Spiral bevel gears carry motion between intersecting shafts through curved teeth that engage gradually along the face width. The curvature of the spiral gives a high contact ratio, smooth transfer of load and considerable torque capacity, which is why the format dominates final drives in vehicles, construction machinery, power transmission trains and aircraft assemblies. Because the flanks are generated by a rotating cutter head on a cradle-type machine, the geometry that actually results depends on a long list of machine settings, and the only practical way to judge whether those settings are right is to look at the contact pattern left on a coated flank after the pair has been run together.
The contact pattern as the controlling acceptance criterion
Two indicators govern the acceptance of a spiral bevel gear pair: the tooth contact pattern and the transmission error. The pattern is the more immediate of the two, because its position, shape and extent reveal whether the load is carried over a controlled region in the middle of the flank or concentrated at an edge where the tooth will spall or break. A pattern that drifts toward the addendum, ends abruptly at both extremities, or tilts strongly relative to the face width signals a pair that will be sensitive to deflection and misalignment in service.
Historically, matching a specified pattern meant trial cutting, measurement, adjustment of the machine settings and another trial cut. Tooth contact analysis (TCA) replaces most of that loop with a numerical simulation: the two generated flanks are placed in a common frame and rotated through mesh, and the instantaneous contact ellipses are accumulated into a predicted pattern. The value of TCA for production is therefore not merely descriptive. It allows a machine setting to be changed on screen, its effect on the pattern quantified, and the correction transferred to the machine only once the predicted pattern is acceptable. The sections below set out the geometric basis for that procedure, the parameterization used to make patterns comparable, the sensitivity of the pattern to individual settings on the pinion, and a verified correction of a 28/33-tooth pair.
Flank generation and meshing simulation
In the generating process the cradle of the machine emulates a virtual gear whose teeth are the cutting edges of the cutter head. The workpiece rotates at a fixed ratio to the cradle, and the cutter head sweeps out the tooth space; the flank produced is the envelope of the cutter surface. A static machine frame Smi, a cutter frame Sgi, a workpiece frame Si, a cradle frame Sci rotating about the machine axis, and two auxiliary frames Shi and Ski define the kinematic chain. The settings that appear in it are the sliding base XBi, the vertical offset Emi, the workpiece mounting angle γi, the horizontal wheel position XPi, the cradle angle φci, the workpiece rotation φi, the angular cutter position qi and the radial setting Sri. Throughout, the subscript i = 1 denotes the pinion and i = 2 the gear.
The cutter head carries straight-edged blades whose cutting edges, rotated about the cutter axis, sweep a conical surface. Its position vector in the cutter frame is
$$\mathbf{r}_{gi}(u_{gi},\theta_{gi})=\begin{bmatrix}\left(r_{Gi}\pm u_{gi}\sin\alpha_{gi}\right)\cos\theta_{gi}\\ \left(r_{Gi}\pm u_{gi}\sin\alpha_{gi}\right)\sin\theta_{gi}\\ -u_{gi}\cos\alpha_{gi}\\ 1\end{bmatrix}$$
where ugi and θgi are the surface parameters, αgi is the cutter profile angle and rGi is the cutter tip radius; the upper sign applies to the concave flank and the lower sign to the convex flank. The unit normal of the cutting surface follows from the cross product of the two partial derivatives:
$$\mathbf{n}_{gi}=\frac{\mathbf{N}_{gi}}{\left|\mathbf{N}_{gi}\right|}=\begin{bmatrix}\cos\alpha_{gi}\cos\theta_{gi}\\ \cos\alpha_{gi}\sin\theta_{gi}\\ \pm\sin\alpha_{gi}\end{bmatrix},\qquad \mathbf{N}_{gi}=\frac{\partial\mathbf{r}_{gi}}{\partial u_{gi}}\times\frac{\partial\mathbf{r}_{gi}}{\partial\theta_{gi}}$$
The envelope condition, expressing that the relative velocity between the cutting edge and the workpiece is orthogonal to the common normal, is
$$\mathbf{n}_{mi}\cdot\mathbf{v}_{mi}=0$$
and with the 4×4 transformation matrix Mgi from the cutter frame to the workpiece frame, the generated flank and its normal are
$$\mathbf{r}_{i}(u_{gi},\theta_{gi},\phi_{ci})=\mathbf{M}_{gi}\,\mathbf{r}_{gi}(u_{gi},\theta_{gi})$$
$$\mathbf{r}_{i}(\theta_{gi},\phi_{ci})=\mathbf{M}_{gi}\,\mathbf{r}_{gi}(u_{gi},\theta_{gi}),\qquad \mathbf{n}_{i}(\theta_{gi},\phi_{ci})=\mathbf{L}_{gi}\,\mathbf{n}_{gi}(u_{gi},\theta_{gi})$$
where Lgi is the 3×3 rotational part of Mgi, obtained by deleting the last row and column.
