A two-dimensional quasi-zero stiffness lattice device and a design method thereof

Abstract

The application discloses a two-dimensional quasi-zero stiffness lattice device, a vibrator unit of which comprises a square frame, fixed blocks, bent slender beams and a center mass block; four fixed blocks are installed at the centers of four edges of the square frame, the outer contour of the center mass block is square, a hollow circle is arranged at the center of the center mass block, and the four edges of the center mass block and the fixed blocks opposite to the four edges are respectively connected rigidly through a group of bent slender beams, each group of bent slender beams comprises two bent slender beams and is arranged symmetrically about a center axis; the application further discloses a design method of the device; the quasi-zero stiffness characteristics of the vibrator unit in x-axis and y-axis directions are realized by designing the geometric parameters of the bent slender beams; the mass of the center mass block is controlled by adjusting the radius of the hollow circle of the center mass block, so that the control of the natural frequency of the vibrator unit is realized. The structure has the quasi-zero stiffness characteristics by the bent slender beams, and the fine geometric design and the topological optimization are easier to realize.

Description

TECHNICAL FIELD

[0001] The present application relates to the field of structural vibration control and impact resistance, and particularly relates to a two-dimensional quasi-zero stiffness lattice device and a design method thereof. BACKGROUND

[0002] In engineering and architectural structures, most environmental noise and vibration sources, such as traffic, wind, mechanical operation, etc., generate energy mainly concentrated in the low-frequency region. Low-frequency waves have stronger penetrating power and carry more energy, easily penetrating through structures and materials, causing widespread vibration and noise problems. Studies have shown that low-frequency waves propagate in structures with low natural frequencies to form a band gap feature, and the frequency range of the band gap prevents the further propagation of low-frequency waves. The vibration and noise reduction characteristics of such low-natural-frequency structures have attracted widespread attention from many scholars.

[0003] The lattice structure form with quasi-zero stiffness has the property of low natural frequency, and its "high static and low dynamic" nonlinear characteristics can better isolate low-frequency and ultra-low-frequency vibrations. Traditional quasi-zero stiffness vibration isolation equipment usually involves complex positive and negative stiffness parallel mechanisms, although the band gap effect in the low-frequency region is achieved, but the complex structure form is difficult to carry out fine geometric design and topological optimization. In addition, once the metamaterial structure is manufactured, the width and center frequency of the band gap cannot be changed, therefore, the traditional quasi-zero stiffness vibration isolation equipment is difficult to achieve precise control of the natural frequency, thereby limiting its promotion in specific applications. SUMMARY

[0004] The main purpose of the present application is to propose a novel two-dimensional quasi-zero stiffness lattice device and a design method thereof, which aims to replace the complex positive and negative stiffness parallel mechanism with a simple bent slender beam, so that the structure has the quasi-zero stiffness characteristic, and it is easier to realize fine geometric design and topological optimization; at the same time, through the programmable parameter design method, the precise control of the natural frequency of the structure can be realized.

[0005] The technical scheme adopted by the present application is:

[0006] A two-dimensional quasi-zero stiffness lattice device is formed by periodically arranging vibrator units in the form of a lattice, the vibrator unit comprising a square frame, a fixed block, a bent slender beam and a center mass block; four fixed blocks are respectively installed at the center positions of the four edges of the square frame, and the center mass block is located at the center position of the square frame, the outer contour of the center mass block is a square, and a hollow circle is arranged at the center; the four edges of the center mass block and the fixed blocks opposite to the four edges are respectively connected by a group of bent slender beams, and the connection forms are all rigid connections, and each group of bent slender beams comprises two bent slender beams which are symmetrically arranged about the center axis; by designing the geometric parameters of the bent slender beams, the vibrator unit formed has quasi-zero stiffness characteristics in the x-axis and y-axis directions; by adjusting the radius of the hollow circle of the center mass block, the mass of the center mass block is controlled, so that the control of the natural frequency of the vibrator unit is realized.

[0007] In the above scheme, the outer contour of the fixed block is a rectangle, and the length of the fixed block is slightly smaller than the length of the edge of the center mass block.

