In modern mechanical transmission systems, helical gears are widely employed due to their high load-carrying capacity and smooth operation. However, under high-speed and heavy-duty conditions, the friction-induced heat generation can lead to significant thermal loads, potentially causing tooth surface scuffing and failure. This study focuses on developing a rapid numerical method for calculating the flash temperature distribution on the tooth surfaces of helical gears, utilizing MATLAB-based computational techniques. The objective is to provide a reliable tool for predicting transient temperatures and preventing scuffing in helical gear transmissions.
The investigation begins with an analysis of the meshing process of helical gears. The tooth surface contact model is established based on the geometrical relationships between the driving and driven gears. The position vector for a point on the tooth surface can be expressed as:
$$ \mathbf{r}_i(u_i, \theta_i) \in C^2 $$
where \( u_i \) is the axial parameter, \( \theta_i \) is the rotation angle during gear cutting, and \( C^2 \) represents the space of twice continuously differentiable functions. The unit normal vector at any point on the tooth surface is given by:
$$ \mathbf{n}_i(u_i, \theta_i) = \frac{(\partial \mathbf{r}_i / \partial u_i) \times (\partial \mathbf{r}_i / \partial \theta_i)}{|(\partial \mathbf{r}_i / \partial u_i) \times (\partial \mathbf{r}_i / \partial \theta_i)|} $$
Here, the subscript \( i \) denotes the driving gear (1) or driven gear (2). By transforming the coordinate systems of both gears into a unified reference frame \( S_f \), the family of tooth surfaces can be described using the matrix equations:
$$ \mathbf{r}_f^i = M_{fi} \mathbf{r}_i $$
$$ \mathbf{n}_f^i = L_{fi} \mathbf{n}_i $$
where \( M_{fi} \) and \( L_{fi} \) are coordinate transformation matrices for the position and unit normal vectors, respectively.
The contact line length in helical gears is influenced by their high contact ratio. For helical gears with different types of contact ratios, the effective contact width \( B_e \) and total contact line length \( L \) vary along the path of contact. Specifically, for helical gears where the transverse contact ratio \( \varepsilon_\alpha \) exceeds the axial contact ratio \( \varepsilon_\beta \), the effective width is calculated as:
$$ B_e = \begin{cases}
\lambda \cot \beta_b, & \lambda < L_\beta \\
B L_\alpha, & L_\alpha \leq \lambda \leq L_\beta \\
B – (\lambda – L_\alpha) \lambda \cot \beta_b, & L_\alpha < \lambda
\end{cases} $$
For helical gears with \( \varepsilon_\alpha < \varepsilon_\beta \), the expression becomes:
$$ B_e = \begin{cases}
\lambda \cot \beta_b, & \lambda < L_\alpha \\
\cot \beta_b L_\alpha, & L_\alpha \leq \lambda \leq L_\beta \\
B – (\lambda – L_\alpha) \lambda \cot \beta_b, & L_\beta < \lambda
\end{cases} $$
The total contact line length is then derived as:
$$ L = \frac{B_e}{\cos \beta_b} $$
where \( \lambda \) is the projection length of the contact line on the front face, \( L_1 \) and \( L_2 \) are the lengths of the actual transverse path of contact and the exit path, respectively, \( \beta_b \) is the base circle helix angle, and \( B \) is the actual face width. The variation in contact line length for a single tooth and the total contact line during meshing is summarized in Table 1.
| Parameter | Symbol | Variation Trend |
|---|---|---|
| Single Tooth Contact Length | \( L_{\text{single}} \) | Increases then decreases, peaking at mid-mesh |
| Total Contact Length | \( L_{\text{total}} \) | Oscillates due to multiple tooth pairs in contact |
The analysis of the local contact area involves determining the relative sliding velocity and the comprehensive curvature radius at the contact points. For a point \( M_0 \) on the contact ellipse, the position vector is given by:
$$ \mathbf{r}_{M_0} = \mathbf{r}_M + \overline{MM_0} \cdot \mathbf{n}_L $$
The points \( M_1 \) and \( M_2 \) on the driving and driven gear surfaces, respectively, aligned with the normal vector through \( M_0 \), have position vectors and normal vectors defined as:
$$ \mathbf{r}_{M_i} = \mathbf{r}_M – \overline{M_0 M_i} \cdot \mathbf{n}_f $$
$$ \mathbf{n}_{M_i} = \frac{(\partial \mathbf{r}_{M_i} / \partial u_1) \times (\partial \mathbf{r}_{M_i} / \partial \theta_1)}{|(\partial \mathbf{r}_{M_i} / \partial u_i) \times (\partial \mathbf{r}_{M_i} / \partial \theta_i)|} $$
The absolute velocities at these points are calculated using the angular velocities of the gears:
$$ \mathbf{v}_{M_1} = \boldsymbol{\omega}_1 \times \mathbf{r}_{M_1} $$
$$ \mathbf{v}_{M_2} = \boldsymbol{\omega}_2 \times \mathbf{r}_{M_2} $$
The tangential velocities, which exclude the component along the normal direction, are:
$$ \mathbf{v}_{t,M_1} = \mathbf{v}_{M_1} – (\mathbf{v}_{M_1} \cdot \mathbf{n}_{M_1}) \mathbf{n}_{M_1} $$
$$ \mathbf{v}_{t,M_2} = \mathbf{v}_{M_2} – (\mathbf{v}_{M_2} \cdot \mathbf{n}_{M_2}) \mathbf{n}_{M_2} $$
Thus, the relative sliding velocity at the contact point is:
$$ \mathbf{v}_c = \mathbf{v}_{t,M_1} – \mathbf{v}_{t,M_2} $$
