The central drive bevel gear of an aero-engine accessory gearbox is a high-speed, heavily loaded element whose web is thin compared with its diameter. Because it runs at elevated temperature, at speeds of the order of 104 r/min and under alternating tooth loads, the web behaves as a disc in bending rather than as a rigid hub. Its nodal-diameter modes appear in a stationary reference frame as a pair of counter-rotating waves, and when a tooth-mesh harmonic matches one of these wave frequencies the structure responds in traveling wave resonance. The damage signature is characteristic: a circumferential crack that runs around the web root and can release a block of teeth together with the web.
A failure of this kind in the driven wheel of a central bevel pair motivated a combined modal, dynamic and experimental investigation of the parameters that a designer can actually adjust. A reproduction test on the same gear set showed the two regimes clearly. At a fourth-nodal-diameter resonance speed, defect-free gears ran with adequate life, whereas gears carrying a seeded initial defect fractured across twelve consecutive teeth together with the web. Resonance speed alone therefore does not determine failure; the stress level that the resonant mode produces at the web and root does. The relevant engineering questions are where the third- and fourth-nodal-diameter resonances fall relative to the operating speed band, how far the adjustable parameters move them, and which locations on the gear carry the resulting stress.
Nodal-diameter modes and the resonance condition
An unstretched disc with a fixed centre axis supports three families of modes: nodal circles, nodal diameters, and compound shapes containing both. Disc-like bodies are dominated by the nodal-diameter family, and a spiral bevel gear is geometrically close enough to a disc to follow the same rule. The gear is not symmetric about the mid-plane of its web, so the vibration stresses on the two faces of the web differ in magnitude; this asymmetry is the reason both faces, and not only the web, must be tracked alongside the tooth root.
For a nodal-diameter mode of order m, the transverse displacement of a point on the web, expressed in coordinates fixed to the gear, is
$$z(r,\theta,t)=A(r)\cos(m\theta)\cos(\omega t)$$
where A(r) is the radial amplitude distribution in mm, θ the circumferential angle, ω the angular frequency of the sector vibration in rad/s, and t time in s. Expanding the product of cosines separates the standing pattern into two waves of equal amplitude that rotate in opposite directions:
$$z=\tfrac{1}{2}A(r)\left[\cos(m\theta-\omega t)+\cos(m\theta+\omega t)\right]$$
Each wave carries the same nodal pattern, but in a stationary frame their frequencies are split by the rotation of the gear. With fd the nodal-diameter natural frequency in Hz, n the driving gear speed in r/min and i = Z1/Z2 the gear ratio,
$$f_f=f_d+\frac{inm}{60},\qquad f_b=f_d-\frac{inm}{60}$$
so that the forward wave, which travels in the direction of rotation, is raised in frequency and the backward wave is lowered. Resonance requires that the excitation coincide with one of these wave frequencies in the same reference frame. The tooth-mesh excitation produced by a driving gear with Z1 teeth is
$$f_e=\frac{nZ_1}{60}=f_f$$
provided the excitation does positive work on the wave; the corresponding driving-gear speed follows as
$$n=\frac{60f_d}{i\left(Z_2\mp m\right)}$$
with the minus sign for the forward wave and the plus sign for the backward wave. Because the backward branch is the lower of the two, it normally sets the speed at which a given nodal-diameter resonance is first encountered as the engine spools up.
Finite element model and modal correlation
The driven wheel (35 teeth against 47 on the driving wheel, gear ratio i = 47/35 = 1.343) was modelled in a CAD environment together with its mating pinion, and the modal analysis was restricted to the driven wheel because that is the member in which web fracture occurs. The geometry and material data used are listed below.
| Parameter | Driving gear | Driven gear |
|---|---|---|
| Number of teeth Z | 47 | 35 |
| Large-end module (mm) | 3.875 | 3.875 |
| Hand of spiral | Left | Right |
| Normal pressure angle (°) | 20 | 20 |
| Spiral angle (°) | 35 | 35 |
| Normal backlash (mm) | 0.10–0.15 | |
| Normal meshing clearance (mm) | 0.15–0.35 | |
| Temperature (°C) | Young’s modulus (GPa) | Poisson’s ratio | Density (kg/m³) |
|---|---|---|---|
| 20 | 215 | 0.38 | 7.86 × 10³ |
| 100 | 213 | 0.39 | |
| 200 | 210 | 0.41 |
Constraints were applied to the four surfaces that correspond to the ball-bearing seats in service, with no external load, and the 20 °C material set was used. The mesh was hexahedral-dominant, containing 13,120 SOLID187 and 65,034 SOLID186 elements. Because high rotational speed brings high-order modes into the excitation range, the first twelve modes were extracted; the four lowest nodal-diameter shapes behaved as disc modes, as expected.
