1. Introduction
Gears are among the most critical components in mechanical transmission systems, serving as the primary means for transferring power and torque through tooth engagement. Their application spans a broad spectrum of industrial fields, including automotive transmissions, aerospace engines, marine propulsion systems, and agricultural machinery, among others. When compared with common belt drives and hydraulic transmission systems, gear drives exhibit several distinct technological advantages: notably higher transmission efficiency, lower operational noise, precise transmission ratios, and excellent control accuracy. Moreover, well-designed gear pairs offer an extended service life, which substantially enhances the overall reliability of mechanical systems.
Within the operational envelope of spur gear transmission systems, tooth surface wear emerges as a pervasive and detrimental fault phenomenon. According to statistical data from developed countries, roughly one-third of global primary energy consumption is dissipated through friction and wear, and more than 60% of mechanical accidents are attributable to wear-related failures. The adverse effects of tooth surface wear on mechanical systems can be categorized into three principal aspects. Initially, tooth surface wear constitutes a common failure mode that progressively amplifies system vibration and noise levels, ultimately precipitating gear system failure. Secondly, wear induces geometrical alterations in tooth profiles, which consequently compromises transmission efficiency and system performance. Thirdly, intensive wear generates substantial quantities of frictional heat, elevating gear temperatures and potentially causing material softening, deformation, embrittlement, or even catastrophic failure.
Given the aforementioned considerations, investigating the evolution mechanisms of tooth surface wear and the associated quantitative evaluation methodologies is of paramount importance for intelligent condition monitoring and condition-based maintenance of mechanical transmission systems. Therefore, this thesis focuses on establishing a comprehensive tooth surface wear model for spur gears and developing a vibration-signal-based quantitative wear assessment technique.
2. Research Status and Analysis
2.1 Tooth Surface Wear Mechanism and Computational Methods
Early research on gear tooth wear primarily concentrated on material improvements to mitigate wear rates. In 1953, Archard formulated the foundational adhesive wear law, arguing that wear particles are produced when contacting asperities adhere and subsequently fracture during relative sliding motion. Later, Andersson developed sliding distance calculations for gear teeth and integrated the Archard model into spur gear wear prediction. Wu and Cheng subsequently proposed an enhanced wear model grounded in mixed lubrication theory that incorporated surface roughness and lubricant film effects.
Flodin and Andersson advanced the field significantly by discretizing involute gear tooth profiles and adopting the Winkler pressure model. Their approach recalculated the tooth profile after every N mesh cycles, thereby establishing an early-stage wear simulation framework for spur gears that was subsequently extended to helical gear configurations. With the advancement of computational technologies in the twenty-first century, finite element analysis (FEA) methods became prevalent. Lundvall applied FEA to compute contact pressure and wear distributions, while Brauer formulated an analytical expression for sliding distance incorporating manufacturing errors.
In the Chinese research community, extensive contributions have also been made. Pan et al. analyzed the relationship between gear meshing states and meshing angles, employing the Flodin model coupled with load and speed considerations to predict gear wear under varying operating conditions. Liu et al. developed a mathematical wear model for helical gears by integrating Hertzian contact theory with the Archard wear equation, revealing non-uniform wear distribution along the face width and substantial sensitivity to load and wear cycles. Zhang et al. proposed a micro-scale mixed-lubrication wear model that accounts for flash temperature, surface roughness, and lubricant properties. Recent years have witnessed growing interest in multi-factor coupling analyses, including dynamic load distribution, center distance errors, surface roughness, and lubricant characteristics.
A significant limitation of many existing computational models is their reliance on simplified assumptions, including dry contact conditions or quasi-static loading. These simplifications often fail to capture the intricate interplay among lubrication, thermal effects, dynamic loading, and progressive surface evolution.
2.2 Time-Varying Meshing Stiffness and Dynamic Characteristics under Wear
Time-varying meshing stiffness (TVMS) serves as a crucial excitation source for gear vibration. Common computational methods include the Ishikawa formula, ISO standards, the potential energy method, and FEA. For non-uniform tooth wear, the potential energy method is particularly advantageous because it accommodates local tooth profile modifications.
Shen et al. incorporated Archard wear depths into the energy-method stiffness model, deriving analytical corrections for TVMS considering wear effects. Feng et al. adopted the ISO approach for uniformly worn gears, while Yang et al. employed material mechanics principles to analyze non-uniform wear and profile modifications. He et al. coupled wear depth distributions with potential-energy-based stiffness computations, comparing cases where the base circle radius is larger or smaller than the root circle radius. Liu et al. generated involute profiles via generating theory and computed TVMS under non-uniform wear using Archard wear depths.
