Simple Mechanisms - site.iugaza.edu.ps

Chapter 5

Simple Mechanisms

September 29, 2014

Dr. Mohammad Suliman Abuhaiba, PE 1

Homework

All questions at the end of chapter

1st quiz will be held on Monday

13/10/2014 @ 13:00 pm

September 29, 2014

Dr. Mohammad Suliman Abuhaiba, PE

2

../../Robotics/Videos/Chapter 2/19 سبتمبر، 2013 08_34 صباحاً _ فيس بوك.mp4

5.2. Kinematic Link or Element

kinematic link (link) or element: Each part of a machine, which

moves relative to some other

part.

A link need not to be a rigid body

It must be a resistant body

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5.2. Kinematic Link or Element

A resistant body: capable of

transmitting required forces with

negligible deformation.

A link has two characteristics:

1. It should have relative motion

2. It must be a resistant body

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5.3. Types of Links

1. Rigid link: does not undergo any deformation

while transmitting motion

2. Flexible link: partly deformed in a manner not to

affect the transmission of motion

Example: belts, ropes, chains and wires

3. Fluid link: formed by having a fluid in a

container and motion is transmitted through the

fluid by pressure or compression only

Example: hydraulic presses, jacks and brakes

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5.4. Structure and a Machine

Structure: assembly of a number

of resistant bodies having no

relative motion between them

and meant for carrying loads

having straining action.

Examples: A railway bridge, a roof

truss, machine frames

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10610068_762548127141198_1972163806_n.mp4

10610068_762548127141198_1972163806_n.mp4

10610068_762548127141198_1972163806_n.mp4

Machine Structure

Parts move relative to one

another

members do not move relative

to one another

A machine transforms

available energy into some

useful work

no energy is transformed into

useful work

links of a machine may

transmit both power & motion

members of a structure transmit

forces only

5.4. Structure and a Machine

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5.6. Kinematic Pair

A pair: two links of a machine, in

contact with each other.

kinematic pair: the relative motion between the links is completely or

successfully constrained

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5.7. Types of Constrained Motions

1. Completely constrained motion

2. Incompletely constrained motion

3. Successfully constrained motion

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5.7. Types of Constrained Motions Completely constrained motion

Motion between a pair is limited to a definite

direction irrespective of direction of force applied

Example: piston and cylinder form a pair

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5.7. Types of Constrained Motions Incompletely constrained motion

Motion between a pair can take place in

more than one direction.

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5.7. Types of Constrained Motions Successfully constrained motion

Motion between elements, forming a pair is such that the constrained motion is not completed by itself, but by some other means

Fig. 5.5: shaft may rotate in a bearing or it may move upwards.

a case of incompletely constrained motion.

If load is placed on shaft to prevent axial upward movement of shaft, then motion of pair is said to be successfully constrained motion.

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5.8. Classification of Kinematic Pairs

1. According to type of relative motion between elements

a. Sliding pair

b. Turning pair

c. Rolling pair

d. Screw pair

e. Spherical pair

2. According to type of contact between elements

a. Lower pair

b. Higher pair

3. According to type of closure

a. Self closed pair

b. Force - closed pair

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5.8. Classification of Kinematic Pairs According to type of relative motion between elements

a. Sliding pair

one pair can only slide relative to the

other

completely constrained motion

Examples:

Piston & cylinder

Tail stock on lathe bed

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b. Turning pair

One pair can only turn or revolve about a

fixed axis of another link

Completely constrained motion

Examples:

Crankshaft in a journal bearing in an engine

Lathe spindle supported in head stock

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c. Rolling pair

One pair rolls over another fixed link

Example: Ball and roller bearings

d. Screw pair

One element can turn about the

other by screw threads

Example: Lead screw of a lathe with

nut

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e. Spherical pair

One element (with spherical shape)

turns or swivels about the other fixed

element

Examples:

Ball and socket joint

Attachment of a car mirror

Pen stand

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5.8. Classification of Kinematic Pairs According to type of contact between the elements

a. Lower pair

The two elements of a pair have a

surface contact when relative motion

takes place

Surface of one element slides over

surface of the other

Examples: Sliding pairs, turning pairs

and screw pairs

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5.8. Classification of Kinematic Pairs According to type of contact between the elements

a. Higher pair

The two elements of a pair have a line or point

contact when relative motion takes place

The motion between the two elements is

partly turning and partly sliding

Examples:

A pair of friction discs

Toothed gearing

Belt and rope drives

Ball and roller bearings, and cam and follower

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5.8. Classification of Kinematic Pairs According to type of closure

a. Self closed pair

The two elements of a pair are

connected together mechanically in

such a way that only required kind of

relative motion occurs.

