Spiral bevel gears transmit motion between intersecting shafts and are used in vehicle drivelines, construction machinery, and aerospace transmissions because of their high transmission efficiency, load capacity, and low vibration. Their curved tooth flanks are cut with a rotating cutter head, so the local cutting geometry changes continuously along the tooth. Residual stress left by this operation is a surface-integrity variable: tensile stress near the machined surface can promote crack initiation and distort the finished tooth, while compressive stress below the surface can retard crack growth. Controlling the stress profile therefore requires a link between cutting parameters and the resulting stress distribution, not merely a final dimensional check.
Cutting Geometry and Equivalent Oblique Model
In form-cutting of spiral bevel gears, the blank is indexed between tooth slots while a rotating cutter head removes material. The cutter edge is not perpendicular to the cutting velocity: the edge orientation and the velocity vector form an inclination angle, so the local removal process is oblique rather than orthogonal. Modelling the complete tooth slot, cutter head, indexing motion, and transient contact would be computationally expensive. A micro-element approach is therefore used, in which a representative segment of the cutting zone is represented as a three-dimensional oblique cut.
The equivalent model used a rake angle of 12°, a clearance angle of 6°, a tool-nose radius of 0.05 mm, and an inclination angle of 3°. The cutting speed v, feed f, and equivalent depth of cut a were treated as the process variables. This simplification preserves the inclined edge geometry and the local thermo-mechanical loading while avoiding a full five-axis simulation of the gear-cutting cycle.
Finite-Element Representation
The cutting tool was represented as a rigid carbide body. The workpiece was AISI 4340, corresponding to the Chinese grade 40CrNiMoA, a high-strength alloy structural steel used for gears and shafts subjected to cyclic loads. Its density was 7.83 × 10−9 t/mm3, elastic modulus 2 × 105 MPa, Poisson’s ratio 0.29, yield strength 792 MPa, and tensile strength 980 MPa.
The workpiece response was described with the Johnson–Cook constitutive model, which combines strain hardening, strain-rate sensitivity, and thermal softening:
$$
\sigma=\left[A+B(\varepsilon_{pl})^n\right]
\left[1+C\ln\left(\frac{\dot{\varepsilon}_{pl}}{\dot{\varepsilon}_0}\right)\right]
\left[1-\left(\frac{T-T_0}{T_m-T_0}\right)^m\right]
$$
Here, \(\sigma\) is the equivalent stress, \(\varepsilon_{pl}\) the equivalent plastic strain, \(\dot{\varepsilon}_{pl}\) the plastic strain rate, \(\dot{\varepsilon}_0\) the reference plastic strain rate, \(T\) the current temperature, \(T_0\) the ambient temperature, and \(T_m\) the workpiece melting temperature. The constants used were \(A=792\) MPa, \(B=510\) MPa, \(C=0.014\), \(n=0.26\), and \(m=1.03\).
Chip formation was represented with the Johnson–Cook failure criterion. Failure was initiated when the accumulated damage parameter \(\omega\) exceeded unity:
$$
\omega=\sum\frac{\Delta\varepsilon_{pl}}{\varepsilon_{pl}^{f}},\qquad \omega>1
$$
The failure parameters were \(D_1=0.05\), \(D_2=3.44\), \(D_3=-2.12\), \(D_4=0.002\), and \(D_5=0.61\). Contact between chip and rake face was modelled with a modified Coulomb friction law, using a friction coefficient of 0.3 and a limiting shear stress in the sticking region.
Residual stress was extracted after a four-stage simulation sequence: cutting, unloading, constraint conversion, and cooling to room temperature. The normal stress \(S_{11}\) along the X direction was evaluated. Edge regions affected by boundary conditions were excluded, and a layered sampling grid was used with 60 points per layer; the layer mean was taken as the representative stress at that depth.
Residual Stress Profile
Single-factor simulations varied cutting speed from 75 to 235 m/min while the other parameters were held fixed. The calculated profile had a consistent shape: the machined surface carried residual tensile stress, the stress changed to compression with increasing depth, reached a compressive peak, and then decayed toward zero. The surface tensile layer results from incompatible plastic deformation and elastic recovery of the underlying material; below the surface, the mechanical loading dominates and produces compression before the effect decays with depth.
