Spiral acoustic black hole dynamic vibration absorption calculation method based on reduced multi-body system transfer matrix method

Abstract

The invention discloses a spiral acoustic black hole dynamic vibration absorption calculation method based on a reduced multi-body system transfer matrix method. Based on a reduced multi-body system transfer matrix method, a dynamic model of a combined system formed after a main structure is additionally provided with a spiral acoustic black hole vibration absorber is established, and therefore the characteristic frequency and steady-state response of the dynamic model are obtained through solving. Compared with a finite element method, the method has the advantages of high calculation precision and high calculation speed.

Description

Technical Field

[0001] This patent belongs to the technical field of dynamic analysis of spiral acoustic black hole structures, and specifically relates to a method for calculating the performance of a spiral acoustic black hole vibration absorber based on the reduced multi-body system transfer matrix method. Background Art

[0002] Traditional acoustic black holes need to increase the length of the acoustic black hole structure to reduce the threshold frequency at which they become effective, and cannot be applied when the installation space is small. The spiral acoustic black hole compresses the structural space by bending the baseline, significantly improving the space utilization rate while maintaining the equivalent wave absorption performance, providing a new idea for low-frequency vibration reduction and noise reduction. Regarding the calculation methods for spiral acoustic black hole structures, there are the wave impedance analysis method and the finite element method. The former is mainly for semi-infinite structures and cannot handle the dynamic problems of finite-length actual structures. The latter has defects such as high model complexity and long calculation time when analyzing the coupled dynamic problems of the spiral acoustic black hole as a vibration absorber and the main structure, resulting in deficiencies in efficient calculation and optimization for existing methods. There is still a lack of an efficient theoretical calculation method for the dynamic problems of spiral acoustic black hole dynamic absorbers. Summary of the Invention

[0003] The purpose of the present invention is to provide a method for calculating the dynamics of a spiral acoustic black hole dynamic absorber based on the reduced multi-body system transfer matrix method.

[0004] The technical solution for achieving the purpose of the present invention is as follows: A method for calculating the dynamics of a spiral acoustic black hole dynamic absorber based on the reduced multi-body system transfer matrix method, comprising the following steps:

[0005] (1) Consider the spiral acoustic black hole dynamic absorber as an additional vibration reduction device on the main structure, and the two form a combined system; where the spiral acoustic black hole dynamic absorber is a two-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 then determine the state vectors at the input and output ends of the components.

[0007] (3) Segment the spiral acoustic black hole components according to the spiral angle, calculate the geometric parameters of the beam during the transfer process; calculate the material parameters for the parts with composite structures; and derive the transfer matrix for the spiral acoustic black hole components.

[0008] (4) Derive the transfer equation characterizing the relationship between the state vectors at the input and output ends of the multi-input single-output components and the motion coordination equation characterizing the relationship between the state vectors at its multiple input ends; derive the transfer equation for the single-input single-output components.

[0009] (5) Obtain the recurrence equations of the reduced transfer matrix S at the input and output ends of the component and the column matrix e related to the load function according to the transfer equation and the motion coordination equation of the component; combine the boundary conditions to obtain the characteristic equation and the total transfer equation of the combined system, and solve them respectively to obtain the characteristic frequency and the steady-state response.

[0010] In the said step (3), the helical variable cross-section part is regarded as a helical curved beam with a thickness power-law change formed by connecting N straight beams with equal cross-section thickness in sequence. The transfer matrix of the helical acoustic black hole component is obtained through the following steps: (3a) The helical variable cross-section part is regarded as N beams with a power-law change in cross-section thickness. The cross-section thickness of the k-th beam after discretization is:

[0011] h(s k ) = εs k m +h d (1)

[0012] where ε and m are constants; s k is the length of the baseline helix θ k is the helix angle corresponding to the end of the k-th beam, and θ Start is the starting angle of the helical acoustic black hole part;

[0013] The length of the k-th beam after discretization is:

[0014]

[0015] where b is the Archimedes spiral coefficient, and θ k-1 is the coefficient corresponding to the beginning of the k-th beam, that is, the angle corresponding to the end of the (k - 1)-th beam.

