Quasi-zero stiffness metamaterial structure based on titanium alloy cosine beam

Abstract

The invention belongs to the technical field of vibration isolation, and particularly relates to a quasi-zero stiffness metamaterial structure based on a titanium alloy cosine beam. The metamaterial structure is composed of a metamaterial unit cell one-dimensional array, each metamaterial unit cell comprises an upper unit cell unit, a lower unit cell unit and a connecting block arranged between the unit cell units, and the connecting block is used for connecting the upper unit cell unit and the lower unit cell unit; each unit cell unit comprises a positive stiffness cosine beam, a negative stiffness cosine beam, a plate body and a supporting frame; the two positive stiffness cosine beams are vertically arranged in the supporting frame in parallel, and the negative stiffness cosine beam is divided into two sections through a connecting block, wherein the two sections are arranged between the outer ends of the supporting frame in a bilateral symmetry mode. According to the metamaterial structure, titanium alloy serves as a base material, the quasi-zero stiffness metamaterial with positive stiffness and negative stiffness functions is obtained by adopting a cosine beam structure in the same form, the design can keep the uniformity of the structural form to improve the integration level, the stiffness cooperation requirement can be met through parameter regulation and control, and low-frequency longitudinal waves can be effectively shielded.

Description

TECHNICAL FIELD

[0001] The application belongs to the technical field of vibration isolation, and particularly relates to a quasi-zero stiffness metamaterial structure based on a titanium alloy cosine beam. BACKGROUND

[0002] With the rapid development of aerospace technology (such as deep space exploration, gravitational wave detection, advanced aircraft), the demand for low-frequency and ultra-low-frequency vibration isolation of equipment is becoming more and more demanding. Low-frequency impact during the launch phase of the spacecraft, micro-vibration during on-orbit operation, and ground pulsation interference of ground precision experiments (such as gravitational wave detection) will seriously affect the instrument precision, component life and mission reliability, so efficient isolation of low-frequency vibration has become a core technical problem in the field of aerospace. The traditional linear vibration isolator has a fundamental contradiction that "reducing stiffness sacrifices bearing capacity and stability, and increasing mass violates the lightweight requirement". The quasi-zero stiffness (QZS) vibration isolator realizes stiffness cancellation at the static balance position by connecting a negative stiffness mechanism and a positive stiffness mechanism in parallel, thereby combining the superior characteristics of "high static stiffness (high bearing capacity)" and "low dynamic stiffness (ultra-low frequency vibration isolation)", and becoming an effective solution to the low-frequency vibration isolation problem.

[0003] Currently, the metamaterial based on the local resonance (LR) mechanism is the mainstream direction to solve the low-frequency vibration isolation, and the band gap center frequency only depends on the natural frequency of the local resonator, which can realize low-frequency band gap in small-scale structures. However, the existing local resonance metamaterial has two major defects: 1. Material system limitation: current researches mostly use resin, TPU, PLA and other high polymer soft materials, which have low bearing capacity, poor environmental temperature adaptability, weak long-term service stability, and are difficult to meet the harsh requirements of the aerospace field.

[0004] 2. Complex structure design: in order to realize positive and negative stiffness, different principles and forms of components are often used, such as linear springs for positive stiffness and buckling beams or lever mechanisms for negative stiffness, resulting in complex structure, difficult assembly, and poor integration performance.

[0005] As the patent "A kind of quasi-zero stiffness vibration isolator" 2019101487489, it represents a typical mechanical quasi-zero stiffness scheme, the vibration isolator is mainly composed of a negative stiffness mechanism and a positive stiffness spring bearing weight in parallel, through stiffness offset makes the system dynamic stiffness close to zero. Its negative stiffness mechanism is composed of cross arm structure, tensile spring, T-shaped slide plate, baffle, connecting block and guide rail and other special-shaped components, the structure is complex, the parts are numerous, the space occupancy is large, not only leads to processing and assembly difficulty, also makes its application limited in the space limited scene (such as inside the spacecraft). In addition, its positive and negative stiffness elements are completely different components in principle and form, lack of integration, performance control is complex. Patent application "A carbon fiber panel sandwich structure with quasi-zero stiffness characteristics" 2024116882528 belongs to quasi-zero stiffness scheme based on metamaterial, which adopts periodic unit cell array as core layer, realizes quasi-zero stiffness characteristics through positive and negative stiffness element coupling, and strives to realize light weight and high bearing. Its negative stiffness element (orthogonal Euler beam) requires TPU flexible material, while the positive stiffness element (semicircular arc support) and the bearing panel adopt carbon fiber composite material. This "soft-hard" mixed material system must be formed through complex processes such as secondary curing, and there is a risk of material interface failure. The viscoelasticity of TPU material also leads to unstable mechanical behavior, complex dynamic performance prediction, and fundamental limitations on bearing capacity and environmental adaptability (such as temperature resistance), which cannot meet the high requirements of aerospace and other extreme working conditions on material consistency and reliability.

