Why the flank and the machine settings have to be recovered together
Spiral bevel gears carry motion between intersecting shafts and are standard equipment in aerospace, marine, vehicle and construction-machinery drivetrains, where compact packaging, smooth running and high load capacity are required simultaneously. The same curved-tooth geometry that delivers those properties is what makes the parts difficult to specify. A flank is not given by a closed-form algebraic surface; it is the envelope traced by a cutter or grinding wheel moving through a prescribed kinematic chain. Curvature, spiral-angle distribution and ease-off are therefore implicit functions of cutter geometry and machine settings, and so are the contact pattern, the transmission error and the sensitivity to misalignment.
That coupling defines the classical development loop. In Gleason practice the gear member is cut first by the generating or forming method, a reference point is selected on its flank, and the pinion curvature is corrected against it using the local conjugate principle; the pair is then rolled and the contact pattern inspected. If the pattern is unacceptable, the settings are adjusted and another pair is cut. Each iteration consumes a machine setup, a workpiece and experienced judgement, and convergence depends on the operator rather than on a deterministic procedure.
The loop becomes a real obstacle when the settings are not available. High-performance transmissions are often procured as imported matched pairs, with the adjustment card withheld, so there is no model against which to evaluate performance and no path to improvement; because the two members are manufactured as a pair, replacing a single failed gear means buying another pair. Reverse design addresses this gap directly: measure the finished flank, recover the blank geometry, recover the settings that would generate that flank, and use the recovered model for analysis and correction.
What makes the inverse problem tractable is the architecture of modern CNC bevel gear machines. From Gleason’s No. 16 (1919) and No. 116 (1954, with a cutter-tilt mechanism for pressure-angle correction and modified-roll cutting of the pinion), Oerlikon’s PLC-equipped S17 (1973), Gleason’s GMAXX2010 (1986) and Phoenix II (2003, three linear and three rotary axes coordinated without cradle, cutter tilt or modified-roll mechanisms), to Oerlikon’s vertical dry-cutting machines (2005), the mechanical cradle was progressively replaced by software-defined motion. Chinese machine building followed with the seven-axis CNC grinder YK2045 in 1999 and the YKA2260 dry-cutting machine in 2012. On such machines the flank is generated by a small set of universal motion parameters — three locating the cutter (radial setting, cutter tilt, swivel angle) and four locating the workpiece (vertical offset, horizontal setting, sliding-base position, machine root angle), plus the roll ratio. A moderate, well-defined unknown vector is precisely what an inverse identification procedure requires.
Acquisition: structured light instead of point-by-point probing
Touch probing on coordinate measuring machines and gear measuring centers remains the reference for accuracy, but it acquires one point at a time, requires collision-free path planning around a twisted flank, and is slow enough that a full-flank map becomes expensive. Optical methods capture an entire flank in a handful of views without contacting the surface. Within that family, structured-light fringe projection offers a simple optical layout, high accuracy, short acquisition time and good immunity to ambient light, which is why it is widely used for three-dimensional inspection of complex tooth surfaces.
The measurement system used here is a GOM ATOS Triple Scan instrument. The head integrates a high-precision fringe projector with two stereo CCD cameras; a stand, turntable and robot arm provide multi-view access, and the companion GOM Scan / ATOS Professional software triangulates the captured data into a mesh and exports STL or PLY. The system is appropriate for the object class considered — gear pairs within a 150 × 150 × 150 mm³ to 300 × 300 × 300 mm³ envelope, with an overall data accuracy requirement no worse than 0.01 mm/m.
| Parameter | Value |
|---|---|
| Measuring field, single scan | 100 × 75 mm² |
| Overall accuracy within the scan volume | ≤ 0.01 mm |
| CCD resolution | 3692 × 2472 px |
| Point spacing | 0.01–0.61 mm |
| Working distance | 490–2000 mm |
| Scanner head dimensions | 570 × 340 × 240 mm |
| Turntable load capacity | 5 kg |
| Tilt range | ±115° |
| Maximum rotation speed | 60 °/s |
Four mechanisms govern the result. The triple-scan arrangement captures the workpiece from three directions in a single shot. Triangulation converts the projected pattern into depth. Fourier-transform profilometry recovers the phase information that encodes surface relief: the projected grating is distorted by the flank topography, and the deformed fringe image carries the geometric signature of the surface. Finally, reference-point stitching allows the individual views to be merged into one coordinate frame; adhesive markers are numbered during the first shot and, once at least three numbered points are recognized in a later view, the software aligns the new data automatically.
