The web is the first compliant element on the vibration path
Aeronautical gearboxes are dimensioned by power density, and one structural consequence is that spiral bevel gears are built with thin spoke plates: material is kept where the teeth and the shaft coupling require it, and the web carries a ring of lightening holes. Those holes are normally selected for mass removal and manufacturability. Dynamically, however, they sit directly on the transmission path that runs from the mesh to the airframe — tooth flank, web, shaft, bearing, bearing housing, casing — and each element along that chain filters and amplifies what the next one receives. The web is the first component with enough compliance to alter the spectrum that the rest of the chain is forced with.
The influence of web topology on gear dynamics has been examined from several directions. Fang Zongde and co-workers showed in 1992 that the flexibility of a spoke-plate web changes the dynamic characteristics of the whole gear system, and Shen Yunwen’s group (1995) quantified how support form and structural dimensions of the web affect dynamic load in helical gear transmissions. Cai Xianxin et al. (2009) formulated an optimization model for gear webs built on CAD geometry, which made parametric web design practical. Wu Bin and Ge Wenjie (2011) mapped how the number and size of web holes shift the natural characteristics of spiral bevel gears. Zhao Ning and Lin Yanhu (2013) combined orthogonal design with a hybrid genetic algorithm to obtain a parameter set that moves the wheel body away from travelling-wave resonance under operating conditions. Wang Dayong (2016) computed natural frequencies and mode shapes of thin-web aviation gears and plotted dynamic stress at both resonant and operating speeds, providing a basis for web stress checking.
On the excitation and response side, Carmignani et al. (2007) simulated the dynamic response of bevel gears and extracted stress distributions associated with travelling-wave behaviour. Wang P Y et al. (2011) included bearing stiffness, gear errors and tooth-flank friction in a contact and tooth-impact analysis of spiral bevel gears, and Luan Xiaochi et al. (2019) combined transient contact dynamics with experimental verification to characterise the resonant behaviour and stress state of a driven bevel gear during travelling-wave resonance.
What has received comparatively little quantitative treatment is the shape of the lightening hole itself. Hole patterns are still largely chosen by experience, so the distribution of removed material around the web is rarely the outcome of a dynamics criterion. The case examined below treats a single hole ring as a design variable at the source end of the transmission path: four hole patterns of equal number and equal total area are compared on a 1,900 kW, 20,900 r/min spiral bevel gear pair, and the best-performing pattern is then optimized over five geometric parameters using an L16 orthogonal array with signal-to-noise ratio as the objective.
Modal superposition as the analysis framework
The response of the gear body to a time-varying mesh force is obtained by superposition of modal contributions. The discretized equation of motion is
$$ \mathbf{M}\ddot{\mathbf{u}} + \mathbf{C}\dot{\mathbf{u}} + \mathbf{K}\mathbf{u} = \mathbf{F} $$
where \(\mathbf{M}\), \(\mathbf{C}\) and \(\mathbf{K}\) are the mass, damping and stiffness matrices, \(\mathbf{u}\) is the displacement vector and \(\mathbf{F}\) the load vector. For undamped free vibration the solution of a single mode is harmonic, \(\mathbf{u} = \boldsymbol{\Phi}_i \cos \omega_i t\), with \(\boldsymbol{\Phi}_i\) the \(i\)-th mode shape and \(\omega_i\) the corresponding natural frequency. The load vector is split into a time-varying nodal part and a scaled static part, \(\mathbf{F} = \mathbf{F}_{nd} + s\mathbf{F}_{s}\).
Introducing modal coordinates \(y_i\), the physical displacement is expanded as
$$ \mathbf{u} = \sum_{i=1}^{n} \boldsymbol{\Phi}_i y_i $$
Substituting this expansion into the equation of motion and pre-multiplying by \(\boldsymbol{\Phi}_j^{\mathrm{T}}\) gives, for mass-, stiffness- and damping-orthogonal modes,
$$ \boldsymbol{\Phi}_j^{\mathrm{T}}\mathbf{M}\boldsymbol{\Phi}_i = \boldsymbol{\Phi}_j^{\mathrm{T}}\mathbf{K}\boldsymbol{\Phi}_i = \boldsymbol{\Phi}_j^{\mathrm{T}}\mathbf{C}\boldsymbol{\Phi}_i = 0 \qquad (i \neq j) $$
so that the system decouples into independent single-degree-of-freedom equations
$$ \boldsymbol{\Phi}_j^{\mathrm{T}}\mathbf{M}\boldsymbol{\Phi}_j\,\ddot{y}_j + \boldsymbol{\Phi}_j^{\mathrm{T}}\mathbf{C}\boldsymbol{\Phi}_j\,\dot{y}_j + \boldsymbol{\Phi}_j^{\mathrm{T}}\mathbf{K}\boldsymbol{\Phi}_j\,y_j = \boldsymbol{\Phi}_j^{\mathrm{T}}\mathbf{F} $$
Only a truncated set of low-order modes is retained, because these are the modes that external excitation can realistically drive; higher modes contribute little to the response of a stiff, compact gear body. The truncation is justified quantitatively through the effective modal mass, which is the criterion used to size the extraction range in the next section.
