Spur Gear Tooth Pitting Propagation Assessment Using Model-based Analysis

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.

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.

According to the American Society for Metals (ASM) handbook [3], 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 [4,5,6,7] created man-made tooth pitting on gears to experimentally explore fault symptoms of a gearbox. For example, in Ref. [8], 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 gearbox.

Figure 1

Man-made tooth pitting on gears [8]

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 [24,25,26,27]. 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

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 [6, 7]. 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:

Figure 2

Schematics of pitting damage levels (slight to severe – from top to bottom) on the nth tooth of the pinion and its neighboring teeth

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 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.

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 derivation [29]. Each circular pit is expressed by three variables [18]: ($u$, r, $\delta$), 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 $\delta$ is the pitting depth.

Figure 3

A gear tooth with a circular pit [18]

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 k _h , bending stiffness k _b , shear stiffness k _s and axial compressive stiffness k _a can be expressed as follows [18]:

$$k_{h} = \frac{{\pi E(L - \sum\limits_{1}^{N} {\Delta L_{xj} } )}}{{4(1 - \nu^{2} )}},$$

(1)

$$\begin{aligned} \frac{1}{{k_{b} }} &= \frac{{\left[ {1 - \frac{{(Z - 2.5)\cos \alpha_{1} \cos \alpha_{3} }}{{Z\cos \alpha_{0} }}} \right]^{3} - (1 - \cos \alpha_{1} \cos \alpha_{2} )^{3} }}{{2EL\cos \alpha_{1} \sin^{3} \alpha_{2} }} + \hfill \\ & \quad \times \; \int_{{ - \alpha_{1} }}^{{\alpha_{2} }} {\frac{{3\left\{ {1 + \cos \alpha_{1} \left[ {(\alpha_{2} - \alpha )\sin \alpha - \cos \alpha } \right]} \right\}^{2} (\alpha_{2} - \alpha )\cos \alpha }}{{E\left( {2L[\sin \alpha + (\alpha_{2} - \alpha )\cos \alpha ]^{3} - 3\sum\limits_{1}^{N} {\frac{{\Delta I_{xj} }}{{R_{b}^{3} }}} } \right)}}} {\text{d}}\alpha , \hfill \\ \end{aligned}$$

(2)

$$\begin{aligned} \frac{1}{{k_{s} }} = \frac{{1.2(1 + \nu )\cos^{2} \alpha_{1} \left( {\cos \alpha_{2} - \frac{Z - 2.5}{{Z\cos \alpha_{0} }}\cos \alpha_{3} } \right)}}{{EL\sin \alpha_{2} }} + \hfill \\ \int_{{ - \alpha_{1} }}^{{\alpha_{2} }} {\frac{{1.2(1 + \nu )(\alpha_{2} - \alpha )\cos \alpha \cos^{2} \alpha_{1} }}{{E\left( {L\left[ {\sin \alpha + (\alpha_{2} - \alpha )\cos \alpha } \right] - 0.5\sum\limits_{1}^{N} {\frac{{\Delta A_{xj} }}{{R_{b} }}} } \right)}}} {\text{d}}\alpha , \hfill \\ \end{aligned}$$

(3)

$$\begin{aligned} \frac{1}{{k_{a} }} = \frac{{\sin^{2} \alpha_{1} \left( {\cos \alpha_{2} - \frac{Z - 2.5}{{Z\cos \alpha_{0} }}\cos \alpha_{3} } \right)}}{{2EL\sin \alpha_{2} }} + \hfill \\ \int_{{ - \alpha_{1} }}^{{\alpha_{2} }} {\frac{{(\alpha_{2} - \alpha )\cos \alpha \sin^{2} \alpha_{1} }}{{E\left( {2L\left[ {\sin \alpha + (\alpha_{2} - \alpha )\cos \alpha } \right] - \sum\limits_{1}^{N} {\frac{{\Delta A_{xj} }}{{R_{b} }}} } \right)}}} {\text{d}}\alpha , \hfill \\ \end{aligned}$$

(4)

where $E$, $L$, $\nu$ 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; $\alpha_{0}$ is the pressure angle; $\alpha_{ 1}$ denotes the angle between action force $F$ and its decomposition component $F_{b}$; $\alpha_{2}$ indicates the half tooth angle on the base circle; $\alpha_{3}$ describes the approximated half tooth angle on the root circle; $\Delta L_{xj}$, $\Delta A_{xj}$ and $\Delta I_{xj}$ (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 $x$; $\Delta L_{xj}$, $\Delta A_{xj}$ and $\Delta I_{xj}$ are expressed as follows [18]:

$$\Delta L_{x} = \left\{ {\begin{array}{*{20}l} {2\sqrt {r^{2} - (u - x)^{2} } , \quad x \in [u - r, u + r],} \hfill \\ {0 , \quad {\text{others,}}} \hfill \\ \end{array} } \right.$$

