The design problem: a thin web in a high-speed, hot gear train
The central drive of an aero-engine transfers power from the engine shaft to the accessory gearbox through a pair of spiral bevel gears. These gears run at high speed, carry heavy load and sit in a hot environment, and the driven wheel in particular is a disc-like body: a toothed rim carried on a comparatively thin web that connects to the hub. That geometry is mass-efficient, but it also creates a vibration path. Tooth-mesh excitation acting on the rim can drive nodal-diameter modes of the wheel, and once the wheel rotates each such mode splits into a forward and a backward traveling wave. When one of these waves coincides in frequency with the mesh excitation, the wave is continuously fed and the alternating stress concentrates at the web root. In-service failures of central-drive bevel wheels follow this route and appear as a complete circumferential fracture along the web root — the toothed ring separates from the hub.
Earlier work has examined both the structural and the dynamic side of the problem. Arai and Harada varied the fillet radius between web and rim and quantified its effect on tooth-root stress. Marunic analysed thin-rimmed gears over a range of rim and web thicknesses and evaluated the resulting stress on the inner rim surface. Qiu and Xu used strain gauges on driven helical gears to locate traveling-wave resonance in speed and frequency, and identified the forward wave as a damage mechanism capable of generating induced force components that may grow into self-excitation. Talbert and Gockel showed that traveling waves modulate the strong strain of the meshing cycle, so that local stress can exceed the nominal value. Shen Yunwen and co-workers compared different web arrangements in helical gear transmissions and found structural size and web support to have little influence on low-order natural frequencies but a substantial one on higher orders. Yang Rong and co-workers investigated a fractured central-drive bevel wheel from an accessory gearbox and attributed the damage mainly to traveling-wave resonance excited by axial vibration. Li Shuting and co-workers proposed a structural dynamic design route for thin, light aero-engine gear webs that combines theoretical calculation, structural sensitivity analysis and dynamic modification of parameters. The present problem belongs to this line of work: which geometric parameter moves the traveling-wave resonance of a spiral bevel wheel, and by how much.
Nodal-diameter modes and the forward–backward wave split
An axisymmetric disc has three classical families of mode shapes: nodal circles, nodal diameters, and combinations of the two. Thin discs, including gear webs, respond preferentially in nodal-diameter modes, in which $m$ diametral lines remain stationary while the remaining sectors oscillate out of phase. Because a spiral bevel wheel is geometrically close to a disc, the same family dominates its dynamic behaviour.
For a wheel observed in a frame fixed to the structure, the transverse displacement of a point on the web is
$$z(r,\theta,t) = A(r)\cos(m\theta)\cos(\omega t)$$
where $z$ is the transverse displacement, $A(r)$ the radial amplitude distribution, $m$ the number of nodal diameters, $\theta$ the circumferential angle, $\omega$ the angular frequency of the mode and $t$ time. Rewriting the standing wave as a pair of counter-rotating components gives
$$z(r,\theta,t) = \tfrac{1}{2}A(r)\left[\cos(m\theta-\omega t) + \cos(m\theta+\omega t)\right]$$
so a nodal-diameter mode of a stationary disc is equivalent to two waves of equal amplitude travelling in opposite directions around the circumference. In a rotating wheel the two components are no longer degenerate: in the frame of the rotating gear they appear at split frequencies
$$f_f = f_d + \frac{m\,n_g}{60}, \qquad f_b = f_d – \frac{m\,n_g}{60}$$
with $f_d$ the nodal-diameter natural frequency in Hz, $n_g$ the rotational speed of the wheel in r·min−1, and $f_f$, $f_b$ the forward and backward wave frequencies. The splitting is proportional to both the nodal-diameter number and the speed; at 15 000 r·min−1 and $m = 4$ the two families are already about 2 kHz apart.
