A multi-resonant body combination structure low-frequency broadband vibration suppression method

Abstract

The application provides a low-frequency broadband vibration suppression method for a multi-resonant body combined structure, and comprises the following steps: S1, embedding a local resonance structure in a four-edge simply supported infinite rigid barrier, and constructing a radiation sound field sound pressure function by using a Rayleigh integral formula; S2, constructing a substrate transverse bending displacement function for the case that the four edges of the substrate are simply supported on the infinite rigid barrier, and the substrate transverse bending displacement is expressed as a linear superposition of vibration mode functions; S3, establishing an acoustic-vibration coupling model of the multi-resonant body combined local resonance periodic structure considering the influence of fluid load; and S4, substituting the radiation sound field sound pressure function and the substrate transverse bending displacement function into the acoustic-vibration coupling model of the local resonance periodic structure, and calculating to obtain a matrix form of the acoustic-vibration coupling model of the underwater multi-resonant body combined local resonance periodic structure. The low-frequency characteristic can be considered, the vibration suppression frequency band can be greatly widened, and the vibration reduction demand in actual engineering can be met.

Description

Technical Field

[0001] This invention relates to the field of low-frequency vibration and noise control technology for underwater structures, and in particular to a low-frequency broadband vibration suppression method for a multi-resonator composite structure. Background Technology

[0002] Low-frequency vibration and noise, due to its long wavelength, has always been a difficult problem in the field of vibration and noise control due to the poor effectiveness of conventional control methods. Periodic structures based on the local resonance mechanism have elastic wave band gaps. The generation of these band gaps depends on the interaction between the resonant characteristics of the resonator and the bending waves in the matrix. Therefore, the band gap frequency is closely related to the resonant frequency of the resonator. Band gaps can be formed at low frequencies, thus providing a new technical approach for low-frequency vibration and noise control. However, there are currently few designs for underwater structures.

[0003] A local resonant low-frequency bandgap vibration suppression periodic structure, disclosed in CN106678271A, includes periodically distributed mass elements, elastic elements, and a matrix. The periodic unit includes a metal structure, i.e., a mass element, and a ring-shaped structure of rubber or silicone material, i.e., an elastic element, surrounding the mass element on its side. The circumferential side of the mass element is a curved surface with a certain shape, and the elastic element and the mass element are in close contact with each other. The elastic element surrounding the mass element is arranged periodically as a whole in the elastic structure matrix, wherein the elastic element and the matrix are also in close contact with each other.

[0004] However, existing technologies generally suffer from "too narrow bandgap" in local resonant periodic structures under relatively small added mass ratios. Furthermore, the influence of fluid added mass in underwater environments makes it difficult to obtain a wide damping band at lower frequencies. Multi-resonator combined local resonant periodic structures can improve this situation to some extent, but the effect is limited. While increasing the number of resonators within a cell increases the number of bandgap segments, this does not guarantee a wider damping bandgap. This is mainly because although the number of bandgap segments increases, the width of each individual bandgap often narrows, resulting in a decrease in the overall bandgap width. Firstly, the vibration damping effect is not as expected. Secondly, multiple band gaps are often discretely distributed in the frequency domain, resulting in a discrete vibration damping area and failing to achieve a truly broadband vibration damping effect. Furthermore, there is no fixed rule for how to combine the frequencies of multiple resonators or multi-degree-of-freedom resonators. Designers often determine this based on the modal characteristics of the controlled structure and engineering experience, thus failing to fully utilize the broadband vibration damping advantage of local resonant structures with multiple resonators or multi-degree-of-freedom resonators. This makes it difficult for current local resonant periodic structures to simultaneously achieve low-frequency and broadband vibration damping effects, making it difficult to meet the vibration reduction requirements in practical engineering. Summary of the Invention

[0005] In view of this, the present invention proposes a low-frequency broadband vibration suppression method for multi-resonator composite structures, which can realize the flexible design of local resonance structures and significantly broaden the vibration suppression frequency band while taking into account low-frequency characteristics.

