Spiral bevel gears transmit rotation between intersecting shafts and are standard components in precision machine tools, high-speed rail drives and aerospace transmissions. Because the tooth flanks carry the entire contact load and dominate the acoustic signature of the drive, their surface quality is a functional requirement rather than a cosmetic one. Grinding with a cup-shaped wheel is the finishing operation that establishes this quality: the wheel rotates at high speed while the workpiece executes a generating motion, and the convex flank is formed by the outer side of the wheel rim while the concave flank is formed by the inner side. The same kinematic setting therefore produces two flanks with different roughness, and both must be controlled simultaneously.
Physics-based roughness models for gear grinding have become quite detailed. They represent the trajectories of individual abrasive grains, the stochastic distribution of grain protrusion heights, wheel wear and the discretized generating motion, and they can explain why roughness varies along the involute. Their weakness is not the kinematics but the unmodelled physics: thermal distortion of the workpiece, spindle and fixture vibration, dressing-induced topography scatter and coolant effects all shift the achieved roughness away from the calculated value. A data-driven mapping from process parameters to measured roughness is an attractive complement, provided that it can be built from the small number of trials that a gear shop can realistically run on a production machine.
Grinding trials and roughness measurement
The analysis considered here rests on 12 grinding trials on a Gleason-600G CNC gear grinding machine, using a Norton straight cup ceramic alumina (SG) wheel of grit size 46. The workpiece material was 18Cr2Ni4WA case-hardening steel. Before grinding, the gears were carburized at 900 °C for 420 min to a case depth of approximately 1.5 mm, heated to 800 °C and held for 80 min before oil quenching, then tempered at 200 °C for 200 min and air cooled. This sequence fixes the hardness of the flank that the wheel subsequently cuts, and hardness in turn shifts the roughness obtained at a given set of grinding parameters.
Roughness was measured with a CCILITE-M112 white-light interferometer. The quantity used throughout is the areal roughness parameter VSa, obtained by integrating the absolute height deviation over the measured area:
$$VS_{a}=\frac{1}{A}\iint_{A}\left|z(x,y)\right|\,\mathrm{d}x\,\mathrm{d}y$$
where A is the area of the measurement region and |z(x,y)| is the absolute deviation of the surface height from the reference plane. Three process variables were varied: grinding speed vs, generating speed and grinding depth ap. Trials 1–9 form an L9 orthogonal array and were used as the training set; trials 10–12 are single-factor tests at a fixed grinding speed of 20 m/s and were reserved for checking the fitted models, giving a 3:1 split.
| Trial | vs (m/s) | Generating speed (°/s) | ap (mm) | Convex VSa (µm) | Concave VSa (µm) |
|---|---|---|---|---|---|
| 1 | 15 | 8 | 0.05 | 0.657 | 0.450 |
| 2 | 15 | 10 | 0.03 | 0.567 | 0.423 |
| 3 | 15 | 12 | 0.07 | 0.633 | 0.439 |
| 4 | 20 | 8 | 0.03 | 0.527 | 0.416 |
| 5 | 20 | 10 | 0.07 | 0.586 | 0.429 |
| 6 | 20 | 12 | 0.05 | 0.520 | 0.414 |
| 7 | 25 | 8 | 0.07 | 0.550 | 0.429 |
| 8 | 25 | 10 | 0.05 | 0.502 | 0.392 |
| 9 | 25 | 12 | 0.03 | 0.451 | 0.352 |
| 10 | 20 | 6 | 0.03 | 0.585 | 0.435 |
| 11 | 20 | 10 | 0.03 | 0.485 | 0.399 |
| 12 | 20 | 10 | 0.05 | 0.547 | 0.415 |
Convex-flank roughness spans 0.451–0.657 µm and concave-flank roughness 0.352–0.450 µm. The convex flank is rougher in every trial, by 0.086–0.207 µm, which is consistent with the different contact geometry of the wheel’s outer and inner sides. The single-factor comparisons show the expected directions. At 10 °/s and 0.03 mm, raising the grinding speed from 15 to 20 to 25 m/s lowers convex VSa from 0.567 to 0.485 to 0.451 µm. At 20 m/s and 10 °/s, increasing the depth from 0.03 to 0.05 mm raises it from 0.485 to 0.547 µm. Increasing the generating speed from 8 to 10 °/s at 20 m/s and 0.03 mm reduces it from 0.527 to 0.485 µm, a smaller change than the speed or depth effects.
