Spiral bevel gears are used in intersecting-shaft transmissions where the tooth flank shape governs the contact path, motion quality, and load distribution. In batch production, the theoretical flank generated from tooth contact analysis (TCA) is rarely reproduced exactly. Machine-setting errors, cutter variation, fixture effects, and other process shifts produce a repeatable deviation field across the batch. Correcting that field requires a reliable estimate of the actual machined surface, but measuring every part is costly. A small-sample statistical route is therefore attractive: estimate the batch mean deviation from a limited set of measurements, reconstruct a smooth equivalent surface, and convert that surface into machine-setting corrections.
From theoretical flank to measured deviation field
Let \(H_L\) denote the theoretical design tooth surface obtained from TCA-based machine settings, and let \(H_R\) denote an equivalent actual machined surface represented by a measured or estimated mean-deviation surface. The machine-setting vector is \(\boldsymbol{\Phi}=(\Phi_1,\ldots,\Phi_m)\), where \(m\) is the number of adjustable parameters. At a surface point described by design variables \((\phi,\theta)\), the signed normal deviation is
\[
\delta_i(\phi_i,\theta_i;\boldsymbol{\Phi})=
\left[\mathbf{r}_L(\phi_i,\theta_i;\boldsymbol{\Phi})-\mathbf{r}_R(u_i,v_i)\right]\cdot
\mathbf{n}_L(\phi_i,\theta_i;\boldsymbol{\Phi}),
\]
where \(\mathbf{r}_L\) and \(\mathbf{r}_R\) are position vectors on \(H_L\) and \(H_R\), respectively, \(\mathbf{n}_L\) is the unit normal on \(H_L\), \(i\) indexes the sampled surface points, and \(u,v\in[0,1]\) are surface reconstruction parameters. The sign follows the normal direction.
The correction problem is to find machine-setting increments that move the theoretical surface toward the measured mean surface. A useful geometric construction is to introduce \(H_D\), the surface symmetric to \(H_R\) with respect to \(H_L\). The corresponding position vectors satisfy
\[
\mathbf{r}_R(u,v)=\mathbf{r}_L(\phi,\theta;\boldsymbol{\Phi})+\delta\,\mathbf{n}_L(\phi,\theta;\boldsymbol{\Phi}),
\]
\[
\mathbf{r}_D(u,v)=\mathbf{r}_L(\phi,\theta;\boldsymbol{\Phi})-\delta\,\mathbf{n}_L(\phi,\theta;\boldsymbol{\Phi}),
\]
and therefore
\[
\mathbf{r}_D(u,v)=\mathbf{r}_R(u,v)-2\delta\,\mathbf{n}_L(\phi,\theta;\boldsymbol{\Phi}).
\]
Equation (3) gives an intermediate correction target rather than a direct machine-axis command. The actual compensation is obtained by solving the inverse relation between the target surface and the machine settings, then adding the resulting parameter increments to the theoretical TCA settings. Because the mapping is nonlinear and the fitted surface is approximate, the process is iterative: adjust the machine parameters, re-evaluate the residual deviation, and stop when the deviation is small enough for the specified tooth accuracy.
Bootstrap estimation from a small batch
Bootstrap resampling, introduced by Efron in 1979, estimates the sampling behaviour of a statistic by repeatedly resampling with replacement from the observed data. It does not require a assumed parametric distribution, which is useful when the available batch sample is small. In tooth-surface correction, the statistic of interest is the mean normal deviation at each measurement point.
In the reported batch, 20 spiral bevel gear sets were measured on a Klingelnberg P65 gear measuring instrument using a multi-function 3D scanning probe. The pinion convex flank is used to illustrate the procedure: 20 measurement groups were available, each containing a 225-point grid. Bootstrap resampling of these groups produced a pointwise mean deviation estimate over the grid. The resulting field represents the repeatable batch bias rather than the random scatter of an individual part.
This distinction matters. A batch mean surface is appropriate for pre-control compensation, where the aim is to correct a systematic offset shared by the batch. It does not remove part-to-part random variation, and it cannot compensate for a sample that is not representative of the batch. The bootstrap estimate is therefore a statistical input to the correction model, not a substitute for process control.
NURBS reconstruction of the mean difference surface
The pointwise bootstrap means are discrete and may contain measurement noise. A smooth surface representation is needed before the deviation field can be mapped to machine parameters. Non-uniform rational B-spline (NURBS) surfaces are suitable because they represent free-form flanks with a compact control net, provide \(C^2\) continuity, and remain invariant under translation, rotation, and scaling.