[0006] The technical solution of this invention is implemented as follows: This invention provides a low-frequency broadband vibration suppression method for a multi-resonator composite structure, characterized by comprising the following steps:

[0007] S1, When the local resonant structure is embedded in a four-sided simply supported infinite rigid baffle, the sound pressure function of the radiated sound field is constructed by Rayleigh integral formula;

[0008] S2, For the case where the four sides of the substrate are simply supported by an infinitely large rigid barrier, construct the lateral bending displacement function of the substrate. The lateral bending displacement of the substrate is expressed as a linear superposition of vibration mode functions.

[0009] S3. Establish an acoustic-vibration coupling model for a multi-resonator combined local resonant periodic structure that considers the influence of fluid load.

[0010] S4. Substitute the radiated sound field pressure function and the substrate lateral bending displacement function into the acoustic-vibration coupling model of the local resonant periodic structure to calculate the matrix form of the acoustic-vibration coupling model of the underwater multi-resonator combined local resonant periodic structure.

[0011] S5, substitute the preset frequency into the acoustic-vibration coupling model of the matrix-form underwater multi-resonator combination local resonant periodic structure, obtain the vibration coefficient corresponding to the vibration mode function, and substitute the vibration coefficient into the substrate lateral bending displacement function to obtain the substrate lateral bending displacement.

[0012] S6, obtain the corresponding vibration velocity response based on the lateral bending displacement of the substrate;

[0013] S7, obtain the vibration transmissibility of the local resonant periodic structure based on the vibration velocity response;

[0014] S8. Define the vibration suppression frequency band fusion bandwidth based on the vibration transmissibility of the local resonant periodic structure to obtain the optimization objective function;

[0015] S9 uses a genetic algorithm to optimize the resonant frequency of the resonator contained in the vibration transmissibility in order to obtain the widest vibration suppression frequency band.

[0016] Based on the above technical solutions, preferably, in step S1, when the local resonant structure is embedded in a four-sided simply supported infinite rigid baffle, the sound pressure function of the radiated sound field is constructed using the Rayleigh integral formula, and the expression is:

[0017]

[0018] In the formula, ρ is the vibration velocity on the substrate surface; ω is the angular frequency; f Let R be the density of water, R be the distance from the field point r′=(x′,y′,z′) to a point r=(x,y,0) on the substrate surface, a be the length of the substrate, b be the width of the substrate, k be the wave number, x and y be the position coordinates on the substrate, and j be the imaginary unit.

[0019] Based on the above technical solution, preferably, in step S2, for the case where the four sides of the substrate are simply supported by an infinitely large rigid barrier, a lateral bending displacement function of the substrate is constructed. The lateral bending displacement w(x, y) of the substrate is expressed as a linear superposition of vibration mode functions, and the expression is:

[0020]

[0021] Among them, W mn Let A be the (m, n) order vibration mode function; mn Let be the vibration coefficients corresponding to the vibration mode functions; the expression for its (m, n)th order vibration mode function is:

[0022] W mn (x, y) = sin(k) m x)sin(k n y);

[0023] In the formula, k m =mπ / a; k n = nπ / b, where a is the length of the substrate and b is the width of the substrate.

[0024] Based on the above technical solution, preferably, in step S3, an acoustic-vibration coupling model of a multi-resonator combination local resonant periodic structure considering the influence of fluid load is established, and the expression is:

[0025]

[0026] Among them, D P The bending stiffness of the substrate is expressed as: D p =E p h p 3 / 12(1-ν p 2 In the formula, E p v is the Young's modulus of the substrate. p The Poisson's ratio of the substrate; The expression is: ω is the angular frequency; ρ p h is the density of the substrate. pThe thickness of the substrate is given by ; w(r) is the lateral bending displacement of the substrate; F0 is the harmonic point force acting at position (x0, y0) on the substrate; p(x, y, 0) is the sound pressure on the substrate surface; F s The reaction force of each resonator on the substrate is expressed as: For a single-degree-of-freedom resonator ZF s =ω 2 m s k s / (k s -ω 2 m s ), where m s Let k be the mass of the s-th resonator within the cell. s R is the stiffness of the s-th resonator within the cell; t = (x t y t R represents the vector pointing from the origin to the center point of the t-th cell, indicating the position of the cell. s =(F 1s ,r 2s ) represents the vector pointing from the center point of the cell to the s-th resonator within the cell, representing the relative position of the resonators within the cell.

