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Systematic conceptual engineering design using graph
representations .
Research Objectives
Development of Systematic design methods to facilitate conceptual engineering design using discrete mathematical models called combinatorial representations that are based on graph theory as a medium for knowledge transfer.
• Design through Common Graph Representation.
• Design through Dual Graph Representation.
• Identification and usage of special properties obtained by graphs.
Satellite communications
Different problems from different domainsNot Really!
Chessboard problem
All can be represented by a common bipartite graph
Problem solving with Graph Representations
Tensegrity
Satellite communications
Chessboard problem
Common Graph Representation
Solving one of the problems in its domain solves the analogous problems using the graph to transfer the solution.
Special properties of the graph are reflected in the domains represented.
Tensegrity solved
Satellite problem solved
Chessboard problem solved
Special Properties
Tensegrity
Design using Common Graph Representations
It was found that the same type of graph representations, say G can
be associated with more than one engineering domain, say D1 and D2.
In this case, G can be used to transfer solution from D1 to D2 and vice-versa.
Original engineering
domain
Step 1: Defining engineering problem in original domain.•Function Definition – What it does. •Use of “Black Box” Function Definition (Pahl and Wallace, 1996)
Design Problem
Alternating angular velocity drive
V
t
Rectified angular velocity output
Design Problem
t
V
Design using Common Graph Representations
Original engineering
domain
CGRCommon Graph Representation
Step 2: Transforming problem to Graph Representation level .•Use of “common language” to describe system function.•Flow or Potential variables to describe system.
Design Problem
Alternating Potential = input
Rectified Potential =
output
t t
Design Problem
Alternating angular velocity drive
Rectified angular velocity output
V V
t t
Design using Common Graph Representations
Original engineering
domain
CGRCommon Graph Representation
Secondary engineering
domain
Step 3: Locate a solution in another engineering domain .•Engineering domain must share common representation.•Flow or Potential variables translated to corresponding terminology of secondary engineering domain.
Design Problem
Alternating Potential = input
Rectified Potential =
output
t t
Secondary engineering domain – Electrical engineeringElectric circuit is found that rectifies an alternating
voltage source: The Full Wave rectifier
Design Problem
Alternating voltage source
Rectified voltage output
V V
t t
Design using Common Graph Representations
Original engineering
domain
CGRCommon Graph Representation
Secondary engineering
domain
Step 4: Transfer solution from engineering domain to Graph Representation level .•Each structure element in the engineering level is translated into it’s equivalent element representation in the graph through deterministic steps.•Graph topology insures proper representation of properties and system behavior.
2
4
C
A
3
BB
1
0
Design using Common Graph Representations
Original engineering
domain
CGRCommon Graph Representation
Secondary engineering
domain
Original engineering
domain
CGRCommon Graph Representation
Secondary engineering
domain
Step 5: Building new design at the engineering level using the graph solution .•Each element in the graph representation is represented at the engineering level as an equivalent element through deterministic steps.
•Graph topology again insures that proper representation of properties and system behavior is transferred to engineering solution.
This structural procedure on the graph representation ensures:•Each edge corresponds to an element in the mechanical system.•Each vertex corresponds to a point in the mechanical system where velocity is measured.
C
Design using Common Graph Representations
Step 5: Building new design at the engineering level using the graph solution .
2
4
C
A
3
BB
1
0
Original engineering
domain
CGRCommon Graph Representation
Secondary engineering
domain
C
A
1
0AA
AC
AC
2A
C
A
CAC AC AC
Design using Common Graph Representations
Step 5: Building new design at the engineering level using the graph solution .
2
4
C
A
3
BB
1
0
Original engineering
domain
CGRCommon Graph Representation
Secondary engineering
domain
3
BB
A
C
B
AB C 4
BB
C
BC
•C elements both possess the same potential.
C
C
C
Design using Common Graph Representations
Step 5: Building new design at the engineering level using the graph solution .
2
4
C
A
3
BB
1
0
Original engineering
domain
CGRCommon Graph Representation
Secondary engineering
domain
A
BC
A
BC
Linear to Angular Design
Mechanical Design process can be made simpler by first designing linear systems and then converting to angular systems.
2
4
C
A
3
BB
1
0
Original engineering
domain
CGRCommon Graph Representation
Secondary engineering
domain
A
BC
•Potential ( ) can be represented as tangential velocity with edges possessing angular velocity.•Flow (F) can be represented as force acting around an axis (Moment).
