Theory of Machines - Module 1 Notes

8/10/2019 Theory of Machines - Module 1 Notes

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Theory of Machines Module 1

1 | R a j a n e e s h . R . C h a n d r a n - M a r i a n E n g i n e e r i n g C o l l e g e , K a z h a k u t t o m

08.503 : Theory of Machines

Module 1

1. MECHANISM

A combination of rigid or restraining bodies which are so shaped and connected that they

move upon each other with definite relative motion is known as mechanism. ( Rigid body means a

body with no deformation when the force is transmitted and restraining body means a body

capable of transmitting force with negligible deformation).

The above mechanism is known as slider-crank mechanism. It is a combination of rigid or

restraining bodies namely crank (2), connecting rod (3) and slider (4). They are so shaped and

connected that they move upon each other with definite relative motion.

2. MACHINE

A machine is a mechanism or combination of mechanisms which not only imparts definite

motions to the parts but also transmits and modifies the available mechanical energy into some

kind of useful energy. This useful energy may be some kind of desired work.

The slider crank mechanism will become a machine when it is used with automobile engine

by adding some other mechanisms. In that case it will convert the available energy (force on the

piston) into the torque on the crank shaft which is the desired energy which will move the vehicle.

3. THEORY OF MACHINES

Theory of machine is that branch of science which deals with the study of relative motion

between the various parts of a machine and forces acting on them.

4. KINEMATIC LINK OR ELEMENT

Each part of a machine, which moves relative to some other part, is known as a kinematic

link (or simply link) or element. A link may consist of several parts, which are rigidly fastened

together, so that they do not move relative to one another. A slider crank mechanism consists offollowing four links 1. Frame 2. Crank 3. Connecting rod 4. Slider. The slider reciprocates in a guide

which is connected to the frame. Hence guide also becomes link 1


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5. 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 :A rigid link is one which does not undergo any deformation while transmitting

motion. Strictly speaking, rigid links do not exist. However, as the deformation of a

connecting rod, crank etc. of a reciprocating steam engine is not appreciable; they can

be considered as rigid links.

2. Flexible link : A flexible link is one which is partly deformed in a manner not to affect the

transmission of motion. For example, belts, ropes, chains and wires are flexible links and

transmit tensile forces only.

3. Fluid link : A fluid link is one which is formed by having a fluid in a receptacle and the

motion is transmitted through the fluid by pressure or compression only, as in the case of

hydraulic presses, jacks and brakes.

6. KINEMATIC PAIR

A joint of two links having relative motion between them is known as a kinematic pair. In a

slider crank mechanism link 2 rotates relative to link 1 and hence link 1 and 2 is a kinematic pair.

Similarly link 2 is having relative motion with link 3 and hence link 2 and 3 is also a kinematic pair.

7. TYPES OF CONSTRAINED MOTIONS

In a kinematic pair, if one element has got only one definite motion relative to the other,

then the motion is called constrained motion. Following are the three types of constrained motions

1. Completely constrained motion:When the motion between a pair is limited to a definite

direction irrespective of the direction of force applied, then the motion is said to be a completely

constrained motion. For example, the piston and cylinder (in a steam engine) form a pair and the

motion of the piston is limited to a definite direction (i.e. it will only reciprocate) relative to the

cylinder irrespective of the direction of motion of the crank, as shown in Fig.

Square bar in a square hole. Shaft with collars in a circular hole.

The motion of a square bar in a square hole, as shown in Fig. and the motion of a shaft with

collars at each end in a circular hole, as shown in Fig. are also examples of completely

constrained motion.

2. Incompletely constrained motion:When the motion between a pair can take place in

more than one direction, then the motion is called an incompletely constrained motion. The

change in the direction of impressed force may alter the direction of relative motion between the


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pair. A circular bar or shaft in a circular hole, as shown in Fig. is an example of an incompletely

constrained motion as it may either rotate or slide in a hole. These both motions have no

relationship with the other.

Shaft in a circular hole. Shaft in a foot step bearing.

3. Successfully constrained motion: When the motion between the elements, forming a

pair, is such that the constrained motion is not completed by itself, but by some other means, then

the motion is said to be successfully constrained motion. Consider a shaft in a foot-step bearing as

shown in Fig. The shaft may rotate in a bearing or it may move upwards. This is a case ofincompletely constrained motion. But if the load is placed on the shaft to prevent axial upward

movement of the shaft, then the motion of the pair is said to be successfully constrained motion.

The motion of an I.C. engine valve (these are kept on their seat by a spring) and the piston

reciprocating inside an engine cylinder are also the examples of successfully constrained motion.

8. CLASSIFICATION OF KINEMATIC PAIRS

The kinematic pairs may be classified according to the following considerations:

1.

According to the type of relative motion between the elements: The kinematic pairsaccording to type of relative motion between the elements may be classified as discussed

below:

(a)Sliding pair.When the two elements of a pair are connected

in such a way that one can only slide relative to the other, the

pair is known as a sliding pair. The piston and cylinder, cross-

head and guides of a reciprocating steam engine, ram and

its guides in shaper, tail stock on the lathe bed etc. are the

examples of a sliding pair. A little consideration will show that

a sliding pair has a completely constrained motion.