Regarding dynamic behavior under wear, Wang et al. combined Archard’s model with a two-degree-of-freedom dynamic model to investigate how dynamic loads influence wear progression. Ren et al. incorporated gear backlash and transmission errors into a wear-coupled dynamic model, highlighting the roles of input speed and wear cycles. Shen et al. integrated TVMS and transmission errors into dynamic equations, analyzing the dynamic response of worn spur gear pairs. Zhang et al. developed a six-degree-of-freedom model incorporating surface wear, bearing clearance, and gear backlash, revealing how gradual wear alters system dynamics.
Despite these contributions, the majority of studies treat wear as a quasi-static parameter and subsequently substitute the resulting wear depth into dynamics models, thereby neglecting the feedback coupling between dynamics and wear evolution. A comprehensive model that simultaneously addresses dynamic loads and wear coupling remains relatively underexplored. This research gap motivates the present study.
3. Tooth Surface Wear Computational Model of Spur Gears
3.1 Gear Meshing Parameters
The object of study is an involute spur gear pair with a contact ratio between 1 and 2. Let m represent module, z1 and z2 the tooth numbers of the driving and driven gears, respectively, α the pressure angle, and B the face width. For a gear pair under standard center distance A, the following relations hold:
$$ r_1 = \frac{m z_1}{2}, \quad r_2 = \frac{m z_2}{2}, \quad A = \frac{m(z_1+z_2)}{2} $$
The base circle radii rb1, rb2, addendum circle radii ra1, ra2, and base pitch pb are expressed as:
$$ r_{b1} = r_1 \cos\alpha, \quad r_{b2} = r_2 \cos\alpha, \quad r_{a1} = \frac{m(z_1+2h_a)}{2}, \quad r_{a2} = \frac{m(z_2+2h_a)}{2}, \quad p_b = \pi m \cos\alpha $$
For an arbitrary meshing point k on the line of action, the instantaneous radii of curvature R1k and R2k and the equivalent curvature radius Rk are:
$$ R_{1k} = r_{b1}\tan\alpha_{1k}, \quad R_{2k} = r_{b2}\tan\alpha_{2k}, \quad \frac{1}{R_k} = \frac{1}{R_{1k}} + \frac{1}{R_{2k}} $$
3.2 Contact Force and Load Distribution
Based on Hertzian contact theory, the maximum contact pressure pmax and average contact pressure pavg are:
$$ p_{max} = \frac{2W}{\pi b}, \quad p_{avg} = \frac{4}{\pi} p_{max} $$
Here, W is the normal load per unit face width, b is the Hertzian contact half-width:
$$ b = \sqrt{\frac{8 W R}{\pi E_z}} $$
where Ez is the equivalent elastic modulus.
For load sharing in the double-tooth engagement zone, the load W on one tooth pair is computed based on the time-varying stiffness of the two pairs and the geometric backlash caused by wear.
3.3 Sliding Velocity and Sliding Distance
The tangential velocities of the driving and driven gear surfaces are:
$$ v_1 = \omega_1 R_1, \quad v_2 = \omega_2 R_2 $$
Thus, the sliding velocity vs and entrainment velocity v are:
$$ v_s = |v_1 – v_2|, \quad v = \frac{v_1 + v_2}{2} $$
The sliding distances L1 and L2 of the driving and driven gears are expressed as:
$$ L_1 = 2b\lambda_1, \quad L_2 = 2b\lambda_2 $$
where λ1 and λ2 are the sliding coefficients.
