Acoustic black hole structure sound radiation calculation method based on hybrid method of reduced multi-body system transfer matrix method and surface element method

Abstract

The invention discloses an acoustic black hole structure sound radiation calculation method based on a hybrid method of a reduced multi-body system transfer matrix method and a surface element method. Based on a reduced multi-body system transfer matrix method, establishing a total transfer equation of a combined system formed after the main structure is additionally provided with the acoustic black hole dynamic vibration absorbers, and solving the total transfer equation to obtain a steady-state response of the total transfer equation; and solving the radiation sound power of the main structure in the semi-infinite space by combining a surface element method and utilizing the speed distribution of the main structure. The method has the advantages of being high in calculation speed, flexible in modeling and wide in applicability when the acoustic radiation of the acoustic black hole dynamic vibration absorber and main structure combined system is calculated.

Description

Technical Field

[0001] The present invention relates to the dynamics and acoustics modeling and analysis of acoustic black hole structures, and specifically to a method for calculating the radiated sound power of acoustic black hole structures based on a hybrid method of the reduced multibody system transfer matrix method and the element radiator method (Reduced Multibody System Transfer Matrix Method-Element Radiators Method, hereinafter referred to as RMSTMM-ERM). Background Technique

[0002] In recent years, acoustic black hole structures have received increasing attention in the field of vibration reduction and noise control due to their advantages such as light weight, wide operating frequency band, and simple and flexible implementation. Previous acoustic black hole calculation methods mainly focused on vibration characteristics. Among them, the multibody system transfer matrix method has been proven to be an efficient calculation method for calculating the vibration response of acoustic black hole structures. There is relatively little research on the sound radiation of acoustic black hole structures. The current theoretical analysis methods for the sound radiation of acoustic black hole structures mainly include the analytical method, impedance analysis method, wavenumber analysis, etc. However, these theoretical analysis methods are only applicable to acoustic black hole structures with relatively simple geometric shapes. For the study of the sound radiation of more complex acoustic black hole structures, more reliance is placed on numerical simulation and experimental methods, and there is still a lack of effective theoretical calculation methods, especially for an efficient theoretical calculation method for the sound radiation of a combined system composed of an additional acoustic black hole structure as a dynamic vibration absorber and a main structure. Summary of the Invention

[0003] The purpose of the present invention is to provide a method for calculating the sound radiation of acoustic black hole structures based on a hybrid method of the reduced multibody system transfer matrix method and the element radiator method.

[0004] The technical solution to achieve the purpose of the present invention is: A method for calculating the sound power of an acoustic black hole based on a hybrid method of the reduced multibody system transfer matrix method and the element radiator method, comprising the following steps:

[0005] (1) Regarding the acoustic black hole dynamic vibration absorber as an additional vibration reduction device on the main structure, and the two constitute a combined system; wherein the acoustic black hole dynamic vibration absorber is a one-dimensional acoustic black hole structure, and the main structure adopts a beam structure form; then determine the size parameters and material parameters of each part of the combined system.

[0006] (2) Split the combined system into each component according to its structural form, number each component, and segment the main structure, and then determine the state vectors at the input and output ends of the components.

[0007] (3) Derive the transfer equation characterizing the relationship between the state vectors at the input and output ends of a multi-input single-output component and the motion coordination equation characterizing the relationship between the state vectors at its multiple input ends; derive its transfer equation for a single-input single-output component.

