Spur Gear Tooth Pitting Propagation Assessment Using Model-based Analysis
Chin. J. Mech. Eng.
Spur Gear Tooth Pitting Propagation Assessment Using Model-based Analysis
Xi-Hui Liang 0 1 2 3
Zhi-Liang Liu 0 1 2 3
Jun Pan 0 1 2 3
Ming Jian Zuo 0 1 2 3
Gear Dynamic 0 1 2 3
0 School of Mechatronics Engineering, University of Electronic Science and Technology , Chengdu 611731 , China
1 & Ming Jian Zuo
2 Supported by Natural Science and Engineering Research Council of Canada (Grant No. RGPIN-2015-04897), International S&T Cooperation Program of China (Grant No. 2015DFA71400), National Key Research and Development Program of China (Grant No. 2016YFB1200401), and National Natural Science Foundation of China , Grant No. 51375078, 51505066
3 Key Laboratory of Reliability Technology for Mechanical and Electrical Product of Zhejiang Province, Zhejiang Sci- Tech University , Hangzhou 310018 , China
Tooth pitting is a common failure mode of a gearbox. Many researchers investigated dynamic properties of a gearbox with localized pitting damage on a single gear tooth. The dynamic properties of a gearbox with pitting distributed over multiple teeth have rarely been investigated. In this paper, gear tooth pitting propagation to neighboring teeth is modeled and investigated for a pair of spur gears. Tooth pitting propagation effect on time-varying mesh stiffness, gearbox dynamics and vibration characteristics is studied and then fault symptoms are revealed. In addition, the influence of gear mesh damping and environmental noise on gearbox vibration properties is investigated. In the end, 114 statistical features are tested to estimate tooth pitting growth. Statistical features that are insensitive to gear mesh damping and environmental noise are recommended.
Mesh stiffness; Mesh damping dynamics; Vibration; Statistical feature simulation
Department of Mechanical Engineering, University of
Alberta, Edmonton T6G 1H9, Canada
Gearbox is one of the most widely used transmission
systems in the world. However, due to high service load, harsh
operating conditions or fatigue, faults may develop in gears
]. Through observations at gearboxes used in Syncrude
Canada Ltd, tooth pitting was a common failure mode [
When tooth pitting appears on gears, gear mesh stiffness
reduces and correspondingly the vibration properties of
According to the American Society for Metals (ASM)
], gear pitting damage can be classified into
five levels according to pitted areas as follows:
1. Some micro-pitting (pits with dimensions in the order
of millimeters) and a few macro-pits on the pinion. No
pitting on the gear.
2. Micro-pitting and appreciable macro-pitting on the
pinion. Almost no pitting on the gear.
3. Micro-pitting and considerable macro-pitting on the
pinion with one or more gross pits. Damage to both the
pinion and the gear.
4. Macro-pitting over 50%–100% of the pinion tooth
surface. Removal of metal thins the teeth and disrupts
load sharing between teeth. Gear unit has greatly
increased noise and vibration.
5. Macro-pitting all over the teeth with considerable gross
pitting. Teeth are thinned so much by wear that the tips
are becoming sharp like a knife.
Some researchers [
] created man-made tooth pitting
on gears to experimentally explore fault symptoms of a
gearbox. For example, in Ref. , three pitting levels were
created using the electro discharge machining, namely,
slight pitting, moderate pitting and severe pitting as shown
in Figure 1. In these methods, vibration sensors are
generally installed on the casing of bearings or the housing of
gearboxes to measure the vibration responses. In the first
step, all the gears are in perfect condition and signals are
collected. Then, a damaged gear is installed in the gearbox
and signals are collected. The fault symptoms are
investigated by comparing the signals from the healthy gearbox
with those from the gearbox with a damaged gear.
However, these experimental signals are polluted by noise. The
fault symptoms may be submerged by the noise and hard to
be observed. More importantly, the above comparison
between signals can hardly reveal the fault physics of a
Feng and Zuo [
] proposed a mathematical model to
investigate fault symptoms of a planetary gearbox with
tooth pitting. In their model, amplitude modulation and
frequency modulation caused by pitting damage are
considered. However, their model cannot be used to model
pitting growth. In addition, their mathematical model lacks
the connection with physical parameters of a gearbox, like
gear mesh stiffness and damping [
Several researchers investigated dynamic properties of a
gearbox with tooth pitting through dynamic simulation.