The comprehensive curvature radius at the contact point is derived from the principal curvatures of the gear surfaces. If the principal curvatures of the driving gear at the contact point are \( k_{M_{11}} \) and \( k_{M_{12}} \) with principal directions \( \mathbf{e}_{M_{11}} \) and \( \mathbf{e}_{M_{12}} \), and those of the driven gear are \( k_{M_{21}} \) and \( k_{M_{22}} \) with directions \( \mathbf{e}_{M_{21}} \) and \( \mathbf{e}_{M_{22}} \), then the curvature radii are:
$$ \rho_{M_1} = \frac{1}{k_{M_{11}} \cos^2 \alpha_{M_{11}} + k_{M_{12}} \sin^2 \alpha_{M_{11}}} $$
$$ \rho_{M_2} = \frac{1}{k_{M_{21}} \cos^2 (\alpha_{M_{11}} + \beta_{M_{11}}) + k_{M_{22}} \sin^2 (\alpha_{M_{11}} + \beta_{M_{11}})} $$
where \( \alpha_{M_{11}} \) is the angle between \( \mathbf{e}_{M_{11}} \) and the major axis of the instantaneous contact ellipse, and \( \beta_{M_{11}} \) is the angle between \( \mathbf{e}_{M_{21}} \) and \( \mathbf{e}_{M_{11}} \). The comprehensive curvature radius is then:
$$ \rho_{M_{\text{red}}} = \frac{1}{(\rho_{M_1})^{-1} + (\rho_{M_2})^{-1}} $$
Numerical results obtained through MATLAB indicate that the relative sliding velocity between the driving and driven helical gears increases with distance from the pitch point, while the comprehensive curvature radius is smallest at the initial contact point, leading to high contact stresses. The trends for these parameters are summarized in Table 2.
| Parameter | Symbol | Behavior Along Contact Path |
|---|---|---|
| Relative Sliding Velocity | \( v_c \) | Increases away from pitch point |
| Comprehensive Curvature Radius | \( \rho_{M_{\text{red}}} \) | Minimum at initial contact, then increases |
Based on Blok’s flash temperature theory, the instantaneous contact temperature, or flash temperature, is calculated for discrete points along the contact path. The discrete flash temperature equation implemented in MATLAB is:
$$ T_f = 1.11 \frac{\mu_m^k w_k |v_{t1k} – v_{t2k}|}{(B_1 (v_{t1k})^{0.5} + B_2 (v_{t2k})^{0.5}) (2 b_k)^{0.5}} $$
where \( \mu_m^k \) is the local average friction coefficient at the \( k \)-th contact point, \( w_k \) is the contact load, \( v_{t1k} \) and \( v_{t2k} are the tangential velocities of the driving and driven gears, respectively, \( b_k \) is the semi-width of the contact, and \( B_1 \) and \( B_2 \) are the thermal contact coefficients of the gears. The average friction coefficient is given by:
$$ \mu_m^k = 0.12 \frac{(w_k \cos \alpha R_a)^{0.25}}{(\eta_a v_{\tau} R_k)^{0.25}} $$
Here, \( R_a \) is the surface roughness, \( \eta_a \) is the dynamic viscosity of the lubricant at bulk temperature, \( v_{\tau} \) is the sum of tangential velocities, \( \alpha \) is the pressure angle, and \( R_k \) is the comprehensive curvature radius at the contact point.
The MATLAB numerical method involves discretizing the contact ellipse and solving the flash temperature equation iteratively along the contact path. The results show that the flash temperature distribution for helical gears peaks at the initial contact point and decreases towards the pitch point, with a slight increase thereafter. A comparison with traditional ISO calculations reveals that the MATLAB-based method provides more reliable results, especially at the pitch point where the ISO method predicts zero temperature due to assumed zero sliding velocity, whereas the numerical approach accounts for contact deformation and挤压生热, yielding a non-zero value of approximately 0.48°C. The deviation between the two methods is within 4.08%, validating the accuracy of the numerical approach.
Further analysis of the flash temperature distribution under various operating conditions for helical gears is presented in Table 3, highlighting the influence of key parameters.
| Condition | Load (N) | Speed (rpm) | Max Flash Temperature (°C) |
|---|---|---|---|
| High Load, Low Speed | 5000 | 1000 | 95.3 |
| Medium Load, Medium Speed | 3000 | 3000 | 87.6 |
| Low Load, High Speed | 1000 | 5000 | 78.9 |
The findings demonstrate that the tangential velocities of helical gears exhibit non-smooth variations with singularities, directly influencing the flash temperature curve. The comprehensive curvature radius is minimal at the initial meshing point, contributing to high contact stresses and increased risk of scuffing. The MATLAB-based numerical method offers a fast and reliable computational tool for predicting flash temperatures in helical gears, facilitating the design and analysis of gear transmissions under demanding conditions. This approach enhances the understanding of thermal behavior in helical gears and aids in preventing scuffing failures, thereby improving the reliability and lifespan of gear systems.
In conclusion, the study establishes a robust framework for flash temperature calculation in helical gears using MATLAB, emphasizing the importance of accurate meshing analysis and local contact parameters. The method’s validity is confirmed through comparison with established standards, and its application can be extended to other types of gears and operating scenarios. Future work may involve incorporating thermal-elastohydrodynamic lubrication models and experimental validation to further refine the predictions for helical gears in practical applications.