In the field of precision mechanical transmissions, internal gear drives play a critical role in applications requiring high torque density and compact design, such as planetary gear systems. As an internal gear manufacturer, we often encounter challenges related to thermal deformation and efficiency losses due to frictional heating during operation. This study focuses on analyzing the steady-state temperature field of meshing surfaces in internal gears, which is essential for minimizing thermal errors and enhancing the durability of gear systems. Internal gears are commonly used in reducers like the 2K-V planetary transmission, where heat generation from sliding friction can significantly impact performance. Understanding the temperature distribution helps internal gear manufacturers optimize designs and select appropriate materials and lubrication strategies.
The theoretical foundation for this analysis is based on gear meshing principles and heat transfer theory. The steady-state heat conduction equation for a solid structure, derived from Fourier’s law and energy conservation, is given by:
$$ \rho c \frac{\partial T}{\partial t} = \frac{\partial}{\partial x} \left( k \frac{\partial T}{\partial x} \right) + \frac{\partial}{\partial y} \left( k \frac{\partial T}{\partial y} \right) + \frac{\partial}{\partial z} \left( k \frac{\partial T}{\partial z} \right) + \dot{Q} $$
where $\rho$ is the density of the gear material, $c$ is the specific heat capacity, $k$ is the thermal conductivity, $t$ is time, $T$ is temperature, and $\dot{Q}$ is the heat generated per unit volume per unit time. For steady-state conditions, the temperature does not change with time, so $\frac{\partial T}{\partial t} = 0$. This simplifies the equation for the gear tooth working surface to:
$$ -k \frac{\partial T}{\partial n} = h_k (T – T_0) + Q $$
where $\frac{\partial T}{\partial n}$ is the temperature gradient along the outward normal direction of the heat transfer surface, $h_k$ is the heat transfer coefficient between the tooth surface and the surrounding lubricant, $T_0$ is the lubricant temperature, and $Q$ is the boundary heat flux density. Similar equations apply to non-working surfaces, tooth ends, tip, and root surfaces, with respective heat transfer coefficients $h_k’$, $h_n$, $h_d$, and $h_g$. These coefficients are crucial for accurate thermal analysis and are derived based on the gear’s operating conditions and lubricant properties.
Frictional heat flux on the meshing surfaces arises from relative sliding between internal and external gears during engagement. The heat flux $q$ at any meshing point depends on the average contact stress $\sigma_k$, relative sliding velocity $v_k$, friction coefficient $f$, and thermal energy conversion coefficient $\gamma$ (typically 0.9 to 0.95), expressed as:
$$ q = \sigma_k v_k f \gamma $$
The friction coefficient $f$ varies with lubrication regimes, such as elastohydrodynamic, mixed, or boundary lubrication. For mixed lubrication, which is common in gear operations, the friction coefficient can be calculated using:
$$ f = 0.012 \times \frac{1.13}{1.13 – S_{av}} \lg \left( \frac{29,700 F_N / b}{\eta v_s v_e^2} \right) $$
where $S_{av}$ is the average surface roughness, $\eta$ is the dynamic viscosity, $v_s = |v_{K1t} – v_{K2t}|$ is the relative sliding velocity, and $v_e = v_{K1t} + v_{K2t}$ is the entrainment velocity. Here, $v_{K1t}$ and $v_{K2t}$ represent the tangential velocities of the external and internal gears, respectively. The relative sliding velocity $v_k$ at different meshing positions along the line of action is given by:
$$ v_k = \overline{PK} (\omega_{H5} – \omega_{H4}) $$
where $\overline{PK}$ is the distance from the meshing point $K$ to the pitch point $P$, and $\omega_{H4}$ and $\omega_{H5}$ are the angular velocities of the external and internal gears in the converted gear train, respectively. These angular velocities are derived from the gear train kinematics; for instance, in a 2K-V planetary system, they can be expressed as:
$$ \omega_{H4} = \frac{2Z_1 (1 + Z_4)}{2Z_2 – Z_2 (2 + Z_4)} \omega_1 $$
$$ \omega_{H5} = \frac{Z_1 Z_4}{2Z_2 – Z_2 (2 + Z_4)} \omega_1 $$
where $Z_1$, $Z_2$, and $Z_4$ are the tooth numbers of the sun gear, planet gear, and external gear, respectively, and $\omega_1$ is the input angular velocity. The variation of relative sliding velocity along the meshing path shows higher values near the root and tip of the tooth, with zero sliding at the pitch point. This non-uniform distribution directly influences the frictional heat generation.
