A porous structure with wide range of in-situ adjustable stiffness

CN122129521BActive Publication Date: 2026-08-14HUAQIAO UNIVERSITY
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Patent Information

Authority / Receiving Office
CN · China
Patent Type
Patents(China)
Current Assignee / Owner
Filing Date
2026-05-06
Publication Date
2026-08-14

AI Technical Summary

Technical Problem

[0003]①材料驱动型依赖智能材料(如形状记忆合金、磁流变液、电活性聚合物)或外置能量输入实现刚度切换,但存在成本高、响应滞后、能耗大等问题,且智能材料的疲劳寿命与长期可靠性有限,难以满足高频次调节需求;

Benefits of technology

[0027]1. 双维度调控,刚度调节精准灵活:通过参数(α、β)与位置的双调控,实现刚度宽范围无间断连续变化,可适配宽范围的刚度需求,解决了现有结构调节维度单一、范围窄的问题;

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Abstract

This application relates to the field of variable stiffness structural design, providing a porous structure with a wide range of in-situ adjustable stiffness, including a pressure-bearing component and a supporting component. The pressure-bearing component has symmetrical elastic ribs on both sides of its transverse direction, with at least two openwork structures on each side of the elastic rib. The supporting component includes a prismatic member that slides between the elastic ribs on both sides when nested. The ratio of the inner and outer diameters of the pressure-bearing component is defined as α = d / D, and the ratio of the height of the prismatic member to the pressure-bearing component is defined as β = h / H, where both α and β are positively correlated with the structural stiffness. The supporting component slides with the pressure-bearing component, and stiffness adjustment is achieved by changing the overlap area between the prismatic member and the elastic ribs through longitudinal adjustment of the supporting component's position. This structure achieves the beneficial effects of simple structure, wide stiffness control range, and multi-dimensional adjustability through the coordinated cooperation of dimensional ratio parameters and the adjustment of the supporting component's position.
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Description

Technical Field

[0001] This application relates to the field of variable stiffness structure design, and more particularly to a porous structure with a wide range of in-situ adjustable stiffness. Background Technology

[0002] Variable stiffness structures can adjust their mechanical properties in real time according to external loads or environmental changes, showing broad application prospects in aerospace (such as morphing wings), soft robotics (such as grasping operations), and biomedical engineering (such as rehabilitation exoskeletons). Existing variable stiffness structures are mainly divided into two categories: material-driven and parametrically adjustable.

[0003] ① Material-driven methods rely on smart materials (such as shape memory alloys, magnetorheological fluids, and electroactive polymers) or external energy input to achieve stiffness switching, but they have problems such as high cost, slow response, and high energy consumption. Moreover, the fatigue life and long-term reliability of smart materials are limited, making it difficult to meet the needs of high-frequency adjustment.

[0004] ② Parametric adjustment type achieves stiffness adjustment by changing geometric parameters (such as structural length, cross-sectional dimensions, and support position), but it is mostly a single-dimensional control, adjusting only one of the length / cross-section / support position. The fineness of stiffness adjustment is low and the dynamic range is limited, making it difficult to adapt to the wide range and continuous adjustability requirements of structural stiffness under complex working conditions.

[0005] In summary, existing variable stiffness structures each have their own applicable boundaries and limitations: material-driven structures are limited by the intrinsic properties of the materials and the power supply system, while parameter-adjustable structures are constrained by the single adjustment dimension. Although some studies have attempted to optimize stiffness characteristics by adjusting the size ratio, they have not combined this with the spatial position adjustment of the supporting components to achieve multi-dimensional coordinated control. This results in a limited range of applicable scenarios for stiffness adjustment and insufficient dynamic response capabilities, making it difficult to meet the complex requirements for dynamic stiffness matching in different scenarios such as aerospace variant structures and human-computer interaction software robots. Summary of the Invention

[0006] In view of the above practical problems and the shortcomings of the existing technology, the main technical problem to be solved by the present invention is to provide a porous structure with a wide range of in-situ adjustable stiffness. By adjusting the structural size ratio parameters and cooperating with the multi-position adjustment of the structural support components, a variable stiffness structure with simple overall structure, wide range of stiffness requirements, and more dimensions of control compared with pure structural parameter adjustment can be achieved.

