Spiral bevel gears carry power between intersecting shafts in vehicle drivelines, helicopter main reducers and machine-tool spindles. Their curved, gradually engaging teeth give a high contact ratio and smooth transfer of load, which is why they dominate right-angle, high-torque applications. The same geometry becomes a liability as shaft speeds rise: mesh frequencies move into the kilohertz range, thin lightweight wheels radiate noise efficiently, and the dynamic component of the mesh force can exceed the steady component during transients. In helicopter main reducers, pinion speeds above 20,000 r/min are routine, and the resulting dynamic loads accelerate tooth distress and shorten bearing life.
Two families of countermeasures are available. Source control acts on the excitation itself — flank modification, machining-parameter tuning, improved accuracy, tooth-tip relief — and is most effective when applied at the design stage. Path control acts downstream, converting part of the vibration energy into heat through a damping element. In a geared rotor the transmission path runs from the tooth flank through the web (spoke plate), the shaft, the bearing housing and finally the casing; attenuation applied near the end of that chain is inefficient because the energy has already spread over a large radiating area. The web is therefore the natural place to intervene. It carries no direct tooth load but accounts for a large share of wheel mass, and it is usually perforated with lightening holes whose number, shape and surrounding profile are often chosen by experience rather than analysis.
This article examines a 27/65 spiral bevel pair rated for 20,000 r/min and 1500 kW, and follows three design levers applied to the driven wheel: the planform of the lightening holes, the meridional curve of the web, and a C-type dry-friction ring inserted in a groove at the wheel rim. Simulation results are then checked against a back-to-back test stand and against hammer impact measurements on the two wheel variants.
Flank design and mesh excitation of the 27/65 pair
The gear set was generated with the Gleason GEMS environment, using the CAGE module for blank data and the TCA/LTCA modules for contact evaluation. Face milling was selected, the shaft angle fixed at 90°, the outer transverse module at 3.25 mm and the mid-face spiral angle at 35° with a left-hand pinion and right-hand wheel. A 76.2 mm cutter head with standard taper was used, and the contact pattern was tuned through the blade angle, pattern-length factor, bias factor, profile-separation factor, contact-position factor and offset until the pattern sat in the middle of the tooth both lengthwise and depthwise, with transmission-error peaks below 100 μrad across the small-end, mid-face and large-end contact positions.
| Parameter | Pinion (z = 27) | Wheel (z = 65) |
|---|---|---|
| Outer transverse module (mm) | 3.25 | 3.25 |
| Mid-face spiral angle (°) | 35 | 35 |
| Normal pressure angle (°) | 20 | 20 |
| Face width (mm) | 23 | 23 |
| Spiral hand | left | right |
| Radial modification coefficient | +0.32 | −0.32 |
| Tangential modification coefficient | +0.04 | −0.04 |
| Outer pitch diameter (mm) | 87.75 | 211.25 |
| Tip diameter (mm) | 94.7732 | 212.5715 |
| Outer cone distance (mm) | 114.3750 | 114.3750 |
| Pitch cone angle (°) | 22.5572 | 67.4428 |
| Face cone angle (°) | 24.7671 | 68.6115 |
| Root cone angle (°) | 21.3885 | 65.2329 |
| Addendum (mm) | 3.8025 | 1.7225 |
| Dedendum (mm) | 2.3335 | 4.4135 |
| Whole depth (mm) | 6.136 | 6.136 |
Tooth-flank point clouds exported from the TCA were imported into UG as curve sets, lofted into a single tooth slot, filleted at 0.5 mm and patterned to the full tooth count; the two bodies were then constrained at 90° with a rolling kinematic check at 10°/s. The resulting instantaneous contact ellipse was well formed at mid-face, with no interference. This solid model was passed to ADAMS through the Parasolid interface, and the mesh was represented by a penalty contact between the two flanks rather than by an ideal gear joint, so that the actual force history could be extracted.
Both bodies were assigned 20CrMnTi properties (7800 kg/m³, E = 2.07×10⁵ MPa, ν = 0.25). The pinion speed was ramped to 20,000 r/min over 0.03 s with a step function and held constant; a braking torque of 1724.306 N·m, corresponding to 1500 kW at the gear ratio, was applied to the wheel. The contact used an impact-function formulation with a stiffness of 8.1956×10⁵ N/mm, damping of 2×10³ N·s/mm, force exponent 1.5, penetration limit 0.01 mm, dynamic and static friction coefficients of 0.06 and 0.08, and friction transition velocities of 10 mm/s and 0.1 mm/s. The simulation ran for 0.2 s at a 10⁻⁵ s step; the computed wheel speed of 49,846.154°/s matched the kinematic ratio.
