Rotary bistable structure and configuration design method

Abstract

The invention discloses a rotary bistable structure and a configuration design method, relates to the field of mechanical metamaterials and protection structures, and has excellent energy absorption and dissipation performance. The rotary bistable paper-cut structure mainly comprises a rigid platform, a fixed surface, a supporting surface and a frame. According to the design method, the geometric contour of a rotary unit is determined through analytical derivation and geometric construction, and the space layout of cutting creases is optimized to construct the rotary paper-cut structure with the function meeting the expectation. By means of the design method, digital design of the rotating unit under various initial geometric parameters and different boundary shapes can be achieved.

Description

TECHNICAL FIELD

[0001] The present application relates to the field of paper-cut structure design, in particular to a design method of a rotating bistable paper-cut structure configuration. BACKGROUND

[0002] Paper-cut structure is a deployable structure based on the principle of origami, often with stronger morphological adaptability and deformation ability than origami structure, and can unlock more diverse deformation modes, such as stretch-induced buckling, negative Poisson's ratio, adjustable stiffness and other complex nonlinear mechanical behaviors. This "on-demand shaping" capability from two-dimensional to three-dimensional makes it show great potential in metamaterial design with adjustable performance. It will be widely used in aerospace, medical devices and flexible robots, etc. This study focuses on a special class of bistable paper-cut structures-rotating bistable paper-cut structures. This structure uses the coordination of cutting marks and folding marks to achieve multi-stable behavior dominated by rotating lifting motion (spin-up motion). However, it is extremely challenging and of great application value to design cutting marks and folding marks on a plane so that the structure can achieve spin-up motion around a specific axis under simple external driving, and remain stable in multiple different stable states after removing the driving. Therefore, in-depth and systematic study of rotating multi-stable paper-cut structures not only has important theoretical significance, but also shows broad engineering application prospects.

[0003] Currently, the design of rotating bistable paper-cut structures mainly relies on experience and trial-and-error method. Designers usually determine the positions and shapes of cutting marks and folding marks by repeatedly experimenting and adjusting based on existing origami or paper-cut structures. For example, in some simple rotating bistable paper-cut structure designs, designers will first determine a basic geometric shape, such as a square or a circle, and then randomly arrange cutting marks and folding marks on the shape. Through multiple folding and rotating experiments, the motion behavior of the structure is observed, and the layout of cutting marks and folding marks is gradually optimized to achieve the desired rotating bistable function. However, this experience-based and trial-and-error design method has many shortcomings. First, this method lacks systematic theoretical guidance, and the design process is tedious and time-consuming, making it difficult to quickly and accurately determine the optimal layout of cutting marks and folding marks. Second, due to the lack of in-depth understanding of the relationship between structural geometric parameters and motion behavior, the designed structures often have limitations in performance, such as limited rotation angle range and insufficient structural stability.

[0004] With the increasing complexity of structural performance requirements in engineering fields, the traditional design method of rotating bistable paper-cut structures has been difficult to meet the needs of practical applications. Therefore, there is an urgent need for a systematic and theoretical design method of rotating bistable paper-cut structures, which can achieve digital design of structures under various initial geometric parameters and different boundary shapes through precise geometric analysis and optimization, thereby improving design efficiency, enhancing structural performance, and meeting the needs of different application scenarios. Summary of the Invention

[0005] Purpose of the invention: The purpose of this invention is to provide a method for designing a rotationally bistable paper-cutting structure and the rotationally bistable paper-cutting structure obtained by the method. Through geometric derivation and numerical analytical derivation verification, the method achieves the determination of the non-stretched boundary of the structure, the layout of the cutting lines and fold lines, and finally forms an overall unit structure.

[0006] Technical solution: A method for designing the configuration of a rotationally bistable paper-cutting structure, characterized by comprising the following steps: Step 1: Based on the relative rotation angle α of the inner polygon and the outer polygon, the border width t, and the crease angle θ, first determine the non-stretched area on the outer polygon corresponding to one side of the inner polygon. Within the non-stretched area, select any two points, define a point connecting the center of the inner polygon as the upper boundary point, and define a point connecting the vertices of the inner polygon as the lower boundary point. Step 2: Create a symmetrical circular array of line segments connecting the determined upper and lower boundary points, circumferentially around the structure's center. The number of copies is equal to the number of sides of the inner polygon. E n To obtain the complete outer polygon boundary outline design drawing required for the entire rotating paper-cutting structure; Step 3: Based on all the determined upper and lower boundary points, determine the cut lines, valley folds, and mountain folds between the inner and outer polygons; the line segment connecting each vertex of the inner polygon to the lower boundary point closest to that vertex is called a mountain fold; the line segment connecting the two endpoints of each edge of the inner polygon to the corresponding lower boundary point of the outer polygon on that edge is called a cut line; the line segment connecting the point where the line segment between the upper boundary point of the outer polygon and the center point of the inner polygon intersects the cut line directly to the upper boundary point is called a valley fold; Step 4: Based on the determined cut lines, valley folds, and mountain folds, the inner and outer polygons move relative to each other under external drive to achieve rotational ascent, thereby obtaining the rotational bistable paper-cutting structure configuration.