The lower pairs are self closed pair

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5.8. Classification of Kinematic Pairs According to type of closure

b. Force - closed pair

The two elements of a pair are not

connected mechanically but are kept

in contact by the action of external

forces

Example: The cam and follower, as it is

kept in contact by the forces exerted

by spring and gravity

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5.9. Kinematic Chain

A kinematic chain:

A combination of kinematic pairs, joined

in such a way that each link forms a part

of two pairs

Relative motion between the links or

elements is completely or successfully

constrained

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Example:

Crankshaft of an engine forms a kinematic pair

with the bearings which are fixed in a pair

Connecting rod with crank forms a second

kinematic pair,

Piston with connecting rod forms a third pair

Piston with cylinder forms a fourth pair

Total combination of these links is a kinematic

chain

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If each link is assumed to form two

pairs with two adjacent links, the

relation between number of pairs (p)

forming a kinematic chain and

number of links (l) may be expressed as:

l = 2 p – 4


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Relation between number of links (l)

and number of joints (j) which constitute a kinematic chain is given

by:


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The above Equations are applicable

only to kinematic chains, in which

lower pairs are used

These equations may also be applied

to kinematic chains, in which higher

pairs are used.

each higher pair may be taken as

equivalent to two lower pairs with an

additional element or link

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5.9. Kinematic Chain Examples

Arrangement of three links AB, BC and CA with pin joints at A, B and C

It is not a kinematic chain and hence no relative motion is

possible.

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Examples – constrained kinematic chain of one DOF

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5.9. Kinematic Chain Examples

It is not a kinematic chain

Unconstrained chain

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5.9. Kinematic Chain Examples – Compound Kinematic Chain

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A chain having more than

four links is known as

compound kinematic chain

5.10. Types of Joints in a Chain 1. Binary joint

Two links are joined at the same connection

4 links and 4 binary joins at A, B, C and D.

To determine whether chain is a locked chain (structure) or kinematic chain or unconstrained chain, the following relation, as given by A.W. Klein, may be used :

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j = Number of binary joints

h = Number of higher pairs

l = Number of links

5.10. Types of Joints in a Chain 2. Ternary joint

Three links are joined at the same

connection

Fig. 5.11:

6 links

3 binary joints at A, B & D

2 ternary joints at C & E

One ternary joint ≈ two binary

joints

Equivalent binary joints = 3 + 2 × 2

= 7

l = 6 and j = 7

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5.10. Types of Joints in a Chain 3. Quaternary joint

Four links are joined at the same

connection

It is equivalent to three binary joints

When l number of links are joined at

same connection, joint is equivalent to (l – 1) binary joints.

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Fig. 5.12 (a):

one binary joint at D

4 ternary joints at A, B, E and F

equivalent to 8 binary joints

2 quaternary joints at C & G

equivalent to 6 binary joints

total number of binary joints in

chain are

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Since LHS is greater than RHS, chain, is not a kinematic chain

locked chain: forms a rigid frame or structure.


Total number of binary

joints are 1 + 2 × 6 = 13

Since LHS is equal to RHS,

chain is a kinematic chain

or constrained chain

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5.11. Mechanism

Mechanism: one of the links of a kinematic chain is

fixed

It may be used for transmitting or transforming

motion e.g. engine indicators, typewriter etc.

