Tom[unit 1]

THEORY OF MACHINES

By-ranjeetnitj@gmail.com

Unit 1
Mechanism:-If a no. of bodies assembled in such a way that the motion of one
causes constrained & predictable motion to the others, it is known as mechanism.
Eg. Slider Crank Mechanism

Figure a illustrates a cross section of a machine—an internal combustion engine. The assembly
of the piston, connecting rod, and crankshaft is a mechanism, termed a slider-crank
mechanism. The basic schematic drawing of that mechanism, Fig. b, called a skeleton outline,
shows only its fundamental structure without the technical details explaining how it is constructed.

Machine := A machine is a mechanism or a combination of mechanism which
apart from imparting motion to the part , also transmit & modifies the available
mechanical energy into some kind of desired work.

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THEORY OF MACHINES


Note :There is a difference between a machine and a mechanism: All machines transform energy to
do work, but only some mechanisms are capable of performing work. The term machinery
means an assembly that includes both machines and mechanisms, means a machine would be
always mechanism(s) but for a mechanism it is not always true, that it would be a machine.

Resistant & Rigid Body:Rigid Bodies: - if a body does not suffer any distortion under the action of force or
the distance between any two points on it remains constant on applying force or
with time, then it would be a Rigid Body. Even though such an object cannot
physically exist
Resistant Bodies: - A body which only behave as Rigid in some circumstances or
bodies which are rigid for the purpose they have to serve for.
E.g. Belt Drive, where belt is rigid when subjected to a tensile force.
Link:- A link is defined as a Rigid/Resistant body is having two or more pairing
elements which connect it to other bodies for the purpose of transmitting force
or motion. Means to say a Rigid/Resistant body only would not be a link until it is
not attached to another body by a joint.
OR
A Resistant Body or a group of resistant bodies with rigid connections preventing
their relative movement is known as link. A Link may be consisting of one or more
resistant bodies.

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The links are classified depending on number of joints:-

1.Singular Link
A link which is connected to only one other link is called a singular link

Figure : Singular Link
2. Binary Link
A link which is connected to two other links is called a binary link

Figure : Binary Link
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3. Ternary Link
A link which is connected to three other links is called a ternary link

Figure : Ternary Link

4. Quaternary Link
A link which is connected to four other links is called quaternary link.

Figure : Quaternary Link

DEGREE OF FREEDOM
The degree of freedom of a body is equal to the number of independent
coordinates required to specify the movement. For a cricket ball when it is in air,
six independent coordinates are required to define its motion. Three independent
displacements coordinates along the three axes (x, y, z) and three independent
coordinates for rotations about these axes are required to describe its motion in
space. Therefore, degree of freedom for this ball is equal to six. If this cricket ball
moves on the ground, this movement can be described by two axes in the plane.
When the body has a plane surface to slide on a plane, the rotation about x and yaxes shall be eliminated but it can rotate about an axis perpendicular to the plane,
i.e. z-axis. At the same time, while executing plane motion, this body undergoes
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displacement which can be resolved along x and y axis. The rotation about z-axis
and components of displacement along x and y axes are independent of each
other. Therefore, a sliding body on a plane surface has three degrees of freedom.
These were the examples of unconstrained or partially constrained bodies. If a
cylinder rolls without sliding along a straight guided path, the degree of freedom
is equal to one only because rotation in case of pure rolling is dependent on linear
motion.
Degree of Freedom = 6- No. of Restrains
The No. of Restrains can never be Zero or Six.

Analyzing Degree of Freedom in Mechanisms:1. Kutzbach criterion:-In the Kutzback criteria the DOF analyze by a Simple
FormulaF=3(N-1)-2j-h-fr
Where,
F: Degree of freedom
N: Number of links
j: Number of binary(Lower) pairs or binary joints (i.e. single degree of freedom
pairs)
h: Number of higher pairs (i.e. two degree of freedom pairs),
fr : Redundant Degree of Freedom
Redundant Degree of Freedom:- a redundant degree of freedom is happen when
one or more links of a mechanism can be moved without causing the motion of
the rest, the effect of this Redundant term is ;it will cause of decrease in resultant
DOF

