Design method of quasi-zero stiffness morse metamaterial and metamaterial

Abstract

The application belongs to the technical field of mechanical design, and relates to a quasi-zero stiffness moire metamaterial design method and a metamaterial. The design method comprises the following steps: S1) rotating the initial periodic plane grid clockwise around the off-plane axis to form a first periodic plane grid; rotating the initial periodic plane grid counterclockwise around the off-plane axis to form a second periodic plane grid; S2) overlapping the first periodic plane grid and the second periodic plane grid to form a third plane grid; S3) cutting a regular polygon region with a moire form in the middle region of the third plane grid to form a basic unit of the metamaterial, and stretching the basic unit in the thickness direction to form a basic structure. The moire metamaterial designed based on the quasi-zero stiffness moire metamaterial design method provided in the application can not only obtain a wide platform curve topology with different load requirements, but also realize the integration of low-frequency vibration isolation and high-efficiency impact energy absorption.

Description

Technical Field

[0001] This invention belongs to the field of mechanical design technology, and relates to a design method and metamaterial of quasi-zero stiffness Mohr metamaterial. Background Technology

[0002] In the field of high-end equipment (such as precision machining equipment, spacecraft, shipborne electronic systems, etc.), vibration and shock control has always been a core bottleneck restricting its reliability and accuracy. In particular, low-frequency and even ultra-low-frequency vibrations, due to their long wavelengths and slow attenuation, are difficult to be effectively suppressed by traditional passive vibration isolators (such as rubber and metal springs); while impact loads (such as launch, landing, and impact) can easily cause plastic deformation or fatigue failure in critical parts of the structure.

[0003] While existing metamaterials can achieve low-frequency vibration isolation through quasi-zero stiffness design, most configurations rely on specific geometries (such as buckling beams and inclined springs), resulting in weak load-bearing capacity. The quasi-zero stiffness mechanical platform has a limited width and usually focuses only on single vibration isolation performance, lacking system integration of multiple functions such as load-bearing, vibration isolation, and energy absorption, making it difficult to meet practical engineering needs. Summary of the Invention

[0004] To address the problems of weak load-bearing capacity, limited width of quasi-zero stiffness mechanical platform, and single performance of existing vibration isolators that are difficult to meet practical engineering needs, this application provides a design method and metamaterial for a multifunctional quasi-zero stiffness moiré metamaterial with load-bearing, vibration isolation, and energy absorption capabilities. This will help achieve multifunctional protection for high-end equipment under complex service conditions.

[0005] The technical solution of this application is as follows: In a first aspect, this application provides a design method for quasi-zero stiffness Mohr metamaterials, comprising the following steps: S1) Rotate the initial periodic planar mesh clockwise around the outer axis to form a first periodic planar mesh; rotate the initial periodic planar mesh counterclockwise around the outer axis to form a second periodic planar mesh; wherein, the initial periodic planar mesh is a regular polygon and is composed of rods and nodes, wherein the rods are the sides of the regular polygon, and the nodes connect the two rods corresponding to adjacent sides; S2) Overlap the first periodic planar grid with the second periodic planar grid to form a third planar grid; S3) Extract a regular polygonal region with a moiré shape from the middle region of the third planar grid to form the basic unit of the metamaterial; stretch the basic unit in the thickness direction to form the basic structure.

[0006] In step S1), the initial state of the periodic planar mesh can be a square, a regular hexagon, an octagon, etc. The nodes can be circular or other shapes. The rods can be cuboids or any shape with symmetrical cross-sections and longitudinal sections.

[0007] The shape of the regular polygonal region in step S4) can be consistent with the shape of the periodic planar grid in the initial state, so as to obtain a more uniform moiré pattern.

[0008] Mohr's morphology refers to a new interference pattern with greater periodicity that arises when two or more patterns with periodic or quasi-periodic structures (such as grids, gratings, and lattices) are superimposed due to differences in spatial frequencies. This pattern manifests as alternating bright and dark, wavy, or distorted stripes, the shape of which depends on the period, angle, and shape of the original structure.

[0009] The mechanism of this application lies in the fact that the basic structure of the moiré-shaped metamaterial is composed of multiple elastic rods and nodes, exhibiting a spatially non-uniform stiffness distribution. During axial compression, the weaker rod regions undergo spatial bending and torsional deformation first, locally producing buckling behavior similar to a cosine beam, thus displaying significant negative stiffness characteristics in specific sections of the force-displacement curve; simultaneously, the unbuckled rods and node constraints continuously provide a positive stiffness response. This spatial distribution and competition of positive and negative stiffness enable the overall structure to achieve a flat mechanical plateau, i.e., quasi-zero stiffness characteristics, over a wide displacement range.

