1. Introduction and Research Background
Gears are among the most critical components in modern industrial machinery due to their structural simplicity, ease of maintenance, accurate transmission ratios, high load-carrying capacity, and long operational life. They serve as fundamental transmission elements in sectors ranging from automotive and aerospace to heavy engineering equipment. Despite these advantages, gear failures frequently lead to severe production accidents, considerable economic losses, and even threats to human safety. The operational environment of gear transmission systems is often harsh, involving high speeds, heavy loads, and poor lubrication, which significantly accelerates the probability of gear system faults. Consequently, studying gear system failure mechanisms, discovering faults promptly, and preventing accidents are of profound engineering importance.
The excitation sources in a gear transmission system include external excitations like input torque and load torque, and internal excitations such as time-varying meshing stiffness (TVMS). TVMS is a direct indicator of gear health, reflecting the comprehensive deformation of the gear teeth and the gear body. A high-precision calculation of TVMS is therefore essential for reliable gear fault analysis. In the present work, I focus on spur gears as the research object. I improve the formula for calculating the TVMS of healthy spur gears using material mechanics models and finite element analysis. Moreover, I derive the TVMS expressions for gears with single faults and for those with crack-pitting compound faults of varying degrees. Additionally, an 8-degree-of-freedom (8-DOF) gear-rotor system is established to discuss the dynamic response behavior of the gear system under compound fault conditions, and statistical indicators are employed for quantitative analysis.
The present thesis is structured around four primary research pillars: (i) improving healthy spur gear TVMS calculation with a nonlinear Hertzian contact stiffness and a modification for gear body fillet stiffness repetition; (ii) deriving TVMS formulas for single faults including gear cracks of varying depths and gear pitting of varying severity; (iii) establishing a crack-pitting compound fault gear model and deriving its TVMS; and (iv) building an 8-DOF gear-rotor dynamic model to analyze time-domain and frequency-domain responses under different compound fault combinations. Throughout the work, the results of the improved analytical formulas are validated via the finite element method, which confirms the accuracy and effectiveness of the proposed formulations.
2. Improved Time-Varying Mesh Stiffness Calculation of Healthy Spur Gears
2.1 Enhanced Gear Tooth Cantilever Beam Model
Traditional gear cantilever beam models often consider the gear body as a rigid base, which introduces errors in mesh stiffness calculations. In particular, stored energy is neglected or overcounted in the region between the base circle and the dedendum circle. Based on mechanical principles, when the number of teeth equals roughly 42, the base circle and dedendum circle coincide. Therefore, I classify the stiffness calculations based on the tooth number boundary of 42. When the tooth number is less than 42, the base circle radius is greater than the dedendum circle radius, so the energy stored between these two circles is typically omitted. Conversely, when the tooth number exceeds 42, the dedendum circle surpasses the base circle, and including the energy between them is incorrect. I therefore adopt an improved analytical model that accounts for these differences, applying a piecewise approach based on the 42-tooth limit.
In the following, modified expressions for the geometric parameters are introduced, where the geometric center line, contact point, and tooth root cross-section are all defined along the tooth height. The variable \(h_x\) appears as the half-tooth thickness, \(d\) denotes the distance from the root circle to the contact point, \(R_b\) is the base circle radius, \(R_r\) is the dedendum circle radius, \(\alpha\) is the gear angular position, \(\alpha_1\) is the angle between the line of action and the tooth centerline, and \(\alpha_2\), \(\alpha_3\) are the half-tooth angles on the base and dedendum circles, respectively.