[0016] (3b) The surface of the helical acoustic black hole unit is covered with a damping material with a specific thickness h t . For the composite beam structure with a damping layer applied, its equivalent flexural rigidity can be expressed in complex form:

[0017]

[0018] The linear density of the composite beam is:

[0019]

[0020] In equations (3) and (4), i is the complex number i 2 = -1, the elastic modulus of the composite beam is E, the moment of inertia of the cross-section is I, and the flexural rigidity is denoted as EI; while the flexural rigidity of the beam without the damping layer is denoted as E a I a , and the material damping loss factor η of the composite beam is; I a 、E a 、ηa , h a , ρ a , A a are the moment of inertia, elastic modulus, loss factor, thickness, density, and cross-sectional area of the beam structure part; E b , η b , h b , ρ b , A b are the elastic modulus, loss factor, thickness, density, and cross-sectional area of the damping material; in addition, the ratio of the elastic moduli E b / E a is defined as e, and the ratio of the thicknesses h b / h a is h;

[0021] (3c) When analyzing the spiral acoustic black hole, the Euler-Bernoulli beam model is used for modeling; through the above calculations, the transfer matrix of the k-th segment of the beam in its local coordinate system x k -y k is obtained. After coordinate transformation, the transfer matrix of the overall spiral acoustic black hole element in the reference coordinate system is obtained. Then, the coordinate transformation matrix R′ k -y k between the local coordinate system x k and the reference coordinate system is:

[0022]

[0023] where A′ k is the direction cosine matrix between the local inertial coordinate system of the k-th segment of the beam and the reference inertial system where the spiral acoustic black hole is located; O is the zero matrix; θ' k is the angle between the neutral axis of the k-th segment of the beam when it is undeformed, that is, the angle between the local coordinate system and the x-axis of the reference coordinate system:

[0024]

[0025] Then, the transfer matrix of the k-th segment of the beam in the local coordinate system of the overall spiral acoustic black hole element is:

[0026]

[0027] The transfer matrix of the spiral acoustic black hole element can be obtained:

[0028]

[0029] For the above beam element, its transfer equation U′ k in the local coordinate system of the k-th segment of the beam is:

[0030]

[0031] Among them, \(l_3\) is the length of the beam, \(EI\) is the flexural rigidity of the beam, is the linear mass density of the beam, and \(S(x)\), \(T(x)\), \(U(x)\), \(V(x)\) are the Kroloff functions;

[0032]

[0033] Compared with the prior art, the present invention has the following remarkable advantages: 1. A general theoretical model applicable to the dynamics of spiral acoustic black holes is constructed, solving the problem that existing methods cannot handle the dynamics of acoustic black holes with complex topological structures with continuously varying curvatures, and providing a general method for the design of spiral acoustic black holes. 2. The method of the present invention uses a low-order transfer equation to replace the system dynamics equation for two-dimensional spiral acoustic black holes, and the order of the matrix involved in the solution is independent of the degrees of freedom of the system, reducing the complexity of the dynamics model and making the calculation speed faster, enabling rapid analysis and optimization of the spiral acoustic black hole structure. Brief Description of the Drawings

[0034] Figure 1 is a schematic diagram of the main beam of the additional spiral acoustic black hole dynamic vibration absorber.

[0035] Figure 2 is the dynamic topology diagram of the combined system.

[0036] Figure 3 is a diagram defining the direction of the state vector of the Euler-Bernoulli beam.

[0037] Figure 4 is a simplified schematic diagram of the spiral acoustic black hole structure with a damping layer.

[0038] Figure 5 is a schematic diagram of the unit beam in the local coordinate system.

[0039] Figure 6 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.

[0040] Figure 7 is a comparison diagram of the steady-state response calculation results between the reduced multi-body system transfer matrix method and the finite element simulation. Specific Embodiments

[0041] The technical solution adopted by the present invention is as follows: determining the dimensional parameters and material parameters of the spiral acoustic black hole dynamic vibration absorber and the main structure; splitting the combined system into each component, numbering each component and determining the state vector of the component; deriving the transfer equation and the motion coordination equation of each component; deriving the total transfer equation of the combined system and solving it in combination with the boundary conditions to obtain the characteristic frequency and steady-state response of the system.

