In electric drive systems, helical gears are widely used due to their superior transmission performance and high load-bearing capacity. However, during the meshing process of helical gears, significant heat is generated from sliding friction in both axial and radial directions, especially under high-speed and heavy-load conditions. This heat can lead to gear scuffing failure and thermal deformation, which alters the involute tooth profile and affects transmission characteristics, causing vibration and noise. Therefore, studying the steady-state temperature field and thermal deformation field of electric drive helical gears is crucial for improving their reliability and performance. In this article, I will explore the theoretical foundations, computational methods, and finite element analysis of these fields, with a focus on how different gear models impact the results.
The meshing principle of helical gears involves complex interactions where multiple teeth are in contact simultaneously, leading to distributed loads and friction. Based on heat transfer and tribology theories, I derive the average friction heat flux density on the meshing surface and the convective heat transfer coefficients for other surfaces. The heat conduction differential equation for the steady-state temperature field is derived from energy conservation and Fourier’s law of heat conduction, as shown below:
$$ \lambda \left( \frac{\partial^2 t}{\partial x^2} + \frac{\partial^2 t}{\partial y^2} + \frac{\partial^2 t}{\partial z^2} \right) = 0 $$
where \( \lambda \) is the thermal conductivity and \( t \) is the temperature. The boundary conditions include the meshing surface, which combines the second and third types, and non-meshing surfaces like the tooth tip, end faces, and root fillets, which follow the third type. For the meshing surface, the boundary condition is:
$$ -\lambda \frac{\partial t}{\partial n} = s_1 (t_c – t_o) – q_w $$
where \( n \) is the normal vector, \( s_1 \) is the convective heat transfer coefficient for the meshing surface, \( t_c \) and \( t_o \) are the initial gear surface and lubricant temperatures, respectively, and \( q_w \) is the average friction heat flux density. For other surfaces, it simplifies to:
$$ -\lambda \frac{\partial t}{\partial n} = s_2 (t_c – t_o) $$
To compute the average friction heat flux density, I use the following formula, which accounts for the friction coefficient, Hertzian contact pressure, and sliding velocities:
$$ q_j = \frac{\pi k_f \gamma P_M f \tau_0 |v_1 – v_2|}{2T_1} $$
Here, \( k_f \) is the heat flux distribution coefficient between teeth, \( f \) is the friction coefficient, \( \gamma \) is the heat generation coefficient (taken as 0.95), \( P_M \) is the maximum Hertzian contact pressure, \( \tau_0 \) is the temporal contact half-width, \( v_1 \) and \( v_2 \) are the tangential velocities at the meshing point for the driving and driven gears, and \( T_1 \) is the meshing period of the driving gear. The tangential velocities are calculated as:
$$ v_1 = \frac{R_1 \pi n_1}{30}, \quad v_2 = \frac{R_2 \pi n_2}{30} $$
where \( R_1 \) and \( R_2 \) are the radii of the driving and driven gears, and \( n_1 \) and \( n_2 \) are their rotational speeds. The maximum Hertzian pressure and contact half-width are derived from:
$$ P_M = \sqrt{\frac{w E}{2 \pi R}}, \quad \tau_0 = \frac{b_0}{v_1}, \quad b_0 = \sqrt{\frac{8 w R}{\pi E}} $$
with \( w \) as the unit line load, \( E \) as the equivalent elastic modulus, and \( R \) as the comprehensive curvature radius. The unit line load for helical gears is given by:
$$ w = \frac{9549 P \eta K}{L r_{b1} n_1 \cos \beta_b} $$
where \( P \) is the input power, \( \eta \) is the transmission efficiency, \( K \) is the load factor, \( L \) is the contact line length, \( r_{b1} \) is the base circle radius of the driving gear, and \( \beta_b \) is the base helix angle.
