Why the mounting distance decides whether a gear pair survives
Spiral bevel gears carry motion between intersecting shafts and are chosen wherever a drive must combine a high torque density with quiet, smooth running: vehicle axles, machine-tool spindles, helicopter transmissions and heavy industrial gearboxes. Their load capacity depends less on the nominal tooth geometry than on how that geometry is localized in space. A face-milled pair is designed so that, under a light test load, contact concentrates in a compact elliptical or rectangular patch located toward the toe of the large wheel, spanning roughly 25–45 % of the face width and staying clear of the tooth edges. As torque rises the patch spreads, but it must still not run off the ends or over the tip, because edge contact is what produces spalling, scoring and the characteristic howl of a badly set axle.
The variable most often used to place that patch is the axial mounting distance of each member: the position of the pinion along its own axis and of the wheel along its own axis, adjusted in practice with shims behind the pinion bearing cage and the wheel hub. Two signed offsets describe the departure from the nominal setting, the wheel offset $\Delta G$ and the pinion offset $\Delta T$. On the shop floor a pair is rarely assembled at the theoretical position; it is assembled somewhere slightly wrong, checked by rolling it under a thin film of marking compound, and then brought in by successive shim changes until the pattern is acceptable. Understanding how the pattern migrates during that sequence — from a misaligned state toward a workable one, with the gears carrying load — is therefore more useful to the fitter than a description of the ideal nominal contact alone.
Gear pair and tooth surface generation
The study object is a Gleason-type face-milled spiral bevel pair with 33 teeth on the wheel and 28 on the pinion, a transverse module of 4.8 mm, a 37 mm face width and an outer cone distance of 123.3 mm. The pitch cone angles of 49.4° and 40.2° sum to 89.6°, i.e. a shaft angle close to but not exactly 90°. Full data are listed in Table 1; the pinion (28 teeth) is the driving member, so its pitch cone angle $\delta_1 = 40.2°$ enters the offset relation used later.
| Parameter | Wheel (33 teeth) | Pinion (28 teeth) |
|---|---|---|
| Transverse module (mm) | 4.8 | 4.8 |
| Outer cone distance (mm) | 123.3 | 123.3 |
| Face width (mm) | 37 | 37 |
| Addendum (mm) | 2.2 | 2.3 |
| Dedendum (mm) | 3.3 | 2.9 |
| Pitch cone angle (°) | 49.4 | 40.2 |
| Root cone angle (°) | 47.5 | 39.3 |
| Face cone angle (°) | 52.5 | 43.1 |
Because the flanks of a spiral bevel tooth are not a simple swept surface, the solid model is built point by point. Discrete flank points are first computed in each blank coordinate system from the generating relationship between the cutter and the workpiece,
$$\mathbf{r} = \mathbf{M}\,\mathbf{r}_c(\mu,\theta), \qquad \mathbf{n}\cdot\mathbf{v} = f(\mu,\theta,\varphi) = 0,$$
where $\mathbf{M}$ is the transformation matrix between the cutter and blank frames, $\mathbf{r}_c$ the cutter disc equation, $\varphi$ the rotation angle, $\mu$ and $\theta$ the conical-surface coordinates of the cutting edge, and $\mathbf{n}$ and $\mathbf{v}$ the unit normal and relative velocity at a flank point. The resulting point cloud is fitted to surfaces in a CAD environment, solidified, and assembled into the pair.
Finite element model and loading
The assembled model is meshed with tetrahedral elements, with local refinement on the flanks expected to enter contact, so that resolution is high where the pressure distribution is computed while the total element count stays tractable. Both members are assigned the same case-hardening gear steel, 18Cr2Ni4WA, with the elastic properties given in Table 2.
| Material | Elastic modulus $E$ (GPa) | Poisson ratio $\nu$ | Density $\rho$ (g·cm⁻³) |
|---|---|---|---|
| 18Cr2Ni4WA | 202 | 0.273 | 7.91 |
Contact is defined only over the flank zones that can plausibly touch within the simulated rotation, identified from the kinematics of the pair, and friction is included with a coefficient of 0.2 representative of normal running. Each member rotates about its own axis through a revolute joint of the body-ground type; all remaining degrees of freedom are suppressed. The pinion is driven at 600 r·min⁻¹ and a resisting torque of 200 N·m is applied to the wheel. The transient dynamic solution runs over 0.0125 s of mesh rotation with an initial substep count of 50, a minimum of 20 and a maximum of 3000 substeps, which is sufficient to resolve the passage of individual teeth through the contact zone.