[0007] To address the aforementioned technical problems, this application provides a porous structure with a wide range of in-situ adjustable stiffness, employing the following technical solution:

[0008] A porous structure with a wide range of in-situ adjustable stiffness includes a pressure-bearing component and a support component;

[0009] The pressure-bearing component is symmetrically provided with elastic ribs on both sides of the transverse direction. At least two hollow structures are opened on the elastic rib on one side, and the hollow structures on both sides are symmetrically arranged. The support component includes a prismatic member. When the support component is nested in the pressure-bearing component, the prismatic member slides between the elastic ribs on both sides.

[0010] The outer diameter of the pressure-bearing component is D, and the inner diameter of the pressure-bearing component is d, which is the minimum distance between the elastic ribs on both sides; the ratio of the inner and outer diameters of the pressure-bearing component is α = d / D; the height of the pressure-bearing component is H, and the height of the prismatic member is h, with a ratio β = h / H; the ratios α and β are used to control the structural stiffness, and the ratios α and β are positively correlated with the structural stiffness;

[0011] The supporting component and the pressure-bearing component are in a sliding fit. The supporting component can slide relative to the pressure-bearing component in the longitudinal direction to adjust the position of the supporting component in the longitudinal direction of the pressure-bearing component. By adjusting the position of the supporting component, the overlap area between the prism-shaped component and the elastic rib is adjusted to achieve structural stiffness adjustment.

[0012] In a preferred embodiment, the elastic rib has an arcuate shape and is disposed on the side of the pressure-bearing component in an inwardly concave arcuate structure.

[0013] The prismatic member has an inwardly concave arc structure along its two transverse side walls in conjunction with the elastic rib; the elastic rib and the prismatic member are fitted together through the inwardly concave arc structure.

[0014] In a preferred embodiment, the elastic rib on one side is divided into at least three rib structures by the at least two openwork structures;

[0015] The supporting component is provided with at least three prismatic members corresponding to at least three rib structures. The spacing between adjacent prismatic members is equal to the grid spacing of the openwork structure. The grid spacing is the unit distance of the openwork structure along the longitudinal direction of the pressure-bearing component.

[0016] In a preferred embodiment, when the overlap area between the prismatic member and the elastic rib is 0%, the prismatic member and the rib structure are misaligned, and at least two of the prismatic members are located in the area where the at least two hollow structures are located;

[0017] When the overlap area between the prismatic member and the elastic rib is 100%, the overlap area is mainly composed of the area of ​​the elastic rib. The prismatic member and the hollow structure are misaligned, and at least two of the prismatic members completely overlap with the rib structure.

[0018] In a preferred embodiment, the position adjustment of the support component is provided with five position adjustment positions, which are set at 0%, 25%, 50%, 75%, and 100% along the longitudinal direction according to the overlap rate of the overlapping area, wherein the overlapping area is based on the overlapping area of ​​the elastic ribs.

[0019] In a preferred embodiment, a sliding component and a positioning component are provided longitudinally at the bottom of the pressure-bearing component. The sliding component includes a groove and a slider, the slider being slidably engaged in the groove and disposed at the bottom of the support component.

[0020] The positioning component includes a positioning groove and an elastic buckle. The positioning groove is disposed on the side wall / bottom of the slide groove, and the number of positioning grooves corresponds to the five-position adjustment gear setting.

[0021] The elastic buckle is disposed on the slider. When the support component slides to adjust its position, the elastic buckle is compressed and deformed. When the specified position is reached, the elastic buckle springs into the corresponding positioning groove.

[0022] In a preferred embodiment, the support member includes a support member for supporting the prismatic member, the support member being fixed to the slider or the support member being integrally formed with the slider.

[0023] In a preferred embodiment, the pressure-bearing component and the support component are independently molded parts.

[0024] In a preferred embodiment, the pressure-bearing component is a ribbed structure with an overall hollow frame structure and an axially symmetrical distribution; the elastic ribs and the pressure-bearing component are integrally formed.

[0025] In a preferred embodiment, the ratio α ≤ 1; the ratio β ≤ 1.

[0026] In summary, this application has the following beneficial effects:

[0027] 1. Dual-dimensional control, precise and flexible stiffness adjustment: Through dual control of parameters (α, β) and position, the stiffness can be continuously changed over a wide range without interruption, which can adapt to a wide range of stiffness requirements and solve the problem of single adjustment dimension and narrow range of existing structures.