The steady-state mesh force had a mean of 1.8655×10⁴ N in the vertical direction, 1.6968×10⁴ N horizontally and 6.5991×10³ N axially. Starting transients produced peaks far above these levels, which is the relevant load case for sizing a damping treatment. With 27 pinion teeth at 20,000 r/min the mesh frequency follows from
$$f_m=\frac{n_1 z_1}{60}=9000\ \text{Hz}$$
and the spectra showed clear peaks at 9000 Hz and its harmonics, each flanked by sidebands spaced at the pinion rotational frequency of 333.33 Hz (the wheel rotating at 138.46 Hz). The sidebands were traced to the keyway in the pinion, which offsets its mass centre and becomes increasingly visible as speed rises.
Modal behaviour and the frequency-margin check
Both members were meshed with hexahedral reduced-integration elements (C3D8R) in Hypermesh. The wheel was split into tooth ring, web and shaft, each meshed separately over shared faces so that nodes matched across the interfaces; the single-tooth block was revolved 64 times and the duplicate nodes merged. The tooth contact was replaced by a bonded constraint for the modal step, since contact nonlinearity is not admitted by a linear eigen-solution. A frequency range of 14,000 Hz captured effective modal masses of 99.82%, 99.32% and 99.90% of the structural mass in the three translational directions.
| Mode | Gear pair (Hz) | Driven wheel (Hz) | Margin relative to 9000 Hz |
|---|---|---|---|
| 1 | 67.6 | 100.32 | 98.89% |
| 2 | 1519.9 | 1719.2 | 80.90% |
| 3 | 1537.1 | 1719.2 | 80.90% |
| 4 | 2054.3 | 2126.6 | 76.37% |
| 5 | 2081.6 | 2126.2 | 76.37% |
| 6 | 2514.0 | 2673.6 | 70.29% |
The margin was evaluated as
$$M_f=\frac{f_N-f}{f}$$
with f = 9000 Hz and fN the N-th natural frequency; a margin above 15% is the usual design threshold. Modes 1 and 6 are nodal-circle (umbrella) modes of the wheel with equal displacement around the rim, modes 2 and 3 are one-diameter modes and modes 4 and 5 two-diameter modes. The six lowest modes all sit far below the mesh frequency, but two higher modes lie at 8790.5 Hz and 10,700 Hz. The 8790.5 Hz mode is only about 2% below the mesh frequency, so the margin criterion alone is not conclusive; resonance additionally requires the harmonic order of the excitation to equal the nodal-diameter count of the mode, and on that basis the design was accepted as non-resonant. The closeness of that mode is nevertheless the reason a damping treatment was pursued in parallel with geometry changes.
Four web-hole shapes under harmonic mesh load
The web was perforated with equal-area cut-outs of four planforms: circular, drum-shaped (top and bottom edges formed by outward-convex arcs), pillow-shaped (top and bottom edges formed by inward-concave arcs) and triangular. Equal area was enforced so that the comparison isolated shape rather than the amount of material removed. A mode-superposition steady-state dynamic step was then run with the ADAMS mesh force applied along the contact line, and the acceleration was read at two points on the shaft, downstream of the web in the transmission path.
| Hole planform | 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 shape gave the lowest response at both locations, about 2% below the circular baseline at point 1 and 10.2% below at point 2, and was carried forward. Its concave edges lengthen the load path between the rim and the hub and add local compliance in the radial direction without removing more material than the other planforms.
