Spiral bevel gears carry torque between intersecting shafts in aircraft engine accessories, turboprop reduction boxes and helicopter main transmissions. Their teeth are case-carburized, quenched and ground, so the flank that survives millions of contact cycles is a hard, curved, low-roughness surface (62 HRC in the case examined here). Shot peening is applied after grinding precisely because that hard surface is fatigue-critical: a stream of small steel spheres plastically stretches a thin near-surface layer, and the constraint of the undeformed substrate converts part of that stretch into a compressive residual stress field that suppresses crack initiation at the flank and root.
The same impact events that generate the beneficial stress field also sculpt the surface. Each indentation is a permanent dimple; overlapping dimples turn a smoothly ground flank into a cratered one, and the areal roughness parameter Sa can rise from roughly 0.4 µm on the ground tooth to about 1.0 µm once coverage is complete. At the contacting flank this matters: roughness perturbs lubricant film formation, raises local contact pressure and changes running-in behaviour. Pushing the process further does not simply add stress — excessive peening can nucleate surface microcracks that offset the fatigue gain. Selecting pressure, shot size, flow rate and exposure time therefore becomes a multi-objective problem in which residual stress and topography must be optimized together, and solving it by trial peening of real gears consumes both hardware and time.
Why a flat coupon is not enough for this geometry
Most published numerical studies of peening-induced roughness replace the workpiece with a plane or a cylinder and impinge a random cluster of shots on it. On such surfaces the probability that two shots collide in flight is low enough to be ignored. The tooth space of a spiral bevel gear is the opposite case: a narrow, curved channel bounded by two flanks and a root fillet, into which the nozzle fires at an oblique angle. Shots rebound from one flank directly into the path of incoming shots, and the collision rate inside the channel is high. Because those collisions change both the speed and the direction with which a shot actually reaches the surface, they change the indentation depth, the crater density and hence the predicted roughness.
A second geometric complication is kinematic. The gear rotates on the worktable while the nozzle translates along the face width, so the impact velocity of a given shot must be expressed in a frame attached to the tooth flank before it can be used as a load in a deformation model. The workflow developed for this problem therefore splits the physics: a discrete element method (DEM) stage resolves the flight and collision history of the shot stream, and a finite element (FE) stage resolves the plastic indentation of the flank by individual impacts whose velocity, direction and frequency are transferred from the DEM stage.
The experimental reference case
The workpiece was a spiral bevel gear in AISI 9310 high-strength alloy steel, carburized and quenched to a flank hardness of 62 HRC, with the composition given in Table 1. Peening used S110 cast steel shot of 0.3 mm nominal diameter at 55–62 HRC, an Almen intensity of 0.196 A, a shot flow of 9 kg/min and an air pressure of 4.5 bar (0.45 MPa). The nozzle translated at 150 mm/min along the face width with its axis normal to the tooth root and its exit 180 mm from the root, while the gear rotated with the table at 10 r/min.
| Element | C | Si | Mn | P | S | Cr | Ni |
|---|---|---|---|---|---|---|---|
| wt.% | 0.12 | 0.25 | 0.55 | 0.003 | 0.002 | 1.25 | 3.34 |
After peening, the gear was wire-cut and ultrasonically cleaned, and the flank topography was measured by white-light interferometry (Wyko NT9100, 20× objective). Nine 0.5 mm × 0.5 mm patches were taken, three near the heel, three at mid-face and three near the toe, which provided both a mean level and a measure of scatter across the tooth. Areal roughness was evaluated as the arithmetic mean of the absolute height deviations over the sampled points:
$$ S_a = \frac{1}{n}\sum_{i=1}^{n}\left|Z_i\right| $$
where n is the number of data points and Zi the height of the i-th point relative to the reference plane.
Stage one: shot flight in the DEM model
The nozzle-to-flank trajectory was simulated in EDEM with the tooth surface imported as a meshed shell, so that every contact could be tagged by the element number of the flank on which it occurred. Nozzle standoff (180 mm), orientation normal to the root and translation along the face width reproduced the experimental set-up. The initial shot velocity was set from the empirical nozzle relation used for pressure-fed systems:
$$ v = \frac{16.35\,P^{1.53}}{q_m + P} + \frac{29.5\,P^{0.598}}{d + P} + 4.83P $$
with P the air pressure in bar, qm the shot flow in kg/min, d the shot diameter in mm and v in m/s. The relation makes pressure the dominant term, which is why shop-floor adjustments to peening intensity are usually made at the nozzle rather than through the media specification.
