Acoustic black hole sandwich beam with low-frequency broadband vibration attenuation characteristic and bending wave propagation relation construction method and vibration characteristic analysis method thereof

Abstract

The invention provides a vibration characteristic analysis method for an acoustic black hole sandwich beam, which comprises the following steps of: 1) establishing a transmission matrix T and a reflection matrix R for a discontinuous core sandwich beam coupling section; 2) establishing a transmission matrix TABH containing an acoustic black hole structure and a reflection matrix RABH containing an acoustic black hole structure by using an iterative relationship of waves; 3) combining a transmission matrix TABH and a reflection matrix RABH of the upper beam embedded with the acoustic black hole, a propagation matrix F of the uniform lower beam and a reflection matrix R and a transmission matrix T of the coupling section to establish a transmission matrix and a reflection matrix containing two core bodies; 4) obtaining a transmission matrix and a reflection matrix containing N core bodies through an iteration relation; and 5) assuming that one end of the structure is a unit displacement excitation end and the other end is a free end, calculating the vibration transmission loss of the designed structure by combining the obtained transmission matrix and reflection matrix, and analyzing the vibration characteristics of the designed structure.

Description

Technical Field

[0001] The present invention relates to a discontinuous core sandwich beam, specifically, an acoustic black hole sandwich beam with low-frequency broadband vibration attenuation characteristics, and a method for constructing the bending wave propagation relationship and an analysis method for the vibration characteristics of the acoustic black hole sandwich beam. Background Art

[0002] A sandwich structure generally consists of upper and lower face sheets and an intermediate core layer. The core layer structure is diverse. It can be a continuous material, such as a strongly damped rubber layer, or a lightweight pine wood, foam layer, etc., or a discontinuous core layer, such as honeycomb, corrugation, truss structure, etc., or discrete core units. In practical engineering applications, sandwich structures are often subjected to dynamic loads, which easily cause structural vibrations, and then lead to problems such as premature fatigue failure of the structure, decline in product quality and production efficiency, and in severe cases, major production accidents will occur. Therefore, with the improvement of the requirements for vibration reduction and isolation performance of engineering structures, the design of sandwich structures and the research on their dynamic vibration models have become research hotspots in the engineering field.

[0003] Regarding the vibration characteristics of traditional continuous core sandwich beams, scholars at home and abroad have conducted a large number of studies and formed relatively mature theoretical models. However, for discontinuous core sandwich beams, such as honeycomb cores, periodic cores, etc., due to the complexity of their core structures, the analysis of vibration characteristics is more complex. Researchers usually use the equivalent elastic constant method to equivalent the discontinuous core to a homogeneous continuous medium and analyze it in combination with the anisotropic sandwich theory of continuous core sandwich beams. Although this method simplifies the dynamic analysis of complex cores, the treatment of local structures is relatively rough. In addition, some scholars have explored the direct analytical method for analyzing the vibration characteristics of simple core structures (such as honeycomb, orthogonal stiffening, etc.). This method fully considers the dynamic relationship between the face sheet and the core layer and establishes an analytical calculation model. For complex core structures, the finite element method is widely used, but its calculation amount is large, especially for large-scale complex structures, the calculation efficiency is very low when the mesh is divided finely. Relevant research shows that the above research methods cannot clearly describe the propagation relationship of bending waves in sandwich beams and the contribution degree of each component in the sandwich beam to the vibration response. Therefore, it is impossible to better realize the vibration control of sandwich beams. Summary of the Invention

[0004] The object of the present invention is to provide a method for constructing the bending wave propagation relationship of a coupling section in a sandwich beam, which has universality and can obtain the reflection matrix and transmission matrix when the bending wave enters the coupling section, and is the basis for the design optimization and bending wave control analysis of the sandwich beam structure.

[0005] Another object of the present invention is to provide an acoustic black hole sandwich beam with low-frequency broadband vibration attenuation characteristics, which is light in weight, simple in structure, easy to process, and integrates multiple bending wave attenuation mechanisms such as the acoustic black hole effect, the periodic structure effect, and the vibration coupling effect of the upper and lower beams, and can achieve excellent low-frequency broadband vibration attenuation characteristics.

[0006] Another object of the present invention is to provide a method for constructing the bending wave propagation relationship of an acoustic black hole sandwich beam, which can clearly describe the bending wave propagation relationship in the sandwich beam and the contribution degree of each component in the sandwich beam to the vibration response.

[0007] Another object of the present invention is to provide a method for analyzing the vibration characteristics of an acoustic black hole sandwich beam.

[0008] To achieve the above object, the present invention adopts the following technical solutions:

[0009] A method for constructing the bending wave propagation relationship of a coupling section in a sandwich beam, the sandwich beam is composed of an upper beam, a lower beam and a discrete core, and the sandwich beam includes a coupling section with a core and a non-coupling section without a core;

[0010] The method for constructing the bending wave propagation relationship of the coupling section includes the following steps:

[0011] a) Establish the dynamic equation of the coupling section

[0012] The coupling section is subjected to the force of the core. Based on the Kelvin linear viscoelastic model, it is modeled as a distributed spring damping force model. Then the displacements w u (x,t), w d (x,t) satisfy the following equation:

[0013]

[0014] Among them, E u , E d are the Young's moduli of the upper and lower beams of the coupling section, I u , I d are the moments of inertia of the upper and lower beams of the coupling section, ρ u , ρ d are the material densities of the upper and lower beams of the coupling section, A u , A d are the cross-sectional areas of the upper and lower beams of the coupling section, w u (x,t), w d (x,t) are the amplitudes of the upper and lower beams of the coupling section; E is the complex Young's modulus of the core, E = E0(1 + jη), E0 is the pure Young's modulus, η is the damping dissipation factor of the core material, representing the damping dissipation ability of the material; b and h are the width and thickness of the core respectively; x is the position coordinate of a point on the beam along the length direction of the beam;