[0010] Low-frequency vibration isolation mechanism: The essence of the quasi-zero stiffness platform is the force balance after positive and negative stiffness elements are connected in series. After yielding, the sparse members enter the negative stiffness region (the load-bearing capacity decreases with increasing displacement), while the bending of the dense members provides positive stiffness (the load-bearing capacity increases with increasing displacement). After being connected in series, the total load-bearing capacity is approximately constant over a wide displacement range, while the total stiffness approaches zero. Static pre-compression causes the system to operate at the midpoint of the platform, where the static stiffness still maintains a high value to support the load, while the dynamic stiffness is close to zero, thus reducing the vibration isolation initiation frequency to a very low level. At the same time, the different rotation angles of the two grids introduce chirality, generating compressive-torsional coupling under axial compression: axial compression causes structural torsion, and the torsional component intervenes to change the main load path, converting part of the axial stress into shear stress, delaying the local buckling of the sparse members, and expanding the positive and negative stiffness balance range to a wider displacement range, ultimately achieving ultra-wideband vibration isolation.

[0011] Impact energy absorption mechanism: The constant force characteristic of the quasi-zero stiffness plateau region originates from the series force balance between the positive and negative stiffness regions. After buckling, sparse members enter the negative stiffness stage, and their load-bearing capacity decreases with increasing displacement; the bending of dense members provides positive stiffness, and the load-bearing capacity increases with increasing displacement. After the two are connected in series, the total load-bearing capacity is the sum of the two, which is approximately constant over a wide displacement range. When the impact kinetic energy is converted into the elastic potential energy of the structure, the constant force plateau ensures that the peak acceleration is limited within a preset threshold, avoiding impact overload. Energy dissipation mainly comes from three aspects: ① Buckling hysteresis energy dissipation: During the buckling process, the force-displacement curve of sparse members forms a hysteresis loop, converting kinetic energy into internal energy; ② Compression-torsion coupling energy dissipation: Chiral structures convert part of the axial compressive energy into torsional strain energy, which is dissipated through internal friction and nodal contact during torsional recovery.

[0012] In summary, the moiré metamaterials designed using the quasi-zero stiffness moiré metamaterial design method provided in this application can not only obtain wide plateau curve topologies with different load requirements, but also achieve the integration of low-frequency vibration isolation and efficient impact energy absorption.

[0013] In one possible design, step S4 is included after step S3): the geometry, clockwise rotation angle, counterclockwise rotation angle and side length of the basic element of the initial planar mesh are optimized to maximize the width of the quasi-zero stiffness mechanical platform and obtain the corresponding force-displacement curve.

[0014] In one possible design, step S4) specifically includes: S41) Using the initial planar mesh geometry, clockwise rotation angle, counterclockwise rotation angle, and edge length of the basic element as input variables, and two objective functions Φ1 and Φ2 as output responses, a Kriging surrogate model is constructed and validated; where the first objective function Φ1 is the average value of the absolute deviation between the actual force and the target load; and the second objective function Φ2 is the average value of the square of the relative error between the actual force and the target load. S42) Based on the Kriging surrogate model, the NSGA-II multi-objective genetic algorithm is introduced to minimize two objective functions Φ1 and Φ2 as the optimization objective. The dual-objective constraint optimization is performed to find the non-dominated solution set that can simultaneously minimize Φ1 and Φ2. S43) For the non-dominated solution set output in step S42), the TOPSIS method is used to screen the optimal compromise solution. The optimal solution that takes into account both the accuracy and flatness of the mechanical platform is selected as the optimal compromise geometric characteristic parameter combination for the target load, and the corresponding force-displacement curve is output.

[0015] Based on the design method of quasi-zero stiffness moiré metamaterials provided by this implementation method, the geometric characteristic parameters of the quasi-zero stiffness moiré metamaterials are optimized through efficient Kriging-NSGA-II-TOPSIS combination, realizing a multi-functional integration of load-bearing, vibration isolation, and energy absorption. The target load of the basic structure is 10~30N, and the target load of the quasi-zero stiffness moiré metamaterials is 40~120N. The quasi-zero stiffness plateau ratio can reach 85%, achieving ultra-wideband vibration isolation (1-20000Hz) under dynamic conditions. The absorption ratio reaches 66.4% within the plateau's operating range, and it still maintains about 30% dissipation capacity beyond the plateau. This can solve the problem of simultaneously achieving load-bearing capacity, low-frequency vibration isolation, and impact protection in high-end equipment.

[0016] Kriging+NSGA-II+TOPSIS is a mature combination for handling costly black-box multi-objective optimization, especially suitable for scenarios where there is no explicit analytical expression between geometric parameters and mechanical response, and where finite element computation is costly. Other methods, such as neural network surrogate models, Bayesian optimization, or multifidelity optimization, can also be used as alternatives or complements. The specific choice requires a trade-off between sample efficiency, model accuracy, and computational resources.

[0017] In one possible design, step S43) is followed by: S44) Substitute the optimal compromise geometric feature parameter combination obtained in step S43) back into the complete finite element model of the basic structure for high-fidelity numerical calculation to verify whether the two objective functions Φ1 and Φ2 under this parameter combination match the predicted values ​​of the Kriging surrogate model in step S41). If they match, output the parameter combination as the final optimization result. If they do not match, add the newly obtained finite element simulation results to the sample points in step S41), reconstruct the Kriging surrogate model, and repeat steps S41) to S44) until convergence.

