Spiral bevel gears are widely used in intersecting-shaft drives, where the contact path and meshing quality depend directly on tooth-surface accuracy. The flank is not cut to its final geometry in a single operation: it is generated by a cutter and a set of machine motions whose nominal values are usually established by tooth contact analysis (TCA). In batch production, small errors in those settings produce a distributed normal deviation over the flank. The practical problem is therefore not simply to measure one gear, but to estimate the batch-level deviation from a limited sample and convert it into machine-setting corrections before the next batch is cut.
A useful correction procedure must connect three quantities: measured point deviations, a continuous representation of the as-machined flank, and a set of machine-setting increments. The approach considered here combines bootstrap resampling of a small measured batch, NURBS fitting of a mean-deviation surface, and optimization of machine-setting increments. The objective is pre-control compensation: correct the settings, cut a trial batch, remeasure, and determine whether the aggregate deviation has fallen.
From contact design to a normal-deviation field
TCA provides a theoretical design flank \(H_L\) and a nominal machine-setting vector \(\Phi_j\), with \(j=1,\ldots,m\). In the experimental case discussed below, seven settings were used: cutter tilt angle, cutter swivel angle, workpiece mounting angle, horizontal wheel position, vertical wheel position, bed position, and radial cutter position. The actual mean flank produced by the batch is denoted \(H_R\). At a measured point \(i\), the signed normal deviation is
$$ \delta_i(\phi_i,\theta_i;\Phi_j)=[\mathbf H_L(\phi_i,\theta_i;\Phi_j)-\mathbf H_R(u_i,v_i)]\cdot \mathbf n_L(\phi_i,\theta_i;\Phi_j) $$
where \(\mathbf n_L\) is the unit normal on the theoretical flank, \(\phi_i,\theta_i\) are the design parameters at point \(i\), and \(u_i,v_i\) are the corresponding surface parameters on the reconstructed actual flank. The sign of \(\delta_i\) indicates on which side of the theoretical surface the measured point lies. The correction problem is to find increments \(\Delta\Phi_j\) such that the compensated machined flank approaches \(H_L\) over the measured point set.
Because the relationship between machine settings and flank geometry is nonlinear, the procedure is organized as a measurement–fit–solve–cut loop. The measured batch supplies \(H_R\); the compensation model supplies a set of increments; a trial cut tests whether those increments reduce the deviation field. The correction is local to the nominal TCA settings rather than a global redesign of the tooth geometry.
Bootstrap estimation from a small measured batch
The bootstrap was introduced in 1979 as a resampling method for estimating the sampling behavior of a statistic without requiring a large sample. In this application, each of the 225 measured points on the pinion flank has 20 measured deviation values, one from each sampled pinion. Resampling those 20 values with replacement generates bootstrap replicates at each point; the bootstrap mean estimates the expected deviation at that point. Repeating this at all 225 points produces a dense mean-deviation field from a relatively small batch.
The bootstrap step does not add physical information that was not measured. Its role is statistical: it estimates the batch mean and reduces the influence of a single anomalous part on the fitted surface. The key assumption is that the 20 measured pinions are representative of the production batch. If the batch mean shifts because of a tool change, fixture change, or different blank lot, the correction derived from the earlier sample no longer applies. The procedure also targets the batch mean deviation; it is not by itself a method for reducing random part-to-part scatter.
NURBS reconstruction of the effective machined flank
The discrete bootstrap means must be converted into a continuous surface before they can be used in a machine-setting calculation. NURBS fitting is suitable because it provides \(C^2\) continuity and remains invariant under scaling, rotation, and translation. A NURBS surface is written as
$$ \mathbf p(u,v)=\frac{\sum_{i=0}^{n}\sum_{j=0}^{m}N_{i,k}(u)N_{j,l}(v)W_{i,j}\mathbf V_{i,j}}{\sum_{i=0}^{n}\sum_{j=0}^{m}N_{i,k}(u)N_{j,l}(v)W_{i,j}} $$
where \(N_{i,k}(u)\) and \(N_{j,l}(v)\) are B-spline basis functions of degrees \(k\) and \(l\), \(\mathbf V_{i,j}\) are control points, \(W_{i,j}\) are weights, and \(0\le u,v\le 1\). A third-degree representation was used for the fitting. The 225 measured points on the pinion convex flank served as type points, and the fitted surface became the effective as-machined mean surface \(H_R\).
