1. Proceedings of the ASME 2012 International Design Engineering Technical Conferences & Computers and Information in Engineering Conference IDETC/CIE 2012 August 12-15, 2012, Chicago, IL, USA DETC2012-70904 APPLYING RIGIDITY THEORY METHODS FOR TOPOLOGICAL DECOMPOSITION AND SYNTHESIS OF GEAR TRAIN SYSTEMS Terushkin Maria School of Mechanical Engineering Tel Aviv University Tel Aviv, Israel Shai Offer School of Mechanical Engineering Tel Aviv University Tel Aviv, Israel

2. Rigidity theory deals mostly with the topological computation. Mechanism theory is mainly concerned with the geometrical analysis and generic statements. In rigidity theory there are two main types of representations: 1.Bar-joint representation. 2. Body-bar representation. Rigidity theory

3. 2d – revolute joint 3d – spherical joint Binary Link/one constraint Bar and joint representation Between any two joints (2d-revolute, 3d-spherical) there exists at most one bar/constraint. 2d – revolute joint 3d – spherical joint

4. Body-bar Representation Bodies connected by bars (constraints). Lower kinematic pair 1 2 Higher kinematic pair 1 2 Spherical joint 12 12 12 Hinge

5. Representing a mechanism by Bar-joint and Body-bar representations Body-bar: Two bodies and three bars. Bar-joint: Nine bars and seven joints.

6. Definition of a Body-Bar Assur Graph Graph G is a 2d/3d Body-Bar Assur Graph IFF 1.G has 3/6 DOF 2. G does not contain any sub-graph (of more than one element) which also has 3/6 DOF. 3.There are at most 2/5 constraints between any two bodies.

7. (a)(b) 2 3 1 2 3 1 2d Body-bar Assur Graph Body-bar Assur Graph Not a body-bar Assur Graph

8. 4 3 2 1 0 Body-Bar graph of a mechanism ( mechanism ) (Body-Bar graph) A 1 2 4 3 C

9. Topology synthesis of ALL the mechanisms through Body Bar Assur Graph Every mechanism is a composition of Body-bar Assur Graphs (building blocks). Thus, knowing ALL the Body-bar Assur Graphs enables deriving ALL the topologies of mechanisms, both in 2d and 3d. ALL the Body-bar Assur Graphs are derived by applying one type of operation, called extension. Next slide we will see the extension operation in 2d.

10. The extension operation in 2d - Add an Atom (body with three bars). - Connect the three bars to two or three existent bodies. - Remove one or two bars and reconnect them with the new Atom. - During the operation take care that any two bodies are not connect by three or more bars.

11. Extension rule in plane

12. Extension - example 9BT12 5 1 3 2 4 9BT3 3 2 1 4 5 6 1 7FB2 2 3 4 1 23 4 5 1 23 4 56 1 23 4 5 1 23 4 5 6 1 23 4 1 23 4 1 23 4 1 23 4 5 5 1 23 4

13. It is easy to derive to ALL the Body-bar Assur Graphs with 7 and 9 elements appearing in: E.E. Peisakh: An algorithmic description of the structural synthesis of planar Assur groups, Journal of Machinery Manufacture and Reliability, 2007, vol. 36, No. 6, 505-514Journal of Machinery Manufacture and Reliabilityvol. 36, No. 6

15. The extension operation in 3d - Add an Atom (body with six bars). - Connect the six bars to: 3, 4, 5 or 6 existent bodies. - Remove 1,2,3,4 or 5 and reconnect them with the new Atom. - During the operation take care that any two bodies are not connect by six or more bars.

16. Extension rule in space

17. Every mechanism is decomposed into Body-bar Assur Graphs, in particular any gear train. The decomposition is done through pebble game.

18. 1 4 3 2 5 3 1 2 4 5 3 1 2 4 5 1 3 2 4 5 1 3 2 2 2 3 1 1 3 1 3 Decompositions of a gear train into Body-bar Assur Graphs Directed cutset

19. Now, that we have ALL the body-bar Assur Graphs it opens a new possibility to go the other way around : We are working towards topological synthesis of mechanisms and gear trains through compositions of Body-bar Assur Graphs (this work is in process).

20. 7 6 5 10 8 9 11 4 1 2 1 2 R 4 R 3 R 6 5 7 R R R 8 9 10 R R R 1 2 R 4 R 3 R 6 5 7 R R R 3 11 8 9 10 R R R 7 6 5 4 3 8 9 11 2 1 Topological synthesis of gear trains through a composition of body-bar Assur Graphs.

21. The determination of generic mobility of body-bar and bar- joint mechanisms is done through : pebble game algorithm. This algorithm also decomposes the mechanisms into Body- bar and bar-joint Assur Graphs.

22. Pebble game Algorithm 1.Determines the generic mobility of the system and each element. 2. Reveals redundancies. 3.Decomposes the system into Assur Graphs (body-bar or bar-joint).

23. The main rule/theorem underlying pebble game : The sum of the pebbles on both two end elements before assigning a pebble to a constraint should be at least: dof of a rigid body + 1. Body Joint Four (1,3), (2,2), (2,3), (3,3) Four (2,2)2 d=3+1 Seven (1,6), (2,5), (3,4)…… Six (3,3)?3 d=6+1

24. Assigning pebbled to the elements in dimension d Body Joint ThreeTwo2 d SixThree3 d Element can be body or joint. To each element we assign pebbles according to its dof.

25. Element i Element j Element i Element j Element i Element j Moving pebbles to the constraints from the elements Each constraint reduces one dof thus we move one pebble from the end element to the constraint. The directed constraint is directed from the element from which the pebble was taken.

26. Bar & Joint 2D – Pebble game example e1e1 e4e4 e3e3 e2e2 e6e6 e5e5 o1o1 o2o2 o3o3 The mechanism has 2 DOFs and one redundant edge 1 DOF 2 DOF

27. 4 2 0 1 3 Body & Bar 2D – Pebble game example A 1 2 4 3 C The mechanism has 1 DOF Directed cut = Assur Graph