[0042] The specific process includes the following 5 steps:

[0043] (1) The spiral acoustic black hole dynamic vibration absorber is used as an additional vibration damping device on the main structure, and the two form a combined system; among them, the spiral acoustic black hole dynamic vibration absorber is a two-dimensional acoustic black hole structure, and the main structure adopts a beam structure form; determine the size parameters and material parameters of each part of the combined system;

[0044] (2) According to the principle of the reduced transfer matrix method and the structural form of the system, the system is split into each component, each component is numbered, the combined system is simplified into a branched multi-body system composed of multiple components, and the state vectors at the input and output ends of the components are determined;

[0045] (3) The spiral acoustic black hole components are segmented according to the spiral angle, and the geometric parameters of the beam during the transmission process are calculated; for the part with a composite structure, its material parameters are calculated; the transfer matrix of the spiral acoustic black hole components is derived;

[0046] (4) For multi-input single-output components, derive the transfer equation characterizing the relationship between the state vectors at its input and output ends and the motion coordination equation characterizing the relationship between the state vectors at its multiple input ends;

[0047] For single-input single-output components, derive its transfer equation;

[0048] (5) According to the component transfer equation and the motion coordination equation, obtain the recurrence equations of the reduced transfer matrix S at the input and output ends and the column matrix e related to the load function; then, combined with the boundary conditions, obtain the system characteristic equation and the total transfer equation, and solve them respectively to obtain the characteristic frequency and steady-state response of the system.

[0049] The size parameters in step (1) include the length L3 and thickness h3 of the uniform section of the spiral acoustic black hole, the starting angle θ start , the termination angle θ Stop , the truncated thickness h d , the Archimedes spiral coefficient b, the thickness h t of the damping layer, the length L and thickness h1 of the main beam, and the overall width b1 of the structure; the material parameters include the Young's modulus E a , density ρ a , loss factor η a of aluminum, the Young's modulus E b , density ρ b , loss factor η b of the damping layer.

[0050] The step (2) is realized through the following steps:

[0051] (2a) At the joints of different transmission paths in the system, massless virtual elements are used as connection components, and both the main beam and the spiral acoustic black hole dynamic absorber adopt the Euler-Bernoulli beam model; number each element, and the combined system is simplified into a multi-component bifurcated multi-body system; (2b) For a linear system, define the generalized displacement and generalized force state vectors of element i 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, and P represents the input end I or the output end O. The state vector in the relevant modal coordinate system can be expressed as z i,p =Z i,p e iωt , ω is the angular frequency. Along the transmission direction, the state vector at the output end of one element serves as the state vector at the input end of the next element.

[0052] In step (3) mentioned above, the transfer matrix of the spiral acoustic black hole element cannot be directly obtained. Therefore, the spiral variable cross-section part can be regarded as a spiral curved beam with a thickness power-law variation formed by connecting N straight beams with equal cross-section thickness in sequence;

[0053] (3a) Regard the spiral variable cross-section part as a spiral curved beam with a thickness power-law variation formed by connecting N straight beams with equal cross-section thickness in sequence, as Figure 4 shown. The cross-section thickness after discretization of its k-th beam is:

[0054] h(s k )=εs k m +h d (1)

[0055] where ε and m are constants; s k is the length of the baseline helix θ k is the spiral angle corresponding to the end of the k-th beam, and θ Start is the starting angle of the spiral acoustic black hole part;.

[0056] The length after discretization of its k-th beam is:

[0057]

[0058] where b is the Archimedean spiral coefficient; θ k-1 is the angle corresponding to the beginning of the k-th beam, that is, the angle corresponding to the end of the k - 1-th beam.

[0059] (3b) Among them, in order to improve the energy dissipation of the spiral acoustic black hole unit, a specific thickness h is covered on its surfacet For the damping material, for the composite beam structure with the damping layer applied, its equivalent flexural stiffness can be expressed in complex form as:

[0060]

[0061] The linear density of the composite beam is:

[0062]

[0063] In equations (3) and (4), i is the complex number i 2 = -1, the elastic modulus of the composite beam is E, the moment of inertia of the cross-section is I, and the flexural stiffness is denoted as EI; while the flexural stiffness of the beam without the damping layer attached is denoted as E a I a is represented by, and the material damping loss factor η of the composite beam is. I a 、E a 、η a 、h a 、ρ a 、A a are the moment of inertia of the cross-section, elastic modulus, loss factor, thickness, density, and cross-sectional area of the beam structure part. E b 、η b 、h b 、ρ b 、A b are the elastic modulus, loss factor, thickness, density, and cross-sectional area of the damping material. In addition, the ratio of the elastic moduli E b / E a is defined as e, and the ratio of the thicknesses h b / h a is h.

