chapter 03 - mechanism and linkages_part b

FMB 20202 Mechanics of Machine

Chapter III – Mechanism and Linkages II

ByEngr. Syed Fawwaz Al-Attas

Kinematic InversionKinematic InversionEvery mechanism has 1 stationary base link.Every mechanism has 1 stationary base link.

All other links may move relative to the fixed link.All other links may move relative to the fixed link.

An An inverted mechanisminverted mechanism is obtained by making the original is obtained by making the original fixed link into a movable link and selecting an original moving fixed link into a movable link and selecting an original moving link to be the fixed linklink to be the fixed link

Example: Inverted mechanisms Example:Inverted mechanismsmotion

4-bar Mechanism - Kinematic 4-bar Mechanism - Kinematic Inversion, ExampleInversion, Example

Kinematic inversions:

Kinematic inversion motions: In general:In general:

A mechanism with “A mechanism with “nn” links ” links can have can have n-1n-1 inversions. inversions.

Crank-rocker

Double-crank

Crank-rocker

Double-rocker

4-bar Mechanism - Kinematic 4-bar Mechanism - Kinematic Inversion, ExampleInversion, Example

Kinematic Inversion mechanism examples

Crank – Rocker (Beam Engine)

Double – Crank (Locomotive Coupling Rod)

Double – Rocker (Watt’s Indicator)

Single slider-crank Mechanism Single slider-crank Mechanism – –

Kinematic Inversion, ExampleKinematic Inversion, Example

Kinematic inversions:

(c) slider fixed

(b) connecting rod fixed

(a) crank fixed

Slider –crank Mechanism

I - Inversion of slider crank mechanism. (Crank I - Inversion of slider crank mechanism. (Crank fixed)fixed)

Rotary engineRotary engine



II - Inversion of slider crank mechanism. II - Inversion of slider crank mechanism. (Connecting rod fixed)(Connecting rod fixed)

Oscillating cylinder engineOscillating cylinder engine



III - Inversion of slider crank mechanism. (Slider III - Inversion of slider crank mechanism. (Slider fixed)fixed)

Pendulum pump or bull enginePendulum pump or bull engine



1010

= cutting stroke crank angle= return stroke crank angle

= time ratio

Used on machine tools to give a slow cutting stroke and a quick return stroke

Should be as large as possible

Crank has constant angular velocity

cutting stroke time = /return stroke time =

Quick-return Motion Quick-return Motion MechanismsMechanisms

1111

Examples ofExamples of quick-return motion mechanismsquick-return motion mechanisms

Crank & Slotted Lever

Drag link

Link 2 – constant angular velocity clockwise

Link 4 – non-constant angular velocity

Link 6 – slow upward stroke – fast downward stroke

Whitworth

Slider-crank with crank fixed

1212

Can be used to overcome a very large resistance P with a small driving force F

P

F

5

Link 5:2 force member

P/cos

P/cos

P/cos

F = 2Psin/cos = 2Ptan

P → as → 0

Mechanism is used in stone crusher

2Psin-------------

cos

Toggle mechanismToggle mechanism

1313

Geneva wheel

Plate 1 rotates continuously

Driving pin P engages in slot on member 2

Member 2 turns through 90 each revolution of 1

Locking plate prevents rotation of 2 when pin is not engaged

Gives smooth action without any shock

Intermittent motion Intermittent motion MechanismsMechanisms

1414

Ratchet mechanismProduces intermittent rotation from an oscillating member (arm 2)

Pawl 3 pushes wheel 4 anti-clockwise as arm 2 rotates anti-clockwise

Pawl 5 prevents clockwise motion of wheel when arm 2 rotates clockwise

Wheel can turn anti-clockwise only


1515

Intermittent gearingDriving wheel has one tooth

Driven wheel has a number of tooth spaces

A locking device must be used to stop 2 turning when tooth is not engaged

Can only be used when loads are light and shock is not important


1616

EscapementThe toothed wheel is allowed to turn in small steps by action of a pendulum

The balance wheel oscillates with a fixed period

Each oscillation allows the wheel to advance by one tooth

Used in clocks and watches


Double slider-crank Mechanism Double slider-crank Mechanism – –


A mechanism consist of 2 sliding pair. The types are:

