Formulation of Generalized Approximate Mathematical Model ...Cylindrical pressure vessel made of...
Transcript of Formulation of Generalized Approximate Mathematical Model ...Cylindrical pressure vessel made of...
3rd International Conference on Multidisciplinary Research & Practice P a g e | 31
Volume IV Issue I IJRSI ISSN 2321-2705
Formulation of Generalized Approximate
Mathematical Model for Cylindrical Pressure Vessel
Made of Composite Material (Glass FRP)
Hemant B Warkad1, Dr. P M Bapat
2, Dr. C N Sakhale
3
1Ph. D Scholar Department of Mechanical Engineering, Priyadarshni College of Engineering, Nagpur, India 2 Assistant Professor, Department of Mechanical Engineering, Lonavala College of Engineering Pune, India
3Assistant Professor, Department of Mechanical Engineering, Priyadarshini College of Engineering Nagpur, India
Abstract:- Among these ages, the one of current importance and
future dominance is the age of composites and Nano-Materials.
Manual layup method for FRP reinforcement is very old and
traditional method. There is no other way to make fibre and
epoxy resin and hardener coated surface on the steel tank, inside
or outside for strength and corrosion free. The same time the
detailed study of present manual hand layup winding of
Filament activity indicates that the process suffers from various
draw back like lack of accuracy which results in cracks, weak
structure and instability in surface and round cylinders, low
production rate E-Glass fibre is one of the essential elements of
reinforced Plastics with epoxy resin in aerospace, Pipe industries,
Pressure vessel and Marin Industries. These Fibre roving and
their reinforcement are used for strengthening pressure vessel,
cylinder of thick and thin structure for increase life and trouble
free maintenance. In order to remove above drawbacks and
formulate an approximate experimental data based model by
using E-Glass fibre, Epoxy resin, and hardener for Filament
winding activity. Design of experimental work is executed for
establishing, formulation of experimental mathematical model
for processing time, Density, fibre volume fraction, weight of
shell, and Ultimate tensile strength of FRP Shell by obtaining
specified result with Filament winding. Experimentation data is
chosen, using methodology of engineering experimentation for
CNC filament winding machine. This research also includes the
design, fabrication and Mass Production of Pressure vessel with
Filament winding along with theory of experimentation. It also
includes formulation of mathematical model and its sensitivity
analysis, reliability, optimization and limiting values and ANN.
Out of which process for formulation of mathematical model
established. Field Data collected from Vendors and In-house for
a prediction model was then developed to predict effect of
parameters. The basic steps used in generating the model
adopted in the development of the prediction model are:
collection of experimental data; analysis of data, pre-processing
and feature extraction of the data, design of the prediction
model, training of the model and finally testing the model to
validate the results and its ability to predict Filament winding
operation. This research work presents an experimental
investigations and sequential classical experimentation technique
used to perform experiments for various independent
parameters. An attempt is made to optimize the process
parameters for processing time, Density, fibre volume fraction,
weight of shell, and Ultimate tensile strength. The test results
proved processing time, Density, fibre volume fraction, weight of
shell, and Ultimate tensile strength are significantly influenced
by changing important five dimensionless π terms.
Keywords: Filament winding, Epoxy Resin, Experimental data
based mathematical model, Dimensional Analysis, Buckingham’s
π theorem, Reliability, regression analysis, Sensitivity, SPSS,
Optimization, ANN
1.1 INTRODUCTION
xis-symmetric thin cylindrical pressure vessel for storage
tank, Pressure Pipe, shell for missile, rockets, Launcher
tube are manufactured with filament winding Technology.
Cylindrical pressure vessel made of composite material (Glass
FRP) having wide applications. A machine already installed
with Defense manufacturing units will perform winding
operation. In this work an approximate generalized data based
model for manufacturing pressure vessel by filament winding
machine by varying some independent parameter during
experimentation. Subsequently, the reliability, optimization,
of the model is established, lastly the Artificial Neural
Network simulation of the behavioral data of the system is
also established. It is the rolled model for the future winding
operation.
Figure No. 1 Schematic line diagram for filament winding machine and
final output.
A
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1.2 OVERVIEW COMPOSITE MATERIAL [58]
“The more thing change, the more they stay the same.”
Composite seem fit this old saying because are new and to
some folks, unknown; Human have been making composites
since the first cave man wrapped stick with vine to make a
stronger handle for his stone axe. We have been combining
two more materials (making composites) ever since, in an
effort to produce a substance that unites the goods properties
of the components into one more useful materials.
Figure No. 2 basic theory of composite Reinforcement
Composite are rapidly replacing many more common
materials for structural components because they offer real
advantages :
High strength to weight ratio
Unusual flexibility and elasticity(or rigidity)
Exceptional thermal oxidative stability
Good wear characteristics
Desirable electrical conductivity(conductor,
insulator)
Easy of manufacture
For these reasons composites are showing up more frequently
than ever in aerospace, structural automotives and other
consumer products. The use of composites will continue to
increase as the design criteria for material become more
severe, more complex, and harder to satisfy with the older,
single component materials.
One of earliest of the “modern” composite material was glass
fibre reinforced plastic. The composite gained favor for use
in products. For airplanes and fishing rods to pressure and
vacuum vessels. Reinforcing fibre can be Aramid,
1.2.1 Applications of Composite material
1. LCA (Light Combat Aircraft)
Among the most significant breakthrough is the use of
advance carbon-fibre composites in 40 per cent of its
structural weight and 95 per cent of its body of the LCA,
including wings, fin and fuselage. Apart from making it much
lighter, there are less joints or rivets making the LCA more
reliable. The use of composites results in a 40 per cent
reduction in the total number of parts (if the LCA were built
using a metallic frame).
2. Brake piston insulator
The MMC material used for this has a matrix of stainless steel
with dispersion of SiO2 particles.
3. Windshield
A four ply composite laminate, consisting of two exterior cast
acrylic sheets of 3mm and 2.5 mm thickness and two exterior
polycarbonate sheets each of 6.4 mm thickness.
4. AGNI Missile System
India has successfully tested the use of composites in the
AGNI missile system. DRDO scientists have been able to
indigenously produce carbon-carbon composite material
which could withstand temperatures up to 3500 deg C during
the flight of the missile. AGNI missile system has a unique
carbon composite re-entry heat shield along with 35% of its
components made from composite materials, and this would
be about 80% coming few years. The missile is fitted with a
single Gentry vehicle employing a carbon-composite ablative
shield that Indian sources claim heats to 3000° C, while
keeping the interior cooled to not more than 40° C.
5. Advanced carbon fibre composite Rocket Motor Casing for
large rocket
motors, such as for the AGNI class systems have been
fabricated with indigenous technology. Advanced Systems
Laboratory (ASL), Hyderabad, set up in 2001, is spearheading
the development of long-range missile systems in the country
with two major programmes AGNI-I and AGNI-II inducted
and the AGNI-III under development. The front-end
technologies being developed include ultra high temperature
composites, high performance composite rocket motor
casings, radome for missiles and aircrafts, all-carbon re-entry
vehicle structure, carbon composite canister technology.
1.3 BACKGROUND OF THE PRESENT RESEARCH
“Rocket Trajectory Correction System’: S.Boguslavsky,
.A.Cherevatsky , H. Dayan, M.Shabtai, F.Olevsky enlightens
on the application like Motor Case For
Rocket Trajectory Correction System. Their work describes
the development of the composite, filament wound
glass/epoxy motor case for the Guidance Rocket Motor
(GRM). This presentation describes the development of the
composite, filament wound motor case for the GRM, as
carried out at IMI. The DTC has been chosen as a
comprehensive process from development through
production. It essentially influenced the base materials choice,
internal geometry of the case, its mechanical properties, stress
distribution under internal pressure, and failure mode.
“Composite process equipment, Glassfibre production
equipment, GRP Pipe production plant Unidirectional prepreg
production equipment,” by F.A system, Biodiesel Plant work
for the continuous filament Wound pipes are designed using
as raw materials Resin and Glass fibres. According to the
disposition of the fibres and layers, it is possible to confer
high mechanical properties for flexural, chemical resistant and
tensile strength. Aboveground and Underground Installations
are permitted. Typical Applications: High Pressure and non-
pressure pipelines, Industry, Oil and Gas sector, Cooling lines,
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Fire fighting systems, Water wells, etc. Installation with
restrained and unrestrained joints
. Crouzeix, M. Torres1,, B. Douchin, J.N. Périé,F.
Collombet, H. Hernández elaborates the idea of applications
about the winding pattern effects on the behaviour of filament
wound pipes by using full field measurements and the
equilibrium gap method. In their work, a filament-winding
pipe is tested to identify the local behaviour of the structure,
observed by using CCD cameras. An orthotropic variant of
the Equilibrium Gap Method is then proposed and applied.
The displacement field is used in order to obtain
heterogeneities map for establishing a relation between local
mechanical properties and the wound pattern.
In the view of this filament winding machine for
manufacturing cylindrical pressure vessel made up of glass
fibre reinforcement developed. Its formulation of
experimental data based model is evolved. This model is
evolved applying methodology of experimentation proposed
by H. Schenk Jr. [69]
1.4 OPERATION OF THE FILAMENT WINDING
Filament winding consists of winding resin
impregnated fibres or ravings of glass, aramid, or carbon on a
rotating mandrel in predetermined patterns. The method
makes void free product possible and gives high fibre volume
ratio up to 80%. In the wet method, the fibre picks up the low
viscosity resin either by passing through a trough or from a
metered application system. In the dry method, the
reinforcement is in the pre impregnated form. In the wet
winding process the matrix in liquid form is placed in a resin
bath and the fibres are dipped in that bath and wound. The
matrix will be in the liquid form or is brought to liquid form
by making a solution. Solids like thermoplastics can be
brought to liquid form by melting also.
Process of FRP Shell Manufacturing in Filament Winding
Machine
Preparation of mandrel and mandrel loading into
machine
Preparation of resin mix ( resin +hardener+
plasticizer +accelerator)
Wet winding with glass fibre + resin mix
Curing of FRP shell in curing oven
Machining and mandrel exraction form cured shell.
Figure No. 3 - 4 axis CNC filament winding machine,2) shell after curing
1.5 NEED FOR FORMULATING EXPERIMENTAL DATA
BASED MODEL
In this present research work of cylindrical pressure
vessel manufactured with filament winding process, it is
obvious that one will have ensure the process of filament
winding machine is new concept of composite material
manufacturing. For achiving desired result, for minimizing
pressure test failure and validating processure selecting
formulation of mathematical modeling. what should be the
bahaviour of input parameter or independent variable in
process. By collecting field data sample by classical method
through experimentation of filament winding. Fibre volume
fraction is the property to be checked for matrix quality of
composite. Processing time also important for
manufacturing the FRP shell.
Ultimate tensile strength and Density of FRP Shell
play important role for pressure vessel/rocket performance.
This would be possible if one can have a quantitative
relationship amongst various dependent and independent
variables of the system. This relationship would be known as
the mathematical model of this filament winding processing
operation. It is well known that such a model for the fiamen
winding machine operation for FRP shell cannot be
formulated applying logic The only option with which one is
left is to formulate an experimental data based model, Hilbert
Sc.(1961) [69] Hence, in this investigation it is decided to
formulate such an experimental data based model. In this
approach all the independent variable are varied over a widest
possible range, a response data is collected and an analytical
relationship is established. Once such a relationship is
established then the technique of optimization can be applied
to deduce the values of independent variables at which the
necessary responses can be minimized or maximized, [62] and
[61]. In fact determination of such values of independent
variables is always the puzzle for the operator because it is a
highly complex phenomenon of interaction of various
independent variables and dependant variables for filament
winding machine is shown in
1.6 OBJECTIVES OF RESEARCH
The objectives of present investigation are given below:
To generate design data for filament winding
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operation for manufacturing cylindrical pressure
vessel by means of winding machine, curing oven,
mandrel exractor and solid mandrel, resin mix, and
E-Glass fibre. Performing experimentation by
varying independent quantities over widest possible
range and gathering the response data generated.
