Generalized vibration model for analyzing vibration characteristics of discontinuous core sandwich beam and vibration transmission loss calculation method

Abstract

The invention provides a generalized vibration model for analyzing vibration characteristics of a discontinuous core sandwich beam and a vibration transmission loss calculation method. The generalized vibration model and the vibration transmission loss calculation method have universality and expandability. According to the generalized vibration model, the interlayer beam is composed of an upper beam, a lower beam and discrete cores, and the interlayer beam comprises a coupling section containing the cores and a non-coupling section not containing the cores; when N cores are contained in the discontinuous core interlayer beam, N is larger than or equal to 2, junction points A, B, C and D of the left end and the right end of the first core and the upper beam and the lower beam from left to right are discontinuous feature points, the A end and the C end are incident ends of bending waves, and reflection matrixes # imgabs0 # of the bending waves at the incident ends and transmission matrixes # imgabs2 # # imgabs3 # # imgabs4 # of the bending waves from left end to right are that the discontinuous core interlayer beam contains N-1 cores. A reflection matrix of the bending waves at the incident end and a transmission matrix of the bending waves from the left end to the right end.

Description

Technical Field

[0001] The invention relates to a vibration model for accurately analyzing the vibration propagation characteristics of a non-continuous core sandwich beam and a vibration transmission loss calculation method. Background Art

[0002] Sandwich structures usually consist of two layers of panels, upper and lower, and a middle core layer. The core layer structure is diverse and can be a continuous material, such as a highly damped rubber layer, or a lightweight pine wood, foam layer, etc. It can also be a discontinuous core layer, such as a honeycomb, corrugated, truss structure, etc., or a discrete core unit. Sandwich structures are often subjected to dynamic loads in actual engineering applications, which can easily cause structural vibrations, leading to premature fatigue damage of the structure, reduced product quality and production efficiency, and in severe cases, major production accidents. Therefore, with the increasing requirements for vibration reduction and isolation performance of engineering structures, the design of sandwich structures and the study of their dynamic vibration models have become a research hotspot in the engineering field.

[0003] Domestic and foreign scholars have conducted extensive research on the vibration characteristics of traditional continuous core sandwich beams and formed a relatively mature theoretical model. However, for non-continuous core sandwich beams, such as honeycomb cores and periodic cores, the vibration characteristics analysis is more complicated due to the complexity of their core structures. Researchers usually use the equivalent elastic constant method to equate the non-continuous core to a homogeneous continuous medium, and combine it with the anisotropic sandwich theory of continuous core sandwich beams for analysis. Although this method simplifies the dynamic analysis of complex cores, it is relatively rough in dealing with local structures. In addition, some scholars have explored direct analytical methods for analyzing the vibration characteristics of simple core structures (such as honeycombs, orthogonal reinforcements, etc.). This method fully considers the dynamic relationship between the panel 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 fine. Related studies have shown that the above research methods cannot clearly describe the propagation relationship of bending waves in sandwich beams, as well as the contribution of each component in the sandwich beam to the vibration response, and therefore cannot effectively achieve vibration control of sandwich beams. In addition, there is currently no universal vibration analysis model for vibration analysis of non-continuous core sandwich beams. Detailed and complex derivation analysis must be performed for specific sandwich beam structures, which is very inconvenient for the design and optimization of sandwich beam structures. Summary of the invention

[0004] The purpose of the present invention is to provide a generalized vibration model for the vibration characteristics analysis of non-continuous core sandwich beams, which has universality and scalability, provides an effective theoretical tool for the design optimization and bending wave control analysis of sandwich beam structures, and has important practical value in the field of engineering applications.

[0005] To achieve the above object, the present invention adopts the following technical solutions:

[0006] A generalized vibration model for analyzing the vibration characteristics of a discontinuous core sandwich beam, the sandwich beam being composed of an upper beam, a lower beam and discrete cores, the sandwich beam including a coupled section with cores and an uncoupled section without cores;

[0007] When the discontinuous core sandwich beam contains N cores, N≥2, the junction points A, B, C, D of the left and right ends of the first core from left to right with the upper and lower beams are discontinuous characteristic points, and the A and C ends are the incident ends of the bending wave. The reflection matrix of the bending wave at the incident end and the transmission matrix of the bending wave from the left end to the right end are:

[0008]

[0009] wherein, are respectively the reflection matrix of the bending wave at the incident end and the transmission matrix of the bending wave from the left end to the right end when the discontinuous core sandwich beam contains N - 1 cores, and I is the identity matrix;

[0010] R and T are respectively the reflection matrix of the bending wave at the incident end and the transmission matrix of the bending wave from the left end to the right end when the discontinuous core sandwich beam contains 1 core;

[0011]

[0012] R x is the reflection matrix at the discontinuous characteristic point x, and T yz is the transmission matrix of the bending wave incident from the discontinuous characteristic point y to the discontinuous characteristic point z, and x, y, z are arbitrary discontinuous characteristic points;

[0013]

[0014] k 0u and k 0d are respectively the natural wave numbers of the upper and lower beams in the uncoupled section, E 0u / 0d are respectively the Young's moduli of the upper or lower beam in the uncoupled section, I 0u / 0d are respectively the moments of inertia of the upper or lower beam in the uncoupled section, ρ 0u / 0d are respectively the material densities of the upper or lower beam in the uncoupled section, A 0u / 0d are respectively the cross-sectional areas of the upper or lower beam in the uncoupled section; L2 is the length of an uncoupled section.