[0064] (3c) When analyzing the spiral acoustic black hole, the Euler - Bernoulli beam model is used for modeling; through the above calculations, the transfer matrix of the k-th segment of the beam in its local coordinate system x k -y k is obtained. To obtain the transfer matrix of the overall spiral acoustic black hole element in the reference coordinate system, coordinate transformation needs to be performed, as Figure 5 shown, then the coordinate transformation matrix R′ k -y k between the local coordinate system and the reference coordinate system is: k is:

[0065]

[0066] where A′ k is the direction cosine matrix between the local inertial coordinate system of the k-th segment of the beam and the reference inertial system where the spiral acoustic black hole is located; O is the zero matrix; θ' kis the angle between the neutral axis when the k-th beam is undeformed, i.e., the local coordinate system where it is located and the x-axis of the reference coordinate system:

[0067]

[0068] Then the transfer matrix of the k-th beam in the local coordinate system where the overall spiral acoustic black hole element is located is:

[0069]

[0070] The transfer matrix of the spiral acoustic black hole element can be obtained:

[0071]

[0072] For the above beam element, its transfer equation U′ in the local coordinate system of the k-th beam k is:

[0073]

[0074] where l3 is the length of the beam, EI is the flexural rigidity of the beam, is the linear mass density of the beam, and S(x), T(x), U(x), V(x) are the Kroloff functions;

[0075]

[0076] The step (4), according to the principle of the reduced transfer matrix method, is composed of a series of elements with a single input end and a single output end, and elements with multiple input ends but a single output end. The connected part is regarded as a massless virtual unit with multiple inputs and a single output. There are two types of equations written for any element with multiple inputs and a single output, namely the transfer equation describing the relationship between the state vectors at the input and output ends of the element and the kinematic coordination equation describing the kinematic relationship between different input ends of the element. A single input and single output element can be regarded as a special case of an element with multiple inputs and a single output. The transfer equations and kinematic coordination equations of the remaining elements of the combined system are obtained by the following steps:

[0077] (4a) For the massless virtual unit with multiple inputs and a single output, its transfer equation and kinematic coordination equation are obtained from the kinematic and dynamic equations at its input and output ends and the kinematic equations at different input ends respectively:

[0078] The kinematic and dynamic equations of M different input ends and a single output end:

[0079]

[0080] Written in matrix form:

[0081]

[0082] It is the transfer matrix that describes the relationship between the generalized displacement and the generalized force state vectors of the j-th input terminal and the output terminal of component i; I3 and 03 are the 3-order identity matrix and the 3-order zero matrix respectively.

[0083] From the motion consistency of different input terminals of the massless virtual element, the motion coordination equation can be obtained:

[0084]

[0085] Written in matrix form:

[0086]

[0087] where

[0088]

[0089] is the coordination matrix that describes the relationship between the generalized displacement state vectors of the j-th input terminal and the 1st input terminal of component i;

[0090] (4b) For beam elements without additional damping layers such as in Equation (9), setting the segmentation N to 1 and the damping layer thickness h t to 0, the transfer equations of each component in its respective reference coordinate system can be obtained Similarly, through coordinate transformation, the transfer matrices of each component in the global coordinate system are obtained:

[0091]

[0092] where R i is the coordinate transformation matrix from the local inertial coordinate system x i -y i to the main inertial coordinate system x-y.

[0093] In step (5) above, in the reduced transfer matrix method, in order to solve the system characteristic frequency and steady-state response, first, the recurrence equations of the reduced transfer matrix S of the input and output terminals of each component and the load function-related column matrix e need to be obtained, and then the characteristic frequency and steady-state response are solved in combination with the boundary conditions. It is achieved through the following steps:

[0094] (5a) Deduction of the recurrence equations of the reduced transfer matrix S of the input and output terminals of the component and the load function-related column matrix e:

[0095] The relationship between the state vectors of the multi-input single-output component i is as follows:

[0096]

[0097] where f iis the load function array. The basic process of the reduction transformation is as follows: According to the system boundary conditions (Table 1), the state vectors of all components are divided into two parts, where Z a and Z b are both column arrays of m / 2×1. At the boundary of the system, Z a contains m / 2 zero elements, and Z b contains the remaining m / 2 unknown elements. The Z a and Z b at the intermediate connection points can be arbitrarily divided.

