Method and system for constructing dispersion relation of underwater lateral localized resonance metamaterial plate structure

Abstract

The present invention belongs to the technical field of underwater meta-plate vibration, and discloses a method and system for constructing the dispersion relationship of an underwater lateral localized resonant meta-material plate structure. The present invention utilizes a plane wave expansion method and a fluid-solid coupling principle to consider the influence of attached water on the meta-plate, thereby obtaining a vibration response equation based on a meta-plate base plate; the wave number is scanned in the first irreducible Brillouin zone to obtain the dispersion relationship of the underwater lateral localized resonant meta-plate; the present invention establishes a control equation of an underwater lateral localized resonant meta-plate and a Helmholtz equation of an acoustic wave in a free field, and utilizes a plane wave expansion method and a fluid-solid coupling principle to consider the influence of a radiated acoustic field on the meta-plate, thereby obtaining an equation based on the vibration response of the meta-plate base plate, and the wave number is scanned in the first irreducible Brillouin zone to obtain the dispersion relationship of the underwater lateral localized resonant meta-plate.

Description

Technical Field

[0001] The present invention belongs to the technical field of acoustic metamaterials, and in particular relates to a method and system for constructing a dispersion relation of an underwater lateral localized resonance metamaterial plate structure. Background Art

[0002] Underwater vehicles are often used in various sea conditions, which inevitably produce some subtle irregular vibrations, thereby generating near-field self-noise, resulting in a decrease in the underwater acoustic detection performance of the equipment. In addition, more and more underwater vehicles are beginning to pursue "stealth" characteristics to reduce the probability of being discovered by enemy sonar. In order to solve these problems, by utilizing the excellent properties of hydroacoustic metamaterials, it is aimed to reduce self-noise interference, improve its hydroacoustic communication and sonar detection performance, and counter active and passive sonar detection.

[0003] Hydroacoustic metamaterials are also artificial periodic structures, with microstructures as their basic building blocks. This structure can suppress waves within a certain frequency range, which is the so-called propagation "bandgap" phenomenon. The frequency range of the bandgap can be adjusted by changing the structure and spatial distribution of the metamaterial microstructure units. This property can be used to improve the signal-to-noise ratio of the sonar system and reduce the near-field self-noise. The present invention aims to reveal the dispersion relation of an underwater lateral localized resonance plate.

[0004] The methods for controlling the vibration of underwater vehicles are very limited, and there are few cases of using underwater metamaterial structures other than acoustic covering layers to control the vibration of underwater vehicles. The present invention lays a foundation for the vibration control of underwater structures.

[0005] Through the above analysis, the problems and defects of the prior art are as follows:

[0006] Underwater vehicles are often used in various sea conditions, which inevitably produce some subtle irregular vibrations, thereby generating near-field self-noise, resulting in a decrease in the underwater acoustic detection performance of the equipment. Summary of the invention

[0007] In view of the problems existing in the prior art, the present invention provides a method for constructing the dispersion relation of an underwater lateral localized resonance metamaterial plate structure.

[0008] The present invention is implemented in this way: a method for constructing a dispersion relation of an underwater lateral localized resonance metamaterial plate structure comprises:

[0009] Step 1: Using the plane wave expansion method and the fluid-solid coupling principle, the influence of attached water on the super-structure plate is considered to obtain the vibration response equation of the super-structure plate foundation plate;

[0010] Step 2: Scan the wave number in the first irreducible Brillouin zone to obtain the dispersion relation of the underwater lateral localized resonant meta-plate.

[0011] Furthermore, the plane wave expansion method and the fluid-solid coupling principle are used to consider the influence of attached water on the super-structure plate, thereby obtaining the vibration response equation of the super-structure plate base plate:

[0012] The band gap of periodic structures is usually obtained by plane wave expansion method, and the unit cell size is characterized by side lengths a and b and thickness h. p ; The control equation of the underwater superstructure can be expressed as

[0013]

[0014] In the formula w(r) represents the bending displacement at point r = (x, y), is the surface density of the plate, ω represents the angular frequency, represents the bending stiffness, p f (r,z) represents the attraction between adjacent fluids, w1(R) and w2(R) represent the displacements of m1 and m2 at R, respectively, and f1(R) refers to the displacement of the resonator at R The force applied by the point to the plate, δ(·) is the two-dimensional Dirac function, and σ = L / D is the aspect ratio of the four-bar mechanism;

