Locally resonant metasurface structures and their design methods for flexural wave guidance and focusing

Abstract

The application provides a local resonance super-structure structure for bending wave guiding and gathering and a design method thereof, and the local resonance super-structure structure comprises a substrate and a plurality of first local resonance unit cells and second local resonance unit cells arranged uniformly and collinearly on the substrate, the first local resonance unit cells and the second local resonance unit cells are located in a first region and a second region of the substrate respectively, the first local resonance unit cells and the second local resonance unit cells each comprise a first mass-spring resonator and a second mass-spring resonator, and the first mass-spring resonator and the second mass-spring resonator are arranged in a sequence in reverse. The first mass-spring resonator and the second mass-spring resonator are different in mass but the same in stiffness. When a bending wave of a specific frequency is generated by firing at a set frequency at the edge of the substrate and close to the unit cell position, a very obvious energy gathering effect can be generated at the interface of the two super-structures. The application has the advantages of simple structure, unit cell size far smaller than the wavelength of the incident wave, no limitation on the form of the incident wave, no need for external energy supply, and strong economy and environmental adaptability.

Description

Technical Field

[0001] The present invention relates to elastic fluctuations in engineering structures, and in particular to a localized resonant metasurface structure for guiding and focusing bending waves and a design method thereof. Background Art

[0002] Elastic wave metamaterials possess functions and properties not found in natural materials, providing new avenues for manipulating elastic waves. They typically create band structures through periodic or quasi-periodic arrangements of unit cells, and manipulate the wave properties of elastic waves within the metamaterial using local resonances or Bragg band gaps.

[0003] With the advent of the generalized Snell's law, metasurfaces, as a type of metamaterial with a more compact configuration, have attracted widespread attention. Compact configurations here refer to metamaterial structures whose periodic dimension is smaller than the dimension of the target wave vector, such as layered structures in three-dimensional space or strip structures in two-dimensional space. Most metasurfaces exhibit phase gradients by introducing a gradient in the unit cell, thereby achieving arbitrary modulation of the wavefront, such as anomalous reflection or refraction, phantom sources, and focusing. These metasurfaces are also known as phase-gradient metasurfaces. In addition, other compact periodic structures without phase gradients are also known as metasurfaces. One type is a periodic or gradient structure on a half-space surface, typically composed of a periodic array of holes or slots, which can achieve surface wave manipulation such as rainbow capture, mode conversion, Umklapp transitions, and topological edge states. Another type is a one-dimensional periodic or gradient structure formed in an infinite or semi-infinite plate, typically constructed with an array of slots, which can guide and manipulate flexural waves. Wave manipulation theories and methods based on elastic metamaterials or metasurfaces have promoted developments in fields such as low-frequency vibration isolation, energy capture, and structural health monitoring.

[0004] The ultimate goal of elastic wave manipulation technology based on metamaterials or metasurfaces is to change the energy distribution or energy transmission path in solids. The former includes local energy enhancement or absorption, and the latter includes anomalous refraction, wave mode conversion, etc. If the energy of the bending wave is to be enhanced at a certain location on a free-standing thin plate, the known means are nothing more than constructing two-dimensional metamaterials with point and line defects, two-dimensional high-order topological insulators, and metasurfaces, superlenses, and phononic crystals for focusing flexible waves. However, two-dimensional metamaterials and topological insulators require resonators to be distributed throughout the entire plate surface, which is equivalent to adding a large amount of mass to the original system (free-standing thin plate), while structures that can be used to focus bending waves require that the waves must be incident in the form of plane waves or cylindrical waves. Both of these greatly hinder the path from methods to applications of metamaterials and metasurfaces. Summary of the Invention

[0005] The applicant's research has found that resonators can produce sharp phase jumps near their natural frequencies. Furthermore, within a metasurface structure, only a small number of unit cells generate phase delays in elastic waves through resonance, while the majority of these delays are generated through the movement of the resonators. Furthermore, the localized resonance mechanism can significantly reduce the size of the structure, enabling the manipulation of larger wavelength elastic waves using a compact structure. Based on this research, the present invention proposes a localized resonant metasurface structure for guiding and focusing flexural waves and its design method.