[0027] Based on the above technical solutions, preferably, in step S4, after substituting the radiated sound field pressure function and the substrate lateral bending displacement function into the acoustic-vibration coupling model of the local resonant periodic structure, the infinite series is truncated and simplified using the orthogonality of sine and cosine function integrals, and finally the acoustic-vibration coupling model of the underwater multi-resonator combined local resonant periodic structure in matrix form is obtained.

[0028] Based on the above technical solutions, preferably, the acoustic-vibration coupling model of the matrix-form underwater multi-resonator combined local resonant periodic structure in step S4 is expressed as follows:

[0029] (K+jωZ-ω 2 MF a A = F;

[0030] In the formula, the elements in matrices K, M, Z, and F are respectively:

[0031]

[0032] Matrix F a Represented as:

[0033]

[0034] The elements in matrix W are:

[0035] W sti，1 =sin(k)m (x t +r 1s sin(k) n (y t +r 2s )).

[0036] Based on the above technical solution, preferably, in step S6, the corresponding vibration velocity response is obtained according to the substrate displacement, wherein the calculation expression is:

[0037] v = jω * w;

[0038] In the formula, v is the vibration velocity response, ω is the angular frequency, and w is the lateral bending displacement of the substrate.

[0039] Based on the above technical solution, preferably, in step S7, the vibration transmissibility of the local resonant periodic structure is obtained according to the vibration velocity response, which further includes the following steps:

[0040] S71, select several points on the substrate and extract the vibration response at each point;

[0041] S72, calculate the vibration transmissibility of the local resonant periodic structure based on the vibration velocity response and vibration transfer function;

[0042] The expression for the vibration transfer function is:

[0043]

[0044] In the formula, v f For the velocity response at the excitation point, v ri The velocity response at each point along the selected transmission path.

[0045] Based on the above technical solutions, preferably, in step S8, the vibration suppression frequency band fusion bandwidth is defined according to the vibration transmissibility of the local resonant periodic structure to obtain the optimization objective function, wherein the expression is defined as:

[0046] Bw m =max(Ω), {TL(Ω)<0, f s ∈Ω};

[0047] In the formula, B wm To optimize the objective function, Ω represents the frequency range where the vibration transmissibility TL < 0. In the vibration transmissibility curve of the local resonant structure, the maximum frequency range where TL < 0 includes the resonant frequencies of all resonators is the vibration suppression frequency band fusion bandwidth.

[0048] Based on the above technical solution, preferably, in step S9, a genetic algorithm is used to optimize the resonant frequency of the resonator included in the vibration transmissibility to obtain the widest vibration suppression frequency band. Specifically, the mass and stiffness of the resonator are used as optimization variables in the genetic algorithm, the mass ratio is used as a constraint, and the maximum width B of the vibration suppression frequency band is used as the optimization factor. Wm The maximum vibration attenuation within the vibration suppression frequency band is used as the objective function for multi-objective optimization. Finally, a set of optimal variables is obtained, which yields the widest vibration suppression frequency band.

[0049] The low-frequency broadband vibration suppression method of the multi-resonator combination structure of the present invention has the following advantages over the prior art:

[0050] (1) By establishing an acoustic-vibration coupling model of a multi-resonator combination local resonant periodic structure that considers the influence of fluid load, the sound pressure function of the radiated sound field and the transverse bending displacement function of the substrate are substituted into the acoustic-vibration coupling model of the local resonant periodic structure. The matrix form of the acoustic-vibration coupling model of the underwater multi-resonator combination local resonant periodic structure is calculated. The preset frequency is substituted into the matrix form of the acoustic-vibration coupling model of the underwater multi-resonator combination local resonant periodic structure. The transverse bending displacement of the substrate is obtained through the transverse bending displacement function of the substrate. The vibration transmissibility of the local resonant periodic structure plate can be obtained. The vibration suppression frequency band fusion bandwidth is defined to ensure the widening of the vibration suppression frequency band. A true broadband vibration suppression effect can be obtained. The frequency combination effect of multiple resonators or multiple degrees of freedom resonators is improved, and the broadband vibration suppression advantage of the local resonant structure of multiple resonators or multiple degrees of freedom resonators is fully utilized.