[0015] The above-mentioned generalized vibration model for analyzing the vibration characteristics of a discontinuous core sandwich beam

[0016] The discontinuous core sandwich beam contains 2 cores. The left and right ends of the 1st and 2nd cores from left to right and the intersection points A, B, C, D, A1, B1, C1, D1 of the upper and lower beams are discontinuous characteristic points.

[0017] The transmission matrix of the bending wave transmitted from points A and C to points B1 and D1 is

[0018]

[0019] The transmission matrix of the bending wave transmitted from points A and C to points A1 and C1 is:

[0020]

[0021] The above-mentioned generalized vibration model for analyzing the vibration characteristics of the discontinuous core sandwich beam

[0022] The discontinuous core sandwich beam contains 3 cores. The left and right ends of the 1st, 2nd, and 3rd cores from left to right and the intersection points A, B, C, D, A1, B1, C1, D1, A2, B2, C2, D2 of the upper and lower beams are discontinuous characteristic points.

[0023]

[0024] The above-mentioned generalized vibration model for analyzing the vibration characteristics of the discontinuous core sandwich beam

[0025] The steady-state solutions of the amplitudes of the upper and lower beams in the uncoupled section, w 0u (x), w 0d (x) are:

[0026]

[0027] c 01 、c 02 、c 03 、c 04 、d 01 、d 02 、d 03 、d 04 are the coefficients of each term.

[0028] The above-mentioned generalized vibration model for analyzing the vibration characteristics of the discontinuous core sandwich beam

[0029] The amplitudes of the upper and lower beams in the coupled section, w u (x,t) = w1(x)e iωt ,w d (x,t) = w2(x)e iωt ,

[0030] The transverse vibration response of the coupled beam is the motion of flexural waves with two wave numbers k1 and k2,

[0031]

[0032] where a1, a2, a3, a4, b1, b2, b3, b4, c1, c2, c3, c4, d1, d2, d3, d4 are the coefficients of each term.

[0033] The present invention also provides a method for calculating the vibration transmission loss of a finite periodic sandwich beam,

[0034] The sandwich beam contains N cores. The left ends A and C are subjected to unit displacement excitation, and the right ends B and D are free ends; O1A, BO2, O3C and DO4 are extended at both ends, and their lengths are set 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.

[0035] 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 flexural wave can be obtained as:

[0036]

[0037] For the free-end boundary condition, from the bending moment M = 0 and the shear force F s = 0, the reflection matrices can be obtained as:

[0038]

[0039] From this, the flexural wave transmission relationship at the right ends O2 and O4 of the sandwich beam satisfies:

[0040]

[0041] where

[0042] In the continuous parts O1A, BO2, O3C and DO4 of the sandwich beam, the flexural wave transmission relationship satisfies:

[0043]

[0044] where F is the identity matrix;

[0045] According to the analysis of the flexural wave transmission relationship of the infinitely long periodic discontinuous cores, the flexural wave transmission relationship of the finite length AB and CD segments is obtained as:

[0046]

[0047] where are the equivalent reflection matrix and transmission matrix of the sandwich beam respectively;

[0048] It can be obtained from equations (45)-(49) that:

[0049]

[0050] where

[0051] The vibration energy attenuation characteristic of passing through N cores is defined by the vibration transmission loss τ as:

[0052]

[0053] where w in , w out are the amplitudes of the beams on the input side and the output side respectively;

[0054] It can be known from equation (50) that the amplitudes w1 and w2 at the right ends O2 and O4 of the sandwich beam are:

[0055]

[0056] Then the transmission loss τ1 of the upper beam = 20log(|w1| / 1), and the transmission loss of the lower beam is τ2 = 20log(|w2| / 1).

[0057] Advantages of the present invention:

[0058] In view of the characteristics of discontinuous core sandwich beams with different structures, the present invention realizes the accurate analysis of the vibration propagation characteristics of specific structures by establishing the transmission matrix and reflection matrix of the local structure and equivalently replacing the transmission matrix, reflection matrix or transfer matrix in the generalized vibration model.

