Distortion as the binding constraint in gear carburizing
Spiral bevel gears transfer torque between intersecting shafts through curved teeth that enter mesh gradually and overlap in contact, which gives high load capacity per unit of volume, smooth running behaviour and a compact final-drive package. Aviation, marine and vehicle drivetrains depend on this tooth form. The service requirement is a hard, wear-resistant flank supported by a core tough enough to resist bending fatigue, and the standard route to that gradient is carburizing followed by quench hardening.
During carburizing the gear is held in the austenite range in a carbon-bearing atmosphere. The carbon potential at the surface fixes the surface carbon content, while the boost–diffuse schedule shapes the carbon profile and, with it, the case depth and the carbon concentration gradient. Quenching converts the carbon-enriched austenite to martensite and produces the hardness that the application demands. The same transformation is what moves the teeth: thermal gradients through the section, the volume change accompanying martensite formation and the carbon gradient across the case generate mutually incompatible local strains, and the part relaxes into a new shape.
The consequences are geometric. Pitch and spiral angle errors, tooth thickness change and a shift of the contact pattern away from its designed position all follow from a distortion of a few hundredths of a millimetre on a gear of 132 mm pitch diameter. Distortion beyond allowance forces extra grinding stock, additional lapping time or outright scrapping, so a process window that delivers the required hardness while keeping dimensional change small has direct economic value.
Five carburizing parameters govern both responses: carburizing temperature, boost carbon potential, boost time, diffusion carbon potential and diffusion time. At four levels each, a full factorial would require 1024 simulation runs, which is impractical for a transient thermomechanical-metallurgical model. The case examined below instead combines a 16-run orthogonal array (five factors, four levels), a finite element carburizing–quench simulation in Deform, entropy-weighted multi-objective scoring of distortion and hardness, level-wise sensitivity analysis, pairwise interaction analysis under the maximum-membership rule, and second-order regression models checked against dedicated confirmation runs.
Gear geometry, material and simulation set-up
The tooth form and blank dimensions define the case-depth-to-section ratio and therefore the thermal and transformation gradients that drive distortion. The analysed wheel is a left-hand spiral bevel gear with the parameters listed in Table 1.
| Parameter | Value |
|---|---|
| Module | 3 mm |
| Number of teeth | 44 |
| Pressure angle | 20° |
| Face width | 20.2 mm |
| Addendum coefficient | 0.85 |
| Pitch diameter | 132 mm |
| Outer cone distance | 69.73 mm |
| Face cone angle | 73.39° |
| Pitch cone angle | 71.1° |
| Root cone angle | 67° |
| Spiral hand | Left |
The material is 18Cr2Ni4WA, a nickel–chromium–tungsten carburizing steel selected for high strength, toughness and hardenability in heavy gear applications. In the Deform material library the closest available dataset, BS655M13, is used as a surrogate for the thermophysical and transformation data. The gear is meshed with 18,218 nodes and 77,031 elements, and the bottom face is constrained to represent the flat-on-charge loading position used in production furnaces.
Quenching is held constant across all runs so that the carburizing variables are not confounded with cooling severity: austenitizing and quench start at 820 °C, quench duration 1200 s, oil at 20 °C, with the oil-side heat transfer coefficient taken from published quench-oil data as the thermal boundary condition. Only the carburizing stage is varied, which is the practical lever available to a process engineer once the quench line and fixture are fixed.
Orthogonal array and simulated responses
The five factors and their four levels are given in Table 2. Levels were chosen to bracket the industrially used window for this steel: 900–930 °C for carburizing, boost carbon potential 0.75–0.90 %, boost time 2.50–4.00 h, diffusion carbon potential 0.80–0.95 % and diffusion time 1.00–2.50 h.