In practice the gear is cleaned of dust and oil, sprayed with a thin developer (titanium powder in alcohol) so that a reflective, sometimes dark ground surface scatters light uniformly, and fitted with reference markers; the tooth flanks and the registration features are coated most carefully. The head then photographs the part from a sequence of angular positions produced by the rotating stand, and the point cloud is exported.
Preprocessing, feature extraction and datum registration
Raw scanner output contains stray points from the environment and from the fixture, and it is far denser than the subsequent geometry work requires. Three operations are applied in sequence. Statistical outlier removal computes, for each point, the mean distance to its neighbors within a set neighborhood and rejects points whose mean distance exceeds a threshold derived from the global mean and standard deviation. Voxel-grid downsampling builds the minimum bounding box of the cloud, divides it into cubic cells of fixed size, and keeps only the original point nearest the centroid of each cell, reducing a multi-million-point cloud to a workable density while preserving geometric detail. Manual segmentation in CloudCompare — polygon selection under free viewpoint control, repeated until only the flank of interest remains — then isolates individual teeth and, separately, the concave and convex flanks.
Registration is the critical step, because point coordinates are acquired in a measuring coordinate system (MCS) defined by the instrument, whereas all subsequent design and meshing analysis is performed in a design coordinate system (DCS) whose origin is the pitch apex. For a gear-shaft part, the pitch apex lies on the axis of the mounting face at a distance equal to the mounting distance; the mounting face is itself a precision-machined datum. The difficulty is that the scanner cannot see the tooth flanks and the mounting face in the same view. The workaround is to use a machined cylindrical surface visible together with the teeth, recover its circular edge and plane, measure the distance from that plane to the mounting face with a height gauge, and propagate the result to the pitch apex.
Extracting the circular edge begins with boundary detection on the segmented patch. A k-d tree is built over the cloud by recursively splitting on the axis of largest variance at the median coordinate, which makes nearest-neighbor queries logarithmic; for each query point the $k$ nearest neighbors are retrieved, their centroid $\mathbf{u}_i$ and the mean neighbor distance $\bar{d}_i$ are computed, and a boundary indicator is compared against a threshold $\lambda$. Points passing the test form the edge set used for fitting.
The edge of a machined face is a circle in space, which has no fixed algebraic form and is treated as the intersection of a sphere and a plane. Fitting therefore proceeds in four stages: a least-squares plane fit $z = \alpha x + \beta y + \gamma$ through the edge points; construction of a local frame $\{P\}$ aligned with the plane normal $\mathbf{n} = (\alpha, \beta, -1)$; projection of the points onto that plane and a least-squares circle fit in two dimensions; and transformation of the center and radius back to the original frame. The output is a center point and a unit normal.
Coordinate transformation then aligns the measured normal with the $z$-axis of the DCS by a rotation $\varphi_1 = \arctan(n_y/n_x)$ about $z$ followed by a rotation $\varphi_2 = \arctan\!\left(\sqrt{n_x^2+n_y^2}/n_z\right)$ about $x$, restores the center to its original position by an inverse translation so that only the orientation is changed, translates by the measured distance from the registration face to the mounting face, and finally translates along the axis by the mounting distance $A_1$. The cloud now sits in the DCS.
Because no closed-form check exists for this sequence, the registration was verified against an independent contact measurement: a coordinate measuring machine, operating natively in the design frame, probed a 5 × 9 grid on the same flank, and the two point sets were overlaid. The positional agreement confirmed that the optically measured cloud had been brought into the design frame to a usable level of accuracy.
Recovering the blank from the axial-section projection
Blank parameters define the outer boundary of the tooth — the pitch cone, face width, addendum and dedendum, face and root angles — and therefore set the load-carrying envelope within which the flank is later shaped. They are recovered from the boundary of the flank, not from its interior curvature. Each flank point $(X_{ij}, Y_{ij}, Z_{ij})$ is rotated onto the axial plane through the gear axis,
$$R_{ij}=\sqrt{X_{ij}^{2}+Y_{ij}^{2}},\qquad L_{ij}=Z_{ij},$$
which converts a three-dimensional surface problem into a two-dimensional one and makes the four boundaries — toe, heel, tip and root — explicit as curves in the $(R, L)$ plane. Boundary points of the projected set are detected with the same neighborhood criterion used for the registration feature and are classified by position into toe, heel, tip and root groups. Because the projection is two-dimensional, each group is fitted by a straight line $L = a + bR$ in the least-squares sense,
$$\min_{a,b}\sum_{i=1}^{N}\left[L_{ij}-(a+bR_{ij})\right]^{2},$$
giving the four lines that form the reconstructed flank boundary. Tip and root lines govern tooth height and hence contact ratio and bending strength; toe and heel lines govern usable face width and the extent of the contact zone.
