This study investigates the lubrication mechanism of cylindrical gears with variable hyperbolic circular-arc-tooth-trace (VH-CATT) under oil injection conditions. A computational fluid dynamics (CFD) model was developed to analyze gas-oil interactions and optimize lubrication efficiency. Key parameters including nozzle angle, injection velocity, spray height, and gear speed were systematically evaluated to enhance lubrication performance.
1. Mathematical Model and Gear Characteristics
The VH-CATT cylindrical gear features a unique tooth geometry combining circular arc traces in the tooth length direction and hyperbolic profiles in cross-sections. The tooth surface equation is derived as:
$$
\begin{cases}
\mathbf{r}_i^{(d)} = \mathbf{M}_{di}\mathbf{M}_{i1}\mathbf{r}_1(\theta_i, u_i) \\
\mathbf{n}_i^{(d)} = \mathbf{L}_{di}\mathbf{L}_{i1}\mathbf{n}_1(\theta_i, u_i)
\end{cases}
$$
Where transformation matrices $\mathbf{M}$ and $\mathbf{L}$ describe coordinate conversions between cutting tools and gear blanks. The contact characteristics exhibit elliptical pressure distribution with semi-axes:
$$
a = \sqrt{\frac{\delta}{A}}, \quad b = \sqrt{\frac{\delta}{B}}
$$
| Parameter | Pinion | Gear |
|---|---|---|
| Teeth Number | 21 | 29 |
| Module (mm) | 4 | 4 |
| Pressure Angle (°) | 20 | 20 |
2. Oil Injection Lubrication Theory
The Eulerian multiphase model with VOF approach was employed to simulate gas-oil interactions. Key governing equations include:
Continuity Equation:
$$
\frac{\partial \rho}{\partial t} + \nabla \cdot (\rho \mathbf{U}) = 0
$$
Momentum Conservation:
$$
\frac{\partial (\rho \mathbf{U})}{\partial t} + \nabla \cdot (\rho \mathbf{UU}) = -\nabla p + \nabla \cdot [\mu(\nabla \mathbf{U} + (\nabla \mathbf{U})^T)] + \rho \mathbf{g}
$$
The standard k-ε turbulence model was adopted for flow prediction:
$$
\mu_t = \rho C_\mu \frac{k^2}{\epsilon}
$$
3. Optimal Nozzle Configuration
CFD simulations revealed significant air barrier effects caused by high-speed gear rotation. The optimal injection angle was determined through streamline analysis:
| Angle (°) | Max Oil Fraction | Pressure Differential (MPa) |
|---|---|---|
| -5 | 0.42 | 0.83 |
| 10.73 | 0.78 | 1.47 |
The optimal 10.73° injection angle demonstrated 86% higher lubrication efficiency compared to conventional tangential injection.
4. Parametric Sensitivity Analysis
Key operational parameters were evaluated for lubrication performance:
4.1 Injection Velocity Effects
$$
\eta = 0.67v^{0.8} \quad (30 \leq v \leq 60 \, \text{m/s})
$$
$$
\eta = 0.92v^{0.2} \quad (v > 60 \, \text{m/s})
$$
4.2 Spray Height Influence
| Height (mm) | Oil Coverage (%) | Max Pressure (MPa) |
|---|---|---|
| 30 | 89.2 | 3.15 |
| 55 | 54.7 | 1.02 |
4.3 Rotational Speed Impact
$$
P_{oil} = 2.34 – 0.0007n \quad (3000 \leq n \leq 8000 \, \text{rpm})
$$
5. Practical Implementation Guidelines
For optimal cylindrical gear lubrication:
- Position nozzles at 10.73° towards driving gear
- Maintain injection velocity between 50-60 m/s
- Limit spray height ≤40 mm
- Control gear speed ≤6000 rpm
This comprehensive analysis provides fundamental insights for designing efficient lubrication systems in cylindrical gear transmissions, particularly for high-speed industrial applications requiring precise thermal management and wear prevention.