Internal spur gear pairs represent a critical form of mechanical transmission, widely employed in various machinery due to their compact center distance, low contact stress, reduced sliding velocity, and high contact ratio. Particularly in wind turbine planetary gearboxes, internal meshing is an indispensable configuration. Wear is a failure mode that accompanies the entire lifecycle of gear systems, and accurate prediction of wear progression is essential for life analysis and reliability assessment. While numerous models for wear calculation exist, many struggle to align closely with practical observations. The Archard wear model, however, has demonstrated reasonable agreement with experimental results in many scenarios. Traditional static or quasi-static wear analyses often overlook the dynamic interactions between meshing forces, lubrication conditions, and the evolving tooth surface geometry. This study focuses on the wear degradation of internal spur gears operating under hybrid elastohydrodynamic lubrication (EHL) conditions, coupling a dynamic model with a wear calculation to investigate the interplay between dynamic meshing behavior and surface degradation over time.
Mathematical Model
1.1 Adhesive Wear Model
The fundamental adhesive wear model proposed by Archard is given by:
$$ \frac{V}{s} = K \frac{F_t}{H} $$
where \( V \) is the wear volume, \( s \) is the sliding distance, \( K \) is the dimensionless wear coefficient, \( H \) is the hardness of the softer contact surface, and \( F_t \) is the contact load.
For a point \( P \) on a surface under dry, mixed, or boundary lubrication, the wear depth evolution can be described differentially:
$$ \frac{dh}{dt} = k p v $$
where \( h \) is the wear depth, \( k \) is the wear coefficient, \( p \) is the contact pressure, \( v \) is the relative sliding velocity, and \( t \) is time. Integrating over time gives the wear depth at point \( P \):
$$ h_P = \int_0^t k p v \, dt $$
For mild wear studies in gears, the tooth profile can be discretized into small segments. Assuming steady-state wear within each segment over a time step, the wear depth at segment \( i \) after the \( j \)-th meshing event can be approximated as:
$$ h_{i,j} = h_{i,j-1} + k \, p_{i,j-1} \, v_j \, \Delta t $$
where \( \Delta t \) is the time step corresponding to the meshing event.
1.2 Dynamic Model of Internal Spur Gears
A two-degree-of-freedom dynamic model is established for an internal spur gear pair, considering the angular displacements \( \theta_1 \) and \( \theta_2 \) of the external (pinion) and internal (gear) members, respectively.
The equations of motion are derived as:
$$
\begin{cases}
I_1 \ddot{\theta}_1 = T_1 – R_{b1} \sum_{i=1}^{n_z} F_i – \sum_{i=1}^{n_z} \Lambda_i \rho_{1i} \mu_i F_i \\
I_2 \ddot{\theta}_2 = R_{b2} \sum_{i=1}^{n_z} F_i + \sum_{i=1}^{n_z} \Lambda_i \rho_{2i} \mu_i F_i – T_2
\end{cases}
$$
where \( I \) is moment of inertia, \( T \) is torque, \( R_b \) is base radius, \( n_z \) is the number of tooth pairs in contact, \( F_i \) is the dynamic meshing force for pair \( i \), \( \Lambda_i \) is the friction direction parameter (±1), \( \rho \) is the radius of curvature at the contact point, and \( \mu \) is the friction coefficient.
Transforming to displacements along the line of action, \( y_1 \) and \( y_2 \), and considering an external load \( F_t \), the equations become:
$$
\begin{cases}
m_1 \ddot{y}_1 = F_t – \sum_{i=1}^{n_z} F_i – \sum_{i=1}^{n_z} \Lambda_i S_{1i} \mu_i F_i \\
m_2 \ddot{y}_2 = \sum_{i=1}^{n_z} F_i + \sum_{i=1}^{n_z} \Lambda_i S_{2i} \mu_i F_i – F_t
\end{cases}
$$
where \( m \) is mass, and \( S_{1i} = \rho_{1i}/R_{b1} \), \( S_{2i} = \rho_{2i}/R_{b2} \). The dynamic meshing force for any pair is:
$$ F_i = K_{ei} (y_r – e) $$
where \( y_r = y_1 – y_2 \) is the relative displacement along the line of action, \( e \) is the composite static transmission error (including initial errors and accumulated wear), and \( K_{ei} \) is the comprehensive meshing stiffness. The force is zero when \( y_r – e < 0 \) (separation).
