1. The failure mode behind the analysis
In a turbofan or turboshaft engine the central transmission takes power from the core and turns it through 90° with a pair of spiral bevel gears. Mass reduction drives the design towards thin webs and thin rims, which lowers the axial bending stiffness of the driven wheel and pushes its nodal-diameter modes into the speed band the engine actually uses. When an excitation order crosses one of these modes, the deflection pattern stops standing still in the wheel: it migrates around the circumference as a traveling wave, and any fixed material point sees a high-frequency alternating stress superimposed on the quasi-static mesh load. Fatigue cracks then start at the tooth root, and the resulting tooth fracture is sudden and destructive.
A documented central-transmission failure of an aeroengine was traced to exactly this mechanism: the driven bevel gear broke into several blocks, and the investigation attributed the initiation to fourth-nodal-diameter traveling wave resonance combined with a poor meshing contact pattern. That case illustrates the two ingredients that matter in practice — a nodal-diameter mode close to an excitation order, and a contact condition that raises the local stress at the root fillet. What has remained less clear is the spatial form of the stress field while the wave travels, and the way the displacement and stress signals are modulated in time at a fixed point on the wheel. Both determine where a crack starts and what a sensor mounted on the gear can be expected to record.
2. Kinematics of a nodal-diameter traveling wave
A gear behaves, to first order, like an annular disc, and its modes fall into nodal-diameter, nodal-circle and combined families. Service fractures are almost always associated with the nodal-diameter family, in which radial nodal lines divide the wheel into sectors that vibrate in antiphase. A nodal-diameter mode can be decomposed into two identical cosine waves traveling in opposite directions around the disc. In a frame rotating with the gear, the forward and backward waves of an m-nodal-diameter mode have equal wavelength, speed and frequency:
$$\lambda_f=\lambda_b=\frac{2\pi}{m} \tag{1}$$
$$\omega_i=\pm\frac{2\pi f_d}{m},\qquad i=f,b \tag{2}$$
$$f_f=f_b=f_d \tag{3}$$
where $\lambda$ is wavelength, $\omega$ the angular wave speed, $f_d$ the dynamic frequency of the gear, and the subscripts $f$ and $b$ denote forward and backward waves. Resonance requires the excitation frequency to equal the dynamic frequency in the same reference frame, and to coincide with either the forward or the backward wave frequency. The mesh force is fixed in space while the wheel rotates, so in the gear frame it sweeps circumferentially at the shaft rotation frequency. Its m-th order component therefore appears at
$$f_j=\frac{kzn}{60}\mp\frac{mn}{60},\qquad j=ef,eb \tag{4}$$
with the minus sign for the forward wave and the plus sign for the backward wave, $z$ the tooth number, $n$ the rotational speed in r/min and $k$ the harmonic order of the mesh force, normally taken as 1. Setting $f_j=f_d$ gives the resonant speeds
$$N_i=\frac{60f_d}{kz\mp m},\qquad i=f,b \tag{5}$$
Using the mesh frequency $f_m=zn/60$ and the shaft frequency $f_z=n/60$, the wave period in the gear frame and the wave speed relative to the rotor follow directly:
$$T_w=\frac{1}{f_d}=\frac{1}{kf_m\mp mf_z} \tag{6}$$
$$\omega_w=\frac{f_m\mp mf_z}{mf_z}\,\omega_g=\frac{z\mp m}{m}\,\omega_g \tag{7}$$
Equation (7) is the key scaling result. For the 35-tooth driven wheel considered here, a third-nodal-diameter forward wave travels at $(35-3)/3=10.67$ times the rotor angular velocity, and the corresponding backward wave at $-(35+3)/3=-12.67$ times. The stress cycle seen by a fixed point is therefore set by the dynamic frequency (about 8.1 kHz), not by the tooth-passing rate of the rotor, and the pattern moves a full 120° in one wave period.