[0059] The derived reflection matrix and transmission matrix of the sandwich beam with N cores belong to a generalized vibration model. This model mainly involves the reflection matrix and transmission matrix of the coupling section and the transfer matrix of the non-coupling section. If a new structural design is carried out on the coupling section or non-coupling based on the structure of the discontinuous core sandwich beam under study, only the reflection matrix and transmission matrix of this part need to be established to replace the corresponding part in the original model, and then the reflection matrix and transmission matrix of the designed discontinuous core sandwich beam can be obtained. Therefore, the established generalized vibration model has good universality and scalability, providing an effective theoretical tool for the design optimization and bending wave regulation analysis of the sandwich beam structure.

[0060] From the frequency-domain curves of the moduli of the elements in the matrix, the reflection and transmission relationships of the energy in the bending wave propagation can be analyzed, and then the attenuation characteristics and bandgap characteristics of the bending wave can be analyzed. On this basis, by adding boundary and external excitation conditions, the analytical solutions of the steady-state amplitude-frequency response and the vibration transmission loss of the upper and lower beams can be further obtained. Description of the Drawings

[0061] Figure 1 It is a structural diagram of a non - continuous discrete core sandwich beam.

[0062] Figure 2 It is a force analysis diagram of a micro - element taken from the coupling section beam.

[0063] Figure 3 It is a diagram of the propagation relationship of bending waves in the local structure of the sandwich beam.

[0064] Figure 4 It is a schematic diagram of the sandwich beam structure considering boundary conditions.

[0065] Figure 5 It is a diagram of the propagation relationship of bending waves in a non - continuous sandwich beam with two cores.

[0066] Figure 6 It is a diagram of the propagation relationship of bending waves in a non - continuous sandwich beam with three cores.

[0067] Figure 7 It is a diagram of the propagation relationship of bending waves in a sandwich beam with a 0.5 - m core.

[0068] Figure 8 It is a diagram of the bending wave transmission relationship of the sandwich beam under boundary and excitation conditions. Detailed Description of the Preferred Embodiments

[0069] The present invention will be described in detail below in conjunction with embodiments.

[0070] 1 Dynamic Modeling of the Local Structure of a Non - continuous Core Sandwich Beam

[0071] The sandwich beam structure under study is a non - continuous core sandwich beam, which is composed of upper and lower beams and discrete cores, as Figure 1 shown. Among them, the structural dynamic modeling of the upper and lower beams is carried out based on the Euler - Bernoulli beam model. The middle core layer of the sandwich beam is composed of discretely arranged cores, and its material is considered as a light - filled viscoelastic material. According to the arrangement and material characteristics of the cores, the transverse shear force is small and not considered here. Instead, the normal force generated by the action of the upper and lower beams on the cores is considered. According to the arrangement of the cores, the sandwich beam can be divided into a part with cores and a part without cores. Among them, in the part with cores, due to the action of the cores, there is a certain coupling relationship between the vibration responses of the upper and lower beams, so it is called the coupling section, such as sections AB and CD in the figure; while the part of the sandwich beam without cores is called the non - coupling section, such as sections O1A, O2B, O3C, and O4D in the figure. The arrangement lengths of the coupling section and the non - coupling beam are L1 and L2 respectively. The thicknesses of the upper and lower beams and the core are h u 、h d and h respectively, and the width of the sandwich beam is b.

[0072] For the upper and lower beams of the uncoupled section, their free vibration equations both satisfy:

[0073]

[0074] Among them, E 0u / 0d is the Young's modulus of the upper or lower beam of the uncoupled section respectively, I 0u / 0d is the moment of inertia of the upper or lower beam of the uncoupled section respectively, ρ 0u / 0d is the material density of the upper or lower beam of the uncoupled section respectively, A 0u / 0d is the cross-sectional area of the upper or lower beam of the uncoupled section respectively, w 0u / 0d (x, t) is the amplitude of the upper or lower beam of the uncoupled section respectively, and x is the position coordinate of a certain point on the beam along the length direction of the beam.

[0075] By separating the time variable t, the steady-state solution of the amplitude of Equation (1), w 0u (x), w 0d (x) can be written in the form of:

[0076]

[0077] Among them, k 0u , k 0d are the natural wave numbers of the upper and lower beams of the uncoupled section respectively, c 01 , c 02 , c 03 , c 04 , d 01 , d 02 , d 03 , d 04 are the coefficients of each term.

[0078]

[0079] ω is the angular frequency;

[0080] For the coupled section of the sandwich beam, such as Figure 1 the AB and CD sections in. Take a differential element dx of the upper beam, and its force analysis is as Figure 2 shown. The dynamic equation of this differential element in the y direction is:

[0081]

[0082] Among them, ρ u is the material density of the upper beam of the coupled section, A u is the cross-sectional area of the upper beam of the coupled section, w u(x, t) is the beam amplitude on the coupling section. Q is the shear force; E is the complex Young's modulus of the core. Considering the viscoelastic properties of the core, based on the Kelvin model, E = E0(1 + jη), where 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 is the width of the core, ε = Δh / h, h is the height of the core, and Δh is the deformation of the core.

