The failure mode behind thin-web bevel gear fractures
Helicopter main transmissions are designed around mass per unit of transmitted torque, and the final stage is where that pressure is felt most. The large driven wheel of a spiral bevel pair is frequently produced with a thin web joining the rim to the hub: removing material there cuts rotating inertia and bearing load, but it also turns the web into an annular plate with a dense family of nodal-diameter (diametral) modes, generally between a few hundred hertz and about 3 kHz.
On a spiral bevel tooth the mesh force is not purely tangential. The spiral angle and the cone angle give it a large axial component, and because the number of teeth in contact alternates between one and two, that axial component is periodic at the mesh frequency and its harmonics. When the mesh frequency, or a harmonic of it, coincides with the natural frequency of one of the web’s nodal-diameter modes, the structure responds as a rotating deflection pattern rather than a standing one. The wave circulates around the gear, and the bending stress at the junction between web and rim reverses once per wave passage. That reversed bending is what initiates the web cracks and, in service, the web fractures reported on helicopter spiral bevel wheels.
Resonance condition and the forward/backward split
For a wheel with z teeth running at N r/min, the mesh frequency is
$$f_m = \frac{zN}{60}$$
A mode with m nodal diameters is degenerate: it supports two waves travelling in opposite directions around the circumference. Written in terms of the shaft speed and the harmonic order K1 of the mesh force, resonance of that mode requires
$$K_1\frac{zN}{60} = f_d \pm m\frac{N}{60}$$
which gives separate speeds for the forward and backward waves:
$$N_f = \frac{60 f_d}{K_1 z – m}, \qquad N_b = \frac{60 f_d}{K_1 z + m}$$
Here fd is the rotating, or dynamic, natural frequency. Centrifugal stiffening raises it above the stationary (static) value obtained from a modal analysis, but measurements on this class of wheel show the increase staying within about 3% across the operating speed range. That figure matters when judging margins: a screening margin of a few percent is the same order as the stiffening correction, so the ±3% band has to be carried through any clearance decision rather than treated as negligible.
The conventional screening tool follows directly. A Campbell diagram plots the mesh-order lines K1zN/60 against the horizontal nodal-diameter frequencies; each intersection is a candidate resonance, and a resonance margin
$$\phi = \left|\frac{f_g – f_j}{f_g}\right| \times 100\%$$
with fg the resonance frequency and fj the excitation frequency, ranks how close a given operating point sits to that intersection.
Constructing a physically consistent excitation
At a nodal-diameter resonance the axial (bending) energy dominates every other vibration form, so only the axial load on the tooth needs to be modelled. With a torque T0 and the pitch diameter at the mid-face dm, the tangential force is
$$F_t = \frac{2T_0}{d_m}$$
and, for a right-hand driven wheel rotating clockwise, the axial force follows from the spiral angle βm, pressure angle α and pitch cone angle δ:
$$F_x = \frac{2T_0}{d_m\cos\beta_m}\left(\tan\alpha\,\sin\delta – \sin\beta_m\cos\delta\right)$$
The load is not constant along the mesh cycle. Load sharing is governed by the total contact ratio, built from the transverse and overlap components through interpolation functions of the equivalent tooth numbers and the mid-face spiral angle at a 20° pressure angle, and of the face width and face-width coefficient for the overlap term:
$$\varepsilon_\alpha = K\left(\varepsilon_{\alpha 1} + \varepsilon_{\alpha 2}\right), \qquad \varepsilon_\beta \approx \frac{1}{1-0.5\phi_R}\cdot\frac{b\tan\beta_m}{\pi m}, \qquad \varepsilon_\gamma = \sqrt{\varepsilon_\alpha^2 + \varepsilon_\beta^2}$$
Assuming the two pairs in contact share the axial force equally, a tooth carries Fx in single contact and Fx/2 in double contact. Representing that alternation as a half-sine pulse of duration equal to one meshing period T = 60/(zN), launched at intervals T/εγ, reproduces the overlap between consecutive teeth automatically: with εγ ≈ 1.92 the pulses overlap, so either one or two teeth are loaded at any instant.