[0018] The design method for quasi-zero stiffness moiré metamaterials provided by this implementation method can improve the reliability of geometric characteristic parameter optimization, thereby ensuring the effective performance of the load-bearing, vibration isolation and energy absorption functions of moiré metamaterials.

[0019] In one possible design, the force-displacement curve includes two curve forms: the first curve form is characterized by a peak before the plateau segment, and the second curve form is characterized by a flat mechanical plateau.

[0020] In one possible design, the initial periodic planar mesh is a square with round dots as nodes.

[0021] Setting the initial periodic planar mesh as a square with nodes located at circular points has several beneficial effects: First, the side length, node radius, and member width of the square mesh are independent of each other, decoupling the geometric parameters and making it easier to clearly distinguish the influence of each parameter on positive and negative stiffness, thus achieving precise control of the target performance. Second, after rotating and superimposing two identical square meshes, periodic moiré fringes are generated, forming a symmetrical and predictable sparse-dense alternation region, providing a structural basis for the generation of a quasi-zero stiffness platform. Third, the member axis passes through the center of the circular node, avoiding eccentric connection, so that the node mainly bears the normal force rather than the bending moment during compression, maintaining symmetrical deformation and uniformly transferring the load, preventing stress concentration. In addition, the square basic unit is isotropic in-plane, and when uniformly stretched out-of-plane to form a three-dimensional structure, it can avoid torsion or warping, ensuring uniform out-of-plane stiffness and making it less prone to out-of-plane instability. At the same time, the square mesh is the simplest space-filling polygon, and the setting of nodes as circular points also simplifies the mathematical expression of geometric relationships, which is beneficial for surrogate model training. Finally, squares and circles are common basic shapes, reducing manufacturing difficulty and improving sample repeatability and batch production consistency. These advantages together create favorable conditions for the formation of a series of positive and negative stiffness and an ultra-wide quasi-zero stiffness platform.

[0022] In one possible design, the geometry of the optimized initial state planar mesh in step S4) includes the width D of the rod, the distance L0 between two nodes on the same side, and the radius R of the node.

[0023] Simultaneous optimization of three parameters—member width D, node spacing L0, and node radius R—offers significant advantages over optimizing only two parameters: D primarily controls the member's bending stiffness and yield load; L0 determines the member's slenderness ratio, thus influencing the critical buckling force and the displacement range where negative stiffness occurs; and R adjusts stress concentration and local rotational constraints at the nodes. Simultaneous optimization of all three completely decouples the effects of each parameter on the positive stiffness amplitude, the starting point of negative stiffness, and the plateau width, avoiding the limitations of two-parameter optimization where performance cannot be independently controlled due to parameter coupling. Furthermore, the synergistic effect of these three parameters significantly expands the design space, allowing for different combinations under a wider range of loads. A flat plateau is formed within the displacement plane (e.g., increasing D to increase the plateau load, increasing L0 to widen the plateau after buckling, and increasing R to enhance nodal constraints and suppress premature failure), thereby maximizing the proportion of the plateau to the compression stroke (experimentally reaching 85%). Simultaneously, three-parameter optimization improves engineering robustness: when one parameter has manufacturing tolerances, the other two parameters can compensate for performance fluctuations and avoid sensitive designs; a larger parameter space can also effectively avoid local optima and release better configurations limited by fixed parameters.

[0024] In one possible design, the basic structure is made of polyurethane. Alternatively, the basic structure of the quasi-zero stiffness moiré metamaterial can also be made of silicone rubber or polyether-based TPU.

[0025] Secondly, based on the same inventive concept, this application also provides a basic structure designed using the above-mentioned design method for quasi-zero stiffness moiré metamaterials.

[0026] Thirdly, based on the same inventive concept, this application also provides a metamaterial, which includes multiple basic structures. The multiple basic structures are basic structures designed using any of the above-mentioned design methods, and the multiple basic structures are symmetrically arranged around a central axis.

[0027] Based on the metamaterial provided by this implementation, firstly, the axisymmetric layout can balance the stress state of each basic structure during compression. When a load is applied to the top of the structure, the symmetrical arrangement ensures that each basic structure bears a similar axial force, avoiding local overload or premature buckling caused by eccentricity, thereby maintaining the smoothness of the force-displacement curve and the stability of the plateau region as a whole. Secondly, the axisymmetric arrangement is conducive to the uniform transmission of compressive and torsional effects: since each basic structure has chirality (α≠β), the axisymmetric arrangement can coordinate the torsional directions of adjacent units, and the torque of the overall structure can cancel each other out or form a symmetrical deformation mode, preventing the overall structure from undergoing out-of-plane rotation or torsion, and ensuring the consistency of compression deformation. Thirdly, from a manufacturing and assembly perspective, the axisymmetric structure facilitates modular design and rapid assembly. Multiple identical basic structures can be arranged radially around a central axis to form cylindrical or annular metamaterials, which are suitable for cylindrical equipment requiring isotropic vibration isolation or impact protection (such as spacecraft sections, pipe supports, and vibration damping bases). Finally, axisymmetric layout also helps simplify boundary conditions in multi-objective optimization, reducing the number of design variables (only the geometric parameters of a single basic structure need to be optimized, and then the overall structure can be obtained by rotation and replication), greatly reducing computational costs. Therefore, axisymmetric setup around a center is an important layout strategy that balances mechanical performance, manufacturing feasibility, and computational efficiency. Attached Figure Description

[0028] Figure 1 This is a schematic diagram of the design method for the basic unit of a quasi-zero stiffness moiré metamaterial.