[0083] According to the Euler - Bernoulli model, from the moment balance condition of the infinitesimal segment and the relationship between the bending moment M and the deflection curve w, it can be known that:

[0084]

[0085] Among them, E is the Young's modulus of the beam, I is the moment of inertia of the beam, and w is the amplitude of the beam.

[0086] From equations (4) and (5), the vibration equations of the upper and lower beams can be obtained as follows:

[0087]

[0088] In the formula, E u 、E d are the Young's moduli of the upper or lower beam on the coupling section, I u 、I d are the moments of inertia of the upper or lower beam on the coupling section, ρ u 、ρ d are the material densities of the upper or lower beam on the coupling section, A u 、A d are the cross - sectional areas of the upper or lower beam on the coupling section, w u (x, t)、w d (x, t) are the amplitudes of the upper or lower beam on the coupling section.

[0089] Assume the solution of equation (6) is w u (x, t) = w1(x)e iωt , w d (x, t) = w2(x)e iωt , and we can get:

[0090]

[0091] Among them, are the wave numbers of the upper or lower beam on the coupling section respectively.

[0092] Equation (7) is a system of higher - order ordinary differential equations. Assume w1(x) = A1e kx , w2(x) = A2e kx , and we can get:

[0093]

[0094] If the equation has non-trivial solutions, it satisfies:

[0095]

[0096] That is:

[0097]

[0098] Equation (10) is the wave number equation of the coupled beam. Solving this equation gives the wave number k 4 There are two solutions:

[0099]

[0100] From this, we can get:

[0101]

[0102] Where a1, a2, a3, a4, b1, b2, b3, b4, c1, c2, c3, c4, d1, d2, d3, d4 are the coefficients of each term.

[0103] Substituting Equation (12) into Equation (7), we get:

[0104]

[0105] Then the solution of Equation (7) can be written as:

[0106]

[0107] From Equation (14), it can be seen that the transverse vibration response of the coupled beam can be regarded as the motion of bending waves with two wave numbers k1 and k2. When the Young's modulus of the core is 0, there is only k u , k d a kind of wave number motion for the upper and lower beams of the coupled beam respectively. At this time, the upper and lower beams are in an uncoupled state.

[0108] 2 Construction of the Bending Wave Propagation Relationship of the Sandwich Beam Local Structure Based on the Wave Vector Method

[0109] 2.1 Bending Wave Component Analysis of the Sandwich Beam Local Structure

[0110] 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. From Equation (2), it can be seen that for the bending 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 For the bending wave in the lower beam of the uncoupled section, it can be regarded as a forward wave 0d with a wave number of k and a backward wave Their wave vectors are respectively:

[0111]

[0112] As can be seen from Equation (14), the flexural waves on the AB beam of the coupling section are regarded as two forward waves with wave numbers k1 and k2 respectively and two backward waves which are respectively

[0113]

[0114] Similarly, according to the transverse vibration amplitude-frequency response of the CD section of the coupling beam, the flexural waves with wave numbers k1 and k2 are respectively α1 and α2 times of the corresponding wave numbers on the AB beam. At this time, its forward waves and backward waves are respectively

[0115]

[0116] For the local structure with a single core in the sandwich beam, as Figure 3 shown, assuming that the 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 coupling section and the non-coupling 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 non-coupling section, the incident wave is only reflected at A, while there are only transmitted waves at B, C, and D without reflected waves. According to the above analysis, the propagation relationship of this local structure is as Figure 3 shown.

[0117] When the flexural wave propagates on the upper and lower beams of the coupling section, the transfer relationships of the forward and backward waves with each wave number satisfy

[0118]

[0119] where j = 1, 2, are respectively the transfer matrices of the flexural waves with wave numbers k1 and k2 in the upper and lower coupling beams.

[0120] In the non-coupling upper and lower beams, the transfer matrices of the waves are respectively

[0121]

[0122] 2.2 Construction of the wave propagation relationship at the discontinuous characteristic points of the local structure

[0123] Analyze the reflection and transmission relationships at the discontinuous characteristic points A, B, C, and D. According to the displacement and rotation angle continuity conditions and the shear force and bending moment balance equations at the discontinuous characteristic points, on both sides of each node, the flexural wave satisfies the following conditions

[0124]

[0125] Among them, 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.

[0126] Combined with Equation (20) and Figure 3 it can be obtained that the bending waves at the discontinuous features A, B, C, and D satisfy the relationship:

[0127]

[0128] Among them,

[0129] 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 as:

[0130]

[0131] From Equations (21) and (22), it can be obtained that:

[0132]

[0133] The details of each matrix in Equation (23) can be found in Appendix A.

