In modern mechanical transmissions, helical gears are widely used in high-speed and heavy-duty applications due to their superior load-carrying capacity and smooth operation. However, under such conditions, the friction-induced heat generation can lead to elevated thermal loads, potentially causing tooth surface scuffing and failure. Understanding the flash temperature distribution on the tooth surface of helical gears is crucial for preventing these issues. This study focuses on developing a rapid numerical method using MATLAB to analyze the flash temperature distribution, based on tooth surface contact analysis and Blok’s flash temperature theory. The approach involves modeling the meshing process, deriving key relationships for contact line length, relative sliding velocity, and comprehensive curvature radius, and discretizing the flash temperature equation for numerical computation. Results are compared with traditional ISO methods to validate the reliability of the proposed approach.
The helical gear meshing process is complex, involving continuous engagement and disengagement of teeth along the contact lines. To analyze this, a tooth surface contact model is established. The position vector for 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 machining, and \( C^2 \) represents the space of twice continuously differentiable functions. The unit normal vector \( \mathbf{n}_i \) 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, \( i = 1, 2 \) distinguishes between the driving and driven gears. By unifying the coordinate systems to a common reference frame \( S_f \), the tooth surface family matrix equations are derived:
$$ \mathbf{r}_f^i = M_{fi} \mathbf{r}_i $$
$$ \mathbf{n}_f^i = L_{fi} \mathbf{n}_i $$
where \( M_{fi} \) and \( L_{fi} \) are coordinate transformation matrices for the position and normal vectors, respectively.
The contact line length in helical gears is influenced by the overlap ratio, which includes the transverse contact ratio \( \varepsilon_\alpha \) and the axial contact ratio \( \varepsilon_\beta \). For helical gears with \( \varepsilon_\alpha > \varepsilon_\beta \), the effective face width \( B_e \) and total contact line length \( L \) are 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 equations adjust accordingly. The total contact line length is then:
$$ L = \frac{B_e}{\cos \beta_b} $$
where \( \lambda \) is the projection length on the front face, \( L_1 \) and \( L_2 \) are engagement distances, \( \beta_b \) is the base helix angle, and \( B \) is the actual face width. The variation in contact line length shows that engagement progresses from point to line and back to point contact, with the total length fluctuating due to the overlap ratio and the number of simultaneously engaged teeth.
To analyze the local contact region, the relative sliding velocity at contact points is critical. In a local coordinate system on the contact ellipse, the position vector of a point \( M_0 \) is:
$$ \mathbf{r}_{M_0} = \mathbf{r}_M + \overline{MM_0} \cdot \mathbf{n}_L $$
Points \( M_1 \) and \( M_2 \) on the driving and driven gear surfaces, respectively, are determined along the normal direction. Their position vectors and normal vectors are:
$$ \mathbf{r}_{M_i} = \mathbf{r}_M – \overline{M_0 M_i} \cdot \mathbf{n}_f $$
$$ \mathbf{n}_{M_i} = \frac{(\partial \mathbf{r}_{M_i} / \partial u_1) \times (\partial \mathbf{r}_{M_i} / \partial \theta_1)}{|(\partial \mathbf{r}_{M_i} / \partial u_1) \times (\partial \mathbf{r}_{M_i} / \partial \theta_1)|} $$
The absolute velocities at these points are:
$$ \mathbf{v}_{M_1} = \boldsymbol{\omega}_1 \times \mathbf{r}_{M_1} $$
$$ \mathbf{v}_{M_2} = \boldsymbol{\omega}_2 \times \mathbf{r}_{M_2} $$
where \( \boldsymbol{\omega}_1 \) and \( \boldsymbol{\omega}_2 \) are the angular velocities. The tangential velocities are derived by subtracting the normal components:
$$ \mathbf{v}_{t,M_1} = \mathbf{v}_{M_1} – (\mathbf{v}_{M_1} \cdot \mathbf{n}_{M_1}) \mathbf{n}_{M_1} $$
$$ \mathbf{v}_{t,M_2} = \mathbf{v}_{M_2} – (\mathbf{v}_{M_2} \cdot \mathbf{n}_{M_2}) \mathbf{n}_{M_2} $$
The relative sliding velocity \( \mathbf{v}_c \) is then:
$$ \mathbf{v}_c = \mathbf{v}_{t,M_1} – \mathbf{v}_{t,M_2} $$
Numerical computation in MATLAB reveals that the tangential velocities of the driving and driven helical gears exhibit opposite trends, with the relative sliding velocity increasing as the distance from the pitch point increases.
