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Research Papers

Mathematical Approach to Model Foldable Conical Structures Using Conformal Mapping

[+] Author and Article Information
Sachiko Ishida

Assistant Professor
Department of Mechanical Engineering,
School of Science and Technology,
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,
Tokyo 1648525, Japan
e-mail: taketoshinojima@gmail.com

Ichiro Hagiwara

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

Contributed by the Design Automation Committee of ASME for publication in the JOURNAL OF MECHANICAL DESIGN. Manuscript received July 9, 2013; final manuscript received May 28, 2014; published online June 26, 2014. Editor: Shapour Azarm.

J. Mech. Des 136(9), 091007 (Jun 26, 2014) (7 pages) Paper No: MD-13-1298; doi: 10.1115/1.4027848 History: Received July 09, 2013; Revised May 28, 2014

A new approach for obtaining the crease patterns of foldable conical structures from crease patterns of cylindrical structures based on the origami folding theory using conformal mapping is presented in this paper. Mapping for flow with circulation, which is the so-called polar conversion, is demonstrated as an example. This mapping can be used to produce similar elements and maintain the regularity of fold lines. This is a significant advantage when the mapping approach is used to produce foldable structures, because it is relatively easy to control angles between fold lines. Thus, this proposed approach enables us to design complex structures from simple original structures systematically, maintaining advanced characteristics particular to origami such as folding up spatial structures onto a plane and expanding them at will. To the best of our knowledge, this study is the first attempt to disclose a comprehensive design approach that can simplify the conventional design process. The proposed design approach can be addressed for further foldable structures such as circular membranes and toroidal tubes to broaden the design possibility of foldable mechanical products.

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Figures

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Fig. 1

Samples of conformal mapping on fluid flows: (a) analytic flow on the Φ–Ψ plane; (b) flow with circulation given by f = i·k log z; (c) flow with a source given by f = k log z; (d) dipole flow given by f = −k/z, and (e) flow in a wedge given by f = −kz2, where k is constant

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Fig. 2

Definition of fold lines, (a) the original fold lines and (b) the new fold lines and transformed lines

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Fig. 3

Crease patterns on the original and transformed planes: (a) crease pattern of cylindrical structures on the original plane and (b)–(e) are crease patterns formed by the transformation shown in Figs. 1(b)1(e), respectively

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Fig. 4

Crease patterns and physical models of conical structures; (a), (d), and (g): original crease patterns of cylindrical structures; (b), (e), and (h): transformed crease patterns of conical structures after angle correction; (c), (f), and (i): physical models of conical structures

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Fig. 5

Difference E defined by Eq. (15) under the condition k = 1 and α = π/6. (a) D = 0, (b) D = 1, and (c) D = 2

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Fig. 6

Condition for flat-foldability with four mountain and two valley fold lines. Here, lines (“—”), dashed–dotted lines (“-⋅-⋅-”), and broken lines (“- - -”) show the mountain fold, valley fold, and subordinate lines, respectively.

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Fig. 7

Three transformed elements to prove flat-foldability

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