Research Papers

Force–Deflection Modeling for Generalized Origami Waterbomb-Base Mechanisms

[+] Author and Article Information
Brandon H. Hanna, Spencer P. Magleby, Larry L. Howell

Department of Mechanical Engineering,
Brigham Young University,
Provo, UT 84602

Robert J. Lang

Lang Origami,
Alamo, CA 94507

Contributed by the Applied Mechanics Division of ASME for publication in the JOURNAL OF APPLIED MECHANICS. Manuscript received September 16, 2014; final manuscript received May 15, 2015; published online June 9, 2015. Assoc. Editor: Glaucio H. Paulino.

J. Appl. Mech 82(8), 081001 (Aug 01, 2015) (10 pages) Paper No: JAM-14-1428; doi: 10.1115/1.4030659 History: Received September 16, 2014; Revised May 15, 2015; Online June 09, 2015

The origami waterbomb base (WB) is a single-vertex bistable mechanism that can be generalized to accommodate various geometric, kinematic, and kinetic needs. The traditional WB consists of a square sheet that has four mountain folds alternating with five valley folds (eight folds total) around the vertex in the center of the sheet. This special case mechanism can be generalized to create two classes of waterbomb-base-type mechanisms that allow greater flexibility for potential application. The generalized WB maintains the pattern of alternating mountain and valley folds around a central vertex but it is not restricted to eight total folds. The split-fold waterbomb base (SFWB) is made by splitting each fold of a general WB into two “half folds” of the same variety as the parent fold. This study develops kinematic, potential energy, and force–deflection models for the rigid-foldable, developable, symmetric cases of the generalized WB and the SFWB, and investigates the relative effects of numbers of folds and split-fold panel size, on device behavior. The effect of selected key parameters is evaluated, and equations are provided to enable the exploration of other important parameters that may be of interest in the design and analysis of specific mechanisms. The similarities and differences between the two general forms are discussed, including tunability of the bistable and force–deflection behavior of each.

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Fig. 1

(a) Traditional n = 4 WB fold pattern, (b) folded base resting on the flat plane, and (c) spherical mechanism representation

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Fig. 2

(a) n = 3 and (b) n = 6 WBs

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Fig. 3

SFWB with n = 4. (a) Fold pattern in a flat state, (b) line drawing in first stable state, and (c) line drawing in second stable state. Links with different folds on each side (one mountain and one valley) are called facets and those with the same type of fold on both sides (mountain–mountain or valley–valley) are split-fold facets.

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Fig. 4

Spherical parameters used in WB analysis

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Fig. 10

Spherical parameters used to analyze SFWBs

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Fig. 9

(a) Potential energy and (b) force deflection plots for n = 3 and n = 10 WBs for which the average of the initial angles γm0 and γv0 is 54 deg. For the n = 3 WB, this average occurs at θ0 = 48 deg and for the n = 10 WB it occurs at θ0 = 54 deg.

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Fig. 8

Boundary conditions for the force–deflection analysis of the n = 4 WB. A vertical force is applied at the vertex while the outermost points of the valley folds are vertically supported.

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Fig. 7

(a) Potential energy and (b) force deflection plots for n = 3, n = 4, and n = 10 WBs where θ0 = 30 deg. The potential energy and force are divided by the stiffness to facilitate comparison.

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Fig. 6

γm and γv plotted against θ for an n = 4 WB

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Fig. 5

Position plots for n = 3, n = 4, and n = 10 WBs. Note the symmetry about the line passing through (0, 0) and (−180, 180).

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Fig. 15

Actuation testbed example (a) flat pattern, (b) metallic glass prototype in nearly flat state, and (c) prototype in erect state. The nearly flat state occurs when all WBs are in one stable state and the erect state occurs when all WBs are switched to the other stable state. Although a cube only has 12 edges, this pattern has 17 links. The extra links enable overlap for additional stability in the erect state.

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Fig. 16

Polycarbonate gripper prototype in the (a) open and (b) closed positions. It was designed to grasp a sphere. Lines are shown on the plastic sheet to identify the crease locations.

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Fig. 11

Position plots for n = 3, n = 4, and n = 10 SFWB with b = 5 deg. Note the symmetry about the line passing through (−100, 100) and (0, 0).

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Fig. 12

(a) Potential energy and (b) force–deflection plots for n = 3, n = 4, and n = 10 SFWBs where b = 5 deg and θ0 = 30 deg. The strain energy is nondimensionalized by dividing out the stiffness k, where k = kγm = kγv.

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Fig. 13

Boundary conditions for the split-fold force–deflection analysis. A vertical force is applied at the vertex while the midpoints of the valley split-fold facets are vertically supported.

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Fig. 14

(a) Position, (b) potential energy, and (c) force–deflection plots for b = 1 deg, b = 10 deg, and b = 20 deg SFWBs where n = 4 and θ0 = 30 deg



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