Optimization of a Two-stage Bevel Helical Gearbox using Multiple
Objectives to Increase Efficiency and Reduce Gearbox Bottom Area
TRAN HUU DANH1, DINH VAN THANH2, BUI THANH DANH3, NGUYEN MANH CUONG4,
LUU ANH TUNG4,*
1Vinh Long University of Technology Education,
Vinh Long,
VIETNAM
2East Asia University of Technology,
Bac Ninh,
VIETNAM
3University of Transport and Communications,
Hanoi,
VIETNAM
4Thai Nguyen University of Technology,
Thai Nguyen,
VIETNAM
*Corresponding Author
Abstract: This study aims to look at multi-target optimization of two-stage bevel helical gearboxes to determine
the best major design factors for reducing gearbox bottom area (GBA) and increasing gearbox efficiency (GE).
Grey relation analysis (GRA) and the Taguchi technique were used to address the problem in two steps.
Prioritizing the closure of the variable level gap, the single-objective optimization problem was addressed,
followed by the multi-objective optimization problem, which identified the ideal primary design variables.
Additionally, the first-stage gear ratio, allowable contact stresses (ACS), and first and second-stage coefficients
of wheel face width (CWFW) were calculated. The outcomes of the study were used to determine the best
values for five essential design features of a two-stage bevel helical gearbox (BHG).
Key-Words: - Bevel helical gearbox, Two-stage gearbox, Optimization, Multi-objective, Gear ratio, Gearbox
efficiency, Gearbox bottom area.
Received: April 2, 2023. Revised: November 14, 2023. Accepted: December 11, 2023. Published: January 16, 2024.
1 Introduction
Extensive studies have been shown in the
optimization of gearboxes. In [1], the author focused
on optimizing gear ratios for a drive system with a
three-stage BLH and chain drive, with the objective
being minimizing the system cross section area. The
authors in [2], analyzed input parameters such as
total gear ratio, face width coefficients, contact
stress, and output torque to minimize system length,
deriving optimal gear ratios for a system with a two-
stage helical gearbox with first stage double gear
sets and a chain drive. Furthermore, in [3], authors
focused on minimizing cross-sectional height,
considering gear pitting resistance and movement
equilibrium, and from that deriving optimal partial
ratio, allowing accurate and efficient calculations.
The authors in [4], developed a prototype of an
active driven knee with BHG. Moreover, in [5],
scientists used a simulation experiment to propose
an equation for optimal gear ratios for three-step
BHG to reduce the height of the gearbox. In [6], the
authors presented a novel approach to determine
optimal partial transmission ratios in a drive system
using a chain drive and two-stage BHG, with the
optimization aimed at minimizing the system’s cross
section dimension. Besides, a study to minimize the
mass of a two-stage BHG was introduced, [7].
Another cost optimization study presented insights
into input factor effects and proposed models for
WSEAS TRANSACTIONS on APPLIED and THEORETICAL MECHANICS
DOI: 10.37394/232011.2024.19.1
Tran Huu Danh, Dinh Van Thanh,
Bui Thanh Danh, Nguyen Manh Cuong,
Luu Anh Tung
E-ISSN: 2224-3429
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Volume 19, 2024
optimal gear ratios for a two-stage BHG, [8]. With
an optimization problem that minimized geabox
volume using eight main input parameters, the best
gear ratios in a three step BHG were proposed in
[9]. Furthermore, the authors in [10], focused on
optimizing the gear ratios of a drive system with a
three-step BHG and a V-belt for minimal system
height.
From the above analysis, it is found that up to
now there have been several studies on optimization
and multi-objective optimization of different
gearboxes. However, up to now, there has been no
research on multi-objective optimization of BHG
with single-objective functions minimal GBA and
maximal GE.
The current study aims to investigate multi-target
optimization learning for a two-step BHG. The work
being done had two distinct goals: lowering GBA
and optimizing GE. Furthermore, five main design
elements were evaluated: the CWFW and the ACS
of steps 1 and 2, and the gear ratio of step 1. Also, a
multi-target optimization task for gearbox design
was tackled in two stages by integrating the Taguchi
technique with the GRA. The ideal values for five
essential design factors were also suggested for
creating a two-step BHG.
