Clearance as the Hidden Link Between Bearing Selection and Mesh Quality
In an aero-engine accessory gearbox, the spiral bevel pair that closes the reduction train carries the highest specific load in the system while being supported by rolling bearings whose internal clearance is never zero in service. Clearance is usually treated as a bearing parameter — a manufacturing tolerance band chosen for lubrication, thermal growth and mounting fit. In a bevel gear train it is also a kinematic parameter. Under the axial and radial forces generated by the mesh, a bearing with clearance allows its shaft to translate relative to the housing, and that translation is transmitted directly to the relative position of the pinion and wheel axes. What the gear teeth experience is not a bearing specification but an installation error.
For spiral bevel and hypoid gears the conjugate action is fixed by the machine settings used to cut the flanks, and the contact pattern is the visible expression of that conjugacy. A displacement of either axis of a few tens of micrometres or a shaft-angle change of a few tenths of a milliradian is enough to move the contact pattern along the tooth, change the local load density, and alter the peak-to-peak transmission error that governs the dynamic excitation passed into the housing. Four error components are conventionally used to describe this displacement:
- Offset error E — displacement of the pinion axis perpendicular to the wheel axis, positive when the pinion moves upward relative to the wheel axis.
- Pinion axial displacement Xp — positive when the pinion moves away from the pitch apex.
- Wheel axial displacement Xg — positive when the wheel moves away from the pitch apex.
- Shaft angle error Σ — positive when the shaft angle between the two axes increases.
Because the four components respond differently to the same bearing clearance, and because their signs can change across the clearance range, the net effect on the mesh cannot be inferred from the clearance value alone. It has to be computed through the full drivetrain: bearing stiffness and clearance, shaft bending, gear loads and the loaded tooth contact of the bevel pair.
System Model of the Accessory Gearbox
The drivetrain studied here is the accessory gearbox of an aero engine, in which a spiral bevel pair transmits power to an offset output. A system-level model of the gearbox was assembled in MASTA, comprising the housing, shafts, rolling bearings and both the bevel pair and the spur gear stages. Bearings were selected from the shaft layout and basic envelope dimensions; shafts and gears were generated from their geometric parameters and connected according to the relative positions of the stages; the three-dimensional housing model was imported and coupled to the shaft system. This model supplies the load-dependent displacement of each shaft centre as a function of the bearing clearance grade, which is then converted into the four installation-error components of the bevel pair.
The bevel pair itself is a 90° shaft-angle, Gleason-generated spiral bevel gear set with the parameters listed in Table 1. Its working flank in the direction of power flow is the pinion concave side mating with the wheel convex side, and this flank carries the load for nearly the whole duty cycle; tooth-cutting parameters for such gears are in practice optimised by adjusting the contact pattern on the pinion, so the analysis is restricted to that flank.
| Parameter | Pinion | Wheel |
|---|---|---|
| Shaft angle | 90° | |
| Number of teeth | 27 | 38 |
| Face width | 15 mm | |
| Pitch diameter | 51.516 mm | 72.504 mm |
| Transverse module | 1.908 mm | |
| Mean spiral angle | 30° | |
| Hand | Right | Left |
Loaded Tooth Contact Analysis of the Bevel Pair
Flank geometry was generated from the Gleason GEMS cutter and machine-tool settings, and the tooth surface point set was computed numerically; solid models of pinion and wheel were then built and assembled for finite element analysis. The contact model uses seven teeth on each member to keep the computation tractable, with 20CrMo gear steel (elastic modulus 210 GPa, Poisson’s ratio 0.3) and hexahedral C3D8R elements throughout.
Two quantities extracted from the loaded contact solution define mesh quality. The first is the peak Hertzian contact stress on the flank, obtained for an elliptical contact patch as
$$\sigma_{\max}=\frac{3P}{2\pi ab}$$
where P is the applied normal load and a, b are the semi-axes of the contact ellipse,
$$a=\alpha\sqrt[3]{\frac{3P}{4A}\left(\frac{1-\mu_1^{2}}{E_1}+\frac{1-\mu_2^{2}}{E_2}\right)},\qquad b=\beta\sqrt[3]{\frac{3P}{4A}\left(\frac{1-\mu_1^{2}}{E_1}+\frac{1-\mu_2^{2}}{E_2}\right)}$$
with μi and Ei the Poisson’s ratio and elastic modulus of each body, A and B the geometric parameters derived from the principal curvatures at the contact point, and the coefficients α and β interpolated from the parameter θ = arccos(B/A). The second quantity is the loaded transmission error. Each contact point corresponds to a pair of rotational positions of pinion and wheel, and with the pinion driving, the transmission error is
$$T_e=(\varphi_w-\varphi_{w0})-(\varphi_p-\varphi_{p0})\,\frac{z_p}{z_w}$$
where φp0, φw0 are the initial rotational angles, φp, φw the corresponding angles at the end of a mesh cycle, and zp, zw the tooth counts of pinion and wheel.