Scotch yoke

Elliptical trammels

Oldham’s coupling

1818

Gives Simple Harmonic Motion

coscos

sinsin

)cos1(

; cos

222

2

rtrdt

xda

rtrdt

dxv

trrx

trrx

yoke velocity

yoke acceleration

yoke position

Scotch YokeScotch Yoke

Hooke’s JointHooke’s Joint

Condition for perfect steeringCondition for perfect steering

Steering gear MechanismSteering gear Mechanism

Ackermann steering gear Ackermann steering gear MechanismMechanism

Types of 4-Bar and Types of 4-Bar and Slider-crank Slider-crank MechanismsMechanisms

4-Bar 4-Bar MechanismsMechanisms

There are 3 basic types of 4-bar There are 3 basic types of 4-bar mechanisms:mechanisms:

1. Crank-rocker mechanism mechanism

2. Double-crank mechanism mechanism

3. Double -rocker mechanism mechanism

Crank-rocker

Double-crank

Double-rocker

Links:Links: CrankCrank: is able to : is able to

make full rotationmake full rotation RockerRocker: oscillates : oscillates

between two limit between two limit points with an points with an amplitude of “amplitude of “ΔΔ44””

Grashof’s CriteriaGrashof’s Criteria If we distinguish the lengths in a 4-bar mechanism as follow:If we distinguish the lengths in a 4-bar mechanism as follow:

ss: the length of the shortest link: the length of the shortest link ll: the length of the longest link: the length of the longest link pp,, qq: the length of the other two links: the length of the other two links

The following relationship (The following relationship (Grashof’s criteriaGrashof’s criteria) must be satisfied ) must be satisfied in order to be able to assemble the kinematic chain:in order to be able to assemble the kinematic chain:

s + p + q l

Class I Kinematic chain Class II Kinematic chain s + l < p + q s + l > p + q

If s = input link The mechanism isThen Crank-rocker (Figs. a+b) Triple-rocker

(see notes for figure)If s = base linkThen Double-crank (Fig. c)

If otherwiseThen Double-rocker (Fig. d)

Examples:Examples:Using Grashof’s CriteriaUsing Grashof’s Criteria

If the following mechanism has the following dimensions what type of If the following mechanism has the following dimensions what type of

4-bar mechanism has been implemented ? 4-bar mechanism has been implemented ?

aa = 30 cm ; = 30 cm ; gg = 20 cm ; = 20 cm ; bb = 10 cm ; = 10 cm ; hh = 5 cm = 5 cm

If If aa is the input link ? is the input link ?

Answer: Triple-Rocker mechanism (Class II)

What if: What if: aa = 30 cm ; = 30 cm ; bb = 7 cm ; = 7 cm ; hh = 15 cm ; = 15 cm ; gg = 25 = 25 cmcm

If If bb is the input link ? is the input link ?

If If bb is the base link ? is the base link ?

Answer: Crank - rocker (Class I)

Answer: Double - crank (Class I)

s + l > p + q h + a > b + g

b + a < h + g ; Shortest link = Input link

b + a < h + g ; Shortest link = Base link

Toggle positions of a crank-rocker mechanism. Links 2 and 3 become collinear.

Limitmin

Limitmax

4-Bar Mechanism Limit Positions4-Bar Mechanism Limit Positions

2828

The range of values for The range of values for is important for the is important for the operation of the mechanismoperation of the mechanism

For best power transmission For best power transmission should be close to should be close to ±± 90˚ 90˚

The acceptable deviation is usually 45˚ <The acceptable deviation is usually 45˚ << < 135˚135˚

: transmission angle

r1

r3

r2r4z

input output


Transmission Angle, Transmission Angle, The transmission angle is an important criterion for the design of The transmission angle is an important criterion for the design of

mechanisms by means of which the quality of motion transmission in a mechanisms by means of which the quality of motion transmission in a mechanism can be judged.mechanism can be judged.

During a mechanism design it helps to decide the “Best” among a During a mechanism design it helps to decide the “Best” among a family of possible mechanisms for most effective force transmission. family of possible mechanisms for most effective force transmission.


