Project Report on

A REPORT SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENT

FOR THE DEGREE OF BACHELOR OF TECHNOLOGY
IN
MECHANICAL ENGINEERING

Under the Supervision of

PROF. DR. SUDIP MUKHERJEE
Submitted by

AKASH SASMAL - 17101103034

ANIK HAZRA - 18101103081
ASIM CHANDRA - 18101103086
BIBEK KUNDU - 18101103092
BISWAJIT PAUL – 18101103082
MOLOY SARKAR – 18101103087

DEPARTMENT OF MECHANICAL ENGINEERING

JALPAIGURI GOVERNMENT ENGINEERING COLLEGE

(A Government Autonomous Engineering College under Maulana Abul Kalam Azad University
of Technology (formerly known as WBUT))
JALPAIGURI – 735102

JULY 2021
 ACKNOWLEDGEMENT
It is our proud privilege to release the feelings of our gratitude to several
persons who helped us directly or indirectly to conduct this project. We
express our heart full indebtedness and owe a deep sense of gratitude to our
project guide Prof. Dr. Sudip Mukherjee sir, without whose valuable
suggestion and comments, we could hardly finish this project.

We would like to express our sincere thanks to Department of Mechanical

Engineering, Jalpaiguri Government Engineering College, for their kind
guidance and encouragement and also like to thank respected principal of our
college for allowing us to do this project.

The study has indeed helped us to explore more knowledge related to our
topic and we are sure that it will help us in my future.
AKASH SASMAL – 17101103034

ANIK HAZRA – 18101103081

ASIM CHANDRA – 18101103086

BIBEK KUNDU – 18101103092

BISWAJIT PAUL – 18101103082

MOLOY SARKAR – 18101103087

Department of Mechanical Engineering

Jalpaiguri Government Engineering College

Page | 2
 FORWARD

I have immense pleasure in certifying that the project entitled “-------”

submitted to the department of Mechanical Engineering, Jalpaiguri
Government Engineering College is bona-fide record of work performance by
the group of students as a result of their combined and untiring effort under
my guidance and supervision. The main purpose of this is to help the students
to activate their skills most effectively, in applying the knowledge they have
already acquired to reach a satisfactory solution of engineering problem,
particularly those involving in the process of “Developing India with
Automation". I think they will be able to deal boldly with such problems in
practical field in future.

I wish them every success in life.

Prof. Dr. Sudip Mukherjee

Dept. Of Mechanical Engineering

Jalpaiguri Government Engineering College

Page | 3
 Government of West Bengal

Department of Mechanical Engineering

Jalpaiguri Government Engineering College

Jalpaiguri-735102, West Bengal

CERTIFICATE
I have immense pleasure in certifying Akash Sasmal (2017-2021), Anik
Hazra, Asim Chandra, Bibek Kundu, Biswajit Paul, Moloy Sarkar , of 2018-
2021 final year student (B-TECH) of Mechanical Engineering Department of
Jalpaiguri Government Engineering College for their project “Robotics In
Indian Automobile Industry” under the supervision of Prof. Dr. Sudip
Mukherjee, a distinguished member faculty in the Department of Mechanical
Engineering of Jalpaiguri Government Engineering College.

This is confided record of the project work submitted by them towards

partial fulfilment for obtaining the degree of “Bachelor of Technology” in
Mechanical Engineering from Jalpaiguri Government Engineering College.
I do think that they will be able to deal efficiently with such problems in
practice. I wish them all success in their future life.

Prof. Subrata Bhattacharya

(Head of the department)

Department of Mechanical Engineering

Jalpaiguri Government Engineering College

Page | 4
 CONTENTS Page No
The problem 7

CHAPTER-1 9-24

1.1. Introduction 9

1.2. Turning operation 10

1.3. Different types of turning operations 11

1.3.1. Plain and step turning 11

1.3.2. Rough and finish turning 11-12

1.3.3. Eccentric turning 13

1.3.4. Facing 14

1.3.5. Taper turning 15

1.4. Cutting Tool Materials 16-17

1.5. Turning machines 18

1.6. Surface roughness 19

1.7. Factors Affecting surface roughness 19-21

1.8. Measurement of Surface Roughness 22-23

1.9. Reduction or Control of Surface Roughness 24

Page | 5
Page | 6
 The Problem
One of the most crucial and determining factor for the successful
maximization of the manufacturing process and its automation in
any typical metal cutting process is surface roughness. Here the work
is concerned with the optimization of surface roughness of a mild
steel specimen undergoing turning operation. The parameters
controlling the surface roughness on the specimen are spindle speed,
feed rate and depth of cut. The experimental data are analyzed with
the help of a regression model and an attempt has been made to
arrive at an optimum combination of the control parameters, within
their working ranges, which may provide the minimum surface
roughness on the specimen undergoing turning operation.

