Spiral bevel gears transmit motion between intersecting or offset shafts and are widely used in automotive drive axles, differentials, and machine-tool drives. Their curved teeth provide gradual contact and high load capacity, but the final mesh quality is sensitive to tooth-surface deviation after heat treatment. Heat treatment introduces non-uniform thermal and microstructural strains, so the resulting distortion varies from tooth to tooth and from gear to gear even when the gear material, NC milling method, and heat-treatment specification are fixed.
Batch Distortion as a Statistical Estimation Problem
Single-gear tooth-surface inspection is well established. CNC gear measuring centers digitize a flank on a defined grid, compare the measured points with the theoretical surface, and provide deviation data for reverse correction of machine settings. For a batch, however, measuring every tooth surface is impractical. Production quality is usually inferred from a small number of sampled gears, and the sampled deviation values have an unknown probability distribution. The practical question is therefore not only how one gear deviates, but what mean deviation and scatter should be expected across the batch, and how much confidence can be placed in those estimates.
The bootstrap method addresses this situation by resampling the observed values with replacement. It uses the empirical distribution of a small sample to approximate the sampling distribution of a statistic, without requiring a prior parametric distribution. Since Efron introduced the method in 1979, bootstrap resampling has been applied to bearing reliability, machining-error assessment, and other small-sample problems. For batch spiral bevel gears, the same logic applies when the material, NC milling method, and heat-treatment specification define a common production population.
Bootstrap Model for a Tooth-Surface Measurement Point
Let \(q\) denote a fixed sequential measurement point on the tooth-surface grid. For \(n\) measured gear surfaces in the batch, the deviation values at this point form the original sample
\[
\boldsymbol{\sigma}=(\sigma_1,\sigma_2,\ldots,\sigma_n)
\]
where \(\sigma_i\) is the deviation of the \(i\)-th measured surface at point \(q\). A bootstrap sample is generated by drawing \(L\) values from \(\boldsymbol{\sigma}\) with replacement. Repeating this procedure \(B\) times gives \(B\) bootstrap samples. The mean of the \(j\)-th bootstrap sample is
\[
\bar{\xi}_b^{(j)}=\frac{1}{L}\sum_{s=1}^{L}\xi_b^{(j)}(s),
\qquad j=1,\ldots,B
\]
and the overall bootstrap mean and variance are
\[
\bar{\xi}_B=\frac{1}{B}\sum_{j=1}^{B}\bar{\xi}_b^{(j)}
\]
\[
s_B^2=\frac{1}{B-1}\sum_{j=1}^{B}
\left(\bar{\xi}_b^{(j)}-\bar{\xi}_B\right)^2
\]
For comparison, the original sample mean and variance at the same measurement point are
\[
\bar{\sigma}=\frac{1}{n}\sum_{i=1}^{n}\sigma_i,
\qquad
s_\sigma^2=\frac{1}{n}\sum_{i=1}^{n}\left(\sigma_i-\bar{\sigma}\right)^2
\]
Confidence intervals are obtained by the percentile method. The \(B\) bootstrap estimates are sorted as \(\hat{\theta}_{(1)}\le\hat{\theta}_{(2)}\le\cdots\le\hat{\theta}_{(B)}\). For significance level \(\alpha\), the lower and upper ranks are
\[
K_1=\left\lfloor B\frac{\alpha}{2}\right\rfloor,
\qquad
K_2=\left\lfloor B\left(1-\frac{\alpha}{2}\right)\right\rfloor
\]
and the \(1-\alpha\) percentile interval is \((\hat{\theta}_{(K_1)},\hat{\theta}_{(K_2)})\). A histogram of the bootstrap means or standard deviations shows the shape of the resampled distribution. A Kolmogorov–Smirnov test can then be used to check whether the bootstrap statistics are approximately normal. When normality is acceptable, the bootstrap mean provides a point estimate of the batch mean, while the percentile limits quantify its uncertainty.
NURBS Reconstruction of the Batch Mean Surface
Bootstrap resampling produces a mean and a standard deviation at each grid point, but a gear flank is a continuous surface. To represent the batch trend geometrically, the point statistics are fitted with a NURBS surface. A NURBS surface is defined on a control grid as
\[
\mathbf{P}(u,v)=
\frac{
\sum_{i=0}^{n}\sum_{j=0}^{m}
B_{i,k}(u)B_{j,l}(v)W_{i,j}\mathbf{V}_{i,j}
}{
\sum_{i=0}^{n}\sum_{j=0}^{m}
B_{i,k}(u)B_{j,l}(v)W_{i,j}
}
\]
where \(B_{i,k}(u)\) and \(B_{j,l}(v)\) are B-spline basis functions of degrees \(k\) and \(l\), \(W_{i,j}\) are weights, and \(\mathbf{V}_{i,j}\) are control points. Cubic NURBS with \(k=l=3\) was used so that the reconstructed surface is continuous up to the second derivative. NURBS is a standard geometry representation in STEP and can represent both free-form and analytic surfaces, which makes it suitable for smooth tooth-flank reconstruction.
Measurement and Resampling Results
The batch tests were performed on a Klingelnberg P65 gear measuring center. After identical NC milling and heat treatment, 5, 10, and 20 pairs of ring and pinion were measured. A \(15\times15\) point grid was used on the tooth surface, and the deviation at each grid point was recorded as a relative value from the measuring center. This grid provides a regular set of sequential measurement points for statistical analysis and surface reconstruction.