To simulate mesh, both flanks are mapped into a common frame in which the pinion rotates with angular velocity ω1 and the gear with ω2. Continuous tangency throughout the engagement requires a common contact point and a common normal at every instant:
$$\begin{cases}\mathbf{r}_{h1}(\theta_{g1},u_{g1},\phi_{1})=\mathbf{r}_{h2}(\theta_{g2},u_{g2},\phi_{2})\\[2pt] \mathbf{n}_{h1}(\theta_{g1},u_{g1},\phi_{1})=\mathbf{n}_{h2}(\theta_{g2},u_{g2},\phi_{2})\end{cases}$$
Projected on the three axes of the meshing frame, these vector equations yield six scalar relations, of which five are independent because both normals are unit vectors; with six unknowns, the system is solved by stepping the pinion rotation φ1. At each contact point the principal directions and curvatures of the cutter surface are known, and the curvature relation between two contacting surfaces, combined with an assumed elastic approach in the range 2.5×10−4 to 6.25×10−4 mm, gives the orientation and size of the instantaneous contact ellipse. Superposing the ellipses computed along the contact path, inside the flank boundary, produces the predicted contact pattern.
Parameterizing the pattern
Judging a pattern by eye gives no quantitative feedback on a small change of a machine setting, so the pattern is reduced to a small set of numbers. The four extremities of the traced region are labelled A(x1, y1), B(x2, y2), C(x3, y3) and D(x4, y4) in a frame whose x axis runs along the face width and whose y axis runs across the tooth height. The long axis L, short axis M, area index S and inclination ζ of the long axis relative to the pitch-cone direction are
$$\begin{cases}L=\sqrt{(x_{2}-x_{1})^{2}+(y_{2}-y_{1})^{2}}\\[2pt] M=\sqrt{(x_{3}-x_{4})^{2}+(y_{3}-y_{4})^{2}}\\[2pt] S=\pi L M\end{cases}\qquad\qquad \zeta=\arctan\frac{y_{2}-y_{1}}{x_{2}-x_{1}}$$
Because an ellipse with semi-axes L/2 and M/2 encloses an area of πLM/4, the quantity S is four times the enclosed area and serves as a proportional index of pattern size rather than an absolute area. Each pattern is recorded as the ordered set [L, M, S, (x0, y0), ζ], where (x0, y0) is the midpoint of the pattern. The target form is a near-elliptical pattern whose midpoint, short axis and long axis fall within specified limits.
Kinematic relations for the correction levers
Position along the face width and across the tooth height
An error in the spiral angle displaces the pattern along the face width and is corrected through the radial cutter setting, which is related to the mid-face spiral angle βm by
$$S_{d1}=\sqrt{L_{m}^{2}+r_{d}^{2}-2L_{m}r_{d}\sin\beta_{m}}$$
with Lm the mean cone distance and rd the cutter radius. In the EPG convention, where E is the offset, P the pinion axial displacement and G the gear axial displacement, a change of offset ΔE is converted into the required change of radial setting by
$$\Delta S_{d1}=\frac{\Delta E\,r_{d}\cos\beta_{m}}{S_{d1}}$$
An error in the pressure angle displaces the pattern across the tooth height and is corrected through the horizontal wheel position. Advancing the pinion horizontal position by ΔP must be accompanied by a retraction of the sliding base:
$$\Delta X_{P1}=-\Delta P,\qquad\qquad \Delta X_{B1}=-X_{P1}\sin\delta_{1}$$
where δ1 is the pinion pitch cone angle.
Shape and diagonal contact
Diagonal contact — a pattern inclined across the flank rather than aligned with the face width — is corrected by altering the rolling pitch cone, principally through the roll ratio, while the horizontal wheel position maintains the pattern height and the sliding base and cutter position preserve the cutting depth. If the new rolling cone departs from the original by δi = δ − δ′, the roll change required for a given horizontal displacement is
$$\Delta j=\frac{j\cos\delta_{i}}{L_{m}}\,\Delta X_{P1}$$
with j the roll ratio and Δj its change.