[0098]

[0099] Table 1: Blocking of state vectors under fixed and free boundary conditions

[0100] Define the relationship between the two parts of the input-end state vector as S i,I and e i,I . Then, based on S i,I and the transfer matrix of component i, recursively derive the reduced transfer matrix S i,O and e i,O at the output end of component i. And S j,O = S j+1,I , e j,O = e j+1,I . The reduced transfer matrix at the input end of component i+1 can be obtained. Therefore, taking the boundary conditions at the input end as the initial values of the state vectors, the reduced transfer matrices at the input and output ends of any component in the entire system can be derived sequentially.

[0101] Redefine the state vectors of components according to the reduced transfer matrix method. Equations (17) and (18) can be rewritten as:

[0102]

[0103] For a multi-input single-output component, the reduction transformation of the state vector of its jth input is expressed as:

[0104]

[0105] Substitute Equation (20) into Equations (19) and (20) to obtain the reduction transformation form of the state vector at the output end of the multi-input single-output component:

[0106]

[0107] Where:

[0108]

[0109] Where E a , E b , e a , eb Satisfy:

[0110]

[0111] Equation (23) 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:

[0112]

[0113] (5b) Solving the steady-state response:

[0114] 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 (23) and (30) along the system transmission path. Then, substitute the boundary conditions at the system output end into Equation (18) 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 (19), (22), (28), and (29) against the transmission path, and thus the steady-state response at any point of the system can be obtained.

[0115] (5c) Solving the characteristic frequency:

[0116] By removing the external excitation in (5a), the characteristic equation of the bifurcated multi-body system can be obtained. However, if the root-finding method based on symbolic transformation such as the bisection method is directly used, the poles will be misidentified as the roots of the equation. To solve this problem, in (4b) Denote as Denote as Obtain the general form of the characteristic equation of the bifurcated multi-body system:

[0117]

[0118] where n is the component number at the system output end, and the expression of |Γ i | is as follows:

[0119]

[0120] Excluding the influence of the denominator being 0 in |S n,0 |, the situation of misidentifying the poles as roots is avoided.

[0121] The following is further illustrated by the following simulation experiment in conjunction with the accompanying drawings of the specification.

[0122] 1. Determine the structural parameters and material parameters of each part:

[0123] The present invention takes the combined system of the two-dimensional spiral acoustic black hole vibration absorber and the main beam shown as an example for illustration. The coating on the spiral 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 2. Figure 1 As shown in the figure, the coating on the spiral 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 2.

[0124]

[0125] Table 2: Material and dimensional parameters of the combined system

[0126] 2. Split the system into each component, number each component, and determine the state vectors of the input and output ends of the components:

[0127] The tip of the spiral acoustic black hole 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 5 is the 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 into the Figure 2 forked multi-body system containing 6 components shown in the figure. Figure 2 where x i -y i represents the local inertial coordinate system where component i is located, and x-y is the main inertial coordinate system of the system. The direction of the state vector of the beam element in its respective local inertial coordinate system is defined as Figure 3 .

[0128] 3. Derivation of the transfer matrix of the spiral acoustic black hole element; (3a) For the spiral acoustic black hole unit with an additional damping layer, it can be simplified as N uniform beams connected in series, as shown in Figure 4 the figure. The cross-sectional thickness and length of each section of the beam are obtained from equations (1) and (2). For the composite beam with an additional damping layer, its equivalent flexural strength and linear density can be obtained from equations (3) and (4). Combining the method in step (3), the transfer matrix of the spiral acoustic black hole unit, i.e., component 1, in the coordinate system x1-y1 can be obtained from equation (8).

[0129] 4. Derive the transfer equation and kinematic compatibility equation according to the dynamic and kinematic equations of each component:

[0130] (4a) The transfer matrix and kinematic compatibility equation of component 5, which is a multi-input single-output component, can be obtained from equations (11) and (13).

