The finishing problem after grinding
Spiral bevel gears are the default choice for right-angle torque transfer in aircraft engine and gearbox transmissions because they combine high load capacity, smooth conjugate motion and a compact axial envelope. Their flanks are generated spatial surfaces, and the contact pattern that carries the load is extremely sensitive to topography: shifts of a few micrometres in flank position or shape redistribute contact pressure, alter transmission error and change the mesh excitation that drives gear whine. A finishing step that removes only micrometres of material can therefore have an effect on noise and durability out of proportion to the amount of material removed.
Profile (form) grinding is the usual final operation on hardened flanks. It produces a regular, well-oriented surface that is geometrically accurate but topographically “peaked”: the asperities are sharp, so the real area of contact is small, local contact stress is high, and the peaks are the first features to be flattened during running-in or to initiate fatigue. Gear honing, lapping, abrasive-flow polishing and electrochemical honing have all been applied to hardened gears to reduce Ra, but each requires dedicated tooling or electrolyte handling, and the achievable uniformity along the face width and around the flanks is often poor.
Barrel finishing (mass finishing with free abrasive media) offers a different route. The workpiece is immersed in a rotating charge of abrasive media and compound; material is removed by the collective action of thousands of uncontrolled contacts rather than by a defined cutting edge. The process is cheap per part, insensitive to component geometry, suitable for batch operation and already established for aeroengine blisks, automotive components and electronic parts. Published gear work, however, has concentrated on involute cylindrical gears — cross-axis machines, the influence of tooth deflection angle, empirical models for root roughness, isotropic surfaces with improved contact fatigue strength, spindle-oscillating variants and horizontal finishing of large gear shafts. Spiral bevel flanks differ in curvature, in taper along the face width and in the way they sit in the media, so those results do not transfer directly.
Working principle of the parallel-spindle configuration
In the parallel-spindle arrangement the workpiece axis and the barrel axis are parallel. The gear is mounted on a vertical spindle and dipped into a barrel filled with abrasive media and a water-based compound; the barrel and the workpiece are driven in opposite senses so that the media slide, roll and press against the flanks. Material removal is a micro-scale consequence of these collisions and indentations, which knock off burrs and grinding marks and progressively blunt the asperity peaks.
The kinematics can be reduced to the relative velocity at a representative point A on the outer circle of the workpiece. Let N be the barrel speed, n the workpiece speed (counter-clockwise positive), Q the distance between the barrel axis and the workpiece axis, q the radius of point A from the workpiece axis, and θ the angle between the line of centres and the radius to A. The two velocity fields, vN from barrel rotation and vn from workpiece rotation, superpose to give the resultant relative velocity v and its cutting angle φ:
$$\mathbf{v}=\mathbf{v}_{N}+\mathbf{v}_{n}$$
with the magnitudes combined geometrically. Writing $k=1-n/N$ for the normalised relative speed ratio, the resultant cutting speed and the cutting angle at A are
$$v=2\pi N\left[Q^{2}+k^{2}q^{2}+2kQq\cos\theta\right]^{1/2}$$
$$\varphi=\arcsin\left(-\frac{2\pi NQ\sin\theta}{v}\right)$$
With N and n in r/min and lengths in metres, v is obtained in m/min. The angle φ is measured between the resultant velocity and the tangent to the workpiece surface at A. When φ is positive the media are driven into the flank and cut; when it is negative the normal component points away from the surface and the contact is non-cutting. This sign behaviour, rather than the magnitude of v, governs where on the component material is actually removed.
Sensitivity of cutting speed and cutting angle
Evaluating the expressions above over the operating envelope of the process — barrel speeds of 32, 42 and 52 r/min, workpiece-to-barrel speed ratios from −3 to 3, a centre distance Q of 410 mm and workpiece radii in the 50–100 mm range — gives four practical rules.
- Rotation sense dominates. Cutting speed is largest when the barrel and the workpiece rotate in opposite senses (n/N < 0), and it grows roughly in proportion to the absolute value of the speed ratio. Counter-rotation is not a refinement; it is the condition under which the process works efficiently.
- Larger radii finish faster. At fixed N and n, v increases with q because the tangential velocity contributed by the workpiece rotation grows towards the outer circle. A wheel of large diameter therefore reaches a given surface quality sooner than a small pinion meshing with it.
- Barrel speed scales the whole field. Raising N increases the resultant cutting speed more or less linearly, so throughput can be tuned through the barrel drive.
- The cutting angle is only weakly sensitive to speed ratio. Over one revolution of the workpiece, φ is negative for roughly half the cycle, so the flanks are machined intermittently; changing the speed ratio alters the magnitude of the velocity far more than it alters this duty cycle.
For the hardware used in the tests described below (Q = 410 mm, q ≈ 85 mm, N = 52 r/min, n = −115 r/min, ratio −2.2), the relative speed at the outermost point varies by roughly a factor of five around a revolution, from about 0.7 m/s to about 3.7 m/s. The intermittent cutting angle and the strongly position-dependent speed together explain the characteristic finishing behaviour: fast initial removal of peaks, then a plateau as the remaining topography requires impacts that the duty cycle rarely delivers.