Combining and including viscous damping yields the governing equation for the relative motion:
$$ \ddot{y}_r + 2 \zeta \omega_n \dot{y}_r + \omega_n^2 y_r = \omega_n^2 y_s $$
where \( \omega_n \) is the natural frequency, \( \zeta \) is the damping ratio, and \( y_s \) is the static transmission error under load.
1.3 Time-Varying Meshing Stiffness Model under Hybrid EHL
The lubrication state is often characterized by the film thickness ratio \( \lambda \). For hybrid EHL conditions (\( \lambda \) typically between 0.5 and 1.5), the contact stiffness is modeled as a parallel combination of the asperity contact stiffness \( K_s \) and the oil film stiffness \( K_l \).
To accurately model the load sharing between the asperities and the lubricant film, load-sharing factors \( \gamma_1 \) and \( \gamma_2 \) are introduced, where \( 1/\gamma_1 + 1/\gamma_2 = 1 \). The comprehensive meshing stiffness \( K_e \) is then:
$$ K_e = \frac{K_l}{\gamma_1} + \frac{K_s}{\gamma_2} $$
The tooth body stiffness \( K_s \) for a pair at contact point parameter \( \xi \) is calculated based on potential energy principles, considering bending, axial compressive, and shear deformation:
$$ K_s(\xi) = \left( \frac{1}{k_{1x}} + \frac{1}{k_{1s}} + \frac{1}{k_{1n}} + \frac{1}{k_{2x}} + \frac{1}{k_{2s}} + \frac{1}{k_{2n}} + \frac{1}{K_H} \right)^{-1} $$
The oil film stiffness \( K_l \) is derived from EHL theory. Starting from the central film thickness formula and its dependence on load, the stiffness is defined as:
$$ K_l = -\frac{dF_l}{dh_c} = \frac{dF_l}{d\delta_{\infty}} $$
where \( F_l \) is the load carried by the lubricant film, \( h_c \) is the central film thickness, and \( \delta_{\infty} \) is the steady-state compression of the film. Using dimensionless groups from EHL empirical formulas, \( K_l \) can be expressed as a function of the dynamic load, material properties, and lubricant parameters.
1.4 Friction and Load-Sharing Factors
The total friction force \( F_f \) in hybrid lubrication is the sum of the friction from the lubricant film \( F_{fl} \) and the asperity contact \( F_{fs} \):
$$ F_f = F_{fl} + F_{fs} = \iint_{A_0} \tau_s \, dA + f_s F_s $$
where \( \tau_s \) is the lubricant shear stress, \( A_0 \) is the Hertzian contact area, \( f_s \) is the boundary friction coefficient, and \( F_s \) is the load carried by asperities.
The total load \( F_t \) is shared:
$$ F_t = F_l + F_s = \frac{F_t}{\gamma_1} + \frac{F_t}{\gamma_2} $$
The load-sharing factors \( 1/\gamma_1 \) and \( 1/\gamma_2 \) are determined by solving the rough surface contact equation, which relates the asperity contact load to the composite surface roughness, material properties, and the nominal film thickness.
Numerical Solution Procedure
The wear process is simulated by discretizing the meshing cycle. The time from the start of a tooth pair’s contact \( t_1 \) to its end \( t_2 \) is divided into steps. The initial static load distribution \( F_{d0} \) is calculated using the load distribution ratio \( R_i(\xi) \) derived from the stiffness:
$$ R_i(\xi) = \frac{F_i(\xi)}{F_t} = \frac{K_s(\xi)}{\sum_{j=0}^{n_z-1} K_s(\xi + j)} $$
This initial distribution serves as the starting point for an iterative coupled solution. In each iteration:
- The dynamic meshing force \( F_{d1} \) is solved from the dynamic equations using a numerical integrator (e.g., 4th-order Runge-Kutta), with periodicity conditions \( y_r(0)=y_r(T_z) \) and \( \dot{y}_r(0)=\dot{y}_r(T_z) \).
