Spiral bevel gears are widely used in aero-engine accessory gearboxes because their curved teeth combine high load capacity with smooth motion between intersecting shafts. Their meshing quality depends on the relative position of the pinion and the gear. In a real gearbox, that position is set by shafts, bearings, and the housing, not by the gear pair alone. Bearing clearance therefore acts as a mesh-alignment parameter: it allows the shaft centre to shift under load, and the resulting displacement is transmitted directly into the gear contact.
The sensitivity is high because a spiral bevel gear mesh is controlled by axial position, offset, and shaft angle. A change of a few micrometres in axial displacement, or a few milliradians in shaft angle, can move the contact pattern from the centre of the tooth toward the toe or the tip, change the local contact ellipse, and alter the load shared by neighbouring teeth. The same displacement also changes backlash and the loaded transmission error. Clearance selection is thus not only a bearing-life issue; it is also a gear-mesh issue.
From Bearing Clearance to Mesh Misalignment
Under an external load, a bearing with clearance reaches an equilibrium position in which the rolling elements, races, shaft, and housing deform together. The shaft can move radially and axially, and its tilting angle can change. For the spiral bevel gear pair, these movements are represented by four installation-error components:
- E, the offset error between the pinion and gear axes. With the gear axis as reference, upward pinion displacement is positive and downward displacement is negative.
- Xp, the axial displacement of the pinion. Movement away from the shaft intersection is positive; movement toward it is negative.
- Xg, the axial displacement of the gear, defined with the same sign convention.
- Σ, the shaft-angle error. An increase in shaft angle is positive; a decrease is negative.
These components are not independent. A change in bearing clearance alters the support stiffness and the loaded equilibrium of the whole shaft system, so offset, axial displacement, and shaft angle change together. The resulting contact pattern is a combined response rather than the effect of a single displacement.
Transmission System and Loaded Contact Model
The analysis is based on the transmission system of an aviation accessory gearbox. Shafts, bearings, the spiral bevel gear pair, a spur gear pair, and the housing are assembled in a system-level transmission model. Bearing positions and basic dimensions define the bearing selection; shaft and gear geometry define the flexible shaft model; the housing is coupled to the shaft-bearing system so that support deformation is included in the loaded equilibrium.
| Parameter | Pinion | Gear |
|---|---|---|
| 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 of spiral | Right | Left |
The tooth surfaces are generated from Gleason cutter and machine settings. Surface point sets are calculated numerically, converted into solid models, and assembled for finite element analysis. The loaded tooth contact model uses seven teeth on each member to reduce computational cost while retaining the neighbouring-tooth support that controls load sharing. The material is 20CrMo, with an elastic modulus of 210 GPa and a Poisson ratio of 0.3. Hexahedral C3D8R elements are used. The contact analysis is restricted to the pinion working flank, which carries most of the operating load.
Two load cases are considered: a rated condition of 32 kW and 7.2 N·m, and an extreme condition of 38.4 kW and 8.64 N·m. Bearing clearance is varied over grades 2, N, 3, 4, and 5; the sequence represents progressively wider clearance ranges, with grade N as the standard grade. A zero-clearance assembly provides the baseline for the meshing metrics.
Tooth contact stress is estimated from Hertzian elliptical contact. For a normal load P and contact-ellipse semi-axes a and b,
$$\sigma_{\max}=\frac{3P}{2\pi ab}.$$
The loaded transmission error is evaluated from the angular positions of the two members. With the pinion as the driving member,
$$T_e=(\varphi_w-\varphi_{w0})-\frac{z_p}{z_w}(\varphi_p-\varphi_{p0}),$$
where φw and φp are the instantaneous gear and pinion angles, φw0 and φp0 are their initial angles, and zw and zp are the numbers of teeth. The peak-to-peak value of Te is used as the transmission-error metric.
Installation Error Response
The calculated installation errors at the rated and extreme conditions are listed below. Increasing clearance from grade 2 to grade 5 raises the offset error and the gear axial displacement, while the pinion axial displacement becomes less negative. The shaft-angle error decreases and reverses sign. The extreme load amplifies all four components.
| Error component | Grade 2 | Grade N | Grade 3 | Grade 4 | Grade 5 |
|---|---|---|---|---|---|
| E (μm) | 35.72 | 47.06 | 54.92 | 62.50 | 69.88 |
| Xp (μm) | −21.88 | −20.38 | −19.69 | −18.08 | −16.86 |
| Xg (μm) | 39.32 | 53.88 | 71.95 | 91.77 | 113.13 |
| Σ (mrad) | 0.02 | 0.04 | −0.01 | −0.07 | −0.15 |
| Error component | Grade 2 | Grade N | Grade 3 | Grade 4 | Grade 5 |
|---|---|---|---|---|---|
| E (μm) | 40.23 | 51.47 | 59.42 | 66.30 | 74.34 |
| Xp (μm) | −26.62 | −24.12 | −22.72 | −21.28 | −19.63 |
| Xg (μm) | 42.25 | 56.76 | 74.65 | 94.42 | 115.74 |
| Σ (mrad) | 0.14 | 0.07 | 0.02 | −0.04 | −0.10 |
From grade 2 to grade 5, the offset error increases by 34.16 μm at the rated condition and by 34.11 μm at the extreme condition. The gear axial displacement increases by 73.81 μm and 73.49 μm, respectively. The pinion axial displacement becomes less negative by 5.02 μm and 6.99 μm. The shaft-angle error changes from +0.02 to −0.15 mrad at rated load and from +0.14 to −0.10 mrad at extreme load. This sign reversal matters because the direction of the shaft-angle error changes the side of the tooth on which the contact pattern is concentrated.
