The Design Problem: Mass in the Web, Vibration in the Train
Spiral bevel gears carry power between intersecting shafts in vehicle drivelines, aerospace reducers and helicopter main gearboxes. In the high-speed, high-power-density regime that now defines these applications, the pinion may run above 20,000 r/min while the pair transmits of the order of a megawatt. Under those conditions the mesh is no longer a quasi-static contact problem: the dynamic component of the meshing force becomes comparable to the mean load, and the resulting structural vibration drives radiated noise, fretting at splines and bearings, and ultimately high-cycle fatigue of the wheel body.
The excitation sources are well known. Time-varying mesh stiffness, loaded transmission error, and the impact that occurs when a tooth pair enters contact with a velocity difference along the common normal all inject energy at the mesh frequency and its harmonics. Manufacturing and assembly errors, plus mass unbalance from features such as keyways, add modulation sidebands around these carriers. What matters for design is the transmission path: vibration generated at the tooth flank passes through the web, the hub, the shaft, the bearings and finally the housing. Attenuation applied late in that chain, for example at the housing, is inefficient because the energy has already been distributed through compliant intermediate structures.
The gear web is the natural place to intervene. It is a non-load-bearing region that nevertheless accounts for a large fraction of the wheel mass, so it controls both the inertia and the local compliance of the structure. Thinning the web saves weight but degrades strength and shifts natural frequencies in ways that are difficult to predict without a full modal model. Two complementary strategies follow: shape the web geometry so that the modes in the operating range are pushed away from the excitation, and add a dissipative element at the rim that converts vibration into frictional heat without altering the primary load path. The discussion below quantifies both for a representative pair: 27/65 teeth, 3.25 mm module, 35° spiral angle, 90° shaft angle, 23 mm face width, 1500 kW at 20,000 r/min input speed.
Reference Gear Pair and Mesh Excitation
The pair was generated in the Gleason GEMS environment using the CAGE design module, with face-milling (fixed-setting) cutting, standard taper, a 76.2 mm cutter radius, 20° normal pressure angle (40° sum), left-hand pinion and right-hand wheel, radial offset coefficients of ±0.32 and tangential coefficients of ±0.04. Tooth surfaces were evaluated through the tooth contact analysis and ease-off modules: the contact ellipse sits near mid-face and mid-height, the contact path is inclined, and the transmission error amplitude at the crossover points remains within 100 μrad, satisfying the usual engineering acceptance band.
| Parameter | Pinion (z1) | Wheel (z2) |
|---|---|---|
| Number of teeth | 27 | 65 |
| Module at outer end (mm) | 3.25 | |
| Spiral angle at mid-face (°) | 35 | |
| Normal pressure angle (°) | 20 | |
| Shaft angle (°) | 90 | |
| Face width (mm) | 23 | |
| Outer pitch diameter (mm) | 87.75 | 211.25 |
| Outer cone distance (mm) | 114.375 | |
| Pitch cone angle (°) | 22.5572 | 67.4428 |
| Addendum / dedendum (mm) | 3.8025 / 2.3335 | 1.7225 / 4.4135 |
| Hand of spiral | Left | Right |
Solid models were built from the calculated tooth-surface point clouds and assembled without interference, then checked by a kinematic rolling simulation that reproduced an elliptical instantaneous contact patch at mid-tooth. For the dynamic analysis the assembly was transferred to a multibody environment and the mesh represented by an impact-type contact law. The contact stiffness of 8.1956×105 N/mm was obtained from the equivalent radii and elastic moduli of the two bodies, with 2×103 N·s/mm damping, a force exponent of 1.5, a maximum penetration of 0.01 mm and friction coefficients of 0.06 (dynamic) and 0.08 (static). The pinion was ramped to 20,000 r/min within 0.03 s and the wheel loaded with 1724.306 N·m, computed from the 1500 kW rating and the 27/65 ratio.
