Spiral bevel gears combine intersecting-shaft power transfer with a large contact ratio and smooth motion, which is why they are widely used in aero-engine central drives. The pitch-point contact is not uniformly severe, but at mesh entry near the pitch circle the sliding speed, normal load, and oil temperature can peak together. Under these conditions the lubricant film is thin enough for elastic deformation, viscosity–pressure response, and viscous heating to act simultaneously. An isothermal Hertzian or line-contact approximation is then inadequate: the film thickness, pressure distribution, and temperature rise must be solved as a coupled point-contact thermal elastohydrodynamic lubrication (EHL) problem.
Point-contact thermal EHL formulation
The tooth pair is represented as two equivalent ellipsoids in contact, reduced to one ellipsoid against a rigid plane. With \(R_x\) and \(R_y\) as the composite radii of curvature in the rolling and transverse directions, the film thickness including elastic deformation is
where \(h_0\) is the central film thickness, \(p\) is the oil-film pressure, and \(E’\) is the composite elastic modulus. The pressure field follows the generalized Reynolds equation for a Newtonian lubricant:
Here \(\rho\) and \(\eta\) are the local density and viscosity, and \(u_s=(u_1+u_2)/2\) is the entrainment velocity in the rolling direction. The viscosity is updated with a Roelands temperature–pressure relation and the density with a Dowson–Higginson relation. The energy equation includes convection, conduction across the film, compressive work, and viscous dissipation; the two solid surfaces are treated as moving heat sources with their own conduction equations. The contact is assumed to be fully flooded and smooth in the EHL solution, with surface roughness introduced only when interpreting the film ratio.
The equations are solved in dimensionless form on a domain of \(-2.5<x<1.5\) \(-2<y<2\).="" \(1\times10^{-6}\).="" according="" after="" and="" are="" at="" backflow="" below="" by="" composite="" convergence="" density="" destabilize="" differencing="" direct="" direction="" does="" each="" errors="" expands="" film="" flow="" high="" inlet="" is="" iteration,="" line="" local="" loop.="" not="" obtained="" p="" pressure="" region,="" relative="" requires="" scanning,="" so="" solution.
The numerical model was checked against optical interferometry measurements for point contacts. For a high-viscosity oil, the calculated central film thickness was 1.523 μm against a measured 1.413 μm, a deviation of about 7.8%. Across two lower-viscosity oils and entrainment speeds of 0.05 and 0.1 m/s, the maximum deviation was below 12%. This level of agreement is sufficient for comparative trends in pressure, film thickness, and temperature, although absolute film values retain the uncertainty of the optical measurements and the numerical discretization.
Reference operating point and film state
The analysis targets an aero-engine central drive. The pinion has 51 teeth and runs at 15,000 r/min; the gear has 38 teeth and runs at 20,132 r/min at the design point. Design power is 1,000 kW, and the oil supply temperature is controlled between 120 and 141 °C. The main geometry, material, and lubricant data are summarized in Table 1.
| Parameter | Value |
|---|---|
| Tooth numbers, pinion/gear | 51/38 |
| Pinion speed at design point | 15,000 r/min |
| Gear speed at design point | 20,132 r/min |
| Design power | 1,000 kW |
| Module | 4.0 mm |
| Pressure angle | 20° |
| Face width | 22 mm |
| Spiral angle at pitch circle | 35° |
| Gear material | 15Cr14Co12Mo5Ni |
| Elastic modulus | 208.4 GPa |
| Surface roughness \(Ra\) | 0.4 μm |
| Oil 4106 at 140 °C: viscosity | 2.345 mPa·s |
| Oil density at 140 °C | 882.3 kg/m³ |
| Oil specific heat at 140 °C | 2166 J/(kg·K) |
| Oil thermal conductivity at 140 °C | 0.139 W/(m·K) |
At 20,132 r/min, 1,000 kW, and 140 °C, the maximum pressure is 0.369 GPa, the minimum film thickness is 0.524 μm, and the maximum temperature rise is 10.829 °C. The pressure profile does not show the two well-separated peaks familiar from classical EHL. The secondary peak produced by film necking moves toward the inlet and merges with the Hertz pressure peak. The central film is also not strongly flattened; it retains a slight depression rather than the usual wide plateau. Pressure and temperature contours follow similar trends, while film thickness varies inversely.