Chaari et al. [
], Cheng et al. [
], and Abouel-seoud
et al. [
] modeled a single tooth pit in the rectangular
shape (all other teeth are perfect) and investigated the
single tooth pit effect on the dynamic properties of a
gearbox. Rincon et al. [
] modeled an elliptical pit on a
single tooth and evaluated the dynamic force of a pair of
gears. Ma et al. [
] studied the effect of tooth spalling on
gear mesh stiffness. A single rectangular spalling was
modeled and the effects of spalling width, spalling length
and spalling location on stiffness were investigated,
respectively. Saxena et al. [
] incorporated the gear tooth
friction effect in modeling a single gear tooth spalling.
Liang et al. [
] evaluated the mesh stiffness of gears with
multiple pits on a single tooth using the potential energy
method. However, all these studies focus on single tooth
pitting modeling. According to the current studies [
pitting propagation to neighboring teeth is a common
phenomenon. This study overcomes the shortcomings of
single tooth pitting modeling. We will model gear tooth
pitting propagation to neighboring teeth and analyze its
effect on gearbox vibration.
Gear dynamic models may provide useful information
for fault diagnosis [
]. Vibration-based time domain,
frequency domain, and time-frequency domain analyses
provide powerful tools for fault diagnosis of rotating
]. One traditional technique is based on
statistical measurements of vibration signals . Many
statistical indicators were proposed for machine fault
]. In Liu et al. , 34 statistical
indicators were summarized and 136 features were generated. In
Zhao et al. [
], 63 statistical indicators were summarized
and 252 features were produced. The features [
used for the classification of gear damage levels of a lab
planetary gearbox. In this study, 36 statistical indicators are
selected from the literature. Then, 114 statistical features
are generated and tested using simulated vibration signals
for the pitting growth estimation of a fixed-axis gearbox.
The effect of gear mesh damping and environmental noise
on the performance of statistical features will be analyzed.
The objective of this study is to simulate vibration
signals of gears with tooth pitting covering multiple teeth,
investigate pitting effects on vibration properties and
provide effective features for pitting growth estimation. The
scope of this paper is limited to a fixed-axis gearbox with a
single pair of spur gears. A dynamic model is used to
investigate the effects of tooth pitting growth on vibration
properties of a gearbox. The tooth pitting propagation to
the neighboring teeth is modeled. Three pitting levels are
modeled: slight pitting, moderate pitting and severe pitting.
The vibration signals of a gearbox are simulated for each of
the three severity levels. The vibration properties are
investigated and fault symptoms are summarized.
Statistical features are tested on simulated vibration signals. These
features are ranked for pitting growth estimation. The
features insensitive to gear mesh damping and
environmental noise are recommended.
This paper is organized as follows. In Section 1, an
introduction of this study is given including literature
review, our research scope and objective. In Section 2, a
pitting propagation model and a method to evaluate mesh
stiffness of gears with tooth pitting are presented. In
Section 3, a dynamic model is utilized to simulate vibration
signals of a spur gearbox with tooth pitting, and pitting
effects on the vibration signals are analyzed. In Section 4,
114 statistical features are tested for estimation of gear
tooth pitting propagation, and gear mesh damping and
environmental noise effect on these features are analyzed.
In the end, a summary and conclusion of this study is
2 Tooth Pitting Propagation Modeling and Mesh
2.1 Tooth Pitting Propagation Modeling
In this study, we assume the pinion (driving gear) has
relatively soft gear tooth surfaces and the gear (driven gear)
has surface-hardened teeth. Tooth pitting only propagates
in the pinion (the gear is always in perfect condition).
