Why web thickness governs resonance risk in a central drive
The central drive of an aero-engine extracts power from the high-pressure spool and delivers it to the accessory gearbox through a spiral bevel gear pair that runs at pitch-line speeds far above those of industrial gearing. Weight targets push the driven member toward a thin annular web: in the configuration examined here the driven spiral bevel gear carries 35 teeth and meshes with a 47-tooth pinion at a contact ratio of 1.41, with a web only 8 mm thick. Removing material from the web lowers the bending stiffness of the disk that carries the teeth, and the nodal-diameter family of modes — in which the rim oscillates as $m$ full waves around the circumference — falls into the 8–13.5 kHz band swept by the mesh harmonic family between idle and maximum speed.
Once a nodal-diameter mode is excited by the mesh, the rim does not vibrate as a pattern fixed in space. The mode decomposes into two counter-rotating waves of equal amplitude, one traveling with the rotation and one against it. Because the mesh excitation is a rotating force field of order $Z$ (the tooth count) rather than a stationary one, the two waves are driven at different excitation orders, $Z-m$ and $Z+m$, and therefore resonate at two different shaft speeds. A single mode can thus appear twice inside the operating envelope, and a gear designed against one intersection of the Campbell diagram can still be caught by the other.
The consequence is well documented in service: failure investigations of an aero-engine central transmission attributed a driven spiral bevel gear fracture to fourth-nodal-diameter backward-wave resonance combined with poor meshing. Such fractures are abrupt. A crack initiates at the root-groove fillet near the small end, propagates axially along the tooth space and radially outward through the web, and terminates in a block-type separation in which a sector of several teeth leaves the wheel. Designing against this failure requires two things that classical practice does not supply: a resonance-speed prediction made with the gear under its real centrifugal, torque and mesh-stiffness preload, and a transient solution that retains the meshing contact so that the dynamic stress field at resonance can be read directly and compared with measurement.
Analytical treatments of disk-like gear vibration, ring dampers for thin-walled gears, and modal identification of forward and backward waves on rotating disks are all established. What has been missing is a transient contact model of an actual thin-web spiral bevel gear pair driven at its own resonance speed, with the resulting displacement and stress fields resolved in space and frequency and checked against strain-gauge data and against the fracture pattern of a test gear.
Kinematics and resonance conditions of nodal-diameter waves
Macroscopically the bevel gear behaves as a disk, and its modes fall into nodal-circle, nodal-diameter, and combined families. The nodal-circle and combined modes of an aviation gear sit at frequencies too high to be reached by the mesh, so the nodal-diameter family dominates the failure behaviour. For a point at radius $r$ and circumferential angle $\theta$, the axial displacement of an $m$-diameter mode is
$$z(r,\theta,t)=A(r)\cos(m\theta)\cos(\omega t)$$
where $A(r)$ is the radial deflection function (mm), $\omega$ the angular frequency of the nodal-diameter mode (rad/s) and $t$ time (s). Nodal lines lie at $\theta=(2j+1)\pi/(2m)$ and antinodes at $\theta=j\pi/m$. Expanding the standing pattern into two traveling components,
$$z=\tfrac{1}{2}A(r)\left[\cos(m\theta-\omega t)+\cos(m\theta+\omega t)\right]$$
gives a forward wave that travels in the direction of increasing $\theta$, and a backward wave that travels against it. Both carry $m$ waves around the circumference and differ only in propagation direction. In one wave period the pattern advances by $2\pi/m$, so the wave angular speed $\omega_w$ is related to the shaft angular speed $\omega_g$ by
$$\omega_w=\frac{2\pi f_d}{m}=\frac{Z\mp m}{m}\,\omega_g$$
with the minus sign for the forward wave and the plus sign for the backward wave, $Z$ being the tooth count. For the 35-tooth driven gear this gives ratios of $(35-3)/3=10.67$ for the third-diameter forward wave and $(35+4)/4=9.75$ for the fourth-diameter backward wave: the wave pattern sweeps around the rim roughly ten times faster than the gear itself turns.