| Mode | Simulated static frequency (Hz) | Measured, gear A (Hz) | Measured, gear B (Hz) | Measured, gear C (Hz) | Maximum deviation |
|---|---|---|---|---|---|
| 1 nodal diameter | 3,474.5 | — | — | — | — |
| 2 nodal diameters | 4,510.9 | — | — | — | — |
| 3 nodal diameters | 7,996.9 | 7,964 | 7,928 | 7,908 | 1.13 % |
| 4 nodal diameters | 12,768 | 13,402 | 13,360 | 13,320 | 4.73 % |
Laboratory modal testing of three driven wheels gave third-nodal-diameter frequencies between 7,908 Hz and 7,964 Hz against a simulated 7,997 Hz, and fourth-nodal-diameter frequencies between 13,320 Hz and 13,402 Hz against a simulated 12,768 Hz. Deviations are below 2 % for the third and below 5 % for the fourth nodal diameter, which is sufficient to treat the model as a valid predictor when thickness and temperature are varied. Since the static and dynamic frequencies differ only slightly at these stiffness-to-mass ratios, static values were used in place of dynamic values throughout the resonance-speed calculations.
Web thickness as the primary tuning parameter
The full-web thickness of the driven wheel is h = 17.28 mm. Seven models were built with web thickness h′, expressed as a fraction ξ = h′/h of the full value: 8.28, 9.28, 9.78, 10.28, 10.78, 11.28 and 12.28 mm, corresponding to ξ = 48 %, 54 %, 57 %, 60 %, 62 %, 65 % and 71 %. Each model was analysed at 20, 60, 100, 150 and 200 °C.
| ξ (%) | h′ (mm) | f3 (Hz) | ff3 (Hz) | fb3 (Hz) | nf3 (r/min) | nb3 (r/min) |
|---|---|---|---|---|---|---|
| 48 | 8.28 | 7,496 | 8,198 | 6,904 | 10,466 | 8,814 |
| 54 | 9.28 | 7,736 | 8,461 | 7,125 | 10,801 | 9,096 |
| 57 | 9.78 | 7,864 | 8,601 | 7,243 | 10,980 | 9,247 |
| 60 | 10.28 | 7,997 | 8,747 | 7,366 | 11,166 | 9,403 |
| 62 | 10.78 | 8,135 | 8,898 | 7,493 | 11,359 | 9,565 |
| 65 | 11.28 | 8,280 | 9,056 | 7,626 | 11,561 | 9,736 |
| 71 | 12.28 | 8,589 | 9,395 | 7,911 | 11,993 | 10,099 |
| ξ (%) | h′ (mm) | f4 (Hz) | ff4 (Hz) | fb4 (Hz) | nf4 (r/min) | nb4 (r/min) |
|---|---|---|---|---|---|---|
| 48 | 8.28 | 12,147 | 13,714 | 10,901 | 17,508 | 13,916 |
| 54 | 9.28 | 12,439 | 14,044 | 11,163 | 17,929 | 14,251 |
| 57 | 9.78 | 12,600 | 14,226 | 11,308 | 18,161 | 14,435 |
| 60 | 10.28 | 12,768 | 14,415 | 11,458 | 18,403 | 14,628 |
| 62 | 10.78 | 12,946 | 14,616 | 11,618 | 18,659 | 14,832 |
| 65 | 11.28 | 13,136 | 14,831 | 11,789 | 18,933 | 15,049 |
| 71 | 12.28 | 13,548 | 15,296 | 12,158 | 19,527 | 15,521 |
At 20 °C, raising ξ from 48 % to 71 % increases the third-nodal-diameter forward-wave frequency by 1,197 Hz and the backward-wave frequency by 1,007 Hz, moving the corresponding speeds by 1,527 r/min and 1,285 r/min. For the fourth nodal diameter the same change in thickness raises the forward-wave frequency by 1,582 Hz and the backward-wave frequency by 1,257 Hz, and shifts the speeds by 2,019 r/min and 1,605 r/min. The monotonic rise in both branches, at all five temperatures, indicates that thickening the web adds stiffness faster than it adds mass, so the frequency ratio that governs the resonance condition moves upward.