[0015] Separate the time variable \(t\), and assume the solution of the equation is \(w\) u (x,t)=w1(x)e iωt , \(w\) d (x,t)=w2(x)e iωt , \(\omega\) is the angular frequency, and we can get:

[0016]

[0017] where \(a1\), \(a2\), \(a3\), \(a4\), \(b1\), \(b2\), \(b3\), \(b4\), \(c1\), \(c2\), \(c3\), \(c4\), \(d1\), \(d2\), \(d3\), \(d4\) are the coefficients of each term,

[0018] It can be seen from Equation (3) that for the transverse vibration responses of the upper and lower beams in the coupled beam, they can be regarded as the motion of flexural waves with two wave numbers \(k1\) and \(k2\);

[0019] b) Analyze the propagation relationship of flexural waves in the coupling section

[0020] The junction points A, B, C, and D of the left and right ends of the core body in the coupling section with the upper and lower beams are discontinuous characteristic points,

[0021] It can be seen from Equation (3) that the flexural waves on the AB beam in the coupling section are regarded as two forward waves with wave numbers \(k1\) and \(k2\) respectively and two backward waves which are respectively:

[0022]

[0023] Similarly, according to the transverse vibration amplitude-frequency response of the CD section of the coupled beam, the flexural waves with wave numbers \(k1\) and \(k2\) are respectively \(\alpha1\) and \(\alpha2\) times the corresponding wave numbers on the AB beam. At this time, the forward waves and backward waves are respectively:

[0024]

[0025] For the local structure with one core body in the sandwich beam, assuming this structure is an infinitely long structure, when the flexural wave is incident from section A, the junction points A, B, C, and D of the coupling section and the non-coupling section are discontinuous characteristic points, and reflection and transmission phenomena will occur for the flexural wave here; due to the characteristics of its infinite structure, for the non-coupling section part, the incident wave only reflects at A, and there are only transmitted waves but no reflected waves at B, C, and D;

[0026] When the flexural wave propagates on the upper and lower beams in the coupling section, the transmission relationships of the forward waves and backward waves with each wave number satisfy:

[0027]

[0028] where \(j = 1, 2\), The transfer matrices of the flexural waves with wave numbers k1 and k2 are transmitted in the upper and lower coupled beams respectively;

[0029] Assume that the reflection matrix of the incident wave at the discontinuous feature point A is R A , and the transmission matrices of the incident wave transmitted to points B, C, and D are T B , T C , T D respectively. Then, the propagation of the flexural waves in the upper and lower beams of the uncoupled section satisfies the relationship:

[0030]

[0031] From Equation (17), the reflection matrix R of the flexural wave incident from end A can be obtained A and the transmission matrices T B , T C , T D . From this relationship, the reverse wave at point A and the transmitted waves at points B, C, and D can be obtained. Similarly, the reflection matrix and transmission matrix can be obtained when the flexural wave is incident from points B, C, and D.

[0032] An acoustic black hole sandwich beam with low-frequency broadband vibration attenuation characteristics, the sandwich beam is composed of an upper beam, a lower beam and a discrete core, and the sandwich beam includes a coupled section with a core and an uncoupled section without a core; the thickness of the upper beam in the coupled section is h u , the length of any uncoupled section is L2, and an acoustic black hole is embedded in the upper beam in the uncoupled section; the thickness h(x) of any acoustic black hole changes to satisfy the power-law relationship:

[0033] h(x) = ε0x m + h0, m ≥ 2 (1)

[0034] where ε0 is a constant coefficient, m is the power-law exponent, h0 is the truncation thickness, and x is the distance from the central axis of symmetry of the acoustic black hole; when x = ±L2 / 2, h(x) = h u .

[0035] A method for constructing the flexural wave propagation relationship of an acoustic black hole sandwich beam includes the following steps:

[0036] 1) Use the aforementioned method to construct the flexural wave propagation relationship of a coupled section to obtain its transmission matrix T and reflection matrix R;

[0037] 2) Use the iterative relationship of the wave to establish the transmission matrix T ABH and reflection matrix R ABH of the upper beam with an acoustic black hole structure;

[0038] 3) Combine the transmission matrix T ABHand the reflection matrix R ABH , the propagation matrix F of the uniform lower beam, and the reflection matrix R and transmission matrix T of the coupling section, to establish the transmission matrix and reflection matrix of the two-core body;

[0039] 4) Through the iterative relationship, obtain the transmission matrix and reflection matrix of the acoustic black hole sandwich beam with N cores.

[0040] The specific process of step 2) in the above method for constructing the bending wave propagation relationship of the acoustic black hole sandwich beam is as follows:

[0041] When an acoustic black hole is embedded in the uncoupled section, the beam part of the acoustic black hole can be divided into N micro-sections, and the junction of each micro-section is a discontinuous point. Take the i-th discontinuous point for analysis, then the amplitude responses on both sides of this point can be written as:

[0042]

[0043] In the formula, p1, p2, p3, p4 and q1, q2, q3, q4 are the coefficient items of the amplitude responses of two adjacent micro-sections at the discontinuous point; the natural wave numbers k 01 , k 02 of the left and right two micro-section beams at the discontinuous point i are respectively:

[0044]

[0045] According to the wave vector method, the forward wave on the left side at the i-th discontinuous characteristic point is The backward wave is

[0046] The forward wave on the right side is The backward wave

[0047]

[0048] Among them: f n is the transfer matrix of the n-th micro-section between the (i - 1)-th section and the i-th section; the forward wave on the left side at the (i - 1)-th discontinuous characteristic point is The backward wave is The forward wave on the right side is The backward wave is After sorting, the transmission and reflection matrices from the bending wave at the (i - 1)-th section to the i-th section are:

