A vibration absorbing device with a slender structure and a calculation method of its energy band structure

Abstract

The present invention discloses a vibration absorbing device of a slender structure and a calculation method of its energy band structure. The vibration absorbing device is an elastic metamaterial with an energy band structure. The vibration absorbing device includes a plurality of periodically arranged primitive cells, which are composed of a spider web resonator and Kevlar tethers on both sides thereof. When calculating the energy band structure, a finite element model of the primitive cell is established based on the absolute node coordinate method. The relationship between the node coordinates and the global position vector of any point in the unit is obtained by using the shape function. The dynamic equation of the system is calculated by the unit strain energy and mass matrix. The Bloch boundary condition is added, and the real part and the imaginary part of the displacement field are separated by combining the Euler formula. The real part and the imaginary part of the displacement field are simulated by the finite element model respectively, and the dispersion curve cluster of the eigenfrequency with respect to the wave vector is obtained, and then the band gap of the energy band structure of the vibration absorbing device is obtained. The present invention adopts the above-mentioned vibration absorbing device, combines the elastic metamaterial with the passive shock absorber, and can be applied to the space tether system.

Description

Technical Field

[0001] The invention relates to the technical field of space tethered systems, and in particular to a vibration absorbing device with a slender structure and a calculation method for its energy band structure. Background Art

[0002] Space tether systems have broad application prospects, but their large swept area will also increase the risk of collision between the system and space debris, so researchers have conducted extensive research on tether vibration control. Fujii et al. studied the application of wave absorption control in suppressing the lateral vibration of the flexible tether of the space tether system, Wen et al. proposed a nonlinear optimal feedback control method for the deployment process of the tethered satellite model, and Kojima et al. proposed an application of smart film for sensing ribbon tethers.

[0003] The above-mentioned vibration control methods for space tether systems are all active vibration control, which requires the consumption of propellant or electrical energy, which will affect the working life of the space tether system. Compared with active vibration control, passive vibration control absorbs or dissipates energy through additional components, does not require additional energy consumption, and is simpler and more reliable. However, there are almost no passive vibration dampers in space tether systems, so research in this field is of great significance.

[0004] Elastic metamaterials, also known as phononic crystals, are composed of periodic artificial structures, which are usually described by unit cells and lattices in crystallography. When elastic waves propagate in phononic crystals, they are affected by the periodicity of the structure, and the propagation in the band gap frequency range is suppressed, while they propagate losslessly in other frequency ranges. Using this characteristic of phononic crystals, specific structures can be designed to block or regulate the propagation of elastic waves. However, elastic metamaterials are rarely used for vibration suppression of space tethers. Summary of the invention

[0005] The object of the present invention is to provide a vibration absorbing device combined with elastic metamaterials for use as a passive vibration absorber in a space tethered system to suppress vibrations and fluctuations of the tether.

[0006] To achieve the above object, the present invention provides the following technical solutions:

[0007] A slender structure vibration absorption device, the vibration absorption device is an elastic metamaterial and has an energy band structure; the vibration absorption device comprises a plurality of periodically arranged spider web resonators, the spider web resonators are all fixedly arranged on a Kevlar tether, and the spider web resonator and the Kevlar tethers in front and behind the spider web resonator together constitute a primitive cell; the spider web resonator comprises a spoke acting as a spring and a ring acting as a mass block, the spokes support the inside of the ring, and a channel for the Kevlar tether to pass through is formed at the intersection of the spokes; the spider web resonator is made of ABS plastic using 3D printing technology.

[0008] Based on the above device, the present invention also discloses a method for calculating the corresponding energy band, as follows:

[0009] S1. Establish a finite element model of the primitive cell; based on the absolute node coordinate method, use the space-reduced beam unit to model the Kevlar tether, and use the thin shell unit to model the spider web resonator; then, define the unit node number and unit degree of freedom number, and store the geometric parameters and material parameters of each unit;

[0010] S2, using shape functions to obtain the relationship between the node coordinates and the global position vector of any point in the unit;

[0011] S3. Obtain unit elastic force by differentiating unit strain energy, obtain Jacobian matrix by differentiating unit elastic force, calculate unit mass matrix according to virtual work principle, then substitute unit mass matrix and Jacobian matrix of elastic force into unit dynamic equation, then assemble each item of unit dynamic equation according to unit degree of freedom number, and obtain system dynamic equation;

[0012] S4. Add Bloch boundary conditions on the upper and lower sides of the primitive cell and combine the Euler formula to separate the real and imaginary parts of the displacement field;

[0013] S5. Use two identical unit cell finite element models to simulate the real and imaginary parts of the displacement field respectively;

[0014] S6. Traverse all wave vectors in the first Brillouin zone to obtain a dispersion curve cluster of eigenfrequency and wave vector, and then obtain the band gap of the energy band structure of the vibration absorbing device based on the dispersion curve cluster.