| Level | A: Carburizing temperature (°C) | B: Boost carbon potential (%) | C: Boost time (h) | E: Diffusion carbon potential (%) | F: Diffusion time (h) |
|---|---|---|---|---|---|
| 1 | 900 | 0.75 | 2.50 | 0.80 | 1.00 |
| 2 | 910 | 0.80 | 3.00 | 0.85 | 1.50 |
| 3 | 920 | 0.85 | 3.50 | 0.90 | 2.00 |
| 4 | 930 | 0.90 | 4.00 | 0.95 | 2.50 |
The 16-run array and the two simulated responses are collected in Table 3. Distortion spans 0.0944–0.116 mm, a spread of roughly 23 % of the minimum, whereas surface hardness spans only 62.3–62.8 HRC. That asymmetry is the central feature of the dataset: within the tested window the hardness response is close to saturated, so the process decision is effectively a distortion decision subject to a hardness floor.
| Run | A (°C) | B (%) | C (h) | E (%) | F (h) | Distortion (mm) | Hardness (HRC) | Composite score |
|---|---|---|---|---|---|---|---|---|
| 1 | 900 | 0.75 | 2.50 | 0.80 | 1.00 | 0.0944 | 62.3 | 0.7170 |
| 2 | 900 | 0.80 | 3.00 | 0.85 | 1.50 | 0.103 | 62.7 | 0.6579 |
| 3 | 900 | 0.85 | 3.50 | 0.90 | 2.00 | 0.110 | 62.8 | 0.4822 |
| 4 | 900 | 0.90 | 4.00 | 0.95 | 2.50 | 0.116 | 62.4 | 0.0566 |
| 5 | 910 | 0.75 | 3.00 | 0.90 | 2.50 | 0.109 | 62.7 | 0.4588 |
| 6 | 910 | 0.80 | 2.50 | 0.95 | 2.00 | 0.104 | 62.6 | 0.5681 |
| 7 | 910 | 0.85 | 4.00 | 0.80 | 1.50 | 0.108 | 62.7 | 0.4920 |
| 8 | 910 | 0.90 | 3.50 | 0.85 | 1.00 | 0.104 | 62.7 | 0.6247 |
| 9 | 920 | 0.75 | 3.50 | 0.95 | 1.50 | 0.106 | 62.7 | 0.5583 |
| 10 | 920 | 0.80 | 4.00 | 0.90 | 1.00 | 0.106 | 62.7 | 0.5583 |
| 11 | 920 | 0.85 | 2.50 | 0.85 | 2.50 | 0.107 | 62.7 | 0.5251 |
| 12 | 920 | 0.90 | 3.00 | 0.80 | 2.00 | 0.107 | 62.7 | 0.5251 |
| 13 | 930 | 0.75 | 4.00 | 0.85 | 2.00 | 0.112 | 62.7 | 0.3592 |
| 14 | 930 | 0.80 | 3.50 | 0.80 | 2.50 | 0.112 | 62.7 | 0.3592 |
| 15 | 930 | 0.85 | 3.00 | 0.95 | 1.00 | 0.102 | 62.6 | 0.6345 |
| 16 | 930 | 0.90 | 2.50 | 0.90 | 1.50 | 0.102 | 62.8 | 0.7477 |
Entropy-weighted scoring of the two objectives
Distortion and hardness are expressed in different units and pull in the same direction only weakly, so they are first mapped onto dimensionless membership values. Distortion is treated as smaller-the-better and hardness as larger-the-better:
$$x_{ij}=\frac{a_{ij}-\min_{k}a_{ij}(k)}{\max_{k}a_{ij}(k)-\min_{k}a_{ij}(k)}$$
$$x_{ij}=\frac{\max_{k}a_{ij}(k)-a_{ij}}{\max_{k}a_{ij}(k)-\min_{k}a_{ij}(k)}$$
where $a_{ij}$ is the element of the evaluation matrix, and the maximum and minimum are taken over the 16 runs for each indicator. The normalised matrix is then converted into an entropy weight, and the objective weights are adjusted by a subjective weight that reflects established practice on which indicator matters more for gear service:
$$L_{ij}=\frac{X_{ij}}{\sum_{j=1}^{n}X_{ij}},\qquad H_{i}=-\frac{1}{\ln n}\sum_{j=1}^{n}L_{ij}\ln L_{ij}$$
$$\omega_{i}=\frac{1-H_{i}}{m-\sum_{i=1}^{m}H_{i}},\qquad W_{i}=\frac{\tau_{i}\omega_{i}}{\sum_{i=1}^{m}\tau_{i}\omega_{i}}$$
with $n=16$ runs, $m=2$ indicators, $\tau_{i}$ the subjective weight and $W_{i}$ the resulting combined weight. The composite score of a run is the weighted sum of its membership values, $b_{j}=\sum_{i=1}^{m}W_{i}x_{ij}$, and is listed in the last column of Table 3.
