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"OPTIMIZATION OF TWISTING PARAMETERS FOR SINGLE-LAYER
RAW SILK IN THE TEXTILE INDUSTRY"
Мuxammadrasulov Sh.
Sattorov N.
Yuldasheva D.
FSTU
https://doi.org/10.5281/zenodo.16741647
In the silk industry, the process of twisting single-layer weft and warp raw
silk threads during the production of woven fabrics is being implemented for the
first time. When a certain amount of twist is applied to the warp and weft
threads, the tensile properties of the raw silk threads improve. Therefore,
determining the optimal parameters of the applied twist is an important task. In
this dissertation, experiments were planned as full factorial experiments to
determine the optimal twisting parameters for single-layer raw silk. These
experiments were conducted based on established standards, i.e., a full factorial
design (FFD) was carried out. [1, 392 s.;2, 224 s.].
To carry out the optimization process, input and output factors were
selected.
The following were chosen as
input factors
:
X₁
– Deviation in linear density, tex
X₂
– Number of twists, twists per meter (br/m)
The following were chosen as
output factors
:
Y₁
– Relative breaking strength, cN/tex
Y₂
– Elongation at break, %
In the rotatable central composite design (RCCD), the regression
coefficients and their variances are determined using the following formulas:
𝑏
0
= 𝑔
1
∑ 𝑌
𝑢
𝑁
𝑢=1
− 𝑔
2
∑∗
𝑀
𝑖=1
∑ 𝑥
𝑖𝑢
2
𝑌
𝑢
𝑁
𝑢=1
(1)
𝑏
𝑖
= 𝑔
3
∑
𝑥
𝑖𝑢
𝑌
𝑢
𝑁
𝑢=1
(2)
𝑏
𝑖𝑗
= 𝑔
4
∑
𝑥
𝑖𝑢
𝑥
𝑗𝑢
𝑁
𝑢=1
𝑌
𝑢
(3)
𝑏
𝑖𝑖
= 𝑔
5
∑
𝑥
𝑖𝑢
2
𝑌
𝑢
𝑁
𝑢=1
− 𝑔
6
∑
∗
𝑀
𝑖=1
∑
𝑥
𝑖𝑢
2
𝑌
𝑢
𝑁
𝑢=1
− 𝑔
7
∑
𝑌
𝑢
𝑁
𝑢=1
(4)
𝑆
2
{𝑏
0
} = 𝑔
1
𝑆
2
{𝑌}
𝑆
2
{𝑏
𝑖
} = 𝑔
3
𝑆
2
{𝑌}
(5)
𝑆
2
{𝑏
𝑖𝑗
} = 𝑔
4
𝑆
2
{𝑌}
𝑆
2
{𝑏
𝑖𝑗
} = 𝑔
7
𝑆
2
{𝑌}
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Based on the experimental results, we seek a second-degree multivariable
regression mathematical model. As a result of this experiment, the following
general form of the regression model can be obtained:
𝑌
𝑅
= 𝑏
0
+ ∑ 𝑏
𝑖
𝑥
𝑖
+
𝑀
𝑖=1
∑ 𝑏
𝑖𝑗
𝑥
𝑖
𝑥
𝑗
+
𝑛
𝑖=𝑗=1
∑ 𝑏
𝑖𝑗
𝑥
𝑖
2
𝑀
𝑖=1
Or, since two factors are involved in our experiment, the above expression
takes the following form:
𝑌
𝑅
= 𝑏
0
+ 𝑏
1
𝑥
1
+ 𝑏
2
𝑥
2
+ 𝑏
12
𝑥
1
𝑥
2
+ 𝑏
11
𝑥
1
2
+ 𝑏
22
𝑥
2
2
In the equation:
b₀,
b₁,
...
–
are
the
regression
coefficients,
x₁, x₂, x₃
– are the coded values of the factors.
The basic (central) levels and variation ranges of the factors are determined
using preliminary trial experiments.
Table 1. Basic Levels and Variation Ranges of the Factors
Factors
Levels of variation
I
i
-1,414
-1
0
1
1,414
X
1
2,20
2,23
2,33
2,43
2,46
0,10
X
2
396,00 400,00
450,00
500,00 504,00
50,00
To obtain a regression model representing the stationary level, a Rotatable
Central Composite Design (RCCD) was carried out using the following matrix.
