Spiral bevel gears are used in aviation drivetrains where shafts intersect and the layout must remain compact. Their curved teeth engage gradually, which spreads contact and supports high torque at low noise. The same geometry, however, can concentrate alternating deformation in the web when the gear runs at high speed. A nodal-diameter mode has radial nodal lines; as the gear rotates, the mode splits into forward and backward traveling waves in the fixed frame. If a wave branch crosses a mesh-order line, the mesh excitation can feed energy into that nodal pattern. The result is not immediate tooth breakage but a cycle of web bending, flank load fluctuation, wear, and possible fatigue cracking. The design question is therefore quantitative: which nodal-diameter families lie in the excitation band, at what speeds do their forward or backward branches intersect the dominant order, and how far is the rated speed from those intersections?
Example gear pair and finite element model
An aviation spiral bevel gear pair with a 27/74 tooth ratio is used as the example. The pinion is left-hand and the driven gear is right-hand; the module is 3.85 mm, the face width is 38.5 mm, and the spiral angle is 35°. Both members are made of 20CrMnTi steel.
| Parameter | Value |
|---|---|
| Tooth ratio, pinion/driven | 27/74 |
| Module | 3.85 mm |
| Face width | 38.5 mm |
| Spiral angle | 35° |
| Hand | Pinion left-hand; driven gear right-hand |
| Material | 20CrMnTi steel |
| Elastic modulus | 207 GPa |
| Poisson’s ratio | 0.25 |
| Density | 7850 kg/m3 |
| Tensile strength | 1080 MPa |
| Yield strength | 885 MPa |
The driven gear carries the modal model. Because the two members are coupled through the mesh, the driven-gear response under external excitation is a practical proxy for the pair’s traveling-wave behavior; the analysis therefore concentrates on that member. The three-dimensional gear is meshed with quadratic tetrahedral elements at a 3 mm nominal size, giving 810,186 elements, an element quality of 0.756, and an aspect ratio of 1.7. A refinement check from about 0.8 million to 1.6 million elements showed a maximum relative frequency difference of no more than 0.541%, so the 3 mm mesh is retained as a balance between accuracy and cost.
The upper shaft end, supported by a cylindrical roller bearing, and the lower end, supported by a ball bearing, are represented by cylindrical constraints at the actual bearing seats; only rotation about the gear axis remains free. A rotational-velocity boundary condition ramps from 0 to 7626 r/min in 1 s and then holds constant, reproducing run-up to rated speed and steady operation. The Coriolis effect is enabled to capture rotation-induced splitting of the nodal-diameter modes. The modal extraction covers 30 modes. The pinion’s 27 teeth set the dominant external excitation order. The meshing excitation frequency of interest is about 8.9 kHz, and the 30th mode at 8891.4 Hz completes the modal coverage of that band.
Nodal-diameter families in the modal spectrum
Modal deformation separates into nodal-circle, nodal-diameter, and composite forms. Nodal-circle modes have concentric nodal lines and symmetric displacement. Nodal-diameter modes have radial lines and are more important for high-speed gears because they bend the web and modulate mesh alignment. Composite modes combine both forms and are less likely to produce a clean resonance. In the first 30 modes, six nodal-diameter families appear: 1 through 6. Each family is represented by a pair of modes, separated only slightly by rotation and Coriolis effects. The pair frequencies are equal for the 1-ND family and nearly equal for the others.
| Nodal diameters | Mode pair | Frequencies / Hz | Modal character |
|---|---|---|---|
| 1 | 2, 3 | 1115.6; 1115.6 | One radial nodal line |
| 2 | 4, 5 | 1261.0; 1261.1 | Two radial nodal lines |
| 3 | 7, 8 | 2573.8; 2574.0 | Three radial nodal lines |
| 4 | 11, 12 | 4405.7; 4411.3 | Four radial nodal lines |
| 5 | 18, 19 | 6393.9; 6394.7 | Five radial nodal lines |
| 6 | 25, 26 | 8416.8; 8417.9 | Six radial nodal lines |
The first mode at 110.6 Hz is a lower-order deformation outside this sequence. The six nodal-diameter families span 1115.6 to 8417.9 Hz, so they occupy the same broad frequency range as the high-frequency mesh excitation. The geometric pattern is simple: N radial nodal lines cross the web, with displacement alternating between adjacent sectors. For a rotating gear, the nodal-diameter mode splits into two wave components: a forward wave whose frequency rises with speed and a backward wave whose frequency falls. This split is the reason a single natural frequency can produce two different resonance speeds.