2 Optimization Problem
2.1 Determining Gearbox Length
The gearbox bottom area can be found in (Figure 1):
(1)
Where, L and B are the gearbox length and
gearbox width which are determined by (Figure 1):
(2)
(3)
In (2), [11]; with is the
initial shaft diameter obtained by [11]:
(4)
is the outside diameter of bevel gear (mm);
can be calculated by [11]:
(5)
Wherein, is the cone distance (mm), [11]:
(6)
In which, (MPa) is a coefficient, [11];
is the contacting load
coefficient of stage 1 [11];
is coefficient of face width; is the torque
on pinion (Nmm).
In (1), and are the pitch diameters of
the pinion and gear of stage 1, [11]:
(7)
(8)
In the above equation, is the center distance
of stage 2 which is found as, [11]:
(9)
In where, is contacting load
ratio of stage 2 [11]; AS2 represents the permitted
contact stress (MPa); denotes a coefficient,
[11]; is the coefficient of wheel face width and
is the torque on the pinion (Nmm) of stage 2:
(20)
(31)
In which, Tout is the output torque (N.mm);
and are the
efficiency of stage 1 and 2, [11]; ηb is the rolling
bearing efficiency (ηh=0.99÷0.995, [11]).
In (3), is determined by, [12]:
(42)
Fig. 1: Calculated schema, [1]
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2.2 Calculating Gearbox Efficiency
The efficiency of the gearbox can be determined by
in the following manner:
(53)
Where, is the overall power loss of the
gearbox, [13]: (64)
In (14) , , and are the total power loss
of gears, bearings, and seals which are calculated
by:
+) The total power loss of gear:
(75)
In which, is the losses of gear power of step
i: (86)
Where, the efficiency of the i step, [16]:
(97)
In which, is gear ratio of the i step, f is
coefficient of friction; and are the arcs of
approach and recess on the i step that can be
determined by, [14]:
+) For stage 1:
(108)
(119)
In which Raev1 and Raev2 are the equivalent pinion
and gear outer radiuses; Rv1 and Rv2 are the
equivalent pinion and gear pitch radiuses; R0v1 and
R0v2 are the equivalent pinion and gear base
radiuses; and α is the pressure angle.
(20)
(21)
In where R1 and R2 are the large pitch radius of
the bevel pinion and gear; and and are the
pitch angles of the bevel pinion and gear,
correspondingly. (22)
(23)
+) For stage 2:
(24)
(25)
In where, 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.
In (17), the friction coefficient is found by, [14]:
- If the sliding velocity v ≤ 0.424 (m/s):
(26)
- If v > 0.424 (m/s):
(27)
+) The bearing power loss can be determined by,
[13]:
(28)
In which, is friction coefficient of a
radical ball bearings, [13]; F is load on bearing (N),
v is the peripheral speed of ith bearing (i = 1÷6).
+) The power loss in seals is determined as,
[13]:
(29)
In which, i =1÷2 is the ordinal seal number;
is the power loss in a single seal (w):
(30)
In which, is the ISO Viscosity Grade
number.
2.3 Objective Function and Constrains
2.3.1 Objectives Functions
The multi-target work has two different objectives:
Minimizing gearbox bottom area:
(31)
Maximizing gearbox efficiency:
(32)
Where, X is the vector indicating variables. As
five input parameters including , , ,
, and were selected as variables, we have:
(33)
2.3.2 Constraints
The multi-objective function must follow the
following constraints:
and (34)
and (35)
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and (36)
3 Methodology
In this study, five input parameters have been
selected for investigation. Table 1 illustrates the
minimum and maximum values for various
parameters. The Taguchi approach and GRA have
been used for solving the optimization work. The
L25 (55) design was used to optimize the total
amount of levels for each parameter. Nevertheless,
u1 has a wide range (ranging from 1 to 6 - Table 1)
among the factors tested. Even with five levels, the
difference in the values of these traits remained
advantageous (in this case, the difference was 1.5
((6-1)/4).