Objective of the research is to design the model for
the low density, iso tropic nature, strength trouble
free and attractive result during the high pressure
involved during firing and achieving target.
The main theme involved in this work is to formulate
approximate generalized experimental data based
model for filament winding machine glass fibre FRP
cylindrical pressure vessel. Glass roving
reinforcements are most widely used in filament
winding applications.
To develop mathematical models for Cycle time of
component processing, Weight of cy. Vessel/Shell,
Ultimate tensile strength of cy. Vessel/Shell,
Density of FRP Shell, Fibre volume ratio required
for cylindrical pressure vessel by filament winding
operation made up of Glass FRP.
1.7 APPROACHES TO PROBLEM DEFINITION
In the present investigation, Independent Variable or
input for process as temperature of resin mix and
curing oven, feed rate of carriage, Modules of
elasticity of glass fibre and viscosity of the resin mix
and wet winding are required past experience.
Hence the approach of the methodology of
experimentation is adopted to generate design data
and validation of performance characteristics of this
complex phenomenon. In the performed
experimentation, the independent physical quantities
are varied over their widest possible range and
generated response data is gathered. Mathematical
models based on response data are formulated
correlating various independent physical quantities to
the responses.
The experimental data based models evolved are the
design data for various responses of filament
winding process. These models evolved are to
represent various responses of filament winding
machine.
Hence the approach of the methodology of
experimentation is adopted to generate design data and
validation of performance characteristics of this complex
phenomenon. In the performed experimentation, the
independent physical quantities are varied over their widest
possible range and generated response data is gathered.
Mathematical models based on response data are formulated
correlating various independent physical quantities to the
responses.
The experimental data based models evolved are the
design data for various responses of filament winding process.
These models evolved are to represent various responses
variables i.e. Cycle time of component processing, Weight
of cy. Vessel/Shell, Ultimate tensile strength of cy.
Vessel/Shell, Density of FRP Shell, Fibre volume ratio.
1.8 BRIEF DESCRIPTION OF APPLICATION OF
THEORY OF EXPERIMENTATION
The approach of methodology of experimentation proposed
by Hilbert Schank Jr.[69] is applied for formulating
experiment data base model for filament winding which given
below.
The basic approach included in following steps:
1. Identification of the need
2. Identification of variables (i.e dependent variables,
independent variables, and extraneous variables.
3. Reduction of independent variables by adopting
dimensional analysis
4. Test planning comprising of determination of test
envelope, test points, test sequence and
experimentation plan.
5. Physical design of an experimental set up.
6. Execution of experimentation.
7. Purification of experimentation data.
8. Formulation of the model.
9. Model optimization.
10. Reliability of the model.
11. ANN simulation of the experimental data.
Identification of variables: The identified variables are shown
in table 1.1.
Table 1.1: Variables related to FRP winding operation by
filament winding Process
S.NO
VARIABLES
SYMBOL
Unit MLT
DEPENDEN
T/ INDEPEND
ENT
1
Cycle time
of component
processing
tp second M0L0T1Ѳ0 Dependent
2 Weight of
cy.
Vessel/Shell
Ws kgf M1L0T0Ѳ0 Dependent
3
Ultimate
tensile strength of
cy.
Vessel/Shell
Es N/mm3 ML-1T-2Ѳ0 Dependent
4 Density of
FRP Shell ρs N/mm3 ML-3T0Ѳ0 Dependent
5
Fibre
volume ratio
Vf % M0L0T0Ѳ0 Dependent
6
Acceleratio
n due to gravity
g m/s2 M0L-1T-2Ѳ0 Independent
7 Dia of
mandrel ds mm M0L1T0Ѳ0 Independent
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8 Viscosity of
hardener µhr N·s/ m2 ML-1T-1Ѳ0 Independent
9 Viscosity of
araldite µar N·s/ m2 ML-1T-1Ѳ0 Independent
10 Elasticity of glass fibre
Es N/mm2 M0L-1T-2Ѳ0 Independent
11 Carriage
feed fc mm/sec M1L0T-1Ѳ0 Independent
12 Rotating mandrel
speed
ωm ω=2πN/6
0 M0L0T-1Ѳ0 Independent
13 Length of
shell Ls mm M0L1T0Ѳ0 Independent
14 Thickness
of shell ts mm M0L1T0Ѳ0 Independent
15 Total
weight of
resin mix
Wr Kg M1L0T0Ѳ0 Independent
16
Oven Curing
(Soaking)
time
tc second M0L0T1Ѳ0 Inependent
17 Temperature of curing
oven
To 00C M0L0T0Ѳ1 Independent
18 Temperature of resin
mix
Tr 00C M0L0T0Ѳ1 Independent
1.9 DESCRIPTION OF EXPERIMENTAL SETUP
The experimental set up was designed for the purpose of
carrying out the experiments to investigate and validate the
phenomenon of filament winding machine.
The experimental set up consists of the following main units:
(i) filament winding processs
It consists of machine and othe accessories attached
i) Filament winding machine
4 Axis CNC for filament winding Contain 12 nos of E-Glass
Fibre rovings are stretched from glass fire stand. Roving are
carefully maintain for separate each other up to mandrel . all
roving passes through Resin impregnating bath having V
shaped Pot with big diameter Drum for roving separation
and roving are to be dipped in resin bath. Temperature of resin
bath are maintained by for maintaining of viscidity of resin
mix.
Figure 4 Schematic view for filament winding machine
Roving are also maintain moisture fee by heating roller where
roving are passes through roller. Speed and feed are controlled
by the PLC of CNC as per programmed. Winding EYE are the
play important role for winding the roving on the mandrel.
Winding EYE are specially designed for Guiding different
winding pattern as Hoop, Helical, and Polar by repeating
movement of forward and backward for carriage movements
during filament wet winding.
ii) Mandrel and mandrel preparation
Mandrel surface are prepared with special Polishing Wax
and very thin Paper overlap on the mandrel for easy extraction
after wet winding and Curing
iii) E-glass fibre creel unit
12 No‟s of E-Glass Fibre Roving are systematically placed in
this stand all roving are separated before winding also
separate each other during winding
IV) Resin impregnation bath
Resin Impregnation bath is V shaped vessel and special
roller drum are placed over the stand for roving are placed
inside the stand for removing air bubble during wet winding
with roving.
v) Curing oven
Curing is specially designed for mandrel rotation during
curing inside the Curing oven with Heating & air circulation
Inside the chamber at 1200 C for 5- 6 Hours for gel effect of
FRP Shell
vi) Mandrel extraction unit
Mandrel extraction machine play important role for extraction
of mandrel from FRP shell after Curing by pushing and
pulling the mandrel with the help of Hydraulic Pressure Piston
cylinder. .
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Figure 5(a) actual movement of spindle, carriage and fibre EYE Figure 5(b) different winding pattern of winding and sequence
1.10 MODELS OBTAINED FOR П01, П02, П03 П04 П05
MATHEMATICAL MODELS
Mathematical modeling is a principled activity that has both
principles behind it and methods that can be successfully
applied. The principles are
over-arching or meta-principles phrased as questions about
the intentions and purposes of mathematical modeling. These
meta-principles are almost philosophical in nature.The
mathematical model is nothing but formulating correlation
between the independent pi terms and a dependent pi term in
the filament winding operation. The mathematical model is
called as generalized experimental data based model as it is
formulated on the data generated through experimentation.
1.11 MODEL FORMULATION:
The mathematical model is called as generalized
experimental data based model as it is formulated on data
generated through experimentation. A classical plan of
experimentation is used to carry out experimentation.
By using the method of dimensional analysis, the relationship
between dependent and independent variables is established
and after experimentation and testing the data , the final
relationship is developed. The mathematical models for all
FIVE dependent variables (Pi) are as shown below:
Model 1: For dependent pi term - 01 (i.e. Cycle time of
component processing, tp):
( √
) f1.
[ (
√ )
(
)
(
)
(
)
(
)
( √
)
( √
)
]
(1.6)
Where the unknown parameters of above equation 1.4 are
calculated as under:
Table 1.2: Equation parameter
K1 3.6814
a1 0.203
b1 0.2999
c1 -0.0169
d1 -0.1258
e1 -0.3761
f1 -17.7729
g1 -0.0406
( )
(√
)
[ (
√ )
(
)
(
)
(
)
(
)
( √
)
( √
)
]
(1.7)
Modal 2 : for Weight of cy. Vessel/Shell (П02)
(
)
[ (
√ )
(
)
(
)
(
)
(
)
( √
)
( √
)
]
(1.8)
Modal 3 : Ultimate tensile strength of cy. Vessel/Shell (П03)
( )
[ (
√ )
(
)
(
)
(
)
(
)
( √
)
( √
)
]
(1.9)
Modal 4 : Ultimate tensile strength of cy. Vessel/Shell (П04)
(
)
[ (
√ )
(
)
(
)
(
)
(
)
( √
)
( √
)
]
(1.10)
Modal 5 : Ultimate tensile strength of cy. Vessel/Shell (П05)
Π05 ( ) 5.457579*
[ (
√ )
(
)
(
)
(
)
(
)
( √
)
( √
)
]
( )
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1.12 CLUBBED MODELS
In this type of combined mathematical model all the
independent Pi terms i.e 1, 2, 3, 4, 5, 6, 7 are multiplied
(clubbed) together and then using regression analysis
mathematical model is formed. The mathematical clubbed
model for filament winding operation for cylindrical pressure
vessel glass FRP is form as below.
1. Cycle time of component processing Π05
(√
)
[ (
√ ) (
) (
) (
)
(
) ( √
) ( √
)]
(1.12)
2. Weight of cy. Vessel/Shell
(
)
[ (
√ ) (
) (
) (
)
(
) ( √
) ( √
)]
(1.13)
3. Ultimate tensile strength of cy. Vessel/Shell–:
( )
[ (
√ )(
) (
) (
)
(
)( √ )( √
)
]
( )
4. Density of FRP Shell
(
)
[ (
√ )(
) (
) (
)
(
)( √ )( √
)
]
( )
5. Fibre volume ratio
( )
[ (
√ ) (
) (
) (
)
(
) ( √
) ( √
)]
( )
1.13 RESPONSE SURFACE METHODOLOGY (RSM)
MODAL
As per the dimensional analysis, seven π terms are
developed. These π terms are dimensionless hence it is very
easily possible to convert into three groups. These three
groups are converted into 3 dimensions in space to develop
response surface. Hence,
, ,
(1.17)
The ranges of input X, Y and output Z are more variant.
Hence by using scaling principle, the above X, Y and Z values
are scaled as follows:
x = X / max (X), y = Y / max (Y), and z = Z / max (Z)
The selection of appropriate model and the development of
response surface models have been carried out by using
statistical software, “MATLAB R2009a”. The best fit
regression equations for the selected model are obtained for
the response characteristics, viz., processing cycle time FRP
Shell/Vessel, weight of FRP shell, Density of FRP shell,
Strength of Shell, Fibre Volume ratio of FRP Shell The
response surface equations are developed using the
experimental data
RSM Model Development
The 54 experiments are conducted, with the process parameter
levels set as given in experimental table to study the effect of
process parameters over the output parameters.
