Optimization of a Two-stage Helical Gearbox with Second Stage Double
Gear Sets to Reduce Gearbox Mass and Increase Gearbox Efficiency
TRIEU QUY HUY1, NGUYEN VAN BINH2, DINH VAN THANH3, TRAN HUU DANH4,
NGUYEN VAN TRANG5*
1University of Economics - Technology for Industries,
Ha Noi,
VIETNAM
2Nguyen Tat Thanh University,
Ho Chi Minh City,
VIETNAM
3East Asia University of Technology,
Bac Ninh,
VIETNAM
4Vinh Long University of Technology Education,
Vinh Long,
VIETNAM
5Thai Nguyen University of Technology,
Thai Nguyen,
VIETNAM
*Corresponding Author
Abstract: - This work aims to identify the key design components for lowering gearbox mass and raising
gearbox efficiency by multi-target optimizing a two-stage helical gearbox (THG) with second-stage double
gear sets (SSDGS). In the study, the Taguchi technique and grey relation analysis (GRA) were applied to
handle optimization work in two phases. The single-target problem was addressed first to narrow the separation
between variable levels, and then multi-target work was addressed to establish the best primary design
variables. The first and second-stage CWFW coefficients, as well as the allowed contact stresses (ACS) and the
gear ratio of the first stage, were computed. The outcomes of the study can be applied to determine the best
values for the main essential design factors of a THG with SSDGS.
Key-Words: - Helical gearbox, Double gear sets, Optimization, Multi-objective, Gear ratio, Gearbox mass,
Gearbox efficiency.
Received: April 9, 2023. Revised: October 22, 2023. Accepted: November 21, 2023. Published: December 31, 2023.
1 Introduction
A gearbox, a chain driver, and a motor can all be
found in a mechanical drive system (Figure 1). The
primary component of the system is a gearbox. Its
purpose is to lessen the torque and speed transfer
from the motor shaft to the working shaft. As a
result, several studies are concentrating on the best
gearbox design.
To date, the optimal gearbox design has been
done on a range of issues. To lower the gearbox
cost, [1], studies the impact of eleven input
parameters on second and third-stage ratios. The
study, [2], outlines a study that optimized the gear
ratios in a system including a gearbox and a chain
drive. The authors in, [3], conducted a simulation
study to design a two-stage gearbox for getting
minimal gearbox volume. The optimal gear ratios
for designing a helical gearbox were proposed in,
[4], to get the minimum area of the cross-section of
the gearbox. For this type of gearboxes, a design
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DOI: 10.37394/232011.2023.18.27
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with the used of hybrid composite gears was
suggested in, [5]. The study, [6], proposed two
methods for determining optimal gear ratios to
minimize helical reducer cross-sectional area in the
same region of concern. Besides, a new design of
gearbox housing was introduced in, [7]. Apart from
that, [8], conducted a simulation experiment to
study the link between partial gear ratios and input
factors, from which models to find the optimum
gear ratio were established. For a three-stage helical
gearbox with SSDGS, [9], proposed models to
determine ideal partial ratios to minimize the length,
the mass, and the cross-section area of the gearbox.
Recognizing the significance of gearbox cost
reduction in both design and construction, [10],
computed cost for helical reducers with SSDGS
utilizing component mass. Furthermore, optimal
gear ratios of a two-step helical reducer with
SSDGS to reduce the system length were introduced
in, [11]. Also, for the same type of gearboxes, [12],
presented the gear ratio model to minimize the
gearbox cross-section area. A variety of multi-target
optimization studies for helical gearboxes, [13], and
bevel helical gearboxes, [14], have recently been
published.
Fig. 1: A belt conveyor drive system, [15].
Note: 1) Motor 2) Coupling 3) Gearbox 4) Chain drive
5) Belt conveyor
Scientists are interested in the ideal design of a
two-step reducer with SSDGS. In, [16], the findings
of optimal partial gear ratios in a two-stage reducer
with SSDGS for obtaining the smallest cross-section
area are presented. The study took into account the
impact of the input variables such as the overall
ratio, the CWFW, the ACF, and the output torque.
Also for this gearbox, the problem of finding
optimal ratios was conducted in, [15], for minimum
gearbox cost. The impact of the key design
characteristics and cost components on the ideal
ratios was explored in this work. A model for
calculating the ideal gear ratio has also been
provided. In, [17], optimal gear ratios were
determined for a drive system with a helical reducer
using SSDGS and a chain drive to find the
minimum system length.
