A vibration isolation metamaterial with a long quasi-zero stiffness plateau and its design method

Abstract

A design method for vibration isolation metamaterials with long quasi-zero stiffness plateaus is disclosed. This method first constructs composite beams with long quasi-zero stiffness plateaus using a beam structure. Then, vibration isolation unit cells with long quasi-zero stiffness plateaus are formed by circularly arraying these composite beams. The underlying mechanism is the combination of Euler buckling of the inclined beams and deflection deformation of the crossbeams within the microscopic vibration isolation unit cells. Second, a theoretical model of the effective load of the vibration isolation unit cells is established, quantifying the relationship between the effective load and the geometric design parameters of the unit cells, and performing finite element simulation. Finally, by adjusting the parameters of the inclined beams within the vibration isolation unit cells to prevent mutual interference between the quasi-zero stiffness plateaus, the unit cells are arranged to achieve flexible multi-level vibration isolation performance. The vibration isolation unit cells designed in this invention exhibit quasi-zero stiffness characteristics and low-frequency vibration isolation capabilities over a long displacement range. Stacking these unit cells to construct metamaterials can also achieve flexible multi-level vibration isolation.

Description

Technical Field

[0001] This invention belongs to the field of metamaterial design and relates to a vibration isolation metamaterial with a long quasi-zero stiffness platform and its design method. This method is applicable to low-frequency vibration isolation design in highly variable vibration environments to control the vibration of protected equipment, such as for whole-satellite vibration isolation systems to achieve two-stage load whole-satellite impact isolation. Background Technology

[0002] Vibration is the periodic or random motion of an object caused by external forces or internal dynamics. Vibration control mainly includes both utilization and suppression. In some situations, clever utilization of vibration can have positive effects, but in engineering applications, uncontrolled vibration often leads to mechanical failures, increased noise, and shortened structural lifespan. Therefore, many engineering structures and precision equipment require isolation from unwanted vibrations, and vibration-isolating metamaterials can achieve vibration control performance that traditional materials cannot. Most vibration-isolating metamaterials are designed to reduce or isolate vibration transmission from external or internal vibration sources, thereby protecting equipment. These metamaterials, based on different working principles and materials, are widely used in construction, industry, and transportation. According to their working principles and designs, vibration-isolating metamaterials can be broadly classified into linear structures and nonlinear structures. Linear vibration-isolating structures are the most common type; their goal is to isolate vibrations by utilizing changes in natural frequency or damping effects, such as intelligent sandwich panel structures designed through topology optimization, suitable for situations where the vibration source and response prediction are relatively certain. Nonlinear vibration-isolating structures, on the other hand, are materials whose stress-strain curves exhibit nonlinear changes during loading, similar to the tensile curve of low-carbon steel.

[0003] Quasi-zero stiffness metamaterials are a special type of nonlinear metamaterial that, through ingenious structural design, can achieve near-zero effective stiffness within specific load and displacement ranges. This unconventional mechanical behavior means that quasi-zero stiffness metamaterials can exhibit extremely low natural frequencies under specific loads, thus effectively isolating vibrations over a wide frequency range. Due to these properties, quasi-zero stiffness metamaterials have shown great potential in designing efficient vibration isolation systems. Generally, quasi-zero stiffness metamaterials are typically achieved by combining negative stiffness mechanisms and positive stiffness elements (such as springs), where positive stiffness can be provided by linear helical springs and negative stiffness by electromagnetic structures, cosine beams, etc. Some optimization methods have also been innovatively used to design metamaterials with quasi-zero stiffness characteristics. For example, Zhang et al. (Tailored mechanical metamaterials with programmable quasi-zero-stiffness features for full-band vibration isolation, advanced functional materials, 31(2021)2101428.) used a genetic optimization algorithm to customize a bending beam with a special geometry, achieving customized quasi-zero stiffness characteristics. These studies all designed quasi-zero stiffness characteristics, achieving low-frequency vibration isolation under effective load. Although the stiffness of quasi-zero stiffness metamaterials is typically close to zero at the static equilibrium point, its stiffness tends to gradually increase with the offset from the static equilibrium point. After reaching a certain offset, the dynamic stiffness of the system will no longer remain at a low level, resulting in a corresponding displacement range for a specific low stiffness value. Within this range, the dynamic stiffness is lower than the preset low stiffness value. Therefore, expanding the width of the low-stiffness displacement range becomes particularly important to achieve optimal vibration isolation. Moreover, structures that maintain quasi-zero stiffness characteristics over a longer displacement range can more flexibly cope with different operating conditions and vibration amplitudes, effectively isolating vibrations over a wider frequency range. Furthermore, a wider quasi-zero stiffness plateau can reduce the risk of system damage or failure due to sudden overloads, thereby increasing the reliability and stability of the vibration isolation structure. This is especially important for dealing with diverse vibration environments and multi-frequency vibration sources. Summary of the Invention

[0004] The technical problem solved by this invention is: in order to overcome the problem that traditional vibration isolation systems have limited frequency range and insufficient flexibility when isolating vibration sources, this invention proposes a vibration isolation metamaterial design method with a long quasi-zero stiffness platform, which is particularly suitable for structures and precision instruments that require low-frequency vibration isolation, and can significantly improve the vibration isolation effect and adaptability.

