1 Introduction
2 Tooth Pitting Propagation Modeling and Mesh Stiffness Evaluation
2.1 Tooth Pitting Propagation Modeling
2.2 Mesh Stiffness Evaluation
Parameter  Pinion (driving)  Gear (driven) 

Number of teeth  19  31 
Module (mm)  3.2  3.2 
Pressure angle 
\(20^{\text{o}}\)

\(20^{\text{o}}\)

Mass (kg)  0.700  1.822 
Face width (m)  0.0381  0.0381 
Young’s modulus (GPa)  206.8  206.8 
Poisson’s ratio  0.3  0.3 
Base circle radius (mm)  28.3  46.2 
Root circle radius (mm)  26.2  45.2 
Bearing stiffness (N/m) 
k
_{1} = k
_{2} = 5.0 × 10^{8}
 
Bearing damping (kg/s) 
c
_{1} = c
_{2} = 4 × 10^{5}
 
Torsional stiffness of shaft coupling (N/m) 
k
_{
p
} = k
_{
g
} = 4.0 × 10^{7}
 
Torsional damping of shaft coupling (kg/s) 
c
_{
p
} = c
_{
g
} = 3 × 10^{4}

Mesh period No.  Doubletoothpair meshing duration  Singletoothpair meshing duration  

Slight  Moderate  Severe  Slight  Moderate  Severe  
1  0  0  0.18  0  0  2.84 
2  0  0.18  1.04  0  2.83  11.61 
3  0.18  1.06  3.27  2.86  11.70  55.78 
4  1.04  3.20  5.38  11.61  55.33  55.37 
5  1.57  4.26  19.02  2.87  11.73  55.83 
6  0.40  1.54  4.17  0  2.87  11.68 
7  0  0.41  1.56  0  0  2.86 
8  0  0  0.40  0  0  0 
3 Dynamic Simulation of a Fixedaxis Gearbox
3.1 Dynamic Modeling
3.2 Numerical Simulation
4 Estimation of Pitting Growth Using Statistical Features
Feature  Name  Definition 

F
_{1}
 Maximum value  The maximum value in x(n), i.e., max(x(n)) 
F
_{2}
 Minimum value  The minimum value in x(n), i.e., min(x(n)) 
F
_{3}
 Mean 
\(\overline{x} = \frac{1}{N}\sum\limits_{n = 1}^{N} {x(n)}\)

F
_{4}
 Peak to peak  max(x(n))−min(x(n)) 
F
_{5}
 Harmonic mean 
\(\frac{N}{{\sum\limits_{n = 1}^{N} {\frac{1}{x(n)}} }}\)

F
_{6}
 Trimmed mean  Mean excluding outliers 
F
_{7}
 Variance 
\(\frac{1}{N}\sum\limits_{n = 1}^{N} {\left( {x(n)  \bar{x}} \right)^{2} }\)

F
_{8}
 Standard deviation 
\(\sqrt {\frac{1}{N}\sum\limits_{n = 1}^{N} {\left( {x(n)  \bar{x}} \right)^{2} } }\)

F
_{9}
 Mean absolute deviation 
\(\frac{1}{N}\sum\limits_{n = 1}^{N} {\left {x(n)  \bar{x}} \right}\)

F
_{10}
 Median absolute deviation 
\(\frac{1}{N}\sum\limits_{n = 1}^{N} {\left {x(n)  x_{\text{median}} } \right}\)

F
_{11}
 Interquartile range  The 1st quartile subtracted from the 3rd quartile 
F
_{12}
 Peak2RMS 
\(\frac{{\hbox{max} (\left {x(n)} \right)}}{{\sqrt {\frac{1}{N}\sum\limits_{n = 1}^{N} {x(n)^{2} } } }}\)

F
_{13}
 Skewness 
\(\frac{{\frac{1}{N}\sum\limits_{n = 1}^{N} {\left( {x(n)  \bar{x}} \right)^{3} } }}{{\left( {\sqrt {\frac{1}{N}\sum\limits_{n = 1}^{N} {\left( {x(n)  \bar{x}} \right)^{2} } } } \right)^{3} }}\)

F
_{14}
 Kurtosis 
\(\frac{{\frac{1}{N}\sum\limits_{n = 1}^{N} {\left( {x(n)  \bar{x}} \right)^{4} } }}{{\left( {\frac{1}{N}\sum\limits_{n = 1}^{N} {\left( {x(n)  \bar{x}} \right)^{2} } } \right)^{2} }}\)

