The central drive of a turbofan engine transfers power from the high-pressure spool to the fan and accessory gearbox through a pair of spiral bevel gears. Weight targets push the driven wheel toward a thin-web architecture: material is removed from the disc between the bore and the rim, the axial bending stiffness of the web falls, and nodal-diameter modes migrate downward into the operating speed range. Once a mesh excitation order coincides with one of those modes, the standing nodal pattern is driven as a traveling wave that sweeps around the rim, and the alternating bending stress at the tooth root can produce high-cycle fatigue fracture with little warning. A failure investigation of an aero-engine central transmission attributed a driven bevel gear fracture specifically to fourth-nodal-diameter forward traveling-wave resonance combined with poor mesh contact, which places the problem in the class of sudden, non-overload failures rather than progressive wear.
The case examined here is the 35-tooth driven wheel of a 47/35 spiral bevel pair operating at a 326 kW design point, with the three-nodal-diameter forward traveling wave predicted near 15,000 r/min. Three questions structure the analysis: what speed excites the wave once rotation, torque and mesh stiffness are accounted for; what the displacement and stress fields actually look like while the wave is present; and whether a strain-gauged component rig reproduces both the resonance speed and the stress magnitude.
Kinematics of nodal-diameter traveling waves
A thin-web bevel gear behaves, to first order, like an annular disc, and its modes fall into nodal-diameter, nodal-circle and combined families. Service fractures concentrate in the nodal-diameter family, where the nodal lines run radially and adjacent sectors separated by a nodal line vibrate in antiphase. In the frame rotating with the gear, an m-nodal-diameter mode is the superposition of two identical cosine waves traveling in opposite directions, so both share wavelength, wave speed and frequency:
$$\lambda_f=\lambda_b=\frac{2\pi}{m},\qquad \omega_{f,b}=\pm\frac{2\pi f_d}{m},\qquad f_f=f_b=f_d$$
where fd is the dynamic (rotating) frequency of the mode, and the subscripts f and b denote the forward (co-rotating) and backward (counter-rotating) waves.
Resonance requires three conditions to be satisfied simultaneously: the excitation frequency equals the natural frequency of the wheel; both are expressed in the same coordinate system; and the excitation frequency equals the forward or backward wave frequency. Viewed from a stationary frame, the mesh at z teeth excites the sidebands of the tooth-pass order kzn/60, and the wave appears at
$$f_j=\frac{kzn}{60}\mp\frac{mn}{60},\qquad j=\mathrm{ef},\ \mathrm{eb}$$
with the minus sign for the forward wave, the plus sign for the backward wave, and the harmonic index k usually taken as 1. Setting fj = fd gives the resonance speeds
$$N=\frac{60f_d}{kz\mp m}$$
so the forward wave resonates at a higher shaft speed than the backward wave of the same mode, since kz − m < kz + m. The pattern also migrates, and its angular rate follows from the relation between the mesh frequency fm = zfz and the shaft rate fz = ωg/2π:
$$\omega_w=\frac{z\mp m}{m}\,\omega_g,\qquad T_w=\frac{1}{f_d}=\frac{1}{f_m\mp m f_z}$$
For z = 35 and m = 3, the ratio (z − m)/m = 10.67, which is the central kinematic result: the wave circulates more than ten times faster than the wheel itself, so a fixed material point experiences the passing wave at kilohertz rates while it completes one revolution in a few milliseconds.
Explicit transient contact model
Because the mesh is a strongly nonlinear contact problem, an implicit formulation risks non-convergence at every load step, and an explicit central-difference scheme was used instead. The solver integrates the dynamic equilibrium
$$M\ddot{u}=P-I$$
where M is the nodal mass matrix, P the applied external force and I the internal element force. At the start of each increment the acceleration is obtained directly,
$$\ddot{u}\big|_t=M^{-1}\left(P-I\right)\big|_t$$
and the velocity and displacement are updated by central differences, with no iteration and therefore no convergence criterion:
$$\dot{u}\big|_{t+\Delta t/2}=\dot{u}\big|_{t-\Delta t/2}+\frac{\Delta t|_{t+\Delta t}+\Delta t|_{t}}{2}\,\ddot{u}\big|_t$$
$$u\big|_{t+\Delta t}=u\big|_t+\Delta t|_{t+\Delta t}\,\dot{u}\big|_{t+\Delta t/2}$$
The solid model was rebuilt from the design parameters, the actual cutter geometry and the tooth-flank modification data. Both wheels were discretized with C3D8R hexahedral elements generated by mapping a two-dimensional mesh into three dimensions: a single tooth sector was meshed in isolation and then arrayed around the axis. The mesh was refined on the working flanks to 0.5 mm × 0.5 mm and at the root groove and the groove-to-rim junction to 0.35 mm × 0.35 mm, with 0.5–1.5 mm elements elsewhere. The pinion model contains 650,668 elements and the driven wheel 586,600, with zero warning elements.
