corner state with vibration localized at the corner point is realized. The application can realize precise manipulation and fixed-point localization effect of elastic waves, and has great application potential in vibration control and energy collection.

Description

Technical Field

[0001] This invention belongs to the field of mechanical topological insulator technology, specifically relating to a double-layer sandwich topological metamaterial plate and its applications. Background Technology

[0002] With the rapid development of mechanical engineering technology, vibration problems have become an important issue that cannot be ignored and urgently needs to be solved. Traditional spring damping vibration reduction technology can only attenuate high-frequency vibrations and is difficult to solve the problem of low-frequency vibrations, which are more harmful. In order to achieve low-frequency vibration reduction, mechanical metamaterials have received widespread attention in recent years. Mechanical metamaterials can suppress the propagation of elastic waves in the bandgap frequency range in the structure by means of bandgap mechanism, thereby achieving low-frequency vibration attenuation and providing a new approach to solving vibration control problems. Vibrations often store a large amount of energy, and under existing vibration reduction methods, a large amount of vibration energy is wasted by damping elements. Realizing the harvesting and utilization of vibration energy has important practical significance for green energy conservation. However, vibration attenuation and vibration energy harvesting are contradictory, and it is difficult to achieve effective vibration energy harvesting under vibration reduction requirements.

[0003] Mechanical topological insulators not only possess the ability to attenuate vibration propagation within a structure's bandgap, but also generate multi-dimensional topological states within the bandgap, enabling the manipulation of elastic wave energy accumulation on structural surfaces, edges, or corners. These topological states exhibit robust topological protection, unaffected by defects and disorder during manufacturing and use, achieving stable and efficient energy accumulation. Therefore, mechanical topological insulators provide a feasible approach to simultaneously achieving vibration attenuation and energy capture, alleviating the contradiction between these two processes. In particular, the vibration problems encountered in engineering structures such as train passenger compartment floors and ship engine floor plates are well-matched by the characteristics of mechanical topological insulators, showing promising application prospects. However, current mechanical topological metamaterial plates are mainly perforated or single-layer plates with added local resonators, resulting in limited load-bearing capacity and vibration reduction effects, making them unsuitable for engineering applications.

[0004] Therefore, a new double-layer sandwich topological metamaterial plate needs to be designed. Summary of the Invention

[0005] The purpose of this invention is to provide a double-layer sandwich topological metamaterial plate to solve the problem mentioned in the background art that current mechanical topological metamaterial plates are mainly perforated or single-layer plates with added local resonators, which have very limited application scenarios.

[0006] To achieve the above objectives, the present invention provides a double-layer sandwich topological metamaterial plate, comprising a unit cell, wherein the unit cell comprises an upper thin plate, a lower thin plate, and a pressure-torsion coupling resonator, the upper thin plate and the lower thin plate having the same shape, and the pressure-torsion coupling resonator being disposed between the upper thin plate and the lower thin plate;

[0007] The pressure-torsion coupling resonator includes inclined rods and a cylindrical mass block. Six inclined rods with a radius of r are arranged above and below the cylindrical mass block. The inclined rods are arranged in a chiral structure around the axis of the cylindrical mass block, which fixes the cylindrical mass block to the upper and lower thin plates.

[0008] In one specific embodiment, the double-layer sandwich topological metamaterial plate is manufactured by integrated 3D printing technology, and the material used to manufacture the double-layer sandwich topological metamaterial plate is TPU material.

[0009] In one specific embodiment, the upper and lower thin plates of the unit cell are both rhombic thin plates; the unit cell includes two compression-torsion coupled resonators, the centers of the two compression-torsion coupled resonators are located on the long diagonal of the rhombic thin plate and are symmetrically distributed, and the distance between the centers of the two compression-torsion coupled resonators is 1 / 3 of the length of the long diagonal of the rhombic thin plate.

[0010] In one specific embodiment, the rhomboid sheet contains an angle of 60°.

[0011] In one specific implementation, the cylindrical mass blocks of the two pressure-torsion coupled resonators in the unit cell have equal radii.

