Dr. Offer Shai and Daniel Rubin Department of Mechanics, Materials and Systems

1. Transforming Knowledge to Structures from Other Engineering Fields by Means of Graph Representations. Dr. Offer Shai and Daniel Rubin Department of Mechanics, Materials and Systems Faculty of Engineering Tel-Aviv University

2. Outline of the Talk • Transforming knowledge through the graph theory duality. • Practical applications • Checking truss rigidity. • Detecting singular positions in linkages. • Deriving a new physical entity – face force. • Further research – form finding problems in tensegrity systems.

3. Constructing the graph corresponding to the kinematical linkage O 1 A Joints  Vertices Links  Edges O2 9 O3 B D O1 4 3 5 1 2 A C 6 7 8 E O4 O4 Kinematical Linkage

4. O 1 A Constructing the graph corresponding to the kinematical linkage 9 B 2 Joints Vertices Links  Edges O2 9 O3 B D O1 4 3 5 1 2 A C 6 7 8 E O4 O4 Kinematical Linkage

5. O 9 1 B 2 A Constructing the graph corresponding to the kinematical linkage 5 3 D 4 C Joints  Vertices Links  Edges O2 9 O3 B D O1 4 3 5 1 2 A C 6 7 8 E O4 O4 Kinematical Linkage

6. O 9 1 B 2 A Constructing the graph corresponding to the kinematical linkage 5 3 D 4 C 7 6 E 8 Joints  Vertices Links  Edges O2 9 O3 B D O1 4 3 5 1 2 A C 6 7 8 E O4 O4 Kinematical Linkage

7. The variables associated with the graph correspond to the physical variables of the system 5 9 1 3 2 4 7 6 Joint velocity  Vertex potential Link relative velocity  Edge potential difference 8 9 4 3 5 1 2 6 7 8

8. The potential differences in the graph representations satisfy the potential law 5 9 Sum of potential differences in each circuit of the graph is equal to zero = polygon of relative linear velocities in the mechanism 1 3 2 4 7 6 8 9 4 3 5 1 2 6 7 8

9. Now, consider a plane truss and its graph representation 9 5 2 3 4 1 6 7 A O 8 Joints  Vertices Rods  Edges 5 9 3 2 1 4 6 7 A O 8 Static Structure

10. 9 5 2 3 4 1 6 7 8 5 9 3 2 1 4 6 7 8 Now, consider a plane truss and its graph representation Rod internal force  Flow through the edge Static Structure

11. 9 5 2 3 4 1 6 7 8 5 9 3 2 1 4 6 7 8 The flows in the graph satisfy the flow law Sum of the flows in each cutset of the graph is equal to zero = force equilibrium Static Structure

12. Constructing the dual graph • Face - circuit forming a non- bisected area in the drawing of the graph. • - For every face in the original graph associate a vertex in the dual graph. • - If in the original graph there are two faces adjacent to an edge – e, then in the dual graph the corresponding two vertices are connected by an edge e’.

13. Constructing the dual graph

14. Consequently there is a duality between linkages and plane trusses

15. The relative velocity of each link of the linkage is equal to the internal force in the corresponding rod of its dual plane truss. Kinematical Linkage Static Structure

16. The equilibrium of forces in the truss == compatibility of the relative velocities in the dual linkage Kinematical Linkage Static Structure

17. 12 12 2 2 4 8 4 3 3 7 5 9 1 7 1 9 5 6 6 10 10 11 11 8 Rigid ???? 2 ’ 12 ’ 4 ’ 8 ’ 3 ’ 2 ’ 1 ’ 7 ’ 5 ’ 9 ’ 12 ’ 8 ’ 6 ’ 4 ’ 10 ’ 3 ’ 9 ’ 7 ’ 1 ’ 1 1 ’ 5 ’ 6 ’ 10 ’ R’ 11 ’ R’ Definitely locked !!!!! Checking system rigidity through the duality Due to links 1 and 9 being located on the same line

18. Using duality relation to detect singular positions Linkages Plane Trusses Kinematical Analysis Statical Analysis Flow == Force Potential difference = = Linear relative velocity Sum of flows at any cutset equals to zero Sum of potential differences at any circuit equals to zero Singular position detection Deformation Analysis Potential Difference = Displacement Flow = Force Sum of flows at any cutset equals to zero Sum of potential differences at any circuit equals to zero

19. C 2 4 7 1 5 6 3 6 A 1 5 B 3 7 4 2 π π π r A' C A B r 2 4 1 3 3 1 C' 5 5 r 6 6 B' 4 2 7 Using duality relation to detect singular positions Forces Deformations Mechanism in singular position

20. Deriving a new entity – face force == dual to the absolute linear velocity in the dual linkage ? Kinematical Linkage Static Structure

21. Face force – a variable associated with each face of the structures The internal force in the element of the structure is equal to the vector difference between the face forces of the faces adjacent to it

22. Face force – a variable associated with each face of the structures Face forces can be considered a multidimensional generalization of mesh currents.

23. It was proved that face forces manifest some properties of electric potentials.

24. The works of Maxwell reaffirm some of the results derived through the graph representations, among them: duality between linkages and trusses, face force, and more …

25. Maxwell Diagramlines in the diagram are associated with the rods of the structure I VI III IV V III IV II O

26. The coordinates of the points in the Maxwell diagram correspond to the face forces in the corresponding faces of the truss I VI III IV V III IV II O

27. Verifying truss-linkage duality through the work of Maxwell I VI IV V III III O3 O2 O1 O4 IV II O

28. Verifying truss-linkage duality through the work of Maxwell III IV

29. Verifying truss-linkage duality through the work of Maxwell III IV

30. 2 O I 3 3 C B B C 4 2 III 1 1 5 5 II 4 D A A D 6 6 4 1 6 3 2 5 Further research – form finding problem in tensegrity systems Tensegrity system at unstable configuration Graph representation of the tensegrity system III 3 4 1 5 I II 6 2 Resulting stable configuration of the tensegrity system Arbitrary chosen faces forces

31. 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