[0051] (2) After obtaining the optimization objective function by defining the vibration suppression frequency band fusion bandwidth, the resonant frequency of the oscillator is optimized by using a genetic algorithm to obtain the widest vibration suppression frequency band, realize the flexible design of the local resonance structure, and greatly broaden the vibration suppression frequency band while taking into account the low frequency characteristics, so as to meet the vibration reduction requirements in actual engineering. Attached Figure Description

[0052] To more clearly illustrate the technical solutions in the embodiments of the present invention or the prior art, the drawings used in the description of the embodiments or the prior art will be briefly introduced below. Obviously, the drawings described below are only some embodiments of the present invention. For those skilled in the art, other drawings can be obtained based on these drawings without creative effort.

[0053] Figure 1 This is a flowchart of the low-frequency broadband vibration suppression method of the multi-resonator combination structure of the present invention;

[0054] Figure 2This is a schematic diagram of the local resonance structure of the multi-resonator combined structure low-frequency broadband vibration suppression method of the present invention, which is simply supported on four sides on an infinitely large rigid baffle.

[0055] Figure 3 This is a flowchart of the genetic algorithm for the low-frequency broadband vibration suppression method of the multi-resonator combination structure of the present invention.

[0056] Figure 4 This is a cell diagram of the low-frequency broadband vibration suppression method of the multi-resonator combination structure of the present invention;

[0057] Figure 5 This is a schematic diagram of the location for extracting the vibration response on the substrate in the low-frequency broadband vibration suppression method of the multi-resonator combination structure of the present invention.

[0058] Figure 6 The average vibration response diagram of the substrate after optimization of the low-frequency broadband vibration suppression method of the multi-resonator combination structure of the present invention is shown.

[0059] Figure 7 The image shows the radiated acoustic power of the substrate after optimization of the low-frequency broadband vibration suppression method of the multi-resonator combination structure of the present invention. Detailed Implementation

[0060] The technical solutions of the present invention will be clearly and completely described below with reference to the embodiments of the present invention. Obviously, the described embodiments are only a part of the embodiments of the present invention, and not all of the embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those of ordinary skill in the art without creative effort are within the scope of protection of the present invention.

[0061] like Figure 1-7 As shown, this invention discloses a low-frequency broadband vibration suppression method using a multi-resonator combination structure. The multi-resonator combination local resonant periodic structure comprises cells, each containing multiple periodically arranged resonators for sound absorption. The resonant frequencies of each resonator are optimized using a genetic algorithm. The specific steps include:

[0062] This embodiment uses a multi-resonator localized resonance structure plate simply supported on four sides on an infinitely large rigid baffle as an example. Figure 2 As shown, the substrate has a length of a, a width of b, and a thickness of h. p In the diagram, the side where z < 0 is the dry surface, and the area where z > 0 is the wet surface, which is an infinitely large body of water.

[0063] S1, when the local resonant structure is embedded in a four-sided simply supported infinite rigid baffle, the sound pressure function of the radiated sound field is constructed by Rayleigh integral formula.

[0064] The expression for the sound pressure function of the radiated sound field is as follows:

[0065]

[0066] In the formula, ρ is the vibration velocity on the substrate surface; ω is the angular frequency; f Let R be the density of water, R be the distance from the field point r′=(x′,y′,z′) to a point r=(x,y,0) on the substrate surface, a be the length of the substrate, b be the width of the substrate, k be the wave number, x and y be the position coordinates on the substrate, and j be the imaginary unit.

[0067] S2, for the case where the four sides of the substrate are simply supported by an infinitely large rigid barrier, construct the lateral bending displacement function of the substrate. The lateral bending displacement of the substrate is expressed as a linear superposition of vibration mode functions.