[0015] Due to the periodicity of the system, the displacement response of the plate can be written as

[0016]

[0017] where k = (k x ,k y ) represents the Bloch wave vector, G represents the reciprocal lattice vector;

[0018] The sound pressure in water satisfies the Helmholtz equation

[0019]

[0020] In the formula represents the sound wave number, c0 is the underwater wave speed;

[0021] The sound pressure in water can be expressed as

[0022]

[0023] in,

[0024]

[0025] And there is

[0026]

[0027] Applying continuity conditions on the interface between the metastructure plate and the fluid yields

[0028]

[0029]

[0030] The properties of the Dirac function can be expressed as

[0031]

[0032] From this we can get

[0033]

[0034] It can be deduced that

[0035]

[0036] therefore

[0037]

[0038] By truncating m and n in the reciprocal lattice vector space, the following formula can be obtained. In order to obtain a non-zero solution, the determinant of its coefficient matrix is ​​set to zero, thereby solving the dispersion relation of the underwater super-structure plate;

[0039]

[0040] The present invention provides a system for constructing a dispersion relation of an underwater lateral localized resonance metamaterial plate structure, the system comprising:

[0041] A calculation unit, configured to calculate the influence of attached water on the superstructure plate by using a plane wave expansion method and a fluid-structure coupling principle, and to generate a vibration response equation of the foundation plate;

[0042] The scanning module is used to perform a wave number scan in the first irreducible Brillouin zone to obtain the dispersion relation of the underwater lateral localized resonant meta-plate.

[0043] The computing unit is further configured to perform the following operations:

[0044] Establish a plane wave expansion method to reflect the band gap of periodic structures;

[0045] According to the governing equations of the meta-plate, the plate's surface density, bending stiffness, gravitational force between adjacent fluids, and the aspect ratio of the four-bar mechanism are considered to express the displacement response of the plate.

[0046] The computing unit is further configured to apply Bloch theory, write the displacement response expression of the plate as a function of the Bloch wave vector and the reciprocal lattice vector, and solve the Helmholtz equation satisfied by the sound pressure in water.

[0047] Wherein the computing unit is further configured as:

[0048] The continuity condition is imposed on the interface between the metastructure plate and the fluid, and the relationship between the acoustic pressure and displacement is derived using the properties of the Dirac function.

[0049] The expression in the reciprocal lattice vector space is truncated to construct the coefficient matrix for solving the non-zero solution and thus obtain the dispersion relation of the underwater meta-plate.

[0050] The scanning module is further configured as follows:

[0051] The coefficient matrix determinant is numerically solved to determine the non-zero solutions at different wave numbers, thereby mapping the dispersion relation;

[0052] The visual representation of the dispersion relation provides an intuitive understanding of the dispersion characteristics of the meta-slab.

[0053] The system further comprises:

[0054] A data processing module that collects dispersion relation data and analyzes the data to identify and predict band gaps and related physical properties;

[0055] A user interface is configured to allow the user to input parameters such as the characteristics of the attached water, the dimensions and material properties of the meta-plate, as well as the ability to export the calculation results.

[0056] In combination with the above technical solutions and the technical problems solved, the advantages and positive effects of the technical solutions to be protected by the present invention are as follows:

[0057] First, the present invention establishes the control equation of the underwater lateral localized resonant meta-plate and the Helmholtz equation of the sound wave in the free field, and uses the plane wave expansion method and the fluid-solid coupling principle to take into account the influence of the radiated sound field on the meta-plate, thereby obtaining an equation based on the vibration response of the meta-plate base plate, and scanning the wave number in the first irreducible Brillouin zone to obtain the dispersion relation of the underwater lateral localized resonant meta-plate, which is then applied to the underwater structure to control the vibration of the underwater structure.

[0058] Second, the present invention establishes the control equation of the underwater lateral localized resonant meta-plate and the Helmholtz equation of the sound wave in the free field, and uses the plane wave expansion method and the fluid-solid coupling principle to take into account the influence of the radiated sound field on the meta-plate, thereby obtaining an equation based on the vibration response of the meta-plate base plate, and scanning the wave number in the first irreducible Brillouin zone to obtain the dispersion relation of the underwater lateral localized resonant meta-plate.

