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 × 108
| |
Bearing damping (kg/s) |
c
1 = c
2 = 4 × 105
| |
Torsional stiffness of shaft coupling (N/m) |
k
p
= k
g
= 4.0 × 107
| |
Torsional damping of shaft coupling (kg/s) |
c
p
= c
g
= 3 × 104
|
Mesh period No. | Double-tooth-pair meshing duration | Single-tooth-pair 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 Fixed-axis 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 | RAW-F
11
| 0.9843 | 12 | RES -F
10
| 0.9718 |
2 | FSB-F
6
| 0.9743 | 13 | FSB -F
8
| 0.9714 |
3 | RES-F
9
| 0.9742 | 14 | FSB -F
21
| 0.9714 |
4 | RES-F
11
| 0.9739 | 15 | FSB -F
4
| 0.9713 |
5 | RAW-F
8
| 0.9737 | 16 | FSB -F
9
| 0.9710 |
6 | RAW-F
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 | DIFF-F
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 | RAW-F
10
| 0.9896 | RAW-F
11
| 0.9885 |
2 | RAW-F
11
| 0.9885 | FSB-F
6
| 0.9747 |
3 | FSB-F
26
| 0.9768 | RES-F
9
| 0.9745 |
4 | FSB-F
6
| 0.9747 | RES-F
11
| 0.9744 |
5 | RES-F
9
| 0.9746 | FSB-F
26
| 0.9742 |
6 | RAW-F
21
| 0.9736 | DIFF-F
10
| 0.9739 |
7 | RAW-F
8
| 0.9736 | RAW-F
9
| 0.9739 |
8 | DIFF-F
11
| 0.9735 | RAW-F
8
| 0.9739 |
9 | RAW-F
9
| 0.9735 | RAW-F
21
| 0.9739 |
10 | DIFF-F
10
| 0.9732 | DIFF-F
11
| 0.9738 |
Feature ranking |
\(\zeta = 0.1\)
|
\(\zeta = 0.15\)
| ||
---|---|---|---|---|
Feature | PCC | Feature | PCC | |
1 | RAW-F
11
| 0.9843 | RAW-F
5
| 0.9903 |
2 | FSB-F
6
| 0.9746 | RAW-F
11
| 0.9859 |
3 | RES-F
11
| 0.9744 | FSB-F
6
| 0.9746 |
4 | RAW-F
8
| 0.9742 | DIFF-F
11
| 0.9744 |
5 | RAW-F
21
| 0.9742 | DIFF-F
10
| 0.9742 |
6 | RES-F
9
| 0.9741 | RES-F
9
| 0.9740 |
7 | RAW-F
9
| 0.9738 | RAW-F
9
| 0.9739 |
8 | DIFF-F
11
| 0.9734 | RAW-F
21
| 0.9739 |
9 | DIFF-F
10
| 0.9733 | RAW-F
8
| 0.9739 |
10 | FSB-F
23
| 0.9720 | RES-F
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 | RAW-F
11
| 0.9843 | DIFF-F
5
| 0.9806 |
2 | FSB-F
6
| 0.9743 | DIFF-F
10
| 0.9791 |
3 | RES-F
9
| 0.9742 | DIFF-F
11
| 0.9790 |
4 | RES-F
11
| 0.9739 | RES-F
11
| 0.9769 |
5 | RAW-F
8
| 0.9737 | RES-F
10
| 0.9769 |
6 | RAW-F
1
| 0.9737 | RAW-F
8
| 0.9756 |
7 | DIFF -F
11
| 0.9737 | RAW-F
21
| 0.9756 |
8 | RAW -F
9
| 0.9736 | RES-F
9
| 0.9746 |
9 | DIFF-F
10
| 0.9735 | RAW-F
9
| 0.9741 |
10 | FSB -F
26
| 0.9727 | RAW-F
10
| 0.9737 |
Feature ranking | 0 db | –10 db | ||
---|---|---|---|---|
Feature | PCC | Feature | PCC | |
1 | FSB-F
22
| 0.9953 | RAW-F
1
| 0.9887 |
2 | FSB-F
1
| 0.9813 | FSB-F
23
| 0.9755 |
3 | RES-F
22
| 0.9803 | DIFF-F
11
| 0.9739 |
4 | FSB-F
21
| 0.9783 | DIFF-F
10
| 0.9736 |
5 | FSB-F
8
| 0.9783 | RAW-F
8
| 0.9727 |
6 | FSB-F
4
| 0.9778 | RAW-F
21
| 0.9727 |
7 | FSB-F
9
| 0.9774 | RAW-F
9
| 0.9720 |
8 | FSB-F
23
| 0.9747 | DIFF-F
23
| 0.9717 |
9 | FSB-F
35
| 0.9743 | RAW-F
23
| 0.9717 |
10 | RES-F
4
| 0.9741 | RES-F
23
| 0.9716 |
Features | PCC | |||
---|---|---|---|---|
No noise | 10 db | 0 db | –10 db | |
RAW-F
11
| 0.9842 | 0.9735 | 0.9697 | 0.9699 |
RES-F
9
| 0.9742 | 0.9746 | 0.9708 | 0.9711 |
RAW-F
8
| 0.9737 | 0.9756 | 0.9727 | 0.9727 |
RAW-F
21
| 0.9737 | 0.9756 | 0.9727 | 0.9727 |
RAW-F
9
| 0.9736 | 0.9740 | 0.9719 | 0.9719 |
DIFF-F
10
| 0.9735 | 0.9791 | 0.9705 | 0.9736 |
RAW-F
7
| 0.9691 | 0.9704 | 0.9681 | 0.9697 |
RAW-F
20
| 0.9673 | 0.9674 | 0.9597 | 0.9548 |
RES-F
28
| 0.9625 | 0.9626 | 0.9505 | 0.9526 |
RES-F
20
| 0.9625 | 0.9626 | 0.9505 | 0.9526 |