Pitch geometry follows from the reconstructed boundary together with the tangency condition of the two pitch cones. With shaft angle $\Sigma$, tooth counts $z_1$ and $z_2$, and $\Sigma=\gamma_1+\gamma_2$,
$$\tan\gamma_2=\frac{\sin\Sigma}{z_1/z_2+\cos\Sigma},$$
which for a 25/37 pair at 90° gives $\gamma_1 = 34.05°$ and $\gamma_2 = 55.95°$. The cone distance $A_0$ is read directly from the projection as the distance from the pitch apex to the heel point; the pitch diameter, module and face width follow as
$$d_g = 2A_0\sin\gamma_g,\qquad m_g=\frac{d_g}{z_g},\qquad F_w = O_gP_b-O_gP_f,$$
with the mid-face cone distance $A_w = A_0 – F_w/2$. Large-end addendum and dedendum are taken from the projected positions of the tip and root lines relative to the pitch line, and the face and root angles follow from the arccosines of the corresponding edge vectors; mid-face values are then obtained by applying the half-difference of face and pitch angles, and of root and pitch angles, over the face width.
Projection, edge detection and line fitting each introduce small errors, so the pinion and gear blanks recovered independently do not satisfy the geometric conditions of a conjugate pair. Three correction rules are imposed: the projected toe and heel lines must be parallel and perpendicular to the pitch line; module, cone distance, face width and whole depth must be identical for both members; and the pinion root angle plus the gear face angle, and the gear root angle plus the pinion face angle, must each equal 90°. Applying these constraints yields the corrected blank set in Table 2, in which the pinion and gear values are consistent and the module becomes exactly 4 mm for both members.
| Parameter | Pinion, projected | Gear, projected | Pinion, corrected | Gear, corrected |
|---|---|---|---|---|
| Number of teeth $z$ | 25 | 37 | 25 | 37 |
| Shaft angle $\Sigma$ (°) | 90 | 90 | ||
| Pitch angle $\gamma$ (°) | 34.05 | 55.95 | 34.05 | 55.95 |
| Module at large end $m$ (mm) | 4 | 3.99 | 4 | 4 |
| Hand | right | left | right | left |
| Cone distance $A_0$ (mm) | 89.24 | 89.15 | 89.30 | 89.30 |
| Pitch diameter $d$ (mm) | 99.93 | 147.73 | 100 | 148 |
| Face width $F_w$ (mm) | 24.92 | 24.98 | 25 | 25 |
| Whole depth $h$ (mm) | 7.70 | 7.58 | 7.60 | 7.60 |
| Addendum at large end $h_{ae}$ (mm) | 4.23 | 2.59 | 4.25 | 2.55 |
| Dedendum at large end $h_{fe}$ (mm) | 3.47 | 4.99 | 3.35 | 5.05 |
| Root angle $\delta_a$ (°) | 31.91 | 52.76 | 31.90 | 52.72 |
| Face angle $\delta_f$ (°) | 37.28 | 58.14 | 37.28 | 58.10 |
| Addendum angle $\theta_a$ (°) | 3.23 | 2.19 | 3.23 | 2.15 |
| Dedendum angle $\theta_f$ (°) | 2.14 | 3.19 | 2.15 | 3.23 |
From blank geometry to initial machine settings
The initial settings are computed from the corrected blank using the generating principle. The gear member is cut by the generating method, in which the cutter represents one tooth of a virtual crown gear and the workpiece rolls with that crown gear; the pinion is cut single-flank by the modified-roll method, in which the roll ratio is deliberately varied during the generating stroke to control curvature. Both flanks of the gear are cut in one pass with a single setting, while the two pinion flanks require separate cutter geometries and settings.