[0059] Improved model accuracy: By considering the fluid-solid coupling effect, the present invention improves the accuracy of the model of the metamaterial plate when it is used underwater, making the model closer to actual application conditions.

[0060] Enhanced understanding of physical phenomena: Accurate dispersion relations can reveal the propagation mechanism of waves in underwater lateral localized resonant meta-plates, which helps to deeply understand related physical phenomena such as band gaps and negative refraction.

[0061] Promoted the design and application of metamaterials: The dispersion relation obtained by this method can guide the design of metamaterials and optimize their structural parameters to meet specific wave control requirements, such as vibration mitigation and sound wave guidance, thus broadening the application field of metamaterials in underwater environments.

[0062] Third, the expected benefits and commercial value after the technical solution of the present invention is transformed as follows: Compared with the finite element method, the present invention realizes the band gap calculation of the metamaterial plate more efficiently and quickly, improves the calculation efficiency, and creates higher commercial value. The present invention is not limited to this structure, and the present invention has guiding significance for the band gap calculation and vibration control design of various underwater metamaterial plates.

[0063] The technical solution of the present invention fills the technical gap in the industry at home and abroad: the present invention uses the plane wave expansion method and the fluid-solid coupling principle to take into account the influence of the radiated sound field on the meta-structure plate, and explains the method for constructing the dispersion equation of the underwater metamaterial plate. It has guiding significance for the construction of dispersion equations and vibration control design of various underwater metamaterial plates.

[0064] The technical solution of the present invention solves a technical problem that people have been eager to solve but have never succeeded in solving: the present invention explains the method for constructing the dispersion equation of an underwater metamaterial plate, which has guiding significance for the construction of dispersion equations and vibration control design of various underwater metamaterial plates.

[0065] The technical solution of the present invention overcomes technical prejudice: the present invention takes into account the influence of the radiated acoustic field on the meta-plate by using the plane wave expansion method and the principle of fluid-solid coupling, thereby obtaining an equation based on the vibration response of the meta-plate base plate, and scanning the wave number in the first irreducible Brillouin zone to obtain the dispersion relation of the underwater lateral local resonance meta-plate. Compared with the finite element method, the band gap calculation of the meta-material plate is realized more efficiently and quickly, and is not limited to this structure. The present invention has guiding significance for the band gap calculation and vibration control design of various underwater meta-material plates.

[0066] Fourth, the present invention proposes a method for constructing the dispersion relation of an underwater lateral localized resonant metamaterial plate structure, which uses the plane wave expansion method combined with the fluid-solid coupling principle, taking into account the influence of attached water on the meta-plate, and thus accurately obtains the vibration response equation of the meta-plate. The core of this method is to accurately calculate the band gap of the periodic structure, and to establish a complete mathematical model using the periodic properties of the meta-plate and the dynamic characteristics of the attached water body.

[0067] Plane wave expansion method: Through the plane wave expansion method, the present invention can accurately describe the dispersion characteristics of the periodic structure. This method can effectively process the periodic boundary conditions and obtain the band gap information of the structure.

[0068] Fluid-solid coupling principle: Taking into account the influence of fluid on the structure, the present invention applies the fluid-solid coupling principle, thereby adding the dynamic characteristics of the attached water body into the model, thereby increasing the accuracy and practicality of the model.

[0069] Vibration response equation: This method provides a detailed vibration response equation that includes factors such as bending displacement, plate surface density, angular frequency, bending stiffness, and gravitational force between adjacent fluids, providing rich parameters for vibration analysis.

[0070] Obtaining dispersion relations: Through the established model, the present invention can solve the dispersion relation of the underwater superstructure, that is, the relationship between the wave number and the frequency, which provides a theoretical basis for analyzing and designing structures with specific vibration and acoustic properties.

[0071] Significant technical effects:

[0072] Accurate vibration control: The dispersion relation model established by the present invention can more accurately predict and control the vibration characteristics of the underwater superstructure, providing an effective tool for designing high-performance underwater noise control structures.

[0073] Acoustic performance optimization: Using the method of the present invention, a meta-plate with excellent acoustic bandgap characteristics can be designed, thereby effectively controlling the propagation of sound waves within a specific frequency range, making it possible to improve the stealth performance of underwater vehicles.