3. Explicit transient contact model of the gear pair
Contact between conjugate flanks is strongly nonlinear, and an implicit solver must iterate on that nonlinearity at every increment, with a real risk of non-convergence once the traveling wave has developed. An explicit formulation avoids the iteration: the state at the end of an increment follows from the displacement, velocity and acceleration at its start through the central-difference integration of the equations of motion,
$$M u” = P – I \tag{8}$$
$$u”\big|^{(t)} = M^{-1}\left(P-I\right)\big|^{(t)} \tag{9}$$
$$u’\big|^{(t+\Delta t/2)} = u’\big|^{(t-\Delta t/2)} + \frac{\Delta t\big|^{(t+\Delta t)}+\Delta t\big|^{(t)}}{2}\,u”\big|^{(t)} \tag{10}$$
$$u\big|^{(t+\Delta t)} = u\big|^{(t)} + \Delta t\big|^{(t+\Delta t)}\,u’\big|^{(t+\Delta t/2)} \tag{11}$$
where $M$ is the nodal mass matrix, $u”$, $u’$ and $u$ are nodal acceleration, velocity and displacement, $P$ is the applied external force and $I$ the internal element force. Stability is conditional, with the increment bounded by the smallest element dimension divided by the dilational wave speed of the material; this makes mesh quality at the contact zone the controlling factor for cost.
The pair modelled here has 47 teeth on the driving pinion and 35 on the driven wheel, and transmits 326 kW. Solid geometry was generated from the design parameters, the actual cutter geometry and the flank modification data, so that the contact pattern of the model reflects the manufactured tooth form rather than an idealised involute-equivalent surface. Both bodies were meshed with C3D8R hexahedral elements using a two-dimensional-to-three-dimensional mapped scheme: one tooth sector was meshed in isolation and then replicated around the axis. The flank contact region uses 0.5 mm × 0.5 mm elements, the root groove and the groove-to-rim junction use 0.35 mm × 0.35 mm elements, and the remaining volume is discretised at 0.5–1.5 mm. This gives 650 668 elements for the pinion and 586 600 for the wheel, with no warning elements. The pinion flank is the master surface and the wheel flank the slave; five degrees of freedom are constrained on each body and only axial rotation is released. Rotation is imposed on the pinion and 206.5 N·m is applied to the wheel, which corresponds to an equivalent flank pressure of 202.4 N/mm². The surface temperature of the gear in service reaches about 200 °C, so the elevated-temperature properties were used, and the response was integrated over 0.08 s with displacement and stress written at fixed intervals.
Table 1. Material parameters of the carburized gear steel
| Temperature / °C | Density / (kg·m⁻³) | Elastic modulus / GPa | Poisson ratio |
|---|---|---|---|
| 20 | 7 860 | 2.15 × 10² | 0.38 |
| 200 | 7 860 | 2.10 × 10² | 0.41 |
4. Modal calibration against a free-free hammer test
Before any dynamic response can be trusted, the finite element model has to reproduce the measured modal frequencies of the real wheel. A free-free boundary condition was approximated by suspending the test gear on elastic cord, and a hammer test with multi-point excitation and single-point response was used to identify the modes. The computed axial displacement mode shapes follow the expected disc pattern: radial nodal lines, antiphase motion of adjacent sectors, and displacement amplitude growing towards the rim. Agreement with the measured frequencies is within 2% for both modes identified, which is adequate for the subsequent resonance prediction.
Table 2. Computed and measured natural frequencies of the driven bevel gear
| Mode | Computed / Hz | Measured / Hz | Relative error / % |
|---|---|---|---|
| 3 nodal diameters | 8 119.9 | 7 964 | 1.96 |
| 4 nodal diameters | 13 554 | 13 420 | 1.00 |
5. Dynamic frequency and resonance speed
The free-free frequency is not the quantity that governs resonance in service. The wheel is located by a spline and a bearing, so a constrained modal analysis was performed with the axial rotation released and the other five degrees of freedom fixed, giving 8 042.5 Hz for the three-nodal-diameter mode. Substituting this value into equations (4)–(5) yields a forward-wave resonance at 15 079.7 r/min. That estimate still ignores the operating loads. Three corrections were applied: the rotational speed was represented as a centrifugal load field; the driving torque was applied as a normal concentrated load over the flank contact area; and the mesh stiffness was set to the mean value of the time-varying mesh stiffness, 5 × 10⁸ N/m. The resulting dynamic frequency is 8 100.5 Hz, and the corresponding forward-wave resonance speed is 15 188.4 r/min, equivalent to 11 310.5 r/min on the pinion through the 35/47 ratio. Centrifugal stiffening therefore accounts for only about 58 Hz, or 0.7%, of the modal frequency, but it shifts the resonant speed by roughly 109 r/min — small, yet comparable to the width of the resonance peak and thus relevant when a speed margin is being argued.
