A method for determining the sound pressure response of a closed acoustic cavity coupled with a perforated plate

Abstract

The application discloses a method for determining the sound pressure response of a coupled closed sound cavity with a perforated plate, which comprises the following steps: a coupled closed sound cavity is formed by two sub-sound fields, a middle perforated plate and a point sound source; an expression form of the sound pressure response in the sub-sound field is given by referring to Chebyshev polynomial series; an energy function of the coupled closed sound cavity is established; the coupling energy of the perforated plate is obtained in combination with the perforated plate theory; the unknown Chebyshev polynomial coefficients in the energy function of the coupled closed sound cavity are taken to extreme values by using the Rayleigh-Ritz method to obtain a control equation of the coupled closed sound cavity; the unknown coefficients in the sound pressure expression are obtained by solving the control equation; and the sound pressure response of the coupled closed sound cavity is obtained by bringing the unknown coefficients into the sound pressure expression. The sound pressure response of the coupled closed sound cavity with the perforated plate can be efficiently and accurately calculated, the calculation time can be effectively reduced, and the demand for computing resources can be reduced while ensuring accuracy.

Description

Technical Field

[0001] This invention pertains to the field of acoustics, specifically to a method for determining the sound pressure response of a perforated plate coupled closed acoustic cavity. Background Technology

[0002] Pipeline structures are widely used in various engineering fields, and their low-noise design and acoustic characteristic control have received widespread attention from scholars. Regarding the low-frequency broadband noise present in pipelines, scholars from various countries have conducted continuous exploration and extensive research on how to develop novel pipeline noise reduction methods with good low-frequency broadband noise suppression performance. Some scholars have proposed (micro)perforated noise reduction structures for low-frequency noise reduction, such as (micro)perforated pipe silencers. In such pipeline silencers, the internal sound field is usually divided into multiple acoustic cavities by perforated plates (pipes). These cavities interact through the perforated structure, thereby achieving noise reduction. Analyzing the acoustic response of the coupled sound field with perforated coupling interfaces is of significant engineering importance. Existing literature shows that the methods for analyzing the coupled acoustic space with perforated coupling interfaces are mostly based on the modal superposition method of wave acoustics (Zhou Zhenwei. Research on the coupling of elastic microperforated plates / membranes with cavities and its application in low-frequency noise reduction [D]. South China University of Technology, 2019.) and the finite element method. The former is not only cumbersome in its processing, but also requires a high number of truncations to ensure the accuracy of the final result, resulting in excessive data processing volume; while the finite element method has a relatively cumbersome process, and its accuracy is affected by the mesh fineness, requiring high computing power resources. Summary of the Invention

[0003] The purpose of this invention is to provide a method for determining the acoustic pressure response of a perforated plate coupled closed acoustic cavity that is simple to process, accurate in results, and highly efficient.

[0004] To achieve the above objectives, the present invention adopts the following technical solution.

[0005] A method for determining the acoustic pressure response of a perforated plate coupled closed acoustic cavity includes the following steps:

[0006] Step 1: A coupled closed acoustic cavity is formed by two sub-sound fields, a perforated plate in the middle, and a point sound source;

[0007] Step 2: Use Chebyshev polynomial series to give the expression of the sound pressure response in the subfield;

[0008] Step 3: Establish the energy function of the coupled closed acoustic cavity; the energy of the coupled closed acoustic cavity consists of the potential energy and kinetic energy of the two sub-sound fields, the work done by the sound source, and the coupling energy generated by the interaction of the sub-sound fields on both sides of the perforated plate; the coupling energy of the perforated plate is obtained by combining the perforated plate theory;

[0009] Step 4: Use the Rayleigh-Ritz method to find the extreme values ​​of the unknown Chebyshev polynomial coefficients in the energy function of the coupled closed acoustic cavity to obtain the control equation of the coupled closed acoustic cavity;

[0010] Step four involves deriving the governing equations for the coupled closed acoustic cavity, including:

[0011] Using the Rayleigh-Ritz method to find the extreme values ​​of each unknown Chebyshev polynomial coefficient in the sound pressure function of energy function (3), it can be expressed as:

