1. Combinatorial Representations for Analysis and Conceptual Design in Engineering Dr. Offer Shai Department of Mechanics, Materials and Systems Faculty of Engineering Tel-Aviv University

2. Solving a problem simply means representing it so as to make the solution transparent Herbert Simon

3. Solving a problem simply means representing it so as to make the solution transparent Herbert Simon It was found that: the proposed research work implements Simon's vision on Conceptual Design and Research.

4. Solving a problem simply means representing it so as to make the solution transparent Herbert Simon Method: Transforming Engineering Design Problem into another Field, where the solution might already exist

5. Solving a problem simply means representing it so as to make the solution transparent Herbert Simon Solid Mathematical Basis: Combinatorial Representations based on Graph and Matroid Theories

6. Current approach employs mathematical models based on graph theory to represent engineering systems Graph Representations - Definition Engineering system Graph Representation Structure and Geometry Voltage, absolute velocity, pressure       Relative velocity, deformation       Force, Current, Moment F F F F F F F

7. Consider two engineering systems from the fields of mechanics and electronics. 4 6 1 2 3 A B D C C D 5 Unidirectional Gear Train  out = |  in | Input shaft Output shaft Overrunning Clutches

8. Graph Representation of the system maps its structure, the behavior and thus also its function 4 6 1 2 3 A B D C C D 5 Building the graph representation of the system

9. 4 6 1 2 3 A B D C C D 5 Consider two engineering systems from the fields of mechanics and electronics. Electronic Diode Bridge Circuit V out = |V in | B A CD Input Source Output

10. Graph Representation of the system maps its structure, the behavior and thus also its function 4 6 1 2 3 A B D C C D 5 B A CD Building the graph representations of the systems

11. The two engineering systems possess identical graph representations 4 6 1 2 3 A B D C C D 5 B A CD Building the graph representations of the systems

12. FR’’={ V out = | V in | } 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

13. FR’’={ V out = | V in | } 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 Electronics B A CD

14. Common Representation Design Technique upon the map of graph representations Trusses (Determinate) (Indeterminate)

15. FR’’ 1 ={ V out = V in } FR’’ 2 ={ I out = kI in } Solving a real design problem through by means of the approach Designing an active torque amplifier FR 1 ={  out =  in } FR 2 ={ T out = kT in ; k>>1 } FR’ 1 ={  out =  in } FR’ 2 ={ F out = kF in } Mechanics Graph Representation Electronics

16. Solving a real design problem by means of the approach Designing an active torque amplifier

17. FR’’ 1 ={ V out = V in } FR’’ 2 ={ I out = kI in } Solving a real design problem by means of the approach Designing an active torque amplifier FR 1 ={  out =  in } FR 2 ={ T out = kT in ; k>>1 } FR’ 1 ={  out =  in } FR’ 2 ={ F out = kF in } Mechanics Graph Representation Electronics

18. The four working modes of the active torque amplifier mechanism Work principle of an active torque amplifier Input shaft Screw thread Output shaft Engine

19. Another Transformation Alternative Graph Representation

20. Same approach can be applied to graph representation Design through mathematically related representations Statics Graph Representation Kinematics Graph Representation of another type

21. Designing a force amplifying beam system Statics Graph Representation Kinematics Graph Representation of another type FR={ P out >>F in } FR’={ F out >>F in } FR’’={  out >>  in } FR’’’={  out >>  in }

22. Known gear train satisfying this requirement is the gear train employed in electrical drills. Statics Graph Representation Kinematics Graph Representation of another type FR={ P out >>F in } FR’={ F out >>F in } FR’’={  out >>  in } FR’’’={  out >>  in } AABB GGCC G 0 43251 IIIIV 0 III GC C ABBA G  out A C B GG A C B 53 1 24  in

23. Current Research Leads

24. 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

25. 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 Representation Dual Robot system 1 2 3 4 5 6 7 8

26. 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 12 2 1 5 8 9 10 6 Due to links 1 and 9 being located on the same line 1 2 3 4 5 6 7 8

27. The dual systems can be employed for detection of special properties of the original system DUALITY RELATIONS Serial Robot The Dual Stewart Platform known singular position Locked configuration 1 2 3 4 5 6 7 8

28. 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 1 2 3 4 5 6 7 8

29. Transforming known engineering concepts from one engineering field through graph representations to another, frequently yields new, useful concepts. DIVISING NEW ENGINEERING CONCEPTS 1 2 3 4 5 6 7 8

30. 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 CONCEPTS FACE FORCE 1 2 3 4 5 6 7 8

31. 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 CONCEPTS FACE FORCE 1 2 3 4 5 6 7 8

32. 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 CONCEPTS Equimomental line KinematicsStatics For 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 1 2 3 4 5 6 7 8

33. 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 CONCEPTS Equimomental line KinematicsStatics 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 1 2 3 4 5 6 7 8

34. 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 2 3 4 5 6 7 8

35. Studying deployable structures requires consideration of both kinematical (during deployment) and statical (in locked position) aspects Topics on the edge between statics and kinematics 1 2 3 4 5 6 7 8

36. Thank you!!! For more information contact Dr. Offer Shai Department of Mechanics, Materials and Systems Faculty of Engineering Tel-Aviv University This and additional material can be found at: http://www.eng.tau.ac.il/~shai