- For the calculated dynamic load, the oil film stiffness \( K_l \) and load-sharing factors \( \gamma_1, \gamma_2 \) are computed.
- The comprehensive stiffness \( K_e \) is updated.
- The dynamic force \( F_{d1} \) is recalculated with the new stiffness. The process iterates until convergence.
- The final dynamic pressure \( p \) and sliding velocity \( v \) are used in the wear model (Eq. 4) to calculate the wear depth for that meshing cycle.
- The accumulated wear depth is added to the static transmission error \( e \) for the next major cycle, simulating profile modification.
The parameters used for the simulation example in this study are summarized in the table below:
| Parameter | Symbol | Value | Unit |
|---|---|---|---|
| Module | \( m_n \) | 2.500 | mm |
| Pinion Teeth | \( Z_1 \) | 21 | – |
| Gear Teeth | \( Z_2 \) | 102 | – |
| Pressure Angle | \( \alpha_0 \) | 0.350 | rad |
| Face Width | \( b \) | 50.000 | mm |
| Pinion Elastic Modulus | \( E_1 \) | 206.0 | GPa |
| Gear Elastic Modulus | \( E_2 \) | 310.0 | GPa |
| Poisson’s Ratio | \( \nu \) | 0.300 | – |
| RMS Roughness | \( \sigma_s \) | 8×10⁻⁶ | m |
| Wear Coefficient | \( k \) | 5×10⁻¹⁶ | m²/N |
| Input Power | \( P \) | 5 | kW |
| Pinion Speed | \( n_1 \) | 500 | rpm |
Results and Analysis
2.1 Tooth Surface Wear Distribution
The calculated wear profiles for the internal and external spur gears after \( j = (0.5, 1.0, 1.5, 2.0, 2.5, 3.0, 3.5, 4.0, 4.5, 5.0) \times 10^6 \) meshing cycles are analyzed. Unlike simple linear wear models, the dynamic coupling results show a non-linear progression.
For the internal gear, the wear distribution develops multiple peaks. Initially (at \( j=0.5 \times 10^6 \)), peak wear depths occur at specific points along the profile. As wear accumulates (at \( j=5.0 \times 10^6 \)), these peak locations shift towards the end of the meshing region (the recess action). The wear depth in the approach (or ingress) segment is consistently higher than in the recess (or egress) segment, aligning with experimental observations for spur gears. This is attributed to the dynamic impact and load-sharing conditions prevalent during the initial contact of a tooth pair.
A similar trend is observed for the external spur gear (pinion), where peak wear points also migrate, and recess segment wear is less severe. This asymmetrical wear distribution between approach and recess is a key characteristic captured by the dynamic hybrid EHL model that static models often miss.
| Component | Segment | j = 0.5×10⁶ | j = 5.0×10⁶ | Increase |
|---|---|---|---|---|
| Internal Spur Gear | Approach | 3.987 | 39.87 | 900% |
| Recess | 1.058 | 10.58 | 900% | |
| External Spur Gear | Approach | 4.617 | 46.17 | 900% |
| Recess | 1.208 | 12.08 | 900% |
The growth is proportional to the number of cycles in this model, but the distribution pattern evolves non-linearly due to the changing dynamic response.
2.2 Degradation of Dynamic Meshing Performance
The static load distribution \( F_{d0} \), calculated from stiffness alone, shows abrupt changes at the single/double tooth contact boundaries. In contrast, the dynamic meshing force \( F_{d1} \), considering inertia and damping, exhibits a smoother, double-peaked profile within a meshing period \( T_z \).
As wear accumulates over millions of cycles, the dynamic meshing force evolves. The primary peak in the recess segment increases significantly (e.g., by ~37.7% from 1264 N to 1740 N in the studied case) and shifts temporally. The force in the approach segment shows more complex changes, with new minor peaks emerging, indicating increased dynamic interaction due to the modified profile of the spur gears. However, the peak-to-peak amplitude of the relative displacement and velocity along the line of action shows only modest changes initially, suggesting the system’s dynamic response adapts to the gradual wear.