The excitation is the mesh force, whose fundamental frequency is
$$f_m=\frac{ZN}{60}$$
measured in hertz for a shaft speed $N$ in r/min, together with its harmonics $kf_m$. Expressed in the frame rotating with the gear, the components that drive the forward and backward waves are
$$f_e=\frac{kZN}{60}-\frac{mN}{60},\qquad f_{eb}=\frac{kZN_b}{60}+\frac{mN_b}{60}$$
and resonance occurs when these coincide with the nodal-diameter natural frequency, $f_e=f_{eb}=f_d$. Solving for shaft speed gives the two resonance conditions used throughout the analysis:
$$N_f=\frac{60f_d}{kZ-m},\qquad N_b=\frac{60f_d}{kZ+m}$$
With $k=1$, $Z=35$ and $m=3$, a dynamic frequency of 8100.6 Hz places the forward-wave resonance at 15188.6 r/min; with $m=4$, a dynamic frequency of 13498 Hz places the backward-wave resonance at 20766.2 r/min. Plotting the order lines $(Z-m)N/60$ and $(Z+m)N/60$ against shaft speed and intersecting them with the horizontal line at $f_d$ produces the rotating-frame Campbell diagram used for screening. Higher harmonics ($k\geq2$) generate additional intersections, but in this gear the first-order crossings are the ones that fall inside the operating range, the third-diameter forward wave near idle and the fourth-diameter backward wave near maximum design speed.
Finite element model and modal calibration
Solid models of the pinion and the thin-web driven gear were meshed with C3D8R hexahedral elements in HyperMesh, using a two-dimensional mapped mesh extruded into three dimensions so that the tooth flanks and the root-groove fillets retained structured element rows. Both gears are made of the carburised aviation gear steel 16Cr3NiWMoVNbE; the temperature-dependent properties used in the analysis are listed in Table 1, with the 200 °C set adopted for all working-condition calculations because the oil-wetted gear surfaces reach that temperature in the central drive.
| Temperature (°C) | Density (kg/m³) | Elastic modulus (Pa) | Poisson’s ratio |
|---|---|---|---|
| 20 | 7860 | 2.15 × 1011 | 0.38 |
| 200 | 7860 | 2.10 × 1011 | 0.41 |
Mesh independence was established over seven models spanning roughly 50 000 to 700 000 elements. Above 347 600 elements the third- and fourth-diameter natural frequencies changed by less than 0.1 %; the production meshes contained 650 668 elements for the pinion and 586 600 for the driven gear, with no warning elements.
Free–free modal testing provided the calibration. Five driven gears from the same production batch were suspended on elastic cords to approximate free boundaries and excited by hammer impact in a multi-point excitation, single-point response scheme. The comparison in Table 2 shows the finite element model reproducing every measured mode: the nodal-circle frequency within 5 %, and the first to fourth nodal-diameter frequencies within 3 %, with the third-diameter error below 2.68 % and the fourth below 1.83 %.
| Mode | Simulation (Hz) | Test range (Hz) | Relative error (%) |
|---|---|---|---|
| 1st nodal diameter | 3437.7 | 3352–3368 | 2.07–2.56 |
| 2nd nodal diameter | 3611.5 | 3656–3664 | 1.22–1.43 |
| 1st nodal circle | 4780.6 | 4556–4560 | 4.84–4.93 |
| 3rd nodal diameter | 8119.9 | 7908–7964 | 1.96–2.68 |
| 4th nodal diameter | 13554 | 13310–13420 | 1.00–1.83 |
Imposing the working boundary condition — axial rotation released, the remaining five degrees of freedom constrained — and switching to the 200 °C property set gives the constrained frequencies that serve as the starting point for the resonance prediction: 8042.5 Hz for the third diameter and 13355 Hz for the fourth.
Dynamic frequency and predicted resonance speeds
Using the static, constrained frequency as the resonance frequency is a convenience that costs accuracy. Three effects shift the frequency of a spinning, loaded gear: centrifugal stiffening of the web, the mean stress field produced by the transmitted torque, and the stiffening introduced by the meshing contact. The mesh stiffness was obtained by dividing the normal contact force by the total elastic deformation of the tooth pair, evaluated separately for single-tooth and double-tooth contact, and applied as an average time-varying stiffness of $5\times10^{8}$ N/m distributed over the contact area. Centrifugal load was applied as a rotational body load at the corresponding resonance speed, and torque was applied as a concentrated normal load on the contact region, derived from the 326 kW working load of the gear: 206.5 N·m (202.4 N/mm² equivalent flank pressure) at the third-diameter forward-wave speed of 15079.7 r/min, and 151.5 N·m (148.5 N/mm²) at the fourth-diameter backward-wave speed of 20546.2 r/min.
