阅读说明：本技术 一种周期非平整界面多孔材料吸声系数的计算方法及装置 (Method and device for calculating sound absorption coefficient of porous material with periodic non-flat interface ) 是由 杨玉真 贾晗 杨军 于 2021-07-23 设计创作，主要内容包括：本发明属于多孔吸声材料的声学测量技术领域,具体地说,涉及一种周期非平整界面多孔材料吸声系数的计算方法及测量装置,包括：获取周期非平整界面多孔材料的等效声学参数；将周期非平整界面多孔材料进行分层处理,每一个薄层材料等效为周期矩形调制的声学材料；获取各个薄层材料等效的周期矩形调制的声学材料中的声压和相关声学参数展开成级数表达形式,并带入声波方程,计算各个薄层材料内声压的本征态形式；通过层间边界连续性条件,采用迭代优化算法,计算反射声压的各阶本征态形式的强度系数向量和透射声压的各阶本征态形式的强度系数向量；根据上述计算结果,确定总反射系数和总透射系数,计算周期非平整界面多孔材料的吸声系数。(The invention belongs to the technical field of acoustic measurement of porous sound-absorbing materials, and particularly relates to a method and a device for calculating the sound-absorbing coefficient of a porous material with a periodic non-flat interface, wherein the method comprises the following steps: obtaining equivalent acoustic parameters of the porous material with the periodic non-flat interface; carrying out layering treatment on the periodic non-flat interface porous material, wherein each thin layer material is equivalent to an acoustic material modulated by a periodic rectangle; acquiring sound pressure and related acoustic parameters in the acoustic material equivalently modulated by the periodic rectangles of each thin-layer material, expanding the sound pressure and the related acoustic parameters into a hierarchical number expression form, substituting the hierarchical number expression form into a sound wave equation, and calculating the eigen state form of the sound pressure in each thin-layer material; calculating intensity coefficient vectors of all orders of eigen state forms of reflected sound pressure and intensity coefficient vectors of all orders of eigen state forms of transmitted sound pressure by adopting an iterative optimization algorithm according to the interlayer boundary continuity condition; and determining the total reflection coefficient and the total transmission coefficient according to the calculation result, and calculating the sound absorption coefficient of the porous material with the periodic non-flat interface.)

1. A method for calculating the sound absorption coefficient of a porous material with a periodic non-flat interface comprises the following steps:

obtaining equivalent acoustic parameters of the porous material with the periodic non-flat interface; wherein the equivalent acoustic parameters include: equivalent refractive index and equivalent density;

carrying out layering treatment on the periodic non-flat interface porous material, wherein each thin layer material is equivalent to an acoustic material modulated by a periodic rectangle;

according to the obtained equivalent acoustic parameters of the porous material, sound pressure and related acoustic parameters in the acoustic material which is equivalently modulated by the periodic rectangles of each thin-layer material are obtained and expanded into a graded number expression form, and are brought into a sound wave equation, and the eigen state form of the sound pressure in each thin-layer material is calculated;

calculating intensity coefficient vectors of all orders of eigen state forms of reflected sound pressure and intensity coefficient vectors of all orders of eigen state forms of transmitted sound pressure by adopting an iterative optimization algorithm according to the interlayer boundary continuity condition;

and determining the total reflection coefficient and the total transmission coefficient according to the calculation result, and calculating the sound absorption coefficient of the porous material with the periodic non-flat interface.

2. The method of calculating the sound absorption coefficient of a periodic non-flat interfacial porous material of claim 1, wherein the periodic non-flat interfacial porous material comprises: a periodic array of non-flat, non-completely filled porous material tips and a completely filled porous material substrate.

3. The method for calculating the sound absorption coefficient of the periodic non-flat interface porous material according to claim 1, wherein equivalent acoustic parameters of the periodic non-flat interface porous material are obtained; the specific process comprises the following steps:

the first thickness obtained by measurement is t₁Reflection coefficient R of planar structure of porous material₁And a second thickness t₂Reflection coefficient R of planar structure of porous material₂Respectively calculating the first thickness as t₁Surface resistance zeta of planar structure of porous material₁And a second thickness t₂Surface resistance zeta of planar structure of porous material₂；

ζ₁＝(1+R₁)/(1-R₁)

ζ₂＝(1+R₂)/(1-R₂)

Therein, ζ₁＝Z_pcoth(γt₁)；ζ₂＝Z_pcoth(γt₂)；

According to ζ₁tanh(γt₁)-ζ₂tanh(γt₂)＝0；

Combined upper, calculated to obtainComplex propagation constant gamma and normalized characteristic impedance Z_p；

Calculating the equivalent refractive index n according to the following formula_pAnd equivalent density ρ_p：

n_p＝ω/(-jγc₀)；

ρ_p＝(-jγρ₀c₀Z_p)/ω；

Wherein, c₀Is the speed of sound of air; rho₀Is the density of air; omega is angular frequency; j is an imaginary unit;

and taking the calculated equivalent refractive index and equivalent density as equivalent acoustic parameters.

4. The method for calculating the sound absorption coefficient of the periodic non-flat interface porous material as claimed in claim 1, wherein the periodic non-flat interface porous material is subjected to layering treatment, and each thin layer material is equivalent to a periodic rectangular modulated acoustic material; the specific process comprises the following steps:

the tip of the periodic non-flat interface porous material is a non-flat, non-completely filled porous material with a plurality of thin layers; carrying out layering treatment on the periodic non-flat interface porous material to obtain a plurality of thin layer materials, wherein each thin layer material is equivalent to an acoustic material modulated by a periodic rectangle;

the substrate of the periodic non-flat interfacial porous material is a completely filled porous material, which is a completely filled, periodic rectangular modulated acoustic material with a fill factor of 1.

