Spiral bevel gears transmit torque between intersecting shafts through curved, conjugate tooth flanks. Because those flanks are spatial surfaces, their roughness and edge condition affect lubricant film formation, contact-pressure distribution, vibration, and fatigue life. As-ground flanks often retain directional grinding marks, burrs at tooth tips and outer edges, and sharp transitions. Barrel finishing is a terminal mass-finishing operation that uses a rotating charge of abrasive media and a liquid compound to remove such defects through repeated impact and sliding contact. It is attractive for aerospace gears because it can treat finished teeth without a defined tool path and can reach regions that are difficult to brush or polish by hand.
For a disc-type spiral bevel gear, the tooth spaces are narrow and the flank curvature is strong. Media flow is therefore nonuniform: some regions are shielded, while others receive concentrated collisions. A parallel-spindle arrangement, in which the workpiece axis is parallel to the drum axis, allows the gear to be immersed and rotated while the drum turns. The relative motion generates a cutting-speed field over the flank. Quantifying this field and the resulting wear is the basis for selecting spindle speed and finishing time.
Parallel-spindle kinematics and cutting speed
Let \(OZ\) be the drum axis and \(O_2Z_2\) the workpiece axis, separated by the center distance \(R\). With the drum taken as fixed, the carrier angle is \(\theta_1=2\pi Nt\), and the workpiece rotates relative to the carrier by \(\theta_2=2\pi nt\). The two rotations are opposite in the experimental arrangement. For a point \(A\) at radius \(r\) from the workpiece axis, the planar position is
$$
\mathbf{p}_A =
\begin{bmatrix}
R\cos\theta_1 + r\cos(\theta_1-\theta_2)\\
R\sin\theta_1 + r\sin(\theta_1-\theta_2)\\
0
\end{bmatrix}
\tag{1}
$$
where \(N\) and \(n\) are rotational frequencies in \(\mathrm{s^{-1}}\). Differentiating Eq. (1) and taking the magnitude gives the local kinematic cutting speed:
$$
v = 2\pi\left[N^2R^2+(N-n)^2r^2+2RrN(N-n)\cos\theta_2\right]^{1/2}
\tag{2}
$$
In Eq. (2), the first term is the carrier motion, the second is the relative rotation of the workpiece, and the third is their geometric coupling. The angle \(\theta_2\) makes the speed position-dependent: a point on the gear sees a cyclic speed as it travels around the drum axis. Equation (2) is a tooling-imposed kinematic field, not the actual abrasive velocity. The real granular charge moves with collisions, shearing, and local stagnation, so the particle-scale velocity and contact force require a discrete-element treatment.
The process window for the aviation gear was \(N=45\ \mathrm{r/min}\), \(n=-45\) to \(-135\ \mathrm{r/min}\), \(R=0.41\ \mathrm{m}\), and \(r=0.05\) to \(0.1\ \mathrm{m}\). Evaluating Eq. (2) over this window shows that the cutting speed increases with the workpiece radius, because the peripheral speed of the gear about its own axis grows with \(r\), and with the spindle speed, because the relative angular speed \(N-n\) increases. The drum speed also contributes directly through the \(N^2R^2\) term. These trends identify spindle speed and radial position as first-order variables. They do not by themselves determine finishing uniformity, which also depends on media accessibility, contact frequency, and the local angle between the relative velocity and the tooth surface.
Discrete-element model of media and wear
The discrete-element method resolves the abrasive charge as a collection of interacting particles. Drum and gear CAD models were imported into EDEM and scaled to keep the particle count and time step tractable. The drum speed was \(N=-50\ \mathrm{r/min}\) and the workpiece speed was \(n=115\ \mathrm{r/min}\) (counterclockwise positive), so the two rotations were opposite. The charge consisted of 200,000 spherical particles with a 2 mm mean diameter and a normal size distribution. The abrasive density was \(4.0\times10^{-9}\ \mathrm{kg/mm^3}\) (\(4.0\times10^3\ \mathrm{kg/m^3}\)), with Poisson ratio 0.25 and shear modulus 129 GPa. The gear steel was modeled with density \(7.8\times10^{-9}\ \mathrm{kg/mm^3}\) (\(7.8\times10^3\ \mathrm{kg/m^3}\)), Poisson ratio 0.3, and shear modulus 70 GPa. Contact parameters are listed in Table 1.
| Contact pair | Rolling friction coefficient | Static friction coefficient | Restitution coefficient |
|---|---|---|---|
| Abrasive–abrasive | 0.15 | 0.50 | 0.50 |
| Abrasive–gear steel | 0.01 | 0.55 | 0.30 |
Particle motion started at 2 s to allow the charge to settle. The total simulated time was 4 s, with a 0.02 s time step and a grid size of \(3R_{\min}\), where \(R_{\min}\) is the smallest particle radius. Wear was computed with the Hertz–Mindlin with Archard wear model. The model accumulates wear from the normal contact force and the relative sliding distance at each contact, so it responds to both local impact intensity and sliding path.