3.4 Line-Contact Mixed Elastohydrodynamic Lubrication Model
The mixed EHL regime is characterized by the coexistence of hydrodynamic lubricant film and asperity contacts. The total contact pressure p is the sum of hydrodynamic pressure ph and asperity contact pressure pa:
$$ p = p_h + p_a $$
The modified Reynolds equation incorporating surface roughness yields the pressure distribution. The average gap between two rough surfaces follows the Gaussian distribution assumption:
$$ \bar{h}_T = \frac{h}{2}\left[1+\mathrm{erf}\left(\frac{h}{\sqrt{2}\sigma}\right)\right] + \frac{\sigma}{\sqrt{2\pi}} e^{-h^2/(2\sigma^2)} $$
The dimensionless minimum film thickness, asperity load ratio, and film thickness ratio are obtained as regression formulas from extensive numerical simulations:
$$ H_{min} = \frac{h_{min}}{R} = 1.652 W^{-0.077} U^{-0.716} G^{-0.695} (1 + 0.026 V)^{0.185} (1 + 0.312 \bar{\sigma})^{0.809} (1 + 0.977 \bar{\sigma})^{-0.977} $$
$$ L_a = 0.005 \ln(1+4470 V) W^{-0.408} U^{-0.088} G^{-0.103} (1 + 1.168 V^{0.485})^{-1} (1 + 3.741 W^{-0.485} U^{-0.103} G^{-0.288})^{0.005} $$
$$ \Lambda = h_{min}/\sigma $$
3.5 Tooth Surface Temperature Calculation
The flash temperature rise is evaluated according to Tian and Kennedy’s model:
$$ \Delta T = \frac{2 b q}{\sqrt{\pi}\left(\frac{K_1}{\sqrt{v_1 k_1}} + \frac{K_2}{\sqrt{v_2 k_2}}\right)} $$
where q is the total heat flux generated at the contact point, comprising both asperity friction heat and lubricant shear heat:
$$ q = f_c v_s p_a + \Lambda_{lim} v_s p_h $$
3.6 Archard Wear Model Under Mixed Lubrication
The classical Archard wear model relates the wear volume W to the normal load F, sliding distance L, material hardness H, and dimensionless wear coefficient K:
$$ W = \frac{K F L}{H} $$
For local wear depth at a point on the gear tooth, the expression becomes:
$$ h = \frac{K L p}{H} $$
In the mixed-lubrication regime, the wear is primarily attributed to asperity contact. Replacing p with the asperity contact pressure pa and introducing the oil film deficit coefficient ψ yields:
$$ h = k L \psi p \frac{L_a}{100} $$
The oil film deficit coefficient ψ is given as:
$$ \psi = 1 – \exp\left[-\frac{X}{v_s t_0 R_g T} \exp\left(-\frac{E_a}{R_g T}\right)\right] $$
3.7 Discretization and Numerical Procedure
The tooth flank is discretized into nodes along the profile. After completing N wear cycles, the cumulative wear at a given node is updated:
$$ h_N = h_{N-1} + k L_{N-1} p_{N-1} \psi_{N-1} \frac{L_a}{100} $$
When the total wear depth reaches 1 μm, the tooth profile is recalculated. The flowchart updates the curvature radius at each worn node, recalculating contact pressure, film thickness, temperature, and wear depth for the subsequent cycle.
3.8 Numerical Results and Influencing Factors
The gear pair parameters utilized for the numerical study are summarized in Table 1.
Table 1 Gear pair parameters
| Parameter | Symbol | Value |
|:—|:—:|:—:|
| Number of teeth | z1/z2 | 18/27 |
| Module | m (mm) | 2 |
| Pressure angle | α (°) | 20 |
| Face width | B (mm) | 30 |
| Addendum coefficient | ha | 1 |
| Tip clearance coefficient | C | 0.25 |
| Elastic modulus | E (GPa) | 207 |
| Poisson’s ratio | δ | 0.29 |
| Hardness | H (GPa) | 2.3 |
| Surface roughness | σ (μm) | 0.5 |
| Standard center distance | A (mm) | 45 |
The operating conditions are listed in Table 2.
Table 2 Operating conditions
| Parameter | Symbol | Value |
|:—|:—:|:—:|
| Input speed | n1 (r/min) | 2000 |
| Load torque | TN2 (N·m) | 300 |
| Lubricant viscosity | η0 (Pa·s) | 0.35 |
| Pressure-viscosity coefficient | α (m²/N) | 2.5×10⁻⁸ |
| Limiting shear stress coefficient | Λlim | 0.0910 |
3.8.1 Wear Distribution Characteristics
Figure presents the wear depth distribution along the profile of the driving and driven gears after 1×10⁶ wear cycles. The key observations are as follows. For the driving gear, the wear depth is greatest at the tooth root and smallest at the pitch point. For the driven gear, the wear depth reaches its maximum at the tooth tip. Both gears exhibit a general decreasing then increasing trend from root to tip. Interestingly, the wear depth undergoes discontinuities at the boundaries between single-tooth and double-tooth engagement zones. Moreover, the total wear of the pinion is larger than that of the gear, since the pinion rotates more times for the same number of wear cycles.
3.8.2 Influence of Number of Cycles
Increasing the number of wear cycles inevitably increases the accumulated wear depth. The wear increment at the tooth root of the driving gear is largest, indicating the highest wear rate near the root. The radius of curvature gradually decreases due to material removal. The sliding distance and contact stress exhibit redistribution: near the root and tip, they decrease, whereas near the single-tooth engagement boundaries, they increase due to dynamic load sharing.
3.8.3 Influence of Input Speed
As the input speed rises, the entrainment velocity increases, promoting the formation of a thicker oil film. Hence, the minimum film thickness increases, while the asperity contact ratio decreases. The lubricating condition is improved. The flash temperature increases with speed due to increased heat generation. The combined influence results in a net decrease of wear depth as the input speed rises for a fixed cycle count.