[0005] The principle underlying this invention is as follows:

[0006] By cleverly combining and configuring horizontal and inclined beams, which traditionally do not exhibit quasi-zero stiffness characteristics, the metamaterial achieves quasi-zero stiffness characteristics over a wide displacement range through the synergistic effect between them. The underlying mechanism is the combination of Euler buckling of the inclined beams and deflection deformation of the horizontal rods within the microscopic unit cell. A theoretical model is established to quantify the relationship between the effective load of the unit cell and its geometric parameters. Furthermore, this invention includes a complete theoretical model and experimental verification mechanism. Numerical and experimental analyses of the metamaterial's mechanical properties demonstrate its quasi-zero stiffness characteristics and low-frequency vibration isolation capabilities over a long displacement range. Based on this, by adjusting the thickness parameters within these unit cells to prevent mutual interference between the quasi-zero stiffness plateaus, and then stacking the vibration-isolation unit cells, flexible multi-stage vibration isolation can be achieved.

[0007] The specific technical solution adopted in this invention is as follows:

[0008] A vibration isolation metamaterial with a long quasi-zero stiffness plateau and its design method are disclosed. The design method employs a combination of horizontal and inclined beams to design the vibration isolation metamaterial. Through the synergistic effect between the horizontal and inclined beams, quasi-zero stiffness characteristics over a long displacement range are achieved. Specifically: First, a composite beam with a long quasi-zero stiffness plateau is constructed by cleverly combining lightweight, high-strength beam structures. Then, a vibration isolation unit cell with a long quasi-zero stiffness plateau is formed by circularly arraying the composite beams. The underlying mechanism is the combination of Euler buckling of the inclined beams and deflection deformation of the horizontal beams within the microscopic vibration isolation unit cell. Second, a theoretical model of the effective load of the vibration isolation unit cell is established, quantifying the relationship between the effective load and the geometric design parameters of the vibration isolation unit cell, and verifying this through finite element simulation. Finally, multi-level vibration isolation performance of the quasi-zero stiffness metamaterial is achieved by changing key geometric parameters. Specifically, by adjusting the parameters of the inclined beams within the vibration isolation unit cell, the quasi-zero stiffness plateaus of the unit cell are made independent of each other. Arranging the vibration isolation unit cells allows for flexible multi-level vibration isolation performance. This invention verifies the quasi-zero stiffness characteristics and vibration isolation performance of the isolation unit cell through static and dynamic tests. Due to its unique structural design and excellent low-frequency vibration isolation capability, the quasi-zero stiffness metamaterial obtained by the designed isolation unit cell array is a promising multi-stage, low-frequency vibration isolation solution. Specifically, it includes the following steps:

[0009] S1: Design of vibration isolation unit cell

[0010] Beam structures, due to their lightweight and high strength, and their ability to effectively absorb and disperse energy in design, have been widely used in earthquake resistance, vibration reduction, and impact protection. In this invention, a vibration-isolation unit cell with a long quasi-zero stiffness displacement range platform is designed using a beam structure. Furthermore, the vibration-isolation unit cell is arrayed to obtain a vibration-isolation metamaterial, such as... Figure 2As shown in (a). The vibration isolation unit cell is square, formed by four circular arrays of composite beams with quasi-zero stiffness characteristics, with a pitch angle of 90 degrees. In the vibration isolation unit cell, the composite beam is formed by combining component one and component two, as shown in (a). Figure 2 As shown in (b). Specifically, component one is designed by combining two horizontal beams and two pairs of symmetrically arranged "<"-shaped corner beams, with the corner beams located in the middle of the two horizontal beams; component two is designed by combining two horizontal beams and two pairs of symmetrically arranged diagonal beams, with the diagonal beams located in the middle of the two horizontal beams. Component two is located between the two components one, with the nodes of the corner beams in the upper part of component one aligned with the upper nodes of the diagonal beams in component two, and the nodes of the corner beams in the lower part of component one aligned with the lower nodes of the diagonal beams in component two. All beams are fixedly connected by integrated 3D printing manufacturing.

[0011] Regarding the rationality of composite beam design in square vibration isolation unit cells, such as Figure 2 As shown in (c), when the center of component two is subjected to vertical compression, the two pairs of inclined beams in component two will experience unstable buckling, and the force-displacement curves will exhibit negative stiffness behavior, with no quasi-zero stiffness plateau appearing. However, when component one and component two are combined to form a composite beam, as shown in (c), the two pairs of inclined beams in component two will experience unstable buckling, and the force-displacement curves will exhibit negative stiffness behavior, with no quasi-zero stiffness plateau appearing. Figure 2 As shown in (d), the design of component one in the composite beam relaxes the boundary constraints of component two, which weakens the negative stiffness behavior of the inclined beam in the composite beam under loading. Then, the inclined beam comes into contact with the cross beam, which increases the stiffness of the composite beam and exhibits a quasi-zero stiffness plateau. Finally, the composite beam becomes denser when further compressed, and its stiffness also increases accordingly.

[0012] S2: Constructing a theoretical model of the effective load of the vibration isolation unit cell

[0013] The effective load of the quasi-zero stiffness plateau of the vibration-isolated unit cell occurs before the inclined beam contacts the horizontal beam. Therefore, to calculate the effective load of the vibration-isolated unit cell at the occurrence of the quasi-zero stiffness plateau, only the load at the quasi-zero stiffness plateau needs to be considered. Figure 3 Theoretical analysis is performed on the inclined beam of the composite beam in the vibration isolation unit cell shown in (a). In the composite beam of the vibration isolation unit cell, the upper surface of the composite beam is subjected to a vertically downward force F, and the lower surface is subject to fixed constraints. When the spacing between the corner beams of component one in the composite beam is small, the strength of the crossbeam is large, which restricts the rotation of the inclined beam of component two in the composite beam. Therefore, it is assumed that the upper end nodes of the inclined beams in component two are all subject to fixed constraints. When the spacing between the corner beams of component one in the composite beam is large, the free boundary allows the outer inclined beam of component two (such as...) to... Figure 3 The constraint on beam CD in (a) is relatively weak, making it difficult to prevent the rotation of the lower node D of the outer inclined beam in component two. Furthermore, the symmetry of the composite beam and its opposite rotation to that of the outer inclined beam in component two increase the internal stiffness of the crossbeam, further restricting the rotation of the inner inclined beam in component two (such as...). Figure 3 Rotation of the lower node of beam AB in (a).