According to the previous study, a lot of studies
have been conducted on optimizing gearbox design
in general, and THG with SSDGS in particular, up
to this point. The research on a THG with SSDGS,
on the other hand, only has a single-target
optimization problem.
The goal of this research is to explore multi-
target optimization learning for a THG utilizing
SSDGS. In this effort, two single goals were
pursued: reducing gearbox mass and increasing
gearbox efficiency. In addition, the CWFW and the
ACS of stages 1 and 2, and the gear ratio for stage 1
were also examined. Furthermore, the multi-target
work in this study was solved in two stages by using
the Taguchi approach with GRA. The optimum key
design parameters were also provided to design
reducers with SSDGS.
2 Optimization Problem
2.1 Gearbox Mass Determination
The gearbox mas mgb is determined by (Figure 1):
(1)
In (1), mg, mgh, and ms are the mass of gears,
gearbox housing, and shafts. The parts that follow
specify these mass elements.
2.1.1 Gear Mass Determination
The mass of gears is determined by:
(2)
In which, mg1 and mg2 denote the mass of gears
of stages 1 and 2:
(3)
(4)
In (3) and (4), ρg is gear density (kg/m3); e1 and
e2 are volume coefficients of pinion and gear; e1=1
and e2 =0.6, [1], bw1 and bw2 are the gear widths of
stage 1 and stage 2; dw11, dw12, dw21, and dw22 are the
pitch diameters of the pinion and the gear sets which
are calculated by, [4]:
(5)
(6)
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(7)
(8)
(9)
(10)
In the above equations, Xba1 and Xba2 are
CWFW of stage 1 and 2; aw1 and aw2 are the center
distances of stages 1and 2, [4]:
(11)
(12)
Where u1 is the gear ratio of stage 1;
is the contacting load ratio; is the
ACS of stage 1; ka=43 (Mpa1/3) is the material
factor, [4]; T11 and T12 are the torque on the pinions
of stage 1 and 2 (Nmm) which can be found by:
(13)
(14)
Where Tout is the output torque (Nmm); ηhg is
the helical gear efficiency ( [4],
ηb is the bearing efficiency (ηb=0.99÷0.995, [4]).
2.1.2 Gear Mass Determination
The mass of the gearbox housing can be determined
by:
(15)
In which Vgh is the mass of gearbox housing
(m3) which is calculated by:
(16)
Where, L, B, and H are found by (Figure 1):
(17)
(18)
(19)
In (17), [4], , , are
gear pitch diameters of stages 1 and 2 which are
found by, [4]:
(20)
(21)
(22)
Fig. 2: Calculated schema
Where, and are the center distances
and bw1 and bw2 are the gear width of stages 1 and 2.
These components can be calculated by, [4]:
(23)
(24)
(25)
(26)
In the above equation, is the material
coefficient, [4], and are the contacting
load ratio of stages 1 and 2; and
[4]. and are ACS
(MPa) and u1 and u2 denote the gear ratios of stages
1 and 2. and are CWFW and and
are the pinion torque (Nmm) of stages 1 and 2:
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(27)
(28)
Where Tout is the output torque (N.mm); ηhg is
the gear efficiency ( ); ηb is the
rolling bearing efficiency (ηh=0.99÷0.995), [4].
2.1.3 Shaft Mass Determination
The shaft mass (kg) is determined by:
(29)
Where
(30)
(31)
(32)
In which, ms1, ms2, and ms3 indicate the mass of
shafts 1, 2, and 3 (kg); ρs is the shaft density
(kg/m3); ls1, ls2, and ls3 denote the length of the shafts
1, 2, and 3 which are determined by (Figure 2):
(33)
(34)
(35)
In Equations (24) to (26), ds1, ds2, and ds3 are the
shaft diameters (mm) which are found by, [4]:
(36)
(37)
(38)
In (36) to (38), [τ]=17 (MPa) is the permissible
shear stress.
