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Rigidity of 2d and 3d Pinned Frameworks
and pebble game".
Dr. Offer ShaiDepartment of Mechanics Materials and Systems,
Tel Aviv University, [email protected]
Rigidity Theory
Mechanical Engineering
Pebble game
Theorems and methods in Engineering
In this talk:
Theorems in engineering underlying rigidity circuits.
Methods for check rigidity of Pinned Frameworks from engineering.
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.
Framework Pinned Framework
used in engineering
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).
( a ) ( b )
A
B
B A
C 1
2
3 4 5
6
1 2
3
4
Assur Graph Not Assur Graph
Pebble game results in Partition into Assur Graphs.
21
Decomposition into two triads.
Application of the
pebble game
First Triad
Second Triad
Each directed Cutset defines a partition.The cyclic (well concected) subgraph are Assur Graphs.
The same applies in 3d
3 2
1
Application of the
pebble gameDecomposition into two triads.
Necessary Condition for Pinned Isostatic Frameworks in d dimension:
• Decomposed into AGs.• There should be at least d+1 ground edges.• Each AG should be connected to the others by at least
d ground vertices.
Example Not rigid because:1. Can not be decomposed into AGs.2. There are d ground (pinned) edges instead of d+1.
Example. Pebble Game reveals the Connection Problem between AGs in 2D.
AG I
AG II
AG I
AG II
Not Rigid – AG I is connected through d-1 vertices.
Rigid
Example. In 3D Pebble Game does not reveal the Connection Problem between the AGs.
3 2
1
Implied Hinge
Decomposition into two 3D
triads.
Application of the
pebble game
AB
C
G
D
O
EF
G
A B
C
G
D
O
EF
D,E,F
G
Application of the
pebble game
Decomposition.
A 2
1
G
D
3
EF
A
Implied Hinges = Implied Hinges + Implied Hinges in the connections in AGs easy ??????????
AIf there is a circuit of size three, we locate the 6 free pebbles on its vertices.
The problem of Implied Hinges
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 Triad
2D
3D
3D Triad
Rigid in 2D
Rigid in 3D
In 3D – Contract the ground vertices into two support vertices and add an edge between the two support vertices.
Adding an edge between the two support vertices – rigidity circuit
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?
A B
CD
FE
G
H
E,F,G
A,B,C,D
A B
CD
FE
G
H
)a(
)a1(
)b(
A B
CD
FE
G
H
)c(
A B
CD
F
E
G
H
)d(
H
E,F,G
A,B,C,D
H A,B,C,D,E,F,G,H A,B,C,D,E,F,G,H
)b1(
)c1()d1(
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
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.
The Map of all AGs in 2DThe Map of all AGs in 2D
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.
u w u w
v
Dyad in dimension d
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.
A mechanismA mechanism
1122
33
44
55
66
77 88
99
1010
1111
AA
BB
CC
DD
EE
FF
JJ
GG
HH
II
A schematic graph of the mechanismA schematic graph of the mechanism
AA
BB CC
DD
EE
JJ
GG FF
HH
II
22
33
44
55
1010
99
88
66
77
1111
FF1010
99
The decomposition of the schematic graph into s-genesThe decomposition of the schematic graph into s-genes
GG
HH
II
88
66
77
1111
EE
AA
BB CC
DD JJ
22
33
44
55
DiadDiadTriadTriadTetradTetrad
1122
33
44
55
66
77 88
99
1010
1111
AA
BB
CC
DD
EE
FF
JJ
GG
HH
II
AA
BB CC
DD JJ
22
33
44
55
EE
FF1010
99
EEGG
HH
II
88
66
77
1111
TetradTetradDecomposition of the mechanismDecomposition of the mechanism
Velocities of inner joints are knownVelocities of inner joints are known
TriadTriadDiadDiad
Nowadays, we use Assur Graphs for
synthesis of linkages, structures and
more.
Thank you!!