Why the contact pattern drives assembly decisions
Spiral bevel gears transmit motion between intersecting shafts and are selected where load capacity, running smoothness and low noise dominate—machine tool spindles, vehicle axles, helicopter and industrial gearboxes. Their flanks are generated by a rotating face-milling cutter, so the theoretical conjugate action holds only at one specific relative position of the two members. Once the pair is mounted in a housing, that position is realized through shimmed mounting distances, and it is almost never reached exactly: the stack-up of housing bore tolerances, bearing width variation and shim selection leaves an axial offset on one or both members.
The tooth contact pattern—the ellipse of marking compound transferred from one flank to the other under a light brake load—remains the standard production acceptance criterion because it folds flank geometry, mounting position and elastic deflection into a single visual record. Its size, shape and location respond far more strongly to axial position than to most other mounting errors, since an axial shift slides the contact band along the tooth height and simultaneously changes where the contact path starts and ends. A pattern pushed to the tooth tip or to the heel produces edge contact, concentrated stress, higher vibration and, in the worst case, spalling of the flank.
Shop-floor adjustment is therefore iterative: a fitter reads the pattern, moves the mounting distance, re-runs the pair, and repeats. What that procedure needs is not a single “correct” pattern at the nominal position but the direction and rate at which the pattern migrates as the offsets are changed, including the intermediate states in which the pair is still meshing incorrectly under load. The analysis summarized here addresses exactly that: finite element contact simulations of one gear pair with a wheel offset, a pinion offset, and both offsets applied together, with instantaneous patterns extracted at meshing-in, mid-mesh and meshing-out, and a rolling-tester check against real marked flanks.
Flank geometry and the finite element model
Because the flanks are not analytic surfaces, geometry was built point by point. Discrete points on both members were computed in their own blank coordinate systems from the generating kinematics, fitted into surfaces in a CAD environment, and assembled into a solid model. The governing relations are the cutter position vector mapped into the blank frame together with the meshing condition:
$$\mathbf{r}=\mathbf{M}\,\mathbf{r}_{c}(\mu,\theta),\qquad \mathbf{n}\cdot\mathbf{v}=f(\mu,\theta,\varphi)=0$$
where $\mathbf{M}$ is the transformation matrix between the cutter and blank coordinate systems, $\mathbf{r}_{c}$ is the cutter disk equation, $\varphi$ the rotation angle, $\mu$ and $\theta$ the cone-surface coordinates of the cutting edge, $\mathbf{v}$ the relative velocity vector and $\mathbf{n}$ the flank normal. The resulting pair has 28 pinion teeth and 33 wheel teeth, a transverse module of 4.8 mm, an outer cone distance of 123.3 mm and a face width of 37 mm; the full parameter set is listed in Table 1.
| Parameter | Wheel | Pinion |
|---|---|---|
| Number of teeth | 33 | 28 |
| 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 (deg) | 49.4 | 40.2 |
| Root cone angle (deg) | 47.5 | 39.3 |
| Face cone angle (deg) | 52.5 | 43.1 |
The assembled solid was meshed with tetrahedral elements and locally refined on the flanks expected to enter contact, a compromise that keeps resolution in the contact zone without inflating the solve time. Both members were assigned the same carburizing gear steel, 18Cr2Ni4WA, with the properties in Table 2. Friction in the contact zone was represented by a Coulomb coefficient of 0.2, taken as representative of the normal operating condition. Both members were given revolute joint conditions about their own axes with all remaining degrees of freedom suppressed, the pinion was driven at 600 r/min, and a resisting torque of 200 N·m was applied to the wheel. The transient dynamic solve used a total step time of 0.0125 s with 50 initial substeps, a minimum of 20 and a maximum of 3000 substeps.
| Material | Elastic modulus $E$ (GPa) | Poisson ratio $\mu$ | Density $\rho$ (g·cm−3) |
|---|---|---|---|
| 18Cr2Ni4WA | 202 | 0.273 | 7.91 |
Defining and scheduling the axial offsets
Two independent offsets are available to the fitter: an axial displacement of the wheel, $\Delta G$, and an axial displacement of the pinion, $\Delta T$. Adjusting one member alone changes the shaft angle geometry and the backlash distribution, so in practice the two are coupled. When the original pitch cones are kept tangent and a working backlash is preserved, the offsets are linked by
$$\Delta G=\Delta T\tan\delta_{1}$$
with $\delta_{1}$ the pinion pitch cone angle, here 40.2°, so that $|\Delta G|\approx 0.845\,|\Delta T|$ with opposite sign. Three series were simulated: the wheel offset alone, swept from −0.10 mm to +0.10 mm in 0.05 mm increments with the pinion fixed; the pinion offset alone, swept from −0.05 mm to +0.05 mm in 0.025 mm increments with the wheel fixed; and the paired schedule of Table 3, in which the offsets satisfy the relation above. In every case the instantaneous contact pattern was extracted on the concave flank of the wheel at the meshing-in, mid-mesh and meshing-out positions, so that both the overall contact area and its migration through the mesh cycle could be read.
| Combined case | $\Delta T$ (mm) | $\Delta G$ (mm) |
|---|---|---|
| 1 | −0.050 | +0.042 |
| 2 | −0.025 | +0.021 |
| 3 | 0 | 0 |
| 4 | +0.025 | −0.021 |
| 5 | +0.050 | −0.042 |
Wheel offset alone: the pattern contracts
With the pinion held at its nominal position and the wheel displaced from −0.10 mm to +0.10 mm, the most conspicuous effect is geometric contraction. As $\Delta G$ increases, the meshing-in pattern retreats from the heel, the meshing-out pattern retreats from the toe, and the mid-mesh pattern climbs from the root toward the tip. The total contact band therefore becomes shorter along the face width and migrates upward along the tooth height, with the meshing-in and meshing-out patterns converging toward mid-face. The physical reason is that the offset shortens the interval during which the pair remains in conjugate contact within one mesh cycle: fewer flank sections share the load at any instant, and the load path crowds toward the middle of the face. For a gearbox this is the least desirable of the three responses, since the loss of contact length is accompanied by a loss of transmission smoothness rather than a simple shift of the loaded zone.