[0008] In the above scheme, the bent slender beam comprises three beams and is in the form of an irregular Z shape, and the eight bent slender beams are completely identical in size.

[0009] In the above scheme, the distance between the two intersection points of the two bent slender beams of the same group and the center mass block is relatively close, and the distance between the two intersection points of the two bent slender beams and the fixed block is relatively far, and the two bent slender beams are in the form of an outward horn.

[0010] In the above scheme, the square frames of adjacent vibrator units are consistent in size and arranged side by side, and the two square frames are rigidly connected.

[0011] Correspondingly, the application also provides a design method of the two-dimensional quasi-zero stiffness lattice device, comprising the following steps:

[0012] S1, a geometric model of the resonator unit is established; structural parameters are extracted, including the geometric parameters of the bent slender beam, the size of the fixed block, the mass of the center mass block and the radius of the hollow circle, wherein the geometric parameters of the bent slender beam include the initial inclination angle of each beam and the node coordinates of the connection between the beams;

[0013] S2, material selection, determine the material parameters; and define the geometric parameters of the bent slender beam as the design variable, wherein the initial inclination angle of each beam of the bent slender beam is defined by a second-order polynomial function; according to the Euler-Bernoulli beam theory, the control equation of the force-displacement of the bent slender beam is obtained;

[0014] S3, define the objective function, define the difference between the maximum and minimum of the y-direction component force in the above control equation as the objective function, and the optimization target is to minimize the difference, that is, in a certain displacement range, the y-axis direction stiffness of the bent slender beam is close to zero;

[0015] S4, determining constraints, including design domain limits, continuity constraints, conditions for avoiding self-intersection when the beam deforms, and maximum stress not exceeding the yield strength of the material;

[0016] S5, solving the objective function to find the minimum value of the objective function, i.e. the minimum stiffness of the vibrator unit corresponding to the bending slender beam geometry optimization result;

[0017] S6, according to the bending slender beam geometry parameters corresponding to the minimum stiffness of the vibrator unit optimized in S5, and the numerical value of the minimum stiffness of the structure, the relationship function of the natural frequency of the vibrator unit structure and the intermediate mass block is obtained, the mass of the center mass block is determined by the hollow circle radius, and further, the functional relationship between the natural frequency of the structure and the hollow circle radius can be obtained; by adjusting the size of the hollow circle of the intermediate mass block, the natural frequency of the vibrator unit is accurately controlled.

[0018] In the above method, step S2 specifically comprises:

[0019] Material selection, determine the material density p of the center mass block and the elastic modulus E and the moment of inertia I of the bending slender beam;

[0020] A second-order polynomial is used to define the inclination angle of each beam cross section of the bending slender beam relative to the horizontal direction, and the second-order polynomial function is:

[0021] θ i (u)=c i0 +c i1 u+c i2 u 2 (1)

[0022] In the formula, u∈[0,1] is a normalized path length variable, coefficients c i0 , c i1 and c i2 are design variables; i=1,2,3, θ i represents the inclination angle of the i-th beam;

[0023] The bending slender beam has four nodes, represented by n j =(n xj ,n yj ), j=1,2,3,4; the four nodes divide the bending slender beam into three sections, and the length of each beam is represented by L i (i=1,2,3); then, the coordinates of the point [L xi (u),L yi (u)] on the center axis of the i-th beam are:

[0024]

[0025] The intersection of the three-segment beam corresponds to the nodes n2=(n x2 ,n y2 ) and n3=(n x3 ,n y3 ) as design variables, and n1 and n4 are fixed nodes; the force-displacement control equation of the folded slender beam is:

[0026]

[0027] In the formula, denotes the inclination angle of the deformation of the i-th segment of the beam, I is the moment of inertia of the folded slender beam, E is the elastic modulus of the folded slender beam, F s and F l are the x-direction component force and y-direction component force at the action point respectively.

