Why distortion resists per-part inspection
Spiral bevel gears carry motion between intersecting or offset axes in vehicle drive axles, reducers and machine tool drives. Their load capacity and running quietness are set by the flank geometry that survives heat treatment, not by the geometry that leaves the cutting machine. Carburizing and quenching, however, always move the flank: residual stress and phase transformation introduce a distortion field that is small in amplitude but distributed over the whole tooth surface, so that a gear cut to nominal machine settings no longer contacts its mate as designed.
The established remedy is digital: measure the flank on a CNC gear measuring center, convert the measured deviation field into corrections of the cutting parameters, and re-cut. Wang Zhonghou and co-workers built digital real tooth surfaces containing both tooth-form error and mounting error and ran tooth contact analysis directly on them; Li Jubo and co-workers embedded measurement and correction in a networked closed-loop manufacturing model for spiral bevel gears; Li Tianxing and co-workers developed a one-dimensional probing strategy for accurate measurement of three-dimensional tooth surfaces on a gear measuring center. All of these address the single flank, and they work well for it.
Batch production poses a different question. Inspecting every flank of every gear on a 15 × 15 grid is prohibitive in time and cost, so batch quality is normally inferred from a few sampled parts. The obstacle is statistical rather than metrological: the probability distribution of the post-heat-treatment deviation at a given flank point is unknown, the sample is tiny, and no parametric form can be justified in advance. What the process engineer actually needs is the batch distortion — the mean deviation field of the population and its spread — because machine-setting compensation must be correct for the batch, not for one gear.
The bootstrap, introduced by Efron in 1979, was designed for exactly this situation. It treats the observed sample as an empirical distribution, draws repeated samples with replacement from it, and uses the spread of the recomputed statistic to approximate its sampling distribution without assuming normality. The method has since been applied to machine tool machining error assessment and reliability evaluation by Xia Xintao and co-workers, and to the construction of prior distributions for bearing life parameters under competing failure modes by Qin Luosheng and co-workers. Its use for gear flank error statistics has remained largely unexplored, and that is the gap addressed by the work summarized here.
Measurement grid and data structure
Deviations were acquired on a Klingelnberg P65 gear measuring center using point-by-point (grid) probing rather than continuous scanning, a choice whose trade-offs against scanning were documented by Xu and co-workers. The flank is discretized into a strictly defined grid with an odd number of rows and columns; a 15 × 15 grid was used, giving 225 sequence points per flank. Each reported value is the deviation of a grid point relative to the flank center reference, so a positive and a negative sign denote material in excess and in deficit respectively.
Gears from one batch — identical material, identical CNC milling settings, identical heat-treatment specification — were measured in groups of 5, 10 and 20 pairs of wheel and pinion. Arranged as a matrix, the data have one row per gear and one column per sequence point. Every column is an independent small sample of the batch at a fixed location on the flank, and it is these columns that the resampling procedure operates on. A single column of n gears at sequence point m is written as
$$\sigma=\left(\sigma_{1},\sigma_{2},\dots,\sigma_{i},\dots,\sigma_{n}\right)$$
where σi is the deviation of the i-th gear at point m.
Resampling a small batch into a distribution
From the column σ, values are drawn with replacement, one at a time, L times, producing one bootstrap sample ξb; the operation is repeated B times. In the reported runs B = 25 000. The mean of the j-th bootstrap sample is
$$\bar{\xi}_{b(j)}=\frac{1}{L}\sum_{s=1}^{L}\xi_{b(j)}(s)$$
and the vector of the B bootstrap means is
$$\bar{\xi}_{B}=\left(\bar{\xi}_{b(1)},\bar{\xi}_{b(2)},\dots,\bar{\xi}_{b(j)},\dots,\bar{\xi}_{b(B)}\right)$$
with the overall bootstrap mean and bootstrap variance
$$\bar{\xi}_{B}=\frac{1}{B}\sum_{j=1}^{B}\bar{\xi}_{b(j)}$$
$$\delta^{2}=\frac{1}{B-1}\sum_{j=1}^{B}\left(\bar{\xi}_{b(j)}-\bar{\xi}_{B}\right)^{2}$$
The corresponding quantities of the original sample are
$$\bar{\sigma}=\frac{1}{N}\sum_{i=1}^{N}\sigma_{i},\qquad \sigma^{2}=\frac{1}{N}\sum_{i=1}^{N}\left(\sigma_{i}-\bar{\sigma}\right)^{2}$$
Because B is large, the 25 000 resampled means can be sorted into M classes and displayed as a histogram, which makes the shape of the sampling distribution visible rather than assumed. Applied point by point over the 225 columns, the procedure yields a mean and a dispersion for every location on the flank, and these two arrays are what the reconstruction step later consumes.
