An acoustic metamaterial equivalent multi-degree-of-freedom vibration modeling method

Abstract

The present application relates to the field of acoustic metamaterial modeling design, in particular to an acoustic metamaterial equivalent multi-degree-of-freedom vibration modeling method, comprising the following: forming a structured parameter table; after substituting the structured parameter table, deriving the relevant transfer matrix between the adjacent interfaces of each sub-domain, and obtaining the normalized surface impedance and sound absorption coefficient calculation formula; mapping the impedance term and the equivalent vibration element one by one to construct an equivalent multi-degree-of-freedom vibration model. The present application analyzes and expands the matrix expression of the multilayer structure under the framework of the classical transfer matrix method, rewrites it into an explicit equivalent impedance form, and constructs an equivalent multi-degree-of-freedom vibration model which explicitly depicts the one-to-one correspondence between the structure parameters and the equivalent mass, damping and stiffness, clearly reveals the coupling mechanism between the structure parameters, and solves the problem of implicit parameter-acoustic performance mapping relationship and low physical transparency in the traditional transfer matrix method.

Description

Technical Field

[0001] This invention relates to the field of acoustic metamaterial modeling and design, and specifically to a method for modeling the equivalent multi-degree-of-freedom vibration of acoustic metamaterials. Background Technology

[0002] Acoustic metamaterials, as artificial acoustic composite structures, can achieve efficient low-frequency sound absorption at the subwavelength scale. They can actively control the reflection, transmission, and absorption behavior of sound waves, providing a new technical approach for small-sized structures to act on large-wavelength low-frequency sound waves, and have become one of the core development directions of the next generation of advanced sound-absorbing structures. Currently, for engineering needs of low-frequency broadband sound absorption, acoustic metamaterials often adopt a periodic configuration of multiple basic units coupled in series and parallel. Through complex coupling effects, sound absorption performance is improved and structural size is reduced. The transfer matrix method is the most widely used modeling method in the acoustic response analysis of such acoustic metamaterials.

[0003] However, while the transfer matrix method is suitable for calculating the acoustic properties of complex layered structures, it condenses the coupling relationships of multi-layered structures into matrix multiplication. The influence of individual geometric or material parameters on macroscopic sound absorption performance cannot be explicitly presented. In parameter inversion and tuning design such as thickness compression, the physical transparency is severely insufficient, making it difficult to clarify the coupling mechanism between structural parameters. At the same time, current thickness compression and lightweight design of acoustic metamaterials often combine the transfer matrix method with numerical optimization algorithms to search for structural schemes that meet the target sound absorption performance within a given parameter constraint range. Such methods not only require a large number of iterations and trial searches in the parameter space, resulting in high computational costs, but are also highly sensitive to parameter value ranges and initial conditions, easily converging to local optima and making it difficult to obtain a globally optimal design scheme.

[0004] For multi-unit acoustic metamaterials with series and parallel coupling, the system degrees of freedom and acoustic modeling complexity are significantly increased due to the coupling effect, making thickness compression a more challenging problem. Traditional design methods, lacking an explicit mapping relationship between structural parameters and acoustic performance, cannot perform targeted dimensional corrections for structures that do not meet requirements. They rely excessively on various auxiliary tools, resulting in low design efficiency and poor physical interpretability of the resulting designs, making it difficult to fully utilize existing design experience for precise parameter tuning. Furthermore, the adjustable degrees of freedom of a single acoustic metamaterial unit are limited, and the contradiction between low-frequency sound absorption performance and subwavelength thickness always exists. Traditional design methods cannot provide reliable dimensional adjustment criteria for thickness compression within an analytical framework, making it difficult to achieve efficient lightweight design while ensuring sound absorption performance. Summary of the Invention

[0005] This invention addresses the technical problems existing in the prior art by providing a method for modeling the equivalent multi-degree-of-freedom vibration of acoustic metamaterials.

[0006] The technical solution of this invention to solve the above-mentioned technical problems is as follows: A method for modeling the equivalent multi-degree-of-freedom vibration of acoustic metamaterials, comprising the following steps: S1. Based on the engineering requirements of low-frequency broadband sound absorption, the series-connected neck-embedded Helmholtz resonator is selected as the target configuration of acoustic metamaterial. The geometric dimensions of each component of the target configuration of acoustic metamaterial are defined and symbolically marked to form a structured parameter table. S2. Based on the transfer matrix method, after completing the interface segmentation and state variable definition according to the direction of sound wave propagation, the general form of the transfer matrix between adjacent interfaces is constructed. The total transfer matrix is ​​solved for the general form of the transfer matrix, and the recursive relationship of the normalized impedance is derived. The expression of the overall normalized surface impedance is derived. After substituting into the structured parameter table, the relevant transfer matrix between adjacent interfaces of each subdomain is derived, and the calculation formulas of normalized surface impedance and sound absorption coefficient are obtained. S3. Explicitly expand and algebraically simplify the relevant transfer matrices between adjacent interfaces of each subdomain to eliminate the matrix form and derive the formula for the ratio of sound pressure to particle velocity at each interface. Transform the matrix relationship into an arithmetic relationship of impedance. Then, eliminate the intermediate variables, including the sound pressure and particle velocity at the intermediate interface, from the formula for the ratio of sound pressure to particle velocity at each interface to obtain the normalized explicit formula for surface impedance. Map the impedance terms one by one with the equivalent vibration elements to construct an equivalent multi-degree-of-freedom vibration model.

[0007] In a preferred embodiment, based on the engineering requirements of low-frequency broadband sound absorption, S1 selects a neck-embedded Helmholtz resonator consisting of a first cavity, a second cavity, a first neck, a second neck, and wall thickness regions at different locations as the target configuration for acoustic metamaterials. The first neck refers to the connection between the first cavity and the external air domain, and the second neck refers to the connection between the first cavity and the second cavity. It should be noted that this application uses CNEHR acoustic metamaterials as the specific implementation subject, and performs systematic modeling and analytical decoupling of its acoustic impedance within the framework of the transfer matrix method. Please refer to the appendix. Figure 2 A CNEHR unit contains two wide cavities and two narrow extension tubes.