[0014] Therefore, assuming the analytical model of the inner inclined beam AB of component two in the composite beam is fixed at both ends, its free body diagram is as follows: Figure 3 As shown in (b); the analytical model of the outer inclined beam CD of component two in the composite beam is fixed at one end and hinged at the other end, and its free body diagram is as follows. Figure 3 As shown in (c), the theoretical model of the effective load of the vibration isolation unit cell is then obtained. The calculation formula for the effective load of the quasi-zero stiffness vibration isolation unit cell is as follows:

[0015]

[0016] Where F0 is the effective load of the quasi-zero stiffness isolation unit cell; E is the Young's modulus of the material; the thickness of all inclined beams in the isolation unit cell is t1, the out-of-plane depth is b, the length is l, and the inclination angle is θ; the moment of inertia of all inclined beams in the isolation unit cell is I1 = (bt1) / ( ... 3 ) / 12; the cross-sectional area is A1=t1b; μ1 is the equivalent length coefficient of the theoretical model of the inclined beam with fixed ends in the vibration isolation unit cell; μ2 is the equivalent length coefficient of the theoretical model of the inclined beam with one fixed end and one hinged end in the vibration isolation unit cell.

[0017] S3: Finite Element Solution and Fabrication of Vibration Isolation Unit Cells

[0018] Based on the theoretical model of the effective load of the vibration isolation unit cell constructed using S2, finite element numerical simulation of the vibration isolation unit cell is performed. It is assumed that both the upper and lower surfaces of the vibration isolation unit cell have rigid plates representing the pressure plate of the testing machine, and that the finite element nodes of the rigid plates are coupled to a reference point located at their center through multi-point constraints for loading and observing the reaction force. Figure 4 As shown in (a), a fixed constraint is applied to the lower rigid plate, and a vertically downward displacement load is applied to the upper rigid plate, which cannot move in other directions. An advanced algorithm is used to mesh the isolation unit cell. In the isolation unit cell, the mesh in the normal direction of all beams exceeds three layers. Furthermore, the overall element size of the meshed isolation unit cell is approximately 0.35 mm.

[0019] In addition, the vibration isolation unit cell designed in step S1 was prepared using mainstream jet melting 3D printing technology.

[0020] S4: Design of Multi-Level Vibration Isolation Metamaterials

[0021] By adjusting the geometric parameters of the inclined beam in the vibration isolation unit cell, including its length, thickness, and tilt angle, vibration isolation unit cells with different quasi-zero stiffness platforms can be realized. These vibration isolation unit cells exhibit quasi-zero stiffness characteristics and low-frequency vibration isolation capability over a long displacement range.

[0022] Furthermore, in many engineering designs, equipment or machinery must withstand various operating conditions, making effective vibration isolation with a single quasi-zero stiffness platform impractical. To obtain multi-layered quasi-zero stiffness platforms for more flexible vibration control of the protected equipment, multiple quasi-zero stiffness isolation cells with different effective loads can be stacked. Specifically, the multiple isolation cells are aligned vertically and then bonded and fixed using a support plate between them. This stacked isolation cell structure achieves a flexible multi-level quasi-zero stiffness platform. When the effective load is light, a lower quasi-zero stiffness platform is used for vibration isolation, while a higher quasi-zero stiffness platform controls vibration when the effective load is large.

[0023] Furthermore, the specific dimensions of components one and two in the S1 composite beam are as follows: the thickness of the corner beam in component one and the thickness of the inclined beam in component two are t1 = 1 mm; the thickness of the crossbeams in both components one and two is t2 = 1 mm; the length of the inclined beam in component two is l = 35 mm; and the inclination angle θ = 45°. The angle between the two ends of the corner beam in component one and the horizontal direction is also θ; the spacing between each pair of corner beams is d = 5.5 mm, which is also the spacing between each pair of inclined beams in component two. The out-of-plane depth of the composite beam is b = 5 mm. The design dimensions of the vibration isolation unit cell can also be selected by proportional scaling.

[0024] Furthermore, in the S1 vibration isolation unit cell, the inclination angle of the inclined beam is 40≤θ≤45. When the inclination angle is θ=45, the quasi-zero stiffness characteristic of the vibration isolation unit cell is optimal.

[0025] Furthermore, in the formula for calculating the effective load of the S2 quasi-zero stiffness isolation unit cell, the equivalent length coefficient μ1 = 0.5, and the equivalent length coefficient μ2 = 0.7.

[0026] Furthermore, the material used to prepare the vibration isolation unit cell in S3 is elastic thermoplastic polyurethane (TPU), nylon, resin, or other preparation materials, preferably elastic thermoplastic polyurethane (TPU). This material is chosen because it has good toughness, which can enhance the flexibility of the design.

[0027] S5: Static and Dynamic Experiments of Isolation Units

[0028] (a) Static Experiment: To test the mechanical properties of the designed quasi-zero stiffness metamaterial, a general-purpose testing machine was used to perform compression tests on the vibration isolation unit cell. To reduce the influence of the boundary effects of the vibration isolation unit cell on the experimental results, a rigid plate with grooves was placed at the upper and lower ends of the prepared vibration isolation unit cell to limit the horizontal displacement at both ends of the unit cell. The experiment was conducted through displacement loading. To reduce the influence of the dynamic effects generated during loading on the experimental results, a constant loading rate of 2 mm / min was set during loading, and the resulting support reaction force was measured by a high-precision sensor.

