1. Systematic conceptual engineering design using graph representations.

2. 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.

3. Satellite communications Different problems from different domains Not Really! Chessboard problem All can be represented by a common bipartite graph Problem solving with Graph Representations Tensegrity

4. 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

5. 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 D 1 and D 2. In this case, G can be used to transfer solution from D 1 to D 2 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

6. Design using Common Graph Representations Original engineering domain CGR Common 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

7. Design using Common Graph Representations Original engineering domain CGR Common 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 engineering Electric 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

8. Design using Common Graph Representations Original engineering domain CGR Common 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

9. Design using Common Graph Representations Original engineering domain CGR Common Graph Representation Secondary engineering domain Original engineering domain CGR Common 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.

10. 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 CGR Common Graph Representation Secondary engineering domain C A 1 0 A C 2 A C A C

11. 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 CGR Common Graph Representation Secondary engineering domain 3 BB A C B C 4 BB C C elements both possess the same potential. C

12. 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 CGR Common Graph Representation Secondary engineering domain A B C A B C

13. 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 CGR Common Graph Representation Secondary engineering domain A B C Potential ( ) can be represented as tangential velocity with edges possessing angular velocity. Flow (F) can be represented as force acting around an axis (Moment). A 1 0

14. B C A C Linear to Angular Design 2 4 C A 3 BB 1 0 Original engineering domain CGR Common Graph Representation Secondary engineering domain 3 B 0 B A 1 A 1 3 C A 1 3 B A B

15. C Linear to Angular Design 2 4 C A 3 BB 1 0 Original engineering domain CGR Common Graph Representation Secondary engineering domain 0 1 3 C 2 0 C 4 C 2 A C Edge 2 subject to Linear element replaced by angular element 4 BB C C 0 C elements both possess the same potential A B C 2 C 4 C C 0 C A B C

16. B 4 C 0 A 2 C B 4 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.

17. 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 CGR Common Graph Representation Secondary engineering domain 2 4 C 3 B 1 0 A Full Wave Rectifier Graph Original engineering domain CGR Common Graph Representation Secondary engineering domain

18. Design using special properties of Graph Representations Self Duality 1 23 45 6 I II III IV 4’ 3’ 2’ 1’ 5’ 6’ IV III III 4’ 3’2’ 1’ 5’ 6’

19. IV I II III IV III III Design using special properties of Graph Representations Self Duality Potential Law: Flow Law: Every cutset has a dual circle and vice-versa 1 23 45 6 4’ 3’2’ 1’ 5’ 6’ Potentials in Graph = Flows in Dual Graph

20. IV III III Design using special properties of Graph Representations Self Duality I Flow Law Broken = Illegal duality operation 1 23 45 Potential Law: 4’ 3’2’ 1’ 5’ Cutset does not have a dual circle and vice-versa Potentials in Graph = Flows in Dual Graph Flow Law:

21. 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. GlGl g1g1 DjDj T s2s2 s1s1 g2g2 Special properties Design using special properties of Graph Representations

22. C A B 0 C A B 0 Special Properties of Dual Graphs 2 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 ≠

23. 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

24. 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

25. 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. BehaviorEquivalent Engineering structure Graph element Y X C

26. C 1 1 A 3 B 4 2 C 5 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 B B 2 4 C 5 C 0 Design domain of concepts

27. 1 0 A A 3 B 4 B B 2 C 5 0 1 3 4 B 2 C 5 A 6 D Mechanisms taken from : Mechanisms and Mechanical Devices Sourcebook By :Nicholas P. Chironis Design domain of concepts