CN104376132A

CN104376132A - Implementing and applying method of paper folding structure

Info

Publication number: CN104376132A
Application number: CN201310354635.7A
Authority: CN
Inventors: 周翔; Y·衷; 汪海
Original assignee: Shanghai Jiao Tong University
Current assignee: Shanghai Jiao Tong University
Priority date: 2013-08-14
Filing date: 2013-08-14
Publication date: 2015-02-25
Anticipated expiration: 2033-08-14
Also published as: CN104376132B

Abstract

A realization and application method of an origami structure in the field of computer image processing technology. After determining the coordinates of m points in the xz plane of a three-dimensional coordinate system and determining the coordinates of n+2 points in the yz plane, all adjacent vertices are Connect them with straight lines, and every three connected connecting line segments form a plane, and all connecting line segments form a three-dimensional origami structure; then map the vertex coordinates in the three-dimensional origami structure to the vertex coordinates in the two-dimensional plane, and then map the two-dimensional plane The adjacent vertices are connected to obtain the plane fold pattern of the three-dimensional origami structure, and the line segments and vertices in the plane fold pattern are projected onto the plane plate, and further processed to obtain the required three-dimensional origami structure. The invention is easy to realize by programming in a computer, and can directly design and obtain precise origami structures of many geometric shapes in three-dimensional space.

Description

Realization and Application of Origami Structure

技术领域 technical field

本发明涉及的是一种计算机图像处理技术领域的方法，具体是一种便于在计算机程序中实现的折纸结构的实现及应用方法。 The present invention relates to a method in the technical field of computer image processing, in particular to a realization and application method of an origami structure that is conveniently realized in a computer program. the

背景技术 Background technique

折纸结构具有非常广泛的用途，例如可以作为吸能材料(Ma J(2011)Thin-walled tubes with pre-folded origami patterns as energy absorption devices.PhD thesis(University of Oxford，Oxford，United Kingdom).)、飞机结构部件填充材料(F，Wolf K，Hauffe A，Alekseev KA，Zakirov IM(2011)Wedge-shaped folded sandwich cores for aircraft applications：from design and manufacturing process to experimental structure validation.CEAS Aeronautical Journal2(1-4)：203-212)、建筑结构部件(Elsayed EA，Basily BB(2004)A continuous folding process for sheet materials.Int J Mater Prod Technol21：217-238.)、可展开式卫星太阳能帆板(Miura K，Natori M(1985)2-D array experiment on board a space flyer unit.Space Solar Power Rev5(4)：345-356)、设计有效的购物袋或地图的折叠方法(Wu W，You Z(2011)A solution for folding rigid tall shopping bags.Proc.R.Soc.A467(2133)：2561-2574.)、植入式医疗器械(Kuribayashi K，et a1.(2006)Self-deployable origami stent grafts as a biomedical application of Ni-rich TiNishape memory alloy foil.Mater Sci Eng A Struct Mater419(1-2)：131-137.)、吸声材料(Wang ZJ，Xu QH(2006)Experimental research on soundproof characteristics for the sandwich plates with folded core.J Vib Eng19：65-69.)、自折式薄膜材料(Pickett GT(2007)Self-folding origami membranes.Europhys Lett78(4)：48003.)或者可折叠式超材料(Schenk M，Gues SD(2013)Geometry of Miura-folded metamaterials.PNAS110(9)：3276-3281.)。上述应用到折纸结构的一个最基本也是最重要的问题是折纸结构的几何设计。只有获得了有效的折纸结构设计，才有可能将这些折纸结构应用到实际工程中。 Origami structures have a very wide range of uses, such as energy-absorbing materials (Ma J(2011) Thin-walled tubes with pre-folded origami patterns as energy absorption devices.PhD thesis(University of Oxford, Oxford, United Kingdom).), Filling materials for aircraft structural parts ( F, Wolf K, Hauffe A, Alekseev KA, Zakirov IM (2011) Wedge-shaped folded sandwich cores for aircraft applications: from design and manufacturing process to experimental structure validation. CEAS Aeronautical Journal2 (1-4): 203-212), Building structural components (Elsayed EA, Basily BB (2004) A continuous folding process for sheet materials. Int J Mater Prod Technol21: 217-238.), deployable satellite solar panels (Miura K, Natori M (1985) 2- D array experiment on board a space flyer unit.Space Solar Power Rev5(4):345-356), Design an effective folding method for shopping bags or maps (Wu W, You Z(2011) A solution for folding rigid tall shopping bags .Proc.R.Soc.A467(2133):2561-2574.), Implantable medical devices (Kuribayashi K, et a1.(2006) Self-deployable origami stent grafts as a biomedical application of Ni-rich TiNishape memory alloy foil. Mater Sci Eng A Struct Mater419 (1-2): 131-137.), sound-absorbing materials (Wang ZJ, Xu QH (2006) Experimental research on soundproof characteristics for the sandwich plates with folded core. J Vib Eng19: 65 -69.), self-folding membrane materials (Pickett GT(2007) Self-folding origami membranes.Europhys Lett78(4):48003.) or foldable metamaterials (Schenk M, Gue s SD (2013) Geometry of Miura-folded metamaterials. PNAS110(9): 3276-3281.). One of the most basic and important issues mentioned above applied to origami structures is the geometric design of origami structures. It is possible to apply these origami structures to practical engineering only when an effective origami structure design is obtained.