[0006] There is an urgent need for a QZS metamaterial structure with high-performance materials and high-integration design to better break through the above limitations. Based on this, the present application proposes a quasi-zero stiffness metamaterial structure based on titanium alloy cosine beam. SUMMARY

[0007] The present application provides a quasi-zero stiffness metamaterial structure based on titanium alloy cosine beam, which uses titanium alloy as base material and adopts the same form cosine beam structure to obtain quasi-zero stiffness metamaterial with positive stiffness and negative stiffness function. This design not only maintains the unity of the structure form to improve the integration, but also meets the stiffness coordination demand through parameter control, effectively shields low-frequency longitudinal waves, and solves the problems in the prior art.

[0008] The present application provides the following technical solutions: A quasi-zero stiffness metamaterial structure based on titanium alloy cosine beam, which is composed of metamaterial unit cells in vertical one-dimensional array, the metamaterial unit cell comprises two unit cells and a connecting block which are mirror images of each other; the connecting block is used to connect the upper and lower unit cells; each unit cell comprises a positive stiffness cosine beam, a negative stiffness cosine beam, a plate body and a support frame arranged on the plate body; the positive stiffness cosine beam in each unit cell comprises two parallel and vertically arranged in the support frame, the two ends of the positive stiffness cosine beam away from the plate body are connected to the two ends of the horizontal end face of the connecting block, the negative stiffness cosine beam is separated by the horizontal end face of the connecting block to form two sections which are symmetrically arranged outside the positive stiffness cosine beam, and the outer end of the negative stiffness cosine beam is connected with the support frame to form a closed frame.

[0009] Further, the positive stiffness cosine beam and the negative stiffness cosine beam are single-wave cosine beams.

[0010] Further, the initial shape of the above cosine beam is as follows: In the formula, is the initial vertex height of the curved beam, is the span of the beam.

[0011] Further, the support frame comprises a horizontal section fixed on the plate body and vertical sections connected to both ends of the horizontal section; the negative stiffness cosine beam is connected between the outer ends of the two vertical sections of the support frame; the negative stiffness cosine beam is curved away from the plate body; the two positive stiffness cosine beams in each unit cell are vertically and mirror image spaced apart, and the middle part of one of the positive stiffness cosine beams is curved to the side of the other positive stiffness cosine beam which is vertically mirror image of it.

[0012] Further, the length of the horizontal section of the support frame does not exceed the outer edge of the plate body.

[0013] Further, the plate body is a circular plate, a square plate or any other shaped plate, which is used to fix the load support frame; like the connecting block, it also plays a role of increasing the mass of the structure in this structure.

[0014] Further, the plate body is a circular plate. The support frame of each unit cell is arranged along the diameter of the circular plate; the length of the horizontal section of the support frame is ≤ the diameter of the circular plate.

[0015] Further, the support frame of each unit cell is an open right-angle frame with outer ends arranged along the diameter of the circular plate, the right-angle frame comprises a horizontal section arranged along the diameter of the circular plate and vertical sections fixed at both ends of the horizontal section; the negative stiffness cosine beam is connected between the two vertical sections of the right-angle frame.

[0016] Further, the connecting block is a rectangular or other regular shape connecting block, which is designed as a regular shape symmetrically up and down.

[0017] Further, assuming that the span of the negative stiffness cosine beam is λ1, the arc height is h1, the in-plane thickness is t1, and the out-of-plane thickness is b1, the span of the positive stiffness cosine beam is λ2, the arc height is h2, the in-plane thickness is t2, and the out-of-plane thickness is b2; in the unit cell, λ1 is the horizontal span of each negative stiffness cosine beam segment separated by the connecting block. Then it satisfies: , In the formula, E is the elastic modulus of the material, I1 is the sectional moment of inertia of the negative stiffness cosine beam: , I2 is the sectional moment of inertia of the positive stiffness cosine beam: ; The semi-empirical formula of the dimensionless coefficient is: .

[0018] Further, λ1 refers to the horizontal distance between the out-of-plane end of the connecting block connected to one end of the negative stiffness cosine beam and the end of the support frame connected to the other end of the negative stiffness cosine beam.

[0019] Further, the connecting block is an integral structure block symmetrically up and down.

