Loss of lubrication as a coupled thermal-tribological problem
Helicopter transmission systems are designed to survive a loss of oil for a defined period: 30 min is a common minimum, and advanced main gearboxes target 60 min or more. The spiral bevel gears at the high-speed end of the main gearbox and in the tail gearbox are critical because they run at 103–104 r/min, carry high specific load, and depend on an oil film only a few micrometres thick. When the oil system is damaged, the film thins; asperity contact raises friction; less oil carries heat away; tooth temperature rises; thermal expansion reduces backlash and further concentrates the contact. The end point is scuffing, severe wear, or seizure.
Predicting the time to that end point requires more than a friction coefficient. The contact is a local point contact with time-varying geometry. The gearbox thermal field sets the oil temperature and the convective boundary. The two are mutually dependent: friction generates heat, heat changes film thickness and friction, and the surrounding oil–air mixture changes the rate at which heat leaves the teeth.
Three-tier coupling architecture
The analysis separates the problem into three domains that exchange only the quantities controlling the next domain. The tooth-flank elastohydrodynamic lubrication (EHL) domain resolves film pressure, film thickness, and local friction. The tooth-meshing domain resolves friction heat, windage, and surface heat-transfer coefficients. The gear-system thermal domain resolves the transient temperature of the gears, oil–air mixture, and housing.
| Domain | Characteristic time | Characteristic length | Information passed to the next domain |
|---|---|---|---|
| Tooth-flank EHL | 10−6–10−5 s | 10−6 m | Film thickness, pressure, friction coefficient, friction heat |
| Tooth meshing | 10−4–10−3 s | 10−4 m | Mesh heat source, windage loss, surface heat-transfer coefficients |
| Gear-system thermal | 10−2–100 s, characteristic 10−1 s | 10−1 m | Oil–air temperature, housing temperature, updated ambient boundary |
The separation is deliberate. The EHL contact cannot be resolved over a 90-min transient, and the full gearbox cannot be resolved at micrometre and microsecond resolution. The coupling passes friction and heat flux upward, and temperature and heat-transfer coefficients downward. The model therefore spans a time scale from 10−6 s to 10−1 s and a spatial scale from 10−6 m to 10−1 m while retaining the structure–force–lubrication–thermal chain that controls dry-running failure.
Full-film EHL on the spiral bevel tooth flank
At each meshing position, the contact is treated as a local steady EHL point contact. The entrainment velocity is not aligned with the contact ellipse, and both the ellipse dimensions and the angle between the entrainment direction and the ellipse axes change along the path. The Reynolds equation for the contact is
$$
\frac{\partial}{\partial x}\left(\frac{\rho h^3}{\eta}\frac{\partial p}{\partial x}\right)
+
\frac{\partial}{\partial y}\left(\frac{\rho h^3}{\eta}\frac{\partial p}{\partial y}\right)
=
12u\frac{\partial}{\partial x}\left(\frac{\rho h}{\eta}\right)
+
12v\frac{\partial}{\partial y}\left(\frac{\rho h}{\eta}\right)
$$
where x and y are the short- and long-axis directions of the contact ellipse, u and v are the entrainment components, p is film pressure, h is film thickness, ρ is oil density, and η is the equivalent viscosity. The film thickness includes the geometric gap and the elastic deformation of the two bodies:
$$
h=h_0+\frac{x^2}{2R_x}+\frac{y^2}{2R_y}+v(x,y)
$$
The viscosity and density are pressure- and temperature-dependent. A Roelands-type viscosity relation and a pressure–temperature density relation are used in the thermal EHL solution. The local friction coefficient is the ratio of the integrated shear stress to the integrated pressure over the contact area:
$$
f_{\mathrm{EHL}}
=
\frac{\iint_{\Omega}\tau(x,y)\,\mathrm{d}x\,\mathrm{d}y}
{\iint_{\Omega}p(x,y)\,\mathrm{d}x\,\mathrm{d}y}
$$
The demonstration case is a Gleason spiral bevel pair with 31 and 46 teeth, a 6.3 mm module, a 57.4 mm face width, a 90° shaft angle, a 35° mid-spiral angle, and a pinion speed of 5000 r/min. Along the contact path, the contact ellipse short axis is 0.82–0.89 mm and the long axis is 1.46–1.53 mm. The entrainment velocity falls from 6.98 to 4.30 m/s, at an angle of about 86° to the short axis. The computed full-film friction coefficient lies between 0.010 and 0.030 over the meshing points.