[0074] Broad application prospects: The present invention is not only suitable for vibration and noise control of underwater vehicles, but can also be extended to other fields that require precise vibration control and acoustic performance optimization, such as architectural acoustics, mechanical vibration control, etc.

[0075] The present invention provides an effective method for controlling the vibration and acoustic performance of underwater lateral localized resonance metamaterial plate structures through precise mathematical models and theoretical analysis. Its significant technical effect lies in precise vibration control and optimization of acoustic performance, providing innovative solutions for related fields. BRIEF DESCRIPTION OF THE DRAWINGS

[0076] Figure 1 It is a flow chart of a method for constructing a dispersion relation of an underwater lateral localized resonance metamaterial plate structure provided by an embodiment of the present invention.

[0077] Figure 2This is the analogy transformation diagram of the lateral localized resonant super-plate equivalent to the homogeneous plate provided by the embodiment of the present invention: (a) front view of the super-plate cell; (b) bird's-eye view of the super-plate cell; (c) super-plate model diagram.

[0078] Figure 3 Schematic diagram of the calculation of the dispersion relation of the lateral local resonance meta-plate provided by the embodiment of the present invention (a) Underwater lateral local resonance meta-plate cell (b) The first irreducible Brillouin zone of the lateral local resonance meta-plate

[0079] Figure 4 This is a graph showing the variation of the band gap of an underwater super-plate with parameters provided by an embodiment of the present invention. DETAILED DESCRIPTION

[0080] In order to make the purpose, technical solution and advantages of the present invention more clearly understood, the present invention is further described in detail below in conjunction with the embodiments. It should be understood that the specific embodiments described herein are only used to explain the present invention and are not used to limit the present invention.

[0081] Two specific application examples of the embodiment of the present invention are:

[0082] Example 1: Underwater Acoustic Stealth Coating

[0083] Application of fluid-structure coupling model: The acoustic stealth coating on the underwater hull surface is first modeled using the plane wave expansion method combined with the fluid-structure coupling principle. Considering the influence of attached water on the superstructure, the vibration response equation of the coating is established to ensure that the model can accurately reflect the physical phenomena in the underwater environment.

[0084] Determination of dispersion relation: The dispersion relation of the acoustic stealth coating is accurately obtained by scanning the wave number of the coating material in the first irreducible Brillouin zone. This step reveals the propagation mechanism of waves in the coating material, especially the behavior at different frequencies, and provides a theoretical basis for designing coatings with specific wave control properties.

[0085] Realization of acoustic stealth performance: Based on the analysis results of the dispersion relationship, the design of the coating structure (such as the size and distribution of the local resonance unit) is optimized to achieve sound wave absorption or reflection control within a specific frequency range, thereby reducing or eliminating the reflection of sound waves by the hull and achieving acoustic stealth effect.

[0086] Example 2: Acoustic wave guide in underwater communication system

[0087] Establishing the Fluid-Structure Interaction Model: Similarly, the metamaterial plate of the acoustic wave guide is first modeled using the plane wave expansion method and the principle of fluid-structure interaction to ensure that the model can accurately describe the interaction between acoustic waves and structures in the underwater environment.

[0088] Obtaining dispersion relations: Obtain the dispersion relations of the acoustic wave guide material through wave number scanning. The analysis of dispersion relations reveals the characteristics of acoustic wave propagation in the material, including information such as propagation speed and wavelength change, which provides an important basis for designing effective acoustic wave guiding paths.

[0089] Design of acoustic wave guiding performance: Based on the analysis of dispersion relations, by adjusting the design parameters of metamaterials (such as the arrangement and geometry of the resonant units), effective guidance of acoustic waves can be achieved. This guide can control the propagation of acoustic waves along a specific path, thereby improving the efficiency of underwater communication systems and the clarity of signals.

[0090] In view of the problems existing in the prior art, the technical solution adopted by the present invention is:

[0091] Considering fluid-solid coupling: This method first uses the plane wave expansion method combined with the principle of fluid-solid coupling to consider the influence of attached water on the meta-structure plate, thereby obtaining the vibration response equation based on the meta-structure plate base plate. This step solves the problem that the influence of fluid action on the vibration characteristics of the meta-structure material plate is often ignored or simplified in the prior art, and improves the accuracy of the model.

