Spiral bevel gears are the standard solution wherever a high-speed shaft has to turn a drive through a right angle. The curved tooth trace produces a gradual, overlapping contact pattern that keeps transmission smooth, load capacity high, noise low and the package compact. The same geometry, however, places the gear rim on a comparatively thin, disk-like web, and at aero-engine speeds that web is not a static ring. It carries bending waves that circulate around the circumference, and when the tooth-mesh excitation keeps pace with one of them, the wave stops travelling relative to the material. The web is then loaded by a stationary pattern of alternating stress while the metal rotates through it — the mechanism behind flank fretting, web cracking and, in the extreme, loss of the web and of the drive.
The mechanism: nodal diameters, wave splitting and resonance margin
Modal deformation of a gear web falls into three categories. Nodal-circle modes have concentric nodal lines and vibrate symmetrically about the axis. Nodal-diameter (ND) modes have radial nodal lines; a mode of n nodal diameters shows n equidistant radial lines that do not move. Compound modes combine the two. ND modes are the damaging class: the radial nodal pattern couples efficiently to tooth-mesh forces, produces uneven contact across the face width and concentrates alternating stress at the rim-to-web fillet.
In a rotating part each ND shape splits into a forward travelling wave, whose frequency in the stationary frame rises with speed, and a backward travelling wave, whose frequency falls. Resonance requires more than an intersection of an excitation order with a natural frequency: the spatial pattern of the excitation must match the ND shape, so only the crossings belonging to ND families are treated as resonance points. Each crossing converts into a resonance speed ni, and the distance between that speed and the operating speed n0 is expressed as the resonance margin
$$\delta=\frac{\left| n_{i}-n_{0}\right|}{n_{0}}\times 100\%,$$
with a margin above 10 % treated as the acceptance threshold for smooth, safe running. Equivalently, a resonance at ni contaminates the speed interval in which the margin falls below 10 %, that is ni/1.1 ≤ n ≤ ni/0.9.
The gear pair and its material
The object of the study is a high-speed spiral bevel gear pair of an aero-engine accessory drive, with the structural and material data listed in Tables 1 and 2. The driven wheel carries 74 teeth against 27 on the pinion, a 3.85 mm module, a 38.5 mm face width and a 35° spiral angle, with left-hand and right-hand helix respectively. Both members are cut from 20CrMnTi case-hardening steel.
Table 1. Structural parameters of the gear pair
| Parameter | Driving gear | Driven gear |
|---|---|---|
| Number of teeth | 27 | 74 |
| Module / mm | 3.85 | 3.85 |
| Face width / mm | 38.5 | 38.5 |
| Spiral angle / (°) | 35 | 35 |
| Hand of helix | Left | Right |
Table 2. Properties of 20CrMnTi
| Property | Value |
|---|---|
| Elastic modulus / GPa | 207 |
| Poisson’s ratio | 0.25 |
| Density / (kg·m−3) | 7 850 |
| Tensile strength / MPa | 1 080 |
| Yield strength / MPa | 885 |
Although the two members exchange vibration through the tooth contact, the response of the driven wheel under external excitation governs the travelling-wave behaviour of the pair, so the numerical work is confined to the driven wheel. Its solid model was built in CATIA and transferred to the Modal module of ANSYS Workbench.
Finite element model and boundary conditions
The mesh uses quadratic tetrahedral elements. A convergence check over the range from 0.8 million to 1.6 million elements changed the computed frequencies by no more than 0.541 %, so a 3 mm element size was retained, giving 810 186 elements with an average element quality of 0.756 and an aspect ratio of 1.7. The shaft runs on a cylindrical roller bearing at the upper end and a ball bearing at the lower end. Because bearing compliance contributes little to the global deformation, each bearing was replaced by a cylindrical constraint applied at the actual mounting position, leaving only rotation about the gear axis free.
A velocity boundary condition ramps the speed from 0 to the rated 7 626 r/min within 1 s and then holds it constant, reproducing the run-up and steady operating condition. The Coriolis effect is switched on in the solver environment so that the gyroscopic splitting of the ND pairs is captured rather than suppressed, and a rotational constraint prevents rigid-body yaw of the wheel in the rotating frame.
Nodal-diameter families inside the excitation band
Because the external excitation is generated by the mating pinion, the excitation order used to build the Campbell diagram is the pinion tooth count of 27, and the tooth-mesh excitation reaches about 8 900 Hz at the rated speed. The first 30 modes, which extend to 8 891.4 Hz, therefore cover the frequency range of interest completely. Their frequencies and ND assignments are given in Table 3.
