Calculation method of acoustic black hole vibration isolation based on transfer matrix method of multi-body system

Abstract

This invention discloses a calculation method for acoustic black hole vibration isolation based on the multibody system transfer matrix method. Based on the multibody system transfer matrix method, the overall transfer equation of an embedded double-leaf stiffened acoustic black hole vibration isolator is established, from which its natural frequency and steady-state response are obtained. This method has the advantages of fast calculation speed and wide applicability when calculating the natural frequency and steady-state response of acoustic black hole vibration isolators.

Description

Technical Field

[0001] The invention relates to the field of dynamic modeling and analysis of acoustic black hole structures, and in particular to a method for calculating characteristic frequencies and vibration isolation steady-state responses of an acoustic black hole vibration isolator based on a multi-body system transfer matrix method. Background Art

[0002] Acoustic black holes have been proven to reduce broadband vibration transmission of structures through energy focusing effects, thus forming broadband vibration isolation devices. In the past, the design of acoustic black hole isolators was mainly based on analytical and semi-analytical methods or finite element analysis. Analytical and semi-analytical methods are mainly applicable to relatively simple ABH vibration isolation structures. With the further requirements for the performance of vibration isolation devices, a series of complex acoustic black holes with embedded double leaves and double leaf reinforcement types have been developed, which can enhance the static stiffness of the structure while enhancing the acoustic black hole effect. For such complex acoustic black hole structures, numerical methods such as finite elements must be used, but these methods are often too time-consuming to handle complex structures. At present, there is still a lack of effective theoretical calculation methods to design complex acoustic black hole isolators with actual required performance. Summary of the invention

[0003] The object of the present invention is to provide an acoustic black hole vibration isolation calculation method based on a multi-body system transfer matrix method.

[0004] The technical solution to achieve the purpose of the present invention is: a method for calculating acoustic black hole vibration isolation based on a multi-body system transfer matrix method, comprising the following steps:

[0005] (1) The double-blade reinforced acoustic black hole is embedded in the main structure to form an embedded double-blade reinforced acoustic black hole vibration isolation structure; the vibration isolation structure is used as a two-dimensional plane structure; and then the size parameters and material parameters of each part of the vibration isolator are determined.

[0006] (2) According to the structural form of the acoustic black hole isolator, it is divided into various components, each component is numbered, and then the state vector of the input and output ends of the component is determined, and the topological diagram of the isolator system is established according to the connection relationship of the components;

[0007] (3) For multi-input single-output elements and single-input multi-output elements, derive the transfer equations that characterize the relationship between the state vectors of their input and output terminals and the geometric equations that characterize the relationship between the state vectors of their multiple input terminals / multiple output terminals; for single-input single-output elements, derive their transfer equations;

[0008] (4) The total transfer equation of the system is obtained based on the transfer equation and geometric equation of each component and the relationship between the output state vector of the previous component and the input state vector of the next component; the boundary conditions are substituted into the total transfer equation to obtain the characteristic equation of the system, and the natural frequency of the system is obtained by solving it; the load and boundary conditions are substituted into the total transfer equation, and the steady-state response of the system is obtained by solving it.

[0009] Compared with the prior art, the present invention has the following significant advantages: 1. Compared with the previous theoretical analysis methods, the method of the present invention is not limited to the dynamics and vibration transmission problems of planar acoustic black hole structures, but provides a general method for the vibration transmission research of acoustic black hole structures with various structural components and more complex structures. 2. Compared with finite element simulation, the method of the present invention replaces the system dynamics equation with a low-order transfer equation, and the matrix order involved in the solution is independent of the system's degree of freedom, and the calculation speed is faster, which is conducive to the design and optimization of acoustic black hole vibration isolation structures. BRIEF DESCRIPTION OF THE DRAWINGS

[0010] Figure 1 It is a schematic diagram of the embedded double-leaf reinforced acoustic black hole structure.

[0011] Figure 2 It is a topological diagram of a planar multi-body system with double closed loops.

[0012] Figure 3 It is the force analysis diagram of rigid thin film.

[0013] Figure 4 It is the equivalent treatment of the wedge segment.

[0014] Figure 5 It is a comparison chart of the calculation results of the steady-state response of vibration transfer using the transfer matrix method of multi-body systems and finite element simulation. DETAILED DESCRIPTION

[0015] The technical solution adopted by the present invention is: determining the size parameters and material parameters of the acoustic black hole isolator structure; splitting the combined system into various components, numbering each component and determining the component state vector; deriving the transfer equation and geometric equation of each component; deriving the total transfer equation of the combined system, and solving it in combination with boundary conditions to obtain the system characteristic frequency and steady-state response.