2.2 Gearbox Efficiency Determination
The gearbox efficiency can be found by:
(39)
With is the overall power loss of the gearbox,
[18]:
(40)
Where, is overall gear power loss; is
bearing power loss; is seal power loss which is
determined by:
+) The power loss in all gears:
(41)
With is the gear power losses of i stage:
(42)
In which, is the anticipated gear efficiency
of the i stage, [19]:
(43)
In (43), is the gear ratio of i stage; f is the
friction coefficient; and are the arcs of
approach and retreat on the i stage which can be
determined by, [19]:
(44)
(45)
In which, and are the outer radiuses;
and are the pitch radiuses and and
are the base-circle radiuses of the pinion and gear; α
is the pressure angle.
The friction coefficient f in Equation (43) depends
on sliding velocity v, [14]:
- If v ≤ 0.424 (m/s):
(46)
- If v > 0.424 (m/s) :
(47)
+) The power loss in bearings can be calculated by,
[18]:
(48)
Wherein, F denotes the bearing load (N);
represents the bearing friction coefficient,
[18], v indicates the peripheral speed; i mean the
bearing ordinal number (i = 1÷6).
+) The total power loss in seals can be found by
[18]:
(49)
In which is the single seal power loss (w):
(50)
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With indicates the ISO Viscosity Grade
number.
2.3 Target Function and Constraints
2.3.1 Target Functions
The following are the objectives of the multi-target
optimization problem:
Reducing the mass of the gearbox:
(51)
Enhancing the gearbox efficiency:
(52)
Where X denotes the design variable vector. As
variables in this study, five primary design
parameters were chosen: , , , , and
. As a result, we have:
(53)
2.3.2 Constraints
The constraints on the multi-objective function are
as follows:
and (54)
and (55)
and (56)
3 Problem Solution
In this study, five main design variables were
chosen for consideration. These variables and their
minimum and maximum values are shown in Table
1. Compared to single-objective optimization issues,
multi-objective optimization (MOO) problems are
more complex because they require simultaneously
maximizing several competing objectives. In reality,
MOO problems can be solved using a variety of
strategies and tactics; the approach you choose will
rely on the specifics of the situation as well as the
trade-offs you wish to make. Multi-objective
optimization (also known as issues optimization)
can be solved in several ways. For instance, swarm
intelligence, evolutionary algorithms, and Pareto-
based techniques. In this paper, the MOO issue was
solved using GRA and the Taguchi approach. To
optimize the number of levels for each variable, the
L25 (55) design was employed. Among the variables
investigated, however, u1 has a nearly large range
(u1=1÷9, Table 1). Even with five levels, the
difference in the values of these qualities remained
significant (in this case, the difference is ((9-1)/4 =
2).
The 2-stage multi-objective optimization
problem solution technique was proposed in, [13],
(Figure 3). In this method, the first stage of this
method addresses a single-target optimization
problem to reduce the gap between variable values
scattered throughout a wide range while the second
addresses a multi-target optimization work to find
the best core design features.
Table 1. Key design parameters and their maximum
and lowest restrictions
Factor
Notation
Lower limit
Upper limit
Gear ratio of stage 1
u1
1
9
CWFW of stage 1
Xba1
0.25
0.3
CWFW of stage 2
Xba2
0.25
0.4
ACS of stage 1 (MPa)
350
420
ACS of stage 2 (MPa)
350
420
Fig. 3: Method to solve multi-objective problem,
[13]
4 Single-target Optimization
As noted above, the single-target optimization
problem was solved to reduce the gap between
variable values scattered throughout a wide range.
Also, in this work, the direct search technique was
applied to deal with the single-target optimization
problem. In addition, a Matlab-based computer
program was developed to address two single-target
problems: lowering gearbox mass and increasing
gearbox efficiency. Based on the program's findings,
Figure 4 displays a connection between the
optimum gear ratio of stage 1 u1 and the overall
ratio ut for both two single objectives. From the
results of the figure, new limitations for the variable
u1 have been added (Table 2).
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Fig. 4: Relation between u1 and ut
Table 2. New constraints of u1
ut
u1
Lower limit
Upper limit
15
1.09
3.07
20
1.63
3.82
25
2.14
4.53
30
2.52
5.2
35
2.86
5.83
40
3.16
6.45
5 Multi-objective Optimization
The aim of this study's multi-target optimization
work for a THG with SSDGS is to find the best
primary design factors with a given total gear-box
ratio that fulfills two single-target functions:
reducing gearbox mass and optimizing gearbox
efficiency. To accomplish this, a computer
experiment was carried out. Table 3 shows the key
design elements and their values for ut=20. The
experimental design was created using the Taguchi
technique and the L25 (55) design, and the data was
analyzed using Minitab R18 software. The design
and results of the experiment for ut=20 are shown in
Table 4 (Appendix).