虽然目前有数种折纸结构在文献中被提出(Nojima T(2002)Modelling of folding patterns in flat membranes and cylinders by origami.JSME Int J C45(1)：364-370.；Nojima T(2003)Modelling of compact folding/wrapping of flat circular membranes.JSME Int J C46(4)：1547-1553.；Khaliulin VI(2003)Classification of regular row-arranged folded structures.Izvestiya VUZ Aviatsionnaya Tekhnika46(1)：7-12.；Khaliulin VI(2005)A technique for synthesizing the structures of folded cores of sandwich panels.Izvestiya VUZ Aviatsionnaya Tekhnika48(1)：7-12.；Zakirov IM， Alekseev KA(2010)Design of a wedge-shaped folded structure.Journal of Machinery Manufacture and Reliability39(5)：412-417.)，然而这些折纸结构种类有限，并且都是在平面折纹图案的基础上提出的，因此在实际应用中具有诸多不便。可见，目前缺少一种可以让工程师或者研究人员针对某一实际问题很方便地利用计算机获得大量潜在的折纸结构设计方案，从而从这些设计方案中选出适用于该问题的最佳折纸结构。 Although several origami structures have been proposed in the literature (Nojima T(2002)Modelling of folding patterns in flat membranes and cylinders by origami.JSME Int J C45(1):364-370.; Nojima T(2003)Modelling of compact folding/wrapping of flat circular membranes. JSME Int J C46(4): 1547-1553.; Khaliulin VI(2003) Classification of regular row-arranged folded structures. Izvestiya VUZ Aviatsionnaya Tekhnika46(1): 7-12.; Khaliulin (2005) A technique for synthesizing the structures of folded cores of sandwich panels. Izvestiya VUZ Aviatsionnaya Tekhnika 48(1): 7-12.; Zakirov IM, Alekseev KA(2010) Design of a wedge-shaped Manufacturing folded achnury structure of J and Reliability39(5):412-417.), however, these origami structures are limited in type and are all proposed on the basis of plane fold patterns, so there are many inconveniences in practical applications. It can be seen that currently there is a lack of a method that allows engineers or researchers to easily use a computer to obtain a large number of potential origami structure design solutions for a practical problem, so as to select the best origami structure suitable for the problem from these design solutions. the

发明内容 Contents of the invention

本发明针对现有技术存在的上述不足，提出一种折纸结构的实现及应用方法，易于在计算机中编程实现，可在三维空间中直接设计得到诸多几何外形的精确折纸结构。利用本发明可以方便计算得到大量折纸结构设计方案，从而免去了通过查阅大量文献寻找合适折纸结构的工作。 Aiming at the above-mentioned deficiencies in the prior art, the present invention proposes a realization and application method of an origami structure, which is easy to program and realize in a computer, and can directly design and obtain precise origami structures of various geometric shapes in three-dimensional space. The invention can conveniently calculate and obtain a large number of origami structure design schemes, thereby saving the work of searching for a suitable origami structure by consulting a large number of documents. the

本发明是通过以下技术方案实现的，本发明包括以下步骤： The present invention is realized through the following technical solutions, and the present invention comprises the following steps:

步骤1，在一个三维坐标系的x-z平面中确定m个点坐标，即 $V_{i}^{x} = {[\begin{matrix} x_{i}^{x} & 0 & z_{i}^{x} \end{matrix}]}^{T};$ 并在y-z平面确定n+2个点坐标，即 $V_{j}^{y} = {[\begin{matrix} 0 & y_{j}^{y} & z_{j}^{y} \end{matrix}]}^{T}$ 表示，其中：i＝1，2，...，m，j＝0，2，...，n+1；然后根据m+n+2个坐标确定m×n个顶点坐标，作为目标折纸结构的顶点，即： $V_{i, j} = [\begin{matrix} x_{i, j} \\ y_{i, j} \\ z_{i, j} \end{matrix}] = V_{j}^{y} + [A_{j}] V_{i}^{x},$ i＝1，2，...，m；j＝1，2，...，n，其中：矩阵[A_j]是一个3×3的矩阵， $[A_{j}] = [\begin{matrix} 1 & 0 & 0 \\ 0 & 0 & {(- 1)}^{j} \frac{{\cos θ}_{j - 1} + \cos θ_{j}}{\sin (θ_{j - 1} - θ_{j})} \\ 0 & 0 & {(- 1)}^{j} \frac{\sin θ_{j - 1} + \sin θ_{j}}{\sin (θ_{j - 1} - θ_{j})} \end{matrix}],$ ${\sin θ}_{j} = \frac{i_{z} \cdot (V_{j + 1}^{y} - V_{j}^{y})}{| | V_{j + 1}^{y} - V_{j}^{y} | |},$ $\cos θ_{j} = \frac{i_{y} \cdot (V_{j + 1}^{y} - V_{j}^{y})}{| | V_{j + 1}^{y} - V_{j}^{y} | |},$ i_y＝[0 1 0]^T为y坐标轴的单位向量，i_z＝[0 0 1]^T为z坐标轴的单位向量，||■||表示对向量取模。 Step 1, determine the coordinates of m points in the xz plane of a three-dimensional coordinate system, namely $V_{i}^{x} = {[\begin{matrix} x_{i}^{x} & 0 & z_{i}^{x} \end{matrix}]}^{T};$ And determine n+2 point coordinates on the yz plane, namely $V_{j}^{the y} = {[\begin{matrix} 0 & {the y}_{j}^{the y} & z_{j}^{the y} \end{matrix}]}^{T}$ Represent, wherein: i=1, 2,..., m, j=0, 2,..., n+1; then determine m×n vertex coordinates according to m+n+2 coordinates, as the target origami The vertices of the structure, namely: $V_{i, j} = [\begin{matrix} x_{i, j} \\ {the y}_{i, j} \\ z_{i, j} \end{matrix}] = V_{j}^{the y} + [A_{j}] V_{i}^{x},$ i=1, 2,..., m; j=1, 2,..., n, wherein: matrix [A _j ] is a 3×3 matrix, $[A_{j}] = [\begin{matrix} 1 & 0 & 0 \\ 0 & 0 & {(- 1)}^{j} \frac{{\cos θ}_{j - 1} + \cos θ_{j}}{\sin (θ_{j - 1} - θ_{j})} \\ 0 & 0 & {(- 1)}^{j} \frac{\sin θ_{j - 1} + \sin θ_{j}}{\sin (θ_{j - 1} - θ_{j})} \end{matrix}],$ ${\sin θ}_{j} = \frac{i_{z} &Center Dot; (V_{j + 1}^{the y} - V_{j}^{the y})}{| | V_{j + 1}^{the y} - V_{j}^{the y} | |},$ $\cos θ_{j} = \frac{i_{the y} &Center Dot; (V_{j + 1}^{the y} - V_{j}^{the y})}{| | V_{j + 1}^{the y} - V_{j}^{the y} | |},$ i _y =[0 1 0] ^T is the unit vector of the y coordinate axis, i _z =[0 0 1] ^T is the unit vector of the z coordinate axis, and |||||