| Preload case | 3rd diameter $f_d$ (Hz) | Shift (%) | 4th diameter $f_d$ (Hz) | Shift (%) |
|---|---|---|---|---|
| None (constrained static) | 8042.5 | — | 13355 | — |
| $T=206.5$ N·m, $N_1=15079.7$ r/min, $k_m=5\times10^{8}$ N/m | 8100.6 | +0.72 | 13490.8 | +1.02 |
| $T=151.5$ N·m, $N_1=20546.2$ r/min, $k_m=5\times10^{8}$ N/m | 8105.5 | +0.78 | 13498 | +1.07 |
The shifts are small in percentage terms but not in speed terms. Recomputing the resonance conditions with the static values of 8042.5 and 13355 Hz gives 15079.7 and 20546.2 r/min; using the preloaded values of 8100.6 and 13498 Hz gives 15188.6 and 20766.2 r/min. The difference, about 110 r/min for the third diameter and 220 r/min for the fourth, is comparable to the sweep rate of a slow resonance search and therefore matters when a prediction is to be compared with a measured crossing. Converting through the 47/35 ratio gives pinion speeds of 11310.5 and 15464.2 r/min for the two resonant conditions imposed in the transient simulation below.
Transient explicit solution: the displacement field at resonance
The transient problem was solved with an explicit central-difference scheme, which avoids the factorisation of a stiffness matrix at every step and handles the moving contact between the pinion’s convex flank and the gear’s concave flank as a contact pair. The semi-discrete system at time $t$,
$$M\ddot{\mathbf{x}}_t+C\dot{\mathbf{x}}_t+K\mathbf{x}_t=\mathbf{Q}_t$$
is integrated with the classical central-difference recurrence
$$\left(\frac{1}{\Delta t^{2}}M+\frac{1}{2\Delta t}C\right)\mathbf{x}_{t+\Delta t}=\mathbf{Q}_t-\left(K-\frac{2}{\Delta t^{2}}M\right)\mathbf{x}_t-\left(\frac{1}{\Delta t^{2}}M-\frac{1}{2\Delta t}C\right)\mathbf{x}_{t-\Delta t}$$
so that $\mathbf{x}_{t+\Delta t}$ follows directly from the two previous configurations. Rotation was released on both gears and the remaining five degrees of freedom constrained; the pinion carried the rotational speed and the driven gear the load torque. Each case was run for 0.08 s, long enough for the resonant response to settle into a clear periodic state, and field output was written at an interval satisfying the Nyquist criterion with respect to the highest mesh frequency of interest (8.86 kHz at the third-diameter resonance, 12.11 kHz at the fourth).
Contour plots of the axial displacement, magnified 200 times for legibility, reproduce the modal shapes exactly. At the third-diameter forward-wave resonance the displacement field forms three peaks alternating with three troughs, six extrema dividing the wheel into six sectors, with adjacent sectors moving in antiphase. At the fourth-diameter backward-wave resonance the same construction gives four peaks and four troughs in eight sectors. In both cases the pattern migrates: tracking a reference crest through frames spaced 20 μs apart — a fraction of the 123 μs wave period at 8100 Hz — shows the sector pattern rotating continuously, and the direction of rotation identifies the wave type. The forward wave advances in the same sense as the gear rotation; the backward wave advances against it. The measured advance per period confirms the kinematic relation, the pattern moving $2\pi/3$ and $\pi/2$ per period for $m=3$ and $m=4$ respectively.
Two features of the migrating field are relevant to design. First, when a crest passes through the meshing zone the local axial displacement rises sharply, so the waveform extracted along a circumference is a sinusoid that steps upward each time the mesh is traversed — the mesh acts as a once-per-revolution modulation superimposed on the traveling wave. Second, the ratio of wave speed to shaft speed, 10.67 and 9.75 for the two cases, means that any point on the rim sees the full wave pattern pass many times per revolution, which is precisely why the dynamic stress at a fixed fillet accumulates rapidly.