[0020] Further, the metamaterial unit cell is periodically arrayed in the vertical direction, and two adjacent metamaterial unit cells share a plate body.

[0021] Further, the out-of-plane thicknesses of the positive and negative stiffness cosine beams are consistent with the out-of-plane thickness of the support frame.

[0022] Further, the metamaterial structure is made of a plate body, a support frame, a positive stiffness cosine beam, a negative stiffness cosine beam, and a connecting block, which are integrally formed by 3D printing of titanium alloy material; or, after the support frame, the positive stiffness cosine beam, and the negative stiffness cosine beam are integrally formed, they are fixed with the plate body and the connecting block.

[0023] The beneficial effects of the present application are: 1. The metamaterial structure of the present application is highly integrated and simplified, adopts a cosine beam structure form with an initial shape of a cosine curve, and can simultaneously produce the required adjustable positive stiffness and negative stiffness through different spatial arrangements (vertical / horizontal placement), greatly simplifying the structure design, reducing the manufacturing and assembly complexity, and improving the compactness and reliability of the structure.

[0024] 2、The material performance of the application is superior, titanium alloy material is adopted, high specific strength, high bearing capacity, wide temperature range stability and excellent fatigue life are given, the application limitation of the soft material QZS structure is broken, and the application is especially suitable for the aerospace field.

[0025] 3、The application realizes the quasi-zero stiffness characteristic in a wide displacement range through the accurate cooperative design of the positive and negative stiffness cosine beams, and an adjustable ultra-low frequency band gap is induced in the metamaterial, and the extremely low frequency vibration below 8Hz can also be effectively isolated.

[0026] 4、The geometric parameters of the cosine beams in the positive and negative stiffness units are designed differently, which provides clear and effective design dimensions for performance regulation and control, and facilitates customized optimization for different application scenarios.

[0027] The above design of the application avoids the complex assembly of different forms of components in the prior art, and simplifies the parameter cooperative regulation through unified structure form, significantly improves the structure integration and stability, and ensures the realization of stable and wide-range quasi-zero stiffness characteristics. BRIEF DESCRIPTION OF DRAWINGS

[0028] The drawings described herein are used to provide further understanding of the application, and form a part of the application, the illustrative embodiments of the application and the description thereof are used to explain the application, and do not constitute improper limitation on the application. In the drawings: Figure 1 It is a schematic diagram of the metamaterial unit of the titanium alloy QZS metamaterial structure of the application; Figure 2 It is a schematic diagram of the titanium alloy QZS metamaterial structure of the application; Figure 1 Figure 3 It is a schematic diagram of the titanium alloy QZS metamaterial structure of the application; Figure 4 It is a three-dimensional finite element model and grid convergence analysis of the titanium alloy QZS metamaterial structure of the application; Figure 5 It is a quasi-zero stiffness characteristic verification diagram of the unit cell of the titanium alloy QZS metamaterial structure of the application; Figure 6 It is a band gap characteristic analysis diagram of the titanium alloy QZS metamaterial structure of the application; Figure 7 It is a comparison diagram of the vibration modes inside and outside the band gap of the titanium alloy QZS metamaterial structure of the application; Figure 8 It is a comparison of the normalized band gap width of the titanium alloy QZS metamaterial structure of the application and six existing QZS structures; Figure 9 It is a geometric parameter optimization and mechanical property analysis diagram of the negative stiffness cosine beam.​ Figure 10 Figure is a buckling characteristic analysis diagram of the positive stiffness cosine beam geometric parameter optimization and buckling.

[0029] In the drawing, 1 is a connecting block, 2 is a round plate, 3 is a supporting frame, 4 is a positive stiffness cosine beam, and 5 is a negative stiffness cosine beam.

[0030] In the formula, a is the positive stiffness cosine beam, b is the negative stiffness cosine beam, c is the supporting frame, and d is the connecting block. Figure 3 In the figure, (a) is the overall configuration of the titanium alloy QZS metamaterial, the metamaterial unit constituting the same, and the constituent parts of the metamaterial unit; and (b) is a design strategy diagram.

[0031] Figure 4 In the figure, (a) is a three-dimensional finite element model of the titanium alloy QZS unit, the negative stiffness cosine beam unit (NS model), and the positive stiffness cosine beam unit (PS model); and (b) is a mesh convergence analysis curve.

[0032] Figure 5 In the figure, (a) is the relationship between the restoring force and displacement of the QZS unit; (b) is the relationship between the stiffness and displacement of the QZS unit; and (c) is a comparison of the force-displacement curves between the finite element simulation and three experiments.

