Formulation of Generalized Approximate Mathematical Model ...Cylindrical pressure vessel made of composite material (Glass FRP) having wide applications. A machine already installed

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



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,



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



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



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. .



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)


(

)

[ (

√ )

(

)

(

)

(

)

(

)

( √

)

( √

)

]

(1.10)


Π05 ( ) 5.457579*

[ (

√ )

(

)

(

)

(

)

(

)

( √

)

( √

)

]

( )



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



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 :



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)


0.5484, RMSE: 0.01168

(iii) For Strength of Shell:



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


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:



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


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:



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


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:


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



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



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



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

http://www.businessdictionary.com/definition/method.html

http://www.businessdictionary.com/definition/variability.html

http://www.businessdictionary.com/definition/factor.html

http://www.businessdictionary.com/definition/relationship.html

http://www.businessdictionary.com/definition/trend-analysis.html

http://www.businessdictionary.com/definition/value.html

http://www.businessdictionary.com/definition/percent.html

http://www.businessdictionary.com/definition/failure-in-time-FIT.html

http://www.businessdictionary.com/definition/square.html

http://www.businessdictionary.com/definition/coefficient-of-correlation-r.html



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



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



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


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



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)



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



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



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




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



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)



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



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



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




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)



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



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


% 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


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


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


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



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


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,


parameters are noted mostly i.e.

temperature of curing oven, Carriage

feed, speed of mandrel

Density of FRP Shell/Vessel


parameters are noted mostly i.e. weight of

resin mix, viscosity of resin mix, speed of

mandrel

Fiber volume ratio


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



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



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



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



0.1587%, -0.006%, 0.1816%, -0.3658 respectively.

(iv) For Density of FRP Shell/Vessel , the coefficient of

determination (R2



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



(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


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


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


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



(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


are found to be 0.0004, 0.1033, 2E-06, 0.0392 respectively.

(iii) For strength of FRP Shell/Vessel , Mean Error



are found to be 0.8158, 2.4416, 4E-06, 3.239 respectively

(iv) For Dendity of FRP Shell/Vessel , Mean Error



are found to be 1E+06, 2E+07, 740.74, 1E+07 respectively.

(v) For fibre volume ratio FRP Shell/Vessel , Mean Error



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





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


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



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



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


0.0064, 0.0156 respectively.

(iii) For strength of FRP Shell/Vessel , M.S.E. compared

with Experimental Data for mathematical model , clubbed


394.5647566, 471.83, 383.84, 640.56 respectively

(iv) For Dendity of FRP Shell/Vessel , M.S.E. compared



3.14324E+15, 2E+16, 3E+15, 3E+15 respectively.

(v) For fibre volume ratio FRP Shell/Vessel , M.S.E.



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



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



56064608, 1.4E+08, 55627534, 58417484 respectively.

(v) For fibre volume ratio FRP Shell/Vessel , ROOT

MEAN SQUARE ERROR (RMSE) compared with



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



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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Documents

Formulation of Generalized Approximate Mathematical Model ...Cylindrical pressure vessel made of composite material (Glass FRP) having wide applications. A machine already installed