CN113640388A - A method and device for calculating sound absorption coefficient of periodic non-flat interface porous materials - Google Patents

Abstract

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.

Description

A method and device for calculating sound absorption coefficient of periodic non-flat interface porous materials

technical field

The invention belongs to the technical field of acoustic measurement of porous sound-absorbing materials, and in particular relates to a method and a measuring device for calculating the sound-absorbing coefficient of periodic non-flat interface porous materials.

Background technique

Porous materials are widely used in the field of noise control due to their ability to convert acoustic energy into thermal energy. A number of porous materials with periodic corrugated structures with wedge-shaped or semi-elliptical surfaces are installed on the wall to achieve sound absorption. Compared with planar porous materials, porous materials with periodic corrugated structures have better sound absorption performance in a wider frequency range.

At present, the research on the sound absorption performance of periodic non-flat interface porous materials is mainly based on experimental tests. To study and optimize the sound absorption performance of porous materials through experimental tests, it is usually necessary to prepare a large number of samples with different material parameters, different geometric shapes and different structural parameters, carry out a large number of repeated measurements, and then summarize all the test data and summarize the rules. . Such an experimental test process is cumbersome, time-consuming and labor-intensive, which is not conducive to rapid optimization design in engineering applications.

For the calculation of the sound absorption coefficient of porous materials with periodic non-flat interfaces, the existing calculation methods mainly include theoretical calculation methods such as finite element, boundary element and time domain finite difference. The existing methods mainly require geometric modeling and mesh division. The computational complexity and difficulty are very large, which greatly reduces the measurement efficiency and measurement accuracy.

SUMMARY OF THE INVENTION

In order to solve the above-mentioned defects in the prior art, the present invention proposes a method for calculating the sound absorption coefficient of periodic non-flat interface porous materials, the method comprising:

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

The periodic non-flat interface porous material is layered, and each thin layer material is equivalent to a periodic rectangular modulated acoustic material;

According to the obtained equivalent acoustic parameters of the porous material, the sound pressure and related acoustic parameters in the periodic rectangular modulated acoustic material equivalent to each thin-layer material are obtained and expanded into a series expression, and brought into the acoustic wave equation to calculate each thin-layer The eigenstate form of the sound pressure in the material;

Through the interlayer boundary continuity condition, the iterative optimization algorithm is used to calculate the intensity coefficient vector of each order eigenstate form of reflected sound pressure and the intensity coefficient vector of each order eigenstate form of transmitted sound pressure;

According to the above calculation results, 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.

As one of the improvements of the above technical solutions, the periodic non-planar interface porous material includes: periodically arranged, non-planar, incompletely filled porous material tips and completely filled porous material bases.

As one of the improvements of the above technical solutions, the acquisition of the equivalent acoustic parameters of the periodic non-flat interface porous material; the specific process is:

According to the measured reflection coefficient R ₁ of the planar structure of the porous material with the first thickness t ₁ and the reflection coefficient R ₂ of the planar structure of the second porous material with thickness t ₂ , the plane of the porous material with the first thickness t ₁ is calculated respectively. the surface impedance ζ ₁ of the structure and the surface impedance ζ ₂ of the second thickness t ₂ planar structure of the porous material;

ζ ₁ =(1+R ₁ )/(1-R ₁ )

ζ ₂ =(1+R ₂ )/(1-R ₂ )

Wherein, ζ ₁ =Z _p coth(γt ₁ ); ζ ₂ =Z _p coth(γt ₂ );

According to ζ ₁ tanh(γt ₁ )-ζ ₂ tanh(γt ₂ )=0;

Combined with the above formula, the complex propagation constant γ and the normalized characteristic impedance Z _p are obtained by calculation;

The equivalent refractive index n _p and the equivalent density ρ _p are calculated according to the following formulas:

n _p =ω/(-jγc ₀ );

ρ _p =(-jγρ ₀ c ₀ Z _p )/ω;

Among them, c ₀ is the sound speed of air; ρ ₀ is the density of air; ω is the angular frequency; j is the imaginary unit; .

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

As one of the improvements of the above technical solutions, the periodic non-flat interface porous material is layered, and each thin layer material is equivalent to a periodic rectangular modulated acoustic material; the specific process is as follows:

The tip of the periodic non-planar interface porous material is a non-planar, non-completely filled porous material with multiple thin layers; the periodic non-planar interface porous material is layered to obtain a plurality of thin layer materials, each thin layer The material is equivalent to an acoustic material with periodic rectangular modulation;

The base of the periodic non-planar interface porous material is a completely filled porous material, and the porous material is a completely filled, periodic rectangularly modulated acoustic material with a filling rate of 1.

As one of the improvements of the above technical solutions, according to the equivalent acoustic parameters of the porous material, the sound pressure and related acoustic parameters in the acoustic material equivalent to the periodic rectangular modulation of each thin-layer material are obtained and expanded into a series expression, and Bring in the acoustic wave equation to calculate the eigenstate form of the sound pressure in each thin-layer material; the specific process is as follows:

The expression of the sound pressure level in the equivalent periodic rectangular modulated acoustic material of each thin layer material is:

Among them, P _I is the expression form of the sound pressure level in the incident and reflection areas; P _II is the expression form of the sound pressure level in the transmission area, and P _l is the periodic rectangular modulation of the acoustic material equivalent to the lth thin layer of the porous material. The expression form of sound pressure level in ; where, l=1,2,...,L-1,L;

k _x0 = k ₀ sinθ, k _z0 = k ₀ cosθ, k _xi = k ₀ sinθ+iK, the index i is a positive integer greater than 0, indicating the i-th order number expansion; K=2π/T, K is the inverse space base vector;