[0049]

[0050] By analogy, the recurrence formula can be obtained as:

[0051]

[0052] The forward and reverse equivalent reflection and transmission matrices from the 1st to the i-th variable cross-section are as follows:

[0053]

[0054] By calculating from the 1st variable cross-section to the last variable cross-section of the acoustical black hole after segmented processing according to Equation (25), the forward transmission matrix of the flexural wave in an uncoupled section of an embedded acoustical black hole can be obtained Reflection matrix And the reverse transmission matrix Reflection matrix Since the acoustical black hole has a symmetric structure, so

[0055] The specific process of step 3) in the method for constructing the flexural wave propagation relationship of the above-mentioned acoustical black hole sandwich beam is as follows:

[0056] First, analyze the flexural wave propagation characteristics of the sandwich beam with two cores. The junction points A, B, C, D, A1, B1, C1, and D1 between the left and right ends of the 1st and 2nd cores from left to right and the upper and lower beams are discontinuous characteristic points. The BA1 section is the upper beam of the uncoupled section of the sandwich beam with an acoustical black hole, and DC1 is the uniform lower beam of the uncoupled section of the sandwich beam. Assume that the flexural wave is incident from sections A and C.

[0057] The relationships at the discontinuous characteristic points A1 and C1 are as follows:

[0058]

[0059] Among them, the incident waves at points A and C are The reflected waves at points A and C are The forward waves at points B and D are The reverse waves at points B and D are The forward waves at points A1 and C1 are The reverse waves at points A1 and C1 are The transmitted waves at points B1 and D1 are R A1 、R C1 Are the reflection matrices of the flexural wave at points A1 and C1 respectively, and T C1A1 、T A1CA Are the transmission matrices of the flexural wave transmitted from C1 to A1 and from A1 to C1 respectively, and O is the zero matrix; these matrices can be obtained through Equation (17);

[0060] From Equation (26), it can be obtained that:

[0061]

[0062] Among them, I is the identity matrix;

[0063] The bending waves at the discontinuous feature points B and D satisfy the relation:

[0064]

[0065] where R B and R D are the reflection matrices of the bending wave at B and D respectively, and T xy are the transmission matrices of the bending wave transmitted from x to y respectively. These matrices can be obtained through Equation (17);

[0066] The bending waves at the discontinuous feature points B1 and D1 satisfy the relation:

[0067]

[0068] The discontinuous feature points A and C satisfy the relation:

[0069]

[0070] From Equations (26)-(30), the equivalent transmission matrix and the reflection matrix of the bending wave in the sandwich beam structure of the acoustic black hole between two cores can be obtained as:

[0071]

[0072] For the acoustic black hole sandwich beam with N cores, the above method can be used for iterative calculation to obtain the equivalent reflection matrix and the transmission matrix

[0073]

[0074] where

[0075] Calculation method for the vibration transmission loss of a finite periodic sandwich beam. Assume that the sandwich beam contains N cores, the left ends A and C are subjected to a unit displacement excitation, and the right ends B and D are free ends; extend O1A, BO2, O3C, and DO4 at both ends respectively, and set their lengths to 0. Then, it can be regarded that a unit excitation is applied at O1 and O3, and the right ends O2 and O4 are free ends.

[0076] The incident waves at points O1 and O3 are The reflected waves at points O1 and O3 are The forward waves at points A and C are The backward waves at points A and C are The forward waves at points B and D are The backward waves at points B and D are The forward waves at points O2 and O4 are The reverse waves at points O2 and O4 are

[0077] Since the displacement excitation ends O1 and O3 are subjected to a unit displacement excitation, the rotation angle here satisfies the relationship θ = 0, and the propagation relationship of the bending wave can be obtained as:

[0078]

[0079] For the free-end boundary condition, from the bending moment M = 0 and the shear force F s = 0, the reflection matrix R is all:

[0080]

[0081] From this, the bending wave transmission relationship at the right ends O2 and O4 of the studied beam satisfies:

[0082]

[0083] Among them,

[0084] In the continuous parts O1A, BO2, O3C, and DO4 of the sandwich beam, the transmission relationship of the bending wave satisfies:

[0085]

[0086] Among them, F is the unit matrix;

[0087] According to the analysis of the bending wave transmission relationship of the infinitely long periodic discontinuous core, the bending wave transmission relationship of the finite-length AB and CD segments can be further obtained as:

[0088]

[0089] It is assumed that from equations (33)-(37), it can be obtained:

[0090]

[0091] Among them,

[0092] The vibration energy attenuation characteristic of passing through N cores is defined by the vibration transmission loss τ as:

[0093]

[0094] Among them, w in 、w out are the amplitudes of the beams on the input side and the output side respectively;

[0095] As can be seen from Equation (38), the amplitudes \(w_1\) and \(w_2\) at the right ends \(O_2\) and \(O_4\) of the sandwich beam are as follows:

[0096]

[0097] Then, the vibration transmission loss \(\tau_1\) of the upper beam is \(\tau_1 = 20\log(|w_1| / 1)\), and the vibration transmission loss of the lower beam is \(\tau_2 = 20\log(|w_2| / 1)\).

[0098] For the analysis method of the vibration characteristics of the acoustic black hole sandwich beam, it is assumed that one end of the sandwich beam is the unit displacement excitation end and the other end is the free end. Combining the transmission matrix and reflection matrix of the acoustic black hole sandwich beam obtained by the foregoing method, the vibration transmission loss of the acoustic black hole sandwich beam is calculated according to the foregoing method, and its vibration characteristics are analyzed.

