Why grease turns lubrication into a moving-target problem
Oil-powered heavy-lift quadrotor UAVs impose a transmission layout that differs sharply from manned aircraft practice. The gearbox cooling circuit is removed to save mass, and the spiral bevel gears are lubricated with grease rather than with circulating oil. Grease has an obvious operational advantage in this setting: it does not depend on sump orientation and therefore tolerates the continuous attitude changes of a multirotor. The cost is that the lubricant can no longer be treated as a Newtonian fluid with a fixed inlet condition. Grease is a two-phase material — a thickener network carrying base oil — whose apparent viscosity depends on shear rate, pressure and temperature, and which yields only above a limiting shear stress.
The duty cycle sharpens the problem. Steady cruise accounts for more than 90% of operating time, and the manoeuvre is executed by changing propeller blade pitch, so the gearbox torque is modulated rather than constant. Three representative levels frame the design envelope: 180 N·m for steady cruise, 200 N·m for slow climb, and 230 N·m for evasive manoeuvring. Within a single tooth engagement at any of these torques, the contact radius, entraining velocity, sliding velocity and shared load all vary, which means the grease itself changes character between the start and the end of mesh. A lubrication assessment based on one representative contact point cannot capture this behaviour.
Contact geometry and kinematics of the spiral bevel pair
Because the tooth flanks are generated by a cradle-type cutter, the Hertzian contact ellipse is inclined with respect to the tooth trace, and the entraining velocity is aligned with neither principal axis of the ellipse. Two coordinate systems fixed to the gear and the pinion are used to describe the contact, with the pitch apex as the assembly intersection. Writing $\mathbf{P}_r$, $\mathbf{P}_l$ for the unit vectors from the apex to the contact point and $\mathbf{R}_{br}$, $\mathbf{R}_{bl}$ for the vectors from the apex along the pitch cones, the sliding and entraining velocities follow from the angular velocity of the pinion $\omega_l$ and the tooth numbers $N_g$ and $N_p$:
$$\mathbf{U}_s=\omega_l(\mathbf{P}_l\times\mathbf{R}_{bl})-\frac{N_g}{N_p}\,\omega_l(\mathbf{P}_r\times\mathbf{R}_{br})$$
$$\mathbf{U}_e=\frac{1}{2}\left[\omega_l(\mathbf{P}_l\times\mathbf{R}_{bl})+\frac{N_g}{N_p}\,\omega_l(\mathbf{P}_r\times\mathbf{R}_{br})\right]$$
Both vectors are then projected onto the major and minor axes of the contact ellipse, which are themselves obtained from the extremal values of the relative normal curvature; the inclination $\Delta\tau$ of the ellipse follows from the normal curvatures $K_{xl}$, $K_{yl}$ and the twist $G_{xl}$ of the pinion flank and the corresponding quantities of the gear. This projection matters for the flow calculation, because the pressure-flow coefficients in the Reynolds equation are defined along the sliding direction, not along the ellipse axes.
| 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 |
| Pitch angle (°) | 18.81 | 71.20 |
| Root angle (°) | 17.001 | 67.93 |
| Face angle (°) | 21.001 | 73.930 |
| Outside diameter (mm) | 101.0 | 259.6 |
| Mean cutter radius (mm) | 228.251 | 225.980 |
| Cradle angle (°) | 352 | 131 |
| Shaft intersection angle (°) | 90 | |
At 3 000 r/min the entraining speed rises monotonically from the meshing-in point to the meshing-out point, spanning roughly 7.2 to 8.7 m/s. The relative sliding speed behaves differently: it falls to a minimum at the pitch point and then rises again, so the two locations with the highest sliding speed are the two ends of the contact path. Load sharing follows the contact ratio rather than the sliding speed — a single tooth carries its maximum load near mid-mesh and its minimum near meshing-out, and the whole distribution scales upward with torque.
| Flight condition | Gearbox torque (N·m) |
|---|---|
| Steady cruise | 180 |
| Slow climb | 200 |
| Evasive manoeuvre | 230 |
Rheology and the roughness-aware Reynolds equation
Three constitutive laws are available for grease. The Ostwald model, $\tau=\eta^{*}\dot{\gamma}^{n}$, the Bingham model, $\tau=\tau_s+\eta^{*}\dot{\gamma}$, and the Herschel–Bulkley model, $\tau=\tau_s+\eta^{*}\dot{\gamma}^{n}$, differ in whether a yield stress $\tau_s$ is retained. Once the shear stress exceeds the yield limit, Herschel–Bulkley collapses onto the Ostwald form, so a power-law description is adequate for the high-shear conditions that dominate a cruising gearbox. The two greases considered here share a lithium-soap thickener, an NLGI grade 2 consistency and a base oil of 32 mm²/s at 40 °C and 6 mm²/s at 100 °C; they differ mainly in consistency at 70 °C (0.336 Pa·s against 0.788 Pa·s) and in flow index (0.822 against 0.791).
