The input stage of a helicopter main reducer is usually a spiral bevel pair: it turns the drive through 90° and transmits high torque in a small envelope. Jet lubrication supplies both the separating film and most of the cooling. When the supply is interrupted—by pump failure, line damage, or combat damage—the pair does not stop transmitting power. The oil inventory on the teeth falls, the film collapses, and the contact passes from full-film elastohydrodynamic lubrication through mixed and boundary lubrication to dry contact. Friction heat rises while convective cooling falls, so tooth temperature climbs until scuffing or seizure ends the run. The interval between oil loss and failure is therefore a thermal transient, and predicting it is a survivability requirement rather than a refinement of steady-state rating.
A steady-state thermal rating cannot describe this interval. Oil fraction, friction coefficient, heat generation, and heat-transfer coefficient all drift together, and the thermal time constant of the gear bodies is comparable to the duration of the event. Two temperatures matter. The body temperature, governed by the heat balance of the gear mass, sets the general thermal level and clearance. The flash temperature, governed by local contact pressure and sliding speed, controls scuffing initiation. A useful model must resolve both, and must let the heat partition between pinion and gear change as the lubricant disappears.
Why starvation is a transient problem
Under normal jet lubrication, the mesh runs with a lubricant film thick enough to separate most asperities. Power loss is modest, and the convection coefficient at the mesh faces is high because oil impinges directly on the contact zone. Once the supply stops, the residual oil is progressively displaced, flung off, and replaced by hot air. The medium in contact with the teeth becomes an oil–air mixture whose thermophysical properties depend on the instantaneous oil–air ratio. That single variable couples three otherwise separate calculations: mixture viscosity and pressure–viscosity coefficient alter film thickness and friction; mixture density, conductivity, and viscosity alter convection; and the declining film shifts the contact into boundary and dry regimes, where friction rises steeply.
Spiral bevel geometry adds a further complication. The contact ellipse migrates across the tooth surface during mesh, curvature and sliding velocity vary along the contact line, and the load distribution is non-uniform. Heat flux is therefore applied over a moving, non-uniform contact rather than a single line. The calculation below resolves the contact line into discrete points and follows each point through the starvation history.
Heat generation from a degrading oil–air mixture
The mixture properties are interpolated between oil and hot air using an instantaneous oil–air ratio bt:
$$\tau_h=b_t\,\tau_o+(1-b_t)\,\tau_a,\qquad b_t:1\rightarrow0$$
where τo is the oil property, treated as temperature-insensitive, and τa is the hot-air property, allowed to vary with time. In the calculation, bt is assumed to decrease linearly from 1 to 0. This is a modelling assumption, not a measured drain curve; the actual decay depends on oil retention, rotational speed, and housing geometry.
Sliding friction dominates the heat input; rolling friction and windage are small enough to be neglected. The heat flux entering each body is split by a partition coefficient ξ:
$$q_1=\xi\,\gamma\,f\,p\,v_s,\qquad q_2=(1-\xi)\,\gamma\,f\,p\,v_s$$
Here f is the friction coefficient, p the contact pressure, vs the relative sliding velocity, and γ the mechanical-to-thermal energy conversion factor, taken as 0.92 (typical range 0.90–0.95). The contact pressure and the shape of the contact ellipse come from a loaded tooth contact analysis; the resulting heat flux is applied over the transient contact ellipse on the tooth surface.
The friction coefficient is obtained from a line-contact elastohydrodynamic film-thickness relation and a mixed-lubrication correlation. The minimum film thickness is expressed through dimensionless load, speed, modulus, hardness, and roughness groups:
$$h_{\min}=1.652\,W^{-0.077}U^{0.716}G^{0.695}\left(1+0.026\,\sigma_M^{1.120}V^{0.185}W^{-0.312}U^{-0.809}G^{-0.977}\right)R_m$$
with Rm the equivalent radius of curvature at the contact point. The fully lubricated friction coefficient and the mixed-lubrication value are:
$$\mu_t=0.0607\left(\frac{\omega}{v_sR_m}\right)^{0.2}\nu_o^{-0.05}\left(\frac{\sigma_m}{d_1}\right)^{0.25}$$
$$\mu_f=e_p^{1.2}\mu_t+(1-e_p)\mu_b,\qquad e_p=\frac{1.21\lambda^{0.64}}{1+0.37\lambda^{1.26}}$$
where ω is the unit contact-length load, σm the mean asperity height, d1 the equivalent pitch diameter, and λ the film-thickness ratio (minimum film thickness divided by mean roughness). For λ > 2 the contact is fully lubricated and μ = μt; for 0.2 < λ < 2 the mixed value μf applies; below that, boundary conditions hold. The correlation carries a nominal boundary value μb = 0.0859, but the endpoint of the starvation trajectory was set by direct tribometry on 9310 gear steel with an MMV-1 friction and wear tester (Table 1).
