Thoery of Machines-Mechanisms

Mechanisms

Commonly used Mechanisms

Pantograph:

Pantographs were traditionally used to scale drawings.

It can exactly magnify or reduce a drawing.

In the days before CAD packages, pantograph was a vital drawing tool.

Pantograph are also used in some sensors, to magnify a small motion.

A

B

D

C

E

O

Pantograph (6 links, 5 turning joints)

Tracingpoint

Drawingpoint (pen)

Magnifiedby the ratioOB/OA

o

A

B

C

E(pen)

Here scaling ratio is 2 (i.e OB/OA)

D

Simple pantograph

A

B

D

C

E

Specifications:1. AB=AD=DC=BC (they must form a Parallelogram).

O

Pantograph

A

B

D

C

E

At any position of pantograph, triangle OAD is similar to OBE.Hence OB/OA =BE/AD.

O

Pantograph

The scaling ratio the Pantograph is OB/OA or BE/AD

Toggle Mechanism:

A device for producing large mechanical advantage.Used in applications such as stone crushers.

Small force

Large force

α

α

tan2

2tan

PFP

F

=∴

=

For small values of α,F << P .

A small force F (obtained by turning the wheel) equals a large force P at thecrushing end.

F

P

Various Straight Line Motion mechanisms:

These are mechanisms which have only turning joints, and produce a straight line motion at some point in a link (i.e. coupler point).

The straight line may be exact or approximate, or only a small portion of the path traced by the point. But it is still useful.

Mechanisms with sliding joint also produce straight line motion, but get worn out soon. Hence they are not considered here.

Straight line producing mechanisms were historically important. Because,

(a)Difficulty of producing exact straight lines using sliding parts(sliding joints have to be accurately machined first).

(b) To guide certain machine parts to move in a straight line.

Using “synthesis of mechanisms” concept, four bar mechanisms can bedesigned to produce portions of straight lines.

Level luffing crane.‘P’ traces approximatelya straight line.

Film advance mechanism in a camera

Approximate Straight Line Mechanisms

1. Watts Mechanism (4 links, 4 turning joints)(originally used to guide the piston/slider in early steam engines)

oA

P

BO1

For limited rotation of O, Point ‘P’ will trace an approximate vertical straight line.

Specification: PA.OA = PB.O1B


oA

P

BO1


oP’

O1

A

BP


oP’

O1A”

B”

P

P”


o

O1A”

B”

P”

It could guide the imperfectly made pistons of the old steam enginesin a straight line.

Used in car suspensionsIt will restrain horizontal movements (i.e swaying)

Scott Russell Mechanism(exact straight line motion)

P

θslider

P moves in a straight line when the slider C moves back and forth .1. Used to operate hammers or lifting water at P.2. Also used in automobile suspensions (for arresting horz. motion)

Specification: AB = PB=BC

A

BC


P

θslider

A

BC


P’

θslider

A

B

C

Grass Hopper Mechanism - A modification of Scot Russell Mechanismand is an Approximate Straight line motion mechanism

A

B

C

Q

P

θ

For small angles θ, the point P will follow an approximate straight line, when C moves in an approximate straight line.Specification: BC2=PB.AB

C’ Instead of a slider at C, itis coupled toa long link CQ

A

B

C

Q

P

CQ is the driving link

B’

Q

P’

C’

B

CQ is the driving link

P

A

Tchebicheffs Mechanism(4 links, 4 turning joints)

P

O

AB

O1

Specifications:

It has crossed equal links OA and O1B.

P is mid point of AB.AB:OO1:OA=1:2:2.5

Straight line traced by P


P

O

AB

O1

Straight line traced by P


O O1

P’


O O1

P”

Tchebicheffs Mechanism is used to guide Saw blades in wood mill

O O1O O1

Only Horizontal motion

allowed

Exact Straight Line Mechanisms

They produce exact straight line motion when another link is rotatedthrough a limited angle.

The two well known devices are,

1. Peaucellier Mechanism2. Harts Mechanism

They are used where precise linear motion is needed. Eg. Mechatronics applications, Cameras etc

Peaucellier Mechanism (8 links, 6 turning joints)

Was invented in 1864 by Peaucellier and was considered a milestone at that time.

P

O

Q

A

B

C OQ is the rotating link.

Then P will move in a straight line perpendicular to OA.

Specifications:1. OQ=OA2. AB=AC3. PBQC is a parallelogram (i.e links are equal)


P

O

Q

A

B

C


P’

OA


P”OA

A

B

C

D

PE (pivoted to the ground)

O

crank

Needs 6 links and 7 turning joints.More compact than Peaucellier Mechanism.

Harts Mechanism

Specifications: It is basically a crossed paralellogram. So CD=AB and AD=BC.

OP is the crank. Link AB is pivoted to the ground at E.

Also, OP=OE.

A

B

C

D

PE

O

Draw lines through AC and DB.They will be always parallel to each other.

crank

A

B

C

D

P

O

Q

Now locate the point Q (on BC) by drawing a parallel line thru P.

This is the straight line generation point.

Ecrank

A

B

C

D

P

O

Q

It can be shown that, when the crank OP rotates, Q will move in a straight line perpendicular to OE .

900

E

Some other important Mechanisms

1. Hookes Joint (Universal Joint)

This is used when two shafts have a misalignment between them.