The consistency of the prediction can be checked arithmetically. At 15 188.4 r/min the shaft frequency is $f_z = 253.1$ Hz and the mesh frequency is $f_m = 35 f_z = 8 859.6$ Hz. The third-order forward-wave excitation then falls at $f_m – 3f_z = 8 100.3$ Hz, within 0.2 Hz of the computed dynamic frequency. Plotted as a Campbell diagram, the order line $f_{ef}=(z-m)n/60$ intersects the horizontal line $f_d$ at 15.19 × 10³ r/min, while the mesh line $f_m=zn/60$ and the backward-wave line $f_{eb}=(z+m)n/60$ cross the same frequency level at higher speeds.
6. Axial displacement field under forward-wave resonance
The response was extracted at 630 nodes on a circumferential line at the small end close to the root groove, numbered 2 301 000 to 2 301 629. Snapshots of the axial displacement field taken at 20 μs intervals show the characteristic three-lobed pattern: three peaks alternating with three valleys around the circumference, six extrema in total, with adjacent sectors of opposite sign. The pattern is not stationary. Tracking a reference crest from frame to frame shows that it migrates in the same direction as the gear rotation, and completing one full wave period in about 123 μs, which is consistent with $T_w = 1/f_d$ from equation (6) and with the 10.67× speed ratio of equation (7). At the simulated speed the peak-to-peak displacement range is −35.9 μm to +30.9 μm, with an RMS value of 18.0 μm.
Unwrapping the circumferential line into a single trace clarifies the propagation. The waveform is essentially harmonic, and the crest translates along the trace at a constant rate except where it crosses the meshing zone, where the local constraint of the mating flank disturbs the amplitude. In the frequency domain the dominant component sits 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 — an independent check that the resonance point was identified correctly. Shaft-order components and their harmonics are also present, but at much lower amplitude; they arise because each sector passes through the mesh once per revolution.
7. Stress field and its modulation
The equivalent stress field at the same instants has a three-petal form, with the number of petals equal to the number of nodal diameters, and it rotates with the displacement field. Two concentrations dominate. The first is at the meshing flank, where the contact pressure plus the wave-induced bending gives a peak of 670.8 MPa. The second is at the root groove, with a peak of 260.7 MPa; this is the location where a fatigue crack would be expected to initiate, and it is consistent with the root-origin fractures observed in service. Elevated stress also appears at the bore and its neighbourhood, but this is the local effect of the torque applied at the bore boundary and decays rapidly with distance, so it does not affect the stress state at the teeth.
At the root-groove node the Mises stress fluctuates between 15.3 and 188.9 MPa with an RMS value of 102.8 MPa, and the time history again shows a dense high-order oscillation with a visible increase each time the sector passes the mesh. The spectrum peaks at 8 100 Hz. Around that peak, and also around its second and third harmonics, a family of sidebands appears, spaced at the shaft frequency and its multiples up to about the sixth order. These are modulation sidebands, not separate modes: as the wave sweeps around the wheel it is alternately stiffened and released by the meshing contact, so the resonance carrier is amplitude-modulated at the rate at which the wheel rotates through the mesh. For diagnostic purposes this is a useful signature — a resonance that is genuinely a traveling wave will always carry shaft-order sidebands, whereas a purely local vibration will not.