| Condition | Run 1 | Run 2 | Run 3 | Mean |
|---|---|---|---|---|
| Boundary lubrication | 0.12784 | 0.10427 | 0.08821 | 0.10677 |
| Dry contact | 0.41851 | 0.43534 | 0.35873 | 0.40419 |
The dry value is almost four times the boundary value. That ratio, applied over a contact that is already losing its cooling, is the mechanism behind the rapid temperature excursion.
Heat removal and the thermal network
Convection at the mesh faces is described by a forced-convection correlation for the lubricant in contact with the tooth surface:
$$\theta_n=0.228\,Re_n^{0.731}Pr^{1/3}\frac{K_o}{L_n}$$
where Ln is the characteristic length, taken as the tooth height, and the Reynolds number, Prandtl number, and thermal conductivity all follow the mixture properties. The end faces are treated as a rotating disk, with a transition between laminar and turbulent flow at Red = 2.5 × 105:
$$\theta_d=\begin{cases}0.308\,K_o\,(m+2)^{0.5}Pr^{0.5}\left(\dfrac{w}{\nu_o}\right)^{0.5}, & Re_d\le 2.5\times10^{5}\\[6pt] 0.0197\,K_o\,(m+2.6)^{0.2}Pr^{0.6}\left(\dfrac{w}{\nu_o}\right)^{0.8}r_c^{0.6}, & Re_d>2.5\times10^{5}\end{cases}$$
with w the rotational speed, m = 2 for the radial temperature distribution, and rc the radius of the chosen location. The root, tip, and non-meshing flanks are assigned one third of the end-face coefficient. Radiation is neglected because its share of the total heat exchange is small.
Convective and conductive resistances connect the lumped nodes of the transmission:
$$R_h=\frac{1}{\theta A},\qquad R_p=\frac{\delta_p}{K_pA_p},\qquad R_y=\frac{\ln\left(d_{y2}/d_{y1}\right)}{2\pi K_yl_y}$$
where the first expression is the convective resistance, the second a plane-wall conduction resistance, and the third a cylindrical-wall resistance. Conduction resistances are treated as constant because temperature has little effect on them. Assembling the nodes under energy conservation gives, for each node i:
$$C_i\frac{dT_i}{dt}=Q_i+\sum_j\frac{T_j-T_i}{R_{icj}}+\sum_k\frac{T_k-T_i}{R_{ivk}}$$
Here Ci is nodal heat capacity, Qi the nodal heat flow, and the two sums represent conduction and convection links. In a normal steady-state calculation the left-hand side vanishes; in starvation it does not, and the time derivative carries the transient. The equation set was integrated in MATLAB with an ODE45 solver using 1 s steps over 2,700 s (45 min), starting from the steady-state field computed for normal lubrication.
Flash temperature with a time-varying heat partition
The instantaneous contact temperature is the sum of the body temperature and the flash temperature:
$$T_{ij}=T_{ib}+T_{is}$$
For dry running, the Blok formulation gives the flash temperature on each surface and, by requiring equal flash temperatures, a heat partition coefficient β:
$$T_{s1}=\frac{1.11\,\beta\,q\,l}{\sqrt{k_1\rho_1c_1v_1}},\qquad T_{s2}=\frac{1.11\,(1-\beta)\,q\,l}{\sqrt{k_2\rho_2c_2v_2}}$$
$$\beta=\frac{1}{1+\sqrt{\dfrac{k_2\rho_2c_2v_2}{k_1\rho_1c_1v_1}}}$$
where q is heat flux, l the contact width, k conductivity, ρ density, c specific heat, and v tangential velocity. This dry-contact form ignores the oil film and cannot represent a starved but not yet dry mesh.