5. The method for calculating the sound absorption coefficient of the porous material with the periodic non-flat interface according to claim 1, wherein the sound pressure and the related acoustic parameters in the acoustic material which is obtained according to the equivalent acoustic parameters of the porous material and is subjected to equivalent periodic rectangular modulation of each thin-layer material are expanded into a graded number expression form and are brought into a sound wave equation to calculate the eigen-state form of the sound pressure in each thin-layer material; the specific process comprises the following steps:

the sound pressure series expression form in the acoustic material with equivalent periodic rectangular modulation of each thin layer material is as follows:

wherein, P_IThe sound pressure series expression form of the incident and reflection areas; p_IIIs a sound pressure level expression form of the transmission region, P_lIs a sound pressure series expression form in the acoustic material with the equivalent periodic rectangular modulation of the first thin material of the porous material; wherein, L is 1,2,. said, L-1, L;

k_x0＝k₀sinθ，k_z0＝k₀cosθ，k_xi＝k₀sin theta + iK, and the subscript i is a positive integer greater than 0, which represents the ith order number expansion; k is 2 pi/T, and K is an inverted space basis vector;

when k is_xi＜k₀When there is

When k is_xi＞k₀When there is

Wherein j is an imaginary unit; k is a radical of₀Is the wave number in air; k is a radical of_xiA wave vector representing the x direction; k is a radical of_ziA wave vector representing the z direction; θ is the angle of incidence; t represents the period of the porous material; d represents the total thickness of the porous material; r_iAnd T_iRespectively representing a reflection coefficient and a transmission coefficient of the normalized ith-order sound pressure; s_li(z) is the l-th layer periodThe ith order sound pressure coefficient of the porous material; exp () represents an exponential function with a natural constant e as the base;

the series expression of the relevant acoustic parameters is in the form of:

where ρ is_l(x) Is the equivalent density of the l layer of periodic porous material; n is_l(x) Is the equivalent refractive index of the l layer of periodic porous material;

wherein f is_lIs the duty cycle of the porous material in the l-th layer of periodic porous material; n is_p，ρ_pThe equivalent refractive index and the equivalent density of the porous material, respectively; n is₀，ρ₀Refractive index and density of air, respectively; j is an imaginary unit; m is the expansion order; k is an inverse spatial basis vector;

the obtained sound pressure of each thin-layer material and the series expression form of the related acoustic parameters are brought into a sound wave equation; in the one-dimensional periodic acoustic structure, an acoustic wave equation in a density non-uniform medium is as follows:

wherein the content of the first and second substances,is a Laplace operator; p is a sound pressure series expression form; p is the sound pressure level expression form P of the incident and reflection regions_ISound pressure level expression form P of transmission region_IIOr sound pressure level order expression form P in periodic rectangular modulated acoustic material equivalent to porous material_l(ii) a n (x) is a series expression of the equivalent refractive index of the porous material of the periodic porous material;

obtaining a coupling equation of each order of coefficient:

writing the above coupling equation in matrix form:

[S_l″]＝[A_l][S_l]

wherein S is_lIs formed by S_liA column vector of components; s_l"is composed of²S_li/dz²A column vector of components; matrix array

Wherein, the matrix X_lAnd Y_lAre respectivelyAnd wherein m1 is the number of rows; n1 is the number of columns;andare all coefficients; diagonal matrix K_xElement K of_i，i＝k_xi；

Computing the matrix A_lEigenvalue and eigenvector to obtain square root of mth eigenvalueAnd corresponding feature vectors

Then the sound pressure coefficient S_li(z) expression:

wherein, l is 1;is a positive eigenstate form of sound pressure in the first thin-layer material;is the reverse eigenstate form of sound pressure in the first thin layer material;an intensity coefficient in the form of a forward propagating eigenstate;intensity coefficients in the form of counter-propagating eigenstates; d₁Is the thickness of the first thin layer; z is the position in the z direction;

wherein, L is 2, 3.., L; d_pIs the thickness of the pth thin layer;

according to the sound pressure coefficient S_li(z) determining the positive eigenstate form of the acoustic pressure in the first thin-layer materialAnd the inverse eigenstate form of the sound pressure in the first thin-layer material

6. The method for calculating the sound absorption coefficient of the porous material with the periodically non-flat interface as claimed in claim 1, wherein the intensity coefficient vector of each order of eigenstates of reflected sound pressure and the intensity coefficient vector of each order of eigenstates of transmitted sound pressure are calculated by an iterative optimization algorithm according to the continuity condition of the boundary between layers; the specific process comprises the following steps:

the boundary continuity condition comprises that the sound pressure and the normal direction particle vibration speed are continuous;

normal direction particle vibration velocity v_zExpression is

Wherein j is an imaginary unit; omega is angular frequency;is the inverse of the normalized density;

expanding the reciprocal of the normalized density into a corresponding series form

Where ρ is₀Is the air density; rho_l(x) Is the equivalent density of the l lamellae;is the expansion coefficient;

i-th order particle vibration velocityThe method comprises the following steps:

wherein, l is 1;

wherein the content of the first and second substances,an expansion coefficient that is the inverse of the normalized density;

the interfacial continuity conditions for the incident, reflective regions and the first layer of periodic porous material are:

the continuity conditions for the interfaces in between are:

the interfacial continuity conditions for the last layer of periodic porous material and the transmissive region are:

wherein R is selected from the group consisting ofCoefficient of radiation intensity R_iA column vector of components; t is the transmission intensity coefficient T_iA column vector of components;intensity coefficients in the form of forward propagating eigenstatesA column vector of components; andintensity coefficients in the form of counter-propagating eigenstatesA column vector of components; w_lIs a feature vectorA constructed matrix, I being an identity matrix; delta_i0Is a column vector with 0 order corresponding coefficient of 1 and other terms of 0; first thin layer matrix V_l＝Z_lW_lQ_lAnd matrix Z_lIs an element ofFirst thin diagonal matrix Q_l，E_lAnd K_zAre respectively an elementElement(s)And element k_ziA matrix of compositions; w_LAs feature vectorsA matrix of formations; e_LAs feature vectorsA matrix of formations; l thin layer V_L＝Z_LW_LQ_L；

