International Journal of Application or Innovation in Engineering & Management (IJAIEM)
Web Site: www.ijaiem.org Email: editor@ijaiem.org
Volume 4, Issue 6, June 2015
ISSN 2319 - 4847
Formulation of Mathematical Models
for establishing Processing time of Stirrup
making operation using human powered
flywheel motor
S.N. Waghmare1, Dr. C.N. Sakhale2
1
Assistant Professor, Mechanical Engg. Deptt.,Priyadarshini College of Engineering, Nagpur:440019, India
2
Associate Professor, Mechanical Engg. Deptt.,Priyadarshini College of Engineering, Nagpur:440019, India
ABSTRACT
In this present work of research the design of experimental work to be executed for formulation of experimental
data based model for stirrup making operation by using human powered flywheel motor. This paper presents an
experimental investigations and sequential classical experimentation technique has been used to perform
experiments to establishment of stirrup making operation by using HPFM. An Attempt of minimum and maximum
principle has been made to optimize the range bound process parameters for minimize the processing time,
maximise the number of bends and maximize the processing torque. The influence of bending operation was studied
experimentally by performing 144 experimental tests. By using experimental data various model is formed and
comparison of these model with help of reliability, coefficient of determinate and sensitivity analysis is done.
Keywords: Bar bending, Stirrup, Buckingham’s π theorem, Regression analysis, Optimization, Sensitivity, ANN
1. INTRODUCTION
Stirrups are the lateral ties which are used to bind the steel framework together. It is an essential element of reinforced
cement concrete in civil construction.In small construction sites workers bend stirrup using traditional method. There is
no other way to make stirrup with less human effort and the same time the detailed study of present manual stirrup
making activity indicates that the process suffers from various draw back like lack of accuracy which results in weak
structure and instability in column and beams. These stirrups are used for strengthening column and beams for
avoiding buckling of long slender column and also avoiding sagging of horizontal beam. Traditionally stirrups are
made on a wooden platform provided with pins and rod is bend with the help of tommy .The force is applied on tommy
and the pin works as a fulcrum point for bending the rod. The details study of present manual stirrup making activity
indicates that the process suffers from various drawback like low production rate and the operator not only subject his
hands to hours of repetitive motion but also utilizing high human energy expenditure results internal injury to his body
organ like carpel tunnel syndrome, spondylisis, muskulo skeletal disorder etc.
Uses of Stirrup Strengthening columns and beams, Avoiding buckling of long slender columns, Avoiding sagging of
horizontal beams.
Figure 1 Various types of stirrup
2. BACKGROUND OF THE PRESENT RESEARCH WORK
Present machine system comprises of three subsystems
(1) Energy Unit (2) Mechanical power transmission system (3) process unit [1-2].
2.1 Energy Unit: Energy unit comprises of an arrangement similar to bicycle, a speed raising gear pair and a flywheel
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Mechanical power transmission system Mechanical transmission comprises of spiral jaw clutch or other clutches [3]
and torque amplification gear pair.
Process unit : The process unit is stirrup making unit which comprises of two spur gear having 3 / 4 and ¼ teeth
over it ,rectangular helical spring ,circular disk it having fixed pin and rotating pin over it and engagement and
disengagement clutch with lever and coupling also slot is provided for feeding and supporting the rod.
2.2 Stirrup making operation by using HPFM: The operator drives the bicycle by pedaling the bicycle mechanism
while clutch is in disengage position. The human power operated flywheel motor is as energy source. This energy
source energizes the process unit i.e. stirrup making unit through clutch and transmission. The flywheel is accelerated
and energized which stores some energy inside it. When the pedaling is stopped, clutch is engaged and stored energy in
the flywheel is transferred to the process unit input shaft by means of clutch and following step are fallow for complete
one stirrup
2.3 Steps for Stirrup Making
The stirrups are made by five bending operation [7-13]. The stirrup rod is first cut in definite length and marking by
chock then the five bending operation are performed as fallows.
i) First the small length of rod is bend by inserting the rod in the guiding slot and put it on centre position of disc then
by press and left the lever with help of foot and first bend is made which is called as anchorage length.
ii) Then the second bend is made according to size of stirrup by forwarding the rod till next marking by press and left
the foot lever.
iii) Similarly third bend is made to make the stirrup square in shape according to each side of stirrup size.
iv) Then turn the rod of three bend and second side small bend is made i.e called as second anchorage length and
finally the fifth bend is made by forwarding the rod till next marking and finally tied with the first anchor and stirrup is
prepared.