(5)

$$\Delta A_{x} = \left\{ {\begin{array}{*{20}l} {\Delta L_{x} \delta , \quad x \in [u - r, u + r],} \hfill \\ {0 , \quad {\text{others,}}} \hfill \\ \end{array} } \right.$$

(6)

$$\Delta I_{x} = \left\{ {\begin{array}{*{20}l} {\frac{1}{12}\Delta L_{x} \delta^{3} { + }\frac{{A_{x} \Delta A_{x} (h_{x} - \delta /2)^{2} }}{{A_{x} - \Delta A_{x} }} ,\;x \in [u - r, u + r],} \hfill \\ {0 , \quad{\text{others}} .} \hfill \\ \end{array} } \right.$$

(7)

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.

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 [18, 28]:

$$k_{t} = \frac{1}{{\frac{1}{{k_{h} }} + \frac{1}{{k_{b1} }} + \frac{1}{{k_{s1} }} + \frac{1}{{k_{a1} }} + \frac{1}{{k_{b2} }} + \frac{1}{{k_{s2} }} + \frac{1}{{k_{a2} }}}},$$

(8)

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 [18, 28]:

$$\begin{aligned} & k_{t} = k_{t1} + k_{t2} \hfill \\ & \quad = \sum\limits_{i = 1}^{2} {\frac{1}{{\frac{1}{{k_{h,i} }} + \frac{1}{{k_{b1,i} }} + \frac{1}{{k_{s1,i} }} + \frac{1}{{k_{a1,i} }} + \frac{1}{{k_{b2.i} }} + \frac{1}{{k_{s2,i} }} + \frac{1}{{k_{a2,i} }}}}} , \hfill \\ \end{aligned}$$

(9)

where $i = 1$ for the first pair and $i = 2$ 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 $\delta_{ - } k_{t}$ is used to quantify the mesh stiffness reduction. It is defined as follows [18]:

Table 1 Physical parameters of a spur gearbox [11]

Figure 4

Pitting effect on gear mesh stiffness

Table 2 Averaged mesh stiffness reduction (%) caused by tooth pitting

$$\delta_{ - } k_{t} = \left( {{{\left| {\frac{{\sum\limits_{s = 1}^{S} {k_{t1s} } }}{S} - \frac{{\sum\limits_{m = 1}^{M} {k_{t2m} } }}{M}} \right|} \mathord{\left/ {\vphantom {{\left| {\frac{{\sum\limits_{s = 1}^{S} {k_{t1s} } }}{S} - \frac{{\sum\limits_{m = 1}^{M} {k_{t2m} } }}{M}} \right|} {\frac{{\sum\limits_{m = 1}^{M} {k_{t2m} } }}{M}}}} \right. \kern-0pt} {\frac{{\sum\limits_{m = 1}^{M} {k_{t2m} } }}{M}}}} \right) \times 100\% ,$$

(10)

where $k_{t1}$ and $k_{t2}$ 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 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

A dynamic model (see Figure 5) reported in Bartelmus [30] 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 mass-spring-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 $M_{1}$ and loaded with torque $M_{2}$. 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 y-direction vibration [31]. We pay our attention to the y-direction vibration since this direction is along with the direction of gear dynamic force.

Figure 5

One stage gearbox system [30]

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.

Only consider the motion in the y-direction. The motion equations are given as follows [30]:

$$m_{1} \ddot{y}_{1} + c_{1} \dot{y}_{1} + k_{1} y_{1} = - F_{k} - F_{c} ,$$

(11)

$$m_{2} \ddot{y}_{2} + c_{2} \dot{y}_{2} + k_{2} y_{2} = F_{k} + F_{c} ,$$

(12)

$$I_{1} \ddot{\theta }_{1} = k_{p} (\theta_{m} - \theta_{1} ) + c_{p} (\dot{\theta }_{m} - \dot{\theta }_{1} ) - R_{b1} (F_{k} + F_{c} ),$$

(13)

$$I_{2} \ddot{\theta }_{2} = R_{b2} (F_{k} + F_{c} ) - k_{g} (\theta_{2} - \theta_{b} ) - c_{g} (\dot{\theta }_{2} - \dot{\theta }_{b} ),$$

(14)

$$I_{m} \ddot{\theta }_{m} = M_{1} - k_{p} (\theta_{m} - \theta_{1} ) - c_{p} (\dot{\theta }_{m} - \dot{\theta }_{1} ),$$

(15)

$$I_{b} \ddot{\theta }_{b} = - M_{2} + k_{g} (\theta_{2} - \theta_{b} ) + c_{g} (\dot{\theta }_{2} - \dot{\theta }_{b} ),$$

(16)

$$F_{k} = k_{t} (R_{b1} \theta_{1} - R_{b2} \theta_{2} + y_{1} - y_{2} ),$$

(17)

$$F_{c} = c_{t} (R_{b1} \dot{\theta }_{1} - R_{b2} \dot{\theta }_{2} + \dot{y}_{1} - \dot{y}_{2} ).$$

(18)

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.