[0131] (4b) By setting the number of segments N to 1 and the thickness of the damping layer to 0, the other uniform beam components 2, 3, 4, and 6 without damping layers can be obtained in their respective local inertial coordinate systems x i -y i(i = 2, 3, 4, 6) Transfer matrix Then, the transfer equations of the above components in the system's principal inertial coordinate system are obtained:

[0132] 5. Derivation of the recurrence equations for S and e at the input and output ends of each component. Combining the boundary conditions, the system characteristic equation and the total transfer equation are obtained, and the system characteristic frequency and steady-state response are solved respectively:

[0133] (5a) Substitute the transfer equations and motion coordination equations of each component obtained in 3 into Eqs. (17) - (22) to obtain the recurrence equations for S and e (23) or (30) at the input and output ends of each component. Then, substitute the boundary conditions S 1,I = 0, e 1,I = 0, S 4,I = 0, e 4,I = 0 into the recurrence equations of each component, and then S and e at each connection point of the system can be obtained sequentially along the transfer path. Then, substitute the boundary condition Z a,6,O = 0 at the output end of the system into Eq. (22) to solve for the unknown variable Z in the state vector at the output end of the system b,6,O , and then substitute the state vector at the output end of the system into Eqs. (19), (21), (27), (29) to recursively obtain the state vector at any connection point of the system. The entire recursive process is as shown in Figure 6 .

[0134] (5b) Remove the load-related array in step (5a), and obtain the characteristic equation of the combined system from Eqs. (31) and (32):

[0135]

[0136] Then, the bisection method or recursive eigenvalue search algorithm can be used to solve the system characteristic frequency.

[0137] 6. Verification of the reduced transfer matrix method

[0138] To prove the correctness and rapidity of the calculation of the reduced transfer matrix method, a two-dimensional finite element model of the combined system shown in Figure 1 is also established in the COMSOL solid mechanics module. As the mesh density gradually increases, when the errors of the first 10-order characteristic frequencies of the system are all within 2%, convergence is achieved.

[0139] (6a) First, use the reduced transfer matrix method (RMSTMM) and finite element simulation (FEM) to calculate the characteristic frequencies of the system respectively. The results are shown in Table 3, where N is the number of segments of the spiral acoustic black hole element part, t is the calculation time, and σ = (f RMSTMM - f FEM ) / f FEM represents the relative error.

[0140]

[0141] Table 3: Comparison of the calculated results of the reduced transfer matrix method and the finite element simulation for the characteristic frequencies

[0142] (6b) Then, the steady-state responses calculated by the two methods are compared. To ensure the convergence of the calculation results of the two methods, the wedge section of the spiral acoustic black hole element is divided into 150 segments, and the calculation frequency interval is taken as 1 Hz. The steady-state responses of the velocity admittance (Mobility = 20log|jωY| / F) at the excitation point of the system are calculated by using RMSTMM and FEM respectively, and the results are as Figure 7 .