[0033] Figure 6 In the figure, (a) is a dispersion curve without pre-compression; (b) is a curve of the start and end frequencies of the band gap changing with the pre-compression amount; and (c) is a comparison of the theoretical band gap and the finite element (FEM) band gap when the pre-compression is 1 mm.

[0034] Figure 7 In the figure, (a) is the overall vibration mode outside the band gap f =21.07Hz, f =44.15Hz （f =90.30Hz 、f =143.50Hz ） of the center mass local vibration mode inside the band gap; Figure 8 In the figure, the normalized band gap width of the QZS metamaterial structure of the present application is compared with that of six kinds of existing publicly disclosed QZS structures, and the red pentagram indicates the structure of the present application.

[0035] Figure 9 , Figure 10 In the figure, (a)-(d) are mechanical property analysis under the changes of the four parameters of the arc height h, the out-of-plane thickness b, the in-plane thickness t, and the span λ. DETAILED DESCRIPTION

[0036] The present application will be further described below in combination with specific embodiments, but the protection scope of the present application is not limited to these embodiments.

[0037] In the description of the present application, it is to be understood by those skilled in the art that the terms "upper", "lower", "front", "back", "left", "right", "top", "bottom", "inner", "outer" and the like indicate the orientation or positional relationship based on the orientation or positional relationship shown in the drawings, and are only for the convenience of describing the present application and simplifying the description, and therefore cannot be understood as indicating or implying that the devices or elements referred to must have a particular orientation, be constructed and operated in a particular orientation, and therefore cannot be understood as limiting the present application.

[0038] Unless otherwise defined, all technical and scientific terms used herein have the same meaning as commonly understood by one of ordinary skill in the art to which this application belongs. The terminology used in the present document is for the purpose of describing particular embodiments only and is not intended to be limiting of the present application.

[0039] In the description of the present application, the term "quasi-zero stiffness (QZS)" refers to a mechanical property that the dynamic stiffness of a structure is close to zero within a displacement interval near the static equilibrium position. This property enables the structure to withstand static loads and exhibit extremely low stiffness to dynamic excitations, thereby achieving ultra-low frequency vibration isolation. The term "positive stiffness unit (PS)" refers to a component composed of a cosine beam placed vertically, which generates a restoring force opposite to the displacement direction when subjected to force, providing linear and stable positive stiffness. The term "negative stiffness unit (NS)" refers to a component composed of a cosine beam placed horizontally, which exhibits negative stiffness characteristics within a certain displacement range, as the restoring force decreases with increasing displacement. The term "cosine beam" refers to a beam structure with an initial shape of a cosine curve, with a unified core structure form, and its mechanical behavior (such as buckling mode, stiffness) can be precisely controlled by its geometric parameters (span, arc height, thickness). In the present application, the cosine beam in the unit cell structure is a single-wave cosine beam. The term "bandgap" refers to a specific frequency interval where elastic waves are significantly attenuated when propagating in a periodic metamaterial, and the start frequency, end frequency and width of the bandgap are key indicators of vibration isolation performance. The "out-of-plane thickness" refers to the thickness of the structure perpendicular to the plane, which in the present description is understood to include the thickness of the positive stiffness unit and the negative stiffness unit perpendicular to the plane in the three-dimensional structure.

[0040] Example 1 Referring to Figures 1-3 , the design strategy of the quasi-zero stiffness (QZS) metamaterial structure unit cell and the overall configuration is shown Figure 3 .

[0041] Figure 3 In the overall configuration diagram of (a), the unit cell and one-dimensional array of the quasi-zero stiffness metamaterial structure are shown, and the geometric parameters marked in the figure, such as the cosine beam span λ, arc height h, and thickness t, are key parameters for subsequent examples. Figure 3The three graphs in (b) respectively show the force-displacement curves of the negative stiffness cosine beam, the positive stiffness cosine beam, and the quasi-zero stiffness force-displacement curve after the two are connected in parallel, which reflects the principle of "stiffness offset".

[0042] Figure 4 For the three-dimensional finite element model of the above structure and the mesh convergence analysis, Figure 4 In (a), the fixed constraint and the displacement load application position are marked, Figure 4 The analysis curves in (b) show the force-displacement curves under different grid densities such as conventional, refined, and more refined, which proves that "more refined grid" can balance the calculation accuracy and efficiency.

