Thin-web, hole-lightened spiral bevel gears are widely used in weight-critical aerospace transmissions because they combine high torque density with low mass. The same construction makes the web an important part of the vibration path. Mesh excitation generated at the tooth contact travels through the web, shaft, bearings and housing, so reducing the web response close to the source can lower the excitation transmitted to the bearing interface and casing. Lightening holes are usually introduced for mass reduction, but their shape and placement also change local stiffness, modal mass and the way the web accepts mesh force. The practical question is whether a hole pattern can be selected to reduce vibration at the bearing interface without pushing dynamic stress beyond the material limit.
The case considered here is a 20,900 r/min, 1,900 kW aeronautical spiral bevel gear pair with 27/74 teeth, a module of 3.85 mm and a face width of 38.5 mm. The pinion is left-hand and the wheel is right-hand. At the rated speed, the 27-tooth pinion produces a mesh frequency of 9,405 Hz, which is the principal tonal excitation used in the response analysis.
Case Definition and Modal Separation
The driven wheel was modelled with a second-order tetrahedral mesh of 549,466 elements and a nominal element size of 3 mm. The material properties were a Young’s modulus of 207 GPa, a Poisson’s ratio of 0.25 and a density of 7.86 × 103 kg/m3; the wheel mass was 9.86 kg. The central bore was fixed, and the bearing support was represented by elastic elements with prescribed stiffness. Natural frequencies and mode shapes were extracted with the Lanczos method.
For a modal-superposition solution, the truncated modal basis must capture most of the structural response. The basis extended to 10,500 Hz and recovered 99.82%, 99.32% and 99.90% of the effective modal mass in the X, Y and Z directions, respectively. This is sufficient for the subsequent harmonic response calculation.
The frequency margin was defined as
$$M=\frac{\left|f_N-f\right|}{f}\times100\%$$
where \(f_N\) is the \(N\)-th natural frequency and \(f\) is the principal excitation frequency, 9,405 Hz. The design criterion used was a margin above 15%. The first six natural frequencies and their margins are listed in Table 1.
| Order | Natural frequency (Hz) | Frequency margin (%) |
|---|---|---|
| 1 | 291.2 | 96.90 |
| 2 | 1,056.6 | 88.77 |
| 3 | 1,137.8 | 87.90 |
| 4 | 1,426.0 | 84.84 |
| 5 | 1,426.0 | 84.84 |
| 6 | 1,708.5 | 81.83 |
The first six modes lie well below the 9,405 Hz mesh frequency, and the lowest margin is 81.83%. Under the stated excitation, primary resonance with these low-order modes is therefore not expected. The mode shapes are disc-like: the first and sixth modes are nodal-circle-type motions with equal displacement around the rim, the second and third are one-diameter modes, and the fourth and fifth are two-diameter modes. For a disc-like gear, coincidence also requires the excitation harmonic order to match the nodal-diameter number; because the mesh force is dominated by the first harmonic, the one-diameter pair is the most relevant low-order family.
Forced Response to Mesh Excitation
The harmonic response was evaluated by modal superposition using the dynamic mesh force extracted from a multibody model of the same gear pair. The time-domain force showed a start-up peak much larger than the steady-state level, while the steady-state spectrum peaked at the mesh frequency of 9,405 Hz and at its multiples. The steady-state mesh force was applied along the contact line of the driven wheel, and the response was monitored at two points on the shaft–bearing connection.
Four lightening-hole patterns were compared under equal removed area and equal hole count: circular, waist-shaped, pillow-shaped and triangular. The acceleration RMS values at the two points are given in Table 2.
| Hole pattern | RMS at point 1 (106 mm/s2) | RMS at point 2 (106 mm/s2) |
|---|---|---|
| Circular | 1.736 | 1.715 |
| Waist-shaped | 1.541 | 1.511 |
| Pillow-shaped | 1.558 | 1.513 |
| Triangular | 1.770 | 1.746 |
At both measurement points the ranking was waist-shaped < pillow-shaped < circular < triangular. The waist-shaped slot gave the lowest RMS, about 11.2% and 11.9% below the circular pattern at points 1 and 2, respectively, while the triangular pattern was slightly worse than the circular one. Because the removed area and hole count were fixed, the mass reduction was essentially the same; the response difference therefore comes mainly from the redistribution of stiffness and the resulting modal participation of the web. This is a structural-tuning effect rather than an added-damping effect. The elongated waist slot changes the local bending paths around the web and reduces the acceleration transmitted to the bearing-side points under the same mesh excitation.
