- Original Article
- Open Access
Spur Gear Tooth Pitting Propagation Assessment Using Model-based Analysis
- Xi-Hui Liang^{1},
- Zhi-Liang Liu^{2},
- Jun Pan^{3} and
- Ming Jian Zuo^{1}Email authorView ORCID ID profile
https://doi.org/10.1007/s10033-017-0196-z
© The Author(s) 2017
- Received: 17 November 2016
- Accepted: 11 October 2017
- Published: 8 November 2017
Abstract
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.
Keywords
- Mesh stiffness
- Mesh damping
- Gear dynamics
- Vibration
- Statistical feature
- Dynamic simulation
1 Introduction
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 [1]. Through observations at gearboxes used in Syncrude Canada Ltd, tooth pitting was a common failure mode [2]. When tooth pitting appears on gears, gear mesh stiffness reduces and correspondingly the vibration properties of gears change.
- 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.
Feng and Zuo [9] 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 [10, 11].
Several researchers investigated dynamic properties of a gearbox with tooth pitting through dynamic simulation. Chaari et al. [12], Cheng et al. [13], and Abouel-seoud et al. [14] 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. [15] modeled an elliptical pit on a single tooth and evaluated the dynamic force of a pair of gears. Ma et al. [16] 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. [17] incorporated the gear tooth friction effect in modeling a single gear tooth spalling. Liang et al. [18] 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 [3, 19], 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 [20]. Vibration-based time domain, frequency domain, and time-frequency domain analyses provide powerful tools for fault diagnosis of rotating machinery [21, 22]. One traditional technique is based on statistical measurements of vibration signals [23]. Many statistical indicators were proposed for machine fault diagnosis [24–27]. In Liu et al. [25], 34 statistical indicators were summarized and 136 features were generated. In Zhao et al. [26], 63 statistical indicators were summarized and 252 features were produced. The features [25, 26] were 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 given.
2 Tooth Pitting Propagation Modeling and Mesh Stiffness Evaluation
2.1 Tooth Pitting Propagation Modeling
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 mm^{2}. 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 handbook [3].
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 [3].
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%, 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 [3].
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. [18], the potential energy method [28, 29] was used to evaluate mesh stiffness of gears with multiple pits on a single tooth. The gear tooth was modeled as a non-uniform 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 vibration properties.
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 \(\alpha_{ 1}\). Applying these equations iteratively to each gear tooth, the stiffness of each tooth can be obtained. But, the values for \(\Delta L_{xj}\), \(\Delta A_{xj}\) and \(\Delta I_{xj}\) may be different among teeth due to the variance of number and location of pits.
Physical parameters of a spur gearbox [11]
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} |
Averaged mesh stiffness reduction (%) caused by tooth pitting
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 |
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 double-tooth-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 single-tooth-pair meshing duration is larger than that in the double-tooth-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 single-tooth-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 investigated.
3 Dynamic Simulation of a Fixed-axis Gearbox
3.1 Dynamic Modeling
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. [31]. Consequently, the vibration response properties of gears in lateral directions are consistent with those on the gearbox casing.