[0012] (10)

[0013] After processing, the governing equations for the coupled closed acoustic cavity are obtained:

[0014] (11)

[0015] In equation (11), , These are the stiffness and mass matrices of the first subfield, respectively. , These are the stiffness matrix and mass matrix of the second sub-sound field, respectively, and their specific calculation formulas are as follows:

[0016] (12)

[0017] (13)

[0018] in, These represent the positions of elements in the stiffness matrix and mass matrix, respectively, corresponding to rows and columns in the matrix; The i-th sub-sound field Dimensions in three directions, i=1,2; The density of the sound field medium; Speed ​​of sound; for The direction of the first The derivative of a Chebyshev polynomial of order 1 and This is used to distinguish whether the Rayleigh-Ritz method is used to find the extreme value of the unknown coefficient; the latter is the term for which the extreme value is to be found. for The direction of the first Chebyshev polynomial of order 1; for The direction of the first Chebyshev polynomial of order 1; for The direction of the first Chebyshev polynomial of order 1;

[0019] In equation (11), For the coupling matrix submatrix, the specific calculation formula is:

[0020] (14)

[0021] (15)

[0022] (16)

[0023] (17)

[0024] in, These represent the positions of elements in each matrix, corresponding to rows and columns, respectively. The perforated plates are respectively Dimensions in three directions; The perforated acoustic impedance is calculated using equation (8); for The direction of the first The value of a Chebyshev polynomial of order 1, where the variable takes the value 1. and This is used to distinguish whether the Rayleigh-Ritz method is used to find the extreme value of the unknown coefficient; the latter is the term for which the extreme value is to be found. for The direction of the first The value of a Chebyshev polynomial of order -1; for The direction of the first Chebyshev polynomial of order 1; for The direction of the first Chebyshev polynomial of order 1;

[0025] In equation (11), , Let be vectors composed of all unknown coefficients in the first and second sub-sound fields, respectively.

[0026] (18)

[0027] In equation (10), The work vector of the sound source in the coupled system;

[0028] (19)

[0029] in: The imaginary unit, The intensity of the sound source; The coordinates of the point sound source; for The direction of the first Chebyshev polynomial of order 1, and the variables take... The value at time; for The direction of the first Chebyshev polynomial of order 1, and the variables take... The value at time; for The direction of the first Chebyshev polynomial of order 1, and the variables take... The value at time;

[0030] Step 5: Solve the governing equations to obtain the unknown coefficients in the sound pressure expression, and substitute the unknown coefficients into the sound pressure expression to obtain the sound pressure response of the coupled closed acoustic cavity.

[0031] In a further improvement or specific implementation of the aforementioned method for determining the sound pressure response of a perforated plate coupled closed acoustic cavity, in step two, the sound pressure response of the two sub-sound fields of the perforated plate coupled closed acoustic cavity is solved based on the Chebyshev polynomial series method. The sound field is expanded according to the Chebyshev polynomial series in all three directions, and its expression is as follows:

[0032] (1)

[0033] in: Number the sub-sound fields. For coefficients, They are respectively The Chebyshev polynomial series expansion in three directions is as follows:

[0034] (2)

[0035] For a further improvement or specific implementation of the aforementioned method for determining the acoustic pressure response of a perforated plate coupled closed acoustic cavity, step three specifically includes:

[0036] The energy function of the coupled closed acoustic cavity is established using energy principles:

[0037] (3)

[0038] In formula (3), , The potential and kinetic energy of the first sub-sound field; , The potential and kinetic energy of the second sub-sound field are calculated as follows:

[0039] , (4)

[0040] , (5)

[0041] in, The density of the sound field medium, Angular frequency, , For frequency, The sound pressure gradient Sub-sound field volume;

[0042] In formula (3), Let be the coupling energy in the first sub-sound field under coupling effect. This represents the coupling energy in the second sub-sound field under the coupling effect. The calculation method is as follows:

[0043] (6)

[0044] According to the principle of interaction:

[0045] (7)