Increasing cutting speed raised the surface tensile stress and reduced the magnitude of the subsurface compressive stress. The interpretation is thermo-mechanical: higher speed generates more cutting heat, which reinforces the surface tensile component, while thermal softening reduces the transfer of mechanical stress into the subsurface. The profile shape remained similar, but the balance between surface tension and subsurface compression shifted with speed.
Five-Level Parameter Array and Stress Response
A five-level, three-factor array was used to cover cutting speed, feed, and depth. The 25 runs are summarized by the factor levels below.
| Level | Cutting speed v (m/min) | Feed f (mm/r) | Depth a (mm) |
|---|---|---|---|
| 1 | 75 | 0.1 | 1.0 |
| 2 | 115 | 0.2 | 1.5 |
| 3 | 155 | 0.3 | 2.0 |
| 4 | 195 | 0.4 | 2.5 |
| 5 | 235 | 0.5 | 3.0 |
Across the array, the peak residual tensile stress ranged from 217.682 to 443.751 MPa, and the peak compressive stress ranged from −191.149 to −428.412 MPa. Feed had a strong effect on both stress magnitudes: larger feed increased tensile stress and increased the magnitude of compressive stress. Cutting speed increased tensile stress but reduced compressive magnitude. Depth had a smaller and non-monotonic influence. Selected runs show the range and the competing responses.
| v (m/min) | f (mm/r) | a (mm) | Peak tensile S1 (MPa) | Peak compressive S2 (MPa) |
|---|---|---|---|---|
| 75 | 0.1 | 1.0 | 217.682 | −307.883 |
| 75 | 0.5 | 3.0 | 347.117 | −428.412 |
| 155 | 0.5 | 1.5 | 389.451 | −375.771 |
| 235 | 0.1 | 3.0 | 322.502 | −191.149 |
| 235 | 0.5 | 2.5 | 443.751 | −322.149 |
Grey Relational Optimization
Because the two stress measures have different targets, they were combined with grey relational analysis. Peak tensile stress was treated as a cost-type index, for which a smaller value is preferred. Peak compressive stress was treated as a benefit-type index using its magnitude, so a larger compressive magnitude receives the higher normalized score. After normalization, grey relational coefficients were calculated with a resolution coefficient of 0.5, and the mean coefficient gave the grey relational degree \(G\). A larger \(G\) indicates a better combination of low surface tension and high subsurface compression.
| Factor | Level 1 | Level 2 | Level 3 | Level 4 | Level 5 | Range |
|---|---|---|---|---|---|---|
| Cutting speed | 0.6935 | 0.6001 | 0.5226 | 0.4628 | 0.4171 | 0.2764 |
| Feed | 0.5644 | 0.5277 | 0.5180 | 0.5217 | 0.5642 | 0.0464 |
| Depth | 0.5573 | 0.5410 | 0.5283 | 0.5308 | 0.5387 | 0.0290 |
The best combination within the tested window was 75 m/min, 0.1 mm/r, and 1.0 mm, with \(G=0.7480\). The second-ranked run, 75 m/min, 0.5 mm/r, and 3.0 mm, reached \(G=0.7331\); its larger compressive magnitude was offset by higher surface tension. The range analysis ranks cutting speed first, feed second, and depth third for the combined index. The feed range was small: level 1 gave 0.5644 and level 5 gave 0.5642. Thus the composite optimum is not sharply sensitive to feed over this window, even though feed produces the steepest change in the individual tensile and compressive responses. Cutting speed dominates the combined grey relational degree because its level-to-level swing is much larger.