The experiments are designed and conducted by employing
response surface methodology (RSM). The selection of
appropriate model and the development of response surface
models have been carried out by using statistical software,
“MATLAB R2009a”. The best fit regression equations for the
selected model are obtained for the response characteristics,
viz., processing cycle time, Weight of shell, strength of shell,
density of shell and fibre volume ratio. The response surface
equations are developed using the experimental data and are
plotted to investigate the effect of process variables on various
response characteristics. Tables 1.3 show the sample
calculations for RSM models for processing cycle time,
Weight of Shell, Strength of Shell, Density of shell and fibre
volume ratio.
(i) For processing cycle time:
Linear model Poly55:
f(x, y) = p00 + p10*x + p01*y + p20*x2 + p11*x*y + p02*y
2
+ p30*x3 + p21*x
2*y + p12*x*y
2 + p03*y
3 + p40*x
4 +
p31*x3*y + p22*x
2*y
2 + p13*x*y
3 + p04*y
4 + p50*x
5 +
p41*x4*y + p32*x
3*y
2 + p23*x
2*y
3 + p14*x*y
4 + p05*y
5
(1.18)
For Response variable processing cycle time,
response surface equation or Polynomial equation of RSM
model of 5th
order is,
f(x, y) = -242.2 + 594.6 *x + 510.4 *y -523.9*x2 -1092 *x*y
-384.5 *y
2 + 256.8 *x
3 + 605.2 *x
2*y + 895.1 *x*y
2 + 46.45
*y3 -65.32 *x
4 -178.4 *x
3*y -271.4 *x
2*y
2 -370.8 *x*y
3 +
76.53 *y4
+ 6.77 *x5 + 20.97 *x
4*y + 35.58 *x
3*y
2+ 46.01
*x2*y
3 + 62.98 *x*y
4 -27.68 *y
5
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Goodness of fit:
(1.19)
SSE: 0.04207, R-square: 0.7296, Adjusted R-square:
0.5657, RMSE: 0.03571
(ii) For Weight of Shell :
Linear model Poly55:
f(x, y) = p00 + p10*x + p01*y + p20*x2 + p11*x*y + p02*y
2
+ p30*x3 + p21*x
2*y + p12*x*y
2 + p03*y
3 + p40*x
4 +
p31*x3*y + p22*x
2*y
2 + p13*x*y
3 + p04*y
4 + p50*x
5 +
p41*x4*y + p32*x
3*y
2 + p23*x
2*y
3 + p14*x*y
4 + p05*y
5
(1.19)
For Response variable weight of shell, response surface
equation or Polynomial equation of RSM model of 5th
order
is,
f(x, y) = -242.2 + 594.6 *x + 510.4 *y -523.9*x2 -1092 *x*y
-384.5 *y
2 + 256.8 *x
3 + 605.2 *x
2*y + 895.1 *x*y
2 + 46.45
*y3 -65.32 *x
4 -178.4 *x
3*y -271.4 *x
2*y
2 -370.8 *x*y
3 +
76.53 *y4
+ 6.77 *x5 + 20.97 *x
4*y + 35.58 *x
3*y
2+ 46.01
*x2*y
3 + 62.98 *x*y
4 -27.68 *y
5
Goodness of fit:
(1.20)
SSE: 0.004502, R-square: 0.7188, Adjusted R-square:
0.5484, RMSE: 0.01168
(iii) For Strength of Shell:
Linear model Poly55:
f(x, y) = p00 + p10*x + p01*y + p20*x2 + p11*x*y + p02*y
2
+ p30*x3 + p21*x
2*y + p12*x*y
2 + p03*y
3 + p40*x
4 +
p31*x3*y + p22*x
2*y
2 + p13*x*y
3 + p04*y
4 + p50*x
5 +
p41*x4*y + p32*x
3*y
2 + p23*x
2*y
3 + p14*x*y
4 + p05*y
5
(1.21)
For Response variable Strength of shell, response surface
equation or Polynomial equation of RSM model of 5th
order
is,
f(x,y) = 2674 -9642 *x -8600 *y + 1.518e+004 *x2
+
2.225e+004 *x*y + 1.225e+004 *y2 -1.291e+004 *x
3 -
1.291e+004 *x2*y -2.221e+004*x*y
2 -9147 *y
3 + 5595*x
4 +
1.257e+004 *x3*y + 1.281e+004 *x
2*y
2 + 1.148e+004 *x*y
3
+ 3284*y4 -957.6 *x5 -2752*x
4*y -3036*x
3*y
2 -2772*x
2*y
3 -
2458*x*y4 -395*y
5
(1.22)
Goodness of fit: SSE: 0.1064, R-square: 0.3835, Adjusted R-
square: 0.009899, RMSE: 0.05679
(iv) For Density of Shell:
Linear model Poly55:
f(x, y) = p00 + p10*x + p01*y + p20*x2 + p11*x*y + p02*y
2
+ p30*x3 + p21*x
2*y + p12*x*y
2 + p03*y
3 + p40*x
4 +
p31*x3*y + p22*x
2*y
2 + p13*x*y
3 + p04*y
4 + p50*x
5 +
p41*x4*y + p32*x
3*y
2 + p23*x
2*y
3 + p14*x*y
4 + p05*y
5
(1.23)
For Response variable Density of Shell, response surface
equation or Polynomial equation of RSM model of 5th
order
is,
f(x, y) = 704.2 -1554 *x -3122 *y + 893.4 *x2 + 6332*x*y +
5154*y2
+ 400.6 *x3 -4409 *x
2*y -8193 *x*y
2 -4117 *y
3 -
610.5 *x4 + 1096*x
3*y + 4122*x
2*y
2 + 4505*x*y
3 +
1595*y4 + 172.9 *x
5 + 14.62 *x
4*y -705.1 *x
3*y
2 -1119
*x2*y
3 -928.5 *x*y
4 -234.6 *y
5
Goodness of fit: SSE: 0.03458, R-square: 0.2646, Adjusted
R-square: -0.1811, RMSE: 0.03237
(1.24)
(v) For Fibre Volume Ratio:
Linear model Poly55:
f(x, y) = p00 + p10*x + p01*y + p20*x2 + p11*x*y + p02*y
2
+ p30*x3 + p21*x
2*y + p12*x*y
2 + p03*y
3 + p40*x
4 +
p31*x3*y + p22*x
2*y
2 + p13*x*y
3 + p04*y
4 + p50*x
5 +
p41*x4*y + p32*x
3*y
2 + p23*x
2*y
3 + p14*x*y
4 + p05*y
5
(1.25)
For Response variable Fibre volume ratio response surface
equation or Polynomial equation of RSM model of 5th
order
is, f(x,y) = 676.8 -2384 *x -2158 *y + 3381*x2 +
5996*x*y + 2799*y2 -2428 *x3 -6257 *x
2*y -5743 *x*y
2 -
1849 *y3 + 878.4 *x
4 + 2962*x
3*y + 3878*x
2*y
2 +
3878*x*y3 + 618.3 *y
4 -130.1 *x
5 -520 *x
4*y -926.1 *x
3*y
2
-789.1 *x2*y
3 -439.6 *x*y
4 -80.11 *y
5 (1.26)
Goodness of fit:
SSE: 0.002233, R-square: 0.7775, Adjusted R-square:
0.6427, RMSE: 0.008225
1.14 ANALYSIS OF PERFORMANCE OF THE MODELS
BY STATISTICAL PACKAGE FOR SOCIAL SCIENCES
(SPSS)
SPSS is one of the most popular statistical packages
which can perform highly complex data manipulation and
analysis with simple instructions .SPSS is capable of
handling large amounts of data and can perform all of the
above analyses covered in the text and much more. SPSS is
one of the most popular statistical packages which can
perform highly complex data manipulation and analysis with
simple instructions. SPSS is capable of handling large
amounts of data and can perform all of the above analyses
covered in the text and much more.
In this study descriptive statistics (arithmetic mean,
standard deviation, maximum and minimum value of
variables, etc.),data testing (Normality test, Data adequacy,
Reliability and Validity)and final analysis(Internal
consistency, factor analysis, analysis of variance ,multiple
3rd International Conference on Multidisciplinary Research & Practice P a g e | 39
Volume IV Issue I IJRSI ISSN 2321-2705
regression analysis and hypothesis testing) are carried out
through SPSS software version 20.0.
Figure 6 SPSS project workflow
i) Linear Regression: Linear regression is used to specify the
nature of the relation between two variables. Another way of
looking at it is, given the value of one variable (called the
independent variable in SPSS), how can you predict the value
of some other variable (called the dependent variable in
SPSS)
The linear regression command is found at Analyze |
Regression | Linear (this is shorthand for clicking on the
Analyze menu item at the top of the window, and then
clicking on Regression from the drop down menu, and Linear
from the pop up menu.):
ii) Descriptive Statistics: The Descriptive Statistics part of the
output gives the mean, standard deviation, and observation
count (N) for each of the dependent and independent
variables.
iii) Correlations: The Correlations part of the output shows
the correlation coefficients. This output is organized
differently than the output from the correlation procedure. The
first row gives the correlations between the independent and
dependent variables
iv) Variables Entered/Removed: The Variables
Entered/Removed part of the output simply states which
independent variables are part of the equation (extravert in
this example) and what the dependent variable .
v) Model Summary: The Model Summary part of the output is
most useful when you are performing multiple regression
(which we are NOT doing.) Capital R is the multiple
correlation coefficient that tells us how strongly the multiple
independent variables are related to the dependent variable. In
the simple bivariate case (what we are doing) R = | r |
(multiple correlation equals the absolute value of the bivariate
correlation.) R square is useful as it gives us the coefficient of
determination.
vi) ANOVA: The ANOVA part of the output is not very useful
for our purposes. It basically tells us whether the regression
equation is explaining a statistically significant portion of the
variability in the dependent variable from variability in the
independent variables.
vii) Coefficients part: The Coefficients part of the output
gives us the values that we need in order to write the
regression equation. The regression equation will take the
form:
Predicted variable (dependent variable) = slope * independent
variable + intercept
The slope is how steep the line regression line is. A slope of 0
is a horizontal line, a slope of 1 is a diagonal line from the
lower left to the upper right, and a vertical line has an infinite
slope. The intercept is where the regression line strikes the Y
axis when the independent variable has a value of 0.
Developing the SPSS model individual Pi terms Here seven
independent pi terms ( i.e. 1, 2, 3, 4, 5, 6, 7 ) and five
dependent pi terms (01, 02, 03, 04, 05) have been identified
in the design of experimentation and are available for the
model formulation.
Independent terms = (1, 2, 3, 4, 5, 6, 7 )
Dependent terms = (01, 02, 03, 04, 05)
Each dependent is assumed to be function of the available
independent terms
By using the SPSS software version 20.0, the linear regression
carried out, linear regression is used to specify the nature of
the relation between two variables. Another way of looking at
it is, given the value of one variable (called the independent
variable in SPSS), how can you predict the value of some
other variable (called the dependent variable in SPSS).
The models developed using SPSS are as follows:
Y Pi 01 = 1.000 + 0. 190 π1 + 0. .313 π2 - 0.116 π3 – 0.355 π5 -
0.127 π6 (1.27)
Y Pi 02 = 0.981+0 .058π1 + 0.103π2 - 0.032 π3 – 0.155 π5 -
0.031 π7 (1.28)
Y Pi 03 = 1.449 +0 .529 π1 - 0.055 π2 - 0.600 π3 – 0.218 π5 -
0.473 π7 (1.29)
Y Pi 04 = 1.094 - 0 .065 π1 - 0.023 π2 - 0.186 π3 + 0.138 π5 -
0.106 π7 (1.30)
Y Pi 05 = 1.322+ 0.068 π1 + 0.031 π2 - 0.237π3 – 0.015 π5 -
0.275 π7 (1.31)
Model summary shows the value of R, R square,
Adjusted R Square, Std. Error of linear regression by using
SPSS software is found too much better than remaining
model.