The calculation is anchored at the mid-face reference point, where the two flanks of the gear slot are equidistant. Because both flanks must be produced with one setting, the two design points are replaced by a single point obtained by offsetting each flank to the center of the slot. With cutter point width $W_2$, mid-face gear dedendum $h_{f2}$ and cutter profile angle $\alpha_{02}$, the offset is
$$h = 0.5\,W_2\cos\alpha_{02}\pm h_{f2}\sin\alpha_{02},$$
which fixes the cutter radius at the reference point and hence the nominal cutter radius. The remaining settings follow from the triangle formed by the crown-gear center, the cutter center and the pitch apex: a radial setting $S_R$ and angular setting $q$ locate the cutter, a vertical offset $E_M$, horizontal setting $X_D$, sliding-base position $X_B$ and machine root angle $\gamma_m$ locate the workpiece, and the roll ratio $m_C$ follows from the ratio of crown-gear radius to cutter radius and the corresponding pitch angles. For the pair considered here these quantities come out at $S_R$ between 69.6 mm and 70.7 mm, angular settings between 59.1° and 62.5°, machine root angles of 31.93° and 52.75°, and roll ratios of 1.7879, 1.7428 and 1.2050 for pinion concave, pinion convex and gear respectively.
A tooth surface model driven by the machine settings
The cutter is described by a straight-line profile with radius $R_c = R_0 \pm P_W/2$, where $R_0$ is the nominal cutter radius, $P_W$ the point width, and the sign selects the outer or inner blade and hence the concave or convex flank. In the profile frame $(s,\theta)$, with $s$ the profile arc parameter, $\theta$ the cutter rotation parameter and $\alpha_c$ the profile pressure angle, the position vector and unit normal of the profile are
$$\mathbf{r}_c(s,\theta)=\begin{bmatrix}(R_c \pm s\sin\alpha_c)\cos\theta\\ (R_c \pm s\sin\alpha_c)\sin\theta\\ \mp s\cos\alpha_c\end{bmatrix},\qquad \mathbf{n}_c=\frac{\partial \mathbf{r}_c/\partial s \times \partial \mathbf{r}_c/\partial \theta}{\left|\partial \mathbf{r}_c/\partial s \times \partial \mathbf{r}_c/\partial \theta\right|}.$$
Eight universal motion parameters define the relative motion between cutter and workpiece, organized as a kinematic chain from the cutter frame through the machine frames to the workpiece frame. Three parameters place the cutter — radial setting $S_r$, cutter tilt $\sigma$ and swivel angle $\zeta$ — and four place the workpiece — vertical offset $E_M$, horizontal setting $X_D$, sliding-base position $X_B$ and machine root angle $\gamma_m$ — while the roll ratio couples the cutter and workpiece rotations. In the full-generating process an additional helical motion is superimposed, so the sliding base is no longer fixed but varies with the cradle angle $\phi_c$,
$$X_B = X_{B0}+\chi_H\phi_c,\qquad \phi_p = R_a\phi_c – C\phi_c^{2}-D\phi_c^{3},$$
where $X_{B0}$ is the initial sliding-base position, $\chi_H$ the helical correction coefficient, $R_a$ the roll ratio and $C$, $D$ higher-order modification coefficients. This is what allows both flanks of a slot to be finished in one grinding pass.
The surface itself is obtained from the envelope condition. With $\mathbf{r}_b$ the position of a point on the generated flank and $\mathbf{v}_{bc}$ the relative velocity between cutter and workpiece, the equation of meshing is
$$\mathbf{n}_b\cdot\mathbf{v}_{bc}=0,$$
and solving it together with the kinematic chain gives the flank as an implicit function of the settings. For computation the surface is discretized on the standard inspection grid: with the gear axis as $x$ and the radial coordinate in the $yz$ plane, points satisfy
$$y_b^{2}+z_b^{2}=R_i^{2},\qquad x_b=L_i,$$
and the grid is taken as 5 × 9 — five points across the tooth height and nine along the face width — matching the layout used for measurement, so that computed and measured points correspond one-to-one.