[0092] Accurate acquisition of dispersion relation: By scanning the wave number in the first irreducible Brillouin zone, the method of the present invention can accurately obtain the dispersion relation of the underwater lateral localized resonance meta-structure plate. This step solves the problem of how to accurately describe the wave propagation characteristics of the underwater meta-structure material plate, and provides an important basis for further physical phenomenon analysis and engineering application.

[0093] like Figure 1 As shown, the present invention provides a method for constructing a dispersion relation of an underwater lateral localized resonance metamaterial plate structure, comprising the following steps:

[0094] S101, using the plane wave expansion method and the fluid-solid coupling principle, the influence of attached water on the super-structure plate is considered to obtain the vibration response equation of the super-structure plate foundation plate;

[0095] S102, scanning the wave number in the first irreducible Brillouin zone to obtain the dispersion relation of the underwater lateral localized resonant meta-plate.

[0096] The present invention provides a plane wave expansion method and fluid-solid coupling principle, taking into account the influence of attached water on the super-structure plate, thereby obtaining the vibration response equation of the super-structure plate base plate:

[0097] The band gap of periodic structures is usually obtained by plane wave expansion method, and the unit cell size is characterized by side lengths a and b and thickness h. p ; The control equation of the underwater superstructure can be expressed as

[0098]

[0099] In the formula w(r) represents the bending displacement at point r = (x, y), is the surface density of the plate, ω represents the angular frequency, represents the bending stiffness, p f (r, z) represents the attractive force between adjacent fluids, w1(R) and w2(R) represent the displacements of m1 and m2 at R, respectively, f1(R) refers to the force applied by the resonator to the plate at point R, δ(·) is the two-dimensional Dirac function, and σ = L / D is the aspect ratio of the four-bar mechanism;

[0100] Due to the periodicity of the system, the displacement response of the plate can be written as

[0101]

[0102] where k = (k x ,k y ) represents the Bloch wave vector, G represents the reciprocal lattice vector;

[0103] The sound pressure in water satisfies the Helmholtz equation

[0104]

[0105] In the formula represents the sound wave number, c0 is the underwater wave speed;

[0106] The sound pressure in water can be expressed as

[0107]

[0108] in,

[0109]

[0110] And there is

[0111]

[0112] Applying continuity conditions on the interface between the metastructure plate and the fluid yields

[0113]

[0114]

[0115] The properties of the Dirac function can be expressed as

[0116]

[0117] From this we can get

[0118]

[0119] It can be deduced that

[0120]

[0121] therefore

[0122]

[0123] By truncating m and n in the reciprocal lattice vector space, the following formula can be obtained. In order to obtain a non-zero solution, the determinant of its coefficient matrix is ​​set to zero, thereby solving the dispersion relation of the underwater super-structure plate;

[0124]

[0125] The present invention explains the method for constructing the dispersion equation of the underwater metamaterial plate, which is convenient for controlling the vibration of the underwater vehicle. Similarly, the present invention is not limited to one underwater metamaterial plate structure, and can achieve vibration control of underwater vehicles in specific frequency bands through reasonable design.

[0126] Extension: The present invention is not limited to the vibration control of underwater vehicles, but can be used to control the vibration of all underwater structures, such as: the sub-water part of offshore wind turbines, drilling platforms, etc.

[0127] Schematic diagram of the underwater lateral local resonance metamaterial plate model Figure 2 shown.

[0128] Take a unit cell of an underwater lateral local resonance metaplate, such as Figure 3 As shown in (a), the unit cell includes a lateral local resonance meta-plate part and an attached water area part. For this unit cell, its base plate surface is rectangular, so its first irreducible Brillouin zone is as follows: Figure 3 (b) as shown.

[0129] The band gap of periodic structures is usually obtained by plane wave expansion method, and the unit cell size is characterized by side lengths a and b and thickness h. p The governing equation of the underwater superstructure can be expressed as

[0130]

[0131] In the formula w(r) represents the bending displacement at point r = (x, y), is the surface density of the plate, ω represents the angular frequency, represents the bending stiffness, p f (r,z) represents the attraction between adjacent fluids, w1(R) and w2(R) represent the displacements of m1 and m2 at R, respectively, f1(R) refers to the force applied by the resonator to the plate at point R, δ(·) is the two-dimensional Dirac function, and σ = L / D is the aspect ratio of the four-bar mechanism.