Tooth pits are modeled in circular shape [
]. All the
circular pits have the diameter of 2 mm and the depth of
1 mm. Three pitting levels are modeled as shown in
Figure 2. The detailed information of these three pitting
damage levels is given below:
Slight pitting: 9 circular pits on one tooth and 3 circular
pits on each of the two neighboring teeth. All the circular
pits center on the tooth pitch line. The surface area of the
meshing side of a tooth is 194 mm2. This way, the middle
pitted tooth has a pitting area of 14.6% of the tooth surface
area. Each of the two neighboring teeth has a pitting area of
4.87% of the tooth surface area. The purpose of this level
of damage is to mimic slight pitting damage that
corresponds to the level 2 pitting damage defined in ASM
Moderate pitting: 18 circular pits on one tooth, 9
circular pits on each of the two neighboring teeth, and 3
circular pits on each of the next neighboring teeth on
symmetric sides. All the circular pits center on the tooth
pitch line. The pitting areas of the 5 teeth are 4.87%,
14.6%, 29.2%, 14.6% and 4.87%, respectively. We call this
damage level the moderate pitting damage corresponding
to the level 3 pitting damage defined in ASM handbook [
Severe pitting: 36 circular pits on one tooth, 18 circular
pits on each of the two neighboring teeth, 9 circular pits on
each of the next neighboring teeth on symmetric sides and
3 circular pits on each of the teeth after the next
neighboring teeth on symmetric sides. For the gear tooth with 36
circular pits, 18 pits center on the tooth pitch line and
another 18 pits on the tooth addendum. For other teeth,
circular pits all center on the tooth pitch line. The pitting
areas of the 7 teeth are 4.87%, 14.6%, 29.2%, 58.4%,
α hx A pit h
29.2%, 14.6% and 4.87%, respectively. We define this
level of damage as the severe pitting damage
corresponding to the level 4 pitting damage defined in ASM handbook
2.2 Mesh Stiffness Evaluation
Gear mesh stiffness is one of the main internal excitations
of gear dynamics. With the growth of gear tooth pitting,
gear mesh stiffness shape changes and consequently
dynamic properties of gear systems change. Therefore,
accurate gear mesh stiffness evaluation is a prerequisite of
gear dynamics analysis.
In Ref. [
], the potential energy method [
used to evaluate mesh stiffness of gears with multiple pits
on a single tooth. The gear tooth was modeled as a
nonuniform cantilever beam. The total energy stored in a pair
of meshing gears was the sum of Hertzian energy, bending
energy, shear energy and axial compressive energy
corresponding to Hertzian stiffness, bending stiffness, shear
stiffness and axial compressive stiffness, respectively.
Their equations are extended here to evaluate the mesh
stiffness of gears with tooth pitting distributed over
multiple neighboring teeth. The gear system is assumed to be
without friction (perfect lubrication), manufacturing error,
or transmission error, and the gear body is treated as rigid
18, 28, 29
]. The same assumptions will be employed in
this paper as this study only focuses on pitting effect on
Figure 3 shows a gear tooth modeled as a non-uniform
cantilever beam. The tooth fillet curve is approximated
using a straight line for the convenience of equation
]. Each circular pit is expressed by three
]: (u, r, d), where u represents the distance
between the tooth root and the circle center of the pit, r is
the radius of the pit circle, and d is the pitting depth.
If many circular pits show on a surface, as long as the
pits do not overlap with each other and all are within the
tooth surface area, the Hertzian contact stiffness kh,
Root Base circle circle
bending stiffness kb, shear stiffness ks and axial
compressive stiffness ka can be expressed as follows [
where E, L, m denote Young’s modulus, tooth width and
Poisson’s ratio, respectively; Z is the number of gear teeth;
N represents the number of circular pits on a tooth surface;
a0 is the pressure angle; a1 denotes the angle between
action force F and its decomposition component Fb; a2
indicates the half tooth angle on the base circle; a3
describes the approximated half tooth angle on the root
circle; DLxj, DAxj and DIxj (caused by the jth circular pit)
represent respectively the reduction of tooth contact width,
area and area moment of inertia of the tooth section where
the distance to the tooth root is x; DLxj, DAxj and DIxj are
expressed as follows [
DLx ¼ 02; r2otheðrus, xÞ2; x 2 ½u r; u þ r ; ð5Þ
< 1 DLxd3þ
DLxd; x 2 ½u
r; u þ r ;
d=2Þ2 ; x 2 ½u
r; u þ r ;
Eqs. (1)–(4) are derived for a single gear tooth (with
pitting) which is modeled as a non-uniform cantilever
beam. They are all expressed as a function of gear rotation
angle a1. Applying these equations iteratively to each gear
tooth, the stiffness of each tooth can be obtained. But, the
values for DLxj, DAxj and DIxj may be different among teeth
due to the variance of number and location of pits.
For a pair of spur gears with contact ratio between 1 and
2, one pair and two pairs of tooth contact takes place
alternatively. For a single-tooth-pair meshing duration, the
total effective mesh stiffness can be evaluated [
kt ¼ 1 1 1 1 1 1 1 ; ð8Þ
kh þ kb1 þ ks1 þ ka1 þ kb2 þ ks2 þ ka2
where subscripts 1 and 2 represent the driving gear and the
driven gear, respectively.
For a double-tooth-pair meshing duration, there are two
pairs of gears meshing at the same time. The total effective
mesh stiffness can be obtained as [
kt ¼ kt1 þ kt2
1 1 1 1 1 1 1 ;
i¼1 kh;i þ kb1;i þ ks1;i þ ka1;i þ kb2:i þ ks2;i þ ka2;i
where i ¼ 1 for the first pair and i ¼ 2 for the second pair
of meshing teeth.