Multi-objective optimization issues are handled
by the Taguchi and GRA approaches. The main
stages of this approach are as follows:
+) Using the following equations, calculate the
signal-to-noise ratio (S/N):
Table 3. Key parameters and levels for ut = 20
Factor
Notation
Level
1
2
3
4
5
Gear ratio of stage
1
u1
1.63
2.1775
3.725
3.2725
3.82
CWFW of stage 1
Xba1
0.25
0.2625
0.275
0.2875
0.3
CWFW of stage 2
Xba2
0.25
0.2875
0.325
0.3625
0.4
ACS of stage 1
(MPa)
AS1
350
368
386
404
420
ACS of stage 2
(MPa)
AS2
350
368
386
404
420
The higher the S/N, the smaller the gearbox mass:
(57)
The greater the S/N, the more efficient the gearbox:
(58)
Where yi is the output result and m=1 is the
experimental repetition number because this is a
simulation. Table 5 (Appendix) provides the
computed S/N values for the objectives.
The data quantities for the two single-objective
functions differ. To ensure comparability, the data
must be normalized, or brought to a suitable scale.
The normalization value Zij, which is 0 to 1, is used
to normalize the data. This value is calculated using
the following formula:
(59)
With n=25 is the experimental run.
+) The grey relational factor is determined by:
(60)
Where, i=1,2,...,n; k=2 is the objective number;
with Z0(k) and Zj(k) are
the reference and particular comparison sequence;
and are the minimum and maximum
values of i(k); ζ=0.5 is the characteristic coefficient.
+) Determining the coefficient of grey relations by:
(61)
Where yij is the grey relation value (GRV) of the
ith experiment's jth output target. Table 6 (Appendix)
displays the estimated grey relation number yi as
well as the average GRV(
) for each experiment.
It is advised to use a higher average GRV to
bring the output parts into harmony. This enables
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the objective function of the multi-objective issue to
be converted into a single-objective optimization
problem, the mean grey relation value being the
result.
6 Results and Discussions
Table 7 (Appendix) displays the results of an
ANOVA test run to investigate the effect of the
major design factors on the average GRV(
).
According to Table 7 (Appendix), u1 has the most
influence on
(36.36%), followed by AS2
(28.73%), Xba2 (26.39%), Xba1 (3.50), and AS1
(2.79%). Table 8 displays the order of the effect of
the key parameters on (
) using ANOVA analysis.
Table 4. Response table for means
Level
u1
Xba1
Xba2
AS1
AS2
1
0.644
0.5722
0.626
0.5402
0.5013
2
0.5705
0.5745
0.5748
0.5452
0.5172
3
0.5383
0.5504
0.5458
0.5623
0.5549
4
0.5186
0.5489
0.5273
0.5546
0.5904
5
0.5092
0.5346
0.5067
0.5784
0.6168
Delta
0.1349
0.0398
0.1193
0.0382
0.1155
Rank
1
4
2
5
3
Average of grey analysis value: 0.556
+) Determining the best key design parameters:
In theory, the best factor set would incorporate
essential design elements with the highest S/N
values. As a result, the S/N ratio influence of the
primary design features (Figure 5) was evaluated.
Furthermore, the ideal set of multi-objective
parameters (corresponding to the red points) may be
easily calculated using the Figure 4 chart. Table 9
provides the appropriate levels and values for the
important design variables of the multi-target
function.
Fig. 5: Main effects plot for S/N ratios
Table 5. Optimal key factors
No.
Key factors
Code
Optimal
Level
Optimal
Value
1
Gear ratio of stage 1
u1
1
1.63
2
CWFW of stage 1
Xba1
1
0.25
3
CWFW of stage 2
Xba2
1
0.25
4
ACS of stage 1 (MPa)
AS1
5
420
5
ACS of stage 2 (MPa)
AS2
5
420
+) Evaluating proposed modeling: Figure 6
displays the Anderson-Darling approach results,
which are applied to estimate the proposed model's
adequacy. The experimental observations' data
points (shown as blue dots in the graph) fall within
the 95% standard deviation zone specified by the
top and bottom boundaries. Furthermore, the p-
value of 0.105 exceeds the level of significance of α
= 0.05. These results show that the proposed model
is appropriate for evaluation.