The normal-offset representation used in the compensation model is
$$ \mathbf r_R(u,v)=\mathbf r_L(\phi,\theta;\Phi_j)+\delta\,\mathbf n_L(\phi,\theta;\Phi_j) $$
where \(\mathbf r_L\) is a point on the theoretical design flank, \(\mathbf r_R\) is the corresponding point on the effective machined flank, and \(\delta\) is the signed normal offset. The compensation geometry also defines a surface symmetric to \(H_R\) about \(H_L\); this symmetric construction supplies the target normal displacement used in the parameter solve. Because the fitted surface is an approximation, the corrected flank is not expected to coincide exactly with the measured mean surface. The engineering aim is to bring it within the specified tolerance.
Machine-setting increments obtained from the deviation field
Solving the compensation model against the fitted mean-deviation surface gives the increments in Table 1. These values are added to the nominal TCA settings; they are not absolute machine settings. The largest tabulated increments are the radial cutter position at \(-0.2405\) mm and the horizontal wheel position at \(-0.2033\) mm. The angular increments are smaller, but they are not negligible because a small cutter-axis rotation changes the generated flank over its full length.
| Parameter | Increment |
|---|---|
| Cutter tilt angle | \(-0.0698^\circ\) |
| Cutter swivel angle | \(0.0867^\circ\) |
| Workpiece mounting angle | \(-0.0749^\circ\) |
| Horizontal wheel position | \(-0.2033\) mm |
| Vertical wheel position | \(0.0593\) mm |
| Bed position | \(0.0349\) mm |
| Radial cutter position | \(-0.2405\) mm |
Experimental verification on a pinion batch
Twenty pinions from the same batch were measured on a Klingelnberg P65 gear measuring instrument with a multi-function 3D scanning probe. The bootstrap reconstruction used 225 points on the pinion convex flank. After applying the increments in Table 1, a correction cutting trial was performed on the pinion and the flank was remeasured under the same measurement conditions. Table 2 compares representative signed deviations on the convex and concave flanks before and after correction.
| Location | 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 pointwise change occurred at the convex small-end root, where the deviation moved from \(-84.6\) μm to \(-29.1\) μm, a change of \(55.5\) μm. Over the measured point set, the sum of squared normal deviations fell from \(0.4611\) mm² to \(0.1076\) mm², a reduction of \(76.66\%\). The deviation trend after correction was flatter than before, and the largest tabulated residual magnitude was \(39.0\) μm at the convex large-end tip.
The pointwise response was not uniformly better. On the concave large-end tip, the tabulated deviation changed from \(-10.0\) μm to \(20.8\) μm, so its magnitude increased even though the aggregate sum of squares fell. The result therefore supports an aggregate improvement rather than a guarantee that every measured location improves in the same direction. This distinction matters when the correction is evaluated against a tolerance band rather than against a single global error metric.
Scope and limitations
The experimental evidence supports the following limited conclusion: for one batch of 20 pinions, a correction derived from bootstrap means and a NURBS mean-deviation surface reduced the sum of squared pinion flank deviations by \(76.66\%\) after one correction cutting trial. The procedure provides a practical way to convert sparse measured deviations into machine-setting increments and to verify the effect by remeasurement.
It does not establish universal applicability. The correction is tied to the measured batch, the nominal TCA settings, and the particular pinion geometry. The bootstrap assumption of a representative sample is central; a shifted batch mean would invalidate the correction. The trial covers one pinion type and one correction cycle, so repeated cycles, additional batches, and different gear sizes would be needed to assess long-term production stability. Because the method corrects the batch mean, it should be complemented by process controls that address part-to-part variation. Within those limits, bootstrap-based pre-control compensation offers a technically coherent route from small-batch measurement to machine-setting correction for spiral bevel gears.