Through matrix calculation, a matrix equation about the sound pressure reflection coefficient and the transmission coefficient is obtained:

calculating intensity coefficient vectors R of all orders of eigenstates of reflected sound pressure and intensity coefficient vectors T of all orders of eigenstates of transmitted sound pressure by adopting an iterative optimization algorithm; the method comprises the following specific steps:

the last four terms of the above matrix equation are expressed as:

introduces an iteration matrix f_L+1And g_L+1And has f_L+1＝I，g_L+1＝jK_z(ii) a And then introducing the following iteration matrix:

wherein, a_L、b_LRespectively introducing a first column intermediate matrix and a second column intermediate matrix;

is provided withThe following relationships were obtained by collation:

then the iteration matrix expression is: through iterative computation, a matrix equation is obtained:

solving the matrix equation to obtain R and T₁；

Vector of transmission coefficients T₁＝(g₁+jK_zf₁)^-1(jK_zΔ_i0+jk₀cosθΔ_i0)；

Intensity coefficient vector R ═ f of each order eigenstate of reflected sound pressure₁T₁-Δ_i0；

Intensity coefficient vector of each order eigenstate of transmission sound pressure

7. The method for calculating the sound absorption coefficient of the periodic non-flat interface porous material according to claim 6, wherein the total reflection coefficient and the total transmission coefficient are determined, and the sound absorption coefficient of the periodic non-flat interface porous material is calculated; the specific process comprises the following steps:

calculating the total reflection coefficient R:

wherein the content of the first and second substances,is the reflection intensity coefficient R in the intensity coefficient vector R of each order eigenstate of the reflection sound pressure_iTaking conjugation;

calculating the total transmission coefficient T:

wherein the content of the first and second substances,transmission intensity coefficient T in intensity coefficient vector T of each order eigenstate of transmission sound pressure_iTaking conjugation; re () is a real part;

calculating the sound absorption coefficient alpha of the porous material with the periodic uneven interface according to the calculated total reflection coefficient R and the total transmission coefficient T;

α＝1-R-T。

8. a device for measuring sound absorption coefficient of porous material with periodic non-flat interface is characterized in that the device comprises:

the acoustic parameter acquisition module is used for acquiring equivalent acoustic parameters of the porous material with the periodic uneven interface; wherein the equivalent acoustic parameters include: equivalent refractive index and equivalent density;

the layering module is used for layering the periodic non-flat interface porous material, and each thin layer material is equivalent to a periodic rectangular modulated acoustic material;

the eigen-state calculation module is used for acquiring the equivalent acoustic parameters of the porous material, expanding the sound pressure and the related acoustic parameters in the acoustic material which is equivalently modulated by the thin-layer material in a periodic rectangular mode into a hierarchical number expression form, bringing the hierarchical number expression form into an acoustic wave equation, and calculating the eigen-state form of the sound pressure in each thin-layer material;

the intensity coefficient calculation module is used for calculating intensity coefficient vectors of all orders of eigen state forms of the reflected sound pressure and intensity coefficient vectors of all orders of eigen state forms of the transmitted sound pressure by adopting an iterative optimization algorithm according to the interlayer boundary continuity condition; and

and the sound absorption coefficient acquisition module is used for determining the total reflection coefficient and the total transmission coefficient according to the calculation result and calculating the sound absorption coefficient of the porous material with the periodic non-flat interface.

Technical Field

The invention belongs to the technical field of acoustic measurement of porous sound absorption materials, and particularly relates to a method and a device for calculating a sound absorption coefficient of a porous material with a periodic non-flat interface.

Background

Porous materials are widely used in the field of noise control because they convert acoustic energy into heat. A plurality of porous materials with a wedge-shaped or semi-elliptical surface and a periodic corrugated structure are arranged on a wall body to achieve the purpose of sound absorption. Compared with a plane porous material, the porous material with the periodic corrugated structure has better sound absorption performance in a wider frequency range.

At present, the research on the sound absorption performance of the porous material with the periodic non-flat interface mainly takes experimental tests as main points. Through experimental test research and optimization of the sound absorption performance of the porous material, a large number of sample pieces with different material parameters, different geometric shapes and different structural parameters are generally required to be prepared, a large number of repeated measurement works are carried out, and then all test data are summarized and counted and the rule is summarized. Such an experimental test procedure is tedious to operate, time-consuming and labor-consuming, and is not favorable for rapid optimization design in engineering application.

Aiming at the calculation of the sound absorption coefficient of the porous material with the periodic non-flat interface, the conventional calculation method mainly comprises theoretical calculation methods such as finite elements, boundary elements, time domain finite difference and the like, the conventional method mainly needs geometric modeling and grid division, the calculation complexity and difficulty are very high, and the measurement efficiency and the measurement precision are greatly reduced.