Page | 7
 CHAPTER – 1
INTRODUCTION

Page | 8
  1.1. Introduction:
Machining operations have been the core of the manufacturing
industry since the industrial revolution. Increasing the productivity
and the quality of the machined parts are the main challenges of
metal-based industry; there has been increased interest in
monitoring all aspects of the machining process. Turning is the
most widely used among all the cutting processes. The machining
parameters (cutting speed, feed rate, depth of cut) accelerate
surface roughness and it affects the surface finishing also.
Optimum machining parameters, being the objective of this work
is planned for turning machines to minimize surface roughness in
order to improve quality of machined products to improve the
tool life.

In this project, our main focus was to derive the SN Ratio

(Signal to Noise Ratio) so that we can find out what is the factor
affecting the most during turning operation and causing surface
roughness. Due to Covid-19 pandemic lockdown, we could not
experimentally find the data from our college machine shop as it
was closed. So we approached a local workshop and collected the
data that are used in this project.

Before going to surface roughness and SN Ratio, first we will

discuss about the different types of turning operations performed
on a lathe machine.

Page | 9
  1.2. Turning operation:
Turning is a machining process in which a cutting tool, typically
a non-rotary tool bit, describes a helix toolpath by moving more or
less linearly while the workpiece rotates.

The turning processes are typically carried out on a lathe,

considered to be the oldest of machine tools, and can be of
different types such as straight turning, taper turning, profiling or
external grooving. Those types of turning processes can produce
various shapes of materials such as straight, conical, curved, or
grooved workpieces. In general, turning uses simple single-point
cutting tools. Each group of workpiece materials has an optimum
set of tool angles that have been developed through the years.

FIG. 1. Turning operation

Page | 10
  1.3. Different types of turning operations:
1.3.1. Plain and step turning:
Both these operations are simple operations and can be done
by holding the job in many different ways. The common methods
of holding the work are:

(i) Between centers,

(ii) On a face plate,

(iii) In a chuck,

(iv) On a mandrel.

1.3.2. Rough and finish turning:

Rough turning is the term used for the process of heavy stock
removal in order to save machining time. In this process, deeper
cut is taken and heavier feed is employed. However, rigidity of the
machine should be considered before deciding upon the feed rate
and depth of cut. The surface produced will, obviously, be rough.
Just enough removable stock should be left over the workpiece for
finishing cut. A sharp edged heavy turning tool with a strong
cutting edge is employed in this operation so that it is strong
enough to take deep cuts and is capable of bearing the heavy
cutting forces.

Once the larger part of the excess material has been removed
through rough turning, it is followed by Finish turning operation in
order to bring the job to correct size and provide a fine surface
finish on it.

Page | 11
 FIIG. 2. Rough and finish turning operations
The amount of excess material to be removed through this
operation is very less and, therefore, lighter feed and smaller
depth of cut is used and the heavier tool is replaced by a finish
turning tool. This operation is usually performed in two stages.
The first stage consists of setting an approximate depth of cut and
turning the job diameter for a limited length (say 5 to 10 mm) by
hand feed. The turned diameter is measured and the depth of cut
adjusted according to the need. It is called a trial run. More trial
runs may have to be made until the depth of cut is adjusted for
the correct diameter. After this has been done the required job-

Page | 12
 length can be finish turned by employing power feeds. This is the
second stage of this operation. If work material permits, the use
of coolant in abundance helps in producing a superior finish on
the work surface.

1.3.3. Eccentric turning:

Many times it is required to machine such jobs on a centre-
lathe which have two or more cylindrical surfaces to be turned
which are eccentric with each other.

FIG. 3. Eccentric turning operation

Page | 13
 This is known as Eccentric turning. So many different methods
can be employed for such work and their selection is governed by
many factors. The most vital factor is the number of jobs to be
made. Such jobs can be best machined with the help of well-
designed fixtures and proper tools, but their use requires a lot of
economic considerations. Where a large number of similar jobs
are to be machined, such that a quicker and larger production will
compensate for the cost of production of the said fixtures and
tools, it is always advantageous to use fixtures. However, if only a
few pieces are to be machined it would certainly be uneconomical
to have the use of fixtures. In such cases, other methods of
eccentric turning are used.