Table 1.3 Computed values of R, R square, Adjusted R
Square, Std. Error of linear regression by using SPSS for all
response variables
3rd International Conference on Multidisciplinary Research & Practice P a g e | 40
Volume IV Issue I IJRSI ISSN 2321-2705
Model R R
Square
Adjusted R
Square
Std. Error
of the
Estimate
Processing
cycle time-01 .703a .495 .442 .04047
Weight of
Shell -02 .714a .509 .458 .01279
Strength of
Shell - 03 .426a .182 .096 .05426
Density of
Shell - 04 .334a .112 .019 .02950
Fibre
Volume
Ratio - 05
.935a .874 .861 .00513
1.15 ANN SIMULATION
The phenomenon of Filament winding Process is
highly complex and nonlinear , so it is planned to develop the
artificial neural network.It is utmost importance to compare
the data generated throught mathematical models,
experimentally observed data and ANN data to validate the
phenomenon.
.
Figure 7 ANN Simulation flow diagrams
An artificial neural network (ANN) consists of three
layers i.e. the input layer, hidden layer and the output layer.
Its node represents neurons of the brain. The three neurons are
interconnected with nodes as like that of neurons in the brain.
The specific mapping performance depends upon the
architecture and a synaptic weight values between the neurons
of an ANN network. ANN is developed on a concept like a
black box. ANN is trained itself with the help of input and
output data as if like a human brain learn with reception of
similar stimuli. An ANN trains itself within input and output
data usually operating without a prior theory that guide or
restricts a relation between the outputs and inputs. Ultimately
the accuracy of predicted output rather than a specific
description of paths and the relation between the input and
output are the eventual goal of ANN model. The input data is
preprocessed and passed hidden layer nodes.
Figure 8 ANN Simulation Figure 1.14 ANN Graph for processing cycle time
meanexp =254.8148, meanann =258.1880, meanmath =
254.4477
mean_absolute_error_performance_function =7.6833
mean_squared_error_performance_function = 107.1095
perf = 4.4913e+03
Table 1.4 : Comparison of the values of dependent pi terms
computed by experimentation, mathematical model and ANN
Mean /Error
Filament winding operation for Cylindrical Pressure
Vessel made of Composite material (Glass FRP)
tp-Π01 Ws-Π02 Es-Π03 ρs-Π04 Vf-Π05
Mean
experimental
254.814
8 6.3661
311.530
7
1.8820e
+09 62.1306
Mean ANN 258.1880
6.4053 314.769
7 1.8917e
+09 62.1404
Mean Math.
(model)
254.447
7 6.3657
310.723
6
1.8813e
+09 62.1448
MAEPF-
mean_absolu7.6833 0.0662 17.4730 4.6614e 0.1410
0 5 10 15 20 25 30 350
1
2
3
4
5
6
Error
Training
validation
Testing
0 10 20 30 40 50 60230
240
250
260
270
280
290
Experimental
Comparision between practical data, equation based data and neural based data
Practical
Equation
Neural
3rd International Conference on Multidisciplinary Research & Practice P a g e | 41
Volume IV Issue I IJRSI ISSN 2321-2705
te_error_perf
ormance_fun
ction
+07
MSEPF-
mean_square
d_error_perf
ormance_fun
ction
107.1095
0.0070 460.987
4 3.4125e
+15 0.0342
1.16 RELIABILITY AND CO-EFFICIENT OF
DETERMINATION (R2) FOR OF THE MODELS
1.16.1 Reliability: The Reliability of the model can be
established by using the following relation.
Reliability (%) = 100 – percentage mean error
Reliability (%) = ∑( )
∑
(1.32)
Where Mean error = ∑( )
∑
(1.33)
is percentage (%) error and is frequency of occurrence of
the error.
Table 1.5: Comparison of values for Reliability
Reliability compared with Experimental Data
π
terms
Response
Variable
Ri betn
Math and
EXP
Ri
betn
club
and
Exp
Ri betn
SPSS and
Exp
Ri
betn
ANN
and
Exp
Π01 Processing
cycle
Time
97.3518519 95.389 97.2962963 94
Π02 Weight of
shell 99.5 96.315 99.46296296 98.907
Π03 Strength
of Shell 99.5 96.315 95.40740741 93.87
Π04 Density of
Shell 98.1666667 94.074 98.11111111 97.944
Π05 Fibre
Volume
Ratio
99.8518519 99.288 99.96296296 99.375
1.16.2 Co-efficient of Determinant (R2) : A statistical method
that explains how much of the variability of a factor can be
caused or explained by its relationship to another factor .Co-
efficient of Determinant is used in trend analysis. It is
computed as a value between 0 (0 percent) and 1 (100
percent). The Higher the value, the better the fit. Coefficient
of determination is symbolized by r2 because it is square of
the coefficient of correlation symbolized by r.
R2 =1- ∑Yi-fi)
2/∑(Yi-Y)
2
(1.34)
Where, Yi= Observed value of dependant variable for ith
Experimental sets (Experimental data), fi=Observed value of
dependant variable for ith
predicted value sets (Model data),
Y= Mean of Yi and R2 = Co-efficient of Determinant
Table 1.6: Comparison of values for coefficient of
determination
R2 compared with Experimental Data
π
term
s
Response
Variable
R2 betn
Math and
EXP
R2
betn
club
and
Exp
R2
betn
SPSS
and
Exp
R2
betn
ANN
and
Exp
R2 of
SPSS
Mode
l
Π01 Processing
cycle Time 0.504527 0.0563
0.4948
-0.7514
0.495
Π02 Weight of
shell
0.51647537
1 -6.299
0.509
4
-
0.2012 0.509
Π03 Strength of
Shell
0.158728032
-0.006 0.181
6 -
0.3658 0.182
Π04 Density of
Shell
0.09751887
5
-
4.6643
0.111
5 0.0202 0.112
Π05
Fibre
Volume
Ratio
0.78717105
9
-
0.4004
0.873
9 0.9548 0.874
1.17 OPTIMIZATION OF THE MODELS
The models have been developed for the
phenomenon of filament winding process. The ultimate
objective of this work is not merely developing the models but
to find out the best set of independent variables, which will
result in maximization/minimization of the objective
functions. There are three models of dependent variables and
related to these models there are five objective functions as
below.
Table 1.7: Objective Function
Variable Model Objective Function
tp Cycle time of component processing
of cy. Vessel/Shell Minimization
Ws Weight of cy. Vessel/Shell Minimization
Es Ultimate tensile strength Maximization
ρs Density of FRP Shell Minimization
Vf Fibre volume ratio Maximization
Table 1.8: Optimized values of response variables
3rd International Conference on Multidisciplinary Research & Practice P a g e | 42
Volume IV Issue I IJRSI ISSN 2321-2705
Pi
Ter
ms
Π01-Processing
Cycle Time (tp),sec
(Min)
Π02- Weight of
shell, Ws, kg (Min)
Π03- Strength of
Shell, Es , N/mm2
(Max)
Log
values
of π
terms
Antilog
of π
terms
Log
values
of π
terms
Antilo
g of π
terms
Log
values
of π
terms
Antilog
of π
terms
Z 2.3675 233.06 0.79353 6.2163 2.5838 383.555
X1 -1.8027 0.01575 -1.8027 0.0157 -1.773 0.01687
X2 -2.8908 0.00129 -2.8908 0.0013 -2.891 0.00129
X3 0.3827 2.41379 0.20761 1.6129 0.2076 1.6129
X4 -3.3019 0.0005 -3.3019 0.0005 -3.302 0.0005
X5 8.7855 6.1E+08 8.78547 6E+08 8.7342 5.4E+08
X6 -0.1987 0.63286 -0.1987 0.6329 -0.199 0.63286
X7 4.8544 71511.6 4.85438 71512 4.6783 47674.4
Table 1.8: Equation parameter
Pi
Ter
ms
Π04- Density of Shell, ρs ,
Kg/mm3 (Min)
Π05- Fibre Volume Ratio, Vf ,
% (Max)
Log values of
π terms
Antilog of π
terms
Log values of
π terms
Antilog of π
terms
Z 9.26116 1.82E+09 1.82429 66.72446
X1 -1.7728 0.016874 -1.7728 0.016874
X2 -2.8908 0.001286 -2.7447 0.0018
X3 0.3827 2.413793 0.20761 1.612903
X4 -3.3019 0.000499 -3.3019 0.000499
X5 8.73416 5.42E+08 8.73416 5.42E+08
X6 -0.1987 0.632864 -0.1987 0.632864
X7 4.85438 71511.64 4.67829 47674.43
1.18 SENSITIVITY ANALYSES
The influence of the various independent π terms has
been studied by analyzing the indices of the various π terms in
the models. Through the technique of sensitivity analysis, the
change in the value of a dependent π term caused due to an
introduced change in the value of individual π term is
evaluated. In this case, of change of ± 10 % is introduced in
the individual independent π term independently (one at a
time).Thus, total range of the introduced change is ± 20 %.
The effect of this introduced change on the change in the
value of the dependent π term is evaluated .The average
values of the change in the dependent π term due to the
introduced change of ± 10 % in each independent π term. This
defines sensitivity. The total % change in output for ±10%
change in input is shown in Table 1.9
Table 1.9 : Sensitivity Analysis (sample) for filament winding
Independent Pi Terms (Varied by ±10%) % of Change effect on Dependent Pi Terms
Pi 1 Pi 2 Pi 3 Pi 4 Pi 5 Pi 6 Pi 7 Pi01 Pi02 Pi03 Pi04 Pi05
0.016
3
0.001
5
2.001
5
0.000
5 6E+08 0.6329 59593 254.82 6.3701 307.78 2E+09 61.934
0.017
9
0.001
5
2.001
5
0.000
5 6E+08 0.6329 59593 259.8 6.406 327.24 2E+09 62.336
0.014
7
0.001
5
2.001
5
0.000
5 6E+08 0.6329 59593 249.43 6.3307 287.61 2E+09 61.493
% Change 4.0697 1.1816 12.876 1.1984 1.3621
0.016
3 0.001
5
2.001
5
0.000
5 6E+08 0.6329 59593 254.82 6.3701 307.78 2E+09 61.934
0.016
3
0.001
7
2.001
5
0.000
5 6E+08 0.6329 59593 262.21 6.4259 306.14 2E+09 62.098
0.016
3
0.001
4
2.001
5
0.000
5 6E+08 0.6329 59593 246.9 6.309 309.59 2E+09 61.755
% Change 6.01 1.8353 1.1221 0.4133 0.5538
0.016
3
0.001
5 2.001
5
0.000
5 6E+08 0.6329 59593 254.82 6.3701 307.78 2E+09 61.934
0.016
3
0.001
5 2.201
6
0.000
5 6E+08 0.6329 59593 254.41 6.3701 295.24 2E+09 61.178
0.016
3
0.001
5 1.801
3
0.000
5 6E+08 0.6329 59593 255.28 6.3701 322.25 2E+09 62.782
% Change 0.3392 0 8.7773 1.2907 2.5904
0.016
3
0.001
5
2.001
5 0.000
5 6E+08 0.6329 59593 254.82 6.3701 307.78 2E+09 61.934
0.016
3
0.001
5
2.001
5 0.000
5 6E+08 0.6329 59593 251.78 6.016 305.8 2E+09 59.409
0.016
3
0.001
5
2.001
5 0.000
4 6E+08 0.6329 59593 258.22 6.7859 309.98 2E+09 64.851
% Change 2.5261 12.086 1.357 29.104 8.7854
0.016
3
0.001
5
2.001
5
0.000
5 6E+08 0.6329 59593 254.82 6.3701 307.78 2E+09 61.934
0.016
3
0.001
5
2.001
5
0.000
5 6E+08 0.6329 59593 245.85 6.3009 300.19 2E+09 61.802
0.016
3
0.001
5
2.001
5
0.000
5 5E+08 0.6329 59593 265.12 6.4476 316.38 2E+09 62.081
% Change 7.5633 2.3031 5.2591 2.8155 0.4516
0.016
3
0.001
5
2.001
5
0.000
5 6E+08 0.6329 59593 254.82 6.3701 307.78 2E+09 61.934
0.016
3
0.001
5
2.001
5
0.000
5 6E+08 0.6962 59593 46.835 31.804 150.61 3E+06 40.786
0.016
3
0.001
5
2.001
5
0.000
5 6E+08 0.5696 59593 1657.6 1.0769 678.16 2E+12 98.284
% Change 632.11 482.37 171.41 126498 92.837
0.016
3
0.001
5
2.001
5
0.000
5 6E+08 0.6329 59593 254.82 6.3701 307.78 2E+09 61.934
0.016
3
0.001
5
2.001
5
0.000
5 6E+08 0.6329 65552 253.84 6.3683 298.1 2E+09 60.926
0.016
3
0.001
5
2.001
5
0.000
5 6E+08 0.6329 53634 255.91 6.3721 318.84 2E+09 63.069
% Change 0.8149 0.0602 6.7371 0.1806 3.4607
1.19 ESTIMATION OF LIMITING VALUES OF
RESPONSE VARIABLES
The mathematical models have been developed for
the phenomenon. The ultimate objective of this work is not
merely developing the models but to find out the best set of
variables, which will result in maximization/minimization of
the response variables.
