Spiral bevel gears are widely used in helicopter reduction drives because their curved teeth combine high load capacity with smooth, stable conjugate motion. In a compact gearbox, the same geometry that provides this smoothness also produces combined rolling and sliding in a narrow contact zone. Frictional heat is generated within milliseconds, and if the local transient temperature rises above the lubricant’s load-carrying limit, scuffing can initiate. Flash temperature is the peak rise above the bulk tooth temperature and is therefore a direct input to scuffing-capacity assessment.
The classical Blok flash-temperature method is widely used because it is simple and physically transparent. It was developed for dry metal-to-metal contact, however, and closed gearboxes normally operate with an oil film between the teeth. That film changes both the amount of frictional heat reaching each tooth and the way heat is divided between the pinion and the gear. A formulation that retains the structure of the Blok method while treating the oil film as a thermal layer gives a more realistic estimate of the maximum tooth temperature.
The Dry-Contact Assumption in the Blok Method
In the Blok model, the pinion and gear surfaces are in direct contact. Frictional heat per unit contact line, qt, is divided between the two bodies according to their thermal effusivity and tangential velocity. The flash-temperature rise for each body can be written as
$$t_{s1}-t_{b1}=\phi q_t A_1,\qquad t_{s2}-t_{b2}=(1-\phi)q_t A_2$$
where
$$A_i=\frac{1.11}{\sqrt{\lambda_i\rho_i c_i V_i l}},\qquad i=1,2$$
Here, ts is the instantaneous surface temperature, tb is the bulk tooth temperature, φ is the heat-partition coefficient, λ is thermal conductivity, ρ is density, c is specific heat, V is tangential velocity, and l is the contact width. Subscripts 1 and 2 refer to the pinion and gear.
Blok simplified the problem by assuming that the pinion and gear flash temperatures are equal. Setting the two temperature rises equal gives the dry-contact partition coefficient:
$$\phi=\frac{1}{1+\sqrt{\frac{\lambda_2\rho_2c_2V_2}{\lambda_1\rho_1c_1V_1}}}$$
This assumption is a modeling convenience rather than a physical identity. Contact analyses have shown that pinion and gear surface temperatures can differ, especially when the two bodies have different geometry, speed, and thermal properties. The dry-contact model also omits the oil film, which is present in nearly all closed helicopter gearboxes. The Chinese guidance standard GB/Z 6413.1—2003 starts from the Blok method and applies empirical service coefficients to account for lubrication, but the result then depends on the selected coefficient. That dependence introduces a subjective element into scuffing calculations.
Treating the Oil Film as a Layered Thermal Resistance
In an elastohydrodynamic lubrication (EHL) contact, the oil film is subjected to high pressure and high shear stress for a very short residence time. Under these conditions the lubricant can behave as a glassy or solid-like layer rather than a simple Newtonian fluid. For thermal analysis, the film can therefore be treated as a thin, stationary layered conductor attached to the tooth surfaces.
The total film thickness is h. Because the film is sheared between the two moving surfaces, it is convenient to divide it into two halves, each of thickness h/2 and each adsorbed on one tooth surface. Frictional heat is generated within the film. A fraction φ is directed toward the pinion side and the remainder toward the gear side. The film is so thin compared with the tooth dimensions that its temperature across the thickness is treated as uniform. Heat must cross the film before it enters the tooth body, so the heat reaching the tooth is not equal to the total frictional heat.
Rashid and Seireg’s layered-contact analysis provides dimensionless relations for the temperature drop across a surface layer and across a second layer. Applying those relations to the oil film and the tooth gives two thermal-resistance terms for each side: Bi for the film and Ci for the film-to-tooth path. These terms depend on film thickness, contact width, oil thermal properties, tooth thermal properties, and surface velocity. The resulting heat-partition coefficient is
$$\phi=\frac{1}{A_1+B_1+A_2+B_2}\left(\frac{t_{b2}-t_{b1}}{q_t}+A_2+B_2\right)$$
The bulk-temperature difference tb2 − tb1 therefore enters the partition calculation directly. This is a substantive change from the dry-contact model, which suppresses that difference by forcing equal flash temperatures.
