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Combinatorial Representations for Analysis and Conceptual Design in Engineering

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Combinatorial Representations for Analysis and Conceptual Design inEngineering

Dr. Offer Shai

Department of Mechanics, Materials and Systems

Faculty of Engineering

Tel-Aviv University

Solving a problem simply means representing it so as to make the solution transparent

Herbert Simon

Solving a problem simply means representing it so as to make the solution transparent

It was found that:

the proposed research work implements Simon's vision on

Conceptual Design and Research.

Herbert Simon

Solving a problem simply means representing it so as to make the solution transparent

Method:

Transforming Engineering Design Problem into another Field, where the solution might already exist

Herbert Simon

Solving a problem simply means representing it so as to make the solution transparent

Solid Mathematical Basis:

Combinatorial Representations based on Graph and Matroid Theories

Herbert Simon

F

F

F

F

F

F

F

Graph Representations - Definition

Graph Representation

Structure and Geometry

Voltage, absolute velocity, pressure

Relative velocity, deformation

Force, Current, Moment

Engineering system

Current approach employs mathematical models based on graph theory to represent engineering systems

Input shaft

Output shaft

1

4

C

D

A

2

5

3

6

B

D

C

Overrunning Clutches

Unidirectional Gear Train

out= |in|

Consider two engineering systems from the fields of mechanics and electronics.

1

4

C

D

A

2

5

3

6

B

D

C

Building the graph representation of the system

Graph Representation of the system maps its structure, the behavior and thus also its function

A

1

4

Input Source

C

D

A

C

D

2

5

3

6

B

D

C

B

Output

Electronic Diode Bridge Circuit

Vout= |Vin|

Consider two engineering systems from the fields of mechanics and electronics.

A

1

4

C

D

A

C

D

2

5

3

6

B

D

C

B

Building the graph representations of the systems

Graph Representation of the system maps its structure, the behavior and thus also its function

A

1

4

C

D

A

C

D

2

5

3

6

B

D

C

B

Building the graph representations of the systems

The two engineering systems possess identical graph representations

Solving Design Problem

FR’={ out= | in| }

Graph Representation

FR={ out= |in| }

FR’’={ Vout= | Vin| }

Mechanics

Electronics

We shall now consider a hypothetical design problem for inventing the unidirectional gear train

A

C

D

B

Solving Design Problem

FR’={ out= | in| }

Graph Representation

FR={ out= |in| }

FR’’={ Vout= | Vin| }

Mechanics

Electronics

In electronics there is a known device satisfying this functional requirement – diode bridge circuit

Trusses

(Determinate)

(Indeterminate)

Common Representation Design Technique upon the map of graph representations

Designing an active torque amplifier

FR’1={ out= in }

FR’2={ Fout= kFin }

Graph Representation

FR1={ out= in}

FR2={ Tout= kTin; k>>1}

FR’’1={ Vout= Vin }

FR’’2={ Iout= kIin }

Mechanics

Electronics

Solving a real design problem through by means of the approach

Designing an active torque amplifier

Solving a real design problem by means of the approach

Designing an active torque amplifier

FR’1={ out= in }

FR’2={ Fout= kFin }

Graph Representation

FR1={ out= in}

FR2={ Tout= kTin; k>>1}

FR’’1={ Vout= Vin }

FR’’2={ Iout= kIin }

Mechanics

Electronics

Solving a real design problem by means of the approach

Output shaft

Screw thread

Input shaft

Engine

Work principle of an active torque amplifier

The four working modes of the active torque amplifier mechanism

Another Transformation Alternative

Graph Representation

Design through mathematically related representations

Graph Representation

Graph Representation

of another type

Statics

Kinematics

Same approach can be applied to graph representation

Graph Representation

Graph Representation

of another type

FR’’={ out>> in }

FR’={ Fout>>Fin }

FR’’’={ out>> in }

FR={ Pout>>Fin }

Statics

Kinematics

Designing a force amplifying beam system

0

G

B

A

B

A

1

2

3

4

5

B

A

B

A

C

G

C

I

II

III

IV

C

G

C

G

G

0

2

4

3

5

C

C

B

B

wout

1

A

A

G

G

win

Graph Representation

Graph Representation

of another type

FR’’={ out>> in }

FR’={ Fout>>Fin }

FR’’’={ out>> in }

FR={ Pout>>Fin }

Statics

Kinematics

Known gear train satisfying this requirement is the gear

train employed in electrical drills.