Representing the deviation surface with six coefficients
The comparison between the computed flank $r_{ij}$ and the measured flank $p_{ij}$ contains two physically different contributions. An angular mismatch between the two grids, caused by indexing when the part was set up, appears as a rigid rotation and is removed first: the theoretical surface is rotated about the gear axis by an angle $\theta$ chosen so that the mid-points of the two grids coincide, giving $r’_{ij}$. What remains is the form deviation, evaluated along the surface normal,
$$h_{ij}=\left(p_{ij}-r’_{ij}\right)\cdot\mathbf{n}_{ij}.$$
A 45-point grid would normally imply 45 objectives, an over-determined and often ill-conditioned system in which the correction step becomes unstable. Instead the deviation field is compressed into a second-order polynomial in local coordinates, with the origin at the mid-point of the flank, $X$ along the face width, $Y$ along the tooth height and $Z$ along the normal:
$$Z = a_0+a_1X+a_2Y+a_3XY+a_4X^{2}+a_5Y^{2}=h_{ij}.$$
The six coefficients have direct geometric meaning: $a_0$ is a uniform offset and coincides with the indexing error already removed, so it is zero; $a_1$ and $a_2$ tilt the surface along the face width and tooth height; $a_3$ rotates it in the $XY$ plane; $a_4$ and $a_5$ bend it in the two principal directions. The deviation surface of a real flank is shallow enough that a second-order model captures it to high accuracy, and the vector $\{a_1,\dots,a_5\}$ replaces 45 residuals as the target of correction.
Because the polynomial is nonlinear in nothing but its coefficients, the fit is a linear least-squares problem in form yet is solved here with Levenberg–Marquardt for consistency with the sensitivity stage and for robustness against outlier points. The update is
$$\mathbf{a}^{(n)}=\mathbf{a}^{(n-1)}-\left(\mathbf{J}^{\mathsf{T}}\mathbf{J}\right)^{-1}\mathbf{J}^{\mathsf{T}}\Delta\mathbf{h},$$
with the Jacobian assembled numerically by perturbing each coefficient in turn and recomputing the fitted surface, and iteration continuing until the residual falls below 0.001 mm. Applied to the initial pinion-convex deviation field, whose peak absolute value is 16.3 μm, the fit returns $a_1 = 5.44\times10^{-4}$, $a_2 = -2.41\times10^{-3}$, $a_3 = 3.57\times10^{-4}$, $a_4 = -2.13\times10^{-4}$ and $a_5 = -1.61\times10^{-4}$ (mm), with the largest term in the tooth-height tilt.
Sensitivity-guided correction of the machine settings
Correction requires knowing which setting controls which coefficient. Each of the twelve candidate parameters — cutter radius, cutter pressure angle, cutter corner radius, radial setting, angular setting, cutter tilt, swivel angle, vertical offset, machine root angle, horizontal setting, sliding-base position and roll ratio — is perturbed in turn while the others are held fixed. Because a raw perturbation would change the magnitude as well as the shape of the deviation, each parameter is iteratively adjusted, using a secant update with finite differences in place of derivatives, until its deviation surface reaches the same extremum as the initial one; only then are the six coefficients refitted. The result is a matrix of coefficients per unit of each setting, reported in Table 3.
| Setting | $a_1$ | $a_2$ | $a_3$ | $a_4$ | $a_5$ |
|---|---|---|---|---|---|
| Cutter radius $R$ (mm) | 0.004 864 | 0.001 550 | 0.000 573 | 0.000 489 | 0.000 016 |
| Cutter pressure angle $\alpha$ (°) | 0.002 141 | 0.003 331 | 0.000 222 | 0.000 404 | 0.000 311 |
| Cutter corner radius $r_e$ (mm) | 0.000 171 | 0.002 513 | 0.000 015 | 0.000 011 | 0.000 017 |