Utilizing the above equations, mesh stiffness of a pair of
spur gears (gear parameters are given in Table 1) is
evaluated for each of the three pitting severity levels as shown
in Figure 2. The mesh stiffness results are shown in
Figure 4. When angular displacement of the pinion is 0, tooth
n-3 (see Figure 2) starts to mesh. Gear mesh stiffness
reduces when the pitted teeth mesh. Table 2 summarizes
the quantifications of stiffness reduction caused by tooth
pitting. The averaged mesh stiffness reduction d kt is used
2 x 109
tiffsen 1 Mesh
sh0.8 period 1
where kt1 and kt2 are the mesh stiffness calculated for a
damaged gear and a perfect gear, respectively, and S and
M represent the mesh stiffness data points collected during
the same angular displacement for a damaged gear and a
perfect gear, respectively.
Eight mesh periods (see Figure 4) are analyzed in
Table 2 because only the mesh stiffness of these eight mesh
periods may be affected in our model during one revolution
of the pinion. One mesh period is defined as an angular
displacement of the pinion experiencing a double-tooth-pair
meshing duration and a single-tooth-pair meshing duration.
For double-tooth-pair meshing durations, 4, 6 and 8 mesh
periods have mesh stiffness reduction caused by slight
pitting, moderate pitting and severe pitting, respectively. The
maximum averaged mesh stiffness reduction in a
doubletooth-pair meshing duration is 1.57%, 4.26% and 19.02%
corresponding to slight pitting, moderate pitting and severe
pitting, respectively. While for single-tooth-pair meshing
durations, 3, 5 and 7 mesh periods experience mesh stiffness
reduction corresponding to slight pitting, moderate pitting
and severe pitting, respectively. The maximum averaged
mesh stiffness reduction in a single-tooth-pair meshing
duration is 11.61%, 55.33% and 55.83% related to slight
pitting, moderate pitting and severe pitting, respectively.
For each mesh period, the stiffness reduction in the
singletooth-pair meshing duration is larger than that in the
doubletooth-pair meshing duration for two reasons: (a) the pitting
mostly appear around the pitch line and the pitch line lies on
the single-tooth-pair meshing duration, and (b) the perfect
gear has a smaller averaged mesh stiffness in the
singletooth-pair meshing duration than the double-tooth-pair
meshing duration. In the following section, the pitting effect
on the vibration properties of a spur gearbox will be
3 Dynamic Simulation of a Fixed-axis Gearbox
3.1 Dynamic Modeling
A dynamic model (see Figure 5) reported in Bartelmus
] is used directly in this study for gear pitting effect
analysis. This model had been used in dynamic analysis of
gears with tooth crack [
23, 31, 32
]. It is a
Figure 5 One stage gearbox system [
damper (8 degrees of freedom) dynamic model with both
torsional and lateral vibrations considered. The system is
driven by an electric motor with an input torque M1 and
loaded with torque M2. The motor shaft and the shaft that
the pinion mounts on are coupled with a flexible coupling.
Similarly, the shaft of the load and the shaft that the gear
mounts on are coupled with a flexible coupling. In this
model, the x-direction vibration is uncoupled with the
ydirection vibration [
]. We pay our attention to the
ydirection vibration since this direction is along with the
direction of gear dynamic force.
To emphasize gear fault symptoms caused by tooth
pitting, this model ignored transmission errors, the frictions
between gear teeth, and other practical phenomena, such as
backlash. In addition, we assume the gearbox casing is
rigid so that the vibration propagation along the casing is
linear as did in Ref. [
]. Consequently, the vibration
response properties of gears in lateral directions are
consistent with those on the gearbox casing.
