Spiral bevel gears are used in heavy-duty gas-turbine transmissions because their curved, oblique teeth provide gradual meshing, high contact ratio, and high load capacity. In a reversing drive, the convex and concave flanks of the same gear tooth both carry load. The contact pattern—the visible record of the instantaneous contact points—is therefore not a single feature to be centered once. It has to remain inside the tooth boundaries, avoid the tip and root edges, keep a usable area and inclination, and remain similar on both working flanks. Installation error changes the relative position of the pinion and gear after cutting. It can move the pattern toward an edge, distort its shape, and raise transmission error. Correcting such errors by trial and error on a rolling tester is slow and operator-dependent. Error tooth contact analysis, or ETCA, offers a numerical route: hold the cutting geometry fixed, vary the assembly offsets, and solve for the contact path on each flank.
How the Contact Pattern Is Generated and Simulated
The pair considered here is a Gleason face-milled spiral bevel set. The gear is generated by the duplex method; the pinion by the single-side modified-roll method. The cutter head carries inner and outer straight blades. The outer blade cuts the concave flank, the inner blade the convex flank. As the cutter head rotates, each blade sweeps a conical generating surface. The cutter-tip radius of that surface is
$$ r_{ci}=R_{ui}\pm \frac{P_W}{2} $$
where \(R_{ui}\) is the nominal cutter radius and \(P_W\) the point width; the plus sign applies to the concave flank and the minus sign to the convex flank. The tooth surface is obtained by transforming this cutter surface into the blank coordinate system and enforcing the meshing condition
$$ f_i(u_{gi},\theta_{gi},\phi_{ci})=\mathbf n_{mi}\cdot\mathbf v_{mi}=0, $$
where \(\mathbf n_{mi}\) and \(\mathbf v_{mi}\) are the unit normal and relative velocity in the machine frame. For the pinion, the roll ratio varies with cradle angle; the second-order coefficient is 0.035 and the third-order coefficient is zero. This matters because the pinion flank is not a simple copy of the cutter profile.
For contact analysis, the theoretical surfaces of pinion and gear are transformed into a common assembly frame. Installation errors enter as homogeneous transformations: \(\Delta E\) is the offset displacement, \(\Delta P\) the pinion axial displacement, \(\Delta G\) the gear axial displacement, and \(\Delta\Sigma\) the shaft-angle error. The contact point must satisfy equality of position and unit normal in the assembly frame:
$$
\mathbf r_h^{(1)}(\theta_{g1},\phi_{c1},\phi_1)
=\mathbf r_h^{(2)}(\theta_{g2},\phi_{c2},\phi_2),\qquad
\mathbf n_h^{(1)}(\theta_{g1},\phi_{c1},\phi_1)
=\mathbf n_h^{(2)}(\theta_{g2},\phi_{c2},\phi_2).
$$
Five independent scalar equations relate six parameters; stepping the pinion roll angle \(\phi_1\) closes the system. Transmission error follows from the difference between the actual and nominal gear rotations:
$$
\Delta\phi_2=(\phi_2-\phi_{20})-\frac{z_1}{z_2}(\phi_1-\phi_{10}),
$$
where \(z_1\) and \(z_2\) are the pinion and gear tooth numbers and the subscript 0 denotes the initial angular position. The gear axial displacement \(\Delta G\) is included in the assembly matrix but is not used as a production adjustment. The gear is the larger, more expensive member; setup corrections are normally made by changing the pinion offset and axial position and, within a small range, the shaft angle.
Pattern Descriptors and the Gear Pair Used for the Analysis
Contact points are projected onto the axial plane of the gear. The pattern is reduced to four descriptors: midpoint \(x\) along the face width, with positive toward the heel; midpoint \(y\) along the tooth height, with positive toward the tip; area \(s\); and major-axis inclination \(k\) relative to the pitch-cone direction. These descriptors convert a visual trace into quantities that can be fitted and optimized. They also reflect practical requirements: the pattern should sit inside the tooth boundary, avoid the tip and root edges, have sufficient area, and keep its major axis from turning too far across the tooth.
