Topological metamaterial for sheet bending wave isolation and design method thereof

Abstract

The invention discloses a topological metamaterial for sheet bending wave isolation and a design method of the topological metamaterial. The topological metamaterial is composed of basic unit cells (1) and sheets (2). Each basic unit cell (1) comprises a regular hexagon area (1-1) and a regular triangle stand column (1-2) which are arranged on a thin plate. A topological common unit cell (3) is obtained by clockwise rotating for 0-30 degrees along the plane normal of the thin plate, and a topological non-common unit cell (4) is obtained by anticlockwise rotating for 0-30 degrees along the plane normal of the thin plate; the topological common unit cells (3) and the topological non-common unit cells (4) are alternately arranged on the thin plate (2) to form a topological metamaterial structure; the topological metamaterials arranged alternately can generate total reflection on incident thin plate bending waves, so that effective isolation of the thin plate bending waves is realized. The system has the advantages of simple design, remarkable vibration isolation effect, high expandability and the like, and is suitable for the fields of vibration prevention of building structures, vibration isolation of precise instruments, interference resistance of aerospace vehicle structures and the like.

Description

Technical Field

[0001] This invention relates to a topological metamaterial design method for flexural wave isolation of thin plates, belonging to the field of vibration control technology. Background Technology

[0002] Thin plates, as common structural components, are widely used in aerospace, rail transportation, shipbuilding, and precision equipment due to their lightweight and high strength. However, due to their small thickness and insufficient structural stiffness, thin plates are prone to vibration and noise under external excitation, which not only affects their performance but may also lead to structural fatigue damage or even failure. Therefore, how to effectively suppress the vibration of thin plates has always been an important research topic in the engineering field.

[0003] To address the aforementioned problems, existing technologies have proposed various vibration control methods, including passive control, active control, and topological metamaterial methods. Passive control methods dissipate vibration energy by adding damping materials, constraint layers, or local vibration absorbers to thin plates. While structurally simple, these methods increase system weight and have limited vibration suppression effects over a wide frequency range. Active control methods achieve vibration suppression through sensor detection and actuator feedback, but these methods suffer from high system complexity, high energy consumption, and insufficient stability in high-frequency scenarios. Metamaterial methods utilize the topological properties of artificial periodic structures to create topological band gaps within a specific frequency range, thereby preventing the propagation of elastic waves in thin plates to achieve vibration control. These methods exhibit good robustness and directional selectivity and have gradually become a research hotspot in recent years.

[0004] However, existing thin-plate bending vibration control technologies based on metamaterials still have many shortcomings. First, while additional damping materials can dissipate energy over a wide frequency range, their suppression effect on low-frequency bending waves is weak, and they often come at the cost of increased system mass and volume, making it difficult to meet lightweight design requirements. Second, dynamic vibration absorbers are only effective near specific tuning frequencies, with an excessively narrow operating bandwidth, unable to handle complex wide-frequency excitations. Furthermore, the bandgap characteristics of traditional periodic structures based on the Bragg scattering principle are heavily dependent on structural dimensions; obtaining a low-frequency bandgap requires structural unit sizes to be comparable to the wavelength, which is difficult to achieve in many space-constrained practical applications. In addition, the aforementioned traditional methods generally suffer from a fundamental flaw: they primarily focus on attenuating waves already inside the structure, rather than physically blocking their propagation. Therefore, existing technologies lack an effective solution for achieving efficient, wide-frequency, and reliable isolation of low-frequency bending waves within a compact space. Summary of the Invention

[0005] Technical Problem: To address the technical problems existing in related technologies, the present invention aims to propose a topological metamaterial for isolating bending waves in thin plates and its design method. The metamaterial, formed by alternating topologically trivial unit cells and topologically non-trivial unit cells, achieves effective isolation of low-frequency bending waves at the subwavelength scale, possesses high robustness and lightweight characteristics, and can physically achieve complete reflection and isolation of waves, providing a new approach for vibration control of thin plates.