Table 2. Experimental Matrix of the RCCD
№
Working matrix
Y₁ – Relative breaking
strength, cN/tex
Y
2
–
Relative elongation at
break, %
x
1
x
2
x
12
x
1
x
2
x
22
Y
1u
Y
1Ru
(Y
u
- Y
Ru
)
2
Y
u
Y
Ru
(Y
u
- Y
Ru
)
2
1
-1
-1
1 1
1
37,86
37,98 0,02
22,13
22,17 0,00
2
1
-1
1 -1
1
37,05
37,31 0,07
22,75
22,81 0,00
3
-1
1
1 -1
1
37,45
37,54 0,01
23
22,87 0,02
4
1
1
1 1
1
36,05
36,27 0,05
23,42
23,31 0,01
5
-1,414 0
2 0
0
37,97
37,77 0,04
22,7
22,68 0,00
6
1,414
0
2 0
0
36,78
36,39 0,15
23,5
23,45 0,00
7
0
-1,414
0 0
2
38,31
37,99 0,10
22,25
22,09 0,03
8
0
1,414
0
0
2
37,22
36,95 0,07
22,85
22,94 0,01
9
0
0
0
0
0
37,45
37,94 0,24
23,15
23,11
0,00
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10 0
0
0
0
0
38,12
37,94 0,03
22,85
23,11
0,07
11 0
0
0
0
0
37,75
37,94 0,04
22,95
23,11
0,03
12 0
0
0
0
0
38,1
37,94 0,02
23,5
23,11
0,15
13 0
0
0
0
0
38,3
37,94 0,13
23,1
23,11
0,00
Based on this matrix, each experiment is repeated three times to determine
the relative breaking strength of the raw silk yarn, and the results are entered
into column 7 of the above table.
Using the table and the formulas mentioned above, we first calculate the
following values in order to determine the regression coefficients:
∑ 𝑥
1𝑢
𝑌
𝑢
=
-3,8927;
∑ 𝑥
2𝑢
𝑌
𝑢
=
-2,9513;
∑ 𝑥
1𝑢
2
𝑌
𝑢
=
297,91;
∑ 𝑥
1𝑢
𝑥
2𝑢
𝑌
𝑢
=
-0,59;
∑ 𝑥
2𝑢
2
𝑌
𝑢
=
299,47;
∑ 𝑌
𝑢
=
488,4;
∑
(𝑌
𝑢𝑀
− 𝑌
𝑀
𝑁
𝑀
𝑢=1
)
2
=
0,464
𝑌
𝑀
=
37,94
∑
(𝑌
𝑢
− 𝑌
𝑅𝑢
𝑁
𝑢=1
)
2
=
0,966.
Then, we determine the regression coefficients:
𝑏
0
=
0,2*488,4-0,1*597,38=37,9
𝑏
1
=
0,125*(-3,8927) = -0,49
𝑏
2
=
0,125*(-2,95) = -0,37
𝑏
12
=
0,25* (-0,59) = -0,15
𝑏
11
=
0,125*297,91+0,0187*597,38-0,1*488,4 = - 0,43
𝑏
22
=
0,125*299,47+0,0187*597,38-0,1*488,4 = - 0,24
The output parameter variance is calculated using the following formula:
𝑆
2
= {𝑌} = 𝑆
𝑀
2
{𝑌}
=
0,5
5−1
=
0,12
The variances of the regression coefficients are determined using Formula
5:
𝑆
2
{𝑏
0
}
=0,2*0,12= 0,02
S(
b
0
)=0,15
𝑆
2
{𝑏
𝑖
} =
0,125*0,012=0,015
S(
b
0
)=0,15
𝑆
2
{𝑏
𝑖𝑗
} =
0,25*0,015=0,004
S(
b
0
)=0,06
𝑆
2
{𝑏
𝑖𝑗
} =
0,1438*0,004=0,000057
S(
b
0
)= 0,02
We determine the calculated values of the Student's t-test for the estimated
coefficients.
𝑡
𝑅
{𝑏
0
} =
37,9/0,15=253
𝑡
𝑅
{𝑏
1
}
= (-0,49)/0,12=4,08
𝑡
𝑅
{𝑏
2
}
=(-0,37)/0,12=3,08
𝑡
𝑅
{𝑏
12
}
=(-0,15)/0,06=2,5
𝑡
𝑅
{𝑏
11
}
=18,89
𝑡
𝑅
{𝑏
22
}
=10,35
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The calculated value of the Student's t-test is compared with the tabulated
value of this criterion taken from Appendix 3 of U.Kh. Meliboyev's textbook
"Fundamentals of Modeling Technological Processes in the Textile Industry."