[0068] The expression for the lateral bending displacement w(x, y) of the substrate is as follows:

[0069]

[0070] Among them, W mn Let A be the (m, n) order vibration mode function; mn Let be the vibration coefficients corresponding to the vibration mode functions; the expression for its (m, n)th order vibration mode function is:

[0071] W mm (x, y) = sin(k) m x)sin(k n y);

[0072] In the formula, k m =mπ / a; k n = nπ / b, where a is the length of the substrate and b is the width of the substrate.

[0073] S3. Establish an acoustic-vibration coupling model for a multi-resonator combined local resonant periodic structure that considers the influence of fluid load.

[0074] The acoustic-vibration coupling model expression for the locally resonant periodic structure is as follows:

[0075]

[0076] Among them, D P The bending stiffness of the substrate is expressed as: D p =E p h p 3 / 12(1-ν p 2 In the formula, E p v is the Young's modulus of the substrate. p The Poisson's ratio of the substrate; The expression is: ω is the angular frequency; ρ p h is the density of the substrate. p The thickness of the substrate is given by ; w(r) is the lateral bending displacement of the substrate; F0 is the harmonic point force acting at position (x0, y0) on the substrate; p(x, y, 0) is the sound pressure on the substrate surface; F s The reaction force of each resonator on the substrate is expressed as: For a single-degree-of-freedom resonator Where m s Let k be the mass of the s-th resonator within the cell. s R is the stiffness of the s-th resonator within the cell; t =(x t y t R represents the vector pointing from the origin to the center point of the t-th cell, indicating the position of the cell. s =(r 1s r 2s ) represents the vector pointing from the center point of the cell to the s-th resonator within the cell, representing the relative position of the resonators within the cell.

[0077] S4. Substitute the radiated sound field pressure function and the substrate lateral bending displacement function into the acoustic-vibration coupling model of the local resonant periodic structure to calculate the matrix form of the acoustic-vibration coupling model of the underwater multi-resonator combined local resonant periodic structure.

[0078] It should be noted that after substituting the radiated sound field pressure function and the substrate lateral bending displacement function into the acoustic-vibration coupling model of the local resonant periodic structure, the infinite series is then truncated, i.e., the upper limit of m is M and the upper limit of n is N. The model is then simplified using the orthogonality of the integral of the sine and cosine functions. Finally, the acoustic-vibration coupling model of the underwater multi-resonator combined local resonant periodic structure in matrix form is obtained.

[0079] The acoustic-vibration coupling model of the matrix-form underwater multi-resonator combined local resonant periodic structure is expressed as follows:

[0080] (K+jωZ-ω 2 MF a A = F;

[0081] In the formula, the elements in matrices K, M, Z, and F are respectively:

[0082]

[0083] Among them, K i，i In the matrix K, the subscript (i, i) represents the i-th row and i-th column of the matrix K; M i，i In this context, (i, i) represents the i-th row and i-th column of matrix M; Fi，1 In this context, (i, 1) represents the i-th row and 1-th column of matrix F; Z i，s In this context, (i, s) represents the i-th row and s-th column of matrix Z;

[0084] Matrix F a Represented as:

[0085]

[0086] The elements in matrix W are:

[0087] W st i，1 =sin(k) m (x t +r 1s sin(k) n (y t +r 2s )).

[0088] The meanings of the parameters in the calculation formula in this step are the same as the meanings of the corresponding parameters in the calculation formula above, and will not be elaborated further here.

[0089] S5. Substitute the preset frequency into the acoustic-vibration coupling model of the matrix-form underwater multi-resonator combined local resonant periodic structure to obtain the vibration coefficients corresponding to the vibration mode function, and substitute the vibration coefficients into the substrate lateral bending displacement function to obtain the substrate lateral bending displacement.

[0090] It should be noted that by substituting any given frequency into the acoustic-vibration coupling model of a matrix-form underwater multi-resonator combined local resonant periodic structure, the vibration coefficient A corresponding to the vibration mode function can be obtained. mn Then the vibration coefficient A mn Substituting these values ​​into the substrate's lateral bending displacement function yields the substrate's bending displacement.

[0091] S6, obtain the corresponding vibration velocity response based on the lateral bending displacement of the substrate.