Simple mechanism: 4 links

Compound mechanism: > 4 links

Machine: When a mechanism is required to

transmit power or to do some particular type of

work

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5.12. Number of DOF for Plane Mechanisms

DOF: number of input parameters

which must be independently

controlled in order to bring the

mechanism into a useful engineering

purpose

It is possible to determine number of

DOF of a mechanism directly from:

number of links

number & types of joints

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Fig. 5.13 (a)

Only one variable is

needed to define the

relative positions of

all the links

Number of DOF = 1

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../Videos/Chapter 5/Kinematics with MicroStation - Ch01C Grueblers Criteria.avi.flv


Fig. 5.13 (b)

Two variables q1 & q2 are needed to define

completely relative

positions of all links

Number of DOF = 2

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Fig. 5.14 (a): two links AB

& CD in a plane motion

Each link has 3 DOF

before it is connected to

any other link

when CD is connected to

AB by a turning pair at A,

position of link CD is now

determined by a single

variable & thus has one

DOF

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when a link is connected to a fixed link by

a turning pair (lower pair), two DOF are

destroyed

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Consider a plane mechanism with

l number of links

Since in a mechanism, one of the

links is to be fixed, therefore:

number of movable links = (l – 1)

total number of DOF = 3 (l – 1) before they are connected to any other link.

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A mechanism with l number of links

connected by j number of binary joints

or lower pairs (single DOF pairs) and h number of higher pairs (two DOF pairs),

then number of DOF of a mechanism is

given by

Kutzbach criterion for movability of a

mechanism having plane motion

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5.13. Application of Kutzbach

Criterion to Plane Mechanisms

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a structure and no relative motion

between the links is possible

there are redundant constraints in the

chain and it forms a statically

indeterminate structure



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Fig. 5.17 (a): 3 links, 2 binary joints and one higher pair: l = 3, j = 2 & h = 1

Fig. 5.17 (b): 4 links, 3 binary joints and one higher pair: l = 4, j = 3 & h = 1

5.14. Grubler’s Criterion (GC)

for Plane Mechanisms

GC applies to mechanisms with only single DOF

joints

Substituting n = 1 and h = 0 in Kutzbach equation,

1 = 3 (l – 1) – 2 j or 3l – 2j – 4 = 0

A plane mechanism with a movability of 1 and

only single DOF joints can not have odd number

of links.

Simplest possible mechanisms of this type are a

four bar mechanism & a slider-crank mechanism

in which l = 4 and j = 4.

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../../Videos/Chapter 5/Kinematics with MicroStation - Ch01C Grueblers Criteria.avi.flv

5.15. Inversion of Mechanism

We can obtain as many mechanisms as

the number of links in a kinematic chain

by fixing different links in a kinematic

chain

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../../Videos/Chapter 5/Kinematics with MicroStation - Ch01 H Slider Crank Inversion III.flv

../../Videos/Chapter 5/Kinematics with MicroStation - Ch01 H Slider Crank Inversion III.flv


Inversion of the mechanism:

method of obtaining different

mechanisms by fixing different links

in a kinematic chain

Relative motions between various links is

not changed, but their absolute motions

may be changed drastically.

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../Videos/Chapter 5/Kinematics with MicroStation - Ch01G Slider Crank Inversion II.flv



Driver: part of a mechanism which initially moves wrt the frame or fixed link

Follower: part of mechanism to which motion is transmitted.

Most of mechanisms are reversible, so

that same link can play the role of a

driver and follower at different times.

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5.16. Types of Kinematic Chains

The most important kinematic chains

are those which consist of four lower

pairs, each pair being a sliding pair or

a turning pair.

Three types of kinematic chains with

four lower pairs are considered:

1. Four bar chain

2. Single slider crank chain

3. Double slider crank chain

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5.17. Four Bar Chain or

Quadric Cycle Chain

Four links, turning pairs at A,

B, C & D.

Grashof ’s law for a four bar

mechanism: For a

continuous relative motion

between links 2 & 4,

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S shortest & longest link lengths ≤ S remaining two link lengths

../../Videos/Chapter 5/Kinematics with MicroStation - Ch01D Grashoffs Criterion.avi.flv

5.17. Four Bar Chain or

Quadric Cycle Chain Input crank makes a complete

revolution relative to other links

crank or driver: One of the links, in particular the shortest link, will make a complete revolution relative to the other three links, if it satisfies Grashof ’s law

lever or rocker or follower: Link BC (2) makes a partial rotation or oscillates

Connecting rod or coupler: Link CD (3) connects crank & lever

Frame of mechanism: Fixed link AB (1)

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5.18. Inversions of Four Bar Chain 1. Beam engine (crank and lever mechanism)

When crank rotates

about A, lever

oscillates about D.

