Spiral bevel gears transmit high torque between intersecting shafts, and their pitch-point contact is an elliptical point contact subjected to high entrainment speed, high normal load, and elevated oil temperature. In an aeroengine central drive, the lubricant may enter the mesh at 120–141 °C, so the film cannot be treated as isothermal. The relevant problem is thermal elastohydrodynamic lubrication (EHL): elastic deformation, pressure- and temperature-dependent viscosity, shear heating, and film thickness must be solved together.
High-speed optical interferometry has shown that when entrainment speed approaches 50 m/s, the classical flat central film can disappear and become a converging wedge; deviations from isothermal film-thickness predictions can reach 75%. Most gear EHL studies, however, remain below 30 m/s and moderate thermal loads. The aeroengine spiral bevel gear case is therefore a high-speed, high-temperature, heavy-load point-contact problem in which the pressure peak, film shape, and temperature rise are strongly coupled.
Contact Model and Governing Equations
At the pitch point, the tooth pair is reduced to an equivalent ellipsoid on a rigid plane. The composite radii in the entrainment direction \(x\) and the side-leakage direction \(y\) are
$$R_x=\frac{R_{x1}R_{x2}}{R_{x1}+R_{x2}},\qquad R_y=\frac{R_{y1}R_{y2}}{R_{y1}+R_{y2}}$$
The film thickness includes the central separation, the undeformed parabolic gap, and the elastic deflection of the contacting bodies:
$$h(x,y)=h_0+\frac{x^2}{2R_x}+\frac{y^2}{2R_y}+\frac{2}{\pi E’}\iint_{\Omega}\frac{p(s,t)}{\sqrt{(x-s)^2+(y-t)^2}}\,ds\,dt$$
where \(h_0\) is the central film thickness, \(p\) is the film pressure, \(E’\) is the composite elastic modulus, and \(\Omega\) is the contact domain. The Hertz reference pressure is \(p_H=3W/(2\pi ab)\), with \(W\) the normal load and \(a\), \(b\) the contact semi-axes.
The pressure field follows a generalized two-dimensional Reynolds equation for a Newtonian lubricant:
$$\frac{\partial}{\partial x}\left(\frac{\rho h^3}{\eta}\frac{\partial p}{\partial x}\right)+\frac{\partial}{\partial y}\left(\frac{\rho h^3}{\eta}\frac{\partial p}{\partial y}\right)=12u_s\frac{\partial(\rho h)}{\partial x}$$
Here \(\rho\) is density, \(\eta\) is viscosity, and \(u_s=(u_1+u_2)/2\) is the entrainment velocity. The boundary conditions impose \(p=0\) at the inlet and outlet edges and \(p\ge 0\) inside the domain. Viscosity follows the Roelands relation, with \(z=0.68\) and \(s=-1.1\):
$$\eta=\eta_0\exp\left\{(\ln\eta_0+9.67)\left[\left(1+5.1\times10^{-9}p\right)^z\left(\frac{T-138}{T_0-138}\right)^s-1\right]\right\}$$
Density follows the Dowson–Higginson relation, with \(D=-0.00065\ \mathrm{K^{-1}}\):
$$\rho=\rho_0\left[1+\frac{0.6\times10^{-9}p}{1+1.7\times10^{-9}p}+D(T-T_0)\right]$$
The pressure field must also satisfy the load balance:
$$W=\iint_{\Omega}p(x,y)\,dx\,dy$$
The film temperature is obtained from the energy equation, including convection, conduction, compressive work, and viscous dissipation:
$$c_p\rho\left(u\frac{\partial T}{\partial x}+v\frac{\partial T}{\partial y}+w\frac{\partial T}{\partial z}\right)=k\frac{\partial^2 T}{\partial z^2}-\frac{T}{\rho}\frac{\partial\rho}{\partial T}\left(u\frac{\partial p}{\partial x}+v\frac{\partial p}{\partial y}\right)+\eta\left[\left(\frac{\partial u}{\partial z}\right)^2+\left(\frac{\partial v}{\partial z}\right)^2\right]$$
The solid surfaces are treated as moving heat sources, and the interfacial temperature condition couples the film to the gear and pinion bodies. In the numerical solution, the domain is normalized as \(-2.5<x<1.5\) \(-2<y<2\),="" \(1\times10^{-6}\).="" \(p="p/p_H\)." \(x="x/a\)," \(y="y/b\)," a="" according="" and="" appear="" are="" at="" avoid="" below="" between="" by="" can="" composite="" convergence="" density="" differencing="" direct="" direction="" errors="" film="" flow="" high="" in="" inlet;="" instability="" is="" iteration,="" line="" local="" near="" numerical="" p="" pressure="" relative="" requires="" reverse-flow="" scanning="" selected="" solution.