[0134] 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 waves in the uncoupled sections of the upper and lower beams satisfy the relationship:

[0135]

[0136] From Equations (21), (23), and (24), it can be obtained that:

[0137]

[0138] From Equation (25), the reflection matrix R A of the bending wave incident from section 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. Here, the obtained matrices are all 2×2 square matrices. Among them, the first column elements R A in the reflection matrix R A (1,1), R A (2,1) are the reflection coefficients of the forward traveling waves, and the second column elements R A (1,2), R A(2,2) is the reflection coefficient of the forward decaying wave. Similarly, in the transmission matrix at each point, the elements in the first column are the transmission coefficients of the forward traveling wave, and the elements in the second column are the transmission coefficients of the forward decaying wave. Since the energy transfer in the flexural wave is proportional to the wave amplitude, the reflection and transmission relationships of the wave energy in the sandwich beam can be analyzed from the magnitudes of the moduli of the elements.

[0139] The reflection matrix and the transmission matrix when the flexural wave is incident from points B, C, and D can also be obtained by the same method. If the local structure under study is symmetric about the left and right, the reflection matrix and the transmission matrix obtained from the incident wave at point B are the same as those from the incident wave at point A. If it is symmetric about the top and bottom, the reflection matrix and the transmission matrix obtained from the incident wave at point C are the same as those from the incident wave at point A.

[0140] 2.3 Example Verification

[0141] To verify the accuracy of the transmission matrix and the reflection matrix of the flexural wave of the above sandwich beam at the discontinuous point, boundary conditions are added to the structure of the local beam under study, and the structural case in a certain reference is verified and analyzed. The structure is as Figure 4 shown. Among them, points A, B, C, and D are simply supported. For the convenience of research, virtual lengths O1A, O2B, O3C, and O4D are set here, and their lengths are all 0. Then, at this time, the structure can be converted to a simply supported constraint at points O1, O2, O3, and O4, and points A, B, C, and D are discontinuous characteristic points.

[0142] To compare and verify with the first two natural frequencies of the structure calculated in the literature, the natural frequency of the structure is calculated here using Equation (25) and the boundary conditions.

[0143] From the simply supported boundary conditions at points O1, O2, O3, and O4, the reflection matrixes R O1 、R O2 、R O3 、R O4 are as follows:

[0144]

[0145] The wave propagation relationship in the virtual length segments O1A, O2B, O3C, and O4D satisfies:

[0146]

[0147] Among them, f u (0) and f d (0) are identity matrices.

[0148] At the discontinuous characteristic points A, B, C, and D, their reflection and propagation relationships satisfy:

[0149]

[0150] where R x is the reflection matrix at the discontinuous feature point x, and T yz is the transmission matrix of the flexural wave incident from the discontinuous feature point y to the discontinuous feature point z. Here, x, y, and z are any discontinuous feature points A, B, C, and D.

[0151] Combining Eqs. (26)-(28), it can be written in the following matrix form:

[0152] Az = 0 (29)

[0153] where A is a 32×32 coefficient matrix and z is a 32×1 wave vector. When |A| = 0, the natural frequencies can be obtained.

[0154] The verified structural parameters are as follows: The total length of the sandwich beam L = 1, the flexural rigidities of the upper and lower beams E u I u = 1, E d I d = 1, the mass per unit length ρ u A u = 1, ρ d A d = 1, the tensile (compressive) rigidity of the core is Eb / h = 10, and the damping coefficient η = 0.1. Calculated by the wave vector method, the first two natural frequencies are 9.8696 rad / s and 10.8355 rad / s, which are consistent with the results obtained by the modal analysis method in the reference literature.

[0155] Through the verification of the above numerical cases, it is confirmed that the established reflection and transmission relationships of the flexural waves of the local structure are correct, providing a solid theoretical basis for the construction of the generalized vibration model of the sandwich beam with a discontinuous core.

[0156] 3 Construction of the generalized vibration model of the periodic sandwich beam

[0157] 3.1 Construction of the vibration propagation relationship of the sandwich beam with two cores

[0158] To establish the generalized vibration model of the periodic sandwich beam, first, the vibration propagation characteristics of the sandwich beam with a two-core structure are analyzed, as Figure 5 shown. Considering the generality of the model, it is assumed here that the A and C ends are the incident ends of the flexural waves, and the upper and lower beams are infinite beam models. The discontinuous feature points in this model are points A, B, C, D and A1, B1, C1, D1. Among them, the incident waves are reflected at A and C, and transmitted at B1 and D1. Since the beam is an infinite model, there are no reflected waves at both ends of the sandwich beam, while there are reflected and transmitted waves at each discontinuous feature point in the middle.

[0159] For the uncoupled segments BA1 and DC1 of the sandwich beam, the transmission relationship of the flexural wave satisfies:

[0160]

[0161] Among them,

[0162] The propagation relationship of the flexural wave in the coupled segment of the sandwich beam is:

[0163]

[0164] Among them, R x is the reflection matrix at the discontinuous feature point x, and T yz is the transmission matrix of the flexural wave incident from the discontinuous feature point y to the discontinuous feature point z, where x, y, and z are arbitrary discontinuous feature points.

