In helicopter transmission systems, spur gears with a high contact ratio (HCR), defined as having a contact ratio greater than 2, are frequently employed. Compared to standard spur gears, HCR cylindrical gears maintain at least two pairs of teeth in contact during meshing. This increases the total length of the contact lines, reduces the average load per unit length on the tooth surface, and is beneficial for transmission safety and load capacity. However, the power losses generated during the operation of HCR cylindrical gears are released as thermal energy. Under high-speed and heavy-load conditions typical in aerospace applications, inadequate lubrication and cooling can lead to excessive temperatures, potentially causing tooth surface scuffing and other failure modes. Therefore, it is crucial to conduct a thermo-fluid coupling analysis on HCR spur gears to investigate the temperature distribution on the tooth surface. This study has practical significance for enhancing the cooling performance, heat transfer characteristics, and overall reliability of HCR cylindrical gear transmissions.
Aerospace gear transmissions commonly use oil injection for lubrication and cooling. The balanced temperature of the tooth surface and the convective heat transfer coefficient are key metrics for evaluating the cooling effectiveness of the oil injection system. Currently, researchers often employ Computational Fluid Dynamics (CFD) methods to analyze the temperature field characteristics of gears. This paper, based on CFD principles, uses Ansys software to perform a numerical simulation of the thermo-fluid coupling for HCR spur gears under oil injection lubrication. The model obtains the temperature and convective heat transfer coefficient distribution on the gear surfaces. The influence of lubricant parameters, operational conditions, and gear design parameters on the tooth surface temperature is investigated. Furthermore, temperature field tests are conducted on a CL-100 gear testing machine to validate the simulation results.
CFD Theoretical Foundation
Governing Equations
The complex fluid flow inside a gearbox must satisfy the Navier-Stokes governing equations, including the continuity equation, momentum equation, and energy equation.
Continuity Equation (Mass Conservation):
$$ \frac{\partial \rho_f}{\partial t} + \nabla \cdot (\rho_f \mathbf{u}) = 0 $$
Momentum Conservation Equation:
$$ \frac{\partial (\rho_f u_i)}{\partial t} + \nabla \cdot (\rho_f u_i \mathbf{u}) = \nabla \cdot (\mu \, \nabla u_i) – \frac{\partial p}{\partial x_i} + S_i \quad (i=1,2,3) $$
Energy Conservation Equation:
$$ \frac{\partial (\rho_f T)}{\partial t} + \nabla \cdot (\rho_f \mathbf{u} T) = \nabla \cdot \left( \frac{k_f}{C_p} \nabla T \right) + S_T $$
where \( t \) is time (s), \( u_i \) are the velocity components, \( \mathbf{u} \) is the velocity vector (m/s), \( \mu \) is the dynamic viscosity (Pa·s), \( p \) is pressure (Pa), \( S_i \) is a momentum source term (kg/(m²·s²)), \( \rho_f \) is the fluid density (kg/m³), \( T \) is temperature (K), \( k_f \) is the thermal conductivity (W/(m·K)), \( C_p \) is the specific heat capacity (J/(kg·K)), and \( S_T \) is the viscous dissipation term (W).
Multiphase Flow Model
The Volume of Fluid (VOF) model is used to track the interface between the oil and air phases in the gearbox. In this model, the sum of the volume fractions of all phases within a control volume is unity:
$$ \alpha_{\text{air}} + \alpha_{\text{oil}} = 1 $$
where \( \alpha_{\text{air}} \) and \( \alpha_{\text{oil}} \) are the volume fractions of air and oil, respectively.
Gear Heat Generation Calculation
The power loss due to gear friction directly influences the thermal characteristics of the transmission system. The Anderson and Loewenthal method is used to calculate the average rolling power loss \( P_r \), average sliding power loss \( P_s \), and windage power loss \( P_w \).
