Spiral bevel gears transmit torque between intersecting shafts in rolling mills, mining hoists, marine drives and heavy machine-tool spindles. Their flanks are cut on a generating machine whose cradle motion sweeps a curved, oblique tooth across the blank, and the resulting light-load contact pattern is the accepted shop-floor criterion for whether a gear set has been cut and mounted correctly. That pattern is normally read from marking compound pressed between mating flanks — a manual, qualitative check carried out on a bench. In service the same feature, the trace swept by the contact area across the flank, migrates as bearings settle, housings deflect and flanks wear. Migration toward the toe or the heel concentrates load on one or two teeth and can end in fracture of a component that is expensive to replace and dangerous when it fails.
Following the trace on a running set by imaging is difficult for three reasons. The contact region occupies a small fraction of a curved, specular, oil-wetted surface; the trace moves at pitch-line speed; and the question of whether a trace exists at all must be settled before tracking begins, since a tracker initialized on a stray reflection will faithfully follow noise.
A kinematic gate for deciding whether a trace exists
The flank is described by a generating surface and carried into a fixed frame through a chain of homogeneous transformations, rc(u, θ) = MebMbaMa1r1(u, θ, φ1), where u and θ are the two surface parameters of the cutter profile and φ1 is the rotation angle of the gear; the inverse chain M1gMgfMfeMedMde returns the surface to the generating frame. The unit normal follows from the partial derivatives,
$$ \mathbf{q}(u,\theta,\varphi_1)=\frac{\dfrac{\partial \mathbf{r}_c}{\partial u}\times\dfrac{\partial \mathbf{r}_c}{\partial \theta}}{\left|\dfrac{\partial \mathbf{r}_c}{\partial u}\times\dfrac{\partial \mathbf{r}_c}{\partial \theta}\right|} $$
and the meshing condition is written as f(u, θ, φ1) = q · vrel = 0, with vrel the relative motion of the two members at the candidate point. The interpretation attached to this scalar is what gives the method its pre-filter. When the surface velocity of the gear and the velocity of the mating flank coincide, f vanishes, no reverse sliding friction is generated, conformity is at its maximum and a contact trace is deemed present. When f is non-zero the two surfaces are sliding against each other or are partly separated, the conformity is incomplete, and no reliable trace is predicted at that location. Points flagged this way are excluded before the image stage, which lowers the false-detection rate and shortens the search.
The criterion is geometric and kinematic, computed from nominal tooth data and the shaft angle; it says nothing about brightness, wear or lubrication. It functions as a plausibility mask over the flank, not as a measurement.
Separating the trace from the flank by pixel thresholding
Within the masked region, the contact trace must be isolated from the surrounding flank. A global threshold T2 is applied so that a pixel keeps its gray level when that level reaches the threshold and is set to zero otherwise:
$$ g(x,y)=\begin{cases}\xi(x,y), & g(x,y)\ge T_2\\ 0, & g(x,y)<t_2\end{cases} $$
Here (x, y) are pixel coordinates of the contact-area image, g the original image and ξ the segmented result. Selecting T2 by inspection alone is unstable, so the threshold is chosen from the image histogram. With gray levels spanning 0 to Q2 and a candidate cut at Kn, the background occupies 0 to Kn and the target trace Kn+1 to Q2. The class probabilities, class means and total mean are
$$ \omega_0=\sum_{i=0}^{K_n}P(i),\quad \omega_1=1-\omega_0,\qquad \mu_0=\frac{1}{\omega_0}\sum_{i=0}^{K_n}iP(i),\quad \mu_1=\frac{1}{\omega_1}\sum_{i=K_n+1}^{Q_2}iP(i),\qquad \mu_T=\omega_0\mu_0+\omega_1\mu_1 $$
and the cut is placed where the between-class variance is largest:
$$ \sigma_B^2(K_n)=\omega_0\left(\mu_0-\mu_T\right)^2+\omega_1\left(\mu_1-\mu_T\right)^2,\qquad K_n^{*}=\arg\max_{0\le K_n<q_2}\ $$
This is the standard Otsu criterion, and it is the appropriate choice only if the trace and the flank form two reasonably separated gray-level populations. On a curved, specular surface under directional lighting, the flank itself produces a wide spread of intensities, so a single global cut is a weak assumption; the approach is best read as a fast first pass whose output supplies the features used downstream.
Tracking through the contact ridge line
Directional convolution of the segmented image with kernels SX and SY gives, at each pixel, a gradient direction θ1 = arctan(SY/SX), the local linear orientation before any transformation. After edge extraction, the trace boundary is observed to lie essentially on the ridge along which the flank meets its mating surface. This is the practical pivot of the method: instead of following the trace directly, the tracker follows the ridge, which is present on both flanks and at every meshing instant.
The ridge through a contact point p(a1, b1) is characterized by its slope ζ, written in terms of the contact area A at the edge point, the contact-point coordinate a1 and an intercept p0,
$$ \zeta = A\left(a_1-p_0\right) $$
and the ridge is compared with the trace position through the index
$$ F(X)=\zeta\left(X-b_1\right)+p_0 A $$
where X is position along the inspection direction. The decision rule reads: F(X) = 1 indicates a positive correlation, so the intersection lies on the trace and the tracked point is correct; 0 < F(X) < 1 indicates an offset, the numerical shortfall measuring the lateral displacement between ridge and trace, and tracking continues along that offset; F(X) = 0 indicates a negative correlation and the trace is not at the target point.
