A thin web reduces the mass and inertia of an aeroengine central-transmission gear, but it also lowers the gear body’s bending stiffness. The resulting structure behaves much like a toothed disc: its nodal-diameter modes can be excited by the axial component of the meshing force, and at high rotational speed a single forward or backward wave can be selectively amplified. This is traveling-wave resonance. In a thin-walled spiral bevel gear, the consequence is not merely a local vibration peak. The wave sweeps around the circumference, repeatedly loading the tooth root and the meshing region, and can drive high-cycle tooth fracture.
The case considered here is a central-transmission pair with 47 driving teeth and 35 driven teeth. The driven gear is the thin-walled component of interest. Its operating load is 326 kW, and the three-nodal-diameter forward traveling wave is the critical mode. At resonance the driven gear reaches about \(15\,000\ \mathrm{r/min}\), the mesh frequency is about \(8.8\ \mathrm{kHz}\), and the three-nodal-diameter dynamic frequency is about \(8.1\ \mathrm{kHz}\). These closely spaced frequencies make the resonance condition sensitive to speed, tooth contact, and structural stiffness.
Resonance condition for a rotating gear
For a gear vibrating with \(m\) nodal diameters, the gear-fixed frame contains two degenerate waves of the same shape: a forward wave and a backward wave. Their wavelengths are \(2\pi/m\), their angular phase speeds are \(\pm 2\pi f_d/m\), and their frequencies are both equal to the gear dynamic frequency \(f_d\). In a stationary frame, rotation separates the two waves. The meshing force supplies harmonics at the tooth-passing frequency and its multiples; the \(m\)-lobed pattern is also seen as a rotating modulation at \(m\) times the shaft frequency. The excitation frequency associated with the \(m\)-nodal-diameter wave is therefore
$$f_j=\frac{kzn}{60}\mp\frac{mn}{60},\qquad j=\mathrm{ef},\mathrm{eb}$$
where \(z\) is the tooth number, \(n\) is the gear speed in \(\mathrm{r/min}\), \(k\) is the mesh-force harmonic order, and \(m\) is the nodal-diameter order. The minus sign corresponds to the forward wave and the plus sign to the backward wave. Resonance requires the excitation frequency to equal the gear dynamic frequency, \(f_j=f_d\), which gives the forward and backward resonance speeds
$$N_f=\frac{60f_d}{kz-m},\qquad N_b=\frac{60f_d}{kz+m}$$
For the 35-tooth driven gear, the three-nodal-diameter forward wave uses \(m=3\), \(z=35\), and \(k=1\). The denominator is \(35-3=32\), so the forward-wave speed is higher than the backward-wave speed for the same dynamic frequency. The wave pattern also migrates around the gear at an angular speed
$$\omega_w=\frac{z\mp m}{m}\omega_g$$
where \(\omega_g\) is the gear rotational speed. For the three-nodal-diameter forward wave, \(\omega_w/\omega_g=(35-3)/3=10.67\). This ratio is a phase speed of the wave pattern, not a material velocity: the stress and displacement lobes sweep around the circumference much faster than the gear rotates.
Finite element model and modal calibration
The gear geometry was generated from the design tooth parameters, the actual cutter geometry, and the tooth-surface modification dimensions. The mesh used C3D8R hexahedral elements. The tooth contact surface was refined to \(0.5\ \mathrm{mm}\times0.5\ \mathrm{mm}\), the root groove and the groove-to-rim junction to \(0.35\ \mathrm{mm}\times0.35\ \mathrm{mm}\), and the remaining regions to \(0.5\)–\(1.5\ \mathrm{mm}\). The driving gear model contained 650,668 elements and the driven gear 586,600 elements, with zero warning elements. The material is an aviation carburized gear steel; the modal analysis used 20 °C properties and the transient contact analysis used 200 °C properties, corresponding to the approximate surface temperature in service.
| Temperature (°C) | Density (kg/m³) | Elastic modulus (GPa) | Poisson’s ratio |
|---|---|---|---|
| 20 | 7860 | 215 | 0.38 |
| 200 | 7860 | 210 | 0.41 |
Free-boundary modal analysis showed the expected disc-like nodal-diameter behavior. The nodal lines are radial, adjacent sectors move in opposite axial directions, and the axial displacement is largest near the outer rim. A free-boundary modal test used elastic-cord suspension, impact excitation, and multipoint excitation with single-point response. The comparison below validates the finite element stiffness and mass distribution before the rotating, loaded condition is introduced.
| Mode | Simulation (Hz) | Modal test (Hz) | Relative error (%) |
|---|---|---|---|
| Three nodal diameters | 8119.9 | 7964 | 1.96 |
| Four nodal diameters | 13554 | 13420 | 1.00 |
For the operating boundary condition, the driven gear was constrained in five directions with the axial rotation released. The three-nodal-diameter constrained modal frequency was 8042.5 Hz. Using this frequency alone gave a forward-wave speed of \(15\,079.7\ \mathrm{r/min}\). That estimate neglects the stiffening and loading effects of rotation, torque, and tooth contact.