1 -Seating arrangement
2- Small Chain Sprocket
3 - Pedal
4 - Chain
5 - Large Sprocket
6- Shaft B1
7- Bearing for bicycle
8 - Gear I
9-Pinion
17-Shaft B3
10-Shaft B2
18- Shaft B4
11-Bearing
19-Clutch for Process Unit
12- Flywheel
20-3/4th Teeth spur gear
13 -Clutch
21-1/4th Teeth spur gear
14- Lever
22- Rectangular helical spring
15- Gear II
23- Rotating Disk
16- Gear III
24- Rotating Pin
Figure 2 Line diagram of stirrup making machine by HPFM
3. FORMULATION EXPERIMENTAL DATA BASED MODELS
In view of for formulation of experimental data based model it is obvious that one will have to decide what should be
the optimum processing time, maximum number of bends and maximum processing torque. The only option with
which one is left is to formulate an experimental data based model, Hilbert Sc. Jr. (1961) [14]. 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, Singiresu [15]. In
fact determination of such values of independent variables is always the puzzle for the operator because it is a complex
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phenomenon of interaction of various independent variables and dependant variables for optimizing the stirrup bending
operation is shown in table
Table 1 : Various dependant and independent Variables related to stirrup making operation
Sr.
Variables
Unit
MLT
Dependent/
Independent
1
Tr = Resistive Torque
N-m
ML2T-2
Dépendent
2
tp = Processing Time
Sec
T
Dépendent
3
nb = No. of actual bend per cycle
-M0L0T0
Dépendent
4
Ef = Flywheel Energy
N-m
ML2T-2
Independent
5
ωf = Angular speed of flywheel
Rad /sec
T-1
Independent
6
tf = Time to speed up the flywheel
Sec
T
Independent
7
ds = Diameter of stirrup
m
L
Independent
8
s = Size of stirrup
m2
L2
Independent
9
θ = Angle of bend
Degree
Independent
10
Hs = Hardness of stirrup
N/m2
ML-1T-2
Independent
11
r = Distance between pin & center
m
L
Independent
12
G = Gear Ratio
-M0L0T0
Independent
13
k = Stiffness of spring
N/m
MT-2
Independent
14
dr = Diameter of Rotating Disc
m
L
Independent
15
tr = Thickness of Rotating Disc
m
L
Independent
16
g = Acceleration due to Gravity
m/s2
LT-2
Independent
17
Ls = Length of stirrup
m
L
Independent
18
Es= Modulus of Elasticity of stirrup
N/m2
ML-1T-2
Independent
It is necessary to evolve physical design of an experimental set up having provision of setting test points, adjusting test
sequence, executing proposed experimental plan, provision for necessary instrumentation for noting down the
responses. The experimental set up is designed considering various physical aspects of the elements.
Measurement of Important Parameters
i) Measurement of speed and time of flywheel before engagement of the clutch: The selected speed of flywheel is
set during experimentation before the engagement of clutch. After attaining the required set speed, the speed of
flywheel and the time taken by the flywheel to speed up is measured and recorded with the help of embedded sensors in
specially designed kit.
(ii) Measurement of speed of process unit (i.e. stirrup making unit) after engagement of clutch:As soon as the
flywheel reaches the required set speed, the clutch is engaged to the process unit shaft and the varying speeds of the
process unit while the stirrup bending operation is recorded on the computer through the signals given by sensors of the
specially designed kit.
(iii) Measurement of time of process unit after engagement of clutch: After engagement of clutch, the process unit
starts and actual process of bending the rod or wire on the output shaft starts. The time of the process unit / output shaft
is measured and recorded until the process unit shaft reaches the rest position, This time is recorded and measured on
the display of computer by the way of signals through the specially designed electronic kit.
(iv)Measurement of resistive torque: When the energy stored in the flywheel is transferred to the process unit by
engaging the clutch, the values of torques are displayed on the computer for every varying speed of the process unit i.e.
stirrup making unit with the help of electronic kit sensors. The values of the resistive torques are recorded by using
following formulae:
= Equivalent moment of inertia
IF = Moment of inertia of flywheel
IC = Moment of inertia of cutter
NF = Speed of flywheel shaft
NC = Speed of cutter shaft/process unit shaft
α = Angular acceleration of process unit shaft = θ × (π/180)
and
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Figure 6 Software for measurement of Torque
4. DESIGN OF AN EXPERIMENTAL SET UP
The design of experimentation has given a road map of how the experimentation is planned. But implementation of
experimental plan and conduction of actual test run requires a systematic detailing of execution. During
experimentation the stirrup rod of three varying lengths i.e. 968.4mm, 1068.4mm and 1220.4 mm and two type of 6
mm plain , 6 mm TMT having same 6 mm diameter are processed in the stirrup making machine at four different
speeds i.e. 300 rpm, 400 rpm, 500 rpm and 600 rpm and at three different gear ratios 1/2, 1/3, and 1/4. Thus the two
different types material are used during experimentation for monitoring the actual feasibility of the machine. During
experimentation processing time, resistive torque, number of bends and time of flywheel to speed up etc. are measured
using specially designed electronic kit i.e. instrumentation system as described above .