[0015] The vibration absorption device of the present invention applies elastic metamaterials to the space tether system, replaces active vibration reduction with passive vibration reduction, has a significant vibration isolation effect, and can effectively suppress the vibration of the space tether caused by the impact of space debris. Furthermore, a band structure calculation method is proposed, which provides guidance for the study of band gap regulation. BRIEF DESCRIPTION OF THE DRAWINGS

[0016] Figure 1 is a three-dimensional model diagram of the primitive cell in Example 1;

[0017] Figure 2 is the finite element model of the spider web elastic metamaterial primitive cell in Example 2;

[0018] Figure 3 Schematic diagram of the principle of the space reduction beam unit in Example 2;

[0019] Figure 4 Schematic diagram of the principle of the thin shell unit in Example 2;

[0020] Figure 5 is the finite element model of the real part and the imaginary part in Example 2;

[0021] Figure 6 is the modal vibration shape of the primitive cell under the hinge boundary condition in Example 2;

[0022] Figure 7 is the mode vibration shape of the spider web resonator under the free boundary condition in Example 2;

[0023] Figure 8 The spider web elastic metamaterial band structure in Example 2;

[0024] Fig. 9 is the eigenmode of the longitudinal wave and the torsional wave in Example 2;

[0025] Fig.10 is the first bandgap boundary eigenmode in Example 2;

[0026] Fig.11 is the second bandgap boundary eigenmode in Example 2;

[0027] Fig.12 is the third bandgap boundary eigenmode in Example 2;

[0028] Fig.13 is the fourth bandgap boundary eigenmode in Example 2;

[0029] Fig.14 The experimental process and experimental device in Example 3;

[0030] Fig.15 3 is the frequency response function and vibration transmission characteristic curve in Example 3. DETAILED DESCRIPTION

[0031] The technical solution of the present invention is further described below in conjunction with the accompanying drawings and embodiments.

[0032] Example 1

[0033] like Figure 1 As shown, the vibration absorption device of the slender structure includes a number of periodically arranged primitive cells, which are composed of a spider web resonator and flexible tethers on both sides. Specifically, the spider web resonator includes three spokes acting as springs and a ring acting as a mass block. The spokes support the inside of the ring, and the intersection of the spokes forms a channel for the tether to pass through. The tether is a Kevlar tether, and the spider web resonator is made of ABS plastic using 3D printing technology. The vibration absorption device as a whole constitutes a spider web elastic metamaterial with a band structure.

[0034] Example 2

[0035] The structure of spider web elastic metamaterial is complex, and the propagation of elastic waves in the structure cannot be expressed analytically. Several commonly used methods for calculating energy bands cannot solve the energy bands of spider web elastic metamaterials. Therefore, this scheme adopts the finite element method and combines the absolute node coordinate formula to perform mechanical modeling and band structure calculation on the primitive cell.

[0036] S1. Finite element modeling of primitive cell structure

[0037] ANCF space-reduced beam elements are used to model the mechanical properties of the flexible tether, and ANCF thin shell elements are used to model the mechanical properties of the spokes and rings. Compared with the full-parameter beam element, the space-reduced beam element omits the slope vector in the cross-sectional direction, reducing the unit's degrees of freedom from 24 degrees of freedom of the full-parameter beam element to 12 degrees of freedom; compared with the full-parameter plate element, the thin shell element omits the global slope vector in the thickness direction, reducing the unit's degrees of freedom from 48 degrees of freedom to 36 degrees of freedom. This modeling method greatly improves the computational efficiency and avoids the locking problem.

[0038] The finite element model of the spider web elastic metamaterial primitive cell is as follows Figure 2 As shown in the figure, the left side is the original cell structure of the spider web metamaterial, the tether is used as the matrix, and the spokes and rings are spider web resonators. The detailed size parameters and material parameters of the original cell are shown in Table 1.

[0039] Table 1 Design parameters of spiderweb metamaterial

[0040]

[0041] Then, define the element node numbers and element degree of freedom numbers, and store the geometric parameters and material parameters of each element.

