Central-drive spiral bevel gears in aero-engines connect the engine spool to the accessory gearbox and operate under high speed, heavy load, and elevated temperature. The driven gear is a disk-like part: a relatively thin web carries the rim and teeth. This geometry is efficient in mass and stiffness, but it supports nodal-diameter modes whose forward and backward waves can synchronize with gear-mesh excitation. When that synchronization occurs, alternating stress at the web root can grow until block loss or a full-circumferential fracture develops. The design problem is therefore to place the traveling-wave resonance branches outside the operating speed range without sacrificing the static and fatigue requirements of the gear.
Traveling-wave mechanism in a disk-like gear
A stationary disk can vibrate in nodal-circle, nodal-diameter, or combined modes. For a thin web, the nodal-diameter family is the most relevant because the bending deformation is concentrated in the web and the rim remains relatively stiff. With $m$ nodal diameters, the transverse displacement of the web can be written as
$$ z(r,\theta,t)=A(r)\cos(m\theta)\cos(\omega t) $$
where $z$ is the transverse displacement in mm, $A(r)$ is the radial amplitude in mm, $m$ is the number of nodal diameters, $\theta$ is the circumferential angle, $\omega$ is the angular frequency in rad·s-1, and $t$ is time in s.
When the gear rotates, the standing nodal-diameter pattern can be decomposed into two counter-rotating waves:
$$ z(r,\theta,t)=\frac{A(r)}{2}\left[\cos(m\theta-\omega t)+\cos(m\theta+\omega t)\right] $$
In a fixed coordinate system, the forward and backward wave frequencies shift with the rotation speed of the gear:
$$ f_f=f_d+\frac{m n}{60},\qquad f_b=f_d-\frac{m n}{60} $$
Here $f_d$ is the static natural frequency of the nodal-diameter mode in Hz, $n$ is the rotational speed of the gear in r·min-1, and $f_f$ and $f_b$ are the forward- and backward-wave frequencies in Hz. For a gear pair, the relevant $n$ is the speed of the gear whose web is being assessed, obtained from the input speed through the tooth ratio.
The mesh excitation frequency is
$$ f_e=\frac{N_{\mathrm{in}}Z_1}{60} $$
where $N_{\mathrm{in}}$ is the input speed in r·min-1 and $Z_1$ is the tooth count of the input gear. Resonance occurs when $f_e$ coincides with either $f_f$ or $f_b$ and when the mesh force does positive work on that wave. Frequency coincidence alone is a necessary but not sufficient condition for resonant growth. Solving for the input speed gives
$$ N_{\mathrm{in}}=\frac{60f_d}{i\left(Z_2\mp m\right)} $$
where $Z_2$ is the tooth count of the assessed gear, $i=Z_1/Z_2$ is the speed ratio of the assessed gear to the input gear, and the minus sign applies to the forward wave and the plus sign to the backward wave.
For the 47/35 tooth pair considered here, the baseline three- and four-nodal-diameter static frequencies at 20 °C are 7997 Hz and 12768 Hz. These values give the approximate input-speed estimates in Table 1. They are calculated from the static modal frequencies and the tooth ratio; they are not direct measurements of resonance speed.
| Mode, $m$ | $f_d$ (Hz) | Forward-wave input speed (r·min-1) | Backward-wave input speed (r·min-1) |
|---|---|---|---|
| 3 | 7997 | ≈11,170 | ≈9,400 |
| 4 | 12768 | ≈18,400 | ≈14,630 |
Finite-element model and modal validation
The example is a central-drive spiral bevel gear pair with 47 teeth on the driving gear and 35 teeth on the driven gear. The main geometry is listed in Table 2. The driven gear is the focus because service failures occurred at its web, where the web root is subjected to alternating bending. A three-dimensional model of the meshing pair was built and the driven gear was meshed with hexahedral solid elements. The first 12 modes were extracted, and the one-, two-, three-, and four-nodal-diameter mode shapes were identified from the results. The mode shapes follow the expected disk-like pattern: the rim and teeth remain relatively rigid while the web undergoes out-of-plane bending.
| Parameter | Value |
|---|---|
| Driving gear teeth, $Z_1$ | 47 |
| Driven gear teeth, $Z_2$ | 35 |
| Large-end module, $m$ | 3.875 mm |
| Spiral hand | Driving gear left-hand; driven gear right-hand |
| Normal pressure angle, $\alpha$ | 20° |
| Spiral angle, $\beta$ | 35° |
| Normal backlash | 0.10–0.15 mm |
| Normal mesh clearance | 0.15–0.35 mm |
The temperature-dependent elastic properties used in the model are summarized in Table 3. The density is held constant over the range, while Young’s modulus decreases and Poisson’s ratio increases with temperature.