[0016] The present invention will be further described below in conjunction with the accompanying drawings and embodiments.

[0017] The present invention provides an acoustic black hole vibration isolation calculation method based on a multi-body system transfer matrix method, and the specific process includes the following four steps:

[0018] (1) Embed the double-blade reinforced acoustic black hole into the main structure to form an embedded double-blade reinforced acoustic black hole vibration isolation structure; the vibration isolation structure is a two-dimensional plane structure; and then determine the size parameters and material parameters of each part of the complex acoustic black hole vibration isolator;

[0019] (2) According to the structural form of the acoustic black hole isolator, it is divided into various components, each component is numbered, and then the state vector of the input and output ends of the component is determined, and the topological diagram of the system is established according to the connection relationship of the components;

[0020] (3) For multi-input single-output elements and single-input multi-output elements, derive the transfer equations that characterize the relationship between the state vectors of their input and output terminals and the geometric equations that characterize the relationship between the state vectors of their multiple input terminals / multiple output terminals; for single-input single-output elements, derive their transfer equations;

[0021] (4) The total transfer equation of the system is obtained based on the transfer equation and geometric equation of each component and the relationship between the output state vector of the previous component and the input state vector of the next component; the boundary conditions are substituted into the total transfer equation to obtain the characteristic equation of the system, and the natural frequency of the system is obtained by solving it; the load and boundary conditions are substituted into the total transfer equation, and the steady-state response of the system is obtained by solving it.

[0022] The size parameters in step (1) include the total length L, thickness H, rib width L of the acoustic black hole structure. j , cut-off thickness H d , the total length of the acoustic black hole wedge segment L ABH The material parameters include Young's modulus E, density ρ, and loss factor η.

[0023] The step (2) is achieved by the following steps:

[0024] (2a) A massless virtual rigid sheet is used as a connecting element at the connection of different transmission paths in the system, and the rest of the system adopts the Euler Bernoulli beam model. Each element is numbered, and the entire structure is simplified into a general multi-body system composed of multiple elements. Draw its topology diagram and its transmission direction.

[0025] (2b) For a linear system, the generalized displacement and generalized force state vectors of element j 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, and P represents input I or output O. The state vector in the relevant modal coordinate system can be expressed as z j,p =Z j,p e iωt , ω is the angular frequency. Along the transmission direction, the state vector at the output of one element serves as the state vector at the input of the next element.

[0026] In the step (3), in the multi-body system transfer matrix method, for any multi-input single-output element and single-input multi-output element, the equations written are divided into two types, namely, the transfer equation describing the relationship between the state vectors of the element input and output ends and the geometric equation describing the kinematic relationship between different input ends / output ends of the element. The single-input single-output element can be regarded as a special case of the multi-input single-output element. The transfer equation and geometric equation of the complex acoustic black hole as each element of the multi-body system are obtained by the following steps:

[0027] (3a) For a single-input single-output plane vibrating Euler Bernoulli beam with longitudinal vibration on the x-axis and transverse vibration on the y-axis, such as elements 1, 6, and 11, the transfer equation is:

[0028] Z n,I =U n Z n,O (n=1,6,11)(1) The transfer matrix is

[0029]

[0030] in

[0031]

[0032] (3b) For a single-input dual-output massless rigid sheet, such as elements 2 and 7, according to its force analysis Figure 3 And the sign convention of the transfer matrix method of multi-body system, the positive direction of the input force is the same as the coordinate system, and the positive direction of the moment is opposite to the coordinate system; the positive direction of the output force is opposite to the coordinate system, and the positive direction of the moment is the same as the coordinate system.

[0033] In the inertial coordinate system Ixy there is a vector relationship

[0034]

[0035] in is the displacement at the input, is the displacement of output terminal O1, is the radius vector from the input terminal I to the output terminal O1;

[0036] By using the coordinates in Ixy to represent the vector relationship of equation (3), the displacement relationship can be obtained as follows:

[0037]

[0038] From the force balance equation we can get

[0039]

[0040] Arranging equations (4) and (5) yields the transfer equation for a rigid sheet with single-end input and double-end output:

[0041] Z n,I =U n Z n,O (n=2,7)(6)where

[0042]

[0043]

[0044] The geometric equation for the relationship between the output state vectors is

[0045]

[0046] in

[0047]

[0048] (3c) For a rigid sheet with double-end input and single-end output, such as elements 5 and 10, its transfer equation and transfer matrix can be derived in a similar way to (3a). Its transfer equation can be expressed as

[0049] Z n,O =U n Z n,I (n=5,10)(11)where

[0050]