[0006] Technical Solution: The present invention provides a topological metamaterial for thin-plate bending wave isolation, comprising a basic unit cell and a thin plate. The basic unit cell includes a regular hexagonal region and equilateral triangular pillars on the thin plate. Topologically trivial unit cells are obtained by rotating the plate clockwise by 0-30° along the normal to the thin plate plane, and topologically non-trivial unit cells are obtained by rotating the plate counterclockwise by 0-30° along the normal to the thin plate plane. The rotation angle of the equilateral triangular pillars in each row of topologically trivial and topologically non-trivial unit cells that make up the metamaterial is the same. The topologically trivial unit cells are arranged in a row along the transverse direction of the thin plate, and the topologically non-trivial unit cells are also arranged in another row along the transverse direction of the thin plate. In the longitudinal direction of the thin plate, the topologically trivial and topologically non-trivial unit cells are alternately arranged on the thin plate to form a topological metamaterial structure.

[0007] The equilateral triangular column is fixed to the regular hexagonal region of the thin plate by welding or bonding, ensuring that the rotation angle remains unchanged and the structure is stable.

[0008] The thin plate and the equilateral triangular column are made of metal, fiber composite material or polymer material. Different materials are selected as needed to meet the flexural wave isolation requirements in different frequency ranges.

[0009] The design method of the topological metamaterial for thin-plate bending wave isolation of the present invention includes the following steps:

[0010] Step 1: Based on the center frequency of the thin plate vibration, design the metamaterial fundamental unit cell such that the center frequency of the thin plate vibration is equal to the frequency of the Dirac cone of the fundamental unit cell. Determine the geometric dimensions of the equilateral triangular prism;

[0011] Step 2: Rotate the equilateral triangular prism of the basic unit cell clockwise and counterclockwise by 0-30° respectively along the normal of the thin plate plane to open the degeneracy point in the band structure of the Dirac cone basic unit cell and obtain the band gap, and make the band gap include the vibration frequency range of the thin plate. This determines the rotation angle of the equilateral triangular prism and obtains the topologically trivial unit cell and the topologically non-trivial unit cell.

[0012] Step 3: On a thin plate, topologically trivial unit cells are arranged in a row along the transverse direction of the plate, and topologically non-trivial unit cells are also arranged in another row along the transverse direction of the plate. Vertically, topologically trivial and non-trivial unit cells are alternately arranged on the thin plate to form a topological metamaterial structure; the number of alternating rows of the topological metamaterial is greater than 2; determine the bending wave refraction angle θ. t ;

[0013] Step 4: Based on the structural parameters of the thin plate and the actual working conditions, optimize the geometric dimensions, material properties, and number of alternating rows of the metamaterial to improve the isolation performance of the thin plate for bending wave vibration.

[0014] The frequency of the Dirac cone is given by the following formula:

[0015]

[0016] in, The equivalent bending stiffness of a metamaterial unit cell. The lattice constant of the fundamental unit cell of the metamaterial. and Given the density and thickness of the thin plate, the side length of the equilateral triangular prism is... The height is ; , These are the elastic modulus and Poisson's ratio of the thin plate, respectively.

[0017] Based on the wavenumber of the bending wave of an alternating topological metamaterial Determine the angle of refraction of the curved wave :

[0018]

[0019]

[0020] In the formula, The wave number of the bending wave in the metamaterial; and These are the angle of refraction of the curved wave and the angle of incidence, respectively.

[0021] The wave number of the bending wave in the thin plate. It refers to the bending stiffness of the thin plate. ;

[0022] The optimization of the metamaterial's geometry is achieved by adjusting the height h1 and side length L of the equilateral triangular prism, so that the wave number of the bending wave in the topological metamaterial is less than that of the bending wave in the thin plate. The bending wave undergoes total reflection at the topological interface, thereby achieving isolation of the bending wave and control of the thin plate vibration.

[0023] The material properties and the number of alternating rows are determined based on the thin plate bending wave transmittance requirement, i.e., transmitted energy / incident energy ≤ 0.02.