[0165] Combining equations (30) and (31), we can obtain:

[0166]

[0167] That is, the transmission matrix for the wave to transmit from points A and C to A1 and C1 is:

[0168]

[0169] Assume that the transmission matrix for the flexural wave to transmit from points A and C to B1 and D1 is Here, the subscript 2 represents the number of cores contained. Then, combining equation (33) and the propagation relationships of the coupled segments A1B1 and C1D1, we can further obtain:

[0170]

[0171] According to the reflection relationship of the flexural wave, the reflection matrices at points A and C can be obtained as:

[0172]

[0173] Let Then, from equations (30), (32), and (35), the reflection matrix is:

[0174]

[0175] 3.2 Construction of the generalized vibration model of the sandwich beam with N cores

[0176] Based on the analysis of the flexural wave relationship of the sandwich beam with two cores, further consider the vibration propagation relationship of the sandwich beam with N cores. Assume that the number of cores is 3, as Figure 6As shown. From equations (34) and (36), the reflection and transmission relationships between two adjacent cores in this structure can be known. Here, the two core structures on the left can be regarded as a whole, and the third core is iteratively processed with it.

[0177] At this time, the bending waves at the discontinuous feature points B1, D1, A2, C2, B2, and D2 satisfy the relationship:

[0178]

[0179] Among them,

[0180] From equation (37), we can get:

[0181]

[0182] Let Then:

[0183]

[0184] The reflected waves at the discontinuous feature points A and C satisfy the relationship:

[0185]

[0186] From equations (38) and (40), we can get:

[0187]

[0188] Let Then:

[0189]

[0190] When there are N cores in the discontinuous core sandwich beam, according to the above same analysis method, through the iterative relationship, the reflection matrix and the transmission matrix from the left end to the right segment are:

[0191]

[0192] The iterative model established by Equation (43) provides a systematic method for analyzing the propagation of flexural waves, which has good flexibility and scalability. Among them, the transmission matrix T, the reflection matrix R, and the propagation matrix F can be equivalently replaced according to the discontinuous core sandwich beams of different structures. For example, if the designed structure corresponds to the coupled section part, the transmission matrix and the reflection matrix of this coupled section are first derived, and then these matrices are used to replace the transmission matrix T and the reflection matrix R in the generalized vibration model. If the designed structure corresponds to the uncoupled section part, the transmission matrix and the reflection matrix of this uncoupled section are first derived, and they are converted into an equivalent propagation matrix, and then this matrix is used to replace the propagation matrix F in the generalized vibration model. Although this is for periodic structures here, it is also applicable to non-periodic structures. In this case, only the transmission matrix T of each coupled section and uncoupled section needs to be derived i , the reflection matrix R i and the propagation matrix F i , and then iteration is carried out, and the method is the same

[0193] 3.3 Verification of the generalized vibration model and analysis of the flexural wave propagation relationship

[0194] To verify the accuracy of the generalized vibration model established by Equation (43), it is assumed here that a certain section of the sandwich beam is intercepted for analysis. Its length L and the core length L1 are both 0.5 m, and the structure is as shown Figure 7 . The material and dimension parameters of each part are shown in Table 1, where the core dissipation factor η = 0.1. Since the symmetry of the upper and lower beams will cause the modes of the upper and lower beams to be the same under the same input conditions, resulting in no coupling effect on the core, the upper and lower beams are set as asymmetric structures here. Assuming that the discontinuous characteristic points A and C are the incident points of flexural waves, the reflection matrix and the transmission matrix

[0195] Table 1 Structural geometry and material parameters

[0196]

[0197] First, the reflection matrix and the transmission matrix can be obtained from the local structure transfer model Equation (25) as follows

[0198]

[0199] Secondly, iterative calculations are carried out using the generalized vibration model Equation (43). By dividing the sandwich beam into several equally wide coupled sections and setting the length L2 of the uncoupled section to 0, its reflection matrix and the transmission matrices at B and D can be obtained. It is found through verification that It can be seen therefrom the correctness of the established generalized vibration model.

[0200] In this reflection matrix the element represents the reflection relationship generated here by the traveling wave component of the incident wave at A, and represents the relationship of the traveling wave component of incident wave C transmitted to A; similarly, the element in the reflection matrix represents the transmission relationship generated by the traveling wave component of the incident wave at A at C, and represents the reflection relationship of the traveling wave component of incident wave C at C. In the transmission matrix the elements respectively represent the transmission relationships of incident wave A and incident wave C transmitted to B; similarly, the elements respectively represent the transmission relationships of incident wave A and incident wave C transmitted to D. It can be seen therefrom that, from the modulus of each item in the reflection matrix and the transmission matrix, the situation of the flexural wave energy propagating from the incident end to the right-end output end can be further analyzed.