Average Rolling Power Loss:
$$ P_r = 90\,000 \cdot \bar{V}_t \cdot \bar{h} \cdot b \cdot e_p $$
Average Sliding Power Loss:
$$ P_s = f \cdot \bar{F}_n \cdot \bar{V}_s / 1000 $$
Windage Power Loss:
$$ P_w = C \left(1 + 2.3 \frac{b}{R}\right) \rho_{\text{eq}}^{0.8} n^{2.8} R^{4.6} \mu_{\text{eq}}^{0.2} $$
where \( \bar{V}_t \) is the average rolling speed (m/s), \( \bar{V}_s \) is the average sliding speed (m/s), \( \bar{h} \) is the lubricant film thickness (m), \( b \) is the face width (m), \( e_p \) is the contact ratio, \( f \) is the friction coefficient, \( \bar{F}_n \) is the average normal load (N), \( C \) is a proportionality constant (\(C = 2.04 \times 10^{-8}\)), \( R \) is the pitch circle radius (m), \( \rho_{\text{eq}} \) is the equivalent density of the air-oil mixture (kg/m³), \( n \) is the rotational speed (rpm), and \( \mu_{\text{eq}} \) is the equivalent dynamic viscosity of the mixture (Pa·s).
The total power loss \( Q \) is:
$$ Q = P_s + P_r + P_w $$
Introducing a heat partition coefficient \( \gamma \), the heat assigned to the driving and driven gears, \( Q_1 \) and \( Q_2 \), are:
$$ Q_1 = \gamma Q, \quad Q_2 = (1 – \gamma)Q $$
$$ \gamma = \frac{ \sqrt{k_{s1} \rho_1 C_{p1} v_1} }{ \sqrt{k_{s1} \rho_1 C_{p1} v_1} + \sqrt{k_{s2} \rho_2 C_{p2} v_2} } $$
where \( k_s \) is the thermal conductivity of the gear material (W/(m·K)), \( \rho \) is the density (kg/m³), \( C_p \) is the specific heat (J/(kg·K)), and \( v \) is the tangential velocity at the mesh point (m/s).
CFD Analysis Model
Parameters and Conditions
The geometrical and operational parameters for the HCR cylindrical gears are listed in the tables below.
| Parameter | Driving Gear (Large) | Driven Gear (Small) |
|---|---|---|
| Normal Module \( m_n \) (mm) | 3.25 | 3.25 |
| Number of Teeth \( z \) | 32 | 25 |
| Face Width \( b \) (mm) | 16.0 | 16.5 |
| Pressure Angle \( \alpha \) (°) | 20 | 20 |
| Addendum Coefficient \( h_a^* \) | 1.32 | 1.32 |
| Contact Ratio \( e_p \) | 2.2 (High Contact Ratio) | |
| Rotational Speed \( n \) (rpm) | 1500 | 1920 |
| Load Level | Grade 9 | |
| Material | Thermal Conductivity \( k_s \) (W/(m·K)) | Density \( \rho \) (kg/m³) | Specific Heat \( C_p \) (J/(kg·K)) |
|---|---|---|---|
| 20CrMnMoA Steel | 46 | 7850 | 470 |
| Density at 15.6°C \( \rho_{\text{oil}} \) (kg/m³) | Kinematic Viscosity at 37.8°C \( \nu_{\text{oil}} \) (mm²/s) | Kinematic Viscosity at 98.9°C \( \nu_{\text{oil}} \) (mm²/s) |
|---|---|---|
| 993 | 29 | 5.4 |
Computational Model and Mesh
The interaction between the lubricating oil and the high-speed rotating cylindrical gears involves both moving and stationary fluid domains. The Multiple Reference Frame (MRF) method is employed to accurately simulate this interaction. The basic idea is a coordinate transformation, converting the flow equations into a rotating coordinate system for numerical solution. The rotating domains (containing the gears) exchange information with the surrounding stationary fluid domain through interfaces. The speeds for the driving and driven gear rotating domains are set to 1500 rpm and 1920 rpm, respectively.
Ansys Meshing software is used for grid generation. Due to the complex structure of the cylindrical gears, non-conformal, unstructured tetrahedral meshes are applied to the fluid domains to adapt to the geometry. The mesh near the gear surfaces and the啮合 region is refined to capture the flow and heat transfer details accurately.