[0012] In one specific embodiment, the double-layer sandwich topological metamaterial plate includes two types of unit cells with different topological properties, namely unit cell I and unit cell II. In unit cell I, the radius of the cylindrical mass block of the first compression-torsion coupled resonator is smaller than the radius of the cylindrical mass block of the second compression-torsion coupled resonator. In unit cell II, the radius of the cylindrical mass block of the first compression-torsion coupled resonator is larger than the radius of the cylindrical mass block of the second compression-torsion coupled resonator. The two are mirror-symmetric inverse structures.

[0013] In one specific embodiment, the double-layer sandwich topological metamaterial plate includes a finite element structure composed of multiple unit cells I and II with two different topologies; the finite element structure includes an interface, with unit cells I and II located on both sides of the interface; all unit cells in the finite element structure are arranged in a straight line in the same plane as parallelograms, and the rhomboid sides of adjacent unit cells are closely connected; the finite element structure composed of multiple unit cells I and II with different topological properties arranged on both sides of the interface can realize the topological interface state within the band gap.

[0014] In one specific embodiment, the upper and lower thin plates of the unit cell are both regular hexagonal thin plates. The unit cell includes six uniformly distributed pressure-torsion coupled resonators. The centers of the six pressure-torsion coupled resonators are respectively located on the perpendicular bisector of each side of the regular hexagonal thin plate, and the distance D between the centers of the six pressure-torsion coupled resonators and the center of the unit cell is equal.

[0015] In one specific implementation, a unit cell whose distance from the center of the compression-torsion coupled resonator to the center of the unit cell is less than one-third of the side length of the regular hexagonal plate is a shrinking unit cell; a unit cell whose distance from the center of the compression-torsion coupled resonator to the center of the unit cell is greater than one-third of the side length of the regular hexagonal plate is an expanding unit cell; shrinking and expanding unit cells have different topological properties; the double-layer sandwich topological metamaterial plate includes at least one of shrinking and expanding unit cells. A finite element structure composed of one of shrinking and expanding unit cells can realize boundary states and corner states within the bandgap.

[0016] Application of the double-layer sandwich topological metamaterial plate in vibration energy harvesting or low-frequency vibration reduction.

[0017] Compared with the prior art, the present invention has the following beneficial effects:

[0018] When the double-layer sandwich topological metamaterial plate designed in this invention is subjected to compression / tension, the compression-torsion coupling resonator can undergo torsion while translating, i.e., the compression-torsion coupling effect, which is beneficial to opening the low-frequency band gap and realizing low-frequency vibration reduction.

[0019] This invention achieves topological edge and corner states within the bandgap of a double-layer sandwich metamaterial plate. Vibrational energy can be concentrated at desired edges and corners, facilitating further vibrational energy harvesting, while vibration is suppressed within the structure. The topological states possess robust topological protection, immune to moderate defects and disorder during the fabrication process, resulting in stable topological devices. This invention alleviates the conflict between vibration attenuation and vibrational energy capture, achieving both low-frequency vibration attenuation and vibrational energy localization.

[0020] The lattice system in this invention supports both topological acute-angle and topological obtuse-angle states, reducing the dependence of topological angle states on angles and broadening the shape design options for topological metamaterial plates.

[0021] This invention eliminates the need to design complex composite lattice structures composed of two unit cells with different topological properties. A single lattice composed of expanding or contracting unit cells can simultaneously support one-dimensional edge states and zero-dimensional corner states within the topological bandgap, enabling the design of topological structures of different shapes and facilitating the addition of suitable energy harvesting systems to the corners and edges of the structure to collect vibrational energy.

[0022] In addition to the objectives, features, and advantages described above, the present invention has other objectives, features, and advantages. The invention will now be described in further detail with reference to the figures. Attached Figure Description

[0023] The accompanying drawings, which form part of this application, are used to provide a further understanding of the invention. The illustrative embodiments of the invention and their descriptions are used to explain the invention and do not constitute an undue limitation of the invention. In the drawings:

[0024] Figure 1 This is a schematic diagram of a double-layer metamaterial plate with a pressure-torsion coupling core, where a1 and a2 are lattice vectors.

[0025] Figure 2 This is a schematic diagram of a typical compression-torsion coupling structure, where φ represents the compression-torsion angle.

