This study investigates the interaction behavior between abrasive particles and the tooth surface of straight spur gears during spindle finishing. Using the discrete element method (DEM), we simulated the particle flow field and analyzed the effects of gear immersion depth and rotational speeds on contact forces and relative velocities. Experimental validation was performed using strain measurements and surface roughness tests. The results reveal periodic particle actions, uneven force distribution between upper and lower tooth flanks, and distinct influences of processing parameters. Increasing depth predominantly enhances contact forces, while increasing speed primarily raises relative velocities. These findings provide insights for optimizing spindle finishing of straight spur gears.
1. Introduction
Straight spur gears are critical components in mechanical transmissions, requiring high surface integrity for long service life and fatigue resistance. Spindle finishing, a type of mass finishing process, is widely used to reduce surface roughness, remove burrs, and improve residual stress on gear surfaces. The process involves placing a workpiece in a rotating drum filled with abrasive media; the relative motion between media and gear leads to material removal and surface modification. Understanding the particle–gear interaction at the contact interface is essential for predicting and optimizing the finishing outcome.
Previous studies have measured contact forces and particle velocities using sensors and high‑speed cameras, but direct observation of the interface remains challenging. Numerical simulation, particularly the discrete element method, offers a powerful tool to overcome these limitations. In this work, we employ DEM to simulate spindle finishing of straight spur gears, focusing on the behavior of particles in contact with the tooth surfaces. We examine the effects of gear immersion depth and rotational speeds on contact forces and relative motion velocities, and validate the simulations through experiments.
2. Principle of Spindle Finishing
Figure presents the schematic of spindle finishing. A straight spur gear is mounted on a fixture that rotates around its own axis while the drum rotates simultaneously. The gear is partially immersed in granular abrasive media. The combined rotation causes relative motion between the particles and the gear surfaces, generating contact forces and sliding velocities that remove material.
The relative velocity \(V\) between a point on the gear tooth and the surrounding particles can be expressed as:
$$V = 2\pi n_1 \left[ r^2 \left(1 – \frac{n_2}{n_1}\right)^2 + R^2 + 2Rr \left(1 – \frac{n_2}{n_1}\right) \cos\theta \right]^{\frac{1}{2}}$$
where \(n_1\) is drum rotational speed, \(n_2\) is gear rotational speed, \(R\) is center distance between drum and gear, \(r\) is distance from the point to gear axis, and \(\theta\) is the angle between the line connecting the point to the gear axis and the line connecting the two centers. According to Archard’s wear model, material removal is proportional to the product of normal force and sliding velocity:
$$\Delta h = \frac{K P v}{H} \Delta t$$
where \(K\) is wear coefficient, \(P\) is normal pressure, \(v\) is relative velocity, \(H\) is material hardness, and \(\Delta t\) is time. Thus, both contact force and relative velocity are key factors governing finishing efficiency.
3. Discrete Element Simulation Setup
We performed DEM simulations using EDEM software. The contact model was the Hertz–Mindlin (no‑slip) model based on Archard’s wear theory. Table 1 lists the material properties of the particles (brown corundum spheres, diameter 3 mm), the drum (steel), and the straight spur gear (40Cr steel). Table 2 gives the contact parameters.
| Material | Density (kg/m³) | Poisson’s ratio | Shear modulus (MPa) |
|---|---|---|---|
| Drum (steel) | 7850 | 0.300 | 7940 |
| Particles (brown corundum) | 2675 | 0.360 | 1260 |
| Gear (40Cr) | 7870 | 0.277 | 8080 |
| Interaction | Restitution coefficient | Static friction | Rolling friction |
|---|---|---|---|
| Particle–drum | 0.50 | 0.35 | 0.10 |
| Particle–gear | 0.43 | 0.36 | 0.10 |
| Particle–particle | 0.46 | 0.39 | 0.10 |
The straight spur gear had module \(m = 5\) mm, number of teeth \(z = 23\), face width \(b = 40\) mm, and pressure angle 20°. The simulation time was set to three gear rotation periods with a Rayleigh time step of 20%. The gear immersion depth \(h_1\) (distance from the gear top face to the static media top surface) and drum speed \(n_1\) were varied as listed in Table 3. The gear speed \(n_2\) was kept proportional to drum speed with a fixed ratio \(n_1:n_2 = 5:4\), matching the experimental setup.
| Immersion depth \(h_1\) (mm) | Drum speed \(n_1\) (r/min) |
|---|---|
| 80, 110, 140 | 12, 21, 30 |
4. Simulation Results and Analysis
4.1 Particle Flow Field Analysis
We first analyzed the particle flow field around the straight spur gear. The rotating drum creates a parabolic upper interface of the media. When the gear is inserted, it obstructs the flow, leading to particle accumulation in front of the gear and a void region behind it. Figure 3 (a typical snapshot at \(h_1=80\) mm, \(n_1=30\) r/min) shows the media height distribution: the maximum accumulation reached 192.23 mm, while the static height was 140 mm. The height difference between front and rear reached 70 mm.
The velocity vectors at three orthogonal planes (xz, yz, xy) provide insight into particle motion. In the yz plane, particles impacting the gear front rise upward (speeds 0–0.3 m/s), then flow over the gear top and accelerate to 0.4–0.6 m/s as they fall into the wake. At the gear bottom, particles pass quickly underneath. In the xy plane, some particles move laterally at about 0.2 m/s toward the drum wall, while others pass the gear right side at velocities above 0.6 m/s due to combined rotation effects.
