In modern industrial applications, internal gears are critical components in heavy-duty machinery such as tunnel boring machines, where they operate under high torque and low-speed conditions. The reliability and longevity of these systems depend heavily on the lubrication state of the gear teeth. As an internal gear manufacturer, we recognize that unmodified gear profiles often lead to meshing interference, resulting in surface wear and pitting. To mitigate these issues, tooth profile modification techniques, such as the Walker curve method, are employed. This study investigates the effects of Walker curve root modification on the thermoelastohydrodynamic (TEHD) grease lubrication characteristics of large internal gears. We focus on parameterizing the modified tooth profile, establishing a film thickness calculation model, and numerically analyzing the lubrication state under various modification amounts and grease viscosities. Our findings provide insights for internal gear manufacturers aiming to optimize gear performance in demanding environments.
The engagement of internal gears involves complex contact conditions where the pinion and ring gear teeth interact along a dynamically changing path. Standard involute profiles, while efficient, are prone to deformation under load, leading to concentrated stresses and poor lubrication. For internal gears, the pinion’s root region, from the base circle to the pitch circle, is particularly susceptible to interference. We selected this region for modification based on meshing laws and engineering experience. The Walker curve, defined by a power-law function, allows for a smooth transition in the tooth profile, reducing the risk of sudden load changes. The modified profile is described parametrically to facilitate numerical analysis of the lubrication state.
The mathematical model for the standard involute tooth profile is given by:
$$x = r_b (\cos \phi + \phi \sin \phi)$$
$$y = r_b (\sin \phi – \phi \cos \phi)$$
where $r_b$ is the base radius, and $\phi$ is the roll angle. The base radius $r_b$ and tip radius $r_a$ are calculated as:
$$r_b = \frac{m z \cos \alpha}{2}$$
$$r_a = \frac{m (z + 2h_a)}{2}$$
Here, $m$ is the module, $z$ is the number of teeth, $\alpha$ is the pressure angle, and $h_a$ is the addendum coefficient. The roll angle $\phi$ ranges from 0 to $\phi_{\text{max}}$, which is derived from the gear geometry. For modification, we apply the Walker curve to the pinion’s root region, with the modification amount $\delta(x)$ defined as:
$$\delta(x) = \delta_{\text{max}} \left( \frac{x}{l} \right)^e$$
where $\delta_{\text{max}}$ is the maximum modification amount, $l$ is the modification length, and $e$ is the exponent parameter (set to 1.5 for Walker curve). The modified coordinates $(x’, y’)$ are then:
$$x’ = x – \delta(x) \sin \alpha_A$$
$$y’ = y + \delta(x) \cos \alpha_A$$
where $\alpha_A$ is the pressure angle at point A on the tooth profile. Substituting the involute equations, we obtain the parametric form of the modified profile:
$$x’ = r_b (\cos \phi + \phi \sin \phi) – \delta_{\text{max}} \sin \phi \left(1 – \frac{r_b \phi_{\text{max}}}{l} + \frac{r_b \phi}{l}\right)^e$$
$$y’ = r_b (\sin \phi – \phi \cos \phi) + \delta_{\text{max}} \cos \phi \left(1 – \frac{r_b \phi_{\text{max}}}{l} + \frac{r_b \phi}{l}\right)^e$$
This parameterization allows us to model the tooth profile accurately for subsequent lubrication analysis. The modification length $l$ is set to 26.157 mm, covering the region from the pinion’s base circle to the pitch circle. We analyze three modification amounts: $\delta_{\text{max}} = 20\,\mu\text{m}$, $40\,\mu\text{m}$, and $60\,\mu\text{m}$, to evaluate their impact on lubrication.
To assess the lubrication state, we employ a loaded tooth contact analysis (LTCA) to determine key parameters such as curvature radii, contact stress, and entrainment velocity at various meshing points. The instantaneous curvature radii for the pinion and ring gear at any meshing point $K$ are:
$$R_p(K) = r_{bp} \tan \alpha_{rp} – s(t)$$
$$R_r(K) = r_{br} \tan \alpha_{rp} – s(t)$$
where $r_{bp}$ and $r_{br}$ are the base radii of the pinion and ring gear, respectively, $\alpha_{rp}$ is the operating pressure angle, and $s(t)$ is the distance from the pitch point. The sliding velocities are:
$$U_p(t) = \omega_p R_p(t)$$
$$U_r(t) = \omega_r R_r(t)$$
with $\omega_p$ and $\omega_r$ being the angular velocities. The entrainment velocity $U_e$ is then:
$$U_e(t) = \frac{U_p(t) + U_r(t)}{2}$$
The Hertzian contact stress $\sigma_H$ is calculated as:
$$\sigma_H = \sqrt{\frac{F_n \left( \frac{1}{R_p} + \frac{1}{R_r} \right)}{\pi l \left( \frac{1 – \mu_1^2}{E_1} + \frac{1 – \mu_2^2}{E_2} \right)}}$$
where $F_n$ is the normal load, $l$ is the contact length, $E_1$ and $E_2$ are the elastic moduli, and $\mu_1$ and $\mu_2$ are the Poisson’s ratios of the pinion and ring gear materials, respectively.