The related notations are listed as follows: \(c_{1}\) – Vertical damping of the input bearing, \(c_{2}\) – Vertical damping of the output bearing, \(c_{g}\) – Torsional damping of the output shaft coupling, \(c_{p}\) – Torsional damping of the input shaft coupling, \(c_{t}\) –Gear mesh damping, \(c_{x1}\) –x-direction damping of the input bearing, \(c_{x2}\) –x-direction damping of the output bearing, \(f_{m}\) – Gear mesh frequency, \(f_{s}\) – Rotation frequency of the pinion, \(I_{1}\) – Mass moment of inertia of the pinion, \(I_{2}\) – Mass moment of inertia of the gear, \(I_{b}\) – Mass moment of inertia of the load, \(I_{m}\) – Mass moment of inertia of the driving motor, \(k_{g}\) – Torsional stiffness of the output shaft coupling, \(k_{p}\) – Torsional stiffness of the input shaft coupling, \(k_{t}\) –Gear mesh stiffness, \(k_{1}\) –y-direction stiffness of the input bearing, \(k_{2}\) –y-direction stiffness of the output bearing, \(k_{x1}\) –x-direction stiffness of the input bearing, \(k_{x2}\) –x-direction stiffness of the output bearing, \(R_{b1}\) – Base circle radius of the pinion, \(R_{b2}\) – Base circle radius of the gear, \(x_{1}\) –x-direction displacement of the pinion, \(x_{2}\) –x-direction displacement of the gear, \(y_{1}\) –y-direction displacement of the pinion, \(y_{2}\) –y-direction displacement of the gear, \(\theta_{1}\) – Angular displacement of the pinion, \(\theta_{2}\) – Angular displacement of the gear, \(\theta_{b}\) – Angular displacement of the load, \(\theta_{m}\) – 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 [33]. 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 (f _{ s }) and the gear mesh frequency (f _{ m }) is 570 Hz. The torque and speed values come from Refs. [18, 31]. 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.
4 Estimation of Pitting Growth Using Statistical Features
Definition of thirty-six statistical indicators [25]
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^{*} | \(\frac{{\frac{1}{N}\sum\limits_{n = 1}^{N} {\left( {e(n) - \overline{e} } \right)^{4} } }}{{\left( {\frac{1}{{M^{'} }}\sum\limits_{m = 1}^{{M^{'} }} {\left( {\frac{1}{N}\sum\limits_{n = 1}^{N} {\left( {e_{m} (n) - \overline{e}_{m} } \right)^{2} } } \right)} } \right)^{2} }}\) |
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., \(\bar{x}\) is the mean of x(n). The symbols r _{ m }(n), d _{ m }(n) and b _{ m }(n) denote the mth time record of \(r(n)\), 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) + j×H(b(n))|, where H(b(n)) is the Hilbert transform of b(n). e _{ m }(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 Δx(n) = x(n)^{2}–x(n–1)x(n + 1).
Features with a PCC value greater than 0.97
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 |
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-F _{11} (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
Top 10 features for each damping condition
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
Top 10 features for each noise level
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 |
Top 10 stable features
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 |
Time Synchronous Averaging can remove interference frequencies induced by environmental noise and other irrelevant machine components [43]. 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 diagnosis.
5 Conclusions
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.
Notes
Declarations
Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.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 made.
Authors’ Affiliations
References
- X H Liang, M J Zuo, W H Chen. Dynamics based vibration signal modeling for tooth fault diagnosis of planetary gearboxes, Chapter 7 in: M. Demetgul, M. Ünal (Eds.), Fault Diagnosis and Detection, In Tech, 2017: 175–192.Google Scholar
- M R Hoseini, M J Zuo. Literature review for creating and quantifying faults in planetary gearboxes. Reliability Research Lab, Mechanical Department, University of Alberta, May 2009.Google Scholar