[0028] In the above method, step S3 specifically comprises:

[0029] According to the quasi-zero stiffness characteristic of the structure, the component force of the folded slender beam in the y direction is close to a constant, and therefore the ratio of the maximum value and the minimum value of the y-direction component force is selected as the quasi-zero stiffness optimization index:

[0030]

[0031] In the formula, Φ is the quasi-zero stiffness optimization index, F l max and F l min respectively denote the maximum component force and the minimum component force in the y direction;

[0032] The optimization objective is to minimize Φ, which indicates that the y-direction component force changes as little as possible when vibration occurs; the objective function is established as:

[0033]

[0034] In the above method, in step S4, the constraint conditions are defined as follows:

[0035] (1) Design domain restriction: to prevent each beam segment from exceeding the predetermined design domain

[0036] 0≤L xi (u)≤b; (6)

[0037] 0≤L yi (u)≤h; (7)

[0038] In the formula, L xi (u), L yi (u) are respectively the coordinates of the point on the center axis of the i-th segment of the beam in the x direction and the y direction, b is the design domain width, and h is the design domain length;

[0039] (2) Continuity constraint: Ensure the continuity of the folded slender beam at the nodes, especially at the nodes corresponding to the intersection of three beams, which involves ensuring that the angle and slope match at the nodes

[0040] At node n2, we have:

[0041]

[0042] At node n3, we have:

[0043]

[0044] (3) Prevent self-intersection: Limit the inclination angle of the deformed beam to avoid self-intersection of the beam

[0045] max |c i1 u + c i2 u 2 |≤180° (12)

[0046] (4) Stress constraint: Ensure that the maximum stress under each deformed configuration does not exceed the yield strength of the material

[0047] σ max ≤σ y (13)

[0048] where σ max is the maximum stress of the folded slender beam, and σ y is the yield strength of the material of the folded slender beam.

[0049] In the above method, in step S5, based on MATLAB, the optimization function fmincon is used to solve the nonlinear optimization problem with constraints; the optimal solution of the objective function is calculated, and the state equation of the node coordinates n1, n2, n3, n4 of the folded slender beam and the inclination angle of each section of the folded slender beam relative to the horizontal direction, that is, the minimum stiffness value of the vibrator unit and the corresponding geometric parameters of the folded slender beam are calculated.

[0050] In the above method, the specific steps of step 6 are as follows:

[0051] The mass calculation formula of the center mass block with hollow circle is:

[0052] m = ρ (d1 2 - πr 2 )t (14)

[0053] where m is the mass of the center mass block, ρ is the material density, d1 is the side length of the center mass block, r is the hollow circle radius, and t is the thickness of the center mass block;

[0054] According to the natural frequency calculation formula of the vibrator unit with mass block, we have:

[0055]

[0056] In the formula, f represents the natural frequency of the vibrator unit, m is the mass of the center mass, and K is the overall stiffness of the vibrator unit;

[0057] By adjusting the hollow circle radius r, the two-dimensional quasi-zero stiffness lattice device with different natural frequencies is accurately controlled.

[0058] The beneficial effects of the present application are:

[0059] 1. The device of the present application realizes two-dimensional quasi-zero stiffness in x and y directions by simply bending the slender beam and the mass block with a hollow circle, and it is easier to realize fine geometric design and topological optimization. Low-frequency band gaps are generated under simple harmonic vibration to achieve the purpose of shock absorption.

[0060] 2. The design method of the present application takes the geometric parameters of the bent slender beam and the hollow circle radius of the center mass block of the device as variables, and derives and calculates through the control equation and the optimization objective function to obtain the functional relationship between the natural frequency of the vibrator unit and the variables, so that the natural frequency of the structure can be accurately controlled. In practical application, the structure can be designed according to the requirement of any low-frequency band gap to realize the isolation of low-frequency and ultralow-frequency vibration, and has good practicability and economic benefit. BRIEF DESCRIPTION OF DRAWINGS

[0061] In order to more clearly illustrate the technical solutions in the embodiments of the present application or the prior art, the drawings needed to be used in the embodiments or prior art description will be briefly introduced. Obviously, the drawings in the following description are some embodiments of the present application, and other drawings can be obtained by those skilled in the art without creative labor.