(b)Turning pair.When the two elements of a pair are connected in

such a way that one can only turn or revolve about a fixed axis

of another link, the pair is known as turning pair. A shaft with

collars at both ends fitted into a circular hole, the crankshaft in a

journal bearing in an engine, lathe spindle supported in head

stock, cycle wheels turning over their axles etc. are the

examples of a turning pair. A turning pair also has a completely

constrained motion.

(c)Rolling pair.When the two elements of a pair are connected in

such a way that one rolls over another fixed link, the pair is

known as rolling pair. Ball and roller bearings are examples of

rolling pair.


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(d)Screw pair.When the two elements of a pair are connected in

such a way that one element can turn about the other by screw

threads, the pair is known as screw pair. The lead screw of a

lathe with nut, and bolt with a nut are examples of a screw pair.

(e)Spherical pair. When the two elements of a pair are

connected in such a way that one element (with sphericalshape) turns or swivels about the other fixed element, the pair

formed is called a spherical pair. The ball and socket joint,

attachment of a car mirror, pen stand etc., are the examples

of a spherical pair.

2. According to the type of contact between the elements:The kinematic pairs according to

the type of contact between the elements may be classified as discussed below :

(a)Lower pair.When the two elements of a pair have a surface contact when relative motion

takes place and the surface of one element slides over the surface of the other, the pair

formed is known as lower pair. It will be seen that sliding pairs, turning pairs and screw pairs

form lower pairs.

(b)Higher pair.When the two elements of a pair have a line or point contact when relative

motion takes place and the motion between the two elements is partly turning and partly

sliding, then the pair is known as higher pair. A pair of friction discs, toothed gearing, belt

and rope drives, ball and roller bearings and cam and follower is the examples of higher

pairs.


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3. According to the type of closure:The kinematic pairs according to the type of closure between

the elements may be classified as discussed below:

(a)Self closed pair.When the two elements of a pair are connected together mechanically in

such a way that only required kind of relative motion occurs, it is then known as self closed

pair. The lower pairs are self closed pair.

(b)Force - closed pair.When the two elements of a pair are not connected mechanically but

are kept in contact by the action of external forces, the pair is said to be a force-closed

pair. The cam and follower is an example of force closed pair, as it is kept in contact by

the forces exerted by spring and gravity.

9. KINEMATIC CHAIN

When the kinematic pairs are coupled in such a way that the last link is joined to the first link

to transmit definite motion (i.e. completely or successfully constrained motion), it is called a

kinematic chain. In other words, a kinematic chain may be defined as a combination of kinematic

pairs, joined in such a way that each link forms a part of two pairs and the relative motion

between the links or elements is completely or successfully constrained.

The relation between the number of pairs ( P ) forming a kinematic chain and the number

of links ( L ) can be expressed as L = 2P4 . . . (i)

The relation between the number of links ( L ) and the number of joints ( J ) which constitute

a kinematic chain is given by the expression J = 3/2 *L2(ii)

The 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. In

that case each higher pair may be taken as equivalent to two lower pairs with an additional

element or link.

In equations (i) and (ii) if LHS > RHS then chain is locked,

if LHS = RHS then chain is constrained and if LHS < RHS then chain is unconstrained.

1.Consider a three link chainwith three joints A,B,C as in fig.


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Number of links, l = 3; Number of pairs, p = 3;

Number of joints, j = 3

From equation (i), l = 2p4

L.H.S. > R.H.S.

Now from equation (ii),J = 3/2 *L2

L.H.S. > R.H.S.

Since the arrangement of three links, as shown in Figure does not satisfy the equations (i)

and (ii) and the left hand side is greater than the right hand side, therefore it is not a kinematic

chain and hence no relative motion is possible. Such type of chain is called locked chain and

forms a rigid frame or structure which is used in bridges and trusses.

2.Consider a four link chainwith links AB, BC, CD and DA as shown in

Figure

Number of links, L = 4; Number of pairs, P = 4;

Number of joints, J = 4

From equation (i), L = 2P4 = 4

L.H.S. = R.H.S.Now from equation (ii),J = 3/2 *L2 = 4

L.H.S. = R.H.S.

Since the arrangement of four links, as shown in Figure satisfy the equations (i) and (ii),it is a

constrained kinematic chain. A chain in which a single link such as AD in Figure is sufficient to

define the position of all other links, it is then called a kinematic chain of one degree of freedom. A

little consideration will show that in Fig. 5.7, if a definite displacement (say ) is given to the link AD,

keeping the link AB fixed, then the resulting displacements of the remaining two links BC and CD

are also perfectly definite. Thus we see that in a four bar chain, the relative motion is completely

constrained. Hence it may be called as a constrained kinematic chain, and it is the basis of all

machines.

3.Consider a five link chain, as shown in Figure. In this case,

l = 5, p = 5, and j = 5

From equation (i), l = 2 p4 or 5 = 2 54 = 6

i.e. L.H.S. < R.H.S.

From equation (ii),j L.H.S. < R.H.S.