Improved Flash-Temperature Formulation
After the heat has crossed the film, the fraction that actually reaches the tooth body is represented by a second coefficient, γi. The oil-lubricated flash temperatures are then
$$t_{so1}-t_{b1}=\gamma_1 q_t A_1,\qquad t_{so2}-t_{b2}=\gamma_2 q_t A_2$$
with
$$\gamma_1=\phi\frac{A_1-C_1}{A_1},\qquad \gamma_2=(1-\phi)\frac{A_2-C_2}{A_2}$$
Here, tso denotes the surface temperature under oil lubrication. The coefficients γ1 and γ2 represent the fraction of frictional heat that reaches the pinion and gear bodies after the film resistance has been accounted for. The formulation uses the same frictional heat input as the Blok method, but it changes how that heat is divided and attenuated before it enters the teeth.
The improved model therefore depends on oil film thickness, oil density, oil thermal conductivity, oil specific heat, and the bulk-temperature difference between the two teeth. The Blok model depends only on the thermal properties of the two metals. This is the main reason the improved formulation can produce different flash temperatures on the pinion and gear without imposing an artificial equality.
Finite-Element Reference Model
The improved analytical result was compared with a transient thermal finite-element calculation. Because every tooth in the gear pair experiences the same meshing cycle, a single-tooth model is sufficient. The spiral bevel gear is replaced at the mid-face section by an equivalent spur gear with the same mid-face tooth parameters. Tangential velocities at the meshing point are projected onto the normal and tangent directions to obtain the sliding speed Vs and the entrainment speed Vr.
The frictional heat per unit contact line is
$$q_t=fP_cV_sl$$
where f is the friction coefficient, Pc is the Hertzian contact pressure, and l is the contact width. For the bulk-temperature calculation, this heat is averaged over one meshing period:
$$Q_i=\frac{q’_{ti}}{V_iT_mz_i},\qquad q’_{ti}=\gamma_iq_t$$
where Tm is the single-tooth meshing period and zi is the tooth number. The finite-element model uses ANSYS Steady-State Thermal for the bulk temperature and ANSYS Transient Thermal for the instantaneous temperature. The bulk temperature is first obtained under average heat input; the transient temperature is then calculated over one meshing cycle with the instantaneous heat input. Flash temperature is the difference between the transient and bulk temperatures.
The convection boundary conditions are divided into three zones. On the meshing face, the surface heat-transfer coefficient follows the form
$$\alpha=0.228\,Re\,Pr\,\frac{\lambda_o}{L}$$
where Re, Pr, and λo are the Reynolds number, Prandtl number, and thermal conductivity of the oil, and L is a characteristic length taken as the pitch diameter. The end face is treated as a rotating disk. Depending on radius, the oil flow is laminar, transitional, or turbulent, and the corresponding disk correlations are used. The non-working faces, including the root, tip, non-meshing flank, and tooth-section faces, are assigned heat-transfer coefficients between one-third and one-half of the end-face average.
Case Study: A 27/74 Helicopter Gear Pair
The numerical case is a pair of oil-lubricated spiral bevel gears from a helicopter reduction gearbox. The pinion has 27 teeth and the gear has 74 teeth; the module is 3.85 mm and the face width is 38.5 mm. The pair transmits 1000 kW at an input speed of 20,900 r/min. The main thermal and geometric parameters are listed in Table 1.
| Parameter | Value |
|---|---|
| Teeth, pinion/gear | 27/74 |
| Module | 3.85 mm |
| Face width | 38.5 mm |
| Input power | 1000 kW |
| Input speed | 20,900 r/min |
| Gear density | 7850 kg/m³ |
| Oil density | 970.2 kg/m³ |
| Gear thermal conductivity | 36 W/(m·°C) |
| Oil thermal conductivity | 0.147 W/(m·°C) |
| Gear specific heat | 641 J/(kg·°C) |
| Oil specific heat | 2131 J/(kg·°C) |
The mid-face section is used as the comparison location because prior thermal analyses identify it as the hottest region of this gear type. At the mid-face node, the finite-element calculation gives bulk temperatures of 75.36°C for the pinion and 72.03°C for the gear, with transient maxima of 90.64°C and 91.59°C. The corresponding finite-element flash temperatures are 15.28°C and 19.56°C.