The improved treatment replaces the fixed partition with two successive coefficients. The first, φ, uses the instantaneous body temperatures Tb1 and Tb2 taken from the thermal network, so the temperature difference between the two surfaces is allowed to evolve rather than being fixed:
$$\phi=\frac{1}{A_1+B_1+A_2+B_2}\left(\frac{T_{b2}-T_{b1}}{q\,l}+A_2+B_2\right)$$
$$\xi_1=\frac{\phi\left(A_1-C_1\right)}{A_1},\qquad \xi_2=\frac{\left(1-\phi\right)\left(A_2-C_2\right)}{A_2}$$
$$T_{so1}=\xi_1\,q\,l\,A_1,\qquad T_{so2}=\xi_2\,q\,l\,A_2$$
The coefficients A, B, and C are built from the thermophysical properties and contact geometry, and are re-evaluated at every time step. Because the properties, the film thickness, and the body-temperature difference all change during starvation, the partition coefficient becomes a function of time rather than a constant. This is the main difference from earlier starved-lubrication flash-temperature treatments that imposed a fixed temperature difference between the two gears.
Case study: a 27/81 tooth pair and its test box
The configuration examined is an input-stage spiral bevel pair with the parameters in Table 2. The calculation concentrates on the gears; bearings, shafts, and housing are not represented, on the grounds that their influence on the mesh-face temperature is secondary.
| Parameter | Value |
|---|---|
| Teeth (pinion / gear) | 27 / 81 |
| Module | 3.85 mm |
| Face width | 49 mm |
| Pressure angle | 20° |
| Spiral angle | 35° |
| Shaft angle | 90° |
| Surface roughness | 0.5 µm |
For normal lubrication the reference condition was 20 kW at input speeds of 6,000, 5,000, and 4,000 r/min. Oil temperature was 50 °C, the internal air reached 40 °C at steady state, and during starvation the internal air was allowed to rise at 0.02 °C/s. The starved case started from a stable normal-lubrication state at 20 kW and 6,000 r/min, then dropped to 15 kW at the same speed while the oil pump was switched off. The contact line was divided into 45 points for the friction and heat-flux calculation.
Predicted behaviour under normal and starved lubrication
At constant power, lower speed produced higher tooth temperature. Measured driven-gear mesh temperature rose from 45.1 °C at 6,000 r/min to 46.6 °C at 4,000 r/min, and pinion mesh temperature from 47.4 °C to 48.1 °C (Table 3). Two effects act together: lower speed requires higher torque for the same power, which increases friction heat, and lower speed weakens convection at the tooth surface. The normal-lubrication regime therefore already contains a speed penalty that becomes more serious when cooling is degraded.
Once the pump was switched off, the calculated sliding friction coefficient increased at every contact point. Points near the pitch point showed the highest values, because the relative sliding velocity approaches zero there while the curvature radius is small and the contact stress is high. Averaged over the contact line, the friction coefficient and the friction heat rose through the starved interval, with the steepest increase after the contact entered boundary lubrication. The mesh-face convection resistance rose monotonically as the air fraction increased, until the medium was effectively air and the resistance levelled off.
The body temperature did not rise immediately. Because power fell from 20 kW to 15 kW at the start of starvation, the body temperature first decreased, then turned upward as friction heat accumulated. The rate of rise increased before the dry friction value was reached. Flash temperature behaved differently: it accelerated sharply near the boundary regime and then stabilized once the dry friction coefficient was attained. Although the driven gear received the larger share of the friction heat, the pinion had the higher convection resistance, and that resistance dominated the comparison. With the two flash temperatures differing only modestly, the pinion remained the hotter of the two bodies.