Length and width of the pattern
The long axis of the pattern is governed by the difference in pitch-line curvature between the two conjugate flanks and is set through the cutter radius, with the radial setting compensated accordingly:
$$\Delta r=0.1\left(r_{d}\cos\beta_{m}\right)^{2}\left(\frac{1}{l_{1}^{2}}-\frac{1}{l_{2}^{2}}\right),\qquad\qquad \Delta S_{d1}=\Delta r\sin\left(q_{1}-\beta_{m}\right)$$
Here Δr is the change of cutter radius, l1 the measured contact length, l2 the theoretical contact length and q1 the angular cutter position of the pinion. The short axis is governed by the profile curvature of the two flanks, which changes in the opposite sense for pinion and gear, and is adjusted through the vertical offset. Because that offset also perturbs the mid-face spiral angle, the radial setting is corrected simultaneously:
$$\Delta S_{d1}=\Delta E_{m1}\sin q_{1}$$
with ΔEm1 the change of vertical offset. In practice the correction is applied to the pinion, since it has fewer teeth and a shorter machining cycle than the gear; the gear is left as generated.
Case pair and baseline pattern
The procedure was applied to a high-precision pair manufactured by Qinghai Huading Gearbox Co., Ltd. Its basic data are given in Table 1, and the initial pinion settings in Table 2. Computed with these settings, the pattern on the working flanks — the pinion concave and the gear convex — gave the parameter sets [18.1647, 2.9703, 169.4197, (104.6581, 1.1127), 10.12] for the pinion and [18.1716, 4.3057, 245.6795, (104.7635, −1.0245), −9.86] for the gear, with lengths in millimetres, area index in square millimetres, midpoint in millimetres and inclination in degrees.
Both patterns were unsatisfactory. They lay too close to the addendum edge, ended abruptly at both extremities without a transition zone, and departed conspicuously from the target elliptical form. The inclination of the long axis relative to the face width was also increasing along the contact path, an early indication of diagonal contact and a risk to stable running.
| Parameter | Pinion | Gear |
|---|---|---|
| Number of teeth | 28 | 33 |
| Hand of spiral | Left | Right |
| Shaft angle (°) | 90 | 90 |
| Module at outer end (mm) | 5.7 | 5.7 |
| Face width (mm) | 37 | 37 |
| Outer cone distance (mm) | 123.343 | 123.343 |
| Mean cone distance (mm) | 104.841 | 104.841 |
| Inner cone distance (mm) | 86.34 | 86.34 |
| Normal pressure angle (°) | 20 | 20 |
| Mean spiral angle (°) | 35 | 35 |
| Pitch cone angle (°) | 40.314 | 49.686 |
| Face cone angle (°) | 43.317 | 52.18 |
| Root cone angle (°) | 37.82 | 46.683 |
| Addendum (mm) | 4.422 | 3.493 |
| Dedendum (mm) | 4.564 | 5.495 |
| Pitch diameter (mm) | 135.66 | 159.885 |
| Setting | Concave flank | Convex flank |
|---|---|---|
| Cutter profile angle αg1 (°) | 20 | 20 |
| Cutter tip radius rg1 (mm) | 1.8 | 1.8 |
| Radial setting Sd (mm) | 97.8676 | 111.142 |
| Angular cutter position q1 (°) | 63.7692 | 63.7692 |
| Sliding base XB (mm) | 0.55 | −0.65 |
| Roll ratio j | 2.2 | 2.3 |
| Horizontal wheel position XP (mm) | 2.5068 | −2.4762 |
| Vertical offset Em (mm) | −0.4456 | 0.5289 |
| Cutter radius rd (mm) | 114.3 | 114.3 |
Computed sensitivity of the pattern
Position: radial setting and horizontal wheel position
Sweeping the radial setting from negative to positive change moves the pattern midpoint from the heel toward the toe of the face width. Across the tooth height the same sweep moves the midpoint of the pinion concave flank from addendum toward dedendum, and the midpoint of the gear convex flank from dedendum toward addendum. The horizontal coordinate responds far more strongly than the vertical one, so a pattern whose face-width position is wrong should be corrected mainly through the radial setting.
The horizontal wheel position acts in the perpendicular sense. Sweeping it from negative (forward) to positive (backward) moves the midpoint from the toe toward the heel, while the vertical coordinate of the pinion concave flank moves from dedendum toward addendum and that of the gear convex flank from addendum toward dedendum. Here the vertical coordinate is the more sensitive of the two, so a pattern that sits too close to the root or the tip is corrected mainly through the horizontal wheel position.
Shape: roll ratio
Increasing the roll ratio increases the angle between the pattern and the face-width direction, which is the signature of inner diagonal contact on the working flank. When the inclination becomes excessive, the roll ratio is the setting to reduce.
