1. Rigidity of 2d and 3d Pinned Frameworks and pebble game". Dr. Offer Shai Department of Mechanics Materials and Systems, Tel Aviv University, Israel shai@eng.tau.ac.il

2. Rigidity Theory Mechanical Engineering Pebble game Theorems and methods in Engineering In this talk: Theorems in engineering underlying rigidity circuits. Methods for check rigidity of Pinned Frameworks from engineering.

3. Pebble Game and Pinned Frameworks Pebble game results in 1. Pinned Framework (the pinned edges correspond to the free pebbles at the end). 2. Separation of the Pinned Framework into special type of Pinned Framework, referred as Assur Graphs.

4. Assur Graphs (AGs) Definitions (many): 1. Structure with zero mobility that does not contain a simpler substructure of the same mobility. 2. Minimal rigid related to vertices. 3. Cycles of dyads. and more (Servatius et al., 2010). (a) (b) A B B A C 1 2 3 4 5 6 1 2 3 4 Assur Graph Not Assur Graph

5. Pebble game results in Partition into Assur Graphs. 2 1 Decomposition into two triads. Application of the pebble game First Triad Second Triad Each directed Cutset defines a partition. The cyclic (well concected) subgraph are Assur Graphs.

6. The same applies in 3d 3 2 1 Application of the pebble game Decomposition into two triads.

7. Necessary Condition for Pinned Isostatic Frameworks in d dimension: Decomposed into AGs. There should be at least d+1 ground edges. Each AG should be connected to the others by at least d ground vertices.

8. Example Not rigid because: 1. Can not be decomposed into AGs. 2. There are d ground (pinned) edges instead of d+1.

9. Example. Pebble Game reveals the Connection Problem between AGs in 2D. AG I AG II AG I AG II Not Rigid – AG I is connected through d-1 vertices. Rigid

10. Example. In 3D Pebble Game does not reveal the Connection Problem between the AGs. 3 2 1 Implied Hinge Decomposition into two 3D triads. Application of the pebble game

11. A B C G D O E F G A B C G D O E F D,E,F G Application of the pebble game Decomposition. A 2 1 G D 3 E F A

12. Implied Hinges = Implied Hinges + Implied Hinges in the connections in AGs easy ?????????? A If there is a circuit of size three, we locate the 6 free pebbles on its vertices. The problem of Implied Hinges

13. Relation between Assur Graphs and rigidity circuits. Contract the ground vertices into d-1 support vertices and add a d-2 simplex. contract all the ground vertices into one vertex. (Servatius et al., 2010) 2D Triad 2D 3D 3D Triad Rigid in 2D Rigid in 3D In 3D – Contract the ground vertices into two support vertices and add an edge between the two support vertices. Adding an edge between the two support vertices – rigidity circuit

14. Engineering theorems underlying Rigidity Circuits Suppose you have a Pinned Framework and an external force applied on one of the vertices. When will there be forces on all the edges? Theorem: there will be forces on all the edges IFF the vertex,where the external force acts, belongs to the first AG in the decomposition order and there is a directed path from this AG to any AGs in the decomposition graph

15. The topology structure of rigidity circuits (in 2d, 3d and possibley in higher dimensions): The scheme of rigidity circuits as a composition of Assur Graphs and an additional edge.

16. The Map of all AGs in 2D

17. The Map of 3d AGs - The map is NOT complete. - We try to “rephrase” the extension operation. - Extension in dimension d is – adding a ‘d’ dyad. u w uw v Dyad in dimension d

19. The Origin of Assur Graphs (Groups) Assur Graphs (Groups) were developed in Russia, in 1914, for decomposing linkages (mechanisms) into primitive building blocks – for analysis, optimization and more.

20. A mechanism 1 2 3 4 5 6 78 9 10 11AB C D E F J G H I A schematic graph of the mechanism ABC D E J G F H I 2 3 4 5 10 9 8 6 7 11

21. F 10 9 The decomposition of the schematic graph into s-genes G H I 867 11 E ABC D J 2 3 4 5 Diad Triad Tetrad

22. 1 2 3 4 5 6 78 9 10 11 A B C D E F J G H I ABC D J 2 3 4 5 E F 10 9 E G H I 8 6 7 11 Tetrad Decomposition of the mechanism Velocities of inner joints are known Triad Diad

23. Nowadays, we use Assur Graphs for synthesis of linkages, structures and more.

26. Thank you!!