Since the arrangement of five links, as shown in Fig. 5.8 does not

satisfy the equations and LHS is less than right hand side, therefore it is not a kinematic chain. Such

a type of chain is called unconstrained chain i.e. the relative motion is not completely

constrained. This type of chain is of little practical importance.

10.TYPES OF JOINTS IN A CHAIN

The following types of joints are usually found in a chain:

1. Binary joint.When two links are joined at the same connection, the

joint is known as binary joint. For example, a chain as shown in Figure has

four links and four binary joins at A, B,C and D. In order to determine the

nature of chain, i.e. whether the chain is a locked chain (or structure) or

kinematic chain or unconstrained chain, the following relation between

the number of links and the number of binary joints, as given by

A.W. Klein, may be used:


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where j = Number of binary joints, h = Number of higher pairs, and

l = Number of links.

When h = 0, the equation becomes j=3/2*l - 2

Applying this equation to a chain, as shown in Figure, where l = 4 and j = 4, we have

LHS=RHS, the chain is a kinematic chain or constrained chain.

2. Ternary joint.When three links are joined at the same connection, the

joint is known as ternary joint. It is equivalent to two binary joints as one of

the three links joined carry the pin for the other two links. For example, a

chain, as shown in Fig. has six links. It has three binary joints at A, B and D

and two ternary joints at C and E. Since one ternary joint is equivalent to

two binary joints, therefore equivalent binary joints in a chain, , are 3 + 2

2 = 7. We know that l = 6 and j = 7, therefore from j=3/2*l 2 LHS = RHS

therefore the chain, as shown in Figure, is a kinematic chain or

constrained chain.

3. Quaternary joint. When four links are joined at the same connection, the joint is called a

quaternary joint. It is equivalent to three binary joints. In general, when l numbers of links are joinedat the same connection, the joint is equivalent to (l 1) binary joints.

Fig. a Fig. b

For example consider a chain having eleven links, as shown in Fig. a.It has one binary joint

at D, four ternary joints at A, B, E and F, and two quaternary joints at C and G. Since one

quaternary joint is equivalent to three binary joints and one ternary joint is equal to two binary

joints, therefore total number of binary joints in a chain, as shown in Fig. aare 1 + 4 2 + 2 3 = 15.

But Since the LHS> RHS the chain, as shown in Fig.a is not a kinematic chain and such a

type of chain is called locked chain and forms a rigid frame or structure.

For Fig b

therefore the chain, is a kinematic chain or constrained chain.

11.MECHANISM


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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. A

mechanism with four links is known as simple mechanism, and the mechanism with more than four

links is known as compound mechanism. When a mechanism is required to transmit power or to do

some particular type of work, it then becomes a machine. In such cases, the various links or

elements have to be designed to withstand the forces (both static and kinetic) safely. A

mechanism may be regarded as a machine in which each part is reduced to the simplest form to

transmit the required motion.

12.NUMBER OF DEGREES OF FREEDOM FOR PLANE MECHANISMS

Degrees of freedom (Movability) of plane mechanism means the

number of inputs ( ie number of independent co-ordinates) needed to

determine the configuration or position of all the links of the mechanism

with respect to fixed link.

In order to find the degrees of freedom of any plane

mechanism first consider two links 1 and 2 in x-y plane as in the fig. Link1 is fixed along x axis and link 2 is moving in a plane. At any time the

position of link 2 is shown in fig.

The location of point A of link 2 can be completely specified by

three variables ie co-ordinates x and y of the point A and inclination of the link 2 with x axis. Thus

three independent variables ie two translations (x,y co-ordinates) and one rotation () are

necessary to completely locate link 2 in x-y plane. Hence link 2 has 3 degrees of freedom.

But if the point D is joined by a pin to point A on the fixed link 1, then

the two independent variables x and y for the point d are fixed and the

position of the link 2 can be completely locate with the variable only. Hen ce

now the link 2 has only one degree of freedom. This shows two degrees of

freedom (translational) are lost at every pin (or hinge) joint of any link with

fixed link.

From above, we see that when a link is connected to a fixed link by a turning pair (i.e.

lower pair), two degrees of freedom are destroyed. This may be clearly understood from following

fig., in which the resulting four bar mechanism has one degree of freedom (i.e. n = 1).

Now let us consider a plane mechanism with l number of links. Since in a mechanism, one

of the links is to be fixed, therefore the number of movable links will be (l 1) and thus the total

number of degrees of freedom will be 3 (l 1) before they are connected to any other link. In

general, a mechanism with l number of links connected by j number of binary joints or lower pairs

(i.e. single degree of freedom pairs) and h number of higher pairs (i.e. two degree of freedom

pairs), then the number of degrees of freedom of a mechanism is given by n = 3 (l1)2 jh... (i)


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This equation is called Kutzbach criterion for the movability of a mechanism having plane

motion. If there are no two degree of freedom pairs (i.e. higher pairs), then h = 0. Substituting h = 0

in equation (i), we have

n = 3 (l1)2 j

13. APPLICATION OF KUTZBACH CRITERION TO PLANE MECHANISMS

1. The mechanism, as shown in Fig. (a), has three links and three binary joints, i.e. l = 3 and j = 3.

n = 3 (31)2 3 = 0

2. The mechanism, as shown in Fig. (b), has four links and four binary joints, i.e. l = 4 and j = 4.

n = 3 (41)2 4 = 1

3. The mechanism, as shown in Fig. (c), has five links and five binary joints, i.e. l = 5, and j = 5.n = 3 (51)2 5 = 2

4. The mechanism, as shown in Fig. (d), has five links and six equivalent binary joints (because there

are two binary joints at B and D, and two ternary joints at A and C), i.e. l = 5 and j = 6.