Fig. 6: Probability plot of
The Anderson-Darling approach is used to
confirm the suitability of the suggested model
(Figure 6). The data corresponding to the
experimental points (blue dots) are all contained
within the area bounded by two upper and lower
limit lines with a 95% standard deviation limit,
according to the results from this figure.
Additionally, the applied empirical model is
adequate because the P value of 0.105 is higher than
the value = 0.05.
Proceed as with ut=20, but with ut values 15, 25,
30, 35, and 40. Table 10 shows the optimum values
of the five basic design parameters at various ut.
Figure 7 depicts the relationship between the
optimal first-stage gear ratio and the overall gearbox
ratio. To achieve the ideal values of u1, the
following regression formula (with R2=0.9986) was
presented:
(62)
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After having u1, u2=ut/u1 determines the
optimum gear ratio of the second stage.
Table 6. Optimal values of main design parameters
No.
ut
15
20
25
30
35
40
u1
1.09
1.63
2.14
2.52
2.86
3.16
Xba1
0.25
0.25
0.262
5
0.262
5
0.262
5
0.262
5
Xba2
0.25
0.25
0.25
0.25
0.25
0.25
AS1
420
420
420
420
420
420
AS2
420
420
420
420
420
420
Fig. 7: Relation between optimal gear ratio of stage
1 and total ratio
7 Conclusion
This paper discusses the results of a multi-objective
optimization study on optimizing a two-step helical
gearbox with SSDGS to reduce gearbox across
section area and enhance gearbox efficiency. The
first stage of this research improved the gear ratio,
wheel face width efficiency in stages 1 and 2, and
permissible contact stress in steps 1 and 2. To
address this issue, a simulation experiment based on
the Taguchi L25 type was designed and carried out.
The impact of major design elements on the multi-
objective goal was also studied. The gear ratio u1
was discovered to have the greatest influence on
(36.36%), followed by AS2 (28.73%), Xba2
(26.39%), Xba1 (3.50), and AS1 (2.79%).
Additionally, the ideal settings for the important
gearbox features have been recommended. To
determine the ideal first stage u1 gear ratio, a
regression technique (Equation (62) was also
implemented.
Acknowledgement:
The Thai Nguyen University of Technology
supported this work.
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WSEAS TRANSACTIONS on APPLIED and THEORETICAL MECHANICS
DOI: 10.37394/232011.2023.18.27
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Dinh Van Thanh, Tran Huu Danh,
Nguyen Van Trang
E-ISSN: 2224-3429
295
Volume 18, 2023
APPENDIX
Table 7. Experimental plan and output results for ut=20
No.
Input Factors
mgb
ηgb
u1
Xba1
Xba2
AS1
AS2
(kg)
(%)
1
1.6300
0.2500
0.2500
350
350
231.475
95.572
2
1.6300
0.2625
0.2875
368
368
226.429
95.540
3
1.6300
0.2750
0.3250
386
386
222.282
95.534
4
1.6300
0.2875
0.3625
404
404
218.818
95.500
5
1.6300
0.3000
0.4000
420
420
216.390
95.498
6
2.1775
0.2500
0.2875
386
404
207.879
95.381
7
2.1775
0.2625
0.3250
404
420
205.853
95.353
8
2.1775
0.2750
0.3625
420
350
222.889
95.385
9
2.1775
0.2875
0.4000
350
368
223.271
95.326
10
2.1775
0.3000
0.2500
368
386
212.696
95.407
11
2.7250
0.2500
0.3250
420
368
211.044
95.211
12
2.7250
0.2625
0.3625
350
386
212.869
95.211
13
2.7250
0.2750
0.4000
368
404
210.344
95.188
14
2.7250
0.2875
0.2500
386
420
201.561
95.210
15
2.7250
0.3000
0.2875
404
350
216.044
95.276
16
3.2725
0.2500
0.3625
368
420
204.317
95.070
17
3.2725
0.2625
0.4000
386
350
217.962
95.107
18
3.2725
0.2750
0.2500
404
368
207.604
95.116
19
3.2725
0.2875
0.2875
420
386
205.258
95.092
20
3.2725
0.3000
0.3250
350
404
208.465
95.098
21
3.8200
0.2500
0.4000
404
386
208.831
94.972
22
3.8200
0.2625
0.2500
420
404
199.810
94.985
23
3.8200
0.2750
0.2875
350
420
204.252
94.981
24
3.8200
0.2875
0.3250
368
350
216.388
94.995
25
3.8200
0.3000
0.3625
386
368
213.575
94.995
Table 8. S/N values for each run when ut=20
No.