[0201] 4 Calculation method for vibration transmission loss of finite periodic sandwich beams

[0202] Based on this generalized vibration model, if the boundary conditions of the sandwich beam and the external excitation are further considered, the vibration characteristics of the finite periodic sandwich beam can be analyzed. The steps are as follows: according to the boundary characteristics of the upper and lower beams in the finite structure, the reflection matrix at this boundary is obtained from its rotation angle, displacement conditions or mechanical equilibrium conditions. Assume that the studied sandwich beam contains N core bodies, and the A and C ends are subjected to unit displacement excitations, 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 make their lengths 0. Then, at this time, it can be regarded that the unit excitation is applied at O1 and O3, and the right-end O2 and O4 are free ends, as Figure 8 shown.

[0203] Since the displacement excitation ends O1 and O3 are subjected to a unit displacement excitation, the rotation angle here satisfies the relationship θ = 0. From this, the propagation relationship of the flexural wave can be obtained as:

[0204]

[0205] For the free-end boundary condition, from the bending moment M = 0 and the shear force F s = 0, the reflection matrices can be obtained as:

[0206]

[0207] From this, the flexural wave transmission relationships at the right-end O2 and O4 of the studied beam are obtained as:

[0208]

[0209] Among them,

[0210] In the continuous parts O1A, BO2, O3C, and DO4 of the sandwich beam, the transmission relationship of the bending wave satisfies:

[0211]

[0212] Among them, F is the identity matrix.

[0213] Based on the analysis of the bending wave transmission relationship of the infinitely long periodic discontinuous core, the transmission relationship of the bending wave in the finite-length AB and CD segments can be further obtained as:

[0214]

[0215] Among them, are the equivalent reflection matrix and transmission matrix of the studied sandwich beam, respectively.

[0216] It is assumed that from equations (45)-(49), we can obtain:

[0217]

[0218] Among them,

[0219] 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:

[0220]

[0221] Among them, w in , w out are the amplitudes of the beams on the input side and output side, respectively.

[0222] From equation (50), it can be seen that the amplitudes w1 and w2 at the right ends O2 and O4 of the sandwich beam are:

[0223]

[0224] Then the transmission loss τ1 of the upper beam = 20log(|w1| / 1), and the transmission loss of the lower beam is τ2 = 20log(|w2| / 1).

[0225] The present invention first performs segmentation processing on the discontinuous core sandwich beam, dividing it into a coupled section with the action of the core and an uncoupled section without the action of the core. Secondly, dynamic modeling and wave component analysis are carried out on the upper and lower beams of the coupled section and the uncoupled section. Among them, the vibration dynamic models of the upper and lower beams of the coupled section are mainly established. This model considers the discontinuous distribution characteristics of the core and equivalent the coupling force of the core on the upper and lower beams as a linearly viscoelastic distributed force. Through the dynamic models of the upper and lower beams, the wave vector form, wave number equation, and wave amplitude coefficients of the upper and lower beams of their vibration steady-state response are obtained. Then, the continuity conditions and equilibrium conditions between the upper and lower beams in the coupled section and the uncoupled section are constructed to obtain the reflection matrix and transmission matrix of the bending wave in the local structure with one core. Finally, based on the established local structure reflection matrix, transmission matrix, and the wave transmission relationship of the uncoupled section, first analyze the reflection matrix and transmission matrix of the bending wave in the sandwich beam with two cores, and then analyze the sandwich beam with three cores. When analyzing the propagation relationship of the bending wave in the sandwich beam with three cores, the part of the sandwich beam containing the first two cores is regarded as a whole for processing. Since the reflection matrix and transmission matrix of the sandwich beam with two cores and the reflection matrix and transmission matrix of the sandwich beam with one core have been derived previously, through the transfer matrix of the uncoupled section between these two parts, the reflection matrix and transmission matrix of the bending wave of the sandwich beam with three cores can be obtained. And so on, when extending to the reflection relationship and transmission relationship of the bending wave in the sandwich beam with N cores, the part of the sandwich beam containing N - 1 cores is regarded as a whole and iterated with the part of the sandwich beam containing the Nth core. Similarly, through the transfer matrix relationship of the uncoupled section between these two parts, the propagation relationship of the bending wave is established. Through such an iterative method, the reflection matrix and transmission matrix of the bending wave of the sandwich beam with N cores can finally be derived.

[0226] From the frequency-domain curves of the moduli of the elements in the matrix, the reflection and transmission relationships of the energy in the propagation of the bending wave can be analyzed, and then the attenuation characteristics and bandgap characteristics of the bending wave can be analyzed. On this basis, by adding boundary and external excitation conditions, the analytical solution of the steady-state amplitude-frequency response and the vibration transmission loss of the upper and lower beams can be further obtained.

[0227] The derived reflection matrix and transmission matrix of the sandwich beam with N cores belong to a generalized vibration model. This model mainly involves the reflection matrix and transmission matrix of the coupled section and the transfer matrix of the uncoupled section. If, based on the discontinuous core sandwich beam structure under study, a new structural design is carried out on the coupled section or the uncoupled section, only the reflection matrix and transmission matrix of this part need to be established to replace the corresponding part in the original model, and the reflection matrix and transmission matrix of the designed discontinuous core sandwich beam can be obtained. Therefore, the established generalized vibration model has good universality and scalability, providing an effective theoretical tool for the design optimization and bending wave regulation analysis of the sandwich beam structure.