[0029] (b) Dynamic Experiment: To test the vibration isolation performance of the designed quasi-zero stiffness metamaterial, a vibration platform was constructed for vibration testing. The vibration isolation unit cell, held by the experimental device, was placed on a modal exciter. An accelerometer sensor for measuring output data, a support plate, and weights were fixed to the upper end of the prepared vibration isolation unit cell. The weight of these three components constituted the weight carried by the vibration isolation unit cell. During the experiment, only the weight of the weights was changed to alter the load-bearing capacity. An accelerometer sensor for measuring input signals was fixed to the lower end of the vibration isolation unit cell experimental device. Vibration information was generated by a signal generator, amplified by a power amplifier, and transmitted to the exciter. A dynamic data acquisition system measured and acquired the acceleration signals, and finally, data analysis was performed on a computer.

[0030] A vibration isolation metamaterial with a long quasi-zero stiffness platform, obtained using the above design method, is a promising multi-stage, low-frequency vibration isolation solution. It can isolate external low-frequency excitation and perform low-frequency vibration isolation, thereby controlling the vibration of the protected equipment. For example, it can be used in a whole-satellite vibration isolation system to achieve two-stage load whole-satellite impact isolation.

[0031] The design principle of this invention is as follows:

[0032] This invention utilizes the lightweight and high-strength characteristics of beam structures, along with their ability to effectively absorb and disperse energy during design. By combining the Euler buckling of the inclined beam with the bending deformation of the crossbeam, a vibration-isolating metamaterial with a long quasi-zero stiffness plateau is designed. This design enables the metamaterial to exhibit excellent vibration isolation performance in low-frequency applications. This design not only enhances the flexibility of the metamaterial's frequency response but also improves the efficiency of vibration energy absorption, thereby more effectively suppressing unwanted vibrations transmitted from the vibration source. In practical applications, its vibration isolation performance can be further improved by adjusting the geometric parameters of the metamaterial, such as the length, thickness, and tilt angle of the inclined beam. For example, increasing the length of the inclined beam can effectively increase the quasi-zero stiffness range. Furthermore, this invention includes a complete theoretical model and experimental verification mechanism for predicting and testing the performance of the metamaterial unit cell in actual vibration environments. This includes quasi-static compression tests and dynamic vibration tests on the unit cell to ensure accurate prediction of material performance during the design and manufacturing stages, and to adjust design parameters based on test results to achieve optimal vibration isolation.

[0033] The beneficial effects of this invention are as follows:

[0034] This invention, based on combining the Euler buckling of an inclined beam with the bending deformation of a crossbeam, designs a metamaterial capable of providing quasi-zero stiffness over a long displacement range. This long quasi-zero stiffness platform of the metamaterial effectively isolates low-frequency vibrations, significantly improving the performance and reliability of precision instruments and sensitive equipment requiring precise vibration control. The metamaterial design allows for flexible stacking designs by adjusting geometric parameters, such as the length and tilt angle of the inclined beam, to adapt to different vibration isolation requirements. Utilizing the lightweight and high-strength characteristics of beam structures, this invention not only improves the load-bearing capacity of the structure but also maintains a low mass, which is significant for reducing the overall system load and improving energy efficiency. This invention not only provides a theoretically innovative design but also includes an experimental verification mechanism to ensure the feasibility and effectiveness of the design. This combination of experimental and theoretical methods enhances the scientific rigor and practicality of the invention. Attached Figure Description

[0035] Figure 1 This is a flowchart illustrating the implementation of the present invention.

[0036] Figure 2A schematic diagram of a mechanical metamaterial design with a long quasi-zero stiffness platform provided for an example of the present invention. (a) Metamaterial and its vibration isolation unit cell; (b) Design of the vibration isolation unit cell combined beam and its parameters: the thickness of the corner beam in component one and the thickness of the inclined beam in component two are t1, the thickness of the crossbeams in component one and component two, the length l of the inclined beam in component two, the inclination angle θ, the spacing d of each pair of corner beams, and the out-of-plane depth b = 5 mm; (c) Force-displacement curve of component two under load; (d) Force-displacement curve of the combined beam under load; (e) Achieving a multi-level force platform through the stacking of vibration isolation unit cells; (f) When the mass of the object is on the metamaterial quasi-zero stiffness platform, it will be isolated from vibration in almost the entire frequency domain.

[0037] Figure 3 Buckling analysis of the stress model of the vibration-isolated unit cell composite beam. (a) Loads and boundary conditions of the stress model: initial state and critical buckling state; (b) Buckling mode of the inner inclined beam AB; (c) Buckling mode of the outer inclined beam CD.

[0038] Figure 4 The finite element model and experimental model of the designed vibration isolation unit cell are shown. (a) Finite element model; (b) Mesh convergence analysis of the finite element model; (c) Vibration isolation unit cell 1 and vibration isolation unit cell 2 prepared by 3D printing, and the vibration isolation unit cell formed by stacking the two.

[0039] Figure 5 For experimental testing apparatus. (a) Quasi-static uniaxial compression test apparatus; (b) Vibration test apparatus.

[0040] Figure 6 The results are quasi-static test results for a single cell. (a) Measured force-displacement curves and theoretically predicted effective loads; (b) Deformation during finite element simulation and experiment; (c) Stiffness-displacement relationship from numerical simulation results; (d) Stress contour plot under compacted conditions.

[0041] Figure 7 The vibration test results of the isolation unit cell 1 under different loads are shown. (a) Acceleration time history curve when M = 160g; (b) Acceleration time history curve when M = 200g; (c) Acceleration time history curve when M = 240g; (d) Transmissibility curve when M = 160g; (e) Transmissibility curve when M = 200g; (f) Transmissibility curve when M = 240g.

