Spiral bevel gears in aero-engine central transmissions carry high torque through relatively thin, lightweight webs. At high rotational speed, a nodal-diameter bending mode of the gear can separate into forward and backward traveling waves. When the tooth-mesh excitation frequency coincides with either wave, the gear enters traveling-wave resonance (TWR). The resulting alternating stress can drive a crack from a fillet or web stress concentration and end in block fracture, often with little prior visible warning. A practical monitoring question follows: can the resonance frequency itself be used to track crack growth before final fracture?
Why nodal-diameter resonance leaves a measurable signature
Axial bending vibration of a gear can be described by nodal circles, nodal diameters, or a combination of both. Modes with nodal circles have relatively high natural frequencies and are less easily excited. Nodal-diameter modes, in which the vibration displacement is zero along one or more diameters, are more likely to fall near the mesh excitation frequency. For a mode with m nodal diameters, the forward wave travels in the direction of rotation and the backward wave travels against it. In a stationary frame, the two wave frequencies shift away from the natural frequency by the rotational speed:
$$ f’_f=f+\frac{mn}{60},\qquad f’_b=f-\frac{mn}{60} $$
where f is the nodal-diameter natural frequency in Hz, n is the rotational speed in r/min, and m is the number of nodal diameters. The tooth-mesh excitation frequency is
$$ f_e=\frac{zn}{60} $$
with z the number of teeth. Resonance occurs when fe equals either traveling-wave frequency. Solving for speed gives the forward and backward resonance speeds:
$$ n_f=\frac{60f}{z-m},\qquad n_b=\frac{60f}{z+m} $$
and the corresponding resonance frequencies are fr,f=zf/(z−m) and fr,b=zf/(z+m). The static natural frequency is commonly used in place of the dynamic value because the difference is small for the present gear. This frequency shift is the basis of acoustic monitoring: a waveguide near the gear web captures the resonant tone, and the drive speed can be adjusted to keep the gear on the resonance peak. As a crack grows, the gear stiffness falls and the resonant frequency moves downward; as the crack also breaks the symmetry of the gear body, a pair of nominally identical nodal-diameter modes can split into two distinct frequencies.
Test configuration and baseline behavior
The fatigue tests used a central-transmission spiral bevel gear pair with a driven gear of z2=47 teeth and a pinion of z1=35 teeth. Two critical operating points were examined: the three-nodal-diameter forward wave at 8,730 Hz and 11,145 r/min, and the four-nodal-diameter backward wave at 11,975 Hz and 15,288 r/min. A 1,000 mm acoustic waveguide was inserted through the gearbox side plate, with its open end about 10 mm above the small-end web of the driven gear. Two measurement ports were placed at a radius of 38 mm from the gear center and separated by 45°. The acoustic signal was used to track the resonance peak as the crack developed.
Modal analysis of the uncracked gear located the highest stress at the small-end convex fillet. Relative to a reference point, the modal stress ratio was 10.33 for the three-nodal-diameter mode and 33.74 for the four-nodal-diameter mode. This hot spot was therefore selected for the initial pre-crack: 0.6 mm deep and 0.27 mm wide. The normal gear was first run at both resonance conditions. It survived 2.6 h at the three-nodal-diameter forward resonance and 128 min at the four-nodal-diameter backward resonance, the latter corresponding to about 9.2×107 vibration cycles. Both runs exceeded the design fatigue requirement, and inspection found no crack. Resonance alone, in a defect-free gear, was not sufficient to cause fracture.
| Item | Value |
|---|---|
| Gear pair | Driven spiral bevel gear z2=47; pinion z1=35 |
| Critical resonances | Three nodal diameters, forward: 8,730 Hz at 11,145 r/min; four nodal diameters, backward: 11,975 Hz at 15,288 r/min |
| Acoustic access | 1,000 mm waveguide, 10 mm above small-end web; two ports at 38 mm radius, 45° apart |
| Normal gear life | 2.6 h at three-nodal-diameter forward resonance; 128 min at four-nodal-diameter backward resonance; no crack found |
| Initial pre-crack | 0.6 mm deep, 0.27 mm wide, at small-end convex fillet |
Acoustic evolution during crack growth
The pre-cracked gear behaved very differently. Under three-nodal-diameter forward resonance, the 8,730 Hz peak fell from 333.1 Pa to 263.7 Pa and then to 236.8 Pa as the crack advanced. Its second harmonic at 17,460 Hz fell from 74.9 Pa to 42 Pa and then to 32.6 Pa. In the middle-to-late stage, the main peak moved toward lower frequency, reaching 8,720 Hz at 158 Pa and later 55.1 Pa, while a backward-wave component near 7,230 Hz also appeared. Eventually the three-nodal-diameter resonance could no longer be tracked, and the gear fractured almost immediately.