F
_{15}
 Shape factor 
\(\frac{{\sqrt {\frac{1}{N}\sum\limits_{n = 1}^{N} {x(n)^{2} } } }}{{\frac{1}{N}\sum\limits_{n = 1}^{N} {\left {x(n)} \right} }}\)

F
_{16}
 Crest factor 
\(\frac{\hbox{max} (x(n))}{{\sqrt {\frac{1}{N}\sum\limits_{n = 1}^{N} {x(n)^{2} } } }}\)

F
_{17}
 Clearance factor 
\(\frac{\hbox{max} (x(n))}{{\frac{1}{N}\sum\limits_{n = 1}^{N} {x(n)^{2} } }}\)

F
_{18}
 Impulse factor 
\(\frac{\hbox{max} (x(n))}{{\frac{1}{N}\sum\limits_{n = 1}^{N} {\left {x(n)} \right} }}\)

F
_{19}
 Third order central moment 
\(\frac{1}{N}\sum\limits_{n = 1}^{N} {\left( {x(n)  \bar{x}} \right)^{3} }\)

F
_{20}
 Fourth order central moment 
\(\frac{1}{N}\sum\limits_{n = 1}^{N} {\left( {x(n)  \bar{x}} \right)^{ 4} }\)

F
_{21}
 Root mean square 
\(\sqrt {\frac{1}{N}\sum\limits_{n = 1}^{N} {x(n)^{2} } }\)

F
_{22}
 Energy operator 
\(\frac{{\frac{1}{N}\sum\limits_{n = 1}^{N} {\left( {\Delta x(n)  \Delta \overline{x} } \right)^{4} } }}{{\left( {\frac{1}{N}\sum\limits_{n = 1}^{N} {\left( {\Delta x(n)  \Delta \overline{x} } \right)^{2} } } \right)^{2} }}\)

F
_{23}
 Mean frequency 
\(\frac{1}{K}\sum\limits_{k = 1}^{K} {X(k)}\)

F
_{24}
 Frequency center 
\(\frac{{\sum\limits_{k = 1}^{K} {\left( {f(k) \times X(k)} \right)} }}{{\sum\limits_{k = 1}^{K} {X(k)} }}\)

F
_{25}
 Root mean square frequency 
\(\sqrt {\frac{{\sum\limits_{k = 1}^{K} {\left( {f(k)^{2} \times X(k)} \right)} }}{{\sum\limits_{k = 1}^{K} {X(k)} }}}\)

F
_{26}
 Standard deviation frequency 
\(\sqrt {\frac{{\sum\limits_{k = 1}^{K} {\left( {\left( {f(k)  F_{28} } \right)^{2} \times X(k)} \right)} }}{{\sum\limits_{k = 1}^{K} {X(k)} }}}\)

F
_{27}
 NA4 
\(\frac{{\frac{1}{N}\sum\limits_{n = 1}^{N} {\left( {r(n)  \overline{r} } \right)^{4} } }}{{\left( {\frac{1}{M}\sum\limits_{m = 1}^{M} {\left( {\frac{1}{N}\sum\limits_{n = 1}^{N} {\left( {r_{m} (n)  \overline{r}_{m} } \right)^{2} } } \right)} } \right)^{2} }}\)

F
_{28}
 NA4^{*}

\(\frac{{\frac{1}{N}\sum\limits_{n = 1}^{N} {\left( {r(n)  \overline{r} } \right)^{4} } }}{{\left( {\frac{1}{{M^{'} }}\sum\limits_{m = 1}^{{M^{'} }} {\frac{1}{N}\sum\limits_{n = 1}^{N} {\left( {r_{m} (n)  \overline{r}_{m} } \right)^{2} } } } \right)^{2} }}\)

F
_{29}
 FM4 
\(\frac{{\frac{1}{N}\sum\limits_{n = 1}^{N} {\left( {d(n)  \overline{d} } \right)^{4} } }}{{\left( {\frac{1}{N}\sum\limits_{n = 1}^{N} {\left( {d(n)  \overline{d} } \right)^{2} } } \right)^{2} }}\)

F
_{30}
 FM4^{*}

\(\frac{{\frac{1}{N}\sum\limits_{n = 1}^{N} {\left( {d(n)  \overline{d} } \right)^{4} } }}{{\left( {\frac{1}{{M^{'} }}\sum\limits_{m = 1}^{{M^{'} }} {\left( {\frac{1}{N}\sum\limits_{n = 1}^{N} {\left( {d_{m} (n)  \overline{d}_{m} } \right)^{2} } } \right)} } \right)^{2} }}\)

