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Systematic conceptual engineering design using graph representations.

Systematic conceptual engineering design using graph representations. Research Objectives.

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Systematic conceptual engineering design using graph representations.

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

  4. Chessboard problem Satellite communications Common Graph Representation Special Properties Tensegrity solved Chessboard problem solved Satellite problem solved Tensegrity 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.

  5. V V Design Problem Design Problem t t Design using Common Graph Representations It was found that the same type of graph representations, say Gcan 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. • Step 1: Defining engineering problem in original domain. • Function Definition – What it does. • Use of “Black Box” Function Definition (Pahl and Wallace, 1996) Alternating angular velocity drive Rectified angular velocity output Original engineering domain

  6. Alternating Potential = input Rectified Potential = output Design Problem t t Alternating angular velocity drive Rectified angular velocity output V V Design Problem t t Design using Common Graph Representations • Step 2: Transforming problem to Graph Representation level . • Use of “common language” to describe system function. • Flow or Potential variables to describe system. CGR Common Graph Representation Original engineering domain

  7. Alternating Potential = input Rectified Potential = output Design Problem t t Alternating voltage source Rectified voltage output V V Design Problem t t Design using Common Graph Representations • 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. Secondary engineering domain – Electrical engineering Electric circuit is found that rectifies an alternating voltage source: The Full Wave rectifier CGR Common Graph Representation Original engineering domain Secondary engineering domain

  8. 4 2 A B 0 1 C 3 B Design using Common Graph Representations • 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. CGR Common Graph Representation Original engineering domain Secondary engineering domain

  9. CGR Common Graph Representation CGR Common Graph Representation Original engineering domain Secondary engineering domain Secondary engineering domain Design using Common Graph Representations 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 edgecorresponds to an element in the mechanical system. • Each vertex corresponds to a point in the mechanical • system where velocity is measured. Original engineering domain

  10. C A 4 2 2 A A C B A 0 0 1 1 C C C C A 3 CGR Common Graph Representation B Original engineering domain Secondary engineering domain Design using Common Graph Representations Step 5: Building new design at the engineering level using the graph solution .

  11. C 4 4 2 A C B B B B A C 0 1 C C B 3 3 CGR Common Graph Representation B B Original engineering domain Secondary engineering domain Design using Common Graph Representations Step 5: Building new design at the engineering level using the graph solution . • C elements both possess the same potential.

  12. 4 2 A B A B A B C C 0 1 C 3 CGR Common Graph Representation B Original engineering domain Secondary engineering domain Design using Common Graph Representations Step 5: Building new design at the engineering level using the graph solution .

  13. 4 2 A A B B A C 0 0 1 1 C 3 CGR Common Graph Representation B Original engineering domain Secondary engineering domain Linear to Angular Design Mechanical Design process can be made simpler by first designing linear systems and then converting to angular systems. • Potential ( ) can be represented as tangential velocity with edges • possessing angular velocity. • Flow (F) can be represented as force acting around an • axis (Moment).

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

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

  16. Looking at the complete mechanical rectifier where the driving input gear is subject to direction change: 2 2 C C C Rotates Anti-clock wise. A A 0 0 B B 4 4 C C C Rotates Clock wise.

  17. A B 0 C A C 0 3 B 4 2 1 Design using Common Graph Representations The same systematic process resulted in design through knowledge transfer of another available solution from the electronic engineering domain. Diode Bridge Graph Full Wave Rectifier Graph CGR Common Graph Representation CGR Common Graph Representation Original engineering domain Secondary engineering domain Original engineering domain Secondary engineering domain

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

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

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

  21. g2 s2 g1 s1 Design using special properties of Graph Representations 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 Dj Special properties Special properties T

  22. A B A B A 0 0 0 II II I I C C C IV III III I = III II IV B Special Properties of Dual Graphs 2 types of “rectifier” graphs Graph 1: Diode Bridge Graph 2: Full Wave rectifier Dual to itself Potential Source can be automatically exchanged for Flow Source Not Dual to itself Potential Source cannot be automatically exchanged for Flow Source Resulting Graph is Illegal ≠

  23. A B A B 0 0 C C A A B C C B 0 0 Special Properties of Dual Graphs Graph 1: Diode Bridge Graph 2: Full Wave rectifier Not Dual to itself Dual to itself Dual Statically Valid Dual Statically Non-Valid

  24. A B A B 0 0 C C B 0 0 Special Properties of Dual Graphs Graph 1: Diode Bridge Graph 2: Full Wave rectifier Not Dual to itself Dual to itself Dual Statically Valid A Dual Statically Non-Valid A B C C B

  25. C X Y 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.

  26. Design domain of concepts B A B B A A D A A B 2 2 2 C C 1 1 1 2 5 5 5 5 4 0 0 0 1 C C C C 6 3 3 3 3 4 4 4 B 0 0 5 2 1 5 3 4

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

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