Under four-nodal-diameter backward resonance, the initial 11,985 Hz peak at 159.3 Pa fell to 96.3 Pa and then 51.1 Pa, while a characteristic peak near 8,900 Hz also weakened. The tracked resonance moved to 11,960 Hz and finally 11,930 Hz at only 9.2 Pa before tracking failed. In both cases the crack started at the prefabricated fillet defect. The three-nodal-diameter fracture produced a block containing eight full teeth and two half teeth; the four-nodal-diameter fracture produced a block of twelve full teeth. The crack paths ran from the small end into the web, through the shaft bore and lightening holes toward another tooth, and also along the tooth slot toward the large end. These paths are consistent with a resonance-driven crack that changes the stiffness and symmetry of the gear body as it extends.
| Stage | Three nodal diameters, forward | Four nodal diameters, backward |
|---|---|---|
| Initial | 8,730 Hz at 333.1 Pa; second harmonic 17,460 Hz at 74.9 Pa | 11,985 Hz at 159.3 Pa; characteristic peak near 8,915 Hz at 85.1 Pa |
| Early to middle | 8,730 Hz falls to 263.7 Pa, then 236.8 Pa; second harmonic falls to 42 Pa, then 32.6 Pa | 11,985 Hz to 11,960 Hz; 96.3 Pa to 51.1 Pa; characteristic peak near 8,900 Hz at 71.4 Pa |
| Late | 8,720 Hz at 158 Pa, then 55.1 Pa; resonance tracking lost | 11,930 Hz at 9.2 Pa; resonance tracking lost |
| Final failure | Block fracture with eight full teeth and two half teeth | Block fracture with twelve full teeth |
The downward frequency shift is expected from loss of bending stiffness. The simultaneous drop in peak sound pressure is also consistent with increasing asymmetry: as the crack grows, the resonant frequency fluctuates with gear rotation, so the gear cannot remain at a single stable resonance condition. The acoustic peak therefore broadens and weakens even before it can no longer be followed.
Modal simulation: frequency drop and mode splitting
A finite-element model of the same spiral bevel gear was used to examine the evolution of the nodal-diameter modes with crack size. The material properties represented carburized gear steel at 200 °C: density 7,860 kg/m³, elastic modulus 2.1×105 MPa, and Poisson’s ratio 0.41. Radial and axial displacements on the inner shaft surface were fixed, while rotation about the axis was free. The mesh used tetrahedral elements with a global size of 1.5 mm and converged at about 600,000 elements. The initial simulated crack had a total length L=1.84 mm (1.00 mm plus 0.84 mm), depth H=0.6 mm, and width 0.27 mm, with depth taken normal to the web. Twenty-two crack stages were analyzed for each resonance case, extending L to 122.02 mm for the three-nodal-diameter path and 122.24 mm for the four-nodal-diameter path; H increased from 0.6 mm to 17.55 mm and 18.17 mm, respectively.
For the uncracked gear, the three-nodal-diameter pair already differed slightly: 8,026.4 Hz and 8,053.7 Hz, a split of 27.3 Hz. The four-nodal-diameter pair was degenerate at 13,380 Hz. The difference in the three-nodal-diameter case arises because the nodal lines can pass through, or avoid, the lightening holes, producing two different web stiffness conditions. A four-nodal-diameter pattern cannot form two equivalent limiting configurations in the same way, so the uncracked pair remains degenerate.
