Spiral bevel gears transmit torque between intersecting shafts and are widely used in high-speed, high-load drivetrains, including aeroengine systems. Their meshing action combines rolling with sliding. The sliding component does work against friction and introduces heat directly at the contact, so the tooth surface and the material beneath it do not remain at a uniform temperature. The contact zone becomes hotter than the tooth body, while the end faces remain comparatively cool. This gradient matters because it produces thermal stress and changes the operating backlash. If heat input exceeds the cooling capacity of the gear and its surroundings, scuffing, adhesive wear, and seizure become possible.
Under continuous lubrication, oil removes a substantial part of the frictional heat. After lubricant loss, the gear must reject heat through the tooth body, the end faces, and the surrounding air. The transient period is therefore critical: the bulk temperature can continue to rise after the oil film has failed. A transient finite-element analysis can resolve this period and separate the influence of operating conditions from that of tooth geometry.
Transient Heat Balance and Boundary Conditions
Because lubrication loss is time-dependent, the temperature field is governed by the transient conduction equation rather than by a steady-state balance. For a homogeneous gear material, the governing equation is
$$\rho C \frac{\partial T}{\partial t}=k\left(\frac{\partial^2 T}{\partial x^2}+\frac{\partial^2 T}{\partial y^2}+\frac{\partial^2 T}{\partial z^2}\right)$$
where k is the thermal conductivity of the gear material, ρ is its density, C is its specific heat, T is temperature, and t is time.
The tooth is divided into several thermal boundary regions. On the meshing flank, region n, the boundary condition includes both convective exchange with ambient air and the frictional heat flux entering the surface:
$$-k\frac{\partial T}{\partial n}=\alpha_t^{\prime}(T-T_0)-q$$
On the tooth root, the tooth tip, and the non-meshing flanks, region t, only convective exchange is imposed:
$$-k\frac{\partial T}{\partial n}=\alpha_t^{\prime}(T-T_0)$$
On the gear end faces, region s, the condition is
$$-k\frac{\partial T}{\partial n}=\alpha_s^{\prime}(T-T_0)$$
The lower boundary, region d, is treated as adiabatic:
$$\frac{\partial T}{\partial n}=0$$
The two cut planes, regions p and q, satisfy temperature and normal-gradient continuity:
$$T|_{p}=T|_{q},\qquad \left.\frac{\partial T}{\partial n}\right|_{p}=\left.\frac{\partial T}{\partial n}\right|_{q}$$
The initial condition is
$$T(x,y,z,0)=\phi(x,y,z)$$
Here n is the outward normal direction, q is the frictional heat flux supplied during meshing, 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 and Partitioning
The heat source is dominated by sliding friction because the rolling component produces a smaller frictional resistance. The frictional heat flux generated at the contact is estimated as
$$Q=fP_0v$$
where f is the sliding friction coefficient, P0 is the maximum Hertzian contact stress, and v is the sliding velocity. For steel gear contacts, the friction coefficient is commonly taken in the range of 0.045 to 0.065.
The two meshing members do not absorb equal shares of the generated heat. A partition coefficient ξ is therefore introduced:
$$Q_1=\xi Q,\qquad Q_2=(1-\xi)Q$$
The corresponding input heat fluxes for the pinion and the gear are
$$q_1=\frac{b_1\xi Q\omega_1}{2\pi v_1},\qquad q_2=\frac{b_1(1-\xi)Q\omega_2}{2\pi v_2}$$
where ω1 and ω2 are the angular velocities of the pinion and gear, v1 and v2 are their linear velocities at the mesh, and b1 is the contact half-width. The Hertzian contact half-width is obtained from
$$b_1=\sqrt{\frac{8P_0R}{\pi E}},\qquad \frac{1}{E}=\frac{1-\mu_1^2}{E_1}+\frac{1-\mu_2^2}{E_2}$$
with R as the equivalent radius of curvature, E1 and E2 as the elastic moduli of the two members, and μ1 and μ2 as their Poisson ratios.