Contact Stress and Contact Pattern
With no bearing clearance, the maximum tooth contact stress is 636.51 MPa at the rated condition and 698.45 MPa at the extreme condition. Introducing bearing clearance increases the stress and moves the contact pattern toward the toe and tip of the pinion working flank. At rated load, the maximum contact stress reaches 683.40 MPa at grade 4, about 7.4% above the zero-clearance baseline. At extreme load, it reaches 748.5 MPa at grade 4, about 7.2% above the baseline. When the clearance grade exceeds 4, the peak contact stress falls from this maximum.
The contact-area ratio remains within 13%–14% at rated load and 14.5%–16% at extreme load. The stress increase is therefore not caused by a simple loss of contact area. It is associated with a shift in contact position: the misalignment moves the loaded ellipse toward a region of less favourable curvature and changes the local pressure distribution. At larger clearance, the contact pattern and the load distribution over the neighbouring teeth redistribute the load again, which lowers the peak stress from its grade-4 value.
| Metric | Rated baseline | Rated extremum | Extreme baseline | Extreme extremum |
|---|---|---|---|---|
| Maximum contact stress (MPa) | 636.51 | 683.40 at grade 4 | 698.45 | 748.5 at grade 4 |
| Maximum root bending stress (MPa) | 79.12 | 95.41 at grade 4 | 90.41 | 116.86 at grade 4 |
| Peak-to-peak transmission error (μrad) | 165.57 | 133.78 at grade 5 | 144.51 | 127.31 at grade 5 |
Root Bending Stress
The root bending stress follows the same non-monotonic trend as the contact stress. At the rated condition, the maximum root bending stress is 79.12 MPa without clearance and rises to 95.41 MPa at grade 4, an increase of about 20.6%. At the extreme condition, it rises from 90.41 MPa to 116.86 MPa, an increase of about 29.3%. As the clearance grade increases, the location of the maximum root stress moves toward the toe of the tooth.
The movement of the contact load toward the toe changes the bending moment arm and the local tooth stiffness. The extreme condition produces a larger response because the external load and the installation errors increase together. The root stress therefore reflects both the nominal load and the clearance-induced shift in the load path.
Transmission Error
The zero-clearance transmission error is 165.57 μrad peak-to-peak at rated load and 144.51 μrad at extreme load. With bearing clearance, the transmission error decreases. At the rated condition, the minimum within the tested grades is 133.78 μrad at grade 5, about 19.2% below the baseline. At the extreme condition, the minimum is 127.31 μrad at grade 5, about 11.9% below the baseline.
Within the tested range, the transmission error falls monotonically as the clearance grade increases. The minimum therefore lies at grade 5, the largest clearance analysed; the data do not show a subsequent rise inside this range. The reduction is a loaded-contact effect: the misalignment changes the sequence of contact and the local compliance along the path, which can flatten the transmission-error curve even while the contact and root stresses increase. A low transmission error is thus not sufficient evidence of a favourable mesh.
Design Implications and Model Scope
Bearing clearance changes the mesh through the shaft system. The dominant chain is clearance → loaded shaft displacement → offset, axial displacement, and shaft-angle error → contact-pattern shift → stress and transmission-error change. For this accessory gearbox, the offset error and gear axial displacement grow steadily with clearance grade, while the shaft-angle error reverses direction. The stress response peaks at grade 4, whereas the transmission-error response reaches its minimum at grade 5. Clearance selection therefore involves a trade-off between contact and root strength on one side and motion smoothness on the other.
A clearance chosen only for low transmission error would move toward the upper end of the tested range. A clearance chosen only for stress would avoid the grade-4 peak. Because the contact pattern at large clearance is displaced toward the toe and tip, the choice also affects the usable contact reserve under further load increases or manufacturing scatter. The extreme condition amplifies both the installation errors and the stress penalties, so rated-condition data alone are not sufficient for clearance selection.
The results apply to steady loaded contact of the pinion working flank, with seven teeth modelled on each member and with the housing and bearing supports represented in the system model. Transient dynamics, thermal expansion, lubricant-film effects, and run-in wear are outside this scope. Within that scope, the analysis shows that bearing clearance is a first-order variable for the meshing behaviour of aviation spiral bevel gears, not merely a bearing-internal tolerance.