[0028] 2. High performance controllability: Dimensionless parameter quantitative design is adopted to establish a parameter-position-stiffness mapping relationship. Parameter combinations can be pre-selected or adjusted on-site according to actual needs to ensure that the structural performance meets expectations.

[0029] 3. Simple structure and easy assembly / disassembly: The overall structure is simple, using nested assembly with no unnecessary connecting parts. At the same time, the structure can be modularly spliced, which is convenient for disassembly, assembly and maintenance, reducing the cost of use;

[0030] 4. High structural stability: The pressure-bearing components and support components fit tightly together without splicing gaps or swaying, and the force is transmitted evenly, which can effectively improve the overall strength and deformation stability of the structure, and take into account the dual functions of buffering and support. Attached Figure Description

[0031] Figure 1 This is a schematic diagram of the overall structure of the porous structure with adjustable stiffness in this embodiment;

[0032] Figure 2 This is a schematic diagram illustrating the parameter definition of the adjustable stiffness porous structure in this embodiment;

[0033] Figure 3 This is a schematic diagram of the five-position adjustment of the porous structure with adjustable stiffness in this embodiment.

[0034] Explanation of reference numerals in the attached drawings: 1. Pressure-bearing component; 1-1. Elastic rib; 1-2. Hollowed-out structure; 1-3. Rib structure; 1-4. Slide groove; 2. Support component; 2-1. Prismatic component; 2-2. Support component; 2-3. Sliding block; 2-4. Connecting block; 3. Concave arc structure. Detailed Implementation

[0035] The technical solutions of the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some embodiments of the present invention, and not all embodiments. All other embodiments obtained by those skilled in the art based on the embodiments of the present invention without creative effort are within the scope of protection of the present invention.

[0036] In the description of this invention, it should be noted that the terms "upper," "lower," "inner," "outer," "top / bottom," etc., indicate the orientation or positional relationship based on the orientation or positional relationship shown in the accompanying drawings. They are used only for the convenience of describing the invention and for simplifying the description, and do not indicate or imply that the device or element referred to must have a specific orientation, or be constructed and operated in a specific orientation. Therefore, they should not be construed as limitations on the invention. Furthermore, the terms "first" and "second" are used for descriptive purposes only and should not be construed as indicating or implying relative importance.

[0037] In the description of this invention, it should be noted that, unless otherwise explicitly specified and limited, the terms "installed", "equipped", "sleeved / connected", "connected", etc., should be interpreted broadly. For example, "connection" can be a wall-mounted connection, a detachable connection, or an integral connection; it can be a mechanical connection or an electrical connection; it can be a direct connection or an indirect connection through an intermediate medium; it can be a connection within two components. For those skilled in the art, the specific meaning of the above terms in this invention can be understood according to the specific circumstances.

[0038] The following is in conjunction with the appendix Figures 1-3 This application will be described in further detail.

[0039] This embodiment provides a porous structure with a wide range of in-situ adjustable stiffness. Specifically, it is a variable stiffness structure that can be used in scenarios such as flexible support and buffering vibration reduction, and achieves continuous variation of stiffness over a wide range through parameter control and assembly position adjustment.

[0040] The adjustable stiffness porous structure provided in this embodiment is composed of two core units: the core pressure-bearing component 1 and the structural support component 2. It adopts a nested assembly structure with no splicing gaps, which can improve the overall strength and deformation stability of the structure. The support component 2 is an independently molded part, which does not require additional connecting parts when it is used with the pressure-bearing component 1. This ensures the reliability of the connection after installation and facilitates disassembly and position adjustment.

[0041] The pressure-bearing component 1, serving as the main load-bearing element, adopts a hollow frame structure (rib structure) with an overall axially symmetrical distribution. The elastic ribs 1-1 are integrally formed with the pressure-bearing component 1, and have uniform thickness. Elastic ribs 1-1 are symmetrically arranged on both sides of the pressure-bearing component 1. The elastic ribs 1-1 have a regular arc and are arranged on the side of the pressure-bearing component 1 with concave arc-shaped structures 3. At least two hollow structures 1-2 are opened on each side of the elastic rib 1-1. The hollow structures 1-2 on both sides of the pressure-bearing component 1 are symmetrically arranged, and the elastic ribs 1-1 on each side are divided into at least three rib structures 1-3 by the at least two hollow structures 1-2.