[0008] In a preferred embodiment, S1 further includes: The geometric dimensions and symbols of each component of the acoustic metamaterial target configuration are defined and marked. Specifically, the characteristic dimensions, height, inner diameter of the neck, length of each tube segment, and wall thickness at different positions of the cylindrical and / or square cavities are marked. The air physical property parameters, including air density, sound velocity, aerodynamic viscosity coefficient, and the calculation parameters required for viscosity and heat dissipation, are determined and integrated into a structured parameter table.

[0009] In a preferred embodiment, S2 divides the acoustic metamaterial target configuration into x1-x5 interfaces perpendicular to the direction of sound wave propagation, from the incident end to the end of the acoustic metamaterial. In this application, this is physically expressed as dividing the CNEHR into five interfaces (x1-x5) perpendicular to the direction of sound wave propagation, and defining the sound pressure p of each interface. i and particle velocity v i Wherein, sound pressure refers to the pressure exerted on the medium by the sound wave, and particle velocity refers to the vibration velocity of the medium particles under the action of the sound wave. After completing the interface segmentation and defining the state variables, the linear mapping relationship of the state variables between adjacent interfaces is derived according to the acoustic wave equation, resulting in the general form of the transfer matrix: Where N is the total number of interfaces, i.e., the number of interfaces divided by the target configuration of the acoustic metamaterial, p i v i M represents the sound pressure and particle velocity at the i-th interface, respectively. i M is the transfer matrix between the i-th interface and the (i+1)-th interface. i (1,1), M i (1,2) are M i The elements in.

[0010] In a preferred embodiment, step S2 then multiplies the transfer matrices of all adjacent interfaces sequentially according to the sound wave propagation order to complete the general form solution of the transfer matrix, obtaining the formula for the total transfer matrix: .

[0011] Where T is the total transfer matrix after multiplying all transfer matrices. In a preferred embodiment, S2 further includes: According to the common definition in acoustic metamaterial theoretical models, the ratio of sound pressure to particle velocity divided by the characteristic impedance of air is used as the normalized impedance, and its mathematical expression is: Expanding the general form of the obtained transfer matrix and substituting the normalized impedance into the general form of the transfer matrix, we obtain the recursive formula for the normalized impedance of adjacent interfaces: Where p0 and c0 are air density and speed of sound, respectively. For the existence of a definite functional relationship, since the function f in equation (3) iThe coefficients are determined by the elements in the transfer matrix, and all elements are uniquely determined by the geometric dimensions of the acoustic metamaterial and the physical properties of air. Therefore, equation (3) represents that there is a definite functional relationship between the normalized impedance on the adjacent interfaces of the acoustic metamaterial. Extending the recursive relationship of the recursive formula to the entire target configuration of the acoustic metamaterial, we obtain the overall normalized surface impedance expression: z1=f1(z2)=f1(f2(z3))=……=f1(f2(……fN-1(zN))), (4); Where z1 is the normalized surface impedance of the acoustic metamaterial, z N The normalized backplane impedance, i.e., z N =+∞; Substituting the structured parameter table obtained in S1, the relevant transfer matrices of adjacent interfaces of each subdomain are derived, specifically: Where M a1 Z is the transfer matrix of interface x1-x2. a1 For the first neck impedance, p c1 c c1 k c1 These represent the complex density, complex speed of sound, and complex wave number of the air within the first neck region, respectively, where η is the aerodynamic viscosity coefficient, and A is the complex density, complex speed of sound, and complex wave number. a1 This is the cross-sectional area of ​​the first neck. The transfer matrix M of interface x2-x3 v1 for: Among them, A v1 Z is the cross-sectional area of ​​the first cavity. r1 For the impedance of the toroidal air domain; Furthermore, since the effects of viscosity and heat dissipation within the cavity have a relatively weak impact on the acoustic response of the system, this application neglects the corresponding dissipation mechanism in this region, assuming that the density, sound velocity, and wavenumber of the air medium can all be approximated as real constants, and the transfer matrix M of the interface x3-x4 is... a2 for: Among them, Z a2 For the second neck impedance, Z r2 For the impedance of the toroidal air domain; The transfer matrix M of interface x4-x5 v2 for: Among them, A v2 Z is the cross-sectional area of ​​the second cavity. r3 It represents the impedance of the toroidal air domain.

[0012] In a preferred embodiment, S2 further includes: Multiplying the relevant transfer matrices between each interface yields the total transfer matrix for interfaces x1-x5, specifically: Where interface x5 is the bottommost backplate of the CNEHR acoustic metamaterial target configuration, it is usually regarded as a hard acoustic field boundary, i.e., v5=0, from which the normalized surface impedance and sound absorption coefficient calculation formulas are obtained: Where z is the normalized surface impedance and α is the sound absorption coefficient. It should be noted that these two formulas are the core basis for all subsequent performance calculations, thickness compression and simulation experiments in this application. α directly characterizes the sound absorption performance of the metamaterial. The closer the value is to 1, the better the sound absorption effect.

[0013] In a preferred embodiment, step S3 explicitly expands and algebraically simplifies the correlation transfer matrix between adjacent interfaces of each subdomain to derive the formula for the ratio of sound pressure to particle velocity at each interface, specifically: (15); (18); Among them, Z part1 Z part2 Z represents the impedance of the cylindrical air domain within the first cavity. v2 This represents the impedance of the cylindrical air domain within the second cavity.

[0014] In a preferred embodiment, the formula for the ratio of sound pressure to particle velocity at each interface is modified by eliminating the intermediate variables of sound pressure and particle velocity at the intermediate interface, resulting in the normalized explicit formula for the surface impedance of the incident end x1: .