Finite element model of the gear pair
The object of the study is an aeronautical spiral bevel gear pair operating at 20,900 r/min and transmitting 1,900 kW, with a 27/74 tooth combination. The pinion has a left-hand spiral and the wheel a right-hand spiral. The main design and material data are collected in Table 1.
| Parameter | Value |
|---|---|
| Rated speed | 20,900 r/min |
| Transmitted power | 1,900 kW |
| Number of teeth, pinion / wheel | 27 / 74 |
| Module | 3.85 mm |
| Face width | 38.5 mm |
| Hand of spiral, pinion / wheel | left / right |
| Mass of driven wheel | 9.86 kg |
| Young’s modulus | 207 GPa |
| Poisson’s ratio | 0.25 |
| Density | 7.86 × 10³ kg/m³ |
| Element type | quadratic tetrahedra |
| Number of elements | 549,466 |
| Nominal element size | 3 mm |
The driven wheel is the component of interest because it carries the larger web. The central bore is fixed, and the bearing is represented by spring elements of assigned stiffness acting on the shaft, so the model reproduces the gear–shaft interface rather than a complete casing. Modal extraction uses the Lanczos algorithm over a frequency range of 10,500 Hz, a limit selected by checking the effective modal mass in the three coordinate directions. At the chosen range, the effective modal mass reaches 99.82 %, 99.32 % and 99.90 % of the structural mass in X, Y and Z respectively, which is sufficient for the subsequent modal-superposition harmonic response to be reliable.
Frequency margin against the mesh excitation
Resonance of the gear body requires two conditions to be satisfied simultaneously: the excitation frequency must coincide with a natural frequency, and the harmonic order \(K\) of the excitation must match the number of nodal diameters \(m\) of the mode. Since the mesh force is dominated by its first harmonic, the modes of practical interest are those with a single nodal diameter. The mesh frequency follows directly from the tooth count and the shaft speed,
$$ f = \frac{20\,900}{60} \times 27 = 9\,405\ \text{Hz} $$
and the separation of each natural frequency \(f_N\) from this excitation is expressed as a relative margin,
$$ M = \frac{\left| f_N – f \right|}{f} $$
A margin of at least 15 % is normally required to keep a component clear of resonance. The first six natural frequencies and the corresponding margins are listed in Table 2.
| 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 |
All six margins exceed 80 %, roughly five times the customary 15 % threshold, so the wheel does not approach resonance at the rated operating point. The mode shapes are those of a disc-like body: modes 1 and 6 are nodal-circle forms in which the circumference displaces uniformly, modes 2 and 3 carry one nodal diameter, and modes 4 and 5 carry two nodal diameters. The repeated frequency at 1,426.0 Hz for modes 4 and 5 is the degenerate pair expected for a diametrically symmetric structure, and the regularity of the mode set confirms that the web behaves as a plate in bending rather than as a locally dominated shell.
Harmonic response of four equal-area hole patterns
The modal solution provides the basis for a harmonic response analysis. The dynamic mesh force was first extracted from a multibody simulation of the gear pair. Its time history shows a start-up transient whose peak amplitude substantially exceeds the steady-state level, a consequence of the load ramp at engagement, while the frequency spectrum peaks at the mesh frequency of 9,405 Hz and at its multiples. The mesh force components were then applied along the contact line of the driven wheel in the finite element model.
Four web patterns were compared under a controlled condition: the same number of holes and the same total removed area, so that the comparison isolates shape rather than mass. The patterns were circular, waist-shaped (elongated slot with rounded ends), pillow-shaped and triangular. Vibration acceleration was recorded at two points on the shaft–bearing connection. The root-mean-square values are given in Table 3.
| Hole pattern | RMS acceleration, point 1 (10⁶ mm/s²) | RMS acceleration, point 2 (10⁶ mm/s²) |
|---|---|---|
| Waist (slot) | 1.541 | 1.511 |
| Pillow | 1.558 | 1.513 |
| Circular | 1.736 | 1.715 |
| Triangular | 1.770 | 1.746 |
The two measurement points rank the patterns identically, which indicates that the ordering reflects a property of the web rather than a local artefact of the sampling location. With equal removed area, the response decreases in the order triangular, circular, pillow, waist. The waist slot lowers the transmitted acceleration by about 11 % relative to the circular pattern and by about 13 % relative to the triangular one, and it is therefore the natural candidate for further parameter optimization.
A plausible interpretation is that, for a fixed amount of removed material, the waist slot spreads its boundary over a longer perimeter and orients that boundary more nearly along the circumferential direction. The web is then interrupted along a longer arc of the radial load path between rim and hub, and the slot ends act as local compliant hinges that redistribute modal strain energy instead of concentrating it at a small cut-out. The triangular pattern, by contrast, places sharp corners where the stress flow has to turn, and produces the highest response of the four despite removing the same area.