[0042] S2. Use shape functions to obtain the relationship between the node coordinates and the global position vector of any point in the unit.

[0043] 1) Space reduction beam element

[0044] like Figure 3 As shown, using the separation variable method, any point P on the beam under the current configuration can be expressed as

[0045] (1)

[0046] In the formula, is the global position vector, is the global velocity vector, is the global acceleration vector, is the shape function, is the node coordinate vector.

[0047] Among them, the shape function , specifically:

[0048] (2)

[0049] In the formula, , are the local coordinates of the beam element, is the length of the beam in the reference configuration.

[0050] The node coordinate vector of the beam element is

[0051] (3)

[0052] 2) Thin shell element

[0053] like Figure 4 As shown, the node coordinate vector of the thin shell element can be expressed as

[0054] (4)

[0055] In the formula, is the global position vector of the four nodes, is the global slope vector within the four node faces.

[0056] The global position vector, global velocity vector, and global acceleration vector of any point on the neutral layer of a shell element are similar to those of a beam element.

[0057] S3. Establish the dynamic equation of the system

[0058] After the unit node coordinates and the unit global position vector are established, according to the principle of virtual work,

[0059] (5)

[0060] is the virtual work done by the unit inertia force, which can be expressed as

[0061] (6)

[0062] In the formula, is the mass matrix, Represents the integration region as the volume of the cell.

[0063] is the virtual work done by the unit elastic force and can be expressed as

[0064] (7)

[0065] In the formula, is the unit generalized elastic force, is the strain energy of the element.

[0066] The strain energy of the beam element can be expressed as

[0067] (8)

[0068] In the formula, is the Young's modulus of the material, is the section moment of inertia of the beam, A is the cross-sectional area of ​​the beam, is the axial strain of the beam, is the curvature of the beam, It can be expressed as , is the global slope vector, is the second derivative of the global position vector with respect to the local coordinates.

[0069] The strain energy of the shell element can be expressed as

[0070] (9)

[0071] In the formula, is the Green-Lagrange strain of the shell element mid-surface, is the bending strain of the shell element, is the fourth-order elasticity tensor for the plane stress problem.

[0072] The elastic force can be obtained by taking the first derivative of the unit strain energy with respect to the node coordinates, and the Jacobian matrix of the elastic force can be obtained by taking the first derivative of the elastic force , which is the tangent stiffness matrix.

[0073] According to formulas (6) and (7), the virtual work done by the external force is:

[0074] (10)

[0075] In the formula, is the generalized external force acting on the unit.

[0076] The unit dynamics equation is

[0077] (11)

[0078] Assemble the unit dynamics equations according to the unit degree of freedom number to obtain the system dynamics equation

[0079] (12)

[0080] In the formula, M is the mass matrix of the system, J is the stiffness matrix of the system, The generalized external force acting on the system.

[0081] Furthermore, the calculation problem of the band structure can be reduced to the generalized eigenvalue problem under the Bloch boundary conditions of a given wave vector, so the generalized external force on the right side of the system dynamics equation is should be zero. Using the Lagrange multiplier method and introducing the Bloch boundary condition, we have

[0082] (13)

[0083] In the formula, is the Jacobian matrix of the constraint equation , is the constraint equation, is the Lagrange multiplier.

[0084] S4. Add Bloch boundary conditions

[0085] Bloch's theorem states that, considering simple harmonic waves in periodic structures, the displacement field r (or other field quantities such as stress field and strain field) in the medium can be expressed in the form of an amplitude modulated plane wave (also called Bloch wave), that is,

[0086] (14)

[0087] In the formula, is the amplitude modulation function, are material coordinates - representing the position of a point on the component in the reference configuration, r is the spatial coordinate - indicating the position of the point on the component under the current configuration, is the frequency of the wave, It is the wave vector.

[0088] According to Bloch's theorem, the amplitude modulation function It also has translation periodicity, that is, according to the positive lattice vector It remains unchanged after translation, that is

[0089] (15)

[0090] Pick Amplitude at point , combined with formula (14) and (15), we can get

[0091] (16)

[0092] It can be seen that the amplitudes of the corresponding points of two different primitive cells are the same, and the vibrations differ by a phase factor . From equation (16), we can see that the wave function of the entire infinite periodic structure can be expressed as the wave function of a unit cell multiplied by a phase factor, and the entire structure is compressed into a primitive cell. When calculating the energy band, a set of eigenfrequencies can be obtained for each given wave vector. The wave vector takes values ​​continuously, and then takes the wave vector as the independent variable, and each eigenfrequency becomes a function Furthermore, since there are no new eigenvalues ​​outside the first Brillouin zone, a band gap can be obtained when the wave vector takes all the values ​​of the first Brillouin zone.