[0042] Figure 8 Stacking behavior of quasi-zero stiffness metamaterials. (a) Measured force-displacement curves of isolation unit cell 2; (b) Force-displacement curves of the stacked isolation unit cell of isolation unit cell 1 and isolation unit cell 2. Detailed Implementation

[0043] The specific embodiments of the present invention are described in detail below with reference to the technical solutions and accompanying drawings:

[0044] This invention first utilizes a lightweight and high-strength beam structure to design a vibration-isolation unit cell with a long quasi-zero stiffness platform, and then conducts theoretical analysis, finite element simulation, and experimental verification on it. Figure 1 The description is a flowchart of the implementation of the present invention, namely: designing vibration isolation unit cells, constructing a theoretical model of the effective load of vibration isolation unit cells, finite element solution and preparation of vibration isolation unit cells, static and dynamic experiments of vibration isolation unit cells, performance verification and design of multi-level vibration isolation metamaterials.

[0045] S1: Design of vibration isolation unit cell structure

[0046] Figure 2 The description illustrates the construction of a vibration-isolation metamaterial, which consists of vibration-isolation unit cells with long quasi-zero stiffness plateaus, such as... Figure 2 As shown in (a). The vibration isolation unit cell consists of Figure 2 (b) The composite beam is formed by four circular arrays with a pitch angle of 90 degrees. The composite beam consists of two parts: Component One and Component Two. The specific dimensions of Component One and Component Two are as follows: the thickness of the corner beam in Component One and the thickness of the inclined beam in Component Two are t1 = 1 mm; the thickness of the crossbeams in both Component One and Component Two is t2 = 1 mm; the length of the inclined beam in Component Two is l = 35 mm; in the vibration isolation unit cell, the inclination angle of the inclined beam is 40 ≤ θ ≤ 45°; the quasi-zero stiffness characteristic of the vibration isolation unit cell is optimal when the inclination angle is θ = 45°. The angle between the two ends of the corner beam in Component One and the horizontal direction is also θ; the spacing between each pair of corner beams is d = 5.5 mm, which is also the spacing between each pair of inclined beams in Component Two. The out-of-plane depth of the composite beam is b = 5 mm. The design dimensions of the vibration isolation unit cell can also be selected by proportional scaling. Figure 2 As shown in (c), when the center of the upper crossbeam of component two in the composite beam is subjected to vertical compression, the inclined beam in component two will exhibit unstable buckling, and the force-displacement curve will show negative stiffness behavior, with no quasi-zero stiffness plateau appearing. However, when component one and component two are combined to form a composite beam, as shown in (c), Figure 2 As shown in (d), the design of component one in the composite beam relaxes the boundary constraints of component two, thus reducing the negative stiffness behavior of the inclined beam under loading. Then, the inclined beam contacts the transverse beam, increasing the stiffness of the composite beam and exhibiting a quasi-zero stiffness plateau. Finally, the composite beam becomes denser under further compression, further increasing its stiffness. Moreover, by adjusting the parameters of the inclined beams in the vibration isolation unit cell to prevent the quasi-zero stiffness plateaus of the unit cells from affecting each other, and then arranging the unit cells, it is possible to achieve... Figure 2 (e) illustrates the multi-stage vibration isolation performance. When the weight of the object requiring vibration control is comparable to the effective load of the isolation unit cell, the dynamic input signal applied to the isolation unit cell will be isolated from the protected object. Figure 2 (f)).

[0047] S2: Theoretical model of effective load of vibration isolation unit cell

[0048] Since the effective load of the quasi-zero stiffness plateau of the vibration isolation unit cell occurs before the inclined beam and the cross beam come into contact, in order to calculate the effective load of the vibration isolation unit cell when the quasi-zero stiffness plateau appears, only the load at the point of quasi-zero stiffness plateau is considered. Figure 3 Theoretical analysis is performed on the inclined beam of component two in the composite beam of the vibration isolation unit cell shown in (a). Due to the symmetry of the composite beam, the relationship between the compressive force at the upper end and the critical buckling force of the inclined beam in component two under different constraint types is as follows:

[0049] F = 2(F B -F D )#(1)

[0050] In the formula, F B F is the vertical force at the upper end of the inner inclined beam AB in component two. D The force is the vertical force at the lower end of the outer inclined beam CD in component two.

[0051] (a) Force analysis of the inner inclined beam AB of component two in the vibration isolation unit cell: in free body Figure 3 In (b), M B Q represents the bending moment generated by restricting the rotation of the upper node B of the inner inclined beam AB. B The force at node B on the upper end of the inner inclined beam AB is in the horizontal direction. Assuming no horizontal displacement at point B, the axial displacement component δ of the inner inclined beam AB in component two is... N and normal displacement component δ M Projecting horizontally, we have:

[0052] δ N =δ M tanθ#(2)

[0053] Among them, the displacement of the inner inclined beam AB of component two is mainly generated by compressive deformation and bending deformation, and the displacement components can be further expressed as:

[0054]

[0055] In the formula, A1 = t1b is the cross-sectional area of ​​the inner inclined beam AB of component two, and I1 = (bt1) 3 ) / 12 is its moment of inertia, and E is the Young's modulus of the material.