[0046] in, These represent the pressure on both sides of the perforated plate; Angular frequency; The area of ​​the perforated plate; The vibration velocity of the particles inside the perforation of the perforated plate; The perforation rate of a perforated plate is the ratio of the perforated area to the area of ​​the perforated plate. The acoustic impedance of a perforated structure is the ratio of the acoustic pressure difference across the perforation to the vibration velocity of the particles inside the perforation. It is calculated as follows:

[0047] (8)

[0048] in, The imaginary unit, Angular frequency, and These are the dynamic viscosity coefficient and density of the fluid medium, respectively. For the thickness of the perforated plate, The diameter of the perforation. The perforation constant;

[0049] In formula (3), The term representing the work done by a point sound source on the sound field is calculated as follows:

[0050] (9)

[0051] in, The sound pressure level at the point source is... For the intensity of the sound source, The imaginary unit, Angular frequency, Let be the volume of the subfield where the point sound source is located.

[0052] For a further improvement or specific implementation of the aforementioned method for determining the sound pressure response of a perforated plate coupled closed acoustic cavity, step five specifically refers to calculating the unknown coefficients in the sound pressure expression (1) according to the control equation (11), substituting the obtained unknown coefficients and the sound pressure observation point into equation (1), and calculating the sound pressure response at the sound pressure observation point.

[0053] Its beneficial effects are as follows:

[0054] This invention employs a perforated plate coupled closed acoustic cavity model, expressing the sound pressure response based on the Chebyshev polynomial series form. Building upon the energy function of the perforated plate coupled closed acoustic cavity constructed using energy principles, it innovatively utilizes the Rayleigh-Ritz method to obtain the sound pressure control equation of the perforated plate coupled closed acoustic cavity by taking the extreme values ​​of the unknown coefficients in the energy function. The sound pressure response in the coupled model is obtained by solving for the unknown coefficients, without requiring additional assumptions about the model. The physical meaning is clear, and the process is simple. Compared with existing methods, this invention does not require any meshing, converges rapidly while ensuring accuracy, requires less computational resources, and is faster. Attached Figure Description

[0055] Figure 1 A schematic diagram of the acoustic space coupled by the perforated plate;

[0056] Figure 2 Diagram of the coupled acoustic field model;

[0057] Figure 3 The convergence plots of the sound pressure response at different cutoff numbers are shown.

[0058] Figure 4 Let be the sound pressure response at observation point 1 (1.4, 0.3, 0.6) under the excitation of sound source 1 (0.1, 0.1, 0.1). The parameters of the perforated plate are ( );

[0059] Figure 5 Let be the sound pressure response at observation point 2 (0.3, 0.8, 0.2) under the excitation of sound source 1 (0.1, 0.1, 0.1). The parameters of the perforated plate are ( );

[0060] Figure 6 Let be the sound pressure response at observation point 3 (1.7, 0.5, 0.6) under the excitation of sound source 2 (0.4, 0.3, 0.6). The parameters of the perforated plate are ( );

[0061] Figure 7 Let be the sound pressure response at observation point 4 (0.1, 0.7, 0.4) under the excitation of sound source 2 (0.4, 0.3, 0.6). The parameters of the perforated plate are ( ). Detailed Implementation

[0062] The present invention will be described in detail below with reference to specific embodiments.

[0063] This invention provides a method for determining the acoustic pressure response of a perforated plate coupled closed acoustic cavity, the specific steps of which are as follows:

[0064] Step 1: A coupled closed acoustic cavity is formed by two sub-sound fields, a perforated plate in the middle, and a point sound source;

[0065] Among them, the spatial structure of the coupled enclosed acoustic cavity and the sound field model are as follows: Figure 1 , Figure 2 As shown.