[0132] Due to the periodicity of the system, the displacement response of the plate can be written as

[0133]

[0134] where k = (k x ,k y ) represents the Bloch wave vector, and G represents the reciprocal lattice vector.

[0135] The sound pressure in water satisfies the Helmholtz equation

[0136]

[0137] In the formula represents the sound wave number, and c0 is the underwater wave speed.

[0138] The sound pressure in water can be expressed as

[0139]

[0140] in,

[0141]

[0142] And there is

[0143]

[0144] Applying continuity conditions on the interface between the metastructure plate and the fluid yields

[0145]

[0146]

[0147] The properties of the Dirac function can be expressed as

[0148]

[0149] From this we can get

[0150]

[0151] It can be deduced that

[0152]

[0153] therefore

[0154]

[0155] By truncating m and n in the reciprocal lattice vector space, the following formula can be obtained. In order to obtain a non-zero solution, the determinant of its coefficient matrix is ​​made zero, thereby solving the dispersion relation of the underwater super-structure plate.

[0156]

[0157] It should be noted that the embodiments of the present invention can be implemented by hardware, software, or a combination of software and hardware. The hardware part can be implemented using dedicated logic; the software part can be stored in a memory and executed by an appropriate instruction execution system, such as a microprocessor or dedicated design hardware. It can be understood by a person of ordinary skill in the art that the above-mentioned devices and methods can be implemented using computer executable instructions and / or contained in a processor control code, such as a carrier medium such as a disk, CD or DVD-ROM, a programmable memory such as a read-only memory (firmware), or a data carrier such as an optical or electronic signal carrier. Such code is provided on the carrier medium. The device and its modules of the present invention can be implemented by hardware circuits such as very large-scale integrated circuits or gate arrays, semiconductors such as logic chips, transistors, etc., or programmable hardware devices such as field programmable gate arrays, programmable logic devices, etc., can also be implemented by software executed by various types of processors, and can also be implemented by a combination of the above-mentioned hardware circuits and software, such as firmware.

[0158] Figure 4 That is the band gap result calculated by the dispersion equation of the underwater metamaterial plate of the present invention. Through discussions with different parameters, the band gap situation of the present invention under different structural parameters is explained, and it is further explained that it has a vibration control effect underwater.

[0159] Figure 4 The differences in the band gaps of the metaplate underwater and in vacuum are shown: (1) Due to the fluid-structure interaction, the heavy fluid acts as an additional mass to the base plate, resulting in a narrower band gap underwater. The second band gap is much narrower than the first band gap. (2) The first band gap shows an opposite trend with the increase of attached mass. The addition of mass 1 widens the first band gap, while mass 2 narrows it. The second band gap maintains its width and shifts slightly to a lower frequency range with increasing mass 1, while it widens significantly with increasing mass 2. This shift and widening can be attributed to the reduction of resonances and the increase in the mass ratio of the resonator to the substrate. The band gap narrows with increasing side length and increasing thickness due to a similar mechanism of decreasing mass ratio of the resonator to the base plate. (3) Both vertical and transverse springs lift the band gap to a higher frequency range. The vertical spring helps to widen the first band gap and squeeze the second band gap. However, the transverse spring plays the opposite role. Since the interaction occurs between the vertical resonator and the transverse resonator, it is difficult to clearly distinguish the main band gaps of the two resonators. Therefore, in actual engineering implementation, advanced calculation of the natural frequency of the target structure and ingenious design of metamaterial parameters are the guarantee for obtaining satisfactory results.

[0160] The above description is only a specific implementation mode of the present invention, but the protection scope of the present invention is not limited thereto. Any modifications, equivalent substitutions and improvements made by any technician familiar with the technical field within the technical scope disclosed by the present invention and within the spirit and principle of the present invention should be covered by the protection scope of the present invention.

