High-speed aeroengine central transmissions are pushed toward lower weight, higher speed, and higher load. A thin web is an effective way to reduce gear mass, but it also lowers bending stiffness and brings nodal-diameter modes into the operating speed range. When the mesh excitation matches a nodal-diameter dynamic frequency, a forward or backward traveling wave can be amplified around the gear rim. The resulting stress concentration at the tooth root groove is a recognized cause of sudden tooth fracture in spiral bevel gears. The technical problem is therefore not only to identify the resonance speed, but also to resolve how the displacement field migrates, where the stress concentrates, and how the response is modulated by the meshing position.
Resonance Mechanism in Thin-Web Spiral Bevel Gears
A thin-web spiral bevel gear behaves, at the rim level, much like an annular disk. Its modal family includes nodal-circle modes, nodal-diameter modes, and combined nodal-circle/nodal-diameter modes. For thin aeroengine gears, the nodal-circle frequencies are usually high and difficult to excite, whereas nodal-diameter modes can fall near the mesh-frequency order and become the relevant fatigue risk. In a nodal-diameter mode, the rim deflects alternately in adjacent sectors, and the pattern rotates as a traveling wave.
For a mode with m nodal diameters, the axial displacement can be written as
$$z(r,\theta,t)=A(r)\cos(m\theta)\cos(\omega t)$$
where A(r) is the radial deflection function, θ is the circumferential coordinate, and ω is the modal angular frequency. Expanding the product separates the field into two counter-rotating components:
$$z=\frac{1}{2}A(r)\left[\cos(m\theta-\omega t)+\cos(m\theta+\omega t)\right]$$
The first term is the forward traveling wave, which moves in the direction of gear rotation; the second is the backward traveling wave, which moves opposite to rotation. The mesh frequency for a gear with Z teeth running at N r/min is
$$f_m=\frac{ZN}{60}$$
In the rotating frame, the excitation frequencies associated with the forward and backward waves are
$$f_{e}=k\frac{ZN_f}{60}-m\frac{N_f}{60},\qquad f_{eb}=k\frac{ZN_b}{60}+m\frac{N_b}{60}$$
where k is the harmonic order, normally taken as 1 in this resonance check. Resonance occurs when these excitation frequencies equal the nodal-diameter dynamic frequency fd, giving the forward and backward resonance speeds
$$N_f=\frac{60f_d}{kZ-m},\qquad N_b=\frac{60f_d}{kZ+m}$$
The traveling-wave angular speed and its ratio to shaft speed follow from the wave period:
$$\omega_w=\frac{2\pi f_d}{m},\qquad \frac{\omega_w}{\omega_g}=\frac{Z\mp m}{m}$$
with the minus sign for the forward wave and the plus sign for the backward wave. This ratio explains why the visible displacement pattern can migrate much faster than the gear itself rotates. For the 35-tooth driven spiral bevel gear analyzed here, the predicted ratios are 10.67 for the three-nodal-diameter forward wave and 9.75 for the four-nodal-diameter backward wave.
Finite Element Model and Modal Correlation
The gear pair represents an aeroengine central drive: 47 driving teeth, 35 driven teeth, contact ratio 1.41, and a web thickness of 8 mm. The material is the carburized gear steel 16Cr3NiWMoVNbE, with density 7860 kg/m³ and Poisson’s ratio 0.38 at 20 °C and 0.41 at 200 °C. The finite element mesh used C3D8R hexahedral elements, with 650,668 elements for the driving gear and 586,600 for the driven gear. A mesh-independence check showed that above about 347,600 elements, the third- and fourth-nodal-diameter frequencies changed by less than 0.1%.
Free modal testing used five driven gears from the same production batch. The gears were suspended by elastic cords to approximate free boundary conditions and excited by a hammer with multi-point excitation and single-point response. The correlation between simulation and test is summarized in Table 1. The first nodal-circle frequency agreed within about 5%, and the nodal-diameter frequencies within about 3%. The third-nodal-diameter error was below 2.68%, and the fourth-nodal-diameter error below 1.83%.
| Mode | Simulation (Hz) | Test range (Hz) | Relative error (%) |
|---|---|---|---|
| First nodal diameter | 3437.7 | 3352–3368 | 2.07–2.56 |
| Second nodal diameter | 3611.5 | 3656–3664 | 1.22–1.43 |
| First nodal circle | 4780.6 | 4556–4560 | 4.84–4.93 |
| Third nodal diameter | 8119.9 | 7908–7964 | 1.96–2.68 |
| Fourth nodal diameter | 13554 | 13310–13420 | 1.00–1.83 |
For the operating condition, a constrained modal analysis was performed at 200 °C, close to the oil-wetted gear surface temperature. The axial rotation degree of freedom was released and the remaining five were constrained. The resulting third-nodal-diameter frequency was 8042.5 Hz and the fourth-nodal-diameter frequency was 13355 Hz.