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 6

Displacement signals of the pinion for four health conditions of gearbox (the arrows point at weak fault symptoms)

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 (f _m ) 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 7

Frequency spectrum of the simulated vibration signals (f _m is gear mesh frequency)

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 (f _s , rotation frequency of the pinion). We can observe that the gear mesh frequency is modulated by the characteristic frequency of the pitted pinion (f _s ). 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 severity.

Figure 8

Zoomed-in frequency spectrum of simulated vibration signals (f _m is gear mesh frequency)

4 Estimation of Pitting Growth Using Statistical Features

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 (F ₁ to F ₂₆) are applied to each of the four type signals. Therefore, 104 (26 × 4) features can be generated. The statistical indicators F ₂₇ and F ₂₈ are only applied to RES since they were originally proposed for RES [34]. Similarly, the statistical indicators F ₂₉ to F ₃₄ are only applied to DIFF since they are specially designed for DIFF [27]. The statistical indicators F ₃₅ and F ₃₆ are only applied to FSB [27]. In total, we generate 114 (104 + 10) features for estimation of pitting growth.

Table 3 Definition of thirty-six statistical indicators [25]

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).

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 [–1, 1]. The value closer to one indicates a better statistical feature for pitting growth estimation. The expression of PCC [35] is given below:

$$\rho = \frac{{\sum\limits_{i = 1}^{N} {(x_{i} - \bar{x})(y_{i} - \bar{y})} }}{{\sqrt {\sum\limits_{i = 1}^{N} {(x_{i} - \bar{x})^{2} } } \sqrt {\sum\limits_{i = 1}^{N} {(y_{i} - \bar{y})^{2} } } }},$$

(19)

where x _i and y _i represent a feature value and a label value, respectively, for the ith health condition of a gearbox; $\bar{x}$ and $\bar{y}$ are the means of x (x = [x ₁, x ₂,…, x _N ]^T) and y (y = [y ₁, y ₂,…, y _N ]^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 pitting, respectively.

Test results give that only one feature (RAW-F ₈) 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-F ₁₁, FSB-F ₆, RES-F ₉, RES-F ₁₁, RAW-F ₈, and RAW-F ₁. Their PCC values are 0.9843, 0.9743, 0.9742, 0.9739, 0.9737, and 0.9737, respectively. RAW-F ₁₁ means the statistical indicator F ₁₁ (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:

Table 4 Features with a PCC value greater than 0.97

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 feature values)

$${\text{V}}\_normalized = \, \left( {{\text{V}}\_current - {\text{ V}}\_perfect} \right)/\left( {{\text{V}}\_severe - {\text{ V}}\_perfect} \right),$$

(20)

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-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

In many studies, gear mesh damping is ignored or considered to be constant [36,37,38]. Li and Kahraman [39] demonstrated that it was not proper to use a constant gear mesh damping. Amabili and Rivola [33] 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 [33] is used directly in this study.

$$c_{t} = 2\zeta \sqrt {\frac{{m_{1} m_{2} }}{{m_{1} + m_{2} }}k_{t} } ,$$

(21)

where m ₁ and m ₂ represents the mass of the pinion and the gear, respectively, k _t and c _t denotes the gear mesh stiffness and damping, respectively, and $\zeta$ is gear mesh damping ratio.

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 μm, 1.09 μm, 1.06 μm and 1.04 μm 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.

Figure 10

Gear mesh damping effect on gear vibration under severe pitting condition

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-F ₁₁, FSB-F ₆, RES-F ₉, RAW-F ₂₁, RAW-F ₈, DIFF-F ₁₁, RAW-F ₉ and DIFF-F ₁₀. Damping has negligible effect on the performance of these eight features.

Table 5 Top 10 features for each damping condition

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 [41, 42]. Four 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.

Figure 11

Environmental noise effect on gear vibration under severe pitting condition

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 6 Top 10 features for each noise level

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-F ₁₁, RES-F ₉, RAW-F ₈, RAW-F ₂₁, RAW-F ₉, DIFF-F ₁₀. 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.

Table 7 Top 10 stable features

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.