$$F(t) = F_x\sin\frac{\pi t}{T}, \qquad \frac{(l-1)T}{\varepsilon_\gamma} \le t < \frac{(l-1)T}{\varepsilon_\gamma} + T$$
Expanding this train in a Fourier series with the mesh frequency as fundamental,
$$F(t) = \frac{a_0}{2} + \sum_{k=1}^{\infty}\left[a_k\cos(k\omega t) + b_k\sin(k\omega t)\right]$$
only the component whose frequency and spatial pattern match the target nodal-diameter mode is retained as the excitation input, since that is the component that pumps the resonant mode. Before using it, the excitation is checked against the energy the mode can absorb. Treating the gear as a disk of thickness h with the load distributed over the tooth face, the work done over one period,
$$W_1 = \int_{\theta_0}^{\theta_t}\int_0^t\int_{r_1}^{r_2} f(t)\,w(r,\theta,t)\,\mathrm{d}r\,\mathrm{d}t\,\mathrm{d}\theta$$
is compared with the vibration energy of the disk, obtained by integrating ½ dm′ (dw/dt)² with dm′ ≈ ρhr dr dθ. Because f(t) > 0 and the displacement is taken in the travelling-wave form w(r,θ,t) = ½A(r)cos(mθ ± ωt), the input work exceeds the vibrational energy over the same period, confirming that the synthesized excitation can sustain resonance rather than merely perturb the structure.
Finite element solution of the whole structure
The structural solve avoids a rotating contact simulation. The gear is held stationary and the synthesized load is applied at the tooth nodes and migrated circumferentially across the model, so that the loaded sector sweeps the wheel at the resonance frequency while the mesh remains fixed in space. The gear and its hollow shaft are bonded at the interface, meshed with tetrahedral elements (288 203 elements for the reference case), restrained at the shaft boundary, and solved with an implicit dynamic step. Equivalent elastic strain is then sampled at rim and web nodes for time–frequency analysis.
Reference case: a 2 MW twin-input helicopter gear
The reference wheel is the driven gear of a twin-input spiral bevel stage, run at 1 193.8 r/min input speed with 2 000 kW total input power. Its geometry is summarized in Table 1.
| Parameter | Value |
|---|---|
| Number of teeth | 101 |
| Module | 3.85 mm |
| Spiral angle | 25° |
| Pitch cone angle | 81.852° |
| Pressure angle | 22.5° |
| Face width | 72 mm |
| Addendum coefficient | 1 |
Modal analysis of the whole structure returns the nodal-diameter ladder in Table 2’s first column: 530.28 Hz (1 nodal diameter), 555.28 Hz (2), 1 122.1 Hz (3), 1 891.7 Hz (4) and 2 694.3 Hz (5). At the operating point the mesh frequency is 101 × 1 193.8/60 ≈ 2 010 Hz, and the shaft frequency is 19.9 Hz.
| Nodal diameters | Wave | Resonance frequency / Hz | Resonance speed / (r·min⁻¹) | Margin / % |
|---|---|---|---|---|
| 3 | Backward | 1 089.7 | 647 | 45.77 |
| 3 | Forward | 1 156.4 | 687 | 42.45 |
| 4 | Backward | 1 819.6 | 1 081 | 9.45 |
| 4 | Forward | 1 969.7 | 1 170 | 1.98 |
Only one intersection is operationally relevant. The 3-nodal-diameter crossings sit at 647 and 687 r/min, 42–46% away from the operating speed. The 4-nodal-diameter forward wave crosses the first mesh order at 1 170 r/min, and the operating point at 1 193.8 r/min leaves a margin of 1.98% — the case the subsequent analysis is built around.