1.3.4. Facing:
It is an operation which enables the production of a flat surface
through machining at the end of a job. In this operation the tool is
fed at right angles to the axis of the job. Since no longitudinal feed
is needed, the carriage is usually clamped to the bed so that it
remains stationary during the operation. Feed to the tool is
provided by moving the cross-slide by hand. Depending upon the
diameter of the work piece the power feed can also be used. Two
methods of holding the tools are usually employed. One is to set
the tool slightly inclined to the exact right angle position so that
only the tip of the cutting edge remains in contact with the job
face and the remainder of the tool is free of the job. Another
method is to set the tool parallel to the axis of the job. This is
done when the job is held in a chuck or a face plate. In both the
cases the tool can either be fed outwards, i.e., away from the
centre, or inwards, i.e., towards the centre. Normally the selection

Page | 14
 of any one of these methods is according to convenience, still the
latter method is more commonly employed.

FIG. 4. Facing operation

1.3.5. Taper turning:
A large variety of components used in engineering practice is
found to have conical shapes or, if flat, having a gradual reduction
in its width or thickness along their length. Such components are
known as tapered for conical pieces, the difference between the
diameters of their ends is
known as taper and for flat
pieces the difference
between the widths or
thicknesses of their ends is
known as taper Parts may
have external or
internal taper according to
the requirement. A
lathe centre, shank of a twist
drill or reamer and sockets, etc. have external taper whereas a tail
stock barrel or the nose of lathe spindle has internal taper. A
Page | 15
 cotter is a good example of flat tapered pieces. However, so far as
the lathe work is concerned, we will confine our study to external
and internal tapers of conical parts only. It is not sufficient to
mention only the amount of this taper.

To specify the taper fully the length over which this taper
occurs should also be mentioned. The common methods of
expressing the taper are

1. Taper per foot, i.e., the difference in inches of end diameters

per foot length of the job.

2. Taper per inch, i.e., the difference in inches of end diameters

per inch length of the job.

3. Taper 1 in x; for this, the units should be uniform, such as a

taper 1 in 20 means either a taper of 1 inch on 20 inches length or
a taper of 1 foot over 20 feet length of the job.

In metric units we usually express the taper in 1 in x form, all

measurements being in millimetres.

1.4. Cutting Tool Materials:

 Essential Properties
The vast progress in industrial manufacturing has brought about
several remarkable improvements in cutting tool materials and
geometry

1. To meet the ever-growing demands for enhanced productivity,

high quality and overall economy of machining.

Page | 16
 2. To enable effective and efficient machining of the so-called
difficult-to-machine exotic materials which are rapidly and
widely coming up with the progress of the industrial world.
3. To accomplish precision and ultra-precision machining as per
demand of the day and future.
4. For micro-machining as its demand is increasing.

The service life and overall performance of cutting tools, for a

given job, are governed by

1. Material of the cutting tools.

2. Geometry of the cutting tools.

3. Proper selection and use of the cutting tools,

4. The condition of machining and cutting fluid application.

Of these, the most vital role is played by the tool material. We

have discussed some of the commonly used tool materials.

1. High Carbon Steels: These steels contain carbon from 0.8 to

1.2%, have good hardenability and can successfully withstand high
temperatures upto 300°C. They are mostly used for wood working
tools, hand tools and some striking tools like hammers.

2. High Speed Steels: It is an alloy steel manufactured in

different compositions. But, the most popular composition is the
one containing 18% W. 4% Cr. and 1% V. It retains its red hardness
upto about 600°C. Yet another variety of H.S.S. is also commonly
used which carries an addition of cobalt upto 12%. H.S.S. is
commonly used for manufacturing a large number of cutting tools
like lathe tools, drills, reamers, milling cutters, planer and shaper
tools, slotting tools, hobs, broaches, etc.

Page | 17
 3. Stellite: It is a non-ferrous alloy and is capable of retaining its
red hardness upto about 920°C. It facilitates cutting at twice the
cutting speeds used with H.S.S. tools. Hard bronzes, cast iron and
malleable iron are some of the common materials machined by
these tool materials.

4. Cemented Carbides: They are used as inserts and

manufactured in standard shapes through powder metallurgy
process. They are capable of retaining their red hardness upto
1200°C. They are extremely hard, wear resistant and brittle. They
facilitate application of cutting speeds of the order of 3 to 4. times
higher than those used with H.S.S. tools.