Input Factors
mgb
ηgb
u1
Xba1
Xba2
AS1
AS2
(kg)
S/N
(%)
S/N
1
1.6300
0.2500
0.2500
350
350
231.475
-47.2901
95.572
39.6066
2
1.6300
0.2625
0.2875
368
368
226.429
-47.0986
95.540
39.6037
3
1.6300
0.2750
0.3250
386
386
222.282
-46.9381
95.534
39.6032
4
1.6300
0.2875
0.3625
404
404
218.818
-46.8017
95.500
39.6001
5
1.6300
0.3000
0.4000
420
420
216.390
-46.7047
95.498
39.5999
6
2.1775
0.2500
0.2875
386
404
207.879
-46.3562
95.381
39.5892
7
2.1775
0.2625
0.3250
404
420
205.853
-46.2711
95.353
39.5867
8
2.1775
0.2750
0.3625
420
350
222.889
-46.9618
95.385
39.5896
9
2.1775
0.2875
0.4000
350
368
223.271
-46.9766
95.326
39.5842
10
2.1775
0.3000
0.2500
368
386
212.696
-46.5552
95.407
39.5916
11
2.7250
0.2500
0.3250
420
368
211.044
-46.4875
95.211
39.5737
12
2.7250
0.2625
0.3625
350
386
212.869
-46.5622
95.211
39.5737
13
2.7250
0.2750
0.4000
368
404
210.344
-46.4586
95.188
39.5716
14
2.7250
0.2875
0.2500
386
420
201.561
-46.0881
95.210
39.5737
15
2.7250
0.3000
0.2875
404
350
216.044
-46.6908
95.276
39.5797
16
3.2725
0.2500
0.3625
368
420
204.317
-46.2061
95.070
39.5609
17
3.2725
0.2625
0.4000
386
350
217.962
-46.7676
95.107
39.5642
18
3.2725
0.2750
0.2500
404
368
207.604
-46.3447
95.116
39.5651
19
3.2725
0.2875
0.2875
420
386
205.258
-46.2460
95.092
39.5629
20
3.2725
0.3000
0.3250
350
404
208.465
-46.3807
95.098
39.5634
21
3.8200
0.2500
0.4000
404
386
208.831
-46.3959
94.972
39.5519
22
3.8200
0.2625
0.2500
420
404
199.810
-46.0123
94.985
39.5531
23
3.8200
0.2750
0.2875
350
420
204.252
-46.2033
94.981
39.5527
24
3.8200
0.2875
0.3250
368
350
216.388
-46.7047
94.995
39.5540
25
3.8200
0.3000
0.3625
386
368
213.575
-46.5910
94.995
39.5540
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DOI: 10.37394/232011.2023.18.27
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Dinh Van Thanh, Tran Huu Danh,
Nguyen Van Trang
E-ISSN: 2224-3429
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Volume 18, 2023
Table 9. Values of and
No
S/N
Zi
i (k)
GRVyi
mgb
ηgb
Mgb
ηgb
Reference values
Mgb
ηgb
mgb
ηgb
1.000
1.000
1
-47.2901
39.6066
0.0000
1.0000
1.000
0.000
0.333
1.000
0.667
2
-47.0986
39.6037
0.1498
0.9468
0.850
0.053
0.370
0.904
0.637
3
-46.9381
39.6032
0.2755
0.9369
0.725
0.063
0.408
0.888
0.648
4
-46.8017
39.6001
0.3823
0.8803
0.618
0.120
0.447
0.807
0.627
5
-46.7047
39.5999
0.4581
0.8770
0.542
0.123
0.480
0.803
0.641
6
-46.3562
39.5892
0.7309
0.6823
0.269
0.318
0.650
0.612
0.631
7
-46.2711
39.5867
0.7975
0.6357
0.203
0.364
0.712
0.579
0.645
8
-46.9618
39.5896
0.2569
0.6890
0.743
0.311
0.402
0.617
0.509
9
-46.9766
39.5842
0.2453
0.5908
0.755
0.409
0.399
0.550
0.474
10
-46.5552
39.5916
0.5752
0.7256
0.425
0.274
0.541
0.646
0.593
11
-46.4875
39.5737
0.6282
0.3991
0.372
0.601
0.573