\[
d = R_b \left[ \cos \alpha_1 + \cos \alpha_2 – \cos(\alpha_3 – \alpha_1) – \frac{\alpha}{\cos \alpha_2} \right]
\]
\[
h_x = \begin{cases}
R_b \sin \alpha, & 0 \le \alpha \le \alpha_1 \\
R_b \left[ \sin \alpha_2 – (\alpha – \alpha_2)\cos \alpha_2 \right], & \alpha_1 < \alpha \le \alpha_d
\end{cases}
\]
For gear teeth counts above 42, the formula for the half-tooth thickness changes slightly:
\[
h_x = R_b \left[ (\alpha – \alpha_2)\cos \alpha_2 – \sin \alpha_2 \right]
\]
2.2 Nonlinear Hertzian Contact Stiffness and Gear Fillet Foundation Stiffness
A significant improvement in my TVMS methodology is the replacement of the conventional linear Hertzian contact stiffness with a nonlinear counterpart. This modification is grounded in the fact that gear meshing produces nonlinear contact forces, which naturally result in a nonlinear Hertzian stiffness. The nonlinear stiffness expression is given as:
\[
k_{hi} = \frac{1.275 E^{0.9} L^{0.8} F_i^{0.1}}{(1 – \nu^2)^{0.9}}
\]
where \(k_{hi}\) is the nonlinear Hertzian contact stiffness of the \(i\)-th tooth pair, \(E\) is Young’s modulus, \(L\) is the tooth face width, \(F_i\) is the mesh force of the \(i\)-th pair, and \(\nu\) is Poisson’s ratio. This nonlinear formulation has been validated against finite element and experimental data.
In addition, the gear body deformation must be considered in the overall tooth stiffness. When a pair of teeth is in the double-tooth engagement zone, the two meshing tooth pairs share the same gear body. Traditional computational methods often sum the fillet foundation stiffness of each tooth pair independently, leading to an overestimation of the stiffness in the double-tooth engagement region. To resolve this issue, I introduce a correction coefficient \(\lambda\) that quantifies the ratio of the fillet-foundation stiffness at the transition between single and double tooth engagement. The modified fillet-foundation stiffness \(K_f\) is expressed as:
\[
K_f = \begin{cases}
k_f, & \text{single-tooth engagement} \\
\lambda k_f, & \text{double-tooth engagement}
\end{cases}
\]
where \(k_f\) is the fillet-foundation stiffness computed by the original formula. Through finite element-based calculations, the correction parameter is precisely calibrated, and the modified model shows excellent agreement with the finite element results.
The total mesh stiffness for a gear pair can be expressed as:
\[
k_{total} = \sum_{i=1}^{2} \frac{1}{\frac{1}{k_{hi}} + \frac{1}{k_{bi}} + \frac{1}{k_{si}} + \frac{1}{K_{fi}} + \frac{1}{k_{ai}}}
\]
Here, \(k_{bi}\), \(k_{si}\), and \(k_{ai}\) represent the bending, shear, and axial compressive stiffnesses of the \(i\)-th tooth pair, respectively. To verify the improved method, I established a three-dimensional finite element model in COMSOL. With the gear parameters listed in Table 1, healthy gear stiffness values were computed, and the comparison results are summarized in Table 2.
Table 1: Parameters of the Spur Gear Pair
| Parameter | Pinion | Gear |
|---|---|---|
| Number of teeth | 19 | 48 |
| Module (mm) | 3.175 | 3.175 |
| Pressure angle (°) | 20 | 20 |
| Face width (mm) | 16 | 16 |
| Young’s modulus (Pa) | 2.068 × 10¹¹ | 2.068 × 10¹¹ |
| Poisson’s ratio | 0.3 | 0.3 |
| Hub bore radius (mm) | 17 | 43 |
Table 2: Comparison of TVMS at Representative Points for the Healthy Gear
| Point | Proposed Method (×10⁸ N/m) | FEM (×10⁸ N/m) | Error (%) |
|---|---|---|---|
| Q (θ=1.94°) | 3.65 | 3.64 | 0.27 |
| P (θ=5.24°) | 2.16 | 2.17 | 0.46 |
The results in Table 2 demonstrate that the proposed modified TVMS methodology provides a significant improvement in accuracy, with errors well below 1% when compared with finite element results. This confirms the viability of the improved nonlinear contact stiffness and the correction of the fillet foundation stiffness repetition issue.