From Clearance Grade to Installation Error
Bearing clearance in the drivetrain model is classified into the standard catalogue grades 2, N (normal), 3, 4 and 5, each covering a progressively wider clearance band. Two operating points were examined: a rated condition of 32 kW transmitted power with 7.2 N·m applied torque, and an extreme condition of 38.4 kW with 8.64 N·m. For each grade the resulting installation errors of the bevel pair were extracted, Table 2.
| Condition | Clearance grade | E (µm) | Xp (µm) | Xg (µm) | Σ (mrad) |
|---|---|---|---|---|---|
| Rated, 32 kW | 2 | 35.72 | −21.88 | 39.32 | 0.02 |
| N | 47.06 | −20.38 | 53.88 | 0.04 | |
| 3 | 54.92 | −19.69 | 71.95 | −0.01 | |
| 4 | 62.50 | −18.08 | 91.77 | −0.07 | |
| 5 | 69.88 | −16.86 | 113.13 | −0.15 | |
| Extreme, 38.4 kW | 2 | 40.23 | −26.62 | 42.25 | 0.14 |
| N | 51.47 | −24.12 | 56.76 | 0.07 | |
| 3 | 59.42 | −22.72 | 74.65 | 0.02 | |
| 4 | 66.30 | −21.28 | 94.42 | −0.04 | |
| 5 | 74.34 | −19.63 | 115.74 | −0.10 |
Three features of this mapping matter. First, the errors are not monotonic in the same direction: as the grade moves from 2 to 5, the offset error and the wheel axial displacement grow steadily — Xg almost triples, from 39.32 µm to 113.13 µm under rated load — while the pinion axial displacement and the shaft angle error decrease in magnitude, and Σ crosses zero between grades N and 3 at rated load and between grades 3 and 4 at extreme load. Second, the sign reversal of the shaft angle error means that the two ends of the clearance range do not merely differ in magnitude but tilt the mesh in opposite senses. Third, every error component is larger at the higher torque for the same clearance grade, because the additional elastic deflection of shafts and bearings adds to the clearance-induced displacement; clearance effects in this gearbox are therefore load-dependent rather than a fixed property of the bearing.
Rated Condition: Stress Rises While Transmission Error Falls
With no bearing clearance the bevel pair develops a peak contact stress of 636.51 MPa, a peak root bending stress of 79.12 MPa, and a peak-to-peak transmission error of 165.57 µrad. Introducing clearance changes all three at once. The contact pattern migrates from its nominal position toward the toe and the addendum of the working flank, and the peak contact stress climbs with increasing clearance up to grade 4, where it reaches 683.40 MPa, an increase of about 7.4% over the clearance-free case. Beyond grade 4 the trend reverses and the peak stress falls again. The share of the flank covered by the contact pattern remains nearly constant at 13–14% across all grades, so the stress rise is not a reduction in contact area but a relocation of that area into a less favourable part of the flank.
Root bending follows the same pattern with a larger penalty. The peak stress rises from 79.12 MPa to 95.41 MPa at grade 4, about 20.6% higher, and the location of the maximum migrates toward the toe as the grade increases; beyond grade 4 the bending stress declines. Transmission error behaves in the opposite sense to stress: clearance lowers the peak-to-peak value, which reaches its minimum of 133.78 µrad at grade 5, roughly 19% below the clearance-free baseline.
Extreme Condition: The Same Trends, Amplified
At 38.4 kW and 8.64 N·m the baseline stresses are higher, as expected — 698.45 MPa contact and 90.41 MPa root bending, about 9.7% and 14.3% above their rated-condition counterparts — while the baseline transmission error is lower, at 144.51 µrad. The clearance response repeats the rated-condition pattern with greater amplitude. Peak contact stress rises to 748.5 MPa at grade 4 (+7.2%), then declines; the contact pattern again moves toward the toe and addendum, with the covered fraction in the range 14.5–16%. Peak root bending rises to 116.86 MPa at grade 4, a 29.2% increase over the clearance-free case, and the location of the maximum again shifts toward the toe. Peak-to-peak transmission error falls to a minimum of 127.31 µrad at grade 5, about 12% below baseline.