Percentile confidence intervals
Interval estimates are obtained by the percentile method, without any normal-theory correction. The B bootstrap estimates θ̂ are ordered,
$$\hat{\theta}_{1}\le\hat{\theta}_{2}\le\dots\le\hat{\theta}_{B}$$
and for a significance level α the interval bounds are located at the order statistics
$$K_{1}=\left\lfloor B\frac{\alpha}{2}\right\rfloor,\qquad K_{2}=\left\lfloor B\left(1-\frac{\alpha}{2}\right)\right\rfloor$$
$$P\left\{\hat{\theta}_{K_{1}}<\theta<\hat{\theta}_{K_{2}}\right\}=1-\alpha$$
so that (θ̂K1, θ̂K2) is the 1 − α confidence interval of the estimated parameter. The interval converts a point estimate into a statement about reliability: it expresses the probability that the true value of the characteristic lies between the two bounds.
Distribution shape, K–S test and the choice of estimate
For a representative sequence point with an original sample of n = 20 gears, the 25 000 resampled means and the 25 000 resampled standard deviations were each binned into histograms. Both distributions are unimodal and close to symmetric, and a Kolmogorov–Smirnov test did not reject normality. This result matters for interpretation rather than for procedure: because the resampled statistics are approximately normally distributed, their mean can be taken as the estimated true value of the characteristic, and the interval endpoints read directly as a confidence statement. No normality assumption was required to generate the distribution — the test merely confirms that the usual summary measures are appropriate for it.
How sample size affects bias and reported precision
The practical question is how few gears are enough. The comparison below uses samples of 10 to 35 elements at the monitored point; the original-sample and bootstrap estimates of the mean and of the dispersion δ (the standard deviation of the deviations) are listed with their relative differences.
| Sample size n | Original mean σ̄ | Bootstrap mean ξ̄B | Relative error | Original δ | Bootstrap δ | Relative error |
|---|---|---|---|---|---|---|
| 10 | −1.610 0 | −1.599 4 | 0.66% | 3.769 3 | 3.579 8 | 5.03% |
| 15 | −0.713 3 | −0.717 2 | 0.55% | 3.516 1 | 3.397 2 | 3.38% |
| 20 | −0.905 0 | −0.908 2 | 0.35% | 3.274 4 | 3.190 8 | 2.64% |
| 25 | 0.420 0 | 0.421 0 | 0.24% | 4.010 0 | 3.926 9 | 2.10% |
| 30 | 0.856 7 | 0.858 0 | 0.15% | 3.824 7 | 3.761 0 | 1.66% |
| 35 | 0.502 9 | 0.503 0 | 0.019% | 4.007 2 | 3.949 3 | 1.44% |
Two trends stand out. The bootstrap mean converges on the sample mean rapidly: the discrepancy falls from 0.66% at n = 10 to 0.019% at n = 35, at which point the two values are indistinguishable at the reported precision. The dispersion converges more slowly, from 5.03% to 1.44%, which is expected — a variance estimator is intrinsically less stable than a mean estimator and needs more data to settle. The important conclusion is that a ten-gear sample already reproduces the sample mean to better than 1%, so the resampling step itself introduces negligible bias at the sample sizes that are affordable in production.
| Sample size n | Interval for the mean | Interval for δ |
|---|---|---|
| 10 | (−1.609 3, −1.589 4) | (3.572 8, 3.586 9) |
| 15 | (−0.716 6, −0.701 3) | (3.394 3, 3.406 5) |
| 20 | (−0.913 3, −0.903 1) | (3.187 2, 3.194 4) |
| 25 | (0.415 4, 0.426 6) | (3.922 9, 3.930 9) |
| 30 | (0.852 6, 0.863 3) | (3.757 2, 3.764 8) |
| 35 | (0.498 2, 0.507 8) | (3.946 0, 3.952 7) |
These intervals are narrow relative to the dispersion δ of the underlying values, which is a property of percentile intervals computed from the empirical distribution of a single given sample: they quantify how precisely the resampled statistic reproduces the sample it came from, not how widely individual gears in the population scatter. They should therefore be read as a stability measure for the batch estimate, while gear-to-gear variability is carried by δ itself.