[0062] Figure 1 is a structural schematic diagram of the two-dimensional quasi-zero stiffness lattice device of the present application;

[0063] Figure 2 is a structural schematic diagram of the vibrator unit of the two-dimensional quasi-zero stiffness lattice device of the present application;

[0064] Figure 3 is a three-dimensional structural schematic diagram after the vibrator unit eliminates the square frame;

[0065] Figure 4 is a geometric parameter schematic diagram after the vibrator unit eliminates the square frame;

[0066] Figure 5 is Figure 4Geometric parameters of the bent slender beam in the lower right corner of the middle vibrator unit;

[0067] Figure 6 is the design flowchart of the two-dimensional quasi-zero stiffness lattice device of the application.

[0068] In the figure: 10, square frame; 20, fixed block; 30, bent slender beam; 40, center mass. DETAILED DESCRIPTION

[0069] In order to make the purpose, technical scheme and advantages of the application clearer, the application will be further described in detail below in combination with the drawings and examples. It should be understood that the specific examples described herein are only used to explain the application and do not limit the application.

[0070] It should be noted that the diagrams provided in the embodiments of the application only illustrate the basic concept of the application in a schematic manner, and therefore only the components related to the application are shown in the diagrams, not the number, shape and size of the components when actually implemented. The actual implementation of each component may be a random change in shape, number and proportion, and the layout pattern of the components may be more complex.

[0071] In the application, it should also be noted that, if the terms "center", "upper", "lower", "left", "right", "vertical", "horizontal", "inner", "outer" and the like appear, the indicated orientation or position relationship is based on the orientation or position relationship shown in the drawings, and is only for the convenience of describing the application and simplifying the description, and therefore cannot be understood as indicating or implying that the indicated device or element must have a particular orientation, be constructed and operated in a particular orientation, and therefore cannot be understood as limiting the application. In addition, if the terms "first", "second" appear, they are only for description and distinction purposes, and cannot be understood as indicating or implying relative importance.

[0072] As Figure 1 shown, the application proposes a new two-dimensional quasi-zero stiffness lattice device, which is periodically arranged in the form of a lattice by a vibrator unit, and has a low frequency bandgap characteristic. As Figures 2-4As shown, the vibrator unit includes a square frame 10, four fixed blocks 20, eight bent slender beams 30 and a center mass block 40. The fixed block 20 is rectangular in outline, and the four fixed blocks 20 are respectively installed at the center positions of the four edges of the square frame 10; the center mass block 40 is located at the center position of the square frame 10, and the center mass block 40 is square in outline and has a hollow circle at the center; the four edges of the center mass block 40 and the fixed blocks 20 opposite to them are respectively connected by a group of bent slender beams 30, and the connection forms are all rigid connections, and each group of bent slender beams 30 includes two and is symmetrically arranged about the center axis. By designing the geometric parameters of the eight bent slender beams 30, the vibrator unit formed can have quasi-zero stiffness characteristics in both x-axis and y-axis directions. The mass of the center mass block 40 can be controlled by adjusting the radius of the hollow circle of the center mass block 40, so as to further realize the adjustment of the natural frequency of the vibrator unit.

[0073] Further optimization, the length of the fixed block 20 is slightly smaller than the edge length of the center mass block 40, and the deformation range of the bent slender beam 30 does not exceed the edge length of the center mass block 40.

[0074] Further optimization, as shown in the figure, Figure 5 As shown, the bent slender beam 30 includes three sections of beams and is irregularly Z-shaped, and the eight bent slender beams 30 are completely identical in size.

[0075] Further optimization, the distance between the two intersection points of the two bent slender beams 30 in the same group and the center mass block 40 is relatively close, and the distance between the two intersection points and the fixed block 20 is relatively far, and it is a horn-shaped outward, which can form quasi-zero stiffness in two directions.

[0076] Further optimization, the square frames 10 of the adjacent vibrator units are consistent in size and arranged side by side, and the two are rigidly connected.