[0008] (4) Obtain the reduced transfer matrix S and the recurrence equation of e at the input and output ends of the component according to the transfer equation and the motion coordination equation of the component; combine the boundary conditions to obtain the steady-state response of the entire system, and substitute the velocity array of the main structure into the panel method to solve for the radiated sound power;

[0009] The step (4) is implemented through the following steps:

[0010] (4a) Deduction of the recurrence equation of the reduced transfer matrix S and the load function-related array e at the input and output ends of the component:

[0011] The relationship between the state vectors of the multi-input single-output component j is as follows:

[0012]

[0013] In Formulas 7 and 8: j is the multi-input multi-output component number; I k represents the k-th input end of the component; O represents the output end; N is the total number of input ends; Z j,O and are the state vectors of the output end O and the I k -th input end of the component j in the modal coordinate system, respectively; is the transfer matrix when the component j inputs from the I k input end, f j is the load acting on the component j; is the geometric matrix, which is determined by the motion coordination equation between different input ends. After re-dividing the state vector of the component, Formulas (7) and (8) can be written as:

[0014]

[0015] where Z a , Z b are both m / 2×1 arrays, T aa , T ab , T ba , T bb , H a and H b are sub-matrices formed by re-arranging the elements in the transfer matrix and the geometric matrix respectively, f a , f b are the arrays composed of m / 2 elements in the load function array f corresponding to Z a , Z b respectively; the reduced transformation of the state vector of the k-th input end of the component is:

[0016]

[0017] In Equation (11), and are respectively the reduced transfer matrices of the k-th input terminal of component j. and are respectively a column matrix composed of m / 2 zero elements in and a column matrix composed of m / 2 non-zero elements in

[0018] Substituting Equation (11) into Equations (9) and (10) gives the reduced transformation form of the output terminal state vector of a multi-input single-output component:

[0019]

[0020] In Equation (12), Z a,j,O and Z b,j,O are respectively a column matrix composed of m / 2 zero elements in Z j,O and a column matrix composed of m / 2 non-zero elements in Z j,O . In the formula, the matrix also satisfies:

[0021]

[0022]

[0023] E a , E b , e a , e b satisfy:

[0024]

[0025] Equation (13) is the recurrence equation of S and e for a multi-input single-output component. The recurrence equation of S and e for a single-input single-output component can be simplified by removing the redundant inputs and terms of the motion coordination equation, and the result is as follows:

[0026]

[0027] (4b) Solving the steady-state response:

[0028] According to the principle of re-partitioning of the state vector and the system boundary conditions, at the system input point, S = 0 and e = 0; the S and e at each intermediate connection point and the output terminal of the system can be obtained from the recurrence equations (13) and (20) along the system transfer path; then, substituting the boundary conditions at the system output terminal into Equation (12) to solve the unknown variables in the system output terminal state vector; after obtaining the system output terminal state vector, the state vectors of all connection points of the system can be recursively obtained from Equations (9), (11), (17), and (19) against the transfer path, and thus the steady-state response at any point in the system is obtained;

[0029] (4c) Solution of Radiated Sound Power

[0030] After obtaining the steady-state response of any point of the system by using the reduced transfer matrix method of multi-body system, the vibration displacements of all elements of the main structure are further solved. By differentiating the vibration displacements, the vibration velocities of all elements of the main structure are obtained. Combining with the surface element method, the vibration velocity amplitudes of all elements of the main structure are formed into a column matrix:

[0031]

[0032] where R is the number of segments of the main structure;

[0033] Similarly, the sound pressure amplitudes of all elements of the main structure are formed into a column matrix:

[0034]

[0035] When the lengths of all elements on the main structure are much smaller than the minimum wavelength of the sound wave, the sound pressure of the elements on the main structure is:

[0036]

[0037] where A e is the area of the element, ρ0 is the air density, R ij is the distance from the center of the i-th beam segment to the center of the j-th beam segment, ω is the angular frequency, and k is the wave number of the sound wave;

[0038] Equation (22) can be rewritten as:

[0039]

[0040] where is a symmetric matrix, and its matrix elements are:

[0041] By summing up the sound powers of each element on the main structure, the total sound power of the main structure can be obtained:

[0042]

[0043] where [R] is a symmetric matrix:

[0044]

[0045] In Equations (25) and (26): S is the area of the main structure, which is the product of the length L and the width b; c is the speed of sound in air.

