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Research Papers

A New Approach to the Generation of Retractable Plate Structures Based on One-Uniform Tessellations

[+] Author and Article Information
Aylin Gazi Gezgin

Department of Architecture,
Izmir Institute of Technology,
Gulbahce Koyu Kampusu,
Urla, Izmir 35430, Turkey
e-mail: aylingazi@msn.com

Koray Korkmaz

Department of Architecture,
Izmir Institute of Technology,
Gulbahce Koyu Kampusu,
Urla, Izmir 35430, Turkey
e-mail: koraykorkmaz@iyte.edu.tr

1Corresponding author.

Manuscript received August 12, 2016; final manuscript received March 22, 2017; published online May 24, 2017. Assoc. Editor: Jian S. Dai.

J. Mechanisms Robotics 9(4), 041015 (May 24, 2017) (13 pages) Paper No: JMR-16-1233; doi: 10.1115/1.4036570 History: Received August 12, 2016; Revised March 22, 2017

Retractable plate structure (RPS) is a family of structures that is a set of cover plates connected by revolute joints. There exists wide range of possibilities related with these structures in architecture. Configuring the suitable shape of rigid plates that are able to be enclosed without any gaps or overlaps in both closed and open configurations and eliminating the possibility of contact between the plates during the deployment have been the most important issues in RPS design process. Many researchers have tried to find the most suitable shape by using kinematical or empirical analysis so far. This study presents a novel approach to find the suitable shape of the plates and their assembly order without any kinematical or empirical analysis. This approach is benefited from the one-uniform mathematical tessellation technique that gives the possibilities of tiling a plate using regular polygons without any gaps or overlaps. In the light of this technique, the shape of the plates is determined as regular polygons and two conditions are introduced to form RPS in which regular polygonal plates are connected by only revolute joints. It should be noted that these plates are not allowed to become overlapped during deployment and form gaps in closed configuration. Additionally, this study aims to reach a single degree-of-freedom (DoF) RPS. It presents a systematic method to convert multi-DoF RPS into single DoF RPS by using the similarity between graph theory and the duality of tessellation.

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Figures

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Fig. 1

One-uniform regular tessellations

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Fig. 2

One-uniform semiregular tessellations

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Fig. 3

Examples of k(2) and k(3) uniform tessellations

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Fig. 4

Dual of regular tessellations

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Fig. 5

Dual of semiregular tessellations

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Fig. 6

Movement capability of regular polygons around a single vertex of every one-uniform tessellations

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Fig. 7

Expansion of RPS based on 44 tessellation

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Fig. 8

Expansion of RPS based on (3.6.3.6) tessellation

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Fig. 9

Simplest module of the RPS based on square tessellation with an excessive plate

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Fig. 10

Simplest module of the RPS based on square tessellation with and without the excessive plate

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Fig. 12

Unpredictable expansion of single DoF RPS based on (3.4.6.4) tessellation

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Fig. 13

Unpredictable expansion of RPS based on (34.6) tessellation

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Fig. 14

M = 1 parallelogram loop

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Fig. 15

Singular case of M = 1 parallelogram

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Fig. 17

Parallelogram mod-change

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Fig. 18

Retraction and dead center position for RPS based on 44 tessellation

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Fig. 19

Retraction and dead center position for RPS based on (3.6.3.6) tessellation

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Fig. 20

(a) Sides of single loop reaches dead center position before fully closed configuration and (b) unpredictable movement of the parallelogram loops after dead center

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Fig. 21

(a) Structural representation and (b) graph representation

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Fig. 22

(a) Structural representation and (b) graph representation

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Fig. 23

Similarity between graph representation of RPS and dual of 44 tessellation

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Fig. 24

Similarity between graph representation of RPS and dual of 3.6.3.6 tessellation

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Fig. 25

Similarity between graph representation of RPS and dual of 34.6 tessellation

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Fig. 26

Similarity between graph representation of RPS and dual of 36 tessellation

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Fig. 27

34.6 tessellation and its dual

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Fig. 28

Modification of the dual of base tessellation

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Fig. 29

Graph representation of a subchain creation

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Fig. 30

Structural representation a subchain

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Fig. 31

Graph representation of an over constraint subchain creation

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Fig. 32

Structural representation with over constrained subchain

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Fig. 33

Graph representation

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Fig. 34

Structural representation

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Fig. 35

Graph representation of the RPS module without any subchain

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Fig. 36

Structural representation of RPS module

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Fig. 37

Expansion of a RPS module based on 34.6 tessellation

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Fig. 38

(a) Graph representation with six subchains and (b) with six extra 2R joints

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Fig. 39

Graph representation based on 34.6 tessellation by generating overconstrained subchains

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Fig. 40

Structural representation based on 34.6 tessellation by generating overconstrained subchains

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Fig. 41

Expansion of iterated RPS based on 34.6 tessellation by generating overconstrained subchain

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Fig. 42

Attempt to draw graph representation of RPS based on 36 tessellations

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Fig. 43

Addition of extra points to the dual

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Fig. 44

Drawing tetragons on the dual

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Fig. 45

Graph representation of RPS based on 36 tessellation

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Fig. 46

Expansion of RPS based on 36 tessellation

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Fig. 47

Application of a RPS to a building

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