Only consider the motion in the y-direction. The motion
equations are given as follows [
m1y€1 þ c1y_1 þ k1y1 ¼
m2y€2 þ c2y_2 þ k2y2 ¼ Fk þ Fc;
I1h€1 ¼ kpðhm
h1Þ þ cpðh_m
I2h€2 ¼ Rb2ðFk þ FcÞ
Imh€m ¼ M1
Fk ¼ ktðRb1h1
Fc ¼ ctðRb1h_1
Rb2h2 þ y1
Rb2h_2 þ y_1
M2 þ kgðh2
hbÞ þ cgðh_2
Rb1ðFk þ FcÞ;
The related notations are listed as follows: c1 – Vertical
damping of the input bearing, c2 – Vertical damping of the
output bearing, cg – Torsional damping of the output shaft
coupling, cp – Torsional damping of the input shaft
coupling, ct –Gear mesh damping, cx1 –x-direction damping of
the input bearing, cx2 –x-direction damping of the output
bearing, fm – Gear mesh frequency, fs – Rotation frequency
of the pinion, I1 – Mass moment of inertia of the pinion, I2
– Mass moment of inertia of the gear, Ib – Mass moment of
inertia of the load, Im – Mass moment of inertia of the
driving motor, kg – Torsional stiffness of the output shaft
coupling, kp – Torsional stiffness of the input shaft
coupling, kt –Gear mesh stiffness, k1 –y-direction stiffness of
the input bearing, k2 –y-direction stiffness of the output
bearing, kx1 –x-direction stiffness of the input bearing, kx2 –
x-direction stiffness of the output bearing, Rb1 – Base circle
radius of the pinion, Rb2 – Base circle radius of the gear, x1
–x-direction displacement of the pinion, x2 –x-direction
displacement of the gear, y1 –y-direction displacement of
the pinion, y2 –y-direction displacement of the gear, h1 –
Angular displacement of the pinion, h2 – Angular
displacement of the gear, hb – Angular displacement of the
load, hm – Angular displacement of the driving motor.
3.2 Numerical Simulation
To investigate pitting effects on vibration properties,
vibration signals are simulated for a gearbox of which
physical parameters are given in Table 1. Gear mesh
damping is considered to be proportional to gear mesh
stiffness as did in Amabili and Rivola [
]. The gear mesh
damping ratio is selected to be 0.07. Four health conditions
are considered: perfect condition and three pitting severity
levels as shown in Figure 2. A constant torque of 11.9 N m
is generated by the driving motor and the shaft speed of
load is constrained to be 18.4 Hz. Correspondingly, the
theoretical rotation speed of the pinion is 30 Hz (fs) and the
gear mesh frequency (fm) is 570 Hz. The torque and speed
values come from Refs. [
]. Numerical results are
obtained using MATLAB ode15 s solver with sampling
frequency of 100000. The time duration of simulated
signals covers four revolutions of the pinion.
Figure 6 gives y-direction displacement signals of the
pinion for perfect condition, slight pitting condition,
moderate pitting condition and severe pitting condition,
respectively. The time duration of all signals given in this
figure is 0.033 s corresponding to one revolution of the
pinion. In one revolution of the pinion, 19 gear meshes go
through as the pinion has 19 teeth. We can see 19 big
spikes in the perfect condition corresponding to these 19
gear meshes. The amplitude of these spikes are almost the
same because all the teeth are in perfect condition. For the
slight pitting, the fault symptom is very weak, but careful
observation can see one spike (pointed by an arrow) is
slightly bigger than others. This spike is mainly generated
by the nth tooth (9 pits, pitted area 14.6%) as indicated in
Figure 2. For the moderate pitting, one bigger spike is
mainly induced by the nth tooth (18 pits, pitted area 29.2%)
and two slightly bigger spikes (pointed by arrows) are
mainly caused by the (n-1)th tooth (9 pits, pitted area
14.6%) and the (n ? 1)th tooth (9 pits, pitted area 14.6%),
respectively. For the severe pitting, three bigger spikes are
mainly caused by the (n-1)th tooth (18 pits, pitted area
29.2%), the nth tooth (36 pits, pitted area 58.4%) and the
(n ? 1)th tooth (18 pits, pitted area 29.2%), respectively.
In addition, two slightly bigger spikes (pointed by arrows)
are mainly generated by the (n-2)th tooth (9 pits, pitted area
14.6%) and the (n ? 2)th tooth (9 pits, pitted area 14.6%),
respectively. Overall, when the pitted area is below about
15%, the fault symptom is very weak for visual
observation, while when the pitted area reaches about 30%, the
fault symptom is obvious.
Figure 7 shows frequency spectrum of the simulated
vibration signals obtained using the fast Fourier transform.
Four health conditions (perfect, slight pitting, moderate
pitting and severe pitting) are analyzed. Sizable amplitudes
show at gear mesh frequency (fm) and its harmonics under
all the four health conditions. In the perfect and slight
pitting conditions, the gear mesh frequency and its
harmonics dominate the frequency spectrum (the sideband
amplitude is very small). While in the moderate pitting and
severe pitting conditions, a large number of sidebands
appear around the gear mesh frequency and its harmonics.
