Spiral bevel gears transmit torque between intersecting shafts in a compact package, which makes them attractive for aircraft and heavy-lift drone gearboxes. In an oil-powered heavy-lift four-rotor UAV, removing the gearbox cooling system saves weight, but it also removes the heat path that conventional oil lubrication relies on. Grease lubrication becomes a practical option, especially when flight-control and packaging constraints limit the oil system. The design problem is not simply whether grease can form a film. The relevant question is how the film, asperity contact, friction, and meshing efficiency vary along the curved contact path of a spiral bevel gear under cruise, climb, and maneuvering torque.
The curved tooth trace of a spiral bevel gear is the reason the lubrication problem is path-dependent. Contact radius, entraining velocity, sliding velocity, and load all change from mesh-in to mesh-out. A grease does not respond to those changes like a Newtonian oil: its yield stress, shear thinning, and temperature sensitivity alter the balance between fluid-film load support and asperity contact. A path-resolved thermal elastohydrodynamic (EHL) model is therefore more appropriate than a single central-film estimate.
Contact geometry and kinematics set the lubrication problem
The analysis uses two body-fixed coordinate frames, one for the pinion and one for the gear, with the shaft axes and tooth-surface normals defined at the assembly intersection. For a pinion angular velocity \(\boldsymbol{\omega}_l\), pinion and gear tooth numbers \(N_p\) and \(N_g\), and position vectors from the cone apex to the contact point, the sliding and entraining velocities are
\[
\mathbf{U}_s=\boldsymbol{\omega}_l(\mathbf{P}_l\times\mathbf{R}_{bl})-\frac{N_g}{N_p}\boldsymbol{\omega}_l(\mathbf{P}_r\times\mathbf{R}_{br})
\]
\[
\mathbf{U}_e=\frac{1}{2}\left[\boldsymbol{\omega}_l(\mathbf{P}_l\times\mathbf{R}_{bl})+\frac{N_g}{N_p}\boldsymbol{\omega}_l(\mathbf{P}_r\times\mathbf{R}_{br})\right]
\]
These vectors are not generally aligned with the major and minor axes of the Hertzian contact ellipse. Their components along the ellipse axes determine the entraining direction and the sliding direction used in the Reynolds equation and the friction calculation. The contact-ellipse orientation itself follows from the relative normal curvature of the two tooth surfaces, including the curvature and twist terms produced by the gear-cutting adjustment. That is why the contact geometry cannot be reduced to a single equivalent radius for the whole tooth.
The gear pair considered here has a 90° shaft angle, a 17-tooth pinion, and a 45-tooth gear. The main geometry is listed in Table 1. Both members are steel with an elastic modulus of 211.11 GPa and Poisson’s ratio of 0.3. The tooth surfaces are represented with a Gaussian roughness distribution: an asperity radius of 0.2 μm and a composite root-mean-square roughness of 0.3 μm.
| Parameter | Pinion | Gear |
|---|---|---|
| Number of teeth | 17 | 45 |
| Normal module (mm) | 5 | 5 |
| Face width (mm) | 42 | 42 |
| Pressure angle (°) | 20 | 20 |
| Helix angle (°) | 30 | 30 |
| Face angle (°) | 21.00173 | 73.930 |
| Pitch angle (°) | 18.81 | 71.20 |
| Root angle (°) | 17.001 | 67.93 |
| Outside diameter (mm) | 101.0 | 259.6 |
| Axis intersection angle (°) | 90 | 90 |
| Mean cutter radius (mm) | 228.251 | 225.980 |
| Cradle angle (°) | 352 | 131 |
At 3 000 r/min, the calculated entraining speed rises monotonically from about 7.2 to 8.7 m/s between mesh-in and mesh-out. The relative sliding speed first falls to a minimum near the pitch point, or node, and then rises again toward mesh-out. Load sharing also changes along the path: as the input torque increases from 180 N·m for stable cruise, to 200 N·m for slow climb, and to 230 N·m for maneuver avoidance, the contact load on the active tooth rises. Cruise is taken to represent more than 90% of the operating profile, so the 180 N·m case is not a minor off-design point; it is the condition that dominates energy loss over a mission.
Grease rheology and the mixed thermal EHL framework
Grease is a two-phase lubricant: a base oil carried in a thickener structure. Its apparent viscosity depends on shear rate, temperature, and pressure. At low shear, the thickener network gives the grease a yield stress. Once the shear stress exceeds that yield limit, the response becomes strongly shear-thinning. A Herschel–Bulkley relation captures this transition:
\[
\tau=\tau_s+\eta^*\dot{\gamma}^{\,n}
\]
where \(\tau_s\) is the shear yield stress, \(\eta^*\) is a consistency parameter, \(\dot{\gamma}\) is the shear rate, and \(n\) is the rheological index. In the high-shear, high-pressure contact of a gear tooth, the grease cannot be treated as a fixed-viscosity fluid. The model therefore combines a power-law shear response with a limiting-shear-stress description for the friction calculation. The same framework allows the grease to behave more like a plastic solid at low shear and more like a high-viscosity oil when shear heating and high shear rate break down the thickener structure.