The average contact stress $\sigma_k$ at the meshing point is calculated using Hertzian contact theory, where the gears are modeled as equivalent cylinders with radii equal to the equivalent curvature radii at the contact point. The normal load $F_n$ on the tooth surface is:
$$ F_n = \frac{K T}{r_k \cos \alpha_K} $$
where $T$ is the input torque on the external gear, $r_k$ is the distance from the meshing point to the gear center, $\alpha_K$ is the pressure angle at the contact point, and $K$ is the load distribution coefficient (taken as 0.5). The average contact pressure is then:
$$ \sigma_k = \frac{1}{4} \sqrt{ \frac{\pi F_n}{b} \left( \frac{1}{R_1} – \frac{1}{R_2} \right) \left( \frac{1 – \mu_1^2}{E_1} + \frac{1 – \mu_2^2}{E_2} \right) } $$
where $R_1$ and $R_2$ are the equivalent curvature radii of the external and internal gears, $b$ is the tooth width, and $\mu_1$, $\mu_2$, $E_1$, $E_2$ are the Poisson’s ratios and elastic moduli of the gears, respectively. The semi-width of the contact area $L$ is given by:
$$ L = \sqrt{ \frac{4 F_n}{\pi b} \left( \frac{1 – \mu_1^2}{E_1} + \frac{1 – \mu_2^2}{E_2} \right) \left( \frac{1}{R_1} – \frac{1}{R_2} \right) } $$
To account for the distribution of frictional heat between the internal and external gears, a heat partition coefficient $\beta$ is introduced:
$$ \beta = \frac{\lambda_1 \rho_1 c_1 v_{K1t}}{\lambda_1 \rho_1 c_1 v_{K1t} + \lambda_2 \rho_2 c_2 v_{K2t}} $$
where $\lambda$, $\rho$, and $c$ are the thermal conductivity, density, and specific heat capacity of the gear materials, respectively. The heat fluxes on the external and internal gears are then:
$$ q_1 = q \beta $$
$$ q_2 = q (1 – \beta) $$
Since the actual meshing time $t_1$ is only a fraction of the total meshing cycle time $t$, the average heat fluxes over one cycle are calculated as:
$$ \bar{q}_1 = \frac{Z_4 L \omega_2}{\pi v_{K1t}} q_1 $$
$$ \bar{q}_2 = \frac{Z_4 L \omega_2}{\pi v_{K2t}} q_2 $$
where $\omega_2$ is the angular velocity of the planet gear. These average values are used in the steady-state analysis to represent the continuous heat input. For internal gears, the heat flux distribution typically shows higher values near the tooth root due to increased sliding and contact stress.
The convective heat transfer coefficients for different gear surfaces are determined based on the gear geometry and operating conditions. For the tooth ends, the flow is often laminar, and the heat transfer coefficient $h_n$ is given by:
$$ h_n = \frac{0.6 \lambda_f \Pr}{(0.56 + 0.26 \Pr^{1/2} + \Pr)^{2/3}} \left( \frac{\omega_{H4}}{\nu_f} \right) $$
where $\lambda_f$ is the thermal conductivity of the lubricant, $\Pr = \frac{\rho \nu_f c}{\lambda_f}$ is the Prandtl number, and $\nu_f$ is the kinematic viscosity. For the tooth flanks (working and non-working surfaces), the heat transfer coefficient $h_k$ is:
$$ h_k = \frac{0.228 \text{Re}^{0.731} \Pr^{1/3} \lambda_f}{d} $$
where Re is the Reynolds number based on the pitch diameter $d$. For the tooth root and tip surfaces, which can be approximated as slender plates, the heat transfer coefficient $h_g$ is:
$$ h_g = 0.664 \lambda_f \Pr^{1/3} \left( \frac{\omega_{H4}}{\nu_f} \right)^{1/2} $$
These coefficients are essential for modeling the heat dissipation to the surrounding lubricant. For example, using lubricant SCH632 with properties such as density $\rho = 870 \, \text{kg/m}^3$, kinematic viscosity $\nu_f = 92.5 \times 10^{-6} \, \text{m}^2/\text{s}$, specific heat capacity $c = 2000 \, \text{J/(kg·K)}$, and thermal conductivity $\lambda_f = 0.14 \, \text{W/(m·K)}$ at 70°C, the convective coefficients can be computed accurately.
In the finite element analysis, a single-tooth model of the external gear is created and imported into ANSYS Workbench. The meshing surface is divided into multiple strip regions along the tooth width to apply the non-uniform friction heat flux. Each strip region is assigned an average heat flux value based on the local meshing position, allowing for a precise representation of the thermal load. The number of strip regions influences the accuracy of the temperature field; as the number increases, the solution converges. For instance, when the number exceeds 50, the temperature field stabilizes, indicating that further refinement has minimal impact.
The steady-state temperature distribution reveals that the highest temperatures occur near the tooth root and along the center of the tooth width, where heat generation is maximal and cooling is less effective due to reduced convection. The temperature field is symmetric about the tooth width center, with lower temperatures at the ends due to better heat dissipation. This pattern is consistent across different paths along the involute profile, as shown in the analysis of three distinct paths on the tooth surface. The results emphasize the importance of adequate cooling and material selection for internal gears to prevent thermal damage.
The following table summarizes key parameters used in the analysis for a typical internal gear drive:
| Parameter | External Gear | Internal Gear |
|---|---|---|
| Module (mm) | 2 | 2 |
| Number of Teeth | 64 | 66 |
| Pressure Angle (°) | 20 | 20 |
| Tooth Width (mm) | 25 | 25 |
| Input Torque (N·m) | 500 | — |
| Input Angular Velocity (rad/s) | 300 | — |
Another table illustrates the variation of relative sliding velocity and heat flux at different meshing positions:
| Meshing Position | Relative Sliding Velocity (m/s) | Heat Flux (W/m²) |
|---|---|---|
| Tooth Root | High | High |
| Pitch Point | 0 | Low |
| Tooth Tip | High | High |
The analysis demonstrates that for internal gear manufacturers, optimizing the tooth profile and cooling system can mitigate thermal issues. Internal gears, due to their enclosed design, may experience different heat dissipation patterns compared to external gears, necessitating tailored approaches. The use of advanced materials with high thermal conductivity and efficient lubricants can further enhance performance. In planetary systems, the interaction between multiple gears amplifies the need for comprehensive thermal management.
In conclusion, the steady-state temperature field analysis of internal gear meshing surfaces provides valuable insights for design and optimization. By accurately modeling frictional heat generation and convective cooling, internal gear manufacturers can develop more reliable and efficient transmission systems. The finite element method, combined with precise boundary conditions, offers a robust tool for predicting thermal behavior. Future work could explore dynamic thermal effects or the impact of different lubricants on internal gears under varying operating conditions.