[0015] In a preferred embodiment, S3 is based on the correspondence between vibration theory and acoustic impedance, and according to the obtained normalized surface impedance explicit formula, the real part of the impedance of the neck corresponds to the equivalent damping element, the imaginary part corresponds to the equivalent mass element, the impedance of the annular air domain corresponds to the equivalent stiffness element, and the impedance of the cavity air domain corresponds to the equivalent stiffness element, thus finally constructing an equivalent multi-degree-of-freedom vibration model.

[0016] The beneficial effects of this invention are: under the framework of the classical transfer matrix method, the matrix expression of multi-layer structures is analytically expanded and rewritten into an explicit equivalent impedance form. The constructed equivalent multi-degree-of-freedom vibration model explicitly describes the one-to-one correspondence between structural parameters and equivalent mass, damping, and stiffness, clearly revealing the coupling mechanism between various structural parameters, and solving the problems of implicit parameter-acoustic performance mapping relationship and low physical transparency in the traditional transfer matrix method. An explicit analytical formula for the normalized surface impedance of acoustic metamaterials was derived. When using this model for thickness compression and lightweight design, the target sound absorption performance constraint can be directly transformed into analytical conditions for equivalent mass, stiffness, and damping parameters. This allows for the direct derivation of dimensional adjustment schemes that meet performance requirements. Furthermore, the complex acoustic wave-dissipation process is abstracted into a discrete vibration system composed of mass, stiffness, and damping elements. The physical meaning of each impedance term and the contribution of each part of the impedance to the overall normalized surface impedance are clarified. This enables designers to directly evaluate the influence of each structural parameter on the surface impedance and sound absorption characteristics, thereby making targeted corrections and tuning of the structural parameters. It fully utilizes existing design experience and solves the problems of traditional design methods lacking precise parameter tuning ideas and having poor physical interpretability. Based on the equivalent multi-degree-of-freedom vibration model, the thickness compression design of acoustic metamaterials can be carried out. Only a few key structural parameters need to be adjusted in a coordinated manner to meet the constraint of constant total impedance. Under the premise of basically maintaining the sound absorption performance, sound absorption peak position and peak value at the target resonant frequency, the acoustic metamaterials can be effectively lightweighted. This breaks through the inherent contradiction between the low-frequency sound absorption performance and subwavelength thickness of a single acoustic metamaterial unit. Furthermore, the model prediction accuracy is high and the design scheme has good engineering feasibility, as verified by finite element simulation and physical experiments. Attached Figure Description

[0017] Figure 1 This is a flowchart of the present invention; Figure 2 The diagram shows the structural unit of the CNEHR acoustic metamaterial in a specific implementation, where (a) is a schematic diagram of the model and (b) is a cross-sectional view with dimension annotations. Figure 3 This is the equivalent multi-degree-of-freedom vibration model corresponding to the CNEHR acoustic metamaterial unit of the present invention; Figure 4 This is a contour plot showing the variation of the sound absorption coefficient of the CNEHR acoustic metamaterial unit with frequency and the height of the second cavity after compression in a specific implementation. Figure 5 In a specific implementation, when the CNEHR acoustic metamaterial unit is compressed... l v2-c right L , l a3-c The influence curve; Figure 6 The image shows the sound absorption curves and cross-sectional views of the CNEHR acoustic metamaterial unit before and after compression in a specific implementation. Figure 7 This is a schematic diagram of the finite element simulation model in a specific implementation method; Figure 8 The simulation results are for the CNEHR acoustic metamaterial unit before and after compression. Detailed Implementation

[0018] The technical solutions of the embodiments of this application will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some embodiments of this application, and not all embodiments. Based on the embodiments of this application, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of this application.

[0019] As attached Figure 1-8 As shown, this embodiment provides a method for modeling the equivalent multi-degree-of-freedom vibration of acoustic metamaterials, including the following steps: S1. Based on the engineering requirements of low-frequency broadband sound absorption, the series-connected neck-embedded Helmholtz resonator is selected as the target configuration of acoustic metamaterial. The geometric dimensions of each component of the target configuration of acoustic metamaterial are defined and symbolically marked to form a structured parameter table. Based on the engineering requirements of low-frequency broadband sound absorption, S1 selected a neck-embedded Helmholtz resonator consisting of a first cavity, a second cavity, a first neck, a second neck, and wall thickness regions at different locations as the target configuration for acoustic metamaterials. The first neck refers to the connection between the first cavity and the external air domain, and the second neck refers to the connection between the first cavity and the second cavity. It should be noted that this application uses CNEHR acoustic metamaterials as the specific implementation subject, and performs systematic modeling and analytical decoupling of its acoustic impedance within the framework of the transfer matrix method. Please refer to the appendix. Figure 2 A CNEHR unit contains two large cavities and two narrow extension tubes. The geometric dimensions and symbols of each component of the neck-embedded Helmholtz resonator are defined and marked. Specifically, the characteristic dimensions, height, inner diameter of the neck, length of each tube segment, and wall thickness at different positions of the cylindrical and / or square cavities are marked. The air physical property parameters, including air density, sound velocity, aerodynamic viscosity coefficient, and the calculation parameters required for viscosity and heat dissipation, are determined and integrated into a structured parameter table. Please refer to the appendix. Figure 2 Cavity-1 represents the first cavity, cavity-2 represents the second cavity, neck-1 represents the first neck, neck-2 represents the second neck, and so on. Figure 2Five interfaces, x1-x5, were selected within a CNEHR unit based on cross-sectional variations. The characteristic dimension of the cylindrical cavity is the diameter of the cross-sectional circle, and the characteristic dimension of the square cavity is the larger side length of the cross-sectional rectangle. These two characteristic dimensions and their heights are denoted as dv1, lv1, dv2, and lv2, respectively. The extension tube connecting the first cavity and the external air domain is the first neck, with its inner diameter and length denoted as da1 and la1, respectively. The extension tube connecting the first and second cavities is the second neck, with an inner diameter of da2. The length of the portion extending into the first cavity is denoted as la2, and the remaining portion as la3. The structural parameters in the following derivations are all defined according to the above definitions.