Rebuilding the batch surface with cubic NURBS
With a mean and a dispersion available at each of the 225 grid points, the batch distortion becomes a surface that can be reconstructed and compared with the design flank. A NURBS surface is the tensor-product extension of a NURBS curve over an (m + 1) × (n + 1) control net:
$$P(u,v)=\frac{\sum_{i=0}^{n}\sum_{j=0}^{m}B_{i,k}(u)B_{j,l}(v)W_{i,j}V_{i,j}}{\sum_{i=0}^{n}\sum_{j=0}^{m}B_{i,k}(u)B_{j,l}(v)W_{i,j}}$$
where Vi,j are the control vertices, Wi,j the weights, and Bi,k, Bj,l the B-spline basis functions of degree k and l in the two parametric directions. The basis functions follow the recursive de Boor–Cox definition:
$$B_{i,0}(x)=\begin{cases}1, & s_{i}\le x\le s_{i+1}\\ 0, & \text{otherwise}\end{cases}$$
$$B_{i,k}(x)=\frac{x-s_{i}}{s_{i+k}-s_{i}}B_{i,k-1}(x)+\frac{s_{i+k+1}-x}{s_{i+k+1}-s_{i+1}}B_{i+1,k-1}(x),\quad k\ge 1$$
with the convention 0/0 = 0, s the knot vector and i the basis index. Cubic basis functions (k = 3) were selected so that the fitted surface is twice continuously differentiable, which keeps the reconstructed flank smooth enough for subsequent contact analysis. NURBS is also the only surface representation in the STEP standard and can describe both free-form and analytic geometry without changing its mathematical form under coordinate transformation, so the reconstructed batch surface stays exchangeable between measurement, design and manufacturing software.
Five versus ten measured gears
The reconstruction was tested on the wheel of an automotive drive-axle hypoid gear set from a single production batch. Two bootstrap runs were compared, one based on 5 gears and one on 10 gears, each producing a bootstrap mean-difference surface — the deviation of the reconstructed flank from the theoretical design flank, computed from the resampled means at every grid point — alongside the actual mean-difference surface obtained directly from the measured gears.
The comparison was made at two locally magnified locations of the 15 × 15 grid, point 1 at row 8, column 4 and point 2 at row 10, column 11, with identical magnification for both runs. At both points the gap between the bootstrap mean-difference surface and the actual mean-difference surface is visibly larger for the 5-gear reconstruction than for the 10-gear reconstruction, while the position of the actual mean-difference surface relative to the theoretical design surface is, of course, unchanged — it is the fixed reference against which both reconstructions are judged. The 10-gear surface therefore reproduces the batch distortion field more faithfully, and the improvement is consistent with the error trend in Table 1: the resampling error of the mean is already small, but a ten-gear sample captures the shape of the distortion field better than a five-gear sample because more of the spatial variability is sampled.
What the model supports in production, and where its limits lie
The procedure gives a batch-level description of heat-treatment distortion from a handful of measured gears: a mean deviation field with a 95% confidence interval at each flank point, a dispersion field that quantifies gear-to-gear scatter, and a smooth NURBS reconstruction suitable for direct comparison with the design flank. Because the mean field represents the systematic part of the distortion, it is the quantity that should be compensated in the machine settings; the dispersion field sets the residual error that compensation cannot remove and therefore defines the tolerance capability of the process. This is the practical link between flank metrology and the closed-loop correction of cutting parameters, and it replaces the practice of chasing a distortion trend through repeated orthogonal cutting trials with a computation performed on data that a sampling inspection already produces.
The scope of the result should be stated as clearly as its content. All statistics were derived from one batch with fixed material, fixed CNC milling settings and a fixed heat-treatment specification, so the estimated mean field characterizes that process window and would have to be re-established after any change of material, fixture, cutting parameters or quench schedule. The bootstrap resamples the measured gears and cannot create information that the sample does not contain: if the sampled gears are unrepresentative, the confidence intervals will be confidently wrong. The method assumes exchangeability of gears within the batch, and it does not model the physical mechanism of distortion, so it interpolates and summarizes observed behaviour rather than predicting the effect of a process change. Within those boundaries — a stable process, a small inspected sample, an unknown distribution — the resampling approach converts an inspection constraint into a quantified estimate, and it does so at a computational cost that is negligible next to the price of measuring additional gears.