[0024] Beneficial effects: Compared with the prior art, the present invention has the following beneficial effects:

[0025] 1. The present invention proposes a design method for topological metamaterials for thin-plate bending wave isolation, which can achieve total reflection of incident thin-plate bending waves through physical mechanisms, thereby forming an extremely efficient wave isolation effect within the target frequency band. Its isolation efficiency and bandwidth far exceed those of traditional damping vibration reduction or Bragg scattering type vibration isolation structures.

[0026] 2. The present invention proposes a topological metamaterial design method for thin-plate bending wave isolation, which can change the equivalent bending stiffness of the metamaterial unit cell so that the refraction angle of the bending wave has no real solution at different angles, thereby realizing wide-angle total reflection of bending waves in a wide frequency band.

[0027] 3. The present invention proposes a design method for topological metamaterials for thin-plate bending wave isolation. By adjusting the structural parameters of the metamaterial unit (such as the shape, size, and distribution of holes or protrusions), the topological phase transition point and isolation frequency band can be flexibly controlled to meet the vibration control requirements of different engineering scenarios.

[0028] 4. The present invention proposes a design method for topological metamaterials for thin-plate bending wave isolation, which can be directly fabricated on the basic thin-plate structure (such as cutting, milling, 3D printing, etc.) without the need for an excessively heavy mass block or a complex external control system. It is easy to integrate with existing equipment and conforms to the trend of lightweight and integrated design. Attached Figure Description

[0029] Figure 1 This is a schematic diagram of the metamaterial structure of the present invention.

[0030] Figure 2 This is a schematic diagram of a four-layer alternating topological metamaterial structure according to an embodiment of the present invention.

[0031] Figure 3 The band structure of a topological metamaterial unit cell is shown in this embodiment of the invention.

[0032] Figure 4 This is a bending displacement field diagram of a four-layer alternating topological metamaterial plate capable of achieving bending wave isolation, according to an embodiment of the present invention.

[0033] Figure 5 The displacement frequency response curve of a four-layer alternating topological metamaterial plate capable of flexural wave isolation is shown in an embodiment of the present invention.

[0034] Figure 6The mean reflectivity of a 4-layer alternating topological metamaterial plate capable of flexural wave isolation is given in this embodiment of the invention.

[0035] Figure 7 The average transmittance of flexural waves for a four-layer alternating topological metamaterial plate capable of flexural wave isolation according to an embodiment of the present invention.

[0036] The diagram contains: basic unit cell 1, regular hexagonal region 1-1 on a thin plate, equilateral triangular prism 1-2, thin plate 2, topologically trivial unit cell 3, and topologically non-trivial unit cell 4. Detailed Implementation

[0037] The present invention will now be described in more detail with reference to the accompanying drawings, which illustrate preferred embodiments of the invention. It should be understood that those skilled in the art can modify the invention described herein while still achieving its advantageous effects. Therefore, the following description should be understood as being of general knowledge to those skilled in the art and is not intended to limit the invention.

[0038] The following description, in conjunction with the accompanying drawings, further illustrates an embodiment of the present invention of a topological metamaterial for thin-plate bending wave isolation and its design method.