[0026] Figure 3 (a) is a schematic diagram of a rhombic unit cell, where R1 is the radius of the cylindrical mass block in the first compression-torsion coupling resonator, R2 is the radius of the cylindrical mass block in the second compression-torsion coupling resonator, T is the thickness of the thin plate, t is the thickness of the cylindrical mass block, r is the radius of the inclined rod, H is the overall height of the rhombic unit cell, and h is the height between the lower surface of the upper thin plate and the upper end face of the cylindrical mass block. Figure 3 (b) is a top-view schematic diagram of a rhombic unit cell with the upper plate hidden. c α is the distance between the center of the end of the inclined bar and the center of the end face of the cylindrical mass block, and α is the side length of the rhomboid plate;

[0027] Figure 4 It is a band structure of a rhombic unit cell with equal radius of cylindrical mass blocks (R1 = R2 = 11 mm). The illustration shows the irreducible Brillouin zone of the rhombic unit cell, where Г, M, and K are the high symmetry points of the irreducible Brillouin zone.

[0028] Figure 5 It is the band structure of rhombic unit cell I (R1= 11mm, R2= 13mm) with unequal radii of cylindrical mass blocks and rhombic unit cell II (R1= 13mm, R2= 11mm) with interchanged radii of the two cylindrical mass blocks; R1 is the radius of the cylindrical mass block in the first compression-torsion coupled resonator, and R2 is the radius of the cylindrical mass block in the second compression-torsion coupled resonator; S1 and S2 mark the upper and lower boundary points of the topological bandgap, respectively.

[0029] Figure 6 It is the mode reversal at the upper and lower boundary points S1 and S2 of the topological band gap of rhombic unit cell I and rhombic unit cell II;

[0030] Figure 7(a) is a strip-shaped supercell composed of 7 unit cell II and 7 unit cell I placed on the left and right sides of the interface respectively; Figure 7 (b) is a strip-shaped supercell in k x The directional band structure shows a dispersion curve corresponding to the topological interface state that spans the entire band gap in the topological band gap of the strip-shaped supercell.

[0031] Figure 8 (ab) is the wave vector k x =π. Top view and side view of the eigenmodes of the topological interface state of the strip-shaped supercell. In the topological interface state, the energy is highly localized at the interface and decays rapidly in the direction away from the interface, which has the ability to localize the vibrational energy at the interface.

[0032] Figure 9 (ab) is a schematic diagram of a finite element structure consisting of 7 unit cell II and 7 unit cell I placed on the left and right sides of the interface, and a diagram of the transport characteristics of the finite element structure.

[0033] Figure 10 (ab) are top and side views of the out-of-plane displacement distribution cloud map of the finite element structure under the topological interface state;

[0034] Figure 11 This is a schematic diagram of a regular hexagonal unit cell with a compression-torsion coupling core. One regular hexagonal unit cell contains 6 compression-torsion coupling resonators. D represents the distance between the center of the compression-torsion coupling resonator and the center of the unit cell.

[0035] Figure 12 It is the band structure of the initial unit cell (D0=a / 3) of a regular hexagonal double-layer metamaterial plate;

[0036] Figure 13 It is the band structure of a shrinking unit cell (D=D0+∆D,∆D=-3mm) of a regular hexagonal double-layer metamaterial plate; the color legend represents the out-of-plane polarization factor.

[0037] Figure 14 It is a band structure of an expanded unit cell (D=D0+∆D,∆D=3mm) of a regular hexagonal double-layer metamaterial plate; the color legend is the out-of-plane polarization factor, and the in-plane coupled modes with smaller polarization factors do not affect the bandgap characteristics of out-of-plane bending waves;

[0038] Figure 15 It represents the out-of-plane displacement distribution of intrinsic modes at the upper and lower boundary points of the topological bandgap in contracting and expanding unit cells. Figure 15 (ab) represents the two dipole modes at the upper boundary of the topological bandgap of the contracted unit cell. Figure 15 (cd) represents two quadrupole modes at the lower boundary of the topological bandgap of the contracted unit cell; Figure 15(ef) represents two dipole modes at the upper boundary of the topological bandgap of the extended unit cell. Figure 15 (gh) represents the quadrupole mode at the lower boundary of the topological bandgap of the expanded unit cell; mode reversal occurs at the upper and lower boundaries of the topological bandgap of the contracted and expanded unit cells.