In this section attempt is made to find out the limiting values
of four response variables viz.processing cycle time of
vessel/shell, weight of shell, Density of shell, Ultimate
Strength of FRP shell, Fibre volume ratio of FRP shell To
achieve this, limiting values of independent π term viz. π1, π2,
π3, π4, π5, π6, π7, are put in the respective models. In the
process of maximization, maximum value of independent π
term is substituted in the model if the index of the term was
3rd International Conference on Multidisciplinary Research & Practice P a g e | 43
Volume IV Issue I IJRSI ISSN 2321-2705
positive and minimum value is put if the index of the term
was negative. The limiting values of these response variables
are compute for filament winding operation
Table 1.10 Limiting Values of Response Variables
Max
and
Min.
of
Resp
onse
π
terms
Filament winding operation for Cylindrical Pressure Vessel
made of Composite material (Glass FRP)
Π01-Processing
Cycle Time (tp),sec
Π02-
Weight
of shell,
Ws, kg
Π03-
Strengt
h of
Shell,
Es ,
N/mm2
Π04-
Density
of
Shell,
ρs ,
Kg/mm3
Π05-
Fibre
Volum
e
Ratio,
Vf , %
Maxi
mum 279.7559
6.53247
9
383.555
2
383.555
2
66.724
46
Mini
mum 233.0597
6.21625
8 255.597 255.597
58.094
58
1.20 COMPARISON OF PHENOMENAL RESPONSES BY
CONVENTIONAL APPROACH AND ANN SIMULATION
The comparison between Experimental, ANN and by
Model for all the three response/dependent variables viz.
processing cycle time FRP Shell/Vesse(П01),, weight of FRP
shell(П02),, Density of FRP shell(П03),, Strength of
Shell(П04),, Fibre Volume ratio of FRP Shell(П05), is made
as shown in the following sections. The figures 9, 10, 11 and
Table 1.11 depicts that the comparison made by Experimental,
ANN and mathematical Models gives the response data which
is overlapping. The overlapping curves are due to less
percentage error between Experimental, ANN and
mathematical Models. This proves the authenticity of the
responses predicted.
From the comparisons of all models for response
variables shown that, In the previous research work all the
researcher have worked on only comparison between the
mathematical model , clubbed model and ANN model values,
they have formulated the model and compare its values. No
one has work on the analysis of performance of the models by
statistical package for social sciences (SPSS), Response
Surface Methodology and comparion of all these models.
Here in this present research, the model are formulated for all
the response variable and Comparison between the models on
the basis of Mean value, Mean Error, Percentage Error, Root
Mean Square Error (RMSE), Mean Square Error (MSE),
Coefficient of Determination (R2) and Reliability (Ri) of the
Model with experimental data was done.
Here in the present research work, the models are
formulated for All five response variables and the results are
tested for sensitivity, reliability, and values of coefficient of
determinant (R2 ) is calculated for Response variable and for
all the developed models of Response Variables., RSM, SPSS
and ANN, and compared it.
Table 1.11 : Comparison of the values of dependent pi terms
computed by experimentation, mathematical model and ANN
Mean /Error
Filament winding operation for
Cylindrical Pressure Vessel made of
Composite material (Glass FRP)
tp-Π01 Ws-
Π02
Es-
Π03 ρs-Π04
Vf-
Π05
Mean experimental 254.8148
6.3661
311.5307
1.8820e+09
62.1306
Mean ANN 258.1
880
6.40
53
314.7
697
1.8917
e+09
62.1
404
Mean Math. (model) 254.4477
6.3657
310.7236
1.8813e+09
62.1448
MAEPF-
mean_absolute_error_perfor
mance_function
7.6833
0.0662
17.4730
4.6614e+07
0.1410
MSEPF-
mean_squared_error_perfor
mance_function
107.1095
0.0070
460.9874
3.4125e+15
0.0342
Sample calculation of mean value, mean error, percentage
error ,square error and root mean square error Model
Developed By Mathematical Model, Clubbed Model, SPSS
Regression Model And ANN for are shown in tables 1.12 to
table 1.23.
Table 1.12: Sample values of dependent pi term computed by
experimentation, math. model, clubbed model, RMS modal,
Linear regression using SPSS and ANN for Processing Cycle
Time, tp-Π01
S.N. tp
(Expe)
tp
(Math.
Model)
tp
(Clubbed)
tp(Linear
regre.
Model
using
SPSS)
tp (ANN)
1 240 243.5198 255.0799 243.8268 258.8476
2 240 240.1266 255.2294 240.1364 267.2203
: : : : : :
53 260 270.1922 257.1698 270.662 266.7394
54 270 273.1135 256.7677 273.8876 265.1101
Sum 13760 13745.13 13835.67 13760 13942.15
AVG 254.8148 254.5394 256.2161 254.8149 258.188
Table 1.13: calculation for Mean Error between
experimentation, mathematical model, clubbed model, Linear
regression using SPSS and ANN for Processing Cycle Time,
tp-Π01
S.N.
Error betn
Math and
EXP
Error betn
club and Exp
Error betn
SPSS and Exp
Error betn
ANN and
Exp
1 3.519764 15.07986 3.8268 18.8476
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2 0.126593 15.22941 0.1364 27.2203
3 5.099275 14.68052 5.1484 22.8643
4 4.720839 6.287875 3.9628 22.2212
5 7.341966 3.653179 8.3024 3.6334
: : : : :
: : : : :
51 10.48276 8.753823 9.8052 3.5567
52 14.67048 22.48098 13.4316 19.7586
53 10.19216 2.830172 10.662 6.7394
54 3.113485 13.23229 3.8876 4.8899
AVG 0.275372 1.401324 6.67E-05 3.373183
Table: 1.14 : Sample calculation for percentage error between
experimentation, mathematical model, clubbed model, Linear
regression using SPSS and ANN for Processing Cycle Time,
tp-Π01
S.N. % Error
betn Math
and Exp
% Error
betn club
and Exp
% Error
betn SPSS
and Exp
% Error
betn ANN
and Exp
1 1.466568 6.283276 1.5945 7.853167
2 0.052747 6.345588 0.056833 11.34179
3 2.124698 6.116883 2.145167 9.526792
4 1.888336 2.51515 1.58512 8.88848
5 2.823833 1.405069 3.193231 1.397462
: : : : :
: : : : :
51 3.95576 3.303329 3.700075 1.342151
52 5.239458 8.02892 4.797 7.056643
53 3.92006 1.088528 4.100769 2.592077
54 1.153143 4.900847 1.439852 1.811074
Avg 0.108068 0.549938 2.62E-05 1.323778
Table: 1.15 : Sample calculation for Mean Square error and
Root Mean Square Error between experimentation,
mathematical model, clubbed model, Linear regression using
SPSS and ANN for Processing Cycle Time, tp-Π01
S.N.
Mean Square
Error betn
Math and
EXP
Mean Square
Error betn
club and Exp
Mean
Square
Error betn
SPSS and
Exp
Mean
Square
Error betn
ANN and
Exp
1 12.38874 227.4022 14.6444 355.232
2 0.016026 231.9349 0.018605 740.9447
3 26.0026 215.5176 26.50602 522.7762
4 22.28633 39.53737 15.70378 493.7817
5 53.90446 13.34572 68.92985 13.2016
: : : : :
: : : : :
51 109.8883 76.62941 96.14195 12.65011
52 215.223 505.3942 180.4079 390.4023
53 103.88 8.009873 113.6782 45.41951
54 9.693787 175.0934 15.11343 23.91112
MSE MSE MSE MSE
MSE 111.9232 213.1833 114.1152 395.6324
RMSE RMSE RMSE RMSE
RMSE 10.57938 14.6008 10.68247 19.89051
Figure 9 Graph comparisons of Values of Model Developed By Math. Model,
Clubbed Model, RSM Model, SPSS Regression Model and ANN for all
Dependant Pi Terms for Processing Cycle Time, tp-Π01
Table 1.16: Sample values of dependent pi term computed by
experimentation, math. model, clubbed model, RMS modal,
Linear regression using SPSS and ANN for Weight of Shell,
Ws-Π02
S.N. Ws
(Expem)
Ws(Math
Model)
Ws
(club
Model)
Ws(Linear
regre. Model
using spss) Ws(ANN)
1 6.25 6.284287 6.731771 6.282607 6.5394
2 6.255 6.262278 6.69606 6.260331 6.504
3 6.26 6.288516 6.82817 6.286639 6.4451
4 6.3 6.371113 6.449242 6.361332 6.4027
5 6.41 6.360234 6.435796 6.350029 6.2876
: : : : : :
: : : : : :
51 6.45 6.357784 6.458772 6.360803 6.45
52 6.6 6.447374 6.17478 6.455458 6.6
53 6.4 6.488182 6.251286 6.493267 6.4
0
50
100
150
200
250
300
1 4 7 101316192225283134374043464952
tp(Expem)
tp(Math Model)
tp(club Model)
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54 6.5 6.496036 6.340692 6.505165 6.5
Sum 343.767 343.7437 349.3442 343.7671 343.767
AVG 6.366056 6.365624 6.469337 6.366058 6.366056
Table 1.17: calculation for Mean Error between
experimentation, mathematical model, clubbed model, Linear
regression using SPSS and ANN for Weight of Shell, Ws-Π02
S.N. Error betn
Math and
EXP
Error betn
club and Exp
Error betn
SPSS and
Exp
Error betn
ANN and
Exp
1 0.034287 0.481771 0.032607 0.2894
2 0.007278 0.44106 0.005331 0.249
3 0.028516 0.56817 0.026639 0.1851
4 0.071113 0.149242 0.061332 0.1027
5 0.049766 0.025796 0.059971 0.1224
: : : : :
: : : : :
51 0.092216 0.008772 0.089197 0.0571
52 0.152626 0.42522 0.144542 0.2378
53 0.088182 0.148714 0.093267 0.0672
54 0.003964 0.159308 0.005165 0.0207
AVG 0.000431 0.103282 2.39E-06 0.03922
Table: 1.18 : Sample calculation for percentage error between
experimentation, mathematical model, clubbed model, Linear
regression using SPSS and ANN for Weight of Shell, Ws-Π02
S.N. % Error
betn Math
and Exp
% Error
betn club
and Exp
% Error
betn SPSS
and Exp
% Error betn
ANN and Exp
1 0.548598 7.708331 0.521707 4.6304
2 0.116349 7.051323 0.085228 3.980815
3 0.455532 9.07619 0.42554 2.956869
4 1.128776 2.368917 0.973521 1.630159
5 0.77638 0.40244 0.93559 1.909516
: : : : :
: : : : :
51 1.429701 0.135996 1.382899 0.885271
52 2.312509 6.442733 2.190027 3.60303
53 1.377843 2.323663 1.457303 1.05
54 0.060979 2.450894 0.079468 0.318462
Avg 0.006777 1.622382 3.76E-05 0.616086
Table: 1.19 : Sample calculation for Mean Square error and
Root Mean Square Error between experimentation,
mathematical model, clubbed model, Linear regression using
SPSS and ANN for Weight of Shell, Ws-Π02
S.N.