Figure 3 Fabricated and CAD model of stirrup making machine by using HPFM
Figure 5 Experimental arrangement of stirrup making operation by HPFM
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4.1 Design of Experiments
In this experimentation 144 experiments (Sample readings are shown in Table 2 were designed on the basis of
sequential classical experimental design technique that has been generally proposed for engineering applications,
Hilbert Schank [16]. The basic classical plan consists of holding all but one of the independent variables constant and
changing this one variable over its range. The main objective of the experiments consists of studying the relationship
between 12 independent process parameters with the 03 dependent responses for stirrup making operation
optimization. Simultaneous changing of all 12 independent parameters was cumbersome and confusing. Hence all 12
independent process parameters were reduced by dimensional analysis. Buckingham π theorem was adapted to develop
dimensionless π terms for reduction of process parameters .This approach helps to better understand how the change in
the levels of any one process parameter of a π terms affects 07 dependant responses for stirrup making operation.
Table 2: Sample of Experimental observations
4.2 Formulation of Experimental Data Base Model by Dimensional Analysis
As per dimensional analysis, for processing time was written in the function form as:
tp =f1 f (Ef,, ωf, tf, , ds, s, , θ, Hs, , r, G, k, dr, tr, g ,Ls, Es)
(1)
By selecting Mass (M), Length (L), and Time (T) as the basic dimensions, the basic dimensions of the forgoing
quantities were mentioned in table 1:According to the Buckingham’s - theorem, (n- m) number of dimensionless
groups are forms. In this case n is 12 and m=3, so π1 to π12 dimensionless groups were formed. By choosing ‘g’, ‘Ls’
and ‘Es’ as a repeating variable, twelve π terms were developed as follows:
tp = f1 (Ef,, ωf, tf, , ds, s, , θ, Hs, , r, G, k, dr, tr, g ,Ls, Es)
Π01 = f1 (Π1, Π2, Π3, Π4, Π5, Π6, Π7, Π8, Π9, Π10, Π11, Π12)
(2)
4.3 Reduction of independent variables by dimensional analysis
Deducing the dimensional equation for a phenomenon reduces the number of independent variables in the experiments.
The exact mathematical form of this dimensional equation is the targeted model. This is achieved by applying
Buckingham’s π theorem when n (no. of variables) is large even by applying Buckingham’s π theorem number of π
terms will not be reduced significantly than number of all independent variables. Thus, much reduction in number of
variables is not achieved. It is evident that, if we take the product of the π terms it will also be dimensionless number
and hence a π term. This property is used to achieve further reduction of the number of variables. Thus few π terms are
formed by logically taking the product of few other π terms and final mathematical equations are given below:
(3)
4.4 Test planning
This comprises of deciding test envelope, test points, test sequence and experimentation plan for deduced set of
dimensional equations. Table 3 shows Test envelope, test points for stirrup making operation.
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Table 3 Test envelope, test points for stirrup making operation
5. MODEL FORMULATION
The relationship between various parameters was unknown. The dependent parameter Π01 i.e. relating to tp was be an
intricate relationship with remaining terms (ie. π1 to π7) evaluated on the basis of experimentation. The true
relationship is difficult to obtain. The possible relation may be linear, log linear, polynomial with n degrees, linear with
products of
independent πi terms. In this manner any complicated relationship can be evaluated and further investigated for error.
Hence the relationship for tp- processing time was formulated as:
Model of dependent pi term for D1 (i.e. Processing Time, tp):
01 = f (1, 2, 3, 4, 5, 6, 7 )
Approximate generalized experimental models for predicting processing time for bamboo sliver cutting by human
powered flywheel motor has been established.
01 = k1 x (1)a1 x (2)b1 x (3)c1 x (4)d1 x (5)e1 x (6)f1 x (7)g1
The values of exponential a1, b1, c1, d1, e1, f1, g1 are established, considering exponential relationship between
dependent pi term tp and Independent  terms 1, 2, 3, 4, 5, 6, 7, independently taken one at a time, on the
basic of data collected through classical experimentation. Thus corresponding to the seven independent pi terms one
have to formulate seven pi terms from the set of observed data for processing time. From these models values of
dependent pi term is computed.
01= k1 x (1)a1 x (2)b1 x (3)c1 x (4)d1 x (5)e1 x (6)f1 x (7)g1
(4)
There are eight unknown terms in the equation 7. These are curve fitting constant K1 and indices a1, b1, c1, d1, e1, f1,
g1. To get the values of these unknown we need minimum eight sets of values of (1, 2, 3, 4, 5, 6, 7 ).
