Resonance mechanism and modeling chain
Spiral bevel gears are used in aero-engine and vehicle drivetrains where axes intersect and the mesh must carry high torque at high speed. Their curved teeth give a high contact ratio, high load capacity, and relatively low noise, but the same disk-like web and rim that make the design compact can support nodal-diameter (ND) modes. In a rotating gear, a nodal-diameter mode splits into forward and backward waves. When a harmonic of the mesh frequency coincides with one of those wave frequencies in the fixed frame, the response can grow into a traveling-wave resonance. The resulting alternating stress at the tooth root, web fillet, or rim can initiate high-cycle fatigue, so the useful design question is not only “what are the natural frequencies?” but “which nodal-diameter families can be excited, and at what shaft speeds?”
The assessment described here addresses that question for an aero-engine spiral bevel gear system. The test article contains one spiral bevel gear pair, two gear shafts, and six bearings. The gear pair and shafts are modeled in three dimensions; the casing and bearings are represented by support stiffness. The chain proceeds from an accurate tooth surface to static contact, prestressed modal analysis, traveling-wave speed prediction, and correlation with a rotating rig. The system-level representation matters because shaft bending and bearing stiffness shift the gear modes relative to a standalone tooth or wheel model.
Tooth-surface and support representation
The pair has 40 pinion teeth and 41 gear teeth. Its main design data are summarized below.
| Parameter | Value |
|---|---|
| Teeth, pinion / gear | 40 / 41 |
| Module | 3.24 mm |
| Mid-point spiral angle | 35° |
| Normal pressure angle | 20° |
| Shaft angle | 80° |
| Face width | 26 mm |
| Hand, pinion / gear | right / left |
| Outer cone distance | 102.077 mm |
| Pitch cone angle, pinion / gear | 39.4065° / 40.5935° |
| Face cone angle, pinion / gear | 41.3227° / 42.4514° |
| Root cone angle, pinion / gear | 37.5486° / 38.6773° |
| Dedendum (root height), pinion / gear | 3.298 / 3.428 mm |
| Addendum, pinion / gear | 2.819 / 2.689 mm |
The flanks were generated from both design and machining data rather than from nominal cone surfaces. Macro geometry was defined in Masta, and the manufacturing module was used to introduce the cutter radius, roll ratio, swivel angle, tilt angle, and second- and third-order modified-roll coefficients. Typical manufacturing inputs included a cutter radius of 82.156 mm for the gear and 89.422/79.983 mm for the pinion concave/convex flanks. A 7 × 9 grid of tooth-surface points was exported, imported into CATIA, fitted with splines, and used to build a mesh patch; the tooth solid was then trimmed and arrayed. This preserves the local curvature and contact path that control mesh stiffness.
The gear pair and shafts were modeled in 3D. All gear shafts use 9310 steel with an elastic modulus of 204 GPa, density of 7,850 kg/m3, and Poisson’s ratio of 0.293. Contact between flanks used a frictional law with a coefficient of 0.1. The bearings and casing were replaced by stiffness at six support surfaces, A–F.
| Bearing location | kx (N/mm) | ky (N/mm) | kz (N/mm) |
|---|---|---|---|
| A | 319,550 | 319,550 | 1 |
| B | 69,973 | 69,973 | 34,243 |
| C | 338,250 | 338,250 | 1 |
| D | 448,380 | 448,380 | 1 |
| E | 307,540 | 307,540 | 1 |
| F | 262,930 | 262,930 | 139,850 |
The table lists the stiffness values applied at the six bearing locations. The 1 N/mm axial entries are the low-stiffness values used at those support points; they do not represent a rigid axial constraint.
Static contact and prestressed modal response
With the convex pinion flank driving the concave gear flank, a 100 N·m torque was applied to the pinion and the driven gear was constrained in rotation. The computed contact pattern was normal, and the maximum stress was 34.081 MPa. This static contact state was then used as the prestress for the modal calculation.
A standard modal solve cannot retain frictional contact nonlinearity, while bonding the flanks would overstiffen the mesh. The prestressed modal solve therefore imported the static contact state, linearized the contact, and extracted modes to 22 kHz. The mode set includes shaft modes and strongly coupled gear/shaft modes. A high-order coupled mode at 16,245 Hz (mode 23) contains pinion 4ND and gear high-order 2ND character. The pinion and gear 3ND family is especially close: 9,594.8, 9,612.9, and 9,629.5 Hz. This separation by component and nodal diameter makes modes easy to label, but it also makes frequency tuning difficult, because shifting one component’s family leaves the other family nearby.