所述的点坐标的参数方程采用但不限于以下任意一种： The parameter equation of the point coordinates adopts but is not limited to any of the following:

① $V_{i}^{x} = {[\begin{matrix} \frac{2 i - 1}{2} a \cos α & 0 & \frac{{(- 1)}^{i - 1}}{2} a \sin α \end{matrix}]}^{T}, i = 1, . . ., m;$ ① $V_{i}^{x} = {[\begin{matrix} \frac{2 i - 1}{2} a \cos α & 0 & \frac{{(- 1)}^{i - 1}}{2} a \sin α \end{matrix}]}^{T}, i = 1, . . ., m;$

${V V}_{j j}^{y the y} = = {[\begin{matrix} 00 & \frac{22 j j - - 11}{22} b b cos cos β β & \frac{{((- - 11))}^{j j - - 11}}{22} b b sin sin β β \end{matrix}]}^{T T},, j j = = 0,1 0,1,, . . . . . .,, n no + + 11 . .$

② $V_{i}^{x} = {[\begin{matrix} \frac{2 i - 1}{2} a \cos α & 0 & \frac{{(- 1)}^{i - 1}}{2} a \sin α \end{matrix}]}^{T}, i = 1, . . ., m;$ ② $V_{i}^{x} = {[\begin{matrix} \frac{2 i - 1}{2} a \cos α & 0 & \frac{{(- 1)}^{i - 1}}{2} a \sin α \end{matrix}]}^{T}, i = 1, . . ., m;$

${V V}_{j j}^{y the y} = = [[r r + + {((- - 11))}^{j j} δ δ]] [\begin{matrix} 00 \\ sin sin ((jβ jβ)) \\ cos cos ((jβ jβ)) \end{matrix}],, j j = = 0,1 0,1,, . . . . . .,, n no + + 11 . .$

③ $V_{i}^{x} = {[\begin{matrix} \frac{2 i - 1}{2} a \cos α & 0 & \frac{{(- 1)}^{i - 1}}{2} a \sin α \end{matrix}]}^{T}, i = 1, . . ., m;$ ③ $V_{i}^{x} = {[\begin{matrix} \frac{2 i - 1}{2} a \cos α & 0 & \frac{{(- 1)}^{i - 1}}{2} a \sin α \end{matrix}]}^{T}, i = 1, . . ., m;$

${V V}_{j j}^{y the y} = = \{\begin{matrix} {[\begin{matrix} 00 & 00 & 00 \end{matrix}]}^{T T},, j j = = 00 \\ {[\begin{matrix} 00 & ((j j + + 11)) b b & 00 \end{matrix}]}^{T T},, j j = = 1,5 1,5,, . . . . . .,, 44 N N - - 33 \\ {[\begin{matrix} 00 & jb jb & ((j j + + 11)) b b tan the tan β β \end{matrix}]}^{T T},, j j = = 2,6 2,6,, . . . . . .,, 44 N N - - 22 \\ {[\begin{matrix} 00 & ((j j + + 11)) b b & jb jb tan the tan β β \end{matrix}]}^{T T},, j j = = 3,7 3,7,, . . . . . .,, 44 N N - - 11 \\ {[\begin{matrix} 00 & jb jb & 00 \end{matrix}]}^{T T},, j j = = 4,8 4,8,, . . . . . .,, 44 N N \\ {[\begin{matrix} 00 & ((j j + + 11)) b b & 00 \end{matrix}]}^{T T},, j j = = 44 N N + + 11 \end{matrix} . .$

步骤2，将所有相邻顶点，即{V_i，j V_i+1，j}以及{y_i，j V_i，j+1}之间用直线连接，每三个收尾相连的连接线段即构成一个平面，所有连接线段构成三维折纸结构； Step 2, connect all adjacent vertices, that is, {V _{i, j} V _{i+1, j} } and {y _{i, j} V _{i, j+1} } with a straight line, and every three connected connecting line segments are Constitute a plane, and all connecting line segments form a three-dimensional origami structure;

步骤3，将三维折纸结构中的顶点坐标V_i，j映射为二维平面中的顶点坐标即 ${\tilde{V}}_{i, j} = {[\begin{matrix} {\tilde{x}}_{i, j} & {\tilde{y}}_{i, j} \end{matrix}]}^{T},$ ${\tilde{V}}_{1,1} = {[\begin{matrix} 0 & 0 \end{matrix}]}^{T},$ i＝2，...，m， ${\tilde{V}}_{i, j} = {\tilde{V}}_{i, j + 1} + | | V_{i, j - 1} - V_{i, j} | | [\begin{matrix} 0 \\ 1 \end{matrix}],$ i＝1，...，m；j＝2，...，n，其中： Step 3, map the vertex coordinates V _{i, j} in the 3D origami structure to the vertex coordinates in the 2D plane Right now ${\tilde{V}}_{i, j} = {[\begin{matrix} {\tilde{x}}_{i, j} & {\tilde{the y}}_{i, j} \end{matrix}]}^{T},$ ${\tilde{V}}_{1,1} = {[\begin{matrix} 0 & 0 \end{matrix}]}^{T},$ i=2,...,m, ${\tilde{V}}_{i, j} = {\tilde{V}}_{i, j + 1} + | | V_{i, j - 1} - V_{i, j} | | [\begin{matrix} 0 \\ 1 \end{matrix}],$ i=1,...,m; j=2,...,n, where:

然后将二维平面中的相邻顶点用直线段连接起来，即得到所述三维折纸结构的平面折纹图案，将该平面折纹图案中的线段及顶点投影至平面板材上，并进一步加工得到所需的三维折纸结构。 Then the adjacent vertices in the 2D plane Connecting with straight line segments can obtain the plane fold pattern of the three-dimensional origami structure, project the line segments and vertices in the plane fold pattern onto the plane plate, and further process to obtain the required three-dimensional origami structure.