3. Time-Varying Mesh Stiffness Calculation of Spur Gears with a Single Fault
3.1 Gear Tooth Root Crack Model and Stiffness Formulation
Tooth root cracks are among the most common and dangerous gear failure modes. The stress concentration at the gear root fillet makes this region highly susceptible to fatigue crack initiation. Once initiated, the crack propagates along a path influenced by the gear’s support ratio. For gears with a support ratio greater than one, the crack typically propagates through the tooth root. In the present study, I model a straight-line crack at a 45° inclination angle for simplicity and engineering practicality, as this simplification introduces minimal error while significantly easing the analytical derivation.
For gears with fewer than 42 teeth, I classify the stiffness formulas into four cases based on the crack depth. Taking a specific case where the crack extends through the tooth centerline, the bending and shear stiffness values are expressed as:
\[
k_b = \frac{4(1-\nu^2) E L z}{R_b} \cdot \frac{1}{\int_{\alpha_1}^{\alpha_2} \left[ \frac{(\alpha – \alpha_2)\cos\alpha_2 – \sin\alpha_2}{\sin\alpha_2 – \sin\alpha \cdot \cos\alpha} \right]^3 \cdot \frac{d\alpha}{I_x^*}}
\]
\[
k_s = \frac{2.4(1+\nu) E L z}{R_b} \cdot \frac{1}{\int_{\alpha_1}^{\alpha_2} \frac{(\alpha – \alpha_2)\cos\alpha_2 – \sin\alpha_2}{A_x^*} d\alpha}
\]
where \(I_x^*\) and \(A_x^*\) are the normalized effective area and area moment of inertia, respectively, reduced by the presence of the crack.
For gears with more than 42 teeth, the formulas are adjusted accordingly. In addition to the tooth stiffness components, the presence of a crack also modifies the gear body fillet-foundation stiffness. The effective fillet-foundation geometry changes, and the modified stiffness is computed based on the effective tooth root arc length and the reduced distance from the root to the loading point. I analyze four crack depths: 10%, 30%, 50%, and 70% of the maximum possible crack length. The corresponding TVMS results are summarized in Table 3, which clearly illustrates a monotonic decrease in mesh stiffness with increasing crack depth.
Table 3: Effect of Crack Depth on TVMS of Spur Gears
| Crack Depth (%) | Crack Length (mm) | Minimum TVMS (×10⁸ N/m) | Reduction vs. Healthy (%) |
|---|---|---|---|
| Healthy | 0 | 2.16 | — |
| 10 | 0.53 | 2.01 | 6.94 |
| 30 | 1.60 | 1.68 | 22.2 |
| 50 | 2.66 | 1.28 | 40.7 |
| 70 | 3.72 | 0.58 | 73.1 |
A notable finding is that during the single-tooth engagement zone, the TVMS reduction is most prominent for deep cracks. This behavior arises because, as the contact point moves from the tooth root toward the tooth tip, the effective cross-sectional area supporting the load decreases substantially, amplifying the crack’s effect. This phenomenon is more pronounced in the second double-tooth engagement zone, where the contact point is at its highest position.
3.2 Gear Tooth Pitting Model and Stiffness Formulation
Pitting is another prevalent gear failure mode, particularly under conditions of poor lubrication and high contact stress. Pitting typically originates near the pitch line, where cyclic contact stresses are maximal, and then propagates across the tooth flank. I employ the circular pitting model, which offers a balance between geometrical realism and analytical tractability. Each pit is modeled as a cylindrical void with a diameter of 2 mm and a depth of 1 mm. The degree of pitting is classified into three categories based on the number of pits:
– Slight pitting: 4 pits arranged along the pitch line.
– Moderate pitting: 7 pits along the pitch line.
– Severe pitting: 14 pits, including an additional row of 7 pits extending toward the tooth tip.
Pitting reduces the effective face width and cross-sectional properties of the tooth. The reduced face width \(\Delta L_x\), cross-sectional area \(\Delta A_x\), and area moment of inertia \(\Delta I_x\) are expressed as functions of the pitting radius \(r\), depth \(h_p\), and position along the tooth face. These corrective terms are incorporated into the bending, shear, axial, and Hertzian stiffness expressions. The modified nonlinear Hertzian contact stiffness in the presence of pitting is:
\[
k_{hi} = \frac{1.275 E^{0.9} (L – \sum \Delta L_{xi})^{0.8} F_i^{0.1}}{(1 – \nu^2)^{0.9}}
\]
The effects of different pitting levels on the TVMS of spur gears are presented in Table 4.