Response-Surface Model and Control Runs
A second-order response surface was fitted to relate the grey relational degree \(Y\) to the three process variables:
$$
Y=0.09575-0.0358a-0.918f-0.00167v+1.227f^2+0.0876fa
$$
The model was significant: the regression mean square was 0.050280 and the error mean square 0.000343, giving \(F=146.45\) with \(p < 0.0001\). The coefficient of determination was \(R^2=97.47\%\), and the adjusted value was \(R^2_{\text{adj}}=96.81\%\). Predicted and calculated values were close, and the residuals were scattered around zero rather than showing a systematic trend.
| Source | Degrees of freedom | Sum of squares | Mean square | F value | p value |
|---|---|---|---|---|---|
| Regression | 5 | 0.251400 | 0.050280 | 146.45 | < 0.0001 |
| Error | 19 | 0.006523 | 0.000343 | ||
| Total | 24 | 0.257923 |
Control simulations compared the optimum with three nearby conditions. The first and third alternatives increased the peak tensile stress and reduced the compressive magnitude relative to the optimum. The second alternative increased the compressive magnitude to −333.543 MPa, but it also raised the peak tensile stress from 217.682 to 242.827 MPa. Because surface tension was weighted as the cost-type response, this trade-off did not improve the combined index.
| Case | v (m/min) | f (mm/r) | a (mm) | Peak tensile S1 (MPa) | Peak compressive S2 (MPa) |
|---|---|---|---|---|---|
| Optimum | 75 | 0.1 | 1.0 | 217.682 | −307.883 |
| Control 1 | 115 | 0.1 | 1.0 | 243.542 | −284.614 |
| Control 2 | 75 | 0.2 | 1.0 | 242.827 | −333.543 |
| Control 3 | 75 | 0.1 | 2.5 | 226.682 | −306.383 |
Experimental Validation
Cutting tests were performed with the predicted parameter set. Residual stress was measured by X-ray diffraction with an Xstress 300G2 instrument. Layer-by-layer material removal was carried out with an FH2016DM electrolytic polisher. The gear was wire-cut with a DK7745 machine into single-tooth specimens, and four teeth were measured at the same position on the concave flank near the pitch line.
| Specimen | Predicted S1 (MPa) | Measured S1 (MPa) | Error (%) | Predicted |S2| (MPa) | Measured |S2| (MPa) | Error (%) |
|---|---|---|---|---|---|---|
| 1 | 217.682 | 207.528 | 4.89 | 307.883 | 294.642 | 4.48 |
| 2 | 217.682 | 205.614 | 5.87 | 307.883 | 291.432 | 5.64 |
| 3 | 217.682 | 208.847 | 4.23 | 307.883 | 287.824 | 6.97 |
| 4 | 217.682 | 202.761 | 7.36 | 307.883 | 293.973 | 4.73 |
The maximum discrepancy was 7.36%, and all differences were below 10%. The model consistently overpredicted both tensile and compressive magnitudes, but the error band supports the use of the finite-element and grey-relational procedure for ranking parameters within the tested range. The validation does not establish predictive accuracy outside AISI 4340, this cutter geometry, and the tested ranges of 75–235 m/min, 0.1–0.5 mm/r, and 1–3 mm.
Process Implications and Limits
Within the tested window, the preferred combination is low cutting speed, low feed, and shallow depth: 75 m/min, 0.1 mm/r, and 1.0 mm. This choice minimizes the surface tensile stress while retaining a subsurface compressive peak of about 308 MPa. Increasing feed to 0.5 mm/r can raise the compressive magnitude to about 428 MPa, but it also raises the surface tensile stress, so it is not automatically better for fatigue performance. The appropriate choice depends on whether surface crack initiation or subsurface crack growth dominates the application.
The result is a local optimization, not a production-rate optimization. Low speed and low feed reduce material removal rate, and the preferred combination should be evaluated against cycle time and cost. The finite-element model also treats the tool as rigid and does not resolve tool wear or microstructural evolution. The equivalent oblique cut represents the local tooth-cutting zone rather than the complete indexing cycle, so edge effects and transient cutter-workpiece engagement are outside the model. The regression equation and grey relational ranking are therefore best used as a screening tool: they identify parameter directions and trade-offs before a narrower experimental campaign on the actual gear-cutting machine.