When k _xi <k ₀ , we have

When k _xi >k ₀ , we have

where j is the imaginary unit; k ₀ is the wave number in air; k _xi is the wave vector in the x direction; k _zi is the wave vector in the z direction; θ is the incident angle; T is the period of the porous material; D is the total thickness; Ri and T _i represent the reflection coefficient and transmission coefficient of the normalized _i -th order sound pressure, respectively; S _li (z) is the i-th order sound pressure coefficient of the l-th layer periodic porous material; exp( ) represents The exponential function with the natural constant e as the base;

The series expression form of the relevant acoustic parameters is:

where ρ _l (x) is the equivalent density of the lth layer of periodic porous material; n _l (x) is the equivalent refractive index of the lth layer of periodic porous material;

Among them, f _l is the duty ratio of the porous material in the lth layer of periodic porous material; n _p , ρ _p are the equivalent refractive index and equivalent density of the porous material, respectively; n ₀ , ρ ₀ are the refractive index of air and Density; j is the imaginary unit; m is the expansion order; K is the inverse space basis vector;

The obtained series expressions of the sound pressure and related acoustic parameters of each thin-layer material are brought into the acoustic wave equation; among them, in the one-dimensional periodic acoustic structure, the acoustic wave equation in the density inhomogeneous medium is:

为拉普拉斯算符；P为声压级数表达形式；P为入射、反射区域的声压级数表达形式P_I，透射区域的声压级数表达形式P_II，或多孔材料第l个薄层材料等效的周期矩形调制的声学材料中的声压级数表达形式P_l；n(x)为周期多孔材料的多孔材料的等效折射率的级数表达形式；in,

is the Laplace operator; P is the sound pressure level expression form; P is the sound pressure level expression form P _{I in the incident and reflection areas, the sound pressure level expression form P II in} the transmission area, or the first The sound pressure series expression P _l in the acoustic material equivalent to the periodic rectangular modulation of the thin layer material; n(x) is the series expression form of the equivalent refractive index of the porous material of the periodic porous material;

The coupling equations for the coefficients of each order are obtained:

Write the above coupling equation in matrix form:

[S _l ″]=[A _l ][S _l ]

Among them, S _l is a column vector composed of S _li ; S _l ″ is a column vector composed of d ² S _li /dz ² ; a matrix

和

其中，m1为行数；n1为列数；

和

均为系数；对角矩阵K_x的元素K_i,i＝k_xi；Among them, the element expressions of matrices X _l and Y _l are respectively

and

Among them, m1 is the number of rows; n1 is the number of columns;

and

are all coefficients; the elements of the diagonal matrix K _x are K _i,i =k _xi ;

和对应的特征向量

Calculate the eigenvalues and eigenvectors of matrix A _l , and get the square root of the mth eigenvalue

and the corresponding eigenvectors

Then the expression of the sound pressure coefficient S _li (z):

为第l薄层材料内声压的正向本征态形式；

为第l薄层材料内声压的反向本征态形式；

为正向传播本征态形式的强度系数；

为反向传播本征态形式的强度系数；d₁为第一薄层的厚度；z为z方向的位置；Among them, l=1;

is the forward eigenstate form of the sound pressure in the first thin layer of material;

is the inverse eigenstate form of the sound pressure in the first thin layer of material;

is the strength coefficient of the forward propagation eigenstate form;

is the intensity coefficient in the form of back-propagating eigenstates; d ₁ is the thickness of the first thin layer; z is the position in the z direction;

Among them, l=2,3,...,L; _dp is the thickness of the pth thin layer;

和第l薄层材料内声压的反向本征态形式

According to the sound pressure coefficient S _li (z), determine the forward eigenstate form of the sound pressure in the material of the first thin layer

and the inverse eigenstate form of the sound pressure in the first thin layer of material

As one of the improvements of the above technical solutions, the iterative optimization algorithm is used to calculate the intensity coefficient vectors of the eigenstates of each order of the reflected sound pressure and the intensity of the eigenstates of each order of the transmitted sound pressure through the interlayer boundary continuity condition. Coefficient vector; its specific process is:

Boundary continuity conditions include sound pressure and normal particle vibration velocity continuity;

The normal particle vibration velocity v _z expression is

为归一化密度的倒数；Among them, j is the imaginary unit; ω is the angular frequency;

is the inverse of the normalized density;

Expand the reciprocal of the normalized density into the corresponding series form

为展开系数；Among them, ρ ₀ is the air density; ρ _l (x) is the equivalent density of the l thin layer;

is the expansion coefficient;

是：Vibration velocity of i-th order particle

Yes:

Among them, l=1;

为归一化密度的倒数的展开系数；Among them, l=2,3,...,L;

is the expansion coefficient of the reciprocal of the normalized density;

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

The continuity conditions for the intermediate interfaces are:

The interfacial continuity conditions for the last layer and the transmission region of periodic porous materials are:

是正向传播本征态形式的强度系数

组成的列向量；和

是反向传播本征态形式的强度系数

组成的列向量；W_l是特征向量

构成的矩阵，I是单位矩阵；Δ_i0是0阶对应系数为1，其他项为0的列向量；第l薄层矩阵V_l＝Z_lW_lQ_l，且矩阵Z_l的元素为

第l薄层对角矩阵Q_l，E_l和K_z分别是元素

元素

和元素k_zi组成的矩阵；W_L为特征向量

构成的矩阵；E_L为特征向量

构成的矩阵；第L薄层V_L＝Z_LW_LQ_L；where R is a column vector composed of reflection intensity coefficients R _i ; T is a column vector composed of transmission intensity coefficients T _i ;

is the strength coefficient of the forward-propagating eigenstate form

a column vector consisting of; and

is the strength coefficient of the backpropagating eigenstate form

composed of column vectors; W _l is the eigenvector

The matrix formed, I is the unit matrix; Δ _i0 is the column vector with the 0-order corresponding coefficient of 1 and other terms of 0; the lth thin layer matrix V _l =Z _l W _l Q _l , and the elements of the matrix Z _l are