5. Ceramics: Their main ingredients are aluminum oxides and

boron nitrides, which are sintered and compacted into standard
shapes. They are also used as throw-away type tips and facilitate
application of 2 to 3 times cutting speeds of those used with
cemented carbides.

6. Diamond: It is the hardest tool material so far known. Because

of extremely high cost and brittleness they are not in common
use, except for machining hard to machine materials, as dressing
and trueing tools for grinding wheels, as abrasive grains in laps,
cutting points for glass and marbles, etc.

1.5. Turning machines:

The turning machines are, of course, every kind of lathes. Lathes
used in manufacturing can be classified as engine, turret,
automatics, and numerical control etc. The lathe can be defined
as a machine tool which holds the work between two rigid and
strong supports, called centres, or in a chuck or face plate while
the latter revolves. The chuck or the face plate is mounted on
the projected end of the machine spindle. The cutting tool is

Page | 18
 rigidly held and supported in a tool post and is fed against the
revolving work. While the work revolves about its own axis the
tool is made to move either parallel to or at an inclination with
this axis to cut the desired material. In doing so it produces a
cylindrical surface, if it is fed parallel to the axis, or will produce a
tapered surface if it is fed at an inclination.

FIG. 4. Components of a Lthe machine

1.6. Surface roughness:

This usually refers to the macro and micro irregularities developed

on the product surfaces produced by the various manufacturing
processes, such as preforming processes like casting, forging,
rolling, extrusion, powder metallurgy, etc. and even the semi-
finishing and finishing processes like machining, grinding, honing,
lapping, etc. However, the pattern and extent of the surface

Page | 19
 irregularities depend upon the type and working condition of the
manufacturing process undertaken. Crude surface roughness
becomes visible even with the naked eye. Finer roughness is
observed under different microscopes of varying resolution.
Surface roughness can be quantitatively measured sufficiently
precisely by different instruments and techniques.

1.7. Factors Affecting surface roughness:

A. Depth of cut: When a job is turned on lathe a certain amount
of material is removed continuously by the tool, in a cut, in the
form of chips. The total thickness of the material to be removed
depends upon the amount of machining allowance provided on
the job and the final dimensions required. Thickness of the metal
removed in a cut by the tool is known as depth of cut 't' (say). In
other words we can define the term 'depth of cut' as the distance,
measured normal to the work axis, by which the point of the tool
penetrates into the job surface in a cut. Increasing the depth of
cut increases the cutting resistance and the amplitude of
vibrations. As a result, cutting temperature also rises. Therefore, it
is expected that surface roughness rate will be more.

B. Feed: The feed of the tool denotes the advancement of the

tool for each revolution of the work. The tool can be moved in
three directions with respect to the axis of the work and the
corresponding feeds are named after these directions of the tool
movement as described below.

1. Longitudinal feed - tool moves parallel to the axis of the work;

such as in plain turning, step turning, threading, etc.

Page | 20
 2. Cross feed - tool moves normal to the axis of the work; such as
in facing, undercutting, parting off, etc.

3. Angular feed - tool moves at an inclination to the axis of the

work; such as in taper turning.

Experiments show that as the feed rate increases, surface

roughness also increases due to the increase in cutting force and
vibration.

C. Cutting Speed: An important term to be considered is the

cutting speed of the tool. It is the distance travelled per minute by
a point on the circumference of the work in a direction parallel to
the direction of the feed of the tool. It is expressed in metres per
minute and is given by the formula.

V= (лDN/1000) mpm.

Where, V = Cutting speed in metres per minute.

D = Diameter of the work surface in mm.

And, N = Speed of the work in rpm.

It is found that an increase of cutting speed generally increases

surface roughness rate.
D. Cutting fluids: The use of lubricants is indispensable in all
metal working operations in the workshops. Particularly, in metal
machining these fluids play a very significant role. As we are
aware, a lot of heat energy is generated during the machining
processes which is either wasted unnecessarily or proves
harmful to the tool or work or both. These fluids help in
minimising these adverse effects and, thus, help to increase the
tool life and decrease the surface roughness on the job.

Page | 21
 E. Tool geometry: Some geometric factors which affect
achieved surface roughness include:
I. Nose radius
II. Rake angle
III. Side cutting edge angle, and
IV.Cutting edge.

Other factors are-

F. Work piece and tool material combination and their

mechanical properties.

G. Quality and type of the machine tool used.

1.8. Measurement of Surface Roughness:

The randomly distributed micro-surface irregularities, that is,

the roughness of the machined surface is measured by several
methods and instruments. The measured or monitored surface
roughness is characterized mainly by its-

1. Magnitude, generally in average.

Page | 22
 2. Pattern of distribution of irregularities.
3. Texture or lay of the surface roughness.

