In a helicopter main reducer, the input-stage spiral bevel gear pair is exposed to a rapid loss of lubrication after oil-system damage. The contact then passes from a fully elastohydrodynamic (EHL) film through starved lubrication toward oil-free operation. This is a thermal transient problem: the oil fraction, the share of load carried by asperities, the friction coefficient, and the convective medium all change with time. The relevant design question is not only the final bulk temperature but the location of the hot spot and the sensitivity of that hot spot to power and speed during the emergency running interval.
A transient conduction model for this situation combines an EHL–boundary friction law with a loaded tooth contact analysis and a finite-element solution of the tooth body. For a 27/74-tooth spiral bevel gear pair operating at 1,800 kW and 20,900 r/min, the model gives a peak tooth temperature of 407.54 °C at the end of a 5 min starved interval. Power and speed shift this maximum in opposite directions: at constant speed, raising power increases the maximum, whereas at constant power, raising speed decreases it.
Lubrication Transition and Friction-Heat Input
Starved lubrication is intermediate between a fully established EHL film and dry contact. As the oil supply falls, the asperity peaks carry an increasing share of the normal load. The effective tooth-surface friction coefficient is represented as a load-share mixture:
$$\mu_f=c_b\mu_b+c_t\mu_t,\qquad c_b+c_t=1$$
where μb and μt are the boundary-lubrication and EHL friction coefficients, and cb and ct are the load shares carried by asperity contacts and the EHL film. In the calculation, the transition is approximated as linear over the 300 s starvation period: ct falls and cb rises while their sum remains unity. For the gear pair considered, the endpoint friction coefficients are μt = 0.0561 and μb = 0.1886.
The local friction heat flux at a mesh point is
$$q=\alpha\,\zeta\,\mu_f\,p\,v_s$$
where p is the contact pressure, vs is the relative sliding speed, ζ is the fraction of friction work converted to heat, and α is the heat-partition coefficient. Because the two gears have the same isotropic material, α is taken as 0.5. The flux is not spread uniformly over the whole tooth flank. It is applied over the contact ellipse at each mesh position, because p, vs, and the ellipse dimensions vary along the path of contact.
For a contact width a in the rolling direction and tangential velocities v1 and v2 at the mesh point, the residence times of the flux on the two surfaces are
$$T_1=\frac{a}{1000\,v_1},\qquad T_2=\frac{a}{1000\,v_2}$$
and the revolution-averaged fluxes follow from the rotation periods T11 = 60/n1 and T22 = 60/n2:
$$\bar q_1=\frac{T_1}{T_{11}}q,\qquad \bar q_2=\frac{T_2}{T_{22}}q$$
Here n1 and n2 are the rotational speeds in r/min. This pulse-averaging step is necessary because a mesh point is heated only while the contact ellipse passes over it.
Transient Conduction Model and Boundary Conditions
The tooth-body temperature t is obtained from the three-dimensional transient conduction equation
$$\rho_g c_g\frac{\partial t}{\partial \tau}=\kappa_g\left(\frac{\partial^2 t}{\partial x^2}+\frac{\partial^2 t}{\partial y^2}+\frac{\partial^2 t}{\partial z^2}\right)$$
where τ is time, κg is the gear-material thermal conductivity, ρg is density, and cg is specific heat. The initial condition is the steady EHL temperature field of the meshing pair, so the temperature rise is relative to a lubricated steady state rather than a cold start. The transient calculation runs from 0 to 300 s in 1 s steps.
On the meshing face, the boundary condition combines the friction heat input with convection to the surrounding oil–air mixture. The tooth end faces lose heat by convection. The tip, root, and non-working faces also exchange heat with the mixture; their surface coefficients are approximated as one-third to one-half of the end-face coefficient. At the tooth-slice boundaries, temperature and normal heat flux are matched to represent continuity within the gear body.
The convective medium changes during starvation. The oil fraction decreases and the air fraction increases, so density, specific heat, dynamic viscosity, and thermal conductivity are evaluated as time-dependent oil–air mixture properties. The air temperature is taken as 20 °C at the start and 69.5 °C at the end of the interval. Reynolds- and Prandtl-number correlations provide the meshing-face and end-face heat-transfer coefficients; the mixture properties are updated at each time step.