Unwrapping the axial displacement along a circumference 2 mm above the root groove (630 nodes) and at the root-groove centres (35 nodes) gives the quantitative picture in Table 4. The root-groove nodes consistently show larger excursions and higher RMS values than the nodes slightly above the groove, and the fourth-diameter backward-wave case shows the larger RMS stress despite the lower displacement amplitude at the resonance frequency.
| Condition | Node set | Axial displacement range (μm) | RMS displacement (μm) | Stress range (MPa) | RMS stress (MPa) |
|---|---|---|---|---|---|
| 3rd diameter, forward wave, 15188.6 r/min | 2 mm above root groove | −34.4 to 31.6 | 16.8 | 15.3–188.9 | 102.8 |
| 3rd diameter, forward wave, 15188.6 r/min | Root groove centre | −36.1 to 34.3 | 18.0 | 16.7–315.6 | 164.2 |
| 4th diameter, backward wave, 20765.4 r/min | 2 mm above root groove | −29.9 to 4.2 | 17.2 | 3.9–226.7 | 133.9 |
| 4th diameter, backward wave, 20765.4 r/min | Root groove centre | −33.0 to 4.6 | 18.2 | 4.3–386.5 | 223.9 |
| Off-resonance, 17457 r/min | 2 mm above root groove | −16.4 to 9.4 | 11.0 | 38.4–141.2 | 73.5 |
The frequency content identifies the excitation mechanism unambiguously. Under the third-diameter forward-wave condition the dominant peak sits at $f_m-3f_z\approx8100$ Hz, matching the computed dynamic frequency of 8100.6 Hz, with side peaks at the shaft frequency $f_z$ and its multiples $2f_z$ and $3f_z$. Under the fourth-diameter backward-wave condition the dominant peak sits at $f_m+4f_z\approx13500$ Hz, matching 13498 Hz. The time histories are dense with high-order harmonic content and show a clear low-frequency envelope repeating at the shaft period, the signature of the once-per-revolution mesh passage. Off resonance at 17457 r/min the same node responds predominantly at shaft-order frequencies with a much smaller excursion, confirming that the high-frequency content is a resonance phenomenon and not a meshing artefact.
Stress field: petal structure and root-groove concentration
The stress field inherits the symmetry of the displacement field but not its smoothness. In the circumferential direction the von Mises stress forms a petal pattern whose petal count equals the nodal-diameter number: three petals under the third-diameter forward wave, four under the fourth-diameter backward wave. The petals are separated by nodal lines of low stress, and each petal contains a high-stress lobe that spans several teeth.
Peak values occur at two distinct sites. On the meshing flank the maximum reaches 670.8 MPa for the third-diameter case and 655.9 MPa for the fourth, values governed by the contact load rather than by the resonance itself. At the root-groove fillet the maximum reaches 260.7 MPa for the third-diameter case and 358.3 MPa for the fourth, a 37 % increase that reverses the ranking suggested by the displacement field: the higher-order backward wave, despite its lower displacement amplitude at the resonance frequency, produces the more severe fillet stress because its petal pattern concentrates deformation closer to the rim. Elevated stress also appears near the bore where the torque is applied, an artefact of the boundary condition that decays with distance according to Saint-Venant’s principle and does not affect the web or the teeth.
Spectral analysis of the stress at the two node sets confirms where the resonance energy is deposited. At the third-diameter resonance the amplitude at $f_m-3f_z$ is 57.9 MPa 2 mm above the groove and 106.9 MPa at the groove centre; at the fourth-diameter resonance the amplitude at $f_m+4f_z$ is 92.5 MPa and 175.7 MPa respectively. Both the fundamental and its harmonics $2(f_m\mp mf_z)$ and $3(f_m\mp mf_z)$ appear, together with shaft-order components up to $6f_z$, giving the characteristic dense high-order spectrum of a resonant thin disk driven through a meshing contact.
Strain-gauge measurements and fatigue fracture evidence
Validation used a central-drive component test rig with the driven gear instrumented by resistance strain gauges. Because the root fillet is too small to host a gauge, four gauges were bonded 2 mm above the root groove at the small end, spaced 8 and 9 teeth apart around the circumference. The gauges were of the BA120-1AA type, 120.2 ± 0.1 Ω, gauge factor 2.16 ± 1 %, range ±2000 με, rated from −30 °C to +210 °C, with a gauge-induced uncertainty of about 3 %. Leads were routed through the hollow gear shaft to a high-speed slip-ring unit, air-cooled to remove the heat generated at speed, and thence to a strain amplifier, data acquisition system and computer. All channels were calibrated before each run.