Orthogonal optimisation of the pillow-shaped hole
Five parameters define the pillow cut-out once its area is fixed: hole count, radial position of the centre, the distance d between the arc centres, the included angle θ₁ and the tilt angle θ₂. Each was varied over four levels and an L16 array was simulated, with the two reference-point RMS accelerations as responses. Because smaller vibration is better, the signal-to-noise ratio used was the smaller-the-better form
$$\eta=-10\log_{10}\!\left(\frac{1}{n}\sum_{i=1}^{n}y_i^{2}\right)$$
| Level | A: count | B: centre radius (mm) | C: distance d (mm) | D: θ₁ (°) | E: θ₂ (°) |
|---|---|---|---|---|---|
| 1 | 5 | 65.5 | 5 | 5 | 0 |
| 2 | 6 | 66.5 | 6 | 6 | 5 |
| 3 | 7 | 67.5 | 7 | 7 | 10 |
| 4 | 8 | 68.5 | 8 | 8 | 15 |
| Trial | A | B | C | D | E | RMS 1 (10⁵ mm/s²) | RMS 2 (10⁵ mm/s²) |
|---|---|---|---|---|---|---|---|
| 1 | 5 | 65.5 | 5 | 5 | 0 | 14.367 | 6.4883 |
| 2 | 5 | 66.5 | 6 | 6 | 5 | 7.7425 | 4.3009 |
| 3 | 5 | 67.5 | 7 | 7 | 10 | 6.2776 | 3.8622 |
| 4 | 5 | 68.5 | 8 | 8 | 15 | 6.9630 | 4.2148 |
| 5 | 6 | 65.5 | 6 | 7 | 15 | 6.2183 | 3.9799 |
| 6 | 6 | 66.5 | 5 | 8 | 10 | 6.7694 | 4.1403 |
| 7 | 6 | 67.5 | 8 | 5 | 5 | 5.5492 | 3.4615 |
| 8 | 6 | 68.5 | 7 | 6 | 0 | 5.8661 | 3.6568 |
| 9 | 7 | 65.5 | 7 | 8 | 5 | 5.8121 | 4.1126 |
| 10 | 7 | 66.5 | 8 | 7 | 0 | 5.9680 | 4.1483 |
| 11 | 7 | 67.5 | 5 | 6 | 15 | 7.8735 | 4.5239 |
| 12 | 7 | 68.5 | 6 | 5 | 10 | 6.7348 | 4.0924 |
| 13 | 8 | 65.5 | 8 | 6 | 10 | 9.1970 | 5.3728 |
| 14 | 8 | 66.5 | 7 | 5 | 15 | 6.3070 | 4.2353 |
| 15 | 8 | 67.5 | 6 | 8 | 0 | 6.2248 | 4.2018 |
| 16 | 8 | 68.5 | 5 | 7 | 5 | 6.0099 | 3.9775 |
Level-mean signal-to-noise ratios pointed to the combination A2B4C3D3E2: six holes, centre radius 68.5 mm, arc-centre distance 7 mm, included angle 7° and tilt 5°. The trials with five holes and small centre radius were consistently the worst, confirming that a few large cut-outs near the hub concentrate stress and lower local stiffness. The verified optimum gave 5.5155×10⁵ mm/s² at point 1 and 3.4396×10⁵ mm/s² at point 2, below every trial in the array and 13.95% and 19.77% below the equal-area circular-hole baseline. Peak dynamic stress in the optimised web was 216.6 MPa, within the allowable range for 20CrMnTi.
Shaping the web profile
With the hole planform fixed, the meridional curve of the rear web face was varied. Two analytic profiles were used as building blocks: a rapid-drop curve and an inverse-proportional curve, both entered as expressions and generated as law curves in UG, then used to trim or pad the solid. Eleven combinations of front/rear profile, fillet treatment and inward offset were simulated.
| Web configuration | RMS 1 (mm/s²) | RMS 2 (mm/s²) |
|---|---|---|
| Orthogonal-test optimum (flat web) | 5.5155×10⁵ | 3.4396×10⁵ |
| Rapid-drop, both sides | 6.8485×10⁵ | 3.7994×10⁵ |
| Inverse-proportional, both sides | 1.5065×10⁶ | 6.4650×10⁵ |
| Inverse-proportional, front only | 3.8689×10⁷ | 1.6883×10⁷ |
| Inverse-proportional, rear only | 9.2704×10⁵ | 6.0993×10⁵ |
| Rapid-drop, front only | 6.2172×10⁵ | 3.5820×10⁵ |
| Rapid-drop, rear only | 5.9316×10⁵ | 3.6837×10⁵ |
| Rapid-drop rear, large fillet | 5.9558×10⁵ | 3.6920×10⁵ |
| Rapid-drop rear, no fillet | 5.9242×10⁵ | 3.6857×10⁵ |
| Rapid-drop rear, offset 1 mm inward | 4.4437×10⁵ | 3.5092×10⁵ |
| Rapid-drop rear, offset 2 mm inward | 7.5328×10⁵ | 5.1560×10⁵ |
The rapid-drop curve on the rear face, offset 1 mm toward the hub, gave the lowest response of the eleven. The front-only inverse-proportional variant was an order of magnitude worse, which shows how sensitive the wheel is to where the section change is placed: a curve that thins the web near the rim reduces local stiffness exactly where the tooth load enters. Offsetting the same curve by 2 mm overshot the optimum and raised the response again, so the benefit is bounded rather than monotonic.
