combinatorial representations for analysis and conceptual design in engineering

Combinatorial Representations for Analysis

and Conceptual Design inEngineering

Dr. Offer Shai Department of Mechanics, Materials and Systems

Faculty of EngineeringTel-Aviv University

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

solution transparentHerbert Simon



It was found that:the proposed research work

implements Simon's vision onConceptual Design and Research.



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



Solid Mathematical Basis:Combinatorial Representations

based on Graph and Matroid Theories

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

systems

Graph Representations - Definition

Engineering system

Graph Representation

Structure and GeometryVoltage, absolute velocity, pressure

Relative velocity, deformation

Force, Current, Moment

FF

F

FF

FF

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

4

6

12

3

A

B

D

C

C

D

5

Unidirectional Gear Trainout= |in|

Input shaftOutput shaft

Overrunning Clutches

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

4

6

12

3

A

B

D

C

C

D

5

Building the graph representation of the system

4

6

12

3

A

B

D

C

C

D

5

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

Electronic Diode Bridge CircuitVout= |Vin|

B

A

C DInput Source

Output

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

4

6

12

3

A

B

D

C

C

D

5

B

A

C D

Building the graph representations of the systems

The two engineering systems possess identical graph representations

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6

12

3

A

B

D

C

C

D

5

B

A

C D

Building the graph representations of the systems

FR’’={ Vout= | Vin | }

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

Solving Design Problem

FR={ out= |in| }

FR’={ out= | in | }

Mechanics


Electronics

FR’’={ Vout= | Vin | }

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

Solving Design Problem

FR={ out= |in| }

FR’={ out= | in | }

Mechanics


ElectronicsB

A

C D

Common Representation Design Technique upon the map of graph representations

Trusses(Determinate)

(Indeterminate)

FR’’1={ Vout= Vin }FR’’2={ Iout= kIin }

Solving a real design problem through by means of the approach

Designing an active torque amplifier

FR1={ out= in }FR2={ Tout= kTin; k>>1 }

FR’1={ out= in }FR’2={ Fout= kFin }

Mechanics


Electronics

Solving a real design problem by means of the approach


FR’’1={ Vout= Vin }FR’’2={ Iout= kIin }

Solving a real design problem by means of the approach


FR1={ out= in }FR2={ Tout= kTin; k>>1 }

FR’1={ out= in }FR’2={ Fout= kFin }

Mechanics


Electronics

The four working modes of the active torque amplifier mechanism

Work principle of an active torque amplifier

Input shaft

Screw thread Output shaft

Engine

Another Transformation Alternative


Same approach can be applied to graph representation

Design through mathematically related representations

Statics


Kinematics

Graph Representationof another type

Designing a force amplifying beam system

Statics


Kinematics


FR={ Pout>>Fin }

FR’={ Fout>>Fin } FR’’={ out>> in }

FR’’’={ out>> in }

Known gear train satisfying this requirement is the geartrain employed in electrical drills.

Statics


Kinematics


FR={ Pout>>Fin }

FR’={ Fout>>Fin } FR’’={ out>> in }

FR’’’={ out>> in }

A AB B

GG CCG

0

432 51

I II IV

0

IIIG CC

A BB AG

out

A

CB

G GA

CB

53

1

2 4

in

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

I II

O

P1 A B D C F

G

E

H

I

L K

J

P2

7

5

8

6

4 3 2

0

1

12

9 11 10

P1

P2

L K

L K

H I J

7

8

9

12 10

11

P2

A

B

C D

F

Applying the graph theoretical duality principle to the graph representations yielded new relations between

systems belonging to different engineering fields

DUALITY RELATIONS

Statical platform system


Dual Graph RepresentationDual

Robotsystem

1

2

3

4

5

6

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

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

1

2

3

4

5

6

7

8

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

DUALITY RELATIONS

A

C

B

A’

C’

B’

1’

2’ 3’ 4’

5’

6’

1

2 3 4

5 6

P

(a)

(b)

Serial Robot

The Dual Stewart Platform

known singular position

Locked configuration

1

2

3

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

IDENTIFICATION OF SINGULAR CONFIGURATIONS

Given mechanism topology

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2

3

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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

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 CONCEPTSFACE FORCE

1

2

3

4

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 CONCEPTSEquimomental line

Kinematics StaticsFor 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

1

2

3

4

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 CONCEPTSEquimomental line

Kinematics Statics

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

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.

Multidisciplinary engineering education 1

2

3

4

5

6

7

8

http://www.cvc.uab.es/shared/projects/ivp/pc/big/pcbs.jpg

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

aspects

Topics on the edge between statics and kinematics

1

2

3

4

5

6

7

8

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

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

combinatorial representations for analysis and conceptual design in engineering

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