Relations%20Between%20Engineering%20Fields

1
Relations Between Engineering Fields
2
Duality between engineering systems

• Since graph representations are mathematical
  entities, mathematical relations can be
  established between them, such as duality
  relations
• (gi?G1)Tdual(gj ?G2)

G1
G2
Tdual
gj
gi
3
Duality between engineering systems

• Duality between graph representations yield the
  duality relations between the represented
  engineering systems.

G1
G2
Db
Da
D
T
gj
gi
sj
T
si
4
Dual Representation Design Technique
5
Duality between Linkages and Structures
Duality between Stewart platforms and Robots
Duality between Planetary and Beam Systems
6
Duality relation between linkages and structures
7
Consider a kinematical linkage and its graph
representation
Kinematical Linkage
8
Consider a kinematical linkage and its graph
representation
Kinematical Linkage
9
Now, consider a static structure and its graph
representation

Static Structure
10
There exist a mathematical relation between the
representations of the two systems

Kinematical Linkage
Static Structure
11
Therefore there is the duality relation between
the represented engineering systems

Kinematical Linkage
Static Structure
12
The relative velocity of each link of the linkage
is equal to the internal force in the
corresponding rod of the structure

Kinematical Linkage
Static Structure
13
The equilibrium of forces in the structure is
thus equivalent to compatibility of relative
velocities in the linkage

Kinematical Linkage
Static Structure
14
Checking system stability through the duality
Due to links 1 and 9 being located on the same
line
15
Duality relation between Stewart platform and
serial robot
16
Consider a Stewart platform system.
17
BUILDING THE GRAPH REPRESNTATION OF THE STEWART
PLATFORM
P
P
3
3
2
2
4
4
1
1
5
5
6
6
18
BUILDING THE DUAL GRAPH REPRESNTATION
P
1
3
6
2
4
5
P
3
2
4
1
5
6
19
BUILDING THE DUAL SPATIAL LINKAGE (SERIAL ROBOT)
P
1
6
3
5
2
4
20
BUILDING THE DUAL SPATIAL LINKAGE (SERIAL ROBOT)
P
1
6
3
5
2
4
21
VERIFYING THE DUALITY RELATION BETWEEN THE TWO
SYSTEMSThe correspondence in geometric positions
of the kinematical pairs of the linkage and the
legs of the Stewart platform.
P
22
VERIFYING THE DUALITY RELATION BETWEEN THE TWO
SYSTEMSEach force in the Stewart platform leg is
directed in the same direction as the relative
angular velocity of the corresponding kinematical
pair in the dual serial robot.
P
23
Duality relation between planetary and beam
systems
24
Consider a simple two-gear system.
25
BUILDING THE GRAPH REPRESNTATION OF THE GEAR
SYSTEM
C
A
B
2
C
rC
B
1
rB
A
rA
26
VERIFYING THE ISOMORPHISM OF THE REPRESENTATION
C
A
B
2
C
w1/0
rC
B
rA x w1/0
1
rB
A
rA
27
VERIFYING THE ISOMORPHISM OF THE REPRESENTATION
C
A
B
2
C
w1/0w2/1
rC
B
rA x w1/0rB x w2/1
1
rB
A
rA
28
VERIFYING THE ISOMORPHISM OF THE REPRESENTATION
C
A
B
2
C
w1/0w2/1w0/20
rC
B
rA x w1/0rB x w2/1rC x w0/20
1
rB
A
rA
29
VERIFYING THE ISOMORPHISM OF THE REPRESENTATION
C
A
B
2
C
w1/0w2/1w0/20
rC
B
rA x w1/0rB x w2/1rC x w0/20
1
rB
A
rA
30
VERIFYING THE ISOMORPHISM OF THE REPRESENTATION
C
A
B
2
C
w1/0w2/1w0/20
rC
B
rA x w1/0rB x w2/1rC x w0/20
1
rB
A
rA
31
DEDUCING BEHAVIORAL EQUATIONS FROM THE GRAPH
KNOWLEDGE
C
A
B
2
C
DAw2/1w0/20
rC
B
rA x DArB x w2/1rC x w0/20
1
rB
A
rA
32
DEDUCING BEHAVIORAL EQUATIONS FROM THE GRAPH
KNOWLEDGE
C
A
B
2
C
DADBw0/20
rC
B
rA x DArB x DBrC x w0/20
1
rB
A
rA
33
DEDUCING BEHAVIORAL EQUATIONS FROM THE GRAPH
KNOWLEDGE
C
A
B
2
C
DADBDC0
rC
B
rA x DArB x DBrC x DC0
1
rB
A
rA
34
DEDUCING BEHAVIORAL EQUATIONS FROM THE GRAPH
KNOWLEDGE
C
A
C
A
B
B
2
C
DADBDC0
rC
B
rA x DArB x DBrC x DC0
1
rB
A
rA
35
BUILDING THE DUAL GRAPH REPRESENTATION
C
C
A
A
C
A
B
B
B
2
C
DADBDC0
rC
B
rA x DArB x DBrC x DC0
1
rB
A
rA
36
DEDUCING BEHAVIORAL EQUATIONS FROM THE DUAL
REPRESENTATION
C
A
C
A
B
B
2
C
FADBDC0
rC
B
rA x FArB x DBrC x DC0
1
rB
A
rA
37
DEDUCING BEHAVIORAL EQUATIONS FROM THE DUAL
REPRESENTATION
C
A
C
A
B
B
2
C
FAFBDC0
rC
B
rA x FArB x FBrC x DC0
1
rB
A
rA
38
DEDUCING BEHAVIORAL EQUATIONS FROM THE DUAL
REPRESENTATION
C
A
C
A
B
B
2
C
FAFBFC0
rC
B
rA x FArB x FBrC x FC0
1
rB
A
rA
39
DEDUCING BEHAVIORAL EQUATIONS FROM THE DUAL
REPRESENTATION
C
A
C
A
B
B
2
C
FAFBFC0
rC
B
rA x FArB x FBrC x FC0
1
rB
A
rA
40
BUILDING THE DUAL BEAM SYSTEM
C
A
C
A
B
B
2
C
C
FAFBFC0
rC
B
B
rA x FArB x FBrC x FC0
1
rB
A
A
rA
41
VERIFYING THE DUALITY RELATION BETWEEN THE TWO
SYSTEMS
C
A
C
A
B
B
2
C
C
PAFBFC0
rC
B
B
rA x PArB x FBrC x FC0
1
rB
A
A
rA
42
VERIFYING THE DUALITY RELATION BETWEEN THE TWO
SYSTEMS
C
A
C
A
B
B
2
C
C
PARBFC0
rC
B
B
rA x PArB x RBrC x FC0
1
rB
A
A
rA
43
VERIFYING THE DUALITY RELATION BETWEEN THE TWO
SYSTEMS
C
A
C
A
B
B
2
C
C
PARBRC0
rC
B
B
rA x PArB x RBrC x RC0
1
rB
A
A
rA
44
VERIFYING THE DUALITY RELATION BETWEEN THE TWO
SYSTEMS
C
A
C
A
B
B
2
C
C
PARBRC0
rC
B
B
rA x PArB x RBrC x RC0
1
rB
A
A
rA