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European Journal of Applied Sciences – Vol. 11, No. 6
Publication Date: December 25, 2023
DOI:10.14738/aivp.116.16104
Hung, T. Q., & Tung, L. A. (2023). Application of Taguchi Technique and Grey Relational Analysis for Multi-Target Optimization of
Two Stage Helical Gearboxes. European Journal of Applied Sciences, Vol - 11(6). 372-387.
Services for Science and Education – United Kingdom
Application of Taguchi Technique and Grey Relational Analysis
for Multi-Target Optimization of Two Stage Helical Gearboxes
Tran Quoc Hung
Ha Noi University of Industry, Hanoi, Vietnam
Luu Anh Tung
Thai Nguyen University of Technology, Thai Nguyen, Vietnam
ABSTRACT
This paper describes an investigation on multi-target optimization of two-stage
helical gearboxes with the aim to identify the optimal main design factors to
minimize gearbox cross section area and maximize gearbox efficiency. The Taguchi
method and grey relation analysis (GRA) were utilized to solve the problem in two
processes: first, the single-objective optimization problem was solved to close the
gap within variable levels, and then the multi-objective optimization problem was
handled to find the optimal main design factors. The coefficients of wheel face width
(CWFW) of the first and second stages, the permissible contact stresses (ACS) of the
first and second stages, and the first stage's gear ratio were also selected the
outcomes of the research have been utilized to figure out the best values for five
major design variables when designing two-stage helical gearboxes.
Keywords: Helical gearbox, Two-stage gearbox, multi-target optimization, Gear ratio,
Across-section area, gearbox efficiency.
INTRODUCTION
Mechanical drive systems are commonly used to reduce the speed of the motor shaft and
increase the torque of the working device shaft. Many scientists are interested in optimal
gearbox design because the gearbox is the primary part of the mechanical drive system.
Helical gearbox has been the subject of various studies. [1] investigates the optimal gear ratios
for mechanical driven systems, incorporating a two-stage helical gearbox with double gear sets
in the first stage and a chain drive. Another study by [2] proposes several equations to calculate
the best gear ratios of a system using two-stage helical gearbox, which the approach is to
minimize system length. Within the same field of interest, [3] presents a study on optimal gear
ratios for a two-stage helical gearbox, with the objective function of the optimization problem
being the cross-sectional area. In addition, [4] aims to enhance quasi-static behavior of a power
transmission gearbox using a modified particle swarm algorithm. Another study presents
calculation of the optimal gear ratios for a two-stage helical gearbox with second stage double
gear sets [5]. Using an optimization problem to find the ideal gear ratios, it also put forth
effective and straightforward models for their determination. Moreover, [6] investigated the
best design parameters for minimizing the volume of a two-stage helical gearbox. Studies on
vibrations of helical gearbox are also conducted. [7] runs Finite Element Analysis for a single-
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Services for Science and Education – United Kingdom 374
European Journal of Applied Sciences (EJAS) Vol. 11, Issue 6, December-2023
In which, L, and H can be calculated by:
L =
dw11
2
+ aw1 + aw2 +
dw22
2
+ 20 (2)
H = max(dw21, dw22) + 8.5 ∙ SG (3)
SG = 0.005 ∙ L′ + 4.5 [11] (4)
In (2) and (3), bw1, dw12, dw21, and dw22 are the gear width of the first stage, the pitch diameters
of the pinion and the gear sets, correspondingly. These elements can be determined by [12]:
dw21 = 2 ∙ aw1 ∙ u1/(u1 + 1) (5)
dw12 = 2 ∙ aw2/(u2 + 1) (6)
dw22 = 2 ∙ aw2 ∙ u2/(u2 + 1) (7)
bw1 = Xba1 ∙ aw1 (8)
In the above Equations, u1 and u2 are the gear ratios of the first and the second stages; u2 ∙ u1 =
ug; with ug is the total gearbox ratio; aw1 and aw2 are the center distances of the first and the
second stages which are found by [12]:
aw1 = ka ∙ (u1 + 1) ∙ √T11 ∙ kHβ/(AS1
2
∙ u1 ∙ Xba1)
3
(9)
aw2 = ka ∙ (u2 + 1) ∙ √T12 ∙ kHβ2/(AS2
2
∙ u2 ∙ Xba2)
3
(10)
Where, kHβ = 1.05 ÷ 1.27 is contacting load ratio for pitting resistance [12]; AS1 and AS2 is
permissible contact stress of the first and the second stages (MPa); ka = 43 is the material
coefficient [12]; Xba1 and Xba2 are the coefficients of the wheel face width of the first and the
second stages; T11 and T12 are the torques on the pinions of the first and the second stages
(Nmm):
T11 = Tout/(ug ∙ ηhg
2
∙ ηb
3
) (11)
T12 = Tout/(u2 ∙ ηhg ∙ ηbe
2
) (12)
In which, Tout is the output torque (N.mm); ηhg is the helical gear efficiency ((ηhg = 0.96 ÷
0.98 [12]); ηb is the rolling bearing efficiency (ηh=0.99÷0.995 [12]).
Calculating the Mass of Gearbox
The gearbox mass mgb is calculated by (Fig. 1):