[0043] As a specific embodiment, the unit cell structure of the titanium alloy QZS metamaterial structure designed above specifically refers to Figure 1 、 2 The QZS metamaterial unit cell includes two unit cells that are mirror images of each other and a connecting block 1 (i.e., a central mass block) arranged between the two unit cells. Each unit cell is mainly composed of a positive stiffness unit, a negative stiffness unit, a circular plate 2, and a support frame 3. Specifically, each unit cell includes a circular plate 2, a support frame 3, a positive stiffness cosine beam 4, and a negative stiffness cosine beam 5. In each unit cell, the positive stiffness cosine beam 4 includes two parallel and vertical positive stiffness cosine beams arranged in the support frame 3. The two ends of the positive stiffness cosine beams away from the circular plate 2 are connected to the two ends of a horizontal end face of the connecting block 1. The negative stiffness cosine beam 5 is separated by the aforementioned horizontal end face of the connecting block 1 to form two segments that are symmetrically arranged outside the two positive stiffness cosine beams. The outer ends of the negative stiffness cosine beam 5 are connected to the support frame 3 to form a closed frame.

[0044] The support frame 3 of each unit cell is an open-ended right-angle frame arranged along the diameter of the circular plate 2. The right-angle frame includes a horizontal segment arranged along the diameter of the circular plate and vertical segments fixed to the two ends of the horizontal segment. The negative stiffness cosine beam 5 is connected between the two vertical segments of the right-angle frame. The negative stiffness cosine beam bends away from the circular plate 2. The two positive stiffness cosine beams 4 in each unit cell are vertically and mirror-imaged spaced apart. The middle part of one positive stiffness cosine beam bends towards the other positive stiffness cosine beam. The horizontal segment of the right-angle frame is ≤ the diameter of the circular plate.

[0045] The horizontal lengths of the upper and lower horizontal end faces of the connecting block 1 are equal to the distance between the end parts of the two positive stiffness cosine beams in each unit cell. The two ends of the upper and lower horizontal end faces of the connecting block 1 are respectively connected to the negative stiffness cosine beams 5 of the upper and lower unit cells.

[0046] The connecting block 1 and the circular plate 2 are also used as mass blocks to increase the mass of the overall structure. The shape of the circular plate is not specifically required, and the shape of the connecting block is set to avoid the surface of the connecting block from being touched during the deformation of the positive stiffness cosine beam and the negative stiffness cosine beam, so as to avoid the change of the stress of the cosine beam and affect the change of the structural stiffness.

[0047] As an embodiment, in the present embodiment, the connecting block 1 adopts a rectangular connecting block. According to the horizontal distance between the upper and lower horizontal end faces of the connecting block, the connecting positions of the two positive stiffness cosine beams in the upper and lower unit cells are determined. Specifically, one end of the two positive stiffness cosine beams 4 is connected and fixed at both ends of a horizontal end face of the connecting block, and the two symmetrical two-segment negative stiffness cosine beams 5 divided into two segments are respectively connected at the left and right of the horizontal end face of the connecting block 1. Then, the support frame 3 is surrounded to realize the fixed connection of the support frame with the negative stiffness cosine beam and the positive stiffness cosine beam. In the support frame 3, the other end of the positive stiffness cosine beam is fixed to the middle of the horizontal section of the support frame. Then, the circular plate 2 is arranged outside the horizontal section of the support frame. Finally, the aforementioned entire QZS unit cell structure is obtained through the upper and lower horizontal mirror image.

[0048] The cosine beam core structure forms of the above-mentioned positive stiffness cosine beam and negative stiffness cosine beam are unified, and are all cosine curve initial shapes. The initial shape formula is as follows: In the formula, h is the initial vertex height of the cosine curve beam, is the initial vertex height of the cosine curve beam, is the span of the beam.

[0049] The above structure all adopts titanium alloy material. The key performance parameters of the titanium alloy material are as shown in Table 1.

[0050] Table 1 The implementation principle of the quasi-zero stiffness characteristic of the above-mentioned quasi-zero stiffness (QZS) metamaterial unit cell structure is described in detail in Figure 5 . Figure (a) compares the relationship between the positive stiffness (F P ), the negative stiffness (F N ), the overall restoring force (F Q ) and the displacement. It can be seen that the positive stiffness unit (PS) composed of the positive stiffness cosine beam provides an approximately linear restoring force (F P ), and the negative stiffness unit (NS) provides a nonlinear negative restoring force (F N ) after buckling. Through parallel superposition, the overall restoring force (F Q ) changes gently in a specific region (QZS region), and the corresponding overall stiffness (F Q ) is close to zero.