For convective heat transfer coefficients, different surfaces require distinct calculations. The tooth tip coefficient is:
$$ h_d = 0.664 \lambda_o P_o^{0.333} \left( \frac{\omega}{\nu_o} \right)^{0.5} $$
where \( \lambda_o \) is the lubricant thermal conductivity, \( P_o \) is the Prandtl number of the lubricant, \( \omega \) is the angular velocity, and \( \nu_o \) is the kinematic viscosity. The tooth face and root fillet coefficient is:
$$ h_a = 0.228 R_e^{0.731} P_o^{0.333} \frac{\lambda_o}{L_d} $$
with \( R_e \) as the Reynolds number, calculated as \( R_e = \omega r_{nk} / \nu_o \), where \( r_{nk} \) is the radius at any meshing point. The end face coefficient depends on the flow regime:
$$ h_t = \begin{cases}
0.308 \lambda_{mix} (m_z + 2)^{0.5} P_{mix}^{0.5} \left( \frac{\omega}{\nu_{mix}} \right)^{0.5}, & \text{for } Re \leq 2 \times 10^5 \\
10^{-19} \lambda_{mix} \left( \frac{\omega}{\nu_{mix}} \right)^4 r_{nk}^7, & \text{for } 2 \times 10^5 \leq Re \leq 2.5 \times 10^5 \\
0.0197 \lambda_{mix} (m_z + 2.6)^{0.2} \left( \frac{\omega}{\nu_{mix}} \right)^{0.8} r_{nk}^{0.6}, & \text{for } Re \geq 2.5 \times 10^5
\end{cases} $$
Here, \( \lambda_{mix} \), \( \nu_{mix} \), and \( P_{mix} \) are properties of the oil-air mixture, derived from linear combinations based on the mixing ratio \( \alpha_{mix} \).
To facilitate the analysis, I introduce a dimensionless coordinate \( \Gamma \) to represent any point on the meshing line. For an external meshing gear, \( \Gamma \) is defined as the distance from the point to the pitch point divided by the length from the theoretical start to the pitch point. This helps in standardizing the position along the meshing path:
$$ \Gamma = \frac{\tan \alpha_{nk}}{\tan \alpha_{nc}} – 1 $$
where \( \alpha_{nk} \) is the pressure angle at the meshing point and \( \alpha_{nc} \) is the pitch circle pressure angle.
Using APDL language, I develop a parametric thermal analysis model for helical gears. The model includes a full-tooth representation to accurately capture the temperature and deformation fields. The mesh is generated with 8-node hexahedral elements (SOLID70) and mapped meshing techniques. Surface effect elements (SURF152) are applied to impose the friction heat flux on the meshing surfaces. The material properties and operating conditions are summarized in the tables below.
| Parameter | Value |
|---|---|
| Driving gear speed \( n_1 \) (rpm) | 2000 |
| Number of teeth \( z_1 / z_2 \) | 23 / 30 |
| Module \( m \) (mm) | 3 |
| Pressure angle \( \alpha \) (°) | 20 |
| Face width \( b \) (mm) | 20 |
| Helix angle \( \beta \) (°) | 8 |
| Input power \( P \) (kW) | 50 |
| Elastic modulus \( E \) (GPa) | 206 |
| Poisson’s ratio \( \upsilon \) | 0.3 |
| Specific heat capacity \( c \) (J·kg⁻¹·K⁻¹) | 465 |
| Thermal conductivity \( \lambda \) (W·m⁻¹·K⁻¹) | 46 |
| Thermal expansion coefficient \( k \) | 1.13 × 10⁵ |
| Density \( \rho \) (kg·m⁻³) | 7850 |
| Parameter | Value |
|---|---|
| Density \( \rho_o \) (kg·m⁻³) | 870 |
| Kinematic viscosity \( \nu_o \) (cSt) | 320 at 40°C, 38.5 at 100°C |
| Thermal conductivity \( \lambda_o \) (W·m⁻¹·K⁻¹) | 2000 |
| Specific heat capacity \( c_o \) (J·kg⁻¹·K⁻¹) | 0.14 |
From the finite element analysis, I obtain the steady-state temperature field of the driving helical gear. The results show a gradient distribution, with the highest temperatures on the meshing surface and the lowest at the hub. The maximum temperature reaches 92.4488°C. Two distinct high-temperature regions are observed on the meshing surface, located in the double-tooth meshing zones. Along the tooth height direction, the temperature distribution resembles an “M” shape. In the double-tooth meshing zone at the meshing-in end, the peak temperature is near the tooth root, while at the meshing-out end, it is closer to the tooth tip. Along the tooth width direction, the temperature is asymmetrically distributed. For a clockwise-rotating right-hand driving gear, the high-temperature zone at the meshing-in end is nearer to the front end face, and at the meshing-out end, it is closer to the rear end face. This asymmetry arises from the inclined contact lines and varying friction heat flux due to the helix angle.