Surface roughness is measured based on different principles

which include-

1. High resolution microscopy.

2. Suitable profilometry.
3. Replica method.

1. Microscopic Methods: Surface roughness is experimentally

investigated by high resolution optical microscopes generally
for some qualitative assessment. But surface roughness can
be evaluated both qualitatively and partially quantitatively by
using a stereomicroscope and a scanning electron microscope
(SEM). But detailed and precise quantitative measurement of
surface roughness is not possible by any microscopy.

2. Profilometry: Precision measurement of surface roughness

is commonly done by using profilometers working on the basis
of tracing or scanning.
Measurement and characterization of surface roughness by a
profilometer may be broadly classified as-
1. Contact type or non-contact type.
2. 2-D or 3-D.
3. Replica Method: In some complicated and inaccessible
situation a relatively crude but simple method namely cast
replica method is employed. The actual surface profile is
transferred to a plastic sample. A suitable plastic, after
softening by heating or by a solvent, is pressed or pasted on
the surface of interest. After reasonable hardening, the plastic
bead with the surface replica is stripped off. Then the
replicated surface is examined by suitable instrumentation.

Page | 23
 1.9. Reduction or Control of Surface Roughness:

The surface roughness in machining can be reduced by-

1. Proper selection of cutting tool geometry, such as
a) Rounding or radiusing the tip or nose of the single-point
tools (turning, shaping, planning, boring, etc.).

Page | 24
 (b) Reducing the cutting angles (Φ1,Φ2), if not restricted
otherwise, of single-point tools.
(c) Slight flattening of the tool tip.

2. Proper selection of levels of the process parameters without

sacrificing productivity or MRR, such as
(a) Reducing the feed per tooth as far as possible,
(b) Taking as large a depth as admissible.
(c) Raising cutting velocity if not restricted otherwise.

3. Reduction of damage and wear at the tool tips by

(a) Reducing cutting temperature by proper cutting fluid
application.
(b) Selection of proper (chemically stable and wear-resistant)
tool material.
(c) Controlling wear by proper lubrication.
(d) Using edge (cutting) rounded tools or inserts.
(e) Preventing built-up-edge formation.

4. Control of the machining condition by

a) Preventing scratching on the machined surface due the
flowing continuous chips, by proper chip-breaking.
(b) Prevention of BUE formation and its overflow.
(c) Regular cleaning and washing away of the chips and debris
by cutting fluid.

Page | 25
 CHAPTER – 2
THEORETICAL ANALYSIS

Page | 26
 2.1 Introduction:
The current research was carried out in the following stages:

a) Inspecting and prepping the lathe in preparation for the

machining process.

b) Setting up the work piece specimen for the turning process.

c) Carry out the turning procedure.

d) Using a Profilometer, determine the roughness of the surface.

e) Determine the turning operation's surface roughness.

f) Experimental results are produced by using a combination of

process variables to create 27 regression equations. The final
equation for the system is formed by solving the set of equations
for the coefficients.

2.2 Limits and parameters of control:

The parameter working ranges for the subsequent design of
experiment have been chosen to correspond to the standard
working ranges for such an operation. Only spindle speed, feed
rate, and depth of cut were evaluated as process factors in this
experiment because these are the most relevant characteristics
impacting tool surface roughness. Table 1 lists the process
parameters, as well as their limits (and notations):

Page | 27
 Table 1: Limits and levels of control parameters

Limits
Control Low Medium High
Parameters (1) (2) (3)
Spindle speed (v)
220 380 540
(rpm)
Feed rate (f) 0.4 0.6
0.5
(mm/rev)
Depth of cut (d)
0.1 0.2 0.3
(mm)

2.3 Design of experiment:

The experiments were created using a full factorial design of
experiments approach. Design of experiments is a powerful tool
for optimizing and analyzing the effect of process variables on
some specific variable that is an unknown function of these
process variables. It is also one of the most intelligent tools for
optimizing and analyzing the effect of process variables on some
specific variable that is an unknown function of these process
variables. The process of selecting such sites in design space is
known as design of experiments (DOE). The studies in this paper
on turning Mild steel were carried out using three key influencing
process parameters, namely spindle speed, Feed Rate, and Depth
of Cut, at three different levels, namely Low, Medium, and High.
As a result, a three-level full factorial design of trials (3 3 = 27) was
created and carried out based on the parameters chosen. Table 2
shows the level designations for several process parameters.