[0056] from Figure 3 (b) As can be seen from the free body diagram, although on the surface F B Q B and M B It affects the rotation of node B, but the rotation of point B will disappear after the equivalent process:

[0057]

[0058] Combining equations (2-5), we can obtain:

[0059]

[0060] The axial compressive force N of the inner inclined beam AB of component two B It can be represented as:

[0061] N B =F B sinθ+Q B cosθ#(7)

[0062] According to Euler's formula, the critical buckling load of the inner inclined beam AB of component two is:

[0063]

[0064] Where μ1 = 0.5 is the equivalent length coefficient, which depends on the boundary conditions at the support end. Combining equations (6-8), the critical force F for the buckling mode of the inner inclined beam AB of component two can be obtained. B for:

[0065]

[0066] (b) Force analysis of the outer inclined beam CD of component two in the composite beam: The analysis of inclined beam CD is similar to that of AB, except that the crossbeam does not fully restrict the rotation of inclined beam CD at the upper node D. The specific free body diagram is as follows. Figure 3 As shown in (c). Therefore, the displacement components of the inclined beam CD can be expressed as:

[0067]

[0068] Substituting into equation (2), the horizontal force of the outer inclined beam CD of component two can be written as:

[0069]

[0070] Similarly, according to Euler's formula, the upward vertical force on the inclined beam can be expressed as:

[0071]

[0072] In the formula, μ2 is the equivalent length coefficient of the outer inclined beam CD of component two, and its value is 0.7.

[0073] Substituting equations (9) and (13) into equation (1), we can obtain the effective load of the quasi-zero stiffness unit cell composite beam. Multiplying this by the degree of the circular array (4) yields the effective load F0 of the unit cell. The relationship between the effective load of the vibration isolation unit cell and the geometric design parameters of the vibration isolation unit cell is quantified through the theoretical model constructed above.

[0074] S3: Finite Element Solution and Fabrication of Vibration Isolation Unit Cells

[0075] To numerically simulate a vibration isolation unit cell, this invention utilizes commercial finite element software for finite element analysis. A conventional static analysis step is employed for loading to obtain stable calculation results. Due to the large deformation and highly complex contact nonlinearities exhibited by the vibration isolation unit cell during calculation, geometric nonlinearity is enabled. Rigid plates representing the pressure plate of the testing machine are used on both the upper and lower surfaces of the vibration isolation unit cell. The finite element nodes of these rigid plates are coupled to a reference point at their center via multi-point constraints to facilitate loading and observation of reaction forces. Figure 4 As shown in (a), a fixed constraint is applied to the lower rigid plate, and a vertically downward displacement load is applied to the upper rigid plate, which cannot move in other directions. The normal behavior of surface-to-surface contact is considered for all surfaces that come into contact during loading. An advanced algorithm is used to mesh the isolation unit cell, with more than three mesh layers in the normal direction of each beam. Figure 4 The mesh convergence analysis in (b) shows that a unit size of 0.35 mm is sufficient to ensure the accuracy of the calculation results.

[0076] The vibration isolation unit cells designed in step S1 were fabricated using mainstream jet melting 3D printing technology. The material used was TPU (thermoplastic polyurethane) with a Young's modulus of E = 17 MPa. This material was chosen because of its good toughness, which enhances the design flexibility. Specifically, vibration isolation unit cells 1 and 2 were fabricated using 3D printing. Figure 4 This was demonstrated in (c).

[0077] S4: Static and Dynamic Experiments of Isolation Units

[0078] (a) Static Experiment: To test the mechanical properties of the designed quasi-zero stiffness metamaterial, a general-purpose testing machine was used to perform compression tests on the vibration isolation unit cell. To reduce the influence of the boundary effects of the vibration isolation unit cell on the experimental results, a rigid plate with grooves was placed at the upper and lower ends of the prepared vibration isolation unit cell to limit the horizontal displacement at both ends of the unit cell. The experiment was conducted by displacement loading, with a loading displacement of 25 mm. To reduce the influence of the dynamic effects generated during loading on the experimental results, a constant loading rate of 2 mm / min was set during loading, and the resulting support reaction force was measured by a high-precision sensor.

[0079] (b) Dynamic Experiment: To test the vibration isolation performance of the designed quasi-zero stiffness metamaterial, this invention constructed a vibration platform for vibration testing, such as... Figure 5As shown in (b), the vibration isolation unit cell, held by the experimental setup, is placed on a modal exciter. An accelerometer for measuring output data, a support plate, and weights are fixed to the upper end of the prepared vibration isolation unit cell. The weight of these three components constitutes the weight carried by the vibration isolation unit cell. During the experiment, only the weight of the weights is changed to alter the load. An accelerometer for measuring input signals is fixed to the lower end of the vibration isolation unit cell experimental setup. Vibration information is generated by a signal generator, amplified by a power amplifier, and transmitted to the exciter. A dynamic data acquisition system measures and acquires the acceleration signals, and finally, the data is analyzed in a computer. During the vibration experiment, a sinusoidal sweep frequency signal is used as the input signal and applied to the vibration isolation unit cell through the modal exciter. The frequency range is 0.5-35 Hz, and the vibration time is 15 s.

[0080] S5: Performance verification of the vibration isolation unit cell

[0081] (a): Static verification: During loading, the force-displacement curves simulated in the vibration isolation unit cell and measured in the experiment of vibration isolation unit cell 1 are as follows: Figure 6 As shown in (a), although the experimental results showed compaction earlier than the simulation results (possibly due to assembly errors), both still exhibited good consistency on the quasi-zero stiffness platform. The vibration isolation unit cell also presented the ideal quasi-zero stiffness platform as expected and maintained it for a long distance (approximately 55% of the entire experimental loading process). The theoretically predicted effective load is as follows: Figure 6 As shown by the dashed line in (a), the effective load of the theoretical result is 2.02 N, and the maximum deviation from the simulated value and the experimental value is about 5.9% and 18.7%, respectively. Figure 6 (b) shows the deformation of the isolation unit cell 1 during compression. It can be observed that the experimental and numerical results correspond well; in both cases, the inclined beam in the isolation unit cell first begins to deform outward, generating the initial elastic response of force displacement (e.g., ...). Figure 6 (b) in (i)). As loading progresses, the isolation unit cell buckles, and the inclined beam deforms to contact the crossbeam. At this point, the force-displacement curve exhibits quasi-zero stiffness characteristics, and the isolation unit cell gradually densifies (e.g. Figure 6 (ii) in (b). When the vibration isolation unit cell is further loaded, the force-displacement curve will again show an elastic phase (such as...). Figure 6 (iii) in (b). Figure 6 (c) describes the stiffness change of the vibration isolation unit cell during compression, which is derived from... Figure 6 The red curve in (a) is a transformation. Observation reveals that the stiffness of the isolation unit cell first decreases to near 0 and remains there, then increases again, forming a large QZS region (a region with stiffness less than 0.05 N / mm) within the range of 1.30 mm to 15.17 mm. The stress cloud diagram of the isolation unit cell under compaction is shown below. Figure 6As shown in (d), the maximum stress is less than the yield stress of the TPU material, which can be considered as the vibration isolation unit cell not being destroyed, which is consistent with the experimental phenomenon.