Surface roughness is handled with the average-flow concept, in which the pressure-flow coefficient for a Gaussian surface is $\phi_{U_s}=1-0.9\exp(-0.56h/\sigma)$, with $\sigma$ the standard deviation of asperity height. Combining the force balance on a lubricant element with the Ostwald relation and mass continuity gives the grease Reynolds equation in the local ellipse frame:
$$
\frac{n}{2n+1}\left(\frac{1}{2}\right)^{\frac{n+1}{n}}
\left\{
\frac{\partial}{\partial x}\left[\phi_{U_s}\cos\theta_s\,\rho h^{\frac{2n+1}{n}}\left(\frac{1}{\eta}\frac{\partial p}{\partial x}\right)^{\frac{1}{n}}\right]
+
\frac{\partial}{\partial y}\left[\phi_{U_s}\sin\theta_s\,\rho h^{\frac{2n+1}{n}}\left(\frac{1}{\eta}\frac{\partial p}{\partial y}\right)^{\frac{1}{n}}\right]
\right\}
=
U_e\cos\theta_e\frac{\partial(\rho h)}{\partial x}
+
U_e\sin\theta_e\frac{\partial(\rho h)}{\partial y}
+
\frac{\partial(\rho h)}{\partial t}
$$
Here $\theta_e$ and $\theta_s$ are the angles between the entraining and sliding velocity vectors and the minor axis of the ellipse, and the transient term is retained because the contact conditions change continuously along the path of contact. Density and viscosity are strongly temperature-dependent for grease, far more so than for oil, so the pressure–temperature relations are solved together with the energy equation:
$$
\eta=\eta_0\exp\left\{(\ln\eta_0+9.67)\left[\left(1+\frac{p}{p_0}\right)^{z}\left(\frac{T-138}{T_0-138}\right)^{-1.1}-1\right]\right\}
$$
$$
\rho=\rho_0\left[1+\frac{0.6\times10^{-9}p}{1+1.7\times10^{-9}p}-6.5\times10^{-4}(T-T_0)\right]
$$
with $z=\alpha/[5.1\times10^{-9}(\ln\eta_0+9.67)]$, where $\alpha$ is the Barus pressure–viscosity coefficient of the base oil. Film thickness combines the rigid central value, the two principal radii of curvature, the roughness distributions of the two flanks and the elastic deflection produced by the total pressure $p_t$, which is the sum of fluid and asperity contributions:
$$
h=H_0+\frac{x^2}{2R_x}+\frac{y^2}{2R_y}+\delta_g+\delta_p+\frac{2}{\pi E’}\iint_{\Omega}\frac{p_t(x^*,y^*)}{\sqrt{(x-x^*)^2+(y-y^*)^2}}\,\mathrm{d}x^*\mathrm{d}y^*
$$
Aspersity contact is described statistically with the K–E model, in which the contact pressure depends on the separation $h_0$, the asperity height distribution, the asperity radius and a hardness coefficient $K=0.454+0.41\nu$. The critical interference at which an asperity yields is $\omega_c=(\pi K h_d/E’)^2R$, with $h_d$ the surface hardness. Load balance then requires the applied load to equal the integral of the combined pressure:
$$
W=\iint p(x,y)\,\mathrm{d}x\,\mathrm{d}y+\iint p_c(x,y)\,\mathrm{d}x\,\mathrm{d}y
$$
The energy equation for the film contains convection in the rolling and sliding directions, conduction across the film, a compressibility term and two dissipation terms: the conventional viscous heating and an additional contribution from asperity friction, written with a composite shear stress $\tau_c=\mu p_t$. This coupling is what makes the problem thermal rather than isothermal — the local temperature feeds back into viscosity, and viscosity controls how much of the load the fluid can carry.