As per the experimental plan in design of experimentation we have 108 sets of these values. If any arbitrary twenty sets
from table are selected and the values of unknown K1 and indices a1, b1, c1, d1, e1, f1, g1 are computed, then it may
not result in one best unique solution representing a best fit unique curve for the remaining sets of values. To be very
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specific one can find out nCr combinations of r are taken together out of the available n sets of the values. The value
nCr in this case will be 108C7,
Solving these many sets and finding their solutions will be a Herculean task. Hence it is decided to solve this problem
by curve fitting technique . To follow this method it is necessary to have the equations in the form as under.
Z = a + bX + cY +dZ+……
(5)
The equation 4 can be brought in the form of equation 6 by taking log on both sides.
Taking log on the both sides of equation for 01, to get eight unknown terms in the equations,
Putting the values in equations the same can be written as
Z1 = K1+ a1 A + b1 B + c1 C + d1 D + e1 E + f1 F + g1G
(6)
Equation 6 is a regression equation of Z on A, B, C, D, E, F and G in n dimensional co-ordinate system. This
represents a regression hyper plane. To determine the regression hyper plane, determine a1, b1, c1, d1, e1, f1and g1 in
equation 6 so that,
[P1] = [W1] [X1]
Using Mat lab, X1= W1\ P1 , after solving X1 matrix with K1 and indices a1, b1, c1, d1, e1, f1, g1 are as follows
K1
A1
B1
C1
D1
E1
F1
G1
-3.2593
0.3195
0.2209
-1.0589
-4.4256
0.1162
-0.2425
0.0444
But K1 is log value so to convert into normal value taking antilog of K1
Antilog (-3.2593) = 0.0005504273433
Hence the model for dependent term D1
01 = k1 x (1)a1 x (2)b1 x (3)c1 x (4)d1 x (5)e1 x (6)f1 x (7)g1
(7)
5.1 Clubbed Mathematical model Term
In this type of model all the Pi terms i.e 1, 2, 3, 4, 5, 6 and 7 are multiplied (clubbed)
together and then using regression analysis mathematical model is formed. The mathematical clubbed
model for stirrup making operation is form as below:
01 = f (1, 2, 3, 4, 5, 6, 7 )
(8)
Min
Max
Mean
Median
Mode
Std
Range
Table 4 Statistics of processing time general and clubbed model
General model
Clubbed Model
Data
Model
Experimental
Model
1
45.42
32
45.33
144
110.1
140
113.9
72.5
75.93
76.71
74.88
72.5
76.63
77
73.45
1
45.42
63
45.33
41.71
19.23
22.2
15.02
143
64.72
108
68.55
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Experimental
32
140
76.71
77
63
22.2
108
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140
Model Tp
Expem TP
Model tp
Experimental tp
O u tp u t o f P i 0 1 – T p (E x p e r i m e n t a l a n d M o d e l )
O u tp u t o f P i 0 1 – P i 0 1 - tp (E x p e r i m e n ta l a n d M o d e l )
Comparison between (clubbed) Experimental and Model for Processing Time (Pi01-tp)
Comparison between Experimental and Model for Processing Time (Pi01-tp)
140
ISSN 2319 - 4847
120
100
80
60
40
20
0
50
100
Experiment Number
150
120
100
80
60
40
20
0
Figure 7 (a) comparison between experimental and Math. Model
50
Experiment Number
100
150
(b) clubbed and Math. Modal
6. R2 = CO-EFFICIENT OF DETERMINATION OF MODEL FOR PROCESSING TIME
Co-efficient of Determinant is a statistical method that explains how much of the variability of a factor can be caused or
explained by its relationship to another factor. Coefficient of determination is used in trend analysis. It is computed as a
value between 0 (0 percent) and 1 (100 percent). 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. The coefficient of determination
is an important tool in determining the degree of linear-correlation of variables ('goodness of fit') in regression analysis
and also called r-square. It is calculated using relation shown below:
R2 =1- ∑Yi-fi)2/∑(Yi-Y)2
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 = C0-efficient of Determination
From calculation the value of R2 for general Model is 0.767916 and clubbed model is 0.529713. A value of General
Model indicates a nearly perfect fit, and therefore, a reliable model for future forecasts. A value of clubbed model, on
the other hand, would indicate that the model moderate to accurately model the data set. This shows that General
Model gives better accuracy results as compared to clubbed model.
7. RELIABILITY OF MODEL
Reliability of model is established using relation Reliability =100% mean error and Mean error = Σfi*xi/Σfi where, xi is
% error and fi is frequency of occurrence. Therefore the reliability of General model and Clubbed Model are equal to
89.4375 and 84.6737 respectively. Figure shown 4 graphs between % of Error and frequency occurrence of error for
general and clubbed model.
(a) (b)
Figure 8 Graph between % of Error and frequency occurrence of error for general and clubbed model.