Because DEM output is expressed in the machine tool frame, whereas the FE model needs impact data in the gear frame, each recorded impact position Cm and velocity vm was rotated by the angle swept during the flight-to-impact interval, α = ωt, for a table rotation ω:
$$ \mathbf{M}_{mg} = \begin{bmatrix} \cos\alpha & \sin\alpha & 0 \\ -\sin\alpha & \cos\alpha & 0 \\ 0 & 0 & 1 \end{bmatrix}, \qquad \mathbf{v}_g = \mathbf{M}_{mg}\,\mathbf{v}_m $$
A second transformation, built from three adjacent nodes of the flank element, defined a local surface frame whose third axis is the outward normal, so that the normal velocity component of every impact could be isolated. Impacts with very low normal velocity contribute negligibly to indentation and were filtered out rather than carried into the FE stage; this step is what keeps the number of simulated indentations tractable without biasing the crater population toward grazing events.
Stage two: indentation in the finite element model
The indentation model in Abaqus/CAE used a 1 mm × 1 mm × 0.5 mm target block with C3D8R elements. The central impact zone was meshed at 10 µm × 10 µm × 10 µm, a transition zone let the element size grow outward, and the block was surrounded by CIN3D8 infinite elements to absorb the stress waves that would otherwise reflect from the artificial boundaries and contaminate the crater profile. Contact between shot and target was surface-to-surface, with hard normal behaviour and penalty friction; a friction coefficient of 0.15 was adopted, consistent with the observation that values between 0.1 and 0.5 have little influence on the predicted plastic strain.
The shot was treated as an elastoplastic body with isotropic hardening, and both materials used E = 210 GPa, ν = 0.3 and ρ = 7 800 kg/m³, with a shot yield strength of 1 400 MPa. The flank response was described by the Johnson–Cook model,
$$ \sigma = \left(A + B\varepsilon^{n}\right)\left(1 + C\ln\frac{\dot{\varepsilon}}{\dot{\varepsilon}_0}\right)\left[1 – \left(T^{*}\right)^{m}\right], \qquad T^{*} = \frac{T – T_r}{T_q – T_r} $$
with A = 1 234.38 MPa, B = 881 MPa, C = 0.018, n = 0.238 and m = 0.686, a reference temperature of 25 °C and a melting point of 1 510 °C for the 9310 steel. Mass-proportional Rayleigh damping was applied to the target to suppress residual oscillation of the stress waves.
Two details distinguish this stage from a generic single-impact calculation. First, the pre-peening topography was not assumed flat: the measured Sa ≈ 0.4 µm surface was introduced as a predefined field, so the simulation accumulates roughness on a realistic starting surface rather than on an ideal plane. Second, the sequence of impacts delivered to the target was assembled by drawing impact positions at random within the refined central zone, with the count and velocity distribution taken from the DEM statistics. The random placement preserves the stochastic character of a real shot stream, while the DEM stage supplies the correct impact frequency for a given nozzle path and exposure time.
Agreement with measured topography
Running the coupled model with the experimental process parameters and comparing the resulting flank topography with the interferometric measurements on the convex side gives the results in Table 2. The measured Sa ranged from 0.741 µm to 0.938 µm across the three patches, and the simulated values from 0.883 µm to 1.09 µm. Every patch fell within 20% of measurement, with errors of 13.5% to 19.2%.
| Patch position on convex flank | Measured Sa (µm) | Simulated Sa (µm) | Error (%) |
|---|---|---|---|
| a — near heel | 0.938 | 1.090 | 16.2 |
| b — mid-face (main contact zone) | 0.817 | 0.927 | 13.5 |
| c — near toe | 0.741 | 0.883 | 19.2 |
The bias is systematic rather than random: the model over-predicts roughness at all three positions, by 0.09–0.16 µm. Part of this offset is expected, since the simulated field samples a finite number of discrete indentations whereas the real surface has been smoothed by overlapping impacts and by the residual elastic recovery that follows them. For process-selection purposes the size of the bias matters less than its consistency, because parameter trends can be compared on the same basis. The mid-face patch, being the principal contact region of the working flank, was adopted as the reference area for the parameter study that follows; its measured and simulated roughness differ by 0.11 µm.