n = 3 (51)2 6 = 0

5. The mechanism, as shown in Fig. (e), has six links and eight equivalent binary joints (because

there are four ternary joints at A, B, C and D), i.e. l = 6 and j = 8. n = 3 (61)2 8 =1

It may be noted that

(a) When n = 0, then the mechanism forms a structure and no relative motion between the

links is possible, as shown in Fig. (a) and (d).


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(b) When n = 1, then the mechanism can be driven by a single input motion, as shown in

Fig. (b).

(c) When n = 2, then two separate input motions are necessary to produce constrained

motion for the mechanism, as shown in Fig. (c).

(d) When n = 1 or less, then there are redundant constraints in the chain and it forms a

statically indeterminate structure, as shown in Fig. (e).

The application of Kutzbachs criterion applied to mechanisms with a higher pair or two

degree of freedom joints is shown in Fig.

In Fig. (a), there are three links, two binary joints and one higher pair, i.e. l = 3, j = 2 and h = 1.

n = 3 (31)2 21 = 1

In Fig. (b), there are four links, three binary joints and one higher pair, i.e. l = 4, j = 3 and h = 1

n = 3 (41)2 31 = 2

Here it has been assumed that the slipping is possible between the links (i.e. between the

wheel and the fixed link). However if the friction at the contact is high enough to prevent slipping,

the joint will be counted as one degree of freedom pair, because only one relative motion will be

possible between the links.

14.GRUBLERS CRITERION FOR PLANE MECHANISMS

The Grublers criterion applies to mechanismswith only single degree of freedom joints

where the overall movability of the mechanism is unity. Substituting n = 1 and h = 0 in Kutzbach

equation, we have

1 = 3 (l1)2 j or 3l2j4 = 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 degree of freedom 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.

15.INVERSION OF MECHANISM

Mechanism is a kinematic chain in which one link is fixed. By fixing the links of a kinematic

chain one at a time, we get as many different mechanisms as the number of links in the chain. This

method of obtaining different mechanisms by fixing different links of the same kinematic chain is

known as inversion of the mechanism. In the process of inversion, the relative motions of the links of

the mechanism produced remain unchanged. A particular inversion of a mechanism may give

rise to different mechanisms of practical utility when the proportions of the lengths of the links are

changed.

16. FOUR BAR CHAIN OR QUADRIC CYCLE CHAIN


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The simplest and the basic kinematic chain is a four bar chain or quadric cycle chain, as

shown in Fig.. It consists of four links, each of them forms a turning pair at A, B, C and D. The four

links may be of different lengths. A very important consideration in designing a mechanism is to

ensure that the input crank makes a complete revolution relative to the other links. The mechanism

in which no link makes a complete revolution will not be useful. For the four bar linkage there is a

very simple test to check this case.

17.GRASHOFS LAW

Grashofs law states that for a planar four-bar linkage

the sum of the shortest and longest link lengths cannot be

greater than the sum of the remaining two link lengths if there

is to be continuous relative rotation between two members.

When the longest link has length L, the shortest link

has link S and the other two links have lengths P and Q.

Thus according to Grashofs Law one of the links, in particularthe shorter link will rotate continuously relative to the other three links if and only if S + L P + Q. If

this inequality is not satisfied no link will make a complete revolution relative to another. But

Grashofs Law does not specify the order in which the links are connected or which link of the four

bar chain is fixed.

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

bar chain, one of the links, in particular the shortest link, will make a complete revolution relative

tothe other three links, if it satisfies the Grashof s law. Such a link is known as crank or driver. In Fig.

link AD (link 4 ) is a crank. The link BC (link 2) which makes a partial rotation or oscillates is known as

lever or rocker or follower and the link CD (link 3) which connects the crank and lever is called

connecting rod or coupler. The fixed link AB (link 1) is known as frame of the mechanism.

18. INVERSIONS OF FOUR BAR CHAIN OR QUADRIC CYCLE

CHAIN

Kinematically all the four inversions of a four bar chain are

identical. It can be observed from the schematic representation of a

four bar chain that the four mechanisms obtained by inversion or by

fixing links 1,2,3 and 4 one by one are identical as in the figure. Howeverby suitably altering the proportions of lengths of the links of the links 1,2,3

and 4 , several forms of the four bar chain can be obtained.

1. CRANK - LEVER OR CRANKROCKER MECHANISM


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If a link (l or q) adjacent to the shortest link s is fixed then we get a mechanism known as

crank lever or crank rocker mechanism as in the figure. In this link s is the crank because it is able to

rotate completely and the link p which can only oscillate between the limits is the crank.