The central flat film reappears only when the lubricant becomes substantially more viscous. At an inlet temperature of 80 °C, where the viscosity is 7.108 mPa·s, the central region becomes noticeably flatter. At 140 °C and 2.345 mPa·s, the high-speed, heavy-load contact remains sharp-edged and dominated by the merged pressure peak.
With both surfaces at \(Ra=0.4\) μm, the composite roughness is \(\sigma=\sqrt{0.4^2+0.4^2}=0.566\) μm. The film ratio is therefore
giving \(\lambda\approx0.93\) at the design point and \(\lambda\approx0.71\) at 60% design speed. Both values are below unity, so the film cannot fully separate the asperities under the smooth-surface, fully flooded assumption. This is a boundary or mixed-contact warning, not proof of inlet starvation. At 120 °C, where the minimum film is 0.649 μm, \(\lambda\approx1.15\), placing the contact in the mixed-lubrication range.
Speed and power effects
At fixed power, increasing speed from 12,079 to 20,132 r/min raises the minimum film from 0.402 to 0.524 μm, an increase of about 30.4%. The maximum pressure rises from 0.277 to 0.369 GPa, about 33.4%, and the maximum temperature rise rises from 3.581 to 10.829 °C, about 202%. The speed increase is 66.7%, so the thermal response is much stronger than the film response.
| Speed (r/min) | \(h_{\min}\) (μm) | \(p_{\max}\) (GPa) | \(\Delta T_{\max}\) (°C) |
|---|---|---|---|
| 12,079 | 0.402 | 0.277 | 3.581 |
| 14,092 | 0.432 | 0.303 | 5.161 |
| 16,106 | 0.463 | 0.327 | 6.640 |
| 18,118 | 0.494 | 0.349 | 8.855 |
| 20,132 | 0.524 | 0.369 | 10.829 |
The pressure rise with speed may appear counterintuitive because, at fixed power, higher speed means lower torque and lower contact load. The reported maximum pressure is not controlled by the Hertzian load alone. At high speed, the dynamic pressure and the necking-induced secondary peak become dominant, and their merger with the Hertz peak raises the maximum. At lower speed, the secondary peak weakens and moves toward the outlet, allowing the first Hertz peak to emerge. Because both pressure and speed increase heat generation, the maximum temperature rise follows the pressure trend and occurs near the merged pressure peak.
At fixed speed, increasing power from 600 to 1,000 kW has a much weaker effect on the peak values. The minimum film decreases only from 0.536 to 0.524 μm, the maximum pressure decreases from 0.389 to 0.369 GPa, and the maximum temperature rise decreases from 11.892 to 10.823 °C. Higher load suppresses the film-necking instability, so the secondary peak becomes less pronounced even though the contact load is higher. This does not mean that higher power improves load capacity; the minimum film still declines, and the load is carried over a more heavily stressed contact.
| Power (kW) | \(h_{\min}\) (μm) | \(p_{\max}\) (GPa) | \(\Delta T_{\max}\) (°C) |
|---|---|---|---|
| 600 | 0.536 | 0.389 | 11.892 |
| 700 | 0.531 | 0.384 | 11.614 |
| 800 | 0.528 | 0.378 | 11.340 |
| 900 | 0.526 | 0.374 | 11.077 |
| 1,000 | 0.524 | 0.369 | 10.823 |
Lubricant temperature and elastic modulus
Viscosity is varied through the inlet oil temperature. At 80, 100, 120, 140, and 160 °C, the corresponding viscosities are 7.108, 4.595, 3.191, 2.345, and 1.789 mPa·s. Lowering the inlet temperature from 140 to 80 °C more than triples the viscosity, raises the minimum film from 0.524 to 1.104 μm, raises the maximum pressure from 0.369 to 0.790 GPa, and raises the maximum temperature rise from 10.827 to 55.712 °C. The pressure peak is not located at the minimum-film point; it sits near the film depression, where local elastic recovery and necking control the film shape.