3.8.4 Influence of Load Torque
A higher load torque amplifies the Hertzian contact half-width, sliding distance, and contact stress. Although the asperity contact ratio decreases slightly, the minimum film thickness and film thickness ratio decrease, indicating a deteriorated lubrication condition. Consequently, the wear depth increases monotonically with increasing load torque.
3.8.5 Influence of Surface Roughness
Surface roughness plays a dual role. On one hand, rougher surfaces increase the average gap, enhancing the storage of lubricant and slightly increasing the minimum film thickness. On the other hand, the asperity contact ratio increases significantly, and the film thickness ratio decreases, worsening the lubrication state. Additionally, rougher interfaces generate more heat, increasing the flash temperature and the oil film deficit coefficient. The net effect is that wear depth increases considerably with increasing surface roughness.
3.8.6 Influence of Lubricant Properties
The lubricant properties were investigated by comparing SAE30 and PAO oils, as summarized in Table 3.
Table 3 Lubricant properties
| Lubricant | η0 (Pa·s) | y0 (m²/N) | Λlim |
|:—|:—:|:—:|:—:|
| SAE30 | 0.35 | 2.5×10⁻⁸ | 0.0910 |
| PAO | 0.04 | 1.5×10⁻⁸ | 0.0434 |
SAE30, with higher viscosity and pressure-viscosity coefficient, yields a substantially larger minimum film thickness and lower asperity contact ratio, resulting in notably lower wear than PAO. The dry contact condition produces the largest wear, confirming that proper lubrication effectively reduces tooth surface wear.
3.8.7 Influence of Center Distance
Variations in center distance alter the contact ratio, pressure angle at the pitch point, and the positions of the engagement boundaries. When the center distance increases, the contact ratio decreases, the single-tooth engagement zone becomes longer, and the engagement start point moves away from the tooth root. As a consequence, the wear zone shrinks, the maximum wear depth (found near the tooth root) and the total wear both decrease. Thus, a properly increased center distance can reduce the wear of spur gears.
3.8.8 Comparative Sensitivity Analysis
Sensitivity computation indicates that when the input speed, load torque, surface roughness, and center distance vary by 1%, the average total wear changes by approximately 0.71%, 2.69%, 2.23%, and 133.77%, respectively. The center distance has the greatest influence, followed by load torque and surface roughness, with input speed exerting the least influence.
4. Time-Varying Meshing Stiffness and Dynamic Characteristics under Tooth Surface Wear
4.1 TVMS Calculation Using the Energy Method
The potential energy stored in a meshing gear tooth includes bending, shear, axial compressive, Hertzian contact, and fillet-foundation components. The bending stiffness, shear stiffness, and axial compressive stiffness for a gear tooth with non-uniform wear are expressed as follows:
$$ M_b = \frac{E_z B}{\int_{-\alpha_1}^{\alpha_2} \frac{2.5 \cos^2\alpha_0 (1+\cos\alpha_1 \cos\alpha_2)}{E_z B \cos^3\alpha_0} d\alpha} + \cdots $$
$$ M_s = \frac{E_z B}{\int_{-\alpha_1}^{\alpha_2} \frac{1.2(1+\delta)\cos^2\alpha_0 \cos^2\alpha}{E_z B \cos\alpha_0 \sin^2\alpha} d\alpha} + \cdots $$
$$ M_a = \frac{E_z B}{\int_{-\alpha_1}^{\alpha_2} \frac{\sin^2\alpha_0 \cos^2\alpha}{E_z B \cos\alpha_0} d\alpha} + \cdots $$
The Hertzian contact stiffness is given by:
$$ M_h = \frac{\pi E_z B}{4(1-\delta^2)} $$
The fillet-foundation stiffness is expressed as:
$$ M_f = \frac{E_z B \cos^2\alpha_1}{L^* \frac{u_f}{S_f} + M^* \frac{u_f}{S_f} + P^* (1+Q^* \tan^2\alpha_1)} $$
For a gear pair, the combined meshing stiffness is:
$$ M = \frac{1}{\frac{1}{M_h} + \sum_{i=1}^{2} \left(\frac{1}{M_{b,i}} + \frac{1}{M_{s,i}} + \frac{1}{M_{a,i}} + \frac{1}{M_{f,i}}\right)} $$
4.2 Finite Element Verification
A three-dimensional model of the gear pair was developed using UG12.0, discretized using HYPERMESH, and analyzed in ABAQUS. The driven gear was fixed in all degrees of freedom except rotation, and a torque was applied to the driving gear. The contact force F and deformation δ at each meshing position were extracted to compute the stiffness M = F/δ. The gear was rotated incrementally through the entire meshing cycle.
