Life Distribution Transformation Model of Planetary Gear System
Li et al. Chin. J. Mech. Eng.
Life Distribution Transformation Model of Planetary Gear System
Ming Li 0
LiY‑ang Xie 0
HaiY‑ang Li 0
Jun‑Gang Ren 0
0 Key Laboratory of Vibration and Control of Aviation Propulsion System, Ministry of Education, Northeastern University , Shenyang 110819 , China
Planetary gear systems have been widely used in transportation, construction, metallurgy, petroleum, aviation and other industrial fields. Under the same condition of power transmission, they have a more compact structure than ordinary gear train. However, some critical parts, such as sun gear, planet gear and ring gear often suffer from fatigue and wear under the conditions of high speed and heavy load. For reliability research, in order to predict the fatigue probability life of planetary gear system, detailed kinematic and mechanical analysis for a planetary gear system is firstly completed. Meanwhile, a gear bending fatigue test is carried out at a stress level to obtain the strength information of specific gears. Then, a life distribution transformation model is established according to the order statistics theory. Transformation process is that, the life distribution of test gear is transformed to that of single tooth, and then the life distribution of single tooth can be effectively transformed to that of the planetary gear system. In addition, the effectiveness of the transformation model is finally verified by a processing method with random censoring data.
Planetary gear system; Reliability modeling; Probabilistic life; Random censoring data
Planetary gear system has the advantages of light weight,
small size, large transmission ratio, powerful bearing
capacity and high transmission efficiency [1–3].
Therefore, it has been widely used in a variety of mechanical
equipment. Actual situation and experience data have
showed that the reliability of key gears (such as sun gear,
planet gear and ring gear) has a significant impact on the
reliability of the whole transmission system .
Therefore, some scholars have predicted the life or reliability of
the gear system consisting of these gears. On the basis of
considering time-varying meshing stiffness of gears, Qin
et al.  studied the dynamic reliability of wind turbine
gear system. Hu et al.  established a reliability model
for closed planetary gear system, which considered the
effects of load, tooth width and load sharing on reliability
of the gear system. Zhou et al.  considered
reliabilitybased sensitive factors to conduct the reliability analysis
for the planetary gear system in shearer mechanism. Li
et al.  studied the influence of unequal load sharing
on the reliability of planetary gear system. Wu et al. 
established a typical two stage planetary gear
transmission reliability model based on the product rule of system
reliability. Huang et al.  presented a novel method to
evaluate the reliability of the kinematic accuracy of gear
mechanisms with truncated random variables. Nejad
et al.  presented a long-term fatigue damage
reliability analysis method for tooth root bending in wind
turbine drive trains. Shang et al.  combined fuzzy
mathematics with reliability theory, and adopted
multiobjective optimization design to study the design
technology of high reliability and high power density for
planetary gear transmission in large energy installations.
Meanwhile, Hao  carried out a multi-objective fuzzy
reliability optimization design for the multistage
planetary gear system. Zhang et al.  applied an improved
genetic algorithm to the reliability optimization design of
NGW planetary gear transmission to improve the
stability and transmission efficiency of the transmission
system. In addition, many other researchers have conducted
relevant studies [15–18].