Table 4: Effect of Pitting Level on TVMS of Spur Gears
| Pitting Level | Number of Pits | Minimum TVMS (×10⁸ N/m) | Reduction vs. Healthy (%) |
|---|---|---|---|
| Healthy | 0 | 2.16 | — |
| Slight | 4 | 1.88 | 12.9 |
| Moderate | 7 | 1.41 | 34.7 |
| Severe | 14 | 1.02 | 52.8 |
The presence of pitting causes significant local drops in the TVMS, particularly when the contact point traverses the pitted zones near the pitch line. The influence of pitting extends across the entire single-tooth engagement region, leading to a global reduction in stiffness.
4. Time-Varying Mesh Stiffness Calculation of Spur Gears with Crack-Pitting Compound Faults
4.1 Single-Tooth Model of Compound Fault
In practical engineering scenarios, gears operating under heavy loads and with insufficient lubrication frequently develop cracks and pitting simultaneously. The tooth root crack reduces the local load-bearing capacity, which in turn intensifies contact stress on the tooth flank, accelerating the formation and propagation of pitting. Conversely, the presence of pitting alters the load distribution, potentially accelerating crack propagation. Thus, the crack-pitting compound fault is a realistic and dangerous condition that warrants dedicated investigation.
I propose a compound fault model in which a straight crack originating at the tooth root and multiple circular pits positioned along the pitch line coexist on the same tooth. The interaction between the two fault types creates a compound fault region with a characteristic staircase-like effective cross-section. As the crack propagates, three distinct stages of the compound fault region are observed:
– Stage one: The crack is shallow; the compound fault region is small, and only a portion of the pitting region is affected by the crack.
– Stage two: The crack tip plane intersects the bottom plane of the pitting; the compound fault region expands and includes the entire pitting region plus an additional crack-dominated zone.
– Stage three: The crack is deep; its failure region fully covers the pitting failure region, and the bending and shear stiffnesses are dominated by the crack, while pitting only affects the Hertzian and axial compressive stiffnesses.
The angular parameter \(a_p\), which denotes the gear angle corresponding to the intersection of the crack tip plane with the pitting bottom plane, is calculated using the formula:
\[
a_p = \frac{2 \cdot \cos(\alpha – \alpha_p) \cdot \left[ (\alpha – \alpha_p) \cos(\alpha – \alpha_p) – \sin(\alpha – \alpha_p) \right]}{f(a_p, L_p)}
\]
where \(L_p\) is the half-tooth thickness at the intersection point. For the gear geometry studied here, \(a_p\) is calculated as 0.3163 rad.
4.2 Stiffness Formulas for Compound Fault Combinations
I derive the stiffness formulas for 12 possible compound fault combinations by combining four crack depths (10%, 30%, 50%, and 70%) with three pitting levels (slight, moderate, and severe). The derivations are performed separately for gears with fewer than 42 teeth and those with greater than 42 teeth.
For the case of a gear with fewer than 42 teeth, a crack depth of 10%, and slight pitting, the bending stiffness in the compound fault region is expressed as:
\[
k_{bc-10} = \frac{1}{\int_{a_{h1}}^{a_l} \frac{12 \left[ 1 + \cos\alpha_1 (\alpha_1 – \alpha_2) \sin\alpha_1 – \cos(\alpha_1 – \alpha_2) \cos\alpha_1 \right]^2}{E L \left\{ 2 \left[ \sin\alpha_1 – (\alpha – \alpha_2)\cos\alpha_2 \right]^3 – 3 \frac{\Delta I_x}{R_b^3} \right\}} d\alpha}
\]
For the same crack depth but with severe pitting, the presence of two rows of pits necessitates the use of modified pitting-induced reductions \(\Delta A’_x\) and \(\Delta I’_x\), reflecting the additional row of pits positioned closer to the tooth tip. For crack depths of 50% and 70%, the crack failure region completely covers the pitting region, and the bending and shear stiffnesses are computed using the crack model alone.