C
A
1
0
C
B
C
AC
Linear to Angular Design
2
4
C
A
3
BB
1
0
Original engineering
domain
CGRCommon Graph Representation
Secondary engineering
domain
3
B
0
B
A
1
A
1
3
C
A
1in
in 3
B
A
B
C
Linear to Angular Design
2
4
C
A
3
BB
1
0
Original engineering
domain
CGRCommon Graph Representation
Secondary engineering
domain
0
1
3
C
2
0
C4
C
2A
C
•Edge 2 subject to •Linear element replaced by angular element
2
4BB
C
4
C0
•C elements both possess the same potential
A
B
2
C
AC
2C
4
C
C
0
C
A
B
C
B4
C
0
A
2
C
B4
C
0
A
2
C
Looking at the complete mechanical rectifier where the driving input gear is subject to direction change:
C Rotates Anti-clock wise.
C Rotates Clock wise.
Design using Common Graph Representations
The same systematic process resulted in design through knowledge transfer of another available solution from the electronic engineering domain.
C
A
B
0
Diode Bridge Graph
Original engineering
domain
CGRCommon Graph Representation
Secondary engineering
domain
2
4
C
3
B
10
A
Full Wave Rectifier Graph
Original engineering
domain
CGRCommon Graph Representation
Secondary engineering
domain
Design using special properties of Graph Representations
Self Duality
1
23
4 5
6
III
III
IV
4’3’ 2’
1’
5’
6’IV
II I
III4’
3’ 2’
1’
5’
6’
IV
III
III
IV
II I
III
Design using special properties of Graph Representations
Self Duality
Potential Law: Flow Law:
Every cutset has a dual circle and vice-versa
11
23
4 5
6
4’
3’ 2’
1’
5’
6’
5
2
0521
'1F
0521 ''' FFF
'2F'5F
0431
0431 ''' FFF
0632
0632 ''' FFF
0654 0654 ''' FFF
Potentials in Graph = Flows in Dual Graph
IV
II I
III
Design using special properties of Graph Representations
Self Duality
I
Flow Law Broken = Illegal duality operation
1
23
4 55
4
23
03254 Potential Law:
4’
3’ 2’
1’
5’
Cutset does not have a dual circle and vice-versaPotentials in Graph = Flows in Dual Graph
Flow Law:
'4F '5F
0'3'2'5'4 FFFF
'2F'3F
Two Engineering systems in the Engineering Domain are transformed to graphs in the Graph Domain.
The Graph Domain reveals properties that were not discovered at the Engineering level.
These special properties may be transferred back to the Engineering Domain where they reflect the special properties in the Graph Domain .
Gl
g1
Dj
Ts2
s1
g2 Special properties
Special properties
Design using special properties of Graph Representations
C
A
B
0
C
A
B
0
Special Properties of Dual Graphs2 types of “rectifier” graphs
Graph 1: Diode Bridge
Dual to itself
Potential Source can be automatically exchanged for Flow
Source
I
II
III
IV
II
I
III=
IV
Graph 2: Full Wave rectifier
Not Dual to itself
Potential Source cannot be automatically exchanged for Flow
Source
Resulting Graph is Illegal
II
III
I
C
A
B
0≠
C
A
B
0
C
A
B
0
Graph 1: Diode Bridge
Dual to itself
Graph 2: Full Wave rectifier
Not Dual to itself
C
0
A
B
C
0
A
Dual Statically Valid
B
Dual Statically
Non-Valid
Special Properties of Dual Graphs
C
A
B
0
C
A
B
0
Graph 1: Diode Bridge
Dual to itself
Graph 2: Full Wave rectifier
Not Dual to itself
C
0
A
B
C
0
A
Dual Statically Valid
B
Dual Statically
Non-Valid
Special Properties of Dual Graphs
B
Design domain of concepts
•Each element in the graph representation is represented at the engineering level as an equivalent element through deterministic steps.
•A graph element can be represented by different structures possessing the same behavior.
Graph element
Equivalent Engineering structure
Behavior
Y
XX
Y
X Y
CXY
XYXY .1
XY .2
C
1
1
A
3
B4
2
C5
0
B
A
C
3
2
4
5
5
0
1
3
4
B
2
C
5
A
6
D
0
0
1
3
4
2
5
1
AA
3
BB
2
4
C5
C0
Design domain of concepts
1
0
AA
3
B
4BB
2
C5
01
3
4
B
2
C
5A
6
D
Mechanisms taken from :
Mechanisms and Mechanical Devices Sourcebook
By :Nicholas P. Chironis
Design domain of concepts