Proceedings of the ASME 2012 International Design Engineering Technical Conferences &

Computers and Information in Engineering ConferenceIDETC/CIE 2012

August 12-15, 2012, Chicago, IL, USADETC2012-70904

APPLYING RIGIDITY THEORY METHODS FOR TOPOLOGICAL DECOMPOSITION AND SYNTHESIS OF GEAR TRAIN SYSTEMS

Terushkin MariaSchool of Mechanical Engineering

Tel Aviv UniversityTel Aviv, Israel

Shai OfferSchool of Mechanical Engineering

Tel Aviv UniversityTel Aviv, Israel

Rigidity theory deals mostly with the topologicalcomputation.

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

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 mostone bar/constraint.

2d – revolute joint 3d – spherical joint

Body-bar Representation Bodies connected by bars (constraints).

Lower kinematicpair

1 2

Higher kinematic pair1 2

Spherical joint

1 2

1 2

1 2

Hinge

Representing a mechanism by Bar-joint and

Body-bar representations

Body-bar: Two bodies and three bars.

Bar-joint: Nine bars and seven joints.

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.

(a) (b)

2 3

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2d Body-bar Assur Graph

Body-bar Assur Graph Not a body-bar Assur Graph

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Body-Bar graph of a mechanism

(mechanism) (Body-Bar graph)

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Topology synthesis of ALL the mechanisms through Body Bar

Assur GraphEvery 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.

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.

Extension rule in plane

Extension - example

9BT12

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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, 505-514

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.

Extension rule in space

Every mechanism is decomposed into Body-bar Assur Graphs, in particular any gear train.

The decomposition is done through pebble game.

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Decompositions of a gear train into Body-bar Assur Graphs

Directed cutset Directed cutset

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).

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Topological synthesis of gear trains through a composition of body-bar Assur Graphs.

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.

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).

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

Assigning pebbled to the elements in dimension d

Body Joint Three Two 2 dSix Three 3 d

Element can be body or joint.To each element we assign pebbles according to its dof.

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.

Bar & Joint 2D – Pebble game example

e1

e4 e3

e2

e6

e5

o1 o2 o3

The mechanism has 2 DOFs and one redundant edge

1 DOF1 DOF1 DOF2 DOF

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Body & Bar 2D – Pebble game example

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The mechanism has 1 DOF

Directed cut = Assur

Graph

Directed cut = Assur

Graph