| Radial setting $S_R$ (mm) | 0.000 212 | 0.004 002 | 0.000 187 | 0.000 044 | 0.000 499 |
| Angular setting $q$ (°) | 0.001 146 | 0.001 878 | 0.000 018 | 0.000 121 | 0.000 002 |
| Cutter tilt $\sigma$ (°) | 0.000 968 | 0.001 926 | 0.000 081 | 0.000 106 | 0.000 049 |
| Swivel angle $\zeta$ (°) | 0.002 831 | 0.000 718 | 0.000 011 | 0.000 216 | 0.000 056 |
| Vertical offset $E_M$ (mm) | 0.000 773 | 0.000 016 | 0.000 446 | 0.000 240 | 0.000 222 |
| Machine root angle $\gamma_m$ (°) | 0.001 896 | 0.001 344 | 0.000 106 | 0.000 154 | 0.000 050 |
| Horizontal setting $X_D$ (mm) | 0.000 012 | 0.003 102 | 0.000 064 | 0.000 026 | 0.000 173 |
| Sliding-base position $X_B$ (mm) | 0.003 620 | 0.000 265 | 0.000 273 | 0.000 348 | 0.000 001 |
| Roll ratio $m_C$ | 0.002 319 | 0.001 577 | 0.000 102 | 0.000 183 | 0.000 043 |
Five settings dominate: cutter radius (strong in $a_1$), cutter pressure angle and radial setting (strong in $a_2$), horizontal setting (strong in $a_2$) and sliding-base position (strong in $a_1$). Since five coefficients must be cancelled, selecting exactly these five is not merely convenient but matched to the dimension of the target vector. Treating the response as linear in small increments, the compensation $\Delta\boldsymbol{\xi}$ solves
$$\begin{bmatrix}\partial a_1/\partial\xi_{k1} & \cdots & \partial a_1/\partial\xi_{k5}\\ \vdots & & \vdots\\ \partial a_5/\partial\xi_{k1} & \cdots & \partial a_5/\partial\xi_{k5}\end{bmatrix}\begin{bmatrix}\Delta\xi_{k1}\\ \vdots\\ \Delta\xi_{k5}\end{bmatrix}=-\begin{bmatrix}a_1(\boldsymbol{\xi}^{(0)})\\ \vdots\\ a_5(\boldsymbol{\xi}^{(0)})\end{bmatrix},$$
and the linearization is repeated rather than trusted once. For the pinion convex flank, the first correction required $\Delta R = 2.795$ mm, $\Delta\alpha = 0.25°$, $\Delta S_R = 0.4809$ mm, $\Delta X_D = 1.2090$ mm and $\Delta X_B = 0.6373$ mm.
The effect on the peak deviation is rapid. After one iteration the maximum fell from 16.3 μm to 5.7 μm, a reduction of 65%; after two iterations it reached 1.5 μm, a cumulative reduction of 91%; third and fourth iterations left it at 1.4 μm, indicating convergence. Applying the same procedure to the other flanks gave peak deviations of 3.6 μm for the pinion concave flank, 2.3 μm for the gear concave flank and 3.7 μm for the gear convex flank. The resulting setting set is listed in Table 4; the largest changes from the initial values are the cutter radius, the pressure angles and the horizontal setting, which is consistent with the sensitivity ranking.
| Setting | Pinion, concave | Pinion, convex | Gear |
|---|---|---|---|
| Cutter diameter $D$ (mm) | 156.9720 | 151.8920 | 154.5590 / 150.2410 |
| Cutter pressure angle $\alpha$ (°) | 18.75 | 21.25 | 18 / 22 |
| Main-profile modification radius $r_c$ (mm) | 0 | 0 | 0 |
| Toprem radius $r_t$ (mm) | 0 | 0 | 0 |
| Toprem height $h_t$ (mm) | 0 | 0 | 0 |
| Cutter corner radius $r_e$ (mm) | 0.8000 | 0.8000 | 0.8128 |
| Radial setting $S_R$ (mm) | 72.8443 | 70.1027 | 70.6535 |
| Angular setting $q$ (°) | 65.54 | 59.11 | 62.06 |
| Cutter tilt $\sigma$ (°) | 0 | 0 | 0 |
| Swivel angle $\zeta$ (°) | 25.58 | 27.74 | 207.06 |
| Vertical offset $E_M$ (mm) | -0.1016 | -0.1016 | 0 |
| Machine root angle $\gamma_m$ (°) | 31.93 | 31.93 | 52.72 |
| Horizontal setting $X_D$ (mm) | 3.4513 | -1.4947 | 0 |
| Sliding-base position $X_B$ (mm) | -1.8207 | 0.7926 | -0.0053 |
| Roll ratio $m_C$ | 1.7879 | 1.7428 | 1.2050 |
Verification against contact measurement and against running contact
Two independent checks were performed: whether the recovered settings reproduce the physical flank, and whether the resulting pair behaves correctly when it rolls.