Pinion offset alone: the pattern translates
An important baseline precedes the pinion sweep. Under light load, the acceptable pattern sits near mid-face, slightly toward the toe, as an ellipse or a rounded rectangle occupying roughly 25% to 45% of the face width; as the working load rises the marked area extends outward and covers most of the flank without reaching the edges. With the wheel fixed, increasing $\Delta T$ from −0.05 mm to +0.05 mm translates this pattern rather than shrinking it: all three meshing states move toward the heel along the face width and toward the tip along the tooth height. The offsets at both ends of the sweep produce visibly misplaced contact, and the nominal zero position returns the most reasonable pattern of the series. Translation without contraction means the pinion offset is comparatively benign for contact ratio but aggressive for edge loading—at +0.05 mm the contact band approaches the heel and the tip simultaneously, which is the classic route to tooth-end breakage in service.
Both offsets together
When the paired schedule of Table 3 is applied, the two single-offset responses combine rather than cancel. As $\Delta T$ increases and $\Delta G$ decreases correspondingly, all three instantaneous patterns migrate toward the upper-left region of the flank—from toe to heel along the face width, and from root to tip along the tooth height. The contact band keeps approximately its original length, so the paired adjustment behaves as a controlled translation in the diagonal direction. That is precisely what makes the coupling useful during assembly: the diagonal trajectory allows a pattern that is misplaced in both directions to be corrected with a single coordinated change of the two mounting distances, instead of chasing one direction at the expense of the other.
Pattern trajectories side by side
Plotting the pattern centre for each offset series—face-width position measured from the heel as abscissa, tooth-height deviation as ordinate—condenses the three responses into three distinguishable trajectories, summarized in Table 4. The wheel-only series produces a converging, contracting trajectory; the pinion-only series produces a nearly straight translation toward heel and tip; the paired series produces a diagonal translation in the same general direction as the pinion-only case but with the contact length preserved.
| Offset condition | Face-width direction | Tooth-height direction | Contact length |
|---|---|---|---|
| Wheel offset $\Delta G$: −0.10 → +0.10 mm | Meshing-in moves toward the toe, meshing-out toward the heel; both converge to mid-face | All states move from root toward tip | Shortens |
| Pinion offset $\Delta T$: −0.05 → +0.05 mm | All states move from toe toward heel | All states move from root toward tip | Broadly unchanged |
| Paired offsets: $\Delta T$ up, $\Delta G$ down | Toe toward heel | Root toward tip | Broadly unchanged |
Verification on a rolling inspection machine
The simulated trends were checked against marked flanks on a production pair manufactured by Qinghai Huading Gearbox Co., whose parameters match the modelled geometry. Testing used a CNC-retrofitted Y9550 bevel gear rolling inspection machine, which reduces the scatter that manual assembly of the test pair would introduce. The driven-side load was kept at 40 N·m—enough to seat the flanks and register a pattern without overloading the fixture. The flanks were coated with marking compound, the pair was set to a deliberately non-ideal mounting distance (wheel offset +0.05 mm with the corresponding pinion offset of −0.042 mm from the relation above), and the machine was run before the offsets were stepped through the schedule of Table 3.
The initial setting produced the expected unreasonable pattern, with the contact zone biased toward the tooth tip and the marked area pressed against the upper part of the flank. After the offsets were brought to the correct mounting distance, the marked area returned to a mid-face position close to the toe, with an extent consistent with the light-load ideal. The direction of migration observed between the two states—upward along the tooth height and toward the heel along the face width as the pinion offset increased—agrees with the finite element predictions, supporting the use of the model as a qualitative guide. The comparison is not exact: the simulation carried 200 N·m while the tester carried 40 N·m, and the tester introduces fixture compliance, compound thickness and running-in effects that the model does not represent.
What the trends mean for mounting adjustment
Read as a diagnostic chart, the results give the fitter two levers with different signatures. A pattern that is too short along the face width, with meshing-in and meshing-out crowding toward mid-face, points to a wheel mounting-distance error; the correction is a wheel shim change, and the pattern length recovers as the offset is removed. A pattern that is the right size but sitting too close to the toe and the root points to a pinion mounting-distance error, and the correction is a pinion shim change combined with the compensating wheel movement of the paired schedule, which translates the band diagonally without shortening it. Because the pinion-side response moves the pattern toward heel and tip together, an over-correction shows up first as edge contact at the heel–tip corner, the condition most likely to initiate tooth-end failure.
The scope of these conclusions should be stated plainly. They derive from one gear pair geometry, one friction coefficient and one load level in the simulation, with rigid shaft and bearing assumptions; thermal distortion, housing deflection and shaft bending under load are not represented, and all three can shift the absolute pattern position. The value of the result therefore lies in the sign and the coupling of the offsets—which way the pattern moves, and how much wheel offset must accompany a given pinion offset—rather than in absolute pattern coordinates transferable to another gear set. Within that scope, the three-state extraction adds a practical detail that single-position TCA does not provide: a pattern that looks acceptable at mid-mesh can still be contracting at meshing-in and meshing-out, which is exactly the condition that shortens service life while passing a static inspection.