[0092] The expression for calculating the vibration velocity response is as follows:

[0093] v = jω * w;

[0094] In the formula, v is the vibration velocity response, ω is the angular frequency, and w is the lateral bending displacement of the substrate.

[0095] S7, obtain the vibration transmissibility of the local resonant periodic structure based on the vibration velocity response.

[0096] In step S7, the vibration transmissibility of the local resonant periodic structure is obtained based on the vibration velocity response. This step also includes the following steps:

[0097] S71, select several points on the substrate and extract the vibration response at each point;

[0098] S72, calculate the vibration transmissibility of the local resonant periodic structure based on the vibration velocity response and vibration transfer function;

[0099] The expression for the vibration transfer function is:

[0100]

[0101] In the formula, v f For the velocity response at the excitation point, v ri The velocity response at each point along the selected transmission path.

[0102] S8 defines the vibration suppression frequency band fusion bandwidth based on the vibration transmissibility of the local resonant periodic structure, and obtains the optimized objective function.

[0103] The expression is defined as follows:

[0104] Bw m =max(Ω), {TL(Ω)<0, f s ∈Ω};

[0105] In the formula, B Wm To optimize the objective function, Ω represents the frequency range where the vibration transmissibility TL < 0. In the vibration transmissibility curve of the local resonant structure, the maximum frequency range where TL < 0 includes the resonant frequencies of all resonators is the vibration suppression frequency band fusion bandwidth.

[0106] S9 uses a genetic algorithm to optimize the resonant frequency of the resonator contained in the vibration transmissibility in order to obtain the widest vibration suppression frequency band.

[0107] It should be noted that the mass and stiffness of the resonator are used as optimization variables in the genetic algorithm, the mass ratio is used as a constraint, and the maximum width B of the vibration suppression frequency band is used as the optimization factor. Wm The maximum vibration attenuation within the vibration suppression frequency band is used as the objective function for multi-objective optimization. Finally, a set of optimal variables is obtained, which yields the widest vibration suppression frequency band.

[0108] The specific steps are as follows: given the substrate structure and material parameters, lattice constant, and stiffness parameters of the resonator [k1, k2, ... k] n ] and mass parameters [m1, m2, ... m n The binary encoding method is used to optimize the design variables [k1, k2, ... k]. n [m1, m2, ... m] nThe chromosomes are encoded and used as chromosomes in the genetic algorithm. The first generation population is randomly generated, and the fitness is calculated. It is determined whether the chromosomes meet the optimization rules. If they do, the optimal solution for the resonator parameter distribution is output. If they do not meet the rules, the fitness is calculated in each evolution process, and the fitness is sorted from largest to smallest. The corresponding oscillator parameter group is evolved using a roulette wheel selection mechanism until the fitness no longer changes. During the evolution process, the crossover probability of the resonator parameter group can be set to 0.8, and the mutation probability can be set to 0.1.

[0109] It should be noted that the computation process of the genetic algorithm is as follows: Figure 3 As shown, the computation process within the genetic algorithm is existing technology and will not be elaborated upon here.

[0110] Understandably, by optimizing the mass and stiffness parameters of the resonator using a genetic algorithm, the optimal resonator parameter distribution can be obtained, thereby achieving the widest vibration suppression frequency band. This enables flexible design of local resonant structures and significantly broadens the vibration suppression frequency band while taking into account low-frequency characteristics, thus meeting the vibration reduction requirements in practical engineering.

[0111] Example

[0112] like Figure 4 and Figure 5 As shown, this embodiment uses a hexagonal cell structure composed of six resonators. Four points on the substrate are selected as examples, and the relevant parameters of the substrate and cell structure are set as detailed in the table below:

[0113]

[0114]

[0115] The table also includes ρ f The density of water is 1000 kg / m³. 3 The speed of sound propagation in water at room temperature is 1500m / s. In order to save computation, the mass of each resonator is set to be the same and the damping coefficient is kept consistent. At the same time, in order to facilitate the optimization of the resonator's resonant frequency and without changing the mass ratio of the resonator to the substrate, the resonator mass is given in the form of mass ratio in the calculation process. The resonator stiffness is calculated based on the resonator's resonant frequency and mass.