[0046] Compared with the prior art, the present invention has the following remarkable advantages: 1. Compared with the previous theoretical analysis methods of acoustic radiation, the method of the present invention has the characteristics of wide applicability and flexible modeling. 2. Compared with finite element simulation, the method of the present invention has a faster calculation speed, which not only provides an efficient theoretical calculation method for the noise reduction performance analysis of the acoustic black hole structure, but also provides a new tool for the optimization design of its noise reduction performance. Description of the Drawings

[0047] Figure 1 It is a schematic diagram of the combined system formed by the acoustic black hole dynamic absorber and the main beam.

[0048] Figure 2 It is a planar bifurcated multi-body system.

[0049] Figure 3 It is a diagram defining the direction of the state vector of the Euler-Bernoulli beam.

[0050] Figure 4 It is a simplified schematic diagram of the acoustic black hole structure with a damping layer.

[0051] Figure 5 It is a schematic diagram of the composite beam cross-section.

[0052] Figure 6 It is a schematic diagram after segmenting the main beam.

[0053] Figure 7 It is a diagram of the calculation process of the steady-state response of the combined system based on the reduced multi-body system transfer matrix method of the present invention.

[0054] Figure 8 It is a schematic diagram of the finite element model of the combined system formed by the acoustic black hole dynamic absorber and the main beam.

[0055] Figure 9 It is a convergence analysis diagram of the sound power of the combined system formed by the acoustic black hole dynamic absorber and the main beam calculated by RMSTMM-ERM.

[0056] Figure 10 It is a comparison diagram of the calculation results between RMSTMM-ERM and FEM. Detailed Embodiments

[0057] The present invention provides an acoustic black hole structure sound radiation calculation method based on a hybrid method of the reduced multi-body system transfer matrix method and the surface element method, which determines the size parameters and material parameters of the acoustic black hole dynamic absorber and the main structure; splits the combined system into each component, numbers each component, segments the main structure and determines the component state vector; derives the transfer equations and motion coordination equations of each component; and solves to obtain the steady-state response of the system in combination with the boundary conditions; then uses the steady-state response of the main structure and combines the surface element method to solve the radiated sound power.

[0058] The specific process includes the following 4 steps:

[0059] (1) Take the acoustic black hole dynamic vibration absorber as an additional vibration damping device on the main structure. The acoustic black hole dynamic vibration absorber is a one-dimensional acoustic black hole structure, and the main structure adopts the form of a beam structure; determine the size parameters and material parameters of each part of the combined system;

[0060] (2) Split the system into each component according to the structural form of the system, number each component, segment the main structure, and determine the state vectors at the input and output ends of the components;

[0061] (3) Derive the transfer equation characterizing the relationship between the state vectors at the input and output ends of the multi-input single-output component and the motion coordination equation characterizing the relationship between the state vectors at its multiple input ends; derive its transfer equation for the single-input single-output component;

[0062] (4) Obtain the recurrence equations of the reduced transfer matrices S and e at the input and output ends of the component according to the transfer equation and motion coordination equation of the component; combine the boundary conditions to obtain the steady-state response of the entire system; substitute the velocity array of the main structure into the panel method to solve for the radiated sound power.

[0063] The size parameters in step (1) include the acoustic black hole length L1, the root thickness h1, the truncated thickness h0, and the damping layer thickness h d , the main beam length L, the thickness h, the connector length L c , and the thickness h c ; the material parameters include the Young's modulus E1, density ρ1, and loss factor η1 of aluminum, and the Young's modulus E2, density ρ2, and loss factor η2 of the damping layer.

[0064] Step (2) is implemented through the following steps:

[0065] (2a) Use massless virtual elements as connection components at the joints of different transfer paths of the system. Both the main beam and the acoustic black hole dynamic vibration absorber adopt the Euler-Bernoulli beam model; segment the main structure, number each component, and the combined system is simplified into a branched multi-body system composed of multiple components;

[0066] (2b) For a linear system, define the generalized displacement and generalized force state vectors of component j as where x and y represent the linear displacements in the x and y directions respectively; θ z represents the angular displacement; q x , q y and m z represent force and moment respectively, and P represents the input end I or output end O. The state vector in the relevant modal coordinate system can be expressed as z j,p =Z j,p eiωt , where ω is the angular frequency. Along the transmission direction, the state vector at the output end of one component serves as the state vector at the input end of the next component.