These values are not constant. The variation with meshing position is the reason a single friction coefficient is inadequate for thermal prediction: the heat source moves, changes magnitude, and changes direction relative to the tooth surface as the contact travels across the flank.
Starved contact: film retention and friction transition
Once oil supply is lost, the contact does not jump from full film to dry friction. The film is depleted by side flow and repeated contact, and the lubrication state passes through mixed and boundary regimes. The state is classified by the film ratio
$$
\Lambda=\frac{h_{\min}}{R_a}
$$
with full-film lubrication at Λ > 3, mixed lubrication at 0.8 < Λ < 3, and boundary lubrication at Λ < 0.8. To follow the transient, a film-retention parameter is introduced:
$$
R(n)=\frac{h_c}{h_{\mathrm{EHL}}}=n^{-1/\gamma},
\qquad n=\omega t
$$
where hc is the central film thickness after n rotations, hEHL is the full-film central thickness, ω is angular speed, t is time, and γ is an oil-film persistence exponent. For the present contact, γ ≈ 3.
For a surface roughness Ra = 0.2 μm, the central film at the boundary-lubrication threshold is about 0.16 μm. The full-film central thickness is about 0.5 μm, so R = 0.32 marks the onset of boundary lubrication and R < 0.01 is treated as dry contact.
The friction coefficient is blended between the EHL, boundary, and dry limits using the EHL load share ε:
$$
\varepsilon=\frac{1.21\Lambda^{0.64}}{1+0.37\Lambda^{1.26}}
$$
$$
f=
\begin{cases}
f_{\mathrm{EHL}}\varepsilon+(1-\varepsilon)f_b, & R>0.32,\\[4pt]
f_b\varepsilon+(1-\varepsilon)f_d, & 0.01<r<0.32,\\[4pt] $$="" &="" <p="" \end{cases}="" f_d,="" rwith the boundary-film friction coefficient fb = 0.11 and the dry-friction coefficient fd = 0.6. As R falls, ε falls and the friction coefficient rises from the EHL range toward 0.11 and then toward 0.6. The resulting time-dependent heat flux becomes the thermal boundary condition for the meshing model. The friction history is specific to the assumed roughness, oil retention, and load case; it is not a universal dry-friction law.
Meshing-scale CFD: windage, heat transfer, and heat sources
At the meshing tier, a steady CFD model is used. The domain contains the gear and the adjacent oil–air mixture, and the volumetric heat source is obtained from the friction power at the contact, summed over the contact points and averaged over a mesh period. The model also computes windage, which becomes significant at high rotational speed.
The windage calculation uses a moving-reference-frame formulation with a thin fluid region near the teeth. The dimensionless drag torque is
$$
C_m=\frac{M}{0.5\rho_{\mathrm{ref}}v_{\mathrm{ref}}^2 A_{\mathrm{ref}}L_{\mathrm{ref}}}
$$
and the windage power is
$$
P_w=M\omega
$$
where M is drag torque, ρref is reference density, vref is reference velocity, Aref is reference area, Lref is reference length, and ω is angular speed. The CFD solution provides drag coefficients for the convex flank, concave flank, other tooth surfaces, and the complete gear over 2000–8000 r/min. It also extracts surface heat-transfer coefficients for the meshing face, tooth flanks, and other gear surfaces.
These quantities are not merely corrections to the friction heat. At high speed, windage can be a comparable heat source, and the surface heat-transfer coefficients determine whether heat is carried into the bulk gear, into the oil–air mixture, or into the housing. The meshing model therefore supplies both a heat-source distribution and a convective boundary to the system model.