[0082] (b): Dynamic verification: In vibration test experiments, whether vibration excitation can be effectively isolated depends mainly on the mechanical properties of the isolation unit cell. Figure 7 (ac) shows the vibration test results under supports of different masses (160, 200 and 240g). By observing the input and output acceleration-time curves, it can be found that when the mass of the support differs significantly from the effective load ( Figure 7 (a) and Figure 7 (c)), although in the high-frequency range, the output acceleration (A) out () is less than the input acceleration (A) in However, the opposite is true in the low-frequency range, indicating that the vibration isolation unit cell does not achieve complete vibration isolation. When the mass of the support is comparable to the effective load of the vibration isolation unit cell, the output acceleration is less than the input acceleration in the measured frequency range, indicating that the designed vibration isolation unit cell basically achieves vibration control. Furthermore, to further demonstrate the low-frequency vibration isolation performance of the proposed vibration isolation unit cell, we evaluated the transmittance of the vibration test results and... Figure 7 The transmittance is shown in (df). The magnitude of transmittance is determined by the input acceleration and the output acceleration, specifically defined as 20*log 10 (A out / A in In the obtained transmittance curves, when the mass of the support is 200g, the transmittance curve is almost always below zero, indicating that the vibration isolation unit cell can effectively shield low-frequency vibrations. In contrast, when the measured support is not near the effective load, peaks with higher transmittance are observed in the measured low-frequency range, indicating that vibration control is not achieved in these cases.

[0083] S6: Design of Multi-Level Vibration Isolation Metamaterials

[0084] A multi-level vibration isolation metamaterial was designed by stacking vibration isolation unit cells with different quasi-zero stiffness platforms. To demonstrate the effectiveness of this design strategy, vibration isolation unit cell 2 with a beam thickness of t1 = 1.2 mm was further measured. The measurement results are as follows: Figure 8 As shown in (a), a quasi-zero stiffness platform with a wide displacement range is also demonstrated, and the effective load is close to the theoretical prediction, at 3.51 N. Next, the two isolation unit cells are aligned vertically and then bonded and fixed using a support plate located between them. The mechanical behavior of the stacked isolation unit cells is then verified using a quasi-static analysis method, with specific results as follows: Figure 8As shown in (b), observation reveals two step-like force plateaus with large displacement ranges in both the experimental and simulated force-displacement curves. The deformation of the isolation unit cell also shows that the thinner inclined beam first becomes denser, followed by the thicker inclined beam, ensuring the multi-condition vibration isolation capability of the stacked isolation unit cells. In summary, the stacked isolation unit cells achieve multi-level quasi-zero stiffness plateaus. When the effective load is light, a lower force plateau is used for vibration isolation, while when the effective load is large, a higher force plateau controls the vibration.

[0085] The essence of this invention lies in leveraging the potential mechanism of the combination of Euler buckling of inclined beams and bending deformation of horizontal beams to design and theoretically and experimentally verify a vibration-isolation unit cell composed of beam structures that can achieve quasi-zero stiffness on a long displacement platform, thereby enabling the design of a vibration-isolation metamaterial. This method not only produces metamaterials with excellent performance in low-frequency vibration isolation, but also allows for adjustment of their response characteristics by modifying the beam configuration and parameters, enabling them to exhibit superior adaptability and effectiveness in a wider range of applications. It should be noted that the mechanical properties of the designed metamaterial depend on the geometric design and can therefore be realized using different 3D printing materials. Due to its unique structural design and excellent low-frequency vibration isolation capability, the designed vibration-isolation metamaterial with a long quasi-zero stiffness platform represents a promising multi-stage, low-frequency vibration isolation solution.

[0086] The above-described embodiments are merely illustrative of the implementation methods of the present invention, but should not be construed as limiting the scope of the present invention. It should be noted that those skilled in the art can make various modifications and improvements without departing from the concept of the present invention, and these modifications and improvements all fall within the protection scope of the present invention.