Temperature as a secondary but systematic effect
Holding the web at ξ = 60 %, the resonance data across the 20–200 °C range are as follows.
| t (°C) | ff3 (Hz) | fb3 (Hz) | nf3 (r/min) | nb3 (r/min) | ff4 (Hz) | fb4 (Hz) | nf4 (r/min) | nb4 (r/min) |
|---|---|---|---|---|---|---|---|---|
| 20 | 8,747 | 7,366 | 11,166 | 9,403 | 14,415 | 11,458 | 18,403 | 14,628 |
| 60 | 8,721 | 7,344 | 11,133 | 9,375 | 14,373 | 11,424 | 18,348 | 14,584 |
| 100 | 8,695 | 7,322 | 11,099 | 9,347 | 14,330 | 11,390 | 18,293 | 14,541 |
| 150 | 8,645 | 7,280 | 11,036 | 9,293 | 14,247 | 11,325 | 18,188 | 14,457 |
| 200 | 8,612 | 7,252 | 10,994 | 9,258 | 14,193 | 11,282 | 18,119 | 14,402 |
Heating from 20 °C to 200 °C at this thickness lowers the third-nodal-diameter static frequency by 123 Hz, the forward-wave frequency by 135 Hz and the backward-wave frequency by 114 Hz, with speed reductions of 172 r/min and 145 r/min. For the fourth nodal diameter the reductions are 197 Hz, 222 Hz and 176 Hz, and 284 r/min and 226 r/min respectively. The mechanism is the softening of the material: Young’s modulus falls from 215 GPa to 210 GPa and Poisson’s ratio rises from 0.38 to 0.41 over the same interval. The trend is uniform in ξ; a hotter gear resonates at a slightly lower speed than a cold one, so the hot condition is the conservative case for clearance against the operating band.
Comparing the two levers on a common basis, the entire 4 mm thickness design space shifts the fourth-nodal-diameter forward-wave speed by about 2,000 r/min, whereas the whole 180 K temperature range shifts it by less than 300 r/min. Thickness is therefore the dominant structural variable and temperature a correction on top of it. Within the requirement envelope of the gear, adjusting the web thickness is a practical route to push a nodal-diameter resonance outside the operating speed range or to move its frequency away from a strong excitation order.
Transient contact model and its validation
Stress under resonance was computed with a transient dynamic analysis of the meshing pair, using Hertzian contact at the flanks. The contacting surfaces were treated as frictionless in the classical Hertz sense for the pressure calculation, while the finite element contact itself used a friction coefficient of 0.05, an augmented Lagrangian formulation and a contact stiffness of 1 established by trial. The driving gear received a rotational speed, the driven gear a resisting torque
$$T_1=\frac{9.55\times10^{6}\,P}{iN}$$
with P the load power (initially 200 kW) and N the driving speed in r/min. The maximum Hertzian pressure itself follows the standard line-contact result for two cylinders of radii ρ1 and ρ2, width w, elastic moduli E1, E2 and Poisson’s ratios μ1, μ2:
$$\sigma_{H\,\max}=\sqrt{\frac{F\left(\dfrac{1}{\rho_1}+\dfrac{1}{\rho_2}\right)}{\pi w\left(\dfrac{1-\mu_1^{2}}{E_1}+\dfrac{1-\mu_2^{2}}{E_2}\right)}}$$
A representative run used ξ = 60 %, 200 °C and the fourth-nodal-diameter backward-wave resonance at 14,402 r/min, with a load of 200 kW giving a resisting torque of 98,759 N·mm; the solution time was 9 × 10−5 s with an initial substep of 24. The choice is self-consistent: at 14,402 r/min the 47th-order excitation is 11,282 Hz, which is exactly the computed backward-wave frequency of the fourth nodal diameter at 200 °C in the table above.
| Measurement location | Measured stress (MPa) | Simulated stress (MPa) | Simulated/measured |
|---|---|---|---|
| 1 (web front face) | 235.5 | 353.9 | 1.50 |
| 2 (web front face) | 214.1 | 353.9 | 1.65 |
The predicted stress at the web front face exceeded the measured values by factors of 1.50 and 1.65, which is inside the 1.7× band normally accepted for engineering stress prediction of this kind. The location of the maximum predicted stress also coincided with the fracture site observed in the reproduction test, in which the driven wheel lost twelve consecutive teeth together with the web. The transient contact model is therefore adequate for comparing stress levels between parameter sets, which is the use made of it below.