Finite-Element Model and Operating Point
A finite-element study of a 25-tooth spiral bevel pinion meshing with a 60-tooth gear used ANSYS to solve the transient conduction problem. The pinion geometry and operating point are listed in Table 1. The material properties assigned in the model correspond to 45 steel and are listed in Table 2. The ambient temperature was set to 52 °C.
| Parameter | Value |
|---|---|
| Pinion tooth count | 25 |
| Mating gear tooth count | 60 |
| Module | 4 mm |
| Face width | 28 mm |
| Pressure angle | 20° |
| Addendum coefficient | 1 |
| Clearance coefficient | 0.25 |
| Spiral hand | Left |
| Transmitted power | 210 kW |
| Pinion speed | 4400 r/min |
| Ambient temperature | 52 °C |
| Gear material | 45 steel |
| 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) |
The transient calculation begins from a lubricated state at t = 0 s and follows the transition to lubricant-starved conditions at 60, 120, and 180 s. The finite-element model resolves the three-dimensional temperature field in the pinion tooth under these conditions.
Temperature Evolution during Lubrication Loss
The computed temperature field is strongly localized at the meshing contact. The maximum temperature occurs at the center of the contact and rises from 121.248 °C at 0 s to 134.888 °C at 60 s and 162.235 °C at 180 s. Relative to the 52 °C ambient, these values correspond to temperature rises of 69.248 K, 82.888 K, and 110.235 K.
The high-temperature zone expands along the tooth height toward the tip as time progresses. The end face remains the coolest region. The tooth body is cooler than the meshing surface but warmer than the end face. Along the face-width direction, the temperature profile follows a parabolic distribution rather than a uniform plateau.
These trends show that lubrication loss does not simply raise the entire gear to a higher uniform temperature. It creates a growing thermal gradient between the contact, the tooth body, and the end faces. That gradient is the source of additional thermal stress and clearance change. If the local temperature exceeds the material’s scuffing resistance limit, adhesive wear can progress to seizure or loss of transmission.
Effects of Face Width and Module
The geometry study compared face widths of 18, 22, and 30 mm and modules of 3, 4, and 5 mm under otherwise fixed conditions. The percentages in Table 3 are relative changes in the maximum body-temperature rise over the 180 s transient; they are not absolute temperatures.
| Factor | Value | Change relative to reference |
|---|---|---|
| Face width | 18 mm | Reference |
| Face width | 22 mm | +357.81% |
| Face width | 30 mm | +1289% |
| Module | 3 mm | Reference |
| Module | 4 mm | −50.98% |
| Module | 5 mm | −64.24% |
Increasing the face width raises the computed maximum body-temperature rise. Relative to an 18 mm face width, the 22 mm case gives a 357.81% increase, and the 30 mm case gives a 1289% increase. A wider face extends the frictional contact and increases the heat input, while heat generated in the interior has a longer path to the end faces, which are the main cooling boundaries. The result is a higher bulk temperature and a stronger axial thermal gradient.
Increasing the module has the opposite effect under the modeled conditions. The 4 mm module produces a maximum body-temperature rise 50.98% lower than the 3 mm module, and the 5 mm module produces a rise 64.24% lower than the 3 mm module. The analysis attributes this reduction to the larger tooth volume and the associated change in heat conduction, with the influence on convective exchange being smaller than the change in frictional heat input.
Design Implications and Model Scope
The thermal response of a spiral bevel gear after lubrication loss is governed by the balance between frictional heat input and heat removal through the tooth body and end faces. The contact center is the critical location, and the hot zone grows toward the tooth tip with time. Under the modeled conditions, a wider face increases the maximum temperature rise, whereas a larger module reduces it. These trends are relevant to geometry selection in high-speed, high-load spiral bevel gears, but they must be combined with contact-ratio, sliding-velocity, mass, and strength constraints.
The formulation uses prescribed friction coefficients, a heat partition coefficient, and convective boundary conditions. It does not resolve the oil film, lubricant distribution, or asperity-scale contact. The computed temperatures therefore represent the thermal response under those boundary assumptions, and calibration against measured tooth-body temperatures would be needed before applying the absolute values to a specific transmission.