F
_{31}
 M6A 
\(\frac{{\frac{1}{N}\sum\limits_{n = 1}^{N} {\left( {d(n)  \overline{d} } \right)^{6} } }}{{\left( {\frac{1}{N}\sum\limits_{n = 1}^{N} {\left( {d(n)  \overline{d} } \right)^{2} } } \right)^{3} }}\)

F
_{32}
 M6A^{*}

\(\frac{{\frac{1}{N}\sum\limits_{n = 1}^{N} {\left( {d(n)  \overline{d} } \right)^{6} } }}{{\left( {\frac{1}{{M^{'} }}\sum\limits_{m = 1}^{{M^{'} }} {\left( {\frac{1}{N}\sum\limits_{n = 1}^{N} {\left( {d_{m} (n)  \overline{d}_{m} } \right)^{2} } } \right)} } \right)^{3} }}\)

F
_{33}
 M8A 
\(\frac{{\frac{1}{N}\sum\limits_{n = 1}^{N} {\left( {d(n)  \overline{d} } \right)^{8} } }}{{\left( {\frac{1}{N}\sum\limits_{n = 1}^{N} {\left( {d(n)  \overline{d} } \right)^{2} } } \right)^{4} }}\)

F
_{34}
 M8A^{*}

\(\frac{{\frac{1}{N}\sum\limits_{n = 1}^{N} {\left( {d(n)  \overline{d} } \right)^{8} } }}{{\left( {\frac{1}{{M^{'} }}\sum\limits_{m = 1}^{{M^{'} }} {\left( {\frac{1}{N}\sum\limits_{n = 1}^{N} {\left( {d_{m} (n)  \overline{d}_{m} } \right)^{2} } } \right)} } \right)^{4} }}\)

F
_{35}
 NB4 
\(\frac{{\frac{1}{N}\sum\limits_{n = 1}^{N} {\left( {e(n)  \overline{e} } \right)^{4} } }}{{\left( {\frac{1}{M}\sum\limits_{m = 1}^{M} {\left( {\frac{1}{N}\sum\limits_{n = 1}^{N} {\left( {e_{m} (n)  \overline{e}_{m} } \right)^{2} } } \right)} } \right)^{2} }}\)

F
_{36}
 NB4^{*}

\(\frac{{\frac{1}{N}\sum\limits_{n = 1}^{N} {\left( {e(n)  \overline{e} } \right)^{4} } }}{{\left( {\frac{1}{{M^{'} }}\sum\limits_{m = 1}^{{M^{'} }} {\left( {\frac{1}{N}\sum\limits_{n = 1}^{N} {\left( {e_{m} (n)  \overline{e}_{m} } \right)^{2} } } \right)} } \right)^{2} }}\)

Ranking  Feature  PCC  Ranking  Feature  PCC 

1  RAWF
_{11}
 0.9843  12  RES F
_{10}
 0.9718 
2  FSBF
_{6}
 0.9743  13  FSB F
_{8}
 0.9714 
3  RESF
_{9}
 0.9742  14  FSB F
_{21}
 0.9714 
4  RESF
_{11}
 0.9739  15  FSB F
_{4}
 0.9713 
5  RAWF
_{8}
 0.9737  16  FSB F
_{9}
 0.9710 
6  RAWF
_{1}
 0.9737  17  FSB F
_{1}
 0.9708 
7  DIFF F
_{11}
 0.9737  18  RES F
_{7}
 0.9707 
8  RAW F
_{9}
 0.9736  19  DIFF F
_{9}
 0.9704 
9  DIFFF
_{10}
 0.9735  20  FSB F
_{10}
 0.9704 
10  FSB F
_{26}
 0.9727  21  FSB F
_{25}
 0.9701 
11  FSB F
_{23}
 0.9721 
4.1 Gear Mesh Damping Effect on the Effectiveness of Statistical Features
Feature ranking 
\(\zeta = 0\)

\(\zeta = 0.05\)
 