The 2-stage technique for solving the multi-
target optimization problem was used to help
decrease the difference between values of a variable
spread across a wide range (Figure 2), [15]. This
technique's first stage addresses a single-target
optimization prob-lem, while the second stage deals
a multi-target optimization work to determine the
optimal primary design features.
4 Optimization problem
4.1 Single-objective Optimization
In this paper, the single-objective optimization issue
is solved using the direct search strategy. Two
single-objective challenges were also solved using a
Matlab-based computer program: maximizing
gearbox efficiency and minimizing gearbox bottom
area. Figure 3 shows the connection between the
total gear-box ratio ut and the ideal gear ratio of the
first stage u1, based on the program's results.
Furthermore, as Table 2 shows, new constraints
have been developed for the variable u1.
4.2 Multi-objective Optimization
The purpose of this work is to find the best primary
design variables for a given total gear-box ratio
while meeting two single-target functions:
decreasing gearbox bottom area and optimizing
gearbox efficiency. A computer experiment was
constructed to address the given multi-objective
optimization issue. Table 3 displays the key design
components and their values for ut = 15. The
experimental design was created using the Taguchi
technique using L25 (55) design, and the data was
analyzed using Minitab R18 software. The
experimental design and results for ut = 15 are
shown in Table 4.
Table 1. Main design factors and their maximum and lowest limits
Factor
Notation
Lower limit
Upper limit
Gear ratio of step 1
u1
1
6
CWFW of step 1
kbe
0.25
0.3
CWFW of step 2
Xba
0.25
0.4
ACS of step 1 (MPa)
350
420
ACS of step 2 (MPa)
350
420
Fig. 2: Method for solving multi-objective problem, [16]
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Fig. 3: Relation between optimal values of u1 and ut
Table 2. New constraints of u1
ut
u1
Lower limit
Upper limit
10
1.17
2.16
15
1.76
3
20
2.34
3.78
25
2.93
4.52
30
3.52
5.24
35
4.1
5.93
Table 3. Input parameters and their levels for ut = 15.
Factor
Notation
Level
1
2
3
4
5
Gear ratio of step 1
u1
1.76
2.33
2.90
3.47
4.04
CWFW of step 1
kbe
0.25
0.2625
0.275
0.2875
0.3
CWFW of step 2
Xba
0.25
0.2875
0.325
0.3625
0.4
ACS of step 1 (MPa)
AS1
350
368
386
404
420
ACS of step 2 (MPa)
AS2
350
368
386
404
420
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Table 4. Experimental plan and results when ut = 15
Exp.
No.
Input parameters
Ab
ηgb
u1
Kbe
Xba
AS1
AS2
(dm2)
(%)
1
1.76
0.2500
0.2500
350
350
5.520
95.188
2
1.76
0.2625
0.2875
368
368
5.413
95.073
3
1.76
0.2750
0.3250
386
386
5.308
95.043
4
1.76
0.2875
0.3625
404
404
5.204
94.917
5
1.76
0.3000
0.4000
420
420
5.132
94.882
6
2.07
0.2500
0.2875
386
404
4.732
95.047
7
2.07
0.2625
0.3250
404
420
4.685
95.013
8
2.07
0.2750
0.3625
420
350
5.761
94.990
9
2.07
0.2875
0.4000
350
368
5.824
94.943
10
2.07
0.3000
0.2500
368
386
4.897
95.091
11
2.38
0.2500
0.3250
420
368
5.165
94.958
12
2.38
0.2625
0.3625
350
386
5.270
95.000
13
2.38
0.2750
0.4000
368
404
5.161
94.930
14
2.38
0.2875
0.2500
386
420
4.394
94.991
15
2.38
0.3000
0.2875
404
350
5.379
95.026
16
2.69
0.2500
0.3625
368
420
4.756
94.923
17
2.69
0.2625
0.4000
386
350
5.778
94.913
18
2.69
0.2750
0.2500
404
368
4.861
95.012
19
2.69
0.2875
0.2875
420
386
4.793
94.892
20
2.69
0.3000
0.3250
350
404
4.947
94.992
21
3.00
0.2500
0.4000
404
386
5.165
94.871
22
3.00
0.2625
0.2500
420
404
4.387
94.932
23
3.00
0.2750
0.2875
350
420
4.587
94.962
24
3.00
0.2875
0.3250
368
350
5.548
94.964
25
3.00
0.3000
0.3625
386
368
5.425
94.930
Table 5. S/N values of each experiment when ut=15
Exp.