Stress distribution and sensitivity to load, temperature and damping
In both the third- and fourth-nodal-diameter resonant states the three monitored locations rank consistently: the tooth root carries the highest stress, the back face of the web the next, and the front face the least. Stresses at the fourth nodal diameter exceed those at the third at every location, which is the quantitative reason why high-speed operation near the fourth-nodal-diameter backward wave is the more damaging condition and why the observed fractures were associated with that mode.
Three parameters were swept around a base condition of ξ = 60 %: load power from 150 kW to 250 kW at 100 °C, temperature from 20 °C to 200 °C at 210 kW, and damping factor from 0.03 to 0.17 at 100 °C and 210 kW. The resulting changes in stress are summarised below.
| Parameter | Range | Mode | Δ web front (MPa) | Δ web back (MPa) | Δ tooth root (MPa) |
|---|---|---|---|---|---|
| Load power | 150 → 250 kW | 3 nodal diameters | +1.2 | +49.1 | +32.6 |
| Load power | 150 → 250 kW | 4 nodal diameters | +17.7 | +71.1 | +60.2 |
| Temperature | 20 → 200 °C | 3 nodal diameters | +2.0 | +42.3 | +42.4 |
| Temperature | 20 → 200 °C | 4 nodal diameters | +13.1 | +50.0 | +38.7 |
| Damping factor | 0.03 → 0.17 | 3 nodal diameters | −4.0 | −11.4 | −33.8 |
| Damping factor | 0.03 → 0.17 | 4 nodal diameters | −6.4 | −63.4 | −32.7 |
Load power and temperature both raise the stress, and damping lowers it, but the three locations respond very differently. The web front face is almost insensitive to every parameter, moving by only 1.2 MPa to 17.7 MPa across the full load range, 2.0 MPa to 13.1 MPa across the full temperature range and 4.0 MPa to 6.4 MPa across the full damping range. The web back face and the tooth root, by contrast, change by tens of megapascals in every sweep.
A damping factor change of 0.14 produces a 63.4 MPa swing on the fourth-nodal-diameter back face, comparable to the 71.1 MPa produced by a 100 kW load change and larger than the 50.0 MPa produced by the whole 180 K temperature range. On the scale of the parameter intervals examined, damping is thus at least as influential as load power and more influential than temperature, which matters because damping is also the parameter most amenable to being raised deliberately through coatings, damping rings or joint design. Because the front face carries the smallest stress and the smallest variation, it is the least informative location for either stress prediction or health monitoring; instrumentation and assessment should be aimed at the web back face and the tooth root.
Implications for design and for monitoring
Three conclusions follow directly from the numbers. First, the third- and fourth-nodal-diameter branches are the ones that matter for this class of thin-web spiral bevel gear, and both the forward and backward branches must be checked, since the backward branch always falls at the lower speed and therefore constrains the upper end of the operating range. Second, web thickness is the effective design lever: a 4 mm change in web thickness over the 48–71 % range moves the fourth-nodal-diameter backward-wave resonance by 1,605 r/min, which is enough to relocate a resonance clear of the operating band without altering the tooth geometry. Third, temperature shifts the resonance downward by a few hundred r/min, so the hot end of the envelope should be used when demonstrating avoidance.
For stress, the practical consequences differ from those for frequency. Making the web thicker or the gear hotter both raise the resonant stress at the root and back face, so a design change made to move a resonance can increase the damage potential of a resonance that remains inside the band; the frequency and stress effects have to be evaluated together, which is why the transient contact analysis, validated to within 1.7× against measured web stresses, is needed alongside the modal calculation. Damping acts in the opposite direction and acts most strongly exactly where the stress is highest, offering a complementary measure when a resonance cannot be moved far enough. Finally, the coincidence of the predicted maximum stress with the fracture site of the defective gear set shows that the failure sequence at the fourth nodal diameter is stress-driven once a defect is present, so inspection and monitoring effort is best concentrated on the web back face and the root fillet of the driven wheel.