[0051]

[0052] The geometric equation for the relationship between the input state vectors is

[0053]

[0054] in

[0055]

[0056] (3d) For a double beam system, such as elements 3 and 4 or elements 8 and 9, denoted by SL k , k = 1, 2, the transfer equation and transfer matrix of the system can be expressed as

[0057]

[0058] Among them U n is the transfer matrix of the wedge-shaped beam, which converts a wedge-shaped section with continuously changing thickness into a step beam with j sections of uniform thickness, such as Figure 4 , the transfer matrix of the tapered beam is equal to the multiplication of the transfer matrices of j uniform step beams. When using the left input and right output, the transfer matrix can be expressed as

[0059] U n=U j U j-1 U j-2 …U1(n=3,4,8,9) (16)

[0060] Among them U j U1 is the transfer matrix of the plane Euler Bernoulli beam considering the longitudinal vibration of the x-axis and the transverse vibration of the y-axis (1).

[0061] The step (4), derivation of the total transfer equation, solution of the natural frequency and steady-state response can be achieved by the following steps respectively:

[0062] (4a) According to the relationship that the output point of each element and the input of the next element are the same point, their state vectors are equal, and the following relationship can be obtained:

[0063]

[0064] Combining the transfer equations of each component with the geometric equation, the total transfer equation of the system can be obtained:

[0065]

[0066] (4b) Natural frequency solution:

[0067] Substitute the boundary conditions into the state vector of the boundary element, then substitute the state vector of the boundary element into the total transfer equation of the system, extract the homogeneous linear equations, and set the coefficient determinant of the homogeneous linear equations to 0 to obtain the characteristic equation of the system. Solve the characteristic equation using the bisection method to obtain the natural frequency of the system.

[0068] (4c) Steady-state response solution:

[0069] For vibration isolation problems, the load or excitation point is located at the input end. At this time, the load and boundary conditions need to be simultaneously brought into the system's total transfer equation to solve the steady-state response of the system. Taking the lateral vibration of the y-axis as an example, the vibration transmissibility of the system can be calculated by the following formula:

[0070]

[0071] where Y I and Y O is the lateral displacement of the input and output ends of the system. The smaller T is, the better the vibration isolation effect of the structure is.

[0072] The following simulation experiment is used to further illustrate the invention in conjunction with the accompanying drawings.

[0073] 1. Determine the structural parameters and material parameters of each part:

[0074] The present invention is as follows Figure 1The embedded reinforced acoustic black hole structure shown in FIG. 1 is used as an example to illustrate the structure. The material and size parameters of each part are shown in Table 1.

[0075]

[0076] Table 1: Structural parameters of embedded reinforced acoustic black hole

[0077] 2. Split the structure into components, number each component and determine the state vector of the component input and output:

[0078] The left end of the structure is the system input, and the right end is the output. The single-end input and double-end output massless rigid sheet 2 (7) and the double-input and single-input massless rigid sheet 5 (10) are the connecting elements between different transmission paths. Considering the low-frequency vibration, the uniform part and the wedge-shaped part of the acoustic black hole both adopt the Euler Bernoulli beam model. The combined system can be simplified as Figure 2 The general multi-body system with 11 elements and two closed loops shown in the figure has a total transmission direction as Figure 2 shown.

[0079] 3. According to the dynamics and kinematics equations of each component, derive its transfer equation and geometric equation:

[0080] (3a) For single-input single-output elements 1, 6, and 11, their transfer equations and transfer matrices can be obtained from equations (1) and (2).

[0081] (3b) For the single-input dual-output elements 2 and 7, their transfer equation and transfer matrix can be obtained from equations (6) and (8).

[0082] (3c) For the dual-input single-output elements 5 and 10, their transfer equations and transfer matrices can be obtained from equations (11) and (12).

[0083] (3d) For a wedge-shaped beam, it can be equivalent to a stepped beam with j segments of uniform thickness, such as Figure 4 , the transfer matrix of the tapered beam is equal to the multiplication of the transfer matrices of j uniform step beams, which can be obtained by equation (16). Therefore, for the dual beam element systems 3 and 4 and 8 and 9 containing dual tapered beams, their transfer equations and transfer matrices can be obtained by equation (15).

[0084] 4. According to the transfer direction, the state vector of the output end of one component is equal to the state vector of the input end of the next component to obtain the total transfer equation of the system. Combined with the boundary conditions, the system characteristic equation is obtained, and the system characteristic frequency is obtained by solving; combined with the load and boundary conditions, the steady-state response of the system is obtained by solving:

[0085] (4a) According to the relationship (17) that the output point of each component and the input of the next component are the same point, and their state vectors are equal, the total transfer equation of the system (18) can be obtained by combining the transfer equations of each component and the geometric equation.