- ASM Committee. Friction, lubrication, and wear technology. American Society for Microbiology (ASM) International, 1992.Google Scholar
- L Gelman, R Zimroz, J Birkel, et al. Adaptive vibration condition monitoring technology for local tooth damage in gearboxes. Insight - Non-Destructive Testing & Condition Monitoring, 2005, 47: 461–464.Google Scholar
- S K Lee, J S Shim, B O Cho. Damage detection of a gear with initial pitting using the zoomed phase map of continuous wavelet transform. Key Engineering Materials, 2006, 306–308: 223–228.Google Scholar
- F Combet, L Gelman. Optimal filtering of gear signals for early damage detection based on the spectral kurtosis. Mechanical Systems and Signal Processing, 2009, 23: 652–668.Google Scholar
- H Öztürk, M Sabuncu, I Yesilyurt. Early detection of pitting damage in gears using mean frequency of scalogram. Journal of Vibration and Control, 2008, 14: 469–484.Google Scholar
- M R Hoseini, Y G Lei, D V Tuan, et al. Experiment design of four types of experiments: pitting experiments, run-tofailure experiments, various load and speed experiments, and crack experiments. Reliability Research Lab, Mechanical Department, University of Alberta, January 2011.Google Scholar
- Z P Feng, M J Zuo. Vibration signal models for fault diagnosis of planetary gearboxes. Journal of Sound and Vibration, 2012, 331: 4919–4939.Google Scholar
- X H Liang, M J Zuo, T H Patel. Evaluating the time-varying mesh stiffness of a planetary gear set using the potential energy method. Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science, 2014, 228(3): 535–547.Google Scholar
- X H Liang, M J Zuo, M R Hoseini. Vibration signal modeling of a planetary gear set for tooth crack detection. Engineering Failure Analysis, 2015, 48: 185–200.Google Scholar
- F Chaari, W Baccar, M S Abbes, et al. Effect of spalling or tooth breakage on gearmesh stiffness and dynamic response of a one-stage spur gear transmission. European Journal of Mechanics - A/Solids, 2008, 27: 691–705.Google Scholar
- Z Cheng, N Q Hu, F S Gu, et al. Pitting damage levels estimation for planetary gear sets based on model simulation and grey relational analysis. Transactions of the Canadian Society for Mechanical Engineering, 2011, 35: 403–417.Google Scholar
- S A Abouel-seoud, E S Dyab, M S Elmorsy. Influence of tooth pitting and cracking on gear meshing stiffness and dynamic response of wind turbine gearbox. International Journal of Science and Advanced Technology, 2012, 2: 151–165.Google Scholar
- A F Rincon, F Viadero, M Iglesias, et al. Effect of cracks and pitting defects on gear meshing. Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science, 2012, 226: 2805–2815.Google Scholar
- H Ma, Z W Li, M J Feng, et al. Time-varying mesh stiffness calculation of spur gears with spalling defect. Engineering Failure Analysis, 2016, 66: 166–176.Google Scholar
- A Saxena, A Parey, M Chouksey. Time varying mesh stiffness calculation of spur gear pair considering sliding friction and spalling defects. Engineering Failure Analysis, 2016, 70: 200–211.Google Scholar
- X H Liang, H S Zhang, L B Liu, et al. The influence of tooth pitting on the mesh stiffness of a pair of external spur gears. Mechanism and Machine Theory, 2016, 106: 1–15.Google Scholar
- C K Tan, P Irving, D Mba. A comparative experimental study on the diagnostic and prognostic capabilities of acoustics emission, vibration and spectrometric oil analysis for spur gears. Mechanical Systems and Signal Processing, 2007, 21: 208–233.Google Scholar
- X H Liang, M J Zuo, Z P Feng. Dynamic modeling of gearbox faults: A review. Mechanical Systems and Signal Processing, 2018, 98: 852–876.Google Scholar
- T H Patel, A K Darpe. Experimental investigations on vibration response of misaligned rotors. Mechanical Systems and Signal Processing, 2009, 23: 2236–2252.Google Scholar
- T H Patel, A K Darpe. Coupled bending-torsional vibration analysis of rotor with rub and crack. Journal of Sound and Vibration, 2009, 326: 740–752.Google Scholar