For grease lubrication, we use the Ostwald-de Waele model to describe the non-Newtonian behavior of grease. The generalized Reynolds equation for thermal elastohydrodynamic lubrication (TEHD) is:
$$\frac{n}{2n+1} \left( \frac{1}{2} \right)^{\frac{n+1}{n}} \left\{ \frac{\partial}{\partial x} \left[ \rho h^{\frac{2n+1}{n}} \left( \frac{1}{\phi} \frac{\partial p}{\partial x} \right)^{\frac{1}{n}} \right] + \frac{\partial}{\partial y} \left[ \rho h^{\frac{2n+1}{n}} \left( \frac{1}{\phi} \frac{\partial p}{\partial y} \right)^{\frac{1}{n}} \right] \right\} = u_s \frac{\partial (\rho h)}{\partial x} + \frac{\partial (\rho h)}{\partial t}$$
where $n$ is the flow behavior index (taken as 0.8), $\rho$ is the density, $\phi$ is the viscosity, $h$ is the film thickness, and $u_s$ is the sliding velocity. The film thickness equation accounts for elastic deformation:
$$h(x) = h_0 + \frac{x^2}{2R_p} + \frac{y^2}{2R_r} – \frac{2}{\pi E’} \iint_\Omega \frac{p(s,t)}{\sqrt{(x-s)^2 + (y-t)^2}} \, ds \, dt$$
Here, $h_0$ is the central film thickness, and $E’$ is the effective elastic modulus. The pressure-viscosity and pressure-density relationships are given by the Dowson-Higginson equations:
$$\phi = \phi_0 \exp \left\{ (\ln \phi_0 + 9.67) \left[ (1 + 5.1 \times 10^{-9} p)^{0.68} \left( \frac{T – 138}{T_0 – 138} \right)^{-0.042} – 1 \right] \right\}$$
$$\rho = \rho_0 \left[ \frac{1 + 0.6 \times 10^{-9} p}{1 + 1.7 \times 10^{-9} p} – 0.00065 (T – T_0) \right]$$
where $\phi_0$ and $\rho_0$ are the initial viscosity and density, and $T_0$ is the initial temperature. The energy equation accounts for heat generation due to shear and compression:
$$\rho c u \frac{\partial T}{\partial x} = k \frac{\partial^2 T}{\partial z^2} – \frac{T}{\rho} \frac{\partial \rho}{\partial T} u \frac{\partial p}{\partial x} + \phi \left( \frac{\partial u}{\partial z} \right)^2$$
with boundary conditions for temperature on the tooth surfaces. The load balance equation ensures that the integral of pressure over the contact area equals the applied load:
$$\iint_\Omega p(x,y) \, dx \, dy = W$$
We solve these equations numerically using a finite difference method with a grid size of 129 × 129 points. The convergence criterion for pressure is set to a relative error of $10^{-6}$. The analysis focuses on four sampling points along the path of contact: $l_1$ (meshing-in point), $l_2$, $l_3$, and $l_4$ (pitch point), to evaluate the lubrication state at critical positions.
The basic parameters of the internal gear pair are listed in Table 1. These parameters are typical for large internal gears used in heavy machinery, and they form the basis for our numerical simulations.
| Parameter | Value |
|---|---|
| Pinion teeth number, $z_1$ | 15 |
| Ring gear teeth number, $z_2$ | 105 |
| Module, $m$ (mm) | 22 |
| Pinion width, $b_1$ (mm) | 250 |
| Ring gear width, $b_2$ (mm) | 240 |
| Pressure angle, $\alpha$ (°) | 20 |
| Elastic modulus, $E’$ (GPa) | 211 |
| Poisson’s ratio, $\nu$ | 0.33 |
| Pinion speed, $n$ (r/min) | 1.8 |
| Pinion torque, $T$ (N·m) | 173,610 |
| Grease viscosity, $\phi_1$ (Pa·s) | 0.3 |
| Grease viscosity, $\phi_2$ (Pa·s) | 0.8 |
| Grease viscosity, $\phi_3$ (Pa·s) | 2.3 |
| Grease specific heat, $c$ (J/(kg·K)) | 2000 |
| Gear specific heat, $c_1$, $c_2$ (J/(kg·K)) | 470 |
| Grease thermal conductivity, $k$ (W/(m·K)) | 0.14 |
| Gear thermal conductivity, $k_1$, $k_2$ (W/(m·K)) | 47 |
| Gear material density, $\rho_1$, $\rho_2$ (kg/m³) | 7850 |
| Grease density, $\rho$ (kg/m³) | 890 |
The LTCA results for the unmodified and modified profiles are summarized in Table 2, showing the curvature radii, contact stress, and entrainment velocity at the sampling points. The modification significantly alters the contact conditions, especially near the root region.
