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DESIGN OPTIMIZATION OF A THREE-STAGE PLANETARY GEAR
REDUCER USING GENETIC ALGORITHM
Yanbiao FENG
University of Victoria
Victoria, BC Canada
Wenming ZHANG
University of Science and Technology Beijing
Beijing, China
Jue YANG
University of Science and Technology Beijing
Beijing, China
Zuomin DONG
University of Victoria
Victoria, BC Canada
ABSTRACT
The multi-stage reducer, especially the planetary gear
reducer, usually serves in heavy-duty machinery such
as shield tunneling machine, tracked excavator,
mining truck and crusher. Those application areas
require great load capacity, long life, and high
geometrical mechanical performance, and the high
ratio so on. This paper first presents a novel
architecture of three-stage reducer. To achieve those
objectives collectively, this paper presents an
optimization methodology based on genetic algorithm
(GA). The geometrical volume is set as objective
function. The gear module, teeth number, and gear
face width are chosen as design variables, taking the
life, geometrical spacing, efficiency and load capacity,
etc. as constraints. The optimization results are
satisfactory and can help designer to employ novel
architecture by fulfilling requirements.
INTRODUCTION
The planetary based gear reducer can support a higher
and constant speed ratio in the compact space, higher
reliability, higher transmission efficiency, great load
capacity and longer life, but much more complex and
higher in cost. Therefore, its implementation has
attracted researchers’ attention, including
optimization design, kinematic analysis, fatigue
prediction, and fault diagnostics [1-4].
In general, a single stage planetary gear set consists of
a sun gear, several planetary gears that externally
meshed with sun gear, a carrier that supports the
planetary gears, and a ring gear that internally meshed
with planetary gears. The multi-stage reducer is a
combination of two or three single stage reducer. No
matter how to combine these or how many single-
stage reducers combined, these constant speed ratio
gear reducer has one input and one output shaft. All of
these components are interrelated and interact with
each other, which means each component parameter
alternation leading to the system status and output
characteristics change. Hereafter, a multi-stage
planetary gear reducer design is a highly integrated,
complicated, and fused with non-linear characteristics
constraints task. Furthermore, non-linear constraints,
as well as increasing demand for compact, efficient,
and reliable force designers utilize optimization based
design methodology.
Thompson [5] investigated a multi-stage gear reducer
optimization. Based on the MATLAB optimization
toolbox, constraint minimization function, the total
gear volume and surface fatigue life factor were
chosen as multi-objectives. At the same time, they
further examined the number of stages. However, this
case was a fixed axle with spur gear. Savsani [6]
presented a weight oriented optimal design using
particle swarm optimization (PSO) and simulated
annealing (SA) algorithms. This research concerned
one pair of externally meshed gear train, aimed at
minimizing the gear train weight, considering the gear
shaft radius and gear topology. In the second study
case, the hardness was also taken into design variables.
Although this paper presented a method to minimize
the gear train weight, it focused on the algorithms
comparison considering the integer, continuous and
discrete variables, especially the genetic algorithm [7].
For the spur gear optimal design application,
Marjanovic [8] and Golabi [9] presented the optimal
gear train design, separately. Marjanovic gave an
overall view of different stage architecture selection in
various ratio ranges. But this paper didn’t consider the
1
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Proceedings of the ASME 2019
International Design Engineering Technical Conferences
and Computers and Information in Engineering Conference
IDETC/CIE2019
August 18-21, 2019, Anaheim, CA, USA
DETC2019-97387
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planetary gear sets. Golabi considered the 1-stage, 2-
stage, and 3-stage gear train selection and gearbox
parameter optimal selection considering the
transmission power, material hardness, gearbox ratio.
