simple mechanisms -...

Chapter 5

Simple Mechanisms

September 28, 2013

Dr. Mohammad Suliman Abuhaiba, PE 1

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.

A resistant body: capable of transmitting required forces with negligible deformation.

Thus a link should have the following two characteristics: 1. It should have relative motion

2. It must be a resistant body

September 28, 2013

Dr. Mohammad Suliman Abuhaiba, PE

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5.3. Types of Links In order to transmit motion, the driver and the follower may be connected by the following three types of links:

1. Rigid link: does not undergo any deformation while transmitting motion.

2. Flexible link: is partly deformed in a manner not to affect the transmission of motion. Example: belts, ropes, chains and wires

3. Fluid link: is formed by having a fluid in a container and the 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

(members) having no relative motion between them and meant for carrying loads having straining action. Examples: A railway bridge, a roof truss, machine frames

Differences between a machine and a structure:

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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

in a structure no energy is

transformed into useful work

links of a machine may transmit

both power & motion

members of a structure transmit

forces only

5.6. Kinematic Pair

a pair: two links of a machine, in contact

with each other.

If the relative motion between them is

completely or successfully constrained

(i.e. in a definite direction), the pair is

known as kinematic pair.

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5.7. Types of Constrained Motions

Three 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 (in a steam engine) form a pair and motion of piston is limited to a definite direction relative to cylinder irrespective of direction of motion of crank.

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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

When motion between elements, forming a pair is such that the constrained motion is not completed by itself, but by some other means, then motion is said to be successfully constrained motion.

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.

Motion of piston reciprocating inside an engine cylinder.

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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.

A sliding pair has a completely constrained motion.

Examples:

The piston & cylinder

tail stock on lathe bed

b. Turning pair.

one pair can only turn or revolve about a fixed axis of another

link.

A turning pair has a completely constrained motion.

Examples:

A shaft with collars at both ends fitted into a circular hole

Crankshaft in a journal bearing in an engine

lathe spindle supported in head stock

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

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.

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.

b. 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.

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.

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. The total combination of these links is a kinematic chain.

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5.9. Kinematic Chain If each link is assumed to form two pairs with two

adjacent links, then the relation between number of pairs (p) forming a kinematic chain and number of links (l) may be expressed as:

l = 2 p – 4

Relation between number of links (l) and number of joints (j) which constitute a kinematic chain is given by:

Equations (i) and (ii) 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

Fig. 5.6: 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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5.9. Kinematic Chain

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. Fig. 5.10: four links and four binary joins at A, B, C and D.

To determine nature of chain, i.e. whether chain is a locked chain (structure) or kinematic chain or unconstrained chain, the following relation between number of links and number of binary joints, 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:

six links

three binary joints at A, B & D

two ternary joints at C & E

One ternary joint is equivalent to

two binary joints

Equivalent binary joints = 3 + 2 ×

2 = 7

l = 6 and j = 7

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kinematic chain or

constrained chain

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.

Fig. 5.12 (a):

one binary joint at D

four ternary joints at A, B, E and F equivalent to

8 binary joints

two 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 and forms a rigid frame or structure.

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

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

When one of the links of a kinematic chain is

fixed, the chain is known as mechanism.

It may be used for transmitting or transforming

motion e.g. engine indicators, typewriter etc.

simple mechanism: four links

compound mechanism: more than four links

When a mechanism is required to transmit

power or to do some particular type of work, it

becomes a machine.

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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 and 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 of a

four bar chain is one.

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


Fig. 5.13 (b).

Two variables q1 and q2

are needed to define

completely the relative

positions of all the links.

Number of DOF is two.

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Fig. 5.14 (a): two links AB and CD in a plane motion,

each link of a mechanism has 3 DOF before it is connected to any other link

when link CD is connected to link AB by a turning pair at A, the position of link CD is now determined by a single variable and thus has one DOF

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when a link is connected to a fixed link by a

turning pair (i.e. 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.

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): three links, two binary joints and one higher pair, i.e. l = 3,

j = 2 and h = 1.

Fig. 5.17 (b): four links, three binary joints and one higher pair, i.e. l = 4, j

= 3 and h = 1

5.14. Grubler’s Criterion for

Plane Mechanisms Grubler’s criterion applies to mechanisms with only

single DOF joints

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

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

This equation is known as the Grubler's criterion for plane mechanisms with constrained motion.

A plane mechanism with a movability of 1 and only single DOF joints can not have odd number of links.

The simplest possible mechanisms of this type are a four bar mechanism and a slider-crank mechanism in which l = 4 and j = 4.