Dynamic Frequency and Resonance-Speed Prediction
Using the static frequency in place of the dynamic frequency is convenient but can shift the predicted resonance speed. Centrifugal loading, transmitted torque, and mesh stiffness all modify the dynamic frequency. The mesh stiffness was represented by an average time-varying value of 5 × 108 N/m. At the 326 kW operating load, the third-nodal-diameter forward-wave case corresponded to 206.5 N·m and an equivalent tooth-surface load of 202.4 N/mm²; the fourth-nodal-diameter backward-wave case corresponded to 151.5 N·m and 148.5 N/mm².
| Prestress condition | Third ND dynamic frequency (Hz) | Change vs static (%) | Fourth ND dynamic frequency (Hz) | Change vs static (%) |
|---|---|---|---|---|
| No prestress | 8042.5 | 0 | 13355 | 0 |
| T = 206.5 N·m, N1 = 15079.7 r/min, km = 5 × 108 N/m | 8100.6 | 0.72 | 13490.8 | 1.02 |
| T = 151.5 N·m, N1 = 20546.2 r/min, km = 5 × 108 N/m | 8105.5 | 0.78 | 13498 | 1.07 |
The final dynamic frequencies used for resonance prediction were 8100.6 Hz for the third nodal diameter and 13498 Hz for the fourth. A Campbell diagram in the rotating frame placed the third-nodal-diameter forward-wave resonance near idle speed, at a driven-gear speed of 15188.6 r/min, and the fourth-nodal-diameter backward-wave resonance near maximum design speed, at 20766.2 r/min. The corresponding driving-gear speeds were 11310.5 r/min and 15464.2 r/min.
| Mode | m | Dynamic frequency (Hz) | Driven speed (r/min) | Driving speed (r/min) |
|---|---|---|---|---|
| Third ND forward wave | 3 | 8100.6 | 15188.6 | 11310.5 |
| Fourth ND backward wave | 4 | 13498 | 20766.2 | 15464.2 |
Transient Contact Model and Load Cases
The transient response was solved with an explicit dynamics algorithm using the central-difference scheme. This avoids iterative convergence inside the contact problem and is suitable for the short-duration, high-frequency response of a thin-web gear. The analysis used 200 °C material properties, a contact-pair formulation between the driving concave flank and the driven convex flank, a released axial rotation for both gears, and constraints on the remaining five degrees of freedom. Rotational speed was applied to the driving gear and torque to the driven gear.
Each load case was run for 0.08 s to allow the resonance response to stabilize and to resolve several shaft-rotation cycles. The output interval satisfied the Nyquist requirement of more than twice the highest mesh frequency. Stress and displacement were written at uniform intervals so that the time-domain modulation and the frequency-domain peak at the nodal-diameter dynamic frequency could both be recovered.
Axial Displacement Field and Wave Migration
Under third-nodal-diameter forward-wave resonance, the axial displacement field formed three peaks and three valleys around the circumference. Six extrema divided the disk into six sectors, with adjacent sectors moving in opposite axial directions. Under fourth-nodal-diameter backward-wave resonance, the field formed four peaks and four valleys, eight extrema, and eight sectors. In both cases the sector count matched the nodal-diameter number, confirming that the intended mode was excited. The displacement contours were magnified 200 times for visualization.