The motion of particles directly in contact with a single tooth surface was further investigated. Over one gear rotation cycle, three distinct phases were identified: particle filling, stable filling, and particle outflow. During stable filling (about 0.3–0.7 T), the number of contacting particles remains nearly constant (30–40), and their velocities are low (0.01–0.05 m/s). This stage is the main contributor to finishing action.
4.2 Contact Force Analysis on Tooth Surface
We extracted normal contact forces on the working tooth surface (data block 1 in the model). Figure 8 shows the evolution of contact force magnitude and number of contacting particles over one cycle for different drum speeds at \(h_1=80\) mm. The stable‑filling phase exhibits the highest forces, averaging 22.45 times that of the filling phase and 26.24 times that of the outflow phase. Increasing drum speed from 12 to 30 r/min (150% increase) raised the average normal force by only 18%. In contrast, increasing immersion depth from 80 to 140 mm (75% increase) increased the force by 76% (as shown in Figure 9). Thus, immersion depth dominates the contact force magnitude.
We also examined force distribution across the tooth surface. Dividing the tooth into 10 segments along the axial direction and 11 segments along the profile (from root to tip), Figure 10 presents the normal force contour. Forces are highest near the pitch circle, increasing from root to tip. The upper flank (the side facing upward relative to gear rotation) experiences 1.52–1.88 times the force on the lower flank. Axially, the lower end of the tooth (near the bottom face) receives slightly higher forces than the upper end (ratio 1.01–1.15), but this axial non‑uniformity diminishes at larger immersion depths (140 mm).
4.3 Particle Velocity Analysis on Tooth Surface
Focusing on the stable‑filling phase, we calculated the mean relative velocity between contacting particles and the tooth surface. Figure 11 shows the results. At \(h_1=80\) mm, increasing drum speed from 12 to 30 r/min increased the mean relative velocity on the tooth face by about 148%, with the upper flank velocity being 1.35–1.40 times that of the lower flank. At constant speed \(n_1=30\) r/min, increasing immersion depth from 80 to 140 mm increased the relative velocity by only 4%, while the upper/lower ratio remained 1.35–1.45. Hence, rotational speed is the primary factor affecting relative velocity, whereas immersion depth has a minor effect.
These findings are summarized in Table 4.
| Parameter change | Contact force increase | Relative velocity increase |
|---|---|---|
| Immersion depth: +75% (80→140 mm) | +76% | +4% |
| Drum speed: +150% (12→30 r/min) | +18% | +148% |
5. Experimental Verification
To validate the simulation, we constructed a test platform based on an X1400 vertical spindle finishing machine. Strain gauges (BX120‑10AA) were attached to three positions on the gear: position 1 (upper tooth face), position 2 (upper end face), and position 3 (lower end face). The gear was subjected to the same processing parameters as in the simulation. Strain signals were recorded at 10 kHz, filtered with a 200 Hz low‑pass filter, and the negative (compressive) parts were retained. Figure 13 shows typical stress curves: position 1 exhibits clear periodicity corresponding to gear rotation, while positions 2 and 3 show no obvious periodicity. The average stress at position 3 (lower end) was significantly larger than at position 2, consistent with the simulated force distribution.
We compared the measured average stress over one rotation with the simulated average normal contact force at the same locations. Figure 14 shows the trends: both stress and simulated force increased only slightly with speed (17–28% stress increase vs. 5–26% force increase for a 150% speed rise), but significantly with immersion depth (35–59% stress increase vs. 32–40% force increase for a 75% depth rise). These results confirm that immersion depth is the dominant factor for contact force, in agreement with the simulations.
Additionally, we measured surface roughness \(R_a\) before and after 2‑hour finishing tests. Figure 15 shows the roughness reduction percentages. As speed increased from 12 to 30 r/min, the reduction at position 3 (lower end) rose from about 11% to 58%. As depth increased from 80 to 140 mm, the reduction at the same position increased from 28% to 57%. The axial uniformity was also evaluated: at shallow depth (80 mm), the roughness reduction on the upper part of the tooth face was only 17% while on the lower part it was 36%; at 140 mm depth, both parts achieved about 55–62% reduction, indicating improved axial homogeneity (Figure 16). This behavior matches the simulation prediction that increasing immersion depth reduces axial variability.
6. Conclusions
We have systematically studied the particle interaction behavior on straight spur gear tooth surfaces during spindle finishing using DEM simulations and experiments. The main conclusions are:
- Particle motion and contact on a tooth surface exhibit three periodic phases: filling, stable filling, and outflow. The stable‑filling phase dominates the finishing effect, with average contact force 22.5 times that of the filling phase and 26.2 times that of the outflow phase.
- Increasing gear immersion depth primarily enhances the normal contact force (76% increase for 75% depth rise), while increasing rotational speed primarily raises the relative particle velocity (148% increase for 150% speed rise).
- The upper flank of the tooth experiences 1.52–1.88 times the contact force and 1.35–1.45 times the relative velocity of the lower flank, indicating inherent asymmetry.
- Increasing immersion depth significantly reduces axial non‑uniformity of finishing: the difference in roughness reduction between upper and lower axial positions decreased from 19% at 80 mm depth to 7% at 140 mm depth. Changing speed or depth does not significantly affect profile‑wise variability.
- Experimental strain and roughness tests validate the simulation trends, confirming that depth controls force magnitude and speed controls velocity magnitude for straight spur gear finishing.