For gears with more than 42 teeth, analogous formulas are derived. As an example, the bending stiffness for a gear with 30% crack and severe pitting is:
\[
k_{bc-30} = \frac{1}{\int_{a_p}^{a_{h2}} \frac{12 \left[ 1 + \cos\alpha_1 (\alpha_1 – \alpha_2) \sin\alpha_1 – \cos(\alpha_1 – \alpha_2) \cos\alpha_1 \right]^2}{E L \left\{ 2 \left[ \sin\alpha_1 – (\alpha – \alpha_2)\cos\alpha_2 \right]^3 – 3 \frac{\Delta I_x}{R_b^3} \right\}} d\alpha + \int_{a_{h2}}^{a_l} \frac{12 \left[ 1 + \cos\alpha_1 (\alpha_1 – \alpha_2) \sin\alpha_1 – \cos(\alpha_1 – \alpha_2) \cos\alpha_1 \right]^2}{E L \left\{ 2 \left[ \sin\alpha_1 – (\alpha – \alpha_2)\cos\alpha_2 \right]^3 – 3 \frac{\Delta I’_x}{R_b^3} \right\}} d\alpha}
\]
The complete set of stiffness values for the 12 compound fault combinations at two representative angular positions is summarized in Table 5. The error between the proposed analytical method and the finite element method is uniformly below 3%, confirming the correctness and applicability of the developed formulas.
Table 5: TVMS Values and Percentage Differences for Compound Faults
| Fault Combination | Q Point (θ=1.94°) | P Point (θ=5.24°) | ||||
|---|---|---|---|---|---|---|
| Proposed (×10⁸ N/m) | FEM (×10⁸ N/m) | Error (%) | Proposed (×10⁸ N/m) | FEM (×10⁸ N/m) | Error (%) | |
| 4 pits + 10% crack | 3.62 | 3.59 | 0.84 | 1.93 | 1.91 | 1.05 |
| 4 pits + 30% crack | 3.54 | 3.55 | 0.28 | 1.71 | 1.74 | 1.72 |
| 4 pits + 50% crack | 3.36 | 3.39 | 0.88 | 1.30 | 1.28 | 1.56 |
| 4 pits + 70% crack | 2.92 | 2.89 | 1.04 | 0.60 | 0.59 | 1.69 |
| 7 pits + 10% crack | 3.63 | 3.59 | 1.11 | 1.41 | 1.39 | 1.44 |
| 7 pits + 30% crack | 3.54 | 3.55 | 0.28 | 1.29 | 1.31 | 1.52 |
| 7 pits + 50% crack | 3.34 | 3.38 | 1.18 | 1.05 | 1.07 | 1.87 |
| 7 pits + 70% crack | 2.92 | 2.87 | 1.74 | 0.56 | 0.57 | 1.75 |
| 14 pits + 10% crack | 3.63 | 3.60 | 0.83 | 1.41 | 1.39 | 1.44 |
| 14 pits + 30% crack | 3.54 | 3.56 | 0.56 | 1.29 | 1.32 | 2.27 |
| 14 pits + 50% crack | 3.33 | 3.37 | 1.19 | 1.05 | 1.08 | 2.78 |
| 14 pits + 70% crack | 2.92 | 2.88 | 1.39 | 0.56 | 0.55 | 1.82 |
5. Dynamic Characteristics Analysis of Spur Gears with Crack-Pitting Compound Faults
5.1 Establishment of the 8-DOF Gear-Rotor Dynamic Model
To investigate the dynamic behavior of spur gears with crack-pitting compound faults, I construct an idealized 8-DOF spring-damper gear-rotor dynamic model. This model accounts for the torsional degrees of freedom of the input motor (\(\theta_m\)), the driving gear (\(\theta_1\)), the driven gear (\(\theta_2\)), and the load (\(\theta_b\)), as well as the transverse degrees of freedom in the \(x\) and \(y\) directions for both the driving gear (\(x_1, y_1\)) and the driven gear (\(x_2, y_2\)). The governing differential equations of motion for this system are derived using Newton’s second law. The equations can be expressed in matrix form as:
\[
\mathbf{M}\ddot{\mathbf{q}} + \mathbf{C}\dot{\mathbf{q}} + \mathbf{K}(t)\mathbf{q} = \mathbf{F}
\]
where \(\mathbf{M}\) is the mass matrix, \(\mathbf{C}\) is the damping matrix, \(\mathbf{K}(t)\) is the time-varying stiffness matrix containing the TVMS of the gear pair, \(\mathbf{q}\) is the displacement vector, and \(\mathbf{F}\) is the external force vector. The time-varying mesh stiffness of the spur gears serves as the internal excitation and is directly substituted into the stiffness matrix. The key material and system parameters utilized in the dynamic model are presented in Table 6.