[0066] Step 2: Using Chebyshev polynomial series, give the expression for the sound pressure response within the subfield:

[0067] The sound pressure response of the two sub-sound fields coupled by the perforated plate in the closed acoustic cavity was solved using the Chebyshev polynomial series method. The sound field was expanded using Chebyshev polynomial series in all three directions:

[0068] (1)

[0069] in: Number the sub-sound fields. For unknown coefficients, They are respectively The Chebyshev polynomial series expansion in three directions is as follows:

[0070] (2)

[0071] Step 3: Establish the energy function of the coupled closed acoustic cavity; the energy of the coupled closed acoustic cavity consists of the potential energy and kinetic energy of the two sub-sound fields, the work done by the sound source, and the coupling energy generated by the interaction of the sub-sound fields on both sides of the perforated plate; the coupling energy of the perforated plate is derived by combining the perforated plate theory:

[0072] The energy function of the coupled closed acoustic cavity is established using energy principles:

[0073] (3)

[0074] In formula (3), , The potential and kinetic energy of the first sub-sound field; , The potential and kinetic energy of the second sub-sound field are expressed as follows:

[0075] , (4)

[0076] , (5)

[0077] in, The density of the sound field medium, Angular frequency, , For frequency, The sound pressure gradient Sub-sound field volume;

[0078] In formula (3), Let be the coupling energy in the first sub-sound field under coupling effect. Let be the coupling energy in the second sub-sound field under coupling, and its expression is:

[0079] (6)

[0080] According to the principle of interaction:

[0081] (7)

[0082] in, These represent the pressure on both sides of the perforated plate; Angular frequency; The area of ​​the perforated plate; The vibration velocity of the particles inside the perforation of the perforated plate; The perforation rate of the perforated plate is the ratio of the perforated area to the area of ​​the perforated plate. The acoustic impedance of the perforation is the ratio of the sound pressure difference across the perforation to the particle velocity inside the perforation. Its calculation expression is:

[0083] (8)

[0084] in, The imaginary unit, Angular frequency, and These are the dynamic viscosity coefficient and density of the fluid medium, respectively. For the thickness of the perforated plate, The diameter of the perforation. The perforation constant;

[0085] In formula (3), Let be the term representing the work done by the point sound source on the sound field, and its expression is:

[0086] (9)

[0087] in, The sound pressure level at the point source is... For the intensity of the sound source, The imaginary unit, Angular frequency, Let be the volume of the subfield where the point sound source is located.

[0088] Step 4: The Rayleigh-Ritz method is used to find the extreme values ​​of the unknown Chebyshev polynomial coefficients in the energy function of the coupled closed acoustic cavity to obtain the governing equation of the coupled closed acoustic cavity. The method for obtaining the governing equation of the coupled closed acoustic cavity is as follows:

[0089] The Rayleigh-Ritz method is used to find the extreme values ​​of the unknown Chebyshev polynomial coefficients in each term of the sound pressure function:

[0090] (10)

[0091] After processing, the governing equations for the coupled closed acoustic cavity are obtained:

[0092] (11)

[0093] In equation (11), , These are the stiffness and mass matrices of the first subfield, respectively. , These are the stiffness matrix and mass matrix of the second sub-sound field, respectively, and their specific calculation formulas are as follows:

[0094] (12)

[0095] (13)

[0096] in, Number the sub-sound fields; These represent the positions of elements in the stiffness matrix (mass matrix), corresponding to rows and columns in the matrix, respectively. Sub-fields Dimensions in three directions; The density of the sound field medium; Speed ​​of sound; for The direction of the first The derivative of a Chebyshev polynomial of order 1 and This is used to distinguish whether the Rayleigh-Ritz method is used to find the extreme value of the unknown coefficient; the latter is the term for which the extreme value is to be found. for The direction of the first Chebyshev polynomial of order 1; for The direction of the first Chebyshev polynomial of order 1; for The direction of the first Chebyshev polynomial of order 1;

[0097] In equation (11), For the coupling matrix submatrix, the specific calculation formula is:

[0098] (12)

[0099] (13)

[0100] (14)

[0101] (15)

[0102] in, These represent the positions of elements in each matrix, corresponding to rows and columns, respectively. The perforated plates are respectively Dimensions in three directions; The perforated acoustic impedance is calculated using equation (8); for The direction of the first The value of a Chebyshev polynomial of order 1, where the variable takes the value 1. and This is used to distinguish whether the Rayleigh-Ritz method is used to find the extreme value of the unknown coefficient; the latter is the term for which the extreme value is to be found. for The direction of the first The value of a Chebyshev polynomial of order -1; for The direction of the first Chebyshev polynomial of order 1; for The direction of the first Chebyshev polynomial of order 1;

[0103] In equation (11), , These are vectors composed of all the unknown coefficients in the first and second sub-sound fields, respectively.