[0007] Beneficial effects: The rotationally bistable paper-cutting structure designed using the above method has the following advantages: 1. The determination of the absence of tension boundaries avoids material stretching during rotation, thus improving structural durability; 2. The systematic layout of cut lines and creases ensures the geometric compatibility and freedom of movement of the structure; 3. This design method is highly effective and widely applicable, suitable for bistable rotating paper-cutting structures of various sizes and proportions. Attached Figure Description

[0008] Figure 1A rotational bistable paper-cut structure configuration.

[0009] Figure 2 A design method of a rotational bistable paper-cut structure configuration.

[0010] Figure 3 Definition of initial state configuration geometry parameters.

[0011] Figure 4 Definition of non-stretching boundary parameters, where a is the definition of the outer quadrilateral S point, and b is the definition of the outer quadrilateral E point.

[0012] Figure 5 Definition of position angle and distance before and after deformation.

[0013] Figure 6 Definition of the relationship between position angle β and rotation angle θr.

[0014] Figure 7 Design of cutting marks and folding marks.

[0015] Figure 8 Different parameter geometry design diagrams and corresponding paper models. DETAILED DESCRIPTION

[0016] The design method will be further described below in combination with specific examples and with reference to the accompanying drawings: As shown in Figure 1 , the basic configuration of the rotational bistable paper-cut structure provided by the present embodiment is a rotational bistable paper-cut structure based on a concentric quadrilateral plane. The configuration consists of four key parts: rigid platform 1, fixed surface 2, support surface 3, and frame 4. Among them, the rigid platform 1 serves as the improved core area, which will undergo a rotational lifting motion; the fixed surface 2 is connected with the frame 4 and the support surface 3, providing the structural basis; the support surface 3 connects and supports the rigid platform 1 during rotation, and connects it with the fixed surface 2; the frame 4 defines the overall outline of the structure. This design allows the rigid platform 1 to rotate, while the support surface 3 deforms, thereby realizing non-rigid motion.

[0017] The design method of the rotational bistable paper-cut structure based on a concentric quadrilateral plane provided by the present invention, as shown in Figure 2 , is as follows: The planar geometric state of the rotational bistable paper-cut structure is defined by Figure 3Composition. The main geometric parameters of this configuration in its initial state include: the number of sides En of the inner and outer polygons (En=4 for the current configuration), the side length b of the inner quadrilateral, the side length B of the outer quadrilateral, the relative rotation angle α between the inner and outer quadrilaterals, the width t of the frame, and the crease angle θ. The rigid platform is composed of the inner quadrilateral and four connecting triangles. The relative rotation angle α is the angle by which the overall orientation of the inner polygons deflects relative to the overall orientation of the outer polygons in the two-dimensional initial state before deformation (i.e., before the rigid platform is raised and rotated). The crease angle θ is the angle between adjacent mountain creases and valley creases.

[0018] During rotation, to maintain the geometric integrity of the structure, non-tensile constraints must be applied to the supporting surfaces connecting the rigid platform and the fixed surface. Therefore, in this design method, it is necessary to determine the non-tensile outer boundary region of each rotational edge of the rigid platform. Figure 4 One side of the inner quadrilateral shown A 0 B Taking 0 as an example, we assume that in the initial planar state, there are no cut marks in the structure. A 0 B Let P be any point on side 0, and let γ be the interior angle corresponding to the side of the inner quadrilateral A0B0. For the current configuration, the inner quadrilateral is a square, and γ = 90°. Assume the rotation angle of the rigid platform is... θ r， That is, in Figure 4 Under the conditions shown, the position angles βs and βe can be specifically expressed by the following formulas: (1) That is, in Figure 5 In the diagram, the unstretched region corresponding to the side of the inner quadrilateral A0B0 is composed of the blue bold line segments QS and QE (points S and E are the endpoints of the blue bold line segments). The position angle of point S relative to the positive x-axis is βs, and the position angle of point E relative to the positive x-axis is βe.

