In a helicopter main transmission, the input-stage spiral bevel gears carry engine power into the main reducer. If the oil supply is interrupted, the gears do not pass directly from full elastohydrodynamic lubrication (EHL) to dry contact. They enter a starved interval in which the oil film collapses, asperity load rises, and frictional heat accumulates. The resulting tooth temperature can produce thermal growth, loss of backlash, scuffing, or seizure, and can ultimately interrupt rotor and tail-rotor drive. The key thermal questions are therefore how fast the tooth surface heats, where the hot spot forms, and how power and speed change the temperature margin.
For a helicopter main-reducer pair with 74 and 27 teeth, a transient thermal analysis shows that this interval is dominated by the competition between rising friction heat and changing convective cooling. The model couples the EHL and boundary-lubrication contributions to tooth friction, applies the resulting heat flux to the actual elliptical contact zones, and solves the tooth-body temperature field in time. The results give both the spatial hot spots and the sensitivity to input power and rotational speed.
The starved-lubrication heat source
Starved lubrication is treated as a load-sharing transition. The tooth-surface friction coefficient is a weighted combination of the EHL and boundary values:
$$
\mu_f(t)=c_b(t)\mu_b+c_t(t)\mu_t,\qquad c_b(t)+c_t(t)=1
$$
Here \(c_t\) is the load share carried by the EHL film and \(c_b\) is the share carried by asperity contact. The model lets both shares vary over the starvation interval; as oil is depleted, the load shifts from the film to asperity contact. The local lubrication mode is classified by film thickness relative to a critical value of about 10 nm: above this value the contact is treated as EHL, below it as boundary lubrication.
For the example pair, the EHL friction coefficient is 0.0561 and the boundary friction coefficient is 0.1886. The boundary value is more than three times the EHL value, so the progressive loss of film load share is a strong thermal driver. The load-share coefficients are advanced through 300 discrete steps over 5 min, allowing the friction coefficient to rise as the oil film collapses.
The local heat flux generated by sliding is
$$
q=\alpha\zeta\mu_f p v_s
$$
where \(q\) is heat flux, \(\alpha\) is the heat-partition coefficient, \(\zeta\) is the energy-conversion coefficient, \(p\) is contact stress, and \(v_s\) is relative sliding velocity. Because both gears are made of the same material, the heat is divided equally between the two surfaces, so \(\alpha=0.5\). The heat is not distributed uniformly over the whole flank. A loaded tooth contact analysis (LTCA) supplies the contact ellipses along the mesh path, and the flux is applied within those ellipses. In the example, 43 contact ellipses cover one complete mesh cycle from entry to exit.
Because the contact moves periodically, the tooth surface receives a time-averaged flux rather than a continuous local flux. If \(a\) is the contact width, \(v_1\) and \(v_2\) are the tangential velocities, and the rotational periods are \(\tau_{p1}\) and \(\tau_{p2}\), the average fluxes on the two surfaces are
$$
\tau_1=\frac{a}{1000v_1},\quad \tau_2=\frac{a}{1000v_2},\quad \bar q_1=q\frac{\tau_1}{\tau_{p1}},\quad \bar q_2=q\frac{\tau_2}{\tau_{p2}}
$$
This averaging matters for a gear tooth: the local flash temperature and the body temperature are not the same. The finite-element model resolves the tooth body, while the contact-ellipse heat input represents the local frictional source.
Transient thermal model and boundary conditions
The tooth-body temperature field is obtained from the transient conduction equation
$$
\rho_g c_g \frac{\partial T}{\partial t}=\kappa_g\left(\frac{\partial^2 T}{\partial x^2}+\frac{\partial^2 T}{\partial y^2}+\frac{\partial^2 T}{\partial z^2}\right)
$$
where \(T\) is gear-body temperature, \(\kappa_g\) is thermal conductivity, \(\rho_g\) is density, and \(c_g\) is specific heat. The meshing face receives the frictional heat flux and loses heat by convection to the surrounding oil-air mixture. The end faces, tooth tip, root, and non-meshing faces are convective boundaries; the heat-transfer coefficient on the tip, root, and non-meshing faces is taken as one-third to one-half of the end-face value. The cut planes of the single-tooth sector use a symmetry condition: equal temperature and equal normal gradient on the two sides.