本发明涉及上述具有光滑曲线折纹的折纸结构的应用，用于制作板材填充材料。 The present invention relates to the application of the above-mentioned origami structure with smooth curved folds for making board filling materials. the

所述的板材包括但不限于用于汽车前后保险杠的吸能结构、飞机尾翼、垂直翼或副翼、建筑物室外及室内的墙板等。 The panels include, but are not limited to, energy-absorbing structures for front and rear bumpers of automobiles, aircraft empennages, vertical wings or ailerons, outdoor and indoor wall panels of buildings, and the like. the

技术效果 technical effect

与现有技术相比，利用本发明方法可以通过计算机中编程方便实现在三维空间中直接设计的诸多几何外形的精确折纸结构，从而免去了通过查阅大量文献寻找合适折纸结构的工作。 Compared with the prior art, the method of the present invention can conveniently realize the precise origami structure of many geometric shapes directly designed in three-dimensional space through programming in the computer, thereby eliminating the work of searching for a suitable origami structure by consulting a large number of documents. the

附图说明 Description of drawings

图1为点坐标示意图。 Figure 1 is a schematic diagram of point coordinates. the

图2为相邻顶点的连接示意图。 Figure 2 is a schematic diagram of the connection of adjacent vertices. the

图3为实施例1点坐标示意图。 Fig. 3 is a schematic diagram of point coordinates in Example 1. the

图4为实施例1得到的目标折纸结构示意图。 4 is a schematic diagram of the target origami structure obtained in Example 1. the

图5为实施例1中一个重复单元的平面折纹示意图。 FIG. 5 is a schematic diagram of planar creases of a repeating unit in Example 1. FIG. the

图6实施例2点坐标示意图。 Figure 6 is a schematic diagram of the coordinates of 2 points in the embodiment. the

图7实施例2得到的目标折纸结构示意图。 Fig. 7 is a schematic diagram of the target origami structure obtained in Example 2. the

图8实施例2三维折纸结构的平面折纹图。 Fig. 8 is a plane crease diagram of the three-dimensional origami structure in Example 2. the

图9实施例3得到的目标折纸结构示意图。 Fig. 9 is a schematic diagram of the target origami structure obtained in Example 3. the

图10实施例3三维折纸结构的平面折纹图。 Fig. 10 is the plane crease diagram of the three-dimensional origami structure of Example 3. the

图11实施例4得到的目标折纸结构示意图。 Fig. 11 is a schematic diagram of the target origami structure obtained in Example 4. the

图12实施例4三维折纸结构的平面折纹图。 Fig. 12 Plane crease diagram of the three-dimensional origami structure of Example 4. the

具体实施方式Detailed ways

下面对本发明的实施例作详细说明，本实施例在以本发明技术方案为前提下进行实施，给出了详细的实施方式和具体的操作过程，但本发明的保护范围不限于下述的实施例。 The embodiments of the present invention are described in detail below. This embodiment is implemented on the premise of the technical solution of the present invention, and detailed implementation methods and specific operating procedures are provided, but the protection scope of the present invention is not limited to the following implementation example. the

实施例1 Example 1

本实施例设计得到Miura结构，首先确定点坐标和如图3所示，其参数方程为： This embodiment design obtains the Miura structure, first determine the point coordinates and As shown in Figure 3, its parameter equation is:

${V V}_{i i}^{x x} = = {[\begin{matrix} \frac{22 i i - - 11}{22} a a cos cos α α & 00 & \frac{{((- - 11))}^{i i - - 11}}{22} a a sin sin α α \end{matrix}]}^{T T},, i i = = 11,, . . . . . .,, m m;;$

由此可得到顶点坐标： $V_{i, j} = [\begin{matrix} \frac{2 i - 1}{2} a \cos α \\ \frac{2 j - 1}{2} b \cos β - \frac{{(- 1)}^{i - 1}}{2} a \frac{\sin α}{\sin β} \\ \frac{{(- 1)}^{j - 1}}{2} b \sin β \end{matrix}],$ i＝1，2，...，m；j＝1，2，...，n； The vertex coordinates can be obtained from this: $V_{i, j} = [\begin{matrix} \frac{2 i - 1}{2} a \cos α \\ \frac{2 j - 1}{2} b \cos β - \frac{{(- 1)}^{i - 1}}{2} a \frac{\sin α}{\sin β} \\ \frac{{(- 1)}^{j - 1}}{2} b \sin β \end{matrix}],$ i=1,2,...,m; j=1,2,...,n;

根据相邻顶点连接线段得到折纹后，该折纸结构的三维图如图4所示，参数a，b，α，β，m和n分别取为1，1，π/4，π/4，9和9。其中：区域a为该折纸结构的一个重复单元。 After the creases are obtained according to the connecting line segments of adjacent vertices, the three-dimensional diagram of the origami structure is shown in Figure 4, and the parameters a, b, α, β, m and n are respectively set to 1, 1, π/4, π/4, 9 and 9. Among them: area a is a repeating unit of the origami structure. the

对顶点坐标映射至 ${\tilde{V}}_{i, j} = [\begin{matrix} \frac{(i - 1) a}{\sin β} \sqrt{\cos^{2} {α \sin}^{2} β + \sin^{2} α - \cos^{2} β} \\ \frac{{(- 1)}^{i} + 1}{2} a \cot β + (j - 1) b \end{matrix}],$ i＝1，...，m；j＝1，...，n；如图5所示，绘出了一个重复单元的平面折纹图案，其中：实线表示脊(hill)折纹，虚线表示谷(valley)折纹。 Map the vertex coordinates to ${\tilde{V}}_{i, j} = [\begin{matrix} \frac{(i - 1) a}{\sin β} \sqrt{\cos^{2} {α \sin}^{2} β + \sin^{2} α - \cos^{2} β} \\ \frac{{(- 1)}^{i} + 1}{2} a \cot β + (j - 1) b \end{matrix}],$ i=1,...,m; j=1,...,n; As shown in Figure 5, a planar crease pattern of a repeating unit is drawn, wherein: the solid line represents the ridge (hill) crease, Dashed lines indicate valley creases.

本实施例得到折纸结构设计可用作复合板夹层结构的夹层材料。 The origami structure design obtained in this example can be used as a sandwich material for a sandwich structure of a composite board. the

实施例2 Example 2

弯曲折纸结构的设计过程。 Design process for curved origami structures. the

确定点坐标和图6所示，其参数方程为： Determine point coordinates and As shown in Figure 6, its parameter equation is:

${V V}_{j j}^{y the y} = = [[r r + + {((- - 11))}^{j j} δ δ]] [\begin{matrix} 00 \\ sin sin ((jβ jβ)) \\ cos cos ((jβ jβ)) \end{matrix}],, j j = = 00,, 11,, . . . . . .,, n no + + 11 . .$

可以看到，所有下标j为偶数的位于一个半径为r-δ的圆周上，所有下标j为奇数的位于一个半径为r+δ的圆周上。 It can be seen that all subscript j is even Located on a circle with radius r-δ, all subscripts j are odd numbers lies on a circle of radius r+δ.