No.
Input Factors
Ab
ηgb
u1
Kbe
Xba
AS1
AS2
(dm2)
S/N
(%)
S/N
1
1.76
0.2500
0.2500
350
350
5.520
-14.8388
95.188
39.5716
2
1.76
0.2625
0.2875
368
368
5.413
-14.6688
95.073
39.5611
3
1.76
0.2750
0.3250
386
386
5.308
-14.4986
95.043
39.5584
4
1.76
0.2875
0.3625
404
404
5.204
-14.3267
94.917
39.5469
5
1.76
0.3000
0.4000
420
420
5.132
-14.2057
94.882
39.5437
6
2.07
0.2500
0.2875
386
404
4.732
-13.5009
95.047
39.5588
7
2.07
0.2625
0.3250
404
420
4.685
-13.4142
95.013
39.5557
8
2.07
0.2750
0.3625
420
350
5.761
-15.2100
94.990
39.5536
9
2.07
0.2875
0.4000
350
368
5.824
-15.3044
94.943
39.5493
10
2.07
0.3000
0.2500
368
386
4.897
-13.7986
95.091
39.5628
11
2.38
0.2500
0.3250
420
368
5.165
-14.2614
94.958
39.5506
12
2.38
0.2625
0.3625
350
386
5.270
-14.4362
95.000
39.5545
13
2.38
0.2750
0.4000
368
404
5.161
-14.2547
94.930
39.5481
14
2.38
0.2875
0.2500
386
420
4.394
-12.8572
94.991
39.5536
15
2.38
0.3000
0.2875
404
350
5.379
-14.6140
95.026
39.5568
16
2.69
0.2500
0.3625
368
420
4.756
-13.5448
94.923
39.5474
17
2.69
0.2625
0.4000
386
350
5.778
-15.2356
94.913
39.5465
18
2.69
0.2750
0.2500
404
368
4.861
-13.7345
95.012
39.5556
19
2.69
0.2875
0.2875
420
386
4.793
-13.6121
94.892
39.5446
20
2.69
0.3000
0.3250
350
404
4.947
-13.8868
94.992
39.5537
21
3.00
0.2500
0.4000
404
386
5.165
-14.2614
94.871
39.5427
22
3.00
0.2625
0.2500
420
404
4.387
-12.8434
94.932
39.5483
23
3.00
0.2750
0.2875
350
420
4.587
-13.2306
94.962
39.5510
24
3.00
0.2875
0.3250
368
350
5.548
-14.8827
94.964
39.5512
25
3.00
0.3000
0.3625
386
368
5.425
-14.6880
94.930
39.5481
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Table 6. Values of and
No.
S/N
Zi
i (k)
Grey relation
value yi
Ab
ηgb
Ab
ηgb
Reference values
Ab
ηgb
Ab
ηgb
1.000
1.000
1
-15.3387
39.4711
0.2853
1.0000
0.715
0.000
0.412
1.000
0.706
2
-15.1828
39.4590
0.3496
0.6200
0.650
0.380
0.435
0.568
0.501
3
-15.0240
39.4582
0.4150
0.5970
0.585
0.403
0.461
0.554
0.507
4
-14.8639
39.4451
0.4810
0.1876
0.519
0.812
0.491
0.381
0.436
5
-14.7534
39.4391
0.5266
0.0000
0.473
1.000
0.514
0.333
0.423
6
-14.1616
39.4549
0.7705
0.4933
0.229
0.507
0.685
0.497
0.591
7
-14.0847
39.4466
0.8022
0.2366
0.198
0.763
0.717
0.396
0.556
8
-15.8478
39.4510
0.0754
0.3722
0.925
0.628
0.351
0.443
0.397
9
-16.0088
39.4524
0.0091
0.4154
0.991
0.585
0.335
0.461
0.398
10
-14.4708
39.4564
0.6431
0.5394
0.357
0.461
0.583
0.520
0.552
11
-14.9808
39.4448
0.4328
0.1789
0.567
0.821
0.469
0.378
0.424
12
-15.2145
39.4538
0.3365
0.4587
0.663
0.541
0.430
0.480
0.455
13
-15.0394
39.4449
0.4087
0.1818
0.591
0.818
0.458
0.379
0.419
14
-13.6049
39.4492
1.0000
0.3174
0.000
0.683
1.000
0.423
0.711
15
-15.3401
39.4524
0.2847
0.4154
0.715
0.585
0.411
0.461
0.436
16
-14.3584
39.4493
0.6894
0.3203
0.311
0.680
0.617
0.424
0.520
17
-16.0308
39.4500
0.0000
0.3405
1.000
0.660
0.333
0.431
0.382
18
-14.4921
39.4521
0.6343
0.4068
0.366
0.593
0.578
0.457
0.517
19
-14.3800
39.4419
0.6805
0.0895
0.320
0.911
0.610
0.354
0.482
20
-14.7072
39.4441
0.5456
0.1587
0.454
0.841
0.524
0.373
0.448
21
-15.0732
39.4416
0.3947
0.0779
0.605
0.922
0.452