[0051] The quasi-zero stiffness characteristic needs to satisfy the positive stiffness (K P ) and the negative stiffness (KN ) Parameter matching relationship in QZS region. Based on the buckling theory of beam and energy principle, both satisfy the core relationship: . This relationship explicitly shows the key design logic that positive and negative stiffness need to be approximately canceled in value, which is the theoretical basis for realizing the quasi-zero stiffness characteristic of the structure in the embodiment. Figure 5 The mechanical principle of "stiffness cancellation" in the above theory completely matches the above theory. Figure 5 Figure 2 (b) shows the relationship between the overall stiffness (k Q ) and displacement of the aforementioned structure in the embodiment. There is a quasi-zero stiffness interval of "stiffness close to zero"; the "platform stage" (i.e. quasi-zero stiffness region) marked in (c) verifies the effectiveness of the structure model.

[0052] Specifically, the stiffness expression of the negative stiffness cosine beam of the titanium alloy QZS metamaterial unit cell in the embodiment is: In the formula, is the elastic modulus of the material, is the sectional moment of inertia: ; According to the beam theory, the stiffness of the cosine beam providing positive stiffness is inversely proportional to the cubic power of the elastic modulus of the material , the sectional moment of inertia and the beam height . Its stiffness can be expressed as: Where, is the sectional moment of inertia of the positive stiffness cosine beam: ; The semi-empirical formula of the dimensionless coefficient is: Therefore, the cosine beam structure combination with quasi-zero stiffness characteristic in the embodiment satisfies the following relationship: .

[0053] Embodiment 2 The aforementioned QZS unit cell in Embodiment 1 is arranged in one-dimensional periodicity in the vertical direction, and the upper and lower adjacent two QZS unit cells share a circular plate 2, so as to obtain a QZS metamaterial structure. As shown in Figure 3 (a) right side overall structure diagram. The bandgap characteristics are analyzed.

[0054] A certain pre-compression displacement is applied to the QZS metamaterial, so that its working point is in the QZS region. As shown in Figure 6As shown, at this time, the metamaterial will produce a significant band gap in the ultralow frequency range for the propagation of longitudinal waves, i.e., the elastic wave cannot effectively propagate in this frequency range, thereby achieving effective isolation of ultralow frequency vibration. By adjusting the pre-compression amount, the start and end frequencies of the band gap can be continuously regulated to adapt to different vibration isolation requirements.

[0055] Specific analysis, Figure 6 In (a), the pink band gap interval 82.0 Hz-224.2Hz corresponds to no pre-compression; from Figure 6 In (b), it can be seen that when the pre-compression amount is 3mm, the band gap can be reduced to 8.8Hz-24.1Hz, and the elastic wave cannot effectively propagate in this frequency range; Figure 6 In (c), the comparison of the theoretical band gap and the finite element (FEM) band gap when the pre-compression is 1mm verifies the accuracy of the band gap prediction of the structure.

[0056] Figure 7 The comparison chart of the vibration modes inside and outside the band gap of the QZS metamaterial structure is shown. Referring to Figure 7 In (a), when the excitation frequency is outside the band gap range, the structure presents a global deformation mode, the vibration energy is continuously transmitted in the entire structure, each component undergoes obvious displacement deformation, and the elastic wave can continuously propagate along the structure; Figure 7 In (b), when the frequency is within the band gap range, the central mass block undergoes significant localized vibration, most of the elastic wave energy is bound near the central mass block (connecting block), and it is difficult to propagate to the far end of the structure, thereby achieving effective attenuation of vibration. In addition, by comparing the normalized band gap width of the existing QZS structure in the literature, see Figure 8 , it shows the advantage of the QZS metamaterial structure of the embodiment in the low start frequency area.

[0057] Figure 8 Each structure marked in is derived from the following literature [1]-[6]: [1] Xiao, L., Sun, X., Cheng, L., & Yu, X. (2024). A 3D-printed quasi-zero-stiffness isolator for low-frequency vibration isolation: Modelling and experiments. Journal of Sound and Vibration, 577, 118308.

[0058] [2] Shu, Y. Q., Wang, K., Yin, H. F., Zhou, J. X., Chen, T. T., Pan, H. B., & Deng, Y. P. (2025). A quasi-zero-stiffness metastructure for concurrent low-frequency vibration attenuation and energy harvesting. Thin-Walled Structures, 214, 113371.

[0059] [3] Liu, X., Chen, S., Wang, B., & Tan, X. J. (2025). A multi-dimensional quasi-zero-stiffness mechanical metamaterial with different directional vibration isolation capabilities. Smart Materials and Structures, 34(4), 045013.

[0060] [4] Li, Y. L., Tijani, M. Z., Jiang, X. D., & Ahmed, J. O. (2022). Bandgap mechanism and vibration attenuation of a quasi-zero stiffness metastructure. International Journal of Structural Integrity, 13(6), 1041-1059.