[0099] Advantages of the present invention:

[0100] The acoustic black hole sandwich beam structure proposed by the present invention has the following characteristics: 1) The core layer of the sandwich beam is a discontinuous core body, which is arranged periodically. Therefore, the acoustic black holes embedded in the corresponding uncoupled sections are also arranged periodically. 2) The force exerted by the core body of the sandwich beam on the upper and lower beams is a linearly viscoelastic distributed coupling force. Therefore, there is a coupling relationship between the vibration responses of the upper and lower beams. 3) For the uncoupled section of the sandwich beam, it includes the upper beam and the lower beam, and the acoustic black hole structure is embedded in the beam with a larger bending stiffness among them.

[0101] The proposed acoustic black hole sandwich beam structure has the characteristics of light weight, simple structure, and easy processing, and integrates various bending wave attenuation mechanisms such as the acoustic black hole effect, the periodic structure effect, and the vibration coupling effect of the upper and lower beams, and can achieve excellent low-frequency broadband vibration attenuation characteristics. Through the design of parameters such as the core body parameters and the acoustic black hole size, the regulation of the bending wave attenuation band can also be realized. This structure provides a new solution for the vibration attenuation and suppression of the sandwich beam.

[0102] The present invention:

[0103] (1) A structural scheme of an acoustic black hole sandwich beam is proposed, and a power-law function of the acoustic black hole thickness of the uncoupled section is designed.

[0104] (2) An analysis method for the vibration characteristics of the acoustic black hole sandwich beam is proposed. The specific steps are as follows: 1) For the coupled section of the sandwich beam with a discontinuous core body, its transmission matrix \(T\) and reflection matrix \(R\) are established; 2) The transmission matrix \(T\) of the structure containing one acoustic black hole is established by using the iterative relationship of waves ABH and reflection matrix \(R\) ABH ; 3) Combining the transmission matrix \(T\) ABH of the upper beam embedded with the acoustic black hole and the reflection matrix \(R\) ABH, the propagation matrix F of the uniform lower beam, and the reflection matrix R and transmission matrix T of the coupling section, to establish the transmission matrix and reflection matrix of the structure with two cores; 4) Through the iterative relationship, obtain the transmission matrix and reflection matrix of the structure with N cores; 5) Assume that one end of the structure is the unit displacement excitation end and the other end is the free end. Combine the transmission matrix and reflection matrix obtained in the previous step to calculate the vibration transmission loss of the designed structure and analyze its vibration characteristics. Compare the vibration transmission loss of the acoustic black hole sandwich beam with that of the sandwich beam with a uniform structure of the upper and lower beams to illustrate that the acoustic black hole sandwich beam proposed in the present invention has the characteristics of low-frequency broadband bending wave attenuation. Description of the Drawings

[0105] Figure 1 is the structural schematic diagram of the embedded acoustic black hole panel sandwich beam;

[0106] Figure 2 is the schematic diagram of the acoustic black hole structure of the discontinuous core sandwich beam;

[0107] Figure 3 is the bending wave propagation relationship in the local structure of the sandwich beam;

[0108] Figure 4 is the bending wave relationship between micro-segments of the acoustic black hole;

[0109] Figure 5 is the bending wave propagation relationship of the acoustic black hole sandwich beam with two cores;

[0110] Figure 6 is the bending wave transmission relationship of the sandwich beam under boundary and excitation conditions;

[0111] Figure 7 is the vibration transmission loss comparison curve. Detailed Implementation Manner

[0112] The present invention will be described in detail below with reference to the embodiments.

[0113] 1 Structural Scheme of the Acoustic Black Hole Sandwich Beam

[0114] The proposed acoustic black hole sandwich beam structure is as Figure 1 shown, which includes upper and lower beams and an intermediate dispersed viscoelastic core. The unit mass of the core can be ignored compared to the mass of the upper and lower beams. The part between the upper and lower beams containing the core is called the coupling part of the sandwich beam, and the corresponding length of this part is denoted as L1. The coupling effect of the core on the upper and lower beams is considered in this section. The part without the core is called the non-coupling part, and the length of this part is L2. In the non-coupling section of the sandwich beam, the upper beam with a larger bending stiffness is embedded with an acoustic black hole.

[0115] As Figure 2 shown, the thickness h(x) of the embedded acoustic black hole changes to satisfy the power-law relationship:

[0116] h(x) = ε0x m + h0, m ≥ 2 (1)

[0117] Where ε0 is a constant coefficient, m is the power-law exponent, h0 is the truncation thickness, and x is the distance from the central axis of symmetry of the relative acoustic black hole. When x = ±L2 / 2, h(x) = h u .

[0118] 2 Vibration analysis method of acoustic black hole sandwich beam

[0119] 2.1 Dynamic equations of coupling section and non-coupling section

[0120] The coupling section is subjected to the force of the core. Here, since the core is a discrete structure, based on the Kelvin linear viscoelastic model, it is modeled as a distributed spring-damping force model. Then the displacements w u (x, t), w d (x, t) of the upper and lower beams in the coupling section satisfy the following equations:

[0121]

[0122] Where E u , E d are the Young's moduli of the upper or lower beam in the coupling section, I u , I d are the moments of inertia of the upper or lower beam in the coupling section, ρ u , ρ d are the material densities of the upper or lower beam in the coupling section, A u , A d are the cross-sectional areas of the upper or lower beam in the coupling section, w u (x, t), w d (x, t) are the amplitudes of the upper or lower beam in the coupling section. E is the complex Young's modulus of the core, E = E0(1 + jη), E0 is the pure Young's modulus, and η is the damping dissipation factor of the core material, representing the damping dissipation ability of the material; b and h are the width and thickness of the core respectively.