Disclosure of Invention

In order to solve the defects in the prior art, the invention provides a method for calculating the sound absorption coefficient of a porous material with a periodic non-flat interface, which comprises the following steps:

obtaining equivalent acoustic parameters of the porous material with the periodic non-flat interface; wherein the equivalent acoustic parameters include: equivalent refractive index and equivalent density;

carrying out layering treatment on the periodic non-flat interface porous material, wherein each thin layer material is equivalent to an acoustic material modulated by a periodic rectangle;

according to the obtained equivalent acoustic parameters of the porous material, sound pressure and related acoustic parameters in the acoustic material which is equivalently modulated by the periodic rectangles of each thin-layer material are obtained and expanded into a graded number expression form, and are brought into a sound wave equation, and the eigen state form of the sound pressure in each thin-layer material is calculated;

calculating intensity coefficient vectors of all orders of eigen state forms of reflected sound pressure and intensity coefficient vectors of all orders of eigen state forms of transmitted sound pressure by adopting an iterative optimization algorithm according to the interlayer boundary continuity condition;

and determining the total reflection coefficient and the total transmission coefficient according to the calculation result, and calculating the sound absorption coefficient of the porous material with the periodic non-flat interface.

As an improvement of the above technical solution, the periodic non-flat interface porous material comprises: a periodic array of non-flat, non-completely filled porous material tips and a completely filled porous material substrate.

As one improvement of the technical scheme, the equivalent acoustic parameters of the porous material with the periodic non-flat interface are acquired; the specific process comprises the following steps:

the first thickness obtained by measurement is t₁Reflection coefficient R of planar structure of porous material₁And a second thickness t₂Reflection coefficient R of planar structure of porous material₂Respectively calculating the first thickness as t₁Surface resistance zeta of planar structure of porous material₁And a second thickness t₂Surface resistance zeta of planar structure of porous material₂；

ζ₁＝(1+R₁)/(1-R₁)

ζ₂＝(1+R₂)/(1-R₂)

Therein, ζ₁＝Z_pcoth(γt₁)；ζ₂＝Z_pcoth(γt₂)；

According to ζ₁tanh(γt₁)-ζ₂tanh(γt₂)＝0；

Combining the above formula, calculating to obtain a complex propagation constant gamma and a normalized characteristic impedance Z_p；

Calculating the equivalent refractive index n according to the following formula_pAnd equivalent secretDegree rho_p：

n_p＝ω/(-jγc₀)；

ρ_p＝(-jγρ₀c₀Z_p)/ω；

Wherein, c₀Is the speed of sound of air; rho₀Is the density of air; omega is angular frequency; j is an imaginary unit; .

And taking the calculated equivalent refractive index and equivalent density as equivalent acoustic parameters.

As one improvement of the above technical solution, the periodic non-flat interface porous material is subjected to a layering treatment, and each thin layer material is equivalent to a periodic rectangular modulated acoustic material; the specific process comprises the following steps:

the tip of the periodic non-flat interface porous material is a non-flat, non-completely filled porous material with a plurality of thin layers; carrying out layering treatment on the periodic non-flat interface porous material to obtain a plurality of thin layer materials, wherein each thin layer material is equivalent to an acoustic material modulated by a periodic rectangle;

the substrate of the periodic non-flat interfacial porous material is a completely filled porous material, which is a completely filled, periodic rectangular modulated acoustic material with a fill factor of 1.

As one improvement of the above technical solution, the sound pressure and related acoustic parameters in the acoustic material, which is obtained according to the equivalent acoustic parameters of the porous material and is equivalently modulated by the periodic rectangle, of each thin layer material are expanded into a hierarchical number expression form and are brought into a sound wave equation, and the eigen state form of the sound pressure in each thin layer material is calculated; the specific process comprises the following steps:

the sound pressure series expression form in the acoustic material with equivalent periodic rectangular modulation of each thin layer material is as follows:

wherein, P_IThe sound pressure series expression form of the incident and reflection areas; p_IIIs a sound pressure level expression form of the transmission region, P_lIs a sound pressure series expression form in the acoustic material with the equivalent periodic rectangular modulation of the first thin material of the porous material; wherein, L is 1,2, …, L-1, L;

k_x0＝k₀sinθ，k_z0＝k₀cosθ，k_xi＝k₀sin theta + iK, and the subscript i is a positive integer greater than 0, which represents the ith order number expansion; k is 2 pi/T, and K is an inverted space basis vector;

when k is_xi<k₀When there is

When k is_xi>k₀When there is

Wherein j is an imaginary unit; k is a radical of₀Is the wave number in air; k is a radical of_xiA wave vector representing the x direction; k is a radical of_ziA wave vector representing the z direction; θ is the angle of incidence; t represents the period of the porous material; d represents the total thickness of the porous material; r_iAnd T_iRespectively representing a reflection coefficient and a transmission coefficient of the normalized ith-order sound pressure; s_li(z) is the ith order sound pressure coefficient of the layer l periodic porous material; exp () represents an exponential function with a natural constant e as the base;

the series expression of the relevant acoustic parameters is in the form of:

where ρ is_l(x) Is the equivalent density of the l layer of periodic porous material; n is_l(x) Is the equivalent refractive index of the l layer of periodic porous material;

wherein f is_lIs the duty cycle of the porous material in the l-th layer of periodic porous material; n is_p，ρ_pThe equivalent refractive index and the equivalent density of the porous material, respectively; n is₀，ρ₀Refractive index and density of air, respectively; j is an imaginary unit; m is the expansion order; k is an inverse spatial basis vector;

the obtained sound pressure of each thin-layer material and the series expression form of the related acoustic parameters are brought into a sound wave equation; in the one-dimensional periodic acoustic structure, an acoustic wave equation in a density non-uniform medium is as follows:

wherein the content of the first and second substances,is a Laplace operator; p is a sound pressure series expression form; p is the sound pressure level expression form P of the incident and reflection regions_ISound pressure level expression form P of transmission region_IIOr sound pressure level order expression form P in periodic rectangular modulated acoustic material equivalent to porous material_l(ii) a n (x) is a series expression of the equivalent refractive index of the porous material of the periodic porous material;

obtaining a coupling equation of each order of coefficient:

writing the above coupling equation in matrix form:

[S_l″]＝[A_l][S_l]

wherein S is_lIs formed by S_liA column vector of components; s_l"is composed of²S_li/dz²A column vector of components; matrix array

Wherein, the matrix X_lAnd Y_lAre respectivelyAnd wherein m1 is the number of rows; n1 is the number of columns;andare all coefficients; diagonal matrix K_xElement K of_i,i＝k_xi；

Computing the matrix A_lEigenvalue and eigenvector to obtain square root of mth eigenvalueAnd corresponding feature vectors

Then the sound pressure coefficient S_li(z) expression:

wherein, l is 1;is a positive eigenstate form of sound pressure in the first thin-layer material;is the reverse eigenstate form of sound pressure in the first thin layer material;an intensity coefficient in the form of a forward propagating eigenstate;intensity coefficients in the form of counter-propagating eigenstates; d₁Is the thickness of the first thin layer; z is the position in the z direction;

wherein L is 2,3, …, L; d_pIs the thickness of the pth thin layer;

according to the sound pressure coefficient S_li(z) determining the positive eigenstate form of the acoustic pressure in the first thin-layer materialAnd the inverse eigenstate form of the sound pressure in the first thin-layer material

As one improvement of the above technical solution, the intensity coefficient vector of each order of eigen state of the reflected sound pressure and the intensity coefficient vector of each order of eigen state of the transmitted sound pressure are calculated by using an iterative optimization algorithm according to the interlayer boundary continuity condition; the specific process comprises the following steps:

the boundary continuity condition comprises that the sound pressure and the normal direction particle vibration speed are continuous;

normal direction particle vibration velocity v_zExpression is

Wherein j is an imaginary unit; omega is angular frequency;is the inverse of the normalized density;

expanding the reciprocal of the normalized density into a corresponding series form

Where ρ is₀Is the air density; rho_l(x) Is the equivalent density of the l lamellae;is the expansion coefficient;

i-th order particle vibration velocityThe method comprises the following steps:

wherein, l is 1;

wherein L is 2,3, …, L;an expansion coefficient that is the inverse of the normalized density;

the interfacial continuity conditions for the incident, reflective regions and the first layer of periodic porous material are:

the continuity conditions for the interfaces in between are:

the interfacial continuity conditions for the last layer of periodic porous material and the transmissive region are:

wherein R is the reflection intensity coefficient R_iA column vector of components; t is the transmission intensity coefficient T_iA column vector of components;intensity coefficients in the form of forward propagating eigenstatesA column vector of components; andintensity coefficients in the form of counter-propagating eigenstatesA column vector of components; w_lIs a feature vectorA constructed matrix, I being an identity matrix; delta_i0Is 0 order with a corresponding coefficient of 1, and the other terms areA column vector of 0; first thin layer matrix V_l＝Z_lW_lQ_lAnd matrix Z_lIs an element ofFirst thin diagonal matrix Q_l，E_lAnd K_zAre respectively an elementElement(s)And element k_ziA matrix of compositions; w_LAs feature vectorsA matrix of formations; e_LAs feature vectorsA matrix of formations; l thin layer V_L＝Z_LW_LQ_L；

Through matrix calculation, a matrix equation about the sound pressure reflection coefficient and the transmission coefficient is obtained:

calculating intensity coefficient vectors R of all orders of eigenstates of reflected sound pressure and intensity coefficient vectors T of all orders of eigenstates of transmitted sound pressure by adopting an iterative optimization algorithm; the method comprises the following specific steps:

the last four terms of the above matrix equation are expressed as:

introduces an iteration matrix f_L+1And g_L+1And has f_L+1＝I，g_L+1＝jK_z(ii) a And then introducing the following iteration matrix:

wherein, a_L、b_LRespectively introducing a first column intermediate matrix and a second column intermediate matrix;

is provided withThe following relationships were obtained by collation:

then the iteration matrix expression is:through iterative computation, a matrix equation is obtained:

solving the matrix equation to obtain R and T₁；

Vector of transmission coefficients T₁＝(g₁+jK_zf₁)^-1(jK_zΔ_i0+jk₀cosθΔ_i0)；

Intensity coefficient vector R ═ f of each order eigenstate of reflected sound pressure₁T₁-Δ_i0；

Intensity coefficient vector of each order eigenstate of transmission sound pressure

As one improvement of the technical scheme, the total reflection coefficient and the total transmission coefficient are determined, and the sound absorption coefficient of the porous material with the periodic non-flat interface is calculated; the specific process comprises the following steps:

calculating the total reflection coefficient R:

wherein the content of the first and second substances,is the reflection intensity coefficient R in the intensity coefficient vector R of each order eigenstate of the reflection sound pressure_iTaking conjugation;

calculating the total transmission coefficient T:

T＝∑_iT_iT_i*Re(k_zi/k_z0)；

wherein, T_i ^*Transmission intensity coefficient T in intensity coefficient vector T of each order eigenstate of transmission sound pressure_iTaking conjugation; re () is a real part;

calculating the sound absorption coefficient alpha of the porous material with the periodic uneven interface according to the calculated total reflection coefficient R and the total transmission coefficient T;

α＝1-R-T。

the invention also provides a device for measuring the sound absorption coefficient of the porous material with the periodic non-flat interface, which comprises:

the acoustic parameter acquisition module is used for acquiring equivalent acoustic parameters of the porous material with the periodic uneven interface; wherein the equivalent acoustic parameters include: equivalent refractive index and equivalent density;

the layering module is used for layering the periodic non-flat interface porous material, and each thin layer material is equivalent to a periodic rectangular modulated acoustic material;

the eigen-state calculation module is used for acquiring the equivalent acoustic parameters of the porous material, expanding the sound pressure and the related acoustic parameters in the acoustic material which is equivalently modulated by the thin-layer material in a periodic rectangular mode into a hierarchical number expression form, bringing the hierarchical number expression form into an acoustic wave equation, and calculating the eigen-state form of the sound pressure in each thin-layer material;

the intensity coefficient calculation module is used for calculating intensity coefficient vectors of all orders of eigen state forms of the reflected sound pressure and intensity coefficient vectors of all orders of eigen state forms of the transmitted sound pressure by adopting an iterative optimization algorithm according to the interlayer boundary continuity condition; and

and the sound absorption coefficient acquisition module is used for determining the total reflection coefficient and the total transmission coefficient according to the calculation result and calculating the sound absorption coefficient of the porous material with the periodic non-flat interface.