[0077] Correspondingly, the application also proposes a design method of the two-dimensional quasi-zero stiffness lattice device, defines a second-order polynomial function containing the geometric parameter information of the bent slender beam in the vibrator unit, establishes the control equation of force and deformation, and then defines the objective function to realize the fine design and topological optimization of the geometric parameters of the bent slender beam; further, the functional relationship between the natural frequency of the vibrator unit and the radius of the hollow circle of the center mass block is derived, and the natural frequency of the vibrator unit is accurately controlled by adjusting the radius of the hollow circle of the center mass block. Specifically, as shown in the figure, Figure 6 The method comprises the following steps:

[0078] S1, CAD aided design is used to establish a geometric model of the resonator unit; structural parameters are extracted, including geometric parameters of the bent slender beam, size of the fixed block (length T1, width T2), mass of the center mass block (m) and hollow circle radius (r), wherein the geometric parameters of the bent slender beam include initial inclination angle of each beam segment and node coordinates of connections between segments; thickness of the fixed block and the center mass block are both t.

[0079] S2, material selection is performed to determine material parameters; the geometric parameters of the bent slender beam are defined as design variables, and the initial inclination angle of each beam segment of the bent slender beam is defined by a second-order polynomial function; according to the Euler-Bernoulli beam theory, a control equation of force-displacement of the bent slender beam is obtained. The specific steps are as follows:

[0080] Material selection is performed to determine material density ρ of the center mass block and elastic modulus E and moment of inertia I of the bent slender beam;

[0081] Due to the axial symmetry of the overall structure of the vibrator unit, the bent slender beam located at the lower right corner of the center mass block is selected as the analysis object, and a local coordinate system is established as shown in Figure 5 The coordinate origin is located at the midpoint of the fixed block, the y-axis direction is perpendicular to the long side of the fixed block and points to the center, and the x-axis direction is parallel to the long side of the fixed block. A second-order polynomial is used to define the inclination angle of the cross section of each beam segment of the bent slender beam relative to the horizontal direction, and the second-order polynomial function is:

[0082] θ i (u)=c i0 +c i1 u+c i2 u 2 (1)

[0083] In the formula, u∈[0,1] is a normalized path length variable, coefficients c i0 , c i1 and c i2 are design variables; i=1, 2, 3, and θ i represents the inclination angle of the i-th beam segment.

[0084] The bent slender beam has four nodes, which are represented by n j =(n xj ,n yj ), j=1, 2, 3, 4; the four nodes divide the bent slender beam into three segments, and the length of each beam segment is represented by K i (i=1, 2, 3); then, the coordinates of the point [L xi (u), L yi (u)] on the center axis of the i-th beam segment are:

[0085]

[0086] The intersection of the three-segment beam corresponds to the node n2 = (n x2 ,n y2 ) and n3 = (n x3 ,n y3 ) as design variables, and n1 and n4 as fixed nodes; the force-displacement control equation of the folded slender beam is:

[0087]

[0088] wherein represents the inclination angle of the deformation of the i-th segment of the beam (i.e., the angle difference before and after deformation), I is the moment of inertia of the folded slender beam, E is the elastic modulus of the folded slender beam, F s and F l are the x-direction component force and y-direction component force at the action point, respectively.

[0089] S3, define the objective function, define the difference between the maximum and minimum values of the y-direction component force in the above control equation as the objective function, and the optimization goal is to minimize this difference, i.e., in a certain displacement range, the y-axis direction stiffness of the folded slender beam is close to zero. The specific steps are as follows:

[0090] From the quasi-zero stiffness characteristics of the structure, Figure 5 the y-direction component force of the folded slender beam shown is close to a constant. Therefore, the ratio of the maximum value to the minimum value of the y-direction component force is selected as the quasi-zero stiffness optimization index:

[0091]

[0092] wherein Φ is the quasi-zero stiffness optimization index, F l max and F l min represent the maximum component force and the minimum component force in the y-direction, respectively;

[0093] The optimization goal is to minimize Φ, which means that the y-direction force component changes as little as possible when vibration occurs; the objective function is established as:

