Spiral bevel gears are a natural choice for helicopter main reducers: they transmit high torque across intersecting shafts in a compact package and engage gradually, which keeps vibration low. The same contact geometry that produces smooth engagement also creates high sliding speeds and Hertzian pressures near the tooth surface. Under these conditions, the lubricant film is sheared at high pressure, and the local tooth temperature can rise well above the bulk temperature of the gear body. If that transient rise is large enough, the film can break down and scuffing can begin. The design quantity is the flash temperature, defined as the instantaneous surface temperature minus the bulk tooth temperature at the contact.
Blok’s classical flash-temperature method is attractive because it requires only the friction heat, the contact width, and the thermal inertia of the two gear bodies. It was developed for dry contact, however, and it assumes that the pinion and gear have equal flash temperatures. In a closed, oil-lubricated transmission neither assumption is exact. The oil film both generates frictional heat and adds a thermal resistance between the heat source and each tooth. An improved Blok formulation addresses this by treating the film as a thin layered conductor and solving for the heat split without forcing the two flash temperatures to be equal.
Why the Dry-Contact Blok Model Is Incomplete
For dry contact, Blok divides the frictional heat \(q_t\) per unit contact-line length between pinion (subscript 1) and gear (subscript 2). The surface temperature rises are
$$
t_{s1}-t_{b1}=\frac{1.11\,\phi q_t}{\sqrt{\lambda_1\rho_1 c_1 V_1 l}},
\qquad
t_{s2}-t_{b2}=\frac{1.11\,(1-\phi) q_t}{\sqrt{\lambda_2\rho_2 c_2 V_2 l}}
$$
where \(t_s\) is the surface temperature, \(t_b\) is the bulk temperature, \(\phi\) is the heat partition coefficient, \(\lambda\) is thermal conductivity, \(\rho\) is density, \(c\) is specific heat, \(V\) is the tangential surface velocity, and \(l\) is the contact width. The factor 1.11 comes from the moving heat-source solution. To close the problem, Blok set the two flash temperatures equal, \(t_{s1}-t_{b1}=t_{s2}-t_{b2}\), which gives
$$
\phi=\frac{1}{1+\sqrt{\dfrac{\lambda_2\rho_2 c_2 V_2}{\lambda_1\rho_1 c_1 V_1}}}
$$
This partition depends only on the metal properties and speeds. It contains no information about the lubricant, the film thickness, or the fact that the pinion and gear usually run at different bulk temperatures. For a dry gear it is a reasonable first estimate. For an elastohydrodynamic lubrication (EHL) contact it is incomplete: the film carries shear, generates heat, and imposes a temperature drop before heat reaches the metal. Wang and Cheng’s EHL analysis also showed that pinion and gear flash temperatures are generally unequal, so the equal-flash closure is not physically required.
Treating the Oil Film as a Layered Thermal Resistance
In a high-pressure EHL contact, the lubricant is not a simple Newtonian fluid. Its viscosity rises sharply with pressure, and experiments have shown liquid-to-solid-like transitions in thin films under EHL conditions. During the short meshing cycle, the film can therefore be modeled as a thin, stable conducting layer attached to the tooth surfaces. Because the film is sheared between the two surfaces, it is convenient to split it into two halves of thickness \(h/2\), one moving with the pinion and one with the gear.