Mean Square
Error
betnMath and
EXP
Mean Square
Error
betnclub and
Exp
Mean
Square
Error betn
SPSS and
Exp
Mean
Square
Error betn
ANN and
Exp
1 0.001176 0.232103 0.001063 0.083752
2 5.3E-05 0.194534 2.84E-05 0.062001
3 0.000813 0.322817 0.00071 0.034262
4 0.005057 0.022273 0.003762 0.010547
5 0.002477 0.000665 0.003597 0.014982
: : : : :
: : : : :
51 0.008504 7.69E-05 0.007956 0.00326
52 0.023295 0.180812 0.020892 0.056549
53 0.007776 0.022116 0.008699 0.004516
54 1.57E-05 0.025379 2.67E-05 0.000428
MSE MSE MSE MSE
MSE 0.006263 0.094547 0.006356 0.01556
RMSE RMSE RMSE RMSE
RMSE 0.079141 0.307486 0.079722 0.12474
Figure .10 Graph comparisons of Values of Model Developed By Math. Model, Clubbed Model, RSM Model, SPSS Regression Model and ANN for
all Dependant Pi Terms for Weight of Shell, Ws-Π02
Table 1.20: Sample values of dependent pi term computed by
experimentation, math. model, clubbed model, RMS modal,
Linear regression using SPSS and ANN for Strength of Shell,
Es-Π03
S.N.
Es(Expem)
Es(Math
Model)
Es(club
Model)
Es(Linear
regre. Model
using spss) Es(ANN)
1 292.04 303.1866 318.052 304.8718 330.7855
2 307.25 306.7528 317.508 305.477 305.0063
3 306.23 305.4772 319.5108 302.2674 316.3206
5.6
5.8
6
6.2
6.4
6.6
6.8
7
1 4 7 10 13 16 19 22 25 28 31 34 37 40 43 46 49 52
Ws (Expem)Ws(Math Model)Ws (club Model)Ws(Linear regre. Model using spss)
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4 277.15 298.0918 313.6933 299.8852 316.6794
5 287.22 302.8389 313.4826 301.6929 317.4592
: : : : : :
: : : : : :
51 291.32 318.2458 313.8424 320.8085 306.9417
52 333.65 313.5504 309.3332 318.7096 313.0291
53 323.24 324.425 310.5615 324.0027 293.8266
54 323.23 320.8931 311.9842 323.0222 295.4306
Sum 16822.66 16778.6 16954.51 16822.66 16997.57
AVG 311.5307 310.7149 313.9723 311.5308 314.7697
Table 1.21: calculation for Mean Error between
experimentation, mathematical model, clubbed model, Linear
regression using SPSS and ANN for Strength of Shell, Es-Π03
S.N.
Error betn
Math and
EXP
Error betn
club and
Exp
Error betn
SPSS and
Exp
Error
betn ANN
and Exp
1 11.14656 26.01203 12.83183 38.7455
2 0.497193 10.25804 1.77303 2.2437
3 0.75276 13.28081 3.96257 10.0906
4 20.94177 36.54328 22.73517 39.5294
5 15.61892 26.26265 14.47293 30.2392
: : : : :
: : : : :
51 26.92583 22.52239 29.48846 15.6217
52 20.09961 24.31684 14.94038 20.6209
53 1.185034 12.67847 0.76268 29.4134
54 2.336897 11.24582 0.2078 27.7994
AVG 0.815838 2.441578 4.07E-05 3.239
Table: 1.22 : Sample calculation for percentage error between
experimentation, mathematical model, clubbed model, Linear
regression using SPSS and ANN for Strength of Shell, Es-Π03
S.N. % Error
betn Math
and Exp
% Error
betn club
and Exp
% Error
betn SPSS
and Exp
% Error
betn ANN
and Exp
1 3.816793 8.90701 4.39386 13.26719
2 0.16182 3.338662 0.577064 0.730252
3 0.245815 4.336875 1.293985 3.295105
4 7.556113 13.18538 8.2032 14.26282
5 5.437965 9.143739 5.03897 10.52824
: : : : :
: : : : :
51 9.242698 7.731152 10.12236 5.362385
52 6.024161 7.28813 4.47786 6.180399
53 0.366611 3.922309 0.235949 9.099555
54 0.722983 3.4792 0.064289 8.600501
Avg 0.26188 0.783736 1.31E-05 1.039705
Table: 1.23 : Sample calculation for Mean Square error and
Root Mean Square Error between experimentation,
mathematical model, clubbed model, Linear regression using
SPSS and ANN for Strength of Shell, Es-Π03
S.N.
Mean Square
Error betn
Math and
EXP
Mean Square
Error betn
club and Exp
Mean
Square
Error betn
SPSS and
Exp
Mean
Square
Error betn
ANN and
Exp
1 124.2459 676.6258 164.6559 1501.214
2 0.247201 105.2274 3.143635 5.03419
3 0.566648 176.38 15.70196 101.8202
4 438.5577 1335.411 516.888 1562.573
5 243.9508 689.7267 209.4657 914.4092
: : : : :
: : : : :
51 725.0003 507.2582 869.5693 244.0375
52 403.9945 591.3089 223.215 425.2215
53 1.404304 160.7436 0.581681 865.1481
54 5.461089 126.4685 0.043181 772.8066
MSE MSE MSE MSE
MSE 394.5648 471.8314 383.8403 640.564
RMSE RMSE RMSE RMSE
RMSE 19.86365 21.72168 19.59184 25.30937
Figure 11Graph comparisons of Values of Model Developed By Math.
Model, Clubbed Model, RSM Model, SPSS Regression Model and ANN for all Dependant Pi Terms for Fibre Volume Ratio, Vf-Π05
0
100
200
300
400
500
1 4 7 101316192225283134374043464952
Es(Expem)
Es(Math Model)
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Volume IV Issue I IJRSI ISSN 2321-2705
Comparison of developed model of mathematical,
clubbed ,RSM, SPSS and ANN are compared with
experimental values of experimentation are shown in table
1.24.The developed models for all response variables are
shown in table 1.25. From their comparison it is found that
mathematical model is most superior and SPSS models are
superior than other models. As its values are closer with less
percentage of error. It is also concluded on the basis of
reliability, coefficient of determination and statically
comparison.
Table 1.24 Mathematical Model Equation for Dependent
variables
Response
Variable Model Mathematical Equation (Model)
01,
(Processin
g Cycle
Time, tp)
Math.
Model
(√ )
[ (
√ )
( )
( )
(
)
(
)
( √ )
( √
)
]
Clubbed
Model (√
)
[ (
√ )(
) ( ) (
)
(
)( √ )( √
)
]
RSM
f(x,y) = -6521 + 2.481e+004 *x + 1.983e+004 *y -3.586e+004*x2 -
6.336e+004*x*y -2.269e+004*y2 + 2.619e+004 *x3 + 6.754e+004 *x2*y
+ 6.135e+004 *x*y2 + 1.09e+004 *y3 -9691*x4 -3.22e+004 *x3*y -
4.308e+004 *x2*y2 -2.656e+004 *x*y3 -1129 *y4 + 1464 *x5 +
5778*x4*y + 1.017e+004 *x3*y2 + 9194*x2*y3 + 4346*x*y4 -457.1*y5
SPSS
Model Y Pi 01 = 1.000 + 0. 190 π1 + 0. .313 π2 - 0.116 π3 – 0.355 π5 - 0.127 π6
02
(Weight
of shell,
Ws):
Math.
Model
(
)
[ (
√ )
(
)
( )
(
)
(
)
( √ )
( √
)
]
Clubbed
Model (
)
[ (
√ )(
)( ) (
)
(
)( √ )( √
)
]
RSM
f(x, y) = -242.2 + 594.6 *x + 510.4 *y -523.9*x2 -1092 *x*y -384.5*y2
+ 256.8 *x3 + 605.2 *x2*y + 895.1 *x*y2 + 46.45 *y3 -65.32 *x4 -178.4
*x3*y -271.4 *x2*y2 -370.8 *x*y3 + 76.53 *y4 + 6.77 *x5 + 20.97 *x4*y
+ 35.58 *x3*y2+ 46.01 *x2*y3 + 62.98 *x*y4 -27.68 *y5
SPSS
Model Y Pi 02 = 0.981+0 .058π1 + 0.103π2- 0.032 π3 – 0.155 π5 - 0.031 π7
03
(Strength
of Shell,
Es)
Math.
Model
( )
[ (
√ )
(
)
( )
(
)
(
)
( √ )
( √
)
]
Clubbed
Model ( )
[ (
√ ) (
) ( ) (
)
(
)( √ )( √
)
]
RSM
f(x,y) = 2674 -9642 *x -8600 *y + 1.518e+004 *x2 + 2.225e+004 *x*y +
1.225e+004 *y2 -1.291e+004 *x3 -1.291e+004 *x2*y -2.221e+004*x*y2
-9147 *y3 + 5595*x4 + 1.257e+004 *x3*y + 1.281e+004 *x2*y2 +
1.148e+004 *x*y3 + 3284*y4 -957.6 *x5 -2752*x4*y -3036*x3*y2 -
2772*x2*y3 -2458*x*y4 -395*y5
SPSS
Model Y Pi 03 = 1.449 +0 .529 π1 - 0.055 π2 - 0.600 π3 – 0.218 π5 - 0.473 π7
04
(Density
of Shell,
ρs)
Math.
Model
(
)
[ (
√ )
(
)
( )
(
)
(
)
( √ )
( √
)
]
Clubbed
Model (
)
[ (
√ )(
) ( ) (
)
(
)( √ )( √
)
]
RSM
f(x, y) = 704.2 -1554 *x -3122 *y + 893.4 *x2 + 6332*x*y + 5154*y2 +
400.6 *x3 -4409 *x2*y -8193 *x*y2 -4117 *y3 -610.5 *x4 + 1096*x3*y
+ 4122*x2*y2 + 4505*x*y3 + 1595*y4 + 172.9 *x5 + 14.62 *x4*y -
705.1 *x3*y2 -1119 *x2*y3 -928.5 *x*y4 -234.6 *y5
SPSS
Model Y Pi 04 = 1.094 - 0 .065 π1 - 0.023 π2- 0.186 π3 + 0.138 π5 - 0.106 π7
05 (Fibre
Volume
Ratio, Vf)
Math.