[0086] (4b) Substitute the free-free boundary condition into the state vectors of boundary elements 1 and 11, and then substitute the state vectors of boundary elements 1 and 11 into the total transfer equation (18) of the system, and extract the homogeneous linear equation system therein, and set the coefficient determinant of the homogeneous linear equation system to 0, and the characteristic equation of the system can be obtained. The natural frequency of the system can be obtained by solving the characteristic equation using the bisection method.

[0087] (4c) The y-axis unit displacement load applied at one end and the free boundary condition applied at the other end are simultaneously substituted into the system total transfer equation and the steady-state response of the system is obtained by solving it. Taking the lateral vibration of the y-axis as an example, the vibration transmissibility of the system can be calculated by equation (19).

[0088] 5. Verification of transfer matrix method for multi-body system

[0089] In order to prove the correctness and speed of the multi-body system transfer matrix method (MSTMM) calculation, the following two methods were established in COMSOL: Figure 1 The embedded reinforced acoustic black hole structure shown is based on the Euler beam model and the finite element (FEM) model based on the solid mechanics model, where j is the number of wedge segments. When the grid density is gradually increased to j = 300, convergence is achieved when the errors of the first 5 eigenfrequencies of the system are within 5%.

[0090] (5a) First, the characteristic frequencies of the system are calculated based on the MSTMM-Euler beam model, FEM-Euler beam model and FEM-solid mechanics model, respectively. The results are shown in Table 2. δ represents the relative error of the first two Euler beam models relative to the latter solid mechanics model. It can be seen that at the first five natural frequencies, the maximum relative error of MSTMM is only 4.8%, which also verifies the calculation results of MSTMM. At the same time, the calculation time of MSTMM is shorter than that of FEM, indicating that the former has higher calculation efficiency.

[0091]

[0092]

[0093] Table 2: Comparison of natural frequency calculation results of multi-body system transfer matrix method and finite element simulation

[0094] (5b) Then, the steady-state responses calculated by the MSTMM-Euler beam model and the FEM-solid mechanics model are compared. In order to ensure the convergence of the calculation results of the two methods, the wedge segment of the acoustic black hole unit is divided into 300 segments in MSTMM, and the calculation frequency interval is 1Hz; the same frequency interval is taken in the solid mechanics model, the grid is encrypted twice, and the calculation results do not change. The steady-state response of the vibration transmissibility is calculated using the MSTMM-Euler beam model and the FEM-solid mechanics model respectively. The results are shown in Figure 5 It can be seen that the calculation results of the two methods are highly consistent in the frequency range of 20Hz-20kHz, which verifies the correctness of using MSTMM to calculate the steady-state response.

Claims

1. An acoustic black hole vibration isolation calculation method based on the multi-body system transfer matrix method, characterized in that: The following steps are involved: (1) Embed the double-blade reinforced acoustic black hole into the main structure to form an embedded double-blade reinforced acoustic black hole vibration isolator; the vibration isolator is a two-dimensional plane structure; and then determine the size parameters and material parameters of each part of the vibration isolator; (2) According to the structural form of the acoustic black hole isolator, it is divided into various components, each component is numbered, and then the state vector of the input and output ends of the component is determined, and the topological diagram of the isolator system is established according to the connection relationship of the components; (3) For multi-input single-output elements and single-input multi-output elements, derive the transfer equations that characterize the relationship between the state vectors of their input and output terminals and the geometric equations that characterize the relationship between the state vectors of their multiple input terminals / multiple output terminals; for single-input single-output elements, derive their transfer equations; (4) Obtain the total transfer equation of the system based on the transfer equation and geometric equation of each component and the relationship between the output state vector of the previous component and the input state vector of the next component; Substitute the boundary conditions into the total transfer equation to obtain the characteristic equation of the system, and solve it to obtain the natural frequency of the system; Substitute the load and boundary conditions into the total transfer equation, and solve it to obtain the steady-state response of the system; In step (4), the derivation of the total transfer equation, the solution of the natural frequency and the steady-state response can be achieved by the following steps respectively: (4a) According to the relationship that the output point of each element and the input of the next element are the same point, their state vectors are equal, and the following relationship can be obtained: Among them, Z j,p represents the state vector of element j at point p along the transmission direction, where p represents the input end I or the output end O; Combining the transfer equations of each component with the geometric equation, the total transfer equation of the system can be obtained: Among them, U j represents the transfer matrix of element j; O n×m The matrix represents an n-row and m-column zero matrix; H1 and H2 are the geometric constraint matrices of the state vectors of the massless rigid thin-sheet element at the same end; (4b) Natural frequency solution: Substitute the boundary conditions into the state vector of the boundary element, then substitute the state vector of the boundary element into the total transfer equation of the system, and extract the homogeneous linear equations therein, and set the coefficient determinant of the homogeneous linear equations to 0, and the characteristic equation of the system can be obtained; the natural frequency of the system can be obtained by solving the characteristic equation using the bisection method; (4c) Steady-state response solution: For vibration isolation problems, the load or excitation point is located at the input end. At this time, the load and boundary conditions need to be simultaneously introduced into the system's total transfer equation to obtain the steady-state response of the system; considering the lateral vibration of the system in the y direction, the vibration transmissibility of the system can be calculated by the following formula: where Y I and Y O is the lateral displacement of the input and output ends of the system.