However, these two studies weighed more on gear
train parameters optimization than spur gear
optimization. Ref. [10] conducted a two-stage
planetary gear transmission optimization in self-
developed software. This software, based on C# and
VBA, exhausted all the feasible results. This kind of
exhaustive method could be convenient when: 1) the
number of variables is really small; 2) the engineering
problem is relatively simple; 3) and the constraints are
not complicated. Those three requirements seem to be
impossible into a three-stage planetary reducer
optimization design, which have more than 10 design
variables, plenty of nonlinear constraints. To optimize
a three-stage planetary gear reducer, ZHANG et al [11]
implemented the Particle Swarm Optimization (PSO)
into the tunneling machine three-stage planetary
reducer optimization process. Aimed at enhancing the
third stage bending strength and contact fatigue
strength, this study also improved the total volume,
overall efficiency. Nonetheless, this study optimized
each stage separately. An integrated overall reducer
optimization had not been conducted. Furthermore,
Genetic Algorithms (GA) had already conducted in
overall optimization in other reducer applications,
such as helical gear reducer [12], cycloid reducer [13].
Little has been done for a three-stage reducer system-
level integrated optimization.
Therefore, this paper primarily presents a novel three-
stage reducer architecture. To make this architecture
into an application, we implement the GA based global
optimization, considering the speed ratio distribution.
NOVEL REDUCER ARCHITECTURE
Planetary based multi-stage reducer takes over the
heavy duty application mainstream due to its space-
saving, higher speed ratio, higher capacity in torque,
lower noise, and flexible in architecture so on. In
mining truck application area, there are different types,
as shown in Figure 1.
(a) Type Ⅰ
(b) Type Ⅱ
(c) Type Ⅲ
Figure 1 Different types of 3-stage reducer
To explain those different types, few definitions are set:
For all the reducer types, power flows from left to
right, as the arrow shows.
Three planetary sets in each type are defined as
the first stage, second stage and third stage,
respectively.
zr_1 refers to the number of gear teeth, where the
subscript r refers to ring gear (s refers to sun gear,
c refers to carrier), subscript 1 represents the first
stage (2 for second stage, 3 for third stage)
The reducer torque losses are neglected.
Equation 1
1
r
s
s r c
z
kz
n kn k n
Based on the planetary gear set speed relation equation
(Equation 1), three types of reducer speed ratio can be
calculated (Table 1, the negative value means the
rotation direction change in output shaft.).
From the Table 1, all of the three types of reducers can
generate higher speed ratio. According to the power
flows in the planetary gear reducer, the Type Ⅰ differs
from the other two types in that the first and second
planetary gear set act as the torque amplifier, and
power distributer as well. In type Ⅰ, part of power
directly outputs to ring gear in first two stages, and the
part power with the amplified torque inputs to the next
stage sun gear. And the power contribution of each
stage can be found in Table 2. In the other two types,
each stage only amplifies the torque and delivers to
next stage. Some researchers [14] [15] have proven
that the power distribution can reach a high ratio, and
overall efficiency and better power balancing.
From the Table 2, the third stage contributes the most
power. Even if we assume that k1=k2=k3, the power
delivered from third stage is (1+k1)(1+k2) times over
the first stage, (1+k2) times over the second stage. To
2
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further balancing the power contribution at third stage,
we present a novel planetary reducer (Figure 2).
Similar to the type Ⅱ in Figure 1(b), the first stage acts
the torque amplifier, and delivers the power to next
stage. In the second stage, different with type Ⅰ, the
carrier exports part of power to output shaft, and the
ring gear delivers the amplified torque to the third
stage, which is totally same with type Ⅰ third stage.
By eliminating the first stage, this novel reducer can
transfer to a two-stage reducer, which covers more
widely speed range. The speed ratio is
(1+k1)(1+k2+k2k3), and power distribution can be
found in Table 3.