Experimental validation
The test box was driven by an electric motor through couplings instrumented with torque and speed meters, loaded by an eddy-current brake, and supplied by a lubrication system that sprayed oil onto the gears and bearings, with return flow to the tank. Type Pt100 resistance sensors inside the gearbox measured local temperatures; signals from the rotating gear were brought out through a slip ring to a multi-channel temperature meter. An infrared imager viewed the mesh region. Because the instantaneous contact temperature cannot be measured directly, readings taken as the teeth left mesh were treated as the experimental instantaneous mesh temperature—an approximation that necessarily samples the surface slightly after the peak contact condition.
| Speed (r/min) | Surface | Measured (°C) | Thermal network (°C) | Error (%) | Finite element (°C) | Error (%) |
|---|---|---|---|---|---|---|
| 6,000 | Gear mesh | 45.1 | 43.0 | 4.7 | 44.7 | 0.9 |
| 6,000 | Pinion mesh | 47.4 | 43.3 | 8.6 | 45.4 | 4.2 |
| 5,000 | Gear mesh | 45.7 | 43.2 | 5.5 | 44.9 | 1.8 |
| 5,000 | Pinion mesh | 47.8 | 43.5 | 8.9 | 45.6 | 4.6 |
| 4,000 | Gear mesh | 46.6 | 43.4 | 6.9 | 45.0 | 3.4 |
| 4,000 | Pinion mesh | 48.1 | 43.8 | 8.9 | 45.9 | 4.5 |
| Speed (r/min) | Measured (°C) | Thermal network (°C) | Error (%) | Finite element (°C) | Error (%) |
|---|---|---|---|---|---|
| 6,000 | 44.3 | — | — | 44.1 | 0.4 |
| 5,000 | 45.1 | — | — | 44.5 | 1.3 |
| 4,000 | 46.0 | — | — | 44.7 | 2.8 |
Under normal lubrication, all relative errors stayed below 10%, with the thermal-network result consistently on the low side and the finite-element result closer to measurement. In the starved run, the comparison at the end of 45 min is given in Table 5. The largest discrepancy was 10.3%, for the driven-gear mesh temperature predicted by the thermal network. The finite element model, which inherited its thermal boundary conditions from the same heat-generation calculation, was closer for that surface but less accurate for the pinion mesh.
| Surface | Measured (°C) | Thermal network (°C) | Error (%) | Finite element (°C) | Error (%) |
|---|---|---|---|---|---|
| Gear end face | 49.2 | — | — | 51.3 | 4.2 |
| Gear mesh | 51.1 | 56.4 | 10.3 | 51.5 | 0.8 |
| Pinion mesh | 55.4 | 60.1 | 8.5 | 59.1 | 6.7 |
The starved run left visible damage: black marks and wear appeared in the contact zone, and the contact pattern lost robustness. This is consistent with the calculated friction history, in which the boundary regime is reached before the end of the test and the flash temperature rises fastest at that stage.
Engineering implications and limits of the approach
The measured bulk temperatures tell a misleadingly mild story. After 45 min without oil, the driven-gear mesh had risen only from 45.1 °C to 51.1 °C and the pinion mesh from 47.4 °C to 55.4 °C, yet the teeth were already damaged. Bulk temperature alone is therefore a poor scuffing indicator: the damage is driven by the local flash temperature and by the collapse of the film, which a single embedded sensor cannot resolve. A monitoring scheme based on bulk temperature should be paired with a flash-temperature margin, or with a friction-sensitive signal, if it is to give useful warning.
The pinion–gear asymmetry also has a design consequence. Heat partition gave the driven gear the larger share of friction heat, but the pinion ran hotter because its convection resistance was higher. Improvements aimed at the pinion cooling path—oil delivery to the smaller member, or additional convective area—would therefore buy more thermal margin than an equal effort applied to the gear.
Several limitations bound the conclusions. The oil–air ratio is assumed to fall linearly from 1 to 0, whereas a real drain depends on retention, speed, and housing geometry. Rolling friction, windage, and radiation are neglected. Bearings, shafts, and housing are excluded, and in a real reducer they act as heat sinks, so the gear temperatures reported here may be higher than those of an assembled unit. The experimental validation covers one starved condition (15 kW, 6,000 r/min, 45 min) and three normal-lubrication speeds, and the infrared reading at mesh exit is only a proxy for the instantaneous contact temperature. The convection and friction correlations are empirical, and their uncertainty propagates into the predicted transient. Within those bounds, the combination of a time-varying oil–air mixture, a thermal network for the body temperature, and an improved flash-temperature calculation with a time-dependent heat partition reproduces the measured thermal history to within about 10%, and it reproduces the trends that matter for design: the initial dip after power reduction, the accelerating rise as the boundary regime is approached, and the pinion running hotter than the gear.