[0020] S2. Based on the transfer matrix method, after completing the interface segmentation and state variable definition according to the direction of sound wave propagation, the general form of the transfer matrix between adjacent interfaces is constructed. The total transfer matrix is ​​solved for the general form of the transfer matrix, and the recursive relationship of the normalized impedance is derived. The expression of the overall normalized surface impedance is derived. After substituting into the structured parameter table, the relevant transfer matrix between adjacent interfaces of each subdomain is derived, and the calculation formulas of normalized surface impedance and sound absorption coefficient are obtained. Based on the direction of sound wave propagation, the acoustic metamaterial target configuration is divided into x1-x5 interfaces perpendicular to the direction of sound wave propagation, from the incident end to the end end of the acoustic metamaterial. In this application, its physical expression is that the CNEHR is divided into 5 interfaces (x1-x5) perpendicular to the direction of sound wave propagation, and the sound pressure p of each interface is defined. i and particle velocity v i Wherein, sound pressure refers to the pressure exerted on the medium by the sound wave, and particle velocity refers to the vibration velocity of the medium particles under the action of the sound wave. After completing the interface segmentation and defining the state variables, the linear mapping relationship of the state variables between adjacent interfaces is derived according to the acoustic wave equation, resulting in the general form of the transfer matrix: Where N is the total number of interfaces, i.e., the number of interfaces divided by the target configuration of the acoustic metamaterial, p i v i M represents the sound pressure and particle velocity at the i-th interface, respectively. i M is the transfer matrix between the i-th interface and the (i+1)-th interface. i (1,1), M i (1,2) are M i Elements in; Then, multiply the transfer matrices of all adjacent interfaces sequentially according to the sound wave propagation order to complete the general form of the transfer matrix, obtaining the formula for the total transfer matrix: .

[0021] Where T is the total transfer matrix after multiplying all transfer matrices.

[0022] According to the common definition in acoustic metamaterial theoretical models, the ratio of sound pressure to particle velocity divided by the characteristic impedance of air is used as the normalized impedance, and its mathematical expression is: Expanding the general form of the obtained transfer matrix and substituting the normalized impedance into the general form of the transfer matrix, we obtain the recursive formula for the normalized impedance of adjacent interfaces: Where p0 and c0 are air density and speed of sound, respectively. For the existence of a definite functional relationship, since the function f in equation (3) i The coefficients are determined by the elements in the transfer matrix, and all elements are uniquely determined by the geometric dimensions of the acoustic metamaterial and the physical properties of air. Therefore, equation (3) represents that there is a definite functional relationship between the normalized impedance on the adjacent interfaces of the acoustic metamaterial. Extending the recursive relationship of the recursive formula to the entire target configuration of the acoustic metamaterial, we obtain the overall normalized surface impedance expression: z1=f1(z2)=f1(f2(z3))=……=f1(f2(……fN-1(zN))), (4); Where z1 is the normalized surface impedance of the acoustic metamaterial, representing the acoustic properties of the entire acoustic metamaterial, and is the ultimate goal of the transfer matrix method. N The normalized backplate impedance represents the acoustic properties of the bottommost backplate of the acoustic metamaterial structure, and is usually considered as the hard acoustic field boundary, i.e., z N =+∞, because all functions f in equation (4) i All coefficients are uniquely determined by the geometric dimensions of the acoustic metamaterial and the physical properties of air. Therefore, for any acoustic metamaterial configuration that can be characterized by a transfer matrix, the corresponding transfer matrix can be analytically decoupled into an explicit equivalent impedance expression according to equation (4). Based on this explicit impedance framework, the corresponding equivalent multi-degree-of-freedom vibration model can be further deduced, the physical meaning of each impedance term can be clarified, and the one-to-one correspondence between the impedance term and the geometric parameters can be quantitatively characterized. After substituting the structured parameter table obtained from S1, the relevant transfer matrices of adjacent interfaces of each subdomain are derived, as follows: Where M a1 Z is the transfer matrix of interface x1-x2. a1 For the first neck impedance, p c1 c c1 k c1These represent the complex density, complex speed of sound, and complex wave number of the air within the first neck region, respectively, where η is the aerodynamic viscosity coefficient, and A is the complex density, complex speed of sound, and complex wave number. a1 This is the cross-sectional area of ​​the first neck. The transfer matrix M of interface x2-x3 v1 for: Among them, A v1 Z is the cross-sectional area of ​​the first cavity. r1 For the impedance of the toroidal air domain; Furthermore, since the effects of viscosity and heat dissipation within the cavity have a relatively weak impact on the acoustic response of the system, this application neglects the corresponding dissipation mechanism in this region, assuming that the density, sound velocity, and wavenumber of the air medium can all be approximated as real constants, and the transfer matrix M of the interface x3-x4 is... a2 for: Among them, Z a2 For the second neck impedance, Z r2 For the impedance of the toroidal air domain; The relevant physical quantities follow the calculation form in equation (6), only the total length l of the first neck needs to be included. a1 Replace with the second total neck length l a2 +l a3 and p c1 c c1 k c1 The first neck parameter with subscript 1 is replaced with the corresponding value of the second neck parameter, where the transfer matrix M of interface x4-x5 is... v2 for: Among them, A v2 Z is the cross-sectional area of ​​the second cavity. r3 The impedance of the annular air domain is the same as that of the first cavity. To highlight the impact of overall thickness compression on macroscopic acoustic performance and to simplify the model, the viscosity and heat dissipation effects inside the cavity are ignored here. Multiplying the relevant transfer matrices between each interface yields the total transfer matrix for interfaces x1-x5, specifically: Where interface x5 is the bottommost backplate of the CNEHR acoustic metamaterial target configuration, it is usually regarded as a hard acoustic field boundary, i.e., v5=0, from which the normalized surface impedance and sound absorption coefficient calculation formulas are obtained: Where z is the normalized surface impedance and α is the sound absorption coefficient. It should be noted that these two formulas are the core basis for all subsequent performance calculations, thickness compression and simulation experiments in this application. α directly characterizes the sound absorption performance of the metamaterial. The closer the value is to 1, the better the sound absorption effect.