| Indicator | Entropy $H_i$ | Objective weight $\omega_i$ | Subjective weight $\tau_i$ | Combined weight $W_i$ |
|---|---|---|---|---|
| Distortion | 0.9447 | 0.6042 | 0.624 | 0.717 |
| Hardness | 0.9638 | 0.3958 | 0.376 | 0.283 |
The entropy of the distortion indicator is the lower of the two, because distortion differentiates the runs more strongly than hardness does; the objective weighting therefore already favours distortion, and the subjective weighting reinforces it. The final split of 0.717 to 0.283 means that a 1 % relative improvement in distortion is worth roughly 2.5 times a 1 % relative improvement in hardness on the normalised scale — a reasonable stance for a gear whose hardness is already above specification across the whole array.
Level sensitivity: which parameter dominates, and when
To compare parameter levels, the composite scores of all runs sharing a given level are summed for each factor, and the four sums are normalised to a fuzzy subset over the levels. The normalised value $({\sum}b_{gh})’$ expresses how strongly that factor at that level contributes to the composite response, and the largest value in each row identifies the level where the factor is most influential.
| Factor | Level 1 | Level 2 | Level 3 | Level 4 | Peak value |
|---|---|---|---|---|---|
| A: Carburizing temperature | 1.9137 (0.2299) | 2.1436 (0.2575) | 2.1670 (0.2603) | 2.1006 (0.2523) | 0.2603 |
| B: Boost carbon potential | 2.0933 (0.2515) | 2.1436 (0.2575) | 2.1338 (0.2563) | 1.9542 (0.2347) | 0.2575 |
| C: Boost time | 2.5580 (0.3073) | 2.2764 (0.2734) | 2.0244 (0.2432) | 1.4661 (0.1761) | 0.3073 |
| E: Diffusion carbon potential | 2.0933 (0.2515) | 2.1670 (0.2603) | 2.2470 (0.2699) | 1.8176 (0.2183) | 0.2699 |
| F: Diffusion time | 2.5350 (0.3040) | 2.4560 (0.2950) | 1.9350 (0.2320) | 1.4000 (0.1680) | 0.3040 |
Ranking the peak values gives the order of influence: boost time (0.3073) > diffusion time (0.3040) > diffusion carbon potential (0.2699) > carburizing temperature (0.2603) > boost carbon potential (0.2575). The two time variables are almost equally dominant and clearly separated from the three atmosphere variables, which cluster within 0.013 of one another. In physical terms, the schedule duration controls how much of the section is thermally and transformationally loaded at once, whereas the atmosphere settings shift the carbon profile within a case that is already deep enough to saturate hardness.
The relative importance is not constant across levels, and the shift is informative. At level 1, boost time and diffusion time dominate, with the two carbon potentials tied and temperature last: at short, low-intensity schedules the carbon supply kinetics are the rate-limiting mechanism, and the surface carbon build-up is what determines the response. At level 2 diffusion time overtakes boost time, because once the boost stage has delivered sufficient surface carbon the redistribution of carbon towards the core becomes decisive. At level 3 diffusion carbon potential takes the lead ahead of temperature, which is consistent with carbon-potential control of the concentration gradient being the core requirement once the profile is deep. At level 4, the highest settings, carburizing temperature becomes dominant, followed by the two carbon potentials; at 930 °C with high potentials the thermal gradient and the total carbon charge are both at their maximum, and temperature control of the austenite condition governs the outcome. The practical reading is that the parameter that must be held tightly depends on where in the window the process is operated.