[0094]

[0095] S4, determine the constraint conditions. In order to ensure the feasibility of the optimization problem and the integrity of the structure, a series of constraint conditions are set, including design domain limitation (i.e., length limitation of the beam segment), continuity constraint (i.e., limitation of node coordinates), condition for avoiding self-intersection when the beam deforms, and maximum stress not exceeding the yield strength of the material. The constraint conditions are defined as follows:

[0096] (1) Design domain limitation: prevent each beam segment from exceeding the predetermined design domain

[0097] 0 ≤ L xi (u) ≤ b; (6)

[0098] 0≤L yi (u)≤h; (7)

[0099] where L xi (u), L yi (u) are the coordinates of the point on the center axis of the ith segment of the folded slender beam in the x and y directions, respectively, b is the width of the design domain, and h is the length of the design domain.

[0100] (2) Continuity constraint: Ensuring the continuity of the folded slender beam at the nodes, especially at the nodes corresponding to the intersection of three segments of the beam, which involves ensuring that the angles and slopes match at the nodes

[0101] At node n2, we have:

[0102]

[0103] At node n3, we have:

[0104]

[0105] (3) Self-intersection prevention: Restricting the inclination angle of the deformed beam to avoid self-intersection of the beam

[0106] max |c i1 u + c i2 u 2 |≤180° (12)

[0107] (4) Stress limitation: Ensuring that the maximum stress under each deformed configuration does not exceed the yield strength of the material

[0108] σ max ≤σ y (13)

[0109] where σ max is the maximum stress of the folded slender beam, and σ y is the yield strength of the material of the folded slender beam.

[0110] S5, use MATLAB-based optimization function to solve the objective function, find the minimum value of the objective function, which is the minimum stiffness of the vibrator unit corresponding to the folded slender beam geometric parameter optimization result. The specific steps are as follows:

[0111] Based on MATLAB, use the optimization function fmincon to solve the nonlinear optimization problem with constraints. Calculate the optimal solution of the objective function corresponding to the folded slender beam node coordinates n1, n2, n3, n4 and the state equation of the inclination angle of each segment of the folded slender beam relative to the horizontal direction, that is, the minimum stiffness value of the vibrator unit and the corresponding folded slender beam geometric parameters can be calculated. The following is the basic steps of using fmincon to solve the optimization problem:

[0112] 5.1 Define the objective function: First, a MATLAB function needs to be defined, which accepts the design variables as input and returns the value of the objective function. In this embodiment, the objective function refers to formula (5).

[0113] 5.2 Define the constraints: Define the algebraic constraints and inequality constraints in the form of function handles, which should return the value corresponding to the constraint condition. In this embodiment, the constraint conditions refer to the algebraic constraints (8) (9) (10) (11) and the inequality constraints (6) (7) (12) (13).

[0114] 5.3 Call the fmincon function: Use the fmincon function to iteratively solve the objective function. Get the minimum value of the difference between the maximum and minimum values of the y-direction component force and the corresponding bending slender beam geometric parameters.

[0115] S6, according to the optimized vibrator unit minimum stiffness corresponding to the bending slender beam geometric parameters, and the numerical value of the structure minimum stiffness, the relationship function of the vibrator unit structure natural frequency and the intermediate mass block is obtained, the mass of the center mass block is determined by the hollow circle radius, further, the function relationship between the structure natural frequency and the hollow circle radius can be obtained; By adjusting the size of the hollow circle of the intermediate mass block, the natural frequency of the vibrator unit is accurately controlled. The specific steps are as follows:

[0116] The mass calculation formula of the center mass block with hollow circle is:

[0117] m = p (d1 2 - p r 2 ) t (14)

[0118] In the formula, m is the mass of the center mass block, p is the material density, d1 is the side length of the center mass block, r is the hollow circle radius, and t is the thickness of the center mass block;

[0119] According to the natural frequency calculation formula of the vibrator unit with mass block, we have:

[0120]

[0121] In the formula, f represents the natural frequency of the vibrator unit, m is the mass of the center mass block, and K is the overall stiffness of the vibrator unit;

[0122] By adjusting the hollow circle radius r, the two-dimensional quasi-zero stiffness lattice device with different natural frequencies is designed.