In planetary gear system, movement state of each gear
is very complicated, and their working environment
is generally worse . Therefore, for the validity and
simplicity of forecasting method, how to establish model,
and how to get input variable for the model should be
a focus. A large number of scholars have established
the reliability model of planetary gear system based on
dynamics theory, which makes the form of the models
too complex or difficult to ensure the accuracy. Field or
accelerated gear system tests can effectively obtain the
life or reliability information, but for planetary gear
system, long time and high cost make it difficult to realize
. In order to predict the probability life of planetary
gear system simply and effectively, this paper combines
a simple gear pair meshing test with a life distribution
transformation model. Specifically, test result of the
special gears (whose parameters are the same as those
of service gears) is used as input variables for the
transformation model in order to fully reflect the influence
of service gear performance on the gear system life. The
gear pair meshing test can effectively simulate the
running state of gears in planetary gear system to a certain
extent, so that the test data will contain a large number
of the gear system life information. Therefore, prediction
accuracy of the model is guaranteed while its complexity
2 Kinematic and Mechanical Analysis
2.1 Kinematic Analysis
In the same time t, the operating speed of each gear in
the planetary gear system is different. In order to relate
their motion relationships, kinematics analysis of the
planetary gear train is required. On the basis of previous
research , the motion law of planetary gear drive
system is as follows
ωs + RRsr ωr −
1 + Rr
ωc = 0,
ωp − RRss −+ RRrr ωc +
Rs − Rr
ωr = 0,
where ωs, ωp, ωr, ωc are respectively the absolute angular
velocity of sun gear, planet gear, ring gear and planet
carrier and, Rs and Rr are the radii of pitch circle of sun gear
and ring gear respectively. The absolute angular velocity
of each gear, angular velocity relative to planet carrier,
and the number of loads on a gear (if the gear is loaded
once, this means that each tooth of the gear is loaded
once) in time t can be known by Eq. (
). And they are
listed in Table 1, in which, minus sign shows opposite
rotation direction, and np is the number of planet gears
in the system.
2.2 Calculation of Root Bending Stress
Bending fatigue failure of tooth root is the most common
and dangerous failure form for gears, so this paper takes
it as the failure mode of reliability analysis. When
calculating the root bending stress, the tooth can be regarded
as a cantilever beam, and its dangerous section is
determined by the 30° tangent method. As shown in Figure 1,
hF is the length of arm, SF is the thickness of dangerous
section, ρr is the fillet radius at root, αa is the direction
of normal force at addendum. The angle between AB and
the symmetrical center line of tooth is 30, and AB is
tangent to the fillet radius at root, and AC is the same as AB.
The cross section that connects the two tangent points B
and C, and is parallel to the axis of gear is just the
dangerous section, which is the main location of the bending
fatigue crack of teeth.
In the condition of a pair of teeth are meshed, the
maximum bending stress on the dangerous section is
calculated based on the full load acting on the top of tooth.
The normal force acting on the top of tooth is
d1 cos α
where T1 is the torque of driving gear, and d1 is its pitch
diameter, and α is the pressure angle.
If the friction between teeth is neglected, the normal
force acting on the top of tooth can be divided into two
forces Fn cos αa and Fn sin αa, as shown in Figure 2.
Since the shear stress produced by Fn cos αa and the
compressive stress produced by Fn sin αa are much
smaller than the bending stress produced by Fn cos αa,
Figure 1 Dangerous section of tooth bending fatigue
thus the minimum values can be ignored. Then, the root
bending stress is
σF = FnhF cos αa ,
where b is tooth width.
) is substituted into Eq. (
σF = 2KT1 6 hF cos αa
bd1m SmF 2 cos α
where K is load factor. And make
6 hF cos αa
σF = 2KT1 YFYsYε,
at the same time, a stress correction coefficient Ys and
a coincidence degree coefficient Yε are introduced, and
finally the root bending stress is expressed commonly
where YF is tooth shape coefficient, which is related to
the shape of tooth profile, regardless of the modulus. Ys is
stress correction coefficient, which can take into account
the stress concentration at tooth root. And Yε is contact
The root bending stress calculated by Eq. (
) is a
maximum tensile stress on the dangerous section at the load
side of tooth (the root bending stress mentioned in this
paper all refers to this maximum stress), as shown in
Figure 2. It calculates the peak value of bending stress at
tooth root during meshing process, so a load history of
the tooth can be given in a series of discrete values. At
the same time, RomaxDesigner software is used to
calculate the stress, which is based on Eq. (
) and can
accurately reflect the effect of centrifugal force on stress. The
simulation model of a NGW planetary gear system is
shown in Figure 3 and, the number of planet gears in the
system is np=3, rated input speed of the system is 2000 r/
min, and rated input power is 10 kW. System parameters
and stress calculation results are presented in Table 2.