[0123] Separate the time variable, and assume the solution of the equation is w u (x, t) = w1(x)e iωt , w d (x, t) = w2(x)e iωt , and we can get:

[0124]

[0125] Where a1, a2, a3, a4, b1, b2, b3, b4, c1, c2, c3, c4, d1, d2, d3, d4 are the coefficients of each term,

[0126] As can be seen from Equation (3), the transverse vibration responses of the upper and lower beams in the coupled beam can be regarded as the motion of flexural waves with two wave numbers k1 and k2.

[0127] For the uncoupled section, first assume that both the upper and lower beams are homogeneous structures, and their free vibration equations are both satisfied:

[0128]

[0129] Among them, E 0u / 0d is the Young's modulus of the upper or lower beam in the uncoupled section respectively, and I 0u / 0d is the moment of inertia of the upper or lower beam in the uncoupled section respectively, ρ 0u / 0d is the material density of the upper or lower beam in the uncoupled section respectively, A 0u / 0d is the cross-sectional area of the upper or lower beam in the uncoupled section respectively, and w 0u / 0d (x, t) is the amplitude of the upper or lower beam in the uncoupled section respectively.

[0130] By separating the time variable, the form of the steady-state solution of the amplitude of Equation (5) can be written as:

[0131]

[0132] Among them, k 0u / 0d is the natural wave number of the upper or lower beam in the uncoupled section respectively.

[0133]

[0134] 2.2 Propagation relationship of flexural waves in the coupled section

[0135] According to the wave vector method theory, the vibration amplitude-frequency response of the beam can be regarded as the motion of forward and backward waves. As can be seen from Equation (5), for the flexural wave in the upper beam of the uncoupled section, it can be regarded as a forward wave 0u with a wave number of k and a backward wave The flexural wave in the lower beam of the uncoupled section can be regarded as a forward wave 0d with a wave number of k and a backward wave Their wave vectors are respectively:

[0136]

[0137] As can be seen from Equation (3), the flexural waves on the coupled section AB beam are regarded as two forward waves with wave numbers of k1 and k2 respectively and two backward waves

[0138]

[0139] Similarly, according to the transverse vibration amplitude-frequency response of the CD section of the coupled beam, the flexural waves with wave numbers k1 and k2 are respectively α1 and α2 times the corresponding wave numbers on the AB beam. At this time, the forward wave and the backward wave are respectively:

[0140]

[0141] For the local structure with one core in the sandwich beam, such as Figure 3 As shown, assuming that this structure is an infinitely long structure, the flexural wave is incident from section A, and the junction points A, B, C, and D of the coupled section and the uncoupled section are discontinuous characteristic points, where the flexural wave will produce reflection and transmission phenomena. Due to the characteristics of its infinite structure, for the uncoupled section, the incident wave only reflects at A, while there are only transmitted waves and no reflected waves at B, C, and D.

[0142] When the flexural wave propagates on the upper and lower beams of the coupled section, the transfer relationships of the forward wave and the backward wave with each wave number satisfy:

[0143] η (x)j + =f j (x)η Aj + , ξ(x) j + =f j (x)ξC j + η (x)j － =f j -1 (x)η Aj － , ξ(x) j － =f j -1 (x)ξC j － (10)

[0144] where j = 1, 2, are the transfer matrices of the flexural waves with wave numbers k1 and k2 in the upper and lower coupled beams respectively.

[0145] In the uncoupled upper and lower beams, the transfer matrices of the waves are respectively:

[0146]

[0147] According to the displacement and rotation angle continuity conditions and the shear force and bending moment equilibrium equations at the discontinuous characteristic points, on both sides of each node, the flexural wave satisfies the following conditions:

[0148]

[0149] Among them, the subscripts f and r respectively represent the left and right sides of the discontinuous feature points, E is the Young's modulus, I is the moment of inertia, ρ is the material density, A is the cross-sectional area, and w is the amplitude.

[0150] Combined with Equation (12) and Figure 3 it can be obtained that the bending waves at the discontinuous features A, B, C, and D satisfy the relationship:

[0151]

[0152] Among them,

[0153] It is assumed that the forward wave and the backward wave at the discontinuous feature point A in the coupling section satisfy the matrix relationship:

[0154]

[0155] From Equations (13) and (14), it can be obtained that:

[0156]

[0157] The matrices in Equation (15) are shown in detail in Appendix A.

[0158] It is assumed that the reflection matrix of the incident wave at the discontinuous feature point A is R A , and the transmission matrices of the incident wave transmitted to B, C, and D are T B , T C , T D respectively. Then, the bending wave propagation of the upper and lower beams in the uncoupled section satisfies the relationship:

[0159]

[0160] From Equations (13), (15), and (16), it can be obtained that:

[0161]

[0162] From Equation (17), the reflection matrix R A of the bending wave incident from end A and the transmission matrices T B , T C , T D can be obtained. From this relationship, the backward wave at A and the transmitted waves at B, C, and D can be obtained. Similarly, the reflection matrix and the transmission matrix when the bending wave is incident from B, C, and D can be obtained.

[0163] 2.3 Bending Wave Propagation Model of Acoustic Black Hole Beam in Uncoupled Section

[0164] When an acoustic black hole is embedded in the non-coupled section, the beam part of the acoustic black hole can be divided into N micro-segments, and the junction of each micro-segment is a discontinuous point. As Figure 4 shown, take the i-th discontinuous point for analysis, then the amplitude responses on both sides of this point can be written as:

[0165]

[0166] In the formula, p1, p2, p3, p4 and q1, q2, q3, q4 are the coefficient items of the amplitude responses of the two adjacent micro-segments at the discontinuous point; the natural wave numbers k 01 、k 02 of the left and right micro-segment beams at the discontinuous point i are respectively:

[0167]

[0168] The displacement continuity, rotation continuity, shear force balance and bending moment balance conditions in Equation (12) are satisfied at the junction of each micro-segment.