Claims

1. A design method of a vibration isolation metamaterial with a long quasi-zero stiffness platform, characterized in that, The design method employs a combination of horizontal and inclined beams to design vibration-isolating metamaterials. Through the synergistic effect between the horizontal and inclined beams, quasi-zero stiffness characteristics over a long displacement range are achieved. First, a composite beam with a long quasi-zero stiffness platform is constructed using a beam structure. Then, a vibration-isolating unit cell with a long quasi-zero stiffness platform is formed by circularly arraying the composite beams. The underlying mechanism is the combination of Euler buckling of the inclined beams and deflection deformation of the horizontal beams within the microscopic vibration-isolating unit cell. Second, a theoretical model of the effective load of the vibration-isolating unit cell is established, quantifying the relationship between the effective load and the geometric design parameters of the vibration-isolating unit cell, and performing finite element simulations to verify this. Finally, by adjusting the parameters of the inclined beams within the vibration-isolating unit cell to prevent mutual interference between the quasi-zero stiffness platforms of the unit cell, the vibration-isolating unit cells are arranged to achieve flexible multi-level vibration isolation performance. The method includes the following steps: S1: Design of vibration isolation unit cell S2: Constructing a theoretical model of the effective load of the vibration isolation unit cell The payload of the quasi-zero stiffness platform of the vibration isolation unit cell appears before the inclined beam contacts the cross beam. To calculate the payload of the vibration isolation unit cell when the quasi-zero stiffness platform appears, only the inclined beam of the combined beam in the vibration isolation unit cell is analyzed; it is assumed that the inner inclined beam of component two in the combined beam has an analytical model with both ends fixed, and the outer inclined beam of component two in the combined beam has an analytical model with one end fixed and one end hinged, thus obtaining the theoretical model of the payload of the vibration isolation unit cell. In S2, the effective load calculation formula for the quasi-zero stiffness vibration isolation unit cell is as follows: wherein, is the effective payload of the quasi-zero stiffness vibration isolation cell; is the Young's modulus of the material; the thickness of all the inclined beams in the vibration isolation cell is , the out-of-plane depth of all the inclined beams is , the length of all the inclined beams is , the inclination angle of all the inclined beams is , the moment of inertia of all the inclined beams in the vibration isolation cell is ; the cross-sectional area of all the inclined beams is ; is the effective length coefficient of the theoretical model of the clamped-clamped inclined beam in the vibration isolation cell; is the effective length coefficient of the theoretical model of the clamped-pinned inclined beam in the vibration isolation cell S3: Finite Element Solution and Fabrication of Vibration Isolation Unit Cells S4: Design of multi-level vibration isolation metamaterials.

2. The design method of a vibration isolation metamaterial with a long quasi-zero stiffness platform according to claim 1, wherein, Includes the following steps: A vibration-isolating unit cell with a long quasi-zero stiffness displacement range platform is designed using a beam structure, and then the vibration-isolating unit cell is arrayed to obtain a vibration-isolating metamaterial. The vibration-isolating unit cell is square, which is formed by four circular arrays of composite beams with quasi-zero stiffness characteristics, with an array pitch angle of 90 degrees. In the vibration-isolating unit cell, the composite beam is composed of component one and component two: component one is designed by combining two horizontal beams and two pairs of symmetrically arranged "<" shaped corner beams, with the corner beams located in the middle of the two horizontal beams; component two is designed by combining two horizontal beams and two pairs of symmetrically arranged diagonal beams, with the diagonal beams located in the middle of the two horizontal beams; component two is located between the two components one, with the nodes of the corner beams in the upper component one aligned with the upper nodes of the diagonal beams in component two, and the nodes of the corner beams in the lower component one aligned with the lower nodes of the diagonal beams in component two; the beams are fixedly connected by integrated 3D printing manufacturing. Based on the theoretical model of the effective load of the constructed vibration isolation unit cell, a finite element numerical simulation is performed on the vibration isolation unit cell. It is assumed that the upper and lower surfaces of the vibration isolation unit cell are represented by rigid plates to represent the pressure plate of the testing machine, and the finite element nodes of the rigid plates are coupled to a reference point located at their center through multi-point constraints for loading and observing the reaction force. Fixed constraints are applied to the lower rigid plate, and a vertically downward displacement load is applied to the upper rigid plate, which cannot move in other directions. An advanced algorithm is used to mesh the vibration isolation unit cell, and the mesh in the normal direction of all beams in the vibration isolation unit cell exceeds three layers. Vibration isolation unit cells were fabricated using jet melting 3D printing technology. By adjusting the geometric parameters of the inclined beam in the vibration isolation unit cell, including length, thickness and tilt angle, vibration isolation unit cells with different quasi-zero stiffness platforms can be realized. To obtain a multi-layered quasi-zero stiffness platform for more flexible control of the vibration of the protected equipment, multiple quasi-zero stiffness isolation unit cells with different effective loads are stacked.

3. The design method of a vibration isolation metamaterial with a long quasi-zero stiffness platform according to claim 2, wherein, The inclination angle of the inclined beam in the vibration isolation cell is The material for preparing the vibration isolation cell in S3 is elastic thermoplastic polyurethane TPU material, nylon, resin or other preparation materials.

4. The design method of a vibration isolation metamaterial with a long quasi-zero stiffness platform according to claim 3, characterized in that, The inclination angle of the inclined beam in the vibration isolation cell is ; the material for preparing the vibration isolation cell in S3 is an elastic thermoplastic polyurethane (TPU) material.

5. The design method of a vibration isolation metamaterial with a long quasi-zero stiffness platform according to claim 1, wherein, The equivalent length coefficient The equivalent length coefficient .

6. The design method of a vibration isolation metamaterial with a long quasi-zero stiffness platform according to claim 2, wherein, The specific method of stacking quasi-zero stiffness isolation unit cells in S4 is as follows: the arrangement direction of multiple isolation unit cells is set to be consistent, and they are aligned vertically. Then, they are bonded and fixed by a support plate located between them. The stacked isolation unit cells realize a flexible multi-level quasi-zero stiffness platform. When the effective load is light, the lower quasi-zero stiffness platform is used to play the role of vibration isolation, while when the effective load is large, the higher quasi-zero stiffness platform controls the vibration.

7. A vibration isolation metamaterial having a long quasi-zero stiffness platform, characterized by, The material obtained by using the design method described in any one of claims 1-6 is a multi-level, low-frequency vibration isolation metamaterial that can isolate external low-frequency excitation while performing low-frequency vibration isolation, thereby controlling the vibration of the protected equipment.

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