Dynamic frequency and predicted resonance point
The dynamic frequency calculation included centrifugal loading at speed, the operating torque, and the mesh stiffness. The three-nodal-diameter forward-wave resonance torque was \(206.5\ \mathrm{N\cdot m}\), corresponding to an equivalent tooth-surface pressure of \(202.4\ \mathrm{N/mm^2}\) at the 326 kW operating load. The average time-varying mesh stiffness was set to \(5\times10^8\ \mathrm{N/m}\). With these effects, the three-nodal-diameter dynamic frequency became 8100.5 Hz. The predicted forward-wave resonance speed was therefore
$$N_f=\frac{60(8100.5)}{35-3}=15\,188.4\ \mathrm{r/min}$$
for the driven gear. Through the 47/35 tooth ratio, the corresponding driving-gear speed is \(11\,310.5\ \mathrm{r/min}\). On the Campbell diagram, the horizontal dynamic-frequency line intersects the forward engine-order line \(f_{\mathrm{ef}}=(z-m)n/60\) at this speed; the mesh line \(f_m=zn/60\) lies above it, and the backward line \(f_{\mathrm{eb}}=(z+m)n/60\) lies higher still. The separation is consistent with a forward wave being excited before the backward wave in the operating speed range.
Transient contact response at the three-nodal-diameter forward-wave resonance
The meshing problem was solved with an explicit central-difference contact formulation. The explicit method updates nodal acceleration from the dynamic equilibrium equation
$$\mathbf{M}\ddot{\mathbf{u}}(t)=\mathbf{P}(t)-\mathbf{I}(t)$$
and advances velocity and displacement without the iterative convergence step required by implicit contact solutions. The driving tooth surface was defined as the master surface and the driven tooth surface as the slave surface. Both gears were released in axial rotation and constrained in the other five directions. The driving gear was given the speed corresponding to the predicted resonance point, and the driven gear was loaded with \(206.5\ \mathrm{N\cdot m}\). The total simulated time was \(0.08\ \mathrm{s}\), with stress and displacement output at fixed intervals.
The axial displacement field at the three-nodal-diameter forward-wave resonance has three peaks and three valleys around the circumference, giving six alternating extrema. Adjacent sectors separated by a nodal line move in opposite axial directions. Successive snapshots 20 μs apart show the entire three-lobed pattern migrating in the same direction as gear rotation. The wave period is about \(123\ \mu\mathrm{s}\), and the pattern moves by \(2\pi/3\) per period, consistent with \(m=3\). The phase speed is 10.67 times the rotor speed, so a fixed point on the gear experiences a rapid succession of wave passages during each revolution.
At the extraction node near the root groove, the axial displacement fluctuates between \(-35.9\ \mu\mathrm{m}\) and \(30.9\ \mu\mathrm{m}\), with an effective value of \(18.0\ \mu\mathrm{m}\). The spectrum is dominated by a component at \(f_m-3f_z=8100\ \mathrm{Hz}\), with an amplitude of \(22.3\ \mu\mathrm{m}\), essentially equal to the predicted dynamic frequency of 8100.5 Hz. Shaft-order components and their multiples are also present. The time history is densely populated with high-order oscillation, and the response increases as the wave passes the meshing region. This is the signature of a resonant traveling wave rather than a simple once-per-revolution load cycle.
The stress field is three-lobed, with the number of petals equal to the nodal-diameter order. The von Mises stress reaches 670.8 MPa on the meshing tooth surface and 260.7 MPa at the root groove. The root groove and the meshing region are therefore the critical locations: the former is a likely fatigue-crack initiation site, while the latter carries the local contact stress. Elevated stress also appears near the shaft bore because the torque is applied at that boundary; by Saint-Venant’s principle this local boundary effect decays away from the bore and does not govern the tooth-root field.
At the same root-groove node, the von Mises stress ranges from 15.3 MPa to 188.9 MPa, with an effective value of 102.8 MPa. Its spectrum peaks at the three-nodal-diameter dynamic frequency, and the peak is surrounded by sidebands spaced at the shaft frequency and its harmonics, up to about the sixth shaft order. These sidebands arise because the rotating stress lobes pass through the meshing region once per revolution; the contact condition modulates the resonant wave. The central frequency identifies the nodal-diameter mode, while the sideband spacing identifies the shaft order. That combination is useful for separating traveling-wave resonance from other high-frequency excitation.
