Thin-web spiral bevel gears are attractive for aeroengine central transmissions because they carry high torque with low mass. The same thin web that reduces weight also lowers the bending stiffness of the gear body, so a nodal-diameter mode can be excited by the periodic axial component of the meshing force. At high rotational speed, the resulting traveling wave can produce alternating stress at kilohertz rates and initiate fatigue cracks at the tooth root. The failure mode is therefore not simple overload but high-cycle fatigue driven by a resonance that moves around the gear.
In a stationary frame, a nodal-diameter mode appears as a standing pattern. In the rotating frame of the gear, the same mode separates into a forward wave and a backward wave. When a meshing harmonic matches the forward- or backward-wave frequency, the wave is sustained and the displacement pattern migrates around the circumference. For a thin-web spiral bevel gear, this migration is much faster than the shaft rotation, and the stress at a fixed point on the gear alternates at the wave frequency rather than at the shaft frequency.
Resonance condition in a rotating gear
For a gear with \(z\) teeth rotating at \(n\) r/min, the mesh frequency and shaft frequency are
$$ f_m=\frac{zn}{60},\qquad f_z=\frac{n}{60}. $$
For an \(m\)-nodal-diameter mode with dynamic frequency \(f_d\), the forward- and backward-wave excitation frequencies produced by the \(k\)-th mesh harmonic are
$$ f_{\mathrm{ef}}=k f_m-m f_z,\qquad f_{\mathrm{eb}}=k f_m+m f_z. $$
Resonance occurs when the excitation frequency equals the dynamic frequency. The corresponding forward- and backward-wave speeds are
$$ N_f=\frac{60f_d}{kz-m},\qquad N_b=\frac{60f_d}{kz+m}. $$
The harmonic order \(k\) is usually taken as 1. For a 35-tooth driven gear and a three-nodal-diameter mode, the forward-wave condition becomes \(f_m-3f_z=f_d\), so the resonance speed is \(N_f=60f_d/(35-3)\). The wave itself moves in the fixed frame at
$$ \omega_w=\frac{z-m}{m}\omega_g. $$
With \(z=35\) and \(m=3\), \(\omega_w=10.67\omega_g\). A point on the gear therefore experiences a displacement and stress cycle that sweeps past at more than ten times the shaft angular speed. This kinematic amplification is central to the fatigue problem: even a moderate shaft speed can generate a kilohertz stress history at the tooth root.
Transient contact model and modal basis
The representative case is a 47-tooth driving gear and a 35-tooth driven spiral bevel gear made of carburized gear steel. The transient contact calculation used 200 °C material properties because the tooth surface in service can reach approximately that temperature. The elastic and inertial properties are summarized in Table 1.
| Temperature (°C) | Density (kg/m³) | Elastic modulus (GPa) | Poisson’s ratio |
|---|---|---|---|
| 20 | 7,860 | 215 | 0.38 |
| 200 | 7,860 | 210 | 0.41 |
The finite-element mesh used C3D8R hexagonal elements. Contact surfaces were refined to \(0.5\ \mathrm{mm}\times0.5\ \mathrm{mm}\), and the root groove and rim transition were refined to \(0.35\ \mathrm{mm}\times0.35\ \mathrm{mm}\). Element sizes elsewhere ranged from 0.5 to 1.5 mm. The driving gear mesh contained 650,668 elements and the driven gear mesh contained 586,600 elements. The explicit central-difference solver was used because the tooth contact is highly nonlinear and an implicit iteration can struggle to converge. The simulation covered 0.08 s of meshing, with displacement and stress output at fixed intervals.
Only the axial rotation was released on each gear; the other five degrees of freedom were constrained. A master-slave contact pair was defined between the driving and driven tooth surfaces. The driving gear was rotated at 11,310.5 r/min, corresponding to the predicted driven speed through the 47/35 ratio. A torque of 206.5 N·m was applied to the driven gear, equivalent to a working load of 326 kW and a normal tooth-surface load of 202.4 MPa. The average time-varying mesh stiffness was set to \(5\times10^8\ \mathrm{N/m}\).
