Transforming Knowledge to Structures from Other Engineering Fields by

Means of Graph Representations.

Dr. Offer Shai and Daniel RubinDepartment of Mechanics, Materials and Systems

Faculty of EngineeringTel-Aviv University

Outline of the Talk

• Transforming knowledge through the graph theory duality.

• Practical applications – Checking truss rigidity. – Detecting singular positions in linkages.– Deriving a new physical entity – face force.

• Further research – form finding problems in tensegrity systems.

Kinematical Linkage

Constructing the graph corresponding to the kinematical linkage

Joints VerticesLinks EdgesO1

O2

O3

O4

A

B

C

D

E

1234

5

6 7

8

O4

O

1

A

9

Kinematical Linkage

Joints VerticesLinks EdgesO1

O2

O3

O4

B

C

D

E

1234

5

6 7

8

O4

A

O

1

A

B2

9

9

Constructing the graph corresponding to the kinematical linkage

Kinematical Linkage

Joints VerticesLinks EdgesO1

O3

O4

B

C

D

E

1234

5

6 7

8

O4

A

9

A

O

1B

2

9

C

3D4

5

O2

Constructing the graph corresponding to the kinematical linkage

Kinematical Linkage

Joints VerticesLinks Edges

O1

O2

O3

O4

B

C

D

E

1234

5

6 7

8

O4

A

9

A

O

1B

2

9

C

3D4

5

E

76

8

Constructing the graph corresponding to the kinematical linkage

The variables associated with the graph correspond to the physical variables of the system

Joint velocity Vertex potential

Link relative velocity Edge potential difference

1234

5

6 7

8

9

1

2

9

3

4

5

76

8

The potential differences in the graph representations satisfy the potential law

1234

5

6 7

8

9

1

2

9

3

4

5

76

8

Sum of potential differences in each Sum of potential differences in each circuit of the graph is equal to zero = circuit of the graph is equal to zero = polygon of relative linear velocities in polygon of relative linear velocities in

the mechanismthe mechanism

Static Structure

Now, consider a plane truss and its graph representation

Joints VerticesRods Edges

12

9

34

5

76

8

12

9

34

5

76

8O

A

OA

Static Structure

Rod internal force Flow through the edge

12

9

34

5

76

8

12

9

34

5

76

8

Now, consider a plane truss and its graph representation

Static Structure

12

9

34

5

76

8

12

9

34

5

76

8

The flows in the graph satisfy the flow law

Sum of the flows in each cutset Sum of the flows in each cutset of the graph is equal to zero = of the graph is equal to zero =

force equilibriumforce equilibrium

Constructing the dual graph

- Face - circuit forming a non- bisected area in the drawing of the graph.

- For every face in the original graph associate a vertex in the dual graph.

- If in the original graph there are two faces adjacent to an edge –

e, then in the dual graph the corresponding two vertices are connected by an edge e’.

Constructing the dual graph

Consequently there is a duality between linkages and plane trusses

Kinematical Linkage Static Structure

The relative velocity of each link of the linkage is equal to the internal force in the corresponding rod of its dual plane truss.

Kinematical Linkage Static Structure

The equilibrium of forces in the truss = = compatibility of the relative velocities in the

dual linkage

Definitely locked !!!!!

Rigid ????

8

12’

2’

1’

11’

10’6’

7’

3’

5’

’

9’

R’

4’12’

9’

10’

R’11’

6 ’

7’

8’

2’

3’

5’1’

4’

8

5 9

2

4

7

10

11

1

12

6

3

11

7

3

4

122

1 5

8

9

106

Due to links 1 and 9 being located on the same line

Checking system rigidity through the duality

Using duality relation to detect singular positions

Linkages Plane Trusses

Potential difference = = Linear relative velocity

Flow == Force

Sum of potential differences at any circuit equals to zero

Sum of flows at any cutset equals to zero

Flow = Force Potential Difference = Displacement

Sum of flows at any cutset equals to zero

Sum of potential differences at any circuit equals to zero

Kinematical Analysis Statical Analysis

Deformation AnalysisSingular position detection

1

2

3

4

56

7

A'

B'

C'

Aπ

Bπ

Cπ

4

2

61

3 5

7

AB

C

1 3

2

4

56

7

DeformationsForces

Mechanism in singular position

Using duality relation to detect singular positions

2

15

6

43

Kinematical Linkage Static Structure

Deriving a new entity – face force == dual to the absolute linear velocity in the dual linkage

?

Face force – a variable associated with each face of the structures

The internal force in the element of the structure is equal to the

vector difference between the face forces of the faces adjacent to it

Face force – a variable associated with each face of the structures

Face forces can be considered a multidimensional generalization of mesh

currents.

It was proved that face forces manifest some properties of electric potentials.

pathrodFF ii

rodface i

• The works of Maxwell reaffirm some of the results derived through the graph representations, among them: duality between linkages and trusses, face force, and more …

Maxwell Diagramlines in the diagram are associated with the

rods of the structure

II

VI

IIIV

I

IV

O

III

IV

The coordinates of the points in the Maxwell diagram correspond to the face forces in the

corresponding faces of the truss

II

VI

IIIV

I

IV

O

III

IV

Verifying truss-linkage duality through the work of Maxwell

II

IIIIV

O

V

IVI

O1O2O3O4

III

IV

Verifying truss-linkage duality through the work of Maxwell

III

IV

Verifying truss-linkage duality through the work of Maxwell

III

IV

Further research – form finding problem in tensegrity systems

A

B C

D

12

3

45

6

Tensegrity system at unstable configuration

III

I

II

O

1

2

3

4

5

6

C

DA

B

Graph representation of the tensegrity system

43

5

2

1

6

II

III

I

Arbitrary chosen faces forces

4

5

61

3 2

Resulting stable configuration of the tensegrity system

Thank you !!!

For more information contact Dr. Offer Shai

Department of Mechanics, Materials and SystemsFaculty of Engineering

Tel-Aviv University

This and additional material can be found at:

http://www.eng.tau.ac.il/~shai