[0042] refer to Figure 2 The outer diameter of the pressure-bearing component 1 is D, and the inner diameter is d. The inner diameter d is the minimum distance between the two elastic ribs 1-1. The ratio of the inner and outer diameters of the pressure-bearing component 1 is defined as α = d / D, the ratio α ≤ 1, and the ratio α is positively correlated with the structural stiffness. The larger the value of α, the closer the inner diameter d of the rib is to the overall outer diameter D. The effective deformation area of ​​the rib is compressed, the deformation under load is reduced, and the stiffness of the rib is improved.

[0043] The supporting component 2 includes a prismatic member 2-1. When the supporting component 2 is nested in the pressure-bearing component 1, the prismatic member 2-1 slides between the elastic ribs 1-1 on both sides. The prismatic member 2-1 is provided with an inwardly concave arc-shaped structure 3 along the transverse side walls in conjunction with the elastic ribs 1-1. The elastic ribs 1-1 and the prismatic member 2-1 form a tight fit through the inwardly concave arc-shaped structure 3. After the two are nested, they fit tightly together without gaps or shaking.

[0044] The supporting component 2 corresponds to at least three rib structures 1-3 of the pressure-bearing component 1 and is provided with at least three prismatic members 2-1. The spacing between adjacent prismatic members 2-1 is equal to the grid spacing of the openwork structure 1-2 (the grid spacing is the unit distance of the openwork structure 1-2 along the longitudinal direction of the pressure-bearing component 1).

[0045] refer to Figure 2 The height of the pressure-bearing component 1 is H. The ratio of the height h of the prismatic member 2-1 in the supporting component 2 to H is defined as β = h / H. The ratio β ≤ 1, and the ratio β is positively correlated with the structural stiffness. The larger the value of β, the closer the effective height h of the prismatic member 2-1 is to the overall structural height H. The effective support area of ​​the prismatic member 2-1 and the pressure-bearing component 1 is expanded, and the load is jointly borne by the pressure-bearing component 1 and the prismatic member 2-1, thereby improving the overall structural stiffness.

[0046] Furthermore, the supporting component 2 and the pressure-bearing component 1 are in a sliding fit. The supporting component 2 can slide relative to the pressure-bearing component 1 in the longitudinal direction to adjust the position of the supporting component 2 in the longitudinal direction of the pressure-bearing component 1. By adjusting the position of the supporting component 2, the overlap area between the prism-shaped component 2-1 and the elastic rib 1-1 is changed, thereby realizing the adjustment of the structural stiffness.

[0047] Specifically, the position adjustment of the support component 2 is provided with five position adjustment positions, which are set at 0%, 25%, 50%, 75%, and 100% along the longitudinal direction according to the overlap rate of the overlapping area. The overlapping area is based on the area of ​​the elastic rib 1-1, for reference. Figure 3 :

[0048] When the overlap area is 0%, the prism member 2-1 is misaligned with the rib structure 1-3, and at least two prism members 2-1 are located in the area where at least two hollow structures 1-2 are located. At this time, the structural stiffness is the smallest and the deformation capacity is the strongest.

[0049] When the overlap area is 100%, the prism member 2-1 is misaligned with the openwork structure 1-2, and at least two prism members 2-1 completely overlap with the rib structure 1-3. At this time, the structure has the greatest stiffness and the strongest load-bearing capacity.

[0050] The 25%, 50%, and 75% settings are gradient stiffness adjustment settings, which, together with the α and β parameters, enable continuous control of structural stiffness.

[0051] This embodiment achieves dual-dimensional variable stiffness through parameter ratio (α, β) adjustment and support component 2 position adjustment. The parameter combination is selected in advance according to the operation scenario to determine the basic stiffness of the structure. When the environment changes, the position of support component 2 can be quickly switched manually or through a small drive module to achieve active adaptation of structural stiffness and meet the dynamic needs of different scenarios.