Compared with the prior art, the invention has the beneficial effects that:

1. according to the method, equivalent acoustic parameters of the porous material to be tested are obtained, namely only two samples of the planar porous material with different thicknesses are required to be measured, sample pieces with different geometric shapes and structural sizes are not required to be prepared, and a large number of repeated experimental tests are not required, so that the measurement efficiency is greatly improved, and the sound absorption coefficient of the porous material with a periodic uneven interface can be measured more quickly;

2. the method considers the influence of periodic modulation on the sound absorption performance, wherein the influence comprises a high-order mode including evanescent waves, and the accuracy of the measured sound absorption coefficient is improved;

3. after the geometric parameters of the structure are determined, the method can accurately predict the broadband sound absorption coefficient of the porous material with the periodic non-flat interface only by bringing in equivalent acoustic parameters of each frequency, simplifies the calculation complexity, saves the time cost, greatly improves the measurement efficiency, and has important significance for the application in the field of noise control.

Drawings

FIG. 1 is a flow chart of a method for calculating the sound absorption coefficient of a periodic non-flat interfacial porous material according to the present invention;

FIG. 2 is a schematic diagram of a structure for measuring the reflection coefficient R of the porous material in a two-microphone impedance tube;

FIG. 3 is a schematic structural view of a tip and a substrate of a periodic non-planar interfacial porous material in the method of FIG. 1;

FIG. 4 is a schematic structural view of a single layer periodically modulated porous material after delamination of the periodic non-planar interfacial porous material in the method of FIG. 1;

FIG. 5a is a graphical representation of the equivalent refractive index of the porous material over a broad frequency range in the method of FIG. 1;

FIG. 5b is a graphical representation of the equivalent density of the porous material over a wide frequency range in the method of FIG. 1;

FIG. 6a is a schematic structural diagram of the geometric shape of the acoustic absorbent cotton with triangular periodic modulation as a sample to be measured;

FIG. 6b is a schematic structural diagram of the geometric shape of the acoustic absorbent cotton with rectangular periodic modulation of the sample to be measured;

fig. 7 is a result of calculation of the sound absorption coefficient of the sample to be tested and an experimental test result in the method of fig. 1.

Detailed Description

The invention will now be further described with reference to the accompanying drawings and examples.

As shown in figure 1, the invention provides a method for calculating the sound absorption coefficient of the porous material with the periodic non-flat interface, and the method can accurately predict the broadband sound absorption coefficient of the porous material with the periodic non-flat interface only by bringing in equivalent acoustic parameters of each frequency after determining the geometric parameters of the structure, thereby simplifying the calculation complexity, saving the time cost and having great significance for the application in the field of noise control.

The method comprises the following steps:

obtaining equivalent acoustic parameters of the porous material with the periodic non-flat interface; wherein the equivalent acoustic parameters include: equivalent refractive index and equivalent density;

specifically, as shown in fig. 2,3 and 4, in the two-microphone impedance tube, a planar structure of a porous material having a thickness t is mounted on a rigid backing, and the reflection coefficient R of the porous material is measured;

ζ＝(1+R)/(1-R)

wherein ζ is the surface resistance of the porous material, which is a known value; wherein, Z_pcoth(γt)；

Wherein gamma is a complex propagation constant of the periodic non-flat interface porous material; z_pNormalized characteristic impedance of the periodic non-flat interface porous material;

based on the above formula, the first thickness obtained by measurement is t₁Reflection coefficient R of planar structure of porous material₁And a second thickness t₂Reflection coefficient R of planar structure of porous material₂Respectively calculating the first thickness as t₁Surface resistance zeta of planar structure of porous material₁And a second thickness t₂Surface resistance zeta of planar structure of porous material₂；

ζ₁＝(1+R₁)/(1-R₁)

ζ₂＝(1+R₂)/(1-R₂)

Therein, ζ₁＝Z_pcoth(γt₁)；ζ₂＝Z_pcoth(γt₂)；

According to ζ₁tanh(γt₁)-ζ₂tanh(γt₂)＝0；

Combining the above formula, calculating to obtain a complex propagation constant gamma and a normalized characteristic impedance Z_p；

Calculating the equivalent refractive index n according to the following formula_pAnd equivalent density ρ_p：

n_p＝ω/(-jγc₀)；

ρ_p＝(-jγρ₀c₀Z_p)/ω；

Wherein, c₀Is the speed of sound of air; rho₀Is the density of air; omega is angular frequency; j is an imaginary unit; .

And taking the calculated equivalent refractive index and equivalent density as equivalent acoustic parameters.

Carrying out layering treatment on the periodic non-flat interface porous material, wherein each thin layer material is equivalent to an acoustic material modulated by a periodic rectangle;

specifically, the tip of the periodic non-flat interfacial porous material is a non-flat, non-completely filled porous material having a plurality of thin layers; carrying out layering treatment on the periodic non-flat interface porous material to obtain a plurality of thin layer materials, wherein each thin layer material is equivalent to an acoustic material modulated by a periodic rectangle;

the substrate of the periodic non-flat interfacial porous material is a completely filled porous material, which is a completely filled, periodic rectangular modulated acoustic material with a fill factor of 1.