[0039] like Figure 1As shown in the figure, this embodiment presents a four-layer alternating topological metamaterial structure. The topological metamaterial unit consists of a substrate thin plate and periodically arranged equilateral triangular column local resonant units. Different topological phase unit cells are alternately arranged on the thin plate along the wave propagation direction. The alternation method and number of layers can be determined according to different vibration isolation environments. In the region of the thin plate where bending waves need to be isolated, the above-mentioned alternating topological metamaterial array is introduced along the wave propagation path to form a blocking region. The two sides of the blocking region are ordinary thin plate structures, thereby achieving the isolation effect of total reflection. The topological metamaterial for thin-plate bending wave isolation consists of a basic unit cell 1 and a thin plate 2. The basic unit cell 1 includes a regular hexagonal region 1-1 and equilateral triangular pillars 1-2 on the thin plate. Topologically trivial unit cells 3 are obtained by rotating them clockwise by 0-30° along the normal to the plane of the thin plate, and topologically non-trivial unit cells 4 are obtained by rotating them counterclockwise by 0-30° along the normal to the plane of the thin plate. The rotation angle of the equilateral triangular pillars in each row of topologically trivial unit cells 3 and topologically non-trivial unit cells 4 that make up the metamaterial is the same. The topologically trivial unit cells 3 are arranged in a row along the transverse direction of the thin plate 2, and the topologically non-trivial unit cells 4 are also arranged in another row along the transverse direction of the thin plate 2. In the longitudinal direction of the thin plate 2, the topologically trivial unit cells 3 and topologically non-trivial unit cells 4 are arranged alternately on the thin plate 2 to form a topological metamaterial structure. The equilateral triangular pillars 1-2 are connected to the regular hexagonal region 1-1 of the thin plate to ensure that the rotation angle remains unchanged and the structure is stable. The equilateral triangular pillars 1-2 are fixed to the regular hexagonal region 1-1 on the thin plate by welding or bonding, ensuring that the rotation angle remains unchanged and the structure is stable. The thin plate 2 and the equilateral triangular pillars 1-2 are made of metal, fiber composite materials or polymer materials, and different materials are selected as needed to meet the bending wave isolation requirements in different frequency ranges.

[0040] Achieving total internal reflection using alternating topological metamaterials involves the following design steps:

[0041] Step 1: Based on the vibration frequency and unit cell structure of the thin plate, establish a periodic unit cell model in the finite element software and calculate the dispersion relation of the bending wave.

[0042] Step 2: Using the band reversal principle: By adjusting the element geometry parameters, the modal order of the upper and lower boundaries of the band gap is reversed; determine the topological non-trivial unit cell and the topological trivial unit cell.

[0043] Step 3: By alternating the arrangement, a topological interface is formed, where the curved wave cannot propagate and undergoes total reflection.

[0044] Step 4: Adjust the number of unit cells, array thickness, and resonant parameters to expand the isolation bandwidth.

[0045] Specifically, the steps include the following:

[0046] Step 1: Based on the center frequency of the thin plate vibration, design the metamaterial fundamental unit cell 1 such that the center frequency of the thin plate vibration is equal to the frequency of the Dirac cone of the fundamental unit cell 1. Determine the geometric dimensions of the equilateral triangular prism 1-2;

[0047] Step 2: Rotate the equilateral triangular prism of the basic unit cell clockwise and counterclockwise along the plane normal of the thin plate by 0-30° respectively, open the degeneracy point in the band structure of the Dirac cone basic unit cell 1 to obtain the band gap, and make the band gap include the vibration frequency range of the thin plate 2. Thus, determine the rotation angle of the equilateral triangular prism 1-2 to obtain the topologically trivial unit cell 3 and the topologically non-trivial unit cell 4.

[0048] Step 3: On thin plate 2, topologically trivial unit cells 3 are arranged in a row along the transverse direction of thin plate 2, and topologically nontrivial unit cells 4 are also arranged in another row along the transverse direction of thin plate 2. The topologically trivial unit cells 3 and topologically nontrivial unit cells 4 are arranged alternately along the longitudinal direction on thin plate 2, forming a topological metamaterial structure; the number of alternating rows of the topological metamaterial is greater than 2; determine the bending wave refraction angle θ. t ;

[0049] Step 4: Based on the structural parameters of the thin plate and the actual working conditions, optimize the geometric dimensions, material properties, and number of alternating rows of the metamaterial to improve the isolation performance of the thin plate for bending wave vibration.

[0050] The frequency of the Dirac cone is given by the following formula:

[0051]

[0052] in, The equivalent bending stiffness of a metamaterial unit cell. The lattice constant of the fundamental unit cell of the metamaterial. and Given the density and thickness of thin plate 2, and the side length of equilateral triangular prism 1-2, respectively. The height is ; , These are the elastic modulus and Poisson's ratio of the thin plate, respectively.