Eg: Beam Engine

In this mechanism, when the crank rotates about

the fixed centre A, the lever oscillates about a fixed

centre D. The end E of the lever CDE is connected to a

piston rod which reciprocates due to the rotation of the

crank. In other words, the purpose of this mechanism is

to convert rotary motion into reciprocating motion.

2. DOUBLE CRANK MECHANISM

When the shortest link s, is fixed as the frame we will get

another mechanism called double Crank or Drag Link mechanism.In this inversion both the links adjacent to the s can rotate

continuously and both are properly described as cranks, the shorter

of the two generally used as driver.

Eg: Coupling rod of a locomotive

The mechanism of a coupling rod of a locomotive

(also known as double crank mechanism) which consists

of four links is shown in Fig.

In this mechanism, the links AD and BC (having

equal length) act as cranks and are connected to the

respective wheels. The link CD acts as a coupling rod and

the link AB is fixed in order to maintain a constant centre to centre distance between them. This

mechanism is meant for transmitting rotary motion from one wheel to the other wheel.

3. DOUBLE LEVER MECHANISM

If the link opposite to the shortest link ie link p is fixed and shortest

link d is made as connecting rod then the other two links l and q would

oscillate. This mechanism is known as Double Lever or Double Rocker orrocker-rocker mechanism or oscillating-oscillating convertor.

Eg:Watts indicator mechanism (Double levermechanism).

A Watts indicator mechanism (also known as

Watt's straight line mechanism or double lever

mechanism) which consists of four links, is shown in Fig.

The four links are fixed link at A, link AC, link CE and link

BFD.

It may be noted that BF and FD form one link

because these two parts have no relative motion

between them. The links CE and BFD act as levers. The


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displacement of the link BFD is directly proportional to the pressure of gas or steam which acts on

the indicator plunger. On any small displacement of the mechanism, the tracing point E at the

end of the link CE traces out approximately a straight line.

The initial position of the mechanism is shown in Fig. 5.21 by full lines whereas the dotted

lines show the position of the mechanism when the gas or steam pressure acts on the indicator

plunger.

Eg:Watts Straight-line Mechanism

It is a crossed four bar chain

mechanism and was used by Watt for his early

steam engines to guide the piston rod in a

cylinder to have an approximate straight line

motion.

In Fig., OBAO1 is a crossed four bar

chain in which O and O1 are fixed. In the

mean position of the mechanism, links OB and

O1A are parallel and the coupling rod AB isperpendicular to O1A and OB. The tracing point P traces out an approximate straight line over

certain positions of its movement, if PB/PA = O1A/OB.

19. SINGLE SLIDER CRANK CHAIN

A single slider crank chain

is a modification of the basic four

bar chain. It consists of one sliding

pair and three turning pairs. The

link 1 corresponds to the frame of

the engine, which is fixed. The link

2 corresponds to the crank; link 3

corresponds to the connecting

rod and link 4 corresponds to the

slider. As the crank rotates, the cross-head reciprocates in the guides and thus the piston

reciprocates in the cylinder.

20. INVERSIONS OF SINGLE SLIDER CRANK CHAIN

FIRST INVERSION

When the link 1 is fixed, link 2 is made crank and link 4 is

made slider, then the first inversion of single slider crank chain is

obtained. This inversion is used in reciprocating engine and

reciprocating compressor. In case of reciprocating engine the

link 4 (piston) becomes driver whereas in case of reciprocating

compressor, the link 2 (crank) is the driver.

SECOND INVERSION

When the link 2 is fixed, the second inversion of single

slider crank chain is obtained as in the figure. Then the link 3


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along with the slider at its end C becomes a crank. Hence link 3 along with slider (link 4) rotates

about B. By doing so, the link 1 rotates about A along with the slider (link 4) which reciprocates on

the link 1.

Eg: 1.Whitworth Quick Return Motion Mechanism

This mechanism is mostly used in shaping and slotting machines. In this mechanism, the link

CD (link 2) forming the turning pair is fixed, as shown in Fig. The link 2 corresponds to a crank in a

reciprocating steam engine. The driving crank CA (link 3) rotates at a uniform angular speed. The

slider (link 4) attached to the crank pin at A slides along the slotted bar PA (link 1) which oscillates

at a pivoted point D. The connecting rod PR carries the ram at R to which a cutting tool is fixed.

The motion of the tool is constrained along the line RD produced, i.e. along a line passing through

D and perpendicular to CD.

When the driving crank CA moves from the position CA1 to CA2 (or the link DP from the

position DP1 to DP2) through an angle in the clockwise direction, the tool moves from the left

hand end of its stroke to the right hand end through a distance 2 PD. Now when the driving crank

moves from the position CA2 to CA1 (or the link DP from DP2 to DP1) through an angle in the

clockwise direction, the tool moves back from right hand end of its stroke to the left hand end.

The time taken during the left to right movement of the ram (i.e. during forward or cutting

stroke) will be equal to the time taken by the driving crank to move from CA1 to CA2. Similarly, the

time taken during the right to left movement of the ram (or during the idle or return stroke) will be

equal to the time taken by the driving crank to move from CA2 to CA1. Since the crank link CA

rotates at uniform angular velocity therefore time taken during the cutting stroke (or forward

stroke) is more than the time taken during the return stroke. In other words, the mean speed of theram during cutting stroke is less than the mean speed during the return stroke. The ratio between

the time taken during the cutting and return strokes is given by

EG 2.Rotary Internal Combustion Engine or Gnome

Engine.