[0169] According to the wave vector method, the forward wave on the left side at the i-th discontinuous characteristic point is The reverse wave is The forward wave on the right side is The reverse wave Then, from Equation (12), we can get:

[0170]

[0171] Among them, the coefficient α = k 02 / k 01 , β = EI 01 k 02 2 / (EI 02 k 01 2 ).

[0172] If the beam under study is considered as a finite beam, then there are reflected waves and transmitted waves on both sides, and the reflection matrix and transfer matrix are:

[0173]

[0174] From the above formula, the propagation relationship of waves between the (i - 1)-th section and the i-th section can be obtained as:

[0175]

[0176] Among them: f n is the transfer matrix of the n-th micro-segment between the (i - 1)-th section and the i-th section.

[0177] After arrangement, the transmission and reflection matrices of the bending wave from the (i - 1)-th section to the i-th section can be obtained as:

[0178]

[0179] By analogy, the recurrence formula can be obtained as follows:

[0180]

[0181] where

[0182] Then, the forward and backward equivalent reflection and transmission matrices from the 1st to the i-th variable cross-section are:

[0183]

[0184] By calculating from the 1st variable cross-section to the last variable cross-section of the acoustical black hole after segmented processing according to Equation (25), the forward transmission matrix of the flexural wave in an uncoupled section of an embedded acoustical black hole can be obtained Reflection matrix and the backward transmission matrix Reflection matrix Since the designed acoustical black hole has a symmetric structure, so

[0185] 2.4 Vibration Propagation Model of Acoustical Black Hole Sandwich Beam

[0186] First, the flexural wave propagation characteristics of a sandwich beam with two cores are analyzed. It is assumed that the flexural wave is incident from sections A and C, and its transmission and reflection relationships are as Figure 5 shown.

[0187] At the discontinuous characteristic points A1 and C1, the relationships are satisfied:

[0188]

[0189] where, R A1 , R C1 are the reflection matrices of the flexural wave at A1 and C1 respectively, T C1A1 , T A1CA are the transmission matrices of the flexural wave transmitted from C1 to A1 and from A1 to C1 respectively, and O is the zero matrix. These matrices can be obtained through Equation (17).

[0190] From Equation (26), it can be obtained that:

[0191]

[0192] where, I is the identity matrix.

[0193] At the discontinuous characteristic points B and D, the flexural wave satisfies the relationships:

[0194]

[0195] Among them, R B , R D are respectively the reflection matrices of the flexural wave at B and D, and T xy are respectively the transmission matrices of the flexural wave transmitted from x to y. These matrices can be obtained through Equation (17).

[0196] The flexural waves at the discontinuous feature points B1 and D1 satisfy the relationship:

[0197]

[0198] The discontinuous feature points A and C satisfy the relationship:

[0199]

[0200] From Equations (26)-(30), the equivalent transmission matrix and the reflection matrix of the flexural wave in the sandwich beam structure of the acoustic black hole between two cores can be obtained as:

[0201]

[0202] For the sandwich beam of the acoustic black hole with N cores, the above method can be used for iterative calculation to obtain the equivalent reflection matrix and the transmission matrix

[0203]

[0204] Among them

[0205] 2.5 Vibration Transmission Loss Model

[0206] Assume that the A and C ends of the studied sandwich beam of the acoustic black hole are subjected to unit displacement excitation, and the B and D ends are free ends. For the convenience of research, extend O1A, BO2, O3C, and DO4 at both ends here, and set their lengths to 0. The number of cores in the sandwich beam is 10, and the number of acoustic black holes in the uncoupled section is 9, as Figure 6 shown.

[0207] The rotation angles at the displacement excitation ends O1 and O3 satisfy the relationship θ = 0. From this, the propagation relationship of the flexural wave can be obtained as:

[0208]

[0209] For the free-end boundary condition, from the bending moment M = 0 and the shear force F s = 0, the reflection matrix R is obtained as:

[0210]

[0211] Thus, the bending wave transmission relationships at the right ends O2 and O4 of the studied beam are satisfied as follows:

[0212]

[0213] Among them,

[0214] In the continuous parts O1A, BO2, O3C, and DO4 of the sandwich beam, the bending wave transmission relationships are satisfied as follows:

[0215]

[0216] Among them, F is the identity matrix.

[0217] According to the analysis of the bending wave transmission relationship of the infinitely long periodic discontinuous core, the bending wave transmission relationships of the finite-length AB and CD segments can be further obtained as:

[0218]

[0219] It is assumed that from Eqs. (33)-(37), it can be obtained that:

[0220]

[0221] Among them,

[0222] In order to quantify the energy transmission process of the bending wave in the sandwich beam, the vibration energy attenuation characteristics after passing through N cores are analyzed by the vibration transmission loss τ, and its definition is:

[0223]

[0224] Among them, w in and w out are the amplitudes of the beams on the input side and the output side, respectively.

[0225] It can be seen from Eq. (38) that the amplitudes w1 and w2 at the right ends O2 and O4 of the sandwich beam are:

[0226]

[0227] Then, the vibration transmission loss of the upper beam τ1 = 20log(|w1| / 1), and the vibration transmission loss of the lower beam is τ2 = 20log(|w2| / 1).

[0228] 3 Vibration characteristic analysis

[0229] According to the above vibration analysis method, the vibration transmission losses of the sandwich beam with uncoupled upper and lower beams embedded with acoustic black holes and the sandwich beam with uniform upper and lower beams are compared respectively, as Figure 7 shown. It can be seen from Figure 7 that when the uncoupled section of the upper beam with a large bending stiffness is embedded with an acoustic black hole, the vibration attenuation characteristics of low-frequency broadband can be achieved. From the results of the above analytical method, it can be seen that the designed structure has good low-frequency vibration attenuation performance.