[0093] From equation (16), we can see that since the Bloch boundary condition involves complex number operations, it is necessary to use a Bloch boundary condition addition method suitable for standard finite elements proposed by Åberg et al. to separate the real part and the imaginary part, thereby converting the operations in the complex domain to the real domain.

[0094] The global position vector of any point on the structure in equation (16) is , divided into real and imaginary parts.

[0095] (17)

[0096] In the formula, for The real part of for The imaginary part of Is an imaginary unit.

[0097] Using Euler's formula, write the complex exponential in the form of a trigonometric function, making the real and imaginary parts equal, then

[0098] (18)

[0099] The constraint equation is further refined into the relationship between the coordinates of each node, then

[0100] (19)

[0101] In the formula, is the global position vector at the top of the real grid, is the global position vector at the bottom of the real grid, is the global position vector at the top of the imaginary grid, is the global position vector at the bottom of the imaginary grid. are the global slope vectors of the four points respectively. is the wave vector, or the wave number in the case of one-dimensional elastic metamaterials. is the positive lattice vector and, in the case of one-dimensional elastic metamaterials, the lattice constant.

[0102] S5. Simulate the real and imaginary parts of the displacement field

[0103] Two identical finite element models are established. These two models have the same mesh, the same material parameters, and the same initial configuration. The only difference is the node degree of freedom number. These two identical models are used to simulate the solution domain of the real part and the imaginary part respectively. Figure 5 As shown in the figure, the finite element model on the left is used to simulate the real part solution domain, and the finite element model on the right is used to simulate the imaginary part solution domain. For the designed one-dimensional periodic elastic metamaterial primitive cell, it is only necessary to add Bloch boundary conditions to the nodes at both ends of the primitive cell tether.

[0104] S6. Obtain the dispersion curve cluster of eigenfrequency with respect to wave vector

[0105] Given a wave vector , a set of eigenfrequencies of the wave vector can be obtained through modal analysis. Finally, by traversing all wave vectors in the first Brillouin zone, a cluster of dispersion curves of the eigenfrequencies with respect to the wave vector is obtained, and then the band gap of the band structure of the vibration absorber can be obtained based on the cluster of dispersion curves.

[0106] Based on the above method, this embodiment provides a simulation diagram of the vibration absorbing device.

[0107] The modal vibration shape of the primitive cell under the boundary conditions of hinged supports at both ends is as follows: Figure 6 As shown. The first and second order modal vibration frequency is 11.72Hz, the third and fourth order modal vibration frequency is 33.52Hz, the fifth and sixth order modal vibration frequency is 85.46Hz, the seventh order modal vibration frequency is 110.37Hz, the eighth order modal vibration frequency is 186.77Hz, the ninth and tenth order modal vibration frequency is 213.02Hz, and the eleventh and twelfth order modal vibration frequency is 316.12Hz.

[0108] The mode shape of the spider web resonator under the free boundary is as follows: Figure 7 As shown, the first six orders are rigid body modes, so the vibration modes given in the figure start from the seventh order. The vibration mode frequencies of the seventh and eighth order modes are 85.30 Hz, the vibration mode frequency of the ninth order mode is 186.77 Hz, the vibration mode frequency of the tenth and eleventh order modes is 212.82 Hz, and the vibration mode frequency of the twelfth and thirteenth order modes is 319.29 Hz.

[0109] The band structure is calculated using the finite element method based on ANCF. Figure 8 As shown in the figure, the dispersion curve ④ represents the longitudinal wave in the metamaterial, and the dispersion curve ⑤ represents the torsional wave in the spider web metamaterial. When , the eigenmode of longitudinal wave propagation in the metamaterial structure is as follows Fig. 9 ( a), the vibration phases of adjacent cells are opposite, the longitudinal waves in the tether are coupled with the spider web resonator, and the system reaches a dynamic equilibrium; when the wave number When , the eigenmodes of torsion wave propagation in the metamaterial structure are as follows: Fig. 9 (b) and 9(c), When the vibration phase of adjacent cells is the same, The vibration phases of adjacent cells are opposite.