The numerical case is a 28-tooth left-hand pinion meshing with a 33-tooth right-hand gear. Table 1 lists the basic geometry.
| Parameter | Pinion | Gear |
|---|---|---|
| Number of teeth | 28 | 33 |
| Hand | Left | Right |
| Shaft angle (°) | 90 | |
| Large-end module (mm) | 6.1 | |
| Face width (mm) | 39.6 | |
| Outer cone distance (mm) | 131.998 | |
| Mid-point cone distance (mm) | 112.198 | |
| Normal pressure angle (°) | 20 | |
| Mid-point spiral angle (°) | 35 | |
| Pitch cone angle (°) | 40.314 | 49.686 |
| Face cone angle (°) | 43.317 | 52.18 |
| Root cone angle (°) | 37.82 | 46.683 |
| Addendum (mm) | 5.771 | 4.599 |
| Dedendum (mm) | 5.746 | 6.917 |
| Pitch diameter (mm) | 145.18 | 171.105 |
The error sweeps were ±0.3 mm for \(\Delta E\) and \(\Delta P\), and ±10 arcmin for \(\Delta\Sigma\). For each setting, ETCA generated the contact path, and the descriptors were fitted with sinusoidal functions of the error variables. These fits are local response surfaces used by the optimizer, not a general law for all spiral bevel gears. Outside the tested ranges, the contact may leave the tooth or the response may change qualitatively.
Installation-Error Sensitivity: The Two Flanks Do Not Move Together
Forward rotation is defined so that the gear convex flank is the working surface; reverse rotation makes the gear concave flank the working surface. This definition is important because the contact pattern on a gear is not a single entity: the two working flanks are on opposite sides of the tooth and are cut by different blades. A given assembly shift therefore does not produce the same pattern shift on both flanks.
Within the tested ranges, positive \(\Delta E\) moves the convex pattern from the toe toward the heel and from the root toward the tip. On the concave flank, the same positive offset moves the pattern from the heel toward the toe and from the tip toward the root. Positive \(\Delta P\) moves the convex pattern from the heel toward the toe and from the tip toward the root; on the concave flank, the vertical position moves from the root toward the tip while the face-width position changes only weakly. Positive \(\Delta\Sigma\) moves the convex pattern from the heel toward the toe and from the root toward the tip; on the concave flank it moves from the toe toward the heel and from the tip toward the root.
The major-axis inclination follows the same opposing logic. \(\Delta E\) raises \(k\) on the convex flank but lowers it on the concave flank; \(\Delta P\) lowers \(k\) on the convex flank but raises it on the concave flank; \(\Delta\Sigma\) lowers \(k\) on the convex flank and raises it on the concave flank. The pattern area behaves differently: for all three errors, \(s\) reaches a maximum near the unshifted setting and decreases as the error approaches the ends of the tested range. In this geometry, \(\Delta E\) and \(\Delta P\) produce larger changes in \(s\) and \(k\) than \(\Delta\Sigma\) over the ranges examined.
| Error | Forward rotation, gear convex flank | Reverse rotation, gear concave flank |
|---|---|---|
| Offset \(\Delta E\) (mm) | \(x\) toe→heel; \(y\) root→tip; \(k\) increases | \(x\) heel→toe; \(y\) tip→root; \(k\) decreases |
| Pinion axial \(\Delta P\) (mm) | \(x\) heel→toe; \(y\) tip→root; \(k\) decreases | \(y\) root→tip; \(x\) changes little; \(k\) increases |
| Shaft angle \(\Delta\Sigma\) (arcmin) | \(x\) heel→toe; \(y\) root→tip; \(k\) decreases | \(x\) toe→heel; \(y\) tip→root; \(k\) increases |
| Pattern area \(s\) | Non-monotonic: \(s\) rises toward a maximum near the unshifted setting and then falls as the error grows toward the ends of the tested range. | |
These opposite slopes are the central difficulty in correcting a bi-directional gear pair. An adjustment chosen only from the forward pattern can degrade the reverse pattern. The correction therefore has to be posed as a multi-output problem: find the assembly offsets that bring both flank patterns close to a target set of descriptors, rather than optimizing one flank and checking the other afterwards.
