Spiral bevel gears in aero-engine central drives must carry high torque through a thin, lightweight web. The mesh applies a once-per-tooth load, but the web and rim have nodal-diameter modes in the low-kilohertz range. When the mesh frequency coincides with one of these modes, the deflection pattern does not stand still in the rotating frame: it appears as a wave traveling around the gear. The alternating stress is then repeated at the resonance frequency rather than at shaft speed, and it can load a tooth-root fillet or web transition far more severely than a conventional tooth-load calculation suggests. A small initial defect can grow under this hidden high-frequency cycle.
Two consequences matter for monitoring. A crack removes stiffness and lowers the modal frequency, so the resonance peak should move downward. A crack also breaks the cyclic symmetry of the gear: the two orthogonal standing modes that combine into a forward or backward wave no longer have the same frequency. The wave can no longer remain a pure traveling wave. The acoustic spectrum therefore contains both a frequency shift and a growing mode split. These two effects are quantified below for a 35-tooth/47-tooth spiral bevel pair.
Forward and backward waves in a rotating gear
Axial bending modes of a gear are classified by nodal circles and nodal diameters. Nodal-circle and compound modes have relatively high frequencies and are difficult to excite. Nodal-diameter modes are lower and are the ones that can coincide with mesh frequency. For a stationary gear with \(m\) nodal diameters, the forward and backward wavelengths, wave speeds and frequencies are
\[
\lambda_f=\lambda_b=\frac{2\pi}{m},\qquad
v_{f,b}=\pm\frac{2\pi f}{m},\qquad
f_f=f_b=f,
\]
where \(f\) is the stationary natural frequency of the \(m\)-diameter mode. When the gear rotates at \(n\) r/min, the wavelength in the gear is unchanged, but the wave speed relative to the ground gains a rotation term. The observed forward and backward frequencies become
\[
f’_{f,b}=f\pm\frac{mn}{60}.
\]
The mesh excitation frequency is
\[
f_e=\frac{zn}{60},
\]
where \(z\) is the tooth number. Resonance occurs when \(f_e=f’_{f,b}\). Solving for speed and frequency gives
\[
n_{f,b}=\frac{60f}{z\mp m},\qquad
f_{f,b}=f\pm\frac{mf}{z\mp m}.
\]
For the forward branch, the minus sign is used in the denominator; for the backward branch, the plus sign is used. These relations define the intersection of the mesh-order line with the forward- and backward-wave lines on a Campbell diagram. For a 47-tooth gear, the three-nodal-diameter forward branch and the four-nodal-diameter backward branch were the two operating risks addressed by the fatigue tests.
Test arrangement and resonance targets
The stage contained a 35-tooth member and a 47-tooth member. The speed of the 47-tooth member is denoted \(n_2\), so the mesh excitation was \(f_e=47n_2/60\). A rigid-wall acoustic waveguide, 1,000 mm long, was inserted through the gearbox wall. Its open end sat about 10 mm above the small-end web. Two microphone channels viewed points 38 mm from the gear center, separated by 45°. The waveguide and a semi-infinite attenuation tube kept the microphones away from oil mist and the hot gearbox interior while preserving the pressure fluctuation at the gear.
Hammer-impact modal testing of the free gear gave stationary three-nodal-diameter and four-nodal-diameter frequencies of 7,904 Hz and 13,340 Hz. These values defined the resonance targets. Table 1 compares the predicted resonance points with the peaks measured during the fatigue runs. The measured three-nodal-diameter forward peak was 8,730 Hz at 11,145 r/min; the measured four-nodal-diameter backward peak was 11,975 Hz at 15,288 r/min. The close agreement confirms that the acoustic waveguide was observing the intended nodal-diameter wave rather than a tooth-mesh harmonic.
| Branch | Predicted speed \(n_2\) (r/min) | Predicted frequency (Hz) | Measured speed \(n_2\) (r/min) | Measured frequency (Hz) |
|---|---|---|---|---|
| 3 nodal diameters, forward | 11,036 | 8,645 | 11,145 | 8,730 |
| 4 nodal diameters, backward | 15,283 | 11,972 | 15,288 | 11,975 |
An undamaged gear ran for 2.6 h on the three-nodal-diameter forward branch and for 128 min on the four-nodal-diameter backward branch. At 8,730 Hz and 11,975 Hz, these durations correspond to about \(8.2\times10^7\) and \(9.2\times10^7\) vibration cycles, respectively. No crack was found after either run. Stable traveling-wave resonance alone, at least for this geometry and surface condition, did not initiate fracture. The cracked-gear tests therefore started from a deliberately introduced defect at the location where the modal stress was highest.