[0110] The following focuses on the vibration isolation of bending waves, so the longitudinal wave dispersion curve and the torsional wave dispersion curve are ignored. From the band structure diagram, it can be seen that the first band gap of the spider web metamaterial appears at 11.72Hz~19.78Hz, the second band gap appears at 33.52Hz~83.05Hz, the third band gap appears at 85.46Hz~188.20Hz, and the fourth band gap appears at 214.30Hz~316.12Hz.

[0111] The first band gap is Fig.10 shown. Fig.10 (a) At the starting frequency, the spider web resonator is basically not deformed, and its rigid body motion in the plane is coupled with the bending wave of the tether. In the primitive cell, the tether and the spider web resonator move in the same direction, but the motion phases of the two adjacent primitive cells are exactly opposite, so that the structure is in a dynamic equilibrium. At the boundary of the primitive cell, due to the anti-phase motion, the tether of the next primitive cell exerts a pulling force in the opposite direction on the tether of the previous primitive cell, resulting in the generation of a band gap. Fig.10 (b) At the cutoff frequency, the rigid body swinging motion of the spider web resonator is coupled with the bending wave of the tether, and the motion phases of adjacent primitive cells in this mode are still opposite. In fact, the starting frequency of the first band gap is the same as the first order natural frequency of the primitive cell with hinged branches at both ends, and the eigenmode is also the same as Figure 6 (a) Match.

[0112] The second band gap is Fig.11 shown. Fig.11 (a) The rigid body swinging motion of the spider web resonator at the starting frequency is coupled with the tether vibration. The motion phases of the two adjacent primitive cells in this mode are the same, but at the junction of the adjacent primitive cells, the motion direction of the tether is opposite, so that the latter primitive cell generates a reverse restoring force on the previous primitive cell, thereby suppressing the propagation of vibration. Fig.11 (b) The phase relationship between the vibrations of adjacent primitive cells at the cutoff frequency is also in-phase vibration. The bending wave in the tether clearly shows the characteristics of a long-wave traveling wave, and the overall motion direction of the tether is opposite to that of the spider web resonator. The structure is in a resonant state. It can be seen that the eigenmode at the start frequency of the second band gap is similar to the third-order modal vibration mode. Figure 6 (b) are exactly the same. And compare Figure 7 ( a)It can be seen that the seventh-order natural vibration of the spider web resonator is excited.

[0113] The third band gap is Fig.12 shown. Fig.12 (a) The eigenmode of the primitive cell at the starting frequency and the fifth-order mode vibration shape of the primitive cell with hinged branches at both ends Figure 6 (c) is the same, although the vibration phases of the tether and the spider web resonator in the primitive cell are opposite, due to the seventh-order natural vibration of the spider web resonator Figure 7 (a) When excited, the deformation of the spiderweb resonator produces a reverse restoring force on the tether, thereby generating a band gap. Fig.12 (b) At the third band gap cutoff frequency, the spider web resonator produces out-of-plane deformation, and the vibration phases of adjacent units are opposite. Figure 7 (c) It can be found that since the cutoff frequency is close to the tenth-order frequency of the spider web resonator, the natural vibration mode of this order is excited. The starting frequency of the third band gap is the same as the fifth-order frequency of the two-end hinged primitive cell, and the eigenmode of the starting frequency is also completely consistent with its natural vibration mode.

[0114] The fourth band gap Fig.13 shown. Fig.13 (a) At the starting frequency, unlike the eigenmodes of the first three band gap starting frequencies, the tethers at the boundaries of adjacent primitive cells of this eigenmode are no longer antisymmetric, but symmetric. This is because as the frequency increases, the wavelength of the bending wave gradually becomes shorter, and the elastic wave in the tether gradually no longer exhibits the characteristics of a long-wave traveling wave. It can also be seen from the figure that the bending wave in the tether is related to the tenth-order natural vibration of the spider web resonator. Figure 7 (c) Strong coupling occurs and the energy is concentrated in the spiderweb resonator, which opens a band gap. Fig.13 (b) The bending waves in the tether and the twelfth-order natural vibration of the spider web resonator at the cutoff frequency Figure 7 (d) Coupling occurs, and the vibration phases of adjacent cells are opposite. The starting frequency of the fourth band gap is the same as the ninth-order frequency of the two-end hinged cells.

[0115] Example 3

[0116] Furthermore, the present solution also provides a data acquisition system for the above-mentioned vibration absorbing device.