[0029] Figure 2 It is the first design target curve for the mechanical response of quasi-zero stiffness Mohr metamaterials.

[0030] Figure 3 This is the second design target curve for the mechanical response of quasi-zero stiffness Mohr metamaterials.

[0031] Figure 4 This is the basic structural optimization process for metamaterials.

[0032] Figure 5 This is a schematic diagram of the finite element model of the basic structure of metamaterials.

[0033] Figures 6 to 10 The diagram shows the basic structure of the metamaterial after optimization under different target loads and the corresponding two force-displacement curves.

[0034] Figure 11 This is a schematic diagram of a finite element model of a quasi-zero stiffness moiré metamaterial formed by four basic structures connected in parallel.

[0035] Figure 12 A schematic diagram of the compressive mechanical response of a quasi-zero stiffness molar metamaterial formed by connecting four basic structures that satisfy the first type of force-displacement curve in parallel.

[0036] Figure 13 A schematic diagram of the compressive mechanical response of a quasi-zero stiffness moiré metamaterial formed by connecting four basic structures that satisfy the second type of force-displacement curve in parallel.

[0037] Figure 14 Vibration transmissivity curves of quasi-zero stiffness moiré metamaterials that meet the first force-displacement curve were obtained from vibration isolation experiments.

[0038] Figure 15 Vibration transmissivity curves of quasi-zero stiffness moiré metamaterials that satisfy the second type of force-displacement curve were obtained from vibration isolation experiments.

[0039] Figure 16 This is a schematic diagram showing the energy variation of M1_40 supermolar material at different impact object heights.

[0040] Figure 17 This is a schematic diagram showing the energy variation of the M2_40 supermolar material at different impact object heights.

[0041] Figure 18 This is a schematic diagram showing the energy changes of the M1_40 supermolar material under different impact object masses.

[0042] Figure 19 This is a schematic diagram showing the energy changes of the M2_40 supermolar material under different impact object masses. Detailed Implementation

[0043] To facilitate understanding of the present invention, a more complete description will be given below with reference to the accompanying drawings. Preferred embodiments of the invention are shown in the drawings. However, the invention can be implemented in many different forms and is not limited to the embodiments described herein. Rather, these embodiments are provided to provide a thorough and complete understanding of the disclosure of the invention.

[0044] The following detailed description of one embodiment of this application is based on the accompanying drawings.

[0045] See Figure 1 The quasi-zero stiffness Mohr metamaterial design method provided by this invention includes the following steps: S1) Rotate the initial periodic planar mesh clockwise around the outer axis to form a first periodic planar mesh; rotate the initial periodic planar mesh counterclockwise around the outer axis to form a second periodic planar mesh; wherein, the initial periodic planar mesh is square and consists of rods and nodes, wherein the rods are the sides of the square, and the nodes connect the two rods corresponding to adjacent sides; S2) Overlap the first periodic planar grid with the second periodic planar grid to form a third planar grid; S3) A square region with a moiré shape is cut out in the middle region of the third planar grid to form the basic unit of the metamaterial; the basic unit is stretched in the thickness direction to form the basic structure.

[0046] S4) By using a combined optimization method based on the Kriging surrogate model, NSGA-II multi-objective genetic algorithm and TOPSIS decision-making, the geometric characteristic parameters of the basic unit of the metamaterial can be controlled to obtain two curved topologies for different target loads. The target loads of the basic structure are 10N, 15N, 20N, 25N and 30N.

[0047] Optimization objective: To maximize the width of the quasi-zero stiffness mechanical platform, a multi-objective optimization framework is established. By simultaneously minimizing the normalized residual and variance between the response curve and the target load, a metamaterial configuration with a wide and flat platform within a preset load range is obtained. The optimization involves six geometric characteristic parameters: node radius R, member width D, side length L0, rotation angle α of the first-layer planar mesh, rotation angle β of the second-layer planar mesh, and side length L1 of the basic element.

[0048] The first objective function Φ1 is the average value of the absolute deviation between the actual force and the target load, i.e., the mean absolute error.

[0049]

[0050] Where n is the number of sampling points, F i F is the actual force value at the i-th discrete point. target The first objective is a predefined target load. This index directly quantifies the overall deviation of the response curve relative to the target load; a smaller value indicates that the actual force on the platform segment is closer to the target load. The second objective function Φ2 is the average of the squared relative errors, i.e., the mean square relative error.

