Rigidity of 2d and 3d Pinned Frameworks

and pebble game".

Dr. Offer ShaiDepartment of Mechanics Materials and Systems,

Tel Aviv University, Israelshai@eng.tau.ac.il

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!!