The results presented in Table 4 show that the energy method agrees well with the FEA results, confirming the validity of the analytical approach.
Table 4 TVMS comparison between energy method and FEA
| Active gear angle (°) | Energy method (10⁸ N/m) | FEA (10⁸ N/m) |
|:—:|:—:|:—:|
| 0 | 7.14 | 7.26 |
| 4.5 | 7.12 | 7.25 |
| 9.0 | 4.10 | 4.15 |
| 13.5 | 4.91 | 4.93 |
| 18.0 | 4.88 | 4.95 |
| 22.5 | 7.08 | 7.52 |
4.3 TVMS under Tooth Wear
With increasing wear cycles, the TVMS of the gear pair decreases. The greatest reduction occurs at the root of the driving gear, where wear is maximal, and in the double-tooth engagement zones, where the contribution of both meshing pairs is affected. The percentage reduction of TVMS increases with wear depth and is more pronounced in double-tooth zones than in the single-tooth zone.
4.4 Gear Dynamics Model
The torsional vibration model of the spur gear pair is established. The dynamic transmission error x is defined as:
$$ x = r_{b1}\theta_1 – r_{b2}\theta_2 – e(t) $$
The corresponding differential equation of motion is:
$$ m_e \ddot{x} + c_m \dot{x} + M_m f(x) = W_0 – m_e \ddot{e}(t) $$
The dynamic meshing force is:
$$ F = c_m \dot{x} + M_m f(x) $$
The tooth backlash after wear is:
$$ \Delta_n = \Delta_0 + 2(h_1 + h_2) $$
The comprehensive gear error includes long-period, short-period, and wear-induced errors:
$$ e(t) = e_{10}\sin(2\pi f_p t) + e_{20}\sin(2\pi f_m t) + h_1 + h_2 $$
4.5 Dynamic Characteristics of Healthy and Worn Gear Pairs
For a healthy gear pair, the dynamic transmission error oscillates between approximately 22.6 μm and 48.8 μm, with a mean of roughly 40.9 μm in the single-tooth zone and 30.2 μm in the double-tooth zone. The dynamic meshing force fluctuates with a mean of about 11831.9 N, slightly lower than the external load due to the gear error.
The frequency response reveals that the dominant frequencies match the mesh frequency and its harmonics. The simulated mesh frequency is approximately 606.89 Hz, which deviates by only 1.15% from the theoretical value of 600 Hz, validating the dynamic model.
As wear cycles increase to 5×10⁶, 1×10⁷, and 1.5×10⁷, the peak-to-peak dynamic transmission error respectively increases by 2.12%, 38.82%, and 100.16% relative to the healthy gear pair. The dynamic meshing force amplitude also increases significantly. This is attributed to the rising wear-induced error, which enhances the dynamic excitation.
4.6 Coupling Effect of Dynamic Load and Wear
By substituting the dynamic meshing force into the wear model, the wear depth under dynamic conditions is obtained. Results show that dynamic load intensifies vibration and impact, leading to substantially higher wear compared with quasi-static conditions. The comparisons are summarized in Table 5.
Table 5 Comparison of wear under dynamic and quasi-static loads after 1.5×10⁷ cycles
| Parameter | Quasi-static (mm) | Dynamic (mm) | Increase (%) |
|:—|:—:|:—:|:—:|
| Max wear of pinion | 2.650×10⁻³ | 1.220×10⁻² | 360.4 |
| Mean wear of pinion | 2.570×10⁻⁴ | 4.329×10⁻⁴ | 68.4 |
| Total wear of pinion | 5.130×10⁻² | 8.658×10⁻² | 68.8 |
| Max wear of gear | 3.806×10⁻⁴ | 1.957×10⁻³ | 414.1 |
| Mean wear of gear | 9.315×10⁻⁵ | 1.265×10⁻⁴ | 35.8 |
| Total wear of gear | 1.863×10⁻² | 2.531×10⁻² | 35.8 |
| Total wear of both gears | 6.993×10⁻² | 1.119×10⁻¹ | 60.0 |
The dynamic load model captures the accelerating wear rate with increasing cycles, which agrees with physical observations, whereas the quasi-static model predicts a nearly linear wear progression. This finding underscores the necessity of considering dynamics-wear coupling in gear wear prediction.