[0006] To achieve the above-mentioned object, the technical solution adopted by the present invention is as follows: a local resonant metasurface structure for guiding and focusing bending waves includes a substrate with four free sides and a first local resonant unit cell and a second local resonant unit cell fixed on the substrate;

[0007] The substrate includes a first region and a second region, a plurality of the first local resonant unit cells and a plurality of the second local resonant unit cells are uniformly arranged in the first region and the second region, respectively, and the first local resonant unit cells and the second local resonant unit cells are collinearly arranged; and a spacing between adjacent first local resonant unit cells and a spacing between adjacent second local resonant unit cells are both defined as a first spacing;

[0008] The first local resonant unit cell includes a first mass-spring resonator and a second mass-spring resonator arranged in sequence; the second local resonant unit cell includes the second mass-spring resonator and the first mass-spring resonator arranged in sequence; the first mass-spring resonator and the first mass-spring resonator have the same stiffness but different masses; the spacing between the first mass-spring resonator and the second mass-spring resonator in the first local resonant unit cell and the second local resonant unit cell is defined as a second spacing;

[0009] The second spacing is equal to the first spacing;

[0010] When a bending wave is generated at a set frequency at the edge of the substrate and near the first local resonant unit cell or the second local resonant unit cell, the bending wave can propagate along the substrate and generate energy concentration at the interface between the first region and the second region.

[0011] Furthermore, the substrate is made of metal material.

[0012] Furthermore, when the local resonant metasurface structure is mounted on an excitation source, the excitation source is located at an edge of the substrate and close to the first local resonant unit cell or the second local resonant unit cell.

[0013] The present invention also provides a method for designing a localized resonant metasurface structure for guiding and focusing bending waves, comprising the following steps:

[0014] Step 1: Obtain the material density ρ, Young's modulus E, Poisson's ratio υ, and thickness h of the substrate, and calculate the bending wave dispersion curve;

[0015] Step 2, according to the set bending wave effective working frequency band and the bending wave dispersion curve obtained in step 1, the distance between the first mass spring resonator and the second mass spring resonator in the first local resonance unit cell is determined, so that the first band edge frequency of the first Brillouin zone of the first local resonance unit cell is located in the set bending wave effective working frequency band.

[0016] Step 3, set the bending wave effective working frequency, and the bending wave effective working frequency is located in the set bending wave effective working frequency band.

[0017] Step 4, a finite element model including the first local resonance unit cell and the substrate corresponding to the first local resonance unit cell is established, a characteristic frequency analysis is performed based on the finite element model, a first local resonance unit cell band structure is obtained, whether the band gap of the first local resonance unit cell band structure corresponds to the bending wave effective working frequency set in step 3 is judged, if yes, the stiffness and mass of the first mass spring resonator and the second mass spring resonator are obtained, if not, the stiffness and mass of the first mass spring resonator and the second mass spring resonator are adjusted until the band gap of the first local resonance unit cell band structure corresponds to the set bending wave effective working frequency.

[0018] Step 5, taking the distance between the first mass spring resonator and the second mass spring resonator in the first local resonance unit cell as the unit cell spacing of the adjacent first local resonance unit cell and the adjacent second local resonance unit cell.

[0019] Step 6, according to the actual bending wave energy gathering interface position requirement, the number of unit cells of the first local resonance unit cell and the second local resonance unit cell is determined.

[0020] Step 7, according to the determined number of unit cells of the first local resonance unit cell and the second local resonance unit cell, the unit cell spacing and the distance between the first mass spring resonator and the second mass spring resonator, the length of the substrate is determined.

[0021] Further, in step 6, the number of unit cells of the first local resonance unit cell and the second local resonance unit cell is not less than 5.

[0022] Further, in step 1, the inherent loss is ignored when calculating the bending wave dispersion curve.

[0023] Further, in step 4, the spring mass in the first mass spring resonator and the second mass spring resonator is ignored.

[0024] The present application has the following beneficial effects:

[0025] 1. The present invention discloses a locally resonant metasurface structure for guiding and focusing bending waves. Multiple spring oscillator units are symmetrically distributed on a first metasurface and a second metasurface on a free-standing substrate. Each spring oscillator unit cell includes two oscillator spring resonators: a first mass-spring resonator and a second mass-spring resonator. The two oscillator spring resonators have the same spring stiffness but different oscillator masses. At the interface between the first and second metasurfaces, energy from bending waves is concentrated. By employing a simple, compact metasurface structure, flexural wave guidance and focusing are achieved with high robustness.