[0053] Based on the wavenumber of the bending wave of an alternating topological metamaterial Determine the angle of refraction of the curved wave :

[0054]

[0055]

[0056] In the formula, The wave number of the bending wave in the metamaterial; and These are the angle of refraction of the curved wave and the angle of incidence, respectively.

[0057] The wave number of the bending wave in the thin plate. It refers to the bending stiffness of the thin plate. ;

[0058] The optimization of the metamaterial's geometry is achieved by adjusting the height h1 and side length L of the equilateral triangular prism 1-2, so that the wave number of the bending wave in the topological metamaterial is less than that of the bending wave in the thin plate. The bending wave undergoes total reflection at the topological interface, thereby achieving the isolation of the bending wave and the control of the thin plate vibration.

[0059] The material properties and the number of alternating rows are determined based on the thin plate bending wave transmittance requirement, i.e., transmitted energy / incident energy ≤ 0.02.

[0060] like Figure 2 As shown in the figure, this embodiment provides a schematic diagram of a four-layer alternating topological metamaterial structure, wherein the thin plate is selected as a steel plate with a density of =7850kg / m 3 Elastic modulus E = 210 GPa, Poisson's ratio =0.2, thickness h=2mm, the material of the equilateral triangular column is steel, the side length L=5mm, and the height is... The thickness is 7mm, and the vibration frequency range of the thin plate is 42 kHz.

[0061] like Figure 3 As shown, by rotating the angle of the triangular prism, the topological trivial unit cell bandgap and the topological non-trivial unit cell bandgap can be obtained, and the bandgap range covers the vibration frequency range of 38-48 kHz.

[0062] like Figure 4 As shown, the bending displacement field distribution of a four-layer alternating topological metamaterial plate when a bending wave is incident normally is illustrated, with units in mm. The bending wave is reflected upon reaching the alternating topological metamaterial, and only a small portion of the bending wave is transmitted into the vibration isolation region of the thin plate.

[0063] like Figure 5 As shown, the bending displacement frequency response curves of four alternating layers of topological metamaterial plates are displayed when bending waves of different frequencies are incident. Within the vibration frequency range, the frequency response curves are all less than -20 dB, indicating that the alternating topological metamaterials can effectively isolate bending waves.

[0064] like Figure 6 and Figure 7 As shown, the average transmittance of flexural waves in a four-layer alternating topological metamaterial plate is as follows when flexural waves of different frequencies are incident: The average reflectance (1.0 - transmittance) of the bending wave is only 0.02 and the average reflectance is as high as 0.98 within the vibration frequency range. That is, the four layers of alternating topological metamaterial plates can make bending waves totally reflective, and have excellent bending wave isolation performance.

[0065] Finally, it should be noted that the accompanying drawings are for illustrative purposes only, representing schematic diagrams rather than actual physical objects, and should not be construed as limiting the invention. To better illustrate the embodiments of the invention, some components in the drawings may be omitted, enlarged, or reduced, and do not represent the actual product dimensions. It is understandable to those skilled in the art that some well-known structures and their descriptions may be omitted in the drawings. The examples of this invention are merely descriptions of preferred embodiments and are not intended to limit the concept and scope of the invention. Various modifications and improvements made to the technical solutions of this invention by those skilled in the art without departing from the inventive concept should fall within the protection scope of this invention.

Claims

1. A topological metamaterial for flexural wave isolation in thin plates, characterized in that: The topological metamaterial is composed of a basic unit cell (1) and a thin plate (2). The basic unit cell (1) includes a regular hexagonal region (1-1) and an equilateral triangular prism (1-2) on the thin plate. A topologically trivial unit cell (3) is obtained by rotating it clockwise by 0-30° along the normal of the thin plate plane, and a topologically non-trivial unit cell (4) is obtained by rotating it counterclockwise by 0-30° along the normal of the thin plate plane. The equilateral triangular prisms in each row of topologically trivial unit cells (3) and topologically non-trivial unit cells (4) that make up the metamaterial are rotated at the same angle. The topologically trivial unit cells (3) are arranged in a row along the transverse direction of the thin plate (2), and the topologically non-trivial unit cells (4) are also arranged in another row along the transverse direction of the thin plate (2). In the longitudinal direction of the thin plate (2), the topologically trivial unit cells (3) and the topologically non-trivial unit cells (4) are arranged alternately on the thin plate (2) to form a topological metamaterial structure.

