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Research Papers: Design of Mechanisms and Robotic Systems

Symmetry-Based Transformable and Foldable Plate Structures

[+] Author and Article Information
Valentina Beatini

Department of Architecture,
Abdullah Gul University,
Sumer Campus,
Kayseri 38280, Turkey
e-mail: valentina.beatini@agu.edu.tr

Contributed by the Mechanisms and Robotics Committee of ASME for publication in the JOURNAL OF MECHANICAL DESIGN. Manuscript received June 21, 2016; final manuscript received March 29, 2017; published online May 18, 2017. Assoc. Editor: Dar-Zen Chen.

J. Mech. Des 139(7), 072301 (May 18, 2017) (12 pages) Paper No: MD-16-1457; doi: 10.1115/1.4036648 History: Received June 21, 2016; Revised March 29, 2017

This paper presents a novel family of modular flat-foldable rigid plate structures composed by assemblies of 4R-linkages. First, in the field of foldable plates, the proposed system is characterized by being not only foldable but also transformable: the slope of one module over the other is capable of changing not only magnitude but also sign. This transformable behavior extends the range of application of foldable plates from simply larger–smaller configurations to substantially different configurations and usages. The transformable curve is obtained by means of symmetry operations on the spherical length of links. For each module, three configurations can be designed. Various examples are illustrated.

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Figures

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Fig. 1

Linear assemblies of doubly symmetrical 4R-linkages: (a) planar linkages, (b) spherical linkages, and (c) the plate structure correspondent to (b)

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Fig. 2

A minimal transformable mechanism where a2;11<a2;12 and its spherical representation during motion: (a) starting position: fully flat-folded state, (b) generic unfolded position, and (c) end position: partially flat-folded state

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Fig. 3

(a) Folding phases of a linear assembly. The centers of mixed linkages are indicated. (b) Spherical representation of the same assembly. Different color tones refer to different 4R-linkages (see figure online for color).

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Fig. 4

Generic unfolded position of a spatial assembly: (a) the plate structure and (b) the spherical links

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Fig. 5

The plate structure, planar angles αi;js, fold angles ϑi;js and τi;js between sides of plates at vertexes i, j, degree of curvature ωs,s+1, and inclination angles οs,s+1;t and οs;t,t+1 defining the slope of modules

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Fig. 6

A mechanism whose planar angles within the double symmetrical linkages are in the form αi;js=π-αi;js+1 : (a) one module, (b) folding phases, and (c) variation of transversal angle τi;js≡τi;js+1 between plates at the mixed linkage as a function of the input degree of curvature ωs,s+1

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Fig. 7

A mechanism whose planar angles within the double symmetrical linkages are in the form αi;js=αi;js+1 : (a) one module, (b) folding phases, and (c) variation of transversal angle τi;js≡τi;js+1 between plates at the mixed linkage as a function of the input degree of curvature ωs,s+1

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Fig. 8

Another mechanism whose planar angles within the double symmetrical linkages are in the form αi;js=αi;js+1 : (a) one module, (b) folding phases, and (c) variation of transversal angle τi;js≡τi;js+1 between plates at the mixed linkage as a function of the input degree of curvature ωs,s+1

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Fig. 9

A folding spiral. The spherical length of links within the quad-based doubly symmetrical linkages changes throughout the assembly in order to fit the variable curvature of the target profiles: (a) axonometric views, (b) detail, and (c) front view at the partially flat-folded state.

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Fig. 10

A folding spiral. The spherical length of links within the mixed linkages changes throughout the assembly in order to fit the variable curvature of the target profiles: (a) axonometric views, (b) detail, and (c) front view at the partially flat-folded state.

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Fig. 11

A folding spiral. The spherical lengths of links are constant throughout, while the linear length of plates changes throughout the assembly in order to fit the variable curvature of the target profiles: (a) axonometric views, (b) detail, and (c) front view at the partially flat-folded state.

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Fig. 12

Folding phases of a surface defined by a circular generatrix and a straight directrix

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Fig. 13

Folding phases of a surface defined by a circular generatrix and an arched directrix

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Fig. 14

A plate structure composed by three modules, two of which are composed by plates of different linear length; degrees of curvature ωs,s+1 and ωs;t,t+1=0 deg; inclination angles οs,s+1;t and οs;t,t+1

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Fig. 15

(a) Any assembly of quad plates has mobility one. A possible input joint is highlighted. (b) An assembly composed by triangular and quadrangular plates. Highlighted joints are in the necessary number and possible position to achieve a mechanism with mobility one. Moving from left to right, and setting the selected joints, the hatched areas become fully defined. (c) The mobility of an assembly composed by triangular and quadrangular plates can be studied focusing on the border longitudinal strips of plates.

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