[0061] [5] Lin, Q. D., Zhou, J. X., Wang, K., Xu, D. L., Wen, G. L., & Wang, Q. (2023). Three-dimensional quasi-zero-stiffness metamaterial for low-frequency and wide complete band gap. Composite Structures, 307, 116656.

[0062] [6] Lin, Q. D., Zhou, J. X., Wang, K., Xu, D. L., Wen, G. L., Wang, Q., & Cai, C. Q. (2022). Low-frequency locally resonant band gap of the two-dimensional quasi-zero-stiffness metamaterials. International Journal of Mechanical Sciences, 222, 107230.

[0063] Example 3 The quasi-zero stiffness characteristics of the quasi-zero stiffness metamaterial unit cell structure of Example 1 are regulated and investigated. The mechanical response of the positive and negative stiffness components directly determines whether the QZS system can maintain a stable force balance within the elastic range. Once the stress exceeds the elastic limit and causes plastic deformation, the stiffness characteristics will be destroyed (the negative stiffness unit will cause the QZS platform to fail, and the positive stiffness unit will destroy the force balance relationship). Therefore, according to the functional requirements of the positive and negative stiffness units, the geometric parameters of the cosine beam, such as thickness t, arc height h, span λ, in-plane thickness t, and out-of-plane thickness b, are first determined to affect the product, and then parameter optimization is performed.

[0064] In the quasi-zero stiffness metamaterial unit cell structure of this example, the negative stiffness cosine beam parameters include arc height h1, out-of-plane thickness b1 (i.e., the side dimension of the negative stiffness cosine beam), in-plane thickness t1, and span λ1, and the positive stiffness cosine beam parameters include arc height h2, out-of-plane thickness b2, in-plane thickness t2, and span λ2.

[0065] The initial basic parameter size of the titanium alloy QZS unit cell structure is: the circular plate diameter D = 180 mm, the circular plate thickness t3 = 6 mm; the support frame (horizontal section) length a = 174 mm, the support frame out-of-plane thickness b = 10 mm, the connecting block length l = 44 mm, the QZS element height H = 44 mm; the negative stiffness cosine beam λ1 = 106 mm, h1 = 4.8 mm, t1 = 0.52559 mm, b1 = 10 mm; the positive stiffness cosine beam λ2 = 35 mm, h2 = 4.8 mm, t2 = 0.6 mm, b2 = 10 mm.

[0066] Reference Figure 9 To optimize and analyze the above geometric parameters of the negative stiffness cosine beam. Figure 9 The static characteristics reflected in (a) show that when the arc height h1 of the negative stiffness cosine beam changes from 2 mm to 7 mm, the force amplitude gradually increases, and the deformation range is adjusted, the maximum stress (black circle) is positively correlated with h 1; Figure 9(b), as the out-of-plane thickness b2 of the negative stiffness cosine beam increases from 8 mm to 14 mm, the force amplitude is approximately proportional to the increase, the deformation range is basically unchanged, and the maximum stress (black circle) is almost unchanged with the increase of b 1, that is, it has no obvious correlation with b 1; Figure 9 (c), when the in-plane thickness t1 of the negative stiffness cosine beam changes from 0.35 mm to 0.6 mm, the force amplitude gradually increases, and the deformation range is adjusted, the maximum stress (black circle) is positively correlated with t 1; Figure 9 (d), as the span λ1 of the negative stiffness cosine beam changes from 8 mm to 14 mm, the force amplitude is approximately proportional to the decrease, the deformation range is basically unchanged, and the maximum stress (black circle) is negatively correlated with Lambda 1. The above results show that the camber height h1 and the in-plane thickness t1 of the negative stiffness cosine beam significantly affect the negative stiffness force amplitude and the nonlinear deformation range; the out-of-plane thickness b1 increases approximately proportional to the carrying capacity; the span λ1 increases, and the force response is significantly weakened. The maximum stress is positively correlated with h1 and t1, and negatively correlated with λ1, and has no obvious correlation with b1.

[0067] Referring to Figure 10 , the above several geometric parameters of the positive stiffness cosine beam are optimized and analyzed. Figure 10 The results show that the force amplitude decreases with the increase of the camber height h2, and the maximum stress is negatively correlated with h 2 (a); with the increase of the out-of-plane thickness b2, the force amplitude gradually increases, and the maximum stress is almost unchanged (b); with the increase of the in-plane thickness t2, the force amplitude gradually increases, and the maximum stress is positively correlated with t 2 (c); with the increase of the span λ2, the force amplitude first increases and then decreases, and the maximum stress is negatively correlated with Lambda 2 (d). The results show that the force amplitude decreases with the increase of the camber height h2 of the positive stiffness cosine beam, and increases with the increase of the span λ2, the out-of-plane thickness b2 and the in-plane thickness t2; the maximum stress is negatively correlated with h2 and λ2, and positively correlated with t2, and has no obvious correlation with b2.