[0067] In step (3), in the transfer matrix method for multi-body systems, the equations written for any multi-input single-output component are of two types, namely the transfer equation describing the relationship between the state vectors at the input and output ends of the component and the kinematic coordination equation describing the kinematic relationship between different input ends of the component. A single-input single-output component can be regarded as a special case of a multi-input single-output component. The transfer equations and kinematic coordination equations of each component of the combined system are obtained through the following steps:

[0068] (3a) For a massless virtual unit with multiple inputs and a single output, its transfer equation and kinematic coordination equation are obtained respectively from the kinematic and dynamic equations at its input and output ends and the kinematic equations at different input ends:

[0069] The kinematic and dynamic equations for N different input ends and a single output end:

[0070]

[0071] Written in matrix form:

[0072]

[0073] is the transfer matrix describing the relationship between the generalized displacement and generalized force state vectors at the k-th input end and the output end of component j; I3 and 03 are the 3rd-order identity matrix and the 3rd-order zero matrix respectively.

[0074] The kinematic equations for different input ends:

[0075]

[0076] Written in matrix form:

[0077]

[0078] where

[0079]

[0080] is the geometric matrix describing the relationship between the generalized displacement state vectors at the k-th input end and the first input end of component j.

[0081] (3b) For a single-input single-output beam component, its transfer equation can be obtained from its flexural rigidity and linear density:

[0082] For the acoustic black hole unit with an additional damping layer, it is evenly divided into n segments of uniform beams, and then the equivalent flexural rigidity of the composite beam composed of the uniform beam and the damping layer is obtained by using the complex Young's modulus method:

[0083]

[0084] At the same time, considering the influence of the additional mass of the damping layer, the equivalent linear density of the composite beam is obtained:

[0085]

[0086] In the formula, EI and E i I i are the flexural rigidities of the composite beam and the uniform beam respectively; η is the material loss factor of the composite beam; η i , E i , h i , ρ i , A i and η l , E l , h l , ρ l , A l are the material loss factors, Young's moduli, thicknesses, densities and cross-sectional areas of the uniform beam and the damping layer respectively; e = E l / E i is the ratio of the Young's moduli of the composite beam and the uniform beam, and H = h l / h i is the ratio of the thicknesses of the composite beam and the uniform beam.

[0087] Considering the transverse bending and longitudinal rigid body motion of the beam, the transfer matrix of the acoustic black hole unit can be obtained as follows:

[0088] Z n,n+1 = U n …U i …U2U1Z 0,1 = UZ 0,1 (6)

[0089] In the formula, Z 0,1 、...、Z n,n+1 represent the state vectors of each connection point, and U i (i = 1, 2,...n) is the transfer matrix of the i-th composite beam with a length of l:

[0090]

[0091] For the beam element without an additional damping layer, set the segmented n to 1 and set the parameters related to the damping layer to 0 to obtain its transfer equation.

[0092] In step (4), in the reduced transfer matrix method, to solve the system characteristic frequency and steady-state response, first, the recurrence equations of the reduced transfer matrices S and e at the input and output ends of each component need to be obtained. Then, combined with the boundary conditions, the steady-state response of the entire system is solved, and further, the velocity amplitude array of the main structure is obtained and substituted into the panel method to calculate the sound power. It is realized through the following steps:

[0093] (4a) Derivation of the recurrence equations of the reduced transfer matrices S and e at the input and output ends of the component:

[0094] The relationship between the state vectors of the multi-input single-output component j is as follows:

[0095]

[0096] where f j is the load function array. The reduced transfer matrix method re-partitions the state vectors of the component. Equations (7) and (8) can be rewritten as:

[0097]