The lth lamella diagonal matrices Q _l , E _l and K _z are elements of

element

A matrix composed of elements k _zi ; W _L is the eigenvector

The matrix formed; E _L is the eigenvector

The matrix formed; the Lth thin layer _VL =Z _L W _{L QL} ;

Through matrix calculation, the matrix equations of sound pressure reflection coefficient and transmission coefficient are obtained:

The iterative optimization algorithm is used to calculate the intensity coefficient vector R of each order eigenstate of reflected sound pressure and the intensity coefficient vector T of each order eigenstate of transmitted sound pressure; the details are as follows:

Express the last four terms of the above matrix equation as:

The iterative matrices f _L+1 and g _L+1 are introduced, and f _L+1 =I, g _L+1 =jK _z ; and the following iterative matrices are introduced:

Among them, a _L and b _L are the first column intermediate matrix and the second column intermediate matrix introduced respectively;

通过整理得到如下关系：Have

The following relationship is obtained by sorting:

通过迭代计算，获得矩阵方程：Then the iterative matrix expression is:

Through iterative calculation, the matrix equation is obtained:

Solve the matrix equation to get R and T ₁ ;

Transmission coefficient vector T ₁ =(g ₁ +jK _z f ₁ ) ^-1 (jK _z Δ _i0 +jk ₀ cosθΔ _i0 );

The intensity coefficient vector R=f ₁ T ₁ -Δ _i0 of each order eigenstate of the reflected sound pressure;

Intensity coefficient vector of each order eigenstate of transmitted sound pressure

As one of the improvements of the above technical solutions, 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 is:

Calculate the total reflection coefficient R:

为反射声压的各阶本征态的强度系数向量R中的反射强度系数R_i取共轭；in,

Take the conjugate for the reflection intensity coefficient R _i in the intensity coefficient vector R of each order eigenstate of the reflected sound pressure;

Calculate the total transmission coefficient T:

T=∑ _i T _i T _i *Re(k _zi /k _z0 );

Among them, T _i ^* is the conjugate of the transmission intensity coefficient T _i in the intensity coefficient vector T of the eigenstates of each order of the transmitted sound pressure; Re() is the real part;

According to the calculated total reflection coefficient R and total transmission coefficient T, calculate the sound absorption coefficient α of the periodic non-flat interface porous material;

a=1-R-T.

The present invention also provides a device for measuring the sound absorption coefficient of a periodic non-flat interface porous material, the device comprising:

an acoustic parameter acquisition module, used for acquiring equivalent acoustic parameters of the periodic non-flat interface porous material; wherein, the equivalent acoustic parameters include: equivalent refractive index and equivalent density;

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

The eigenstate calculation module is used to obtain the equivalent acoustic pressure and related acoustic parameters of each thin-layer material in the periodic rectangular-modulated acoustic material according to the obtained equivalent acoustic parameters of the porous material. Enter the sound wave equation to calculate the eigenstate form of the sound pressure in each thin-layer material;

The intensity coefficient calculation module is used to calculate the intensity coefficient vector of each order eigenstate form of reflected sound pressure and the intensity coefficient vector of each order eigenstate form of transmitted sound pressure by using the iterative optimization algorithm through the continuity condition of the interlayer boundary ;and

The sound absorption coefficient acquisition module is used to determine the total reflection coefficient and the total transmission coefficient according to the above calculation results, and calculate the sound absorption coefficient of the periodic non-flat interface porous material.

The beneficial effects of the present invention compared with the prior art are:

1. The method of the present invention obtains the equivalent acoustic parameters of the porous material to be tested, that is, only two samples of planar porous materials with different thicknesses are required to be measured, and there is no need to prepare samples of different geometric shapes and structural dimensions, There is no need to carry out a large number of repeated experimental tests, which greatly improves the measurement efficiency, so that the sound absorption coefficient of porous materials with periodic non-flat interfaces can be measured more quickly;

2. The method of the present invention considers the influence of periodic modulation on the sound absorption performance, and includes high-order modes including evanescent waves, which improves the accuracy of the measured sound absorption coefficient;

3. After determining the structural geometric parameters, the method of the present invention only needs to bring in the equivalent acoustic parameters of each frequency to accurately predict the broadband sound absorption coefficient of the periodic non-flat interface porous material, which simplifies the calculation complexity and saves money. The time cost greatly improves the measurement efficiency, which is of great significance to the application in the field of noise control.