3 Gear Fatigue Test
Bending fatigue fracture of teeth is the most severe
failure type for gears [22, 23], which may lead to a direct
collapse of the power transmission system. More seriously,
in the field of aviation, fracture of the teeth can lead to
serious accidents . Therefore, a gear bending fatigue
test is carried out to collect the life data of specific gears
and to use the statistical result as input variable for the
3.1 Gear Sample
The gear samples are standard cylindrical helical gears
designed according to the parameters of gear in the
planetary gear system. The sample parameters and
appearance are shown in Table 3 and Figure 4, respectively.
The manufacturing process begins with rough
forging, followed by roughing, high temperature
normalizing, semi-finishing, hobbing, shaving, carburizing and
final grinding. Specifically, carburizing temperature is
920‒930 °C, oil quenching temperature is 820 °C and
tempering temperature is 180 °C. All the gears are made
of the same equipment and process.
3.2 Experimental Equipment
The main body of experimental equipment is a power flow
closed gear rotating tester, as shown in Figure 5. Load is
added by the mechanical levers and weights, and motor
speed remains constant at 3460 r/min. Center distance of
gear pair on the tester is 100 mm, and transmission ratio is
1:1. A vibration monitoring instrument is used to achieve
automatic shutdown when any tooth fracture occurs. Oil
injection position and flow can be adjusted to ensure good
lubrication. Circulating water system keeps the return oil
temperature below 60 °C. The error of circulation recording
device is not more than ± 0.1%. Before starting experiment,
the equipment is calibrated based on the requirements of
GB/T 14230 so as to meet the test accuracy.
3.3 Test Method and Result
According to the stress calculation of three kinds of gears
in Table 2, we need to get the life distributions of gear
specimens under the stresses of 420 MPa, 388 MPa, and
362 MPa. The life data of ten pairs of gear specimens are
obtained under the stress level of 420 MPa and are fitted
with Weibull distribution function. Because the life at the
other two stress levels is too long, their distributions can
be obtained according to the life distribution of 420 MPa
and the life probability mapping principle proposed in
Ref. , the results are shown in Figure 6.
4 Life Distribution Transformation Model
The concept of minimum order statistics can be applied
to the establishment of transformation model, the
concept can be described as, n samples are extracted from
a population and then the minimum sample is chosen,
repeating this action, the final distribution from these
minimum samples is just the minimum order statistics
distribution of the population. If the probability
density function of X is f(x), and the cumulative distribution
function is F(x), the probability density function of the
minimal order statistics is given by Eq. (
g (x) = z[1 − F (x)]z−1f (x).
For the probabilistic life prediction of gear, we
consider a single gear as a sequential system, with each
tooth as a component in the system. If any teeth failed,
the gear would not be able to fulfill its function of power
or motion transmission, thus resulting in failure of the
sequential system. According to the definition of order
statistics, the gear probabilistic life equals the minimal
order statistics of the single tooth probabilistic life .
If the number of teeth on a gear is z, then by
integrating both sides of Eq. (
), we can derive the cumulative
distribution function of minimal order statistics G(n),
which is just the cumulative distribution function of gear
G(n) = 1 − [1 − F (n)]z,
where F (n) is the cumulative distribution function of the
By transforming Eq. (
) the cumulative distribution
function of the tooth life expressed by gear life can be
F (n) = 1 − [1 − G(n)]1/z.
The fatigue life distribution of the gear can be expressed
as a two-parameter Weibull distribution , and its
cumulative distribution function is given by Eq. (
G(n) = 1 − exp −(n/θ0)β0 , n > 0,
where β0 is the geometric parameter of gear life
distribution, and θ0 is corresponding scale parameter.
By substituting Eq. (
) into Eq. (
), then Eq. (
produced, Eq. (
) has the same meaning as Eq. (
probabilistic life of the gear with z teeth is converted
into the probabilistic life of single tooth. From the
Life , N
pattern of the distribution function the life distribution
of tooth also conform to the two-parameter Weibull
distribution, where the geometric parameter of tooth life
distribution is β1 = β0, and the scale parameter changes
to θ1 = θ0z1/β0.