At this speed the mesh frequency is 9000 Hz:
$$f_m=\frac{n_1 z_1}{60}=\frac{20000\times 27}{60}=9000\ \text{Hz}$$
while the pinion and wheel rotation frequencies are 333.33 Hz and 138.46 Hz. The simulated meshing force reproduces the expected structure. The start-up transient, dominated by the acceleration ramp, produces amplitudes far above the steady state. In the steady regime the mean force is largest in the vertical direction (1.8655×104 N), intermediate in the horizontal direction (1.6968×104 N) and smallest along the axis (6.5991×103 N). The spectrum peaks at the mesh frequency and its harmonics, and each peak carries sidebands spaced at the pinion rotation frequency — a signature of the keyway-induced unbalance that becomes pronounced at 20,000 r/min. This force history is the input for all subsequent structural work.
Modal Behaviour and the Resonance Margin
Hexahedral meshes of the wheel and pinion were generated by partitioning the tooth, web and shaft into blocks and sweeping a mapped two-dimensional mesh through each, using reduced-integration first-order brick elements throughout so that nodes are shared across part interfaces. Modal analysis in the frequency domain then solves the undamped eigenproblem
$$(\mathbf{K}-\omega_i^{2}\mathbf{M})\,\boldsymbol{\Phi}_i=\mathbf{0}$$
with the mesh contact treated as a tie constraint, since contact nonlinearity cannot be retained in a linear modal extraction. A cutoff of 14,000 Hz captured 99.82%, 99.32% and 99.90% of the effective modal mass in the three translational directions, which is sufficient for a subsequent mode-superposition solution.
For the assembled pair, the first six natural frequencies are 67.6, 1519.9, 1537.1, 2054.3, 2081.6 and 2514.0 Hz. The corresponding mode shapes are dominated by the wheel: axisymmetric axial motion for the first and sixth, one-nodal-diameter motion for the second and third, and two-nodal-diameter motion for the fourth and fifth. Isolating the wheel raises these frequencies because the assembly carries additional mass and interface stiffness.
| Order | Natural frequency (Hz) | Mode type | Frequency margin Mf vs. 9000 Hz |
|---|---|---|---|
| 1 | 100.32 | Axisymmetric axial | 98.89% |
| 2 | 1719.2 | One nodal diameter | 80.90% |
| 3 | 1719.2 | One nodal diameter | 80.90% |
| 4 | 2126.6 | Two nodal diameters | 76.37% |
| 5 | 2126.2 | Two nodal diameters | 76.37% |
| 6 | 2673.6 | Axisymmetric axial | 70.29% |
with the margin defined as
$$M_f=\frac{\left|f_N-f_m\right|}{f_m}$$
All six margins exceed the 15% design threshold, and the resonance criterion used here is deliberately dual: a travelling-wave resonance requires the excitation harmonic order to match the nodal diameter of the mode as well as the frequency. Two higher-order modes at 8790.5 Hz and 10,700 Hz bracket the 9000 Hz mesh frequency, and their proximity is the reason the nodal-diameter condition, rather than frequency separation alone, governs the assessment; a full speed sweep would be the appropriate way to confirm this in service.
Hole Geometry as a Design Variable
Lightening holes are standard practice in thin webs, and their shape changes the local bending stiffness of the web without touching the tooth flank or the load path. Four hole geometries were compared at equal hole area, so that the comparison isolates shape rather than mass removal: circular, drum-shaped and pillow-shaped (both built from four circular arcs, the drum bulging outward at top and bottom and the pillow curving inward), and triangular. Harmonic response was computed by mode superposition with the ADAMS mesh force applied along the contact line, and acceleration was sampled at two reference points — one on the shaft and one on the web — following the transmission path.
| Hole shape | RMS at point 1 (105 mm/s2) | RMS at point 2 (105 mm/s2) |
|---|---|---|
| Circular | 6.4098 | 4.2870 |
| Drum | 6.6169 | 3.9802 |
| Pillow | 6.2796 | 3.8486 |
| Triangular | 6.6485 | 3.9432 |
The pillow profile gives the lowest response at both locations, and its advantage is consistent across the two points rather than being an artifact of a single measurement location. The ranking at point 1 is pillow < circular < drum < triangular; at point 2 it is pillow < triangular < drum < circular.