Resonance condition and the four critical speeds
As the wheel turns, it is loaded once per engagement of a mating-pinion tooth. The excitation frequency at the wheel is therefore the tooth-passing frequency of the pinion,
$$f_e = \frac{N Z_p}{60} = \frac{n_g Z_g}{60}$$
where $N$ is the pinion speed in r·min−1, $Z_p$ the number of pinion teeth and $Z_g$ the number of wheel teeth, related by $n_g = N Z_p / Z_g$. Resonance occurs when a wave frequency equals the excitation frequency, $f_f = f_e$ or $f_b = f_e$. Solving for the pinion speed gives the critical speeds
$$N = \frac{60\,f_d\,Z_g}{Z_p\,(Z_g \mp m)}$$
with the minus sign for the forward wave and the plus sign for the backward wave. Two consequences follow directly. First, each nodal-diameter mode produces two critical speeds, so a wheel monitored over three and four nodal diameters has four critical speeds to clear rather than two. Second, the forward-wave critical speed always lies above the backward-wave critical speed of the same mode, because the rotation adds to the wave frequency in one case and subtracts in the other. Forward-wave resonance is therefore encountered at the higher shaft speed, which is the regime where the mesh excitation is strongest.
For the baseline wheel considered here, the room-temperature static frequencies of 7 997 Hz (three nodal diameters) and 12 768 Hz (four nodal diameters) place the four critical pinion speeds in the range 9.4×103 to 18.4×103 r·min−1 (Table 4). These values are computed directly from the relations above; the resonance map discussed later differs from them because temperature and web thickness are varied there.
Finite element model, material data and experimental validation
A three-dimensional model of the meshing gear pair was built in a CAD environment from the parameters in Table 1 and meshed with hexahedral elements for modal analysis. Because service damage occurs at the web of the driven wheel, the modal work concentrated on that component. Twelve modes were extracted, and the computed shapes at one, two, three and four nodal diameters reproduced the expected disc-like pattern, confirming that the finite element representation captures the modal family of interest. Elastic properties were specified as functions of temperature (Table 2); between 20 °C and 200 °C the Young’s modulus falls from 215 GPa to 210 GPa while Poisson’s ratio rises from 0.38 to 0.41, and density is held at 7.86×103 kg·m−3.
| Parameter | Value |
|---|---|
| Number of teeth, pinion $Z_p$ | 47 |
| Number of teeth, wheel $Z_g$ | 35 |
| Outer module $m$ | 3.875 mm |
| Hand | Pinion left-hand, wheel right-hand |
| Normal pressure angle $\alpha$ | 20° |
| Spiral angle $\beta$ | 35° |
| Normal backlash | 0.10–0.15 mm |
| Normal meshing clearance | 0.15–0.35 mm |
| $T$ / °C | $E$ / GPa | $\mu$ | $\rho$ / (kg·m−3) |
|---|---|---|---|
| 20 | 215 | 0.38 | 7.86×103 |
| 100 | 213 | 0.39 | 7.86×103 |
| 200 | 210 | 0.41 | 7.86×103 |
The model was checked against modal tests on three wheels of the same design (Table 3). The simulated frequencies are identical for all three because a single finite element model was used, whereas the measured frequencies scatter slightly from wheel to wheel. Agreement is within 1.13% at three nodal diameters and within 4.73% at four nodal diameters, with the four-diameter prediction consistently lower than measurement by roughly 4–5%. That bias is small enough for the model to be used as a trend tool for parameter studies, but it also indicates that absolute frequency predictions at higher nodal diameters carry a systematic offset of a few per cent.