Table 2 compares the finite-element reference with the improved Blok method and Joselito’s flash-temperature method. The same bulk temperatures are used for all three methods, so the differences arise from the flash-temperature calculation.
| Method | Bulk temperature (°C) | Flash temperature (°C) | Maximum temperature (°C) |
|---|---|---|---|
| Improved Blok | 75.36/72.03 | 19.11/18.41 | 94.47/90.44 |
| Joselito | 75.36/72.03 | 26.62/27.97 | 101.98/100.00 |
| Finite element | 75.36/72.03 | 15.28/19.56 | 90.64/91.59 |
Comparison and Sensitivity
For scuffing assessment, the maximum absolute tooth temperature is more relevant than the flash rise alone, because the bulk temperature provides a large common offset. At the pinion, Joselito’s maximum temperature is about 12% above the finite-element value, while the improved Blok method is about 4% above. At the gear, the corresponding deviations are about 9% and about 1%. The improved formulation therefore lies closer to the finite-element reference at both members, although the pinion flash value itself differs more than the maximum temperature because the bulk temperature dominates the sum.
The comparison is numerical rather than experimental. The finite-element result serves as a consistent reference, not as direct proof of field accuracy. Experimental verification remains necessary before the method is used for certification or for setting absolute scuffing limits. Within the present case, however, the improved method reproduces the finite-element maximum temperature more closely than Joselito’s method.
To test whether this agreement persists away from the design point, calculations were repeated at input powers of 800, 1000, and 1200 kW. The flash temperature increases with input power for all three methods, and the improved Blok method remains closer to the finite-element result than Joselito’s method across the range. Replacing the lubricant with two other oils also leaves the ranking unchanged. The tested oils cover densities of 883 and 900 kg/m³, thermal conductivities of 0.144 and 0.134 W/(m·°C), and specific heats of 1600 and 1960 J/(kg·°C). The improved method is therefore not tied to a single oil formulation.
A separate sensitivity study scaled the original oil properties by 50% and 150%. Density was varied between 485.1 and 1455.3 kg/m³, thermal conductivity between 0.0735 and 0.2205 W/(m·°C), and specific heat between 1065.5 and 3196.5 J/(kg·°C). In the improved model, the calculated flash temperature increases with oil thermal conductivity and decreases with oil density and specific heat. When the scaling amplitude is the same, density and specific heat have a much stronger effect than thermal conductivity, and their effects are similar in magnitude. This is consistent with the way density and specific heat enter the flash-temperature expression at the same order.
Design Implications and Limitations
The improved formulation provides a physically consistent way to estimate flash temperature in oil-lubricated spiral bevel gears. It retains the input structure of the Blok method—frictional heat per contact line, contact width, and tangential velocity—but adds the oil film as a thermal resistance and allows the pinion and gear to reach different surface temperatures. For preliminary scuffing screening, this is more defensible than applying a single empirical coefficient to a dry-contact formula.
The model still contains simplifying assumptions. The oil film is treated as a stationary layered conductor with uniform temperature across its thickness. The contact width is fixed, and the film thickness must be supplied as an input. Asperity contact, film collapse, starved lubrication, dynamic load sharing, and thermal expansion are not represented. These omissions matter most when the gear operates near the boundary between full-film and mixed lubrication. Under such conditions, the dry-contact Blok limit may become more relevant, and the improved model should be used with an estimate of how much film remains.
Within the stated scope, the case study shows that the improved Blok method reduces the maximum-temperature deviation from about 12% to about 4% at the pinion, and from about 9% to about 1% at the gear, relative to the finite-element reference. The ranking holds as input power and lubricant type change. These results support the use of the improved formulation for design-stage flash-temperature estimation and for further development of scuffing-capacity calculations for helicopter spiral bevel gears.