2. The acoustic black hole vibration isolation calculation method based on the multi-body system transfer matrix method according to claim 1 is characterized in that: The size parameters described in step (1) include the total length L, thickness H, rib width L of the acoustic black hole structure. j , cut-off thickness H d 、Total length of acoustic black hole wedge L ABH ; Material parameters include Young’s modulus E, density ρ, and loss factor η.

3. The acoustic black hole vibration isolation calculation method based on the multi-body system transfer matrix method according to claim 1 is characterized in that: The step (2) is achieved by the following steps: (2a) A massless virtual rigid sheet is used as a connecting element at the connection of different transmission paths of the vibration isolator system, and the rest of the parts adopt the Euler Bernoulli beam model; each element is numbered, and the entire structure is simplified into a general multi-body system composed of multiple elements, and its topology diagram and transmission direction are drawn; (2b) For a linear system, the generalized displacement and generalized force state vectors of element j 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 the force and moment in the x and y directions respectively, P represents the input terminal I or the output terminal O; let z j,p =Z j,p e iωt , we can get z j,p The state vector in the relevant modal coordinate system is ω is the angular frequency, t is the time, and i is the imaginary unit; Along the transmission direction, the state vector of one component output terminal serves as the state vector of the next component input terminal.

4. The acoustic black hole vibration isolation calculation method based on the multi-body system transfer matrix method according to claim 1 is characterized in that: In the step (3), the transfer equations and geometric equations of the embedded double-blade reinforced acoustic black hole as each element of the multi-body system are obtained by the following steps: (3a) For a single-input single-output plane vibrating Euler Bernoulli beam with longitudinal vibration on the x-axis and transverse vibration on the y-axis, the transfer equation is: WITH n,I =U n WITH n,O (1) The transfer matrix is in EA is the tensile stiffness of the beam, is the linear mass density of the beam, EI is the bending stiffness of the beam, (3b) For a massless rigid sheet with single input and dual output, according to the sign convention of the transfer matrix method of multi-body system for force analysis, the positive direction of the input force is the same as that of the coordinate system, and the positive direction of the moment is opposite to that of the coordinate system; the positive direction of the output force is opposite to that of the coordinate system, and the positive direction of the moment is the same as that of the coordinate system; In the inertial coordinate system Ixy there is a vector relationship in is the displacement at the input, is the displacement of output terminal O1, is the radius vector from the input terminal I to the output terminal O1; By using the coordinates in Ixy to represent the vector relationship of equation (3), the displacement relationship can be obtained as follows: From the force balance equation we can get Arranging equations (4) and (5) yields the transfer equation for a rigid sheet with single-end input and double-end output: n,I =U n Z n,O (6) The geometric equation for the relationship between the output state vectors is in (3c) For a rigid sheet with double-end input and single-end output, the transfer equation and transfer matrix can be derived in the same way as (3a). The transfer equation can be expressed as Z n,O =U n Z n,I (11) Among them The geometric equation for the relationship between the input state vectors is in (3d) For the double beam system, denoted as SL k , k = 1, 2, the transfer equation and transfer matrix of the system can be expressed as Among them U n is the transfer matrix of the wedge-shaped beam. A wedge-shaped section with continuously changing thickness is equivalent to a step beam with j sections of uniform thickness. The transfer matrix of the wedge-shaped beam is equal to the multiplication of the transfer matrices of j uniform step beams. When using the left input and right output, the transfer matrix can be expressed as U n =U j U j-1 U j-2 ...U1(16) where U j U1 is the transfer matrix (1) of the plane Euler Bernoulli beam considering the longitudinal vibration of the x-axis and the transverse vibration of the y-axis.

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