Friction from shear and asperity contact instead of an empirical correlation
Empirical friction coefficient formulas calibrated on oil-lubricated gears do not reflect the yield behaviour of grease. Instead, the shear stress in the film is obtained from the B–W relation, which limits the shear stress to a pressure- and temperature-dependent ceiling:
$$
\dot{\gamma}=-\frac{\tau_{\lim}}{\tau_f}\ln\left(1-\frac{\tau_f}{\tau_{\lim}}\right),
\qquad
\tau_{\lim}=(\tau_0+\gamma_l p)\exp\left[\beta_l\left(\frac{1}{T}-\frac{1}{T_0}\right)\right]
$$
Typical values are $\tau_0$ between 1 and 8 MPa and a pressure coefficient $\gamma_l$ between 0.03 and 0.15. Because the tooth surface is only partly separated at low speed, the instantaneous friction coefficient must combine the fluid shear stress with dry contact of the asperities:
$$
\mu=\frac{\iint\tau_f\,\mathrm{d}x\,\mathrm{d}y+\mu_c\iint p_c\,\mathrm{d}x\,\mathrm{d}y}{W}
$$
This composite definition is the key difference from conventional gear efficiency estimates: the friction coefficient becomes an output of the lubrication solution rather than an input.
From friction to instantaneous meshing efficiency
Excluding vibration and other secondary effects, the dominant loss in a spiral bevel mesh is sliding friction at the contact. The sliding force is $F_s=\mu W$. Rolling traction is estimated with a line-contact lubrication approximation, $F_r=4.318(GU)^{0.658}W^{0.0126}R_x/\alpha$, in which $G=\alpha E’$, $U=\eta_0U_e\cos\theta_e/(E’R_x)$ and $W=W/(\alpha E’R_x)$ are the dimensionless material, speed and load parameters. Because the sliding speed on a bevel flank is large, frictional heating modifies the rolling traction, and a thermal factor $\phi_T$ is applied to give a corrected traction $F_{r1}=\phi_TF_r$; the factor depends on the dimensionless thermal loading $L_s=\partial\eta/\partial T\,(V_s)^2/K_f$, the hydrodynamic pressure and the slide-to-roll ratio. The instantaneous efficiency is then
$$
\eta_e=1-\frac{1}{T_g\omega_g}\left(|F_s\cdot V_s|+|2F_{r1}\cdot U_e|\right)
$$
where $T_g$ and $\omega_g$ are the torque and angular speed of the gear.
Numerical treatment
The Reynolds equation is discretised by finite differences on a domain of $-2.5<x<2$ $-2<y<2$,="" $0.001$.
| Parameter | Base oil | Grease 1 | Grease 2 |
|---|---|---|---|
| NLGI grade | — | 2 | 2 |
| Thickener | — | 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 |
| Flow index | — | 0.822 | 0.791 |
The gear pair is made of a single steel grade with an elastic modulus of 211.11 GPa and a Poisson ratio of 0.3. Surface roughness is generated as a Gaussian distribution with an asperity radius of 0.2 μm and a composite root-mean-square roughness of 0.3 μm, consistent with the manufacturing accuracy assumed for the gearbox.
Validation against measured film thickness
The roughness-aware grease Reynolds equation was checked against published film-thickness measurements and against an established plasto-elastohydrodynamic grease model over an entraining speed range of 0 to 40 m/s. Agreement is close at low entraining speed, where the two numerical approaches are almost indistinguishable. The divergence appears in the medium- and high-speed range, where the maximum difference between the two models reaches 44.335 nm; against the measurements, the roughness-aware formulation tracks the data more closely.
The physical reason is that grease is not an ideal plastic body. At low shear rates the plastic description holds, but under high pressure and temperature the thickener network also stores an elastic component. In the high-shear regime, shear heating makes the grease behave increasingly like a high-viscosity oil. The measured and computed entraining speeds for the UAV gearbox, 7.2 to 8.7 m/s, fall inside the range where the formulation reproduces the experimental trend, which supports its use for this application.
Film thickness along the contact path
Because asperity peaks are randomly distributed inside the contact, local film thickness is a noisy indicator of lubrication state; the comparisons are therefore made on film thickness averaged along the minor axis of the Hertzian ellipse and projected onto the major axis. Three positions are examined — meshing-in, the pitch point and meshing-out — for the base oil and both greases at the three torque levels.
Three trends hold throughout. Film thickness decreases as torque increases, since the higher contact pressure flattens the film. A higher-viscosity grease builds a thicker film than a lower-viscosity one. And the three positions behave quite differently. At meshing-in, grease 1 and the base oil give similar central film thickness, particularly at the lowest torque. At the pitch point, the two greases remain close to each other while the base oil film drops markedly: its lower viscosity and higher mobility allow a hydrodynamic film to spread over a larger area at lower thickness. At meshing-out, the high entraining speed lifts all three films, and the base oil film actually overtakes that of grease 1; as torque rises further, the two converge again. The explanation lies in the contrasting constitutive behaviour: the oil, modelled as an ideal viscoelastic fluid, is drawn into the conjunction in quantity at high entraining speed and builds thickness quickly, whereas the grease, with its plastic character, flows less readily and gains less. Under a pressure rise, the oil film then collapses faster, which is why the ordering reverses and then partially re-establishes itself as torque increases.