Description of drawings

1 is a flow chart of a method for calculating the sound absorption coefficient of a periodic non-flat interface porous material of the present invention;

Fig. 2 is a structural schematic diagram of measuring the reflection coefficient R of the porous material in a microphone impedance tube;

3 is a schematic structural diagram of the tip and the base of the periodic non-planar interface porous material in the method of FIG. 1;

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

5a is a schematic diagram of the equivalent refractive index of the porous material in the method of FIG. 1 in a wide frequency range;

FIG. 5b is a schematic diagram of the equivalent density of the porous material in the broad frequency range in the method of FIG. 1;

6a is a schematic structural diagram of the geometry of the sound-absorbing cotton whose sample to be tested is a triangular periodic modulation;

Fig. 6b is the structural schematic diagram of the geometric shape of the sound-absorbing cotton whose sample to be tested is a rectangular periodic modulation;

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

Detailed ways

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

As shown in FIG. 1, the present invention provides a method for calculating the sound absorption coefficient of periodic non-flat interface porous materials. The method of the present invention only needs to bring in the equivalent acoustic parameters of each frequency after determining the structural geometric parameters, namely The broadband sound absorption coefficient of periodic non-flat interface porous materials can be accurately predicted, which simplifies the computational complexity and saves time and cost, which is of great significance for the application in the field of noise control.

The method includes:

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

Specifically, as shown in Figures 2, 3 and 4, in a microphone impedance tube, a planar structure of a porous material with 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)

where ζ is the surface impedance of the porous material, which is a known value; where ζ=Z _p coth(γt);

Among them, γ is the complex propagation constant of the periodic non-planar interface porous material; Z _p is the normalized characteristic impedance of the periodic non-planar interface porous material;

Based on the above formula, from the measured reflection coefficient R ₁ of the planar structure of the porous material with the first thickness t ₁ and the reflection coefficient R ₂ of the planar structure of the second porous material with thickness t ₂ , the first thickness is calculated as t ₁ , respectively. the surface impedance ζ ₁ of the planar structure of the porous material and the surface impedance ζ _{2 of the planar structure of the porous material with a second thickness t 2 ;}

ζ ₁ =(1+R ₁ )/(1-R ₁ )

ζ ₂ =(1+R ₂ )/(1-R ₂ )

Wherein, ζ ₁ =Z _p coth(γt ₁ ); ζ ₂ =Z _p coth(γt ₂ );

According to ζ ₁ tanh(γt ₁ )-ζ ₂ tanh(γt ₂ )=0;

Combined with the above formula, the complex propagation constant γ and the normalized characteristic impedance Z _p are obtained by calculation;

The equivalent refractive index n _p and the equivalent density ρ _p are calculated according to the following formulas:

n _p =ω/(-jγc ₀ );

ρ _p =(-jγρ ₀ c ₀ Z _p )/ω;

Among them, c ₀ is the sound speed of air; ρ ₀ is the density of air; ω is the angular frequency; j is the imaginary unit; .

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

The periodic non-flat interface porous material is layered, and each thin layer material is equivalent to a periodic rectangular modulated acoustic material;

Specifically, the tip of the periodic non-planar interface porous material is a non-planar, incompletely filled porous material with multiple thin layers; the periodic non-planar interface porous material is layered to obtain a plurality of thin-layer materials, each of which is Each thin-layer material is equivalent to a periodic rectangular modulated acoustic material;

The base of the periodic non-planar interface porous material is a completely filled porous material, and the porous material is a completely filled, periodic rectangularly modulated acoustic material with a filling rate of 1.

Wherein, the periodic non-planar interface porous material includes: periodically arranged, non-planar, non-completely filled porous material tips and completely filled porous material bases.

According to the obtained equivalent acoustic parameters of the porous material, the sound pressure and related acoustic parameters in the periodic rectangular modulated acoustic material equivalent to each thin-layer material are obtained and expanded into a series expression, and brought into the acoustic wave equation to calculate each thin-layer The eigenstate form of the sound pressure in the material;

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

Among them, P _I is the expression form of the sound pressure level in the incident and reflection areas; P _II is the expression form of the sound pressure level in the transmission area, and P _l is the periodic rectangular modulation of the acoustic material equivalent to the lth thin layer of the porous material. The expression form of sound pressure level in ; where, l=1,2,...,L-1,L;

k _x0 = k ₀ sinθ, k _z0 = k ₀ cosθ, k _xi = k ₀ sinθ+iK, the index i is a positive integer greater than 0, indicating the i-th order number expansion; K=2π/T, K is the inverse space base vector;

When k _xi <k ₀ , we have

When k _xi >k ₀ , we have

where j is the imaginary unit; k ₀ is the wave number in air; k _xi is the wave vector in the x direction; k _zi is the wave vector in the z direction; θ is the incident angle; T is the period of the porous material; D is the total thickness; Ri and T _i represent the reflection coefficient and transmission coefficient of the normalized _i -th order sound pressure, respectively; S _li (z) is the i-th order sound pressure coefficient of the l-th layer periodic porous material; exp() represents The exponential function with the natural constant e as the base;

The series expression form of the relevant acoustic parameters is:

where ρ _l (x) is the equivalent density of the lth layer of periodic porous material; n _l (x) is the equivalent refractive index of the lth layer of periodic porous material;

Among them, f _l is the duty ratio of the porous material in the lth layer of periodic porous material; n _p , ρ _p are the equivalent refractive index and equivalent density of the porous material, respectively; n ₀ , ρ ₀ are the refractive index of air and Density; j is the imaginary unit; m is the expansion order; K is the inverse space basis vector;

The obtained series expressions of the sound pressure and related acoustic parameters of each thin-layer material are brought into the acoustic wave equation; among them, in the one-dimensional periodic acoustic structure, the acoustic wave equation in the density inhomogeneous medium is:

为拉普拉斯算符；P为声压级数表达形式；P为入射、反射区域的声压级数表达形式P_I，透射区域的声压级数表达形式P_II，或多孔材料第l个薄层材料等效的周期矩形调制的声学材料中的声压级数表达形式P_l；n(x)为周期多孔材料的多孔材料的等效折射率的级数表达形式；in,

is the Laplace operator; P is the sound pressure level expression form; P is the sound pressure level expression form P _{I in the incident and reflection areas, the sound pressure level expression form P II in} the transmission area, or the first The sound pressure series expression P _l in the acoustic material equivalent to the periodic rectangular modulation of the thin layer material; n(x) is the series expression form of the equivalent refractive index of the porous material of the periodic porous material;