Compared with the initial circular-hole wheel, the final configuration lowered the mesh-frequency peak 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 8662 Hz the reduction was 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. Both structures showed pinion-shaft-frequency modulation of the mesh peak, consistent with the keyway unbalance seen in the dynamic simulation.
C-type damping rings: where the contact pressure comes from
Geometry changes redistribute stiffness; they do not add dissipation. A ring seated in a circumferential groove at the wheel rim does. Three variants are used industrially: a closed ring, a split (C-shaped) ring and a helical ring. The closed ring is fitted with a shrink fit — the wheel is heated, the ring cooled — and thereafter relies on the interference plus centrifugal growth to maintain contact. The C-ring is simply compressed into the groove and presses outward through its own elastic tension. In both cases rotation tightens rather than loosens the joint, so retention is not a concern. The helical form is threaded in and develops the highest friction, again assisted by centrifugal load.
Energy is dissipated by relative sliding in the contact patch. Predicting how much requires the normal pressure and the contact stiffness. For the closed ring, the assembly is treated as a shrink-fitted composite thick-walled cylinder under a centrifugal body force; the Lamé solution with an interference δ and radii a (groove), b (wheel outer) and c (ring bore) yields a contact pressure that rises with the square of rotational speed through a term in ρω²(3+ν)/8, so the centrifugal contribution cannot be neglected at 20,000 r/min. For the C-ring, half the ring is treated as a curved cantilever, fixed at the symmetry plane and free at the split, and the opening Δ is related to the distributed pressure q by the unit-load (Castigliano) integral over the semi-circle,
$$\Delta=\int_{0}^{\pi}\frac{M M_1}{EJ}R_0\,d\phi+\int_{0}^{\pi}\frac{K_y Q Q_1}{GA}R_0\,d\phi+\int_{0}^{\pi}\frac{N N_1}{EA}R_0\,d\phi$$
where M, Q, N are the bending moment, shear and normal force produced by q, the subscripted quantities are those produced by a unit load at the split, J = bh³/12 for a rectangular section of width b and thickness h, and Ky = 1.2. In rotation, the centrifugal field ρAω²R₀² adds directly to these resultants, again increasing the pressure with speed.
Contact stiffness follows from Hertz theory for two equivalent spheres of radius R, with the equivalent modulus E* combining both bodies. The normal approach and its derivative give
$$K_n=\frac{dp}{d(\Delta z)}=2E^{*}\sqrt{R\,\Delta z},\qquad K_t\approx(0.67\text{–}0.86)\,K_n$$
and a tangential-to-normal ratio of 0.75 was adopted for the ring–groove pair. Applying these relations to each candidate geometry produced the stiffness set below; the equivalent sphere radius was taken as half the contact radius, and both parts are 20CrMnTi.
| 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 |
Ring geometry versus vibration
The wheel with ring groove was analysed with an explicit dynamic step. A rigid-body reference point at the centre of mass carried the rotational inertia, torque was ramped in over 0–0.005 s and speed over 0.005–0.01 s, and the run lasted 0.05 s. Contact was defined between the two tooth flanks and between ring and groove, with spring elements of the tabulated stiffness bridging the ring–groove interface and a friction coefficient of 0.05. The response metric was the mean of the three-direction resultant RMS acceleration at a point on the shaft, downstream of the web.
Increasing the ring width from 3 mm to 6 mm first lowered and then raised the response, with a minimum at b = 5 mm. The opening behaved the same way, minimising at Δ = 15 mm. Thickness was non-monotonic with two extrema, and h = 3 mm was selected. Over the ranges swept, the width produced the largest excursion, 1.366×10⁶ mm/s², against 0.406×10⁶ mm/s² for the opening and 0.205×10⁶ mm/s² for the thickness. Width dominates because it sets the contact area, and therefore the friction force available at a given pressure; thickness and opening change the pressure and stiffness but leave the sliding interface roughly the same size.