[0026] 2. The substrate in the local resonant metasurface structure of the present invention is integrated and made of the same material, which is convenient for production and processing. In addition, since the substrate is not cut, the rigidity of the substrate is guaranteed. The guidance and focusing of the bending waves are achieved without destroying the original structure of the substrate. The substrate can be designed into any shape according to actual needs and has strong environmental adaptability.

[0027] 3. The local resonant metasurface structure of the present invention has no restrictions on the excitation frequency. It only needs to adjust the parameters of the first mass-spring resonator and the second mass-spring resonator according to the frequency band of the incident wave. It is universally applicable, especially for low-frequency meter-scale substrates or high-frequency micro-nanoscale substrates.

[0028] 4. The local resonant metasurface structure of the present invention has no restrictions on the incident form of bending waves. It only needs to apply excitation at the edge of the substrate and close to the first mass-spring resonator or the second mass-spring resonator. BRIEF DESCRIPTION OF THE DRAWINGS

[0029] The above and / or additional aspects and advantages of the present invention will become apparent and readily understood from the following description of the embodiments with reference to the accompanying drawings, in which:

[0030] Figure 1 Schematic diagram of the localized resonant metasurface structure for guiding and focusing bending waves according to the present invention;

[0031] Figure 2 is a schematic diagram of the first local resonance unit cell and the corresponding substrate on which it is placed;

[0032] Figure 3 This is a comparison diagram of the wave energy distribution along the local resonant metasurface structure of the present invention and along a single metasurface at different excitation frequencies;

[0033] Figure 4 This is a diagram of the regional displacement amplitude field distribution of the local resonance metasurface structure of the present invention when the excitation frequency is 1.79 kHz.

[0034] Explanation of the reference numerals: 1 - substrate, 2 - first metasurface, 3 - second metasurface, 4 - first local resonant unit cell, 5 - second local resonant unit cell, 6 - first mass-spring resonator, 7 - second mass-spring resonator. DETAILED DESCRIPTION

[0035] The following describes in detail embodiments of the present invention. The embodiments are exemplary and intended to explain the present invention, but are not to be construed as limiting the present invention.

[0036] Reference Figure 1 and Figure 2 This embodiment provides a localized resonant metasurface structure for guiding and focusing bending waves, comprising a substrate 1 and a first localized resonant unit cell 4 and a second localized resonant unit cell 5 fixed to the substrate 1. The substrate comprises a first region 2 and a second region 3, a plurality of the first localized resonant unit cells 4 and a plurality of the second localized resonant unit cells 5 are uniformly arranged in the first region 2 and the second region 3, respectively, with the first localized resonant unit cells 4 and the second localized resonant unit cells 5 being collinearly arranged. The spacing between adjacent first localized resonant unit cells and the spacing between adjacent second localized resonant unit cells are both defined as a first spacing.

[0037] The first local resonant unit cell 4 and the second local resonant unit cell 5 each include two oscillator spring resonators, namely a first mass spring resonator 6 and a second mass spring resonator 7. The two oscillator spring resonators in the first local resonant unit cell 4 and the second local resonant unit cell are arranged in opposite order. The spacing between the first mass spring resonator 6 and the second mass spring resonator 7 in the unit cell is defined as a second spacing. The second spacing is equal to the first spacing. The mass of the first mass spring resonator 6 and the mass of the second mass spring resonator 7 are different, but the stiffness is the same.

[0038] The first region on the substrate and a plurality of first locally resonant unit cells arranged in the region constitute a first metasurface, and the second region on the substrate and a plurality of second locally resonant unit cells arranged in the region constitute a second metasurface. The first metasurface and the second metasurface have the same dispersion curve, but the wave modes corresponding to the different-order curves of the first metasurface and the second metasurface are opposite, that is, the two metasurfaces have opposite topological properties. When a bending wave is generated on the substrate at a set frequency, when the bending wave is incident on the locally resonant metasurface composed of the first metasurface and the second metasurface, a special frequency point will appear within the band gap range. The bending wave at this frequency point will gather at the interface between the first metasurface and the second metasurface, resulting in energy concentration, and forming a one-dimensional topological interface state of the bending wave.

