In mechanical transmission systems, internal gears play a critical role due to their compact design and high efficiency. As an internal gear manufacturer, we often focus on optimizing the lubrication performance to enhance durability and reduce energy losses. The thermo-elastohydrodynamic lubrication (TEHL) characteristics of internal gears significantly influence the overall system behavior, including friction, wear, and thermal stability. This analysis explores the TEHL properties of internal gears, considering various transmission types and modification coefficients, to provide insights for design improvements.
The geometry of internal gear meshing involves complex interactions between the planetary gear and the internal gear. The curvature radii at any meshing point can be expressed as:
$$ R_p(K) = r_{bp} \tan \alpha_{rp} – s(t) $$
$$ R_r(K) = r_{br} \tan \alpha_{rp} – s(t) $$
where \( r_{bp} \) and \( r_{br} \) are the base circle radii of the planetary and internal gears, respectively, \( \alpha_{rp} \) is the meshing angle, and \( s(t) \) is the distance from the pitch point to the instantaneous meshing point over time \( t \). The comprehensive curvature radius is given by:
$$ R(K) = \frac{R_p(K) R_r(K)}{R_r(K) – R_p(K)} $$
The tangential velocities of the gear surfaces are:
$$ U_p(t) = \omega_p R_p(t) $$
$$ U_r(t) = \omega_r R_r(t) $$
with \( \omega_p \) and \( \omega_r \) as the angular velocities. The entrainment velocity is:
$$ U(t) = \frac{U_p(t) + U_r(t)}{2} $$
For the TEHL model, the Reynolds equation for a line contact considering time-varying effects is:
$$ \frac{\partial}{\partial x} \left[ \left( \frac{\rho}{\eta} \right)_e h^3 \frac{\partial p}{\partial x} \right] = 12 \frac{\partial (\rho^* U h)}{\partial x} + 12 \frac{\partial (\rho_e h)}{\partial t} $$
where \( h \) is the film thickness, \( p \) is the pressure, \( \rho \) is the density, and \( \eta \) is the viscosity. The film thickness equation accounting for elastic deformation is:
$$ h = h_0 + \frac{x^2}{2R(t)} – \frac{2}{\pi E} \int_{-\infty}^{x} p(s, t) \ln (x – s)^2 ds $$
Here, \( h_0 \) is the rigid central film thickness, \( R(t) \) is the comprehensive curvature radius, and \( E \) is the composite elastic modulus. The Ree-Eyring fluid model describes the shear stress:
$$ \frac{\partial u}{\partial z} = \frac{\tau_0}{\eta} \sinh \left( \frac{\tau}{\tau_0} \right) $$
with the equivalent viscosity defined as:
$$ \eta^* = \eta \frac{\tau}{\tau_0} / \sinh \left( \frac{\tau}{\tau_0} \right) $$
The viscosity-pressure-temperature relationship is:
$$ \eta = \eta_0 \exp \left\{ (\ln \eta_0 + 9.67) \left[ (1 + 5.1 \times 10^{-9} p)^{Z_0} \left( \frac{T – 138}{T_0 – 138} \right)^{-S_0} – 1 \right] \right\} $$
and the density equation is:
$$ \rho = \rho_0 \left[ 1 + \frac{0.6 \times 10^{-9} p}{1 + 1.7 \times 10^{-9} p} – 0.00065 (T – T_0) \right] $$
The load balance equation is:
$$ \int_{-\infty}^{x} p(x) dx = w $$
where \( w \) is the load per unit width. The oil film stiffness is derived from the central film thickness \( h_c \) as:
$$ k_h = \frac{\partial w}{\partial h_c} $$
To solve these equations, dimensionless forms are discretized using multi-grid methods, with grid layers ranging from 31 to 961 nodes, ensuring convergence with relative errors below 0.001.
The geometric model of internal gear meshing illustrates the engagement between the planetary gear and the internal gear, highlighting critical points such as the pitch point and meshing boundaries. This visualization aids in understanding the curvature variations and their impact on lubrication.
In our analysis, we consider different gear transmission types: standard, positive transmission, negative transmission, and equal modification transmission. The modification coefficient sums for the internal gear and planetary gear are defined as \( x_{\Sigma} = x_3 – x_2 \), where \( x_3 \) and \( x_2 \) are the modification coefficients of the internal gear and planetary gear, respectively. The meshing angle \( \alpha_{rp} \) is derived from the non-backlash meshing equation:
$$ x_{\Sigma} = \frac{z_3 – z_2}{2 \tan \alpha} (\text{inv} \alpha_{rp} – \text{inv} \alpha) $$
where \( \alpha = 20^\circ \) is the standard pressure angle, and \( z_3 \) and \( z_2 \) are the tooth numbers. The following table summarizes the parameters for different transmission types:
| Transmission Type | Modification Sum \( x_{\Sigma} \) | Meshing Angle \( \alpha_{rp} \) (°) | Center Distance (mm) |
|---|---|---|---|
| Standard | 0 | 20 | 102 |
| Positive | 0.2545 | 20.754 | 102.5 |
| Negative | -0.2 | 19.37 | 101.6 |
| Equal Modification | 0 | 20 | 102 |
The load spectrum along the meshing line shows variations due to single and double tooth contact zones, affecting the lubrication conditions. For internal gears, the entrainment velocity and slide-to-roll ratio change during meshing, influencing the film thickness and temperature rise. The comprehensive curvature radius and entrainment velocity are higher in positive transmission, leading to improved lubrication.
The central film thickness \( h_c \) and minimum film thickness \( h_{\text{min}} \) are critical parameters. In positive transmission, the film thickness is maximized due to increased entrainment velocity, while negative transmission results in thinner films. The friction coefficient \( \mu \) and maximum temperature rise \( \Delta T_{\text{max}} \) are minimized in positive transmission, reducing the risk of thermal scuffing. The oil film stiffness \( k_h \) is lower in positive transmission because of the thicker film, which is easier to compress.
For internal gear manufacturers, optimizing the modification coefficients is essential. Increasing the modification sum \( x_{\Sigma} \) in positive transmission further enhances the film thickness and reduces friction, but it decreases the oil film stiffness, potentially affecting dynamic stability. The following table presents results for different modification sums in positive transmission:
| Modification Sum \( x_{\Sigma} \) | Central Film Thickness (μm) | Friction Coefficient | Max Temperature Rise (K) | Oil Film Stiffness (GN/m²) |
|---|---|---|---|---|
| 0.0502 | 0.45 | 0.032 | 45 | 1.8 |
| 0.1516 | 0.52 | 0.028 | 42 | 1.6 |
| 0.2545 | 0.58 | 0.025 | 38 | 1.4 |
The relationship between modification sum and comprehensive curvature radius is expressed as:
$$ R(K) \propto x_{\Sigma} $$
which directly affects the entrainment velocity and film thickness. The central film pressure decreases with increasing film thickness, as shown by the Reynolds equation. The oil film stiffness derivation relies on the load-film thickness relationship, and for internal gears, it is crucial to balance lubrication improvement with stiffness requirements.
In conclusion, positive transmission in internal gears offers the best lubrication performance, with thicker films, lower friction, and reduced temperature rise. Increasing the modification sum in positive transmission further improves these characteristics but compromises oil film stiffness. Internal gear manufacturers should consider these factors in design to enhance thermal scuffing resistance and overall efficiency. Future work could explore dynamic loads and surface roughness effects on TEHL behavior.