[0051]

[0052] This dimensionless index applies a higher penalty to large deviations through squaring, focusing on suppressing potential large local fluctuations or outliers in the force-displacement curve. Φ1 ensures accurate average platform height, while Φ2 ensures the curve remains stable near the target value without drastic jumps. This effectively avoids suboptimal configurations where the solution converges to either "accurate but unstable" (small Φ1 but oscillating curve) or "stable but inaccurate" (small Φ2 but overall deviation from the target).

[0053] like Figure 2 As shown, the first type of curve shape is characterized by a peak before the plateau segment (from positive stiffness to negative stiffness and then to zero stiffness). Figure 3 As shown, the second curve shape is a flat mechanical plateau (from positive stiffness to zero stiffness). The two curve shapes can be defined as a discriminant of the ratio of peak force to the average plateau force.

[0054] (Curve Shape 1) (Curve Shape 2) The constraint problem of the above-mentioned mechanical response platform is then integrated into:

[0055] λ: The threshold value for the ratio of peak force to target platform force, used to distinguish between two force-displacement curve shapes: when When it is in form 1 (with a sudden jump peak), when The timeframe is morphology 2 (smooth transition without peaks). Typically, λ is taken as 1.05–1.20. μ: friction coefficient, set as a constant of 0.2 in finite element simulations, used to evaluate the influence of internal self-contact on the mechanical response. X is the design variable vector, containing six geometric parameters: node radius R, member width D, side length L0, rotation angle α of the first-layer planar mesh, rotation angle β of the second-layer planar mesh, and side length L1 of the basic element. refer to Figure 4 The specific process is as follows: S41) Using the node radius R, bar width D, side length L0, rotation angle α of the first layer of planar mesh, rotation angle β of the second layer of planar mesh, and side length L1 of the basic unit as input variables, and two objective functions Φ1 and Φ2 as output responses, construct the Kriging surrogate model and verify it. Specifically, in step S41), six geometric feature parameters are used as input variables. The Latin hypercube sampling (LHS) method is used to generate a batch of uniformly distributed sample points. The two objective function values ​​Φ1 and Φ2 corresponding to each sample are obtained through finite element simulation, and the initial input matrix X and output matrix Y are obtained.

[0056] Latin hypercube sampling ensures that samples are uniformly covered in the variable space, avoiding the clustering problem of random sampling and providing high-quality basic data for subsequent model fitting.

[0057] Next, by selecting a suitable variogram model, the empirical semivariogram between sample points is calculated, and then the theoretical variogram is fitted to describe the spatial correlation between variables, thus providing a foundation for solving the Kriging equations.

[0058] Based on the well-fitted variogram, a system of Kriging equations is constructed and solved to obtain the regression coefficients and parameters of the Gaussian process, thus establishing a complete Kriging surrogate model. This model represents the response value as the superposition of a global trend term and a local bias term:

[0059] In the Kriging surrogate model, x represents the design variable vector, i.e., the geometric parameters that influence the structural mechanical response. Specifically, in the optimization problem of quasi-zero stiffness Mohr metamaterials, x includes six geometric characteristic parameters: f(x) is the regression basis function, β is the regression coefficient, and z(x) is a Gaussian process with zero mean and non-zero covariance, which can simultaneously capture the overall trend and local fluctuation characteristics of the sample.

[0060] After the model is built, the model accuracy is verified by the coefficient of determination R2 (the threshold is usually set to 0.9): if R2 < 0.9, it means that the model accuracy is insufficient and the variogram model needs to be readjusted, sample points need to be added, and the model needs to be fitted again; if R2 > 0.9, the model passes the verification and can be used for subsequent prediction of unknown points.

[0061] Once validated, the model can replace the original finite element model and quickly predict the objective function value corresponding to any combination of geometric parameters, providing an efficient evaluation tool for subsequent multi-objective optimization.

[0062] S42) Based on the Kriging surrogate model, the NSGA-II multi-objective genetic algorithm is introduced to minimize two objective functions Φ1 and Φ2. Bi-objective constrained optimization is performed to find the non-dominated solution set (Pareto front) that simultaneously minimizes Φ1 and Φ2. The detailed process is as follows: First, set the core parameters of the algorithm (population size, crossover probability, mutation probability, maximum number of iterations), and randomly generate an initial population. Each individual in the population represents a combination of geometric feature parameters.

[0063] By utilizing the validated Kriging surrogate model, the two objective function values ​​corresponding to each individual in the population can be quickly predicted to complete the fitness assessment, avoiding repeated calls to finite element simulation and significantly improving optimization efficiency.

[0064] Based on the two objective function values, the individuals in the population are stratified and sorted to distinguish between dominated and non-dominated solutions, with non-dominated solutions being retained first. The crowding distance for each non-dominated solution is calculated using the following formula:

[0065] In the formula, M represents the number of targets. , These are the function values ​​on the m-th objective side adjacent to the i-th solution. The crowding distance is used to measure the distribution density of solutions in the objective space, ensuring the diversity of the Pareto front and avoiding solutions concentrating in local regions.