[0098] where Z a and Z b are both m / 2×1 arrays. At the boundary of the system, Z a contains m / 2 zero elements, Z b contains the remaining m / 2 unknown elements, and Z a and Z b at the intermediate connection point can be arbitrarily partitioned. For the multi-input single-output component, the reduced transformation of the state vector of its k-th input is expressed as:

[0099]

[0100] Substituting Equation (11) into Equations (9) and (10) gives the reduced transformation form of the output-end state vector of the multi-input single-output component:

[0101]

[0102] where:

[0103]

[0104] where E a ,E b ,e a ,e b satisfy:

[0105]

[0106] Equation (13) is the recurrence equation of S and e for a multi-input single-output element. The recurrence equation of S and e for a single-input single-output element can be simplified by removing the redundant inputs and the terms of the motion coordination equation, and the result is as follows:

[0107]

[0108] (4b) Solving the steady-state response:

[0109] According to the principle of re-partitioning the state vector and the system boundary conditions, each input of the system corresponds to Z a = 0, Z b ≠ 0, then at the system input point, S = 0 and e = 0. The S and e at each intermediate connection point and the output end of the system can be obtained from the recurrence equations (13) and (20) along the system transfer path. Then, substitute the boundary conditions at the system output end into Equation (12) to solve the unknown variables in the state vector at the system output end. After obtaining the state vector at the system output end, the state vector at any connection point of the system can be recursively obtained from Equations (9), (11), (17), and (19) against the transfer path, and thus the steady-state response at any point of the system can be obtained.

[0110] (4c) Solving the radiated sound power

[0111] After obtaining the steady-state response at any point of the system, the velocity amplitude array of the entire main structure can be obtained:

[0112]

[0113] Similarly, the sound pressure amplitudes of all the units of the main structure can also form an array:

[0114]

[0115] When the lengths of all the units on the main structure are much smaller than the minimum wavelength of the sound wave, the sound pressure of the units on the main structure is:

[0116]

[0117] Equation (22) can be rewritten as:

[0118]

[0119] Summing up the sound power of each unit on the main structure, the total sound power of the main structure can be obtained:

[0120]

[0121] where [R] is a symmetric matrix:

[0122]

[0123] The following is further illustrated through the following simulation experiments in conjunction with the accompanying drawings of the specification.

[0124] 1. Determine the structural parameters and material parameters of each part:

[0125] The present invention takes the combined system of a one-dimensional acoustic black hole vibration absorber and a main beam as shown in Figure 1 as an example for illustration. The coating on the acoustic black hole unit is a viscoelastic material, and the rest of the system is aluminum. The material parameters and dimensional parameters of each part are shown in Table 1.

[0126]

[0127] Table 1: The material and dimensional parameters of the combined system. The width b of all components in the combined system is uniformly set to 20 mm.

[0128] 2. Split the system into each component, number each component, segment the main structure, and determine the state vectors at the input and output ends of the components:

[0129] The tip of the acoustic black hole structure and the free end of the main beam are used as the system input, and the fixed end of the main beam is used as the system output. The massless virtual unit 3 is a connecting element between two different transmission paths. Considering low-frequency vibration, both the main beam and the acoustic black hole dynamic absorber adopt the Euler-Bernoulli beam model. The combined system can be simplified to Figure 2 the bifurcated multi-body system shown in Figure 2 where x j -y j represents the local inertial coordinate system where component j is located, and x-y is the main inertial coordinate system of the system. Further segment the main structure, and the schematic diagram is as shown in Figure 3 ; the direction of the state vector of the beam element in its respective local inertial coordinate system is defined as shown in Figure 4 .