Claims

1. A method for constructing the dispersion relation of an underwater lateral localized resonance metamaterial plate structure, characterized in that: The following steps are involved: Step 1: Using the plane wave expansion method and the fluid-solid coupling principle, the influence of attached water on the super-structure plate is considered to obtain the vibration response equation of the super-structure plate foundation plate; Step 2, scanning the wave number in the first irreducible Brillouin zone, that is, obtaining the dispersion relation of the underwater lateral localized resonant meta-plate; The plane wave expansion method and the fluid-solid coupling principle are used to consider the influence of attached water on the super-structure plate, so as to obtain the vibration response equation of the super-structure plate foundation plate: The band gap of periodic structures is usually obtained by plane wave expansion method, and the unit cell size is characterized by side lengths a and b and thickness h. p ; The control equation of the underwater superstructure is expressed as In the formula w(r) represents the bending displacement at point r = (x, y), is the surface density of the plate, ω represents the angular frequency, represents the bending stiffness, w1(R) and w2(R) represent the displacements of m1 and m2 at R, respectively, f1(R) refers to the force applied by the resonator to the plate at point R, δ(·) is the two-dimensional Dirac function, and σ = L / D is the aspect ratio of the four-bar mechanism; Due to the periodicity of the system, the displacement response of the plate is written as where k = (k x ,k y ) represents the Bloch wave vector, G represents the reciprocal lattice vector, and its formula is G = mb1 + nb2, m and n are both integers, b1 = (2π / a, 0), b2 = (0, 2π / b); p in formula (40) f The (r,z) term represents the sound pressure in water, which satisfies the Helmholtz equation for sound pressure in water: In the formula represents the sound wave number, c0 is the underwater wave speed; The sound pressure in water can be expressed as in, And there is Applying continuity conditions on the interface between the metastructure plate and the fluid yields Where ρ0 is the liquid density; using the orthogonal relationship, we get: The periodic condition also implies the following relationship: The properties of the Dirac function are expressed as From this we get Now introduce a function, which is defined as and Considering the following relationship So we get By truncating m and n in the reciprocal lattice vector space, the following formula is obtained. In order to obtain a non-zero solution, the determinant of its coefficient matrix is ​​set to zero, thereby solving the dispersion relation of the underwater superstructure plate; In the formula, W = {W(G1)W(G2)...W(G N×N )} T The form of equation (55) is similar to the standard eigenvalue problem; for each given Bloch vector k = (k x ,k y ), determine its characteristic frequency; and by scanning k along the boundary of the irreducible Brillouin zone, obtain the dispersion relation between wave number and frequency.

2. A system for constructing the dispersion relation of an underwater lateral localized resonance metamaterial plate structure using the method for constructing the dispersion relation of an underwater lateral localized resonance metamaterial plate structure according to claim 1, characterized in that: The system includes: A calculation unit, configured to calculate the influence of attached water on the superstructure plate by using a plane wave expansion method and a fluid-structure coupling principle, and to generate a vibration response equation of the foundation plate; The scanning module is used to perform a wave number scan in the first irreducible Brillouin zone to obtain the dispersion relation of the underwater lateral localized resonant meta-plate.

3. The system according to claim 2, characterized in that The computing unit is further configured to perform the following operations: Establish a plane wave expansion method to reflect the band gap of periodic structures; According to the governing equations of the meta-plate, the plate's surface density, bending stiffness, gravitational force between adjacent fluids, and the aspect ratio of the four-bar mechanism are considered to express the displacement response of the plate.

4. The system according to claim 3, characterized in that The computing unit is further configured to apply Bloch theory, write the displacement response expression of the plate as a function of the Bloch wave vector and the reciprocal lattice vector, and to solve the Helmholtz equation satisfied by the sound pressure in water.

5. The system according to claim 4, characterized in that The computing unit is further configured as: The continuity condition is imposed on the interface between the meta-plate and the fluid, and the relationship between the acoustic pressure and displacement is derived using the properties of the Dirac function. The expression in the reciprocal lattice vector space is truncated to construct the coefficient matrix for solving the non-zero solution and thus obtain the dispersion relation of the underwater meta-plate.

6. The system according to claim 5, characterized in that The scanning module is further configured as follows: The coefficient matrix determinant is numerically solved to determine the non-zero solutions at different wave numbers, thereby mapping the dispersion relation; The visual representation of the dispersion relation provides an intuitive understanding of the dispersion characteristics of the meta-slab.

7. The system according to claim 6, characterized in that The system further comprises: A data processing module that collects dispersion relation data and analyzes the data to identify and predict band gaps and related physical properties; A user interface is configured to allow the user to input parameters including the characteristics of the attached water, the dimensions and material properties of the meta-plate, as well as the ability to export the calculation results.

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