Orthogonal Optimization of the Pillow Hole
Five parameters fully determine the pillow-hole pattern: hole count, radial position of the pattern centre, the arc separation distance d, the arc angle θ1 and the pattern inclination θ2. Each was assigned four levels and arranged in an L16 orthogonal array, giving sixteen harmonic-response runs.
| Factor | Level 1 | Level 2 | Level 3 | Level 4 |
|---|---|---|---|---|
| A: hole count | 5 | 6 | 7 | 8 |
| B: pattern centre radius (mm) | 65.5 | 66.5 | 67.5 | 68.5 |
| C: distance d (mm) | 5 | 6 | 7 | 8 |
| D: angle θ1 (°) | 5 | 6 | 7 | 8 |
| E: inclination θ2 (°) | 0 | 5 | 10 | 15 |
Because both responses are of the smaller-is-better type, the runs were scored by signal-to-noise ratio,
$$\eta=-10\lg\!\left(\frac{1}{n}\sum_{i=1}^{n} y_i^{2}\right)$$
and the level averages of η were used to rank the factors. Across the sixteen runs the point-1 RMS spanned 5.5492×105 to 14.367×105 mm/s2, a factor of about 2.6, showing that hole geometry is a strong lever rather than a marginal refinement. The optimum combination was A2B4C3D3E2: six holes, pattern centre at 68.5 mm, d = 7 mm, θ1 = 7°, θ2 = 5°. A confirmation run gave 5.5155×105 mm/s2 at point 1 and 3.4396×105 mm/s2 at point 2, below every run in the array and 13.95% and 19.77% below the original circular-hole design. The peak dynamic stress in the optimized web was 216.6 MPa, within the allowable for the 20CrMnTi blank.
Spoke Profile Curves
With the hole pattern fixed, the transition curve between hub and rim becomes the remaining geometric freedom. Two families were tested: a rapid-decline (exponential-type) curve and an inverse-proportional curve, applied to the front side, the rear side, or both, together with variants in which the rear curve was offset inward by 1 mm or 2 mm, or blended with a large fillet, or left unfilleted.
| Web configuration | RMS at point 1 (mm/s2) | RMS at point 2 (mm/s2) |
|---|---|---|
| Optimized hole pattern only | 5.5155×105 | 3.4396×105 |
| Decline curve, both sides | 6.8485×105 | 3.7994×105 |
| Inverse curve, both sides | 1.5065×106 | 6.4650×105 |
| Inverse curve, front side | 3.8689×107 | 1.6883×107 |
| Inverse curve, rear side | 9.2704×105 | 6.0993×105 |
| Decline curve, front side | 6.2172×105 | 3.5820×105 |
| Decline curve, rear side | 5.9316×105 | 3.6837×105 |
| Decline curve, rear side, large fillet | 5.9558×105 | 3.6920×105 |
| Decline curve, rear side, offset 1 mm inward | 4.4437×105 | 3.5092×105 |
| Decline curve, rear side, offset 2 mm inward | 7.5328×105 | 5.1560×105 |
The inverse-proportional family is clearly unsuitable when applied to the front side, where it raises the response by nearly two orders of magnitude — a reminder that web curvature changes load sharing between the two sides of the blank rather than acting as a purely local smoothing operation. The best configuration is the decline curve on the rear side offset 1 mm inward, and the offset is a non-monotonic parameter: 2 mm of offset is worse than no offset at all.
Comparing this configuration with the original circular-hole design in the frequency domain, the response peaks at the mesh frequency and at the first shaft-frequency sidebands. The mesh-frequency peak falls from 9.571×106 to 8.064×106 mm/s2 at point 1 (−15.75%) and from 7.823×106 to 6.788×106 mm/s2 at point 2 (−13.23%). At the sideband near 8664 Hz the reduction is larger: 2.545×106 to 1.695×106 mm/s2 (−33.40%) at point 1 and 1.743×106 to 1.024×106 mm/s2 (−41.25%) at point 2. Geometric shaping thus attenuates the modulated components of the response more effectively than the carrier itself, which is consistent with its action on the structural transfer path rather than on the excitation.
Damping Ring Mechanics
A ring seated in a groove at the wheel rim adds a dry-friction interface in parallel with the structure. Three configurations are conventional: a closed ring, which is installed by heating the wheel and cooling the ring so that the interference fit generates preload; a C-shaped (split) ring, which is compressed during assembly and bears on the groove through its own elastic tension; and a helical ring threaded into the groove. All three gain contact pressure from centrifugal loading at speed, so retention is not a concern at high rotational speed. The C-shaped ring is the most practical for thin-web aerospace wheels because it is light, simple to manufacture and insensitive to small groove tolerances.