Table 6: Material Properties of the Gear-Rotor System
| Parameter | Symbol | Value |
|---|---|---|
| Driving gear mass (kg) | m₁ | 2.88 |
| Driven gear mass (kg) | m₂ | 0.96 |
| Input torque (N·m) | Tₚ | 30.06 |
| Load torque (N·m) | T_g | 11.9 |
| Input motor frequency (Hz) | f₁ | 20 |
| Gear mesh frequency (Hz) | fₘ | 960 |
| Input motor inertia (kg·m²) | Iₘ | 0.0021 |
| Driving gear inertia (kg·m²) | I₁ | 4.3659 × 10⁻⁴ |
| Driven gear inertia (kg·m²) | I₂ | 8.3602 × 10⁻⁴ |
| Load inertia (kg·m²) | I_b | 0.0105 |
| Bearing equivalent stiffness (N/m) | k_r | 6.56 × 10⁷ |
| Shaft equivalent torsional stiffness (N·m/rad) | k_s | 4.4 × 10⁴ |
| Bearing damping (N·s/m) | c_r | 1.8 × 10⁵ |
| Shaft damping (N·s/m) | c_s | 5.0 × 10⁵ |
5.2 Time-Domain Response Analysis
The governing differential equations of the 8-DOF spur gear system are solved using the MATLAB ode15s solver, with the time-varying mesh stiffness curves for each compound fault combination incorporated as internal excitations. The time-domain responses—specifically, the vibration displacement and vibration acceleration signals—of both healthy and faulty spur gears are analyzed in detail.
For the healthy spur gear, the time-domain signals are uniform and smooth, with no abnormal pulses observed. In contrast, the compound fault gears exhibit distinct periodic disturbances in the vibration signals. The maximum time-domain response values for all fault combinations are summarized in Table 7. It is evident from the data that, when one fault type remains constant, an increase in the severity of the other fault leads to a monotonic increase in both the displacement and acceleration amplitudes.
Table 7: Comparison of Maximum Time-Domain Vibration Signals
| Fault Condition | Slight Pitting | Moderate Pitting | Severe Pitting | |||
|---|---|---|---|---|---|---|
| Displacement (μm) | Acceleration (m/s²) | Displacement (μm) | Acceleration (m/s²) | Displacement (μm) | Acceleration (m/s²) | |
| Healthy | 0.06 | 8.43 | — | — | — | — |
| 10% crack | 0.07 | 9.25 | 0.16 | 18.82 | 0.17 | 19.65 |
| 30% crack | 0.12 | 15.22 | 0.25 | 29.49 | 0.27 | 30.81 |
| 50% crack | 0.35 | 36.70 | 0.3794 | 38.84 | 0.3796 | 39.76 |
| 70% crack | 0.65 | 56.14 | 1.08 | 59.75 | 1.20 | 62.90 |
A particularly interesting observation from Table 7 is that the difference in peak amplitudes between the moderate pitting and severe pitting cases is relatively small for the same crack depth. This is because the severe pitting condition adds pits closer to the tooth tip, while the compound fault region near the pitch line remains dominated by the same number of pitting cavities. Consequently, the mesh stiffness reduction caused by the additional pits is less significant than that caused by the pitch-line pits.