Size: cutter radius, vertical offset and radial setting
Increasing the cutter radius steadily shortens the long axis of the pattern, and increasing the vertical offset shortens the short axis. The area index behaves non-monotonically with the radial setting: as the radial setting increases from negative to positive, the area first grows and then falls, reaching its maximum at small changes of the radial setting. Since the same setting also positions the pattern along the face width, its adjustment has to be treated conservatively if pattern area is to remain stable.
| Setting | Corrected concave flank | Corrected convex flank |
|---|---|---|
| Radial setting Sd (mm) | 98.2244 | 110.7405 |
| Angular cutter position q1 (°) | 66.6537 | 61.3159 |
| Sliding base XB (mm) | 0.3312 | −0.2157 |
| Roll ratio j | 2.3163 | 2.1836 |
| Horizontal wheel position XP (mm) | 2.9081 | −2.8842 |
| Vertical offset Em (mm) | −0.5515 | 0.6207 |
| Cutter radius rd (mm) | 113.5475 | 114.9689 |
Corrected pattern
The computed sensitivities translate directly into a correction strategy for this pair: increase the radial setting in the positive direction to shift the pattern toward the toe, move the horizontal wheel position in the negative direction to draw the pattern away from the addendum, increase the roll ratio to suppress the diagonal tendency, and reduce both the vertical offset and the cutter radius to enlarge the short and long axes so that a transition zone appears at the ends of the pattern. The resulting settings are listed in Table 3, and the recalculated pattern parameters are compared with the baseline in Table 4.
| Flank and state | L (mm) | M (mm) | S (mm²) | Midpoint (x0, y0) (mm) | ζ (°) |
|---|---|---|---|---|---|
| Pinion concave, before | 18.1647 | 2.9703 | 169.4197 | (104.6581, 1.1127) | 10.12 |
| Gear convex, before | 18.1716 | 4.3057 | 245.6795 | (104.7635, −1.0245) | −9.86 |
| Pinion concave, after | 15.5496 | 5.5747 | 272.1915 | (102.4529, −0.5043) | 2.56 |
| Gear convex, after | 16.8383 | 4.9006 | 259.1057 | (103.0009, −0.5322) | −2.79 |
After correction the patterns on both flanks had moved clear of the addendum edge, showed a visible transition at each end and had acquired the intended elliptical form. The inclination of the long axis fell from 10.12° to 2.56° on the pinion concave flank and from −9.86° to −2.79° on the gear convex flank, removing the diagonal contact tendency. The change in the two axes is complementary rather than uniform: on the pinion the long axis shortened from 18.16 mm to 15.55 mm while the short axis grew from 2.97 mm to 5.57 mm, so the area index increased from 169.4 to 272.2 despite the shorter pattern. The midpoint shifted about 2.2 mm along the face width toward the toe and closed to within 0.5 mm of the pitch line.
Rolling inspection
Verification used a Y9550 bevel gear rolling inspection machine retrofitted with numerical control, which reduces the scatter that manual assembly of the pair would introduce. Both flanks were coated with marking compound, the pair was mounted to the specified running position, and the machine was run until the mesh was smooth and free of abnormality before the pattern was photographed and measured.
The inspected pattern was located at the centre of the flank, slightly toward the toe, as predicted. Its length occupied 50% to 60% of the face width and its width 60% to 70% of the tooth height, again in agreement with the simulation. The one visible discrepancy was in shape: the simulated pattern was near-elliptical, whereas the pattern produced on the inspection machine was date-seed shaped. Grinding error and the ambiguity of locating the pattern boundary during image capture account for a difference of this kind, and the measured pattern still satisfied the design criterion for contact.
Taken together, the position, shape and size of the pattern predicted after adjustment of the pinion settings matched the rolling inspection result closely enough to confirm the correction rules on which the adjustment was based.
Practical correction sequence
Three conclusions follow from the computed sensitivities and their experimental check.
- Pattern position is controlled by the radial setting and the horizontal wheel position, with a clear division of labour: the radial setting displaces the pattern along the face width, the horizontal wheel position along the tooth height.
- Pattern shape is dominated by diagonal contact and is best held to a near-elliptical form by adjusting the roll ratio, which reduces the angle between the pattern and the face width. Pattern size is controlled by the cutter radius and the vertical offset, which act on the long and short axes respectively.
- Because several settings act on more than one attribute — the radial setting, for instance, changes both position and area — a correction must follow a fixed order rather than tuning all settings at once. Position is settled first, then shape, then size, with the radial setting recompensated whenever the cutter radius or vertical offset is altered so that the mid-face spiral angle is preserved.
The practical significance is a shorter path from a measured pattern to a set of machine settings. Rather than iterating trial cuts, the correction is computed from parameterized pattern data and transferred to the machine in a single step, which matters most for the high-precision pairs in which the tolerance on pattern position, area and shape is tightest.