[0023] S3. Explicitly expand and algebraically simplify the relevant transfer matrices between adjacent interfaces of each subdomain to eliminate the matrix form and derive the formula for the ratio of sound pressure to particle velocity at each interface. Transform the matrix relationship into an arithmetic relationship of impedance. Then, eliminate the intermediate variables, including the sound pressure and particle velocity at the intermediate interface, from the formula for the ratio of sound pressure to particle velocity at each interface to obtain the normalized explicit formula for surface impedance. Map the impedance terms one by one with the equivalent vibration elements to construct an equivalent multi-degree-of-freedom vibration model.

[0024] The relevant transfer matrices between adjacent interfaces of each subdomain are explicitly expanded and algebraically simplified, that is, the matrix forms of formulas (5), (7), (9), and (11) are eliminated to derive the formulas for the ratio of sound pressure to particle velocity at interfaces x1-x5 in CNEHR, specifically: (15); (18); Among them, Z part1 Z part2 Z represents the impedance of the cylindrical air domain within the first cavity. v2 The impedance of the cylindrical air domain inside the second cavity; all three are stiffness-type impedances characterizing the elasticity of the medium.

[0025] By eliminating the intermediate variables of sound pressure and particle velocity from the formulas for the ratio of sound pressure to particle velocity at each interface, the normalized explicit formula for the surface impedance at the CNEHR incident end x1 is obtained: Please refer to the appendix. Figure 3 For the purpose of illustrating formula (19), this application uses the appendix. Figure 3 The equivalent multi-degree-of-freedom vibration model corresponding to the acoustic metamaterial unit is revealed. Equation (19) contains multiple impedance expressions and is computationally complex, but in essence, there are only two operations between each impedance: one is the summation of impedances representing series connection, and the other is the summation of the reciprocals of impedances representing parallel connection and then taking the reciprocal. This means that there are only two connection methods between impedances: series and parallel. Therefore, the equivalent multi-degree-of-freedom vibration model corresponding to the CNEHR acoustic metamaterial unit can be constructed. In this model, Z a1 Z a2The imaginary part corresponds to the equivalent mass component, reflecting the inertial effect of the air column inside the neck; the real part corresponds to the equivalent damping component, used to characterize the energy loss caused by viscosity and heat dissipation, and the two are connected in series. The annular air domain formed by the insertion of the extension tube corresponds to the impedance Z. r1 Z r2 and Z r3 In the equivalent vibration model, these impedances are equivalent to stiffness elements connecting adjacent cavities and the neck, connected in parallel with the lower impedance and then in series with the upper impedance. The cylindrical air domain within the first cavity corresponds to impedance Z. part1 and Z part2 In the equivalent vibration model, it is equivalent to three stiffness elements, revealing that there are two vibration modes inside the first cavity of the CNEHR acoustic metamaterial unit; (1) Z part1 This represents a spring model where one end is connected to the air column at the neck, the other end is connected to the wall, and the other end is fixed and free when the medium vibrates.

[0026] (2) Z part2 This represents a spring model where both ends are connected to an air column at the neck during medium vibration, corresponding to a spring model where both ends are free. The connection method of the three is a Z-shaped pattern. part1 With a Z part2 After parallel connection, another Z is connected in series. part1 The cylindrical air domain within the second cavity corresponds to the impedance Z. v2 Its physical essence is the same as Z part1 Z part2 Same. When the x3 interface is replaced with a hard sound field boundary, one of the Z... part1 Failure, remaining Z part1 and Z part2 They form an impedance series relationship, and the sum of the two is: .

[0027] Furthermore, Z is given here. part1 Z part2 The specific expression: ; .

[0028] In summary, S3, based on the correspondence between vibration theory and acoustic impedance, and according to the obtained normalized explicit formula for surface impedance, assigns the real part of the neck impedance to an equivalent damping element, characterizing the energy loss caused by air viscosity and heat dissipation within the narrow neck region; the imaginary part to an equivalent mass element, characterizing the inertial effect of the neck air column under the action of sound waves; assigns the annular air domain impedance to an equivalent stiffness element, characterizing the elastic vibration characteristics of the annular air domain formed after the extension tube is inserted into the cavity; and assigns the cavity air domain impedance to an equivalent stiffness element, characterizing the elastic vibration characteristics of the cylindrical air domain within the cavity. Where Z... part1A spring model with one end fixed and free, Z part2 Corresponding to the spring model with free ends, an equivalent multi-degree-of-freedom vibration model is finally constructed, as shown in the attached figure. Figure 3 As shown.

[0029] In some other specific embodiments, this application experimentally illustrates the application of acoustic metamaterials in specific designs based on the equivalent multi-degree-of-freedom vibration model and its related formulas obtained above, further verifying the feasibility of the equivalent multi-degree-of-freedom vibration model disclosed in this application, as follows: Equation (19) gives an explicit analytical form of the normalized surface impedance of the CNEHR acoustic metamaterial unit, separating the impedance of each structural subdomain from the overall transfer matrix and constructing a set of modular impedances with clear physical meaning. Based on this expression, the contribution of each structural parameter to the surface impedance and sound absorption characteristics can be directly evaluated, providing a clear physical basis for parameter tuning and thickness compression. In contrast, the traditional design process based on the transfer matrix method combined with numerical optimization often requires a large number of iterations and trial searches within a given parameter space, resulting in high computational costs.