2. Gruebler’s Criterion:If in a mechanism one link is fixed then no. of moveable links would be N-1; so for
N-1 links the available Degree of Freedom = 6(N-1) [3 translation & 3 rotation ]
Now from the basic formula the DOF is analyze from only 2 things 1. No of links &
2. No of Restrains so now we will analyze the No. of Restrains for 6(N-1) DOF.
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No. of Restrains:-no. of Restrains means the simply no. of the Restricted Coordinates for Movements; the Restrains are as followsFor the kinematic pair having 1 DOF the Restriction/Restrains 6-1= 5
If there are P1 such pairs so no. of Restrains = 5P1
Same as kinematic pairs having 2,3,4 & 5 DOF will have Restrains 4,3,2 & 1
accordingly. If there are P2,P3,P4,P5 Such pairs then Restrains would be
4P2,3P3,2P4,1P5
SO FROM HERE:DOF F= 6(N-1)-5P1-4P2-3P3-2P4-1P5
P1 :- No. of Kinematic pairs having one DOF
P2:- No. of Kinematic pairs having two DOF
P3:- No. of Kinematic pairs having three DOF
P4:- No. of Kinematic pairs having four DOF
P5:- No. of Kinematic pairs having five DOF
the above criteria is for 3D Mechanism but most of the mechanism are two
dimensional such as four link or a slider crank mechanism in which displacement
is possible along two axes(one Restrains) and rotation about only one axis (two
Restrains); thus there are three general Restrains. Hence
DOF F= 3(N-1)-2P1-1P2
This is known as Gruebler’s criterion for DOF of plane mechanism.

3. When there are Loops & links given:When there is easy to find out loops in a mechanism; then most suitable
formula for finding DOF is
F= N-(2L+1)
N- no. of links; L= No. of loops in the mechanism
KINEMATIC PAIRS
In a mechanism, bodies or links are connected such that each of them moves with
respect to another. The behavior of the mechanism depends on the nature of the
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connections of the links and the type of relative motion they permit. These
connections are known as kinematic pairs. Hence kinematic pair is defined as a
joint of two links having relative motion between them.
Broadly, kinematic pairs can be classified as :
(a) Lower pair,
(b) Higher pair, and
(c) Wrapping pair.
a) Lower Pair:- A Pair of Links Having Surface or Area contact between the
members is known as Lower Pair. Examples are motion of slider in the
cylinder, motion between crank pin and connecting rod at big end, Nut
Turning on a screw , Shaft Rotating in a bearing universal Joint etc.
Types of Lower Pairs
There are six types of lower pairs as given below :
(a) Revolute or Turning Pair (Hinged Joint)
(b) Prismatic of Sliding Pair
(c) Screw Pair
(d) Cylindrical Pair
(e) Spherical Pair
(f) Planar Pair

Revolute or Turning Pair (Hinged Joint)
A revolute pair is shown in Figure. It is seen that this pair allows only one relative
rotation between elements 1 and 2, which can be expressed by a single
coordinate ‘Ɵ’. Thus, a revolute pair has a single degree of freedom.

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Figure : Revolute or Turning Pair
Prismatic or Sliding Pair
As shown in Figure, a prismatic pair allows only a relative translation between
elements 1 and 2, which can be expressed by a single coordinate ‘s’, and it has
one degree of freedom.

Figure : Prismatic or Sliding Pair

Screw Pair
As shown in Figure, a screw pair allows rotation as well as translation but
these two movements are related to each other. Therefore, screw pair has one
degree of freedom because the relative movement between 1 and 2 can be
expressed by a single coordinate Ɵ or s. These two coordinates are related by
the following relation :
Ɵ/2π = s/L
where, L is lead of the screw.
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Figure : Screw Pair
Cylindrical Pair
As shown in Figure, a cylindrical pair allows both rotation and translation
Parallel to the axis of rotation between elements 1 and 2. These relative
Movements can be expressed by two independent coordinates Ɵ or s because
they are not related with each other. Degrees of freedom in this case are equal to
two.

Figure: Cylindrical Pair

Spherical Pair
A ball and socket joint, as shown in Figure, forms a spherical pair. Any
rotation of element 2 relative to 1 can be resolved in the three components.
Therefore, the complete description of motion requires three independent
coordinates. Two of these coordinates α and β are required to specify the
position of axis OA and the third coordinate Ɵ describes the rotation about the
axis of OA. This pair has three degrees of freedom.
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Figure : Spherical Pair
Planar Pair
A planar pair is shown in Figure. The relative motion between 1 and 2 can be
described by x and y coordinates in x-y plane. The x and y coordinates describe
relative translation and Ɵ describes relative rotation about z-axis. This pair has
three degrees of freedom.

Figure : Planar Pair

b)Higher Pair:A higher pair is a kinematic pair in which connection between two elements is
only a point or line contact. The cam and follower arrangement shown in Figure is
an example of this pair. The contact between cam and flat faced follower is
through a line. Other examples are ball bearings, roller bearings, gears, etc. A
cylinder rolling on a flat surface has a line contact while a spherical ball moving on
a flat surface has a point contact.

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Figure : Higher Pair

c) Wrapping Pair
Wrapping pairs comprise belts, chains and such other devices. Belt comes from
one side of the pulley and moves over to other side through another pulley as
shown in Figure

Figure : Wrapping Pair

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. For example, reciprocating
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engine mechanism is a planner mechanism in which link 1 is fixed, link 2 rotates
and link 4 reciprocates. In internal combustion engines, it converts reciprocating
motion of piston into rotating motion of crank. This mechanism is also used in
reciprocating compressors in which it converts rotating motion of crank into
reciprocating motion of piston

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Tom[unit 1]

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