Pairwise interactions and the selected process window
Because the five factors are not independent in their effect on the carbon profile and on transformation strain, the interaction structure is analysed by forming the outer product of the normalised level vectors of each pair. For factors 1 and 2:
$$R_{lp}=\mathbf{c}_{1}^{\mathsf{T}}\mathbf{c}_{2},\qquad r_{lp}=\left(\sum b_{1l}\right)’\left(\sum b_{2p}\right)’$$
where $l$ and $p$ index the levels of the two factors and $r_{lp}$ is the joint membership of that level combination. Ten pairs result from five factors. The maximum-entry combination for each pair is listed in Table 6, with the factor subscripts giving the level.
| Pair | Best level combination | Membership |
|---|---|---|
| C1 × C2 (temperature × boost potential) | C1₃, C2₂ | 0.2575 |
| C1 × C3 (temperature × boost time) | C1₃, C3₁ | 0.2603 |
| C1 × C4 (temperature × diffusion potential) | C1₃, C4₃ | 0.2603 |
| C1 × C5 (temperature × diffusion time) | C1₃, C5₁ | 0.2603 |
| C2 × C3 (boost potential × boost time) | C2₂, C3₁ | 0.2575 |
| C2 × C4 (boost potential × diffusion potential) | C2₂, C4₃ | 0.2575 |
| C2 × C5 (boost potential × diffusion time) | C2₂, C5₁ | 0.2575 |
| C3 × C4 (boost time × diffusion potential) | C3₁, C4₃ | 0.2699 |
| C3 × C5 (boost time × diffusion time) | C3₁, C5₁ | 0.3045 |
| C4 × C5 (diffusion potential × diffusion time) | C4₃, C5₁ | 0.2699 |
The boost time–diffusion time interaction is the strongest at 0.3045, followed by the two interactions linking diffusion carbon potential to the two time variables at 0.2699. Interactions involving carburizing temperature reach 0.2603, and those involving boost carbon potential are the weakest at 0.2575. The dominance of the time–time interaction reflects the fact that boost and diffusion stages are coupled through the carbon profile: shortening one while extending the other largely preserves the surface carbon level, but changes the shape of the profile and the depth over which transformation strain develops. The weak role of boost carbon potential is consistent with the level analysis — with hardness near saturation, raising the boost potential adds carbon without buying hardness, and only adds transformation strain.
Applying the maximum-membership rule level by level to all five factors gives the combination C1₃C2₂C3₁C4₃C5₁: carburizing temperature 920 °C, boost carbon potential 0.80 %, boost time 2.50 h, diffusion carbon potential 0.90 % and diffusion time 1.00 h. The longest schedule in the array, 4.00 h boost plus 2.50 h diffusion, totals 6.5 h; the selected combination totals 3.5 h, so the optimum simultaneously shortens the cycle and reduces the distortion indicator.
Response models and confirmation runs
Second-order response surfaces were fitted by least squares with non-significant terms removed, using A in °C, B and E in %, and C and F in hours. The distortion model in millimetres is
$$D = 0.0843 – 0.000153A + 0.0837B – 0.0418C + 0.1702E + 0.007857F – 0.04B^{2} – 0.0009C^{2} – 0.09E^{2} – 0.0004F^{2} + 0.000058AC$$
and the surface hardness model in HRC is
$$H = -257 + 0.601A + 20.3B + 8.02C + 47.77E + 0.798F – 0.000313A^{2} – 12.5B^{2} – 0.175C^{2} – 27.5E^{2} – 0.225F^{2} – 0.0075AC$$
The distortion fit gives $R^{2}=99.97\ \%$, adjusted $R^{2}=99.92\ \%$, $F=1888.02$ and $p<0.0001$; the hardness fit gives $R^{2}=97.35\ \%$, adjusted $R^{2}=90.06\ \%$, $F=13.36$ and $p<0.0001$. Normal probability plots of the residuals place the data close to the reference line on both sides, so the residuals are consistent with a normal distribution and the fitted surfaces reproduce the simulated responses without systematic bias. Both models are valid only inside the tested ranges: 900–930 °C, 0.75–0.90 % boost potential, 2.50–4.00 h boost, 0.80–0.95 % diffusion potential and 1.00–2.50 h diffusion.