[0066] Based on non-dominated sorting and crowding distance, offspring populations are generated through tournament selection, crossover, and mutation operations, driving the population to evolve in a better direction.

[0067] The parent and offspring populations are merged, and the non-dominated sorting and crowding calculations are performed again to select a new population of the same size as the initial population. This ensures that excellent individuals (elite solutions) are not eliminated, while maintaining the diversity of the population.

[0068] Repeat the above process iteratively until the termination condition is met (the maximum number of iterations is reached). At this point, the algorithm converges and outputs the final Pareto optimal solution set, which contains all compromise solutions that cannot improve one objective without sacrificing the other.

[0069] S43) For the non-dominated solution set output in step S42), the TOPSIS method is used to screen the optimal compromise solution. The optimal solution that balances the accuracy and flatness of the mechanical platform is selected as the optimal compromise geometric characteristic parameter combination for the target load, and the corresponding force-displacement curve is output. The detailed process is as follows: Construct an initial decision matrix with all non-dominated solutions of the Pareto front as rows and two objective functions Φ1 and Φ2 as columns, where the matrix elements are the objective function values ​​corresponding to each solution.

[0070] The decision matrix is ​​standardized using the range method to eliminate differences in the dimensions and orders of magnitude of the objective functions, ensuring that the two objectives are comparable in subsequent calculations.

[0071] The entropy weight method is used to calculate the objective weights of two objectives based on the dispersion of the objective function values, avoiding the bias of subjective weighting. The weights reflect the importance of the objectives in decision-making.

[0072] The standardized decision matrix is ​​multiplied by the weights obtained by the entropy weight method to obtain the weighted standardized matrix, which provides the basis for the subsequent calculation of the ideal solution.

[0073] Define the ideal solution That is, the optimal combination of two objectives; negative ideal solution. That is, the worst combination of the two objectives.

[0074] Calculate the distance from each nondominated solution Aj to the positive ideal solution. Distance to the negative ideal solution :

[0075]

[0076] Calculate the relative similarity of each scheme. :

[0077] The value range is [0,1]. The closer the value is to 1, the closer the scheme is to the positive ideal solution and the further away it is from the negative ideal solution, resulting in better overall performance. Sort by largest to smallest, then select The best solution is the optimal compromise that balances platform accuracy and flatness.

[0078] S44) Substitute the optimal compromise geometric feature parameter combination obtained in step S43) back into the complete finite element model of the basic structure for high-fidelity numerical calculation to verify whether the two objective functions Φ1 and Φ2 under this parameter combination match the predicted values ​​of the Kriging surrogate model in step S41). If they match, output the parameter combination as the final optimization result. If they do not match, add the newly obtained finite element simulation results to the sample points in step S41), reconstruct the Kriging surrogate model, and repeat steps S41) to S44) until convergence.

[0079] The geometric characteristic parameters of the basic structure of the metamaterial under the two different curve shapes and five target loads after optimization are shown in Table 1.

[0080] Table 1

[0081] Performance verification: Using the geometric characteristic parameters in Table 1, the thickness of the basic metamaterial structure is 10 mm, and the material is polyurethane (TPU). Tensile tests were conducted on the basic metamaterial structure using an electronic universal testing machine. The experimental data were fitted using the Odgen (n=3) hyperelastic model, and its dimensionless parameter α... oi Shear modulus μ oi and compressibility parameter D oi (i=1, 2, 3) are respectively: i=1, α oi =-2.12, μoi =4.28, D oi =0; i=2, α oi =8.58, μ oi =-6.69, D oi =0; i=3, α oi =-17.16, μ oi =13.43, D oi =0.

[0082] The experimental results show that D oi All values ​​equal zero, indicating that metamaterials are incompressible materials, and the basic structure of metamaterials has an almost unchanged volume when subjected to uniaxial stretching.

[0083] Using the geometric characteristic parameters in Table 1, the basic structure of the metamaterial has a thickness of 10 mm, is made of polyurethane (TPU), and is constructed as follows: Figure 5 The numerical model of the basic structure of the metamaterial is shown. Compressive mechanical response calculations were performed, yielding the metamaterial's mechanical responses under five target loads with two different curve shapes. These are the optimized quasi-zero stiffness mechanical plateau curves for the ten metamaterial structures, as shown in [reference needed]. Figure 6-10 .

[0084] Depend on Figure 6-10 As can be seen, under each target load, two sets of basic structures of metamaterials with different geometric characteristic parameters can be obtained, resulting in two different curve shapes. The first curve shape is characterized by a peak before the plateau segment (from positive stiffness to negative stiffness and then to zero stiffness), and the second curve shape is characterized by a flat mechanical plateau (from positive stiffness to zero stiffness).

[0085] from Figures 6 to 10 As can be seen, the basic structure of the optimized metamaterial has a wide quasi-zero stiffness mechanical platform, with the platform accounting for up to 85%.