The selected ring (b = 5 mm, h = 3 mm, Δ = 15 mm) has Kn = 817,275.6 N/mm and Kt = 612,956.7 N/mm. Comparing the same wheel with and without it:
| Point | Direction | Without ring (mm/s²) | With ring (mm/s²) | Reduction |
|---|---|---|---|---|
| 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% | |
| 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% |
The web point vibrates roughly 2.4 times more strongly than the shaft point, which is expected because the web is upstream in the transmission path. The axial direction benefits most, since the ring resists out-of-plane motion through its own bending stiffness as well as through friction. In the spectrum, the axial peak at 9000 Hz fell from 4.004×10⁶ to 2.060×10⁶ mm/s², a 48.55% reduction, and the low-frequency peak shifted from 2650 Hz to 2775 Hz — a shift consistent with the added stiffness of the ring–groove joint rather than with mass loading alone.
Test-stand validation at two speeds
Two driven wheels were manufactured, one with circular lightening holes and one with the optimised pillow-shaped holes; both were run against the same pinion so that flank geometry, contact pattern and mounting distance were identical. The contact ellipses of the two wheels were verified to coincide before testing. The rig comprised a 30 kW variable-frequency motor (YP-50-30-4), the test gearbox with eccentric bearing sleeves for setting the three mounting distances, a torque transducer, and a CZ-40 magnetic-particle brake rated at 400 N·m with water cooling. Triaxial accelerometers were mounted on the housing next to the driven shaft, matching the reference points used in the simulations. Signals were acquired with an M+P system at a 32,768 Hz sampling rate and 12.8 kHz bandwidth. Two conditions were run at 100 N·m: 1500 r/min, giving a mesh frequency of 675 Hz, and 2100 r/min, giving 945 Hz.
| Speed | Direction | Circular (10³ mm/s²) | Pillow (10³ mm/s²) | Change |
|---|---|---|---|---|
| 1500 r/min | Vertical | 2.6298 | 1.9947 | −24.15% |
| Axial | 2.2038 | 2.4573 | +11.50% | |
| Horizontal | 4.2908 | 2.7460 | −36.00% | |
| 2100 r/min | Vertical | 5.2553 | 3.7770 | −28.13% |
| Axial | 3.5468 | 2.9631 | −16.46% | |
| Horizontal | 4.7691 | 4.5950 | −3.65% |
The three-direction resultant fell from 4.2888×10³ to 3.5931×10³ mm/s² at 1500 r/min (−16.22%) and from 7.0382×10³ to 6.0382×10³ mm/s² at 2100 r/min (−14.21%). The single axial increase of 11.50% at the lower speed is the one adverse result in the time domain, and it disappeared at 2100 r/min.
| Speed | Direction | Order | Circular (10³ mm/s²) | Pillow (10³ mm/s²) | Change |
|---|---|---|---|---|---|
| 1500 r/min | Vertical | 1× | 0.4772 | 0.6345 | +32.96% |
| Vertical | 2× | 0.6761 | 0.3072 | −54.56% | |
| Vertical | 3× | 1.236 | 0.8549 | −30.83% | |
| Axial | 1× | 1.112 | 1.740 | +56.47% | |
| Axial | 2× | 0.6398 | 0.2849 | −55.47% | |
| Axial | 3× | 0.2718 | 0.7626 | +180.5% | |
| Horizontal | 1× | 2.347 | 1.347 | −42.61% | |
| Horizontal | 2× | 1.935 | 0.4093 | −78.85% | |
| Horizontal | 3× | 1.771 | 0.9010 | −49.12% | |
| 2100 r/min | Vertical | 1× | 2.548 | 1.588 | −37.68% |
| Vertical | 2× | 5.170 | 0.8760 | −83.06% | |
| Vertical | 3× | 1.304 | 0.4410 | −66.18% | |
| Axial | 1× | 1.379 | 0.7380 | −46.48% | |
| Axial | 2× | 3.611 | 1.042 | −71.14% | |
| Axial | 3× | 1.043 | 0.4310 | −58.68% | |
| Horizontal | 1× | 2.451 | 2.743 | +11.91% | |
| Horizontal | 2× | 3.841 | 1.124 | −70.74% | |
| Horizontal | 3× | 0.5820 | 0.4100 | −29.55% |
A detail worth reading carefully: the large percentage increases at 1500 r/min are attached to small absolute values. The axial third harmonic rises by 180.5%, but from 0.2718×10³ to 0.7626×10³ mm/s², which is still the smallest peak among the three directions for the pillow wheel. Per-direction spectra therefore do not tell the whole story; the direction-summed spectra do. Combining the three directions, the pillow wheel reduced the 1500 r/min peaks at 1×, 2× and 3× mesh order from 2.640×10³ to 2.041×10³ mm/s² (−22.69%), from 2.147×10³ to 0.5541×10³ mm/s² (−74.19%) and from 2.176×10³ to 1.457×10³ mm/s² (−33.04%). At 2100 r/min the same comparison gave 3.795×10³ to 3.043×10³ mm/s² (−19.82%), 7.384×10³ to 1.739×10³ mm/s² (−76.45%) and 1.768×10³ to 0.6258×10³ mm/s² (−64.60%). The second-order harmonic benefits most, and the low-frequency content of the pillow wheel is visibly lower, indicating that part of the vibration was redistributed into frequency bands where the structure radiates less.