- S Y Wu, M J Zuo, A Parey. Simulation of spur gear dynamics and estimation of fault growth. Journal of Sound and Vibration, 2008, 317: 608–624.Google Scholar
- X M Zhao, M J Zuo, Z L Liu. Diagnosis of pitting damage levels of planet gears based on ordinal ranking. IEEE Conference on Prognostics and Health Management (PHM), Denver, USA, June 20-23, 2011: 1–8.Google Scholar
- Z L Liu, J Qu, M J Zuo, et al. Fault level diagnosis for planetary gearboxes using hybrid kernel feature selection and kernel fisher discriminant analysis. The International Journal of Advanced Manufacturing Technology, 2012, 67: 1217–1230.Google Scholar
- X M Zhao, M J Zuo, Z L Liu, et al. Diagnosis of artificially created surface damage levels of planet gear teeth using ordinal ranking. Measurement, 2013, 46: 132–144.Google Scholar
- P D Samuel, D J Pines. A review of vibration-based techniques for helicopter transmission diagnostics. Journal of Sound and Vibration, 2005, 282: 475–508.Google Scholar
- D CH Yang, J Y Lin. Hertzian damping, tooth friction and bending elasticity in gear impact dynamics. Journal of Mechanical Design, 1987, 109: 189–196.Google Scholar
- X H Liang, M J Zuo, M Pandey. Analytically evaluating the influence of crack on the mesh stiffness of a planetary gear set. Mechanism and Machine Theory, 2014, 76: 20–38.Google Scholar
- W Bartelmus. Mathematical modelling and computer simulations as an aid to gearbox diagnostics. Mechanical Systems and Signal Processing, 2001, 15: 855–871.Google Scholar
- X H Tian, M J Zuo, K R Fyfe. Analysis of the vibration response of a gearbox with gear tooth faults. ASME International Mechanical Engineering Congress and Exposition, Anaheim, California, USA, November 13-20, 2004: 785–793.Google Scholar
- Z G Tian, M J Zuo, S Y Wu. Crack propagation assessment for spur gears using model-based analysis and simulation. Journal of Intelligent Manufacturing, 2012, 23: 239–253.Google Scholar
- M Amabili, A Rivola. Dynamic analysis of spur gear pairs: steady-state response and stability of the SDOF model with time-varying meshing damping. Mechanical Systems and Signal Processing, 1997, 11: 375–390.Google Scholar
- P Večeř, M Kreidl, R Šmíd. Condition indicators for gearbox condition monitoring systems. Acta Polytech, 2005, 45(6): 35–43.Google Scholar
- Z L Liu, M J Zuo, H Xu. Fault diagnosis for planetary gearboxes using multi-criterion fusion feature selection framework. Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science, 2013, 227: 2064–2076.Google Scholar
- J Lin, R G Parker. Analytical characterization of the unique properties of planetary gear free vibration. Journal of Vibration and Acoustics, 1999, 121: 316–321.Google Scholar
- A Saxena, A Parey, M Chouksey. Dynamic analysis of multi-mesh geared rotor system using modal analysis. Prognostics & System Health Management Conference, Chengdu, China, Oct. 16-22, 2016: 1–5.Google Scholar
- X H Liang, M J Zuo, L B Liu. A windowing and mapping strategy for gear tooth fault detection of a planetary gearbox. Mechanical Systems and Signal Processing, 2016, 80: 445–459.Google Scholar
- S Li, A Kahraman. A spur gear mesh interface damping model based on elastohydrodynamic contact behaviour. International Journal of Powertrains, 2011, 1: 4–21.Google Scholar
- A S Kumar, T S Sankar, M O M Osman. On dynamic tooth load and stability of a spur-gear system using the state-space approach. Journal of Mechanism, Transmissions, and Automation in Design, 1985, 107: 54–60.Google Scholar
- Z P Feng, M J Zuo. Fault diagnosis of planetary gearboxes via torsional vibration signal analysis. Mechanical Systems and Signal Processing, 2013, 36: 401–421.Google Scholar
- Y F Li, X H Liang, M J Zuo. Diagonal slice spectrum assisted optimal scale morphological filter for rolling element bearing fault diagnosis. Mechanical Systems and Signal Processing, 2017, 85: 146–161.Google Scholar
- E Bechhoefer, M Kingsley. A review of time synchronous average algorithms. Annual Conference of the Prognostics and Health Management Society, San Diego, CA, September, 2009: 24–33.Google Scholar