Wherein the periodic non-planar interfacial porous material comprises: a periodic array of non-flat, non-completely filled porous material tips and a completely filled porous material substrate.

According to the obtained equivalent acoustic parameters of the porous material, sound pressure and related acoustic parameters in the acoustic material which is equivalently modulated by the periodic rectangles of each thin-layer material are obtained and expanded into a graded number expression form, and are brought into a sound wave equation, and the eigen state form of the sound pressure in each thin-layer material is calculated;

specifically, the sound pressure level expression in the periodic rectangular modulated acoustic material equivalent to each thin layer material is in the form of:

wherein, P_IThe sound pressure series expression form of the incident and reflection areas; p_IIIs a sound pressure level expression form of the transmission region, P_lIs a sound pressure series expression form in the acoustic material with the equivalent periodic rectangular modulation of the first thin material of the porous material; wherein, L is 1,2, …, L-1, L;

k_x0＝k₀sinθ，k_z0＝k₀cosθ，k_xi＝k₀sin theta + iK, and the subscript i is a positive integer greater than 0, which represents the ith order number expansion; k is 2 pi/T, and K is an inverted space basis vector;

when k is_xi<k₀When there is

When k is_xi>k₀When there is

Wherein j is an imaginary unit; k is a radical of₀Is the wave number in air; k is a radical of_xiA wave vector representing the x direction; k is a radical of_ziA wave vector representing the z direction; θ is the angle of incidence; t represents the period of the porous material; d represents the total thickness of the porous material; r_iAnd T_iRespectively representing a reflection coefficient and a transmission coefficient of the normalized ith-order sound pressure; s_li(z) is the ith order sound pressure coefficient of the layer l periodic porous material; exp () represents an exponential function with a natural constant e as the base;

the series expression of the relevant acoustic parameters is in the form of:

where ρ is_l(x) Is the equivalent density of the l layer of periodic porous material; n is_l(x) Is the equivalent refractive index of the l layer of periodic porous material;

wherein f is_lIs the duty cycle of the porous material in the l-th layer of periodic porous material; n is_p，ρ_pThe equivalent refractive index and the equivalent density of the porous material, respectively; n is₀，ρ₀Refractive index and density of air, respectively; j is an imaginary unit; m is an expansionThe order; k is an inverse spatial basis vector;

the obtained sound pressure of each thin-layer material and the series expression form of the related acoustic parameters are brought into a sound wave equation; in the one-dimensional periodic acoustic structure, an acoustic wave equation in a density non-uniform medium is as follows:

wherein the content of the first and second substances,is a Laplace operator; p is a sound pressure series expression form; p is the sound pressure level expression form P of the incident and reflection regions_ISound pressure level expression form P of transmission region_IIOr sound pressure level order expression form P in periodic rectangular modulated acoustic material equivalent to porous material_l(ii) a n (x) is a series expression of the equivalent refractive index of the porous material of the periodic porous material;

obtaining a coupling equation of each order of coefficient:

writing the above coupling equation in matrix form:

[S_l″]＝[A_l][S_l]

wherein S is_lIs formed by S_liA column vector of components; s_l"is composed of²S_li/dz²A column vector of components; matrix array

Wherein, the matrix X_lAnd Y_lAre respectivelyAnd wherein m1 is the number of rows; n1 is the number of columns;andare all coefficients; diagonal matrix K_xElement K of_i，i＝k_xi；

Computing the matrix A_lEigenvalue and eigenvector to obtain square root of mth eigenvalueAnd corresponding feature vectors

Then the sound pressure coefficient S_li(z) expression:

wherein, l is 1;is a positive eigenstate form of sound pressure in the first thin-layer material;is the reverse eigenstate form of sound pressure in the first thin layer material;an intensity coefficient in the form of a forward propagating eigenstate;intensity coefficients in the form of counter-propagating eigenstates; d₁Is the thickness of the first thin layer; z is the position in the z direction;

wherein L is 2,3, …, L; d_pIs the thickness of the pth thin layer;

according to the sound pressure coefficient S_li(z) determining the positive eigenstate form of the acoustic pressure in the first thin-layer materialAnd the inverse eigenstate form of the sound pressure in the first thin-layer material

Calculating intensity coefficient vectors of all orders of eigen state forms of reflected sound pressure and intensity coefficient vectors of all orders of eigen state forms of transmitted sound pressure by adopting an iterative optimization algorithm according to the interlayer boundary continuity condition;

specifically, the boundary continuity condition includes that the sound pressure and the normal direction particle vibration velocity are continuous;

normal direction particle vibration velocity v_zExpression is

Wherein j is an imaginary unit; omega is angular frequency;is the inverse of the normalized density;

expanding the reciprocal of the normalized density into a corresponding series form

Where ρ is₀Is the air density; rho_l(x) Is the equivalent density of the l lamellae;is the expansion coefficient;

i-th order particle vibration velocityThe method comprises the following steps:

wherein, l is 1;

wherein L is 2,3, …, L;an expansion coefficient that is the inverse of the normalized density;

the interfacial continuity conditions for the incident, reflective regions and the first layer of periodic porous material are:

the continuity conditions for the interfaces in between are:

the interfacial continuity conditions for the last layer of periodic porous material and the transmissive region are:

wherein R is the reflection intensity coefficient R_iA column vector of components; t is the transmission intensity coefficient T_iA column vector of components;intensity coefficients in the form of forward propagating eigenstatesA column vector of components; andintensity coefficients in the form of counter-propagating eigenstatesA column vector of components; w_lIs a feature vectorA constructed matrix, I being an identity matrix; delta_i0Is a column vector with 0 order corresponding coefficient of 1 and other terms of 0; first thin layer matrix V_l＝Z_lW_lQ_lAnd matrix Z_lIs an element ofFirst thin diagonal matrix Q_l，E_lAnd K_zAre respectively an elementElement(s)And element k_ziA matrix of compositions; w_LAs feature vectorsA matrix of formations; e_LAs feature vectorsA matrix of formations;l thin layer V_L＝Z_LW_LQ_L；