Model ( ) 5.457579*
[ (
√ )
(
)
(
)
(
)
(
)
( √
)
( √
)
]
Clubbed
Model ( )
[ (
√ )(
) ( ) (
)
(
)( √ )( √
)
]
RSM
f(x,y) = 676.8 -2384 *x -2158 *y + 3381*x2 + 5996*x*y + 2799*y2 -
2428 *x3 -6257 *x2*y -5743 *x*y2 -1849 *y3 + 878.4 *x4 +
2962*x3*y + 3878*x2*y2 + 3878*x*y3 + 618.3 *y4 -130.1 *x5 -520
*x4*y -926.1 *x3*y2 -789.1 *x2*y3 -439.6 *x*y4 -80.11 *y5
SPSS
Model Y Pi 05 = 1.322+ 0.068 π1 + 0.031 π2 - 0.237π3 – 0.015 π5 - 0.275 π7
Table 1.25 : Comparison between the models on the basis of
Mean value, Mean Error, Percentage Error, Root Mean
Square Error (RMSE), Mean Square Error (MSE),
Coefficient of Determination (R2) and Reliability (Ri)of the
Model compared with experimental data
π
term
s
Response Variable
Mean
Experiment
al
Math.Mod
el
Clubbed
Model SPSS ANN
Π01
Processing cycle
Time 254.814815 254.54
256.216139
1
254.8
1
258.1
9
Π02 Weight of shell 6.36605556 6.3656 6.46933731
6.366
1
6.405
3
Π03 Strength of Shell 311.530741 310.71
313.972318
9
311.5
3
314.7
7
Π04 Density of Shell 1882037037 2E+09
186309375
6
2E+0
9
2E+0
9
Π05 Fibre Volume Ratio 62.1305556 62.145
62.6173815
1
62.13
1 62.14
Mean Error compared with Experimental Data
π
terms Response Variable
Error betn
Math and
EXP
Error
betn
club
andExp
Error
betn
SPSS
andExp
Error
betn
ANN
andExp
Π01 Processing cycle Time 0.2754 1.4013 7E-05 3.3732
Π02 Weight of shell 0.0004 0.1033 2E-06 0.0392
3rd International Conference on Multidisciplinary Research & Practice P a g e | 48
Volume IV Issue I IJRSI ISSN 2321-2705
Π03 Strength of Shell 0.8158 2.4416 4E-05 3.239
Π04 Density of Shell 1E+06 2E+07 740.74 1E+07
Π05 Fibre Volume Ratio 0.0144 0.4868 2E-05 0.0098
% Error compared with Experimental Data
π
terms Response Variable
% Error
betn Math
and EXP
%
Error
betn
club
andExp
% Error betn
SPSS andExp
%
Error
betn
ANN
andExp
Π01 Processing cycle Time 0.10806769 0.5499 2.61628E-05 1.3238
Π02 Weight of shell 0.00677731 1.6224 3.76127E-05 0.6161
Π03 Strength of Shell 0.26188046 0.7837 1.30776E-05 1.0397
Π04 Density of Shell 0.07074532 1.0065 3.93585E-05 0.5133
Π05 Fibre Volume Ratio 0.02319211 0.7836 3.68996E-05 0.0158
M.S.E. compared with Experimental Data
π
terms Response Variable
Mean
Square
Error betn
Math and
EXP
Mean
Square
Error
betn
club
andEx
p
Mean
Square
Error
betn
SPSS
andEx
p
Mean
Square
Error
betn
ANN
andEx
p
Π01 Processing cycle Time 111.9232048 213.18 114.12 395.63
Π02 Weight of shell 0.006263317 0.0945 0.0064 0.0156
Π03 Strength of Shell 394.5647566 471.83 383.84 640.56
Π04 Density of Shell 3.14324E+15 2E+16 3E+15 3E+15
Π05 Fibre Volume Ratio 0.160984928 1.0593 0.0954 0.0342
Reliability compared with Experimental Data
π
terms Response Variable
Ribetn Math
and EXP
Ribetn
club
and
Exp
Ribetn SPSS
and Exp
Ribetn
ANN
and
Exp
Π01 Processing cycle Time 97.3518519 95.389 97.2962963 94
Π02 Weight of shell 99.5 96.315 99.46296296 98.907
Π03 Strength of Shell 99.5 96.315 95.40740741 93.87
Π04 Density of Shell 98.1666667 94.074 98.11111111 97.944
Π05 Fibre Volume Ratio 99.8518519 99.288 99.96296296 99.375
CONCLUSION
1) Present existing filament winding machine
activities is studied critically for cylindrical pressure vessel
and indicates that the process suffers from various drawbacks
and sefects like pressure test failure and lack of dimensional
accuracy, high human energy expenditure and low production
rate. The present research work has role model for similar
Pressure vessel manufacturing Industries by filament winding
method of composite Glass FRP
2) In the present work all the details of proposed
machine has been considering all the design parameter. The
present filament winding machine is robust in construction. It
can be operated by skilled/semiskilled/unskilled operators.
This machine is very useful for missile, Rocket, defense,
aerospace and aviation industries. since it also beneficial for
composite pipe industries for marine and desert area where
long pipe of composite FRP pipes for petroleum and
domestic supply and distribution.
3) The economic viability and feasibility will help
the people to start small scale business in the Pipe and
pressure vessel manufacturing for High strength, light
weight, low maintenance and corrosion free Pipe Industries.
4) The machining properties . Cycle time of
component processing, Weight of cy. Vessel/Shell/Vessel,
Ultimate tensile strength of cy. Vessel/Shell/Vessel,
Density of FRP Shell/Vessel, and Fiber volume ratio of
filament winding machine operation are established through
Theory of experimentation, which was unknown in previously
mentioned literature.
5) The data of filament winding machine operation is
collected by performing actual experimentation. Due to this,
the finding of the present study truly represents the degree of
interaction of various independent variables. This has been
made possible only by the approach adopted in this
investigation. The standard error of estimate of the predicted /
computed values of the dependent variables is found to be
very low. This gives authenticity to the developed
mathematical models and ANN.
6) The models have been formulated mathematically
considering Indian conditions and filament winding
operation. The values of dependent term obtained from
experimental data, mathematical model and ANN are
compared. From the values of percentage errors, it has been
noted that the mathematical models can be successfully used
for the computation of dependent terms for a given set of
independent terms.
7) The sensitivity analysis have been found
complementary to each other. These trends have been found
to be truly justified through some possible physics of
phenomenon.
R2 compared with Experimental Data
π
terms Response Variable
R2betn Math
and EXP
R2betn
club
andExp
R2betn
SPSS
andExp
R2betn
ANN
andExp
Π01 Processing cycle Time 0.504527 0.0563 0.4948 -0.7514
Π02 Weight of shell 0.516475371 -6.299 0.5094 -0.2012
Π03 Strength of Shell 0.158728032 -0.006 0.1816 -0.3658
Π04 Density of Shell 0.097518875 -4.6643 0.1115 0.0202
Π05 Fibre Volume Ratio 0.787171059 -0.4004 0.8739 0.9548
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Volume IV Issue I IJRSI ISSN 2321-2705
8) From sensitivity analysis of filament winding machine
process for cylindrical pressure vessel GFRP, it is analyzed
that–
Cycle time of component processing
o For this influence of the independent
parameters are noted mostly i.e. Geometry
of Shell/Vessel, viscosity of resin mix,
speed of mandrel.
Weight of cy. Vessel/Shell/Vessel
o For this influence of the independent
parameters are noted mostly i.e. viscosity of
resin mix, weight of resin mix, speed of
mandrel,
Ultimate tensile strength of cy.
Vessel/Shell/Vessel,
o For this influence of the independent
parameters are noted mostly i.e.
temperature of curing oven, Carriage
feed, speed of mandrel
Density of FRP Shell/Vessel
o For this influence of the independent
parameters are noted mostly i.e. weight of
resin mix, viscosity of resin mix, speed of
mandrel
Fiber volume ratio
o For this influence of the independent
parameters are noted mostly i.e. weight of
resin mix, curing time (soaking), speed of
mandrel
9) From reliability analysis of filament winding machine
operation, it is analyzed that– The reliability of dependent
variables Cycle time of component processing, Weight of
cy. Vessel/Shell/Vessel, Ultimate tensile strength of cy.
Vessel/Shell/Vessel, Density of FRP Shell/Vessel, and
Fiber volume ratio -total was found to be 97.3513%, 99.5%,
95.59% and 99.85 % respectively.
10) From the optimization of filament winding machine
operation, the optimized values of dependent pi terms –
Cycle time of component processing for
dependent variable- 233.06
o For independent ie. fc=46.667 mm/sec,
Ls=895 mm, ds=206mm, ts=5mm, Tr=58 oC, To=140
oC, Wr=2.954 Kg, Eg=72.5
N/mm2 μar=140080N·s/ mm
2,
μhr=209000N·s/ mm2, ωm=2.092 rad/s
(N=120 rpm), tc=21600 sec
Weight of cy. Vessel/Shell/Vessel for dependent
variable= 6.2163
o fc For independent ie. =46.667 mm/sec,
Ls=895 mm, ds=206mm, ts=5mm, Tr=62 oC, To=100
oC, Wr=2.954 Kg, Eg=72.5
N/mm2 μar=140080N·s/ mm
2, μhr=209000
0N·s/ mm2, ωm=2.092 rad/s (N=120 rpm),
tc=21600 sec
Ultimate tensile strength of cy. Vessel, for
dependent variable= 383.555
o For independent ie. fc=50 mm/sec, Ls=895
mm, ds=206mm, ts=5mm, Tr=62 oC,
To=100oC, Wr=2.954 Kg, Eg=72.5 N/mm
2
μar=1300050N·s/ mm2, μhr=200000 N·s/
mm2, ωm=2.092 rad/s (N=120 rpm),
tc=14400 sec
Density of FRP Shell/Vessel for dependent
variable- 1.82E+09
o For independent ie. fc=50 mm/sec, Ls=895
mm, ds=206mm, ts=5mm, Tr=58 oC,
To=140oC, Wr=2.954 Kg, Eg=72.5 N/mm
2
μar=1300050N·s/ mm2, μhr=200000N·s/
mm2, ωm=2.092 rad/s (N=120 rpm),
tc=21600 sec
Fiber volume ratio for dependent variable =
66.72446
o For independent ie. fc=50 mm/sec, Ls=895
mm, ds=206mm, ts=7mm, Tr=62 oC,
To=100oC, Wr=2.954 Kg, Eg=72.5 N/mm
2
μar=1300050N·s/ mm2, μhr=200000N·s/
mm2, ωm=2.092 rad/s (N=120 rpm),
tc=21600 sec
11) From the calculations and the graphs plotted for
the models of response variables (Cycle time of component
processing, Weight of cy. Vessel/Shell/Vessel, Ultimate
tensile strength of cy. Vessel/Shell/Vessel, Density of FRP
Shell/Vessel, and Fiber volume ratio), comparison for
reliability of mathematical models and clubbed models as well
as value of R2
for mathematical models , clubbed models
,RSM models SPSS models is made from analysis of this
comparison , it is noted that the mathematical models
formulated are the superior than clubbed model, RSM model
12) From reliability analysis, the reliabilities of all
the response variables for clubbed model ,RSM model ,SPSS
model and ANN simulation are compared with mathematical
model and following observations are made.
(i) For processing cycle time, the reliabilities of
mathematical model , clubbed model , R, SPSS model and
ANN simulation are found to be 97.352 %, 95.389 %,
97.296%, 98.907 % , respectively.