Modal analysis of the uncracked gear used carburized gear steel at 200 °C with density \(7,860\ \mathrm{kg\,m^{-3}}\), elastic modulus \(2.1\times10^5\) MPa and Poisson ratio 0.41. Radial and axial displacements of the shaft bore were fixed, while rotation about the shaft axis was free. Modal stresses are relative values, so the useful quantity is the ratio of the maximum-stress point to a fixed reference point. The three-nodal-diameter ratio was 10.33 and the four-nodal-diameter ratio was 33.74 (Table 2). The initial prefabricated crack was placed in the small-end fillet region associated with these maxima.
| Mode | Maximum stress value | Reference stress value | Stress ratio |
|---|---|---|---|
| 3 nodal diameters | 400,974 | 38,813.6 | 10.33 |
| 4 nodal diameters | 695,185 | 20,604.2 | 33.74 |
The initial defect was 0.6 mm deep and 0.27 mm wide. For the finite-element models, the surface crack length was 1.84 mm, divided into 1.00 mm and 0.84 mm segments along the two observed growth directions. This geometry provided a controlled starting condition without removing the essential feature of the service failure: a small fillet crack under a nodal-diameter resonance.
Response of the precracked gear
The three-nodal-diameter test ended in a block fracture with two main fragments; one fragment contained eight full teeth and two half teeth. The four-nodal-diameter test produced a fragment with 12 full teeth. In both cases the main crack started at the prefabricated small-end fillet. It propagated along two routes: from the small end toward the web, through the shaft bore or lightening holes, to the opposite tooth small end and then along the root toward the large end; and from the small end along the tooth slot toward the large end. The final fracture surface was consistent with a crack that had grown around the web before the remaining section separated.
During the three-nodal-diameter forward run, the main peak near 8.73 kHz fell from 333.1 Pa to 263.7 Pa in the early stage, while the second harmonic at 17.46 kHz fell from 74.9 Pa to 42 Pa. In the early-middle stage the main peak fell further to 236.8 Pa and the harmonic to 32.6 Pa. In the middle-late stage the main peak shifted toward lower frequency, the harmonic collapsed, and a backward-wave peak appeared near 7.23 kHz at 51.7 Pa and then 30.8 Pa. In the late stage the main peak was measured at about 8,720 Hz with 158 Pa and then 55.1 Pa; after that point the resonance could no longer be tracked by adjusting speed, and the gear fractured.
During the four-nodal-diameter backward run, the main peak near 11.985 kHz fell from 159.3 Pa to 96.3 Pa in the early stage, and a modulation peak appeared at 8,915 Hz with 85.1 Pa. In the early-middle stage the main peak moved downward and the modulation peak weakened. In the middle-late stage the main peak at about 11,960 Hz fell from 96.3 Pa to 51.1 Pa, while the modulation peak dropped rapidly. In the late stage the main peak was near 11,930 Hz at only 9.2 Pa; tracking was then lost and the gear fractured.
The amplitude decrease is not simply a damping increase. As the crack grows, the gear becomes asymmetric, and the two orthogonal standing modes of the same nodal-diameter family separate. As the gear rotates, the excitation alternately favors one mode and then the other. The tracked acoustic peak therefore fluctuates, broadens and eventually cannot be followed by a speed sweep. Loss of trackability is a physical consequence of mode splitting, not a failure of the acoustic measurement.
Modal splitting with increasing crack length
Finite-element modal analysis was performed on 44 crack geometries: 22 stages for each of the two growth branches. The crack was extended along the two paths observed in the tests. Depth increased during the early stages, was held constant from stage 17 to stage 20, and then increased again in the final two stages. Tetrahedral elements were used with a global size of 1.5 mm; mesh convergence was reached at about 600,000 elements. The boundary conditions and material properties were the same as in the uncracked modal analysis. For each model, only the three- or four-nodal-diameter modes were extracted.
In the uncracked gear, the three-nodal-diameter pair appeared at 8,026.4 Hz and 8,053.7 Hz, while both four-nodal-diameter modes appeared at 13,380 Hz. The small split in the uncracked three-nodal-diameter pair comes from the lightening holes: one standing pattern places the three nodal diameters between the holes, while the other passes them through the holes. The four-nodal-diameter pattern does not form the same two geometric extremes, so its two standing modes are initially degenerate.
As the crack grows, both frequencies fall, but the two modes no longer fall equally. Table 3 gives selected stages. At stage 22, the three-nodal-diameter pair is 7,305.8 Hz and 7,636.7 Hz, a split of 330.9 Hz. The four-nodal-diameter pair is 10,866 Hz and 12,517 Hz, a split of 1,651 Hz. The increasing separation is a direct measure of the loss of cyclic symmetry and explains why a single tracked resonance peak eventually becomes unstable.