8. 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
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response variables. In this section attempt is made to find out the limiting values of four response variables
viz.processing time, number of bends, resistive torque average and resistive torque total. 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 stirrup making operation is as given in Table. 5
Table 5 Limiting Values of Response Variables (Processing time tp in sec)
Max and Min. of Response π terms
Maximum
Minimum
Stirrup Making operation
Processing Time tp (Π01)sec
151.9334984
33.33721357
9. SENSITIVITY ANALYSIS
The influence of the various independent π terms has been studied by analyzing the indices of the various π terms in the
models. 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 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.
Nature of variation in response variables due to increase in the values of independent pi terms is given in Table 6.
Figure 9 Graph of Sensitivity Analysis and Indices for processing time
Table 6 Sensitivity Analysis (sample) for stirrup bending operation by HPFM
Pi 1
1.99E-08
2.19E-08
1.79E-08
Pi 2
1793.07
1
1793.07
1
1793.07
1
Pi 3
1.7E-08
1.7E-08
1.7E-08
Pi 4
1.26871
7
1.26871
7
1.26871
7
Pi 5
Pi 6
8.59E-07
0.045516
8.59E-07
0.045516
8.59E-07
0.045516
Pi 7
0.56571
4
0.56571
4
0.56571
4
% Change
1.99E-08
1.99E-08
1.99E-08
1.99E-08
79.653
82.116
77.016
6.4022
1793.07
1
1972.37
8
1613.76
4
1.7E-08
1.7E-08
1.7E-08
1.26871
7
1.26871
7
1.26871
7
8.59E-07
0.045516
8.59E-07
0.045516
8.59E-07
0.045516
0.56571
4
0.56571
4
0.56571
4
% Change
1.99E-08
Pi01
79.653
81.348
77.82
4.4283
1793.07
1
1793.07
1
1.7E-08
1.86E-08
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1.26871
7
1.26871
7
8.59E-07
0.045516
8.59E-07
0.045516
0.56571
4
0.56571
4
79.653
72.006
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1793.07
1
1.99E-08
1.53E-08
1.26871
7
8.59E-07
0.045516
0.56571
4
% Change
1.99E-08
1.99E-08
1.7E-08
1.7E-08
1.7E-08
1.26871
7
1.39558
8
1.14184
5
8.59E-07
0.045516
8.59E-07
0.045516
8.59E-07
0.045516
0.56571
4
0.56571
4
0.56571
4
% Change
1.99E-08
1.99E-08
1.7E-08
1.7E-08
1.7E-08
1.26871
7
1.26871
7
1.26871
7
8.59E-07
0.045516
9.44E-07
0.045516
7.73E-07
0.045516
0.56571
4
0.56571
4
0.56571
4
% Change
1.99E-08
1.99E-08
1.7E-08
1.7E-08
1.7E-08
1.26871
7
1.26871
7
1.26871
7
8.59E-07
0.045516
8.59E-07
0.050067
8.59E-07
0.040964
0.56571
4
0.56571
4
0.56571
4
% Change
1.99E-08
1.99E-08
126.97
79.653
80.54
78.684
79.653
77.833
81.714
4.8727
1793.07
1
1793.07
1
1793.07
1
1.99E-08
52.241
2.3305
1793.07
1
1793.07
1
1793.07
1
1.99E-08
79.653
93.82
1793.07
1
1793.07
1
1793.07
1
1.99E-08
89.054
21.403
1793.07
1
1793.07
1
1793.07
1
1.99E-08
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1.7E-08
1.7E-08
1.7E-08
1.26871
7
1.26871
7
1.26871
7
8.59E-07
0.045516
8.59E-07
0.045516
8.59E-07
0.045516
% Change
0.56571
4
0.62228
6
0.50914
3
79.653
79.991
79.281
0.8908
10. MODEL OPTIMIZATION
In this case there is model corresponding to processing time for stirrup making operations. This is the objective
functions corresponding to these models. These models have non linear form; hence it is to be converted into a linear
form for optimization purpose. This can be achieved by taking the log of both the sides of the model. The linear
programming technique is applied which is detailed as below for stirrup making operation of processing time.
For processing time:
(9)
Taking log of both the sides of the Equation, we get
Z = K+ K1+ a x X1+ b x X2+ c x X3+d x X4 + e x X5 +f x X6 + g x X7
Z = log(5.50E-04)+log(0.332304) + 0.3195 log


and
1  +0.2209.log  2  -1.0589.log  3  -4.4256 log  4  +0.1162.log
 
7
( 5 )-0.2425. log ( 6 )+0.0444.log
Z = -3.2593 -0.47846 + 0.3195 x X1 +0.2209. x X2 -1.0589. x X3 -4.4256 x X4 +0.1162. x X5 -0.2425 x X6 +0.0444.
x X7
Z (Processing Time: Π01 min) =-3.2593 -0.47846 + 0.3195 x X1 +0.2209. x X2 -1.0589. x X3 -4.4256 x X4 +0.1162.