| Wheel | Mode | Simulated / Hz | Measured / Hz | Error / % |
|---|---|---|---|---|
| A | 3 nodal diameters | 7 997 | 7 964 | 0.41 |
| 4 nodal diameters | 12 768 | 13 402 | 4.73 | |
| B | 3 nodal diameters | 7 997 | 7 928 | 0.87 |
| 4 nodal diameters | 12 768 | 13 360 | 4.43 | |
| C | 3 nodal diameters | 7 997 | 7 908 | 1.13 |
| 4 nodal diameters | 12 768 | 13 320 | 4.14 |
| Nodal diameters $m$ | Wave | Critical pinion speed / (×103 r·min−1) |
|---|---|---|
| 3 | Backward | 9.40 |
| 3 | Forward | 11.17 |
| 4 | Backward | 14.63 |
| 4 | Forward | 18.40 |
How web thickness and temperature move the resonance map
The full web thickness of the wheel is 17.28 mm, and the baseline model uses 10.28 mm, that is 60% of the full thickness. Seven thicknesses were analysed — 8.28, 9.28, 9.78, 10.28, 10.78, 11.28 and 12.28 mm, corresponding to 48%, 54%, 57%, 60%, 62%, 65% and 71% of the full web — at working temperatures of 60 °C, 100 °C, 150 °C and 200 °C. For each case the static frequencies of the three- and four-diameter modes were extracted and used in place of the dynamic frequencies, an approximation justified by the small difference between the two in this speed range, and the forward and backward wave frequencies and their critical speeds were then calculated.
Three regularities emerge from the resulting map. First, at any fixed temperature, increasing the web thickness raises both the resonance frequency and the resonance speed of the three- and four-diameter modes, in the forward as well as the backward branch. Second, at any fixed thickness, raising the working temperature lowers both quantities. Third, comparing the spread produced by the two variables over the ranges examined, web thickness changes the resonance characteristics more strongly than temperature does over 60–200 °C. The temperature trend is consistent with the loss of stiffness implied by Table 2: a 5 GPa reduction in Young’s modulus combined with a rise in Poisson’s ratio from 0.38 to 0.41 softens the web and pulls all wave frequencies, and hence all critical speeds, downward. The thickness trend has a geometrical origin — a thicker web raises the bending stiffness of the disc far more than it raises its mass, so the frequency rises.
An important practical implication follows from the combination of the two trends. A wheel that clears all four critical speeds when cold may approach one of them at the hot end of its operating envelope, because heating shifts the whole resonance map downwards. Thermal margin therefore has to be checked explicitly rather than assumed from a room-temperature modal analysis.
Design consequences and the limits of the approach
Within the constraints that govern the design of the wheel — tooth-root bending strength, rim stress, mass and the stiffness required at the web-to-rim junction — the web thickness is an effective variable for relocating traveling-wave resonance. Because critical speed scales linearly with the nodal-diameter natural frequency, a thickness change that shifts the static frequency by a given percentage shifts every critical speed by the same percentage, which makes the required adjustment predictable. The usual objective is to move the four critical speeds of the three- and four-diameter modes outside the operating speed band, or to reduce the vibration amplitude at the crossings that cannot be avoided. Thinning and thickening are not symmetric in their side effects: a thinner web relieves mass but lowers the critical speeds towards the operating range and increases web stress, whereas a thicker web raises the critical speeds and stiffens the wheel at the cost of mass and of the compliance that the rim-to-web transition needs.
The analysis also has clear limits. Using static frequencies in place of dynamic ones neglects centrifugal stiffening and the stress-stiffening effect of the rotating wheel, so the predicted critical speeds are approximations whose error grows with speed and nodal-diameter number — the same direction in which the four-diameter correlation with measurement already shows a 4–5% offset. Modal analysis supplies frequencies and mode shapes, not stress amplitudes: whether a particular crossing is damaging depends on the excitation amplitude at that order, on the damping of the wheel and on whether the excitation does net positive work on the mode, none of which is contained in an eigenvalue calculation. The step from a resonance map to a service-safe design therefore still requires strain measurement at the web root on a test stand, of the kind used in earlier traveling-wave studies, with the wheel run through the speed range at the temperatures it will actually see. What the parameter study provides is the direction and magnitude of the design lever, so that such a test can be planned around a small number of candidate web thicknesses rather than the full design space.