[0228] Appendix A

[0229]

[0230]

Claims

1. A generalized vibration model for analyzing the vibration characteristics of a discontinuous core sandwich beam, the sandwich beam being composed of an upper beam, a lower beam and discrete cores, and the sandwich beam including a coupled section with cores and an uncoupled section without cores; Characterized in that: When there are N cores in the discontinuous core sandwich beam, N≥2, the intersection points A, B, C, and D of the left and right ends of the first core from left to right with the upper and lower beams are discontinuous characteristic points. The A and C ends are the incident ends of the flexural wave, and the reflection matrix of the flexural wave at the incident end and the transmission matrix of the flexural wave from the left end to the right end are as follows: Among them, They are respectively the reflection matrix of the bending wave at the incident end and the transmission matrix of the bending wave from the left end to the right end when there are N - 1 cores in the discontinuous core sandwich beam, and I is the identity matrix; R and T are respectively the reflection matrix of the bending wave at the incident end and the transmission matrix of the bending wave from the left end to the right end when there is 1 core in the discontinuous core sandwich beam; R x is the reflection matrix at the discontinuous feature point x, T yz is the transmission matrix of the flexural wave incident from the discontinuous feature point y to the discontinuous feature point z, where x, y, and z are arbitrary discontinuous feature points; k 0u and k 0d are the natural wave numbers of the upper and lower beams of the uncoupled section respectively. E 0u / 0d are the Young's moduli of the upper or lower beam of the uncoupled section respectively. I 0u / 0d are the moments of inertia of the upper or lower beam of the uncoupled section respectively. ρ 0u / 0d are the material densities of the upper or lower beam of the uncoupled section respectively. A 0u / 0d are the cross-sectional areas of the upper or lower beam of the uncoupled section respectively; L2 is the length of an uncoupled section.

2. The generalized vibration model for analyzing the vibration characteristics of a discontinuous core sandwich beam according to claim 1, characterized in that: There are 2 cores in the discontinuous core sandwich beam. The junction points A, B, C, D, A1, B1, C1, 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 transmission matrix for the flexural wave transmitted from points A and C to points B1 and D1 is Transmission matrix for the transmission of flexural waves from points A and C to points A1 and C1 is as follows:

3. The generalized vibration model for analyzing the vibration characteristics of a discontinuous core sandwich beam according to claim 1 or 2, characterized in that: There are 3 cores in the discontinuous core sandwich beam. The junction points A, B, C, D, A1, B1, C1, D1, A2, B2, C2, D2 of the left and right ends of the 1st, 2nd and 3rd cores from left to right with the upper and lower beams are discontinuous characteristic points; 4. The generalized vibration model for analyzing the vibration characteristics of a discontinuous core sandwich beam according to claim 1, characterized in that: The steady-state solutions of the amplitudes of the upper and lower beams in the uncoupled section, \(w_{\langle0000009\rangle}(x)\) and \(w_{\langle0000010\rangle}(x)\), are as follows: 0u \((x)\), \(w\) 0d \((x)\) is: c 01 、c 02 、c 03 、c 04 、d 01 、d 02 、d 03 、d 04 are the coefficients for each item.

5. The generalized vibration model for analyzing the vibration characteristics of a discontinuous core sandwich beam according to claim 1, characterized in that: The amplitudes w of the upper and lower beams of the coupling section u (x, t) = w1(x)e iωt , w d (x, t) = w2(x)e iωt , The transverse vibration response of the coupled beam is the motion of bending waves with two wave numbers k1 and k2. In the formula, a1, a2, a3, a4, b1, b2, b3, b4, c1, c2, c3, c4, d1, d2, d3, d4 are coefficients of each item.

6. A calculation method for the vibration transmission loss of a finite periodic sandwich beam, characterized in that: The sandwich beam contains N cores, and the left ends A and C are subjected to unit displacement excitation, and the right ends B and D are free ends; extend O1A, BO2, O3C and DO4 at both ends respectively, and make their lengths 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. 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 matrices can be obtained as follows: From this, the bending wave transmission relationship at the right ends O2 and O4 of the sandwich beam 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: Among them, F is the unit matrix; According to the analysis of the bending wave transmission relationship of the infinite periodic discontinuous core, the bending wave transmission relationship of the finite length AB and CD segments is obtained as: Among them, are the equivalent reflection matrix and transmission matrix of the sandwich beam respectively; From equations (45)-(49), it can be obtained that: 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 (50), it can be known that the amplitudes w1 and w2 at the right ends O2 and O4 of the sandwich beam are: Then the transmission loss of the upper beam τ1 = 20log(|w1| / 1), and the transmission loss of the lower beam is τ2 = 20log(|w2| / 1).

Images