KINEMATIC FUNDAMENTALS

Theory of Machines, MT 317

Engr. Akhtar Khurshid

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Figure below shows a mechanism with one DOF and only full joints in it.

DETERMINING DEGREE OF FREEDOM USING KUTZBACKS EQUATION

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Figure below shows a mechanism with zero DOF which contains full, half and multiple joints.

DETERMINING DEGREE OF FREEDOM USING KUTZBACKS EQUATION

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DETERMINING DEGREE OF FREEDOM OR MOBILITY USING KUTZBACKS EQUATION

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DETERMINING DEGREE OF FREEDOM OR MOBILITY USING KUTZBACKS EQUATION

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DETERMINING DEGREE OF FREEDOM USING KUTZBACKS EQUATION

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MECHANISMS AND STRUCTURES

The degree of freedom of an assembly of links completely predicts its character. There are only three possibilities. If the DOF is positive, it will be a mechanism, and the

links will have relative motion. If the DOF is exactly zero, then it will be a structure,

and no motion is possible. If the DOF is negative, then it is a preloaded structure,

which means that no motion is possible and some stresses may also be present at the time of assembly.

Note: If the sum of the lengths of any two links is less than the length of the third link, then their interconnection is impossible.

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ISOMERS

The word isomer is from the Greek and means having equal parts. Isomers in chemistry are compounds that have the same number and type of atoms but which are interconnected differently and thus have different physical properties. Figure on next slide shows two hydrocarbon isomers, n-butane and iso-butane. Note that each has the same number of carbon and hydrogen atoms (C4H10), but they are differently interconnected and have different properties.

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ISOMERS

Hydrocarbon isomers n-butane and Isobutane

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ISOMERS Linkage isomers are analogous to these chemical compounds in that the links (like atoms) have various nodes (electrons) available to connect to other links nodes. The assembled linkage is analogous to the chemical compound.

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LINKAGE TRANSFORMATION

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

1. Revolute joints in any loop can be replaced by prismatic joints with no change in DOF of the mechanism, provided that at least two revolute joints remain in the loop.

2. Any full joint can be replaced by a half joint, but this will increase the DOF by one.

3. Removal of a link will reduce the DOF by one. 4. The combination of rules 2 and 3 above will keep the original

DOF unchanged. 5. Any ternary or higher-order link can be partially "shrunk" to a

lower-order link by coalescing nodes. This will create a multiple joint but will not change the DOF of the mechanism.

6. Complete shrinkage of a higher-order link is equivalent to its removal. A multiple joint will be created, and the DOF will be reduced.

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LINKAGE TRANSFORMATION Figure below shows a four-bar crank-rocker linkage transformed into the four-bar slider-crank by the application of rule #1. It is still a four-bar linkage. Link 4 has become a sliding block. The Gruebler's equation is unchanged at one DOF because the slider block provides a full joint against link 1, as did the pin joint it replaces.

Transforming a four-bar crank-rocker to a slider-crank

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LINKAGE TRANSFORMATION Figure below shows a four-bar slider-crank transformed via rule #4 by the substitution of a half joint for the coupler.

Transforming the slider-crank to the Scotch yoke also known as slotted link mechanism

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LINKAGE TRANSFORMATION Figure below shows a four-bar linkage transformed into a cam-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.

The cam-follower mechanism has an effective four-bar equivalent

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INTERMITTENT MOTION Intermittent motion is a sequence of motions and dwells. A dwell is a period in which the output link remains stationary while the input link continues to move. There are many applications in machinery which require intermittent motion. The cam-follower variation on the four-bar linkage as shown in previous slide is often used in these situations. The design of that device for both intermittent and continuous output will be addressed in detail in Chapter 8.

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MECHANISMS INVERSION - FOUR BAR Process of obtaining different mechanisms from the same kinematic chain, by fixing

different links in turn, is known as kinematic inversion.

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MECHANISMS INVERSION SLIDER CRANK

Different mechanism by fixing different link of slider crank chain are as

follows :

First inversion: This inversion is obtained when link 1 (ground body) is

fixed. Application- Reciprocating engine, Reciprocating compressor etc...

Second inversion: This inversion is obtained when link 2 (crank) is fixed.

Application- Whitworth quick return mechanism, Rotary engine etc...

Third inversion: This inversion is obtained when link 3 (connecting rod )is

fixed. Application- Slotted crank mechanism, Oscillatory engine etc..,

Fourth inversion: This inversion is obtained when link 4 (slider) is fixed.

Application- Hand pump, pendulum pump etc...

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APPLICATIONS INVERSION SLIDER CRANK

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TYPE OF FOUR BAR MECHANISMS

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GRASHOF CONDITIONS Simplicity is one mark of good design. The fewest parts that can do the job will

usually give the least expensive and most reliable solution. Thus the fourbar

linkage should be among the first solutions to motion control problems to be

investigated. The 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 a linkage is Grashof then at least one link (usually the shortest link) can make a

full revolution without binding. Some definitions:

Crank: one pin is grounded, can make full revolution about grounded pin

Rocker: one pin is grounded, does not make full revolution about grounded pin

Coupler: neither pin is grounded. Experiences complex motion

L

Q

P

S

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GRASHOF CONDITIONS The Grashof condition is used to predict the type of motion that is possible using a

given fourbar linkage. It is a very simple, but powerful tool used in linkage design.

In a planar four bar revolute pair kinematic chain if the sum of the lengths of the

shortest and the longest links is less than or equal to the sum of the lengths of the

other two intermediate links at least one link will have full rotation.

Mechanisms obtained from the kinematic chain satisfying these conditions are

known as Grashofian Mechanisms.

Mechanisms obtained from the kinematic chain which are not obeying these

conditions are known as Non-Grashofian Mechanisms.

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GRASHOF CONDITIONS There are several common situations that we encounter with the Grashof

condition. Examples of Grashof Linkages

L

QP

S

Crank

Coupler

Rocker

L

Q

P

S

Crank

Coupler

Rocker

L

Q

P

S

Cranks

Coupler

L

Q

P

S

Rockers

Coupler

Double-Rocker. Q is grounded (adjacent to L) Double-Crank. S is grounded.

Rocker-Crank. P is grounded (adjacent to short

link).

Crank-Rocker. L is grounded.

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GRASHOF CONDITIONS

Crank-Rocker: When the shortest link is the input link. The

input link has full motion. The output link has limited range of

motion.

Rocker-Crank: When the shortest link is the output link. The

output link has full motion. The input link has a limited range of

motion.

Double-Rocker: this type of Grashof linkage is obtained when

the shortest link is the floating link. Both the input and output

links have limited motion.

Crank-Crank: When the shortest link is ground link. But the

input and output links have full motion.

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NON-GRASHOF CONDITIONS If S + L > P + Q then the linkage is non-Grashof. For linkages of this type continuous relative motion between any two links is not possible.

Rockers

Coupler

Special Case: S + L = P + Q

Cranks

Coupler

Antiparallelogram FormParallelogram Form

Cranks

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ASSIGNMENT # I to be submitted till 10-03-2015 Q-1: Calculate the DOF of the linkages shown in Figures below

Q-2: Describe the difference between a cam-follower (half) joint and a pin

joint.

(e).

(f).

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PRACTICE PROBLEMS

From 2.1 to 2.20