[0039] Figure 16 (a) is a schematic diagram of a rhombic lattice composed of perfectly contracted unit cells. Figure 16 (b) is the characteristic spectrum of a rhombic lattice composed of perfectly contracted unit cells;

[0040] Figure 17 (ad) represents the out-of-plane displacement field distribution corresponding to the bulk, boundary, acute angle, and obtuse angle states of a rhombic lattice composed of perfectly contracted unit cells at intrinsic frequencies of 118Hz, 153Hz, 143Hz, and 145Hz. At the corresponding modal excitation frequencies, the energy is concentrated in the body, edge, acute angle, and obtuse angle of the structure, respectively, thus achieving precise control of elastic waves.

[0041] Figure 18 (a) is a schematic diagram of a rhombic lattice composed of shrinking unit cells after the introduction of defects. Figure 18 (b) is the characteristic spectrum of the rhombic lattice composed of shrinking unit cells after the introduction of defects;

[0042] Figure 19 (ab) shows the out-of-plane displacement field distribution of the acute and obtuse angle states of the rhombic lattice formed by the shrinking unit cell after the introduction of defects. After the introduction of defects, the elastic wave in the antisymmetric angle state is still localized at the corner point, which verifies the topological protection characteristics of the topological angle state.

[0043] Figure 20 (a) is a schematic diagram of a rhombic lattice composed of perfectly expanded unit cells. Figure 20 (b) is the characteristic spectrum of a rhombic lattice composed of perfectly expanded unit cells;

[0044] Figure 21 (ac) is the out-of-plane displacement field distribution of the edge states, obtuse angle states, and acute angle states of a rhombic lattice composed of perfectly expanded unit cells;

[0045] Figure 22 This is a schematic diagram of a square finite element structure under central excitation;

[0046] Figure 23 It is the transmission spectrum calculated numerically, and multiple transmission peaks corresponding to the topological states appear in the bandgap frequency range;

[0047] Figure 24 (ac) is the simulated displacement field distribution of the topological acute angle state, topological obtuse angle state, and edge state under central excitation;

[0048] Figure 25 It is the experimental transmission spectrum of the body, obtuse angle, acute angle and edge obtained from the vibration test of the square lattice structure. Detailed Implementation

[0049] The embodiments of the present invention will be described in detail below with reference to the accompanying drawings. The specific embodiments described herein are merely illustrative of the present invention and are not intended to limit the present invention.

[0050] Example 1

[0051] The present invention proposes a double-layer sandwich topological metamaterial plate, wherein the double-layer sandwich topological metamaterial plate comprises unit cells, each unit cell consisting of upper and lower thin plates and a compression-torsion coupled resonator sandwiched in the middle, such as... Figure 1 As shown. The thickness of the sheet is T=2mm. The double-layer metamaterial sheet can be manufactured using 3D printing technology in a single piece.

[0052] The aforementioned compression-torsion coupling resonator consists of six chiral inclined rods with a radius of r = 1.5 mm at each of the upper and lower ends, and a cylindrical mass block with a radius of R = 11 mm and a thickness of t = 10 mm in the middle. Under compression / tension, this compression-torsion coupling resonator undergoes both translational and torsional motion, i.e., the compression-torsion coupling effect. This compression-torsion coupling effect originates from the chiral arrangement of the inclined rods. By rotating the upper and lower ends, the vertical rod is transformed into a chiral arrangement of inclined rods, as shown below. Figure 2 As shown. The angle of relative rotation between the upper and lower ends is defined as the compression-torsion angle φ. Adjusting this angle can change the strength of the compression-torsion coupling effect. The stronger the compression-torsion coupling effect, the stronger the torsion of the compression-torsion coupled resonator under the same external force.

[0053] The bilayer metamaterial plate can be accessed along the lattice vector. It is constructed by periodically repeating rhomboid unit cells. The rhomboid unit cells are as follows: Figure 3 As shown, the lattice constant a = 60 mm, and one unit cell contains two sandwiched compression-torsion coupled resonators. The rhombic thin plate contains a 60° angle. The centers of the two compression-torsion coupled resonators are located on the long diagonal of the rhombus and are symmetrically distributed, satisfying the C-value of the unit cell. 3v Symmetry. The distance between their centers is 1 / 3 of the length of the long diagonal.