0.352
0.402
22
-13.6121
39.4427
0.9970
0.1126
0.003
0.887
0.994
0.360
0.677
23
-14.0417
39.4527
0.8199
0.4241
0.180
0.576
0.735
0.465
0.600
24
-15.6852
39.4522
0.1425
0.4097
0.858
0.590
0.368
0.459
0.413
25
-15.4976
39.4408
0.2198
0.0549
0.780
0.945
0.391
0.346
0.368
The Taguchi and GRA approaches are used for
dealing with multi-objective optimization problems.
The following are the major steps in this approach:
+) Using the following equations, calculate the
signal-to-noise ratio (S/N):
The better the S/N, the shorter the gearbox
bottom area:
(37)
The greater the S/N ratio, the better for the
gearbox efficiency goal:
(38)
Where yi is the output result and m is the number
of experiment repeats. Since the experiment is a
simulation, m = 1 and no repeats are needed. Table
5 shows the estimated S/N indices of output
objectives.
The data amounts for the two single-target
functions were dissimilar. To guarantee similarity,
the data must be normalized, or brought to a
standard scale. The normalization value Zij, which
changes from 0 to 1, is used to normalize the data.
This value is calculated using the following
formula:
(39)
In which, n=25 is the total test runs.
+) The grey relational (GR) factor can be found
by:
(40)
i=1,2,...,n; k=2 is the
number of objectives; and are the
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minimum and maximum values of i(k),
respectively; and ζ =0.5 is the distinguishing factor.
+) Identifying the level of grey in a situation: It
is calculated by averaging the GR coefficients
associated with the output targets:
(41)
In which yij is the GR value of the ith
experiment's jth output aim. For each trial, Table 6
presents the projected GR number yi as well as the
average GR value
.
A greater average GR value is advised to
promote harmony between the output factors. As a
result, a multi-target task can be reduced to a single-
target work, yielding the mean GR value.
Table 7. Analysis of variance for means
Fig. 4: Main effects plot for S/N ratios
Table 8. Optimum input parameters
No.
Input factors
Code
Optimum Level
Optimum Value
1
Gear ratio of first stage
u1
2
2.07
2
CWFW of first step
kbe
1
0.25
3
CWFW of second step
Xba
1
0.25
4
ACS of first step (MPa)
AS1
1
350
5
ACS of second step (MPa)
AS2
5
420
Table 9. Optimal values of main design factors
ut
10
15
20
25
30
35
u1
1.4175
2.07
2.5
3.1
3.52
4.1
Kbe
0.25
0.25
0.25
0.25
0.25
0.25
Xba
0.25
0.25
0.25
0.25
0.25
0.25
AS1
350
350
350
350
350
350
AS2
420
420
420
420
420
420
WSEAS TRANSACTIONS on APPLIED and THEORETICAL MECHANICS
DOI: 10.37394/232011.2024.19.1
Tran Huu Danh, Dinh Van Thanh,
Bui Thanh Danh, Nguyen Manh Cuong,
Luu Anh Tung
E-ISSN: 2224-3429
8
Volume 19, 2024
Fig. 5: Probability plot of
Fig. 6: Relation between u1 and ut
4.3 Results and Discussion
Table 7 displays the results of an ANOVA test run
to assess the influence of the key input factors on
the average GR value
. According to this table, kbe
has the highest impact on
(27.02 %), followed by
Xba (26.10 %), AS2 (15.62 %), AS1 (12.67 %), and
u1 (10.54 %).