| Metric | Condition | No clearance | Clearance grade | Value | Change |
|---|---|---|---|---|---|
| Peak contact stress | Rated, 32 kW | 636.51 MPa | 4 | 683.40 MPa | +7.4% |
| Peak contact stress | Extreme, 38.4 kW | 698.45 MPa | 4 | 748.5 MPa | +7.2% |
| Peak root bending stress | Rated, 32 kW | 79.12 MPa | 4 | 95.41 MPa | +20.6% |
| Peak root bending stress | Extreme, 38.4 kW | 90.41 MPa | 4 | 116.86 MPa | +29.2% |
| Transmission error, peak-to-peak | Rated, 32 kW | 165.57 µrad | 5 | 133.78 µrad | −19.2% |
| Transmission error, peak-to-peak | Extreme, 38.4 kW | 144.51 µrad | 5 | 127.31 µrad | −11.9% |
| Contact pattern area fraction | Rated, 32 kW | — | 2 to 5 | 13–14% | — |
| Contact pattern area fraction | Extreme, 38.4 kW | — | 2 to 5 | 14.5–16% | — |
Interpreting the Trade-off
The non-monotonic stress behaviour has a plausible mechanical reading. Increasing clearance raises the offset error E and the wheel axial displacement Xg together, and both act to concentrate load on a narrower band of the flank; at the same time the pattern is being driven toward the toe, where the tooth is stiffer in bending and the contact ellipse is oriented differently relative to the root fillet. Up to grade 4 the concentration effect dominates; beyond it, the relocation of the pattern and the accompanying change in shaft-angle error dominate and the peak stresses fall back. The analysis supports this reading but does not isolate the two contributions, since all four error components vary simultaneously with the clearance grade.
The practical consequence is that the clearance grade optimising one metric does not optimise the others. Peak contact and root bending stresses are lowest in the clearance-free configuration and highest at grade 4, whereas transmission error falls monotonically through grade 5 in both load cases — the study’s summary characterises this as a minimum followed by a rise beyond a critical clearance, a reversal that lies outside the grade range computed here. A designer weighting root fatigue life would therefore keep the operating clearance below grade 4, while a designer weighting dynamic excitation of the housing would move in the opposite direction. For the extreme condition, where the bending penalty reaches 29%, the fatigue argument is the stronger of the two.
Two further points follow from the load dependence of the mapping. Because the misalignment produced by a given grade grows with transmitted torque, a clearance band qualified at rated power does not remain equivalent at the limit load, and the sign reversal of the shaft angle error can occur at a different grade in the two conditions — between N and 3 at 32 kW, between 3 and 4 at 38.4 kW. And because the contact pattern retains roughly the same area while its position moves, inspection of pattern size alone would not reveal the stress increase; pattern location, particularly the drift toward the toe and addendum, is the more sensitive indicator.
Scope and Limitations
The conclusions are conditional on the modelling choices. The contact solution is quasi-static and linear-elastic, evaluated on the pinion working flank only, with seven teeth per member; the wheel convex side, the coast flank and the reverse torque direction are not covered. Clearance values follow standard bearing grade bands rather than measurements on a specific bearing set, so the mapping in Table 2 describes a class of assemblies rather than one hardware unit. Thermal growth of shafts and housing, which shifts clearance in running engines, and the stiffness of the housing mount are outside the present model, and the clearance-induced misalignment would be superimposed on both in service.
Conclusions
- Bearing clearance in the accessory gearbox produces a coupled installation error in the spiral bevel pair: offset error and wheel axial displacement increase with clearance grade, pinion axial displacement and shaft angle error decrease, and the shaft angle error changes sign within the grade range. All components grow with transmitted torque.
- Relative to a clearance-free assembly, clearance increases peak contact stress and peak root bending stress and relocates the contact pattern toward the toe and addendum, while reducing peak-to-peak transmission error. Root bending is penalised far more than contact stress: 20.6% at rated load and 29.2% at extreme load, against roughly 7% for contact stress.
- Contact and root bending stresses peak at grade 4 and decline at grade 5, whereas transmission error continues to fall to its minimum at grade 5 — a 19.2% reduction at rated load and 11.9% at extreme load.
- Because the misalignment generated by a given clearance grade depends on load, clearance selection for an aero-engine bevel pair should be validated at the limit condition rather than at rated power, and should be based on pattern location and root stress rather than on contact area alone.