2. The topological metamaterial for thin-plate bending wave isolation according to claim 1, characterized in that: The equilateral triangular column (1-2) is fixed to the regular hexagonal region (1-1) on the thin plate by welding or bonding, ensuring that the rotation angle remains unchanged and the structure is stable.

3. The topological metamaterial design method for thin-plate bending wave isolation according to claim 2, characterized in that: The thin plate (2) and the equilateral triangular column (1-2) are made of metal, fiber composite material or polymer material. Different materials are selected as needed to meet the flexural wave isolation requirements in different frequency ranges.

4. A design method for a topological metamaterial for flexural wave isolation in thin plates as described in claim 1, 2, or 3, characterized in that: Includes the following steps: Step 1: Based on the center frequency of the thin plate vibration, design the basic unit cell (1) of the metamaterial, such that the center frequency of the thin plate vibration is equal to the frequency of the Dirac cone of the basic unit cell (1). Determine the geometric dimensions of the equilateral triangular prism (1-2); Step 2: Rotate the equilateral triangular prism (1-2) of the basic unit cell clockwise and counterclockwise by 0-30° respectively along the normal of the thin plate plane to open the degeneracy point in the band structure of the Dirac cone basic unit cell (1) to obtain the band gap, and make the band gap include the vibration frequency range of the thin plate (2). Thus, determine the rotation angle of the equilateral triangular prism (1-2) to obtain the topologically trivial unit cell (3) and the topologically non-trivial unit cell (4). Step 3: On the thin plate (2), topologically trivial unit cells (3) are arranged in a row along the transverse direction of the thin plate (2), and topologically nontrivial unit cells (4) are also arranged in another row along the transverse direction of the thin plate (2). The longitudinal topologically trivial unit cells (3) and topologically nontrivial unit cells (4) are alternately arranged on the thin plate (2) to form a topological metamaterial structure; the number of alternating rows of the topological metamaterial is greater than 2; determine the bending wave refraction angle θ. t ; Step 4: Based on the structural parameters of the thin plate and the actual working conditions, optimize the geometric dimensions, material properties, and number of alternating rows of the metamaterial to improve the isolation performance of the thin plate for bending wave vibration.

5. The topological metamaterial design method for thin-plate bending wave isolation according to claim 4, characterized in that: The frequency of the Dirac cone is given by the following formula: in, The equivalent bending stiffness of a metamaterial unit cell. The lattice constant of the fundamental unit cell of the metamaterial. and Given the density and thickness of the thin plate (2), the side length of the equilateral triangular prism (1-2) is... The height is ; , These are the elastic modulus and Poisson's ratio of the thin plate, respectively.

6. The topological metamaterial design method for thin-plate bending wave isolation according to claim 5, characterized in that: Based on the wavenumber of the bending wave of an alternating topological metamaterial Determine the angle of refraction of the curved wave : In the formula, The wave number of the bending wave in the metamaterial; and These are the angle of refraction of the curved wave and the angle of incidence, respectively. The wave number of the bending wave in the thin plate. It refers to the bending stiffness of a thin plate. .

7. The topological metamaterial design method for thin-plate bending wave isolation according to claim 6, characterized in that: The optimization of the metamaterial's geometry is achieved by adjusting the height h1 and side length L of the equilateral triangular prism (1-2) so that the wave number of the bending wave in the topological metamaterial is less than that of the bending wave in the thin plate. The bending wave undergoes total reflection at the topological interface, thereby achieving the isolation of the bending wave and the control of the thin plate vibration.

8. The topological metamaterial design method for thin-plate bending wave isolation according to claim 7, characterized in that: The material properties and the number of alternating rows are determined based on the bending wave transmittance requirements of the thin plate, i.e. .

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