[0030] The root cause lies in the fact that the mapping relationship between geometric parameters and sound absorption performance is implicit in matrix multiplication and lacks analytical transparency. Based on an equivalent multi-degree-of-freedom vibration model, this application transforms the target sound absorption performance constraint into analytical conditions for equivalent mass, stiffness, and damping parameters. Thus, while retaining the original sound absorption characteristics, it directly derives a size adjustment scheme that meets the new engineering requirements, avoiding repeated design from scratch and significantly reducing design and testing costs.

[0031] right Figure 2 The CNEHR acoustic metamaterial unit shown is compressed in overall thickness while maintaining its original sound absorption performance as much as possible. According to equations (17) and (18), the ratio of sound pressure to particle velocity at interface x3 can be obtained: To achieve thickness compression of the acoustic metamaterial, the height l of the second cavity is changed. v2 It is necessary to adjust the parameters, according to the aforementioned impedance expression, l v2 When changed, only the impedance Z is affected. v2 The value of l. To maintain the overall sound absorption characteristics of the metamaterial, other impedances need to be adjusted simultaneously to balance l. v2 The change in impedance caused by the alteration. This application selects impedance Z... a2 With Z v2 To achieve a balance, the advantage of this approach lies in Z. a2 l in the expression a3 With l v2 They have the same characteristics, namely l a3 When changed, only the impedance Z is affected.a2 The value.

[0032] Although equations (6), (9), (12) and (18) show l a3 The impedance expression for the influence is Z. a2 Z r3 With Z v2 But in Z r3 With Z v2 The expressions all contain (l a3 The fixed combination of -b2) therefore only requires adjusting l a3 Simultaneously change b2 to make (l a3 -b2) remains unchanged, and l can be achieved. a3 Only affects Z a2 The effect.

[0033] Define the changing impedance Based on the above derivation, in (l a3 -b2) Under constant constraints, Z a3 Only subject to l a3 Adjustment effect, Z a2 Only subject to l a2 Adjustment effect, and these two structural parameters affect Z change Other impedances have no effect. Therefore, it can be deduced that only parameter l needs to be adjusted. a3 b2 and l v2 If the varying impedance is kept constant, then the total impedance will remain unchanged. However, this is a relatively ideal theoretical derivation; in reality, Z... a2 The real part is not zero, and Z r3 With Z v2 Z is a purely imaginary number, and obviously... a2 Since the real part of the impedance does not have a tuning component when it changes, the varying impedance cannot be strictly kept constant in practical applications. Based on this characteristic of varying impedance, this application constructs the following thickness compression scheme for the CNEHR acoustic metamaterial unit: (1) Determine the target resonance frequency f regoal ; (2) Calculate the target resonance frequency f regoal Corresponding Z change ; (3) Given the height l of the compressed second cavity v2-c And calculate the impedance of the second cavity after compression. (4) Calculate the second neck impedance after compression: And decouple the compressed l from it a3-c Due to the problem that the aforementioned varying impedance cannot be strictly maintained constant, the obtained l a3-cIt may be a complex number. After verification, its imaginary part is extremely small. Even if it is ignored, it will have little impact on the final result. Therefore, this application takes the real part.

[0034] Compared to other metamaterial units, CNEHR units have the significant characteristic of forming multiple resonance peaks from a single unit. To evaluate the impact of this compression scheme on other additional resonant frequencies, this application performs a compression design on the CNEHR acoustic metamaterial units shown in Table 1. The target resonant frequency is set to f. regoal =242Hz, the two cavities of the unit are uniformly cylindrical cavities with a diameter of 45mm.

[0035] Table 1

[0036] This application addresses the height l of the compressed second cavity. v2-c Scans were performed at 1mm intervals to obtain the sound absorption curves of the CNEHR acoustic metamaterial unit as a function of l. v2-c The trend of change is as follows Figure 4 As shown. Figure 4 The horizontal axis represents the sound absorption frequency, and the vertical axis represents the height of the second cavity. Different colors represent changes in the sound absorption coefficient. Before compression, i.e., l v2-c When = 50mm, Figure 4 The CNEHR acoustic metamaterial unit exhibits two resonant frequencies: a target resonant frequency of 242Hz and an additional resonant frequency of 96Hz. With l v2-c As the absorption coefficient gradually decreases, the absorption coefficient at the target resonant frequency of 242Hz shows no significant change, and the peak position remains basically unchanged, which is in line with theoretical expectations; however, the absorption coefficient at the additional resonant frequency of 96Hz gradually decreases, and the peak position shifts towards higher frequencies.

[0037] Please refer to the appendix. Figure 5 , Figure 5 The neck length 3 is shown. a3-c Total thickness L varies with l v2-c The changing trend. With l v2-c The increase of l a3-c As L gradually decreases, it first decreases and then increases. The theoretical minimum value of L occurs at l. v2-c =15mm, but Figure 5 The data corresponding to the red X in section 3 shows that this compression scheme shifts the exceptional resonance peak from 96Hz to 146Hz, and the absorption coefficient at 96Hz drops from 1.00 to 0.05 after compression, resulting in severe performance degradation. This is because this section aims to preliminarily discuss the feasibility of using an equivalent multi-degree-of-freedom vibration model to guide acoustic metamaterial compression; therefore, only some parameters of the second cavity were adjusted, without considering the first cavity, which has a greater impact on low-frequency resonance.