+) Identifying the best primary design
parameters: In theory, the best factor set would
include fundamental design elements with the
highest S/N values. Therefore, the influence of the
key input aspects on the S/N ratios was calculated
(Figure 4). From Figure 4, the optimal levels of the
input factors for multi-target work (corresponding to
the red points) have been easily determined. These
optimal levels was described in Table 8.
+) Evaluating the experimental modeling: Figure
5 displays the Anderson-Darling approach findings,
which are used to examine the adequacy of the
suggested model. The data points that match the
findings from the experiment (shown in the graph as
blue points) are among the top 95% standard
deviation zone specified by the top and bottom
limits. Moreover, the significance level of α = 0.05
is significantly lower than the p-value of 0.234.
These results demonstrate the applicability of the
experimental model for assessment in this work.
Continue in the same manner as with ut=15, but
with the lasting ut values including 10, 20, 25, 30,
and 35. Table 9 shows the optimal values for each
of the five key design parameters at different ut.
Figure 6 shows the connection between the proper
WSEAS TRANSACTIONS on APPLIED and THEORETICAL MECHANICS
DOI: 10.37394/232011.2024.19.1
Tran Huu Danh, Dinh Van Thanh,
Bui Thanh Danh, Nguyen Manh Cuong,
Luu Anh Tung
E-ISSN: 2224-3429
9
Volume 19, 2024
first-stage gear ratio and the total gearbox ratio. This
table provided the following results:
- kbe and Xba choose the least value feasible. This
is because these variables were used to optimize the
average grey relation value
.
- Ideal AS1 values are the lowest, while ideal AS2
values are the highest. This is due to the fact that
these modifications increased the average grey
relation value
.
- Figure 6 presents the association between the
suitable first-stage gear ratio and the overall gearbox
ratio. In addition, the following regression formula
(with R2=0.997) is provided to calculate the best
values of u1: (42)
After calculating u1, the optimal value of u2 is
found using u2=ut/u1.
5 Conclusion
This article introduces the results of a multi-
objective optimization work on the optimization of a
two-stage BHG to reduce gearbox bottom area and
boost gearbox efficiency. This study optimized the
gear ratio of step 1, the CWFW for steps 1 and 2,
and the ACS for steps 1 and 2. To address this issue,
a simulation experiment based on the Taguchi L25
type was developed and performed. The impact of
significant design elements on the multi-objective
goal was also investigated. It was noted that Xba has
the highest impact on
(72.15%), followed by AS2
(17.56%), kbe (4.4%), AS1 (2.43%), and u1 (1.02%).
In addition, the ideal settings for the important
gearbox parameters have been recommended. A
regression approach (Equation (42)) for calculating
the appropriate first stage u1 gear ratio was also
described. However, it is necessary to conduct
further research on multi-objective optimization for
the three-stage bevel helical gearbox.
Acknowledgement:
This research has been supported by Thai Nguyen
University of Technology.
References:
[1] Thao, T.T.P., et al., Determining optimum
gear ratios of mechanically driven systems
using three stages bevel heli-cal gearbox and
chain drive, in Advances in Engineering
Research and Application: Proceedings of the
International Conference on Engineering
Research and Applications, ICERA 2019.
2020. Springer, pp. 249-261.
[2] Hung, L.X., et al, Calculation of optimum
gear ratios of mechanically driven systems
using two-stage helical gearbox with first
stage double gear sets and chain drive, in
Advances in Engineering Research and
Application: Proceedings of the International
Conference on Engineering Research and
Applications, ICERA 2019. 2020. Springer,
pp. 170-178.