The Reynolds equation is written for the spiral bevel tooth surface rather than for a simple line or point contact. It includes the direction of the entraining velocity, surface roughness through pressure-flow and shear-flow factors, and squeeze effects. The film thickness includes the rigid-body separation, the local curvature in the \(x\) and \(y\) directions, the roughness distributions of the two surfaces, and the elastic deformation produced by the total contact pressure:
\[
h=H_0+\frac{x^2}{2R_x}+\frac{y^2}{2R_y}+\delta_g+\delta_p+\frac{2}{\pi E’}\iint_\Omega \frac{p_t(x^*,y^*)}{\sqrt{(x-x^*)^2+(y-y^*)^2}}\,dx^*\,dy^*
\]
Here \(H_0\) is the central rigid-body film thickness, \(R_x\) and \(R_y\) are the composite radii of curvature, \(\delta_g\) and \(\delta_p\) are the roughness heights of the two surfaces, \(E’\) is the equivalent elastic modulus, and \(p_t\) is the total pressure carried by the fluid and the asperities. The load balance splits the total load between the hydrodynamic pressure \(p\) and the asperity contact pressure \(p_c\). Asperity contact is evaluated with an elastic–plastic statistical contact model, so the calculation can move continuously from full-film lubrication to mixed lubrication as the film thins.
Temperature is not a passive output. The energy equation includes convection, conduction, compressive heating, viscous dissipation, and friction at asperity contacts. The viscosity and density are allowed to vary with pressure and temperature, which is essential for grease: its viscosity–temperature response is stronger than that of a simple oil. The Reynolds equation is solved on a 275 × 275 grid over the contact domain, with hybrid Gauss–Seidel and Jacobian updates to handle the low- and high-pressure regions. Pressure and temperature are iterated together until the film, viscosity, and contact load are consistent.
The friction coefficient is not taken from an empirical gear-friction formula. It is assembled from the fluid shear stress and the asperity contact pressure:
\[
\mu=\frac{\iint \tau_f\,dx\,dy+\mu_c\iint p_c\,dx\,dy}{W}
\]
where \(\tau_f\) is the local fluid shear stress, \(\mu_c\) is the dry contact friction coefficient at asperities, \(p_c\) is the asperity pressure, and \(W\) is the total normal load. This matters for a spiral bevel gear because the local sliding speed, film thickness, and asperity load ratio change strongly within one mesh cycle. A single empirical friction coefficient cannot represent that variation.
Film thickness: validation and path dependence
The roughness-aware grease Reynolds formulation was checked against independent grease EHL film-thickness measurements over an entraining-speed range of 0 to 40 m/s. At low entraining speeds, the difference between a plastic-fluid Reynolds model and the present formulation is small. At medium and high entraining speeds, the two numerical models diverge, with a maximum film-thickness difference of 44.335 nm. The roughness-aware formulation follows the measurements more closely in that high-shear range. The physical reason is that grease is not a perfect plastic fluid. Under high pressure and temperature it retains part of an elastic-fluid response, and at high shear rate, shear heating makes it behave more like a high-viscosity oil. The gear pair operates at 7.2–8.7 m/s entraining speed, which lies in the range where the improved formulation matches the measurements well.
Two greases and their common base oil were compared. Both greases are NLGI grade 2 lithium-soap products with the same base-oil viscosity, but they differ in grease viscosity and rheological index (Table 2). Grease 1 is the lower-viscosity grease; Grease 2 is the higher-viscosity grease.
| Parameter | Base oil | Grease 1 | Grease 2 |
|---|---|---|---|
| NLGI grade | — | 2 | 2 |
| Thickener | — | Lithium soap | Lithium soap |
| Base-oil viscosity at 40 °C (mm²/s) | 32 | 32 | 32 |
| Base-oil viscosity at 100 °C (mm²/s) | 6 | 6 | 6 |
| Grease viscosity at 70 °C (Pa·s) | — | 0.336 | 0.788 |
| Rheological index \(n\) | — | 0.822 | 0.791 |
The calculated film thickness decreases with increasing torque and increases with grease viscosity. At mesh-in, the central film of Grease 1 can be close to that of the base oil, especially at the lower torque. At the node, the three lubricants separate more clearly: the grease films change relatively little, while the base-oil film drops more. At mesh-out, the entraining speed is high, all films rise, and the base oil can temporarily produce a thicker film than Grease 1. As torque increases, the base-oil and Grease 1 films converge again.
This behavior follows from the different flow responses. The base oil has lower viscosity and flows more readily; under pressure it spreads over a larger area and forms a thinner central film, but at high entraining speed it is carried into the contact quickly and the film grows. The grease has a yield stress and a thickener structure, so its flow is restrained. It resists thinning under load, but it also resists the rapid film buildup that a low-viscosity oil can achieve at high entrainment. The result is a crossover: the base oil can be thinner at the node and thicker at mesh-out, while the grease films are more stable across the path.
