In modern engineering applications, planetary gear systems are widely used due to their compact structure and high load-carrying capacity. However, excessive tooth surface temperatures can lead to lubricant failure and system malfunctions, making the study of dynamic temperature fields crucial. In this paper, I present a comprehensive numerical analysis and experimental investigation of the dynamic temperature field in planetary gear transmission systems, focusing on the effects of gear and bearing friction heat generation and system heat dissipation. By integrating dynamic load analysis with heat transfer principles, I develop a model that accurately predicts temperature distributions under various operating conditions.
The planetary gear system consists of a sun gear, multiple planet gears, and a ring gear, all interacting through meshing actions. The dynamic behavior of these components is complex due to time-varying stiffness and external excitations. To analyze this, I employ a lumped parameter method to establish a translational-torsional coupled multi-degree-of-freedom dynamic model. This model accounts for the interactions between gears and bearings, allowing for the extraction of dynamic loads. The equations of motion are derived as follows:
$$ \mathbf{F}(t) = \mathbf{M} \ddot{\mathbf{x}}(t) + \mathbf{C} \dot{\mathbf{x}}(t) + \mathbf{K} \mathbf{x}(t) $$
where $\mathbf{M}$ is the mass matrix, $\mathbf{C}$ is the damping matrix, $\mathbf{K}$ is the stiffness matrix, and $\mathbf{x}(t)$ represents the displacement vector. The dynamic loads obtained from this model serve as inputs for the thermal analysis.
The heat generation in planetary gear systems primarily arises from friction at gear meshing interfaces and bearing interactions. For gear pairs, the friction heat flux density $q_m(t)$ is calculated based on the contact pressure, sliding velocity, and friction coefficient. The formula is given by:
$$ q_m(t) = \gamma f_m(t) \sigma_m(t) V_s(t) $$
Here, $\gamma$ is the thermal conversion coefficient, $f_m(t)$ is the friction coefficient, $\sigma_m(t)$ is the contact pressure, and $V_s(t)$ is the relative sliding velocity. The contact pressure is derived from the dynamic load $F_K(t)$ and the equivalent radius $R_m(t)$:
$$ \sigma_m(t) = \frac{F_K(t)}{Z_e R_m(t)} $$
The friction coefficient $f_m(t)$ depends on factors like surface roughness and lubrication conditions, expressed as:
$$ f_m(t) = \xi \left[ \frac{F_K(t) R_m(t)}{b (v_{m1}(t) + v_{m2}(t))} \right]^{0.2} $$
where $\xi$ is the lubricant viscosity, $b$ is the gear width, and $v_{m1}(t)$, $v_{m2}(t)$ are the sliding velocities. The heat distribution between meshing gears is determined by the thermal properties, leading to partitioned heat fluxes $q_1(t)$ and $q_2(t)$ for the driving and driven gears, respectively.
For bearings, the friction heat generation is modeled using empirical formulas. The friction torque $M(t)$ is calculated as:
$$ M(t) = 10^{-7} f_0 (v n)^{0.6} D_m^{1.3} + f_1 F_r(t) D_m $$
where $f_0$ and $f_1$ are coefficients based on lubrication and load type, $v$ is the kinematic viscosity, $n$ is the rotational speed, $D_m$ is the mean diameter, and $F_r(t)$ is the dynamic load. The heat generation rate $Q_e(t)$ and heat flux density $q_e(t)$ are then:
$$ Q_e(t) = 1.047 \times 10^{-4} M(t) n $$
$$ q_e(t) = \frac{Q_e(t)}{A} $$
with $A$ being the heat dissipation area. The dynamic loads and heat fluxes exhibit periodic variations, as shown in comparative time-domain analyses.
Heat dissipation in the system involves conduction and convection. The convective heat transfer coefficients vary across different surfaces. For gear end faces, the coefficient $h_d(t)$ is derived from Nusselt number correlations:
$$ h_d(t) = \frac{\lambda_o}{r} \text{Nu}_d(t) $$
$$ \text{Nu}_d(t) = 0.5 \text{Re}_o(t)^{0.5} \text{Pr}_o^{0.66} $$
where $\lambda_o$ is the thermal conductivity of the lubricant, $r$ is the pitch radius, $\text{Re}_o(t)$ is the Reynolds number, and $\text{Pr}_o$ is the Prandtl number. For gear meshing surfaces, a different correlation is used:
$$ h_m(t) = \frac{0.0863 \lambda_o \text{Re}_o(t)^{0.618} \text{Pr}_o^{0.35}}{d} $$
where $d$ is the pitch diameter. Shaft surfaces employ external flow correlations, with the Nusselt number $\text{Nu}_r(t)$ given by:
$$ \text{Nu}_r(t) = 0.3 + \frac{0.62 \text{Re}_r(t)^{0.5} \text{Pr}_o^{0.33}}{[1 + (0.4/\text{Pr}_o)^{0.67}]^{0.25}} \left[1 + \left(\frac{\text{Re}_r(t)}{282000}\right)^{0.625}\right]^{0.8} $$
These coefficients are crucial for setting boundary conditions in the temperature field simulation.
To validate the numerical model, I conduct experiments using thermocouples and thermal imaging. The thermocouples measure temperature at fixed points on the ring gear, while thermal imaging captures the overall temperature distribution. The experimental setup includes a planetary gear test rig driven by an electric motor, with a magnetic brake applying loads. Thermal insulation is used to minimize external heat influences. The results show that the numerical predictions align well with experimental data, with a maximum error of 4.57%. The temperature evolution over time follows a rapid rise followed by stabilization, consistent with heat transfer theory.
The dynamic temperature field analysis reveals significant insights into the behavior of planetary gear systems. Under varying rotational speeds, the temperature response increases logarithmically. For instance, at a constant load of 50 N·m, increasing the speed from 100 to 500 r/min raises the temperature, but the rate of increase decelerates. The table below summarizes the temperature rises for key components at different speeds:
| Component | Speed (r/min) | Temperature Rise (°C) |
|---|---|---|
| Sun Gear | 100 | 45.0 |
| Sun Gear | 300 | 55.0 |
| Sun Gear | 500 | 70.0 |
| Planet Gears | 100 | 42.0 |
| Planet Gears | 300 | 50.0 |
| Planet Gears | 500 | 65.0 |
Load variations also impact the temperature field linearly. At fixed speeds of 100, 150, and 200 r/min, increasing the load from 50 to 150 N·m results in proportional temperature increases. The inclusion of bearing friction heat becomes more pronounced at higher loads, affecting the planet gears’ tooth surface temperatures. The following table illustrates this effect:
| Load (N·m) | Speed (r/min) | Sun Gear Temperature (°C) | Planet Gears Temperature (°C) |
|---|---|---|---|
| 50 | 100 | 40.0 | 45.0 |
| 100 | 100 | 60.0 | 65.0 |
| 150 | 100 | 80.0 | 85.0 |
Spatially, the temperature distribution shows that external meshing interfaces exhibit higher temperatures than internal ones, due to greater sliding velocities and heat generation. The sun gear, with its smaller size and higher heat load, reaches the highest temperatures, while the ring gear remains cooler owing to its larger dissipation area. In radial and axial directions, temperatures decrease from the heat sources outward, with a low-high-low pattern along the tooth width.
In conclusion, the developed numerical model effectively captures the dynamic temperature field in planetary gear systems, accounting for both gear and bearing friction heat. The experiments validate the approach, showing that bearing heat generation significantly influences tooth temperatures under high loads. This study provides a foundation for optimizing lubrication and cooling strategies in planetary gear applications, enhancing reliability and performance. Future work could explore more complex lubrication models and transient operational conditions.