[0116] like Figure 5 As shown, to visually demonstrate the propagation of vibration along the substrate, four points located on a straight line were selected on the substrate surface in this embodiment to extract their vibration responses for analysis. The vibration transmissibility of each point along the selected transmission path was calculated based on the vibration velocity response and vibration transfer function. The fusion bandwidth for each vibration suppression frequency band was defined based on the vibration transmissibility. Figure 6 and Figure 7 As shown, the optimization objective function B is obtained. Wm Then, a genetic algorithm is used to optimize the resonant frequency of the resonator to obtain the widest vibration suppression frequency band.

[0117] The above description is only a preferred embodiment of the present invention and is not intended to limit the present invention. Any modifications, equivalent substitutions, improvements, etc., made within the spirit and principles of the present invention should be included within the protection scope of the present invention.

Claims

1. A low-frequency broadband vibration suppression method for a multi-resonator composite structure, characterized in that, Includes the following steps: S1, When the local resonant structure is embedded in a four-sided simply supported infinite rigid baffle, the sound pressure function of the radiated sound field is constructed by Rayleigh integral formula; S2, For the case where the four sides of the substrate are simply supported by an infinitely large rigid barrier, construct the lateral bending displacement function of the substrate. The lateral bending displacement of the substrate is expressed as a linear superposition of vibration mode functions. S3. Establish an acoustic-vibration coupling model for a multi-resonator combined local resonant periodic structure that considers the influence of fluid load. S4. Substitute the radiated sound field pressure function and the substrate lateral bending displacement function into the acoustic-vibration coupling model of the local resonant periodic structure to calculate the matrix form of the acoustic-vibration coupling model of the underwater multi-resonator combined local resonant periodic structure. S5, substitute the preset frequency into the acoustic-vibration coupling model of the matrix-form underwater multi-resonator combination local resonant periodic structure, obtain the vibration coefficient corresponding to the vibration mode function, and substitute the vibration coefficient into the substrate lateral bending displacement function to obtain the substrate lateral bending displacement. S6, obtain the corresponding vibration velocity response based on the lateral bending displacement of the substrate; S7, obtain the vibration transmissibility of the local resonant periodic structure based on the vibration velocity response; S8. Define the vibration suppression frequency band fusion bandwidth based on the vibration transmissibility of the local resonant periodic structure to obtain the optimization objective function; S9 uses a genetic algorithm to optimize the resonant frequency of the resonator contained in the vibration transmissibility in order to obtain the widest vibration suppression frequency band.

2. The low-frequency broadband vibration suppression method for a multi-resonator composite structure as described in claim 1, characterized in that: In step S1, when the local resonant structure is embedded in a four-sided simply supported infinite rigid baffle, the sound pressure function of the radiated sound field is constructed using the Rayleigh integral formula, and its expression is: In the formula, ρ is the vibration velocity on the substrate surface; ω is the angular frequency; f Let R be the density of water, R be the distance from the field point r′=(x′,y′,z′) to a point r=(x,y,0) on the substrate surface, a be the length of the substrate, b be the width of the substrate, k be the wave number, x and y be the position coordinates on the substrate, and j be the imaginary unit.

3. The low-frequency broadband vibration suppression method for a multi-resonator composite structure as described in claim 2, characterized in that: In step S2, for the case where the four sides of the substrate are simply supported by an infinitely large rigid barrier, a lateral bending displacement function of the substrate is constructed. The lateral bending displacement w(x,y) of the substrate is expressed as a linear superposition of vibration mode functions, and the expression is: Among them, W mn Let A be the (m, n) order vibration mode function; mn Let be the vibration coefficients corresponding to the vibration mode functions; the expression for its (m, n)th order vibration mode function is: W mn (x,y)=sin(k m x)sin(k n and); In the formula, k m =mπ / a; k n = nπ / b, where a is the length of the substrate and b is the width of the substrate.