5. Vibration-Signal-Based Quantitative Evaluation of Tooth Surface Wear of Spur Gears
5.1 Wear Evolution Model of Spur Gears
To quantitatively characterize the evolution of wear at the tooth root of the driving gear, several functional forms were fitted to the wear-cycle data for N < 1×10⁷ and validated against data for N > 1×10⁷. The considered functions are presented below:
Quadratic function:
$$ y_d = a_{d1} N^2 + a_{d2} N + a_{d3} $$
Exponential function:
$$ y_e = a_{e1} e^{a_{e2} N} + a_{e3} $$
Power function:
$$ y_p = a_{p1} N^{a_{p2}} $$
Empirical equation:
$$ y_x = a_{x1} e^{a_{x2} N} + a_{x3} e^{a_{x4} N} $$
The coefficients for each operating condition are listed in Tables 6 through 9.
Table 6 Quadratic function coefficients
| Condition | ad1 (×10⁻¹⁷) | ad2 (×10⁻¹⁰) | ad3 (×10⁻⁴) |
|:—:|:—:|:—:|:—:|
| 1 | 5.672 | -2.029 | 5.545 |
| 2 | 2.579 | 1.596 | 1.657 |
| 3 | 2.670 | 2.558 | 1.010 |
| 4 | 6.920 | -1.667 | 7.672 |
| 5 | 4.831 | -1.564 | 4.124 |
Table 7 Exponential function coefficients
| Condition | ae1 (×10⁻⁴) | ae2 (×10⁻⁷) | ae3 (×10⁻⁴) |
|:—:|:—:|:—:|:—:|
| 1 | 4.202 | 2.188 | 0.1596 |
| 2 | 25.48 | 0.9712 | -24.68 |
| 3 | 31.69 | 0.9784 | -31.30 |
| 4 | 12.33 | 1.672 | -8.200 |
| 5 | 6.277 | 1.840 | -5.095 |
Table 8 Power function coefficients
| Condition | ap1 (×10⁻¹⁶) | ap2 |
|:—:|:—:|:—:|
| 1 | 0.004266 | 2.284 |
| 2 | 1005 | 1.521 |
| 3 | 40350 | 1.301 |
| 4 | 0.2146 | 2.064 |
| 5 | 0.006315 | 2.251 |
Table 9 Empirical equation coefficients
| Condition | ax1 (×10⁻⁴) | ax2 (×10⁻⁷) | ax3 (×10⁻⁴) | ax4 (×10⁻⁷) |
|:—:|:—:|:—:|:—:|:—:|
| 1 | 4.976 | 2.062 | -5.375 | -4.104 |
| 2 | 12.81 | 1.281 | -12.95 | -1.213 |
| 3 | 10.42 | 1.638 | -11.74 | -2.644 |
| 4 | 9.074 | 1.843 | -10.13 | -4.686 |
| 5 | 4.611 | 2.016 | -4.678 | -2.295 |
For Condition 3, the empirical equation is:
$$ y_x = 0.001042 e^{1.638 \times 10^{-7} N} – 0.001174 e^{-2.644 \times 10^{-7} N} $$
The fitting quality was evaluated using the root mean square error (RMSE), coefficient of determination (R²), and relative error.
Table 10 Fitting quality evaluation for Condition 3
| Function | RMSE | R² | Max relative error |
|:—|:—:|:—:|:—:|
| Quadratic | 3.079×10⁻⁵ | 0.9997 | 9.47% |
| Exponential | 1.306×10⁻⁵ | 0.9999 | 3.60% |
| Power | 9.171×10⁻⁵ | 0.9973 | -26.21% |
| Empirical | 4.204×10⁻⁵ | 0.9993 | -6.75% |
The exponential function provides the best fitting accuracy, but its predictive capability for N > 1×10⁷ is inferior to the empirical equation. The empirical equation achieves the best prediction performance, with a maximum relative error of only 1.18%, as shown in Table 11.
Table 11 Prediction relative errors for N > 1×10⁷
| Cycles (×10⁷) | Quadratic (%) | Exponential (%) | Power (%) | Empirical (%) |
|:—:|:—:|:—:|:—:|:—:|
| 1.1 | -2.66 | -1.38 | -6.47 | 0.05 |
| 1.2 | -5.36 | -3.00 | -10.75 | 0.78 |
| 1.3 | -8.98 | -5.37 | -15.78 | 1.18 |
| 1.4 | -13.66 | -8.73 | -21.64 | 0.86 |
| 1.5 | -19.62 | -13.40 | -28.44 | -0.71 |
The empirical equation is therefore adopted for the wear evolution model.
5.2 Vibration Signal Feature Parameter Analysis
The dynamic transmission error signal obtained from the gear dynamics model was processed using the feature parameters listed in Table 12.
