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 EngineeringTel-Aviv University
Solving a problem simply means representing it so as to make the
solution transparentHerbert Simon
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.
Solving a problem simply means representing it so as to make the
solution transparentHerbert Simon
Method:Transforming Engineering Design Problem into another Field, where the solution might already exist
Solving a problem simply means representing it so as to make the
solution transparentHerbert Simon
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
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6
12
3
A
B
D
C
C
D
5
Building the graph representation of the system
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6
12
3
A
B
D
C
C
D
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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
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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
The two engineering systems possess identical graph representations
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3
A
B
D
C
C
D
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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
Graph Representation
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
Graph Representation
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
Graph Representation
Electronics
Solving a real design problem by means of the approach
Designing an active torque amplifier
FR’’1={ Vout= Vin }FR’’2={ Iout= kIin }
Solving a real design problem 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
Graph Representation
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
Graph Representation
Same approach can be applied to graph representation
Design through mathematically related representations
Statics
Graph Representation
Kinematics
Graph Representationof another type
Designing a force amplifying beam system
Statics
Graph Representation
Kinematics
Graph Representationof another type
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
Graph Representation
Kinematics
Graph Representationof another type
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
Graph Representation
Dual Graph RepresentationDual
Robotsystem
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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
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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
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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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Transforming known engineering concepts from one engineering field through graph representations to
another, frequently yields new, useful concepts.
DIVISING NEW ENGINEERING CONCEPTS 1
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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
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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
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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
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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
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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
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Studying deployable structures requires consideration of both kinematical (during deployment) and statical (in locked position)
aspects
Topics on the edge between statics and kinematics
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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
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