The material is an aviation carburizing gear steel. Because the tooth surface reaches roughly 200 °C in service, the transient runs used the elevated-temperature properties rather than the room-temperature set.
| Temperature / °C | Density / (kg·m−3) | Elastic modulus / GPa | Poisson’s ratio |
|---|---|---|---|
| 20 | 7,860 | 215 | 0.38 |
| 200 | 7,860 | 210 | 0.41 |
Boundary conditions released the axial rotation degree of freedom of each wheel and constrained the remaining five. A surface-to-surface contact pair was defined with the pinion flanks as the master surface and the wheel flanks as the slave surface. The pinion carried a rotational speed of 11,310.5 r/min, obtained from the predicted wheel speed through the 47/35 ratio, and the driven wheel carried a torque of 206.5 N·m, the load corresponding to 326 kW at that speed. Total simulated time was 0.08 s, about twenty wheel revolutions, with displacement and stress written at fixed intervals.
Modal validation of the finite element model
Free–free modes of the driven wheel were computed first. The nodal-diameter shapes follow the expected disc pattern: straight radial nodal lines, adjacent sectors in antiphase, and the largest axial displacement near the outer rim. The corresponding experiment suspended the wheel on elastic cords to approximate free boundaries and used hammer excitation with multiple input points and a single response point. Agreement is within about 2%, which is sufficient for the subsequent frequency and stress work.
| Mode | Simulated / Hz | Measured / Hz | Relative error / % |
|---|---|---|---|
| Three nodal diameters | 8,119.9 | 7,964 | 1.96 |
| Four nodal diameters | 13,554 | 13,420 | 1.00 |
Dynamic frequency and resonance-speed prediction
In-service the wheel is located by bearings and a spline, so a constrained modal analysis is the appropriate starting point: the axial rotation is free and the other five degrees of freedom are fixed. This gives a three-nodal-diameter frequency of 8,042.5 Hz and, by the resonance-speed relation, 15,079.7 r/min for the forward wave. That figure is already an approximation, because it neglects the two effects that shift a thin web most: centrifugal stiffening and the dynamic mesh force.
Rotation was represented as an equivalent centrifugal load, and the transmitted torque as a normal concentrated load over the contact region — 206.5 N·m and an equivalent flank load of 202.4 N/mm² at the design power. The mesh was represented by the mean time-varying mesh stiffness, 5 × 108 N/m. The combined effect raises the three-nodal-diameter frequency to 8,100.5 Hz, a stiffening of 0.72%, and moves the forward-wave resonance to 15,188.4 r/min at the wheel, corresponding to 11,310.5 r/min at the pinion. On the Campbell diagram the operating point is the intersection of the (z − 3)n/60 order line with the 8,100.5 Hz branch, well separated from the tooth-pass order zn/60 and from the backward-wave branch (z + 3)n/60 at about 12,790 r/min.
The 0.72% correction matters out of proportion to its size. Nodal-diameter resonance peaks in a lightly damped thin web are narrow, and a prediction based on the static frequency falls 109 r/min below the corrected value — comparable to the width of the peak itself, and enough to misplace a speed sweep or a design margin.
Axial displacement field: three lobes migrating at 10.67 times shaft speed
Snapshots of the axial displacement field were taken at 20 μs intervals through the predicted resonance. The field shows three peaks alternating with three valleys around the circumference — six extrema, with adjacent sectors separated by nodal lines vibrating in antiphase. Tracking a reference crest from frame to frame shows the whole pattern rotating in the same direction as the wheel, clockwise in the model, and completing one full circulation in a small fraction of a revolution. At m = 3 and z = 35 the wave rate is 10.67 times the shaft rate, so the sector-shaped field sweeps past any fixed point roughly eleven times per revolution.
Unwrapping the displacement around a circumferential node path near the root on the small end (nodes 2,301,000–2,301,629) confirms the picture: the spatial trace is close to sinusoidal, the wave travels with the rotation, and the waveform is disturbed where the path crosses the meshing zone. Six consecutive frames spanning one wave period Tw show the pattern advanced by 2π/3, consistent with three nodal diameters.
The time history at a single root-groove node (2,301,000) oscillates between −35.9 μm and +30.9 μm with an RMS value of 18.0 μm. Two time scales are visible in the trace: the wave-passage period 1/fd ≈ 123 μs, which sets the dense high-order ripple, and the shaft period 1/fz ≈ 4.0 ms, at which the mesh engagement recurs and the response amplitude swells. The spectrum is dominated by a component at 8,100 Hz with an amplitude of 22.3 μm, matching the predicted dynamic frequency of 8,100.5 Hz to within the resolution of the analysis; shaft order and its harmonics appear as lower-amplitude content. The coincidence of the measured spectral peak with the independently calculated dynamic frequency is the first internal check on the resonance prediction.
Stress field: three-petal pattern and mesh-induced modulation
The von Mises field under the same condition is organized into three petals around the circumference, the petal count equaling the nodal-diameter number, with maxima at two distinct sites. The meshing flank reaches 670.8 MPa, reflecting the local contact and bending load; the root groove reaches 260.7 MPa, and it is this second site that governs fatigue initiation, since it is a geometric stress raiser remote from the contact. Stress is also elevated around the shaft bore, an artifact of applying the torque as a boundary load: by Saint-Venant’s principle the disturbance decays within a short distance and does not affect the rim stress distribution that is of interest.