As the crack deepened, both frequencies in each pair decreased, but by different amounts. By stage 22, the three-nodal-diameter modes had fallen to 7,305.8 Hz and 7,636.7 Hz, a split of 330.9 Hz. The four-nodal-diameter modes had fallen to 10,866 Hz and 12,517 Hz, a split of 1,651 Hz. The growing separation is a direct measure of increasing structural asymmetry. In operation, the rotating excitation location alternately favors the two split modes, so the gear no longer responds as a single stable traveling-wave system.
| Mode family | Normal, mode 1 (Hz) | Normal, mode 2 (Hz) | Stage 22, mode 1 (Hz) | Stage 22, mode 2 (Hz) | Split at stage 22 (Hz) |
|---|---|---|---|---|---|
| Three nodal diameters | 8,026.4 | 8,053.7 | 7,305.8 | 7,636.7 | 330.9 |
| Four nodal diameters | 13,380 | 13,380 | 10,866 | 12,517 | 1,651 |
Calibrating crack length from the frequency ratio
For early crack growth, the change in modal frequency can be condensed into a dimensionless ratio
$$ \lambda=\frac{f_i}{f_0} $$
where fi is the modal frequency at a given crack stage and f0 is the uncracked frequency. Within the early stage, where the depth-to-length ratio of the crack remains approximately constant, the simulated frequency ratio followed a cubic relation with crack length L in millimetres:
$$ \lambda_3=-3.457\times10^{-9}L^3-1.963\times10^{-5}L^2+1.115\times10^{-4}L+0.99981 $$
$$ \lambda_4=-8.279\times10^{-8}L^3-1.514\times10^{-5}L^2+5.574\times10^{-5}L+0.99994 $$
Here the subscripts 3 and 4 denote three and four nodal diameters. The relation is intended for the early crack-growth stage, before the crack reaches the highly asymmetric late stage. To check it against the fatigue tests, the tracked resonance peaks at the beginning and end of each test were converted to modal-frequency ratios. The three-nodal-diameter test gave λ3=0.99874, corresponding to L=10.74 mm; the four-nodal-diameter test gave λ4=0.99541, corresponding to L=18.25 mm. These lengths are consistent with the final crack paths: 10.74 mm is the sum of 5.82 mm along the web and 4.92 mm along the tooth slot, while 18.25 mm is the sum of 9.99 mm and 8.26 mm along the same two directions.
Cutting the predicted crack sizes into the model and repeating the modal analysis gave λ3=0.99888 and λ4=0.99540. The relative error between the test-derived and simulation-derived frequency ratios was 0.014% for the three-nodal-diameter case and below 0.02% for the four-nodal-diameter case. This agreement supports the use of the frequency ratio as an early-stage crack-size indicator for this gear geometry and material condition.
| Metric | Three nodal diameters, forward | Four nodal diameters, backward |
|---|---|---|
| Measured resonance, initial → final | 8,730 Hz → 8,719 Hz | 11,985 Hz → 11,930 Hz |
| Measured frequency ratio λ | 0.99874 | 0.99541 |
| Predicted total crack length L | 10.74 mm | 18.25 mm |
| Modal validation λ at predicted L | 0.99888 | 0.99540 |
| Relative error | 0.014% | <0.02% |
Mechanism and engineering implications
The combined acoustic and modal results support a two-stage failure process. In the early stage, the gear is held at a traveling-wave resonance. The resonant stress field raises the crack-tip stress intensity factor and drives the crack forward. Because the crack is still small, the resonance peak remains trackable, but its frequency falls as stiffness is lost. In the late stage, the crack is large enough to make the gear strongly asymmetric. The two formerly symmetric nodal-diameter modes separate, are excited alternately as the gear rotates, and no longer support a stable single traveling-wave condition. The acoustic tracking method then loses the resonance point. At the same time, the crack-tip stress intensity factor approaches or exceeds the fracture toughness of the material, so crack growth becomes rapid and the remaining section is unable to carry the load. The final event is an instantaneous block fracture rather than a slowly developing tooth-root fatigue failure.
For spiral bevel gears in high-speed central transmissions, the practical value of this behavior is twofold. First, a calibrated frequency-ratio relation can translate an early resonance shift into an estimated crack length, provided the gear geometry, material state, and crack depth-to-length ratio remain within the validated range. Second, loss of resonance tracking is itself a late-stage warning: it indicates that mode splitting and asymmetry have become large and that the remaining strength may be close to exhaustion. The tests also show that a defect-free gear can survive prolonged traveling-wave resonance, whereas a small initial fillet defect can initiate the sequence that ends in fracture. Inspection and design therefore need to treat fillet and web stress concentrations as critical locations, and resonance monitoring should be interpreted together with the initial defect state rather than as a stand-alone measure of gear health.