Rashid and Seireg’s dimensionless solution for layered concentrated contacts gives the temperature drop across such a layer as a function of speed, density, heat capacity, layer thickness, contact width, and thermal conductivity. Using that solution, the film-side temperature drops can be written as
$$
t_{o1}-t_{s1}=\phi q_t B_1,
\qquad
t_{o2}-t_{s2}=(1-\phi) q_t B_2
$$
where \(t_o\) is the oil-film temperature and \(B_i\) is the film-side thermal resistance. The dry-contact rises retain the Blok form,
$$
t_{s1}-t_{b1}=\phi q_t A_1,
\qquad
t_{s2}-t_{b2}=(1-\phi) q_t A_2,
\qquad
A_i=\frac{1.11}{\sqrt{\lambda_i\rho_i c_i V_i l}}
$$
Because the film is thin relative to the tooth, its temperature is nearly uniform through the thickness. Setting the two film temperatures equal, \(t_{o1}=t_{o2}\), and eliminating the surface temperatures gives the improved heat partition:
$$
\phi=\frac{1}{A_1+B_1+A_2+B_2}
\left(\frac{t_{b2}-t_{b1}}{q_t}+A_2+B_2\right)
$$
Here \(\phi\) is the first split of frictional heat between the two film halves; the final fraction reaching the tooth body is handled separately. Unlike the Blok partition, this expression contains the bulk-temperature difference \(t_{b2}-t_{b1}\). Solving the layered problem again gives the final flash-temperature form
$$
t_{so1}-t_{b1}=\gamma_1 q_t A_1,
\qquad
t_{so2}-t_{b2}=\gamma_2 q_t A_2
$$
$$
\gamma_1=\phi\frac{A_1-C_1}{A_1},
\qquad
\gamma_2=(1-\phi)\frac{A_2-C_2}{A_2}
$$
where \(t_{so}\) is the oil-lubricated surface temperature and \(C_i\) is the film-presence correction from the same layered solution. The coefficient \(\gamma_i\) is the fraction of the frictional heat that reaches the tooth body; the rest is retained in the film or redistributed by conduction. When the film correction is set to zero, \(C_i=0\) and \(\gamma_i\) reduces to the dry-contact partition, so the formulation is consistent with Blok in the dry limit. The important difference is that \(\gamma_i\) depends on oil density, conductivity, specific heat, film thickness, contact width, sliding speed, and the bulk-temperature difference, not only on the metal thermal inertia. The \(B_i\) and \(C_i\) terms are evaluated from the dimensionless layered-contact correlation; they are not fitted to the gear test.
Case Study: A 1,000 kW Helicopter Transmission
The case study uses a spiral bevel gear pair from a helicopter reducer. The pair has 27 and 74 teeth, a module of 3.85 mm, and a face width of 38.5 mm. The input power is 1,000 kW. The gear steel and baseline lubricant properties are listed in the table.
| Parameter | Value |
|---|---|
| Teeth, pinion/gear | 27 / 74 |
| Module | 3.85 mm |
| Face width | 38.5 mm |
| Input power | 1,000 kW |
| Gear density | 7,850 kg/m³ |
| Gear conductivity | 36 W/(m·°C) |
| Gear specific heat | 641 J/(kg·°C) |
| Oil density | 970.2 kg/m³ |
| Oil conductivity | 0.147 W/(m·°C) |
| Oil specific heat | 2,131 J/(kg·°C) |
For velocity and contact calculations, the spiral bevel pair was replaced by an equivalent spur pair at the tooth-width midpoint. The equivalent pair preserves the pitch radius and the pitch-line velocity at that section. The tangential velocities \(V_1\) and \(V_2\) give the sliding speed \(V_s=|V_1-V_2|\) and the entrainment speed \(V_r=|V_1+V_2|/2\). The local friction heat per unit contact-line length is
$$
q_t=f P_c V_s l
$$
where \(f\) is the friction coefficient and \(P_c\) is the Hertzian contact pressure. The heat input is averaged over a meshing cycle for the bulk-temperature analysis and applied instantaneously for the transient analysis.
The finite-element model used a single tooth, since each tooth sees the same cycle. ANSYS Steady-State Thermal gave the bulk temperature; ANSYS Transient Thermal, starting from that bulk field, gave the instantaneous temperature over one meshing period. The flash temperature is the difference between the transient and steady values at the same node. Convection on the meshing face was estimated from a Reynolds–Prandtl correlation. The end face was treated as a rotating disk with laminar, transitional, and turbulent zones. Non-working faces used one-third to one-half of the average end-face coefficient. The comparison below is made at the mid-face meshing node, where the tooth temperature is expected to be highest and where the Joselito method is formulated for the maximum flash temperature.