[0104] in, (18)

[0105] In equation (11), Let this be the work vector done by the sound source in the coupled system.

[0106] (19)

[0107] in: The imaginary unit, The intensity of the sound source; The coordinates of the point sound source; for The direction of the first Chebyshev polynomial of order 1, and the variables take... The value at time; for The direction of the first Chebyshev polynomial of order 1, and the variables take... The value at time; for The direction of the first Chebyshev polynomial of order 1, and the variables take... The value at time.

[0108] Step 5: Solve the governing equations to obtain the unknown coefficients in the sound pressure expression, and substitute the unknown coefficients into the sound pressure expression to obtain the sound pressure response of the coupled closed acoustic cavity.

[0109] The following numerical verification of the main contents of this application is based on the COMSOL finite element model from commercial software:

[0110] Combination Figure 2 A schematic diagram of the sound field in the verification example is shown. Following the aforementioned steps, the sound pressure response of the perforated plate coupled to the closed acoustic cavity is calculated, and the prediction results of this invention are compared with the finite element simulation results. The results are as follows: Figures 3-7 As shown.

[0111] The specific parameters of the two sub-sound fields are as follows: , , air density speed of sound .

[0112] The cutoff numbers for the Chebyshev polynomial series of the sound pressure admissibility function in each sound field are 3-3-3 and 6-6-6, respectively. Based on the COMSOL finite element calculation results, the convergence plot of the sound pressure response as a function of the cutoff number is shown below. Figure 3 As shown, when the truncation number is 6, the calculation results match well, indicating that the convergence speed of the prediction results of this invention is relatively fast.

[0113] from Figures 4-7It can be seen that the prediction results of the method of the present invention for the sound pressure response of the coupled sound field of the perforated plate are in good agreement with the COMSOL simulation results, which verifies the correctness of the present invention in predicting the acoustic characteristics of the coupled sound field of the perforated plate.

[0114] Finally, it should be noted that the above embodiments are only used to illustrate the technical solutions of the present invention, and are not intended to limit the scope of protection of the present invention. Although the present invention has been described in detail with reference to preferred embodiments, those skilled in the art should understand that modifications or equivalent substitutions can be made to the technical solutions of the present invention without departing from the essence and scope of the technical solutions of the present invention.