[0019] The expression for the aforementioned position angle will be derived in detail below. For example... Figure 5 As shown, consider any point P0 on side A0B0 of the inner quadrilateral. Let the position of point P0 relative to the center O of the inner quadrilateral be determined by the polar angle φ (defined as the angle between OB0 and OP0) and the polar radius ρ0 (defined as the distance from any point P0 on side A0B0 to the center O of the inner quadrilateral). In the initial planar state, the polar coordinates of P0 are (ρ0, α+φ), where α is the initial rotation angle of the inner quadrilateral relative to the outer quadrilateral. After the inner quadrilateral rotates about the center O by an angle θr, point P0 rotates to a new position P1, and its polar coordinates become (ρ0, α+φ+θr).

[0020] Meanwhile, consider any point P on the external fixed boundary. B Its polar coordinates are represented as (ρB , β), and thus we can compare the distance from the inner boundary point (P0 or P1) to the outer boundary point P before and after the rotation. B Let the distance before rotation be d1 (from P0 to P). B The distance after rotation is d2 (from P1 to P). B (The distance between d1 and d2). Using the distance formula in polar coordinates, d1 and d2 can be expressed as:

[0021] Therefore, to satisfy the no-stretch condition, i.e., the distance d2 after rotation is no greater than the distance d1 before rotation, this constraint must hold for any point P0 on edge A0B0. In other words, the formula must hold for all defined points P0 at the same angle φ:

[0022] Based on the relationship between the angle β-α-φ and 180°, two cases can be distinguished:

[0023] in: The above equation holds for any φ, and can be further simplified to:

[0024] As can be seen from the above formula, the angle on the left side of the inequality is exactly βs, and the angle on the right side of the inequality is exactly βe. This shows that the stretch-free condition derived analytically is completely equivalent to the position angle limit expression previously obtained based on the geometric construction method.

[0025] Figure 6 The curves show the relationship between the position angles βs and βe of the unstretched boundary and the rotation angle θr of the rigid platform. Therefore, to design the actual cut boundary, two different angle values ​​need to be selected within the position angle range corresponding to the unstretched boundary: βs = 170° and βe = 190°. These two selected angles will define the upper and lower boundaries of the fixed surface region connected to a specific side (e.g., A0B0) of the inner quadrilateral (the upper and lower boundaries have no spatial height meaning; the boundary connecting the vertices of the inner polygon is the lower boundary, and the boundary connecting the center of the inner polygon is the upper boundary). The line segments connecting the determined upper and lower boundaries are then symmetrically copied in a circular array around the center of the structure (the number of copies equal to the number of sides of the inner polygon). E n This process yields the complete fixed-surface boundary outline design drawing required for the entire rotating paper-cutting structure.

[0026] After designing and determining the geometric profile of the fixed surface boundary of the rotating unit, the spatial layout of the cut lines and crease lines is determined through the following steps: (1) Define the cut line: Connect a vertex of the inner quadrilateral (taking A0 as an example) to its corresponding point on the lower boundary of the fixed surface (determined by the pre-selected lower boundary angle), and define this line segment as the cut line, which provides the necessary degrees of freedom for the rotational motion of the inner rigid platform.

[0027] (2) Define the valley crease: Connect the center point O of the inner quadrilateral with its corresponding point on the boundary of the fixed surface (determined by the pre-selected upper boundary angle), and geometrically cut this line at the intersection with the cut line defined in step (1). The remaining line segment from the center point O to the upper boundary is defined as the valley crease.

[0028] (3) Define the mountain crease: connect the center point O of the inner quadrilateral with the vertex. A 0, line segment OA 0 intersects the cut line defined in step (1) at vertex A0, and this line segment is defined as the mountain crease.

[0029] By symmetrically applying steps (1) to (3) of defining the cut lines and crease lines to each vertex of the inner En polygon, a complete set of all mountain creases, valley creases, and cut lines required for a complete structural unit can be generated. All crease lines radiate out from the structural center point O, and the initial continuous plane is divided into different geometries by this network of creases and cut lines. The area, shaped like this, formed a region like Figure 7 The overall unit structure is shown.

[0030] Example 1 To verify the effectiveness and universality of the design method (including boundary definition and the generation of cut lines and crease lines), five configurations with different combinations of geometric parameters were designed and analyzed. The specific parameters are shown in Table 1: Table 1:

[0031] The internal core rotating units employ two types of regular polygons: quadrilaterals and hexagons. To maintain consistent geometric proportions across different configurations, the characteristic length ratio b / B, representing the relative dimensions of the internal and external structures, is set to 0.5 in all configurations. The geometric design drawings and corresponding paper models of the four configurations are shown below. Figure 8 As shown, by comparing and analyzing these configurations, it can be seen that each configuration can be designed with reasonable boundaries and creases, and can enable the rigid platform area to achieve specific rotational lifting movements in the paper model.