Page | 28
 Table 2: Design of Experiment Combination Table
Factorial combination
Sl. No.
V f D
1 1 1 1
2 1 1 2
3 1 1 3
4 1 2 1
5 1 2 2
6 1 2 3
7 1 3 1
8 1 3 2
9 1 3 3
10 2 1 1
11 2 1 2
12 2 1 3
13 2 2 1
14 2 2 2
15 2 2 3
16 2 3 1
17 2 3 2
18 2 3 3
19 3 1 1
20 3 1 2
21 3 1 3
22 3 2 1
23 3 2 2
24 3 2 3
25 3 3 1
26 3 3 2
27 3 3 3

Page | 29
 2.4 Equipment used for specimen
preparation:
High Speed Precision Lathe NH 22 Manufactured by Hindusthan
Machine Tools Limited.
Model NH22

Height of centers mm 220

Distance between
mm 1000
centers

16 from 40-2040 forward, 7 from

Spindle Speed range rpm
60-1430 reverse

Spindle power kW 11

Feed range mm/

60 from 0.04-2.24
(longitudinal) rev

mm/
Feed range (cross) 60 from 0.02-1.12
rev

Tailstock sleeve
mm 200
travel

Main motor power kW 7.5 (std.) / 11.0 (opt.)

2.5 Work piece used:

Mild steel bars with a diameter of 20mm and a length of 100mm
are utilized. According to the full factorial design, 27 jobs of the
same diameter and length are taken for each factorial
combination.

Page | 30
 2.6 Surface Roughness Measurement:

FIG 5.1 Profilometer

FIG 5.2 Surface measurement using Profilometer

FIG 5.3 photographic view of machined work piece

After each turning operation surface roughness is measured

by profilometer.

Page | 31
 Machining conditions for full factorial design of experiments
and the experimental output values for surface roughness at
varying input parameters are listed in table 3.
Table 3: Experimental data

Turning parameter
Actual value
Combination Depth of Surface
EXP. Spindle Feed rate
of process cut Roughness
NO. speed(rpm) (mm/rev.)
parameters (mm) (µm)

1 1 1 1 220 0.4 0.1 8.8

2 1 1 2 220 0.4 0.2 8.83
3 1 1 3 220 0.4 0.3 8.85
4 1 2 1 220 0.5 0.1 9.03
5 1 2 2 220 0.5 0.2 9
6 1 2 3 220 0.5 0.3 9.09
7 1 3 1 220 0.6 0.1 10.4
8 1 3 2 220 0.6 0.2 10.2
9 1 3 3 220 0.6 0.3 10.9
10 2 1 1 380 0.4 0.1 12.52
11 2 1 2 380 0.4 0.2 12.8
12 2 1 3 380 0.4 0.3 12.94
13 2 2 1 380 0.5 0.1 10.28
14 2 2 2 380 0.5 0.2 9.88
15 2 2 3 380 0.5 0.3 9.36
16 2 3 1 380 0.6 0.1 12.74
17 2 3 2 380 0.6 0.2 11.52
18 2 3 3 380 0.6 0.3 11.3
19 3 1 1 540 0.4 0.1 7.5
20 3 1 2 540 0.4 0.2 7.24
21 3 1 3 540 0.4 0.3 7.6
22 3 2 1 540 0.5 0.1 6.16
23 3 2 2 540 0.5 0.2 6.23
24 3 2 3 540 0.5 0.3 7.1
25 3 3 1 540 0.6 0.1 6.75
26 3 3 2 540 0.6 0.2 6.46
27 3 3 3 540 0.6 0.3 6.9

Page | 32
 3.7 Development of Mathematical model:
The general second order polynomial mathematical model,
which analyses the parametric influences on the various
response criteria, can be described as follows:

k
k k

Surface Roughness, T = B0+∑ ❑BiXi+∑ ❑B X + i∑

, j=1
❑B X X
ii i
2
ij i j
i=1 i=1
i≠ j

……………………(1)
Where T is Response and Xi(i, j =1, 2, . . . . , k) are levels of k
quantitative variables. The coefficient B0 is the constant term; the
coefficients Bi,Bij, Bijare for the Linear, Quadratic and Interaction
terms. After putting actual values from the experiments, 27
equations are formed.
The general equation for this experiment,

T=B0+B1v+B2f+B3 d+ B11 v 2+B22f 2+B33d 2+ B12 vf+ B13 vd+ B23fd…….

