mechanisms ppt

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ChapterChapter

1&

2

MEEN 4110 – Mechanisms Design

Fall - 2010

Lecture 02

KINEMATICS FUNDAMENTALS

1. Kinematics FundamentalsMech

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can 2.1 NUMBER SYSTHESIS

The term number synthesis has been coined to mean the determination of the number and order of links and joints necessary to produce motion of a particular DOF. Order in this context refers to the number of nodes perlink, i.e., binary, ternary, quaternary, etc.

The value of number synthesis is to allow the exhaustive determination of all possible combinations of links which will yield any chosen DOF.

This then equips the designer with a definitive catalog of potential linkages to solve a variety of motion control problems.


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Hypothesis: If all joints are full joints, an odd number of DOF requires an even number of links and vice versa.

Proof: Given: All even integers can be denoted by 2m or by 2n, and all odd integers can be denoted by 2m - 1 or by 2n - 1, where n and m are any positive integers. The number of joints must be a positive integer.

Let: L = number of links, J = number of joints, and M = DOF = 2m (i.e., all even numbers)

Then: rewriting Gruebler's equation (Equation 2.1b) to solve for J,


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Try: Substituting M=2m, and L=2n (i.e., both any even numbers)

This cannot result in J being a positive integer as required

Try: M=2m-1, and L=2n-1 (i.e., both any odd numbers)

This also cannot result in J being a positive integer as required


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Try: M=2m-1, and L=2n (i.e., odd-even)

This is a positive integer for m>=1 and n>=2. Try: M=2m, and L=2n-1 (i.e., even-odd)

This is a positive integer for m>=1 and n>2

So, for our example of one-DOF mechanism we can only consider combinations of 2, 4, 6, 8 … links


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Letting the order of the links be represented by:

The total number of links in any mechanism will be:

Since two link nodes are needed to make one joint

and nodes


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Then

Substitute J and L in the Gruebler’s equation

The DOF is independent of number of ternary link in the mechanism. But because each ternary link has three nodes, it can only create or remove 3/2 joints


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can 2.2 PARADOXES

The Gruebler’s criterion pays no attention to link sizes or shapes, it can give misleading results in the face of unique geometric configurations.


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can 2.3 ISOMERS

The word isomer is from the Greek and means having equal parts.

Linkage isomers are analogous to chemical compounds in that the links (like atoms) have various nodes (electrons) available to connect to other link’s nodes.


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can 2.4 LINKAGE TRANSFORMATION

There are several transformation techniques or rules that we can apply to planar kinematic chains


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A fourbar crank-rocker linkage transformed into the fourbar slider-crank by the application of rule #1.


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A fourbar slider-crank transformed via rule #4 by the substitution of a half joint for the coupler.


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A fourbar linkage transformed into a earn-follower linkage by the application of rule #4. Link 3 has been removed and a half joint substituted for a full joint between links 2 and 4.


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(a) shows the Stephenson's sixbar chain transformed by partial shrinkage of a ternary link (rule #5) to create a multiple joint. It is still a one-DOF Stephenson's sixbar.

(b) shows the Watt's sixbar chain from with one ternary link completely shrunk to create a multiple joint.


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can 2.5 INTERMITENT MOTION

Is a sequence of motions and dwells. Dwell; is a period in which the output link remains stationary while the input link continues to move.


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can 2.5 INTERMITENT MOTION


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can 2.6 INVERSION

An inversion is created by grounding a different link in the kinematic chain. Thus there are as many inversions of a given linkage as it has links.


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can 2.6 INVERSION


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can 2.6 INVERSION – All inversions of the Grashof fourbar linkage


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can 2.7 THE GRASHOF CONDITION

The four bar linkage, shown below, is a basic mechanism which is quite common. Further, the vast majority of planar one degree-of-freedom (DOF) mechanisms have "equivalent" four bar mechanisms. The four bar has two rotating links ("levers") which have fixed pivots, (bodies 2 and 4 above). One of the levers would be an input rotation, while the other would be the output rotation. The two levers have their fixed pivots with the "ground link"(body 1) and are connected by the "coupler link" (body 3).


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can 2.7 THE GRASHOF CONDITION - Definitions

Crank- a ground pivoted link which is continuously rotatable.

Rocker- a ground pivoted link that is only capable of oscillating between two limit positions and cannot rotate continuously.

Grashof Condition- is a very simple relationship which predicts the rotation behavior or rotatability of a fourbar linkage's inversions based only on the link lengths

Let: S=length of shortest link L=length of longest link P=length of one remaining link Q=length of other remaining link

Then if: S+L<=P+Q


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The linkage is Grashof and at least one link will be capable of making a full revolution with respect to the ground plane. This is called a Class I kinematic chain. If the inequality is not true, then the linkage is non-Grashof and no link will be capable of a complete revolution relative to any other link. This is a Class II kinematic chain.

For the Class I case, S + L < P + Q: Ground either link adjacent to the shortest and you get a

crank-rocker, in which the shortest link will fully rotate and the other link pivoted to ground will oscillate.

Ground the shortest link and you will get a double-crank, in which both links pivoted to ground make complete evolutions as does the coupler.

Ground the link opposite the shortest and you will get a Grashof double-rocker, in which both links pivoted to ground

oscillate and only the coupler makes a full revolution.


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For the Class II case, S + L > P + Q: All inversions will be triple-rockers in which no link can

fully rotate. For Class III, S+L = P+Q

All inversion will be either double-cranks, or crank-rocker


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can 2.8 LINKAGES OF MORE THAN FOUR BARS

Geared Fivebar Linkages


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can 2.8 LINKAGES OF MORE THAN FOUR BARS

Sixbar Linkages


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can 2.9 EXAMPLES – GRASHOF CONDITION - SOLIDWORKS

http://www.me.unlv.edu/~mbt/320/Grashof.html http://wahyu-tjakraningrat.blogspot.com/2009/02/solidwo

rk-planar-joints.html


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Due, Tuesday, August 31, 2010

Homework2 www.bc.inter.edu/facultad/omeza

mechanisms ppt

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