Table 1Speed ratio of different type reducers
Type Ⅰ
Type Ⅱ
Type Ⅲ
Speed ratio
1-(1+k1)(1+ k2)(1+ k3)
_2 r_ 2
1s_2 _3
1c
c
zz
kzz
(1+k1)(1+k2)(1+k3)
Table 2 Power distribution
Power contribution
Type 1
Type Ⅱ
Type Ⅲ
1st Stage
1
1 2 3
1 1 1 1
k
k k k
0
0
2nd Stage
12
1 2 3
1
1 1 1 1
kk
k k k
0
0
3rd Stage
1 2 3
1 2 3
11
1 1 1 1
k k k
k k k
1
1
Figure 2 Novel power split three-stage reducer
Table 3 Novel reducer distribution
Power contribution
1st Stage
0
2nd Stage
2
2 2 3
1
1k
k k k
3rd Stage
23
2 2 3
1kk
k k k
The distribution ratio between first two stages power
contribution and the third stage power contribution are:
Equation 2 Type Ⅰ:
1 1 2
1 2 3 3 1 2 3
111
1 1 1 1
k k k
k k k k k k k
Equation 3 Novel type:
2
2 3 3 2 3
111
k
k k k k k
As we can see from the distribution ratio results in
Equation 2 and Equation 3, the novel reducer first two
stages delivers more power than the type Ⅰ. Therefore,
it can balance the load in the third stage, which would
be better for the third stage serving life, and reliability.
REALIZATION IN PRACTICE
To introduce this novel into real application, we
implement this novel reducer into a real mining truck,
designed by our department [16-18]. Figure 3 shows
the wheel hub housing space. The motor generates
torque to input shaft, amplified by the reducer, then
flows into the reducer shell, which connects with the
wheel rim via a flange, propelling the mining truck.
The system input parameters and requirements are
listed in Table 4.
Table 4 System parameters and requirements
Parameters
Value
Nominal speed
474.6rpm
3
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Torque @nom speed
22000Nm
Speed ratio
46
Figure 3 Wheel hub reducer housing
In a traditional design process, presented in Figure 4(a),
there are many procedures interdependent with each
other, resulting in a complex system overall design.
According to the flowchart displays, traditional design
process describes as follows. Firstly, the system
designer assigns each stage speed ratio to meet the
system speed ratio. Then, taking the comprehensive
consideration of system requirements and
manufacturing cost and difficulty, the system designer
determines each stage’s materials, thermal treatment
and hardness based on his rich experience in gear
system design and manufacturing. For the speed ratio
examination, this paper applies a relative error, which
should not differ from the requirement more than 2%.
Next step is determination the gear module number
and number of gear teeth. The planetary assembling
requires a balance between the number of sun/ring
gear teeth and the number of planet gears, as well as
the planet gear and sun/ring gear elementary center
distance. In this study, we choose the GB as our design
and examination standards. After checking the contact
and bending strength, the planet gear and carrier
bearing determination is the following step. To
enhance the gear strength and balance the sun gear
abrasion (since sun gear meshes with multi planet
gears), the sun gear usually goes through a
modification. After the adjustment of modification
coefficient, the sun gear and planet gear contact ratio
may decrease. To guarantee the continuity of gear
meshing and the torque transmission smoothness, the
contact ratio examination is necessary. At last, re-
checking all the gears contact and bending strength. If
all the results meet requirements, the next design
section is shaft design. Otherwise, gear system re-
design is needed.
Since shaft fatigue directly related to the shaft material
and thermal treatment, the shaft design process first
step is the shaft material selection. Then determine the
shaft layout, for the shaft layout and support directly
influence its strain and deflection.
Based on the above description, reducer system design
is a complex, interdependent, rich experience required,
and time-consuming process. Moreover, it may not be
optimum. To overcome these drawbacks, we proposed
an optimal method to make the novel three-stage
reducer into realization. In this study, we apply the GA
as design tool to simplify the gear system design. The
simplified, the optimal design process presents in
Figure 4(b).
4
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(a) Traditional design process
(b) Optimal design process in this research
Figure 4 Design process
In this study, we give an optimization design of three-
stage planetary gear reducer system. The bearing
selection refers to author’s advice, and GA optimizes
the gear design process.
GA BASED OPTIMAL DESIGN
Based on the foregone bearing, the first step of GA
implementation is the selection of design variables.