The increment in dynamic force \( \Delta F_{d1} \) relative to the initial worn state (\( j=0.5 \times 10^6 \)) reveals six distinct peaks developing over time. The magnitude of these increments tends to converge as wear progresses, implying that the dynamic excitation may stabilize after an initial running-in period, albeit at a higher vibration level.
| Parameter | j = 0.5×10⁶ | j = 5.0×10⁶ | Change |
|---|---|---|---|
| Max Dynamic Force (N) | 2236 | 2193 | -1.9% |
| Recess Peak Force (N) | 1264 | 1740 | +37.7% |
| Max Rel. Displacement (mm) | 0.00236 | 0.00282 | +19.9% |
| Max Rel. Velocity (mm/s) | 0.3369 | 0.5928 | +76.0% |
2.3 Degradation of Meshing Stiffness Components
The tooth body stiffness \( K_s \) is assumed largely unaffected by mild, uniform wear and remains a periodic function of mesh position. The oil film stiffness \( K_l \), however, is sensitive to the instantaneous dynamic load. Under dynamic conditions, \( K_l \) displays a double-peak profile. With increasing wear, the stiffness peaks, particularly in the recess segment, increase (by ~13.4% for the recess peak in this case), and their positions shift. This contrasts with calculations based on steady-state loads, which show little variation. The increase in \( K_l \) is primarily driven by the increased dynamic load in the recess segment.
The load-sharing factors \( 1/\gamma_1 \) (oil film) and \( 1/\gamma_2 \) (asperities) determine the contribution of each stiffness component to the total. Results show that the oil film load share decreases slightly in the recess segment as wear progresses. More notably, after significant wear (\( j=5.0 \times 10^6 \)), the oil film load share in the approach segment exhibits a sharp transient, indicating reduced film stability. Despite the changes in \( K_l \) and \( \gamma_i \), the combined effect on the comprehensive meshing stiffness \( K_e \) is minimal. This is because the oil film’s contribution to the total stiffness is relatively small under the modeled hybrid EHL conditions, dominated by the much larger tooth contact stiffness \( K_s \). Therefore, for mild wear, the overall time-varying mesh stiffness of the spur gears may not degrade significantly.
Conclusion
This study developed a coupled dynamic wear model for internal spur gears operating under hybrid elastohydrodynamic lubrication. The model integrates adhesive wear theory, a nonlinear dynamic model with time-varying mesh stiffness, and a hybrid lubrication load-sharing approach. The key findings are:
- Wear Distribution: The dynamic model predicts non-uniform wear on internal spur gears, with greater wear occurring in the approach segment compared to the recess segment. Wear peak locations shift towards the end of contact as cumulative cycles increase, demonstrating a non-linear progression not captured by static models.
- Dynamic Performance Evolution: Dynamic meshing forces evolve with wear; the recess segment force increases substantially while new excitations appear in the approach segment. The system’s relative vibration frequency increases, indicating deteriorating meshing conditions, yet the dynamic force increments show a convergent trend, suggesting wear may dampen some initial impact behavior.
- Stiffness and Lubrication Behavior: The oil film stiffness is sensitive to dynamic load changes and increases with wear, particularly in the recess segment. However, the load carried by the oil film decreases slightly, and its stability may be compromised after significant wear. The comprehensive meshing stiffness remains largely unchanged during mild wear due to the dominant effect of the tooth body stiffness.
- Implication for Spur Gears: The study highlights the complex interplay between wear, dynamics, and lubrication in spur gears. The recess segment’s increased load and oil film stiffness may help inhibit wear progression there, potentially preventing rapid degradation of load distribution. The model provides a framework for predicting wear-induced dynamic performance changes, offering valuable insight for the life prediction and design of robust internal spur gear transmissions.
Future work could incorporate more advanced wear mechanisms (e.g., abrasive, fatigue), thermal effects on lubrication, and validation through dedicated experiments on spur gears under controlled hybrid EHL conditions.