[0086] The underlying mechanism can be explained as follows: the basic structure of the moiré-shaped metamaterial consists of multiple elastic rods and circular nodes, exhibiting a spatially non-uniform stiffness distribution. During axial compression, the weaker rod regions undergo spatial bending and torsional deformation first, locally producing buckling behavior similar to a cosine beam, thus displaying significant negative stiffness characteristics in specific sections of the force-displacement curve; simultaneously, the unbuckled rods and node constraints continuously provide a positive stiffness response. This spatial distribution and competition of positive and negative stiffness enable the overall structure to achieve a flat mechanical plateau, i.e., quasi-zero stiffness characteristics, over a wide displacement range.

[0087] Similarly, please refer to Figure 11This example constructs a numerical model by connecting four basic metamaterial structures in parallel between two plates to form a quasi-zero stiffness Mohr's metamaterial. The mechanical response curves of the Mohr's metamaterial under target loads of 40N, 60N, 80N, 100N, and 120N are shown below. Figures 12 to 13 Wherein, M1 indicates that the curve shape of the molar metamaterial is the first type of curve shape, and M2 indicates that the curve shape of the molar metamaterial is the second type of curve shape.

[0088] from Figures 12 to 13 As can be seen, the optimized metamaterial also has a wide quasi-zero stiffness mechanical platform, with a platform ratio of up to 85%.

[0089] Please combine Figure 11 , Figure 14 and Figure 15 A mass body corresponding to the target load was placed on a quasi-zero stiffness moiré metamaterial. A sinusoidal excitation was applied to the lower end of the metamaterial, and the acceleration signals at the lower and upper ends of the metamaterial were detected. Data was collected every 4 Hz, and the vibration isolation performance of the quasi-zero stiffness moiré metamaterial was evaluated by the vibration transmissibility: T = 20log(A out / A in ), where T is the vibration transmissibility, A in and A out These represent the acceleration signals at the input and output terminals, respectively.

[0090] Vibration tests were conducted on six quasi-zero stiffness moiré metamaterials: M1_40, M1_80, M1_120, M2_40, M2_80, and M2_120, with a frequency sweep range of 1-20000 Hz. Vibration transmissibility was calculated, and the vibration transmissibility curves are shown below. Figures 14 to 15 As shown in the diagram. M1 indicates that the molar metamaterial curve is of the first type, and M2 indicates that the molar metamaterial curve is of the second type. 40, 80, and 120 are the target loads, in N.

[0091] Depend on Figure 14 and Figure 15 It can be concluded that the quasi-zero stiffness moiré metamaterial with the M2 curve shape can achieve efficient vibration isolation in an ultra-wide frequency range of 1Hz to 20000Hz; while the M1 curve shape, because it is located in the positive stiffness region after preloading and does not enter the quasi-zero stiffness platform, has an effective vibration isolation starting frequency much higher than that of the M2 shape.

[0092] Impact plates of varying heights and masses are used to freely impact a quasi-zero stiffness moiré metamaterial via a vertical slide rail. The acceleration of the impact plates is measured, with upward acceleration defined as positive. Integrating the acceleration-time curves of the impact data yields the velocity-time relationship, and the energy absorption rate is used to evaluate the energy absorption effect of the quasi-zero stiffness moiré metamaterial. ,inv 0 represents the velocity at which the impact plate impacts the quasi-zero stiffness Mohr metamaterial. v 1 represents the speed of the rebound after the impact plate is struck. This indicates that the acceleration-time data is integrated.

[0093] Impact experiments were conducted on two quasi-zero stiffness moiré metamaterials, M1_40 and M2_40. By changing the mass and height of the impacting object, the acceleration during the entire impact process was recorded, and the input energy, output energy, and energy absorption rate of the two metamaterials under the first impact were calculated.

[0094] like Figures 16 to 19 As shown, the energy absorption rate can reach about 70% within the quasi-zero stiffness mechanical platform range during the impact stroke, while in the overload state, that is, when the impact stroke exceeds the quasi-zero stiffness mechanical platform range, the energy absorption rate can still reach about 30%.

[0095] Please combine Figure 16 and Figure 18 For the M1_40 configuration, the energy absorption rate curve fluctuates significantly regardless of whether the fall height or the added mass changes. This is due to the inherent peak of the M1 curve. At low energy, elastic bending is the main process with limited absorption. At medium energy, although the sparse members begin to yield, the sudden jump causes force instability, and some energy is dissipated in the form of vibration. At high energy, it skips the plateau region and enters the overload region directly, losing the constant force characteristic, but still maintaining an energy absorption rate of about 30%.

[0096] Please combine Figure 17 and Figure 19 The mechanical response curve of the M2_40 configuration has no sudden peaks. It achieves efficient and stable energy absorption through constant force deformation within the platform's working range. Even if the impact energy exceeds the platform's range and enters the overload zone, it still maintains high energy absorption efficiency thanks to the bending of dense members and nodal contact.