| Temperature (°C) | Young’s modulus $E$ (GPa) | Poisson’s ratio $\mu$ | Density $\rho$ (kg·m-3) |
|---|---|---|---|
| 20 | 215 | 0.38 | 7.86 × 103 |
| 100 | 213 | 0.39 | 7.86 × 103 |
| 200 | 210 | 0.41 | 7.86 × 103 |
The finite-element modal results were checked against modal tests on three driven gears. The three-nodal-diameter frequency is predicted within 1.2%, and the four-nodal-diameter frequency within 4.8% (Table 4). The four-diameter mode shows larger scatter, but the error remains below 5%. This level of agreement is sufficient to use the model for trend analysis of web thickness and temperature, provided the results are treated as design guidance rather than as a substitute for hot modal testing.
| Gear sample | Mode, $m$ | Finite-element result (Hz) | Test result (Hz) | Error (%) |
|---|---|---|---|---|
| A | 3 | 7997 | 7964 | 0.41 |
| 4 | 12768 | 13402 | 4.73 | |
| B | 3 | 7997 | 7928 | 0.87 |
| 4 | 12768 | 13360 | 4.43 | |
| C | 3 | 7997 | 7908 | 1.13 |
| 4 | 12768 | 13320 | 4.14 |
The difference between static and dynamic frequency is small for this gear, so static frequencies are used as a proxy for dynamic frequencies when estimating resonance speed and frequency. This simplification is reasonable only where the rotation-induced split is small relative to the separation between the resonance branch and the operating mesh order.
Effect of web thickness
The full-web reference thickness is 17.28 mm. The baseline finite-element model uses a web thickness of 10.28 mm, which is 60% of the full-web thickness. The parametric study varied the web thickness to 8.28, 9.28, 9.78, 10.28, 10.78, 11.28, and 12.28 mm, corresponding to 48%, 54%, 57%, 60%, 62%, 65%, and 71% of the full-web thickness. For each geometry, modal analyses were carried out at working temperatures of 60, 100, 150, and 200 °C, and the three- and four-nodal-diameter frequencies were extracted.
At a fixed temperature, increasing the web thickness raises both the forward- and backward-wave frequencies and the input speeds at which those waves coincide with mesh excitation. The trend holds for $m=3$ and $m=4$. A thicker web increases the bending stiffness of the disk-like structure, so the nodal-diameter natural frequencies rise and the resonance branches move to higher speed. Conversely, at a fixed web thickness, increasing temperature lowers the resonance frequencies and resonance speeds. The elastic modulus falls from 215 GPa at 20 °C to 210 GPa at 200 °C, and the associated softening shifts the modal frequencies downward. The temperature effect is systematic but smaller than the web-thickness effect over the ranges examined.
The practical consequence is that web thickness is a strong tuning variable for the resonance map. In a Campbell-diagram representation, the forward and backward wave branches for $m=3$ and $m=4$ move upward as the web is thickened. The design target is to keep the mesh-order lines from intersecting those branches within the operating speed range. Because the $m=3$ and $m=4$ branches are separated, a thickness change that clears one branch should be checked against the other, as well as against the higher nodal-diameter modes that may enter the same frequency range.
Design implications and limits
For a high-speed central-drive spiral bevel gear, the web root is a traveling-wave resonance location, not merely a static stress concentration. The analysis supports a design sequence in which the modal frequencies of the driven gear are calculated first, the forward and backward wave branches are plotted against mesh order, and the web thickness is then adjusted to move the $m=3$ and $m=4$ resonance speeds outside the operating range. Temperature must be included because a branch that is clear at room temperature can move closer to the operating line at 150–200 °C. The adjustment is bounded by the other design requirements of the gear: tooth root strength, rim stiffness, mass, axial and radial packaging, and manufacturing feasibility.
The results should be read within their scope. The modal validation was performed at room temperature; the elevated-temperature trends come from finite-element material properties rather than from hot modal tests. Only web thickness was varied systematically, while rim geometry, fillet details, support stiffness, damping, and operating load were held fixed. The resonance-speed estimates use static frequencies and a simple tooth-ratio relation, so they identify trends and approximate separation margins rather than exact operating limits. The specific frequencies and speeds apply to the 47/35 central-drive pair studied here and should be transferred to other spiral bevel gears only after remodelling the actual geometry and boundary conditions.
Conclusion
Traveling-wave resonance in spiral bevel gears is controlled by the interaction between nodal-diameter web modes and mesh excitation. For the central-drive pair examined here, increasing web thickness raises the three- and four-nodal-diameter resonance frequencies and the corresponding input speeds, while increasing temperature lowers them. Web thickness is the more sensitive of the two parameters over the studied range. Within the design envelope, adjusting the web thickness can move the resonance branches away from the operating speed range and reduce the risk of circumferential web-root fracture.