The coupling equations for the coefficients of each order are obtained:

Write the above coupling equation in matrix form:

[S _l ″]=[A _l ][S _l ]

Among them, S _l is a column vector composed of S _li ; S _l ″ is a column vector composed of d ² S _li /dz ² ; a matrix

和

其中，m1为行数；n1为列数；

和

均为系数；对角矩阵K_x的元素K_i，i＝k_xi；Among them, the element expressions of matrices X _l and Y _l are respectively

and

Among them, m1 is the number of rows; n1 is the number of columns;

and

are coefficients; the elements K i of the diagonal matrix K _{x , i} =k _xi ;

和对应的特征向量

Calculate the eigenvalues and eigenvectors of matrix A _l , and get the square root of the mth eigenvalue

and the corresponding eigenvectors

Then the expression of the sound pressure coefficient S _li (z):

为第l薄层材料内声压的正向本征态形式；

为第l薄层材料内声压的反向本征态形式；

为正向传播本征态形式的强度系数；

为反向传播本征态形式的强度系数；d₁为第一薄层的厚度；z为z方向的位置；Among them, l=1;

is the forward eigenstate form of the sound pressure in the first thin layer of material;

is the inverse eigenstate form of the sound pressure in the first thin layer of material;

is the strength coefficient of the forward propagation eigenstate form;

is the intensity coefficient in the form of back-propagating eigenstates; d ₁ is the thickness of the first thin layer; z is the position in the z direction;

Among them, l=2,3,...,L; _dp is the thickness of the pth thin layer;

和第l薄层材料内声压的反向本征态形式

According to the sound pressure coefficient S _li (z), determine the forward eigenstate form of the sound pressure in the material of the first thin layer

and the inverse eigenstate form of the sound pressure in the first thin layer of material

Through the interlayer boundary continuity condition, the iterative optimization algorithm is used to calculate the intensity coefficient vector of each order eigenstate form of reflected sound pressure and the intensity coefficient vector of each order eigenstate form of transmitted sound pressure;

Specifically, the boundary continuity conditions include sound pressure and normal particle vibration velocity continuity;

The normal particle vibration velocity v _z expression is

为归一化密度的倒数；Among them, j is the imaginary unit; ω is the angular frequency;

is the inverse of the normalized density;

Expand the reciprocal of the normalized density into the corresponding series form

为展开系数；Among them, ρ ₀ is the air density; ρ _l (x) is the equivalent density of the l thin layer;

is the expansion coefficient;

是：Vibration velocity of i-th order particle

Yes:

Among them, l=1;

为归一化密度的倒数的展开系数；Among them, l=2,3,...,L;

is the expansion coefficient of the reciprocal of the normalized density;

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

The continuity conditions for the intermediate interfaces are:

The interfacial continuity conditions for the last layer and the transmission region of periodic porous materials are:

是正向传播本征态形式的强度系数

组成的列向量；和

是反向传播本征态形式的强度系数

组成的列向量；W_l是特征向量

构成的矩阵，I是单位矩阵；Δ_i0是0阶对应系数为1，其他项为0的列向量；第l薄层矩阵V_l＝Z_lW_lQ_l，且矩阵Z_l的元素为

第l薄层对角矩阵Q_l，E_l和K_z分别是元素

元素

和元素k_zi组成的矩阵；W_L为特征向量

构成的矩阵；E_L为特征向量

构成的矩阵；第L薄层V_L＝Z_LW_LQ_L；where R is a column vector composed of reflection intensity coefficients R _i ; T is a column vector composed of transmission intensity coefficients T _i ;

is the strength coefficient of the forward-propagating eigenstate form

a column vector consisting of; and

is the strength coefficient of the backpropagating eigenstate form

composed of column vectors; W _l is the eigenvector

The matrix formed, I is the unit matrix; Δ _i0 is the column vector with the 0-order corresponding coefficient of 1 and other terms of 0; the lth thin layer matrix V _l =Z _l W _l Q _l , and the elements of the matrix Z _l are

The lth lamella diagonal matrices Q _l , E _l and K _z are elements of

element

A matrix composed of elements k _zi ; W _L is the eigenvector

The matrix formed; E _L is the eigenvector

The matrix formed; the Lth thin layer _VL =Z _L W _{L QL} ;

Through matrix calculation, the matrix equations of sound pressure reflection coefficient and transmission coefficient are obtained:

The iterative optimization algorithm is used to calculate the intensity coefficient vector R of each order eigenstate of reflected sound pressure and the intensity coefficient vector T of each order eigenstate of transmitted sound pressure; the details are as follows:

Express the last four terms of the above matrix equation as:

The iterative matrices f _L+1 and g _L+1 are introduced, and f _L+1 =I, g _L+1 =jK _z ; and the following iterative matrices are introduced:

Among them, a _L and b _L are the first column intermediate matrix and the second column intermediate matrix introduced respectively;

通过整理得到如下关系：Have

The following relationship is obtained by sorting:

通过迭代计算，获得矩阵方程：Then the iterative matrix expression is:

Through iterative calculation, the matrix equation is obtained:

Solve the matrix equation to get R and T ₁ ;

Transmission coefficient vector T ₁ =(g ₁ +jK _z f ₁ ) ^-1 (jK _z Δ _i0 +jk ₀ cosθΔ _i0 );

The intensity coefficient vector R=f ₁ T ₁ -Δ _i0 of each order eigenstate of the reflected sound pressure;

Intensity coefficient vector of each order eigenstate of transmitted sound pressure

According to the above calculation results, 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.