[3] Cam, N.T.H., et al, A study on determination
of optimum partial transmission ratios of
mechanically driven sys-tems using a chain
drive and a three-step helical reducer, in
Advances in Engineering Research and
Application: Proceedings of the International
Conference on Engineering Research and
Applications, ICERA 2018. 2019. Springer,
pp. 91-99.
[4] Budaker, B. and A. Verl, Design,
Development and Realisation of an active
driven knee-prosthesis with bevel helical
gearbox, Advanced Materials Research, 907,
2014, pp. 225-239.
[5] Pi, V.N., et al. Determination of optimum gear
ratios of a three-stage bevel helical gearbox,
in IOP Conference Series: Materials Science
and Engineering, 2019. IOP Publishing.
[6] Pi, V.N., and N.K. Tuan, Determining
optimum partial transmission ratios of
mechanical driven systems using a chain drive
and a two-step bevel helical gearbox.,
International Journal of Mechanical
Engineering and Robotics Research, 8(5),
2019, pp. 708-712.
[7] Hong, T.T., et al., Optimization of
transmission ratios for two-stage bevel helical
gearboxes based on mass function, Int. J. Eng.
Res. Technol, 13, 2020. pp. 1692-1699.
[8] Linh, N.H., et al. Optimization of
Transmission Ratios for Two-Stage Bevel
Helical Gear-boxes Based on Cost Function,
in International Conference on Engineering
Research and Applications, 2021, Springer,
pp. 754-766.
[9] Tuan, T.K., et al. Optimization of Gear Ratios
for Three-Stage Bevel Helical Gearboxes
Based on Gearbox Volume Function, in
International Conference on Engineering
Research and Applications, 2021, Springer,
pp. 708-720.
[10] Tuan, N.K., et al. Optimum calculation of
partial transmission ratios of mechanically
driven systems using a V-belt and a three-step
WSEAS TRANSACTIONS on APPLIED and THEORETICAL MECHANICS
DOI: 10.37394/232011.2024.19.1
Tran Huu Danh, Dinh Van Thanh,
Bui Thanh Danh, Nguyen Manh Cuong,
Luu Anh Tung
E-ISSN: 2224-3429
10
Volume 19, 2024
bevel helical gearbox, in Advances in
Engineering Research and Application:
Proceedings of the International Conference
on Engineering Research and Applications,
ICERA 2018. 2019. Springer, pp. 469-476.
[11] Chat, T. and L. Van Uyen, Design and
calculation of Mechanical Transmissions
Systems, Vol 1, Educational Republishing
House, Hanoi, 2007.
[12] Römhild, I. and H. Linke, Targeted design of
gear drives with minimal mass based on new
calculation methods (Gezielte Auslegung Von
Zahnradgetrieben mit minimaler Masse auf
der Basis neuer Berechnungsverfahren),
Konstruktion, 44(7-8), 1992, pp. 229-236.
[13] Jelaska, D.T, Gears and gear drives, John
Wiley & Sons, 2012
[14] Buckingham, E., Analytical mechanics of
gears, Courier Corporation, 1998.
[15] Chat, T. and L. Van Uyen, Design and
calculation of Mechanical Transmissions
Systems (Tính toán Thiết kế Hệ dẫn động Cơ
khí), vol. 1. Educational Republishing House,
Hanoi, 2007.
[16] Tran, H.-D., et al, Application of the Taguchi
Method and Grey Relational Analysis for
Multi-Objective Optimization of a Two-Stage
Bevel Helical Gearbox, Machines, 11(7),
2023, pp. 716.
Contribution of Individual Authors to the
Creation of a Scientific Article (Ghostwriting
Policy)
The paper's concept was set out by Luu Anh Tung.
Luu Anh Tung and Tran Huu Danh carried out the
optimization and simulation. Luu Anh Tung wrote
the manuscript with assistance from Nguyen Manh
Cuong and Tran Huu Danh. After reading the
manuscript, each author gave their approval.
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.
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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WSEAS TRANSACTIONS on APPLIED and THEORETICAL MECHANICS
DOI: 10.37394/232011.2024.19.1
Tran Huu Danh, Dinh Van Thanh,
Bui Thanh Danh, Nguyen Manh Cuong,
Luu Anh Tung
E-ISSN: 2224-3429
11
Volume 19, 2024