Motor (driving unit)Pump (driven)

Hookes joint

α=angle between shafts

ω1, θDriven shaft -ω2

Driving shaft -

When the driving shaft makes one revolution at constant ω1, driven shaft also makes one revolution.

But ω2 is not constant during the revolution. It varies as a function of θ.

Even if driving shaft runs at constant ω1 , ω2 will fluctuate above and below ω1 .

Shaft-1Sha

ft-2

Hookes Joint

Forked ends of shaft

Cross piece

Hookes Joint

ω1, θ

ω2

α Shaft-1(Driving)

Shaft-

2(D

riven

)

How does ω2 fluctuate with ω1 ?

It can be shown that,

2 1 2 2

2 1

02

2 max 1 2 min 1

cos1 sin cos

when 0 (shafts are perfectly aligned) then .

Also it can be summarized that in a 360 rotation will fluctuate between,

. / cos and . cos .

Usuall

αω ωα θ

α ω ω

ω

ω ω α ω ω α

⎛ ⎞= ⎜ ⎟−⎝ ⎠

→ →

= =

2 10

y the speed fluctuation of must be kept within 5% of .

Hence it is necessary to keep as small as possible (say <15 )

ω ω

α

±

Double Hooke’s Joint(used in vehicle transmissions)

α1

α2

EngineRear Wheel

Driving shaft(ω1)

Here the speed (ω2) of driving shaft depends on α1 and α2.

If α1 = α2 and all are on the same plane, it can be shown that ω2= ω1.

Driven shaft(ω2)

Oldhams coupling

Used to connect two misaligned shafts, parallel to each other.

Flange 1 Flange 2Intermediatepiece

Motion possiblem

otio

n po

ssib

le

Intermediate piece

The intermediate piece will couple the two flanges. It has an eccentricmotion as the shafts rotate.

The angular velocities of the driving and driven shafts are same.

Geneva MechanismThis is used to convert uniform rotary input to intermittent rotaryoutput.

It has many applications where intermittent motion is needed:Eg. movie projector film advance, assembly lines.

90 0

Basic Geneva Mechanism

o

Uniformly rotating link Intermittently

rotating Geneva wheel.

Basic Geneva Mechanism

o

a

b

The actual Geneva mechanism has some refinementsfor smooth operation.

o

ar

p

oa= r; Also there is a circle of radius ‘p’ (p<r)

ω

Uniformly rotating link “oa”is fashioned with a circular portion

Geneva Wheel

p

pr√2 r

This is the center distance for Geneva wheel with 4 slots

Assembled Geneva Wheel

ω Uniform Rotation

Intermittent Rotation

Geneva is Lockedbecause of the radius

Plug enters the next Genevawheel slot, forcing it to rotate.

Applications:

Movie projectors: Each film has to pause 1/24 sec infront of the projection lens. The film roll has to be rotated intermittently.

The film roll is advanced through a Geneva Mechanism.

The Geneva Mechanism is driven by a motor with uniform RPM.

Automobile Steering Mechanism

Simplified representation of Automobile Steering.

Chassis

Rear Wheels, axle direction is fixed

Front wheels, axis can beswivelled.


Instantaneous Center

V


V


V1


V2


For the vehicle to turn smoothly, all wheels must turnabout a common instantaneous center.

V


V1

V2

θ

Φ

Front wheels have to be swivelled thru different angles.

Φθ

Steering Mechanism must turn Inner wheel thru θ andouter wheel thru Φ .

V


V1

V2

θ

Φ

Φθ

Condition for correct steering :

V1

V2

θ

Φ

Φθ

W (track width)

H (wheel base)

From the geometry it can be shown that,cot Φ – cot θ = (W/H) .

Davis Steering Gear:Obeys the law of steering, cot Φ – cot θ = (W/H),

(for all angles)

Two slotted bell crank levers


(for all angles)

Slider bar is movedleft or right


(for all angles)α

b

h

tan α = b/h

Also, tan α = W / (2H)= Track width 2 x Wheel base


(for all angles)


(for all angles)

θΦ

θ > Φ

Davis Steering Gear

Advantage: Obeys law of steering

Disadvantage: Due to sliding joints, the mechanism gets worn outafter a period of use. There will be slackness (‘play’)during steering.

Ackermans Steering Gear:

It is a steering gear with all turning joints.

Only approximately follows law of steering.

It is extensively used as steering gear for most modern vehicles.

It is a compromise between law of steering and practicality.

Steering bar is movedleft or right

Ackermans mechanism is set behind the front wheels

B

A

C

D

αα

B

A

C

D

Usually, the center lines ofLinks CD and AB are designed to meet the rear axle center.

Ackerman steering gear willsatisfy “law of steering” foronly one steering position on each side.

This is usually set as thesmallest turning radius.

For larger turning radiusthe front wheel inst. centerswill intersect above the rearaxle.

For larger turning radii, the instantaneous centers of front wheels will intersect above the rear axle.

For larger turning radius, instantaneous centers would be far awayand the effect of errors are relatively less.

SLIP:

If all the instantaneous centres of all tyres perfectly coincide, the tyrespoint in the direction of motion of the vehicle (i.e concentric circular arcs) -No Slip.

If the instantaneous centres do not coincide there is mismatch betweenwhere each tyre is pointed and the general direction of motion of the vehicle.

This is the slip angle of each tyre, and it must be kept to a minimum. Because of slip angle sidewise forces occur, which force the vehicleoff from its path.

Thoery of Machines-Mechanisms

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