| Inlet temperature (°C) | Viscosity (mPa·s) | \(h_{\min}\) (μm) | \(p_{\max}\) (GPa) | \(\Delta T_{\max}\) (°C) | Inlet + \(\Delta T_{\max}\) (°C) |
|---|---|---|---|---|---|
| 80 | 7.108 | 1.104 | 0.790 | 55.712 | 135.712 |
| 100 | 4.595 | 0.846 | 0.601 | 34.117 | 134.117 |
| 120 | 3.191 | 0.649 | 0.463 | 18.969 | 138.969 |
| 140 | 2.345 | 0.524 | 0.369 | 10.827 | 150.827 |
| 160 | 1.789 | 0.441 | 0.308 | 7.230 | 167.230 |
The inlet-plus-rise index shows the trade-off. The 80 °C case has the largest local temperature rise, 55.712 °C, but the lowest combined value, 135.712 °C. The 160 °C case has a much smaller rise, 7.230 °C, but a higher combined value, 167.230 °C. A colder, more viscous oil therefore thickens the film and lowers the bulk contact temperature, while increasing the peak pressure and the local viscous heating.
Elastic modulus acts mainly on the pressure and temperature peaks. As \(E/E_0\) increases from 0.90 to 1.10 around the 208.4 GPa baseline, the minimum film decreases from 0.539 to 0.513 μm, the maximum pressure rises from 0.321 to 0.421 GPa, and the maximum temperature rise rises from 9.452 to 12.397 °C. A stiffer gear material reduces elastic conformity, concentrates pressure, and increases viscous heating; the film thickness changes by less than 5% over this range.
| \(E/E_0\) | \(h_{\min}\) (μm) | \(p_{\max}\) (GPa) | \(\Delta T_{\max}\) (°C) |
|---|---|---|---|
| 0.90 | 0.539 | 0.321 | 9.452 |
| 0.95 | 0.531 | 0.345 | 10.115 |
| 1.00 | 0.524 | 0.369 | 10.823 |
| 1.05 | 0.518 | 0.395 | 11.581 |
| 1.10 | 0.513 | 0.421 | 12.397 |
Sensitivity ranking and lubrication regime
To compare the influence of speed, power, viscosity, and elastic modulus, each input is normalized at the design point and a dimensionless sensitivity coefficient is calculated:
Here \(a_k\) is an input parameter and \(M_k\) is the corresponding film characteristic. The superscript \(*\) denotes the design reference: 20,132 r/min, 1,000 kW, and 140 °C. A coefficient of \(|S|=1\) means that a 10% input change produces a 10% output change; the sign indicates whether the response is positive or negative. The coefficients are averaged over the operating range to rank the parameters in Table 6.
| Output characteristic | Ranking, strongest to weakest | Dominant behavior |
|---|---|---|
| Minimum film thickness \(h_{\min}\) | Viscosity > speed > elastic modulus > power | Viscosity and speed control film formation; power has a weak effect. |
| Maximum pressure \(p_{\max}\) | Elastic modulus > viscosity > speed > power | Modulus controls conformity and peak pressure; viscosity and speed modify the secondary peak. |
| Maximum temperature rise \(\Delta T_{\max}\) | Viscosity > speed > elastic modulus > power | Viscosity and speed control heat generation; modulus changes local pressure concentration. |
The ranking is not the same for every output. Elastic modulus is the strongest factor for maximum pressure, while viscosity and speed are stronger for minimum film and maximum temperature rise. A 10% change in modulus, viscosity, or speed can alter the temperature rise by more than 10%. Viscosity or speed changes of 10% can alter the film thickness by at least 5%. Power is consistently the weakest parameter in this high-speed regime, with sensitivities below about 0.25.
For film thickness, lowering the oil temperature and increasing speed are effective. For peak pressure and flash temperature, lowering the modulus and avoiding unnecessary speed increases are more effective. Speed is therefore a coupled variable: it thickens the film but also raises the merged pressure peak and the temperature rise. Oil temperature and viscosity are the most consistent levers because they simultaneously affect film formation and heat generation. Modulus is the primary material lever for pressure and temperature, but softer gear materials must still satisfy structural and fatigue requirements.
The reference calculation assumes a fully flooded, smooth contact. Consequently, \(\lambda<1\) at 140 °C should be read as a risk of asperity contact and boundary lubrication, not as a direct prediction of oil starvation. At 120 °C the contact moves into the mixed-lubrication range, with \(\lambda\approx1.15\). The practical implication is that thermal EHL analysis of high-speed spiral bevel gears cannot rely on a single Hertzian pressure peak or on the classical flat central film. The merged peak, the weak central plateau, and the strong viscosity sensitivity all belong to the same high-speed, high-temperature contact regime.
</x