[0039] In this embodiment, the substrate is made of metal. Compared with other materials, the bending wave energy concentration effect on the metal substrate is more obvious.

[0040] Because the unit cell parameters and number of the first and second local resonant unit cells are different, the interface positions of the first and second metasurfaces are different, and the energy collection effect is also different. Next, we need to design the unit cell parameters, number, and spacing of the first and second local resonant unit cells based on the effective operating frequency band of the bending wave and the energy collection position requirements. The specific steps are as follows:

[0041] Step 1: Obtain the material density ρ, Young's modulus E, Poisson's ratio υ, and thickness h of the substrate and calculate the bending wave dispersion curve. Intrinsic loss is ignored when calculating the bending wave dispersion curve.

[0042] Step 2: Determine the spacing between the first mass-spring resonator and the second mass-spring resonator in the first local resonant unit cell based on the set effective bending wave operating frequency band and the bending wave dispersion curve obtained in step 1, so that the first band edge frequency of the bending wave dispersion curve in the first Brillouin zone of the first local resonant unit cell lies within the set effective bending wave operating frequency band.

[0043] Step 3: Setting the effective bending wave operating frequency, wherein the effective bending wave operating frequency is within a set effective bending wave operating frequency band.

[0044] Step 4: Establish a finite element model including the first local resonant unit cell and the substrate corresponding to the first local resonant unit cell. Perform characteristic frequency analysis based on the finite element model to obtain the band structure of the first local resonant unit cell. Determine whether the band gap of the band structure of the first local resonant unit cell corresponds to the effective operating frequency of bending waves set in step 3. If so, determine the stiffness and mass of the first mass-spring resonator and the second mass-spring resonator. If not, adjust the stiffness and mass of the first mass-spring resonator and the second mass-spring resonator until the band gap of the band structure of the first local resonant unit cell corresponds to the set effective operating frequency of bending waves.

[0045] When adjusting the mass of the first mass-spring resonator and the second mass-spring resonator, only the mass of the oscillator is considered and the mass of the spring is ignored.

[0046] In order to further confirm the stiffness and mass of the first mass-spring resonator and the second mass-spring resonator, and whether the second local resonance unit cell band structure meets the requirements, a verification step of the second local resonance unit cell band structure is included before step 5. The specific process is as follows:

[0047] Using the parameters of the first mass-spring resonator and the second mass-spring resonator obtained in steps 2 and 4, a finite element model is established including the second local resonant unit cell and the substrate corresponding to the second local resonant unit cell. Eigenfrequency analysis is performed to obtain the band structure of the second local resonant unit cell. The band structure of the second local resonant unit cell is checked to see if it is consistent with the band structure of the first local resonant unit cell obtained in step 4. If so, it is determined that the parameters of the first mass-spring resonator and the second mass-spring resonator obtained in step 4 meet the band structure design requirements of the first local resonant unit cell and the second local resonant unit cell. Since both the first local resonant unit cell and the second local resonant unit cell include the first mass-spring resonator and the second mass-spring resonator arranged at the same interval, only the arrangement order is different. In theory, if the parameters of the first mass-spring resonator and the second mass-spring resonator in the designed first local resonant unit cell are applied to the second local resonant unit cell, the band structures of the two units will be the same. However, in actual engineering design, verification and confirmation are required to ensure that the band structures of the two units are the same.

[0048] Step 5: Using the spacing between the first mass-spring resonator and the second mass-spring resonator in the first local resonance unit cell as the unit cell spacing between adjacent first local resonance unit cells and adjacent second local resonance unit cells.

[0049] Step 6: Determine the number of first and second local resonant cells based on the actual bending wave energy concentration interface location requirements. To achieve a better energy concentration effect, the number of first and second local resonant cells should be no less than five. Depending on the interface location requirements, the number of first and second local resonant cells can be the same or different.

[0050] Step 7: Determine a substrate length based on the determined number of first and second local resonant unit cells, the inter-cell spacing, and the spacing between the first mass-spring resonator and the second mass-spring resonator. The substrate length is not less than the sum of the lengths of all uniformly arranged first and second local resonant unit cells.