Before the transient calculation, the free-free modal behavior was checked against a hammer-impact test using elastic suspension to approximate free boundary conditions. The three-nodal-diameter frequency was 8,119.9 Hz in the simulation and 7,964 Hz in the test, a difference of 1.96%. The four-nodal-diameter frequency was 13,554 Hz in the simulation and 13,420 Hz in the test, a difference of 1.00% (Table 2). This agreement supports the geometric and mesh representation used for the subsequent contact analysis.
| Mode | Simulation (Hz) | Measurement (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 forward-wave resonance speed
The constrained three-nodal-diameter frequency of the driven gear was 8,042.5 Hz. Using this static value alone would place the forward-wave resonance at 15,079.7 r/min. In operation, however, centrifugal loading, torque, and mesh stiffness shift the dynamic frequency. Including those effects raised the three-nodal-diameter dynamic frequency to 8,100.5 Hz and moved the predicted forward-wave resonance to 15,188.4 r/min. The shift is small in percentage terms, about 0.7% in speed, but it is large enough to matter when a resonance avoidance margin is narrow.
The Campbell construction for the driven gear places the \(f_m-3f_z\) excitation line against the three-nodal-diameter dynamic frequency. Their intersection defines the forward-wave resonance point. At 15,188.4 r/min, \(f_m-3f_z\) is approximately 8,100 Hz, which is the dynamic frequency used in the prediction. The calculation also shows why a static natural frequency is not sufficient for a high-speed thin-web gear: the operating point depends on speed, load, and mesh stiffness, not on the modal frequency alone.
Axial displacement field under forward-wave resonance
At the predicted three-nodal-diameter forward-wave resonance, the axial displacement field of the driven gear has three peaks and three valleys around the circumference, giving six extrema in total. Adjacent sectors separated by a nodal line move in opposite directions. The pattern migrates in the same direction as gear rotation, and the migration is fast: the wave angular speed is 10.67 times the gear angular speed. At 8.1 kHz, one wave period is about 123 µs, and the three-lobed pattern advances by \(2\pi/3\) in that period.
The displacement history at a node below the root groove shows dense high-order harmonic content rather than a simple once-per-revolution sinusoid. The axial displacement ranged from −35.9 to 30.9 µm, with an effective value of 18.0 µm. The spectrum is dominated by the component at \(f_m-3f_z\), approximately 8,100 Hz, with an amplitude of 22.3 µm. That frequency matches the three-nodal-diameter dynamic frequency used for the resonance prediction. Shaft-frequency components and their harmonics are also present, and the response increases when the traveling wave passes through the meshing zone.
The displacement field is therefore not a quasi-static deflection of the web. It is a rapidly moving three-lobed pattern whose local amplitude is modulated by the mesh. This distinction is important for instrumentation: a sensor at a fixed circumferential position sees a high-frequency carrier with mesh-related modulation, not a slowly varying load proportional to shaft speed.
Stress field and fatigue-critical locations
The stress field under the same three-nodal-diameter forward-wave resonance has a three-petal distribution around the circumference, with the number of petals equal to the number of nodal diameters. Stress concentration appears at two locations: the tooth-root groove and the meshing tooth surface. The maximum Mises stress at the meshing tooth surface was 670.8 MPa, while the maximum at the root groove was 260.7 MPa. The inner bore and its vicinity also carry stress from the applied torque, but this is a local boundary effect; according to Saint-Venant’s principle, it has little influence on the far-field tooth and web stress distribution.
At the root-groove node used for response extraction, the Mises stress ranged from 15.3 to 188.9 MPa, with an effective value of 102.8 MPa. The stress spectrum is again dominated by the traveling-wave component near 8,100 Hz. Around that carrier, the spectrum contains sidebands spaced at shaft-frequency harmonics, approximately \(1f_z\) to \(6f_z\). The modulation arises because the traveling wave passes the meshing zone repeatedly, so the local stress is not a pure tone but a carrier with mesh- and shaft-related sidebands.