At the gauge-plane node the stress oscillates between 15.3 and 188.9 MPa with an RMS of 102.8 MPa, again in a dense high-order ripple superimposed on a once-per-revolution envelope that peaks as the sector passes the mesh. The spectrum has its largest component at fm − 3fz = 8,100 Hz, surrounded by sidebands spaced at the shaft frequency and extending through roughly six orders. The physical origin is straightforward: the wave circulates at 10.67 ωg, so any material point passes through the meshing zone once per revolution, and the mesh contact modulates the amplitude of the wave observed at that point. The sideband family is therefore not a numerical artifact but a signature of a rotating wave sampled by a rotating mesh — and it is the feature that makes nodal-diameter resonance identifiable in a measured spectrum.
Rig validation with strain gauges and a slip ring
Dynamic stress was measured on a component rig built around the same central-drive bevel pair. The root fillet itself is too small to instrument directly, so four BA120-1AA elevated-temperature strain gauges (gauge factor 2.16 ± 0.01, usable range −30 to 210 °C) were bonded on the small end approximately 2 mm below the root groove bottom, spaced 8, 9, 9 and 9 teeth apart around the circumference. Leads were routed through the hollow shaft to a high-speed slip-ring assembly, then to a strain amplifier, a data acquisition unit and a display of the stress spectrum.
The rig was swept slowly from 72% to 78% of physical speed while the pinion speed and the measured stress spectra were recorded. All four gauges registered the three-nodal-diameter forward-wave resonance at a dynamic frequency of 8,012 Hz with the pinion at 11,190 r/min, that is 15,026.6 r/min at the driven wheel. The measured amplitudes were 66.3, 67.0, 66.6 and 66.4 MPa, essentially uniform around the circumference — the expected result for a wave whose pattern is axisymmetric in amplitude even though its phase migrates.
| Quantity | Transient simulation | Rig measurement | Deviation |
|---|---|---|---|
| Driven-wheel resonance speed / (r/min) | 15,188.4 | 15,026.6 | 1.1% |
| Pinion speed at resonance / (r/min) | 11,310.5 | 11,190 | 1.1% |
| Dynamic frequency / Hz | 8,100.5 | 8,012 | 1.1% |
| Stress amplitude on the gauge circle / MPa | 57.5–57.9 | 66.3–67.0 | about 13% lower |
To compare stress magnitudes, 35 nodes were extracted on the same circumferential path at the gauge radius (nodes 2,301,000–2,301,629 at intervals of 18). The simulated resonance-frequency amplitudes fall in the narrow band 57.5–57.9 MPa, reproducing the measured circumferential uniformity, but they sit about 13% below the 66.3–67.0 MPa recorded by the gauges. Part of the gap is geometric: the extraction path is a nodal line on the surface, whereas a gauge averages strain over its grid area a short distance below the groove bottom, in a region of steeper gradient. Part may come from damping and from the gauge and adhesive adding local stiffness. Both results are of the same order and show the same spatial distribution, which is the level of agreement the model was built to deliver; the 1.1% speed and frequency error is the stronger validation, because it does not depend on stress amplitude calibration.
Design and diagnostic implications
Three practical consequences follow from the results. First, resonance screening for thin-web bevel gears should be performed on the (kz ∓ m) order family, not on the tooth-pass order alone, and it should use a dynamic frequency computed with centrifugal stiffening, operating torque and mean mesh stiffness; the static free–free value biases the predicted speed low by an amount comparable to the width of the resonance peak. Second, the stress field localizes at the root groove in a three-petal pattern whose orientation rotates with the wave, so a single strain gauge can capture the amplitude but not the spatial phase — circumferential arrays, as used here, are needed to confirm the nodal-diameter number and to separate forward from backward waves. Third, the sideband family around fm ∓ mfz, spaced at the shaft frequency, is a robust diagnostic marker: it appears in both simulated and measured spectra and is absent when the wheel runs off resonance.
Mitigation follows the same logic. Raising the web stiffness or adding a damping ring detunes or damps the nodal-diameter branch; reducing the dynamic mesh force through flank modification lowers the excitation; and where neither is possible, the operating range must be kept clear of the forward-wave crossing with an adequate margin, since the backward-wave crossing of the same mode lies several thousand revolutions per minute lower and is easily overlooked.
The scope of the present evidence should be stated plainly. The validation covers one geometry, one mode and one operating point; the thermal state is treated as a uniform 200 °C rather than a measured gradient; damping is not calibrated against a dedicated measurement; and the explicit formulation, while free of convergence difficulties, constrains the attainable simulated time. The stress comparison is one of magnitude and distribution rather than a point-by-point match. Within those limits, the combination of a dynamic-frequency-based resonance prediction and an explicit transient contact solution reproduces the measured resonance speed to about 1% and the measured root stress to the correct order, which is sufficient to locate a traveling-wave resonance before it produces a fracture.