For the C-ring, the contact pressure follows from the free-end deflection of a curved beam. Treating one half of the ring as a cantilever and applying the unit-load method gives
$$\Delta=\int_{0}^{\pi}\frac{MM_1}{EJ}R_0\,d\phi+\int_{0}^{\pi}\frac{K_y QQ_1}{GA}R_0\,d\phi+\int_{0}^{\pi}\frac{NN_1}{EA}R_0\,d\phi$$
where M, Q, N are the bending moment, shear and normal force produced by the contact pressure q, the subscript 1 denotes the corresponding quantities for a unit load, J = bh3/12 for a rectangular section, and Ky = 1.2. For the closed ring the same problem is solved as a thick-walled composite cylinder, in which the interference and the rotational term both appear, the dynamic part scaling with ρω2 and therefore non-negligible at high speed. In both cases the resulting q is converted into a contact stiffness through the Hertz solution for two curved bodies:
$$K_n=\frac{dp}{d\Delta z},\qquad K_t\approx 0.75\,K_n$$
with the equivalent radius taken as half the contact radius at the ring–groove interface. The stiffness values used in the explicit dynamic model, computed for each ring variant, are listed below.
| b (mm) | h (mm) | Δ (mm) | Kn (N/mm) | Kt (N/mm) |
|---|---|---|---|---|
| 4 | 3 | 10 | 705,943.8 | 529,457.8 |
| 3 | 3 | 10 | 581,021.9 | 435,766.4 |
| 5 | 3 | 10 | 819,448.4 | 614,586.3 |
| 6 | 3 | 10 | 925,356.6 | 694,017.4 |
| 4 | 2 | 10 | 617,895.3 | 463,421.5 |
| 4 | 4 | 10 | 776,034.3 | 582,025.7 |
| 4 | 5 | 10 | 834,684.2 | 626,013.2 |
| 4 | 3 | 5 | 708,038.3 | 531,028.7 |
| 4 | 3 | 15 | 704,306.8 | 528,230.1 |
| 4 | 3 | 20 | 702,419.5 | 526,814.6 |
The ring width dominates the stiffness, since it sets the contact area; thickness acts more weakly, and the opening angle has only a marginal effect on the computed contact stiffness — its influence is felt mainly through the elastic preload.
Ring Geometry, Dissipation and Simulated Response
The wheel with its ring was analysed by explicit dynamics, with the rigid-body reference point at the centre of mass, torque ramped over the first 5 ms and speed over the next 5 ms, a total step of 0.05 s, friction coefficient 0.05 at both the tooth contact and the groove, and spring elements at the ring–groove interface carrying the calculated normal and tangential stiffness. Three ring parameters were swept with the others held fixed at b = 4 mm, h = 3 mm, Δ = 10 mm.
The response is non-monotonic in every parameter. Increasing width or opening first reduces and then increases the shaft-point acceleration, while increasing thickness reduces, then increases, then reduces it again — a behaviour consistent with a friction interface whose dissipation depends on the ratio of contact pressure to relative slip amplitude rather than on stiffness alone.
| Ring parameter | Sweep range | Peak-to-peak variation of RMS at point 1 (mm/s2) |
|---|---|---|
| Width b | 3–6 mm | 1.366×106 |
| Thickness h | 2–5 mm | 0.205×106 |
| Opening Δ | 5–20 mm | 0.406×106 |
Width is therefore the primary design variable, by a factor of roughly three to seven over thickness and opening. The selected geometry is b = 5 mm, h = 3 mm, Δ = 15 mm, giving Kn = 817,275.6 N/mm and Kt = 612,956.7 N/mm.