[0052] refer to Figure 1 The specific structure of the sliding fit between the supporting component 2 and the pressure-bearing component 1 is as follows:

[0053] The pressure-bearing component 1 has a sliding assembly and a positioning assembly along its longitudinal direction at its bottom. The sliding assembly includes a slide groove 1-4 and a slider 2-3. The slider 2-3 slides within the slide groove 1-4, and the support component 2 is connected to the slider 2-3. The positioning assembly includes a positioning groove and an elastic buckle. The positioning groove is located on the side wall or bottom of the slide groove 1-4, and its position and number correspond to the stop position. The elastic buckle is located on the slider 2-3 to achieve positioning and fixation.

[0054] The support component 2 includes a support component 2-2 for supporting the prism-shaped component 2-1. The support component 2-2 is fixed to the slider 2-3 or integrally formed with the slider 2-3. At least three sliders 2-3 are integrally connected by connecting blocks 2-4 to ensure that each prism-shaped component 2-1 slides synchronously, thereby realizing the overall position adjustment and stable positioning of the support component 2.

[0055] It should be noted that the positioning component adopts a conventional structure in the prior art, which is not shown in the accompanying drawings of this application; those skilled in the art can use any existing structure that can achieve the sliding positioning function, including but not limited to spring plungers, elastic buckles, ball positioning pins, magnetic positioning, ratchet pawls or threaded fastening, according to actual needs, and all such structures fall within the protection scope of this application.

[0056] During assembly, align the prism-shaped member 2-1 of the support component 2 with the gap between the elastic ribs 1-1 on both sides of the pressure-bearing component 1, slide the slider 2-3 and the connecting block 2-4 into the groove 1-4 to complete the nested assembly. No additional connecting parts are required, making assembly convenient. When disassembling, the support component 2 can be directly pulled out from the pressure-bearing component 1, making maintenance convenient.

[0057] During stiffness adjustment, pushing the support component 2-2 causes the slider 2-3 and connecting block 2-4 to slide longitudinally along the slide groove 1-4. The elastic buckle deforms under pressure, and when it slides to the designated position, the elastic buckle springs into the corresponding positioning groove, thus fixing the support component 2. By adjusting the position of the support component 2, the overlap area between the prism-shaped component 2-1 and the rib structure 1-3 is changed. Combined with the α and β parameters, accurate stiffness control is achieved. (Reference) Figure 3 :

[0058] 0% position: Prism-shaped component 2-1 is completely misaligned with rib structure 1-3. Two of the three prism-shaped components 2-1 are located in the areas where the two hollow structures 1-2 are located. At this time, the overlap area is 0%, the structural stiffness is the smallest, the deformation capacity is the strongest, and it is suitable for low load and high buffering requirements.

[0059] 25% setting: The overlap area between the prismatic member 2-1 and the elastic rib 1-1 is 25% of the area of ​​the elastic rib 1-1, and the structural stiffness is 25%~50% of the basic stiffness, which is suitable for medium buffering needs.

[0060] 50% setting: The overlap area between the prismatic member 2-1 and the elastic rib 1-1 is 50% of the area of ​​the elastic rib 1-1, and the structural stiffness is 50%~75% of the basic stiffness, which is suitable for conventional load scenarios.

[0061] 75% setting: The overlap area between the prismatic member 2-1 and the elastic rib 1-1 is 75% of the area of ​​the elastic rib 1-1, and the structural stiffness is 75%~90% of the basic stiffness, which is suitable for high load scenarios.

[0062] 100% position: Prism-shaped component 2-1 is completely misaligned with the openwork structure 1-2, and the three prism-shaped components 2-1 and the three rib structures 1-3 completely overlap, with an overlap area of ​​100%. At this position, the structure has the greatest stiffness and the strongest load-bearing capacity, which is suitable for ultra-high load and rigid support requirements.

[0063] If the basic stiffness needs to be adjusted, the support component 2 and the pressure-bearing component 1 with different α and β parameters can be replaced. For example, α can be adjusted to 0.8 and β can be adjusted to 0.8. At this time, the basic stiffness of the structure is significantly improved. With the five-position adjustment, a wider range of stiffness control can be achieved to adapt to different operating scenarios.

[0064] The above description is merely a preferred embodiment of the present invention, but the design concept of the present invention is not limited thereto. Any non-substantial modifications made to the present invention by those skilled in the art within the scope of the technology disclosed in the present invention using this concept shall be deemed as an infringement of the protection scope of the present invention.