Figure 8 presents the zoomed-in frequency spectrum of
the simulated vibration signals under the health conditions
of moderate pitting and severe pitting, respectively. The
frequency interval between sidebands is the characteristic
frequency of the pitted pinion (fs, rotation frequency of the
pinion). We can observe that the gear mesh frequency is
modulated by the characteristic frequency of the pitted
fs fs fs fs fs fs fs fs fs fs fs fs fs fs fs fs fs fs
fs fs fs fs fs fs fs fs fs fs fs fs fs fs fs fs fs fs
Table 3 Definition of thirty-six statistical indicators [
Peak to peak
The 1st quartile subtracted from the
x ¼ N1 P xðnÞ
N1 P ðxðnÞ
N1 PN ðxðnÞ xÞ2
1 P ðxðnÞ xÞ3
N1 Pn¼1 ðxðnÞ xÞ2
1 P ðxðnÞ xÞ4
N1 Pn¼1 ðxðnÞ xÞ2
1 P jxðnÞ
1 P jxðnÞ
1 P xðnÞ2
1 P xðnÞ2
1 P jxðnÞj
1 P xðnÞ2
1 P xðnÞ2
1 P jxðnÞj
1 PN ðdðnÞ dÞ4
1 PN 2
N n¼1 ðdðnÞ dÞ
1 PN ðdðnÞ dÞ4
M10 PmM¼01 N1 Pn¼N1 ðdmðnÞ dmÞ2
1 PN ðdðnÞ dÞ6
1 PN 2
N n¼1 ðdðnÞ dÞ
pinion (fs). The degree of frequency modulation increases
with the degree of damage of the pitting. In addition,
sidebands near gear mesh frequency and its harmonics
increase obviously with the growth of tooth pitting
4 Estimation of Pitting Growth Using Statistical
In Section 3, we analyzed vibration properties of a spur
gearbox for four health conditions (perfect condition and
three pitting severity levels). In this section, we test the
effectiveness of thirty-six statistical indicators (see
Table 3) for estimation of pitting growth. Four types of
signals are used for the tests: raw signals (RAW), residual
signals (RES), difference signals (DIFF), and first-order
sideband signals (FSB). RAW denotes the simulated
vibration signals in the Section 3 with the mean subtracted.
RES is obtained by removing gear mesh frequencies and
their harmonics from the RAW. DIFF is generated by
removing the first-order sidebands from the RES. FSB
represents the signals containing only the first-order
sidebands. Twenty-six statistical indicators (F1 to F26) are
applied to each of the four type signals. Therefore, 104
(26 9 4) features can be generated. The statistical
indicators F27 and F28 are only applied to RES since they were
originally proposed for RES [
]. Similarly, the statistical
indicators F29 to F34 are only applied to DIFF since they
are specially designed for DIFF [
]. The statistical
indicators F35 and F36 are only applied to FSB [
]. In total, we
generate 114 (104 ? 10) features for estimation of pitting
This paragraph explains the symbols used in the
expressions of the 36 statistical indicators as listed in
Table 3. We use x(n), r(n), d(n), and b(n) to represent
RAW, RES, DIFF, and FSB, respectively. The symbol
X(k), k = 1, 2,…, K, represents the kth measure of the
frequency spectrum of a signal. The symbol f(k) denotes
frequency amplitude of the kth spectrum component. The
bar notation represents mean, e.g., x is the mean of x(n).
The symbols rm(n), dm(n) and bm(n) denote the mth time
record of rðnÞ, d(n) and b(n), respectively. The symbol
e(n) represents the envelope of the current time record,
which is expressed as e(n) = |b(n) ? j9H(b(n))|, where
H(b(n)) is the Hilbert transform of b(n). em(n) represents
the envelope of the mth time record of b(n). M describes
the total number of time records up to present. M’
represents the total number of time records for a healthy
gearbox. In this study, M and M’ equal to 1 for simplicity. A
signal x(n) is looped around to calculate Dx(n) = x(n)2–
x(n–1)x(n ? 1).