Campbell diagram and resonance margin
The Campbell diagram plots modal frequency on the vertical axis and rotational speed on the horizontal axis. The 27th-order excitation line is superimposed. A crossing is a candidate resonance only when the mode shape has a nodal-diameter pattern that can be driven by the mesh order; not every line intersection is a traveling-wave resonance. The forward branch moves upward with speed and the backward branch moves downward, so the two branches intersect the order line at different speeds.
Resonance margin is defined as
$$\delta=\frac{\left|n_i-n_0\right|}{n_0}\times 100\%$$
where \(n_i\) is the resonance speed and \(n_0\) is the rated operating speed. A margin greater than 10% is used as the acceptance condition. At the rated speed of 7626 r/min, the critical crossings are the 3-ND backward wave, the 3-ND forward wave, and the 4-ND backward wave. Their resonance speeds are 5218.6, 6330.9, and 8758.6 r/min.
| Mode | Nodal diameters | Wave type | Resonance speed / (r/min) | Resonance margin / % |
|---|---|---|---|---|
| 7 | 3 | Backward | 5218.6 | 31.57 |
| 8 | 3 | Forward | 6330.9 | 16.98 |
| 11 | 4 | Backward | 8758.6 | 14.85 |
All three margins exceed 10%, so the rated speed meets the acceptance condition under the modeled boundary conditions. The lower 1-ND and 2-ND crossings occur below 3000 r/min and are far from rated speed, with margins well above the threshold. The tightest case is the 4-ND backward wave at 8758.6 r/min: the margin is 14.85%, and the 10% boundary lies at 7962.4 r/min, about 336 r/min above the rated point. This is a positive but limited overspeed allowance; a downward shift of the 4-ND frequency or a change in bearing stiffness could reduce it.
Safe speed bands for testing
Using the 10% criterion, the exclusion band around each resonance speed is bounded by \(n_{\text{res}}/1.1\) and \(n_{\text{res}}/0.9\). The three bands are listed below.
| Resonance speed / (r/min) | Exclusion band / (r/min) | Critical wave |
|---|---|---|
| 5218.6 | 4744.2–5798.4 | 3-ND backward |
| 6330.9 | 5755.4–7034.3 | 3-ND forward |
| 8758.6 | 7962.4–9731.8 | 4-ND backward |
The first two bands overlap, so the continuous exclusion region extends from 4744.2 to 7034.3 r/min. The test window is 60%–130% of rated speed, or 4575.6–9913.8 r/min. Removing the exclusion bands leaves three safe intervals: 4575.6 to <4744.2, >7034.3 to <7962.4, and >9731.8 to 9913.8 r/min. The rated speed of 7626 r/min lies in the middle safe interval, consistent with its positive margin. For a test schedule, operating points should be chosen inside these intervals; if an exclusion band must be crossed, the gear should accelerate through it rather than dwell.
Design implications
The case shows that a natural-frequency list is not enough for a high-speed spiral bevel gear. The engineer needs the nodal-diameter family, the forward/backward split, the dominant excitation order, and the margin at the intended operating point. A mode can be close to an order line but harmless if its shape is not a nodal-diameter pattern; conversely, a small frequency shift in a high-N family can move a resonance into the operating band. For this pair, the 3-ND and 4-ND families control the upper speed range. Design changes that alter web stiffness or mass will shift these families; bearing support stiffness and damping also affect the split and the effective margin. Modal testing, strain gauges on the web, or run-up vibration measurements can confirm the predicted bands before they are used as hard limits.
Within the stated model, the rated speed of 7626 r/min is safe: all three critical margins exceed 10%, and the minimum is 14.85%. The result is conditional on the geometry, material, bearing constraints, and Coriolis-enabled modal model; it should not be transferred to a different web design without recalculating the Campbell crossings and exclusion bands.