Spiral bevel gears are widely used in high-speed, heavy-load transmissions because their curved teeth provide gradual meshing, high contact ratio, and large load capacity. In aerospace propulsion and other high-power drivetrains, however, the same contact geometry that improves motion transmission also concentrates frictional heat near the pitch region. If this heat is not removed effectively, the tooth surface temperature rises, thermal stress develops, clearance changes, and the risk of scuffing or seizure increases. The bulk temperature of the tooth body is therefore a key design variable: it governs thermal distortion and clearance variation, although it is distinct from the short-lived flash temperature at individual asperities.
Heat Balance and Boundary Conditions
For a tooth sector of a spiral bevel gear, transient heat conduction in the tooth body is described by
where k is the thermal conductivity of the gear material, ρ is density, C is specific heat, T is temperature, and t is time. The boundary conditions divide the tooth surface into several thermal regions. On the meshing surface, region n, frictional heat enters the tooth while heat is simultaneously lost by convection:
On the tooth root, tooth tip, and non-meshing flanks, region t, the boundary condition is convective:
On the gear end face, region s, the corresponding condition is
The lower boundary of the tooth, region d, is treated as adiabatic:
The cut planes of the tooth sector, regions P and Q, satisfy periodic conditions:
The initial condition is
Here, n is the outward normal direction, q is the frictional heat flux entering the meshing surface, T0 is the ambient air temperature, and αt′ and αs′ are the convective heat-transfer coefficients on the tooth surfaces and end faces, respectively.
Frictional Heat Input at the Mesh
During meshing, the tooth surfaces move with combined rolling and sliding. Rolling friction is smaller than sliding friction, so the thermal analysis considers sliding friction as the dominant heat source. The frictional heat generated at the contact is
where f is the sliding friction coefficient, p0 is the maximum contact pressure, and v is the sliding velocity. For gear contacts, empirical data place f in the range of 0.045 to 0.065. The heat is partitioned between the two meshing gears by a coefficient ξ:
The corresponding heat fluxes into the driving and driven gears are
where ω1 and ω2 are the angular velocities of the driving and driven gears, v1 and v2 are their linear velocities at the mesh, and b1 is the Hertzian contact half-width. The contact half-width is obtained from
where R is the contact radius, E1 and E2 are the elastic moduli of the two gears, and μ1 and μ2 are their Poisson’s ratios.
Finite Element Model and Operating Case
A transient finite element model of the spiral bevel gear pair is solved with ANSYS. The driving gear has 25 teeth, and the mating driven gear has 60 teeth. The transmitted power is 210 kW, the driving-gear speed is 4400 r/min, and the gear material is 45 steel. The ambient temperature is 52 °C. The calculation follows a transition from a lubricated state at t = 0 s to a loss-of-lubrication state at 60, 120, and 180 s.
| Parameter | Value |
|---|---|
| Number of teeth | 25 |
| Gear module | 4 mm |
| Face width | 28 mm |
| Pressure angle | 20° |
| Addendum coefficient | 1 |
| Clearance coefficient | 0.25 |
| Spiral direction | Left-hand |
| Property | Value |
|---|---|
| Thermal conductivity | 50 W/(m·°C) |
| Elastic modulus | 200 GPa |
| Density | 7870 kg/m³ |
| Specific heat | 468 J/(kg·K) |
| Convective heat-transfer coefficient | 1100 W/(m²·°C) |
Transient Temperature Evolution
The highest bulk temperature occurs at the centre of the meshing contact, while the lowest temperature occurs at the gear end face. The tooth body remains cooler than the meshing surface but warmer than the end face. Along the face-width direction, the bulk temperature follows a parabolic profile.
| Time (s) | Maximum bulk temperature (°C) |
|---|---|
| 0 | 121.248 |
| 60 | 134.888 |
| 180 | 162.235 |
From t = 0 s to t = 180 s, the peak bulk temperature rises from 121.248 °C to 162.235 °C, an increase of about 41 K. The high-temperature zone also expands with time and moves toward the tooth tip. Because heat generation continues while the lubricated cooling path is lost, the temperature gradient steepens and thermal accumulation becomes more severe. If the peak temperature reaches the material’s scuffing limit, the tooth surface can no longer maintain reliable operation.
Effect of Face Width
The face-width study compares 18 mm, 22 mm, and 30 mm cases by examining the bulk temperature along the tooth height at the middle of the face and the temperature distribution along the face width. The maximum bulk-temperature rise increases with face width. Relative to the 18 mm case, the rise at the maximum-temperature location is 357.81% higher for 22 mm and 1,289% higher for 30 mm.
The trend is linked to two effects. A wider face increases the frictional heat generated over the meshing region. At the same time, the end-face cooling capacity becomes less effective relative to the larger tooth volume, so heat removal cannot keep pace with heat input. Increasing face width therefore improves contact load capacity but can worsen the bulk thermal state under loss of lubrication.
Effect of Gear Module
The module study compares 3 mm, 4 mm, and 5 mm cases by examining the bulk temperature at the pitch circle along the face width. In contrast to face width, increasing the module reduces the maximum bulk-temperature rise. Relative to the 3 mm case, the rise is 50.98% lower for 4 mm and 64.24% lower for 5 mm.
The analysis attributes this reduction to the combined effect of a larger tooth volume, a changed heat-conduction path, and an altered convective heat-transfer area. Under loss of lubrication, the variation in convective heat-transfer coefficient with module is smaller than the variation in frictional heat flow, so the net result is a lower maximum bulk-temperature rise as the module increases from 3 mm to 5 mm.
| Parameter | Baseline | Comparison | Change in maximum bulk-temperature rise |
|---|---|---|---|
| Face width | 18 mm | 22 mm | +357.81% |
| Face width | 18 mm | 30 mm | +1,289% |
| Gear module | 3 mm | 4 mm | −50.98% |
| Gear module | 3 mm | 5 mm | −64.24% |
Positive values indicate an increase, and negative values indicate a decrease. The percentages refer to the rise at the maximum-temperature location, not to the absolute temperature. The two parameter groups have separate baselines: 18 mm for face width and 3 mm for module.
Design Implications and Scope
The transient analysis shows that the thermal behavior of a spiral bevel gear under loss of lubrication is controlled by the balance between frictional heat input and heat removal through the tooth body, flanks, and end faces. The contact centre is the thermal hot spot, and the hot zone spreads toward the addendum as time progresses. Face width and gear module influence this balance in opposite directions: increasing face width raises the maximum bulk-temperature rise, whereas increasing module lowers it within the studied range.
These results suggest that face width and module should be evaluated not only for contact stress and bending strength but also for their effect on bulk temperature in high-speed, heavy-load spiral bevel gears. A wider face may reduce contact stress while increasing thermal risk under poor lubrication; a larger module may reduce the bulk-temperature rise but must still satisfy other transmission requirements.
The conclusions are tied to the stated geometry, 45 steel, 210 kW transmitted power, 4400 r/min driving speed, 52 °C ambient temperature, and the 0–180 s transition from lubricated to unlubricated operation. They should not be extended to other loads, lubrication states, or gear geometries without recalculation. The model addresses the tooth-body temperature field and does not resolve asperity-scale flash temperature.