Comparison With Finite-Element Results
At 1,000 kW, the finite-element model gave maximum tooth temperatures of 90.64°C on the pinion and 91.59°C on the gear. The improved Blok method gave 94.47°C and 90.44°C, respectively. Joselito’s method gave 101.98°C and 100.00°C. The bulk temperatures were 75.36°C and 72.03°C.
| Method | Bulk temperature, pinion/gear (°C) | Flash temperature, pinion/gear (°C) | Maximum temperature, pinion/gear (°C) | Largest deviation from FE maximum |
|---|---|---|---|---|
| Improved Blok | 75.36 / 72.03 | 19.11 / 18.41 | 94.47 / 90.44 | About 4% |
| Joselito | 75.36 / 72.03 | 26.62 / 27.97 | 101.98 / 100.00 | About 12% |
| Finite element | 75.36 / 72.03 | 15.28 / 19.56 | 90.64 / 91.59 | — |
The bulk temperatures are common inputs from the steady-state finite-element solution, so the comparison isolates the flash-temperature calculation. The improved method is clearly closer to the finite-element maximum than Joselito’s method. The largest deviation is about 4% for the improved formulation and about 12% for Joselito’s. That 4% figure applies to the maximum temperature, not to the flash temperature alone. Because the flash is only 15–20°C on a 72–75°C bulk, a few degrees of error in flash temperature becomes a small percentage of the maximum. For the pinion, the improved flash was 19.11°C versus 15.28°C from the finite-element model; Joselito’s was 26.62°C. The improved method also predicted a slightly lower gear maximum than the finite-element model, so it is not uniformly conservative and should not be used as a guaranteed upper bound.
Sensitivity to Input Power and Lubricant Properties
The comparison was repeated at 800 kW and 1,200 kW. In all three cases, the flash temperature increased with input power, and the improved Blok method remained closer to the finite-element result than Joselito’s method. The ranking did not change. Two alternative lubricants were also tested: oil A with 883 kg/m³, 0.144 W/(m·°C), and 1,600 J/(kg·°C); and oil B with 900 kg/m³, 0.134 W/(m·°C), and 1,960 J/(kg·°C). Changing the oil shifted the absolute flash temperatures, but the improved method stayed closest to the finite-element benchmark and the three methods retained their relative order.
To isolate the effect of individual oil properties, the baseline oil was scaled to 50% and 150% of its density, conductivity, and specific heat, one property at a time. The ranges were 485.1–1,455.3 kg/m³ for density, 0.0735–0.2205 W/(m·°C) for conductivity, and 1,065.5–3,196.5 J/(kg·°C) for specific heat. Within these ranges, the flash temperature increased with oil conductivity and decreased with oil density and specific heat. Higher conductivity lowers the film’s thermal resistance, allowing more heat to reach the tooth; higher density and specific heat increase the oil’s thermal inertia and reduce its temperature rise. The density and specific-heat effects were stronger than the conductivity effect and were similar to each other, consistent with the fact that density and specific heat enter the flash-temperature expression as a product.
Implications for Scuffing-Limited Design
Flash temperature is a screening quantity for scuffing risk, not a complete failure criterion. The improved formulation is useful because it retains the speed and simplicity of the Blok approach while adding the two effects that matter most in a lubricated gearbox: the thermal resistance of the oil film and the unequal thermal response of the pinion and gear. For the 1,000 kW case, the improved method reproduced the finite-element maximum temperature to within about 4%, while the Joselito method was off by about 12%. The improved method was also stable across input powers from 800 to 1,200 kW and across different lubricant formulations.
The sensitivity results show that oil density, conductivity, and specific heat are part of the scuffing margin, in addition to the rheological properties that control film thickness. In this case, higher density and specific heat reduced the flash, while higher conductivity increased it. That does not mean one property should be optimized in isolation; it means that lubricant thermal properties are part of the scuffing margin and should be included when comparing oils for a high-power-density gearbox.
The method still has clear limits. It assumes a fully flooded EHL film and a stable layered film during the meshing cycle. It does not resolve asperity contact, surface roughness, starved lubrication, or loss-of-lubrication conditions. The comparison here is at a single mid-face meshing node, not over the full tooth surface, and the finite-element result is a numerical benchmark rather than an experimental measurement. The next step is experimental validation under controlled torque, speed, and oil-supply conditions. Until then, the improved Blok method is best used for preliminary design, oil screening, and relative comparisons of scuffing load capacity, with an appropriate design margin for absolute temperature limits. For spiral bevel gears in high-power-density transmissions, that combination of computational speed and physical transparency is valuable.