The comprehensive curvature radius at the contact point is essential for stress and temperature analysis. Given the principal curvatures \( k_{M11} \), \( k_{M12} \) for the driving gear and \( k_{M21} \), \( k_{M22} \) for the driven gear, with corresponding principal directions, the curvature radii are:
$$ \rho_{M1} = \frac{1}{k_{M11} \cos^2 \alpha_{M11} + k_{M12} \sin^2 \alpha_{M11}} $$
$$ \rho_{M2} = \frac{1}{k_{M21} \cos^2 (\alpha_{M11} + \beta_{M11}) + k_{M22} \sin^2 (\alpha_{M11} + \beta_{M11})} $$
where \( \alpha_{M11} \) is the angle between the principal direction and the contact ellipse major axis, and \( \beta_{M11} \) is the angle between the principal directions of the two gears. The comprehensive curvature radius \( \rho_{Mred} \) is:
$$ \rho_{Mred} = \frac{1}{(\rho_{M1})^{-1} + (\rho_{M2})^{-1}} $$
MATLAB results show that the comprehensive curvature radius is smallest at the initial engagement point, leading to higher contact stresses, and follows a pattern of rapid increase followed by a gradual decrease.
Based on Blok’s flash temperature theory, the flash temperature \( T_f \) represents the instantaneous contact temperature during meshing. For discretized contact points along the path, the flash temperature equation 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 point \( k \), \( w_k \) is the contact load, \( v_{t1k} \) and \( v_{t2k \) are the tangential velocities, \( b_k \) is the contact semi-width, and \( B_1 \), \( B_2 \) are the thermal contact coefficients for the driving and driven helical gears. The average friction coefficient is calculated as:
$$ \mu_m^k = 0.12 \frac{(w_k \cos \alpha R_a)^{0.25}}{(\eta_a v_{\tau} R_k)^{0.25}} $$
where \( R_a \) is the surface roughness, \( \eta_a \) is the dynamic viscosity 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 point \( k \).
The following table summarizes key parameters used in the numerical analysis for helical gears:
| Parameter | Symbol | Value |
|---|---|---|
| Base Helix Angle | \( \beta_b \) | 20° |
| Face Width | \( B \) | 0.05 m |
| Surface Roughness | \( R_a \) | 0.8 μm |
| Dynamic Viscosity | \( \eta_a \) | 0.08 Pa·s |
| Pressure Angle | \( \alpha \) | 20° |
Numerical results from MATLAB indicate that the flash temperature distribution peaks at the initial engagement point due to high relative sliding velocity and low comprehensive curvature radius. The table below compares the maximum flash temperatures obtained from the proposed numerical method and the traditional ISO method:
| Method | Maximum Flash Temperature (°C) | Deviation |
|---|---|---|
| MATLAB Numerical | 120.5 | – |
| ISO Traditional | 115.8 | 4.08% |
The flash temperature curve decreases initially, reaches a minimum near the pitch point, and then increases. Notably, the ISO method yields zero flash temperature at the pitch point, whereas the numerical approach gives 0.48°C, accounting for contact deformation and extrusion heating effects.
In conclusion, the tangential velocities at contact points in helical gears display non-smooth variations with singularities, influencing the flash temperature distribution similarly. The comprehensive curvature radius is minimal at initial engagement, contributing to high contact stresses and increased risk of scuffing. The MATLAB-based numerical method for flash temperature calculation in helical gears shows a maximum deviation of only 4.08% from ISO methods, with improved reliability by considering non-zero temperatures at the pitch point. This approach provides a rapid and accurate tool for analyzing helical gear performance under high-speed and heavy-duty conditions, aiding in the prevention of scuffing failures.