4. The low-frequency broadband vibration suppression method for a multi-resonator composite structure as described in claim 3, characterized in that: In step S3, an acoustic-vibration coupling model of a multi-resonator combined local resonant periodic structure considering the influence of fluid load is established, and its expression is: Among them, D P The bending stiffness of the substrate is expressed as: D p =E p h p 3 / 12(1-v p 2 In the formula, E p v is the Young's modulus of the substrate. p The Poisson's ratio of the substrate; The expression is: ω is the angular frequency; ρ p h is the density of the substrate. p The thickness of the substrate is given by ; w(r) is the lateral bending displacement of the substrate; F0 is the harmonic point force acting at position (x0, y0) on the substrate; p(x, y, 0) is the acoustic pressure on the substrate surface; F s The reaction force of each resonator on the substrate is expressed as: For a single-degree-of-freedom resonator Where m s Let k be the mass of the s-th resonator within the cell. s R is the stiffness of the s-th resonator within the cell; t =(x t ,y t R represents the vector pointing from the origin to the center point of the t-th cell, indicating the position of the cell. s =(r 1s ,r 2s ) represents the vector pointing from the center point of the cell to the s-th resonator within the cell, representing the relative position of the resonators within the cell.

5. The low-frequency broadband vibration suppression method for a multi-resonator composite structure as described in claim 4, characterized in that: In step S4, the radiated sound field pressure function and the substrate lateral bending displacement function are substituted into the acoustic-vibration coupling model of the local resonant periodic structure. Then, the infinite series is truncated and simplified using the orthogonality of the integral of the sine and cosine functions. Finally, the acoustic-vibration coupling model of the underwater multi-resonator combined local resonant periodic structure in matrix form is obtained.

6. The low-frequency broadband vibration suppression method for a multi-resonator composite structure as described in claim 5, characterized in that: The acoustic-vibration coupling model of the matrix-form underwater multi-resonator combined local resonant periodic structure in step S4 is expressed as follows: (K+jωZ-ω 2 MF a )A=F; In the formula, the elements in matrices K, M, Z, and F are respectively: Matrix F a Represented as: The elements in matrix W are: W sti，1 =sin(k m (x t +r 1s ))sin(k n (and t +r 2s ))。 7. The low-frequency broadband vibration suppression method for a multi-resonator composite structure as described in claim 6, characterized in that: In step S6, the corresponding vibration velocity response is obtained based on the substrate displacement, and the calculation expression is: v = jω * w; In the formula, v is the vibration velocity response, ω is the angular frequency, and w is the lateral bending displacement of the substrate.

8. The low-frequency broadband vibration suppression method for a multi-resonator composite structure as described in claim 7, characterized in that: Step S7, which obtains the vibration transmissibility of the local resonant periodic structure based on the vibration velocity response, further includes the following steps: S71, select several points on the substrate and extract the vibration response at each point; S72, calculate the vibration transmissibility of the local resonant periodic structure based on the vibration velocity response and vibration transfer function; The expression for the vibration transfer function is: In the formula, v f For the velocity response at the excitation point, v ri The velocity response at each point along the selected transmission path.

9. The low-frequency broadband vibration suppression method for a multi-resonator composite structure as described in claim 8, characterized in that: In step S8, the vibration suppression frequency band fusion bandwidth is defined based on the vibration transmissibility of the local resonant periodic structure to obtain the optimization objective function, wherein the expression is defined as: Bw m =max(Ω),{TL(Ω)<0,f s ∈Ω}; In the formula, B Wm To optimize the objective function, Ω represents the frequency range where the vibration transmissibility TL < 0. In the vibration transmissibility curve of the local resonant structure, the maximum frequency range where TL < 0 includes the resonant frequencies of all resonators is the vibration suppression frequency band fusion bandwidth.

10. The low-frequency broadband vibration suppression method for a multi-resonator composite structure as described in claim 9, characterized in that: In step S9, a genetic algorithm is used to optimize the resonant frequencies of the resonator included in the vibration transmissibility to obtain the widest vibration suppression frequency band. The mass and stiffness of the resonator are used as optimization variables in the genetic algorithm, with the mass ratio as a constraint, and the maximum width B of the vibration suppression frequency band as the optimization factor. Wm The maximum vibration attenuation within the vibration suppression frequency band is used as the objective function for multi-objective optimization. Finally, a set of optimal variables is obtained, which yields the widest vibration suppression frequency band.

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