Specifically, the total reflection coefficient R is calculated:

为反射声压的各阶本征态的强度系数向量R中的反射强度系数R_i取共轭；in,

Take the conjugate for the reflection intensity coefficient R _i in the intensity coefficient vector R of each order eigenstate of the reflected sound pressure;

Calculate the total transmission coefficient T:

T=∑ _i T _i T _* ⁱ Re(k _zi /k _z0 );

Among them, T _i ^* is the conjugate of the transmission intensity coefficient T _i in the intensity coefficient vector T of the eigenstates of each order of the transmitted sound pressure; Re() is the real part;

According to the calculated total reflection coefficient R and total transmission coefficient T, calculate the sound absorption coefficient α of the periodic non-flat interface porous material;

a=1-R-T.

The present invention also provides a device for measuring the sound absorption coefficient of a periodic non-flat interface porous material, the device comprising:

an acoustic parameter acquisition module, used for acquiring equivalent acoustic parameters of the periodic non-flat interface porous material; wherein, the equivalent acoustic parameters include: equivalent refractive index and equivalent density;

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

The eigenstate calculation module is used to obtain the equivalent acoustic pressure and related acoustic parameters of each thin-layer material in the periodic rectangular-modulated acoustic material according to the obtained equivalent acoustic parameters of the porous material. Enter the sound wave equation to calculate the eigenstate form of the sound pressure in each thin-layer material;

The intensity coefficient calculation module is used to calculate the intensity coefficient vector of each order eigenstate form of reflected sound pressure and the intensity coefficient vector of each order eigenstate form of transmitted sound pressure by using the iterative optimization algorithm through the continuity condition of the interlayer boundary ;and

The sound absorption coefficient acquisition module is used to determine the total reflection coefficient and the total transmission coefficient according to the above calculation results, and calculate the sound absorption coefficient of the periodic non-flat interface porous material.

As shown in Figures 5a and 5b, a commonly used porous sound-absorbing material, high-density sound-absorbing cotton, was purchased. First, the equivalent acoustic parameters of the porous material were tested through the microphone impedance tube system, and the thickness was 2 cm. and the reflection coefficient of the 4.8cm plane sound-absorbing cotton, and then the equivalent refractive index and equivalent density of the porous material were obtained by numerical calculation, and the results are shown in Figures 5a and 5b.

Using the method for sound absorption coefficient of porous material with periodic non-flat interface of the present invention, the sound absorption coefficients of two kinds of high-density sound-absorbing cotton with periodic non-flat interface are measured, and the two kinds of sound absorption are measured in the two-microphone impedance tube system. Cotton was tested experimentally, and the experimentally tested sound absorption coefficient was compared with the calculated results. The porous materials of the two periodic non-flat interfaces are triangular and rectangular periodically modulated sound-absorbing cottons, respectively, and the specific parameters are shown in Figures 6a and 6b, from which it can be determined that the calculation method of the present invention has higher accuracy.

As shown in Fig. 6a, the geometric parameters of the triangular periodically modulated sound-absorbing cotton include period T _t = 5 cm, base thickness d _t = 1 cm, and triangular tip thickness h _t = 3.7 cm:

As shown in Fig. 6b, the geometric parameters of the rectangular period modulated sound-absorbing cotton include period Tr = 3.3 cm, base thickness _dr = 1.9 cm, rectangular tip thickness hr = _2.8 cm and rectangular tip width _{wr =} 1.5 cm:

The sound absorption coefficient of the porous material measured by the method of the present invention and the experimental test results are shown in Figure 7. The two calculation results are very close, indicating that the calculation method of the sound absorption coefficient of the periodic non-flat interface porous material proposed by the present invention is very good. efficient.

Finally, it should be noted that the above embodiments are only used to illustrate the technical solutions of the present invention and not to limit them. Although the present invention has been described in detail with reference to the embodiments, those of ordinary skill in the art should understand that any modification or equivalent replacement of the technical solutions of the present invention will not depart from the spirit and scope of the technical solutions of the present invention, and should be included in the present invention. within the scope of the claims.

Claims (8)