The expanded circumferential displacement trace was close to a simple harmonic wave. The forward wave propagated in the same direction as gear rotation, while the backward wave propagated opposite to rotation. The pattern did not remain stationary: it migrated as a fan-shaped field, and the migration speed was far higher than the shaft speed. At the meshing position, the axial displacement increased noticeably, producing a local disturbance in the otherwise harmonic wave. The displacement response also showed a low-frequency modulation with the shaft-rotation period, superimposed on the dense high-order harmonic content.
| Case | Location | Range (µm) | RMS (µm) |
|---|---|---|---|
| Third ND forward wave | Root groove bottom, node 2301000 | −34.4 to 31.6 | 16.8 |
| Third ND forward wave | Root groove, node 2400101 | −36.1 to 34.3 | 18.0 |
| Fourth ND backward wave | Root groove bottom, node 2301000 | −29.9 to 4.2 | 17.2 |
| Fourth ND backward wave | Root groove, node 2400101 | −33.0 to 4.6 | 18.2 |
| Non-resonant, 17457 r/min | Root groove bottom, node 2301000 | −16.4 to 9.4 | 11.0 |
The frequency-domain displacement peaks occurred at the nodal-diameter dynamic frequency and at shaft-rotation harmonics. For the third-nodal-diameter forward-wave case, the main peaks were at fm − 3fz, fz, 2fz, and 3fz; the resonance-frequency displacement amplitudes were 19.9 µm at the root groove bottom and 21.5 µm at the root groove. For the fourth-nodal-diameter backward-wave case, the corresponding peaks were at fm + 4fz, fz, 2fz, and 3fz, with amplitudes of 10.8 µm and 11.7 µm. At the non-resonant speed of 17457 r/min, the response was dominated by low-frequency, higher-amplitude shaft-order motion, with no high-frequency resonance peak; the RMS was 11.0 µm, below the resonance cases.
Stress Field and Dynamic Stress Response
The stress field under traveling-wave resonance had a petal-like circumferential pattern. The third-nodal-diameter forward-wave case produced three stress lobes, and the fourth-nodal-diameter backward-wave case produced four lobes, so the lobe count again matched the nodal-diameter number. Stress concentrated at the root groove fillet and at the meshing flank. The maximum von Mises stress at the meshing surface was 670.8 MPa for the third-nodal-diameter forward wave and 655.9 MPa for the fourth-nodal-diameter backward wave. At the root groove, the corresponding maxima were 260.7 MPa and 358.3 MPa.
The root groove is the more relevant location for fatigue initiation because it is a geometric stress raiser under repeated wave passage. Time histories at the root groove showed dense high-order harmonic content, a clear increase when the wave passed the meshing position, and a periodic modulation at the shaft-rotation frequency. The fourth-nodal-diameter backward-wave case produced higher root-groove stress than the third-nodal-diameter forward-wave case, despite its lower displacement amplitude, because the higher nodal diameter and higher speed increased the dynamic stress.
| Case | Location | Range (MPa) | RMS (MPa) |
|---|---|---|---|
| Third ND forward wave | Root groove bottom, node 2301000 | 15.3–188.9 | 102.8 |
| Third ND forward wave | Root groove, node 2400101 | 16.7–315.6 | 164.2 |
| Fourth ND backward wave | Root groove bottom, node 2301000 | 3.9–226.7 | 133.9 |
| Fourth ND backward wave | Root groove, node 2400101 | 4.3–386.5 | 223.9 |
| Non-resonant, 17457 r/min | Root groove bottom, node 2301000 | 38.4–141.2 | 73.5 |
The stress spectra contained peaks at the nodal-diameter dynamic frequency and at shaft-order harmonics. For the third-nodal-diameter forward wave, the highest stress amplitude was at about 8100 Hz, matching the predicted 8100.6 Hz. At fm − 3fz, the amplitude was 57.9 MPa at the root groove bottom and 106.9 MPa at the root groove. For the fourth-nodal-diameter backward wave, the highest amplitude was at about 13500 Hz, matching the predicted 13498 Hz. At fm + 4fz, the amplitude was 92.5 MPa at the root groove bottom and 175.7 MPa at the root groove. These spectral matches provide a quantitative check that the intended forward and backward traveling waves were excited.
At the non-resonant speed of 17457 r/min, the stress response ranged from 38.4 to 141.2 MPa with an RMS of 73.5 MPa. Its spectrum was dominated by shaft-order components, indicating that the non-resonant stress mainly came from mesh impact rather than a nodal-diameter resonance.
Experimental Validation
Strain gauges were placed near the root groove bottom at the small end, about 2 mm from the groove, because the fillet itself is too small for direct gauge bonding. Four gauges were used: gauges 1 and 2 were separated by eight teeth, and the other adjacent gauges by nine teeth. The gauges were BA120-1AA medium-temperature strain gauges with 120.2 ± 0.1 Ω resistance, gauge factor 2.16 ± 1%, range ±2000 µε, and operating range −30 to 210 °C. The estimated gauge-related error was about 3%. Leads were routed through the hollow gear shaft to a high-speed slip ring, then to a strain amplifier and data acquisition system. Air cooling was used to limit slip-ring heating.