Computed response and its experimental check
With a static axial load of 132.17 kN, a total contact ratio of 1.921 4 and a meshing period of 0.498 ms, the synthesized excitation was applied and the dynamic response solved. The resulting strain field has a clean four-lobed pattern, and the strain extremum occurs at the small-end tooth root rather than in the web — the location where aero-engine spiral bevel wheels are in fact observed to fail. Time–frequency analysis of the equivalent elastic strain at a front rim element gives a wave transit time of 0.000 23 s and a migration rate of 123.62 rad/s; combined with the shaft frequency of 19.9 Hz and the mesh frequency of 2 009.9 Hz, this yields a 4-nodal-diameter frequency of about 1 929.98 Hz, within roughly 2% of the modal value of 1 891.7 Hz. The simulated dynamic strain amplitude at the location later instrumented is 226.16 × 10⁻⁶.
Validation used a dynamic response test on the same wheel. Eight resistance strain gauges were placed symmetrically around the circumference: two on the front rim, four on the front web near the rim, and two on the back rim near the large end. Gauges were oriented along the strain gradient to maximize sensitivity, bonded with a 250 °C-rated adhesive system. Order analysis of the recorded signals gave a mean measured strain of 240 × 10⁻⁶ and an identified 4-nodal-diameter frequency of 1 968 Hz.
| Quantity | Simulation | Test | Deviation |
|---|---|---|---|
| 4-nodal-diameter frequency | 1 892 Hz | 1 968 Hz | 4.0% |
| Strain amplitude | 226.16 × 10⁻⁶ | 240 × 10⁻⁶ | 5.77% |
The strain deviation, defined as |α0 − α′|/α′ × 100% with α0 the computed and α′ the measured value, stays at or below 5.77%, with the gauge-location tolerance, adhesive compliance, gauge nonlinearity and system error contributing. The 4% frequency difference between the stationary finite element model and the running wheel is consistent in sign and rough magnitude with centrifugal stiffening, which is why the static value can serve for screening only when the stiffening band is carried alongside it.
Web lightening holes: what they change and what they do not
With the same gear geometry and operating point, the web was perforated with 0, 2, 4 and 6 equally spaced holes and re-analysed (Table 4).
| Number of web holes | 4-ND natural frequency / Hz | Forward-wave resonance frequency / Hz | Resonance speed / (r·min⁻¹) |
|---|---|---|---|
| 0 | 1 891.7 | 1 969.7 | 1 170.0 |
| 2 | 1 879.8 | 1 957.3 | 1 162.8 |
| 4 | 1 879.2 | 1 956.7 | 1 162.4 |
| 6 | 1 878.2 | 1 955.7 | 1 161.8 |
The unperforated wheel has the highest 4-nodal-diameter frequency, and every perforated variant sits lower, but the total spread from zero to six holes is only about 13.5 Hz, or 0.7%, and the resonance speed moves by less than 9 r/min. Detuning by hole count is therefore ineffective at this scale: the operating point remains inside the resonant band, and the change in stiffness is far smaller than the ±3% centrifugal correction already present. What the holes do change is the strain distribution. Peak strain remains at the small-end tooth root regardless of hole number, while the regions adjacent to the holes carry higher local strain than the corresponding solid web when the wheel resonates — the perforations act as local compliance features and stress raisers rather than as a detuning measure.
Implications for design screening
Three points follow from the evidence. First, a Campbell intersection with a small margin is not sufficient evidence of a problem nor of its absence: at 1.98% the reference wheel sits inside the stiffening uncertainty, and the finite element response is what shows whether the resulting alternating strain at the small-end root is tolerable. Second, the critical location is set by the mode shape and the tooth-root geometry, not by the web detail, so mass-reduction features cannot be credited with moving the failure site. Third, because the method freezes the structure and sweeps a synthesized load, it avoids the convergence and run-time cost of a rotating-contact transient simulation, at the price of restricting the excitation to the single frequency component that drives the target mode. The approach is thus a screening and response-estimation tool for nodal-diameter resonance of thin-web bevel wheels, with the load-sharing assumption of equal force in double contact and the static-frequency approximation as its two main modelling limits.