| Reference point | Direction | Without ring (mm/s2) | With ring (mm/s2) | Reduction |
|---|---|---|---|---|
| Point 1 (shaft) | X (axial) | 5.0716×107 | 2.3037×107 | 54.58% |
| Y (vertical) | 5.1418×107 | 3.2507×107 | 36.78% | |
| Z (horizontal) | 5.2336×107 | 3.2328×107 | 38.23% | |
| Point 2 (web) | X (axial) | 12.474×107 | 6.8014×107 | 45.48% |
| Y (vertical) | 12.631×107 | 7.6029×107 | 39.81% | |
| Z (horizontal) | 12.371×107 | 7.4491×107 | 39.79% |
Two features stand out. First, the web point responds roughly 2.5 times more strongly than the shaft point in every direction without the ring, which confirms that the web is the dominant compliant element in the path; the ratio is preserved after the ring is fitted, so the ring reduces the level of the whole chain rather than shifting it elsewhere. Second, the axial direction benefits most (54.58% at the shaft), which is the expected signature of a ring whose bending stiffness resists out-of-plane rim motion. In the frequency domain, the axial peak at the mesh frequency falls from 4.004×106 to 2.060×106 mm/s2, a 48.55% reduction, and the low-frequency peak associated with the eighth pinion-order component shifts from 2650 Hz to 2775 Hz — evidence that the ring also modifies the local stiffness of the rim, not only the damping.
Experimental Validation on a Bevel Gear Rig
A dedicated test stand was built around a variable-frequency 30 kW drive motor, a gearbox accepting the bevel pair, a torque transducer, and a 400 N·m magnetic-particle brake with water cooling. The two wheels — circular-hole and optimized pillow-hole — were run against the same pinion, and the contact patterns were verified to be identical in position and height before testing, so the comparison isolates the web geometry. Both wheels used the same hub and mounting interface. Vibration was measured with a multi-channel acquisition system using three accelerometers on the shaft close to the wheel, in the vertical, axial and horizontal directions, with a 32,768 Hz sampling rate and 12,800 Hz bandwidth. The optimized web-profile variant was not tested; the experiments compare only the hole geometry.
Two operating points were used: 1500 r/min and 2100 r/min, both at 100 N·m, corresponding to mesh frequencies of 675 Hz and 945 Hz.
| Direction | 1500 r/min, 100 N·m | 2100 r/min, 100 N·m | ||||
|---|---|---|---|---|---|---|
| Circular | Pillow | Change | Circular | Pillow | Change | |
| Vertical | 2.6298 | 1.9947 | −24.15% | 5.2553 | 3.7770 | −28.13% |
| Axial | 2.2038 | 2.4573 | +11.50% | 3.5468 | 2.9631 | −16.46% |
| Horizontal | 4.2908 | 2.7460 | −36.00% | 4.7691 | 4.5950 | −3.65% |
| Synthesized RMS | 4.2888 | 3.5931 | −16.22% | 7.0382 | 6.0382 | −14.21% |
Values in 103 mm/s2. The one adverse result, the 11.50% axial increase at 1500 r/min, does not persist at the higher speed, where the axial response falls by 16.46%; the direction of the change is therefore speed dependent, a common outcome when a geometric modification redistributes stiffness between bending directions.
In the frequency domain the pillow-hole wheel reduces the response at the first three mesh harmonics in nearly every direction.
| Direction | 1× (675 Hz) | 2× (1350 Hz) | 3× (2025 Hz) | ||||||
|---|---|---|---|---|---|---|---|---|---|
| Circular | Pillow | Change | Circular | Pillow | Change | Circular | Pillow | Change | |
| Vertical | 0.4772 | 0.6345 | +32.96% | 0.6761 | 0.3072 | −54.56% | 1.236 | 0.8549 | −30.83% |
| Axial | 1.112 | 1.740 | +56.47% | 0.6398 | 0.2849 | −55.47% | 0.2718 | 0.7626 | +180.5% |
| Horizontal | 2.347 | 1.347 | −42.61% | 1.935 | 0.4093 | −78.85% | 1.771 | 0.901 | −49.12% |
Peaks in 103 mm/s2, 1500 r/min at 100 N·m. The large percentage increase at the third axial harmonic is a small-value effect: the absolute peak of 0.7626×103 mm/s2 remains lower than the pillow-hole peaks in the other two directions and lower than the corresponding circular-hole peaks elsewhere. Combining the three directions, the first-harmonic peak falls from 2.64×103 to 2.041×103 mm/s2 (−22.69%), the second from 2.147×103 to 0.5541×103 mm/s2 (−74.19%) and the third from 2.176×103 to 1.457×103 mm/s2 (−33.04%).