[0230] Appendix A

[0231]

[0232]

Claims

1. Method for constructing bending wave propagation relationship of a coupling section in a sandwich beam, the sandwich beam is composed of an upper beam, a lower beam and a discrete core, and the sandwich beam includes a coupling section with a core and a non-coupling section without a core; characterized in that: The method for constructing the bending wave propagation relationship of the coupling section includes the following steps: a) Establish the dynamic equation of the coupling section The coupling section is subjected to the force of the core body. Based on the Kelvin linear viscoelastic model, it is modeled as a distributed spring-damping force model. Then, the displacements w u (x, t) and w d (x, t) satisfy the following equation: Among them, E u and E d are the Young's moduli of the upper and lower beams of the coupling section, I u and I d are the moments of inertia of the upper and lower beams of the coupling section, ρ u and ρ d are the material densities of the upper and lower beams of the coupling section, A u and A d are the cross-sectional areas of the upper and lower beams of the coupling section, w u (x, t) and w d (x, t) are the amplitudes of the upper and lower beams of the coupling section; E is the complex Young's modulus of the core, E = E0(1 + jη), E0 is the pure Young's modulus, η is the damping dissipation factor of the core material, representing the damping dissipation ability of the material; b and h are the width and thickness of the core respectively; x is the position coordinate of a point on the beam along the beam length direction; Separate the time variable t and assume the solution of the equation is w u (x, t) = w1(x)e iωt , w d (x, t) = w2(x)e iωt , where ω is the angular frequency, we can obtain: wherein, a1, a2, a3, a4, b1, b2, b3, b4, c1, c2, c3, c4, d1, d2, d3, d4 are coefficients of respective terms, As can be seen from Equation (3), for the transverse vibration responses of the upper and lower beams in the coupled beam, they can be regarded as the motions of bending waves with two wave numbers k1 and k2. b) Analyze the bending wave propagation relationship of the coupling section The junction points A, B, C, and D of the left and right ends of the core in the coupling section with the upper and lower beams are discontinuous characteristic points. As can be seen from Equation (3), the flexural waves on the AB beam in the coupling section are regarded as two forward waves with wave numbers k1 and k2 respectively and two backward waves which are respectively Similarly, according to the transverse vibration amplitude-frequency response of the CD section of the coupled beam, the bending waves with wave numbers k1 and k2 are respectively α1 and α2 times the corresponding wave numbers on the AB beam. At this time, its forward wave and backward wave are respectively: For the local structure with one core in the sandwich beam, assuming this structure is an infinitely long structure, the bending wave is incident from section A, and the junction points A, B, C, and D of the coupling section and the non-coupling section are discontinuous characteristic points, where the bending wave will produce reflection and transmission phenomena; due to the characteristics of its infinite structure, for the non-coupling section part, the incident wave only reflects at A, and there are only transmitted waves without reflected waves at B, C, and D. When the bending wave propagates on the upper and lower beams of the coupling section, the transfer relationships of the forward wave and backward wave of each wave number satisfy: where j = 1, 2, are the transfer matrices of the flexural waves with wave numbers k1 and k2 propagating in the upper and lower coupled beams, respectively; Suppose the reflection matrix of the incident wave at the discontinuous feature point A is R A , and the transmission matrices of the incident wave transmitted to points B, C, and D are T B , T C , and T D respectively. Then, the bending wave propagation of the upper and lower beams on the uncoupled section satisfies the relationship: The reflection matrix R when the flexural wave is incident from end A can be obtained from Equation (17). A and the transmission matrix T B 、T C 、T D From this relationship, the backward wave at A and the transmitted waves at B, C, and D Similarly, the reflection matrix and transmission matrix when the flexural wave is incident from B, C, and D can be obtained.

2. Acoustic black hole sandwich beam with low-frequency broadband vibration attenuation characteristics, the sandwich beam is composed of an upper beam, a lower beam and a discrete core, and the sandwich beam includes a coupled section with a core and an uncoupled section without a core; the thickness of the upper beam in the coupled section is h u , the length of any uncoupled section is L2, and its characteristics are: in The upper beam in the non-coupling section is embedded with an acoustic black hole; the thickness h(x) of any acoustic black hole changes according to a power law relationship: h(x) = ε0x m + h0, m ≥ 2 (1) where ε0 is a constant coefficient, m is the power-law exponent, h0 is the truncation thickness, and x is the distance relative to the central symmetry axis of the acoustic black hole; when x = ±L2 / 2, h(x) = h u .

3. The method for constructing the bending wave propagation relationship of the acoustic black hole sandwich beam according to claim 2, characterized in that: Including the following steps: 1) Use the method of claim 1 to construct the bending wave propagation relationship of a coupling section to obtain its transmission matrix T and reflection matrix R; 2) Establish the transmission matrix \(T\) and reflection matrix \(R\) of the upper beam with an acoustic black hole structure using the iterative relationship of waves ABH and ABH ; 3) Combine the transmission matrix T of the upper beam embedded with an acoustic black hole ABH and the reflection matrix R ABH , the propagation matrix F of the uniform lower beam, and the reflection matrix R and transmission matrix T of the coupling section to establish the transmission matrix and reflection matrix of a structure with two cores; 4) Through the iterative relationship, obtain the transmission matrix and reflection matrix of the sandwich beam with acoustic black holes containing N cores.