根据上述点坐标得到目标折纸结构的顶点坐标V_i，j。根据相邻顶点连接线段得到折纹，即完成目标折纸结构的三维模型。如图7所示，为该三维折纸结构，其中：a，r，δ，α，β，m和n分别取为10，1，π/4，π/30，5和15。可以看到该三维折纸结构呈空间弯曲形态。区域b为该折纸结构的一个重复单元。 According to the above point coordinates, the vertex coordinates V _i,j of the target origami structure are obtained. The creases are obtained according to the connecting line segments of adjacent vertices, that is, the three-dimensional model of the target origami structure is completed. As shown in Figure 7, it is the three-dimensional origami structure, where: a, r, δ, α, β, m and n are respectively taken as 10, 1, π/4, π/30, 5 and 15. It can be seen that the three-dimensional origami structure is in a spatially curved shape. Region b is a repeating unit of the origami structure.

为了进一步了解该空间曲面的几何特性，进行如下参数化研究。首先，固定参数a，r，δ和α的参数值为10，1和π/4，同时让参数β从π/40变化到π/20，表1列出了在该条件下顶点V_i，j到x坐标轴的最短距离ρ_i，j。其次，固定参数a，δ，α和β的参数值为1，π/4 和π/30，同时让参数r从8变化到12，表2列出了在该条件下顶点V_i，j到x坐标轴的最短距离ρ_i，j。再次，固定参数a，r，α和β的参数值为10，π/4和π/30，同时让参数δ从1变化到3，表3列出了在该条件下顶点V_i，j到x坐标轴的最短距离ρ_i，j。最后，固定参数r，δ和β的参数值为10，1，和π/30，同时让参数(a/2)sinα从0.5变化到1.4，表4列出了在该条件下顶点V_i，j到x坐标轴的最短距离ρ_i，j。 In order to further understand the geometric characteristics of the space surface, the following parametric study is carried out. First, the parameter values of the fixed parameters a, r, δ and α are 10, 1 and π/4, while changing the parameter β from π/40 to π/20, Table 1 lists the shortest distance ρ _i, j from the vertex V _{i, j} to the x coordinate axis under this condition. Second, the parameter values of the fixed parameters a, δ, α and β are 1, π/4 and π/30, while changing the parameter r from 8 to 12, Table 2 lists the shortest distance ρ _i, j from the vertex V _{i, j} to the x coordinate axis under this condition. Again, the parameter values for the fixed parameters a, r, α and β are 10, π/4 and π/30, while changing the parameter δ from 1 to 3, Table 3 lists the shortest distance ρ _i , j from the vertex V _{i, j} to the x coordinate axis under this condition. Finally, the parameter values of fixed parameters r, δ and β are 10, 1, and π/30, and the parameter (a/2) sinα is changed from 0.5 to 1.4 at the same time. Table 4 lists the vertices V _{i under this condition,} The shortest distance ρ _i,j from _j to the x-coordinate axis.

表1 β π/40 π/35 π/30 π/25 π/20 ρ_1，1 11.0508 11.0525 11.0549 11.0588 11.0655 ρ_1，2 9.0660 9.0694 9.0748 9.0840 9.1023 ρ_1，3 11.0508 11.0525 11.0549 11.0588 11.0655 ρ_1，4 9.0660 9.0694 9.0748 9.0840 9.1023 ρ_1，5 11.0508 11.0525 11.0549 11.0588 11.0655 ρ_1，6 9.0660 9.0694 9.0748 9.0840 9.1023 ρ_2，1 11.0508 11.0525 11.0549 11.0588 11.0655 ρ_2，2 9.0660 9.0694 9.0748 9.0840 9.1023 ρ_2，3 11.0508 11.0525 11.0549 11.0588 11.0655 ρ_2，4 9.0660 9.0694 9.0748 9.0840 9.1023 ρ_2，5 11.0508 11.0525 11.0549 11.0588 11.0655 ρ_2，6 9.0660 9.0694 9.0748 9.0840 9.1023 Table 1 beta π/40 π/35 π/30 π/25 π/20 ρ _1,1 11.0508 11.0525 11.0549 11.0588 11.0655 ρ _{1, 2} 9.0660 9.0694 9.0748 9.0840 9.1023 ρ _1,3 11.0508 11.0525 11.0549 11.0588 11.0655 ρ _{1, 4} 9.0660 9.0694 9.0748 9.0840 9.1023 ρ _1,5 11.0508 11.0525 11.0549 11.0588 11.0655 ρ _1,6 9.0660 9.0694 9.0748 9.0840 9.1023 _ρ2,1 11.0508 11.0525 11.0549 11.0588 11.0655 _ρ2,2 9.0660 9.0694 9.0748 9.0840 9.1023 ρ _2,3 11.0508 11.0525 11.0549 11.0588 11.0655 ρ _2,4 9.0660 9.0694 9.0748 9.0840 9.1023 ρ _2,5 11.0508 11.0525 11.0549 11.0588 11.0655 ρ _2,6 9.0660 9.0694 9.0748 9.0840 9.1023