Def.:Def.: Transmission angleTransmission angle is the angle is the angle between the direction of velocity between the direction of velocity difference vector difference vector vvBABA of the driving of the driving

link and the direction of absolute link and the direction of absolute velocity vector velocity vector vvBB of the output link of the output link

both taken at the point of connection both taken at the point of connection ((in a 4-bar linkage it is the angle between the in a 4-bar linkage it is the angle between the follower and the coupler linkfollower and the coupler link).).


4

43

224

231

..2cos

rr

zrr

2212

22

1 cos...2 rrrrz

41

224

211

4 ..2cos

rr

arr

MinMax ,4,44 Limit Angle,

s + p + q l

3131

min

r1

r3

r2

r4

z

z = r1 - r2

max

r1

r3

r2

r4z

z = r1 + r2

Configuration withmaxmax

Configuration withminmin


Max & Min Transmission AnglesMax & Min Transmission Angles

Double-rocker mechanism.

The crank-rocker mechanism Double-crank mechanism

Optimum transmission angle = 900

Transmission angle in planar5-bar mechanism

Transmission angle in planar6-bar mechanism

Slider-crank Slider-crank MechanismsMechanisms

This mechanisms has numerous characteristics:This mechanisms has numerous characteristics:1.1. Offset (rOffset (r1 1 )): distance between the line of action and : distance between the line of action and

the crank’s rotating point (parallel to the line of the crank’s rotating point (parallel to the line of action)action)

2.2. rr22 & r & r33: The lengths of the crank and the coupler: The lengths of the crank and the coupler3.3. Stroke (rStroke (r44)): distance between the limit positions.: distance between the limit positions.

r2 < r3 and r1 r3 – r2

In order for the crank (r2) to have fullrotation, we must satisfy:

Offset 0

I.I. Limit Position:Limit Position: is defined as the point where is defined as the point where the mechanism reaches its maximum or minimum the mechanism reaches its maximum or minimum point within the entire motion of the mechanism.point within the entire motion of the mechanism.

II.II. Stroke:Stroke: is defined as the distance between the is defined as the distance between the limit positions of the mechanism associated with limit positions of the mechanism associated with the corresponding motion (the corresponding motion (e.g., piston in a car e.g., piston in a car engineengine).).

III.III. Time Ratio:Time Ratio: is the time for the part of the is the time for the part of the mechanisms associated with the stroke (mechanisms associated with the stroke (e.g., e.g., piston in a car enginepiston in a car engine) to move in one direction ) to move in one direction between the limit positions over the time it takes between the limit positions over the time it takes to move in the opposite direction between the to move in the opposite direction between the same limit positions. same limit positions.

Imax

Imin

Imax

Imin

Limit Positions & Limit Positions & Time Ratio of Time Ratio of MechanismsMechanisms

In-Line (No Offset) Crank-slider In-Line (No Offset) Crank-slider Mechanism Mechanism

Limit positions & StrokeLimit positions & Stroke

22

CrankCrank

angleangle

4r

23

2123 sinsin180

r

roStroke at any 3322

32

242 cos...2 rrrrr

321 rrS

232 rrS 21 SSStroke

Offset Slider-crank Mechanism Limit Offset Slider-crank Mechanism Limit positions, Stroke,Time ratio & Average positions, Stroke,Time ratio & Average

speedsspeedsstrokestroke = s = s11 – –

ss22

ΔΔ22 = 180 + = 180 + 22 - - 11

11 = sin = sin-1-1((rr11 / ( / (rr22 + + rr33))))

ss22 = [( = [(rr33 – – rr22))22 – – rr1122 ] ] 1/21/2

Time ratioTime ratio = = ΔΔtt1 1 / / ΔΔtt22

ss11 = [( = [(rr22 + + rr33))22 – – rr1122 ] ] 1/21/2

22 = sin = sin-1-1((rr11 / ( / (rr33 – – rr22))))

((vv44,,avgavg))left-motionleft-motion = stroke / = stroke / ΔΔtt11

((vv44,,avgavg))right-motionright-motion = stroke / = stroke / ΔΔtt22

ΔΔtt11 = = ΔΔ22 / / 33

Time taken to move to the left

ΔΔtt22 = [ 2 = [ 2 - - ΔΔ2 2 ] / ] / 33

Time taken to move to the right

3

221123

sinsin180

r

rro

22

CrankCrank

angleangle

chapter 03 - mechanism and linkages_part b

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