# THEORY OF MACHINES

## Introduction

The Theory of Machines may be defined as the branch of engineering science which deals with the study of relative motion between the various parts of a machine and the forces which act on them.

Each part of a machine which moves relative to some other part,is known as kinematic link or simply link or an element.
A link or element need not to be a rigid body but it must be a resistant body.
A body is said to be a resistant body if it is capable of transmitting the required forces with negligible deformation.

## Kinematic pair

The two links or elements of a machine,when in contact with each other,are said to form a pair.If the relative motion between them is completely or successfully constrained ( i.e. in a definite direction) then the pair is known as kinematic pair.

## Types of constrained motion

There are three types of constrained motion.
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. reciprocating only) relative to the cylinder irrespective of the direction of motion of the cranck.
The motion of a square bar in a square hole and the motion of a shaft with collars at each end in a circular hole,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.
A circular bar or shaft in a circular hole is an example of incompletely constrained motion as it may either rotate or slide in the hole.These both motions have no relationship with each other.
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.
The motion of an I.C. engine valve ( kept on their seat by spring) is an example of successfully constrained motion.

## Classification of kinematic pairs

The kinematic pairs may be classified according to following considerations:
1.) According to the type of relative motion between the elements

A.} SLIDING PAIR-
When the two elements of a pair are connected in such a way that one can only slide relative to other,then the pair is known as sliding pair.
The piston and cylinder,cross head and guides of a reciprocating steam engine,ram and it’s 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 to 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 ones rolls over another fixed link ,then the pair is known as rolling pair.
Ball and roller bearings are examples of rolling pair.
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,
then the pair is known as screw pair.
The lead screw of a lathe with nut,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 spherical shape) turns or swivels about the other fixed element,then the pair formed is called as spherical pair.
The ball and socket joint, attachment of a car mirror,pen stand etc.are the examples of spherical pair.
2.) According to the type of contact between the elements

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 other surface,then the pair formed is known as lower pair.
It can observed that the 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,cam and follower are the examples of higher pair.
3.) According to type of closure

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,then it is 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.

## Kinematic chain

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.
For example,the crankshaft of an engine forms a kinematic pair with the bearings which are fixed in a pair,the connecting rod with the cranck forms a second kinematic pair,the piston with the connecting rod forms a third pair and the piston with the cylinder forms a fourth pair.
The total combination of these links is a kinematic chain.
If each link is assumed to form two pairs with two adjacent links,then the relation between the number of pairs (p) forming a kinematic chain and the number of links (l) may be expressed in the form of an equation:
l= 2p-4
Since in a  kinematic chain each link forms a part of two pairs, therefore there will be as many links as the number of pairs.
Another relation between the number of links (l) and the number of joints (j) which constitute a kinematic chain is given by the expression:
j= 3l/2 -2
In order to determine the nature of chain i.e. whether the chain is locked chain, kinematic chain or a constrained chain, the following relation given by A.W. Klein may be used:
j+h/2=3l/2 -2
Note-
When in equation second and third L.H.S. greater than R.H.S.,then the chain is called a locked chain and forms a frame or structure which is used in bridges and trusses.
When the L.H.S.= R.H.S. then the chain is a kinematic chain.
When L.H.S.< R.H.S. then the chain is called unconstrained chain.

## Mechanism and Inversion

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 indicator,typewriter etc.
A mechanism with four links in 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 useful work,then it becomes a machine.
In such cases,the various links or elements have to be designed to withstand both static and kinetic forces safely.

## Types of kinematic chains and their inversions

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:
1.) Four bar chain or Quadric cycle chain

It consists of four links,each of them forms a turning pair.The inversions of four bar chain are as follows-
a } Beam engine ( cranck and lever mechanism)
b.} Coupling rod of a locomotive ( double cranck mechanism)
c.} Watt’s indicator mechanism ( double lever mechanism).
2.) Single slider cranck chain

It is the modification of the four bar chain.
It consists of one sliding pair and three turning pairs.It is usually found in reciprocating Steam engine mechanism.
This type of mechanism converts rotary motion into reciprocating motion.
The inversions of single slider cranck chain are found in following mechanisms-
a.} Pendulum pump or Bull engine
b.} Oscillating cylinder engine
c.} Rotary Internal combustion engine or Gnome engine
d.} Cranck and slotted lever quick return motion mechanism
e.} Whitworth quick return motion mechanism.
3.) Double slider cranck chain

It consists of two sliding pairs and two turning pairs.The inversions of double slider cranck chain are as follows-
a.} Elliptical trammels.
b.} Scotch yoke mechanism
c.} Oldham’s coupling.
That was like a introduction to Theory of Machines.
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