[0123] It should be noted that, according to the needs of implementation, each step / component described in the present application can be split into more steps / components, or two or more steps / components or part operations of the steps / components can be combined into a new step / component, to achieve the purpose of the present application.

[0124] The size of the serial number of each step in the above embodiment does not mean the order of execution, and the execution order of each process should be determined according to its function and inherent logic, and should not constitute any limitation on the implementation process of the embodiments of the present application.

[0125] It should be understood that, for those skilled in the art, improvements or changes can be made according to the above description, and all these improvements and changes shall fall within the protection scope of the appended claims of the present application.

Claims

1. A design method for a two-dimensional quasi-zero stiffness lattice device, wherein the two-dimensional quasi-zero stiffness lattice device is formed by periodically arranging vibrator units in a lattice pattern, wherein each vibrator unit includes a square frame, fixed blocks, bent slender beams, and a central mass block; four fixed blocks are respectively installed at the center of the four sides of the square frame, and the central mass block is located at the center of the square frame, the outer contour of the central mass block being square and having a hollow circle at its center; the four sides of the central mass block and the fixed blocks opposite to it are respectively connected by a set of bent slender beams, all of which are rigid connections, each set of bent slender beams including two beams and arranged symmetrically about the central axis; by designing the geometric parameters of the bent slender beams, the resulting vibrator unit simultaneously possesses quasi-zero stiffness characteristics in both the x-axis and y-axis dimensions; The design method includes the following steps: S1. Establish the geometric model of the resonator unit; extract structural parameters, including the geometric parameters of the bent slender beam, the size of the fixed block, the mass of the central mass block, and the radius of the hollow circle. The geometric parameters of the bent slender beam include the initial inclination angle of each beam segment and the coordinates of the nodes connecting each beam segment. S2. Material selection and determination of material parameters; and definition of geometric parameters of the bent slender beam as design variables, wherein the initial inclination angle of each segment of the bent slender beam is defined by a second-order polynomial function; and the force-displacement control equation of the bent slender beam is obtained according to the Euler-Bernoulli beam theory. S3. Define the objective function. The difference between the maximum and minimum values ​​of the component force in the y-direction in the above control equation is defined as the objective function. The optimization objective is to minimize this difference, that is, within a certain displacement range, the stiffness of the y-axis of the bent slender beam is close to zero. S4. Determine the constraints, including design domain restrictions, continuity constraints, conditions to avoid self-intersection during beam deformation, and the maximum stress not exceeding the material's yield strength. S5. Solve for the objective function and find the minimum value of the objective function, which is the result of optimizing the geometric parameters of the bent slender beam corresponding to the minimum stiffness of the vibrator element. S6. Based on the geometric parameters of the bent slender beam corresponding to the minimum stiffness of the vibrator unit optimized in S5, and the obtained value of the minimum stiffness of the structure, the relationship function between the natural frequency of the vibrator unit structure and the intermediate mass block is obtained. The mass of the central mass block is determined by the radius of the hollow circle. The functional relationship between the natural frequency of the structure and the radius of the hollow circle is obtained. By adjusting the size of the hollow circle of the intermediate mass block, the natural frequency of the vibrator unit is precisely controlled.

2. The method of designing a two-dimensional quasi-zero stiffness lattice device according to claim 1, wherein, The bent slender beam consists of three beam segments in an irregular Z-shape, and the eight bent slender beams are all of the same size.

3. The method of designing a two-dimensional quasi-zero stiffness lattice device according to claim 1, wherein, The two bent slender beams in the same group are relatively close to each other at the two intersection points with the central mass block, and relatively far from each other at the two intersection points with the fixed block. The two bent slender beams form an outward-facing trumpet shape.

4. The method of designing a two-dimensional quasi-zero stiffness lattice device according to claim 1, wherein, The square frames of adjacent vibrator units are of the same size and arranged side by side, with a rigid connection between them.