表2 r 8 9 10 11 12 ρ_1，1 9.0625 10.0582 11.0549 12.0523 13.0503 ρ_1，2 7.0875 8.0801 9.0748 10.0708 11.0678 ρ_1，3 9.0625 10.0582 11.0549 12.0523 13.0503 ρ_1，4 7.0875 8.0801 9.0748 10.0708 11.0678 ρ_1，5 9.0625 10.0582 11.0549 12.0523 13.0503 ρ_1，6 7.0875 8.0801 9.0748 10.0708 11.0678 ρ_2，1 9.0625 10.0582 11.0549 12.0523 13.0503 ρ_2，2 7.0875 8.0801 9.0748 10.0708 11.0678 ρ_2，3 9.0625 10.0582 11.0549 12.0523 13.0503 ρ_2，4 7.0875 8.0801 9.0748 10.0708 11.0678 ρ_2，5 9.0625 10.0582 11.0549 12.0523 13.0503 Table 2 r 8 9 10 11 12 ρ _1,1 9.0625 10.0582 11.0549 12.0523 13.0503 ρ _{1, 2} 7.0875 8.0801 9.0748 10.0708 11.0678 ρ _1,3 9.0625 10.0582 11.0549 12.0523 13.0503 ρ _{1, 4} 7.0875 8.0801 9.0748 10.0708 11.0678 ρ _1,5 9.0625 10.0582 11.0549 12.0523 13.0503 ρ _1,6 7.0875 8.0801 9.0748 10.0708 11.0678 _ρ2,1 9.0625 10.0582 11.0549 12.0523 13.0503 _ρ2,2 7.0875 8.0801 9.0748 10.0708 11.0678 ρ _2,3 9.0625 10.0582 11.0549 12.0523 13.0503 ρ _2,4 7.0875 8.0801 9.0748 10.0708 11.0678 ρ _2,5 9.0625 10.0582 11.0549 12.0523 13.0503

ρ_2，6 7.0875 8.0801 9.0748 10.0708 11.0678 ρ _2,6 7.0875 8.0801 9.0748 10.0708 11.0678

表3 δ 1 1.5 2 2.5 3 ρ_l，1 11.0549 11.5471 12.0434 12.5409 13.0390 ρ_l，2 9.0748 8.5684 8.0686 7.5710 7.0748 ρ_l，3 11.0549 11.5471 12.0434 12.5409 13.0390 ρ_l，4 9.0748 8.5684 8.0686 7.5710 7.0748 ρ_l，5 11.0549 11.5471 12.0434 12.5409 13.0390 ρ_1，6 9.0748 8.5684 8.0686 7.5710 7.0748 ρ_2，1 11.0549 11.5471 12.0434 12.5409 13.0390 ρ_2，2 9.0748 8.5684 8.0686 7.5710 7.0748 ρ_2，3 11.0549 11.5471 12.0434 12.5409 13.0390 ρ_2，4 9.0748 8.5684 8.0686 7.5710 7.0748 ρ_2，S 11.0549 11.5471 12.0434 12.5409 13.0390 ρ_2，6 9.0748 8.5684 8.0686 7.5710 7.0748 table 3 δ 1 1.5 2 2.5 3 ρl _,1 11.0549 11.5471 12.0434 12.5409 13.0390 ρl _,2 9.0748 8.5684 8.0686 7.5710 7.0748 ρl _,3 11.0549 11.5471 12.0434 12.5409 13.0390 ρl _,4 9.0748 8.5684 8.0686 7.5710 7.0748 ρl _,5 11.0549 11.5471 12.0434 12.5409 13.0390 ρ _1,6 9.0748 8.5684 8.0686 7.5710 7.0748 _ρ2,1 11.0549 11.5471 12.0434 12.5409 13.0390 _ρ2,2 9.0748 8.5684 8.0686 7.5710 7.0748 ρ _2,3 11.0549 11.5471 12.0434 12.5409 13.0390 ρ _2,4 9.0748 8.5684 8.0686 7.5710 7.0748 ρ2 _,S 11.0549 11.5471 12.0434 12.5409 13.0390 ρ _2,6 9.0748 8.5684 8.0686 7.5710 7.0748

表4 (a/2)sinα 0.5 0.75 1 1.25 1.5 ρ_i，1 11.0137 11.0309 11.0549 11.0857 11.1231 ρ_l，2 9.0187 9.0421 9.0748 9.1166 9.1674 ρ_i，3 11.0137 11.0309 11.0549 11.0857 11.1231 ρ_l，4 9.0187 9.0421 9.0748 9.1166 9.1674 ρ_l，S 11.0137 11.0309 11.0549 11.0857 11.1231 ρ_l，6 9.0187 9.0421 9.0748 9.1166 9.1674 ρ_2，1 11.0137 11.0309 11.0549 11.0857 11.1231 ρ_2，2 9.0187 9.0421 9.0748 9.1166 9.1674 ρ_2，3 11.0137 11.0309 11.0549 11.0857 11.1231 ρ_2，4 9.0187 9.0421 9.0748 9.1166 9.1674 ρ_2，5 11.0137 11.0309 11.0549 11.0857 11.1231 ρ_2，6 9.0187 9.0421 9.0748 9.1166 9.1674 Table 4 (a/2)sinα 0.5 0.75 1 1.25 1.5 ρi _,1 11.0137 11.0309 11.0549 11.0857 11.1231 ρl _,2 9.0187 9.0421 9.0748 9.1166 9.1674 ρi _,3 11.0137 11.0309 11.0549 11.0857 11.1231 ρl _,4 9.0187 9.0421 9.0748 9.1166 9.1674 ρ _{l, S} 11.0137 11.0309 11.0549 11.0857 11.1231 ρl _,6 9.0187 9.0421 9.0748 9.1166 9.1674 _ρ2,1 11.0137 11.0309 11.0549 11.0857 11.1231 _ρ2,2 9.0187 9.0421 9.0748 9.1166 9.1674 ρ _2,3 11.0137 11.0309 11.0549 11.0857 11.1231 ρ _2,4 9.0187 9.0421 9.0748 9.1166 9.1674 ρ _2,5 11.0137 11.0309 11.0549 11.0857 11.1231 ρ _2,6 9.0187 9.0421 9.0748 9.1166 9.1674

从表1-4的结果可以看到，对于任意一种给定的条件，所有下标j为偶数的顶点V_i，j到x坐标轴的最短距离相同，记该距离为r₁；所有下标j为奇数的顶点V_i，j到x坐标轴的最短距离相同，记该距离为r₂；并始终有，r₁＞r₂。该结果表明，上述得到的三维折纸结构构成一个以x坐标轴为中心轴、以r₁和r₂分别为外径和内径的圆柱壳结构。对数据进一步分析可以得到该壳结构的内外径与输入设计参数之间的关系，即： From the results in Table 1-4, we can see that, for any given condition, the shortest distance between all vertices V _i,j with even subscript j and the x coordinate axis is the same, and this distance is r ₁ ; all the following Vertices V _i,j marked with an odd number have the same shortest distance to the x-coordinate axis, and this distance is denoted as r ₂ ; and always, r ₁ >r ₂ . This result shows that the 3D origami structure obtained above constitutes a cylindrical shell structure with the x-coordinate axis as the central axis and _r1 and _r2 as the outer and inner diameters, respectively. Further analysis of the data can obtain the relationship between the inner and outer diameters of the shell structure and the input design parameters, namely:

${r r}_{11}^{22} = = {((r r - - δ δ))}^{22} + + {((\frac{a a}{22} sin sin α α))}^{22} {((11 + + {((\frac{((r r + + δ δ)) sin sin β β}{((r r + + δ δ)) cos cos β β - - ((r r - - δ δ))}))}^{22}))}^{22},,$ ${r r}_{22}^{22} = = {((r r + + δ δ))}^{22} + + {((\frac{a a}{22} sin sin α α))}^{22} ((11 + +$ $r r - - δ δ sin sin βr βr + + δ δ - - r r - - δ δ cos cos β β 22 twenty two . .$

根据步骤3可进一步得到图7所示的三维折纸结构的平面折纹图案，如图8所示。 According to step 3, the planar crease pattern of the three-dimensional origami structure shown in FIG. 7 can be further obtained, as shown in FIG. 8 . the

本实施例得到的空间折纸结构可用作弯曲复合板夹层结构的夹层材料。 The spatial origami structure obtained in this example can be used as a sandwich material for a sandwich structure of a curved composite plate. the

实施例3 Example 3

楔形(wedge-shape)折纸结构的设计过程。 The design process of wedge-shape origami structures. the

确定点坐标和其参数方程为： Determine point coordinates and Its parameter equation is:

令参数a，b，α，β，m和N分别取1，1，π/4，π/12，7和4。根据步骤1，可以得到7×16个顶点坐标V_i，j，i＝1，2，...，7；j＝1，2，...，14，构成目标折纸结构的顶点。再根据步骤2，得到三维折纸结构，如图9所示。可见，图9所示的折纸结构呈楔形。再根据步骤3，可以进一步得到图9所示三维折纸结构的平面折纹图案，如图10所示。 Let the parameters a, b, α, β, m and N be 1, 1, π/4, π/12, 7 and 4, respectively. According to step 1, 7×16 vertex coordinates V _i,j can be obtained, i=1, 2,...,7; j=1,2,...,14, constituting the vertices of the target origami structure. Then according to step 2, a three-dimensional origami structure is obtained, as shown in Figure 9. It can be seen that the origami structure shown in Figure 9 is wedge-shaped. According to step 3, the planar crease pattern of the three-dimensional origami structure shown in FIG. 9 can be further obtained, as shown in FIG. 10 .

本实施例得到的折纸结构可以作为飞机尾翼、垂直翼或副翼的内部结构填充材料。 The origami structure obtained in this embodiment can be used as an internal structure filling material for an aircraft empennage, vertical wing or aileron. the

实施例4 Example 4

六边形环状折纸结构的设计过程。 The design process of the hexagonal ring origami structure. the

${V V}_{00}^{y the y} = = {[\begin{matrix} 00 & - - b b & \sqrt{33} b b \end{matrix}]}^{T T} - - - - - - ((1717))$

${V V}_{11}^{y the y} = = {[\begin{matrix} 00 & b b & \sqrt{33} b b \end{matrix}]}^{T T} - - - - - - ((1818))$

${V V}_{22}^{y the y} = = {[\begin{matrix} 00 & 22 b b & 00 \end{matrix}]}^{T T} - - - - - - ((1919))$

${V V}_{33}^{y the y} = = {[\begin{matrix} 00 & b b & - - \sqrt{33} b b \end{matrix}]}^{T T} - - - - - - ((2020))$

${V V}_{44}^{y the y} = = {[\begin{matrix} 00 & - - b b & - - \sqrt{33} b b \end{matrix}]}^{T T} - - - - - - ((21 twenty one))$

${V V}_{55}^{y the y} = = {[\begin{matrix} 00 & - - 22 b b & 00 \end{matrix}]}^{T T} - - - - - - ((22 twenty two))$

${V V}_{66}^{y the y} = = {[\begin{matrix} 00 & - - b b & \sqrt{33} b b \end{matrix}]}^{T T} - - - - - - ((23 twenty three))$

${V V}_{77}^{y the y} = = {[\begin{matrix} 00 & b b & \sqrt{33} b b \end{matrix}]}^{T T} - - - - - - ((24 twenty four))$

${V V}_{88}^{y the y} = = {[\begin{matrix} 00 & 22 b b & 00 \end{matrix}]}^{T T} - - - - - - ((1717))$

令参数a，b和α分别取1，3和π/4。根据步骤1，可以得到7×7个顶点坐标V_i，j，i＝1，2，...，7；j＝1，2，...，7，构成目标折纸结构的顶点。再根据步骤2，得到三维折纸结构，如图11所示。可见，图11所示的折纸结构呈楔形。再根据步骤3，可以进一步得到图11所示三维折纸结构的平面折纹图案，如图12所示。 Let the parameters a, b and α be 1, 3 and π/4 respectively. According to step 1, 7×7 vertex coordinates V _i,j can be obtained, i=1, 2,...,7; j=1,2,...,7, constituting the vertices of the target origami structure. Then according to step 2, a three-dimensional origami structure is obtained, as shown in Figure 11. It can be seen that the origami structure shown in Figure 11 is wedge-shaped. According to step 3, the planar crease pattern of the three-dimensional origami structure shown in FIG. 11 can be further obtained, as shown in FIG. 12 .