The absolute values matter for design. Under some of the operating conditions examined, the film thickness at meshing-in falls below the composite roughness of 0.3 μm, while at meshing-out every case remains above it. In other words, the lubrication state can change from mixed to nearly full-film within a single mesh cycle. That is a direct consequence of the kinematics: light and heavy load, high and low shear rate, all occur between meshing-in and meshing-out.
This also explains why a shortcut borrowed from ball-on-disc testing is unsafe here. In rolling elastohydrodynamic contacts, grease central film thickness is often close to that of its base oil once shear thinning is fully developed, and oil film formulas are sometimes used as an approximation. On a spiral bevel flank, however, the contact radius, load and entraining speed all change substantially over the path of contact, so the grease is never held at a single shear-thinning condition. At the pitch point and at meshing-out, base oil and grease films differ clearly. Estimating the lubrication state of the whole flank from an oil-based central film thickness formula is therefore not reliable for this gear type.
Asperity load ratio and friction coefficient
The asperity load ratio provides a direct read-out of the lubrication regime: above 85% the contact is in boundary friction, at 0% it is fully separated, and intermediate values indicate mixed lubrication. At a given entraining speed, a higher grease viscosity lowers the asperity load ratio, while a higher torque raises it. Low rotational speed pushes the contact towards mixed and boundary conditions, which is consistent with the film-thickness results at meshing-in.
The friction coefficient does not follow the same ordering as the film thickness. A high-viscosity grease certainly thickens the film and reduces asperity contact, but its own shear stress is larger, so the net friction coefficient can be higher. Lower-viscosity grease gives the lower friction coefficient in these calculations. Within a mesh cycle the friction coefficient is time-varying, because contact radius, entraining speed and sliding speed all change; its maximum occurs at the pitch point. That behaviour is notably different from the monotonic trends assumed by empirical correlations such as the Benedict formula, which should therefore be applied to grease-lubricated spiral bevel gears only with caution.
Meshing efficiency across the cycle
Instantaneous meshing efficiency stays between about 97.2% and 98.4% over a mesh cycle, but the variation is systematic rather than random. Efficiency is lowest at meshing-in, where the friction force is a combination of grease shear and asperity contact and is least sensitive to torque. As the gear rotates and the sliding speed falls towards the pitch point, asperity friction losses shrink and efficiency rises.
At meshing-out the ordering with torque reverses: the lower the torque, the higher the efficiency. Here the contact is close to full-film, so nearly all loss comes from shearing and squeezing the grease. Shear rate scales with sliding speed, squeezing with load, and the two combine to produce the observed behaviour. Although the sliding speed is highest at meshing-out, the contact ratio means that the load carried there is lowest, so the efficiency loss is highly sensitive to operating conditions. For design purposes this suggests reducing both the sliding speed and the load at meshing-out as a way of limiting mesh loss.
Speed and viscosity interact in a way that is easy to misread. At low rotational speed, the two greases give similar efficiency, because shear rates are low, the plastic character dominates and sliding speeds are small. As speed rises, efficiency increases for both, but the lower-viscosity grease begins to shear-thin first, so the gap between them widens. At higher speed the high-viscosity grease also shear-thins, and as the speed approaches 3 000 r/min the difference narrows again. The general rule remains: at constant speed, a higher grease viscosity produces a larger efficiency loss.
Implications for grease-lubricated bevel gearbox design
The results point to several practical constraints. First, a grease-lubricated spiral bevel pair must be evaluated over the whole mesh cycle, not at a single representative contact point, because the lubrication state can shift between mixed and full-film within one engagement. Second, oil-based central film thickness formulas and empirical friction correlations are not transferable to this configuration; the friction coefficient has to be computed from the fluid shear stress and the asperity contact pressure. Third, the meshing-in region deserves particular attention at low speed and high torque, since it combines the lowest film thickness with the lowest efficiency. Fourth, reducing sliding speed and load at meshing-out offers a direct route to lower loss, and choosing the lowest grease viscosity that still keeps the asperity load ratio acceptable improves efficiency without sacrificing film formation. Finally, because grease viscosity responds strongly to temperature, the energy equation must be solved together with the Reynolds equation; an isothermal treatment would misplace the balance between fluid load support and asperity contact across the entire contact path.
</x