For a sample size of 20 deviation values at a measurement point, 25,000 bootstrap resamples were generated. Histograms of the bootstrap means and standard deviations were approximately normal according to Kolmogorov–Smirnov tests, so the bootstrap mean was used as the point estimate and the percentile limits as the confidence bounds. The table below summarizes the effect of the original sample size on the bootstrap estimates and their relative errors.
| Original sample size \(n\) | Original mean | Bootstrap mean | Mean relative error | Original standard deviation | Bootstrap standard deviation | Standard-deviation relative error |
|---|---|---|---|---|---|---|
| 10 | −1.6100 | −1.5994 | 0.66% | 3.7693 | 3.5798 | 5.03% |
| 15 | −0.7133 | −0.7172 | 0.55% | 3.5161 | 3.3972 | 3.38% |
| 20 | −0.9050 | −0.9082 | 0.35% | 3.2744 | 3.1908 | 2.64% |
| 25 | 0.4200 | 0.4210 | 0.24% | 4.0100 | 3.9269 | 2.10% |
| 30 | 0.8567 | 0.8580 | 0.15% | 3.8247 | 3.7610 | 1.66% |
| 35 | 0.5029 | 0.5030 | 0.019% | 4.0072 | 3.9493 | 1.44% |
At \(n=10\), the bootstrap mean differs from the original mean by 0.66%, while the standard deviation differs by 5.03%. At \(n=35\), the corresponding differences fall to 0.019% and 1.44%. The mean estimate converges faster than the standard-deviation estimate, which is consistent with the greater sensitivity of variance estimation to the tails of the sample. Even at small sample sizes, the bootstrap reproduces the central tendency and spread closely enough to support batch-level evaluation.
| Original sample size \(n\) | 95% confidence interval for mean | 95% confidence interval for standard deviation |
|---|---|---|
| 10 | (−1.6093, −1.5894) | (3.5728, 3.5869) |
| 20 | (−0.9133, −0.9031) | (3.1872, 3.1944) |
| 35 | (0.4982, 0.5078) | (3.9460, 3.9527) |
These intervals are narrow because they describe uncertainty in the estimated point statistic, not the full part-to-part range of the batch. The distinction is important for process control: the confidence interval indicates how precisely the batch mean has been estimated from the available sample, while the standard deviation indicates how widely individual gear surfaces vary around that mean.
Reconstructed Batch Surface and Sample-Size Effect
For an automotive drive-axle spiral bevel gear set, the large member was evaluated after heat treatment using bootstrap samples of 5 and 10 elements. At each grid point, the bootstrap mean defined a mean-deviation value. Fitting these values with cubic NURBS produced a continuous bootstrap mean-deviation surface, which was compared with the actual mean-deviation surface obtained from measurement and with the theoretical design surface.
Local enlargements were examined at grid point (8, 4) and grid point (10, 11). At both locations, the 10-element bootstrap surface lay closer to the actual mean-deviation surface than the 5-element bootstrap surface. The position of the actual mean-deviation surface relative to the theoretical design surface remained unchanged, since that relationship is fixed by the measured geometry. The comparison therefore isolates the effect of sample size on the reconstructed statistical surface: increasing the number of sampled elements improves local fidelity to the measured batch trend.
The result also shows why a single measured gear is not sufficient for batch correction. A single flank describes one realization of the distortion field, whereas the bootstrap mean surface represents the central tendency of the sampled batch. The associated variance or confidence interval indicates whether that central tendency is well constrained. If the scatter is large, adjusting machining parameters to the mean alone may not improve the acceptance rate; tighter process or heat-treatment control is then required.
Practical Implications and Limits
The bootstrap model provides a defensible way to estimate batch mean deviation and its uncertainty when only a small number of tooth surfaces can be measured. The resulting mean-deviation surface can guide compensation of NC milling parameters, while the confidence interval helps determine whether the available sample is adequate for that correction. Because the method works from the empirical distribution, it does not require a prior normal distribution for the original deviation data.
The model is bounded by its production assumptions. It applies to a batch with the same gear material, NC milling method, and heat-treatment specification. If any of these conditions changes, the empirical distribution must be re-established. Bootstrap resampling also cannot replace representative sampling: if the measured gears do not represent the batch, or if the measurement grid is inconsistent, the resampled statistics will inherit that bias. The method quantifies sampling uncertainty; it does not by itself identify the physical cause of heat-treatment distortion.
Conclusion
- Bootstrap resampling of small tooth-surface deviation samples provides estimates of the mean and standard deviation at each measurement point, together with percentile confidence intervals. In the tested cases, a 10-element sample gave a mean relative error of 0.66% and a standard-deviation relative error of 5.03%; a 35-element sample reduced these errors to 0.019% and 1.44%.
- Cubic NURBS reconstruction converts point statistics into a continuous mean-deviation surface. In the automotive drive-axle example, the 10-element bootstrap surface matched the actual mean-deviation surface more closely than the 5-element surface at the inspected local regions, while the actual-to-theoretical geometric relationship remained fixed.
- The bootstrap statistical model supports batch-level evaluation of heat-treatment tooth-surface deviation and provides a basis for correcting tooth-surface machining parameters. Sample size should be selected according to the required confidence interval and the spatial detail needed for compensation.