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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.

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 in

any manner through the process of inversion, but their absolute motions may be changed drastically.

Driver: part of a mechanism which initially moves with respect to 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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../Videos/Chapter 5/Kinematics with MicroStation - Ch01G Slider Crank Inversion II.flv

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

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.

The following three types of kinematic chains

with four lower pairs are important from the

subject point of view :

1. Four bar chain or quadric cyclic 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, each of them forms a turning pair at A, B, C & D.

The four links may be of different lengths.

Grashof ’s law for a four bar mechanism: sum of shortest and longest link lengths should not be greater than the sum of remaining two link lengths if there is to be continuous relative motion between the two links.

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../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.

The mechanism in which no link makes a complete revolution will not be useful.

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

Link BC (link 2) which makes a partial rotation or oscillates is known as lever or rocker or follower and link CD (link 3) which connects crank and lever is called connecting rod or coupler. The fixed link AB (link 1) is known as frame of mechanism.

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

When crank rotates about the fixed center A, the lever oscillates about a fixed center D. The end E of the lever CDE is connected to a piston rod which reciprocates due to the rotation of the crank.

The purpose of this mechanism is to 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 and BC (having equal length) act as cranks and are connected to the respective wheels.

link CD acts as a coupling rod and link AB is fixed in order to maintain a constant center to center distance between them. This mechanism is meant for 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, the 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

It is a modification of the basic four bar chain.

one sliding pair and three turning pairs.

It converts rotary motion into reciprocating motion and vice versa.

Links 1 & 2, links 2 & 3, and links 3 & 4 form three turning pairs while links 4 and 1 form a sliding pair.

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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 (link 2) rotates, connecting rod (link 3) oscillates about a pin pivoted to the fixed link 4 at A and the piston attached to piston rod (link 1) reciprocates.

The duplex pump which is used to supply feed water to boilers have two pistons attached to link 1, as shown in 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

Used to convert reciprocating motion into rotary motion.

Link 3 forming the turning pair is fixed.

When crank (link 2) rotates, the piston attached to piston rod (link 1) reciprocates and cylinder (link 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

It consists of seven cylinders in one plane and all revolves about fixed center D, while crank (link 2) is fixed.

when connecting rod (link 4) rotates, piston (link 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.

Link AC (link 3) is fixed. Link 3 corresponds to the connecting rod of

a reciprocating steam engine.

Driving crank CB revolves with uniform angular speed about fixed center C.

A sliding block attached to crank pin at B slides along the slotted bar AP and thus causes AP to oscillate about

the pivoted point A.

A short link PR transmits motion from AP to ram which carries tool and reciprocates along line of stroke R1 R2. The 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 R1 R2 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 (link 2) is fixed. Link 2 corresponds to a crank in a reciprocating steam engine.

Driving crank CA (link 3) rotates at a uniform angular speed.

Slider (link 4) attached to crank pin at A slides along the slotted bar PA (link 1) which oscillates at a pivoted point 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 taken during left to right movement of ram (cutting stroke)

will be equal to time taken by driving crank to move from CA1 to

CA2.

Time taken during right to left movement of ram (return stroke) will

be equal to 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.

Mean speed of ram during cutting stroke is less than mean speed

during return stroke.

Ratio between time taken during cutting & return strokes is given

by

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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 28, 2013


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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 28, 2013


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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

It is a kinematic chain which consists of

two turning pairs and two sliding pairs.

Link 2 & link 1 form one turning pair and

link 2 & link 3 form the second turning

pair.

Link 3 & link 4 form one sliding pair and

link 1 & link 4 form the second sliding

pair.

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5.21. Double Slider Crank Chain

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../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

Link BA is inclined at an angle q with the

horizontal, Fig. 5.34 (b).

Coordinates of point P on link BA will be

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2. Scotch yoke mechanism

Used for converting rotary motion into a reciprocating motion.

Inversion is obtained by fixing either link 1 or link 3.

When Link2 (crank) rotates about B as center, 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.

The 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).

The shafts to be connected have two

flanges (link 1 and link 3) rigidly fastened at

their ends by forging.

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3. Oldham’s coupling If distance between the axes of the shafts is constant, the center of

intermediate piece will describe a circle of radius equal to the distance between the axes of the two shafts.

Max sliding speed of each tongue along its slot = peripheral velocity of the center of the disc along its circular path.

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


3. Oldham’s coupling

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Homework

All questions at the end of chapter

Homework assignment should be

submitted

However, a quiz will be held after finishing

the chapter

1st quiz will be held on 22/9/2013 @ 14:00

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

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