Claims

1. A spiral acoustic black hole dynamic vibration absorption calculation method based on the reduced multi-body system transfer matrix method, characterized in that: The following steps are involved: (1) The spiral acoustic black hole dynamic vibration absorber is used as an additional vibration reduction device on the main structure, and the two constitute a combined system; the spiral acoustic black hole dynamic vibration absorber is a two-dimensional acoustic black hole structure, and the main structure adopts a beam structure; then the size parameters and material parameters of each part of the combined system are determined; (2) Determine the system's dynamic topology diagram based on the principle of the reduced transfer matrix method and the structural form of the combined system, split the system into components, number each component, simplify the combined system into a bifurcated multi-body system composed of multiple components, and then determine the state vectors of the input and output ends of the components; (3) Divide the spiral acoustic black hole element into sections according to the spiral angle, and calculate the geometric parameters of each beam section during the transfer process; calculate the material parameters of the parts with composite structures, and on this basis derive the transfer matrix of the spiral acoustic black hole element; (4) for a multi-input single-output element, derive a transfer equation that characterizes the relationship between its input and output state vectors and a motion coordination equation that characterizes the relationship between its multiple input state vectors; for a single-input single-output element, derive its transfer equation; (5) Based on the transfer equation and motion coordination equation of the element, the recursive equations of the reduced transfer matrix S and the load function correlation matrix e at its input and output are obtained; the characteristic equation and the total transfer equation of the combined system are obtained by combining the boundary conditions, and the characteristic frequency and steady-state response are obtained by solving them respectively; In the step (3), the spiral variable cross-section part is regarded as a spiral curved beam with power-law thickness variation formed by sequentially connecting N straight beams of equal cross-section thickness, and the transfer matrix of the spiral acoustic black hole element is obtained by the following steps: (3a) The spiral variable cross-section part is regarded as a spiral curved beam with power-law thickness variation formed by connecting N straight beams of equal cross-section thickness in sequence. The cross-sectional thickness of the kth beam after discretization is: h(s k )=εs k m +h d (1) Where ε and m are constants; s k is the length of the baseline helix θ k is the helical angle corresponding to the end of the kth beam, θ Start is the starting angle of the spiral acoustic black hole part; The length of the kth beam after discretization is: Where b is the Archimedean spiral coefficient, θ k-1 is the coefficient corresponding to the beginning of the kth beam, which is also the angle corresponding to the end of the k-1th beam. (3b) The surface of the spiral acoustic black hole unit is covered with a specific thickness h t For the composite beam structure with damping layer, its equivalent bending stiffness can be expressed in complex form: The linear density of the composite beam is: In formula (3) and formula (4), i is a complex number i 2 =-1, the elastic modulus of the composite beam is E, the section moment of inertia is I, and the bending stiffness is EI; while the bending stiffness of the beam without the damping layer is E a I a It means that the material damping loss factor η of the composite beam is: a 、E a , η a 、h a , a , A a is the section inertia moment, elastic modulus, loss factor, thickness, density and cross-sectional area of ​​the beam structure; E b , η b 、h b , b , A b is the elastic modulus, loss factor, thickness, density and cross-sectional area of ​​the damping material; in addition, the ratio of the elastic modulus E b / E a Defined as the ratio of e to thickness h b / h a is h; (3c) When analyzing the spiral acoustic black hole, the Euler-Bernoulli beam model is used for modeling; after the above calculation, it can be obtained that the kth beam segment is in its local coordinate system x k -y k The transfer matrix of the reference coordinate system of the spiral acoustic black hole element is obtained by coordinate transformation, then the local coordinate system x k -y k The coordinate transformation matrix R′ between the reference coordinate system k for: Among them, A′ k is the direction cosine matrix between the local inertial coordinate system of the kth beam and the reference inertial system where the spiral acoustic black hole is located; O is the zero matrix; θ' k is the angle between the neutral axis of the kth beam when it is not deformed, that is, the local coordinate system and the x-axis of the reference coordinate system: Then the transfer matrix of the kth beam in the local coordinate system where the overall spiral acoustic black hole element is located is: The transfer matrix of the spiral acoustic black hole element can be obtained: For the above beam elements, the transfer equation U′ of the kth beam segment in the local coordinate system is k for: Where l3 is the length of the beam, EI is the bending stiffness of the beam, is the mass linear density of the beam, S(x), T(x), U(x), V(x) are Krolev functions; 2. The spiral acoustic black hole dynamic vibration absorption calculation method based on the reduced multi-body system transfer matrix method according to claim 1 is characterized in that: The size parameters in step (1) include the uniform length L3 and thickness h3 of the spiral acoustic black hole, the starting angle θ of the spiral acoustic black hole part, start , end angle θ Stop , cut-off thickness h d , Archimedean spiral coefficient b, damping layer thickness h t , main beam length L, thickness h1, overall width of the structure b1; material parameters include the Young's modulus E of aluminum a , density ρ a , loss factor η a , Young's modulus E of the damping layer b , density ρ b , loss factor η b .

3. The spiral acoustic black hole dynamic vibration absorption calculation method based on the reduced multi-body system transfer matrix method according to claim 1 is characterized in that: The step (2) is achieved by the following steps: (2a) Massless virtual units are used as connecting elements at the connection points of different transmission paths of the combined system. The main beam and the spiral acoustic black hole dynamic vibration absorber both adopt the Euler Bernoulli beam model; each element is numbered, and the combined system is simplified to a bifurcated multi-body system composed of multiple elements; (2b) For a linear system, the generalized displacement and generalized force state vectors of element i are defined as Where x and y represent the linear displacement in the x and y directions respectively; θ z represents angular displacement; q x ,q y and m z Represent force and torque respectively, P represents input terminal I or output terminal O; the state vector in the relevant modal coordinate system is expressed as z i,p =Z i,p e iωt , ω is the angular frequency; along the transmission direction, the state vector of one element output terminal serves as the state vector of the next element input terminal.