[0068] The above rules can provide a basis for parameter matching design of positive and negative stiffness units in actual engineering, to ensure that the linear restoring force output is stable while the stress is always controlled within the elastic limit.

[0069] In this embodiment, by changing one or some of the above key parameters, the parameter influence law is determined according to the effect corresponding to the parameter change, and finally the overall stiffness is ensured to be offset near the static equilibrium position through collaborative control, to realize a stable quasi-zero stiffness range, while ensuring that the structure works stably and has the required carrying capacity within the elastic range, and an optimized product is obtained.

[0070] The above merely provides an example of the present application, and the protection scope of the present application is not limited to these specific examples, but is determined by the claims of the present application. Various modifications and changes can be made to the present application by those skilled in the art. Any modification, equivalent replacement, improvement, etc. within the technical thought and principle of the present application shall be included in the protection scope of the present application.

Claims

1. A quasi-zero stiffness metamaterial structure based on a titanium alloy cosine beam, characterized in that, Composed of a vertical one-dimensional array of metamaterial unit cells, each metamaterial unit cell includes two mirror-image unit cells and a connecting block; the connecting block is used to connect the upper and lower unit cells; each unit cell includes a positive stiffness cosine beam, a negative stiffness cosine beam, a plate, and a support frame disposed on the plate; the positive stiffness cosine beams in each unit cell include two that are arranged in parallel and vertically within the support frame, with the ends of the two positive stiffness cosine beams away from the plate connected to the two ends of a horizontal end face of the connecting block, and the negative stiffness cosine beams are separated by the aforementioned horizontal end face of the connecting block to form two symmetrical segments arranged on the outside of the positive stiffness cosine beams, with the outer ends of the negative stiffness cosine beams connected to the support frame to form a closed frame.

2. The quasi-zero stiffness metamaterial structure based on a titanium alloy cosine beam according to claim 1, characterized in that, The support frame includes a horizontal section fixed to the plate and a vertical section connected to both ends of the horizontal section; the negative stiffness cosine beam is connected between the outer ends of the two vertical sections of the support frame; the negative stiffness cosine beam bends away from the plate; the two positive stiffness cosine beams in each unit cell are arranged vertically in mirror image intervals, wherein the middle part of one positive stiffness cosine beam bends toward the other positive stiffness cosine beam that is vertically mirror image of it.

3. The quasi-zero stiffness metamaterial structure based on a titanium alloy cosine beam according to claim 2, characterized in that, The length of the horizontal section of the support frame does not exceed the outer edge of the plate.

4. The quasi-zero stiffness metamaterial structure based on a titanium alloy cosine beam according to claim 1, characterized in that, Let the span of the negative stiffness cosine beam be λ1, the arc height be h1, the in-plane thickness be t1, and the out-of-plane thickness be b1; let the span of the positive stiffness cosine beam be λ2, the arc height be h2, the in-plane thickness be t2, and the out-of-plane thickness be b2; in the unit cell element, λ1 is the horizontal span of each negative stiffness cosine beam segment divided by the connecting blocks; Then it satisfies: , In the formula, The elastic modulus of the material. Moment of inertia of a negative stiffness cosine beam section: , Moment of inertia of the cosine beam section with positive stiffness: ; dimensionless coefficient The semi-empirical formula is: 。 5. The quasi-zero stiffness metamaterial structure based on a titanium alloy cosine beam according to claim 1, characterized in that, The connecting block is an integrated structural block with upper and lower mirror images.

6. The quasi-zero stiffness metamaterial structure based on a titanium alloy cosine beam according to claim 1, characterized in that, The metamaterial unit cells are periodically arrayed in the vertical direction, and two adjacent metamaterial unit cells share a plate.

7. The quasi-zero stiffness metamaterial structure based on a titanium alloy cosine beam according to claim 1, characterized in that, The metamaterial structure's plate, support frame, positive stiffness cosine beam, negative stiffness cosine beam, and connecting block are integrally formed from titanium alloy, or the metamaterial structure's support frame, positive stiffness cosine beam, and negative stiffness cosine beam are integrally formed and then fixedly connected to the plate and connecting block.

Images