[0130] 3. Derive its transfer equation and motion coordination equation according to the dynamic and kinematic equations of each component:

[0131] (3a) For the acoustic black hole unit with an additional damping layer, it can be simplified to n uniform beams connected in series, as shown in Figure 5 . For each section of the composite beam with an additional damping layer, as shown in Figure 6 , its equivalent flexural strength and linear density can be obtained from equations (4) and (5). Therefore, the transfer matrix of the acoustic black hole unit, i.e., component 1, in the coordinate system x1-y1 can be obtained from equation (6). Setting the number of segments n to 1 and the thickness of the damping layer to 0 can obtain the transfer matrices j -y j (j = 2, 4, 5) of the remaining uniform beam components without damping layers in their respective local inertial coordinate systems Then, the transfer equations of the above components in the system's main inertial coordinate system are obtained: where A j is the coordinate transformation matrix from the local inertial coordinate system x j -y j to the main inertial coordinate system x-y.

[0132] (3b) The transfer matrix and motion coordination equations of Component 3 can be obtained from Eqs. (1) and (2).

[0133] 4. Derivation of the recurrence equations of S and e at the input and output ends of each component. Combining the boundary conditions to obtain the system characteristic equation and the total transfer equation, and solving them respectively to obtain the system characteristic frequency and the steady-state response. Substituting the velocity array of the main structure into the panel method to solve for the radiated sound power:

[0134] (4a) Substituting the transfer equations and motion coordination equations of each component obtained in 3 into Eqs. (7) - (12) can obtain the recurrence equations of S and e (13) or (20) at the input and output ends of each component. Then, substituting the boundary conditions S 1,I = 0, e 1,I = 0, S 4,1,I ×0, e 4,1,I = 0 into the recurrence equations of each component, the S and e at each connection point of the system can be obtained sequentially along the transfer path. Then, substituting the boundary conditions at the output end of the system into Eq. (12) to solve for the unknown variables in the state vector at the output end of the system Then, substituting the state vector at the output end of the system into Eqs. (9), (11), (17), and (19), the state vectors at any connection point of the system can be obtained by recurrence. The entire recurrence process is as Figure 7 .

[0135] (4b) From (4a), the velocity amplitude array of the main structure can be obtained, i.e., Eq. (21). Combining with Eqs. (22), (23), (25), and (26), the sound power of the main structure can be obtained.

[0136] 5. Finite element verification

[0137] To prove the correctness and rapidity of the RMSMM-ERM calculation, a three-dimensional finite element model of the combined system shown in Figure 8 was also established in the COMSOL acoustic-solid coupling module. As the maximum mesh size decreases and the mesh density gradually increases, the sound power calculation results of the finite element model converge.

[0138] (5a) First, a convergence analysis was performed on the results of calculating the sound power by the reduced multibody system transfer matrix method-panel method (RMSMM-ERM). The results are as Figure 9 shown, where N1 is the number of segments of the main structure, N1 = N5 + N4, σ = (|Pi+1 -P i | / P i+1 ) = 100%, P i represents the peak sound power corresponding to the i-th segmentation number, P i+1 represents the peak corresponding to the (i + 1)-th segmentation number.

[0139] When the number of segments reaches 800, the peak errors of the first four-order peaks are all less than 1%, which is sufficient to indicate that the sound power calculation results based on the hybrid method of the reduced multi-body system transfer matrix method and the panel method have converged at this time.

[0140] (5b) Then, the sound power level (SWL = 10lg(W / W ref ) of the combined system is calculated using RMSTMM-ERM and the finite element method (FEM) respectively, where W is the sound power with the unit of W, W ref = 10 -12 W), and the comparison results are as Figure 10 . It takes 4 hours and 20 minutes for FEM to complete the calculation, while RMSTMM-ERM only takes 2 minutes and 3 seconds to complete the whole calculation result, which is far lower than the calculation time required by FEM.