Pearson correlation coefficient (PCC) is utilized to
quantify the effectiveness of these 114 features. It is a
metric to estimate the linear relationship between a single
statistical feature and its corresponding label. Its range is
]. The value closer to one indicates a better statistical
feature for pitting growth estimation. The expression of
] is given below:
Table 4 Features with a PCC value greater than 0.97
2 RES-F9 3
2 RAW-F8 3
Figure 9 Best six features (the numbers 1, 2, 3 and 4 in the x-axis
represent the health conditions of perfect, slight pitting, moderate
pitting and severe pitting, respectively; the y-axis is the normalized
q ¼ sffiffiNffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffisffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ;
P ðxi xÞ2 PN ðyi yÞ2
where xi and yi represent a feature value and a label value,
respectively, for the ith health condition of a gearbox; x and
y are the means of x (x = [x1, x2,…, xN]T) and y (y = [y1,
y2,…, yN]T), respectively; and N is the number of health
conditions. In this study, Nis equal to four, and y = [0,
0.146, 0.292, 0.584]T. The elements in y correspond to the
largest tooth pitted area of the pinion in the health
conditions of perfect, slight pitting, moderate pitting and severe
Test results give that only one feature (RAW-F8)
achieves a PCC value of greater than 0.98 while 21 features
can get a PCC value of greater than 0.97. These 21 features
and their PCC values are given in Table 4. Figure 9
presents performances of the best six features: RAW-F11,
FSB-F6, RES-F9, RES-F11, RAW-F8, and RAW-F1. Their
PCC values are 0.9843, 0.9743, 0.9742, 0.9739, 0.9737,
and 0.9737, respectively. RAW-F11 means the statistical
indicator F11 (see Table 3) is tested on RAW. Other
features can be interpreted in the same way. The feature
values are linearly normalized to values between 0 and 1 in
Figure 9 using the following equation:
V normalized ¼ ðV current ðV severe V perfectÞ= V perfectÞ;
where V_current denotes the feature value in current health
condition of a gearbox, V_perfect is the feature value in
perfect condition, and V_severe represents the feature
value in severe pitting condition.
From Figure 9, we can see that the best six features
share the similar increase trend with the tooth pitting
propagation. The feature values change very slightly from
perfect to slight pitting even for the best feature RAW-F11
(see Figure 9), which indicates the features are not
sensitive to slight pitting. This is because fault symptoms are
very weak under the slight pitting health condition (see
Figure 6). From slight pitting to moderate pitting and from
moderate pitting to severe pitting, a large change of feature
values can be observed from the top six features.
Therefore, it is much easier to detect moderate pitting and severe
pitting than slight pitting.
4.1 Gear Mesh Damping Effect on the Effectiveness of Statistical Features
ct ¼ 2f
In many studies, gear mesh damping is ignored or
considered to be constant [
]. Li and Kahraman [
demonstrated that it was not proper to use a constant gear
mesh damping. Amabili and Rivola [
] modeled the gear
mesh damping proportional to gear mesh stiffness and
investigated damping effect on the steady state response of
a pair of spur gears. In most situations, gear mesh damping
ratio ranges from 0.01 to 0.1 [
33, 39, 40
]. This study does
not intend to propose a new model for gear mesh damping.
Our focus is to test gear mesh damping effect on the
effectiveness of statistical indicators. The gear mesh
damping model reported in Amabili and Rivola [
] is used
directly in this study.
m1 þ m2
where m1 and m2 represents the mass of the pinion and the
gear, respectively, kt and ct denotes the gear mesh stiffness
and damping, respectively, and f is gear mesh damping
Table 5 Top 10 features for each damping condition
Using the dynamic model in Section 3, four damping
conditions are tested: no damping, damping ratio of 0.05,
damping ratio of 0.10 and damping ratio of 0.15. Figure 10
gives an example to illustrate gear mesh damping effect on
gear vibration under severe pitting condition. From visual
observation, there is no big difference. To quantify the
difference, the root mean square is calculated for the
signals to be 1.13 lm, 1.09 lm, 1.06 lm and 1.04 lm
corresponding to no damping, damping ratio of 0.05,
damping ratio of 0.10 and damping ratio of 0.15,
respectively. Signal amplitude decreases slightly with the
increase of damping ratio.
Table 5 gives top 10 features for each damping
condition. Eight features perform very well. They belong to the
top 10 in all damping conditions. These eight features are
RAW-F11, FSB-F6, RES-F9, RAW-F21, RAW-F8,
DIFFF11, RAW-F9 and DIFF-F10. Damping has negligible effect
on the performance of these eight features.
4.2 Environmental Noise Effect on the Effectiveness of Statistical Features
For the simulated signals, there is no environmental noise
involved. In real applications, environmental noise always
exists. White Gaussian noise is added to the simulated
signals to mimic the environmental noise [
noise levels are tested: no noise, signal-to-noise ratio
(SNR) of 10 db, SNR of 0 db and SNR of –10 db.
Figure 11 shows the environmental noise effect on gear
vibration under severe pitting condition. With the increase
of noise level, the fault symptom becomes weaker and
weaker. Under SNR of –10 db, the gear fault symptoms are
not visible any more on the time domain waveform.