[0054] To investigate the band structure and transport characteristics of the designed structure, this invention utilized the finite element analysis software COMSOL Multiphysics to conduct numerical calculations to study the band structure and transport characteristics of the metamaterial plate. The entire structure was constructed using TPU material, which has a Young's modulus E = 96 MPa and a density ρ = 1160 kg / m³. 3Poisson's ratio v = 0.48. The side boundaries of the upper and lower thin plates of the rhombic unit cell are set as periodic boundary conditions for band structure calculations. This invention mainly focuses on out-of-plane bending waves, therefore an out-of-plane polarization factor is defined. ,in u , v , w They are x , y , z Displacement components in the direction, V all This represents the volume of a unit cell. When the value is >0.9, the energy band is identified as being dominated by out-of-plane bending waves.

[0055] When the cylindrical mass blocks of the two compression-torsion coupled resonators have the same radius (R1=R2=11mm), the initial unit cell is rhomboid. The structure's C... 3v Symmetry ensures that a Dirac point appears at the boundary of the first Brillouin zone. Furthermore, due to the local resonance effect of the sandwich-type compression-torsion coupled resonator, a local resonant bandgap is opened in the 227-248Hz range, such as... Figure 4 As shown. Figure 4 The illustration shows the first irreducible Brillouin zone in the rhombic unit cell reciprocal lattice space, with M-G-K-G as a high-symmetry point. The band structure is calculated by solving the characteristic frequency equation and scanning the wave vector k along the path M-G-K-G in the first irreducible Brillouin zone of the rhombic unit cell reciprocal lattice space.

[0056] By designing different cylindrical mass block radii for two pressure-torsion coupled resonators, the quantum valley Hall effect was simulated, resulting in two unit cells, I and II, with different topological properties. In unit cell I, the radius of the cylindrical mass block of the first pressure-torsion coupled resonator is smaller than that of the second pressure-torsion coupled resonator (R1=11mm, R2=13mm). In unit cell II, the radius of the cylindrical mass block of the first pressure-torsion coupled resonator is larger than that of the second pressure-torsion coupled resonator (R1=13mm, R2=11mm). These two are mirror-symmetric inverse structures. The spatial inversion symmetry of the structure is broken, introducing valley degrees of freedom. This allows the opening of a topological bandgap (128-146Hz) at the degenerate Dirac point in the initial unit cell and broadens the local resonant bandgap, as shown in the example. Figure 5 As shown. The two rhombic unit cells have identical band structures, but the modes at the upper and lower boundary points S2 and S1 of the topological band gap are opposite. The upper boundary mode S2 of unit cell I is similar to the lower boundary mode S1 of unit cell II, and the lower boundary mode S1 of unit cell I is similar to the upper boundary mode S2 of unit cell II, indicating band inversion. Figure 6As shown, unit cells I and II have opposite valley Chern numbers and different topological properties. Therefore, a topological interface state occurs at the interface composed of these two rhombic unit cells. In the topological interface state, elastic waves are highly localized at the interface and decay rapidly along both sides of the interface, realizing the localization of vibrational energy at the interface. This characteristic provides robust topological protection, making it immune to structural defects and disorder.

[0057] To verify the existence of the topological interface states, seven unit cells II and seven unit cells I were placed on the left and right sides of the interface to form a strip-shaped supercell, and the state was verified by scanning k. x The wave vector of the direction was used to calculate the band structure of the strip-shaped supercell, such as... Figure 7 As shown, a dispersion curve corresponding to the topological interface states appears across the entire bandgap of the strip-shaped supercell. Figure 8 The wave vector k is given. x =π, the eigenmodes of the topological interface state. It can be seen that in the topological interface state, the energy is highly localized at the interface and decays rapidly in directions away from the interface, possessing the ability to localize vibrational energy at the interface. Simultaneously, this characteristic exhibits topological protection robustness, unaffected by defects and disorder.