Rotary internal combustion engines wereused in aviation. It consists of seven cylinders in one

plane and all revolves about fixed centre D, as

shown in Fig. while the crank (link 2) is fixed. In this


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mechanism, when the connecting rod (link 4) rotates, the piston (link 3) reciprocates inside the

cylinders forming link 1.

THIRD INVERSION

When the link 3 is fixed the third inversion of the slider

crank chain is obtained as in figure. Link 2 acts as a crank

and is rotating about point B and link 4 oscillates.

Eg: 1.Crank and Slotted Lever Quick Return Motion Mechanism.

This mechanism is mostly used

in shaping machines, slotting machines

and in rotary internal combustion

engines. In this mechanism, the link AC

(i.e. link 3) forming the turning pair is

fixed, as shown in Fig. The link 3

corresponds to the connecting rod ofa reciprocating steam engine. The

driving crank CB revolves with uniform

angular speed about the fixed centre

C. A sliding block attached to the

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 the motion from AP to the

ram which carries the tool and

reciprocates along the line of stroke

R1R2. The line of stroke of the ram (i.e.

R1R2) is perpendicular to AC produced.

In the extreme positions, AP1 and AP2 are tangential to the circle and the cutting tool is at

the end of the stroke. The forward or cutting stroke occurs when the crank rotates from the position

CB1 to CB2 (or through an angle ) in the clockwise direction. The return stroke occurs when the

crank rotates from the position CB2 to CB1 (or through angle ) in the clockwise direction. Since

the crank has uniform angular speed, therefore,

Since the tool travels a distance of R1 R2 during cutting and return stroke, therefore travel of the

tool or length of stroke


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From Fig. the angle made by the forward or cutting stroke is greater than the angle described

by the return stroke. Since the crank rotates with uniform angular speed, therefore the return stroke

is completed within shorter time. Thus it is called quick return motion mechanism.

FOURTH INVERSION

When the link 4 of a single slider crank chain is fixed,the fourth inversion is obtained as in fig. Link 3 can oscillate

about the fixed point C on link 4. This makes the end B of link

2 to oscillate about C and end A to reciprocate along the

axis of the fixed link 4.

Eg: Pendulum pump or Bull engine.

In this mechanism, the inversion is obtained by fixing the

cylinder or link 4 (i.e. sliding pair), as shown in Fig.. In this case, when

the crank (link 2) rotates, the connecting rod (link 3) oscillates about

a pin pivoted to the fixed link 4 at A and the piston attached to thepiston 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 Figure.

21. DOUBLE SLIDER CRANK CHAIN

A kinematic chain which consists of two turning pairs and two

sliding pairs is known as double slider crank chain We see that the link 2

and link 1 form one turning pair and link 2 and link 3 form the secondturning pair. The link 3 and link 4 form one sliding pair and link 1 and link 4

form the second sliding pair.

22.INVERSIONS OF DOUBLE SLIDER CRANK CHAIN

FIRST INVERSION :( Elliptical trammels)

The first inversion is obtained by fixing the slotted plate (link 4), as shown in Fig. The fixed

plate or link 4 has two straight grooves cut in it, at right angles to each other. The link 1 and link 3,

are known as sliders and form sliding pairs with link 4. The link AB (link 2) is a bar which forms turning

pair with links 1 and 3.

When the links 1 and 3 slide along their respective grooves, any point on the link 2 such as P

traces out an ellipse on the surface of link 4, as shown in Fig. (a) and AP and BP are the semi-major

axis and semi-minor axis of the ellipse respectively. This can be proved as follows:

Let us take OX and OY as horizontal and vertical axes and let the link BA is inclined at an

angle with the horizontal, as shown in Fig. Now the co-ordinates of the point P on the link BA will

be


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This is the equation of an ellipse. Hence the path traced by point P is an ellipse whose semi

major axis isAP and semi-minor axis is BP.

SECOND INVERSION :( Scotch yoke mechanism.)

The inversion is obtained by fixing either the link 1 or link 3. In Fig. 5.35, link 1 is fixed. In this

mechanism, when the link 2 (which corresponds to crank) rotates about B as centre, the link 4

(which corresponds to a frame) reciprocates. The fixed link 1 guides the frame.This mechanism is

used for converting rotary motion into a reciprocating motion.


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THIRD INVERSION :( Oldhams coupling.)

This inversion is obtained by fixing the link 2, as shown in Fig. 5.36 (a). The shafts to be

connected have two flanges (link 1 and link 3) rigidly fastened at their ends by forging. The link 1

and link 3 form turning pairs with link 2. These flanges have diametrical slots cut in their inner faces,

as shown in Fig. 5.36 (b). The intermediate piece (link 4) which is a circular disc, have two tongues

(i.e. diametrical projections) T1 and T2 on each face at right angles to each other, as shown in Fig.(c). The tongues on the link 4 closely fit into the slots in the two flanges (link 1 and link 3). The link 4

can slide or reciprocate in the slots in the flanges.