System thermal model and closure of the coupling loop
At the gear-system tier, a transient conjugate heat-transfer CFD model is used. To keep the computational cost manageable, the two gears are represented as disks, and the heat sources from the meshing tier are applied at the corresponding gear locations. The housing loses heat to ambient air by natural convection:
$$
h_{\mathrm{conv}}=0.3Re^{0.57}\frac{k}{L}
$$
where L is the housing diameter and k is the air thermal conductivity.
In the representative steady-field calculation used to initialize the coupling, the maximum temperature reached 91.6°C. The pinion was hotter than the wheel, and the internal oil–air temperature was about 58.7°C. The hottest housing region was near the bearing outer ring rather than on the gear teeth.
The loop is then closed. The internal air temperature from the system model replaces the room-temperature ambient in the meshing model, and the updated gear temperature provides the oil initial temperature for the EHL model. This iteration is essential: using room temperature as the ambient under-predicts the gear temperature and distorts the predicted dry-running time.
Predicted temperature history and failure time
With the time-varying friction heat source, the system temperatures rise rapidly in the first minutes, then more slowly, and finally flatten. The pinion is consistently hotter than the wheel. Table 2 gives the maximum surface temperatures at selected times.
| Time (s) | Pinion surface temperature (°C) | Wheel surface temperature (°C) |
|---|---|---|
| 600 | 108 | 95 |
| 1200 | 141 | 122 |
| 1800 | 176 | 152 |
| 2400 | 209 | 180 |
| 3000 | 239 | 205 |
| 3600 | 266 | 227 |
| 4200 | 289 | 246 |
| 4800 | 309 | 263 |
At 600 s the pinion is at 108°C; at 4800 s it reaches 309°C, while the wheel is at 263°C. The temperature rise lags the friction-coefficient rise because of the thermal inertia of the gears and housing. The pinion is the critical part: it has the higher sliding velocity and the smaller thermal mass, so it reaches the boundary-film limit first.
Boundary films provide limited protection. Adsorbed films typically break down near 150°C, while chemically reacted films can survive to about 300°C. When the pinion surface approaches and exceeds this range, the boundary film can no longer separate the asperities. The contact then moves into full dry friction, and wear, scuffing, and seizure follow rapidly. The coupled prediction places this transition at about 1.5 h after oil loss.
The housing remains much cooler because it is heated mainly by internal air and does not receive direct frictional heat. The calculation assumes the bearings remain fully lubricated. In a real helicopter gearbox, an emergency oil reservoir may keep a small amount of oil on the teeth, so the actual dry-running time can be longer than the conservative gear-contact limit.
What the result means for design
The analysis identifies the pinion as the first component to fail, not the housing. Design measures that reduce pinion tooth temperature—additional thermal mass, targeted cooling, surface treatments, or a small emergency oil flow—have more leverage than cooling the housing alone. The result also shows that the internal air temperature must be part of the boundary condition; a room-temperature assumption is not conservative in the right direction and can misplace the failure time.
The scale-coupling approach is useful because it connects quantities that can be computed or measured: film thickness and friction from EHL, windage and heat-transfer coefficients from meshing CFD, and transient temperatures from system CFD. It does not require a destructive loss-of-lubrication test to generate the full temperature history, although such tests remain necessary for validation.
Several limits should be stated. The EHL contact is treated as quasi-steady at each meshing position, so squeeze-film effects during very rapid film collapse are not resolved. The boundary and dry friction coefficients are fixed values, and the film-retention exponent is derived for a particular roughness and oil. Bearing failure, seal damage, oil coking, and structural distortion are outside the present thermal-tribological criterion. The predicted 1.5 h is therefore a contact-limited estimate, not a certification of the entire gearbox.
For spiral bevel gears under loss-of-lubrication conditions, the dominant chain is film thinning, friction rise, heat accumulation, and pinion boundary-film failure. The practical value of the multi-domain model is that it makes each link in that chain explicit and traceable, rather than reducing dry-running endurance to a single empirically assumed friction coefficient.
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