Validation against optical interferometry gave a measured central film of 1.413 µm and a calculated value of 1.523 µm at a lubricant viscosity of 5.41 Pa·s, a difference of about 7.8%. For lower-viscosity cases at entrainment speeds of 0.05 and 0.1 m/s, the maximum difference was below 12%.
Table 1. Validation of the point-contact thermal EHL formulation against optical film-thickness measurements.
| Lubricant viscosity (Pa·s) | Entrainment speed (m/s) | Measured film (µm) | Calculated film (µm) |
|---|---|---|---|
| 0.1765 | 0.05 | 0.087 | 0.090 |
| 0.1765 | 0.10 | 0.151 | 0.162 |
| 0.3028 | 0.05 | 0.133 | 0.150 |
| 0.3028 | 0.10 | 0.222 | 0.197 |
Design-Point Contact and Film State
The analyzed spiral bevel gear pair has 51 pinion teeth and 38 gear teeth, a module of 4.0 mm, a pressure angle of 20°, a face width of 22 mm, and a spiral angle of 35° at the pitch circle. The pinion runs at 15,000 r/min, the gear at 20,132 r/min, and the design power is 1,000 kW. The gear material is 15Cr14Co12Mo5Ni with a Young’s modulus of 208.4 GPa and a surface roughness \(R_a=0.4\ \mu\mathrm{m}\). The lubricant is 4106 oil, with a viscosity of 2.345 mPa·s and a density of 882.3 kg/m³ at 140 °C.
Table 2. Pitch-point contact and lubricant parameters.
| Item | Value |
|---|---|
| Tooth count, pinion / gear | 51 / 38 |
| Module | 4.0 mm |
| Pressure angle | 20° |
| Face width | 22 mm |
| Spiral angle at pitch | 35° |
| Pinion / gear speed | 15,000 / 20,132 r/min |
| Design power | 1,000 kW |
| Oil supply temperature | 120–141 °C; design point 140 °C |
| Gear material | 15Cr14Co12Mo5Ni |
| Young’s modulus | 208.4 GPa |
| Surface roughness \(R_a\) | 0.4 µm |
| Lubricant | 4106 oil |
| Viscosity at 140 °C | 2.345 mPa·s |
| Density at 140 °C | 882.3 kg/m³ |
| Specific heat | 2166 J kg⁻¹ K⁻¹ |
| Thermal conductivity | 0.1392 W m⁻¹ K⁻¹ |
At the design point, the calculated 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 and temperature-rise distributions are qualitatively similar, while the film-thickness distribution is inverse to them: regions of high pressure and high temperature rise correspond to thin film. Unlike classical point-contact EHL, two distinct pressure peaks are not resolved. The secondary pressure peak produced by film necking moves toward the inlet at high speed and merges with the Hertz pressure peak. The central film is also not a pronounced flat plateau; only a slight film dimple remains, and the overall film profile becomes sharper.
With a composite surface roughness \(\sigma=\sqrt{\sigma_1^2+\sigma_2^2}=0.5657\ \mu\mathrm{m}\), the film ratio is
$$\lambda=\frac{h_{\min}}{\sigma}$$
At the design point, \(\lambda=0.93\); at 60% design speed, \(h_{\min}=0.402\ \mu\mathrm{m}\) and \(\lambda=0.71\). Both values are below unity, so the contact is in an asperity-contact regime rather than full-film lubrication. At an oil inlet temperature of 120 °C, \(h_{\min}=0.649\ \mu\mathrm{m}\) and \(\lambda\approx1.1\), which corresponds to mixed lubrication. This shift from asperity contact to mixed lubrication is one of the most practical outcomes of the thermal EHL analysis.