8. Strain-gauge measurement on the gear test rig
Validation was carried out on a dedicated central-transmission component test rig driven by an electric motor through a gearbox, with the driven bevel gear instrumented and connected to the acquisition chain through a high-speed slip-ring system. Mid-temperature strain gauges (type BA120-1AA, gauge factor 2.16 ± 0.01, operating range −30 to 210 °C) were used. The root fillet itself is too small to host a gauge, so four gauges were bonded on the small end about 2 mm below the root groove bottom, spaced at 8, 9, 9 and 9 teeth so that the four positions together span the full 35-tooth circumference. Leads were routed out through the hollow shaft to the slip rings, then to a strain amplifier, a bridge box and a data acquisition unit, with the stress spectra displayed in real time. Air cooling kept the instrumentation within its temperature limit during the run.
The rig was swept slowly from 72% to 78% of physical speed. All four channels recorded a clear three-nodal-diameter forward-wave resonance, with stress amplitudes at the resonant frequency of 66.3, 67.0, 66.6 and 66.4 MPa at positions 1# to 4#. The resonant frequency measured on the wheel was 8 012 Hz at a pinion speed of 11 190 r/min, which corresponds to 15 026.6 r/min on the driven wheel. Against the predicted 15 188.4 r/min this is a relative error of 1.1%.
Table 3. Predicted and measured resonance parameters at the three-nodal-diameter forward wave
| Quantity | Prediction | Measurement | Difference |
|---|---|---|---|
| Dynamic frequency of driven gear / Hz | 8 100.5 | 8 012 | 1.1% |
| Driven gear resonance speed / (r·min⁻¹) | 15 188.4 | 15 026.6 | 1.1% |
| Pinion speed at resonance / (r·min⁻¹) | 11 310.5 | 11 190 | 1.1% |
| Stress amplitude at resonance / MPa | 57.5–57.9 | 66.3–67.0 | 13–14% lower |
For a like-for-like comparison of the stress field, 35 nodes were taken around the circumference of the wheel in the simulation at the same axial station as the gauges, and the resonant-frequency component was extracted at each. The simulated amplitudes lie between 57.5 and 57.9 MPa, so the model reproduces the measured magnitude to within about 14%, and both show the same qualitative feature: the amplitude at the resonant frequency is essentially invariant with circumferential position. This invariance is a direct consequence of the wave kinematics. Because the pattern migrates at 10.67 times the rotor speed, every gauge, wherever it sits, is swept by all three petals within each wave period and therefore records the same envelope. A stationary three-lobed stress field, by contrast, would produce amplitudes that differ strongly between gauge positions.
9. Implications for design and instrumentation
Three conclusions follow from the combination of simulation and measurement, and each has a practical consequence.
- The resonance condition is a speed-margin problem, not a frequency-margin problem. The resonant speed is $60f_d/(kz \mp m)$, so a small change in tooth number or in nodal-diameter order moves the dangerous speed substantially, whereas a small change in dynamic frequency moves it only proportionally. Separation is best achieved by shifting the nodal-diameter frequency of the thin web — through web thickness, rim stiffness or the addition of a damping ring — or by avoiding operating points where an order line crosses that frequency.
- The root groove, not the flank, is the fatigue-critical location. The flank carries the higher instantaneous stress, 670.8 MPa against 260.7 MPa, but it is a contact stress with a steep gradient and a short dwell, while the root groove experiences a fully reversed bending cycle at 8.1 kHz with a concentration that persists around the whole circumference. The fracture origin reported for the service failure is consistent with this ranking.
- Instrumentation bandwidth must follow the dynamic frequency, not the tooth-passing rate. The stress signal at a fixed point is dominated by the 8.1 kHz carrier, with sidebands at shaft-order spacing; a measurement chain sized for the mesh frequency alone will alias or attenuate the very component that drives fatigue. The non-contact acoustic and the strain-gauge routes both remain valid, provided the sampling and the telemetry bandwidth are chosen for the dynamic frequency.
The main limitations of the model are worth stating plainly. The mesh stiffness is represented by its mean value, so the tooth-passing modulation of the contact is only partially captured; mistuning between teeth and sectors, which in practice splits and detunes nodal-diameter doublets, is not included; and damping was not calibrated against measurement, which is the most likely reason the computed stress amplitude falls about 14% below the measured one. Extending the model to include measured damping and sector-to-sector mistuning would therefore be the natural next step, particularly for predicting whether a given wheel will respond at a single frequency or over a broader band.