In modern wind turbine gearboxes, planetary gear stages are widely adopted due to their compact design and high torque density. A critical aspect of these systems is the use of journal bearings to support planetary gears, where the inner bore of the gear acts as the sleeve and the pin shaft serves as the journal. However, the meshing forces from helical gears introduce dynamic bending moments, leading to misalignment between the gear and pin shaft. This misalignment can cause edge contact, increasing wear and reducing operational lifespan. In this analysis, I explore the transient lubrication performance of planetary gear journal bearings under the influence of helical gears meshing bending moments, focusing on dynamic radial loads, bending moments, and rotational speeds. The goal is to understand how these factors affect lubrication and to provide insights for design improvements.
Helical gears are preferred in wind turbine gearboxes for their smooth operation and high load capacity. However, the axial components of meshing forces generate time-varying bending moments on planetary gears. These moments induce tilt in the journal bearings, altering the oil film thickness distribution and potentially leading to mixed lubrication conditions. To address this, I developed a transient tribo-dynamic coupling model for planetary gear journal bearings, incorporating dynamic effects from radial loads, bending moments, and rotational speeds. The model uses inputs from a wind turbine drivetrain dynamic model, including time-varying meshing forces and speeds from helical gears interactions. This approach allows for a comprehensive analysis of lubrication performance under realistic operating conditions.
The transient lubrication model is based on the average flow Reynolds equation, which accounts for surface roughness and mixed lubrication. For helical gears systems, the equation is expressed as:
$$\frac{\partial}{r_j \partial \theta} \left( \phi_\theta \frac{h^3}{\eta} \frac{\partial p}{r_j \partial \theta} \right) + \frac{\partial}{\partial z} \left( \phi_z \frac{h^3}{\eta} \frac{\partial p}{\partial z} \right) = 6 u_s \phi_c \frac{\partial h}{r_j \partial \theta} + 6 u_s \sigma \frac{\partial \phi_s}{r_j \partial \theta} + 12 \phi_c \frac{\partial h}{\partial t}$$
Here, \( h \) is the oil film thickness, \( \eta \) is the lubricant viscosity, \( p \) is the pressure, \( u_s \) is the surface velocity, \( r_j \) is the journal radius, \( \phi_\theta \), \( \phi_z \), \( \phi_s \), and \( \phi_c \) are flow and contact factors, and \( \sigma \) is the composite roughness. The coordinates \( \theta \) and \( z \) represent circumferential and axial positions, and \( t \) is time. This equation is solved with Reynolds boundary conditions to model the transient behavior influenced by helical gears meshing.
The oil film thickness equation combines nominal clearance, tilt due to misalignment, and bearing profiling. For helical gears applications, the thickness is given by:
$$h(\theta, z, t) = h_o(\theta, z, t) + h_c(\theta, z) + h_m(\theta, z, t)$$
where \( h_o \) is the nominal thickness, \( h_c \) is from profiling, and \( h_m \) is from tilt. The nominal thickness is:
$$h_o(\theta, z, t) = c[1 + \epsilon(t) \sin(\theta – \phi(t))]$$
with \( c \) as the radial clearance, \( \epsilon \) as eccentricity, and \( \phi \) as attitude angle. The tilt component due to helical gears bending moment is:
$$h_m(\theta, z, t) = (z – L/2)(-\tan \theta_x \cos(\theta – \phi) – \tan \theta_y \sin(\theta – \phi))$$
where \( \theta_x \) and \( \theta_y \) are tilt angles in orthogonal planes, directly linked to the bending moments from helical gears meshing.
Solid contact pressure is modeled using the Greenwood-Williamson approach, relevant for mixed lubrication in helical gears systems:
$$p_{asp} = \frac{16 \sqrt{2}}{15} \pi (\sigma \beta D)^2 \sqrt{\frac{\sigma}{\beta}} E^* F_{2.5}\left( \frac{h}{\sigma} \right)$$
Here, \( \beta \) is asperity radius, \( D \) is density, \( E^* \) is composite elastic modulus, and \( F_{2.5} \) is a function of film thickness ratio. This accounts for rough surface interactions when oil film thickness is low due to misalignment from helical gears.
The planetary gear dynamics are described by Newton’s second law and angular momentum conservation. For helical gears, the equations are:
$$M^* \ddot{X}^* = F_h^* + F_c^* – W^*$$
where \( M^* \) is mass matrix, \( X^* \) is displacement vector, \( F_h^* \) and \( F_c^* \) are oil film and contact force vectors, and \( W^* \) is the external load vector from helical gears meshing. The forces and moments include contributions from radial loads and bending moments induced by helical gears.