Boundary Conditions and Solution Setup
The heat generated by friction is applied as a volumetric heat source to a thin layer (0.01 mm thick) on the tooth flank surfaces of the cylindrical gears. The calculated heat flux values are shown below.
| Surface | Volumetric Heat Source (W/m³) |
|---|---|
| Driving Gear Meshing Flank | 3.65 × 10⁹ |
| Driven Gear Meshing Flank | 4.00 × 10⁹ |
The boundary conditions for the fluid domain are defined as follows: The oil injection inlet is a velocity inlet (40 m/s). The outlet is a pressure outlet at atmospheric pressure. The gearbox walls are set with a natural convection boundary condition (heat transfer coefficient of 50 W/(m²·K)) to the ambient air at 26.85°C. The initial temperature for the gears and the oil is set to 60°C. The fluid-solid interfaces use a coupled thermal condition. The pressure-velocity coupling uses the SIMPLE algorithm. The simulation runs until the residuals for continuity, momentum, energy, and volume fraction converge.
Simulation Results and Parametric Influence Analysis
Temperature Field Distribution
The temperature distribution on the surfaces of the driving and driven cylindrical gears is similar. The temperature field is symmetric about the mid-plane of the face width. The highest temperature on the tooth flank appears in the region from the pitch line to the addendum. This is because sliding friction is more pronounced in this area, generating more heat. The temperature decreases towards the gear ends due to better heat dissipation conditions and towards the gear body (web) which is not directly involved in heat generation.
The convective heat transfer coefficient (HTC) distribution shows that the HTC on the driving gear surface is generally lower than on the driven gear, primarily due to its lower rotational speed. The HTC on the gear rim increases with radius, reaching a maximum at the tip. The mesh zone exhibits the highest HTC due to the abundant presence of oil.
Influence of Lubricant Parameters
1. Oil Inlet Temperature: The steady-state temperature of the cylindrical gears increases approximately linearly with the increase of the oil inlet temperature. Higher initial oil temperature means higher initial system energy. Although increased temperature reduces oil viscosity, which can slightly increase sliding losses, the dominant effect is the higher baseline energy, leading to a higher final gear temperature. Conversely, the convective heat transfer coefficient increases with oil temperature due to reduced viscosity, improving cooling potential. However, the net effect is a temperature rise.
| Oil Inlet Temp. (°C) | 40 | 50 | 60 | 70 | 80 | 90 |
|---|---|---|---|---|---|---|
| Max. Tooth Temp. (°C) | 95.2 | 102.5 | 110.1 | 117.8 | 125.4 | 133.1 |
| Min. Body Temp. (°C) | 62.3 | 70.1 | 78.0 | 85.9 | 93.8 | 101.7 |
2. Oil Injection Flow Rate: Increasing the oil flow rate initially leads to a significant decrease in gear temperature as more oil enhances convective cooling. However, beyond a certain point (around 1.76 L/min in this study), the temperature reduction rate diminishes. Excessive oil can increase churning losses, generating additional heat within the gearbox. The convective HTC increases monotonically with flow rate.
Influence of Operational Parameters
1. Rotational Speed: For the studied range, the temperature of the cylindrical gears generally increases with rotational speed up to a point (around 2500 rpm for the driving gear). This is because the increase in sliding speed raises frictional heat generation faster than the improvement in convective cooling. At higher speeds, the reduced contact time per mesh cycle can lead to more effective cooling, sometimes causing a slight temperature decrease or leveling off. The increase in the maximum temperature is more pronounced than that of the minimum temperature.
2. Load: Increasing the transmitted load causes a significant rise in gear temperature. Higher loads increase the normal force and friction coefficient, leading to greater frictional heat generation. Since load has a less pronounced effect on the convective HTC, the increase in heat input dominates, resulting in higher temperatures. The rate of temperature increase slows at higher load levels due to the influence of other制约 factors.
| Load Grade | 5 | 6 | 7 | 8 | 9 |
|---|---|---|---|---|---|
| Driving Gear Max Temp. (°C) | 87.5 | 95.8 | 102.3 | 107.1 | 110.1 |
| Driven Gear Max Temp. (°C) | 85.2 | 93.1 | 99.4 | 104.0 | 106.9 |
Influence of Gear Design Parameters
1. Face Width: Increasing the face width of the cylindrical gears leads to an overall decrease in tooth surface temperature. Although a wider gear generates more total frictional heat, the increase in surface area for heat dissipation is more significant, resulting in better cooling.