[0038] Taking into account both the thickness compression and the sound absorption performance of the two resonance peaks, this application ultimately selected the height of the second cavity after compression to be 37mm, and the corresponding neck length 3 to be 12mm (rounded down to an integer value for ease of subsequent sample processing). Other geometric parameters are shown in Table 1. The total unit thickness was compressed from 106mm to 95mm, representing a thickness compression of approximately 10.38%. Figure 6 As can be seen, under this parameter combination, the sound absorption curves of both the pre-compression and post-compression units exhibit two distinct peaks in the sound absorption coefficient. After compression, the sound absorption coefficient at the target resonant frequency of 242Hz remains unchanged at 1.00, indicating that the sound absorption performance is essentially unaffected, and the peak position remains unchanged, consistent with theoretical expectations. However, the sound absorption coefficient at the additional resonant frequency of 96Hz decreases from 1.00 to 0.41, retaining only a portion of the sound absorption capacity, and the peak position shifts from 96Hz to 108Hz.

[0039] In summary, this application completed the acoustic metamaterial compression based on an equivalent multi-degree-of-freedom vibration model. To verify the prediction accuracy of the equivalent multi-degree-of-freedom vibration model and the feasibility of the acoustic metamaterial thickness compression scheme guided by it, this application further established a three-dimensional sound field simulation model based on the finite element method, the geometric representation of which is shown below. Figure 7 As shown. The simulation domain consists of a perfect-match layer, a background sound field, and a pair of internal air domains within the CNEHR acoustic metamaterial unit, corresponding to the acoustic scenes before and after compression, respectively. The calculations utilize a pressure acoustic module, specifically the acoustic metamaterial neck, i.e. Figure 7 The blue area is modeled using the narrow-area acoustic (circular duct) submodule within the pressure acoustics module to simulate the viscosity and heat dissipation effects of the neck region. The air medium in the remaining air domain is treated as an ideal gas. The background sound field is set as plane wave incidence, with the sound wave propagating along the neck axis, and the incident sound pressure amplitude is set to 1 Pa. The perfectly matched layer at the end simulates an infinitely extending external sound field, effectively absorbing reflected waves, suppressing secondary reflections, and avoiding interference from boundary reflections on the extraction of the reflected sound field. Rigid acoustic boundary conditions (zero normal velocity) are applied to all solid walls in the model except for the perfectly matched layer, and structural vibrations and acoustic-structure coupling effects are not considered. Based on these settings, the reflected sound field intensity of the CNEHR acoustic metamaterial unit before and after compression can be obtained, and the sound absorption coefficient curve can be calculated for comparison and verification with the theoretical results of the equivalent multi-degree-of-freedom vibration model.

[0040] The sound absorption curve obtained from finite element simulation is as follows Figure 8 As shown, with Figure 6The theoretical predictions are largely consistent. Simulation results show that the absorption coefficient at the target resonant frequency of 244Hz remains unchanged at 1.00 after compression, indicating that the sound absorption performance is basically unaffected, and the peak position remains unchanged, which is in line with theoretical expectations. However, the absorption coefficient at the additional resonant frequency of 98Hz decreases from 0.99 to 0.45, retaining only a portion of the sound absorption capacity, and the peak position shifts from 98Hz to 108Hz. Comparing the theoretical predictions of the equivalent multi-degree-of-freedom vibration model with the finite element simulation results, it can be seen that the theoretical value of the target resonant frequency before and after compression is approximately 242Hz, while the simulation result is approximately 244Hz, with a deviation of 2Hz. The theoretical value of the additional resonant frequency before compression is approximately 96Hz, while the simulation result is approximately 98Hz, with a deviation of 2Hz. The theoretical value of the additional resonant frequency after compression and the simulation result are both 108Hz, with minimal deviation. Furthermore, the deviation of the peak absorption coefficient at each resonant peak does not exceed 0.05, and the overall error is within an acceptable range, indicating that the established model has high prediction accuracy in the target frequency band.

[0041] Therefore, this application, based on an equivalent multi-degree-of-freedom vibration model, completed the thickness compression design of the CNEHR acoustic metamaterial unit, reducing the total thickness from 106 mm to 95 mm, achieving a geometrical size compression of approximately 10.38%. During this process, the absorption coefficient at the target resonant frequency remained essentially at 1.00, fully preserving the primary resonant absorption capacity. While the absorption coefficient at the additional resonant frequency decreased slightly, it still maintained a certain level of sound absorption. The deviations between the theoretical analytical results and the finite element simulation results at the resonant frequency and peak absorption coefficient were within a small range, verifying the accuracy and engineering feasibility of the proposed method. Overall, the equivalent multi-degree-of-freedom vibration model can provide a reliable dimensional adjustment criterion for thickness compression within an analytical framework. The compression scheme driven by this model achieves effective lightweight design of the metamaterial structure while ensuring key acoustic performance constraints.

Claims

1. A method for modeling the equivalent multi-degree-of-freedom vibration of acoustic metamaterials, characterized in that, Includes the following steps: S1. Based on the engineering requirements of low-frequency broadband sound absorption, select the target configuration of acoustic metamaterials connected in series, and define the geometric dimensions and symbol markings of each component of the target configuration of acoustic metamaterials to form a structured parameter table. S2. Based on the transfer matrix method, after completing the interface segmentation and state variable definition according to the direction of sound wave propagation, the general form of the transfer matrix between adjacent interfaces is constructed. The total transfer matrix is ​​solved for the general form of the transfer matrix, and the recursive relationship of the normalized impedance is derived. The expression of the overall normalized surface impedance is derived. After substituting into the structured parameter table, the relevant transfer matrix between adjacent interfaces of each subdomain is derived, and the calculation formulas of normalized surface impedance and sound absorption coefficient are obtained. S3. Explicitly expand and algebraically simplify the relevant transfer matrices between adjacent interfaces of each subdomain to eliminate the matrix form and derive the formula for the ratio of sound pressure to particle velocity at each interface. Transform the matrix relationship into an arithmetic relationship of impedance. Then, eliminate the intermediate variables, including the sound pressure and particle velocity at the intermediate interface, from the formula for the ratio of sound pressure to particle velocity at each interface to obtain the normalized explicit formula for surface impedance. Map the impedance terms one by one with the equivalent vibration elements to construct an equivalent multi-degree-of-freedom vibration model.