本实施例得到的折纸结构可以用来设计汽车前后保险杠的吸能结构。 The origami structure obtained in this embodiment can be used to design the energy-absorbing structure of the front and rear bumpers of automobiles. the

Claims

1. an implementation method of origami structure, is characterized in that, comprises the following steps:

Step 1, determine the coordinates of m points in the xz plane of a three-dimensional coordinate system, namely

V_{i}^{x} = {[\begin{matrix} x_{i}^{x} & 0 & z_{i}^{x} \end{matrix}]}^{T};

And determine n+2 point coordinates on the yz plane, namely

V_{j}^{the y} = {[\begin{matrix} 0 & {the y}_{j}^{the y} & z_{j}^{the y} \end{matrix}]}^{T}

Express, where: i=1, 2,..., m, j=0, 2,..., n+1; then determine m×n vertex coordinates according to m+n+2 coordinates, as the target origami The vertices of the structure, namely:

V_{i, j} = [\begin{matrix} x_{i, j} \\ {the y}_{i, j} \\ z_{i, j} \end{matrix}] = V_{j}^{the y} + [A_{j}] V_{i}^{x},

i=1, 2,..., m; j=1, 2,..., n, wherein: matrix [A _j ] is a 3×3 matrix,

[A_{j}] = [\begin{matrix} 1 & 0 & 0 \\ 0 & 0 & {(- 1)}^{j} \frac{{\cos θ}_{j - 1} + \cos θ_{j}}{\sin (θ_{j - 1} - θ_{j})} \\ 0 & 0 & {(- 1)}^{j} \frac{\sin θ_{j - 1} + \sin θ_{j}}{\sin (θ_{j - 1} - θ_{j})} \end{matrix}],

{\sin θ}_{j} = \frac{i_{z} &Center Dot; (V_{j + 1}^{the y} - V_{j}^{the y})}{| | V_{j + 1}^{the y} - V_{j}^{the y} | |},

\cos θ_{j} = \frac{i_{the y} \cdot (V_{j + 1}^{the y} - V_{j}^{the y})}{| | V_{j + 1}^{the y} - V_{j}^{the y} | |},

i _y ＝[0 1 0] ^T is the unit vector of the y coordinate axis, i _z ＝[0 0 1] ^T is the unit vector of the z coordinate axis, ||||

Step 2, connect all adjacent vertices, that is, {V _{i, j} V _{i+1, j} } and {V _{i, j} V _{i, j+1} } with a straight line, and every three connected connecting line segments are Constitute a plane, and all connecting line segments form a three-dimensional origami structure;

Step 3, map the vertex coordinates V _{i, j} in the 3D origami structure to the vertex coordinates in the 2D plane Right now

{\tilde{V}}_{i, j} = {[\begin{matrix} {\tilde{x}}_{i, j} & {\tilde{the y}}_{i, j} \end{matrix}]}^{T},

{\tilde{V}}_{1,1} = {[\begin{matrix} 0 & 0 \end{matrix}]}^{T},

i=2,...,m,

{\tilde{V}}_{i, j} = {\tilde{V}}_{i, j + 1} + | | V_{i, j - 1} - V_{i, j} | | [\begin{matrix} 0 \\ 1 \end{matrix}],

i=1,...,m; j=2,...,n, where:

Then the adjacent vertices in the 2D plane

\{\begin{matrix} {\tilde{V}}_{i, j} & {\tilde{V}}_{i + 1, j} \end{matrix}\}

as well as

\{\begin{matrix} {\tilde{V}}_{i, j} & {\tilde{V}}_{i, j + 1} \end{matrix}\}

Connecting with straight line segments can obtain the plane fold pattern of the three-dimensional origami structure, project the line segments and vertices in the plane fold pattern onto the plane plate, and further process to obtain the required three-dimensional origami structure.

2. The method according to claim 1, characterized in that, the parametric equation of the point coordinates adopts but is not limited to any of the following:

①

V_{i}^{x} = {[\begin{matrix} \frac{2 i - 1}{2} a \cos α & 0 & \frac{{(- 1)}^{i - 1}}{2} a \sin α \end{matrix}]}^{T}, i = 1, . . ., m;

{V V}_{j j}^{y the y} = = {[\begin{matrix} 00 & \frac{22 j j - - 11}{22} b b cos cos β β & \frac{{((- - 11))}^{j j - - 11}}{22} b b sin sin β β \end{matrix}]}^{T T},, j j = = 0,1 0,1,, . . . . . .,, n no + + 11;;

②

V_{i}^{x} = {[\begin{matrix} \frac{2 i - 1}{2} a \cos α & 0 & \frac{{(- 1)}^{i - 1}}{2} a \sin α \end{matrix}]}^{T}, i = 1, . . ., m;

{V V}_{j j}^{y the y} = = [[r r + + {((- - 11))}^{j j} δ δ]] [\begin{matrix} 00 \\ sin sin ((jβ jβ)) \\ cos cos ((jβ jβ)) \end{matrix}],, j j = = 0,1 0,1,, . . . . . .,, n no + + 11;;

③

V_{i}^{x} = {[\begin{matrix} \frac{2 i - 1}{2} a \cos α & 0 & \frac{{(- 1)}^{i - 1}}{2} a \sin α \end{matrix}]}^{T}, i = 1, . . ., m;

{V V}_{j j}^{y the y} = = \{\begin{matrix} {[\begin{matrix} 00 & 00 & 00 \end{matrix}]}^{T T},, j j = = 00 \\ {[\begin{matrix} 00 & ((j j + + 11)) b b & 00 \end{matrix}]}^{T T},, j j = = 1,5 1,5,, . . . . . .,, 44 N N - - 33 \\ {[\begin{matrix} 00 & jb jb & ((j j + + 11)) b b tan the tan β β \end{matrix}]}^{T T},, j j = = 2,6 2,6,, . . . . . .,, 44 N N - - 22 \\ {[\begin{matrix} 00 & ((j j + + 11)) b b & jb jb tan the tan β β \end{matrix}]}^{T T},, j j = = 3,7 3,7,, . . . . . .,, 44 N N - - 11 \\ {[\begin{matrix} 00 & jb jb & 00 \end{matrix}]}^{T T},, j j = = 4,8 4,8,, . . . . . .,, 44 N N \\ {[\begin{matrix} 00 & ((j j + + 11)) b b & 00 \end{matrix}]}^{T T},, j j = = 44 N N + + 11 \end{matrix} . .

3. An application of the origami structure according to claim 1 or 2, characterized in that it is used to make board filling materials.

4. The application according to claim 3, characterized in that the plates include: energy-absorbing structures of front and rear bumpers of automobiles, aircraft empennages, vertical wings or ailerons, outdoor and indoor wall panels of buildings.