According to the system architecture in Figure 2, we
choose 12 parameters to describe this reducer, listed in
Table 5.
Table 5 Selected design variables in three-stage reducer
Symbol
Description
zs/r_i
Number of sun/ring gear teeth in ith stage
mi
gear module number of ith stage
bi
gear face width of ith stage
5
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Objective function
In the mining truck, the whole truck lightweight can
not only reduce the production cost and improve the
economy, but also increase coefficient of utilization.
Furthermore, wheel-hub reducer is unspring mass,
whose reduction will benefit the handling performance.
Hence, the weight minimization of this three-stage
reducer is the objective function. Since we choose the
bearing and shaft based on author’s experience, this
study only considers the gear system mass as objective
function.
Equation 4 Object:
3
_ _ _
1s i r i c i
i
f V V n V
where, ρ is steel density, 7.9g/cm3. Subscript i is the
stage number. Subscript s, r, c is sun gear, ring gear
and carrier gear (planet gear), respectively. n is the
number of planet gear. In this study, the first, second
and third stage planet gear number is set as 4, 6, and 6,
respectively. Since the steel density is a constant, the
objective function can be rewritten as follows:
Equation 5 Object:
3
_ _ _
1s i r i c i
i
f V V n V
In this study, we consider the gear is total solid, and
calculate the spur gear volume calculation based on an
engineering, simple method [19].
Constraints groups
Each decision in the flowchart in Figure 4 represents
at least one constraint. Several categories describe all
the constraints.
Basic constraints group: regarding the gearing system
The primary task is the total speed ratio. Hence, the
first constraint is to meet the speed ratio requirement.
In this study, we set the speed ratio fall in a narrow
range.
Equation 6 Speed ratio constraints:
ratio desired
desired
ii
i
where, iratio is the reducer actual speed ratio; idesired is
the desired speed ratio, set as 46 in this study; ∆ is the
relative error, set as 1% in this study.
Another constraint must make sure the number of gear
teeth is an integer value. In addition, gear module
number is a discrete number. Based on our engineering
experience, this application usually uses module
number from 5 to 16 to meet heavy duty requirements.
Therefore, the module number is also integer value
(ignore the module number 6.5). Besides, the module
number variable has the upper/lower bound constraint.
Since the number of gear teeth coupling relationship
in planetary gear system, the number of planet gear
teeth is:
Equation 7:
__
_2
r i s i
ci
zz
z
If the teeth difference between ring gear and sun gear
is odd, additional gear modification is needed. In
planetary gear system, sun gear meshes with multi
planet gears, resulting in sun gear heavier abrasion.
Therefore, the sun gear is modified to meet the gear
center distance when the difference is odd number.
Based on the modification of sun gear, the following
requirements defined 24 constraints:
No gear undercuts;
Tooth thickness of all gears on the addendum
circle must be greater than specific value;
Contact ratio of each pair gear must greater than
specific value.
Strength constraints group
To make sure the reducer service life, the gear contact
and bending pressure should not exceed the limitation
(12 constraints):
Safety factor for gear train contact must be greater
than specific value;
Safety factor for gear train bending must be
greater than specific value.
Spacing constraints group
As shown in Figure 3, the space available in wheel hub
is limited. The total gear face width, including the
connector width between 1st and 2nd, 2nd and 3rd
stage, and the space between each stage for lubrication,
should less than the available width.
In addition, the available maximum diameter in the
wheel hub varies. We separate the 1st stage in the
lower diameter space, and the remaining two stages lie
in the greater diameter space. That is because the latter
stage transmits the higher torque, which requires much
stronger gear. Therefore, the maximum sun gear
diameter in each stage must less than the specific value
for the available wheel hub space.
Furthermore, due to the assigned planet bearing and
pin, some other constraints are also defined.
According to the planet gear bearing size and shaft size,
the length L (Figure 5) should not be less than the tooth
full depth.