Five confirmation runs were simulated: the selected combination plus one-at-a-time departures from it. Results are given in Table 7.
| Case | A (°C) | B (%) | C (h) | E (%) | F (h) | Distortion (mm) | Hardness (HRC) |
|---|---|---|---|---|---|---|---|
| Selected combination | 920 | 0.80 | 2.5 | 0.90 | 1.0 | 0.0966 | 62.6 |
| Control 1 | 930 | 0.80 | 2.5 | 0.90 | 1.0 | 0.0972 | 62.1 |
| Control 2 | 920 | 0.85 | 2.5 | 0.90 | 1.0 | 0.0984 | 61.9 |
| Control 3 | 920 | 0.80 | 3.0 | 0.90 | 1.0 | 0.1020 | 62.2 |
| Control 4 | 920 | 0.80 | 2.5 | 0.85 | 1.0 | 0.0972 | 62.2 |
| Control 5 | 920 | 0.80 | 2.5 | 0.90 | 1.5 | 0.0998 | 62.8 |
Every departure from the selected setting increases distortion or reduces hardness, which supports the choice. Scored with the same membership functions and the array’s extreme values as normalisation limits, the selected combination reaches a composite of about 0.81, above the best of the 16 array runs (run 16, 0.7477) — the interaction-based combination is not among the orthogonal runs and improves on all of them.
Two of the controls deserve comment. Control 3, with boost time extended from 2.5 h to 3.0 h, raises distortion by about 5.6 % to 0.1020 mm and lowers hardness slightly to 62.2 HRC, confirming the sensitivity of the time factor seen in the level analysis. Control 5, with diffusion time extended to 1.5 h, gives the highest hardness of the whole set at 62.8 HRC but at 0.0998 mm of distortion, 3.3 % above the selected combination; under the combined weighting, 0.2 HRC is not worth that distortion, although a shop whose hardness margin is tight might legitimately take the trade. The comparison makes the objective conflict explicit rather than hidden: hardness and distortion respond differently to the same schedule change, and the weighting decides the outcome.
What the result means for process planning
The practical content of the optimization is a shift of emphasis. Hardness across the tested window varies by only 0.5 HRC, which is within what a heat-treatment shop would treat as a single population, while distortion varies by more than 20 %. Within this window, therefore, hardness behaves as a constraint to be satisfied rather than an objective to be maximized, and the useful process lever is schedule length. The selected combination of 920 °C, 0.80 % boost carbon potential, 2.50 h boost, 0.90 % diffusion carbon potential and 1.00 h diffusion delivers 62.6 HRC at 0.0966 mm, near the bottom of the distortion range and at the top of the hardness range, with a 3.5 h total cycle instead of the 6.5 h maximum. Shorter furnace time also means less energy per batch and lower high-temperature exposure of the fixture.
The sensitivity results add a control rule: which parameter needs the tightest tolerances depends on the operating level. In the low-intensity part of the window, atmosphere time control dominates; at high temperature and high carbon potential, temperature control does. A single tolerance stack applied uniformly across the window will therefore either over-constrain the inexpensive variables or under-constrain the decisive one.
The scope of the conclusions should be read against the model’s boundaries. The results apply to one geometry, one steel and one quench condition, with oil at 20 °C and a fixed 820 °C quench start; a different quench severity or fixturing would change the distortion magnitudes and could reorder the interaction strengths. The surrogate material dataset in the simulation library is an approximation of 18Cr2Ni4WA, and the analysis stops at the quench, so tempering, retained austenite conditioning and any stress relief are outside it — relevant for this steel class, where retained austenite in a heavily carburized case can both soften the surface and continue to transform after the quench. Distortion is condensed into a single scalar per run, so flank-specific deviations such as pitch and spiral angle errors are not resolved; a gear that passes the scalar criterion could still show a contact-pattern shift. The regression models are interpolation tools inside the tested ranges and should not be extrapolated. Confirmation by measured distortion on cut gears, rather than by simulated response, remains the step that would settle the industrial applicability of the recommended window.