Gear Pair, Contact Loading, and Finite-Element Setup
The analysis uses a single tooth pair because each tooth in the pair experiences the same mesh cycle. The gear geometry is listed in Table 1. The lubricating-oil properties used for the mixture calculation are listed in Table 2.
| Parameter | Value |
|---|---|
| Number of teeth (driving/driven) | 27 / 74 |
| Face width | 38.5 mm |
| Module | 3.85 mm |
| Shaft angle | 68.77° |
| Pressure angle | 20° |
| Spiral angle | 30° |
| Property | Value |
|---|---|
| Thermal conductivity, κo | 0.14 W/(m·°C) |
| Specific heat, co | 2,000 J/(kg·°C) |
| Density, ρo | 998 kg/m3 |
| Dynamic viscosity, ηo | 0.04 Pa·s |
The loaded tooth contact analysis identifies 43 mesh positions from engagement to disengagement. A contact ellipse is drawn at each position in the finite-element mesh, and the local heat flux is applied over that ellipse. The tooth-body model uses 1,632 eight-node hexahedral elements and 2,275 nodes. The analysis is performed in ANSYS 16.0. The power sweep uses 866, 1,200, 1,500, 1,800, and 2,000 kW; the speed sweep uses 5,000, 8,000, 12,000, 15,000, and 20,000 r/min.
Temperature Field and Hot-Spot Location
For the baseline case of 1,800 kW and 20,900 r/min, the maximum tooth temperature at the end of the 5 min starved interval is 407.54 °C. The hot spot is not located at the same position on the two gears. On the driving gear it appears near the pitch circle but displaced toward the tooth tip; on the driven gear it appears near the pitch circle but displaced toward the tooth root. The controlling quantity is the local sliding speed: regions with higher vs receive a larger friction heat flux, so the temperature maximum follows the contact path rather than the geometric pitch point.
Both the maximum and minimum tooth temperatures increase throughout the 300 s interval. The field is still transient at oil-off; the calculation does not reach a steady thermal state within the 5 min window. The spatial gradient is also significant: the peak is confined to the vicinity of the contact path, while the rest of the tooth body remains cooler. This localization is the reason a single bulk temperature or a uniform-flux assumption can misplace the critical region.
Power and Speed Sensitivity
At constant speed, increasing power raises both the maximum and minimum tooth temperatures. Between 866 and 2,000 kW, the 5 min maximum rises by 65.7 °C on the driving gear and by 45.12 °C on the driven gear. The mechanism is a larger tangential load and contact stress, which increases q; at the same time, the oil fraction is falling, so the boundary-friction share and μf are rising.
At constant power, increasing speed lowers the tooth temperatures. From 5,000 to 20,000 r/min, the 5 min maximum falls by 85.02 °C on the driving gear and by 58.83 °C on the driven gear. At constant power, a higher speed means a lower torque, so the tangential load and friction coefficient decrease. The convective heat-transfer coefficient also rises with speed. Heat generation therefore does not grow in proportion to speed, while cooling improves, and the net effect is a lower tooth temperature.
| Five-minute starved interval | Driving gear maximum temperature change | Driven gear maximum temperature change |
|---|---|---|
| Power increase from 866 to 2,000 kW at constant speed | +65.7 °C | +45.12 °C |
| Speed increase from 5,000 to 20,000 r/min at constant power | −85.02 °C | −58.83 °C |
Design Implications and Validity Limits
These results provide a thermal basis for clearance and backlash design of spiral bevel gears in helicopter main reducers when the oil supply is interrupted. Because the hot spots are offset from the pitch point, the critical region should be defined from the contact-path thermal field, not from a single bulk-temperature estimate. The power and speed trends also show that emergency limits cannot be set from torque alone: at constant power, a higher speed can reduce the tooth temperature, but at constant speed, a higher power raises it sharply.
The quantitative values are numerical predictions for a specific geometry: 27 and 74 teeth, 38.5 mm face width, 3.85 mm module, 68.77° shaft angle, 20° pressure angle, and 30° spiral angle, with the oil properties in Table 2. The model stops at 300 s, when the starved interval reaches full oil-off. It does not predict dry-running life, scuffing threshold, or thermal deformation; those require coupled thermo-mechanical and tribological failure criteria. The linear time interpolation of EHL/boundary load sharing and of the oil–air mixture fraction is a reduced-order assumption. Extrapolation to other reducers, longer emergency intervals, or different lubricants should therefore be validated against dedicated tests.