For the geometric check, the finished parts were probed on a GLOBAL STATUS 575 coordinate measuring machine (indicated error $MPE_E = 2.5 + 4L/1000$ μm, probing error $MPE_P = 2.5$ μm, stroke 500 × 700 × 500 mm, PH10M indexing head and SP600 probe, PC-DMIS and QUINDOS software). The gear was set on the table with its reference face on the $XOY$ plane and its bore axis along $Z$. Sampling followed the Gleason layout of nine points along the face width and five along the tooth height, confined to ±10% in the width direction and ±5% in the height direction around the nominal grid, with stylus diameters of 0.5 or 1 mm. Because a single stylus cannot reach all 45 positions on the pinion without collision, eight to twelve radially arranged styli were used, with each stylus assigned to a defined subset of grid points and the probe retracted through the tooth space between measurements.
Coordinates were then computed at the same 45 nominal positions from the corrected settings, and the normal deviation between the two sets was evaluated. The worst point on each flank is given in Table 5. The largest discrepancy, 6.7 μm on the gear convex flank, is comparable to the uncertainty of the probing itself and small relative to the tooth depth of 7.6 mm, indicating that the identified settings carry real geometric information rather than a curve fit to the sampled points.
| Flank | Maximum deviation (μm) |
|---|---|
| Gear, concave | 3.0 |
| Gear, convex | 6.7 |
| Pinion, concave | 2.3 |
| Pinion, convex | 5.7 |
For the contact check, the instantaneous contact ellipse was computed on the tooth surface. At a candidate contact point $P$ with unit normal $N$, two orthonormal tangents are constructed, for example
$$\mathbf{a}=\left(0,\ \frac{n_z}{\sqrt{n_y^{2}+n_z^{2}}},\ -\frac{n_y}{\sqrt{n_y^{2}+n_z^{2}}}\right),\qquad \mathbf{b}=\mathbf{a}\times\mathbf{N},$$
and rays $\mathbf{R}(\vartheta)=\mathbf{R}’\cos\vartheta+\mathbf{R}”\sin\vartheta+P$ are cast in the tangent plane and intersected with both flanks. The separation at each ray direction is
$$d_j=\left(P_{Pj}-P_{Gj}\right)\cdot N_{Pj},$$
and the locus where the separation falls below the contact threshold defines the instantaneous contact ellipse; sweeping the contact point across the mesh gives the contact pattern and the transmission error simultaneously.
The physical reference was a rolling test on a bevel gear rolling checker at a load of 9.6658 N·m, with the flanks coated in marking compound and contact defined by a gap below 0.00635 mm, so that the wiped regions after running constitute the contact pattern. For this pair the pattern occupied roughly 30% of the available tooth area on each flank, with no edge contact and no evidence of stress concentration. The gear concave pattern sat at mid-face, displaced slightly toward the tooth tip and the toe; the pinion convex pattern sat at mid-face, displaced toward the toe. The computed pattern at the same load reproduced that size, shape and location, and the corresponding pressure distribution was uniform. Agreement between a numerical contact analysis driven purely by recovered settings and a physical roll test is the stronger of the two validations, because it tests the curvature of the flank rather than its position.
What the workflow establishes, and where it stops
The result is a chain that runs from a structured-light point cloud to a usable manufacturing description: registration of the cloud onto the design datum through a fitted circular feature, recovery of blank parameters from the axial-section projection with geometric consistency constraints, computation of initial settings from the generating principle, a machine-setting-driven surface model based on the universal motion concept, and a correction loop that compresses the 45-point deviation field into six coefficients, ranks settings by their influence on those coefficients, and solves for increments. The loop converges in three iterations, and the closed chain reproduces the measured flanks to within 6.7 μm while producing the same contact pattern as a physical roll test.
Three limitations bound the claim. First, the error sources along the chain — scanner accuracy, developer thickness, registration, boundary extraction and line fitting — are not propagated through a quantitative model, so the uncertainty of the recovered blank and setting values is not stated; the 6.7 μm end-to-end figure includes those effects but does not apportion them. Second, the segmentation and feature selection remain interactive, and the manual work on multi-million-point clouds dominates the processing time; automated feature recognition would be needed before the workflow could be applied routinely. Third, the demonstration covers one structured-light instrument and a standard-tooth-form aerospace pair within a defined size range; non-standard tooth forms, large-module and large-diameter gears, and data from laser line scanners or other optical instruments have not been examined, and the transferability of the sensitivity ranking across such cases is unknown. These are the natural next targets: a quantified error budget, learned feature extraction, and validation across instruments and gear classes.