[0051] Finally, the parameters of the first local resonant unit cell and the second local resonant unit cell are obtained. Using the obtained unit cell spacing between adjacent first local resonant unit cells, as well as the mass and stiffness of the first mass-spring resonator and the second mass-spring resonator, the first local resonant unit cell and the second local resonant unit cell are fixed to a substrate to obtain the local resonant metasurface structure of the present invention. When in use, the local resonant metasurface structure is placed on an excitation source, which is located at the edge of the substrate and close to the first local resonant unit cell or the second local resonant unit cell.

[0052] In order to verify that the local resonant metasurface structure of the present invention has the effect of focusing incident waves of a specific frequency, a simulation analysis of a specific local resonant metasurface structure designed using the above design method is performed below. In this example, the substrate is a rectangular aluminum substrate with four free sides. The overall dimensions of the rectangular substrate are: width l = 160mm, length W = 320mm, thickness h = 1.5mm, and the density, Young's modulus, and Poisson's ratio of the rectangular substrate are 2700kg / m 3 , 70GPa and 0.3; the effective operating frequency band of the bending wave is 1.5kHz to 2.2kHz, the effective operating frequency of the bending wave is 1.79kHz, the masses of the first mass spring resonator and the second mass spring resonator are 3g and 5g respectively, and the stiffness is 10 6 N / m. The spacing between the first mass-spring resonator and the second mass-spring resonator in the unit cell is 8 mm, and the spacing between adjacent first local resonant unit cells and adjacent second local resonant unit cells is also 8 mm. The number of first local resonant unit cells and second local resonant unit cells is 10 each.

[0053] We used COMSOL Multiphysics finite element analysis software to establish a finite element model of the entire local resonant metasurface structure, including a rectangular substrate and 10 first and second local resonant cells arranged on the rectangular substrate. In the shell element module of the finite element analysis software, the thickness of the rectangular substrate was set to 1.5 mm. A point load of magnitude 1, directed vertically downward, was set at the edge of the first metasurface and near the first local resonant cell. The rectangular substrate was meshed using a free triangular mesh, with a maximum mesh size of one-twelfth of the current frequency wave field. Simultaneously, finite element models were established containing only the first or second metasurface, with the same material and structural parameters as those for each region of the local resonant metasurface structure.

[0054] Add a frequency domain analysis study and set the sweep frequency range to 1.5kHz to 2.2kHz with a frequency interval of 0.05kHz. Number all resonators in the local resonant metasurface structure in order and extract the displacement amplitude of the substrate below the oscillator spring resonators numbered 5-15 at different frequencies, that is, the distribution of the bending wave energy along the metasurface, such as Figure 3 As shown. Figure 3It can be seen that for the first metasurface alone or the second metasurface alone, the frequency band from 1.6kHz to 2kHz is an obvious bending wave band gap, and the bending wave cannot propagate along the metasurface. The displacement amplitude below the oscillator spring resonators numbered 5-15 is basically 0; while for the local resonant metasurface structure, a clear highlight area appears within the band gap range of 1.6kHz to 2kHz. The frequency corresponding to this area is 1.79kHz, and the energy is strongest at the interface between the two metasurfaces in the local resonant metasurface structure (that is, between the oscillator spring resonators numbered 10-11), forming a topological interface state of the bending wave.

[0055] In order to more clearly demonstrate the rainbow reflection effect of the local resonance metasurface structure in this example, the displacement amplitude wave field of the entire bending wave at an incident frequency of 1.79 kHz is extracted, as shown in Figure 4 From the wave field diagram, it can be found that for the excitation frequency of 1.79 kHz, the bending wave cannot propagate along the first metasurface or the second metasurface alone, but can propagate along the local resonant metasurface and produce an energy concentration effect at the interface.

[0056] It can be seen that the local resonant metasurface structure of the present invention can produce a very ideal energy concentration effect at the interface for bending waves excited by a specific frequency. The unit cell size is much smaller than the wavelength of the incident wave, there are no requirements for the incident form of the incident wave, and no external energy supply is required during use, which is highly economical.