[0032] Table 1 shows the relative rotation angles of the inner and outer quadrilaterals in configuration 1 (γ=90°), with b / B = 0.5. α=10°The parameters are as follows. Comparing configurations 1 and 3, the main difference lies in the setting of the relative rotation angle α between the inner and outer quadrilaterals. The maximum achievable rotation angle θr,max is 140° for both structures. When α increases from 10° to 45°, although the available angle ranges of their boundaries at the maximum rotation angle are different ([170°, 190°] and [115°, 135°] respectively), the width of both ranges is 20°. Table 1 shows the relative rotation angles of the inner hexagon (γ=60°), b / B=0.5, and the inner and outer hexagons for configuration 4. α= 10° With the relative rotation angle α set to 0°, the maximum rotation angle that the structure can achieve is 240°.

[0033] In summary, the design method proposed in this paper can successfully generate effective combinations of cutting lines and creases for various cases with different initial geometric parameters, and construct a rotating paper-cutting structure that meets the expected functions, fully demonstrating the effectiveness and wide applicability of the design method.

[0034] This design method is not limited to the embodiments described above. The above description of specific embodiments is intended to illustrate the technical solution of this design, and the above specific embodiments are merely illustrative and not restrictive. Without departing from the scope of protection of the claims, those skilled in the art can make many specific modifications based on the teachings of this design method, and these modifications all fall within the protection scope of this invention.

Claims

1. A method for designing the configuration of a rotationally bistable paper-cutting structure, characterized in that, Includes the following steps: Step 1: Based on the relative rotation angle α of the inner polygon and the outer polygon, the border width t, and the crease angle θ, first determine the non-stretched area on the outer polygon corresponding to one side of the inner polygon. Within the non-stretched area, select any two points, define a point connecting the center of the inner polygon as the upper boundary point, and define a point connecting the vertices of the inner polygon as the lower boundary point. Step 2: Create a symmetrical circular array of line segments connecting the determined upper and lower boundary points, circumferentially around the structure's center. The number of copies is equal to the number of sides of the inner polygon. E n To obtain the complete outer polygon boundary outline design drawing required for the entire rotating paper-cutting structure; Step 3: Based on all the determined upper and lower boundary points, determine the cut lines, valley folds, and mountain folds between the inner and outer polygons; The line segment connecting each vertex of the inner polygon to the nearest lower boundary point of that vertex is called a mountain fold; the line segment connecting the two endpoints of each edge of the inner polygon to the corresponding lower boundary point of the outer polygon on the edge is called a cut line; the line segment connecting the point where the line segment between the upper boundary point of the outer polygon and the center point of the inner polygon intersects the cut line and the upper boundary point is called a valley fold. Step 4: Based on the determined cut lines, valley folds, and mountain folds, the inner and outer polygons move relative to each other under external drive to achieve rotational ascent, thereby obtaining the rotational bistable paper-cutting structure configuration.

2. The configuration design method for the rotationally bistable paper-cutting structure according to claim 1, characterized in that, In step 1, the method for determining the non-stretched area is as follows: Determine the first position angle βs and the second position βe: Where γ is an interior angle of the quadrilateral; θ r The rotation angle of the inner and outer polygons; The first endpoint S is obtained based on the determined first position angle βs, and the second endpoint E is obtained based on the second position angle βe; the line segment from the first endpoint E through the vertex of the outer quadrilateral to the second endpoint S is the unstretched region.

3. The configuration design method for the rotationally bistable paper-cutting structure according to claim 1, characterized in that, The inner and outer polygons have at least three sides.

4. The configuration design method of the rotationally bistable paper-cutting structure according to claim 1, characterized in that, The relative rotation angle α is 0-180°.

5. The configuration design method for the rotationally bistable paper-cutting structure according to claim 1, characterized in that, Both the inner and outer polygons are quadrilaterals, and the ratio of the side length b of the inner quadrilateral to the side length B of the outer quadrilateral is within a certain range. b / B It is (0, 1).

6. A rotationally bistable paper-cutting structure, characterized in that, The configuration design method of the rotational bistable paper-cutting structure according to any one of claims 1-5 is used to design the structure.

7. The rotationally bistable paper-cutting structure according to claim 6, characterized in that, It includes a rigid platform, a fixed surface, a supporting surface, and a frame. The fixed surface and the frame constitute a first platform, the rigid platform is a second platform, and the supporting surface connects the first platform and the second platform and bends when the first platform and the second platform rotate relative to each other.

8. The rotationally bistable paper-cutting structure according to claim 6, characterized in that, The rigid platform is composed of a regular inner polygon and a triangle extending outward from each side; each border is an outer regular polygon; the number of sides of the outer regular polygon and the regular inner polygon are the same.

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