End E of lever CDE is

connected to a

piston rod which

reciprocates due to

rotation of crank

Purpose: convert

rotary motion into

reciprocating motion

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../Videos/Chapter 5/Kinematics with MicroStation - Ch01 H Slider Crank Inversion III.flv

5.18. Inversions of Four Bar Chain 2. Coupling rod of a locomotive (Double crank mechanism)

Links AD & BC (AD = BC) act as cranks and are connected to the respective wheels

Link CD acts as a coupler and link AB is fixed

Transmitting rotary motion from one wheel to the other wheel

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../Videos/Train Animation.flv

5.18. Inversions of Four Bar Chain 3. Watt’s indicator mechanism (Double lever mechanism)

Watt's straight line mechanism consists of four

links: fixed link at A, link AC, link

CE, link BFD.

Links CE & BFD act as levers

Displacement of link BFD is

directly proportional to

pressure of gas or steam

which acts on the indicator

plunger.

On any small displacement

of mechanism, tracing point

E at end of link CE traces out

approximately a straight line

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../../Videos/Chapter 5/Kinematics with MicroStation - Ch02A Approx Straight Line Mechanisms.flv

5.19. Single Slider Crank Chain

Modification of the basic four bar chain

One sliding pair and three turning pairs

Converts rotary motion into reciprocating

motion and vice versa

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../Videos/Crank-Slider-Mechanism.mp4.flv

5.20. Inversions of Single Slider Crank Chain 1. Pendulum pump or Bull engine

Fixing cylinder or link 4 (sliding

pair)

When crank (2) rotates,

connecting rod (3) oscillates

about a pin pivoted to the

fixed link 4 at A and the

piston attached to piston rod

(1) reciprocates.

duplex pump : used to supply

feed water to boilers have

two pistons attached to link

1, Fig. 5.23

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../Videos/Chapter 5/Kinematics with MicroStation - Ch01I Slider Crank Inversion IV.flv

../Videos/PENDULUM PUMP OR BULL ENGINE.avi.flv




5.20. Inversions of Single Slider Crank Chain 2. Oscillating cylinder engine

Convert reciprocating

motion into rotary

motion.

Link 3 forming the

turning pair is fixed

When crank (2) rotates,

piston attached to piston rod (1)

reciprocates and

cylinder (4) oscillates

about a pin pivoted to the fixed link at A

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../Videos/slider crank mechanism (Oscillating cylinder engine).mp4

5.20. Inversions of Single Slider Crank Chain 3. Rotary internal combustion engine or Gnome engine

Consists of 7 cylinders in one plane and all revolves about fixed center D, while crank (2) is fixed.

when connecting rod (4) rotates, piston (3) reciprocates inside the cylinders forming link 1

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../Videos/How a Rotary Engine Works.flv



../Videos/LeRhone Rotary Engine Inner Workings.flv

5.20. Inversions of Single Slider Crank Chain 4. Crank and slotted lever quick return motion mechanism

Used in shaping

machines, slotting

machines and in rotary

internal combustion

engines.

A short link PR transmits

motion from AP to ram

which carries tool and

reciprocates along line

of stroke R1R2

Line of stroke of ram is

perpendicular to AC

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AP1 & AP2 are tangential to the circle and cutting tool is at the end of the stroke.

Forward or cutting stroke occurs when crank rotates from position CB1 to CB2 (angle b) in the cw direction.

Return stroke occurs when crank rotates from position CB2 to CB1 (angle a) in cw direction.