Hammer impact on the two wheels
Run-up tests cannot separate the wheel from the housing and bearing dynamics. Both wheels were therefore struck with an instrumented hammer at 100 N on the teeth adjacent to the lightening holes, with accelerometers on the wheel in the axial, horizontal and vertical directions, and the impulse responses transformed to the frequency domain.
| Direction | Circular peak | Pillow peak | Reduction |
|---|---|---|---|
| Axial | 2.236×10⁴ mm/s² at 2450 Hz | 7.740×10³ mm/s² at 2403 Hz | 65.38% |
| Axial (second peak) | 4.960×10⁴ mm/s² at 3593 Hz | 1.668×10⁴ mm/s² at 3504 Hz | 66.37% |
| Horizontal | 2.124×10⁴ mm/s² at 2450 Hz | 5.760×10³ mm/s² at 2403 Hz | 72.88% |
| Horizontal (second peak) | 1.205×10⁴ mm/s² at 3842 Hz | 5.027×10³ mm/s² at 3734 Hz | 58.28% |
| Vertical | 2.114×10⁴ mm/s² at 2450 Hz | 5.509×10³ mm/s² at 2403 Hz | 73.94% |
Every isolation test showed a lower peak for the pillow wheel, and the dominant peak also shifted downward from 2450 Hz to 2403 Hz. The reduced amplitude and the slight frequency drop together indicate a more compliant web with better distributed mass rather than a simple stiffening effect: the mode that the hammer excites most strongly is both softer and less responsive at the measurement points.
What the results support
Three conclusions follow from the numerical and experimental data taken together. First, the planform of a lightening hole matters at the same removed area: a pillow-shaped cut-out with concave upper and lower edges outperforms circular, drum and triangular holes at both reference points, and its geometry can be optimised by an orthogonal array with signal-to-noise ranking, reaching 13.95% and 19.77% lower RMS acceleration than the circular baseline. Second, the meridional profile of the web sets how effectively that gain reaches the shaft; a rapid-drop rear profile offset 1 mm inward produced the lowest response of eleven candidates and cut the mesh-frequency peak by 15.75% and 13.23% at the two points, with larger reductions of 33.40% and 41.25% at the sideband. Third, a C-type ring with b = 5 mm, h = 3 mm and Δ = 15 mm adds dissipation that geometry alone cannot supply, lowering the axial vibration by 54.58% at the shaft, 45.48% at the web, and the 9000 Hz axial peak by 48.55%; all three directions improve by more than 35%, and width is the parameter to control first because it governs contact area.
The test campaign supports the hole-shape result independently of the simulation: combined time-domain RMS fell by 16.22% at 1500 r/min and 14.21% at 2100 r/min, direction-summed peaks at the first three mesh orders fell by 22.69%/74.19%/33.04% and 19.82%/76.45%/64.60%, and the hammer tests showed 58% to 74% lower impulse peaks in all three directions. The two validation routes agree on the ranking of the two wheels, which is the meaningful claim; the absolute acceleration levels from an explicit simulation with rigid-body reference points should not be read as quantitative predictions of housing vibration.
Several limits should be stated. The study covers one gear pair at one load, and only the driven wheel was modified; the pinion, shaft, bearings and housing were left unchanged, although the sidebands in the measured spectra show that the pinion keyway contributes measurably to the excitation. Testing stopped at 2100 r/min for rig reasons, well below the 20,000 r/min design point where centrifugal effects on the ring contact pressure are strongest, so the ring’s benefit at full speed rests on the simulation rather than on measurement. The two higher modes at 8790.5 Hz and 10,700 Hz sit close enough to the 9000 Hz mesh frequency that the nodal-diameter argument, rather than the margin alone, carries the non-resonance conclusion; a mistuned or higher-order excitation could invalidate it. Topology optimisation of the web, rather than parameterised hole shaping, would likely yield further mass-neutral gains.