Through matrix calculation, a matrix equation about the sound pressure reflection coefficient and the transmission coefficient is obtained:

calculating intensity coefficient vectors R of all orders of eigenstates of reflected sound pressure and intensity coefficient vectors T of all orders of eigenstates of transmitted sound pressure by adopting an iterative optimization algorithm; the method comprises the following specific steps:

the last four terms of the above matrix equation are expressed as:

introduces an iteration matrix f_L+1And g_L+1And has f_L+1＝I，g_L+1＝jK_z(ii) a And then introducing the following iteration matrix:

wherein, a_L、b_LRespectively introducing a first column intermediate matrix and a second column intermediate matrix;

is provided withThe following relationships were obtained by collation:

then the iteration matrix expression is:through iterative computation, a matrix equation is obtained:

solving the matrix equation to obtain R and T₁；

Vector of transmission coefficients T₁＝(g₁+jK_zf₁)^-1(jK_zΔ_i0+jk₀cosθΔ_i0)；

Intensity coefficient vector R ═ f of each order eigenstate of reflected sound pressure₁T₁-Δ_i0；

Intensity coefficient vector of each order eigenstate of transmission sound pressure

And determining the total reflection coefficient and the total transmission coefficient according to the calculation result, and calculating the sound absorption coefficient of the porous material with the periodic non-flat interface.

Specifically, the total reflection coefficient R is calculated:

wherein the content of the first and second substances,is the reflection intensity coefficient R in the intensity coefficient vector R of each order eigenstate of the reflection sound pressure_iTaking conjugation;

calculating the total transmission coefficient T:

T＝∑_iT_iT_* ⁱRe(k_zi/k_z0)；

wherein, T_i ^*Transmission intensity coefficient T in intensity coefficient vector T of each order eigenstate of transmission sound pressure_iTaking conjugation; re () is a real part;

calculating the sound absorption coefficient alpha of the porous material with the periodic uneven interface according to the calculated total reflection coefficient R and the total transmission coefficient T;

α＝1-R-T。

the invention also provides a device for measuring the sound absorption coefficient of the porous material with the periodic non-flat interface, which comprises:

the acoustic parameter acquisition module is used for acquiring equivalent acoustic parameters of the porous material with the periodic uneven interface; wherein the equivalent acoustic parameters include: equivalent refractive index and equivalent density;

the layering module is used for layering the periodic non-flat interface porous material, and each thin layer material is equivalent to a periodic rectangular modulated acoustic material;

the eigen-state calculation module is used for acquiring the equivalent acoustic parameters of the porous material, expanding the sound pressure and the related acoustic parameters in the acoustic material which is equivalently modulated by the thin-layer material in a periodic rectangular mode into a hierarchical number expression form, bringing the hierarchical number expression form into an acoustic wave equation, and calculating the eigen-state form of the sound pressure in each thin-layer material;

the intensity coefficient calculation module is used for calculating intensity coefficient vectors of all orders of eigen state forms of the reflected sound pressure and intensity coefficient vectors of all orders of eigen state forms of the transmitted sound pressure by adopting an iterative optimization algorithm according to the interlayer boundary continuity condition; and

and the sound absorption coefficient acquisition module is used for determining the total reflection coefficient and the total transmission coefficient according to the calculation result and calculating the sound absorption coefficient of the porous material with the periodic non-flat interface.

As shown in fig. 5a and 5b, a conventional porous sound absorbing material, i.e., high-density sound absorbing cotton, was purchased, the equivalent acoustic parameters of the porous material were first tested by a two-microphone impedance tube system, the reflection coefficients of the planar sound absorbing cotton with thicknesses of 2cm and 4.8cm were experimentally tested, and the equivalent refractive index and the equivalent density of the porous material were obtained by numerical calculation, and the results are shown in fig. 5a and 5 b.

By adopting the method for measuring the sound absorption coefficient of the porous material with the periodic non-flat interface, the sound absorption coefficients of the high-density sound absorption cotton with the two periodic non-flat interfaces are measured, the two sound absorption cotton are subjected to experimental tests in a two-microphone impedance tube system, and the sound absorption coefficients of the experimental tests are compared with the calculation results. The porous materials of the two periodic non-flat interfaces are triangular and rectangular periodic modulated sound absorption cotton respectively, and specific parameters are shown in fig. 6a and 6b, so that the calculation method provided by the invention can be determined to have higher accuracy.

As shown in FIG. 6a, the geometric parameters of the triangular periodically modulated sound absorbing cotton include the period T_t5cm, substrate thickness d_t1cm, triangular tip thickness h_t＝3.7cm：

As shown in FIG. 6b, the geometric parameters of the rectangular periodic modulated acoustic wool include the period T_r3.3cm, substrate thickness d_r1.9cm, rectangular tip thickness h_r2.8cm and a rectangular tip width w_r＝1.5cm：

The sound absorption coefficient of the porous material measured by the method and the experimental test result are shown in fig. 7, and the two calculation results are very close to each other, which shows that the method for calculating the sound absorption coefficient of the porous material with the periodic non-flat interface provided by the invention is very effective.

Finally, it should be noted that the above embodiments are only used for illustrating the technical solutions of the present invention and are not limited. Although the present invention has been described in detail with reference to the embodiments, it will be understood by those skilled in the art that various changes may be made and equivalents may be substituted without departing from the spirit and scope of the invention as defined in the appended claims.