(ii) Weight of Shell/Vessel :, the reliabilities of mathematical
model , clubbed model , SPSS model and ANN simulation are
3rd International Conference on Multidisciplinary Research & Practice P a g e | 50
Volume IV Issue I IJRSI ISSN 2321-2705
found to be 99.5 % , 96.315%, 99.462% , 98.907
respectively.
(iii) For strength of Shell/Vessel :, the reliabilities of
mathematical model, clubbed model, SPSS model and ANN
simulation are found to be 93.444 %, 56.35% , 59.021 % ,
85.451 % , 91.566 % respectively.
(iv) For Density of Shell/Vessel, the reliabilities of
mathematical model, clubbed model , , SPSS model and ANN
simulation are found to be 98.166%, 94.074% , 98.111 % ,
97.044 %, respectively.
(v) For Fiber Volume Ratio, the reliabilities of mathematical
model, clubbed model SPSS model and ANN simulation are
found to be 99.815%, 99.288 % , 99.962 % , 99.375 %,
respectively.
13 ) From the analysis of coefficient of determination
(R2
value), of all the response variables for clubbed model
,RSM model ,SPSS model and ANN simulation are compared
with mathematical model and following observations are
made.
(i) For processing cycle time of FRP Shell/Vessel, the
coefficient of determination (R2
value), of mathematical
model , clubbed model , , SPSS model and ANN simulation
are found to be 0.5045%, 0.0563%, 0.4948 %, 0.7514 %,
respectively.
(ii) For Weight of FRP Shell/Vessel , the coefficient of
determination (R2
value), of mathematical model , clubbed
model , SPSS model and ANN simulation are found to be
0.5164%, -6.2999%, 0.0.5094%, 0.-0.2013% respectively.
(iii) For, Strength Of FRP Shell/Vessel the coefficient of
determination (R2
value), of mathematical model , clubbed
model , SPSS model and ANN simulation are found to be
0.1587%, -0.006%, 0.1816%, -0.3658 respectively.
(iv) For Density of FRP Shell/Vessel , the coefficient of
determination (R2
value), of mathematical model , clubbed
model , SPSS model and ANN simulation are found to be
0.09715%, 4.6646%, 0.1115%, 0.0202% respectively.
(v) For fibre volume ratio of FRP Shell/Vessel , the
coefficient of determination (R2
value), of mathematical
model , clubbed model , SPSS model and ANN simulation
are found to be 0.7871%, 0.4004%, 0.8739%, 0.9548%
respectively.
(14) From the analysis of Mean Value of all the
response variables for clubbed model ,RSM model ,SPSS
model and ANN simulation are compared with mathematical
model and following observations are made.
(i) For processing cycle time, the mean value of
experimental, mathematical model , clubbed model SPSS
model and ANN simulation are found to be 254.814, 254.54,
256.,216, 254.81, 258.19 respectively.
(ii) For weight of Shell/Vessel , the mean value of
experimental, mathematical model , clubbed model SPSS
model and ANN simulation are found to be 6.3660, 6.3656,
6.469, ,6..3661, 6.4053 respectively.
(iii) For strength of Shell/Vessel , the mean value
experimental, mathematical model , clubbed model SPSS
model and ANN simulation are found to be 311.530741,
310.71, 313.9723189, 311.53, 314.77 respectively.
(iv) For dendity of Shell/Vessel , the mean value of
experimental, mathematical model , clubbed model SPSS
model and ANN simulation are found to be 1882037037, ,
2E+09, 1863093756, 2E+09, 2E+09 respectively.
(iv) For fibre volume ratio of Shell/Vessel , the mean value
of experimental, mathematical model , clubbed model SPSS
model and ANN simulation are found to be 62.1305556,
62.145, 62.61738151, 62.131, 62.14 „ respectively.
(15) From , the analysis of MEAN ERROR of all
the response variables for clubbed model ,RSM model ,SPSS
model and ANN simulation are compared with mathematical
model and following observations are made.
(i) For Processing cycle Time for FRP Shell/ Vessel, Mean
Error compared with Experimental Data for
mathematical model , clubbed model , , SPSS model and
ANN simulation are found to be 0.2754, 1.4013, 7E-05,
respectively.
(ii) For weight of FRP Shell/Vessel , Mean Error
compared with Experimental Data for mathematical
model , clubbed model , , SPSS model and ANN simulation
are found to be 0.0004, 0.1033, 2E-06, 0.0392 respectively.
(iii) For strength of FRP Shell/Vessel , Mean Error
compared with Experimental Data for mathematical
model , clubbed model , , SPSS model and ANN simulation
are found to be 0.8158, 2.4416, 4E-06, 3.239 respectively
(iv) For Dendity of FRP Shell/Vessel , Mean Error
compared with Experimental Data for mathematical
model , clubbed model , , SPSS model and ANN simulation
are found to be 1E+06, 2E+07, 740.74, 1E+07 respectively.
(v) For fibre volume ratio FRP Shell/Vessel , Mean Error
compared with Experimental Data for mathematical
model , clubbed model , , SPSS model and ANN simulation
are found to be 0.0144, 0.4868, 2E-05, 0.0098 respectively.
(16) From , the analysis of PERCENTAGE
ERROR of all the response variables for mathematical,
clubbed model , SPSS model and ANN simulation are
compared with Exprimental model and following
observations are made.. (i) For Processing cycle Time for
FRP Shell/ Vessel, Percentage Error compared with
Experimental Data for mathematical model , clubbed
model , , SPSS model and ANN simulation are found to be
0.10806769, 0.5499, 2.61628E-05, 1.3238 respectively.(ii)
For weight of FRP Shell/Vessel , Percentage Error
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Volume IV Issue I IJRSI ISSN 2321-2705
compared with Experimental Data for mathematical
model , clubbed model , , SPSS model and ANN simulation
are found to be 0.00677731, 1.6224, 3.76127E-05, 0.6161
respectively. (iii) For strength of FRP Shell/Vessel ,
Percentage Error compared with Experimental Data for
mathematical model , clubbed model , , SPSS model and
ANN simulation are found to be 0.26188046, 0.7837,
1.30776E-05, 1.0397 respectively(iv) For Dendity of FRP
Shell/Vessel , Percentage Error compared with
Experimental Data for mathematical model , clubbed
model , , SPSS model and ANN simulation are found to be
0.07074532, 1.0065, 3.93585E-05, 0.5133 respectively.(v)
For fibre volume ratio FRP Shell/Vessel , Percentage
Error compared with Experimental Data for
mathematical model , clubbed model , , SPSS model and
ANN simulation are found to be 0.02319211, 0.7836,
3.68996E-05, 0.0158 respectively.
(17) From , the analysis of PERCENTAGE
ERROR of all the response variables for mathematical,
clubbed model , SPSS model and ANN simulation are
compared with Exprimental model and following
observations are made. (i) For Processing cycle Time for
FRP Shell/ Vessel, M.S.E. compared with Experimental
Data for mathematical model , clubbed model , , SPSS model
and ANN simulation are found to be 111.9232048, 213.18,
114.12, 395.63 respectively.(ii) For weight of FRP
Shell/Vessel , M.S.E. compared with Experimental Data
for mathematical model , clubbed model , , SPSS model and
ANN simulation are found to be 0.006263317, 0.0945,
0.0064, 0.0156 respectively.
(iii) For strength of FRP Shell/Vessel , M.S.E. compared
with Experimental Data for mathematical model , clubbed
model , , SPSS model and ANN simulation are found to be
394.5647566, 471.83, 383.84, 640.56 respectively
(iv) For Dendity of FRP Shell/Vessel , M.S.E. compared
with Experimental Data for mathematical model , clubbed
model , , SPSS model and ANN simulation are found to be
3.14324E+15, 2E+16, 3E+15, 3E+15 respectively.
(v) For fibre volume ratio FRP Shell/Vessel , M.S.E.
compared with Experimental Data for mathematical
model , clubbed model , , SPSS model and ANN simulation
are found to be 0.160984928, 1.0593, 0.0954, 0.0342
respectively.
(18) From , the analysis of ROOT MEAN
SQUARE ERROR (RMSE) of all the response variables for
mathematical model clubbed model ,RSM model ,SPSS
model and ANN simulation are compared with experimental
and following observations are made.
(i) For Processing cycle Time for FRP Shell/ Vessel,
ROOT MEAN SQUARE ERROR (RMSE). compared
with Experimental Data for mathematical model , clubbed
model , , SPSS model and ANN simulation are found to be
0.079141, 0.307486, 0.079722, 0.12474respectively.
(ii) For weight of FRP Shell/Vessel , ROOT MEAN
SQUARE ERROR (RMSE) compared with
Experimental Data for mathematical model , clubbed model
, , SPSS model and ANN simulation are found to be
0.079141, 0.307486, 0.079722, 0.12474 respectively.
(iii) For strength of FRP Shell/Vessel , ROOT MEAN
SQUARE ERROR (RMSE) compared with Experimental
Data for mathematical model , clubbed model , , SPSS model
and ANN simulation are found to be 19.86365, 21.72168,
19.59184, 25.30937 respectively
(iv) For Dendity of FRP Shell/Vessel , ROOT MEAN
SQUARE ERROR (RMSE) compared with
Experimental Data for mathematical model , clubbed
model , , SPSS model and ANN simulation are found to be
56064608, 1.4E+08, 55627534, 58417484 respectively.
(v) For fibre volume ratio FRP Shell/Vessel , ROOT
MEAN SQUARE ERROR (RMSE) compared with
Experimental Data for mathematical model , clubbed
model , , SPSS model and ANN simulation are found to be
0.401229 , 1.029221, 0.308802, 0.18481 respectively.
. (19) From the conclusions discussed in Sr. Nos. 13 to
19 , it can be concluded that the mathematical models
developed for filament winding machine for cylindrical
pressure vessel glass FRP operation in this work are superior
in all respects. This is based on the comparative values stated
in tables
20) In this research work of filament winding for all
operation , have work on Exprimental , Mathematical Model ,
Clubbed model , Artificial Neural Network .along with this
has formulated the SPSS model by using linear regression
analysis also
21) With the help of present filament winding
machine for the cylindrical pressure vessel made of Glass
FRP be the roll model for our rocket shell manufacturing and
for other Frp Ammunation Hardware Industries.
LIMITATIONS OF PRESENT WORK
1. As the working conditions and environment
conditions cannot be controlled in the operational
area as per the experimental requirement, whether
the observed response is on lower side or on the
higher side however cannot be predicted.
2. The ANN performance depends on the training. The
comparative lower value of the regression coefficient
for one of the dependent pi term may be due to the
improper training of the network. The ANN has been
unable to predict beyond the range for which it has
been trained.
SCOPE FOR FURTHER RESEARCH WORK
1. The filament winding machine process unit platform
can be design with field data collected nad coduct
3rd International Conference on Multidisciplinary Research & Practice P a g e | 52
Volume IV Issue I IJRSI ISSN 2321-2705
expreimnet for our inhouse work. Its also useful for
other FRP work of Rocket and Missile Group where
simmiller output required with high calibre and
long range.
2. Planned and trouble free operation boost moral of all
as workers/operater to level of mangments and
satisfaction of utilization of public fund/ Defence
budget for country Goal and targets and attract the
for indignization and self sufficiency . More software
for development for polymer matrix structural work
The applicability of other techniques like Genetic
Algorithm, Fuzzy logic etc. for the modelling of the
phenomenon may also be tested.The work can be
extended for making the filament winding for
cylindrical pressure vessel , Aerodynamic Dom, for
other ammunition hardware manufacturing,
Aerospace, Marin, Navel work also..
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http://www.ijaer.com (ijaer) 2012, vol. No. 4, issue no. Ii, august issn: 2231-5152