Table 3. Natural frequencies of the driven gear and nodal-diameter assignment
| Order | Frequency / Hz | Nodal diameters |
|---|---|---|
| 1 | 110.6 | — |
| 2 | 1 115.6 | 1 |
| 3 | 1 115.6 | 1 |
| 4 | 1 261.0 | 2 |
| 5 | 1 261.1 | 2 |
| 6 | 1 470.7 | — |
| 7 | 2 573.8 | 3 |
| 8 | 2 574.0 | 3 |
| 9 | 3 603.3 | — |
| 10 | 3 605.7 | — |
| 11 | 4 405.7 | 4 |
| 12 | 4 411.3 | 4 |
| 13 | 5 661.0 | — |
| 14 | 5 661.7 | — |
| 15 | 5 709.1 | — |
| 16 | 5 993.8 | — |
| 17 | 5 994.5 | — |
| 18 | 6 393.9 | 5 |
| 19 | 6 394.7 | 5 |
| 20 | 6 989.9 | — |
| 21 | 6 990.2 | — |
| 22 | 7 064.9 | — |
| 23 | 8 012.9 | — |
| 24 | 8 015.0 | — |
| 25 | 8 416.8 | 6 |
| 26 | 8 417.9 | 6 |
| 27 | 8 540.8 | — |
| 28 | 8 541.0 | — |
| 29 | 8 890.9 | — |
| 30 | 8 891.4 | — |
Six ND families occur within the mesh excitation band, each with one radial nodal line per nodal diameter: a single line for the 1 ND shape at 1 115.6 Hz, two lines for the 2 ND shape at 1 261.0 Hz, three at 2 573.8 Hz, four at 4 405.7 Hz, five at 6 393.9 Hz and six at 8 416.8 Hz. Apart from the exactly degenerate 1 ND pair, every family appears as two orders separated by only 0.1 to 5.6 Hz, the forward- and backward-travelling partners of the same shape whose small frequency difference reflects the gyroscopic splitting of the rotating wheel. Frequencies not assigned to a family belong to modes of other character — shaft bending, nodal-circle and compound shapes — which do not present the radial pattern needed to couple strongly to tooth-mesh forces.
Resonance points and margins at the rated speed
Plotting mode frequency against shaft speed and superimposing the 27th-order excitation lines produces the Campbell diagram; the intersections with ND families are the candidate resonance points. Excitation lines cross the first eleven orders, but the 1 ND and 2 ND crossings lie below 3 000 r/min, two and a half times away from rated speed, and the margins there are far above the acceptance threshold. Attention is therefore concentrated on the 3 ND forward and backward waves and the 4 ND backward wave, whose crossings fall close to the operating point (Table 4).
Table 4. Resonance speeds and margins at the rated operating speed of 7 626 r/min
| Order | Nodal diameters | Wave | Resonance speed / (r·min−1) | Resonance margin / % |
|---|---|---|---|---|
| 7 | 3 | Backward | 5 218.6 | 31.57 |
| 8 | 3 | Forward | 6 330.9 | 16.98 |
| 11 | 4 | Backward | 8 758.6 | 14.85 |
All three margins exceed 10 %, so the rated speed of 7 626 r/min sits in an admissible region: the nearest resonance, the 4 ND backward wave at 8 758.6 r/min, is 1 132.6 r/min away, and the 3 ND forward wave at 6 330.9 r/min is 1 295.1 r/min below. The 3 ND backward wave is further still. None of the three crossings can be excited strongly enough at rated speed to threaten the web.
Dangerous intervals and a safe test envelope
For bench testing it is not enough to clear the operating point; an overspeed and underspeed envelope from 60 % to 130 % of rated speed — 4 575.6 to 9 913.8 r/min — must also be screened, because the run-up and run-down pass through every resonance in that range. Applying the 10 % criterion to each resonance speed gives the intervals in Table 5, each centred on a crossing and bounded by ni/1.1 at the lower end and ni/0.9 at the upper end.
Table 5. Speed intervals in which the resonance margin falls below 10 %
| Resonance speed / (r·min−1) | Wave | Margin below 10 % over / (r·min−1) |
|---|---|---|
| 5 218.6 | 3 ND backward | 4 744.2–5 798.4 |
| 6 330.9 | 3 ND forward | 5 755.4–7 034.3 |
| 8 758.6 | 4 ND backward | 7 962.4–9 731.8 |
Subtracting these intervals from the 60 %–130 % envelope leaves three admissible windows (Table 6). The widest of them, from just above 7 034.3 to just below 7 962.4 r/min, contains the rated point with 591.7 r/min of headroom below and 336.4 r/min above; the other two are narrow bands of roughly 170 and 180 r/min at the ends of the envelope.
Table 6. Safe test-speed windows within 60 %–130 % of rated speed
| Window | Speed range / (r·min−1) | Width / (r·min−1) |
|---|---|---|
| Lower | 4 575.6–<4 744.2 | 168.6 |
| Central | >7 034.3–<7 962.4 | 928.1 |
| Upper | >9 731.8–9 913.8 | 182.0 |
In practice a test programme should accelerate through the excluded intervals rather than dwell in them, and should place steady-state comparison points inside the three windows. The central window is the natural choice for rated-condition testing; the narrow outer windows are better suited to short transient passes or to verifying that no unexpected mode is excited at the extremes of the envelope.
How the result should be read
Two features of the modal data drive the outcome. First, the ND families come in closely spaced pairs, so every dangerous shape produces two resonance speeds rather than one, one on either side of the family frequency; the 3 ND family alone brackets the rated point at 5 218.6 and 6 330.9 r/min. Second, the resonance speeds creep upward with nodal-diameter number while the family frequencies rise faster, so the highest-order shape in the excitation band, the 4 ND backward wave, ends up closest to the operating speed and sets the tightest margin at 14.85 %. That is the number a designer has least freedom to change: it is fixed by the web stiffness and mass distribution rather than by the tooth geometry.
The scope of these conclusions should be kept in view. The analysis is linear and undamped, the bearings are idealised as rigid supports, the excitation is represented by a single tooth-mesh order at nominal geometry and room-temperature properties, and no mistuning or manufacturing scatter is included. A margin above 10 % establishes that the rated speed is sufficiently far from a modal coincidence; it does not by itself certify acceptable stress amplitudes if the gear is held near a crossing, nor does it cover the response of the pinion, which was omitted on the grounds that the driven wheel governs the pair. Where a design change is contemplated, the quantities to re-examine are the 4 ND family frequency, which controls the upper edge of the central safe window, and the 3 ND pair, which controls its lower edge.