Spiral Bevel Gears in Two-Stage Reducers
Spiral bevel gears transmit torque between intersecting shafts, usually at 90°, through curved teeth that enter mesh gradually rather than along a full face width at once. The Gleason and Klingelnberg systems dominate this class of gearing. Gleason curved-tooth bevel gears can be ground, which supports high load capacity, smooth running, low noise, and reduced sensitivity to mounting error and elastic deflection. They are common in drives with pitch-line velocity above 5 m/s, input speed above 1000 rpm, or heavy-load duty. Equal-clearance spiral bevel gears are often selected when the tooth flanks must be ground for such service.
The same spiral angle that gives smooth meshing also produces a substantial axial force. That force does not disappear as it would in a simple spur gear, and its direction reverses when the direction of rotation or the tooth hand changes. In a two-stage reducer, the large bevel gear and the second-stage helical pinion often sit on the same intermediate shaft. Their axial forces are therefore collinear and combine algebraically before reaching the shaft bearings. A hand and arrangement that make the two forces oppose can remove most of the net axial load; a different combination can make them add.
Force Direction Is Set by Geometry, Hand, and Rotation
For the force analysis, the viewing direction is from the small end of the bevel pinion toward the large end. Clockwise rotation is treated as right-hand, and counterclockwise rotation as left-hand, under the convention used for the tooth hand. Axial force is positive from the small end to the large end and negative in the reverse direction. Radial force is positive toward the shaft axis.
With a spiral angle β = 35°, a pressure angle α = 20°, and pitch-cone angles δ1 and δ2, the axial and radial components of the pinion and wheel can be written as:
$$
F_{a1}=\frac{K_p}{\cos\beta}\left(\tan\alpha\sin\delta_1 \mp \sin\beta\cos\delta_1\right)
$$
$$
F_{r1}=\frac{K_p}{\cos\beta}\left(\tan\alpha\cos\delta_1 \pm \sin\beta\sin\delta_1\right)
$$
$$
F_{a2}=\frac{K_p}{\cos\beta}\left(\tan\alpha\sin\delta_2 \pm \sin\beta\cos\delta_2\right)
$$
$$
F_{r2}=\frac{K_p}{\cos\beta}\left(\tan\alpha\cos\delta_2 \mp \sin\beta\sin\delta_2\right)
$$
Here Kp is the tangential force term, and the upper or lower sign is fixed by the hand and rotation combination. The absolute sign of one term is less important than the relative sign between the bevel gear and the helical gear on the same shaft.
For a 90° shaft angle, the pitch-cone angles are related to the gear ratio i = z2/z1. With the stated β and α, the axial-force term of the pinion passes through zero near i = 0.63, and the corresponding term of the wheel passes through zero near i = 1.57. At those ratios, the axial component changes sign, while the radial component follows a complementary pattern. This is why a bevel pair cannot be treated as having a fixed axial-force direction independent of tooth count, hand, and rotation.
The B3SH12 Coaxial Cancellation Layout
The B3SH12 is a mining belt-conveyor reducer. In the arrangement analyzed here, the input bevel pinion is right-hand, the large bevel gear is left-hand, the second-stage shaft pinion is left-hand, and the second-stage wheel is right-hand. The large bevel gear and the second-stage pinion are mounted on the same intermediate shaft. Their axial forces therefore act along the same axis and combine into one net force on the shaft bearings.
Viewed from the small end of the input pinion, clockwise and counterclockwise rotation reverse the torque direction and the axial-force signs. Because the bevel pair and the helical pair have different hand combinations, one rotation direction places the two axial forces in opposition; the other can place them in the same direction. The design problem is thus not simply to reduce the bevel gear axial force, but to choose the hand and running direction that produce a low resultant on the intermediate shaft.
Calculated Axial-Force Balance
The numerical comparison for the B3SH12 arrangement was made at 50% of the reducer’s rated power. The nominal ratio range considered is 20:1 to 40:1, with second-stage ratios from 1.26 to 2.05. Table 1 lists the bevel-gear axial force, the helical-pinion axial force in the counterclockwise case, and the resultant axial force on the intermediate shaft for both rotation directions. The values are magnitudes; the sign is handled by the direction logic.
Table 1. Axial force balance on the B3SH12 intermediate shaft at 50% rated power.
| Nominal ratio | Second-stage ratio | Large bevel axial force (N) | Helical pinion axial force, counterclockwise (N) | Resultant, counterclockwise (N) | Resultant, clockwise (N) |
|---|---|---|---|---|---|
| 20 | 1.26 | 12,622 | 15,107 | 2,485 | 12,025 |
| 22.4 | 1.26 | 12,622 | 15,107 | 2,485 | 12,025 |
| 25 | 1.39 | 12,238 | 13,251 | 1,013 | 11,714 |
| 28 | 1.39 | 11,798 | 13,006 | 1,208 | 11,880 |
| 31.5 | 1.62 | 12,174 | 12,304 | 130 | 13,182 |
| 35.5 | 1.78 | 11,728 | 11,633 | 95 | 13,256 |
| 40 | 2.05 | 12,275 | 10,884 | 1,391 | 14,439 |
In the counterclockwise case, the two axial forces oppose over the full ratio range. At 20:1 and 22.4:1, a helical-pinion force of 15,107 N acts against a bevel force of 12,622 N, leaving 2,485 N. At 31.5:1 and 35.5:1, the two components are almost equal—12,304 N against 12,174 N, and 11,633 N against 11,728 N—so the resultant falls to 130 N and 95 N, roughly 1% of the individual forces. At 40:1, the residual rises to 1,391 N, but it is still far below either component.
The clockwise case is markedly different. At 20:1 and 22.4:1, the helical contribution is small and the resultant stays close to the bevel force at 12,025 N. At 25:1 the resultant is 11,714 N, slightly below the bevel force. From 28:1 onward, however, the forces begin to add: the resultant becomes 11,880 N at 28:1, 13,182 N at 31.5:1, 13,256 N at 35.5:1, and 14,439 N at 40:1. The worst case is therefore not necessarily the largest bevel force, but the ratio at which the helical force adds to it.
These values were obtained at half of rated power. At full load the absolute forces scale with torque, but the direction logic and the cancellation or superposition pattern remain governed by hand, rotation, and ratio. The second-stage ratio also matters because it sets the torque split and therefore the tangential force in the helical mesh.
Design Implications
The B3SH12 case shows that axial force in a spiral bevel stage should be treated as a vector design variable, not as a fixed penalty. The hand of the bevel gear, the hand of the helical pinion, and the preferred running direction form a set. For a belt conveyor with a dominant loaded direction, the designer can choose the combination that gives the lowest resultant on the intermediate shaft. If the drive must run in both directions, both cases must be checked, because a layout that cancels in one direction may add in the other.
Bearing selection should use the resultant axial load, not the bevel-gear axial force alone. In the favorable direction, the resultant falls from the 12–15 kN range to 0.1–2.5 kN, which removes most of the axial component from the bearing equivalent load. The actual life gain still depends on bearing type, preload, speed, lubrication, and radial load, but the reduction in axial load is a direct and controllable design lever.
The method does not eliminate axial force from the gearbox. It redistributes the axial forces between the bevel stage and the helical stage so that the net force on the intermediate shaft is reduced. That distinction matters: the gear teeth still carry the same tangential load, and the helical mesh still generates its own axial component. The benefit comes from making the two components oppose on the shaft that carries them both.