Since crank has uniform angular speed, therefore,

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Since tool travels a distance of R1R2 during cutting

and return stroke, therefore travel of tool or length

of stroke

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5.20. Inversions of Single Slider Crank Chain 5. Whitworth quick return motion mechanism

Used in shaping and slotting machines

Link CD (2) is fixed

Driving crank CA (3) rotates at a uniform angular

speed

Slider (4) attached to crank pin at A slides along

slotted bar PA (1) which oscillates around D

Connecting rod PR carries ram at R to which a

cutting tool is fixed

Motion of tool is constrained along line RD

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../Videos/Chapter 5/Detail Animation of Whitworth Quick return mechanism with Description..flv


When driving crank CA moves from position CA1

to CA2 (or link DP from position DP2 to DP1)

through an angle a in cw direction, tool moves

from left hand end of its stroke to right hand end

through a distance 2 PD.

When driving crank moves from position CA2 to

CA1 (link DP from DP1 to DP2) through an angle b

in cw direction, tool moves back from right hand

end of its stroke to left hand end.

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Time of cutting stroke = time taken by driving

crank to move from CA1 to CA2

Time of return stroke) = time taken by driving

crank to move from CA2 to CA1.

Since crank link CA rotates at uniform angular

velocity therefore time taken during cutting stroke

is more than time taken during return stroke.

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Example 5.1

A crank and slotted lever mechanism used in a shaper has a center distance of 300 mm between the center of oscillation of the slotted lever and the center of rotation of the crank. The radius of the crank is 120 mm. Find the ratio of the time of cutting to the time of return stroke.

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Example 5.2 In a crank and slotted lever quick return motion mechanism, distance between the fixed centers is 240 mm and the length of driving crank is 120 mm. Find the inclination of the slotted bar with the vertical in the extreme position and the time ratio of cutting stroke to the return stroke.

If the length of the slotted bar is 450 mm, find the length of the stroke if the line of stroke passes through the extreme positions of the free end of the lever.

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Example 5.2 September 29, 2014


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Example 5.3

Fig. 5.30 shows the layout of a quick return mechanism of oscillating link type, for a special purpose machine. The driving crank BC is 30 mm long and time ratio of the working stroke to the return stroke is to be 1.7. If the length of the working stroke of R is 120 mm, determine the dimensions of AC and AP.

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Example 5.3 September 29, 2014


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Example 5.4

In a Whitworth quick return motion mechanism,

Fig. 5.32, distance between the fixed centers is 50

mm and length of driving crank is 75 mm. The

length of slotted lever is 150 mm and length of

connecting rod is 135 mm. Find the ratio of time of

cutting stroke to time of return stroke and also the

effective stroke.

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Example 5.4

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5.21. Double Slider Crank Chain

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Two turning pairs & two sliding pairs

../Videos/Chapter 5/Kinematics with MicroStation - Ch01J Double Slider Mechanism.flv

5.22. Inversions of Double Slider Crank Chain

1. Elliptical trammels

When links 1 and 3 slide along their

respective grooves, any point on

link 2 such as P traces out an ellipse

on the surface of link 4, Fig. 5.34 (a).

AP and BP are semi-major axis and

semi-minor axis of the ellipse

respectively.

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1. Elliptical trammels

Coordinates of point P on link BA will be

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2. Scotch yoke mechanism

Convert rotary motion into a reciprocating motion.

Inversion is obtained by fixing either link 1 or link 3

When Link 2 (crank) rotates about B, link 4 (frame) reciprocates

The fixed link 1 guides the frame.

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../Videos/Scotch Yoke Mechanism.flv

../Videos/Chapter 5/Kinematics with MicroStation - Ch01K Double Slider Inversion II.flv


3. Oldham’s coupling

Used for connecting two parallel shafts

whose axes are at a small distance apart

Shafts are coupled in such a way that if one

shaft rotates, the other shaft also rotates at

the same speed.

This inversion is obtained by fixing link 2, Fig.

5.36 (a).

Shafts to be connected have two flanges

(link 1 & link 3) rigidly fastened at their ends

by forging.

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If distance between axes of shafts is constant, center of intermediate piece

will describe a circle of radius equal to distance between axes of the two

shafts.

Max sliding speed of each tongue along its slot = peripheral velocity of

center of the disc along its circular path.

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../Videos/Chapter 5/Kinematics with MicroStation - Ch01L Oldhams Coupling.flv



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