6
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(a) Bearing spacing
(b) Planet gear spacing
Figure 5 Spacing constraints
In the planet gear system, the number of planet gear
couples with the space between sun gear and ring gear,
as Figure 5 shows. The relatively large planet gear may
interfere with each other, if set to a large number of
planets. At the same time, the margin between two
planet gears should guarantee the lubricant flowing.
Therefore, the planet gear diameter must less than a
specific value, which is coupled with the planet gear
number, sun/ring gear teeth number and module
number.
OPTIMIZATION RESULTS
With the sun gear, ring gear, and planet gear material
as 17CrNiMo6, 20CrNi3, and 17CrNiMo6
respectively, the GA is implemented to give the
optimal design of a real truck wheel hub application
reducer. After optimization process, the rounded
values are listed in Table 6.
Table 6 Comparison of reducer main parameters for both design methods
Traditional design
Rounded after
optimal design
1st stage
Teeth number zs_1/zc_2/zr_3
24/28/80
17/25/67
Module m1
8
10
Face width b1/mm
200
250
Safety factor for contact (SC/CR)
1.60/1.63
1.64/1.86
Safety factor for bending (SC/CR)
5.00/4.89
6.93/6.76
2nd stage
Teeth number zs_1/zc_2/zr_3
28/23/74
25/20/65
Module m2
12
14
Face width b2/mm
250
300
Safety factor for contact (SC/CR)
1.75/1.53
1.86/1.61
Safety factor for bending (SC/CR)
5.92/6.46
7.57/8.27
3rd stage
Teeth number zs_1/zc_2/zr_3
28/23/74
41/25/91
Module m3
12
11
Face width b3/mm
500
400
Safety factor for contact (SC/CR)
1.75/2.10
2.00/1.99
Safety factor for bending (SC/CR)
4.03/4.89
3.54/4.29
where, SC, and CR is the sun gear to carrier (planet) gear contact or bending, and carrier (planet) gear to ring gear contact or
bending, respectively.
According to the main gearing system parameters in
Table 6, the total gear volume in optimal design and
traditional design is 5.3383e8, and 5.9922e8 (unit,
cubic millimeter), resulting in a 10.91% volume
reduction. Apart from the volume optimization,
gearing system safety factors are also improved in
different levels. The contact safety factor is
proportional to gear diameter and face width, while the
bending safety factor is proportional to gear module
number and face width. For the first stage, the increase
in module number and decrease in teeth number lead
to the gear diameter increasing. Together with the gear
face width increasing, different improvements are
achieved in safety factor. This principle can also
explain the 2nd and 3rd stage safety factor
improvement.
Other researcher reported the third stage always
suffers from the strength weakness, especially the
contact safety factor [1] [18]. In this optimal design,
the results fully utilize the wheel hub radial housing.
The gear diameter increasing bring significant
improvement in the contact safety factor, about 12.5%
for SC factor and 5.5% for CR factor. The gear face
width decreasing sacrifices the bending safety factor,
while the result is still acceptable.
CONCLUSIONS
Multi stage planetary gear reducers are the most
common transmission type in mechanical systems,
especially in heavy-duty area. However, little attention
has been paid to this specific area.
In this paper, firstly, we proposed a novel type of
three-stage planetary reducer. The speed ratio and
7
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torque distribution characteristics are analyzed. The
state-of-art current other three types of three-stage
planetary speed reducer are reviewed, as well as the
comparison of speed ratio and torque distribution
analysis.
Based on the traditional design process, this paper uses
the GA to solve the complex design problem. The
optimization is performed for the gearing system, with
the experience assigned bearing and shaft system. The
optimization object is the total volume (hence, the total
mass) of gears. The design variables, in a total of 12,
contain integer variables (teeth number and module
number), with a quantity of highly non-linear
constraints. The optimization result reveals the
proposed reducer type can match the system
requirement; optimal design can balance the three
stage safety factor; GA based optimization design is
an effective way to solve the complex engineer design
problem.
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