Speed and Power Effects
At a fixed power of 1,000 kW, increasing speed from 12,079 to 20,132 r/min raises the minimum film thickness from 0.402 to 0.524 µm, the maximum pressure from 0.277 to 0.369 GPa, and the maximum temperature rise from 3.581 to 10.829 °C. The speed increase is 66.7%, the film-thickness increase is 30.4%, the pressure increase is 33.4%, and the temperature-rise increase is 202.2%.
Table 3. Oil-film characteristic parameters at different rotational speeds and a fixed power of 1,000 kW.
| 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 is counterintuitive if only torque is considered: at fixed power, a higher speed means lower torque and lower nominal contact load. The calculated maximum pressure nevertheless increases because the high entrainment speed strengthens the dynamic pressure and shifts the necking-induced secondary peak toward the inlet, where it merges with the Hertz peak. At lower speed, the secondary peak moves toward the outlet and the first Hertz peak becomes more visible. The temperature rise follows the pressure trend because both high pressure and high shear rate generate heat; the temperature-rise maximum coincides with the secondary pressure region.
The sensitivity coefficient of minimum film thickness to speed is about 0.58 and remains stable, while the pressure sensitivity decreases slightly from 0.63 to 0.55 as speed rises. The temperature-rise sensitivity is always greater than unity, ranging from 1.67 to 1.93, with a maximum near 80% design speed. Speed is therefore a strong lever for film thickness and an even stronger lever for temperature rise.
At a fixed speed of 20,132 r/min, increasing power from 600 to 1,000 kW lowers the minimum film thickness from 0.536 to 0.524 µm, the maximum pressure from 0.389 to 0.369 GPa, and the maximum temperature rise from 11.892 to 10.823 °C.
Table 4. Oil-film characteristic parameters at different powers and a fixed speed of 20,132 r/min.
| 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 |
The pressure decrease with increasing power is a load-dependent film-shape effect. A higher contact load suppresses necking and reduces the secondary pressure peak, which dominates the maximum pressure under these high-speed conditions. The film thickness does not increase as pressure falls because the lower pressure reduces lubricant viscosity and load capacity, while the higher load increases elastic deformation. The net result is a slight film reduction. The sensitivity coefficients are small: about 0.02–0.05 for minimum film thickness, 0.08–0.09 for maximum pressure, and 0.24 for maximum temperature rise. Power is therefore a fine-tuning parameter rather than a primary film-control lever.
Viscosity and Oil Temperature
At fixed speed and power, changing the oil inlet temperature changes viscosity. The inlet temperatures 80, 100, 120, 140, and 160 °C correspond to viscosities of 7.108, 4.595, 3.191, 2.345, and 1.789 mPa·s. Lower temperature and higher viscosity raise the minimum film thickness from 0.441 to 1.104 µm, the maximum pressure from 0.308 to 0.790 GPa, and the maximum temperature rise from 7.230 to 55.712 °C. The total oil temperature, defined as inlet temperature plus temperature rise, moves in the opposite direction: it is 167.230 °C at a 160 °C inlet and 135.712 °C at an 80 °C inlet.
Table 5. Oil-film characteristic parameters at different lubricant viscosities and inlet temperatures.
| Viscosity (mPa·s) | Inlet temperature (°C) | \(h_{\min}\) (µm) | \(p_{\max}\) (GPa) | \(\Delta t_{\max}\) (°C) | Total temperature (°C) |
|---|---|---|---|---|---|
| 7.108 | 80 | 1.104 | 0.790 | 55.712 | 135.712 |
| 4.595 | 100 | 0.846 | 0.601 | 34.117 | 134.117 |
| 3.191 | 120 | 0.649 | 0.463 | 18.969 | 138.969 |
| 2.345 | 140 | 0.524 | 0.369 | 10.827 | 150.827 |
| 1.789 | 160 | 0.441 | 0.308 | 7.230 | 167.230 |
The high-viscosity case produces a thick film and a high local temperature rise, but the bulk oil remains cooler because the inlet temperature is low. Conversely, a high inlet temperature reduces the local temperature rise but raises the total oil temperature, which is unfavorable for coking and oil degradation. This distinction matters in aeroengine lubrication: lowering the inlet temperature improves film thickness and reduces asperity contact, but it also increases shear heating within the contact.