x X5 -0.2425 x X6 +0.0444. x X7
(10)
Equation 10 is the optimization equation for processing time
Subject to the following constraints
1 x X1+ 0 x X2 + 0 x X3 + 0 x X4 + 0 x X5 + 0 x X6 + 0 x X7 ≤ -7.3442
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1 x X1+ 0 x X2 + 0 x X3 + 0 x X4 + 0 x X5 + 0 x X6 + 0 x X7 ≥ -8.26057
0 x X1+ 1 x X2 + 0 x X3 + 0 x X4 + 0 x X5 + 0 x X6 + 0 x X7 ≤ 3.546148
0 x X1+ 1 x X2 + 0 x X3 + 0 x X4 + 0 x X5 + 0 x X6 + 0 x X7 ≥ 2.819149
0 x X1+ 0 x X2 + 1 x X3 + 0 x X4 + 0 x X5 + 0 x X6 + 0 x X7 ≤ -7.7186
0 x X1+ 0 x X2 + 1 x X3 + 0 x X4 + 0 x X5 + 0 x X6 + 0 x X7 ≥ -7.83201
0 x X1+ 0 x X2 + 0 x X3 + 1 x X4 + 0 x X5 + 0 x X6 + 0 x X7 ≤ 0.104735
0 x X1+ 0 x X2 + 0 x X3 + 1 x X4 + 0 x X5 + 0 x X6 + 0 x X7 ≥ 0.10199
0 x X1+ 0 x X2 + 0 x X3 + 0 x X4 + 1 x X5 + 0 x X6 + 0 x X7 ≤ -5.82622
0 x X1+ 0 x X2 + 0 x X3 + 0 x X4 + 1 x X5 + 0 x X6 + 0 x X7 ≥ -6.4041
0 x X1+ 0 x X2 + 0 x X3 + 0 x X4 + 0 x X5 + 1 x X6 + 0 x X7 ≤ -1.32887
0 x X1+ 0 x X2 + 0 x X3 + 0 x X4 + 0 x X5 + 1 x X6 + 0 x X7 ≥ -1.35942
0 x X1+ 0 x X2 + 0 x X3 + 0 x X4 + 0 x X5 + 0 x X6 + 1 x X7 ≤ -0.10474
0 x X1+ 0 x X2 + 0 x X3 + 0 x X4 + 0 x X5 + 0 x X6 + 1 x X7 ≥ -0.40577
On solving the above problem by using MS solver we get values of X1,X2,X3,X4,X5,X6,X7 and Z. Thus Π01 min =
Antilog of Z and corresponding to this value of the Π01min the values of the independent pi terms are obtained by taking
the antilog of X1,X2,X3,X4,X5,X6,X7 and Z.
Table 7 Optimized values of response variables
Processing Time: Π01 min
Log
values
 terms
1.515520444
-8.260567775
2.819148943
-7.718598596
0.104735351
-6.404100022
-1.328869076
-0.405765346
Z
X1
X2
X3
X4
X5
X6
X7
of
Antilog of  terms
32.77332042
5.48823E-09
659.4
1.91162E-08
1.272727273
3.94366E-07
0.046895473
0.392857143
11. DISCUSSION OF 3D AND 2D GRAPHS
It is possible to evaluate the behaviour of any model through graphical presentation in order to justify how the real
phenomena work on account of appropriate interaction of independent  terms. An attempt has been made for the
human powered stirrup making operation which is explained below.
In this model there are seven independent  terms and three dependent  terms. It is very difficult to plot a 3D graph.
To obtain the exact 3D graph dependent  terms is taken on Z-axis where as from seven independent  -terms, three
are combined and a product is obtained which is presented on X-axis. The remaining four independent  terms are
combined by taking product and represented on Y-axis. Figure 10to Figure 11 shows 3D and 2D graphs for first
dependent term i.e. processing time
From 3D and 2D graphs it is observed that the phenomenon is complex because of variation in the dependent pi terms
are in a fluctuating form mainly due to stiffness of spring and hardness of stirrup rod. This in turn is due to linearly
varying speed of rotating disc, hardness of rod, centre distance of pin and material of the stirrup rod.
For processing Time there are15 peaks in graph of torque i.e. tp vs. X (shown in Graph 10.a). There must be in all 30
mechanisms responsible for giving these 15 peaks. Whereas, in graph of tp vs. Y, there are 12 peaks. Hence there must
be in all 24 mechanisms are responsible for giving these12 peaks.