[0117] The connection relationship between the experimental devices is as follows: Fig.14 (a) shows that the elastic metamaterial containing 6 primitive cells is suspended vertically, and both ends of the rope are fixed. The lower end of the rope is wrapped around the top rod of the exciter, as shown in Fig.14(a) As shown at point A in the middle. Then use two nuts to clamp the tether, and tighten the nuts on the top rod to fix the tether and the top rod in the horizontal direction. The PCB accelerometer is installed on the top rod to measure the acceleration of the excitation point. The acceleration signal is transmitted to the data acquisition system through the data line, and the excitation signal to be output is obtained through calculation, so that the excitation signal is fed back to the power amplifier, and the excitation force is output after amplification by the power amplifier. Use 502 glue to stick the Donghua micro accelerometer to the upper end of the tether and reinforce it with tape to measure the acceleration of this point. Its position is shown in point B. Finally, the acceleration signals of the excitation point and the measurement point are collected and output to the workstation through the data acquisition system. After processing, the vibration transmission characteristics of the 6-period metamaterial structure are finally obtained. The experimental setup is as follows Fig.14 (b) as shown.

[0118] In the EDM software of Crystal Instrument data acquisition system, set the sensitivity of the sensor and the input mode and add a sweep task plan. The displacement amplitude of the excitation is 1mm, the frequency range is from 2Hz to 100Hz, and the sweep speed is 0.5oct / min. The results are as follows: Fig.15 shown.

[0119] The peak acceleration of the excitation point varies with frequency as shown in Fig.15 As shown in (a), the acceleration and frequency have a quadratic relationship, which is consistent with the theory. The logarithmic spectrum of the peak acceleration at the measurement point is as follows: Fig.15 As shown in (b), the peak values ​​are at 5.45Hz, 11.47Hz, and 17.14Hz, which correspond to the first, second, and third order modes of the metamaterial structure, respectively. Its vibration mode is similar to the first, second, and third order modes of a vibrating string. The frequency of these three order natural vibrations is very low and the energy is very large, so the elastic metamaterial structure cannot achieve the vibration isolation effect. Fig.15 (c) It can be seen from the vibration transmission characteristic curve that the acceleration generated by the excitation is not attenuated, but amplified at the measurement point. From the observation of the vibration transmission characteristic curve, it can be found that the vibration isolation effect of the metamaterial gradually becomes obvious from 18 Hz, and the vibration isolation effect is very good in the range of 30 Hz to 40 Hz, and the vibration isolation effect is particularly obvious at 32 Hz.

[0120] The above are specific embodiments of the present invention, but the protection scope of the present invention should not be limited thereto. Any changes or substitutions that can be easily thought of by a person skilled in the art within the technical scope disclosed by the present invention should be included in the protection scope of the present invention, so the protection scope of the present invention should be based on the protection scope defined in the claims.

Claims

1. A method for calculating the band structure of a slender vibration absorbing device. It is characterized in that The calculation method of the band structure of the vibration absorbing device includes: S1. Establish a finite element model of the primitive cell; based on the absolute node coordinate method, use the space-reduced beam unit to model the Kevlar tether, and use the thin shell unit to model the spider web resonator; then, define the unit node number and unit degree of freedom number, and store the geometric parameters and material parameters of each unit; S2, using shape functions to obtain the relationship between the node coordinates and the global position vector of any point in the unit; S3. Obtain unit elastic force by differentiating unit strain energy, obtain Jacobian matrix by differentiating unit elastic force, calculate unit mass matrix according to virtual work principle, then substitute unit mass matrix and Jacobian matrix of elastic force into unit dynamic equation, then assemble each item of unit dynamic equation according to unit degree of freedom number, and obtain system dynamic equation; S4. Add Bloch boundary conditions on the upper and lower sides of the primitive cell and combine the Euler formula to separate the real and imaginary parts of the displacement field; S5. Use two identical unit cell finite element models to simulate the real and imaginary parts of the displacement field respectively; S6, traversing all wave vectors in the first Brillouin zone to obtain a dispersion curve cluster of the eigenfrequency with respect to the wave vector, and then obtaining the band gap of the energy band structure of the vibration absorbing device based on the dispersion curve cluster; The vibration absorbing device is an elastic metamaterial with an energy band structure; the vibration absorbing device includes a plurality of periodically arranged spider web resonators, each of which is fixedly mounted on a Kevlar tether, and the spider web resonator and the Kevlar tethers in front and behind it together constitute a primitive cell; the spider web resonator includes spokes acting as springs and a ring acting as a mass block, the spokes support the interior of the ring, and a hole is formed at the intersection of the spokes for the Kevlar tether to pass through; the spider web resonator is made of ABS plastic using 3D printing technology.

Images