When the driving shaftA is rotated, the flange C (link 1) causes the intermediate piece (link

4) to rotate at the same angle through which the flange has rotated, and it further rotates the

flange D (link 3) at the same angle and thus the shaft B rotates. Hence links 1, 3 and 4 have the

same angular velocity at every instant.

An oldham's coupling is 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.

23.Coupler curve

The connecting rod or coupler of a planar four bar linkage may be imagined as an infinite

plane extending in all directions but pin connected to the input and output links. Then during

motion of the linkage any point attached to the plane of the coupler generates some path with

respect to the fixed link, this path is called a coupler curve. By proper choice of link dimensions

useful curves, such as a straight-line or a circular arc, may be obtained.


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24.Transmission Angle

The force and the motion from the input link to the output link is transmitted through the

connecting rod.

or, the transmission angle can be defined as:

For a 4bar linkage, the transmission angle ( ) is defined as the acute angle between the

coupler (AB) and the follower (O4B), as indicated in Fig. If Angle ABO4 is acute then = Angle

ABO4. On the other hand, if Angle ABO4is obtuse, then =180-- ABO4. As explained in this figure, if

= 90, then the entire coupler force is utilized to drive the follower. For good transmission quality,

the minimum value of (min)>300. For a crank-rocker mechanism, the minimum value of occurs

when the crank becomes collinear with the frame as shown in the figure below.


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Clearly, the optimum value of the transmission angle is 900. Since the angle will be

constantly changing during the motion cycle of the mechanism, there will be a position at which

the transmission angle will deviate most from 900. In practice it has been found out that if the

maximum deviation of the transmission angle from 900exceeds 400or 500(depending on the type

of application), the mechanism will lock. In certain cases this maximum deviation must be kept

within 200(e.g. reciprocating pumps) .Transmission can be calculated using the following relation

where a1,a2,a3and a4are link lengths and 12is the angle between link 1 and 2.

25.Mechanical Advantage

Mechanical Advantage of a mechanism is

the ratio of the output torque exerted by the

driven link to the necessary input torque required

at the driver. The mechanical advantage of afour bar mechanism directly proportional to the

sine of the angle between the coupler and

follower ( - Transmission Angle) and inversely

proportional to the sine of the angle between

coupler and the driver (). Since both these angles goes on changes during motion , Mechanical

Advantage will continuously changes as the mechanism moves.

When this becomes zero ,that is the driver is directly in line with the coupler the sine of the

angle also become zero and thus mechanical advantage becomes infinity. Thus only a small input

torque is necessary to overcome a large output torque load.

26.Straight Line Mechanisms

The mechanisms which permits only relative motion of an oscillatory nature along a straight

line is called straight line mechanisms. These mechanisms are of the following two types:

1. Mechanisms in which only turning pairs are used, and

2. Mechanisms in which in which one sliding pair is used.

These two types of mechanisms may produce exact straight line motion or approximatestraight line motion.

27.Exact Straight Line Motion Mechanisms

The principle adopted for a mathematically correct

or exact straight line motion is described in Figure.

Let O be a point on the circumference of a circle of

diameter OP. Let OA be any chord and B is a point on OA

produced, such that OA OB = constant. Then the locus of

a point B will be a straight line perpendicular to the

diameter OP.


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Proof:Draw BQ perpendicular to OP produced. Join AP. The triangles OAP and OBQ are

similar.

But OP is constant as it is the diameter of a circle, therefore, if OA OB is constant, then OQ will be

constant. Hence the point B moves along the straight path BQ which is perpendicular to OP.

1. Peaucellier mechanism.

It consists of a fixed link OO1 and the other

straight links O1A, OC, OD, AD, DB, BC and CA

are connected by turning pairs at their

intersections.

The pin at A is constrained to move along the

circumference of a circle with the fixed

diameter OP, by means of the link O1A.

AC = CB = BD = DA ; OC = OD ; and OO1 = O1A .

Join CD to bisect AB at R. Now from right angled

triangles ORC and BRC, we have

OC2

= OR2

+ RC2

...(i)and BC2= RB2+ RC2...(ii)

Subtracting equation (ii) from (i), we have

OC2BC2= OR2RB2 = (OR + RB) (ORRB) = OB OA

Since OC and BC are of constant length, the product OB OA remains constant and hence the

point B traces a straight path perpendicular to the diameter OP.

2. Harts mechanism.


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It consists six links, a fixed link OO1 and other straight links O1A, FC, CD, DE and EF are

connected by turning pairs.

The pin at A is constrained to move along the circumference of a circle

FC = DE and CD = EF .

The points O, A and B divide the links FC, CD and EF in the same ratio.

From similar triangles CFE and OFB,

(i)

and from similar triangles FCD and OCA

(ii)

Therefore the product OB X OA remains constant. Hence the point B traces a straight path

perpendicular to the diameter OP.


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28.Approximate Straight Line Motion Mechanisms

Some mechanism describes an approximate straight line motion. Usually a four bar chain

or its modified variant gives an approximate straight line motion.

Watts mechanism.

It is a crossed four bar chain mechanism and was used by Watt for his early steam engines

to guide the piston rod in a cylinder to have an approximate straight line motion.