Current Research Leads

1. Duality relations

2. Duality relations for checking truss rigidity

3. Duality relations for finding special properties

4 Identification of singular configurations

5 Devising new engineering concepts – face force

6 Devising new engineering concepts – equimomental lines

7 Multidisciplinary engineering education

8 Topics on the edges between statics and kinematics

L

K

11

7

0

10

A

P

F

2

P

12

2

P

7

J

1

B

12

I

H

8

9

9

11

C

10

8

1

D

6

L

K

5

2

4

3

G

I

II

H

P

A

B

D

E

C

F

1

I

J

K

L

P

2

O

DUALITY RELATIONS

1

Dual Graph Representation

2

Dual

Robot

system

Statical platform system

3

4

5

6

7

8

Graph Representation

Applying the graph theoretical duality principle to the graph representations yielded new relations between systems belonging to different engineering fields

12

12

2

2

4

8

4

3

3

7

5

9

1

7

1

9

5

6

6

10

10

11

11

8

Rigid ????

2

’

12

’

4

’

8

’

3

’

2

’

1

’

7

’

5

’

9

’

12

’

8

’

6

’

4

’

10

’

3

’

9

’

7

’

1

’

1

1

’

5

’

6

’

10

’

R’

11

’

R’

Definitely locked !!!!!

DUALITY RELATIONS

1

2

3

4

5

6

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

7

8

By means of the duality transformation, checking the rigidity of trusses can be replaced by checking the mobility of the dual mechanisms

DUALITY RELATIONS

1

2

Serial Robot

Locked configuration

3

4

5

6

7

known singular position

8

The Dual Stewart Platform

The dual systems can be employed for detection of special properties of the original system

IDENTIFICATION OF SINGULAR CONFIGURATIONS

1

2

3

Given mechanism topology

4

5

6

7

8

One of the results of applying the approach – a new method for finding all dead center positions for a given mechanism topology

DIVISING NEW ENGINEERING CONCEPTS

1

2

3

4

5

6

7

8

Transforming known engineering concepts from one engineering field through graph representations to another, frequently yields new, useful concepts.

DIVISING NEW ENGINEERING CONCEPTS

1

FACE FORCE

2

3

4

5

6

7

8

The concept of linear velocity has been transformed from kinematics to statics. The result: a new statical variable combining the properties of force and potential

DIVISING NEW ENGINEERING CONCEPTS

1

FACE FORCE

2

3

4

5

6

7

8

The concept of linear velocity has been transformed from kinematics to statics. The result: a new statical variable combining the properties of force and potential

DIVISING NEW ENGINEERING CONCEPTS

1

Equimomental line

2

3

Kinematics

Statics

4

For any two bodies moving in the plane there exists a point were their velocities are equal – relative instant center

For any two forces acting in the place there exists a line, so that both forces apply the same moment upon each point on this line

5

6

7

8

The concept of relative instant center from kinematics has been transformed to statics. Result: new locus of points in statics - equimomental line

DIVISING NEW ENGINEERING CONCEPTS

1

Equimomental line

2

3

Kinematics

Statics

4

5

6

Instant center – long known kinematical tool for analysis and synthesis of kinematical systems

Equimomental line – completely new tool for analysis and synthesis of statical systems

7

8

The concept of relative instant center from kinematics has been transformed to statics. Result: new locus of points in statics - equimomental line

Multidisciplinary engineering education

1

2

3

4

5

6

7

8

The students are first taught the graph representations, their properties and interrelations. Only then, on the basis of the representations they are taught specific engineering fields.

Topics on the edge between statics and kinematics

1

2

3

4

5

6

7

8

Studying deployable structures requires consideration of both kinematical (during deployment) and statical (in locked position) aspects

Thank you!!!

For more information contact

Dr. Offer Shai

Department of Mechanics, Materials and Systems

Faculty of Engineering

Tel-Aviv University

This and additional material can be found at:

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