Claims

1. An acoustic black hole structure acoustic radiation calculation method based on a hybrid method of the reduced multi-body system transfer matrix method and the panel method, comprising the following steps: (1) Take the acoustic black hole dynamic absorber as an additional structure on the main structure, and the two form a combined system; The acoustic black hole dynamic absorber is a one-dimensional acoustic black hole structure, and the main structure adopts a uniform beam structure form; then determine the size parameters and material parameters of each part of the combined system; (2) Split the combined system into each component according to its structural form, then number each component, and segment the main structure, and then determine the state vectors at the input and output ends of the components; (3) Derive the transfer equation characterizing the relationship between the state vectors at the input and output ends of the multi-input single-output component and the motion coordination equation characterizing the relationship between the state vectors at its multiple input ends; derive its transfer equation for the single-input single-output component; (4) Obtain the recurrence equations of the reduced transfer matrices S and e at the input and output ends of the component according to the transfer equation and the motion coordination equation of the component; solve the steady-state response of the entire system in combination with the boundary conditions of the system; substitute the velocity array of the main structure into the panel method to solve for the radiated sound power; The step (4) is implemented through the following steps: (4a) Derivation of the recurrence equations of the reduced transfer matrix S and the load function-related array e at the input and output ends of the component: The relationships between the state vectors of the multi-input single-output component j are as follows: In Formulas 7 and 8: j is the multi-input multi-output component number; I k represents the k-th input terminal of the component; O represents the output terminal; N is the total number of input terminals; Z j,O and are the state vectors of the output terminal O and the I k -th input terminal of component j in the modal coordinate system, respectively; is the transfer matrix when component j inputs from the Ik input terminal, f j is the load acting on component j; is the geometric matrix, which is determined by the motion coordination equations between different input terminals; By re-partitioning the state vectors of the components, Formulas (7) and (8) can be written as: Among which Z a and Z b are both column arrays of m / 2×1, T aa and T ab and T ba and T bb and H a and H b are sub-matrices formed by rearranging the elements in the transfer matrix and the geometric matrix respectively. f a and f b are column arrays composed of m / 2 elements in the load function column array f corresponding to Z a and Z b respectively. The reduced transformation of the state vector at the k-th input end of the component is: In formula (11) and are respectively the reduced transfer matrices of the k-th input terminal of component j; and are respectively a column array composed of m / 2 zero elements in and a column array composed of m / 2 non-zero elements in; Substitute Equation (11) into Equations (9) and (10) to obtain the reduced transformation form of the state vector at the output end of the multi-input single-output component: Z in Equation (12) a,j,O and Z b,j,O are respectively an array composed of m / 2 zero elements in Z j,O and an array composed of m / 2 non-zero elements in Z j,O wherein the matrix also satisfies: E a ,E b ,e a ,e b Satisfy: Equation (13) is the recurrence equation of S and e for the multi-input single-output component. The recurrence equations of S and e for the single-input single-output component can be simplified by removing the redundant input and motion coordination equation terms, and the result is as follows: (4b) Solving the steady-state response: According to the principle of re-dividing the state vectors and the boundary conditions of the system, at the input point of the system, S = 0 and e = 0; the S and e at each intermediate connection point and the output end of the system can be obtained from the recurrence equations (13) and (20) along the system transfer path; then substitute the boundary conditions at the output end of the system into Equation (12) to solve for the unknown variables in the state vector at the output end of the system; after obtaining the state vector at the output end of the system, the state vectors at all connection points of the system can be recursively obtained from Equations (9), (11), (17), and (19) against the transfer path, thereby obtaining the steady-state response at any point of the system; (4c) Solving the radiated sound power After obtaining the steady-state response at any point of the system by using the reduced multi-body system transfer matrix method, further solve for the vibration displacements of all units of the main structure, and then obtain the vibration velocities of all units of the main structure by taking the derivative of the vibration displacements. In combination with the panel method, form an array with the vibration velocity amplitudes of all units of the main structure: where R is the number of segments of the main structure; Similarly, form an array with the sound pressure amplitudes of all units of the main structure: When the lengths of all units on the main structure are much smaller than the minimum wavelength of the sound wave, the sound pressure of the units on the main structure is: where A e is the area of the unit, ρ0 is the air density, R ij is the distance from the center of the i-th beam to the center of the j-th beam, ω is the angular frequency, and k is the acoustic wave number; Equation (22) can be rewritten as: where is a symmetric matrix, and its matrix elements are: Sum the sound powers of each unit on the main structure to obtain the total sound power of the main structure: where [R] is a symmetric matrix: In equations (25) and (26): S is the area of the main structure, which is the product of the length L and the width b; c is the speed of sound waves in air.