0.454
0.514
12
-46.5622
39.5737
0.5696
0.3991
0.430
0.601
0.537
0.454
0.496
13
-46.4586
39.5716
0.6507
0.3607
0.349
0.639
0.589
0.439
0.514
14
-46.0881
39.5737
0.9407
0.3974
0.059
0.603
0.894
0.453
0.674
15
-46.6908
39.5797
0.4690
0.5075
0.531
0.493
0.485
0.504
0.494
16
-46.2061
39.5609
0.8484
0.1638
0.152
0.836
0.767
0.374
0.571
17
-46.7676
39.5642
0.4089
0.2255
0.591
0.774
0.458
0.392
0.425
18
-46.3447
39.5651
0.7399
0.2406
0.260
0.759
0.658
0.397
0.527
19
-46.2460
39.5629
0.8171
0.2005
0.183
0.799
0.732
0.385
0.558
20
-46.3807
39.5634
0.7117
0.2105
0.288
0.789
0.634
0.388
0.511
21
-46.3959
39.5519
0.6998
0.0000
0.300
1.000
0.625
0.333
0.479
22
-46.0123
39.5531
1.0000
0.0217
0.000
0.978
1.000
0.338
0.669
23
-46.2033
39.5527
0.8505
0.0150
0.149
0.985
0.770
0.337
0.553
24
-46.7047
39.5540
0.4582
0.0384
0.542
0.962
0.480
0.342
0.411
25
-46.5910
39.5540
0.5471
0.0384
0.453
0.962
0.525
0.342
0.433
Table 10. Analysis of variance for means
Source
DF
Seq SS
Adj SS
Adj MS
F
P
C (%)
u1
4
0.059333
0.059333
0.014833
16.34
0.010
36.36
Xba1
4
0.005714
0.005714
0.001429
1.57
0.336
3.50
Xba2
4
0.043055
0.043055
0.010764
11.86
0.017
26.39
AS1
4
0.004552
0.004552
0.001138
1.25
0.416
2.79
AS2
4
0.046877
0.046877
0.011719
12.91
0.015
28.73
Residual Error
4
0.003630
0.003630
0.000908
2.22
Total
24
0.163161
Model Summary
S
R-Sq
R-Sq(adj)
0.0301
97.77%
86.65%
WSEAS TRANSACTIONS on APPLIED and THEORETICAL MECHANICS
DOI: 10.37394/232011.2023.18.27
Trieu Quy Huy, Nguyen Van Binh,
Dinh Van Thanh, Tran Huu Danh,
Nguyen Van Trang
E-ISSN: 2224-3429
297
Volume 18, 2023
Contribution of Individual Authors to the
Creation of a Scientific Article (Ghostwriting
Policy)
The idea of the paper was proposed by Nguyen Van
Trang. The simulation and the optimization were
conducted by Nguyen Van Trang and Trieu Quy
Huy. The manuscript was written by Nguyen Van
Trang with support from Trieu Quy Huy and Tran
Huu Danh. All authors have read and agreed to the
manuscript.
Sources of Funding for Research Presented in a
Scientific Article or Scientific Article Itself
No funding was received for conducting this study.
Conflict of Interest
The authors have no conflicts of interest to declare
that are relevant to the content of this article.
Creative Commons Attribution License 4.0
(Attribution 4.0 International, CC BY 4.0)
This article is published under the terms of the
Creative Commons Attribution License 4.0
https://creativecommons.org/licenses/by/4.0/deed.en
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DOI: 10.37394/232011.2023.18.27
Trieu Quy Huy, Nguyen Van Binh,
Dinh Van Thanh, Tran Huu Danh,
Nguyen Van Trang
E-ISSN: 2224-3429
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Volume 18, 2023
