Technical Brief

Kinematic Analysis of Congruent Multilayer Tessellations

[+] Author and Article Information
Yves Klett

Institute of Aircraft Design,
University of Stuttgart,
Pfaffenwaldring 31,
Stuttgart 70569, Germany
e-mail: klett@ifb.uni-stuttgart.de

Peter Middendorf

Institute of Aircraft Design,
University of Stuttgart,
Pfaffenwaldring 31,
Stuttgart 70569, Germany
e-mail: middendorf@ifb.uni-stuttgart.de

Manuscript received June 2, 2015; final manuscript received November 30, 2015; published online March 7, 2016. Assoc. Editor: Larry L. Howell.

J. Mechanisms Robotics 8(3), 034501 (Mar 07, 2016) (7 pages) Paper No: JMR-15-1125; doi: 10.1115/1.4032203 History: Received June 02, 2015; Revised November 30, 2015

Rigidly foldable origami tessellations exhibit interesting kinematic properties. Several tessellation types (most prominently Miura-ori) have shown potential for technical usage in aerospace and general lightweight construction. In addition to static (e.g., as core structures for sandwich components) and single-layer kinematic (e.g., deployable) applications, new possibilities arise from the combination of several layers of tessellations with congruent kinematics. This paper presents an analytical description of the kinematics of multilayered, or stacked, globally plane tessellations which retain rigid/isometric foldability by congruent, compatible movement.

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Fig. 1

Multilayer Miura-ori stack [12]

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Fig. 2

Miura unit cell dimensions: Developed cell, 3D state, and stack of two congruent unit cells with different heights

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Fig. 3

Folding sequence of three stacked Miura unit cells

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Fig. 4

Miura stack with plane outer layers. Top: perspective view, each image scaled to maximize detail. Bottom: side views with fixed scale to show height differences.

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Fig. 5

The configuration from Fig. 4 with inverted outer layer crease assignment/folding direction. The first state is identical to the first state from Fig. 4.

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Fig. 6

Miura stack with plane inner layer (darker)

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Fig. 7

Combined normalized stack height Hcn as functions of S and the ratio of outer and inner height Ho/Hi for unit cells with design parameters SD = 1, V D = 1, LD = 1, and HD = 1. All curves pass through the design point.

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Fig. 8

Two stack setups, featuring L=S=V=10  mm and a design height of HD=40  mm. For the upper stack, the height ratio rH=1.3 results in HiD=25  mm and HoD=32.5  mm, for the lower one with rH = 3 in HiD=8  mm and HoD=24  mm.

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Fig. 9

Overall height Hc as a function of the innermost dihedral folding angle θ (see Fig. 2) of two stacks with differing rH as detailed in Fig. 8. The stack unit cells show the corresponding folding state.

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Fig. 10

Two asymmetric cell stacks shown in different folding states with a design height of HD = 1. The conjoined vertices are marked as spheres, and the accumulated distance between corresponding points on the lower and higher layer is shown below each subfigure. The top configuration features 6, the bottom one 128 coinciding vertices.

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Fig. 11

Miura-type stack with differing cell sizes

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Fig. 12

A stack with biarc-based unit cells. In fact, the geometry used here is #6 from Ref. [17].

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Fig. 13

Conical tessellation stack

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Fig. 14

A spiral stack that winds in on itself during folding

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Fig. 15

Another strip-based stack that results in a 3 × 3 hexagonal grid

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Fig. 16

Stacked Barreto-ori. Top to bottom: perspective, front, side, and top (wireframe) view. The stack is not flat-foldable: the rightmost image shows intersection between the outer and inner layers. The major crease lines stay congruent and aligned in the plane.

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Fig. 17

The stack from Fig. 16 with inverted outer layer folding direction. The left and the right folding states both feature planar face sheets (developed versus flat-folded).

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Fig. 18

Yet another Barreto stack, this time shown for the complete possible kinematic range. This configuration exhibits a flat-foldable state for all layers on the left, and a blocking state on the right.



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