Particle Dynamics in Spindle Finishing of Cylindrical Gears: A DEM-Based Investigation

Spindle finishing is a critical process for enhancing the surface integrity of cylindrical gears, particularly in achieving isotropic surface textures and improved fatigue resistance. This study investigates the granular interaction mechanisms at the gear-to-abrasive particle interface using Discrete Element Method (DEM) simulations. By analyzing velocity fields, contact forces, and parametric dependencies, we establish a framework for optimizing cylindrical gear finishing processes.

Numerical Modeling and Simulation

The DEM model replicates industrial spindle finishing conditions with the following key parameters:

Material Density (kg/m³) Poisson’s Ratio Shear Modulus (MPa)
Drum (Steel) 7,850 0.300 7,940
Abrasive (Al₂O₃) 2,675 0.360 1,260
Gear (40Cr) 7,870 0.277 8,080

The contact mechanics follow the Hertz-Mindlin model with Archard wear integration:

$$ \Delta h = \frac{KPv}{H}\Delta t $$

where \( \Delta h \) represents wear depth, \( K \) the dimensionless wear coefficient, \( P \) normal pressure, \( v \) relative velocity, and \( H \) material hardness.

Granular Flow Characteristics

For a cylindrical gear with 23 teeth (m=5mm), three distinct flow phases emerge:

Phase Duration (% cycle) Particle Count Velocity Range (m/s)
Filling 0-30% 0→35 0.10-0.25
Stable Contact 30-70% 35±5 0.01-0.05
Discharge 70-100% 35→0 0.15-0.35

The velocity gradient across tooth profiles follows:

$$ v_{rel} = 2\pi n_1\sqrt{r^2\left(1-\frac{n_2}{n_1}\right)^2 + R^2 + 2Rr\left(1-\frac{n_2}{n_1}\right)\cos\theta} $$

where \( n_1 \) and \( n_2 \) denote drum and gear speeds, respectively.

Parametric Sensitivity Analysis

Key process parameters significantly affect cylindrical gear surface interactions:

Parameter Range Contact Force Δ Velocity Δ
Immersion Depth (mm) 80→140 +76% +4%
Rotational Speed (rpm) 12→30 +18% +148%

Asymmetric loading occurs due to media accumulation:

$$ \frac{F_{upper}}{F_{lower}} = 1.52-1.88 $$
$$ \frac{v_{upper}}{v_{lower}} = 1.35-1.45 $$

Surface Finish Optimization

Experimental validation reveals axial roughness reduction gradients:

Depth (mm) Upper Surface ΔRa Lower Surface ΔRa Axial Uniformity
80 17% 36% 1:2.12
140 62% 55% 1:1.13

This demonstrates that deeper immersion (140mm) reduces axial disparity by 46% compared to shallow conditions.

Conclusion

For cylindrical gear finishing, our DEM analysis recommends:

1. Prioritize immersion depth adjustments for contact force control
2. Use rotational speed modulation for velocity-sensitive applications
3. Maintain n₁:n₂ = 5:4 ratio for optimal media flow
4. Implement >100mm immersion to minimize axial variance

The established correlations between process parameters and surface outcomes enable predictive optimization of cylindrical gear finishing, particularly for high-precision transmission components requiring uniform surface integrity.

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