5. The method of designing a two-dimensional quasi-zero stiffness lattice device according to claim 1, wherein, Step S2 specifically includes: Material selection involves determining the material density ρ of the central mass block and the elastic modulus E and moment of inertia I of the bent slender beam. The inclination angle of each cross section of a bent slender beam relative to the horizontal direction is defined using a second-order polynomial. The second-order polynomial function is: （1） wherein is a normalized path length variable, the coefficient , and are design variables; , denotes the inclination angle of the segment beam; The slender beam being bent has four nodes, using express, These four nodes divide the slender, bent beam into three segments, each segment's length is... ( Indicate; then, the first Points on the central axis of the beam segment [ The coordinates of ] are: (2) Three-segment beam intersection corresponding node And is a design variable, ; the force-displacement control equation of the bent slender beam is: (3) wherein represents the inclination angle of the segment beam deformation, I is the moment of inertia of the bent elongated beam, E is the modulus of elasticity of the bent elongated beam, represents the inclination angle of the segment beam deformation, I is the moment of inertia of the bent elongated beam, E is the modulus of elasticity of the bent elongated beam, and are the x-direction and y-direction components of the force, respectively.

6. The method of designing a two-dimensional quasi-zero stiffness lattice device according to claim 5, wherein, Step S3 specifically includes: Based on the quasi-zero stiffness characteristics of the structure, the component force in the y-direction of the bent slender beam is close to a constant. Therefore, the ratio of the maximum to minimum value of the y-direction component force is chosen as the quasi-zero stiffness optimization index. = (4) where Φ is the quasi-zero stiffness optimization index, and Fy,max and Fy,min represent the maximum and minimum force components in the y direction, respectively. The optimization objective is to minimize Φ, which means minimizing the change in the force component in the y-direction when vibration occurs; the objective function is established as follows: Min = (5)。 7. The method of designing a two-dimensional quasi-zero stiffness lattice device according to claim 5, wherein In step S4, the constraints are defined as follows: (1) Design domain constraint: to prevent each beam segment from exceeding the predetermined design domain. ；(6) ； (7) In the formula, , The first The coordinates of a point on the central axis of the beam segment in the x and y directions, b is the width of the design domain, and h is the length of the design domain; (2) Continuity constraints: Ensure the continuity of the bent slender beam at the nodes, especially at the nodes corresponding to the intersection of three beam segments. This involves ensuring that the angles and slopes match at the nodes. At node There are: (8) (9) At node There are: (10) (11) (3) Prevent self-intersection: Limit the inclination angle of deformable beams to avoid self-intersection of beams. (12) (4) Stress limitation: Ensure that the maximum stress under each deformation configuration does not exceed the yield strength of the material. (13) In the formula, The maximum stress when bending a slender beam. This represents the yield strength of the material used to bend a slender beam.

8. The design method of the two-dimensional quasi-zero stiffness lattice device according to claim 1, characterized in that, In step S5, the constrained nonlinear optimization problem is solved using the optimization function fmincon based on MATLAB. Calculate the nodal coordinates of the bent slender beam corresponding to the optimal solution of the objective function. , , By using the state equations for the inclination angle of each segment of the bent slender beam relative to the horizontal direction, the minimum stiffness value of the vibrator element and the corresponding geometric parameters of the bent slender beam can be obtained.

9. The design method of the two-dimensional quasi-zero stiffness lattice device according to claim 1, characterized in that, The specific steps for step 6 are as follows: Formula for calculating the mass of the center mass block with a hollow circle: t (14) In the formula, m The mass of the central mass block, For material density, Let be the side length of the central mass block. Let t be the radius of the hollow circle and t be the thickness of the central mass block. Then, according to the formula for calculating the natural frequency of a vibrator unit with a mass block, we have: (15) In the formula, This represents the natural frequency of the vibrator unit. m The mass of the central mass block, The overall stiffness of the vibrator unit; By adjusting the radius r of the hollow circle, a two-dimensional quasi-zero stiffness lattice device with different natural frequencies can be designed and precisely controlled.

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