2. The calculation method according to claim 1, wherein: The dimensional parameters in the step (1) include the length L1 and thickness h1 of the uniform section of the acoustic black hole, the length L2 and truncated thickness h0 of the wedge section, and the thickness h of the damping layer. d , the length L and thickness h of the main beam; the material parameters include the Young's modulus E1, density ρ1, and loss factor η1 of aluminum, and the Young's modulus E2, density ρ2, and loss factor η2 of the damping layer.

3. The calculation method according to claim 1, wherein: The step (2) is implemented through the following steps: (2a) Determine the input and output ends of the combined system. Use massless virtual elements as connecting elements at the joints of different transmission paths of the system. Both the main beam and the acoustic black hole dynamic absorber adopt the Euler-Bernoulli beam model. Segment the main structure, number each element, and the combined system is simplified into a multi-component bifurcated multi-body system; (2b) For a linear system, the generalized displacement and generalized force state vectors of component j are defined as where x and y represent the linear displacements in the x and y directions respectively; θ z represents the angular displacement; q x , q y and m z represent the force and moment respectively, P represents the input end I or the output end O; the state vector in the relevant modal coordinate system is expressed as z j,p = Z j,p e iωt , ω is the angular frequency; along the transmission direction, the output end state vector of one component serves as the input end state vector of the next component.

4. The calculation method according to claim 1, characterized in that: In the step (3), the transfer equations and kinematic compatibility equations of each element of the combined system are obtained through the following steps: (3a) For the massless virtual element with multiple inputs and a single output, its transfer equation and kinematic compatibility equation are obtained respectively from the kinematic and dynamic equations of its different input and output ends and the kinematic equations of different input ends: The kinematic and dynamic equations of N different input ends and a single output end: Written in matrix form: It is the transfer matrix that describes the relationship between the generalized displacement and the generalized force state vectors of the k-th input terminal and the output terminal of component j; I3 and 03 are the 3rd-order identity matrix and the 3rd-order zero matrix respectively; The kinematic equations of different input ends: Written in matrix form: Where is a geometric matrix that describes the relationship between the generalized displacement state vectors of the k-th input terminal and the first input terminal of component j; (3b) For the beam element with a single input and a single output, its transfer equation can be obtained from its flexural rigidity and linear density: For the acoustic black hole element with an additional damping layer, it is divided into n segments of uniform beams on average, and the equivalent flexural rigidity of the composite beam composed of the uniform beam and the damping layer is obtained by using the complex Young's modulus method: The equivalent linear density of the composite beam: where EI and E i I i are the flexural rigidities of the composite beam and the homogeneous beam, respectively; η is the material loss coefficient of the composite beam; η i , E i , h i , ρ i , A i and η l , E l , h l , ρ l , A l are the material loss factor, Young's modulus, thickness, density, and cross-sectional area of the homogeneous beam and the damping layer, respectively; e = E l / E i is the ratio of the Young's modulus of the composite beam to that of the homogeneous beam, and H = h l / h i is the ratio of the thickness of the composite beam to that of the homogeneous beam; Considering the transverse bending and longitudinal rigid body motion of the beam, the transfer matrix of the acoustic black hole element is obtained as follows: Z n,n+1 = U n ...U i ...U2U1Z 0,1 = UZ 0,1 (6) where Z 0,1 、...、Z n,n+1 represent the state vectors of each connection point, and U i (i = 1, 2,... n) is the transfer matrix of the i-th composite beam with length l: For the beam element without an additional damping layer, set the number of segments n in equations (4), (5), and (6) to 1 and set the damping layer-related parameters to 0 to obtain its transfer equation.

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