The test speed was swept from 72% to 78% of operating speed, raised to 106%, pulled back to 104%, and then trimmed again to 106%. The measured third-nodal-diameter forward-wave frequency was 8012 Hz at a driving speed of 11190 r/min, corresponding to 15026.6 r/min at the driven gear. The predicted value was 15188.6 r/min, a relative error of 1.09%. The measured fourth-nodal-diameter backward-wave frequency was 13392 Hz at a driving speed of 15340 r/min, corresponding to 20599.4 r/min at the driven gear. The predicted value was 20766.2 r/min, a relative error of 0.79%. These errors support the use of the dynamic-frequency-based Campbell prediction.
At the resonance frequency, the measured stress amplitudes were 66.3, 67.0, 66.6, and 66.4 MPa at gauges 1–4 for the third-nodal-diameter forward wave, and 131.5, 130.4, 130.8, and 131.0 MPa for the fourth-nodal-diameter backward wave. The corresponding simulation values at the same circumferential positions were 57.5–58.0 MPa and 92.4–93.0 MPa. The test/simulation ratios were 1.14–1.17 and 1.40–1.42. The circumferential distribution was consistent between simulation and test, while the absolute amplitude was lower in the simulation, especially for the fourth-nodal-diameter backward-wave case.
| Case | Simulation (MPa) | Test (MPa) | Test/simulation ratio |
|---|---|---|---|
| Third ND forward wave | 57.5–58.0 | 66.3–67.0 | 1.14–1.17 |
| Fourth ND backward wave | 92.4–93.0 | 130.4–131.5 | 1.40–1.42 |
Fatigue fracture testing provided a further check on the stress-field pattern. A gear with a coarse-grain prefabricated defect at the root groove fillet was run at the third-nodal-diameter forward-wave resonance. After 11 min 57 s, the resonance peak disappeared; on restart, a loud impact was heard near the resonance speed, and the gear was found to have fractured. The block fracture contained a small block with eight full teeth and two half teeth. The main crack started at the prefabricated defect at the small-end root groove fillet center, propagated along the tooth groove toward the large end, and also propagated radially through the rim/web junction toward the shaft root before turning tangentially and circumferentially. The simulated high-stress sector covered 7–8 teeth, and its boundary direction at the rim was consistent with the initial crack direction.
In the fourth-nodal-diameter backward-wave test, the gear ran near resonance for 282 s before a loud noise and emergency shutdown. The driven gear had fractured into two large blocks, with the small block containing 12 full teeth. The simulated two adjacent high-stress sectors covered 12–13 teeth, again consistent with the fracture block size and the crack-initiation region. These comparisons support the model’s ability to reproduce the resonance stress pattern, although the absolute stress amplitude still requires test calibration.
Design Implications
The first implication is that static frequency is not sufficient for a thin-web spiral bevel gear running near a nodal-diameter resonance. The dynamic-frequency shift here was modest, 0.72–1.07%, but it moved the predicted resonance speed by enough to matter when the resonance lies close to an operating point. A rotating-frame Campbell diagram based on the dynamic frequency gives a more reliable screening tool than a static modal result alone.
The second implication is that the fourth-nodal-diameter backward wave deserves particular attention near maximum design speed. It produced lower axial displacement amplitude than the third-nodal-diameter forward wave, but higher root-groove stress: 358.3 MPa spatial maximum and 223.9 MPa RMS at the root groove, compared with 260.7 MPa and 164.2 MPa for the third-nodal-diameter forward wave. The root groove fillet is therefore the critical location for crack initiation, and the stress field is not uniform around the circumference; it is organized into petals whose count follows the nodal-diameter number.
The third implication concerns validation strategy. The measured and simulated stress distributions agreed in circumferential pattern and order of magnitude, but the simulation underestimated the absolute amplitude, with test/simulation ratios of 1.14–1.17 and 1.40–1.42. The model is therefore well suited to mode identification, resonance-speed prediction, and relative ranking of load cases. Absolute fatigue margins should be calibrated against strain-gauge data or full-annulus fatigue tests, especially when the design point lies close to a high-nodal-diameter backward-wave resonance.