[0058] Furthermore, this invention tested the transmission characteristics of a finite element structure composed of 7 rhombic unit cells I and 7 rhombic unit cells II through numerical simulation. A schematic diagram of the finite element structure and its transmission spectrum are shown below. Figure 9 As shown. A vibration point excitation is applied at the lower base plate at the left end of the structure, and the vibration response is detected at the upper top plate at the right end of the structure. The transmittance during the transmission process is defined as... , where d out and d in The average displacement is located at the output and excitation ends. The transmission spectrum shows that the transmittance is negative within the frequency range corresponding to the bandgap, indicating that vibrations within the bandgap frequency are well suppressed, verifying the correctness of the bandgap conclusion derived from the band structure. Notably, a transmittance peak corresponding to the topological interface state (marked with a star, 132 Hz) also appears within the bandgap range. Figure 10 Out-of-plane displacement distribution contour maps of the finite element structure at the frequencies corresponding to the topological interface states are shown. Vibrational energy is highly concentrated on both sides of the interface, with the maximum energy concentrated in the nearest rhombic unit cells on both sides of the interface, demonstrating a strong energy localization capability. Quantitatively, the vibrational energy in the eight rhombic unit cells located on the left and right sides of the interface reaches 93.4% of the total energy.

[0059] Example 2

[0060] This invention also provides a regular hexagonal unit cell, thereby achieving lower-dimensional zero-dimensional topological angular states, such as... Figure 11As shown, the lattice constant of a regular hexagonal unit cell is a = 60√3 mm. A regular hexagonal unit cell has 6 compression-torsion coupled resonators. The centers of the 6 resonators are located on the perpendicular bisector of each side of the hexagon, at a distance D from the center of the unit cell, and are evenly distributed around the center of the unit cell, forming a rotationally symmetric structure that satisfies the C... 6v symmetry.

[0061] Based on geometric relationships, the distance between the center of the pressure-torsion coupled resonator and the center of the unit cell in the initial regular hexagonal unit cell is D0 = a / 3. This invention simulates the quantum spin Hall effect based on the concept of a breathing lattice. By changing the distance D = D0 + ∆D between the center of the pressure-torsion coupled resonator and the center of the unit cell, two types of unit cells are constructed: a contracted (∆D = -3 mm) cell and an expanded (∆D = 3 mm) cell.

[0062] The band structure of the initial regular hexagonal unit cell is as follows Figure 12 As shown, due to the band folding mechanism, the single Dirac point appearing at the Brillouin zone boundary K in the rhombic unit cell is folded to the Brillouin zone center Γ to form a double Dirac point. By changing the parameter D, the translational symmetry of the structure is broken, which can alter the band structure, open the quadruple degenerate double Dirac point, and form a topological band gap.

[0063] Figures 13-14 The band structures of contracted and expanded unit cells are given respectively. These two types of regular hexagonal unit cells have similar band structures, but the modes are reversed at the upper and lower band gap boundary points. For example... Figure 15 As shown, the upper and lower boundaries of the band gap of the contracted unit cell correspond to two quadrupole modes, respectively. Figure 15 (cd)) and two dipole modes ( Figure 15 (ab)), while the upper and lower boundaries of the band gap of the expanded unit cell correspond to two dipole modes respectively. Figure 15 (gh)) and quadrupole mode ( Figure 15 (ef) Band inversion implies that the two types of unit cells have different topological properties. Calculations of topological invariants and topological angular charges show that the contracting unit cell is a topologically trivial structure, while the expanding unit cell is a topologically nontrivial structure. A finite element structure composed of either a contracting or expanding unit cell can realize boundary states and angular states within the bulk band gap. Topological theory indicates that topological states must exist at the domain walls composed of unit cells with both topological properties; however, such a composite lattice structure is too complex. Realizing topological states in a single lattice composed of only one type of unit cell has a wider range of applications and higher application value.

[0064] This invention constructs a rhombic lattice composed of 25 perfectly contracted unit cells, such as... Figure 16As shown in (a), this rhomboid configuration contains both a 60° acute angle and a 120° obtuse angle, facilitating the testing of the relationship between angular states and structural configuration angles. The characteristic spectrum of this rhomboid lattice is calculated through numerical simulation, as shown... Figure 16 As shown in (b), multiple characteristic frequency points corresponding to edge states and corner states appear within the topological bandgap. Figure 17 (ad) shows the out-of-plane displacement field distributions of the bulk, boundary, acute-angle, and obtuse-angle states of a rhombic lattice composed of perfectly contracted unit cells at intrinsic frequencies of 118 Hz, 153 Hz, 143 Hz, and 145 Hz. When the bulk state is excited, energy can propagate throughout the entire structure, while the energy in the boundary state is concentrated at the edge of the structure. The 60° acute angle and 120° obtuse angle support four angular states, which can be divided into two groups according to the symmetry about the diagonal: two symmetrical angular states (dashed boxes) and two anti-symmetrical angular states (solid boxes). In the angular states, energy is highly localized at the angle, and the vibrational energy within the structure is almost zero. Precise control of elastic waves can be achieved by exciting these different modes.