| Mode family | Frequency band (Hz) | Character |
|---|---|---|
| 1 ND | 3,074.7–4,293.6 | Pinion and gear 1ND modes, separate and coupled |
| 2 ND | 4,540.1–5,418.9 | 2ND modes of the pinion and gear, separate and coupled |
| Umbrella | 7,225.0–7,328.4; 14,145 | Coupled umbrella modes; a high-frequency pinion umbrella mode |
| 3 ND | 9,594.8–9,944.9 | 3ND modes, several within tens of hertz |
| High-order 2 ND | 11,548–11,604; 16,154–16,245 | High-order 2ND modes, including a coupled gear mode |
| 4 ND | 15,881–16,099 | 4ND modes of the pinion and gear |
Traveling-wave speed prediction
For a stationary gear, the transverse displacement of a nodal-diameter mode can be written as
$$w(r,\theta,t)=A(r)\cos(m\theta)\cos(\omega t)$$
where w is the transverse displacement, A(r) is the radial amplitude, r is radius, m is the number of nodal diameters, θ is the circumferential angle, ω is angular frequency, and t is time. When the gear rotates, the standing pattern splits into two waves:
$$w(r,\theta,t)=\frac{1}{2}A(r)\left[\cos(m\theta-\omega t)+\cos(m\theta+\omega t)\right]$$
The two terms are forward and backward waves. Their frequencies in the fixed frame are
$$f_f=f_d+\frac{z_1}{z_2}\frac{nm}{60},\qquad f_b=f_d-\frac{z_1}{z_2}\frac{nm}{60}$$
where ff and fb are the forward- and backward-wave frequencies, fd is the dynamic frequency of the nodal-diameter mode, z1 and z2 are the pinion and gear tooth counts, and n is the input speed in r/min. Mesh excitation appears at
$$f_e=\frac{n z_1}{60}$$
Resonance occurs when
$$f_e=f_f\quad\text{or}\quad f_e=f_b$$
Solving for the crossing speed gives
$$n=\frac{60 f_d z_2}{z_1(z_2\mp m)}$$
The minus sign in the denominator corresponds to the forward-wave branch, and the plus sign to the backward-wave branch. The crossing speed therefore depends on m and on z1/z2; it is not the simple 60fd/z1 line.
The Campbell diagram was built from the prestressed modal frequencies and mesh-frequency orders. The main predicted intervals were 4,117–6,043 r/min for 1ND, 6,060–7,828 r/min for 2ND, and 12,270–14,754 r/min for 3ND. The 1ND interval contains several close crossings because the pinion and gear 1ND modes are split. The scan also produced crossings outside the three main bands, including umbrella and high-order families; the experimental comparison below is confined to the 1ND–3ND families.
Test correlation
The rig used the same gear system, instrumented with a three-axis accelerometer at the measurement point. Input speed was swept from 0 to 16,000 r/min at 50–100 r/min/s while a constant 30 ± 2 N·m load was applied at the output. RMS acceleration was recorded in three orthogonal directions.
The three directions showed the same overall trend. A discrete peak appeared at 6,002 r/min, a broad 2ND band at 6,836–7,896 r/min, and a 3ND band at 12,270–14,360 r/min. For comparison with the predicted 2ND interval, the measured 2ND envelope is taken from the 6,002 r/min peak to the upper edge at 7,896 r/min.
| Family | Predicted (r/min) | Measured (r/min) | Offset at interval bounds |
|---|---|---|---|
| 1 ND | 4,117–6,043 | Not identified | — |
| 2 ND | 6,060–7,828 | 6,002–7,896 | +58 (+0.96%) at lower bound; −68 (−0.86%) at upper bound |
| 3 ND | 12,270–14,754 | 12,270–14,360 | 0 at lower bound; −394 (−2.74% of measured upper bound) |
Predicted and measured intervals agree closely for 2ND and 3ND. The 2ND predicted interval lies inside the measured envelope, with bound offsets of 0.96% and 0.86%; the 3ND lower bound is identical, and the upper bound differs by 394 r/min, or 2.74% of the measured upper bound. The largest discrepancy is therefore 2.74%. The 1ND band was not identified in the test. A 1ND mode is strongly affected by boundary conditions at the bearing supports and casing; the model uses linear support stiffness, while the rig has joint compliance and damping. The accelerometer location and transfer path can also attenuate a mode. The absence of a 1ND peak is therefore not proof that no local wave exists, but it does limit confidence in the 1ND prediction.
Engineering interpretation and limits
For this system, the 2ND and 3ND traveling-wave bands are the experimentally supported risks inside 0–16,000 r/min. The correlation supports using the predicted intervals to schedule rapid pass-through or to place damping rings. It does not validate response amplitude or tooth-root stress, because those depend on damping, excitation level, and transfer path.
The model has several limits. The modal solve is linearized about one static contact state; contact stiffness and damping are not resolved as functions of amplitude. The casing and bearings are reduced to discrete stiffness, and the static preload (100 N·m) differs from the rig output load (30 ± 2 N·m). The 1ND family remains boundary-sensitive. A strain or tooth-root stress measurement would be needed to connect the speed correlation to fatigue life.
Within those limits, the workflow provides a defensible system-level screen for spiral bevel gear traveling-wave resonance: accurate tooth flanks, shaft and support compliance, prestressed modal data, and the forward/backward-wave relation place the 2ND and 3ND families within 2.74% of the measured bands. The same procedure can be reused for another spiral bevel gear system when its tooth-surface and support data are updated.