Table 12 Feature parameters and calculation formulas
| Feature parameter | Formula |
|:—|:—|
| RMS value | $$ x_{rms} = \sqrt{\frac{1}{N}\sum_{i=1}^{N} x_i^2} $$ |
| Peak-to-peak value | $$ x_{pp} = x_{max} – x_{min} $$ |
| Kurtosis | $$ x_k = \frac{\frac{1}{N}\sum_{i=1}^{N}(x_i-\bar{x})^4}{\sigma_x^4} $$ |
| Crest factor | $$ C = \frac{x_{max}}{\sqrt{\frac{1}{N}\sum_{i=1}^{N} x_i^2}} $$ |
| Clearance factor | $$ M = \frac{x_{max}}{\left(\frac{1}{N}\sum_{i=1}^{N}\sqrt{|x_i|}\right)^2} $$ |
| Impulse factor | $$ I = \frac{x_{max}}{\frac{1}{N}\sum_{i=1}^{N}|x_i|} $$ |
Table 13 presents the feature parameter values as a function of wear cycles and the corresponding wear depth at the tooth root of the driving gear under Condition 3.
Table 13 Feature parameters under Condition 3
| Cycles (×10⁶) | Wear (mm) | RMS (×10⁻⁵ m) | Peak-peak (×10⁻⁵ m) | Kurtosis | Crest factor | Clearance factor | Impulse factor |
|:—:|:—:|:—:|:—:|:—:|:—:|:—:|:—:|
| 0 | 0 | 3.324 | 2.548 | 2.827 | 1.456 | 1.480 | 1.472 |
| 2 | 0.000726 | 3.333 | 2.534 | 2.705 | 1.453 | 1.477 | 1.469 |
| 4 | 0.001570 | 3.344 | 2.560 | 2.552 | 1.449 | 1.475 | 1.467 |
| 6 | 0.002572 | 3.357 | 2.693 | 2.429 | 1.444 | 1.473 | 1.463 |
| 8 | 0.003789 | 3.376 | 2.922 | 2.461 | 1.437 | 1.471 | 1.459 |
| 10 | 0.005317 | 3.405 | 3.309 | 2.774 | 1.424 | 1.470 | 1.454 |
| 12 | 0.007332 | 3.445 | 4.610 | 3.346 | 1.602 | 1.684 | 1.653 |
| 14 | 0.010206 | 3.472 | 6.361 | 4.657 | 1.804 | 1.957 | 1.889 |
| 15 | 0.012225 | 3.482 | 7.080 | 5.176 | 1.870 | 2.039 | 1.964 |
The observations from Table 13 indicate that the RMS value changes only slightly with wear, demonstrating low sensitivity. The crest factor, clearance factor, and impulse factor remain nearly constant during early wear, but increase sharply when wear becomes severe. The peak-to-peak value and kurtosis exhibit substantial changes throughout the entire wear cycle, indicating high sensitivity to tooth surface wear.
5.3 Experimental Verification
To validate the vibration-signal-based wear evaluation of spur gears, experiments were conducted on an HFXZ-I gearbox fault diagnosis test platform. Faults were injected by manufacturing gears with prescribed wear depths of 0.005 mm, 0.010 mm, and 0.015 mm at the tooth root of the driving gear. The gear parameters matched those in Table 1. The input speed was 2000 r/min and the load torque was 20 N·m. Vibration acceleration signals were collected from the gearbox housing.
Table 14 provides the experimental feature parameters for the healthy and worn gears.
Table 14 Experimental feature parameters
| Wear depth (mm) | RMS (m/s²) | Peak-peak (m/s²) | Kurtosis | Crest factor | Clearance factor | Impulse factor | Waveform factor |
|:—:|:—:|:—:|:—:|:—:|:—:|:—:|:—:|
| Healthy | 1.066 | 16.67 | 4.336 | 8.931 | 14.09 | 11.67 | 1.306 |
| 0.005 | 1.796 | 40.48 | 9.589 | 8.504 | 16.09 | 12.61 | 1.483 |
| 0.010 | 1.859 | 50.54 | 15.46 | 14.57 | 30.59 | 23.27 | 1.597 |
| 0.015 | 2.278 | 70.18 | 52.05 | 20.78 | 49.58 | 37.22 | 1.791 |
The normalized experimental feature parameters exhibit the same general trends as the simulation results, confirming that the proposed model reasonably captures the evolution of vibration characteristics under tooth surface wear of spur gears.