Orthogonal Optimization of the Waist Slot
The waist-shaped slot was then optimized over five geometric variables: hole number \(n\), arc-center angle \(\theta_1\), fillet radius \(r\), radial distance from the axis \(R\), and inclination angle \(\theta_2\). Four levels were assigned to each factor, and an \(L_{16}(4^5)\) orthogonal array of simulations was used to limit the number of cases while retaining the main effects. The factor levels are listed in Table 3, and the array with the two RMS responses is given in Table 4.
| Factor | Description | Unit | Level 1 | Level 2 | Level 3 | Level 4 |
|---|---|---|---|---|---|---|
| A | Hole number, \(n\) | — | 5 | 6 | 7 | 8 |
| B | Arc-center angle, \(\theta_1\) | ° | 5 | 6 | 7 | 8 |
| C | Fillet radius, \(r\) | mm | 5 | 6 | 7 | 8 |
| D | Radial distance, \(R\) | mm | 88 | 89 | 90 | 91 |
| E | Inclination angle, \(\theta_2\) | ° | 0 | 5 | 10 | 15 |
| Run | A | B | C | D | E | RMS1 (106 mm/s2) | RMS2 (106 mm/s2) |
|---|---|---|---|---|---|---|---|
| 1 | 1 | 1 | 1 | 1 | 1 | 1.1899 | 1.0773 |
| 2 | 1 | 2 | 2 | 2 | 2 | 1.2426 | 1.2095 |
| 3 | 1 | 3 | 3 | 3 | 3 | 1.4916 | 1.4680 |
| 4 | 1 | 4 | 4 | 4 | 4 | 2.3885 | 1.8757 |
| 5 | 2 | 1 | 2 | 3 | 4 | 1.4893 | 1.4875 |
| 6 | 2 | 2 | 1 | 4 | 3 | 1.2843 | 1.2491 |
| 7 | 2 | 3 | 4 | 1 | 2 | 3.0063 | 2.3311 |
| 8 | 2 | 4 | 3 | 2 | 1 | 2.2574 | 1.9720 |
| 9 | 3 | 1 | 3 | 4 | 2 | 1.8546 | 1.7741 |
| 10 | 3 | 2 | 4 | 3 | 1 | 2.4025 | 1.6622 |
| 11 | 3 | 3 | 1 | 2 | 4 | 1.4438 | 1.4382 |
| 12 | 3 | 4 | 2 | 1 | 3 | 2.2142 | 1.9329 |
| 13 | 4 | 1 | 4 | 2 | 3 | 2.2166 | 1.2904 |
| 14 | 4 | 2 | 3 | 1 | 4 | 2.7875 | 1.9238 |
| 15 | 4 | 3 | 2 | 4 | 1 | 2.2179 | 1.9458 |
| 16 | 4 | 4 | 1 | 3 | 2 | 1.8485 | 1.7629 |
The evaluation used the smaller-is-better signal-to-noise ratio
$$\eta=-10\log_{10}\left(\frac{1}{n}\sum_{i=1}^{n}y_i^2\right)$$
where \(n\) is the number of repeated trials and \(y_i\) is the \(i\)-th response value. For this smaller-is-better criterion, a higher \(\eta\), closer to zero, is preferable. The factor ranking at point 1 was C, A, D, B, E; at point 2 it was B, C, A, D, E. The two points therefore weight the factors differently, and the preferred combination was selected from the S/N curves as a compromise: A1B1C1D2E3, corresponding to five holes, an arc-center angle of 5°, a fillet radius of 5 mm, a radial distance of 89 mm and an inclination angle of 10°.
A confirmation simulation of this combination gave RMS values of 1.1615 × 106 mm/s2 at point 1 and 1.0699 × 106 mm/s2 at point 2, lower than any of the 16 array runs. The maximum dynamic stress was 201.6 MPa, which the design check regarded as acceptable for the selected gear material. Relative to the original circular-hole design, the optimized waist slot reduced the two RMS values by 33.09% and 37.62%, respectively.
Design Implications and Limits
The case shows that lightening holes in a thin gear web should be treated as a dynamic design variable, not only as a mass-reduction feature. Frequency separation is necessary: the first six driven-wheel modes were more than 80% away from the 9,405 Hz mesh frequency, so the low-order modes were not directly excited. Even so, the forced response at the bearing interface changed substantially with hole geometry. At mesh frequency, the response was governed by off-resonance modal superposition rather than by a single low-order resonance.
The equal-area, equal-count comparison is useful because it isolates the effect of shape. With the same removed mass, the waist-shaped slot produced the lowest acceleration at both measurement points, followed by the pillow-shaped slot, the circular hole and the triangular hole. The subsequent orthogonal study then treated the waist slot as a multi-parameter problem: hole number, arc angle, fillet radius, radial position and inclination all influenced the response, and the two measurement points did not rank the factors identically. This is why a single-factor trial-and-error approach is unlikely to find a robust hole pattern.
The results are numerical and apply to the modeled bearing stiffness, web thickness, load case and boundary conditions. The response metric is acceleration at the shaft–bearing connection, not radiated sound pressure or housing vibration. The optimum is therefore case-specific, and the stress check at 201.6 MPa is only one strength indicator. Experimental validation under representative high-speed, heavy-load conditions remains necessary before the pattern is used in service. Within those limits, the design logic is transferable to other thin-web spiral bevel gears: establish the modal margin first, compare equal-mass hole shapes under the same mesh excitation, and then optimize the selected shape with a small orthogonal array rather than relying on experience alone.