F (n) = 1 − exp − n/ θ0z1/β0
, n > 0.
In the planetary gear system, the sun gear, planet gear,
and ring gear engage with each other, so they have the same
modulus and thus have the same carrying capacity. If
material properties of all the gears are also the same, the strength
of the teeth can be considered the same. Therefore, the tooth
life under a given load level can be considered as the
random variable with independent and identical distribution.
Furthermore, based on the probabilistic life of tooth and the
concept of minimal order statistics, a probabilistic life
transformation model of planetary gear system is established.
When the cumulative distribution function of the tooth
life F (n) is determined, then the life cumulative
distribution functions of sun gear, planet gear, and ring gear can
be determined based on Eq. (
Gs(n) = 1 − [1 − Fs(n)]zs ,
Gp(n) = 1 − 1 − Fp(n) zp ,
Gr(n) = 1 − [1 − Fr(n)]zr ,
where z is the number of teeth, and s, p, r represent
different types of gears in planetary gear system and
meanwhile express the cumulative distribution functions
under different stress levels.
According to reliability product theorem, the
cumulative distribution function of the planetary gear system life
is given by Eq. (
Fsys(n) = 1 − [1 − Fs(n)]zs 1 − Fp(n) np·zp [1 − Fr(n)]zr .
By substituting the number of load actions in Table 1
into Eq. (
), then the load acting numbers of each gear
can be unified into time t
Fsys(t) = 1 − 1 − Fs
1 − Fp
1 − Fr
Rs + Rr
Rr2 − R2
Rs + Rr
Weibull distribution function of the tooth life is shown
as Eq. (
When β is solved, the maximum likelihood estimation
θ can be given by Eq. (
F (t) = 1 − exp −(t/θ1)β1 , t > 0.
By substituting Eq. (
) into Eq. (
) we get Eq. (
Fsys(t) = 1 − exp −zs
np · zp
Rr2 − Rs2 θ1
(Rs + Rr)θ1
(Rs + Rr)θ1
5 Model Validation
5.1 Random Censoring Data
In life-cycle testing some products fail, so an accurate
failure time (i.e., product life failure data) can be obtained;
while other products may exit the test before failure due to
some reasons, thereby obtaining a higher life than the
testing time, namely the life censoring data. The failure data and
censoring data are generally called random censoring data.
In order to verify the transformation model, the
processing method with random censoring data is used. The gear
test will stop when any tooth breaks, thereby deriving
failure data by the tooth and censoring data of the other teeth.
By processing the random censoring data of the teeth, the
distribution of tooth life can be obtained, and it can be
compared with the results of the transformation model.
5.2 Estimating Distribution Parameters of Random
Maximum likelihood estimation possesses obvious
advantages in processing censoring data. Due to the
large sample size of random censoring data, this method
is used to estimate the distribution parameters.
Generally the distribution of fatigue life can be represented by
a two-parameter Weibull distribution with the
distribution function given by Eq. (
), where β and θ are the
parameters to be estimated. The random censoring data
are represented by (t1, δ1), (t2, δ2),…, (tn, δn), where ti is
observation datum for the ith product. When δi = 1, ti is
the failure data, and when δi = 0, ti is the censoring data.
The likelihood function of (β, θ) is given by Eq. (
L(β, θ ) =
F ′(ti, β, θ ) δi [1 − F (ti, β, θ )]1−δi . (
The maximum likelihood estimation β is the root of
1 + i=1
δi ln ti
tiβ ln ti
5.3 Parameters Comparison and Model Validation
The concept of minimal order statistics is used to
create the probabilistic life transformation model. First, the
probabilistic life of the tooth is obtained based on that of
the specific gear. Then, the probabilistic life of the
planetary gear system can be obtained based on the
probabilistic life of the tooth. Since the two transformations
used the same principle, only the first transformation is
actually validated. From Eq. (
), we can know the
estimate of the distribution parameters of tooth life. From
the process of random censoring data, we can also derive
the estimate of distribution parameters of tooth life. The
results of the two methods can be compared to validate
the rationality of the transformation model.