A tensor-product NURBS surface is written as
\[
\mathbf{p}(u,v)=
\frac{
\sum_{i=0}^{n_u}\sum_{j=0}^{n_v}
N_{i,k}(u)N_{j,l}(v)W_{i,j}\mathbf{V}_{i,j}
}{
\sum_{i=0}^{n_u}\sum_{j=0}^{n_v}
N_{i,k}(u)N_{j,l}(v)W_{i,j}
},
\]
where \(n_u+1\) and \(n_v+1\) are the numbers of control points in the \(u\)- and \(v\)-directions, \(N_{i,k}(u)\) and \(N_{j,l}(v)\) are B-spline basis functions of degrees \(k\) and \(l\), \(\mathbf{V}_{i,j}\) are control points, and \(W_{i,j}\) are weights. Cubic basis functions \((k=l=3)\) are sufficient for the smooth flank representation used here.
The bootstrap mean values are treated as type-value points, and the NURBS fit produces the mean difference surface. That surface is then used as the equivalent actual machined surface \(H_R\) in the compensation model. The fitted surface is smoother than the raw point data, which is helpful for stable parameter inversion, but the control-net density must still be fine enough to preserve deviation features that matter for contact performance.
Machine-setting compensation
The compensation model converts the mean difference surface into increments of the machine settings. For the pinion, seven parameters were adjusted: cutter tilt angle, cutter swivel angle, blank mounting angle, horizontal wheel position, vertical wheel position, bed position, and radial cutter position. The values in Table 1 are increments relative to the theoretical TCA settings, not absolute machine coordinates.
| Parameter | Correction |
|---|---|
| Cutter tilt angle (°) | −0.0698 |
| Cutter swivel angle (°) | 0.0867 |
| Blank mounting angle (°) | −0.0749 |
| Horizontal wheel position (mm) | −0.2033 |
| Vertical wheel position (mm) | 0.0593 |
| Bed position (mm) | 0.0349 |
| Radial cutter position (mm) | −0.2405 |
The increments are small: below \(0.09^\circ\) in the angular settings and below \(0.25\ \mathrm{mm}\) in the translational settings. Such magnitudes are consistent with a fine correction of the generating motion rather than a major setup change.
Experimental validation on the pinion
Using the corrected machine settings, a pinion was trial-milled on a CNC gear milling machine and measured again by the same procedure. Table 2 compares selected normal deviations at the large and small ends of the convex and concave flanks before and after correction. The values are signed deviations in micrometres.
| Region | Convex flank | Concave flank | ||
|---|---|---|---|---|
| Before | After | Before | After | |
| Large end, root | 55.5 | 21.2 | −25.8 | 15.3 |
| Large end, tip | 81.8 | 39.0 | −10.0 | 20.8 |
| Small end, root | −84.6 | −29.1 | 22.7 | 9.4 |
| Small end, tip | −38.5 | −12.1 | 26.4 | 7.6 |
The largest single-point change occurred at the convex small-end root, where the deviation moved from \(-84.6\ \mu\mathrm{m}\) to \(-29.1\ \mu\mathrm{m}\), a shift of \(55.5\ \mu\mathrm{m}\). Over the measured pinion points, the sum of squared deviations fell from \(0.4611\ \mathrm{mm^2}\) to \(0.1076\ \mathrm{mm^2}\), a reduction of \(76.66\%\). The deviation trend also became flatter than before correction.
Because Table 2 reports signed values at selected points, not every individual entry decreases in magnitude. The \(76.66\%\) figure refers to the aggregate sum of squares over the measured grid, which is the appropriate summary for the overall correction effect. The table is useful for locating where the correction acts most strongly, but it should not be read as a uniform percentage reduction at every point.
Interpretation and scope
The procedure forms a digital pre-control loop: measure a small batch sample, estimate the mean normal deviation field by bootstrap resampling, fit a NURBS mean difference surface, solve for machine-setting increments, cut a trial part, and verify by re-measurement. Its main advantage is that it uses a limited sample to correct a repeatable batch-level deviation without requiring a parametric error model.
The reported validation is limited to the pinion of a 20-set batch and to the seven machine-setting parameters listed in Table 1. The correction targets systematic batch bias; random part-to-part variation remains. The bootstrap estimate also depends on the measured sets being representative of the batch, and the NURBS fit can smooth localized high-frequency features if the control net is too coarse. For production use, the mean surface should be re-estimated when batch conditions, cutter state, or machine setup change enough to alter the systematic deviation field.
Within that scope, the result supports the use of bootstrap estimation plus NURBS reconstruction as a practical basis for spiral bevel gear tooth-surface correction. The aggregate pinion deviation decreased substantially after applying the pre-control machine-setting increments, showing that a small-sample statistical estimate can be converted into an effective machining correction rather than remaining a purely descriptive measurement result.