| Crack stage | 3ND mode 1 (Hz) | 3ND mode 2 (Hz) | 3ND split (Hz) | 4ND mode 1 (Hz) | 4ND mode 2 (Hz) | 4ND split (Hz) |
|---|---|---|---|---|---|---|
| Uncracked | 8,026.4 | 8,053.7 | 27.3 | 13,380 | 13,380 | 0 |
| 1 | 8,027.0 | 8,054.4 | 27.4 | 13,380 | 13,380 | 0 |
| 7 | 7,927.6 | 8,019.9 | 92.3 | 13,193 | 13,339 | 146 |
| 17 | 7,517.3 | 7,862.4 | 345.1 | 12,455 | 13,043 | 588 |
| 20 | 7,504.1 | 7,844.3 | 340.2 | 12,424 | 13,019 | 595 |
| 22 | 7,305.8 | 7,636.7 | 330.9 | 10,866 | 12,517 | 1,651 |
Frequency ratio as a crack-length indicator
Because the two standing modes are excited with equal probability during rotation, the arithmetic mean of the pair is a practical tracking variable. Define the frequency ratio as
\[
\lambda=\frac{\bar f_i}{\bar f_0},\qquad
\bar f_i=\frac{f_{i,1}+f_{i,2}}{2},
\]
where \(f_{i,1}\) and \(f_{i,2}\) are the two modal frequencies at a given crack stage and \(\bar f_0\) is the uncracked mean frequency. In the early crack-growth stage, stages 1 to 7, the crack depth and surface length increased in a fixed proportion. Within this range, a cubic fit describes the relation between the frequency ratio and the total crack length \(L\) in millimetres:
\[
\lambda^{(3)}=-3.457\times10^{-9}L^3-1.963\times10^{-5}L^2+1.115\times10^{-4}L+0.99981,
\]
\[
\lambda^{(4)}=-8.279\times10^{-8}L^3-1.514\times10^{-5}L^2+5.574\times10^{-5}L+0.99994.
\]
The superscripts 3 and 4 denote three- and four-nodal-diameter vibration. These fits apply only to the early stage and to the constant depth-to-length ratio used in that part of the simulation. Within that scope, they convert a measured resonance frequency into an estimate of crack length.
For the three-nodal-diameter test, the initial and final peaks at 8,730 Hz and 8,719 Hz correspond to modal frequencies of 7,981.71 Hz and 7,971.66 Hz. The ratio 0.99874 gives \(L=10.74\) mm. For the four-nodal-diameter test, 11,985 Hz and 11,930 Hz correspond to 13,354.71 Hz and 13,293.43 Hz; the ratio 0.99541 gives \(L=18.25\) mm. The final crack patterns were consistent with these lengths: 5.82 mm along the web plus 4.92 mm along the tooth slot for the three-nodal-diameter case, and 9.99 mm plus 8.26 mm for the four-nodal-diameter case.
Re-meshing those crack geometries gave modal frequencies of 8,031.05 Hz and 13,318.5 Hz, with frequency ratios of 0.99888 and 0.99540. The three-nodal-diameter error was 0.014%; the four-nodal-diameter error was below 0.02%. The agreement supports the use of the frequency ratio as an early-stage crack gauge, but it does not extend the fit to the late stage, where the crack shape, mode splitting and residual strength change too rapidly.
Failure sequence and monitoring implications
Combining the acoustic and modal results gives a consistent three-phase sequence. Direct crack-tip stress-intensity measurements were not made; the sequence is an interpretation of the modal, acoustic and fracture-path evidence.
- Early resonance-driven growth. The crack is small, the two standing modes are still close in frequency, and a traveling wave can be maintained by speed tracking. The crack-tip stress-intensity range remains above the growth threshold, so each resonance cycle extends the crack. The modal frequency falls and the acoustic amplitude decreases, but the peak remains trackable.
- Mode-splitting transition. The crack removes enough stiffness to separate the two standing modes. As the mesh position rotates, the excitation alternately favors one mode and then the other. The gear can no longer maintain a single coherent traveling wave, and the acoustic peak becomes unstable. Speed tracking may still follow the broadened peak for a time, but the amplitude and frequency fluctuate more strongly.
- Residual-strength fracture. In the late stage the remaining section is small. The crack-tip stress-intensity factor approaches or exceeds the fracture toughness of the carburized gear steel, so the crack advances rapidly rather than by a stable fatigue mechanism. The resonance point disappears, the acoustic amplitude collapses, and the gear breaks before another trackable fatigue stage can develop.
The final break is therefore not a conventional fatigue-life limit. It is a residual-strength limit reached after resonance-driven crack growth has reduced the section and destroyed the cyclic symmetry that sustained the traveling wave. This distinction is important for diagnosis: a downward shift in resonance frequency is an early indicator, while the appearance of a companion mode peak and the loss of trackability are late warnings.
For design, the three-nodal-diameter forward and four-nodal-diameter backward branches should be kept away from the operating speed range, or the web and rim should be stiffened and damped so that the modal stress at the fillet is reduced. For monitoring, a rigid-wall acoustic waveguide can track the resonance peak through the gearbox wall without strain gauges on rotating parts. A persistent frequency decrease, a falling peak amplitude and the emergence of a split mode are the observable signatures of crack growth. The frequency-ratio fit is useful for early-stage estimation, but it is specific to this gear geometry, this crack path and the constant depth-to-length condition; it should not be treated as a general life-prediction model.