5.3 Frequency-Domain Response Analysis
Fourier transform is applied to the time-domain signals to convert them into the frequency domain for further analysis. The frequency-domain spectrum of the healthy gear contains the mesh frequency \(f_m\) and its harmonics, with no visible sidebands. In contrast, the frequency spectra of the compound fault gears display prominent sidebands around the mesh frequency and its harmonics. The sideband spacing corresponds exactly to the input shaft rotational frequency, i.e., \(f_i = 20\) Hz.
As the crack depth increases, the sidebands around the mesh frequency and its harmonics increase in both amplitude and number. For deep cracks (70%), the sidebands spread across the entire frequency range, particularly around \(2f_m\) and \(3f_m\). To compare the sideband amplitudes across different fault combinations quantitatively, I extract the amplitude at a specific frequency point within the sideband near \(2f_m\) (1857 Hz). These values are summarized in Table 8.
Table 8: Sideband Amplitude Comparison in Frequency Domain
| Fault Condition | Slight Pitting | Moderate Pitting | Severe Pitting | |||
|---|---|---|---|---|---|---|
| Displacement (μm) | Acceleration (m/s²) | Displacement (μm) | Acceleration (m/s²) | Displacement (μm) | Acceleration (m/s²) | |
| 10% crack | 0.002 | 0.268 | 0.004 | 0.419 | 0.005 | 0.494 |
| 30% crack | 0.003 | 0.439 | 0.006 | 0.741 | 0.007 | 0.795 |
| 50% crack | 0.009 | 0.471 | 0.010 | 1.056 | 0.012 | 1.057 |
| 70% crack | 0.019 | 1.665 | 0.030 | 1.793 | 0.032 | 2.189 |
Similar to the time-domain analysis, the sideband amplitudes for the moderate and severe pitting conditions are close, making it challenging to distinguish these two pitting severities using spectral inspection alone. This limitation highlights the need for more advanced signal processing and statistical analysis tools to achieve reliable fault classification of spur gears.
5.4 Statistical Indicator Analysis of Fault Signals
To overcome the limitations of traditional time-domain and frequency-domain analyses in distinguishing certain compound fault combinations, I introduce three advanced statistical indicators: the crest factor (CF), skewness (Sk), and sideband index (SI). These indicators are computed for both the time-domain displacement signals and the time-domain acceleration signals, as well as for their corresponding frequency-domain spectra.
The crest factor is defined as the ratio of the peak amplitude to the root mean square (RMS) value:
\[
\text{CF} = \frac{\max|x_i|}{\sqrt{\frac{1}{N}\sum_{i=1}^{N} x_i^2}}
\]
The skewness of the signal is a measure of the asymmetry of the probability distribution and is defined as:
\[
\text{Skewness} = \frac{\frac{1}{N}\sum_{i=1}^{N} (x_i – \bar{x})^3}{\left[\frac{1}{N}\sum_{i=1}^{N} (x_i – \bar{x})^2\right]^{3/2}}
\]
The sideband index is a frequency-domain metric that represents the average amplitude of the first-order sidebands around the mesh frequency:
\[
\text{SI} = \frac{1}{M}\sum_{k=1}^{M} |X(k)|
\]
where \(X(k)\) is the amplitude of the \(k\)-th first-order sideband and \(M\) is the total number of sidebands considered.
The percentage changes in the statistical indicators relative to the healthy gear values are presented in Tables 9 and 10 for the vibration displacement and vibration acceleration signals, respectively.