5.4 Quantitative Evaluation Method
Based on the sensitivity analysis, the peak-to-peak value and kurtosis were selected as the most indicative features of tooth surface wear in spur gears. Their evolution models with respect to the number of wear cycles were established using cubic polynomials:
$$ x_p = 0.002503 \left(\frac{N}{10^6}\right)^3 – 0.01902 \left(\frac{N}{10^6}\right)^2 + 0.03619 \left(\frac{N}{10^6}\right) + 2.548 $$
$$ x_k = 0.002094 \left(\frac{N}{10^6}\right)^3 – 0.01842 \left(\frac{N}{10^6}\right)^2 – 0.03167 \left(\frac{N}{10^6}\right) + 2.827 $$
The peak-to-peak evolution model achieved a maximum relative error below 10% and an RMSE of 1.095×10⁻⁶. The kurtosis evolution model achieved a maximum relative error below 3% and an RMSE of 0.0566. Both models possess coefficients of determination exceeding 0.995.
Combining the wear evolution model with the feature parameter evolution models yields the final quantitative evaluation formula:
$$ h_{root} = \frac{0.001042 \exp\left(\frac{1.638 \times 10^{-7}(-0.2644 \times \sqrt{…} + 10 \times …)}{…}\right) – 0.001174 \exp\left(-\frac{2.644 \times 10^{-7}(-0.2644 \times \sqrt{…} + …)}{…}\right)}{…} $$
The performance of the proposed quantitative evaluation method is evaluated by comparing the predicted wear depth with the theoretical values. The results demonstrate that the root mean square error of the proposed model is only 5.535×10⁻⁸, and the coefficient of determination reaches 0.9955. The maximum relative error is 24.69% at N = 1×10⁶, while the minimum relative error is -0.25% at N = 1.2×10⁷. The average relative error is 5.18%, resulting in a model accuracy of 94.82%.
The quantitative evaluation scheme can be summarized in the following steps:
Step 1: Acquire the vibration signal of the gearbox housing.
Step 2: Compute the peak-to-peak value and kurtosis of the signal.
Step 3: Invert the number of wear cycles via the feature evolution models.
Step 4: Substitute the obtained cycle count into the wear evolution model to determine the corresponding wear depth at the tooth root.
6. Conclusions and Future Work
6.1 Conclusions
This thesis investigated the tooth surface wear model and quantitative evaluation method of spur gear transmission systems. The principal conclusions are as follows:
(1) A line-contact mixed elastohydrodynamic lubrication wear model for spur gears was established, integrating gear meshing theory, Hertzian contact theory, load sharing among tooth pairs, flash temperature calculations, and the Archard wear equation. The wear distribution law demonstrates that the pinion exhibits greater total wear than the gear. From the tooth root to the tooth tip, both gears show a decrease-then-increase wear trend, with the maximum wear of the driving gear located at the tooth root and that of the driven gear at the tooth tip. The minimum wear occurs at the pitch point, and discontinuities exist at the boundaries between single-tooth and double-tooth engagement zones.
(2) The influencing factors of tooth surface wear were systematically analyzed. The number of cycles, input speed, load torque, surface roughness, lubricant properties, and center distance affect the magnitude of wear but hardly alter its distribution pattern. Wear increases with increasing cycles, load torque, and surface roughness, while decreases with increasing input speed, lubricant viscosity, pressure-viscosity coefficient, and properly selected center distance.
(3) The TVMS of spur gears decreases with tooth surface wear, and the reduction is more prominent in double-tooth engagement zones. Furthermore, dynamic loads intensify tooth surface wear due to gear vibration and impact, while wear amplifies the comprehensive error and dynamic transmission error, leading to larger dynamic meshing forces and increased vibration amplitudes.
(4) The sensitivity analysis of vibration signal feature parameters indicated that the RMS value exhibits low sensitivity, the crest factor, clearance factor, and impulse factor become sensitive only at severe wear stages, whereas the peak-to-peak value and kurtosis demonstrate strong sensitivity throughout the entire wear process.
(5) A quantitative wear evaluation method was developed by combining the wear evolution model and the peak-to-peak and kurtosis evolution models. The model accuracy reaches 94.82%, demonstrating the effectiveness of the proposed approach.
6.2 Future Work
Several avenues for future research are worth pursuing. First, the present wear model is developed under mixed lubrication conditions. In practice, gear systems may experience various lubrication states under variable operating conditions. Extending the model to accommodate boundary, elastohydrodynamic, and hydrodynamic lubrication regimes would improve its applicability. Second, the influence of the lubricant on the stiffness and damping of the meshing interface should be incorporated into the gear dynamics model to reflect the actual transmission performance more faithfully. Third, experimental wear evolution tests should be designed to further validate and refine the proposed tooth surface wear model for spur gears. Long-duration run-to-failure tests under controlled operating conditions will provide valuable data for verifying the numerical predictions.