[102, 224 s.].
𝑡
жад
[R=0,95; f
{𝑆
𝑀
2
}
=5-1=4]=
2,776
As seen from the comparisons, all regression coefficients except for
b12b_{12}
b12 are significant.
Thus, the following relationship exists between the deviation in linear
density, the number of twists, and the relative breaking strength:
𝑌
𝑅
= 37,9 − 0,49𝑥
1
− 0,37𝑥
2
− 0,43𝑥
1
2
− 0,24𝑥
2
2
We now test the hypothesis regarding the adequacy of the second-degree
multivariable regression model obtained from the RCCD. To do this, the
calculated value of the
Fisher criterion
is determined using the following
formula:
𝑆
над
2
{𝑌} =
0,966 − 0,464
13 − 5 − (5 − 1)
= 0,083
𝐹
𝑅
=
𝑆
над
2
{𝑌}
𝑆
𝑀
2
{𝑌}
=
0,083
0,464
= 0,18
The tabulated value of this criterion is taken from Appendix 4 of U.Kh.
Meliboyev's textbook
"Fundamentals of Modeling Technological Processes in the
Textile Industry"
and, for our example, is equal to the following:
F
j
[
P
d
= 0,95;
f
{
𝑆
над
2
{Y}}=13-5-(3-1)=6;
f
{
𝑆
𝑀
2
}=5-1=4]=4,53
If
FR<FtabF_R < F_{\text{tab}}
FR <Ftab , the model is considered adequate.
In the case under investigation:
F
R
=
0,18
<
4,53
= F
jad
Thus, the obtained regression mathematical model represents the studied
process with sufficient accuracy and is considered adequate.
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Figure 1. Graph showing the relationship between deviation in linear
density, number of twists, and relative breaking strength
As seen from the above graph, the variation of the first input factor (
x1x_1
x1
) and the second input factor (
x2x_2
x2 ) from their minimum (-1) to maximum
(1) values, using the average value of the first factor, illustrates the changes in
the relative breaking strength (
Y1Y_1
Y1 ). The graph visualizes the behavior of
Y1Y_1
Y1 — relative breaking strength — within the range of
x1x_1
x1 (deviation
in linear density, tex) from 2.20 to 2.46 and
x2x_2
x2 (number of twists,
twists/meter) from 396 to 504. It was found that the relative breaking strength
of the yarn reaches its highest values in the range where deviation in linear
density is between -1 and 0, and the number of twists is between -1 and -0.2.
Y₂ – Calculations related to elongation at break:
The planning matrix with the research results is presented in the table
above. Based on the table and using the formulas mentioned earlier, we first
calculate the following values in order to determine the regression coefficients:
∑ 𝑥
1𝑢
𝑌
𝑢
= 2,17
;
∑ 𝑥
2𝑢
𝑌
𝑢
=
2,38;
∑ 𝑥
1𝑢
2
𝑌
𝑢
=
183,7;
∑ 𝑥
1𝑢
𝑥
2𝑢
𝑌
𝑢
=
- 0,2;
∑ 𝑥
2𝑢
2
𝑌
𝑢
=
181,5;
∑ 𝑌
𝑢
=
298,2;
∑
(𝑌
𝑢𝑀
− 𝑌
𝑀
𝑁
𝑀
𝑢=1
)
2
=
0,247
𝑌
𝑀
=
23,11
∑
(𝑌
𝑢
− 𝑌
𝑅𝑢
𝑁
𝑢=1
)
2
=
0,318
Co‘ngra regressiya koeffisiyentlarini aniqlaymiz:
𝑏
0
=
0,2*298,2-0,1*(183,7+181,5)=23,11
𝑏
1
=
0,125*2,17=0,27
𝑏
2
=
0,125*2,38=0,3
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𝑏
12
=
0,25*(-0,2) = - 0,05
𝑏
11
=
0,125*183,7+0,0187*365,2-0,1*298,2= -0,02
𝑏
22
=
0,125*181,5+0,0187*365,2-0,1*298,2= -0,3
The output parameter variance is calculated using the following formula:
𝑆
2
= {𝑌} = 𝑆
𝑀
2
{𝑌}
=
0,247
5−1
=
0,06
The variances of the regression coefficients are determined using Formula
5:
𝑆
2
{𝑏
0
}
=0,2*0,06=0,012
S(
b
0
)=0,11
𝑆
2
{𝑏
𝑖
} =
0,125*0,06=0,008
S(
b
0
)=0,09
𝑆
2
{𝑏
𝑖𝑗
} =
0,25*0,008= 0,002
S(
b
0
)=0,04
𝑆
2
{𝑏
𝑖𝑗
} =
0,1438*0,002=0,0003 S(
b
0
)= 0,02
We calculate the Student’s t-test values for the estimated coefficients.