[0065] To test the topological protection properties of angular states against defects and to distinguish between topological angular states and trivial angular states, this invention introduces defects into a rhombic lattice composed of perfectly contracted unit cells, such as... Figure 18 As shown in (a), the defect is achieved by increasing the radius of all cylindrical mass blocks in a unit cell near the four corners of the rhombus. The corresponding characteristic spectrum obtained from numerical simulation is shown below. Figure 18 As shown in (b). After introducing the defect, the acute and obtuse angles of the rhomboid configuration still support four angular states, and the out-of-plane displacement field distribution of the acute and obtuse angular states is as follows. Figure 19 As shown in (ab), the two symmetrical acute and obtuse angle states are affected by the defect, with energy leaking towards the designed defect (dashed box). In contrast, the two anti-symmetrical acute and obtuse angle states are significantly unaffected by the defect removal (solid box), with energy still strongly localized at the corners, exhibiting robust topology protection. Therefore, this invention can realize topology-protected angle states at both acute and obtuse angles, making the realization of topology angle states unaffected by structural configuration angles, thus enriching the design strategies for topology structures.

[0066] This invention also tested the topological properties of a rhombic lattice composed of 25 topologically nontrivial extended unit cells, such as... Figure 20 As shown, characteristic frequency points corresponding to edge states and angular states appear within the topological bandgap. Specifically, two topologically protected antisymmetric angular states are supported at both acute and obtuse angles, and no trivial angular states appear. The out-of-plane displacement field distributions of the edge states, obtuse angular states, and acute angular states are shown in the figures below. Figure 21 As shown in (ac).

[0067] In practical engineering applications, plate structures are generally square; therefore, this invention also provides a scheme for applying the proposed topological double-layer sandwich metamaterial plate in practice. For example... Figure 22 As shown, a vibrational excitation was applied to the lower base plate at the center of a square structure composed of 39 perfectly contracted unit cells, and the vibrational response was extracted at the upper top plate at the lower left corner of the structure, thus simulating the structural transmission response under actual excitation. The numerically calculated transmission spectrum is shown below. Figure 23 As shown. Within the frequency range corresponding to the topological bandgap, the transmission transmittance is negative, meaning that vibrations within the bandgap can be attenuated, and multiple transmission peaks corresponding to the topological states appear. The simulated displacement field distributions of the topological acute-angled state, topological obtuse-angled state, and edge state at their respective frequencies are shown in the figure. Figure 24 As shown in (ac), the structural displacement field modes under actual excitation show good agreement with the characteristic modes of numerical simulation, verifying the existence of edge and corner states and demonstrating the feasibility of exciting edge and corner states using actual excitation. In addition to excitation at the center, vibration excitation at the corners and edges of the structure can also excite edge and corner states.

[0068] This invention verifies the edge and topological corner states of the proposed topological double-layer sandwich metamaterial plate through vibration transmission experiments. A square sample of the topological double-layer sandwich metamaterial plate, composed of perfectly shrinking unit cells, was obtained through 3D printing. The 3D printing material was TPU, consistent with the numerical simulation. Vibration excitation was applied to the sample via a vibrator (HEV-200), controlled by a function generator (AWA1651) and a power amplifier (HEA-200C). During the vibration experiment, the sample was suspended from a fixed support using an elastic rope to simulate a free boundary. An accelerometer (LC0101E) was located on the lower left base plate of the structure to detect the excitation. Four accelerometers were placed within the upper plate, at obtuse angles (120°), acute angles (60°), and at the edges, respectively, to detect the response signals. Finally, all acceleration data were acquired by a signal acquisition module (Brüel & Kjaer Pulse) and transmitted to a computer for further data processing. Figure 25 The experimental transmission spectra are shown for the interior, obtuse angle, acute angle, and edge of the structure. Within the topological bandgap, the transmittance within the structure is very low, indicating that the propagation of elastic waves is effectively blocked. The transmission spectra at obtuse angles, acute angles, and edges show higher transmittance, verifying the existence of angular and edge states, thus simultaneously achieving vibration attenuation within the structure and energy localization at the edges and corners.