How the two axial offsets are defined and paired
Offsets are introduced by translating one member along its axis while the other is held at its nominal position. Three configurations are examined: wheel offset alone with $\Delta T = 0$, pinion offset alone with $\Delta G = 0$, and both offsets applied together. For each configuration, contact stresses are solved and the instantaneous patterns on the concave flank of the wheel are extracted at three stages of the mesh cycle — entry, mid-mesh and exit — so that the migration of the patch is captured rather than a single snapshot.
In the combined case the two offsets are not independent. With the original pitch cones kept tangent and a working backlash preserved, the wheel correction follows the pinion correction through
$$\Delta G = \Delta T \tan\delta_1 ,$$
with $\delta_1 = 40.2°$ and $\tan\delta_1 \approx 0.845$, so a pinion shift of 0.05 mm corresponds in magnitude to about 0.042 mm at the wheel. In the combined tests the two offsets were applied with opposite signs, so the wheel moves one way while the pinion moves the other. The offset levels used in the three configurations are collected in Table 3.
| Configuration | Offset levels (mm) |
|---|---|
| Wheel offset only, $\Delta T = 0$ | $\Delta G$ = −0.10, −0.05, 0, +0.05, +0.10 |
| Pinion offset only, $\Delta G = 0$ | $\Delta T$ = −0.050, −0.025, 0, +0.025, +0.050 |
| Combined, both offsets | ($\Delta T$, $\Delta G$) = (−0.050, +0.042), (−0.025, +0.021), (0, 0), (+0.025, −0.042), (+0.050, −0.021) |
Wheel offset alone: the pattern shortens and climbs
Holding the pinion at zero, the wheel was displaced in 0.05 mm increments from −0.10 mm to +0.10 mm. With increasing $\Delta G$ the entry-stage patch retreats from the heel (the large end) and the exit-stage patch retreats from the toe (the small end), while the mid-mesh patch moves from the root toward the tip. The net geometric effect is a contact zone that becomes shorter along the face width and migrates upward along the tooth height; because the interval during which the pair remains in contact also contracts, the mesh becomes less continuous and the transmission less steady. Negative $\Delta G$ produces the mirror image: a longer, lower-lying pattern that encroaches progressively on the heel and the root.
Pinion offset alone: the pattern slides toward the heel and the tip
With the wheel fixed at its nominal position and the pitch cones kept tangent with backlash preserved, the pinion was displaced in 0.025 mm increments between −0.050 mm and +0.050 mm. Here the response is more uniform: increasing $\Delta T$ translates the entry, mid-mesh and exit patches alike toward the heel along the face width and toward the tip along the tooth height. At $\Delta T = 0$ the computed pattern satisfies the usual acceptance criterion for light load — an elliptical patch centred slightly toward the toe, covering 25–45 % of the face width — and the departure from that condition grows monotonically in both directions. The pinion offset is thus the stronger of the two single-parameter levers for positioning the pattern along the face, whereas the wheel offset mainly controls its length.
Both offsets together
When the pinion offset was increased while the wheel offset was reduced in the opposite direction according to Table 3, the entry, mid-mesh and exit patches all migrated toward the heel along the face width and toward the root along the tooth height — the height direction being opposite to that obtained with the pinion offset alone. The two adjustments therefore do not act as independent translations of the same patch: the wheel correction carried by the tangent relation modifies the height position while the pinion correction sets the face-width position. Table 4 condenses the three responses.