Claims

1. A porous structure with a wide range of in-situ adjustable stiffness, characterized in that: Including pressure-bearing components and supporting components; The pressure-bearing component is symmetrically provided with elastic ribs on both sides of the transverse direction. At least two hollow structures are opened on the elastic rib on one side, and the hollow structures on both sides are symmetrically arranged. The support component includes a prismatic member. When the support component is nested in the pressure-bearing component, the prismatic member slides between the elastic ribs on both sides. The outer diameter of the pressure-bearing component is D, and the inner diameter of the pressure-bearing component is d, which is the minimum distance between the elastic ribs on both sides; the ratio of the inner and outer diameters of the pressure-bearing component is α = d / D; the height of the pressure-bearing component is H, and the height of the prismatic member is h, with a ratio β = h / H; the ratios α and β are used to control the structural stiffness, and the ratios α and β are positively correlated with the structural stiffness; The supporting component and the pressure-bearing component are in a sliding fit. The supporting component can slide relative to the pressure-bearing component in the longitudinal direction to adjust the position of the supporting component in the longitudinal direction of the pressure-bearing component. By adjusting the position of the supporting component, the overlap area between the prism-shaped component and the elastic rib is adjusted to achieve structural stiffness adjustment.

2. A porous structure with wide-range in-situ adjustable stiffness according to claim 1, characterized in that: The elastic rib has an arcuate shape and is disposed on the side of the pressure-bearing component in an inwardly concave arcuate structure. The prismatic member has an inwardly concave arc structure along its two transverse side walls in conjunction with the elastic rib; the elastic rib and the prismatic member are fitted together through the inwardly concave arc structure.

3. A porous structure with wide-range in-situ adjustable stiffness according to claim 1, characterized in that: The elastic rib on one side is divided into at least three rib structures by the at least two hollow structures; The supporting component is provided with at least three prismatic members corresponding to at least three rib structures. The spacing between adjacent prismatic members is equal to the grid spacing of the openwork structure. The grid spacing is the unit distance of the openwork structure along the longitudinal direction of the pressure-bearing component.

4. A porous structure with wide-range in-situ adjustable stiffness according to claim 3, characterized in that: When the overlap area between the prismatic member and the elastic rib is 0%, the prismatic member and the rib structure are misaligned, and at least two of the prismatic members are located in the area where the at least two hollow structures are located; When the overlap area between the prismatic member and the elastic rib is 100%, the overlap area is mainly composed of the area of ​​the elastic rib. The prismatic member and the hollow structure are misaligned, and at least two of the prismatic members completely overlap with the rib structure.

5. A porous structure with wide-range in-situ adjustable stiffness according to claim 1, characterized in that: The position adjustment of the support component is provided with five position adjustment positions, which are set at 0%, 25%, 50%, 75%, and 100% along the longitudinal direction according to the overlap rate of the overlapping area. The overlapping area is based on the overlapping area of ​​the elastic ribs.

6. A porous structure with wide-range in-situ adjustable stiffness according to claim 5, characterized in that: The bottom of the pressure-bearing component is provided with a sliding component and a positioning component along the longitudinal direction. The sliding component includes a sliding groove and a slider. The slider slides into the sliding groove and is located at the bottom of the support component. The positioning component includes a positioning groove and an elastic buckle. The positioning groove is disposed on the side wall / bottom of the slide groove, and the number of positioning grooves corresponds to the five-position adjustment gear setting. The elastic buckle is disposed on the slider. When the support component slides to adjust its position, the elastic buckle is compressed and deformed. When the specified position is reached, the elastic buckle springs into the corresponding positioning groove.

7. A porous structure with wide-range in-situ adjustable stiffness according to claim 6, characterized in that: The support component includes a support member for supporting the prismatic member, the support member being fixed to the slider or the support member being integrally formed with the slider.

8. A porous structure with wide-range in-situ adjustable stiffness according to claim 1, characterized in that: The pressure-bearing component and the supporting component are independently molded parts.

9. A porous structure with wide-range in-situ adjustable stiffness according to claim 1, characterized in that: The pressure-bearing component has a ribbed structure and adopts a hollow frame structure with an overall axially symmetrical distribution; the elastic ribs are integrally formed with the pressure-bearing component.

10. A porous structure with wide-range in-situ adjustable stiffness according to claim 1, characterized in that: The ratio α ≤ 1; the ratio β ≤ 1.

Citation Information

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