Experimental validation
Dynamic stress was measured on a central-transmission component tester with a motor, gearbox, air cooling, and a high-speed slip-ring system. The strain gauges were BA120-1AA medium-temperature gauges with a sensitivity of \(2.16\pm0.01\) and an operating range of \(-30\) to \(210\ \mathrm{^\circ C}\). Because the root fillet is too small for direct bonding, the gauges were placed about 2 mm from the root-groove bottom at the small end of the gear. Four circumferential positions were used: positions 1 and 2 were separated by eight teeth, and the remaining intervals were nine teeth. Leads were routed through the hollow gear shaft to the slip ring, then to a strain amplifier and data-acquisition system. The driving-gear speed was swept slowly from 72% to 78% of physical speed to cross the resonance.
At the three-nodal-diameter forward-wave resonance, the measured dynamic frequency was 8012 Hz. The driving-gear speed was \(11\,190\ \mathrm{r/min}\), which gives \(15\,026.6\ \mathrm{r/min}\) at the driven gear through the 47/35 ratio. The four measured stress amplitudes at the resonance frequency were 66.3, 67.0, 66.6, and 66.4 MPa, respectively. Their near equality is consistent with a traveling wave: although the instantaneous stress field is three-lobed, the wave sweeps past every circumferential position, so the resonance-frequency amplitude at a fixed radius is nearly uniform around the gear.
| Quantity | Prediction | Test | Difference |
|---|---|---|---|
| Three-nodal-diameter forward-wave frequency | 8100.5 Hz | 8012 Hz | 1.1% |
| Driven-gear resonance speed | 15,188.4 r/min | 15,026.6 r/min | 1.1% |
| Driving-gear speed at resonance | 11,310.5 r/min | 11,190 r/min | 1.1% |
| Resonance stress amplitude at the gauge-like circumferential positions | 57.5–57.9 MPa | 66.3–67.0 MPa | About 13% lower in simulation |
The resonance-speed prediction is therefore supported to about 1.1% in both frequency and speed. The stress comparison is less exact: the simulated amplitudes at 35 circumferential nodes are 57.5–57.9 MPa, while the measured amplitudes are 66.3–67.0 MPa. The two sets are of the same order and show the same circumferential uniformity, but the simulation is about 13% lower on average. Part of this difference is attributable to the comparison itself: the strain gauges were located about 2 mm from the root-groove bottom, whereas the local peak stress at the groove is much higher, 260.7 MPa. The model is consequently stronger evidence for resonance-speed prediction and for the spatial form of the traveling-wave field than for absolute stress amplitude at a specific gauge location.
Implications for design and monitoring
The critical design margin for this thin-walled spiral bevel gear is not set by the static torque alone. It is set by the intersection of the three-nodal-diameter dynamic-frequency line with a forward engine-order line in the operating speed range. Including rotation, torque, and mesh stiffness shifts the predicted resonance from the constrained-frequency estimate of \(15\,079.7\ \mathrm{r/min}\) to \(15\,188.4\ \mathrm{r/min}\). The shift is small in percentage terms, but in a narrow high-speed resonance window it is large enough to matter for speed avoidance and test planning.
The stress field explains why the root groove is the fatigue-critical location. The instantaneous field is three-lobed and rotates with the wave, producing a concentrated root stress of 260.7 MPa even though the gauge location 2 mm away records about 66 MPa at the resonance frequency. The meshing tooth surface carries a higher local stress of 670.8 MPa, but it is the root groove that concentrates the cyclic bending stress associated with the nodal-diameter mode. Design changes that detune the three-nodal-diameter frequency, increase root-groove fatigue resistance, or add damping at the rim must therefore be assessed against the same rotating, loaded boundary condition; a free-free modal shift alone is not sufficient.
For monitoring, the most distinctive feature is the combination of a central peak at the three-nodal-diameter dynamic frequency and sidebands spaced by the shaft frequency and its harmonics. A fixed strain gauge or casing sensor does not see a stationary three-lobed pattern; it sees the wave sweep past the mesh once per revolution. That produces the characteristic modulation. In the measured case, the four circumferential gauges gave nearly equal resonance amplitudes and the same 8012 Hz peak, while the simulation reproduced the uniform circumferential distribution and the correct order of magnitude. The remaining absolute stress difference indicates that the model should be used with a calibration margin when absolute fatigue stress, rather than resonance speed or mode shape, is the decision variable.