2. The method for modeling acoustic metamaterials equivalent to multi-degree-of-freedom vibration according to claim 1, characterized in that, Based on the engineering requirements of low-frequency broadband sound absorption, S1 selects an acoustic metamaterial target configuration consisting of a first cavity, a second cavity, a first neck, a second neck, and wall thickness regions at different locations.

3. The method for modeling acoustic metamaterials equivalent to multi-degree-of-freedom vibration according to claim 2, characterized in that, S1 further includes: The geometric dimensions and symbols of each component of the acoustic metamaterial target configuration are defined and marked. Specifically, the characteristic dimensions, height, inner diameter of the neck, length of each tube segment, and wall thickness at different positions of the cylindrical and / or square cavities are marked. The air physical property parameters, including air density, sound velocity, aerodynamic viscosity coefficient, and the calculation parameters required for viscosity and heat dissipation, are determined and integrated into a structured parameter table.

4. The method for modeling acoustic metamaterials equivalent to multi-degree-of-freedom vibration according to claim 1, characterized in that, S2 divides the acoustic metamaterial target configuration into x1-x5 interfaces from the incident end to the end end of the acoustic metamaterial along the direction perpendicular to the sound wave propagation, according to the direction of sound wave propagation, and defines the sound pressure p of each interface. i and particle velocity v i After completing the interface segmentation and defining the state variables, the linear mapping relationship of the state variables between adjacent interfaces is derived based on the acoustic wave equation, resulting in the general form of the transfer matrix: Where N is the total number of interfaces, i.e., the number of interfaces divided by the target configuration of the acoustic metamaterial, p i v i M represents the sound pressure and particle velocity at the i-th interface, respectively. i M is the transfer matrix between the i-th interface and the (i+1)-th interface. i (1,1), M i (1,2) is M i The elements in.

5. The acoustic metamaterial equivalent multi-degree-of-freedom vibration modeling method according to claim 4, characterized in that, S2 then multiplies the transfer matrices of all adjacent interfaces sequentially according to the sound wave propagation order to complete the general form of the transfer matrix solution, obtaining the formula for the total transfer matrix: Where T is the total transfer matrix after multiplying all transfer matrices.

6. The method for modeling acoustic metamaterial equivalent multi-degree-of-freedom vibration according to claim 5, characterized in that, S2 further includes: The ratio of sound pressure to particle velocity divided by the characteristic impedance of air is used as the normalized impedance, and its mathematical expression is: Expanding the general form of the obtained transfer matrix and substituting the normalized impedance into the general form of the transfer matrix, we obtain the recursive formula for the normalized impedance of adjacent interfaces: Where p0 and c0 are air density and speed of sound, respectively. To establish a definite functional relationship, the recursive formula is extended to the entire acoustic metamaterial target configuration, resulting in a global normalized surface impedance expression: z1=f1(z2)=f1(f2(z3))=……=f1(f2(……fN-1(zN))))(4); Where z1 is the normalized surface impedance of the acoustic metamaterial, z N The normalized backplane impedance, i.e., z N =+∞; Substituting the structured parameter table obtained in S1, the relevant transfer matrices of adjacent interfaces of each subdomain are derived, specifically: Where M a1 Z is the transfer matrix of interface x1-x2. a1 For the first neck impedance, p c1 c c1 k c1 These represent the complex density, complex speed of sound, and complex wave number of the air within the first neck region, respectively, where η is the aerodynamic viscosity coefficient, and A is the complex density, complex speed of sound, and complex wave number. a1 This is the cross-sectional area of ​​the first neck. The transfer matrix M of interface x2-x3 v1 for: Among them, A v1 Let be the cross-sectional area of ​​the first cavity; The transfer matrix M of interface x3-x4 a2 for: Among them, Z a2 For the second neck impedance; The transfer matrix M of interface x4-x5 v2 for: Among them, A v2 This represents the cross-sectional area of ​​the second cavity.

7. The method for modeling acoustic metamaterial equivalent multi-degree-of-freedom vibration according to claim 6, characterized in that, S2 further includes: Multiplying the relevant transfer matrices between each interface yields the total transfer matrix for interfaces x1-x5, specifically: Where interface x5 is the bottommost backplate of the acoustic metamaterial target configuration, i.e., v5=0, from which the normalized surface impedance and sound absorption coefficient calculation formulas are obtained: Where z is the normalized surface impedance and α is the sound absorption coefficient.

8. The method for modeling acoustic metamaterials equivalent to multi-degree-of-freedom vibration according to claim 1, characterized in that, S3 involves explicitly expanding and algebraically simplifying the relevant transfer matrices between adjacent interfaces of each subdomain, deriving the formula for the ratio of sound pressure to particle velocity at each interface, specifically: ，（15）； ，（18）； Among them, Z part1 Z part2 Z represents the impedance of the cylindrical air domain within the first cavity. v2 This represents the impedance of the cylindrical air domain within the second cavity.

9. The method for modeling acoustic metamaterials equivalent to multi-degree-of-freedom vibration according to claim 1, characterized in that, S3 further includes: By eliminating the intermediate variables of sound pressure and particle velocity from the formulas for the ratio of sound pressure to particle velocity at each interface, the normalized explicit formula for the surface impedance at the incident end x1 is obtained: 。 10. The method for modeling acoustic metamaterials equivalent to multi-degree-of-freedom vibration according to claim 9, characterized in that, The S3 model is based on the correspondence between vibration theory and acoustic impedance, and according to the obtained normalized surface impedance explicit formula, the real part of the impedance of the neck corresponds to the equivalent damping element, the imaginary part corresponds to the equivalent mass element, the impedance of the annular air domain corresponds to the equivalent stiffness element, and the impedance of the cavity air domain corresponds to the equivalent stiffness element, thus constructing an equivalent multi-degree-of-freedom vibration model.

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