The speed schedule followed the converted operating range: from 72 % of design speed (idle) slowly to 78 %, up to 106 %, back to 104 %, then trimmed to 106 %. Over this schedule the driven gear exhibited two distinct resonances. At the third-diameter forward-wave resonance the measured dynamic frequency was 8012 Hz at a pinion speed of 11190 r/min, corresponding to 15026.6 r/min at the driven gear, 1.09 % below the predicted 15188.6 r/min. At the fourth-diameter backward-wave resonance the measured frequency was 13392 Hz at a pinion speed of 15340 r/min, or 20599.4 r/min at the driven gear, 0.79 % below the predicted 20766.2 r/min. The four gauges recorded stress amplitudes at the resonance frequency of 66.3, 67.0, 66.6 and 66.4 MPa for the third-diameter case and 131.5, 130.4, 130.8 and 131.0 MPa for the fourth, confirming the circumferential uniformity expected of a traveling wave whose amplitude is set by the mode shape rather than by tooth position.
| Quantity | 3rd diameter, forward wave | 4th diameter, backward wave |
|---|---|---|
| Simulated amplitude (MPa) | 57.5–58.0 | 92.4–93.0 |
| Measured amplitude (MPa) | 66.3–67.0 | 130.4–131.5 |
| Test/simulation ratio | 1.14–1.17 | 1.40–1.42 |
| Measured resonant frequency (Hz) | 8012 | 13392 |
| Error in predicted resonance speed (%) | 1.09 | 0.79 |
The same 35 circumferential locations were sampled in the simulation, and the simulated amplitudes vary by less than 1 % around the circumference in both cases, matching the flat distribution measured by the four gauges. The absolute values are lower than measurement by a factor of 1.14–1.17 for the third-diameter case and 1.40–1.42 for the fourth. Part of that gap is expected from the damping and contact-compliance assumptions of an explicit contact model; the practical reading is that the model is conservative in the wrong direction for qualification purposes and should be calibrated by such a factor when used to set allowable stress.
Two fatigue tests under sustained resonance completed the validation, both on gears carrying a prefabricated coarse-grain defect at the root-groove fillet of the small end. Under the third-diameter forward-wave condition the resonance peak disappeared after 11 min 57 s of dwell; on restart the gearbox emitted a heavy impact noise and inspection revealed a block-type fracture into two large pieces, the smaller carrying eight complete teeth and two half teeth. The main crack started at the prefabricated defect in the fillet centre, then followed two paths: axially from the small-end groove centre toward the large end, and radially from the rim through the rim-web junction, the web, the shaft fillet and then tangentially around the disk to the lightening holes and back to the root groove. The simulated high-stress sectors each covered seven to eight teeth, and the boundary line between sectors at the rim pointed in the direction of the initial crack propagation.
Under the fourth-diameter backward-wave condition the gear failed after only 282 s of dwell, breaking into two pieces with twelve complete teeth on the smaller one. The simulated field for this case gives two adjacent high-stress sectors covering twelve to thirteen teeth, again with the sector boundary aligned to the observed crack path. The 25-fold difference in time to failure between the two resonances is consistent with the 37 % higher fillet stress computed for the fourth-diameter case and with its 1.36-fold higher RMS stress.
Design consequences
Three conclusions follow for the design of thin-web spiral bevel gears in high-speed drives. Resonance screening must use the preloaded dynamic frequency rather than the constrained static value; the centrifugal, torque and mesh-stiffness contributions shift the frequency by 0.7–1.1 %, which translates into 110–220 r/min of shaft speed and is enough to move a predicted crossing away from the speed at which a test actually resonates.
The dangerous resonance is not necessarily the one with the largest displacement. The fourth-diameter backward wave produced lower axial displacement at the resonance frequency than the third-diameter forward wave but higher fillet stress, higher RMS stress, and a fatigue life two orders of magnitude shorter. Since the backward-wave resonance of order $m$ occurs at $60f_d/(Z+m)$ and the forward wave at $60f_d/(Z-m)$, the backward branch is pushed toward the high-speed end of the envelope where centrifugal preload is greatest, and both branches must be screened.
Finally, the root-groove fillet remains the controlling location. The stress field concentrates there under every resonant condition examined, the measured and simulated petal patterns agree in shape and order of magnitude, and the fracture surfaces of both test gears initiated at a fillet defect and followed the sector boundaries predicted by the simulation. Surface integrity at the fillet, and damping treatments such as ring dampers that raise the effective damping of the nodal-diameter modes, therefore act on the same failure path that the traveling wave exploits.