The wear map at a representative instant was nonuniform, with greater loss at the small end of the disc-type gear. This is consistent with the reverse-taper geometry: the small end presents a different area and local flow condition to the media stream, while the large end is partly shielded by adjacent teeth. Three spindle speeds, \(n=115\), \(75\), and \(50\ \mathrm{r/min}\), were compared. The accumulated wear between 2.1 s and 4 s increased with time in all cases. The highest spindle speed produced the largest wear, because the speed difference between the workpiece and drum increased, raising the collision force of the media on the gear. This ranking agrees with the kinematic trend in Eq. (2).
The absolute wear values are not a prediction of production finishing time. The model used scaled geometry and a 4 s window, so the result is a comparative measure of parameter sensitivity rather than a full-cycle stock-removal calculation.
Experimental setup and surface response
The experimental configuration used a parallel-spindle barrel-finishing platform. The workpiece was a disc-type spiral bevel gear from a pair with module \(m=4.4\ \mathrm{mm}\), 41 teeth, and midpoint spiral angle \(\beta=35^\circ\). The material was 20Cr2Ni4A; the gears had been carburized and quenched, form-ground, and shot-peened, with a mean tooth-flank hardness of 59.2 HRC. The drum diameter was 1260 mm. The media were 2 mm spherical corundum-based abrasive, filling 70% of the drum volume. The compound was water mixed with polishing liquid at a fixed ratio and fixed flow rate. The gear was mounted on a fixture with its axis parallel to the drum axis at a center distance of 410 mm, immersed 150 mm into the media. Forward and reverse spindle times were equal, and the drum and workpiece rotated in opposite directions, to reduce left–right asymmetry.
Roughness was measured with a contact profilometer capable of curved-surface measurement. A dedicated fixture and a small indexing plate positioned the same flank repeatedly. Measurements were taken every 6 min; each value is the average of three readings at the same location.
\(R_a\) fell from about \(0.8\ \mu\mathrm{m}\) to about \(0.4\ \mu\mathrm{m}\). The decrease was not uniform across the flank. Before finishing, the convex flank had a higher \(R_a\) than the concave flank. The concave flank showed a steeper initial slope, attributed to local particle accumulation in the narrow tooth space: more particles collided with the concave surface, increasing local wear. The curves flattened as the highest peaks were removed, providing a practical endpoint criterion. The small end also showed a higher roughness-reduction rate than the large end, consistent with the wear distribution in the DEM model.
After finishing, tooth-tip and outer-diameter burrs were removed, edges were rounded, and grinding marks on the flanks and in the tooth spaces were reduced. Under \(140\times\) digital optical microscopy, the surface appeared more uniform and tended toward isotropic topography. Table 2 summarizes the observed changes.
| Characteristic | Before finishing | After finishing |
|---|---|---|
| Tooth-flank \(R_a\) | \(\approx 0.8\ \mu\mathrm{m}\) | \(\approx 0.4\ \mu\mathrm{m}\) |
| Tooth-tip and outer-diameter burrs | Present | Removed |
| Edge condition | Sharp | Rounded |
| Surface texture | Directional grinding marks | Reduced marks; more isotropic |
Process interpretation and limits
Taken together, the kinematic model, DEM wear simulation, and gear tests describe a consistent mechanism. The parallel-spindle motion creates a position-dependent cutting-speed field. The media remove material fastest where access and relative speed are high. The roughness curve saturates when the tallest peaks are gone. For spiral bevel gears, the practical benefit is not only a lower \(R_a\) but also edge rounding and removal of burrs that can act as stress raisers. A more isotropic surface is expected to be less sensitive to sliding direction in the contact, which is relevant to noise and contact fatigue, although those effects require separate testing.
The results are bounded by the test conditions. They cover one aviation disc-type spiral bevel gear geometry, one media size and fill level, and a short DEM window. The roughness data do not quantify residual-stress redistribution, subsurface damage, or contact-fatigue life. The DEM wear ranking is useful for choosing spindle speed and identifying nonuniformity, but absolute material removal in a production cycle would require a longer calibrated simulation or direct stock-removal measurements. Within this scope, the process reduced \(R_a\) by roughly half, removed edge burrs, and produced a more uniform flank topography on the spiral bevel gears.