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Research Papers: Design Automation

Design of Deployable Membranes Using Conformal Mapping

[+] Author and Article Information
Sachiko Ishida

Mem. ASME
Assistant Professor
Department of Mechanical Engineering,
Meiji University,
1-1-1 Higashimita,
Kawasaki, Kanagawa 2148571, Japan
e-mail: sishida@meiji.ac.jp

Taketoshi Nojima

Meiji Institute for Advanced Study of
Mathematical Sciences,
Meiji University,
4-21-1 Nakano,
Nakano-ku, Tokyo 1648525, Japan
e-mail: taketoshinojima@gmail.com

Ichiro Hagiwara

Mem. ASME
Fellow ASME
Professor
Meiji Institute for Advanced Study of
Mathematical Sciences,
Meiji University,
4-21-1 Nakano,
Nakano-ku, Tokyo 1648525, Japan
e-mail: ihagi@meiji.ac.jp

1Corresponding author.

Contributed by the Design Automation Committee of ASME for publication in the JOURNAL OF MECHANICAL DESIGN. Manuscript received July 3, 2014; final manuscript received March 25, 2015; published online April 16, 2015. Assoc. Editor: Kazuhiro Saitou.

J. Mech. Des 137(6), 061404 (Jun 01, 2015) (6 pages) Paper No: MD-14-1395; doi: 10.1115/1.4030296 History: Received July 03, 2014; Revised March 25, 2015; Online April 16, 2015

This paper proposes a new method for designing the crease patterns of deployable membranes that can be wrapped up compactly. The method utilizes conformal mapping and the origami folding technique. The mapping of the flow with circulation can be used to control the angles between the fold lines, produce elements of the same shape, and maintain regularity of the fold lines. The proposed method thus enables the systematic and efficient design of complex patterns based on simple ones. The proposed method was successfully used to produce the patterns of Nojima and other extended new patterns of deployable membranes consisting of discrete equiangular spirals. The patterns were wrapped and used to form pillars such as regular polygonal, rectangular, and diamond pillars. Toward the industrial application of the proposed method, this paper also discusses pattern design for space-saving storage and to reduce the effect of thickness when using versatile materials.

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Copyright © 2015 by ASME
Topics: Design , Membranes
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References

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Figures

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Fig. 1

Conformal mapping of fluid flows and the analogy of origami patterns: (a) Uniform flow ς = ξ+iη [19]; (b) flow with circulation z = x + iy (mapping between (a) and (b) is done using Eq. (1)) [19]; (c) original crease pattern ς = ξ+iη; and (d) crease pattern z = x + iy obtained from (c) using the inverse function of Eq. (1)

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Fig. 2

Crease patterns and physical models of deployable membranes: (a), (d), and (g) are the crease patterns in the original plane; (b), (e), and (h) are the crease patterns, respectively, transformed from (a), (d), and (g); (c), (f), and (i) are the physical models of (b), (e), and (h), respectively; and (j), (k), and (l) show the wrapping and deployment behavior of the membrane in (i), for which D = 16

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Fig. 3

Crease patterns and physical models of deployable membranes: (a), (d), and (g) are the crease patterns in the original plane, and (b), (e), and (h) are the crease patterns, respectively, transformed from them. (c), (f), and (i) are the physical models of (b), (e), and (h), respectively. (j), (k), and (l) show the wrapping and deployment behavior of the membrane (c), for which D = 4

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Fig. 4

Definition of zigzagged fold lines, crease patterns, and physical models of membranes folded in the radial direction: (a) reversed fold lines; (b) fold lines transformed from (a); (c) fold lines formed by the conventional method of mirroring the image (a blank region without fold lines exists around the center); (d) and (e) original and transformed crease patterns; (f) physical origami model of (e); and (g), (h), and (i) wrapping and deploying behavior of membrane (f), for which D = 4

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Fig. 5

Crease patterns and physical models of deployable membranes: (a) original crease pattern with two angles β1 and β2; (b) patterns transformed from (a); (c) physical model of (b); (d) original crease pattern with zigzagged fold lines (β1 ≈ β2); (e) crease pattern transformed from (d) (the fold lines are not zigzagged); and (f) physical model of (e)

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Fig. 6

Crease patterns and physical model of deployable membrane: (a) original crease pattern; (b) pattern transformed from (a); and (c) physical model of (b)

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Fig. 7

Comparison of the sizes of the deployed and wrapped membranes: (a) pattern with a diameter of 185 mm for producing a square pillar; (b) pattern with a diameter of 185 mm for producing a rectangular pillar; (c) wrapped membrane obtained from (a); and (d) wrapped membrane obtained from (b)

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Fig. 8

Physical models made from various materials: (a) plastic film of thickness 0.1 mm; (b) cloth of thickness 1.0 mm; and (c) stainless steel plate of thickness 0.1 mm

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