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Research Papers

Design of Cylindrical and Axisymmetric Origami Structures Based on Generalized Miura-Ori Cell

[+] Author and Article Information
Yucai Hu

CAS Key Laboratory of Mechanical Behavior and Design of Materials,
Department of Modern Mechanics,
University of Science and Technology of China,
Hefei, Anhui 230026, China
e-mail: huyucai@ustc.edu.cn

Haiyi Liang

Professor
CAS Key Laboratory of Mechanical Behavior and Design of Materials,
Department of Modern Mechanics,
University of Science and Technology of China,
Hefei, Anhui 230026, China;
IAT-Chungu Joint Laboratory for Additive Manufacturing,
Anhui Chungu 3D Printing Institute of Intelligent Equipment and Industrial Technology,
Wuhu, Anhui 241200, China
e-mail: hyliang@ustc.edu.cn

Huiling Duan

Professor
Mem. ASME
State Key Laboratory for Turbulence and Complex Systems,
Department of Mechanics and Engineering Science,
BIC-ESAT, College of Engineering,
Peking University,
Beijing 100871, China;
CAPT and HEDPS,
Peking University,
Beijing 100871, China
e-mail: hlduan@pku.edu.cn

1Corresponding author.

Contributed by the Mechanisms and Robotics Committee of ASME for publication in the Journal of Mechanisms and Robotics. Manuscript received August 15, 2018; final manuscript received April 18, 2019; published online July 8, 2019. Assoc. Editor: Xilun Ding.

J. Mechanisms Robotics 11(5), 051004 (Jul 08, 2019) (10 pages) Paper No: JMR-18-1266; doi: 10.1115/1.4043800 History: Received August 15, 2018; Accepted May 15, 2019

Origami has shown its potential in designing a three-dimensional folded structure from a flat sheet of material. In this paper, we present geometric design methods to construct cylindrical and axisymmetric origami structures that can fit between two given surfaces. Due to the symmetry of the structures, a strip of folds based on the generalized Miura-ori cells is first constructed and then replicated longitudinally/circumferentially to form the cylindrical/axisymmetric origami structures. In both designs, algorithms are presented to ensure that all vertexes are either on or strictly within the region between the target surfaces. The conditions of flat-foldability and developability are fulfilled at the inner vertexes and the designs are rigid-foldable with a single degree-of-freedom. The methods for cylindrical and axisymmetric designs are similar in implementation and of potential in designing origami structures for engineering purposes, such as foldcores, foldable shelters, and metamaterials.

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References

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Figures

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Fig. 1

(a) Crease pattern for the standard Miura-ori pattern in which a representative row of collinear/zigzag creases is indicated by the highlighted horizontal/vertical line and α is the sector angle; (b) and (c) illustrate the generalized Miura-ori pattern for cylindrical and axisymmetric designs. All patterns observe rows of collinear and zigzag creases; the collinear creases are parallel for the cylindrical case in (b) while the zigzag creases between the same radial creases are parallel for the axisymmetric case in (c).

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Fig. 2

(a) Crease pattern for the standard Miura-ori cell with the mountain and valley creases indicated by the solid and dashed lines, respectively; (b) the partially folded state

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Fig. 3

(a) Crease pattern for the strip with nh = 3 Miura-ori cells; (b) the partially folded strip fitted to the profiles z = fo(x) and z = fi(x) where the light gray strip is a copy along the axial direction; (c) the input zigzag by connecting alternative points on the inner and outer profiles

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Fig. 4

Design result by fitting Pi to the profiles: (a) the cross-sectional view in which the black zigzag is formed by connecting the Pi and the lower and upper curves are the target surface profiles; (b) the folded form with the target surfaces z = fi(x) and z = fo(x). Design result by the tight fitting method: (c) the cross-sectional view, (d) the folded form with the target surfaces, (e) the crease pattern and (f) the paper model.

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Fig. 5

The design results of the zigzag Pi determined by x = /10: The cross-sectional view of the structure (a), the crease pattern (b) and the folded geometry (c) by fitting Pi to the profiles. (d), (e) and (f) are the counterparts by the tight fitting method.

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Fig. 6

The input Pi on the inner profile are determined by x = /10 while those on the upper profile are chosen to be on the midperpendicular of its neighbors on the inner profile. Design result by fitting Pi to the profiles: (a) the cross-sectional view and (b) the folded form with the target surfaces z = fi(x) and z = fo(x). Design result by the tight fitting method: (c) the cross-sectional view, (d) the 3D folded form, (e) the crease pattern and (f) the paper model.

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Fig. 7

The folding motion of the cylindrical design with quadratic profiles by the tight fitting: (a) the length (difference in the x-coordinates of P0 and B2nh) and width (difference in the y-coordinates of P0 and B0) of the Miura-ori strip; (b) through (f) are the snapshots of the folded form corresponding to different dihedral angles γ

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Fig. 8

(a) Crease pattern for the strip with (nh = 3, nv = 2) Miura-ori cells; (b) the partially folded strip fitted to the profiles z = fo(r) and z = fi(r) with r=x2+y2; (c) the input zigzag by connecting alternative points on the inner and outer profiles

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Fig. 9

Results for the axisymmetric origami with quadratic profiles. Design result by fitting Pi to the profiles: (a) the cross-sectional view and (b) the folded form with the target surfaces z = fi(r) and z = fo(r). Design result by the tight fitting method: (c) the cross-sectional view, (d) the 3D folded form, (e) the crease pattern and (f) the paper model.

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Fig. 10

Results for the axisymmetric origami with sine profiles. Design result by fitting Pi to the profiles: (a) the cross-sectional view and (b) the 3D folded form. Design result by the tight fitting method: (c) the cross-sectional view, (d) the 3D folded form, (e) the crease pattern and (f) the paper model.

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Fig. 11

The motion of the axisymmetric origami with quadratic profiles: (a) The sector angle φB and inner radius R0 versus γ when the origami folds from the flat state (γ = 180°) to the fully folded state (γ = 0°), γ is the dihedral angle at the crease on the x-axis shown in (b); (b) through (g) are the snapshots of the folded form corresponding to different dihedral angles γ

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Fig. 12

The ring structure designs for the axisymmetric examples with quadratic (a) and sine (b) profiles by the tight fitting method. The inputs for (a) are nh = 12, nv = 30, φB = π/nv and γ = 3π/4 while those for (b) are nh = 20, nv = 60, φB = π/nv and γ = 3π/4.

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Fig. 13

The geometry for half of the generalized Miura-ori cell; P0P1P2 is in the x-z plane while B0B1B2 should be in certain ϕ-z plane

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Fig. 14

The geometry of the generalized Miura-ori cell nearest to the z-axis. E and F are the projections of P1 and A1 on the z = 0 plane, respectively.

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