| Direction | 1× (945 Hz) | 2× (1890 Hz) | 3× (2835 Hz) | ||||||
|---|---|---|---|---|---|---|---|---|---|
| Circular | Pillow | Change | Circular | Pillow | Change | Circular | Pillow | Change | |
| Vertical | 2.548 | 1.588 | −37.68% | 5.170 | 0.876 | −83.06% | 1.304 | 0.441 | −66.18% |
| Axial | 1.379 | 0.738 | −46.48% | 3.611 | 1.042 | −71.14% | 1.043 | 0.431 | −58.68% |
| Horizontal | 2.451 | 2.743 | +11.91% | 3.841 | 1.124 | −70.74% | 0.582 | 0.410 | −29.55% |
Peaks in 103 mm/s2, 2100 r/min at 100 N·m. Here the pattern is more consistent: nine of nine harmonic-direction combinations except one decrease, the largest reductions occur at the second harmonic, and the only increase (horizontal, first harmonic, +11.91%) reflects the redistribution of energy into a direction that was previously quiet. Synthesizing the three directions, the peaks at 1×, 2× and 3× fall from 3.795×103 to 3.043×103 mm/s2 (−19.82%), from 7.384×103 to 1.739×103 mm/s2 (−76.45%) and from 1.768×103 to 0.6258×103 mm/s2 (−64.60%).
Because the gearbox test necessarily includes bearing and housing dynamics, the two wheels were also struck directly with an instrumented hammer at 100 N on the tooth land adjacent to the hole pattern, with accelerometers bonded in the same three directions. This isolates the wheel from the assembly.
| Direction | Circular: frequency / peak | Pillow: frequency / peak | Reduction at the dominant peak |
|---|---|---|---|
| Axial | 2450 Hz / 2.236×104; 3593 Hz / 4.960×104 | 2403 Hz / 7.740×103; 3504 Hz / 1.668×104 | 65.38% and 66.37% |
| Horizontal | 2450 Hz / 2.124×104; 3842 Hz / 1.205×104 | 2403 Hz / 5.760×103; 3734 Hz / 5.027×103 | 72.88% and 58.28% |
| Vertical | 2450 Hz / 2.114×104 | 2403 Hz / 5.509×103 | 73.94% |
Peaks in mm/s2. The dominant wheel mode shifts from 2450 Hz to 2403 Hz, a 1.9% shift that confirms the pillow hole changes stiffness as well as damping distribution, and the impulse response is reduced by roughly two-thirds to three-quarters in all three directions. The consistency between the isolated-wheel hammer test and the assembled gearbox measurements supports the interpretation that the improvement originates in the web structure itself rather than in the assembly.
What the Results Support
Three conclusions follow from the combination of simulation and test. First, web geometry is a first-order design variable for high-speed bevel gears: with hole area held constant, the pillow profile alone lowered the harmonic response by 14–20% relative to a circular pattern, and the orthogonal refinement of five geometric parameters produced a further measurable gain, all without altering the tooth flank or the contact pattern. Second, the transition curve between hub and rim is at least as sensitive as the hole shape — an inverse-proportional profile applied to the front side raised the response by nearly two orders of magnitude — so curve selection deserves the same parametric treatment as hole layout rather than being left to convention. Third, a C-shaped ring in a rim groove delivers a 36–55% reduction in simulated response at 20,000 r/min, dominated by the axial direction, with ring width as the controlling parameter.
Two scope limits should be stated plainly. The experimental programme reached 2100 r/min, about one-tenth of the 20,000 r/min design point, because of rig capability; the tests therefore validate the structural modification and its direction of effect, not the full high-speed operating regime. The damping ring results, in contrast, are simulation only, resting on a Hertzian contact stiffness and a Coulomb friction coefficient of 0.05, and the ring’s performance at speed depends on the centrifugal contact pressure, which was computed but not measured. The natural extension of this work is a topology-level optimization of the web, in which the hole pattern and the profile curve are treated as a single continuous design field, and higher-speed testing of the ring to confirm the predicted dissipation.