In the gear test, the gear pairs are engaged to transfer
power. The drive gear has a higher dynamic torque than
the driven gear . It was also found in the test that all
the teeth breaking occurred on the drive gears; thus, the
life information of the drive gears is used for the
validation of the transformation model.
The ten gear life data in Figure 6 can be used. If the
number of teeth in a drive gear is z, failure of any tooth
will cause the test to stop, and the circulating life is
recorded. For the gear, the life data are comprised of the
failure data for one tooth and censoring data for the z − 1
teeth. Given j as the number of “circulating life” obtained
during the test, then the failure data of the j teeth and
censoring data of the j(z − 1) teeth can be obtained.
The statistical result of random censoring data can well
reflect population’s situation due to the large sample size.
And it can be seen from Figure 7 that the overlap ratio of
two curves is very high. Therefore, it illustrates the
rationality of the life distribution transformation model. And their
relative error of life mean is about 7.8%, and that of life
standard deviation is about 8.3%. These means the
transformation model can be well used to predict the reliability for
planetary gear system. In addition, with different sample
sizes, the results of two methods are consistent, it can be
known that the two methods are not sensitive to variation
in sample size, suggesting that the transformation model is
effective in dealing with data of small sample size.
Reliability model result
Random censored data result
) The life information of specific gear is taken as
the input variables for the transformation model,
thereby enabling the model to reflect the large
amount of factors that influence system life, which
not only improved the prediction precision, but
greatly simplified the model as well.
) Through the fatigue test, the P-S-N curves of the
specific gear can be obtained. Based on the
transformation model, the probabilistic life of the planetary gear
system at an arbitrary stress level can be derived.
) The actual load of the planetary gear system can
be first statistically processed, and then the results
are applied to the fatigue test with the specific gear,
then this method can improve the life prediction
precision of the gear system and is more effective
for the life prediction of service gear system.
) The model does not consider any failure correlation
among gears. In practice, the load of gears are
random, thus, the results of the transformation model
tend to be conservative.
) Based on the concept of minimal order statistics,
the model for the probabilistic life
transformation for planetary gear system is established. In the
model the probabilistic life of single tooth is first
calculated based on that of the specific gear. Further,
the probabilistic life of the planetary gear system can
be estimated based on that of the single tooth.
) A gear bending fatigue test is performed at a
constant stress level, and the bending fatigue life data of
ten 20CrMnTi carbonized gears are obtained.
) The validity of the transformation model is
illustrated by the method of random censored data, and
it also indicates that the model has the ability to
process small sample data. A large number of
calculations, analyses and comparisons show that the
model is also valid for data with different scatter.
L‑ YX was in charge of the whole trial; ML wrote the manuscript; H‑ YL and J‑ GR
assisted with sampling and laboratory analyses. All authors read and approved
the final manuscript.
Ming Li, born in 1986, is currently a PhD candidate at School of Mechanical
Engineering and Automation, Northeastern University, China. His research
interests include mechanical transmission design and fatigue reliability. Tel:
+86‑13940397486; E‑mail: .
Li‑ Yang Xie, born in 1962, PhD, is currently a professor and a doctorial
supervisor at Northeastern University, China. His research interest is reliability
theory. Tel: +86‑13804011565; E‑mail: .
Hai‑ Yang Li, born in 1988, is currently a PhD candidate at School of
Mechanical Engineering and Automation, Northeastern University, China. His
research interests include mechanical transmission design and fatigue reli‑
ability. Tel: +86‑15940426127; E‑mail: .
Number of cycles, N
Figure 7 Probability density curves from two methods
The authors declare that they no competing interests.
Ethics Approval and Consent to Participate
Supported by National Key Technology Research and Development Program
of China (Grant No. 2014BAF08B01), and Natural Science Foundation of China
(Grant No. 51335003), and Collaborative Innovation Center of Major Machine
Manufacturing in Liaoning Province of China.
Springer Nature remains neutral with regard to jurisdictional claims in pub‑
lished maps and institutional affiliations.
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