1. A method for calculating the sound absorption coefficient of a periodic non-flat interface porous material, the method comprising: Obtaining equivalent acoustic parameters of the periodic non-flat interface porous material; wherein, the equivalent acoustic parameters include: equivalent refractive index and equivalent density; The periodic non-flat interface porous material is layered, and each thin layer material is equivalent to a periodic rectangular modulated acoustic material; According to the obtained equivalent acoustic parameters of the porous material, the sound pressure and related acoustic parameters in the periodic rectangular modulated acoustic material equivalent to each thin-layer material are obtained and expanded into a series expression, and brought into the acoustic wave equation to calculate each thin-layer The eigenstate form of the sound pressure in the material; Through the interlayer boundary continuity condition, the iterative optimization algorithm is used to calculate the intensity coefficient vector of each order eigenstate form of reflected sound pressure and the intensity coefficient vector of each order eigenstate form of transmitted sound pressure; According to the above calculation results, 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. 2 . The method for calculating the sound absorption coefficient of periodic non-planar interface porous materials according to claim 1 , wherein the periodic non-planar interface porous materials comprise: periodically arranged, non-planar, and incompletely filled porous materials. 3 . Tip and fully filled porous material base. 3. The method for calculating the sound absorption coefficient of periodic non-flat interface porous materials according to claim 1, wherein the acquisition of the equivalent acoustic parameters of the periodic non-planar interface porous materials; the specific process is: According to the measured reflection coefficient R ₁ of the planar structure of the porous material with the first thickness t ₁ and the reflection coefficient R ₂ of the planar structure of the second porous material with thickness t ₂ , the plane of the porous material with the first thickness t ₁ is calculated respectively. the surface impedance ζ ₁ of the structure and the surface impedance ζ ₂ of the second thickness t ₂ planar structure of the porous material; ζ ₁ =(1+R ₁ )/(1-R ₁ ) ζ ₂ =(1+R ₂ )/(1-R ₂ ) Wherein, ζ ₁ =Z _p coth(γt ₁ ); ζ ₂ =Z _p coth(γt ₂ ); According to ζ ₁ tanh(γt ₁ )-ζ ₂ tanh(γt ₂ )=0; Combined with the above formula, the complex propagation constant γ and the normalized characteristic impedance Z _p are obtained by calculation; The equivalent refractive index n _p and the equivalent density ρ _p are calculated according to the following formulas: n _p =ω/(-jγc ₀ ); ρ _p =(-jγρ ₀ c ₀ Z _p )/ω; where c ₀ is the speed of sound of air; ρ ₀ is the density of air; ω is the angular frequency; j is the imaginary unit; Take the calculated equivalent refractive index and equivalent density as equivalent acoustic parameters. 4. The method for calculating the sound absorption coefficient of periodic non-planar interface porous material according to claim 1, wherein the periodic non-planar interface porous material is layered, and each thin layer material is equivalent to a periodic rectangle Modulated acoustic material; the specific process is: The tip of the periodic non-planar interface porous material is a non-planar, non-completely filled porous material with multiple thin layers; the periodic non-planar interface porous material is layered to obtain a plurality of thin layer materials, each thin layer The material is equivalent to an acoustic material with periodic rectangular modulation; The base of the periodic non-planar interface porous material is a completely filled porous material, and the porous material is a completely filled, periodic rectangularly modulated acoustic material with a filling rate of 1. 5 . The method for calculating the sound absorption coefficient of a porous material with a periodic non-flat interface according to claim 1 , wherein, according to the equivalent acoustic parameters of the porous material, the acoustic equivalent periodic rectangular modulation of each thin-layer material is obtained. 6 . The sound pressure and related acoustic parameters in the material are expanded into a series expression, and brought into the acoustic wave equation to calculate the eigenstate form of the sound pressure in each thin-layer material; the specific process is as follows: The expression of the sound pressure level in the equivalent periodic rectangular modulated acoustic material of each thin layer material is:

Among them, P _I is the expression form of the sound pressure level in the incident and reflection areas; P _II is the expression form of the sound pressure level in the transmission area, and P _l is the periodic rectangular modulation of the acoustic material equivalent to the lth thin layer of the porous material. The expression form of sound pressure level in ; where, l=1,2,...,L-1,L; k _x0 = k ₀ sinθ, k _z0 = k ₀ cosθ, k _xi = k ₀ sinθ+iK, the index i is a positive integer greater than 0, indicating the i-th order number expansion; K=2π/T, K is the inverse space base vector; When k _xi <k ₀ , we have

When k _xi >k ₀ , we have

where j is the imaginary unit; k ₀ is the wave number in air; k _xi is the wave vector in the x direction; k _zi is the wave vector in the z direction; θ is the incident angle; T is the period of the porous material; D is the total thickness; Ri and T _i represent the reflection coefficient and transmission coefficient of the normalized _i -th order sound pressure, respectively; S _li (z) is the i-th order sound pressure coefficient of the l-th layer periodic porous material; exp() represents The exponential function with the natural constant e as the base; The series expression form of the relevant acoustic parameters is:

where ρ _l (x) is the equivalent density of the lth layer of periodic porous material; n _l (x) is the equivalent refractive index of the lth layer of periodic porous material;

Among them, f _l is the duty ratio of the porous material in the lth layer of periodic porous material; n _p , ρ _p are the equivalent refractive index and equivalent density of the porous material, respectively; n ₀ , ρ ₀ are the refractive index of air and Density; j is the imaginary unit; m is the expansion order; K is the inverse space basis vector; The obtained series expressions of the sound pressure and related acoustic parameters of each thin-layer material are brought into the acoustic wave equation; among them, in the one-dimensional periodic acoustic structure, the acoustic wave equation in the density inhomogeneous medium is:

in,

is the Laplace operator; P is the sound pressure level expression form; P is the sound pressure level expression form P _{I in the incident and reflection areas, the sound pressure level expression form P II in} the transmission area, or the first The sound pressure series expression P _l in the acoustic material equivalent to the periodic rectangular modulation of the thin layer material; n(x) is the series expression form of the equivalent refractive index of the porous material of the periodic porous material; The coupling equations for the coefficients of each order are obtained:

Write the above coupling equation in matrix form: [S _l ″]=[A _l ][S _l ] Among them, S _l is a column vector composed of S _li ; S _l ″ is a column vector composed of d ² S _li /dz ² ; a matrix

Among them, the element expressions of matrices X _l and Y _l are respectively

and

Among them, m1 is the number of rows; n1 is the number of columns;

and

are coefficients; the elements K i of the diagonal matrix K _{x , i} =k _xi ; Calculate the eigenvalues and eigenvectors of matrix A _l , and get the square root of the mth eigenvalue

and the corresponding eigenvectors

Then the expression of the sound pressure coefficient S _li (z):