4. The spiral acoustic black hole dynamic vibration absorption calculation method based on the reduced multi-body system transfer matrix method according to claim 1 is characterized in that: The step (4) is achieved by the following steps: (4a) For a massless virtual unit with multiple inputs and a single output, the transfer equations and motion coordination equations of the remaining components are obtained from the kinematic and dynamic equations of its input and output ends and the kinematic equations of different input ends: Kinematic and dynamic equations for M different inputs and a single output: Written in matrix form: is the transfer matrix describing the relationship between the generalized displacement and the generalized force state vector of the j-th input and output ends of element i; I3 and 03 are the third-order unit matrix and the third-order zero matrix respectively; The motion coordination equation can be obtained from the motion consistency of different input ends of the massless virtual unit: Written in matrix form: in is the motion coordination matrix describing the relationship between the generalized displacement state vector of the jth input terminal and the first input terminal of element i; (4b) For the beam element without additional damping layer as in equation (9), the segment N is set to 1 and the damping layer thickness h t Set it to 0 to get the transfer equation of each component in the reference coordinate system Similarly, coordinate transformation is used to obtain the transfer matrix of each element in the global coordinate system: Where R i is the local inertial coordinate system x i -y i Coordinate transformation matrix to the principal inertial coordinate system xy.

5. The spiral acoustic black hole dynamic vibration absorption calculation method based on the reduced multi-body system transfer matrix method according to claim 1 is characterized in that: The step (5) is achieved by the following steps: (5a) The reduced transfer matrix S at the input and output of the element and the load function correlation matrix e are derived by recursive equation: The relationship between the state vectors of multi-input and single-output element i is as follows: where f i is the load function array; the basic process of reduction transformation is: according to the system boundary conditions in Table 1, the state vectors of all components are divided into two parts, where Z a , Z b are all 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, the Z of the middle connection point a , Z b Can be divided arbitrarily; Table 1 Define the relationship between the two parts of the input state vector as S i,I 、e i,I , and then according to S i,I The reduced transfer matrix S at the output of element i is recursively obtained by summing the transfer matrix of element i i,O 、e i,O , and S j,O =S j+1,I 、e j,O =e j+1,I , we can get the reduced transfer matrix of the input end of element i+1; taking the boundary conditions of the input end as the initial value of the state vector, we can derive the reduced transfer matrix of the input and output ends of any element in the whole system in turn; The state vector of the element is repartitioned according to the reduced transfer matrix method; equations (17) and (18) can be rewritten as: For a multi-input single-output element, the reduced transformation of the state vector of its j-th input is expressed as: Substituting equation (20) into equations (19) and (20) yields the reduced transformation form of the output state vector of the multi-input single-output element: in: Where E a , E b , e a , e b satisfy: Formula (23) is the recursive equation of S and e for a multi-input single-output element. The recursive equation of S and e for a single-input single-output element can be simplified by removing redundant input and motion coordination equation terms, and the result is as follows: (5b) Steady-state response solution: According to the principle of state vector redivision and system boundary conditions, each input of the system corresponds to Z a =0, Z b ≠0, then at the system input point, S=0, e=0; S and e of each intermediate connection point and output end of the system can be obtained from recursive equations (23) and (30) along the system transfer path; then substitute the boundary conditions of the system output end into equation (18) to solve the unknown variables in the state vector of the system output end. After obtaining the state vector of the system output end, the state vector of any connection point of the system can be obtained by recursively reversing the transfer path from equations (19), (22), (28), and (29), thereby obtaining the steady-state response of any point of the system; (5c) Characteristic frequency solution: By removing the external excitation in (5a), the characteristic equation of the bifurcated multibody system can be obtained; Recorded as Recorded as Obtain the general form of the characteristic equation for a bifurcated multibody system: Where n is the number of the output component of the system, |Γ i The expression of | is as follows:

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