2. Pressure Angle: Increasing the pressure angle from 14° to 23° reduces the maximum and minimum gear body temperatures. A larger pressure angle can reduce the sliding velocity and the associated frictional heat generation, contributing to lower operating temperatures.
3. Contact Ratio (via Addendum Coefficient): Increasing the addendum coefficient raises the contact ratio. The simulation shows that a higher contact ratio leads to an increase in gear temperature. This is attributed to the increased frictional heat generation associated with the modified tooth geometry. However, the convective heat transfer coefficient also increases with the addendum coefficient due to changes in the gear geometry affecting the oil flow. For the driven gear, at certain increments, the improvement in cooling can temporarily outweigh the heat increase, causing a slight temperature dip.
| Addendum Coeff. \(h_a^*\) | 1.00 | 1.08 | 1.16 | 1.24 | 1.32 |
|---|---|---|---|---|---|
| Contact Ratio \(e_p\) | 1.73 | 1.85 | 1.97 | 2.09 | 2.21 |
| Driving Gear Max Temp. (°C) | 105.2 | 106.8 | 108.1 | 109.3 | 110.1 |
| Driven Gear Max Temp. (°C) | 102.5 | 104.0 | 103.8 | 105.5 | 106.9 |
Experimental Verification
Test Setup and Procedure
Temperature field tests were conducted on a CL-100 gear testing machine to validate the CFD simulation results for the HCR cylindrical gears. The test setup included the test gear pair, a lubrication system with controlled oil injection, and a data acquisition system. Thermocouples were embedded approximately 1 mm beneath the tooth surface at strategic locations (closer to the addendum and dedendum where temperatures are higher) on both the driving and driven gears. The wires were routed through the hollow shaft to a wireless signal transmitter rotating with the shaft. The oil was heated to the specified inlet temperature (60°C or 90°C) and injected at 40 m/s from the mesh entry side. Tests were run under varying load grades until a steady-state temperature was reached.
Results and Comparison with Simulation
The experimental data showed that after start-up, the gear temperature increased rapidly for the first 10 minutes, then rose steadily until reaching a stable maximum after about 25-30 minutes of operation. This steady-state temperature was recorded for comparison.
A comparison between the experimental measurements and the CFD simulation results for the maximum tooth temperature under different loads and oil temperatures showed good agreement. The trend of increasing temperature with increasing load was accurately captured by the simulation. The absolute error between simulation and experiment decreased at higher load levels, indicating the reliability of the CFD model for the temperature field analysis of cylindrical gears under oil injection lubrication.
Comparison Between HCR and Standard Cylindrical Gears
Additional tests were conducted on a standard contact ratio (approximately 1.73) gear pair with the same center distance and module for comparison. The results confirmed that the trend of increasing temperature with load holds for both gear types. Crucially, under identical conditions of oil temperature and load, the HCR cylindrical gears exhibited higher tooth surface temperatures than the standard contact ratio gears. This difference became more pronounced as the load increased.
Conclusion
1. A thermo-fluid coupling model for HCR cylindrical gears under oil injection lubrication was successfully established using the MRF method. The simulation revealed that the highest temperature on the tooth flank of cylindrical gears occurs near the addendum region, and the temperature field is symmetric about the mid-plane of the face width.
2. The convective heat transfer coefficient on the surface of the driving cylindrical gear is generally lower than on the driven gear, indicating better cooling for the latter. The heat transfer coefficient increases with radius on the gear rim and is highest in the mesh zone.
3. The temperature of cylindrical gears increases with rising load and rotational speed, although the rate of increase diminishes at higher levels. Increasing oil injection flow rate, gear face width, and pressure angle can reduce tooth surface temperature. Conversely, increasing oil inlet temperature and gear contact ratio lead to higher operating temperatures.
4. Experimental tests on a gear rig validated the CFD simulation approach. The measured trend of increasing gear temperature with load matched the simulation results. The tests also conclusively demonstrated that HCR cylindrical gears operate at higher temperatures than standard contact ratio cylindrical gears under the same conditions, and this temperature difference amplifies with increasing load.
This comprehensive analysis provides valuable insights for the thermal design and cooling system optimization of high-performance cylindrical gear transmissions, particularly those employing high contact ratio designs for improved load capacity.