Table 6 gives top 10 features for each noise level. Noise
has a larger effect on the performance of features than gear
mesh damping. None of the features is the shared top 10
among the four noise levels. Some features perform well
under low noise level condition but perform badly under
strong noise condition. These features cannot be used if
background noise is strong. Some features have a large
fluctuation on performance with the variation of noise
level. These features are not suggested to use since their
performance is not stable. Good features should be
insensitive to noise variation.
Table 7 summarizes the top 10 features with relatively
stable performance. The PCC value of these 10 stable
features is insensitive to noise level variation. To avoid the
influence of both gear mesh damping and environmental
noise, 6 features are suggested to use. They are RAW-F11,
RES-F9, RAW-F8, RAW-F21, RAW-F9, DIFF-F10. These
6 features belong to both the top 8 features selected in
Section 4.1 considering gear mesh damping effect and the
top 10 stable features selected in Section 4.2 considering
environmental noise effect. They are insensitive to neither
gear mesh damping nor environmental noise.
Time Synchronous Averaging can remove interference
frequencies induced by environmental noise and other
irrelevant machine components [
]. But, there is no
environmental noise in the simulated signals. Other
machine components of the simulated spur gear pair are
represented by constant damping and constant stiffness
parameters and as a result, there are no irrelevant frequency
components caused by other machine components. As
shown in Figures 7 and 8, only the gear mesh frequency
and the pinion pitting fault induced sideband frequency
components are present. When a more complex gearbox
system is simulated, TSA may be needed. However, in real
applications, especially for the slight pitting damage
scenario, TSA and subsequent feature extraction may not be
adequate for effective fault detection and diagnosis.
Advanced signal processing techniques, such as empirical
mode decomposition and wavelet analysis, may be used
together with TSA for more effective fault detection and
This study investigates effects of pitting growth on
vibration properties of a spur gearbox and tests the effectiveness
of 114 features to estimate the pitting growth. The pitting
propagation to neighboring teeth is modeled using circular
pits. The potential energy method is applied to evaluate
gear mesh stiffness of a pair of spur gears for each of the
four health conditions: the perfect condition, the slight
pitting, the moderate pitting and the severe pitting. An
eight degrees of freedom torsional and lateral dynamic
model is used to simulate gearbox vibration signals. Pitting
growth effects on vibration properties of a spur gearbox are
analyzed. These properties can give insights into
developing new signal processing methods for gear tooth pitting
diagnosis. At the end, 114 features are tested to estimate
the pitting growth. The features are ranked based on the
Absolute Pearson Correlation Coefficient. The statistical
features insensitive to gear mesh damping and
environmental noise is recommended. However, further
investigation of these selected features based on experimental
signals is still needed before potential field applications.
Our next step is to design and conduct experiments on a lab
gearbox with introduced gear tooth pitting and refine the
features proposed in this paper.
Open Access This article is distributed under the terms of the
Creative Commons Attribution 4.0 International License (http://crea
tivecommons.org/licenses/by/4.0/), which permits unrestricted use,
distribution, and reproduction in any medium, provided you give
appropriate credit to the original author(s) and the source, provide a
link to the Creative Commons license, and indicate if changes were
Zhi-Liang Liu, is an associate professor at University of Electronic
Science and Technology of China, China. His research interests
mainly include machinery fault diagnosis and prognosis by using
signal processing and data mining techniques. He published more
than 30 papers including 10 ? journal papers. He currently held 7
research grants from National Natural Science Foundation of China,
Open Grants of National Key Laboratory, China Postdoctoral Science
Foundation, etc. E-mail:
Ming Jian Zuo, is a professor at the University of Alberta, Canada.
His research interests include system and network reliability
modeling and analysis, reliability based optimal design, multi-state system
performance evaluation, condition monitoring and fault diagnosis,
maintenance optimization, and manufacturing system modeling and
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Xi-Hui Liang , is currently a postdoctoral research fellow at Reliability Research Lab, University of Alberta, Canada. His research interests include dynamic modeling of mechanical systems, condition monitoring, fault diagnostics and prognostics, asset management, reliability analysis, maintenance, solid mechanics, finite element analysis, and intelligent manufacturing. E-mail: Jun Pan, is a professor at ZhejiangSci-Tech University, China. He received the master degree from Zhejiang University, China, in 2002 and the Ph.D. degree from ZhejiangSci-Tech University, China, in 2011 . His research interests include modeling and statistical analysis of accelerated life testing/degradation testing,design of testing plans,and estimation of system reliability. E-mail: panjun@zstu .edu.cn