(2)

Here X1=v, X2=f,X3=d

Putting the value of process parameters in general equation, the

above equations are formed.
k
k k

Surface Roughness , T = B0+∑ ❑Bi Xi +∑ ❑BiiXi + ∑ ❑BijXi Xjwhere i

2
i , j=1
i=1 i=1
i≠ j

& j are from 1 to 3

B0 , Bi , Bii , Bijare the Co-efficient and Xi (X1, X2, X3) are the
Parameter (Spindle Speed, Longitudinal Feed & Cross Feed /
Depth of Cut)

Page | 33
 T1 = 8.8 = B0 + (B1 x 220+ B2 x 0.4+ B3 x 0.1) + (B11 x 2202+ B22 x 0.42+
B33 x 0.12) + (B12 x 220 x 0.4 + B13 x 220 x 0.1+ B23 x 0.4 x 0.1)
T2 = 8.83 = B0 + (B1 x 220+ B2 x 0.4+ B3 x 0.2) + (B11 x 2202+ B22 x 0.42+
B33 x 0.22) + (B12 x 220 x 0.4 + B13 x 220 x 0.2+ B23 x 0.4 x 0.2)
T3= 8.85 = B0 + (B1 x 220+ B2 x 0.4+ B3 x 0.3) + (B11 x 2202+ B22 x 0.42+
B33 x 0.32) + (B12 x 220 x 0.4 + B13 x 220 x 0.3+ B23 x 0.4 x 0.3)
T4 = 9.03 = B0 + (B1 x 220+ B2 x 0.5+ B3 x 0.1) + (B11 x 2202+ B22 x 0.52+
B33 x 0.12) + (B12 x 220 x 0.5 + B13 x 220 x 0.1+ B23 x 0.5 x 0.1)
T5= 9 = B0 + (B1 x 220+ B2 x 0.5+ B3 x 0.2) + (B11 x 2202+ B22 x 0.52+ B33
x 0.22) + (B12 x 220 x 0.50 + B13 x 220 x 0.2+ B23 x 0.5 x 0.2)
T6= 9.09 = B0 + (B1 x 220+ B2 x 0.5+ B3 x 0.3) + (B11 x 2202+ B22 x 0.52+
B33 x 0.32) + (B12 x 220 x 0.5 + B13 x 220 x 0.3+ B23 x 0.5 x 0.3)
T7 = 10.4 = B0 + (B1 x 220+ B2 x 0.6+ B3 x 0.1) + (B11 x 2202+ B22 x 0.62+
B33 x 0.12) + (B12 x 220 x 0.6 + B13 x 220 x 0.1+ B23 x 0.6 x 0.1)
T8= 10.2 = B0 + (B1 x 220+ B2 x 0.6+ B3 x 0.2) + (B11 x 2202+ B22 x 0.62+
B33 x 0.22) + (B12 x 220 x 0.6 + B13 x 220 x 0.2+ B23 x 0.6 x 0.2)
T9= 10.9 = B0 + (B1 x 220+ B2 x 0.6+ B3 x 0.3) + (B11 x 2202+ B22 x 0.62+
B33 x 0.32) + (B12 x 220 x 0.6 + B13 x 220 x 0.3+ B23 x 0.6 x 0.3)
T10 = 12.52 = B0 + (B1 x380+ B2 x 0.4+ B3 x 0.1) + (B11 x 3802+ B22 x
0.42+ B33 x 0.12) + (B12 x 380 x 0.4 + B13 x 380 x 0.1 + B23 x 0.4 x 0.1)
T11= 12.8 = B0 + (B1 x 380+ B2 x 0.4 + B3 x 0.2) + (B11 x 3802+ B22 x
0.42+ B33 x 0.22) + (B12 x 380 x 0.4 + B13 x 380 x 0.2+ B23 x 0.4 x 0.2)
T12= 12.94 = B0 + (B1 x 380+ B2 x 0.4 + B3 x 0.3) + (B11 x 3802+ B22 x
0.42+ B33 x 0.32) + (B12 x 380 x 0.4 + B13 x 380 x 0.3+ B23 x 0.4 x 0.3)
T13 = 10.28 = B0 + (B1 x 380+ B2 x 0.5+ B3 x 0.1) + (B11 x 3802+ B22 x
0.52+ B33 x 0.12) + (B12 x 380 x 0.5 + B13 x 380 x 0.1+ B23 x 0.5 x 0.1)
T14= 9.88 = B0 + (B1 x 380+ B2 x 0.5+ B3 x 0.2) + (B11 x 3802+ B22 x 0.52+
B33 x 0.22) + (B12 x 380 x 0.5 + B13 x 380 x 0.2+ B23 x 0.5 x 0.2)