𝑡
𝑅
{𝑏
0
} =
23,11/0,11= 210;
𝑡
𝑅
{𝑏
1
}
=0,27/0,09=3;
𝑡
𝑅
{𝑏
2
}
=0,3/0,09=3,4;
𝑡
𝑅
{𝑏
12
}
=(-0,05)/0,04=1,25;
𝑡
𝑅
{𝑏
11
}
=(-0,02)/0,02=1;
𝑡
𝑅
{𝑏
22
}
=(-0,3)/0,02=15;
The calculated value of the Student’s t-test is compared with the tabulated
value of this criterion, which is taken from Appendix 3 of U.Kh. Meliboyev’s
textbook
“Fundamentals of Modeling Technological Processes in the Textile
Industry.”
𝑡
жад
[R=0,95; f
{𝑆
𝑀
2
}
=5-1=4]=2,776
As seen from the comparisons, all regression coefficients are
significant except for
b12b_{12}
b12 and
b11b_{11}
b11 .
Thus, the following relationship exists between the deviation in linear
density, the number of twists, and the elongation at break:
𝑌
𝑅
= 23,11 + 0,27𝑥
1
+ 0,3𝑥
2
− 0,3𝑥
2
2
We test the hypothesis regarding the adequacy of the second-degree
multivariable regression model obtained through the RCCD. To do this, the
calculated value of the Fisher criterion is determined using the following
formula:
𝑆
над
2
{𝑌} =
0,318 − 0,247
13 − 5 − (5 − 1)
= 0,018
𝐹
𝑅
=
𝑆
над
2
{𝑌}
𝑆
𝑀
2
{𝑌}
=
0,018
0,247
= 0,072
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The tabulated value of this criterion is taken from Appendix 4 of U.Kh.
Meliboyev’s textbook
“Fundamentals of Modeling Technological Processes in the
Textile Industry”
and, for our example, is equal to the following:
F
j
[
P
d
= 0,95;
f
{
𝑆
над
2
{Y}}=13-5-(5-1)= 4;
f
{
𝑆
𝑀
2
}=5-1=4]=6,39
If
FR<FtabF_R < F_{\text{tab}}
FR <Ftab , the model is considered adequate. In
the case under investigation:
F
R
=
0,072<6,39=
F
jad
Thus, the obtained regression mathematical model represents the
investigated process with sufficient accuracy and is considered adequate.
Figure 2. Graph showing the relationship between deviation in linear
density and number of twists with elongation at break
As seen from the above graph, when the first input factor (
x1x_1
x1 ) and the
second input factor (
x2x_2
x2 ) vary from their minimum (-1) to maximum (1)
values, and the average value of the first factor is used, the changes in elongation
at break (
Y1Y_1
Y1 ) are illustrated. Using the graph, the elongation at break is
shown for values of
x1x_1
x1 (deviation in linear density, tex) ranging from 2.20
to 2.46, and
x2x_2
x2 (number of twists, twists/meter) ranging from 396.0 to
504.0. It was found that the elongation at break reaches its highest values in the
ranges where deviation in linear density is between 0.6 and 1, and the number
of twists is between 0.2 and 1.
References:
1.
Sevostyanov, A.G. – Methods and Means of Studying Mechanical and
Technological Processes in the Textile Industry / A.G. Sevostyanov // Moscow:
Legkaya Promyshlennost, 1982, 392 pages.
ACADEMIC RESEARCH IN MODERN SCIENCE
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2.
Meliboyev, U.Kh. – Fundamentals of Modeling Technological Processes in
the Textile Industry / U.Kh. Meliboyev // Namangan: “Adabiyot Uchqunlari”
Publishing, 2020, 224 pages.