| Sampling Point | Modification Amount ($\mu$m) | Pinion Curvature Radius (mm) | Contact Stress (MPa) | Entrainment Velocity (m/s) |
|---|---|---|---|---|
| $l_1$ | 0 | 45.2 | 2150 | 0.032 |
| 20 | 45.5 | 2180 | 0.033 | |
| 40 | 46.8 | 1950 | 0.034 | |
| 60 | 48.3 | 1850 | 0.035 | |
| $l_2$ | 0 | 50.1 | 1980 | 0.028 |
| 20 | 50.3 | 2000 | 0.029 | |
| 40 | 52.6 | 1750 | 0.030 | |
| 60 | 55.0 | 1600 | 0.031 | |
| $l_3$ | 0 | 55.5 | 1800 | 0.024 |
| 20 | 55.7 | 1820 | 0.025 | |
| 40 | 58.2 | 1650 | 0.026 | |
| 60 | 60.8 | 1500 | 0.027 | |
| $l_4$ | 0 | 60.0 | 1700 | 0.020 |
| 20 | 60.2 | 1720 | 0.021 | |
| 40 | 62.5 | 1550 | 0.022 | |
| 60 | 65.0 | 1400 | 0.023 |
The film thickness distributions at the sampling points are analyzed for different modification amounts and grease viscosities. At $l_1$, the meshing-in point, the film thickness for $\delta_{\text{max}} = 20\,\mu\text{m}$ and $40\,\mu\text{m}$ is similar to the unmodified case, but for $60\,\mu\text{m}$, the lubrication state deteriorates due to excessive curvature change. At $l_2$ and $l_3$, the film thickness improves with higher modification amounts, showing a nonlinear relationship. At $l_4$, the pitch point, the film thickness is generally the lowest, but modification enhances it, especially for $\delta_{\text{max}} = 60\,\mu\text{m}$.
The effect of grease viscosity on the lubrication state is critical. We tested three viscosities: $\phi_1 = 0.3\,\text{Pa·s}$, $\phi_2 = 0.8\,\text{Pa·s}$, and $\phi_3 = 2.3\,\text{Pa·s}$. Higher viscosities improve film thickness and reduce the negative impact of modification curve defects. For instance, at $l_4$, with $\phi = 2.3\,\text{Pa·s}$, the film thickness distribution becomes uniform across the Hertzian contact zone, eliminating the sharp drop at the outlet.
The maximum film pressures at the sampling points are summarized in Table 3. The pressure varies with modification amount and grease viscosity, reflecting the complex coupling between tooth geometry and grease rheology.
| Sampling Point | Modification Amount ($\mu$m) | $\phi = 0.3\,\text{Pa·s}$ | $\phi = 0.8\,\text{Pa·s}$ | $\phi = 2.3\,\text{Pa·s}$ |
|---|---|---|---|---|
| $l_1$ | 0 | 2100 | 2150 | 2200 |
| 20 | 2120 | 2170 | 2220 | |
| 40 | 1900 | 1950 | 2000 | |
| 60 | 1800 | 1850 | 1900 | |
| $l_2$ | 0 | 1950 | 2000 | 2050 |
| 20 | 1970 | 2020 | 2070 | |
| 40 | 1700 | 1750 | 1800 | |
| 60 | 1550 | 1600 | 1650 | |
| $l_3$ | 0 | 1750 | 1800 | 1850 |
| 20 | 1770 | 1820 | 1870 | |
| 40 | 1600 | 1650 | 1700 | |
| 60 | 1450 | 1500 | 1550 | |
| $l_4$ | 0 | 1650 | 1700 | 1750 |
| 20 | 1670 | 1720 | 1770 | |
| 40 | 1500 | 1550 | 1600 | |
| 60 | 1350 | 1400 | 1450 |
The temperature rise on the tooth surfaces is analyzed to assess thermal effects. The results show that grease viscosity has a minimal impact on temperature, with variations within 1°C. However, modification amounts influence temperature distribution, particularly near the root region. For $\delta_{\text{max}} = 60\,\mu\text{m}$, the temperature at $l_1$ increases by up to 11% compared to the unmodified case, indicating localized heating due to altered contact conditions.
In discussion, we emphasize that the nonlinear behavior of film thickness with modification amount is attributed to the interplay between tooth geometry and grease rheology. Unlike small gears, large internal gears exhibit significant changes in curvature radii, which amplify the effects of modification. The Walker curve, while effective, requires careful selection of $\delta_{\text{max}}$ to avoid lubrication degradation at the root. Internal gear manufacturers should consider these factors when designing modified profiles for heavy-duty applications. Additionally, higher grease viscosities can compensate for modification-induced irregularities, ensuring stable film formation across the contact zone.
In conclusion, our study demonstrates that Walker curve root modification significantly influences the TEHD grease lubrication of large internal gears. The modification amount and grease viscosity are key parameters that affect film thickness, pressure, and temperature in a nonlinear manner. For internal gear manufacturers, optimizing these parameters can enhance gear performance and durability. Future work should explore other modification curves and their interactions with different grease types to further improve lubrication in internal gears.