The sensitivity coefficients to viscosity are about 0.55–0.66 for minimum film thickness, 0.56–0.71 for maximum pressure, and 1.40–2.24 for maximum temperature rise. The total-temperature sensitivity is smaller, 0.05–0.46, but increases rapidly as the oil becomes hotter and thinner. The maximum-pressure sensitivity reaches its peak near a 120 °C inlet, so a temperature change around that condition has the largest effect on peak pressure. At lower inlet temperature and higher viscosity, the pressure peak does not coincide exactly with the minimum film location; it aligns more closely with the film dimple because the minimum film is controlled by elastic deformation and necking rather than by pressure alone. The thermal response also lags the pressure response as the oil becomes more viscous.
Elastic Modulus
Varying the gear modulus by ±10% around the design value changes the contact mechanics without changing the lubricant. Increasing the modulus from 0.90 to 1.10 times the reference value raises the maximum pressure from 0.321 to 0.421 GPa and the maximum temperature rise from 9.452 to 12.397 °C, while the minimum film thickness falls from 0.539 to 0.513 µm.
Table 6. Oil-film characteristic parameters at different elastic moduli, expressed relative to the design value.
| \(E/E_0\) | \(h_{\min}\) (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 |
A higher modulus reduces elastic deformation and the effective contact area, so the same load is carried over a smaller region at higher pressure. The higher pressure and shear rate increase heat generation, and the film thins. The sensitivity coefficient is about 0.21–0.29 for minimum film thickness, 1.30–1.41 for maximum pressure, and 1.27–1.45 for maximum temperature rise. Elastic modulus is therefore the strongest single lever for maximum pressure and a strong lever for temperature rise, but only a moderate lever for film thickness.
Sensitivity Ranking and Design Levers
The normalized sensitivity coefficient is defined as
$$S(a_k)=\frac{(M_k-M^*)/M^*}{(a_k-a^*)/a^*},\qquad k=1,\ldots,n$$
where \(M\) is an oil-film characteristic parameter, \(a\) is an input parameter, and the asterisk denotes the design-point reference. An absolute value of 1 means that a 10% change in the input produces a 10% change in the output. The signs indicate positive or negative correlation; the magnitudes give the ranking.
Table 7. Ranges of normalized sensitivity coefficients and the resulting influence ranking.
| Input parameter | \(S(h_{\min})\) | \(S(p_{\max})\) | \(S(\Delta t_{\max})\) | Rank for \(h_{\min}\) | Rank for \(p_{\max}\) | Rank for \(\Delta t_{\max}\) |
|---|---|---|---|---|---|---|
| Viscosity | 0.55–0.66 | 0.56–0.71 | 1.40–2.24 | 1 | 2 | 1 |
| Speed | 0.58–0.59 | 0.55–0.63 | 1.67–1.93 | 2 | 3 | 2 |
| Elastic modulus | 0.21–0.29 | 1.30–1.41 | 1.27–1.45 | 3 | 1 | 3 |
| Power | 0.02–0.05 | 0.08–0.09 | 0.24–0.25 | 4 | 4 | 4 |
The ranking depends on which film metric is being controlled. For maximum pressure, the order is elastic modulus, viscosity, speed, and power. For minimum film thickness and maximum temperature rise, the order is viscosity, speed, elastic modulus, and power. Viscosity and speed each change 10% can alter the temperature rise by more than 10%, and a 10% change in either viscosity or speed can change the film thickness by at least about 5%. Elastic modulus changes of 10% also produce more than 10% changes in maximum pressure and temperature rise. Power has the weakest influence on all three metrics.
For the present spiral bevel gear, the design-point film ratio at 140 °C is 0.93 and the idle-condition film ratio is 0.71, both indicating asperity contact. At 120 °C, the film ratio rises to about 1.1 and the contact enters mixed lubrication. Lowering the oil working temperature, selecting a lower-modulus gear material within strength limits, or increasing speed can improve the lubrication state. These levers are not equivalent: lower temperature raises viscosity and film thickness but increases local shear heating; a softer material lowers pressure and temperature but must satisfy structural requirements; higher speed thickens the film but strongly increases temperature rise. The practical design choice is therefore a balance among film thickness, peak pressure, bulk oil temperature, and coking risk, with viscosity and speed as the primary controls and power as a secondary adjustment.
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