Exposure time and coverage
Extending the exposure increases the number of impacts per unit area, and coverage was evaluated with an area-based procedure applied to the simulated crater field. At an initial shot velocity of 40 m/s, 50 s of peening produced 98% coverage — effectively full coverage — and 100 s produced 200%.
Roughness does not grow monotonically with time. Starting from Sa = 0.4 µm before peening, the flank reaches Sa = 1.01 µm at 100% coverage, then falls back to 0.97 µm at 200% coverage. The initial rise is the expected accumulation of fresh craters on an undisturbed surface. The subsequent decline is a smoothing effect: once every point of the surface has been struck at least once, further impacts preferentially flatten the ridges that earlier impacts raised, and the crater field becomes more uniform. Two practical consequences follow. First, roughness specifications should be tied to a coverage level, not to a time, since the same time produces different coverage at different shot velocities. Second, the dispersion of Sa across the tooth decreased after peening even though the absolute level increased — the process homogenizes a ground surface whose roughness varied from heel to toe.
Shot velocity
Holding diameter and exposure fixed and varying the initial velocity over the range used in media-velocity measurements gives a monotonic increase in roughness: the maximum Sa on the flank rises from 0.90 µm at 30 m/s to 1.01 µm at 40 m/s and 1.10 µm at 50 m/s. Higher velocity means higher kinetic energy per impact and a deeper plastic zone under each crater, so the crater rims stand higher above the mean surface. The sensitivity is moderate — a 67% increase in velocity produces roughly a 22% increase in peak Sa — which is fortunate, because velocity is the parameter most easily changed on the shop floor through nozzle pressure.
Shot diameter
Diameter is the more aggressive lever. At a fixed velocity of 50 m/s, the time evolution of roughness changes qualitatively as the media coarsens (Table 3). With 0.3 mm shot the curve rises slowly and then declines gently. With 0.6 mm shot the flank reaches a maximum of 1.45 µm at about 7 s and settles near 1.23 µm as exposure continues; with 0.9 mm shot it peaks at 2.00 µm after only about 2 s and settles near 1.65 µm.
| Shot diameter (mm) | Time to peak Sa (s) | Peak Sa (µm) | Settled Sa (µm) |
|---|---|---|---|
| 0.3 | — | 1.10 (at 50 m/s) | 0.97 |
| 0.6 | ≈7 | 1.45 | ≈1.23 |
| 0.9 | ≈2 | 2.00 | ≈1.65 |
The peak arrives earlier with larger media because mass, and therefore kinetic energy at a given velocity, scales with the cube of diameter: a 0.6 mm sphere carries eight times the mass of a 0.3 mm sphere and a 0.9 mm sphere twenty-seven times. Each impact removes a larger volume of material, so the saturated crater field that produces the roughness maximum is reached in far fewer impacts. The higher peak itself follows from the same scaling — the plastic strain under an individual impact grows, the crater radius grows, and the surface becomes measurably rougher. Within the parameter space examined, Sa is thus considerably more sensitive to shot diameter than to shot velocity, a ranking that should inform media selection when a roughness limit constrains the intensity that can be specified.
What the coupled model delivers, and what it does not
The value of the DEM–FE sequence lies in the two things it carries across the interface between stages: a physically resolved impact frequency that accounts for shot–shot collisions inside the tooth space, and an impact velocity expressed in the flank frame. Both are prerequisites for predicting roughness on a curved, rotating, hard-to-access surface rather than on a coupon. Applied to a single gear type in AISI 9310, the model reproduces measured areal roughness within 20% and reproduces the non-monotonic dependence of roughness on exposure time that simpler random-shot models on flat targets also exhibit.
Its scope is also clear. Validation covers the convex flank of one gear geometry and one surface-finish condition; the systematic over-prediction of 0.09–0.16 µm is reported rather than corrected, and a calibration factor would be needed before the model could be used as an absolute predictor on a different tooth form. Coverage was derived from the simulated crater field rather than measured directly, so the reported full-coverage time of 50 s is model-consistent rather than independently verified. The output is topography only — the residual stress field, which is the primary reason for peening in the first place, must be evaluated with a separate mechanical model, and the trade-off between the two cannot yet be optimized inside a single simulation. Within those bounds, the workflow replaces a costly trial-peening loop with a computation that predicts how a given combination of pressure, media size and exposure will leave the tooth flank, and identifies shot diameter as the parameter to fix first when surface roughness is the binding constraint.