Claims

1. A method for determining the sound pressure response of a closed acoustic cavity coupled with a perforated plate, characterized by, The method comprises the following steps: Step one: a coupling closed cavity is formed by two sub-acoustic fields, a middle perforated plate and a point sound source; Step two: Chebyshev polynomial series is cited to give an expression form of the sound pressure response in the sub-acoustic field; Step three: an energy function of the coupling closed cavity is established; the energy of the coupling closed cavity is composed of potential energy, kinetic energy of the two sub-acoustic fields, work done by the sound source and coupling energy generated by the interaction of the two sub-acoustic fields on both sides of the perforated plate; the coupling energy of the perforated plate is obtained by combining the perforated plate theory; Step four: unknown Chebyshev polynomial coefficients in the energy function of the coupling closed cavity are taken to extreme values by using Rayleigh-Ritz method to obtain a control equation of the coupling closed cavity; The coupling closed cavity control equation in step four is obtained by: Each unknown Chebyshev polynomial coefficient of the sound pressure function in the energy function (3) is taken to extreme values by using Rayleigh-Ritz method, which can be expressed as: (10) After arrangement, the control equation of the coupling closed cavity is obtained: (11) In formula (11), , are the stiffness matrix and the mass matrix of the first sub-acoustic field, respectively, , are the stiffness matrix and the mass matrix of the second sub-acoustic field, respectively, and are calculated as follows: ，(12) ，(13) in, These represent the positions of elements in the stiffness matrix and mass matrix, respectively, corresponding to rows and columns in the matrix; The i-th sub-sound field Dimensions in three directions, i=1,2; The density of the sound field medium; Speed ​​of sound; for The direction of the first The derivative of a Chebyshev polynomial of order 1 and This is used to distinguish whether the Rayleigh-Ritz method is used to find the extreme value of the unknown coefficient; the latter is the term for which the extreme value is to be found. for The direction of the first Chebyshev polynomial of order 1; for The direction of the first Chebyshev polynomial of order 1; for The direction of the first Chebyshev polynomial of order 1; In formula (11), is a coupling matrix sub-matrix, and the specific calculation formula is: (14) (15) (16) (17) in, These represent the positions of elements in each matrix, corresponding to rows and columns, respectively. The perforated plates are respectively Dimensions in three directions; The perforated acoustic impedance is calculated using equation (8); for The direction of the first The value of a Chebyshev polynomial of order 1, where the variable takes the value 1. and This is used to distinguish whether the Rayleigh-Ritz method is used to find the extreme value of the unknown coefficient; the latter is the term for which the extreme value is to be found. for The direction of the first The value of a Chebyshev polynomial of order -1; for The direction of the first Chebyshev polynomial of order 1; for The direction of the first Chebyshev polynomial of order 1; In formula (11), , are vectors consisting of all unknown coefficients in the first sub- sound field and the second sub-sound field, respectively, wherein, (18) In formula (10), is the work vector of the sound source in the coupling system; (19) where: is the imaginary unit, is the sound source intensity; is the point sound source coordinate; is the first order Chebyshev polynomial in the direction and the variable takes the value when; is the first order Chebyshev polynomial in the direction and the variable takes the value when; is the first order Chebyshev polynomial in the direction and the variable takes the value when; Step five: unknown coefficients in the sound pressure expression are obtained by solving the control equation, and the unknown coefficients are brought into the sound pressure expression to obtain the sound pressure response of the coupling closed cavity.

2. The method for determining the sound pressure response of a perforated panel coupled closed acoustic cavity of claim 1, wherein, In step two, the sound pressure response of the two sub-acoustic fields of the perforated plate coupling closed cavity is solved based on the Chebyshev polynomial series method, and the three directions of the acoustic field are expanded according to the Chebyshev polynomial series, and the expression is: (1) wherein: is a sub- soundfield number, is a coefficient, are respectively Chebyshev polynomial series expansion in three directions, which is specifically in the form of: (2）。 3. The method for determining the sound pressure response of a perforated plate coupled closed acoustic cavity of claim 2, wherein, Step three specifically comprises: An energy function of the coupling closed cavity is established by using the energy principle: (3) In equation (3), , is the potential and kinetic energy of the first sub-acoustic field; , is the potential and kinetic energy of the second sub-acoustic field, which is calculated as ， (4) ， (5) wherein, is the sound field medium density, is the angular frequency, , is the frequency, is the sound pressure gradient, sub-sound field volume; In formula (3), is the coupling energy in the first sub-acoustic field under the coupling effect, is the coupling energy in the second sub-acoustic field under the coupling effect, The calculation method is: (6) According to the interaction principle: (7) where, P1, P2are the pressures on both sides of the perforated panel, respectively; ω is the angular frequency; A is the area of the perforated panel; v is the particle velocity in the perforation of the perforated panel; F is the perforation ratio of the perforated panel, which is the ratio of the perforation area to the area of the perforated panel; Z is the perforation acoustic impedance ratio, which is the ratio of the sound pressure difference on both sides of the perforation to the particle velocity in the perforation, and the calculation method is: (8) wherein is the imaginary unit, is the angular frequency, and are the dynamic viscosity and the density of the fluid medium, respectively, is the perforated plate thickness, is the perforated diameter, is the perforated constant; In equation (3), is the work done by the point source on the sound field, which is calculated as (9) wherein is the sound pressure value at the point source, is the sound source intensity, is the imaginary unit, is the angular frequency, is the volume of the sub sound field in which the point source is located.

4. The method for determining the sound pressure response of a perforated panel coupled closed acoustic cavity of claim 1, wherein, Step five specifically refers to that unknown coefficients in the sound pressure expression (1) are calculated according to the control equation (11), the unknown coefficients and the sound pressure observation point are brought into equation (1), and the sound pressure response at the sound pressure observation point is calculated.

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