Among them, l=1;

is the forward eigenstate form of the sound pressure in the first thin layer of material;

is the inverse eigenstate form of the sound pressure in the first thin layer of material;

is the strength coefficient of the forward propagation eigenstate form;

is the intensity coefficient in the form of back-propagating eigenstates; d ₁ is the thickness of the first thin layer; z is the position in the z direction;

Among them, l=2,3,...,L; _dp is the thickness of the pth thin layer; According to the sound pressure coefficient S _li (z), determine the forward eigenstate form of the sound pressure in the material of the first thin layer

and the inverse eigenstate form of the sound pressure in the first thin layer of material

6. The method for calculating the sound absorption coefficient of periodic non-flat interface porous materials according to claim 1, wherein, according to the interlayer boundary continuity condition, an iterative optimization algorithm is used to calculate the intrinsic properties of each order of reflected sound pressure The intensity coefficient vector of the state and the intensity coefficient vector of each order eigenstate of the transmitted sound pressure; the specific process is: Boundary continuity conditions include sound pressure and normal particle vibration velocity continuity; The normal particle vibration velocity v _z expression is

Among them, j is the imaginary unit; ω is the angular frequency;

is the inverse of the normalized density; Expand the reciprocal of the normalized density into the corresponding series form

Among them, ρ ₀ is the air density; ρ _l (x) is the equivalent density of the l thin layer;

is the expansion coefficient; Vibration velocity of i-th order particle

Yes:

Among them, l=1;

in,

is the expansion coefficient of the reciprocal of the normalized density; The interfacial continuity conditions for the incident, reflective regions and the first layer of periodic porous materials are:

The continuity conditions for the intermediate interfaces are:

The interfacial continuity conditions for the last layer and the transmission region of periodic porous materials are:

where R is a column vector composed of reflection intensity coefficients R _i ; T is a column vector composed of transmission intensity coefficients T _i ;

is the strength coefficient of the forward-propagating eigenstate form

a column vector consisting of; and

is the strength coefficient of the backpropagating eigenstate form

composed of column vectors; W _l is the eigenvector

The matrix formed, I is the unit matrix; Δ _i0 is the column vector with the 0-order corresponding coefficient of 1 and other terms of 0; the first thin layer matrix V _l =Z _l W _l Q _l , and the elements of the matrix Z _l are

The lth lamella diagonal matrices Q _l , E _l and K _z are elements of

element

A matrix composed of elements k _zi ; W _L is the eigenvector

The matrix formed; E _L is the eigenvector

The matrix formed; the Lth thin layer _VL =Z _L W _{L QL} ; Through matrix calculation, the matrix equations of sound pressure reflection coefficient and transmission coefficient are obtained:

The iterative optimization algorithm is used to calculate the intensity coefficient vector R of each order eigenstate of reflected sound pressure and the intensity coefficient vector T of each order eigenstate of transmitted sound pressure; the details are as follows: Express the last four terms of the above matrix equation as:

The iterative matrices f _L+1 and g _L+1 are introduced, and f _L+1 =I, g _L+1 =jK _z ; and the following iterative matrices are introduced:

Among them, a _L and b _L are the first column intermediate matrix and the second column intermediate matrix introduced respectively; Have

The following relationship is obtained by sorting:

Then the iterative matrix expression is:

Through iterative calculation, the matrix equation is obtained:

Solve the matrix equation to get R and T ₁ ; Transmission coefficient vector T ₁ =(g ₁ +jK _z f ₁ ) ^-1 (jK _z Δ _i0 +jk ₀ cosθΔ _i0 ); The intensity coefficient vector R=f ₁ T ₁ -Δ _i0 of each order eigenstate of the reflected sound pressure; Intensity coefficient vector of each order eigenstate of transmitted sound pressure

7 . The method for calculating the sound absorption coefficient of periodic non-flat interface porous materials 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 is: Calculate the total reflection coefficient R:

in,

Take the conjugate for the reflection intensity coefficient R _i in the intensity coefficient vector R of each order eigenstate of the reflected sound pressure; Calculate the total transmission coefficient T:

in,

is the conjugate of the transmission intensity coefficient T _i in the intensity coefficient vector T of each order eigenstate of the transmitted sound pressure; Re() is the real part; According to the calculated total reflection coefficient R and total transmission coefficient T, calculate the sound absorption coefficient α of the periodic non-flat interface porous material; a=1-R-T. 8. A measuring device for the sound absorption coefficient of a periodic non-flat interface porous material, characterized in that the device comprises: an acoustic parameter acquisition module, used for acquiring equivalent acoustic parameters of the periodic non-flat interface porous material; wherein, the equivalent acoustic parameters include: equivalent refractive index and equivalent density; The layering module is used to layer the periodic non-flat interface porous material, and each thin layer material is equivalent to a periodic rectangular modulated acoustic material; The eigenstate calculation module is used to obtain the equivalent acoustic pressure and related acoustic parameters of each thin-layer material in the periodic rectangular-modulated acoustic material according to the obtained equivalent acoustic parameters of the porous material, and expand it into a series expression, with Enter the sound wave equation to calculate the eigenstate form of the sound pressure in each thin-layer material; The intensity coefficient calculation module is used to calculate the intensity coefficient vector of each order eigenstate form of reflected sound pressure and the intensity coefficient vector of each order eigenstate form of transmitted sound pressure by using the iterative optimization algorithm through the continuity condition of the interlayer boundary ;and The sound absorption coefficient acquisition module is used to determine the total reflection coefficient and the total transmission coefficient according to the above calculation results, and calculate the sound absorption coefficient of the periodic non-flat interface porous material.

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