[0069] The above description, in conjunction with specific preferred embodiments, provides a further detailed explanation of the present invention. It should not be construed that the specific implementation of the present invention is limited to these descriptions. For those skilled in the art, various simple deductions and substitutions can be made without departing from the inventive concept, and all such modifications and substitutions should be considered within the scope of protection of the present invention.

Claims

1. A double-layer sandwich topological metamaterial plate, characterized in that, The device includes a unit cell, which comprises an upper thin plate, a lower thin plate, and a pressure-torsion coupling resonator. The upper and lower thin plates have the same shape, and the pressure-torsion coupling resonator is disposed between the upper and lower thin plates. The pressure-torsion coupling resonator includes inclined rods and a cylindrical mass block. Six inclined rods with a radius of r are disposed above and below the cylindrical mass block. The inclined rods are arranged in a chiral structure around the axis of the cylindrical mass block, fixing the cylindrical mass block to the upper and lower thin plates.

2. The double-layer sandwich metamaterial plate according to claim 1, characterized in that, The double-layer sandwich topological metamaterial board is manufactured using 3D printing technology in an integrated manner, and the material used to manufacture the double-layer sandwich topological metamaterial board is TPU material.

3. The double-layer sandwich metamaterial plate according to claim 1, characterized in that, The upper and lower thin plates of the unit cell are both rhombic thin plates; the unit cell includes two compression-torsion coupled resonators, the centers of which are located on the long diagonal of the rhombic thin plate and are symmetrically distributed, and the distance between the centers of the two compression-torsion coupled resonators is 1 / 3 of the length of the long diagonal of the rhombic thin plate.

4. The double-layer sandwich metamaterial plate according to claim 3, characterized in that, The rhomboid sheet contains a 60° angle.

5. The double-layer sandwich metamaterial plate according to claim 3, characterized in that, The cylindrical mass blocks of the two compression-torsion coupled resonators in the unit cell have equal radii.

6. The double-layer sandwich metamaterial plate according to claim 3, characterized in that, The double-layer sandwich topological metamaterial plate includes two types of unit cells with different topological properties, namely unit cell I and unit cell II. In unit cell I, the radius of the cylindrical mass block of the first compression-torsion coupled resonator is smaller than the radius of the cylindrical mass block of the second compression-torsion coupled resonator. In unit cell II, the radius of the cylindrical mass block of the first compression-torsion coupled resonator is larger than the radius of the cylindrical mass block of the second compression-torsion coupled resonator. The two are mirror-symmetric inverse structures.

7. The double-layer sandwich metamaterial plate according to claim 6, characterized in that, The double-layer sandwich topological metamaterial plate includes a finite element structure composed of multiple unit cell I and unit cell II; the finite element structure includes an interface, with unit cell I and unit cell II located on both sides of the interface; all unit cells in the finite element structure are arranged in a parallelogram along a straight line in the same plane, and the rhomboid sides of adjacent unit cells are closely connected.

8. The double-layer sandwich metamaterial plate according to claim 1, characterized in that, The upper and lower thin plates of the unit cell are both regular hexagonal thin plates. The unit cell includes six uniformly distributed pressure-torsion coupled resonators. The centers of the six pressure-torsion coupled resonators are located on the perpendicular bisector of each side of the regular hexagonal thin plate, and the distance D between the centers of the six pressure-torsion coupled resonators and the center of the unit cell is equal.

9. The double-layer sandwich metamaterial plate according to claim 8, characterized in that, A unit cell whose distance between the center of the pressure-torsion coupled resonator and the center of the unit cell is less than one-third of the side length of the regular hexagonal thin plate is a shrinking unit cell; a unit cell whose distance between the center of the pressure-torsion coupled resonator and the center of the unit cell is greater than one-third of the side length of the regular hexagonal thin plate is an expanding unit cell; shrinking unit cells and expanding unit cells have different topological properties; the double-layer sandwich topological metamaterial plate includes at least one of shrinking unit cells and expanding unit cells.

10. The application of the double-layer sandwich topological metamaterial plate according to any one of claims 1 to 9 in vibration energy harvesting or low-frequency vibration reduction.

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