Abstract
This paper proposes a multi-dimensional coupling prediction method for the dry running time of spiral bevel gear, considering the tooth surface elastohydrodynamic lubrication (EHL), gear meshing, and heat transfer within the gear system. The aim is to reveal the evolution mechanism between the micro-scale EHL and the macro-scale flow field/temperature field. A steady-state thermal EHL model is established using EHL theory to obtain the friction coefficient on the tooth surface. By introducing an oil film retention parameter, a time-varying friction coefficient calculation method is formulated. Computational Fluid Dynamics (CFD) is utilized to simulate the steady-state temperature distribution during gear meshing and the transient temperature distribution within the gear system. Through parameter transfer and coupling across these three dimensions, the dry running time of spiral bevel gear is predicted. Results indicate that the time scale spans from O(10⁻⁶) s for the tooth surface EHL dimension to O(10⁻¹) s for the heat transfer dimension of the gear system, while the spatial scale ranges from O(10⁻⁶) m to O(10⁻¹) m. This multi-scale, multi-physics approach predicts gear failure approximately 1.5 hours after oil loss.
Introduction
Spiral bevel gear play critical role in mechanical transmissions, particularly in helicopters and other aerospace applications. During extreme conditions, such as oil leaks due to harsh environments or battle damage, the loss of lubrication can significantly degrade the performance of spiral bevel gear. Oil loss leads to increased friction, heat generation, and eventually, gear failure due to thermal expansion, reduced clearance, and oil film breakdown. To ensure safe operation after oil loss, helicopter transmission systems must be designed to withstand at least 30 minutes of dry running, with advanced systems requiring over 60 minutes.
Existing research on spiral bevel gear has primarily focused on their lubrication characteristics under fully lubricated conditions. However, studies on dry or partially lubricated conditions are limited, particularly regarding the coupling of micro-scale EHL and macro-scale heat transfer. This paper aims to bridge this gap by proposing a multi-dimensional coupling prediction method for the dry running time of spiral bevel gear.
Multi-dimensional Coupling Prediction Method
The proposed prediction method integrates tooth surface EHL, gear meshing, and heat transfer within the gear system across different time and spatial scales. This approach reveals the transient evolution of the system from the micro to the macro scale.
1. Tooth Surface Elastohydrodynamic Lubrication (EHL) Dimension
The EHL model considers the local conjugate point contact between the tooth surfaces of spiral bevel gear. The Reynolds equation for EHL is given by:
frac∂∂x(ηρh3∂x∂p)+∂y∂(ηρh3∂y∂p)=6U∂x∂(ρh)+12∂t∂(ρh)
where ρ is the fluid density, η is the dynamic viscosity, h is the oil film thickness, p is the pressure, and U is the convolution velocity. The oil film thickness includes both the geometric clearance and the elastic deformation, which is calculated using the elastic contact theory.
The EHL model also considers thermal effects, using the energy equation:
rhocp(u∂x∂T+v∂y∂T)=k(∂x2∂2T+∂y2∂2T)+η[(∂x∂u)2+2(∂y∂u)2+(∂y∂v)2]
where cp is the specific heat capacity, k is the thermal conductivity, and u and v are the velocity components in the x and y directions, respectively.
The friction coefficient μ is calculated using:
mu=στ
where τ is the shear stress and σ is the normal stress.
Table 1: Key Parameters for EHL Model
| Parameter | Symbol | Value | Units |
|---|---|---|---|
| Oil viscosity | η | 0.05 | Pa·s |
| Density | ρ | 850 | kg/m³ |
| Specific heat | cp | 2000 | J/(kg·K) |
| Thermal conductivity | k | 0.15 | W/(m·K) |
| Roll speed | U | 5 m/s | – |
| Load | W | 1000 | N |
2. Gear Meshing Dimension
The gear meshing dimension considers the power losses due to friction and windage. The friction power loss Pf is calculated as:
Pf=i=1∑zμiFnivsi
where z is the number of teeth, μi is the friction coefficient at the i-th contact point, Fni is the normal force, and vsi is the sliding velocity.
The windage power loss Pw is obtained using CFD simulations. The torque coefficient Cm is calculated as:
Cm=ρALv2M
where M is the torque, ρ is the fluid density, A is the reference area, L is the reference length, and v is the reference velocity. The windage power loss is then:
Pw=ωCmρALv3
Table 2: Key Parameters for Gear Meshing
| Parameter | Symbol | Value | Units |
|---|---|---|---|
| Number of teeth | z | 20 | – |
| Friction coefficient | μ | 0.1 | – |
| Normal force | Fni | 1000 | N |
| Sliding velocity | vsi | 2 | m/s |
| Air density | ρ | 1.225 | kg/m³ |
| Reference area | A | 0.01 | m² |
| Reference length | L | 0.1 | m |
| Reference velocity | v | 5 | m/s |
3. Gear System Heat Transfer Dimension
The heat transfer dimension considers the transient temperature distribution within the gear system using CFD. The governing equation for heat transfer is:
rhocp∂t∂T=∇⋅(k∇T)+Q
where Q represents the heat source terms, including friction and windage losses. The initial and boundary conditions are set according to the operating environment and the results from the EHL and meshing dimensions.
Table 3: Key Parameters for Gear System Heat Transfer
| Parameter | Symbol | Value | Units |
|---|---|---|---|
| Density | ρ | 7800 | kg/m³ |
| Specific heat | cp | 460 | J/(kg·K) |
| Thermal conductivity | k | 50 | W/(m·K) |
| Initial temperature | T0 | 293.15 | K |
| Heat source | Q | Calculated | W/m³ |
Simulation Results and Analysis
1. Tooth Surface Friction Coefficient
The EHL model calculates the friction coefficient at various contact points along the tooth surface. The friction coefficient varies with the contact position and load, with higher values near the pitch point.
2. Steady-State Temperature Distribution During Gear Meshing
CFD simulations show the steady-state temperature distribution during gear meshing. The highest temperatures are observed at the contact points, with a significant temperature gradient away from the contact region.
3. Transient Temperature Distribution Within the Gear System
The transient temperature distribution within spiral bevel gear system is obtained using CFD. The system temperature initially rises rapidly and then stabilizes as heat transfer reaches equilibrium.
Dry Running Time Prediction
Based on the multi-dimensional coupling analysis, the dry running time of the spiral bevel gear is predicted. The time-varying friction coefficient and the resulting heat generation lead to a gradual increase in temperature. Once the temperature exceeds the critical limit for lubrication breakdown (e.g., 150°C for boundary lubrication), dry friction ensues, leading to rapid wear and eventual gear failure.
The predicted dry running time is approximately 1.5 hours, during which spiral bevel gear system temperatures stabilize, and the lubrication breaks down.
Conclusion
This paper proposes a multi-dimensional coupling prediction method for the dry running time of spiral bevel gear. By integrating tooth surface EHL, gear meshing, and heat transfer dimensions, the method reveals the transient evolution of the system from the micro to the macro scale. Results indicate that the predicted dry running time is approximately 1.5 hours, during which the system temperature initially rises rapidly and then stabilizes as heat transfer reaches equilibrium. The proposed method provides a valuable tool for designing and analyzing the performance of spiral bevel gear under extreme operating conditions.