Feature  PCC  Feature  PCC  
1  RAWF
_{10}
 0.9896  RAWF
_{11}
 0.9885 
2  RAWF
_{11}
 0.9885  FSBF
_{6}
 0.9747 
3  FSBF
_{26}
 0.9768  RESF
_{9}
 0.9745 
4  FSBF
_{6}
 0.9747  RESF
_{11}
 0.9744 
5  RESF
_{9}
 0.9746  FSBF
_{26}
 0.9742 
6  RAWF
_{21}
 0.9736  DIFFF
_{10}
 0.9739 
7  RAWF
_{8}
 0.9736  RAWF
_{9}
 0.9739 
8  DIFFF
_{11}
 0.9735  RAWF
_{8}
 0.9739 
9  RAWF
_{9}
 0.9735  RAWF
_{21}
 0.9739 
10  DIFFF
_{10}
 0.9732  DIFFF
_{11}
 0.9738 
Feature ranking 
\(\zeta = 0.1\)

\(\zeta = 0.15\)
 

Feature  PCC  Feature  PCC  
1  RAWF
_{11}
 0.9843  RAWF
_{5}
 0.9903 
2  FSBF
_{6}
 0.9746  RAWF
_{11}
 0.9859 
3  RESF
_{11}
 0.9744  FSBF
_{6}
 0.9746 
4  RAWF
_{8}
 0.9742  DIFFF
_{11}
 0.9744 
5  RAWF
_{21}
 0.9742  DIFFF
_{10}
 0.9742 
6  RESF
_{9}
 0.9741  RESF
_{9}
 0.9740 
7  RAWF
_{9}
 0.9738  RAWF
_{9}
 0.9739 
8  DIFFF
_{11}
 0.9734  RAWF
_{21}
 0.9739 
9  DIFFF
_{10}
 0.9733  RAWF
_{8}
 0.9739 
10  FSBF
_{23}
 0.9720  RESF
_{11}
 0.9735 
4.2 Environmental Noise Effect on the Effectiveness of Statistical Features
Feature ranking  No noise  10 db  

Feature  PCC  Feature  PCC  
1  RAWF
_{11}
 0.9843  DIFFF
_{5}
 0.9806 
2  FSBF
_{6}
 0.9743  DIFFF
_{10}
 0.9791 
3  RESF
_{9}
 0.9742  DIFFF
_{11}
 0.9790 
4  RESF
_{11}
 0.9739  RESF
_{11}
 0.9769 
5  RAWF
_{8}
 0.9737  RESF
_{10}
 0.9769 
6  RAWF
_{1}
 0.9737  RAWF
_{8}
 0.9756 
7  DIFF F
_{11}
 0.9737  RAWF
_{21}
 0.9756 
8  RAW F
_{9}
 0.9736  RESF
_{9}
 0.9746 
9  DIFFF
_{10}
 0.9735  RAWF
_{9}
 0.9741 
10  FSB F
_{26}
 0.9727  RAWF
_{10}
 0.9737 
Feature ranking  0 db  –10 db  

Feature  PCC  Feature  PCC  
1  FSBF
_{22}
 0.9953  RAWF
_{1}
 0.9887 
2  FSBF
_{1}
 0.9813  FSBF
_{23}
 0.9755 
3  RESF
_{22}
 0.9803  DIFFF
_{11}
 0.9739 
4  FSBF
_{21}
 0.9783  DIFFF
_{10}
 0.9736 
5  FSBF
_{8}
 0.9783  RAWF
_{8}
 0.9727 
6  FSBF
_{4}
 0.9778  RAWF
_{21}
 0.9727 
7  FSBF
_{9}
 0.9774  RAWF
_{9}
 0.9720 
8  FSBF
_{23}
 0.9747  DIFFF
_{23}
 0.9717 
9  FSBF
_{35}
 0.9743  RAWF
_{23}
 0.9717 
10  RESF
_{4}
 0.9741  RESF
_{23}
 0.9716 
Features  PCC  

No noise  10 db  0 db  –10 db  
RAWF
_{11}
 0.9842  0.9735  0.9697  0.9699 
RESF
_{9}
 0.9742  0.9746  0.9708  0.9711 
RAWF
_{8}
 0.9737  0.9756  0.9727  0.9727 
RAWF
_{21}
 0.9737  0.9756  0.9727  0.9727 
RAWF
_{9}
 0.9736  0.9740  0.9719  0.9719 
DIFFF
_{10}
 0.9735  0.9791  0.9705  0.9736 
RAWF
_{7}
 0.9691  0.9704  0.9681  0.9697 
RAWF
_{20}
 0.9673  0.9674  0.9597  0.9548 
RESF
_{28}
 0.9625  0.9626  0.9505  0.9526 
RESF
_{20}
 0.9625  0.9626  0.9505  0.9526 