Table 9: Percentage Changes of Statistical Indicators (Displacement)
| Fault Combination | Crest Factor (%) | Skewness (%) |
|---|---|---|
| 4 pits + 10% crack | 1 | 50 |
| 4 pits + 30% crack | 2 | 550 |
| 4 pits + 50% crack | 7 | 800 |
| 4 pits + 70% crack | 14 | 3100 |
| 7 pits + 10% crack | 2.6 | 300 |
| 7 pits + 30% crack | 4.5 | 750 |
| 7 pits + 50% crack | 7.6 | 1850 |
| 7 pits + 70% crack | 24 | 3600 |
| 14 pits + 10% crack | 3 | 600 |
| 14 pits + 30% crack | 5.2 | 1050 |
| 14 pits + 50% crack | 8.3 | 1900 |
| 14 pits + 70% crack | 26 | 4000 |
Table 10: Percentage Changes of Statistical Indicators (Acceleration)
| Fault Combination | Crest Factor (%) | Skewness (%) |
|---|---|---|
| 4 pits + 10% crack | 8.9 | 3.4 |
| 4 pits + 30% crack | 71.5 | 3.7 |
| 4 pits + 50% crack | 196 | 36.2 |
| 4 pits + 70% crack | 226 | 80.7 |
| 7 pits + 10% crack | 104 | 23.4 |
| 7 pits + 30% crack | 192 | 53.5 |
| 7 pits + 50% crack | 220 | 73.6 |
| 7 pits + 70% crack | 246 | 297 |
| 14 pits + 10% crack | 110 | 46.6 |
| 14 pits + 30% crack | 204 | 59.3 |
| 14 pits + 50% crack | 235 | 81.2 |
| 14 pits + 70% crack | 263 | 150 |
From the results in Tables 9 and 10, the crest factor and the sideband index demonstrate outstanding performance in distinguishing the compound fault combinations. The crest factor exhibits a monotonic increase with the severity of both crack and pitting faults for both vibration displacement and acceleration signals. The sideband index, computed from the frequency domain, also demonstrates a clear upward trend, effectively complementing the time-domain analysis. On the other hand, the skewness indicator, while showing considerable changes for the displacement signals, is less reliable for the acceleration signals and sometimes fails to differentiate certain fault combinations, such as moderate versus severe pitting at a 50% crack depth.
Therefore, I recommend the crest factor and the sideband index as the most suitable statistical indicators for the diagnosis and quantification of crack-pitting compound faults in spur gears. These indicators can serve as robust reference values for engineering practice, providing an effective basis for early fault warning and condition monitoring of gear transmission systems.
6. Conclusions and Future Work
In this work, I systematically investigate the dynamic modeling and dynamic feature analysis of spur gears with crack-pitting compound faults. The main conclusions are summarized as follows:
First, the improved gear cantilever beam model, which incorporates the energy stored between the base circle and the dedendum circle, combined with the nonlinear Hertzian contact stiffness and the corrected fillet-foundation stiffness, yields high-precision TVMS values for healthy spur gears. The results agree exceptionally well with finite element simulations, with errors below 1%.
Second, the formulas for TVMS of spur gears with single faults (crack and pitting) are derived based on the improved healthy gear model. The crack depth and pitting severity have distinct and significant influences on the TVMS of spur gears. Deeper cracks cause larger stiffness reductions, particularly in the single-tooth engagement zone, while pitting induces local stiffness drops where the pits are located.
Third, the established crack-pitting compound fault model successfully captures the interaction between the two fault types. As the crack depth increases, the compound fault region evolves through distinct stages. When the crack is shallow, both faults contribute to the stiffness reduction; when the crack is deep, it dominates the bending and shear stiffnesses of spur gears. The derived stiffness formulas for 12 fault combinations produce results consistent with finite element analysis within a 3% error margin.
Fourth, the 8-DOF gear-rotor dynamic model enables the analysis of time-domain and frequency-domain responses of spur gears under various compound fault conditions. While the vibration signals effectively reflect the severity of the faults, some fault combinations remain challenging to distinguish through simple signal inspection. The crest factor and sideband index prove to be sensitive and robust statistical indicators for the classification and quantification of crack-pitting compound faults in spur gears.
For future research, the following improvements are suggested. The extended tooth contact effect, observed in finite element results, should be incorporated into the analytical stiffness model of spur gears. The pitting model can be refined to account for variable pit sizes and depths that more closely mirror realistic pitting conditions. The dynamic model, currently assuming idealized conditions, should be extended to include nonlinear factors such as tooth friction, gear backlash, and manufacturing errors, thereby enabling a more accurate representation of real-world spur gear transmission systems.