The cooling medium changes during starvation. Its density, specific heat, dynamic viscosity, and thermal conductivity are represented as linear mixtures of oil and air properties using a time-varying lubricant fraction. The oil properties used in the example are thermal conductivity 0.14 W/(m·°C), specific heat 2000 J/(kg·°C), density 998 kg/m³, and dynamic viscosity 0.04 Pa·s. The air-side reference temperature rises from 20 °C at the start of starvation to 69.5 °C at the end. Convection coefficients on the meshing face and end faces are obtained from Reynolds–Prandtl correlations for the mixture.
| Parameter | Large gear (74 teeth) | Small gear (27 teeth) |
|---|---|---|
| Number of teeth | 74 | 27 |
| Face width / mm | 38.5 | 38.5 |
| Module / mm | 3.85 | 3.85 |
| Shaft angle / ° | 68.77 | 68.77 |
| Pressure angle / ° | 20 | 20 |
| Spiral angle / ° | 30 | 30 |
The initial condition is not room temperature. It is the steady-state EHL temperature field of the same meshing pair, so the transient calculation begins from the thermal state reached during normal lubricated operation. The starved interval is then 5 min, solved at 1 s intervals from 0 to 300 s. A single tooth pair is modeled with eight-node hexahedral elements: 1,632 elements and 2,275 nodes. The 43 LTCA contact ellipses are drawn explicitly on the meshing face so that each heat-flux patch follows the actual contact footprint.
Temperature distribution at representative load
At 1,800 kW input power and 20,900 r/min, the computed maximum tooth-surface temperature reaches 407.54 °C after 5 min of starved operation. The hot spot is not at the same location on the two gears. On the driving gear it appears near the pitch circle toward the tooth tip; on the driven gear it appears near the pitch circle toward the tooth root. The analysis attributes this difference to the local sliding-speed distribution along the contact path, which controls the local heat flux.
During the 5-min interval, both the maximum and minimum tooth-surface temperatures rise. The curves continue upward at 300 s rather than leveling off, so the end of the calculation is still a transient state, not a thermal equilibrium. This is an important distinction for design: a short oil-loss event does not produce a fixed steady temperature that can be read from a conventional steady-state thermal map.
Power and speed sensitivity
Power and speed act in opposite directions in this model. The power sweep was run at constant speed and the speed sweep at constant power. The maximum temperatures change as follows:
| Input variable | Range | Change in driving-gear maximum | Change in driven-gear maximum | Trend |
|---|---|---|---|---|
| Input power, constant speed | 866 to 2000 kW | +65.7 °C | +45.12 °C | Maximum temperature rises |
| Rotational speed, constant power | 5000 to 20,000 r/min | −85.02 °C | −58.83 °C | Maximum temperature falls |
At constant speed, increasing power raises the contact stress and the frictional heat input. Because the lubricant is simultaneously being depleted, the friction coefficient also rises with time; the two effects reinforce each other. In the 5-min window, raising power from 866 kW to 2000 kW increases the maximum driving-gear temperature by 65.7 °C and the driven-gear maximum by 45.12 °C.
At constant power, increasing speed lowers the maximum temperature. Higher speed reduces the torque required to transmit the same power, and the convective heat-transfer coefficient increases. In this model, those effects outweigh the higher sliding velocity, so the net surface heat flux falls. Increasing speed from 5000 r/min to 20,000 r/min lowers the maximum driving-gear temperature by 85.02 °C and the driven-gear maximum by 58.83 °C. This result should be read as a thermal response at constant power; it does not imply that high speed is free of other mechanical or tribological risks.
Implications for clearance and emergency operation
For helicopter main-reducer spiral bevel gears, the practical significance of these results is the thermal input to clearance selection. Backlash must accommodate thermal growth without losing the contact pattern or closing the lubricant path. A starved-lubrication event adds a rapidly rising temperature field on top of the normal EHL state, and the hot spots are offset between the driving and driven gears. A clearance design based only on steady lubricated temperatures can therefore underestimate the local thermal distortion during an oil-loss event.
The computed 407.54 °C and the 5-min window are outputs for the stated geometry, load, speed, and boundary assumptions. They are not a dry-running life rating, a material limit, or a scuffing criterion. Extrapolation beyond 300 s would require additional physics: wear and surface roughening, time-dependent material properties, thermal-structural coupling of the contact ellipses, and a failure criterion for film collapse or scuffing. The present thermal model prescribes the contact ellipses from LTCA and does not update them with thermal distortion, so it should be interpreted as the thermal response under those contact conditions rather than a fully coupled thermomechanical prediction. The linear load-sharing and oil-air property schedules are also modeling assumptions that isolate the thermal effect rather than reproduce every starvation transient.
Within those limits, the analysis gives a clear design message: under starved lubrication, input power is a strong thermal lever, rotational speed changes the balance between torque and convective cooling, and the critical temperatures are transient and spatially localized. These are the quantities that must be carried into backlash and emergency-operation assessments for helicopter transmission spiral bevel gears.