[0097] In summary, based on the design method for quasi-zero stiffness Mohr metamaterials provided in this embodiment, the geometric characteristic parameters of the quasi-zero stiffness Mohr metamaterials are optimized through efficient Kriging-NSGA-II-TOPSIS combination, achieving multi-functional integration of load-bearing, vibration isolation, and energy absorption. The target load of the basic structure is 10~30N, and the target load of the quasi-zero stiffness Mohr metamaterials is 40~120N. The quasi-zero stiffness platform accounts for up to 85%, achieving ultra-wideband vibration isolation (1-20000Hz) under dynamic conditions, with good impact energy absorption rate. This can solve the problem of simultaneously achieving load-bearing capacity, low-frequency vibration isolation, and impact protection in high-end equipment.

[0098] The above description is only of the preferred embodiment of the present invention and should not be construed as limiting the scope of the claims. The present invention is not limited to the above embodiments, and variations in its specific structure are permitted. All variations made within the scope of the independent claims of the present invention are also within the scope of protection of the present invention.

[0099] 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 invention pertains. The terminology used herein in the description of the invention is for the purpose of describing particular embodiments only and is not intended to be limiting of the invention. The term "and / or" as used herein includes any and all combinations of one or more of the associated listed items.

Claims

1. A design method for a quasi-zero stiffness moiré metamaterial, characterized in that, Includes the following steps: S1) Rotate the initial periodic planar mesh clockwise around the out-of-plane axis to form the first periodic planar mesh; The initial periodic planar mesh is rotated counterclockwise around the outer axis to form a second periodic planar mesh; wherein the initial periodic planar mesh is a regular polygon and is composed of rods and nodes, wherein the rods are the sides of the regular polygon, and the nodes connect the two rods corresponding to adjacent sides; S2) Overlap the first periodic planar grid with the second periodic planar grid to form a third planar grid; S3) Extract a regular polygonal region with a moiré shape from the middle region of the third planar grid to form the basic unit of the metamaterial; stretch the basic unit in the thickness direction to form the basic structure; After step S3), step S4 is also included: optimizing the geometric dimensions, clockwise rotation angle, counterclockwise rotation angle and side length of the basic unit of the initial planar mesh to maximize the width of the quasi-zero stiffness mechanical platform and obtain the corresponding force-displacement curve.

2. The design method for quasi-zero stiffness Mohr metamaterials according to claim 1, characterized in that, Step S4) specifically includes: S41) Using the geometric dimensions, clockwise rotation angle, counterclockwise rotation angle, and side length of the basic element of the initial planar mesh as input variables, and two objective functions Φ1 and Φ2 as output responses, a Kriging surrogate model is constructed and verified; wherein the first objective function Φ1 is the average value of the absolute deviation between the actual force and the target load; and the second objective function Φ2 is the average value of the square of the relative error between the actual force and the target load. S42) Based on the Kriging surrogate model, the NSGA-II multi-objective genetic algorithm is introduced to minimize two objective functions Φ1 and Φ2 as the optimization objective. The dual-objective constraint optimization is performed to find the non-dominated solution set that can simultaneously minimize Φ1 and Φ2. S43) For the non-dominated solution set output in step S42), the TOPSIS method is used to screen the optimal compromise solution. The optimal solution that takes into account both the accuracy and flatness of the mechanical platform is selected as the optimal compromise geometric characteristic parameter combination for the target load, and the corresponding force-displacement curve is output.

3. The design method for quasi-zero stiffness Mohr metamaterials according to claim 2, characterized in that, Step S43) is followed by: S44) Substitute the optimal compromise geometric feature parameter combination obtained in step S43) back into the complete finite element model of the basic structure for high-fidelity numerical calculation to verify whether the two objective functions Φ1 and Φ2 under this parameter combination match the predicted values ​​of the Kriging surrogate model in step S41). If they match, output the parameter combination as the final optimization result. If they do not match, add the newly obtained finite element simulation results to the sample points in step S41), reconstruct the Kriging surrogate model, and repeat steps S41) to S44) until convergence.

4. The design method for quasi-zero stiffness Mohr metamaterials according to claim 2 or 3, characterized in that, The force-displacement curve includes two curve forms. The first curve form is characterized by a peak before the plateau segment, and the second curve form is characterized by a flat mechanical plateau.

5. The design method for quasi-zero stiffness Mohr metamaterials according to any one of claims 1 to 3, characterized in that, The initial state has a periodic planar grid that is square, and the nodes are round dots.

6. The design method for quasi-zero stiffness Mohr metamaterials according to claim 4, characterized in that, The geometry of the optimized initial state planar mesh in step S4) includes the width D of the rod, the distance L0 between two nodes on the same side, and the radius R of the node.

7. The design method for quasi-zero stiffness Mohr metamaterials according to claim 1, characterized in that, The basic structure is made of polyurethane.

8. A basic structure designed using the design method of the quasi-zero stiffness moiré metamaterial as described in any one of claims 1-7.

9. A metamaterial, characterized in that, It includes multiple basic structures, which are basic structures designed using any one of the design methods described in claims 1 to 7, and the multiple basic structures are arranged symmetrically around a central axis.

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