4. The method for constructing the bending wave propagation relationship of the sandwich beam with an acoustic black hole according to claim 3, characterized in that: The specific process of step 2) is as follows: When an acoustic black hole is embedded in the non-coupling section, the beam part of the acoustic black hole can be divided into N micro-sections, and the junction of each micro-section is a discontinuous point. Take the i-th discontinuous point for analysis, then the amplitude responses on both sides of this point can be written as: wherein, p1, p2, p3, p4 and q1, q2, q3, q4 are respectively the amplitude response coefficient items of two adjacent micro-segments at the discontinuous point; the natural wave numbers k 01 and k 02 of the left and right micro-segment beams at the discontinuous point i are respectively: According to the wave vector method, the forward wave on the left side at the i-th discontinuous feature point is The reverse wave is The right forward wave is The reverse wave Wherein: f n is the transfer matrix of the nth micro-segment between the (i - 1)th cross-section and the ith cross-section; the forward wave on the left side at the (i - 1)th discontinuous feature point is The reverse wave is The forward wave on the right side is The reverse wave is After arrangement, the transmission and reflection matrices of the flexural wave from the (i - 1)th cross-section to the ith cross-section are obtained as follows: And so on, the recurrence formula can be obtained as: Then the forward and backward equivalent reflection and transmission matrices from the 1st to the i-th variable cross-section are: By calculating from the first variable cross-section to the last variable cross-section of the acoustical black hole after segmented processing using Equation (25), the forward transmission matrix of the flexural wave in an uncoupled section of the embedded acoustical black hole can be obtained. Reflection matrix And the backward transmission matrix Reflection matrix Since the acoustical black hole has a symmetric structure, so 5. The method for constructing the bending wave propagation relationship of the acoustic black hole sandwich beam according to claim 3, characterized in that: The specific process of step 3) is as follows: First, analyze the bending wave propagation characteristics of the sandwich beam with two cores. The junction points A, B, C, D, A1, B1, C1, and D1 of the left and right ends of the 1st and 2nd cores from left to right with the upper and lower beams are discontinuous characteristic points. The B A1 section is the upper beam of the non-coupling section of the sandwich beam with an acoustic black hole, and DC1 is the uniform lower beam of the non-coupling section of the sandwich beam. Assume the bending wave is incident from sections A and C. The relationship is satisfied at the discontinuous characteristic points A1 and C1: Among them, the incident waves at points A and C are The reflected waves at points A and C are The forward waves at points B and D are The backward waves at points B and D are The forward waves at points A1 and C1 are The backward waves at points A1 and C1 are The transmitted waves at points B1 and D1 are R A1 and R C1 are the reflection matrices of the flexural wave at A1 and C1 respectively, and T C1A1 and T A1CA are the transmission matrices of the flexural wave transmitted from C1 to A1 and from A1 to C1 respectively, and O is the zero matrix; these matrices can be obtained through Equation (17); It can be obtained from Equation (26): Among them, I is the identity matrix; The bending wave satisfies the relationship at the discontinuous characteristic points B and D: wherein, R B , R D are respectively the reflection matrices of the flexural wave at B and D, and T xy are respectively the transmission matrices of the flexural wave transmitted from x to y, and these matrices can be obtained through Equation (17); The bending wave satisfies the relationship at the discontinuous characteristic points B1 and D1: The relationship is satisfied at the discontinuous characteristic points A and C: The equivalent transmission matrix of flexural waves in the sandwich beam structure of acoustic black holes in two cores can be obtained from Eqs. (26)-(30). and the reflection matrix are as follows: For the acoustic black hole sandwich beam with N cores, the above method can be used for iterative calculation to obtain the equivalent reflection matrix when the flexural wave is incident from ends A and C and the transmission matrix Among them 6. Method for calculating the vibration transmission loss of a finite periodic sandwich beam, characterized in that: Assume that the sandwich beam contains N cores, and the left ends A and C are subjected to unit displacement excitation, while the right ends B and D are free ends; extend O1A, BO2, O3C, and DO4 at both ends respectively, and set their lengths to 0. At this time, it can be regarded that the unit excitation is applied at O1 and O3, and the right ends O2 and O4 are free ends. The incident waves at points O1 and O3 are The reflected waves at points O1 and O3 are The forward waves at points A and C are The backward waves at points A and C are The forward waves at points B and D are The backward waves at points B and D are The forward waves at points O2 and O4 are The backward waves at points O2 and O4 are Since the displacement excitation ends O1 and O3 are subjected to a unit displacement excitation, the rotation angle here satisfies the relationship θ = 0, and the propagation relationship of the bending wave can be obtained as: For the free-end boundary condition, from the bending moment M = 0 and the shear force F s = 0, the reflection matrix R can be obtained as follows: From this, the bending wave transmission relationship at the right ends O2 and O4 of the beam under study can be obtained as: Among them, In the continuous parts O1A, BO2, O3C, and DO4 of the sandwich beam, the transmission relationship of the bending wave satisfies: where F is the identity matrix; Based on the analysis of the bending wave transmission relationship of the infinitely long periodic discontinuous cores, the bending wave transmission relationship of the finite length AB and CD segments can be further obtained as: Assume that it can be obtained from equations (33)-(37): Among them, Define the vibration energy attenuation characteristic of passing through N cores by the vibration transmission loss τ as: where, w in and w out are the amplitudes of the beams on the input side and the output side, respectively; From equation (38), the amplitudes w1 and w2 at the right ends O2 and O4 of the sandwich beam are: Then the vibration transmission loss of the upper beam τ1 = 20log(|w1| / 1), and the vibration transmission loss of the lower beam is τ2 = 20log(|w2| / 1).

7. The method for analyzing the vibration characteristics of the acoustic black hole sandwich beam according to claim 2, characterized in that: Assume that one end of the sandwich beam is a unit displacement excitation end and the other end is a free end. Combine the transmission matrix and reflection matrix of the acoustic black hole sandwich beam obtained from claim 3, and calculate the vibration transmission loss of the acoustic black hole sandwich beam according to the method of claim 6 to analyze its vibration characteristics.

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