Three candidate regressors and their tuning
Three algorithms commonly used for regression on tabular data were compared. A decision tree recursively partitions the parameter space into regions and assigns each region an output value, so its behaviour is fully visible but its resolution is limited by the number of splits the sample can support. Support vector regression fits a linear function inside an ε-tube and penalizes deviations beyond it,
$$f(X)=wX+b,\qquad \left|Y_{k}-(wX_{k}+b)\right|\le\varepsilon$$
with the optimization objective of minimizing ½‖w‖² subject to that constraint. An artificial neural network combines many simple units, each receiving several inputs, weighting them and passing the result through an activation function; errors are back-propagated by the chain rule and the weights updated iteratively.
Hyperparameters were set by combining experience-based ranges, grid search and cross-validation, and the configuration retained for each model is listed in Table 2. Inputs were normalized before training so that all three variables enter the models on a comparable scale:
$$x’=\frac{x_{0}-X_{\min}}{X_{\max}-X_{\min}}$$
| Model | Hyperparameter | Value |
|---|---|---|
| Decision tree | Maximum depth Minimum samples per leaf Minimum samples to split |
3 (searched over 2–5) 2 2 |
| SVR | Kernel function Regularization parameter C |
RBF 1 |
| ANN | Hidden layer sizes Solver Activation Maximum iterations Random state |
(5, 3, 3) L-BFGS Tanh 100 34 |
The network has three input neurons, one per process parameter, one output neuron, and three hidden layers with 5, 3 and 3 neurons. The limited-memory BFGS quasi-Newton solver suits a network of this size, and the tanh activation keeps the output smooth across the parameter window.
Model comparison: fit quality and speed
Each model was run 20 times and scored with the root mean squared error and the coefficient of determination,
$$R^{2}=1-\frac{\sum_{i=1}^{n}\left(y_{i}-\hat{y}_{i}\right)^{2}}{\sum_{i=1}^{n}\left(y_{i}-\bar{y}_{i}\right)^{2}},\qquad RMSE=\sqrt{\frac{1}{n}\sum_{i=1}^{n}\left(y_{i}-\hat{y}_{i}\right)^{2}}$$
where yi, ŷi and ȳi are the measured, predicted and mean values and n is the number of samples. An R² of 1 indicates a perfect fit, 0 indicates that the model does no better than predicting the mean, and negative values indicate a fit worse than the mean.
The three algorithms separated cleanly. The decision tree scored poorly on both the training and the test data — a textbook underfit. With nine training points, a tree of depth 3 cannot resolve the interaction structure, and increasing the depth would only trade bias for variance. SVR fitted the training set well but lost accuracy on the test set, the signature of overfitting; with C = 1 and an RBF kernel, the ε-tube is effectively tuned to noise in a nine-point design, and uninformative or redundant features aggravate this. The ANN scored well on both sets and reproduced the measured roughnesses most closely, which is attributable to its ability to represent smooth nonlinear and interaction terms with relatively few parameters.
Prediction speed behaves in the opposite order to accuracy. Fitting and prediction times for all three models are short because the dataset is small, but the tree and SVR fit fastest while the network is slowest, reflecting its iterative back-propagation training. In a production setting the distinction matters little at inference time — a trained network returns a prediction in negligible time — but retraining the network on an enlarged dataset is the more expensive operation.
A closed-form surrogate derived from the network
For process planning, a transparent expression is often preferable to a black-box model. A multiple linear regression with linear, quadratic and two-factor interaction terms was therefore fitted to the network’s predictions over the ranges vs = 10–25 m/s, generating speed 5–15 °/s and ap = 0–0.1 mm, with coefficients solved by least squares:
$$VS_{a,\mathrm{convex}}=1.4220-0.0522x_{1}-0.0588x_{2}+2.1000x_{3}+0.0007x_{1}^{2}+0.0006x_{2}^{2}+0.0015x_{1}x_{2}+0.0433x_{1}x_{3}+0.1000x_{2}x_{3}$$
$$VS_{a,\mathrm{concave}}=1.2828-0.0230x_{1}-0.0944x_{2}-6.1833x_{3}+0.0030x_{2}^{2}+0.0008x_{1}x_{2}+0.1967x_{1}x_{3}+0.3333x_{2}x_{3}$$
Here x1, x2 and x3 denote grinding speed in m/s, generating speed in °/s and grinding depth in mm, and the outputs are in µm. Because the polynomial is a local approximation, individual coefficient signs — notably the opposite linear depth terms for the two flanks — should not be read as physical mechanisms; the interaction terms absorb much of the curvature within this narrow window.