Experimental conditions
Tests were run on a parallel-spindle finishing platform with a barrel 1 260 mm in diameter, using a mating pair of aviation spiral bevel gears that had already passed carburising, quenching, profile grinding and shot peening. The gears were mounted on dedicated fixtures with the workpiece axis 410 mm from the barrel axis and immersed to a depth of 150 mm.
| Parameter | Value |
|---|---|
| Module | 4.4 mm |
| Pinion / wheel tooth count | 8 / 41 |
| Mean spiral angle | 35° |
| Material | 20Cr2Ni4A, carburised, quenched, ground, shot peened |
| Surface hardness (mean) | 59.2 HRC |
| Barrel diameter | 1 260 mm |
| Abrasive media | Fused alumina, 2 mm × 2 mm oblique triangular chips |
| Barrel fill ratio | 70% of barrel volume |
| Compound | Water and polishing liquid, fixed mixing ratio and flow rate |
| Immersion depth | 150 mm |
| Axis centre distance | 410 mm |
| Barrel speed N | 52 r/min |
| Workpiece speed n | 115 r/min, opposite sense to the barrel |
| Reversal | Equal forward and reverse dwell to balance both flanks |
Roughness was tracked with a portable stylus instrument at 10 min intervals. Four teeth per gear were measured, each on the convex and the concave flank, at three positions along the face width (toe, mid-face and heel), with three repetitions averaged per position. Residual stress was measured by X-ray diffraction on three points per flank, located 5 mm from each end of the flank and 3 mm above the root, with the remaining flank length bisected.
Roughness evolution
The finishing cycle reduced Ra on both members of the pair to roughly half of the ground-and-peened value, and the two members converged towards a common level despite different starting geometry and size.
| Component | Flanks | Ra before (µm) | Ra after (µm) | Reduction |
|---|---|---|---|---|
| Pinion (z = 8) | Convex and concave | 0.8 | ≈0.4 | ≈50% |
| Wheel (z = 41) | Convex and concave | 0.8 | ≈0.3 | ≈62% |
Three features of the Ra–time curves are worth separating. First, the initial state is not uniform. On the pinion the toe region is the roughest and the heel the smoothest, a consequence of the generating method and of the spiral angle, and the convex flank is rougher than the concave one. Second, removal is front-loaded: in the first 20 min the curves are steep, with the convex flank losing height faster than the concave flank at the same axial position, so the convex side is the more efficient of the two under identical conditions. Third, the curves flatten within the test duration; once the change per interval approaches zero the peaks have effectively been removed, and this plateau is the natural stopping criterion. Extending the cycle beyond it buys no further roughness improvement and risks altering the profile and other accuracy attributes.
The wheel reached its plateau in a shorter time than the pinion, consistent with the radius dependence of cutting speed noted above: the larger member sits further from its own axis and therefore sees a higher relative velocity at the same barrel speed. Both members ended with convex and concave flanks at nearly the same roughness, which is the practically important outcome — an asymmetric pair, where one flank is noticeably rougher, biases the mesh excitation towards one direction of rotation.
Surface texture and edge condition
Roughness parameters describe only part of the change. Before finishing the pinion flanks were dull, carried visible grinding marks and shot-peening imprints, and had burrs on the tooth tips and along the flank edges. After finishing the same surfaces were visibly brighter, with no distinguishable tool marks or peening dimples, and the previously sharp edges had become rounded and burr-free.
Profiles confirmed the visual impression: peak amplitudes after finishing were substantially lower than before, with the deep valleys largely preserved rather than smeared over. At 140× magnification the ground surface showed distinct pits and parallel grinding scratches; after finishing the same area appeared as a more uniform, directionally mixed topography with the scratches broken up. This is the signature of free-abrasive action at many random contact angles, and it is the mechanism behind the improved consistency of the flank texture along the face width. A surface that no longer carries a single dominant lay direction is also less likely to produce the strongly directional contact patterns that feed mesh noise.
Residual stress
Compressive residual stress in the flank is a primary determinant of pitting and bending fatigue life, and it is the property most at risk when a finishing operation removes material from a peened surface. In these tests the compressive stress did not merely survive the process; it increased.
| Condition | Residual stress (MPa) | Change |
|---|---|---|
| Before finishing | −605.75 | — |
| After finishing | −899.38 | +48.5% in magnitude |
Negative values denote compression. The 48.5% increase is consistent with the impact component of the process: the media repeatedly press into the flank at shallow angles, superimposing a peening-like layer on top of whatever the shot-peening operation produced, while the material removed is confined to asperity peaks a few tenths of a micrometre high. The two effects are complementary rather than competing — the same contacts that smooth the topography also add compression.
What the process can and cannot deliver
Taken together, the results define a useful but narrow window. Parallel-spindle barrel finishing removes grinding marks and peening imprints, halves Ra, equalises convex and concave flanks, rounds edge burrs and raises compressive stress — all without a defined tool path or part-specific tooling. For spiral bevel gears, where flank topography translates directly into mesh excitation, this is a plausible route to lower meshing noise and better running-in behaviour.
The limitations follow from the same physics. Material removal is intermittent: the cutting angle is negative for about half of each revolution, so the effective machining time is a fraction of the nominal cycle and the removal rate is inherently low. The driving velocity varies strongly with position around the workpiece, which is why a large wheel and a small pinion of the same pair reach the same roughness on different schedules and why uniform finishing along the face width cannot be assumed without measurement. Saturation of Ra marks the end of useful processing; beyond that point the process starts to work on flank form rather than on asperities.
Open questions remain on the quantitative side. The reported data cover one gear pair, one media specification and one speed setting, so the transferability of the 0.4 µm and 0.3 µm end points to other modules, spiral angles and case depths is unestablished. Whether the reduction from roughly 0.8 µm to 0.4 µm is sufficient to produce a measurable change in transmission error or sound pressure level has not been demonstrated here; that requires mesh testing of finished pairs against ground references. And because the process is stochastic, flank form and lead must be verified after finishing, not assumed, if the operation is to be inserted into a production route behind profile grinding.