[0057] The above description is only a specific embodiment of the present invention, but the scope of protection of the present invention is not limited thereto. Any technician familiar with this technical field can easily think of various equivalent modifications or replacements within the technical scope disclosed in the present invention, and these modifications or replacements should all be included in the scope of protection of the present invention.

Claims

1. Localized resonant metasurface structure for flexural wave guiding and focusing, characterized by: It includes a substrate with four free sides and a first local resonance unit cell and a second local resonance unit cell fixed on the substrate; The substrate includes a first region and a second region, a plurality of the first local resonant unit cells and a plurality of the second local resonant unit cells are uniformly arranged in the first region and the second region, respectively, and the first local resonant unit cells and the second local resonant unit cells are collinearly arranged; and a spacing between adjacent first local resonant unit cells and a spacing between adjacent second local resonant unit cells are both defined as a first spacing; The first local resonant unit cell includes a first mass-spring resonator and a second mass-spring resonator arranged in sequence; the second local resonant unit cell includes the second mass-spring resonator and the first mass-spring resonator arranged in sequence; the first mass-spring resonator and the first mass-spring resonator have the same stiffness but different masses; defining the spacing between the first mass-spring resonator and the second mass-spring resonator in the first local resonance unit cell and the second local resonance unit cell to be a second spacing; The second spacing is equal to the first spacing; When a bending wave is generated at a set frequency at the edge of the substrate and near the first local resonant unit cell or the second local resonant unit cell, the bending wave can propagate along the substrate and generate energy concentration at the interface between the first region and the second region.

2. The localized resonant metasurface structure for guiding and focusing bending waves according to claim 1, characterized in that: The substrate is made of metal material.

3. The localized resonant metasurface structure for guiding and focusing bending waves according to claim 1, characterized in that: When the local resonance metasurface structure is mounted on an excitation source, the excitation source is located at the edge of the substrate and close to the first local resonance unit cell or the second local resonance unit cell.

4. The method for designing a localized resonant metasurface structure for guiding and focusing bending waves according to any one of claims 1 to 3, characterized in that: The following steps are involved: Step 1: Obtain the material density ρ, Young's modulus E, Poisson's ratio υ, and thickness h of the substrate, and calculate the bending wave dispersion curve; Step 2: determining a spacing between the first mass-spring resonator and the second mass-spring resonator in the first local resonant unit cell based on the set effective bending wave operating frequency band and the bending wave dispersion curve obtained in Step 1, such that a first band edge frequency of the bending wave dispersion curve in the first Brillouin zone of the first local resonant unit cell is within the set effective bending wave operating frequency band; Step 3, setting a bending wave effective operating frequency, wherein the bending wave effective operating frequency is within a set bending wave effective operating frequency band; Step 4: establishing a finite element model including the first local resonant unit cell and the substrate corresponding to the first local resonant unit cell; performing eigenfrequency analysis based on the finite element model to obtain a band structure of the first local resonant unit cell; determining whether a band gap of the band structure of the first local resonant unit cell corresponds to the effective bending wave operating frequency set in Step 3; if so, obtaining the stiffness and mass of the first mass-spring resonator and the second mass-spring resonator; if not, adjusting the stiffness and mass of the first mass-spring resonator and the second mass-spring resonator until the band gap of the band structure of the first local resonant unit cell corresponds to the set effective bending wave operating frequency; Step 5: Using the spacing between the first mass-spring resonator and the second mass-spring resonator in the first local resonance unit cell as the unit cell spacing between adjacent first local resonance unit cells and adjacent second local resonance unit cells; Step 6: determining the number of the first local resonance unit cell and the second local resonance unit cell according to the actual bending wave energy concentration interface position requirement; Step 7: Determine the length of the substrate according to the determined number of units of the first local resonance unit cell and the second local resonance unit cell, the unit cell spacing, and the spacing between the first mass-spring resonator and the second mass-spring resonator.

5. The design method according to claim 4, characterized in that: In step 6, the number of the first local resonance unit cell and the number of the second local resonance unit cell are both no less than 5.

6. The design method according to claim 4, characterized in that: When calculating the bending wave dispersion curve in step 1, the intrinsic loss is neglected.

7. The design method according to claim 4, characterized in that: In step 4, the spring masses in the first mass-spring resonator and the second mass-spring resonator are ignored.

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