Rigidity of 2d and 3d Pinned Frameworks and pebble game".

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Rigidity of 2d and 3d Pinned Frameworks and
pebble game".

• Dr. Offer Shai
• Department of Mechanics Materials and Systems,
  Tel Aviv University, Israel
• shai_at_eng.tau.ac.il
•  

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Pebble game
Rigidity Theory
Mechanical Engineering
Theorems and methods in Engineering
In this talk Theorems in engineering underlying
rigidity circuits. Methods for check rigidity of
Pinned Frameworks from engineering.
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Pebble Game and Pinned Frameworks

• Pebble game results in
• 1. Pinned Framework (the pinned edges
  correspond to the free
• pebbles at the end).
• 2. Separation of the Pinned Framework into
  special type of Pinned
• Framework, referred as Assur Graphs.

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Assur Graphs (AGs)

• Definitions (many)
• 1. Structure with zero mobility that does not
  contain a simpler
• substructure of the same mobility.
• 2. Minimal rigid related to vertices.
• 3. Cycles of dyads.
• and more (Servatius et al., 2010).

Not Assur Graph
Assur Graph
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Pebble game results in Partition into Assur
Graphs.
Each directed Cutset defines a partition.The
cyclic (well concected) subgraph are Assur
Graphs.
First Triad
Application of the pebble game
Decomposition into two triads.
Second Triad
2
1
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The same applies in 3d
Application of the pebble game
Decomposition into two triads.
3
2
1
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Necessary Condition for Pinned Isostatic
Frameworks in d dimension

• Decomposed into AGs.
• There should be at least d1 ground edges.
• Each AG should be connected to the others by at
  least d ground vertices.

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Example Not rigid because1. Can not be
decomposed into AGs.2. There are d ground
(pinned) edges instead of d1.
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Example. Pebble Game reveals the Connection
Problem between AGs in 2D.
AG I
AG I
AG II
AG II
Not Rigid AG I is connected through d-1
vertices.
Rigid
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Example. In 3D Pebble Game does not reveal the
Connection Problem between the AGs.
Implied Hinge
Application of the pebble game
Decomposition into two 3D triads.
3
2
1
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D,E,F
G
Application of the pebble game
A
Decomposition.
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Implied Hinges Implied Hinges
Implied Hinges in
the connections in AGs
easy
??????????
The problem of Implied Hinges
If there is a circuit of size three, we locate
the 6 free pebbles on its vertices.
A
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Relation between Assur Graphs and rigidity
circuits.Contract the ground vertices into d-1
support vertices and add a d-2 simplex.

• contract all the ground vertices into one
  vertex. (Servatius et al., 2010)

2D
Rigid in 2D
2D Triad

• In 3D Contract the ground vertices into two
  support vertices and add an edge between the two
  support vertices.
•  

3D
Rigid in 3D
3D Triad
Adding an edge between the two support vertices
rigidity circuit
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Engineering theorems underlying Rigidity Circuits

• Suppose you have a Pinned Framework and an
  external force applied on one of the vertices.
  When will there be forces on all the edges?

• Theorem there will be forces on all the edges
  IFF the vertex ,where the external force acts,
  belongs to the first AG in the decomposition
  order and there is a directed path from this AG
  to any AGs in the decomposition graph  

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The topology structure of rigidity circuits (in
2d, 3d and possibley in higher dimensions)
The scheme of rigidity circuits as a composition
of Assur Graphs and an additional edge.
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The Map of all AGs in 2D













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• The Map of 3d AGs
• - The map is NOT complete.
• - We try to rephrase the extension operation.
• - Extension in dimension d is adding a d
• dyad.

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• The Origin of Assur Graphs (Groups)
• Assur Graphs (Groups) were developed in
• Russia, in 1914, for decomposing linkages
  (mechanisms) into primitive building
• blocks for analysis, optimization and
• more.

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A mechanism
A schematic graph of the mechanism
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The decomposition of the schematic graph into
s-genes
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F
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Tetrad
Diad
Triad
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Decomposition of the mechanism
Tetrad
Triad
Diad
Velocities of inner joints are known
A
C
3
B
4
1
2
5
E
J
10
D
F
6
G
9
7
8
H
I
11
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• Nowadays, we use Assur Graphs for synthesis of
  linkages, structures and
• more.

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Thank you!!