OBAO1 is a crossed four bar chain in which O and O1 are fixed.

In the mean position of the mechanism, links OB and O1A are parallel and the coupling rod

AB is perpendicular to O1A and OB.

The tracing point P traces out an approximate straight line over certain positions of its

movement, if PB/PA = O1A/OB.

Proof :

In the initial mean position of the mechanism, the instantaneous centre of the link BA lies at

infinity. Therefore the motion of the point P is along the vertical line BA. Let OB AO1 be the new

position of the mechanism after the links OB and O1A are displaced through an angle and

respectively. The instantaneous centre now lies at I. Since the angles and are very small,

arc B B = arc A A or OB = O1 A ,.. (i)

OB / O1 A = /

Also AP = IP , and BP = IP

AP / BP = / ...(ii)

From equations (i) and (ii),

Thus, the point P divides the link AB into two parts whose lengths are inversely proportional to the

lengths of the adjacent links.


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29.Intermittent motion mechanisms

A motion which starts and stops regularly is called intermittent motion. The mechanisms

which are used to convert continuous motion into intermittent motion are known as intermittent

motion mechanisms.

1. Geneva Mechanisms:

The Geneva drive or Maltese cross is a gear mechanism

that translates a continuous rotation into an intermittent rotary

motion. The rotating drive wheel has a pin that reaches into a slot

of the driven wheel advancing it by one step. The drive wheel also

has a raised circular blocking disc that locks the driven wheel in

position between steps.

The Geneva wheel as shown in the figure consists of a plate

A which rotates continuously and contains a driving pin that

engages in a slot in the driving member B. Member B is turned 1/4 th

of a revolution for each revolution of plate A. The slot in the

member B must be tangential to the path of the pin upon

engagement in order to reduce shock.

In the most common arrangement, the driven wheel has

four slots and thus advances for each rotation of the drive wheel

by one step of 90.If the driven wheel has nslots, it advances by 360/n per full rotation of the

drive wheel.

Applications :

One application of the Geneva drive is inmovie projectors: the film does not runcontinuously through the projector. Instead, the film is advanced frame by frame, each frame

standing still in front of the lens for 1/24 of a second . This intermittent motion is achieved using a

Geneva drive. (Modern film projectors may also use an electronically controlled indexing

mechanism orstepper motor,which allows for fast-forwarding the film.)

Other applications of the Geneva drive include the pen change mechanism inplotters,

automated sampling devices, indexing tables in assembly lines, tool changers for CNC machines,

bank note counting and so on.
http://en.wikipedia.org/wiki/Degree_(angle)http://en.wikipedia.org/wiki/Degree_(angle)http://en.wikipedia.org/wiki/Movie_projectorhttp://en.wikipedia.org/wiki/Secondhttp://en.wikipedia.org/wiki/Stepper_motorhttp://en.wikipedia.org/wiki/Plotterhttp://en.wikipedia.org/wiki/Indexing_(motion)http://en.wikipedia.org/wiki/CNChttp://en.wikipedia.org/wiki/Banknotehttp://en.wikipedia.org/wiki/Banknotehttp://en.wikipedia.org/wiki/CNChttp://en.wikipedia.org/wiki/Indexing_(motion)http://en.wikipedia.org/wiki/Plotterhttp://en.wikipedia.org/wiki/Stepper_motorhttp://en.wikipedia.org/wiki/Secondhttp://en.wikipedia.org/wiki/Movie_projectorhttp://en.wikipedia.org/wiki/Degree_(angle)


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2. Ratchet and Pawl Mechanisms:

A ratchet is a mechanical device that allows continuous

linear or rotary motion in only one direction while preventing

motion in the opposite direction. Ratchets are widely used in

machinery and tools.

A ratchet consists of a roundgearor linearrackwithteeth, and a pivoting, spring loaded finger called a pawlthat

engages the teeth. The teeth are uniform butasymmetrical,with

each tooth having a moderate slope on one edge and a much steeper slope on the other edge.

When the teeth are moving in the unrestricted (i.e., forward) direction , the pawl easily slides up

and over the gently sloped edges of the teeth, with a spring forcing it (often with an audible 'click')

into the depression between the teeth as it passes the tip of each tooth. When the teeth move in

the opposite (backward) direction, however, the pawl will catch against the steeply sloped edge

of the first tooth it encounters, thereby locking it against the tooth and preventing any further

motion in that direction.
http://en.wikipedia.org/wiki/Gearhttp://en.wikipedia.org/wiki/Gearhttp://en.wikipedia.org/wiki/Gearhttp://en.wikipedia.org/wiki/Rack_and_pinionhttp://en.wikipedia.org/wiki/Rack_and_pinionhttp://en.wikipedia.org/wiki/Rack_and_pinionhttp://en.wikipedia.org/wiki/Asymmetryhttp://en.wikipedia.org/wiki/Asymmetryhttp://en.wikipedia.org/wiki/Asymmetryhttp://en.wikipedia.org/wiki/Asymmetryhttp://en.wikipedia.org/wiki/Rack_and_pinionhttp://en.wikipedia.org/wiki/Gear

Theory of Machines - Module 1 Notes

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