Against the three held-out trials, the largest relative error for the convex flank was 7.2%, and the mean relative errors were 3.5% for the convex flank and 6.09% for the concave flank. The concave flank is harder to predict, in line with its smaller absolute roughness and the correspondingly larger weight of measurement and process scatter. These figures support use of the surrogate for ranking candidate parameter sets, not for certifying a flank against a tolerance without further measurement.
Which parameter dominates?
Machine-learning predictions carry no intrinsic physical interpretation, so the contribution of each input was decomposed with SHapley Additive exPlanations. The method borrows the Shapley value from cooperative game theory and assigns each feature a share of the prediction, averaged over all possible feature subsets:
$$\phi_{j}=\sum_{S\subseteq N\setminus\{j\}}\frac{\left|S\right|!\,\left(\left|N\right|-\left|S\right|-1\right)!}{\left|N\right|!}\left[v\left(S\cup\{j\}\right)-v\left(S\right)\right]$$
where N is the full feature set, S is a subset not containing feature j, and v(·) is the model output for that subset. Features are ranked by the spread of their Shapley values across samples: a wide spread away from the centre line means a strong and consistent influence on the prediction.
For both flanks the ranking is identical. Grinding speed dominates, grinding depth is second and generating speed is third. This agrees with the single-factor comparisons in Table 1, where a 10 m/s change in grinding speed shifted convex roughness by about 0.116 µm while a 2 °/s change in generating speed shifted it by about 0.042 µm. The practical reading is that grinding speed is the primary lever for roughness control on this workpiece–wheel combination, depth is the secondary lever and is also constrained by material removal requirements, and generating speed can be adjusted for kinematic reasons with comparatively little roughness penalty.
Scope and limitations
Several boundaries qualify these results. The models were trained on nine orthogonal trials and checked on three single-factor trials from one material, one wheel specification and one machine tool, so the reported errors describe interpolation within that window rather than general predictive capability. The three-point validation set gives a useful but statistically thin estimate of accuracy; a larger factorial design with replicate measurements would allow proper uncertainty bounds, for example through Gaussian-process regression or ensemble variance. The “perfect” fit of the network on such a small dataset should also be read cautiously — a flexible model can fit twelve points closely without having captured the underlying response surface, and independent trials at parameter combinations not present in the design are the appropriate test. Finally, the input set omits wheel wear state, dressing conditions and coolant strategy, all of which are known to shift roughness and would be needed for a model that tracks wheel life.
Conclusions
On a 12-trial grinding dataset for 18Cr2Ni4WA spiral bevel gears, three machine-learning regressors were tuned and compared for their ability to predict convex- and concave-flank areal roughness from grinding speed, generating speed and grinding depth. The decision tree underfitted and SVR overfitted the nine-point training design, while the multilayer network with layers of 5, 3 and 3 neurons, tanh activation and L-BFGS training reproduced both flanks closely, at the cost of the slowest fitting among the three. Mean relative errors were 3.5% on the convex flank and 6.09% on the concave flank, with a maximum convex-flank error of 7.2%.
A polynomial surrogate fitted to the network’s output provides an explicit relation between the three process parameters and roughness on each flank, valid over 10–25 m/s grinding speed, 5–15 °/s generating speed and 0–0.1 mm depth, with accuracy adequate for comparing candidate parameter sets. Shapley decomposition ranks grinding speed as the strongest influence on both flanks, grinding depth second and generating speed third, matching the trends visible in the raw trials. The workflow — small orthogonal design, tuned network, polynomial surrogate, attribution analysis — offers a practical route to roughness prediction where physics-based models cannot capture the scatter of a real grinding process, provided that the model is treated as an interpolator within its calibration window and is extended as more production data become available.