Page | 34
 T15= 9.36 = B0 + (B1 x 380+ B2 x 0.5+ B3 x 0.3) + (B11 x 3802+ B22 x 0.52+
B33 x 0.32) + (B12 x 380 x 0.5 + B13 x 380 x 0.3+ B23 x 0.5 x 0.3)
T16 = 12.74 = B0 + (B1 x 380+ B2 x 0.6+ B3 x 0.1) + (B11 x 3802+ B22 x
0.62+ B33 x 0.12) + (B12 x 380 x 0.6 + B13 x 380 x 0.1+ B23 x 0.6 x 0.1)
T17= 11.52 = B0 + (B1 x 380+ B2 x 0.6+ B3 x 0.2) + (B11 x 3802+ B22 x
0.62+ B33 x 0.22) + (B12 x 380 x 0.6 + B13 x 380 x 0.2+ B23 x 0.6 x 0.2)
T18= 11.3 = B0 + (B1 x 380+ B2 x 0.6+ B3 x 0.3) + (B11 x 3802+ B22 x 0.62+
B33 x 0.32) + (B12 x 380 x 0.6 + B13 x 380 x 0.3+ B23 x 0.6 x 0.3)
T19 = 7.5 = B0 + (B1 x540+ B2 x 0.4+ B3 x 0.1) + (B11 x 5402+ B22 x 0.42+
B33 x 0.12) + (B12 x 540 x 0.4 + B13 x 540 x 0.1+ B23 x 0.4 x 0.1)
T20= 7.24 = B0 + (B1 x 540+ B2 x 0.4+ B3 x 0.2) + (B11 x 5402+ B22 x 0.42+
B33 x 0.22) + (B12 x 540 x 0.4 + B13 x 540 x 0.2+ B23 x 0.4 x 0.2)
T21= 7.6 = B0 + (B1 x 540+ B2 x 0.4+ B3 x 0.3) + (B11 x 5402+ B22 x 0.42+
B33 x 0.32) + (B12 x 540 x 0.4 + B13 x 540 x 0.3+ B23 x 0.4 x 0.3)
T22 = 6.16 = B0 + (B1 x 540+ B2 x 0.5+ B3 x 0.1) + (B11 x 5402+ B22 x 0.52+
B33 x 0.12) + (B12 x 540 x 0.5 + B13 x 540 x 0.1+ B23 x 0.5 x 0.1)
T23= 6.23 = B0 + (B1 x 540+ B2 x 0.5+ B3 x 0.2) + (B11 x 5402+ B22 x 0.52+
B33 x 0.22) + (B12 x 540 x 0.5 + B13 x 540 x 0.2+ B23 x 0.5 x 0.2)
T24= 7.1 = B0 + (B1 x 540+ B2 x 0.5+ B3 x 0.3) + (B11 x 5402+ B22 x 0.52+
B33 x 0.32) + (B12 x 540 x 0.5 + B13 x 540 x 0.3+ B23 x 0.5 x 0.3)
T25 = 6.75 = B0 + (B1 x 540+ B2 x 0.6+ B3 x 0.1) + (B11 x 5402+ B22 x 0.62+
B33 x 0.12) + (B12 x 540 x 0.6 + B13 x 540 x 0.1+ B23 x 0.6 x 0.1)
T26= 6.46 = B0 + (B1 x 540+ B2 x 0.6+ B3 x 0.2) + (B11 x 5402+ B22 x 0.62+
B33 x 0.22) + (B12 x 540 x 0.6 + B13 x 540 x 0.2+ B23 x 0.6 x 0.2)
T27= 6.9 = B0 + (B1 x 540+ B2 x 0.6+ B3 x 0.3) + (B11 x 5402+ B22 x 0.62+
B33 x 0.32) + (B12 x 540 x 0.6 + B13 x 540 x 0.3+ B23 x 0.6 x 0.3)

Page | 35
 To solve these 27 equations and find out the value of co-efficient
the following mathematical calculations needed to be
performed. The flow chart of the calculation is given below:

START

Input Matrix X= A, O,
S, T, F, P, R, W.
Find Transpose of X, Y = XT

Find (XT)*(X) = A

Find inverse of A, C = A-1

Input Matrix T, T represents surface roughness

Find out Y*T = D

Find C*D =B

Print B

END

Page | 36
Page | 37
Page | 38
Page | 39
Page | 40