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Research Papers

Normalized Coordinate Equations and an Energy Method for Predicting Natural Curved-Fold Configurations

[+] Author and Article Information
Jacob C. Badger

Department of Mechanical Engineering,
Brigham Young University,
Provo, UT 84602
e-mail: j_badger@byu.edu

Todd G. Nelson

Department of Engineering,
University of Southern Indiana,
Evansville, IN 47712
e-mail: tgnelson@usi.edu

Robert J. Lang

Lang Origami,
Alamo, CA 94507
e-mail: robert@langorigami.com

Denise M. Halverson

Department of Mathematics,
Brigham Young University,
Provo, UT 84602
e-mail: halverson@math.byu.edu

Larry L. Howell

Department of Mechanical Engineering,
Brigham Young University,
Provo, UT 84602
e-mail: lhowell@byu.edu

1Corresponding author.

Contributed by the Applied Mechanics Division of ASME for publication in the Journal of Applied Mechanics. Manuscript received January 17, 2019; final manuscript received March 21, 2019; published online April 12, 2019. Assoc. Editor: Yihui Zhang.

J. Appl. Mech 86(7), 071006 (Apr 12, 2019) (9 pages) Paper No: JAM-19-1029; doi: 10.1115/1.4043285 History: Received January 17, 2019; Accepted March 21, 2019

Of the many valid configurations that a curved fold may assume, it is of particular interest to identify natural—or lowest energy—configurations that physical models will preferentially assume. We present normalized coordinate equations—equations that relate fold surface properties to their edge of regression—to simplify curved-fold relationships. An energy method based on these normalized coordinate equations is developed to identify natural configurations of general curved folds. While it has been noted that natural configurations have nearly planar creases for curved folds, we show that nonplanar behavior near the crease ends substantially reduces the energy of a fold.

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Figures

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Fig. 1

Illustration of a curved fold formed by folding along a curve f(u) in an unfolded configuration (left) and folded with a fold angle γ (right)

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Fig. 2

Tangent (left), normal (center), and principle (right) planes to a curve embedded in a developable surface. These planes contain the geodesic curvature (κg), the normal curvature (κn), and the principle curvature (κp), respectively.

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Fig. 3

Normalized distance of a point from the edge of regression

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Fig. 4

Solution process for obtaining u, v parameterization of an edge point χ(u0)

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Fig. 5

Physical model of a circular crease of width 0.9 units (top) with the simulated planar-uniform fold and ruling field (left) and simulated minimal-energy uniform fold and ruling field (right). Both simulated configurations have fixed fold-angle γ = π/2.

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Fig. 6

Optimized torsion function (top) and planar deviation (bottom) for a thin circular crease of width 0.9 units with a fixed fold-angle γ = π/2

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Fig. 7

Physical model of a circular crease of width 0.05 units (top) with the simulated planar-uniform fold and ruling field (left) and simulated minimal-energy uniform fold and ruling field (right). Both planar-uniform and minimal-energy uniform folds have fixed fold-angle γ = π/2.

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Fig. 8

Optimized torsion function (top) and planar deviation (bottom) for a thin circular crease of width 0.05 units with a fixed fold-angle γ = π/2

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Fig. 9

A physical model (top) and simulated natural configuration using K = 50 and γ0 = π/2 (bottom) for a circular crease of width 0.8 units

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Fig. 10

Physical model (left) and natural simulated configuration using K = 100 and γ0 = π/2 (right) along with the simulated torsion and fold-angle functions (bottom) for a crease with curvature κg = cos(s) for s ∈ [0, 3π] of width 0.7 units

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Fig. 11

Physical model (left) and natural simulated configuration using K = 100 and γ0 = π/2 (right) along with the simulated torsion and fold-angle functions (bottom) for a crease with curvature κg = 4tan−1(s) for s ∈ [−1.5, 1.5] and variable fold width

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Fig. 12

Planar (top-left) and natural (bottom-left) configurations of a quarter circular crease of width 0.8. The distance to the edge of regression, v0 (center), and ruling energy (right) are given to illustrate the effect that the location of the edge of regression on the total energy of the fold. Note that the ruling energies are small near the ends of the crease in the natural configurations because the length of the ruling lines decreases toward the ends of the crease.

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