Orthogonal optimization of the waist slot
The waist slot is described by five geometric parameters: the number of holes \(n\), the included angle between the two end-centre radii \(\theta_1\), the corner radius \(r\), the radial distance of the slot centre from the axis \(R\), and the inclination angle of the slot axis \(\theta_2\). Four levels were assigned to each parameter within a range consistent with the available web width, giving the factor-level scheme of Table 4 and an L16(4⁵) array.
| Factor | Symbol | Unit | Level 1 | Level 2 | Level 3 | Level 4 |
|---|---|---|---|---|---|---|
| Number of holes | A | — | 5 | 6 | 7 | 8 |
| End-centre included angle | B | ° | 5 | 6 | 7 | 8 |
| Corner radius | C | mm | 5 | 6 | 7 | 8 |
| Radial distance from axis | D | mm | 88 | 89 | 90 | 91 |
| Inclination angle | E | ° | 0 | 5 | 10 | 15 |
The RMS accelerations at the two measurement points, RMS1 and RMS2, serve as the evaluation indices. Each of the sixteen parameter combinations was solved with the same harmonic response procedure, and the results are listed in Table 5.
| 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 spread of the results, from 1.1899 to 3.0063 × 10⁶ mm/s² at point 1, shows that the slot geometry is a strong first-order variable and not a second-order refinement. Because both indices are of the smaller-is-better type, they are converted to signal-to-noise ratios,
$$ \eta = -10\lg\left( \frac{1}{n}\sum_{i=1}^{n} y_i^{2} \right) $$
where \(n\) is the number of repetitions and \(y_i\) the \(i\)-th measured value. Maximizing \(\eta\) is equivalent to minimizing the response. Averaging the signal-to-noise ratio over the runs at each level yields the main-effect curves for the two measurement points.
The two points agree on the least influential factors but not on the dominant one. At point 1 the factors rank C, A, D, B, E — corner radius first, then hole number, radial position, end-centre angle and inclination. At point 2 the ranking is B, C, A, D, E — end-centre angle first, followed by corner radius, hole number, radial position and inclination. In both cases the inclination angle is the weakest factor and the radial position is second weakest, while corner radius and hole number remain among the three most influential. Reading the level averages gives the same optimum from both curves, A1B1C1D2E3: five holes, an end-centre angle of 5°, a corner radius of 5 mm, a radial distance of 89 mm and an inclination of 10°.
Verification, strength check and scope of the result
That combination does not appear among the sixteen array runs, so it was evaluated directly. The harmonic response of the optimized web gives RMS1 = 1.1615 × 10⁶ mm/s² and RMS2 = 1.0699 × 10⁶ mm/s², lower than every run in Table 5, including run 1 (1.1899 and 1.0773) whose levels coincide with the optimum except for radial position and inclination. The improvement is therefore not an artefact of selecting the best existing run from a discrete array.
Measured against the original circular-hole design, the optimized waist slot reduces the transmitted acceleration by 33.09 % at point 1 and 37.62 % at point 2. The maximum dynamic stress extracted from the optimized geometry is 201.6 MPa, which remains within the allowable strength of the gear material, so the vibration gain is not obtained at the cost of a strength violation.
| Configuration | RMS at point 1 (10⁶ mm/s²) | RMS at point 2 (10⁶ mm/s²) | Reduction vs. circular web |
|---|---|---|---|
| Circular holes (baseline) | 1.736 | 1.715 | — |
| Optimized waist slot, A1B1C1D2E3 | 1.1615 | 1.0699 | 33.09 % / 37.62 % |
Several features of this result deserve emphasis for design practice. First, the whole comparison is conducted at constant removed area and constant hole count, which means the reported differences are attributable to shape alone; a design that also changes hole number or total area will superimpose a mass effect on the shape effect and cannot be read off these numbers directly. Second, the modal and frequency-margin assessment was performed on the circular-hole configuration, whereas the shape comparison and the parameter optimization were performed on the harmonic response; a production design iteration would normally repeat the margin check on the final geometry. Third, the boundary conditions idealize the installation, with a fixed bore and spring elements representing the bearing stiffness, so the absolute acceleration levels are model-dependent even though the ranking of the patterns is consistent across both measurement points and across the entire orthogonal array.
The most important limitation is experimental. The conclusions rest on finite element modal and harmonic response analysis, and the high speed and heavy load of an aeronautical gear make the corresponding rig test demanding; validation of the predicted reduction remains outstanding. The workflow itself, however, is transferable: establish the frequency margin against the mesh frequency, rank candidate hole shapes at constant removed area, screen the geometric parameters of the best shape with an orthogonal array and a signal-to-noise objective, then verify the predicted optimum and check the dynamic stress. Applied at the web, the first compliant element after the tooth mesh, this sequence buys a reduction in transmitted vibration without additional mass, additional damping hardware or a change of gear macro-geometry.