To analyze the impact of helical gears, I considered various operating conditions. The table below summarizes key parameters for the journal bearing and planetary gear system, highlighting factors influenced by helical gears design.
| Parameter | Value | Description |
|---|---|---|
| Bearing Radius, \( r_p \) | 150 mm | Inner radius of planetary gear bore |
| Radial Clearance, \( c \) | 150 μm | Critical for lubrication in helical gears systems |
| Bearing Width, \( L \) | 395 mm | Affected by helical gears axial forces |
| Lubricant Viscosity, \( \eta \) | 0.175 Pa·s | At 50°C, for typical wind turbine applications |
| Helical Gears Helix Angle | 9° | Directly influences bending moments |
| Helical Gears Pressure Angle | 22.5° | Affects meshing force distribution |
| Planetary Gear Mass | 379 kg | Dynamic response to helical gears loads |
The dynamic loads from helical gears meshing were extracted from a SIMPACK model of a 6 MW wind turbine drivetrain. The input torque varied from 20% to 100% of the rated torque (6500 kN·m), with corresponding time-varying speeds and meshing forces. These inputs were used to simulate transient lubrication performance, focusing on how helical gears bending moments affect bearing behavior.
Under steady conditions, the bending moment from helical gears significantly increases edge oil film pressure. For example, with a radial load of 1305.23 kN and a bending moment of 41.43 kN·m from helical gears, the maximum oil film pressure rose from 54.96 MPa to 77.39 MPa, a 40.81% increase. This demonstrates the critical role of helical gears in inducing misalignment and pressure concentration. The table below summarizes the effects under different load ratios, emphasizing helical gears influence.
| Load Ratio, \( K_F \) | Eccentricity, \( \epsilon \) | Tilt Angle \( \theta_y \) (deg) | Max Pressure (MPa) | Min Film Thickness (μm) |
|---|---|---|---|---|
| 20% (No Moment) | 0.45 | 0.001 | 22.5 | 85.2 |
| 20% (With Moment) | 0.48 | 0.005 | 28.7 | 78.9 |
| 60% (No Moment) | 0.68 | 0.003 | 45.3 | 52.4 |
| 60% (With Moment) | 0.72 | 0.012 | 61.8 | 44.7 |
| 100% (No Moment) | 0.85 | 0.004 | 78.2 | 25.6 |
| 100% (With Moment) | 0.89 | 0.025 | 105.4 | 18.3 |
As the load increases, the tilt angles \( \theta_y \) and \( \theta_x \) change dynamically due to helical gears bending moments. The relationship can be expressed as:
$$\theta_y(t) = f(M_{sy}(t), \epsilon(t))$$
where \( M_{sy} \) is the bending moment from helical gears meshing. This tilt reduces minimum film thickness and increases solid contact risk. For transient conditions, the orbital motion of the journal center is affected by helical gears dynamics. At 100% rated load, the eccentricity varies cyclically with amplitude up to 0.9, and the tilt angle \( \theta_y \) fluctuates between 0.02° and 0.03°, driven by time-varying helical gears forces.
The radial clearance \( c \) is a key design parameter. I analyzed its impact on lubrication performance under helical gears influences. The formula for pressure distribution with clearance is:
$$p(\theta, z) \propto \frac{1}{c^3} \left( \frac{\partial h}{\partial \theta} \right)$$
indicating that smaller clearances enhance pressure generation but may increase friction. The table below shows how different clearances affect performance under helical gears loading.
| Radial Clearance, \( c \) (μm) | Max Eccentricity, \( \epsilon \) | Max Tilt \( \theta_y \) (deg) | Solid Contact Force \( F_c \) (kN) | Oil Film Pressure Peak (MPa) |
|---|---|---|---|---|
| 110 | 0.75 | 0.018 | 0.0 | 92.5 |
| 130 | 0.80 | 0.022 | 5.3 | 98.7 |
| 150 | 0.89 | 0.025 | 12.8 | 105.4 |
| 170 | 0.93 | 0.028 | 20.1 | 112.6 |
| 190 | 0.96 | 0.030 | 28.5 | 118.9 |
As clearance increases, solid contact forces rise due to reduced film thickness, exacerbated by helical gears bending moments. The transient analysis shows that for clearances above 130 μm, solid contact occurs during high-load phases, highlighting the need for optimal design in helical gears systems.
The numerical model was validated against experimental data from a full-scale test rig for wind turbine journal bearings. Under radial load of 590 kN and bending moment of 20 kN·m from simulated helical gears effects, the oil film pressure distribution matched closely with measurements. The error in maximum pressure was less than 10%, confirming the model’s accuracy for helical gears applications. The test rig applied dynamic loads to replicate helical gears meshing conditions, and sensors measured pressure and film thickness, providing validation for the transient analysis.
In conclusion, helical gears meshing bending moments have a profound impact on the transient lubrication performance of planetary gear journal bearings in wind turbine gearboxes. The dynamic loads from helical gears cause cyclic changes in journal center position and tilt angles, leading to increased oil film pressure and reduced film thickness. Higher input torques exacerbate these effects, with solid contact occurring under severe conditions. Reducing radial clearance can improve lubrication by minimizing misalignment from helical gears, but it must be balanced with other design constraints. This analysis underscores the importance of considering helical gears dynamics in bearing design to enhance reliability and lifespan in wind energy systems.
For future work, I plan to extend the model to include thermal effects and elastic deformations, which are crucial for high-power helical gears applications. Additionally, optimizing helical gears parameters such as helix angle and pressure angle could mitigate bending moment impacts. The integration of real-time monitoring data from wind turbines will further refine the transient lubrication analysis for helical gears systems.