The root groove is a fatigue-critical site because it combines a geometric stress raiser with the alternating stress produced by wave passage. The contact stress at the meshing surface is higher in magnitude, but it is localized at the contact. The root-groove stress is the more relevant quantity for crack initiation under repeated wave cycles, and its concentration should be evaluated together with the dynamic stress field rather than with the static tooth load alone.
Experimental correlation
The experimental measurement used BA120-1AA intermediate-temperature strain gauges with a gauge factor of 2.16 and an operating range of −30 to 210 °C. The gauges were placed at the small end of the driven gear, approximately 2 mm from the root groove, because the root fillet itself is too small for direct bonding. Four gauges were distributed around the circumference at intervals of 8, 9, 9, and 9 teeth. Lead wires were routed through the hollow gear shaft to a high-speed slip-ring system, then to a strain amplifier and data-acquisition system.
The test swept the driving shaft from 72% to 78% of physical speed. At the three-nodal-diameter forward-wave resonance, the measured dynamic frequency was 8,012 Hz at a driving speed of 11,190 r/min. Through the 47/35 ratio, the driven speed was 15,026.6 r/min. The predicted resonance speed was 15,188.4 r/min, a relative difference of 1.1%. The predicted dynamic frequency of 8,100.5 Hz differs from the measured 8,012 Hz by the same 1.1%, which is consistent with the speed comparison.
| Quantity | Simulation or prediction | Experiment | Difference |
|---|---|---|---|
| Dynamic frequency | 8,100.5 Hz | 8,012 Hz | 1.1% |
| Driven-gear speed | 15,188.4 r/min | 15,026.6 r/min | 1.1% |
| Stress amplitude at resonance | 57.5–57.9 MPa | 66.3–67.0 MPa | Same order; simulation about 13% lower at the mean |
The four measured stress amplitudes at the resonance frequency were 66.3, 67.0, 66.6, and 66.4 MPa. The circumferential uniformity is notable: at the same radius, the gauges read nearly the same amplitude. The simulation extracted 35 nodes around the same circumferential region and gave 57.5–57.9 MPa at the resonance frequency, also nearly uniform. The absolute simulated values are about 13% lower than the measured mean, but the order of magnitude and the circumferential distribution agree.
The comparison supports the resonance-speed prediction and the ability of the transient contact model to reproduce the qualitative stress distribution. It does not establish exact amplitude calibration. The residual difference is a reminder that absolute dynamic stress in a thin-web gear depends on contact detail, damping, and the relationship between the gauge location and the extracted nodal stress, so the model is most reliable for resonance placement and relative field distribution unless it is calibrated against the specific test configuration.
Design and test implications
The case shows that traveling-wave resonance in a high-speed thin-web spiral bevel gear is a system-level operating condition, not a single modal property. The following implications follow directly from the measured and computed behavior.
- Resonance avoidance should use the dynamic frequency calculated with speed, torque, and mesh stiffness. A static or constrained modal value can shift the predicted crossing by a measurable amount, and the forward-wave condition \(f_m-mf_z=f_d\) should be checked on a Campbell diagram over the full operating range.
- For a 35-tooth gear with a three-nodal-diameter mode, the forward wave migrates at 10.67 times the shaft speed. Fatigue assessment should therefore use the high-frequency dynamic stress history at critical points, not a once-per-revolution load cycle.
- The root groove is a stress raiser under wave passage. Its geometry, surface condition, and local stress concentration should be evaluated under the traveling-wave stress field, in addition to the contact stress at the meshing surface.
- Strain gauges placed near the root groove can capture a representative circumferential amplitude because the measured and simulated stress fields are nearly uniform at the same radius. Absolute amplitude prediction, however, requires attention to gauge location, damping, and contact representation.
The broader lesson is that thin-web spiral bevel gears require resonance maps that combine modal analysis, transient tooth contact, and rotating-frame wave kinematics. When those elements are consistent, the predicted resonance speed, the three-lobed displacement field, the petal-shaped stress field, and the measured strain response form a coherent picture of a high-cycle fatigue mechanism that would not be visible from static tooth-load analysis alone.