| Offset case | Face-width direction | Tooth-height direction | Pattern length |
|---|---|---|---|
| Wheel $\Delta G$ alone, pinion fixed | Entry patch away from heel, exit patch away from toe; ends draw toward mid-face | Mid-mesh patch root → tip | Shortens as $\Delta G$ increases |
| Pinion $\Delta T$ alone, wheel fixed | All three stages toward heel | All three stages toward tip | Broadly preserved, position shifts |
| Combined, opposite signs | All three stages toward heel | All three stages toward root | Broadly preserved, position shifts |
Plotting the patch centre of each stage in a common frame — distance from the heel along the face width against height deviation along the tooth — makes the trajectories explicit. For the wheel offset alone, the entry and mid-mesh centres move away from the heel and away from the root while the exit centre approaches the heel and leaves the root, so the three trajectories fan apart and the effective contact line rotates as well as translates. For the pinion offset alone the three centres move together toward the heel and the tip, maintaining their relative spacing. For the combined adjustment the three centres again move as a group, toward the heel and away from the root. Grouped motion is what a fitter can work with: a single shim change shifts the whole pattern without destroying its shape, whereas a change that fans the stages apart trades position for contact ratio.
Rolling inspection on a production pair
The computed trends were checked on a high-precision pair manufactured by Qinghai Huading Gearbox Co. Ltd., with parameters essentially identical to the modelled geometry. Testing used a CNC-retrofitted Y9550 bevel gear rolling inspection machine, whose controlled setting of the mounting distances removes much of the scatter that manual assembly would introduce. Because the driven member should not be heavily loaded during inspection, the braking torque was set to 40 N·m, well below the 200 N·m of the simulation; the comparison therefore concerns the direction and relative magnitude of pattern migration rather than absolute patch dimensions.
Flanks were coated with marking compound and the pair was first set with a wheel offset of +0.05 mm, which by the tangent relation corresponds to a pinion offset of −0.042 mm. After a running period the offsets were stepped through the sequence of Table 3 and the pattern was recorded at each setting. The initial, deliberately misaligned setting produced a patch concentrated toward the tip, exactly the height-direction error predicted for the wheel-dominant offset case. Bringing the mounting distances to the correct values produced the expected compact pattern in the middle-to-toe region, clear of both edges. The transition from the unreasonable to the reasonable pattern reproduced the simulated migration in face-width and height directions, which supports the use of the finite element results as a qualitative guide for adjustment. Residual differences arise from elastic deflection of the fixture, from the lower inspection torque, and from the finite thickness of the marking film, and they do not alter the direction of the trends.
Practical reading of the results
Three rules follow directly from the simulations and are consistent with the rolling tests. A pattern that is too short along the face and sits high on the tooth points to an excessive wheel offset; because the entry and exit patches retreat toward mid-face as $\Delta G$ grows, the pattern length is the most sensitive indicator of that error. A pattern displaced bodily toward the toe, with all three stages sharing the displacement, points instead to the pinion offset, which translates the contact without shortening it. When both errors are present, correcting the pinion offset alone shifts the pattern along the face but leaves the height position wrong, because the compensating wheel correction required by the tangent relation acts in the opposite height direction; the two mounting distances must be adjusted together, as is done in practice when a pair is brought in from a rough setting.
The scope of these conclusions should be kept in mind. The results refer to one face-milled geometry with a 33/28 tooth combination and a 4.8 mm module, to a single friction coefficient of 0.2, and to a quasi-steady load of 200 N·m; at much higher torque the pattern spreads toward the edges and the margins available for adjustment shrink. Within that envelope, however, the direction of pattern migration with each offset is a robust function of the kinematics, and it provides a quantitative basis for the shim changes that installation technicians currently make by trial and error.
Conclusions
- With the pinion at its nominal position, increasing the wheel axial offset shortens the contact pattern along the face width and moves it from the root toward the tip; the entry patch retreats from the heel and the exit patch from the toe, so the two ends of the pattern draw toward mid-face and the contact interval contracts.
- With the wheel at its nominal position, increasing the pinion axial offset translates the entry, mid-mesh and exit patterns together toward the heel and the tip, leaving the pattern shape largely intact; $\Delta T = 0$ reproduces the accepted light-load pattern of 25–45 % face-width coverage.
- When both offsets are applied with opposite signs and magnitudes related by $\Delta G = \Delta T\tan\delta_1$, all three stages migrate toward the heel and toward the root, i.e. the height direction reverses relative to the pinion-only case.
- Rolling inspection of a production pair on a CNC-retrofitted Y9550 machine reproduced the simulated migration from a tip-biased, unreasonable pattern to an acceptable mid-to-toe pattern, supporting the use of these trends for setting axial mounting distances during assembly.