One of the practical conclusions is that a single oil-film central-thickness formula is not a reliable proxy for the whole grease-lubricated spiral bevel tooth. In the calculations, some mesh-in films fall below the 0.3 μm composite roughness peak, while mesh-out films remain above it. A designer who estimates the entire tooth from one central-film value can therefore miss a local transition into mixed lubrication. The base-oil approximation is most plausible at mesh-in with a low-viscosity grease under some conditions, not across the full contact path.
Friction, asperity load, and lubrication state
The asperity load ratio provides a direct indicator of lubrication state. A ratio above 85% corresponds to boundary friction; a ratio of 0% corresponds to full-film lubrication; intermediate values indicate mixed lubrication. At low entraining speeds, the asperity load ratio rises and the tooth surface enters mixed lubrication. Higher grease viscosity reduces the asperity load ratio because it builds a thicker film. Higher torque increases the asperity load ratio because the film thins under the larger load.
The friction coefficient does not follow the film thickness monotonically. A higher-viscosity grease lowers the asperity load ratio, but it also requires a larger shear stress to be sheared in the film. In the calculated results, the lower-viscosity grease gives a lower friction coefficient even though its asperity load ratio is higher. The fluid shear penalty of the high-viscosity grease outweighs its film-thickness benefit over the investigated range. This is an important distinction for efficiency: the lubricant that gives the thickest film is not necessarily the lubricant that gives the lowest friction loss.
Within one mesh cycle at 3 000 r/min, the friction coefficient is time-varying. The contact radius, entraining speed, sliding speed, and load all change, and the grease rheology responds to those changes. The model predicts a friction maximum near the pitch point. Empirical gear-friction correlations calibrated for oil-lubricated contacts should therefore be used with caution for grease-lubricated spiral bevel gears: they can miss both the path dependence and the transition between fluid-film and asperity load support.
Meshing efficiency and design implications
The instantaneous meshing efficiency is calculated from the sliding friction force and the rolling traction, using the gear torque and angular speed:
\[
\eta_e=1-\frac{1}{T_g\omega_g}\left(|\mathbf{F}_s\cdot\mathbf{V}_s|+|2\mathbf{F}_{r1}\cdot\mathbf{U}_e|\right)
\]
Here \(T_g\) and \(\omega_g\) are the gear torque and angular speed, \(\mathbf{F}_s\) is the sliding friction force, \(\mathbf{V}_s\) is the relative sliding velocity, and \(\mathbf{F}_{r1}\) is the thermally corrected rolling traction. The expression isolates the meshing-friction loss; it is not a complete gearbox loss model that includes bearings, seals, windage, or churning.
For the cases calculated, the instantaneous meshing efficiency stays roughly in the 97.2–98.4% band. The minimum occurs at mesh-in. At that position the friction has both a grease-shear component and an asperity-contact component, and the efficiency is least sensitive to torque. As the mesh progresses toward the node, the sliding speed falls sharply, asperity friction loss decreases, and efficiency rises. At mesh-out, the behavior depends strongly on torque: lower torque gives higher efficiency because the contact is close to full-film lubrication and the loss is dominated by grease shear and squeeze. The shear rate is linked to sliding speed, while squeeze is linked to load; their coupling produces the torque-dependent efficiency at mesh-out.
The speed dependence is equally important. Meshing efficiency increases with rotational speed. A lower-viscosity grease reduces the efficiency loss because it requires less shear stress, but the difference between greases is small at very low speed. There, the shear rate is low, the grease retains a stronger plastic response, and both greases lose efficiency in a similar way. As speed rises, the lower-viscosity grease begins shear-thinning earlier, so its efficiency advantage becomes clearer. Near 3 000 r/min, the higher-viscosity grease also shear-thins substantially, and the difference between the two greases narrows again.
At constant speed, a higher grease viscosity produces a larger efficiency loss. The design implication is not to choose the highest-viscosity grease available, but to choose the lowest viscosity that still maintains an adequate film at the most critical position, which is generally mesh-in. The mesh-in region combines high sliding speed with a thinner film and a higher asperity load ratio; it is also where the efficiency minimum appears. At mesh-out, reducing sliding speed and load would reduce the shear and squeeze losses, but those variables are constrained by the macrogeometry and the contact ratio.
The broader message for grease-lubricated spiral bevel gears in weight-sensitive UAV drives is that lubrication state must be evaluated along the contact path, not at a single representative point. Film thickness, asperity load ratio, friction coefficient, and meshing efficiency are all path-dependent, and a single oil-film estimate can misrepresent the mesh-in condition. A thermal EHL model that resolves the contact path, grease rheology, surface roughness, and asperity load sharing provides a more defensible basis for choosing grease viscosity, predicting efficiency, and setting torque limits for cruise, climb, and maneuvering flight.