3-D Graph for Pi01-Processing Time-tp
120
Z = P i 0 1 -tp
100
80
60
40
7
6
5
4
-8
x 10
3
Y= Pi4xPi5xPi6xPi7
2
1
0
0
0.5
1
2
1.5
X=Pi1xPi2xPi3
2.5
3.5
3
x 10
-12
Figure 10 3-D Graph of Processing time vs Pi01 terms (X= pi1 x pi2 x pi3 , Y = pi4 x pi5 x pi6 x pi7 , Z= pi 01 - tp)
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Volume 4, Issue 6, June 2015
120
120
2-D Graph for Pi01-Processing Time-tp
110
110
100
100
90
90
Z = P i 01-tp
Z = P i 0 1 -tp
2-D Graph for Pi01-Processing Time-tp
80
80
70
70
60
60
50
50
40
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40
0
0.5
1
1.5
X=Pi1xPi2xPi3
2
2.5
3
3.5
0
1
2
-12
3
4
Y= Pi4xPi5xPi6xPi7
5
6
7
-8
x 10
x 10
Figure 11 (a)2D Graph of (X = pi1 x pi2 x pi3 vs Z= Z= pi 01 - tp ) (b) 2-D Graph of (Y= pi4 x pi5 x pi6 x pi7 Vs Z=
pi 01 - tp )
12. COMPUTATIONS OF THE PREDICTED VALUES BY ‘ANN’
In this research the main issue is to predict the future result. In such complex phenomenon involving non-linear system
it is also planned to develop Artificial Neural Network (ANN). The output of this network can be evaluated by
comparing it with observed data and the data calculated from the mathematical models. For development of ANN the
designer has to recognize the inherent patterns. Once this is accomplished training the network is mostly a fine-tuning
process.
Comparis ion between practic al data, equation based dat a and neural based data
160
Practic al
Equation
140
Neural
120
100
80
60
40
20
0
50
100
150
Experimental
Figure 12 Performance analysis of ANN and Comparison of actual and computed data by ANN (Processing Time-Π01)
13. ANALYSIS OF PROCESSING TIME MODELS DEPENDENT TERM
The model for the dependent pi term 01 is as under:
(11)
Constant and Indices of Response variable
PI TERMS
K1
Π1
Π2
Π3
Π4
Π5
Π6
Π7
PROCESSING
0.00055042
0.3195
0.2209
-1.0589
-4.4256
0.1162
-0.2425
0.0444
TIME (TP)
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Volume 4, Issue 6, June 2015
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Figure 13 Pi 01 indices of response variable
The deduced equation for this pi term is given by,
(12)
It would be seen that the equation 11 and fig.12 is a model of a pi term containing Processing Time tp as a response
variable. The following primary conclusion drawn appears to be justified from the above model.
(i) The absolute index of π1 is highest viz.0.3195. The factor ‘π1 is related to energy of flywheel which is the most
influencing term in this model. The value of this index is positive indicating involvement of energy of flywheel has
strong impact on π01 and π01 is directly varying with respect to π1.
(ii) The absolute index of π7 is lowest viz. 0.0444. Thus π7, the term related to angle of bent and Gear ratio which is
the least influencing term in this model. Low value of absolute index indicates that the factor, geometrical parameters
of the machine needs improvement.
(iii) The influence of the other independent pi terms present in this model is π2 and π5 having absolute index of 0.2209
and 0.1162. The indices of π3, π4 and π6 are negative viz. -1.0589,-4.4256 and -0.2425 respectively. The negative indices
are indicating need for improvement. The negative indices indicating that π01 varies inversely with respect to π3, π4 and
π6.
(iv) The constant in this model is 0.0005504273433. This value being less than one, hence it has no magnification
effect in the value computed from the product of the various terms of the model.
9 CONCLUSIONS
1. The dimensionless π terms have provided the idea about combined effect of process parameters in that π terms. A
simple change in one process parameter in the group helps the manufacturer to maintain the required tp,nb, resistive
torque average and resistive torque total values so that the productivity is increased.
2. In the stirrup making operation the mathematical models developed with dimensional analysis for different
combinations of parameters for processing time, number of bends, resistive torque average and resistive torque total
3. The comparison of values of dependent term obtained from experimental data,
mathematical clubbed model is shown in Table 5. From the values of % errors, it seems that the mathematical models
can be successfully used for the computation of dependent terms for a given set of independent terms.
4. RSM model can be also utilized for estimation of maximum and minimum values of response variables i.e tp, nb,
resistive torque average and resistive torque total.
ACKNOWLEDGMENT
I would like to sincere thanks All India Council of Technical Education (AICTE), New Delhi, for granting us a
Research Promotion Scheme grand of Rs. 9.70 Lacs to my Research project “Formulation of Experimental Data Based
Model for Human Energized Flywheel Motor for Stirrup Making” and others two project”
References
[1] J.P. Modak & Mrs. S. D. Moghe, “Design & Development of a Human Powered Machine for the Manufacture of
Lime-Fly-ash-Sand-Bricks” Human Power Vol. 13, pp 3-8, 1998, Journal of International Human Powered Vehicle
Association, U.S.A.