Next, I analyze the thermal deformation field by applying the steady-state temperature data as a load in the structural analysis. The inner cylinder surface is constrained, and the thermal expansion coefficient is specified. The overall deformation shows that the tooth tip experiences the largest deformation, while the hub has the least. The maximum deformation occurs near the gear end faces. Projecting the deformations in the X and Y directions onto the meshing line direction gives the normal deformation at meshing points:
$$ \delta_K = \delta_{xK} \cos(\omega_K) + \delta_{yK} \sin(\omega_K) $$
where \( \delta_K \) is the deformation along the meshing line, \( \delta_{xK} \) and \( \delta_{yK} \) are deformations in the X and Y directions, and \( \omega_K \) is the load angle. The results indicate that the rear end face has greater deformation than the front end face, consistent with the temperature distribution along the tooth width.
To investigate the impact of different gear models, I compare full-tooth models with partial-tooth models (single-tooth, three-tooth, and five-tooth). The steady-state temperature fields from partial models are similar to the full model, with maximum temperatures of 92.4708°C, 92.6603°C, and 92.7150°C for single, three, and five-tooth models, respectively. The temperature distributions along the tooth height and width show minor deviations. For instance, at random points on the meshing surface, the relative errors for the single-tooth model are below 0.1%, making it a cost-effective alternative for temperature field analysis. However, for thermal deformation fields, only the full-tooth model provides accurate results. Partial models exhibit significant errors due to constraints on the hub sides; for example, the single-tooth model shows up to 19.19% error in deformation values compared to the full model.
| Node | Dimensionless Coordinate \( \Gamma \) | Full-Tooth Temp (°C) | Single-Tooth Temp (°C) | Three-Tooth Temp (°C) | Five-Tooth Temp (°C) |
|---|---|---|---|---|---|
| A | -0.43 | 91.48 | 91.55 | 91.78 | 91.85 |
| B | -0.10 | 90.49 | 90.53 | 90.76 | 90.83 |
| C | 0.24 | 91.26 | 91.28 | 91.51 | 91.57 |
| D | 0.51 | 92.32 | 92.34 | 92.56 | 92.61 |
| Tooth Width (mm) | Full-Tooth Deformation (×10⁻⁵ m) | Single-Tooth Deformation (×10⁻⁵ m) | Three-Tooth Deformation (×10⁻⁵ m) | Five-Tooth Deformation (×10⁻⁵ m) |
|---|---|---|---|---|
| 4 | 1.275 | 1.415 | 1.376 | 1.430 |
| 10 | 1.355 | 1.615 | 1.475 | 1.507 |
| 16 | 1.388 | 1.614 | 1.510 | 1.542 |
I also examine partial-tooth models with complete hubs to assess the influence of the hub on results. The steady-state temperature fields in these models show lower maximum temperatures (e.g., 81.9251°C for single-tooth with hub) due to the hub’s distance from the meshing surface, leading to greater errors. The deformation fields align with the full model in pattern but deviate in magnitude, with relative errors up to 17.32% for single-tooth with hub models. This confirms that for accurate thermal deformation analysis, the full-tooth model is essential.
In conclusion, my analysis of helical gears in electric drive systems reveals that the steady-state temperature field exhibits gradient changes with asymmetric distributions along the tooth width and height. Thermal deformation is cumulative, with the tooth tip deforming the most. While partial-tooth models can approximate temperature fields with high accuracy, they are inadequate for deformation studies due to constraint effects. Therefore, I recommend using full-tooth models for comprehensive thermal-mechanical analysis of helical gears to ensure precision in design and optimization.
This research underscores the importance of model selection in finite element analysis of helical gears. Future work could explore dynamic thermal conditions or different lubrication regimes to further enhance the understanding of helical gear behavior in electric drive applications.