Because F varies continuously with coordinates and slope, treating unity as an exact coincidence test presupposes a normalization that the formulation does not derive. The usable content of the rule is the monotonic relation between F and the ridge–trace offset: F is a proxy for how far the tracked point has drifted, and its sign tells the tracker whether it is looking at contact or at empty flank. The same proxy is what allows concave and convex flanks to be handled by one procedure, since both possess a ridge, even though the concave side presents a broad contact area and the convex side a contact area that contracts toward a single loaded point.
Contact forms and their motion signatures
| Contact form | Appearance of the contact area | Motion direction |
|---|---|---|
| Forward contact | Gear and flank contact areas essentially equal; contact confined to the middle | Large end toward small end |
| Forward “large” contact | Areas essentially equal; enlarged contact area at the middle position | Large end toward small end |
| Forward contact (peripheral) | Areas essentially equal; enlarged contact area at the periphery | Motion present across the whole range |
| Reverse “small” contact | Small contact area, inclined toward the small end | Small end toward large end |
| Reverse “large” contact | Large contact area, inclined toward the large end | Small end toward large end |
| No contact | No contact area | No motion direction |
Test configuration
The evaluation used a pinion and wheel pair whose surfaces were generated in software and stored as image samples in a relational database, with the concave and convex flanks tracked separately. The tooth data are listed below.
| Parameter | Pinion | Wheel |
|---|---|---|
| Number of teeth | 14 | 30 |
| Module (mm) | 3.5 | 3.5 |
| Spiral angle (°) | 38 | 38 |
| Face width (mm) | 4 | 16 |
| Shaft angle (°) | 90 | 90 |
| Hand | Left | Right |
Two entries deserve comment. A 14-tooth pinion meshing a 30-tooth wheel at 90° shaft angle is a low-tooth-count, high-reduction pair, the configuration in which undercut and load concentration at the toe are most critical; a 38° spiral angle is typical of high-speed automotive and industrial drives. The tabulated face widths, 4 mm for the pinion and 16 mm for the wheel, are not equal, which is unusual for a conjugate pair and is not explained in the description, so they should be treated with caution when the geometry is reused.
Trace morphology on the two flanks
Tracking results were plotted as gear inspection distance in the transverse direction (roughly 10–45 mm) against inspection distance in the vertical direction (roughly 4–40 mm), with the trace rendered as a signal.
On the concave flanks the traces formed irregular segments of unequal length. The interpretation is that at high rotational frequency the initial engagement produces a large meshing area whose centre line is the trace; as rotation proceeds the flank separates from the mating concave surface and the trace shortens. Because the concave side keeps regenerating contact as speed, contact count and contact area rise, traces appear repeatedly and begin to cycle, which accounts for the scattered distribution in the plots. The wheel produced a wider trace envelope than the pinion, consistent with its larger contact area and correspondingly longer centre line.
On the convex flanks the behaviour differed markedly: traces varied regularly, with small gaps between adjacent lines. The convex side spreads the contact force, so the meshing area contracts toward a minimum loaded point; the smaller that point, the shorter the trace, and the result is a concentrated band of motion rather than scattered segments. Both morphologies were judged consistent with the expected kinematics of the pair.
Feature-extraction accuracy against two reference methods
Feature extraction is the precondition for tracking: a method that recovers many trace features accurately has little to correct afterwards. The proposed procedure was compared with an installation-adjustment-based contact-trace optimization method and with a straight bevel gear pair flank contact analysis, over a sweep of contact-point counts from about 10 to 100.
The proposed method held the highest normalized extraction accuracy across the whole sweep and its curve was the most stable. The two reference methods started lower and degraded as the number of contact points increased, which is the signature of a method that has not modelled the gear motion in enough detail: sensitivity to running condition is weak, and when trace features appear the algorithm has no learned basis for capturing them. No numerical accuracy values are tabulated for this comparison, so the ranking rather than the margin is what the data support.
What the results establish and what they leave open
Taken together, the work describes a coherent pipeline: a kinematic test that suppresses candidate points where contact cannot occur, a histogram-based segmentation that separates trace pixels from flank pixels, and a ridge-slope proxy that converts a hard tracking problem into a comparison between two lines. Each stage is cheap to compute, which matters for a monitoring task that has to keep up with a rotating gear.
The evidence, however, comes from software-generated images of one gear pair. No physical gearbox, no marking-compound reference pattern and no independent measurement of trace position are reported, so accuracy is assessed against the model that produced the images rather than against an external standard. Illumination, oil film, surface finish and motion blur — the factors that dominate real flank imaging — are absent from that setting. The threshold is global and would need to become adaptive before it could survive the intensity spread of a curved specular flank. The coincidence criterion F(X) = 1 rests on an undemonstrated normalization.
The natural next step, stated as the direction for further work, is to recover kinematic parameters from the tracked trace rather than only its position, which would let the trace serve as an input to transmission-error and misalignment estimation. Before that, the pipeline needs validation on hardware: simultaneous marking-compound patterns as ground truth, repeated runs across speed and load, and a measured frame rate against pitch-line velocity. Until then, the method is best regarded as a well-motivated candidate for in-situ contact-pattern monitoring rather than a demonstrated instrument.