[2] A. R. Bapat, “Experimental Optimization of a Manually Driven Flywheel Motor” M.E. (By Research) Thesis of
Nagpur University 1989, under the supervision of Dr. J. P. Modak.
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International Journal of Application or Innovation in Engineering & Management (IJAIEM)
Web Site: www.ijaiem.org Email: editor@ijaiem.org
Volume 4, Issue 6, June 2015
ISSN 2319 - 4847
[3] J. T. Pattiwar, “Design, Development and Analysis of Torsionally Flexible Clutches for on load Starting of a
Manually Energized Machine” M.E. (by Research) thesis under the supervision of Dr. J. P. Modak, Nagpur
University, 1997.
[4] J.P.Modak, “Manufacture of Lime -Fly-ash Sand Bricks using manually driven brick making machine”, Project
report – project sponsored by MHADA, Bombay, 1982.
[5] S.D.Moghe & J.P.Modak, “Design and Development of a Human Powered Machine for the Manufacture of Lime Fly-ash-Sand Bricks”, Human Power, Vol. 13, pp 3-8, 1998
[6] R.D.Askhedkar & J.P.Modak, “Hypothesis for the Extrusion of Lime-Fly-ash-Sand Bricks Using Manually Driven
Brick Making Machine”, Building Research & Information U.K., Vol. 22, N1, 1994, pp 47-54.
[7] A.V.Vanalkar ,A.R.Bapat , P.M.Padole, “ ergonomic studies of stirrup making processing view of productivity
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[8] A.V.Vanalkar , P.M.Padole ,” Machine synthesis for stirrup making machine”, 4th International conference on
Mechanical Engineering , ICME,BUET,DHAKA, Bangaldesh,2001
[9] A.V.Vanalkar , P.M.Padole ,”Design & development and fabrication of stirrup making machine”, 9th National
conference on machine & mechanism December NACOMM, I.I.T Pawai, Mumbai, India, 1999.pp. 341-352.
[10] A.V.Vanalkar , P.M.Padole , “Design & Development of feeding mechanism for stirrup making machine,” 10th
National conference on industrial automation and application, PCE&A. Nagpur, 2001.
[11] A.V.Vanalkar , P.M.Padole , “feeding system using gearised D.C motor for stirrup making machine,” 10th
National conference on machine & mechanism NaCOMM, IIT. Kharagpur. 2001.
[12] A.V.Vanalkar , P.M.Padole , “Optimal design for stirrup making machine using computer approach”, 10 th
National conference on machine & mechanism NaCOMM, IIT. New delhi. 2003.
[13] A.V.Vanalkar , P.M.Padole , “ Design development of coupled to uncoupled mechanical system for stirrup
making machine”, 12th National conference on machine & mechanism NaCOMM, New delhi .2007.
[14] H.Schenck Jr, (1961) ‘Reduction of variables Dimensional analysis’, Theories of Engineering Experimentation,
McGraw Hill Book Co, New York, p.p 60-81.
[15] Singiresu S. Rao (2004) ‘Engineering Optimization’ Third Edition, New Age International (P) Ltd. Publishers.
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Graw Hill Book Co, New York , P.P. 6.
AUTHOR
Mr. Subhash N. Waghmare: Author, Presently he is working as Assistant Professor in Mechanical
Engineering Department, Priyadarshini College of Engineering, Nagpur. He is pursuing his Ph.D in
Engg. & Tech. from R.T.M.Nagpur University. His specialization is in Mechanical Engg. Design. He
has published 17 papers in International Journal and Conferences. He also received Governments
funding of Rs. 10.70 Lacs under RPS scheme from AICTE, New Delhi. He is a member of various
bodies like AMM,ISTE, ISHRAE.
Dr. Chandrashekhar N. Sakhale: Author, presently he is working as Associate professor and Coordinator of Ph.D and M.Tech. (MED) in Mechanical Engineering Department, Priyadarshini College
of Engineering, Nagpur. He has completed his Ph.D in Engg. & Tech. from R.T.M. Nagpur
University. His specialization is in Mechanical Engg. Design. He has published 67 papers in,
International Journal and Conferences. He is recipient of U.P. Government Award-2015 from ISTE,
New Delhi (India) for outstanding work done in Engineering. He received young delegate award of
$500 in 13th World Congress of IFToMM at Guanjuato, Mexico. He also received three Governments
funding of Rs. 30 Lacs under RPS scheme from AICTE, New Delhi. He is a member of various bodies like AMM,
ISTE, ISHRAE, ISB. He also published books on “Analysis of Automotive Driveline System using Finite Element
Approach”, LAP Lambert Academic Publishing, Germany. He is a Member and working Chairmen of Board of Studies
Aeronautical Engineering R.T.M. Nagpur University, Nagpur. He had also worked as Head of Aeronautical Engg.
Department.
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