Target-Driven Correction and Rolling-Check Validation
The correction model treats \(\lambda=[x,y,s,k]\) as the output and \([\Delta E,\Delta P,\Delta\Sigma]\) as the input. A target pattern \(\lambda_0\) is defined from the production requirement. The major-axis inclination is driven toward its target, while the midpoint displacement is constrained to remain within the face width and tooth height, and the area is kept below the specified maximum. This is an assembly-level correction: the cutting settings and tooth surfaces are unchanged; only the mounting position of the pair is adjusted.
With no installation correction, ETCA gave the baseline patterns in Table 3. The convex and concave traces had opposite inclinations and sat close to the addendum edge; the mesh-transition transmission-error amplitude was 21.4858. The optimizer returned \(\Delta E=0.0113\) mm, about 11 μm; \(\Delta P=0.0095\) mm, about 10 μm; and \(\Delta\Sigma=0\). These are small changes, but they act on a stiff, point-contact mesh and are sufficient to move the pattern.
| State | Working flank | \(x\) | \(y\) | \(s\) | \(k\) |
|---|---|---|---|---|---|
| Before correction | Gear convex, forward | 19.0425 | 0.6631 | 388.884 | −8.8605 |
| Before correction | Gear concave, reverse | 19.1065 | 0.8458 | 342.706 | 11.0183 |
| After correction | Gear convex, forward | 17.064 | 0.0606 | 332.8346 | −3.9825 |
| After correction | Gear concave, reverse | 16.627 | −0.07955 | 360.4073 | 2.8149 |
After correction, the \(y\) coordinates of the two patterns came within about ±0.08 of the mid-height position, the absolute convex–concave difference in \(k\) fell from 19.88 to 6.80, and the area difference fell from 46.2 to 27.6. The patterns became more elliptical and developed transition traces at their ends, while the mesh-transition transmission-error amplitude dropped from 21.4858 to 16.1163, about 25% lower. The improvement is not that the two traces become identical in every descriptor; it is that their height position, area, and inclination move toward a common, centered configuration instead of diverging in opposite directions.
The geometric result was checked on a 500 mm universal bevel gear rolling tester. The gears were ground, red lead powder was spread on the pinion, and the pair was mounted with the optimized offsets. Patterns were taken at 90°, 180°, 270°, and 360° positions of the gear to check repeatability around the circumference. The experimental traces lay near the toe side in the face-width direction and near mid-height, with a lens-like shape, consistent with the ETCA patterns. This is a rolling-contact geometric check; it does not by itself establish full-load fatigue life, thermal behavior, or lubricant-film performance.
What the Correction Procedure Means for Setup
The practical value of ETCA here is not a universal correction formula but a procedure for generating a setup chart for a specific gear pair. Once the tooth surfaces and cutter settings are fixed, the error response can be computed over the allowed adjustment range. Such a chart shows which combination of \(\Delta E\), \(\Delta P\), and \(\Delta\Sigma\) centers both flanks, and the rolling test confirms that the computed pattern is physically achievable. For the 28/33 pair, the correction required only about 0.01 mm changes in pinion position and no shaft-angle change—small relative to the 39.6 mm face width.
The result should not be extrapolated blindly. The sensitivity coefficients depend on tooth count, module, spiral angle, pressure angle, face width, and the chosen cutter geometry. A 28/33 pair with a 35° spiral angle and 39.6 mm face width is not a universal surrogate for other heavy-duty gears. The ±0.3 mm and ±10 arcmin ranges are also local; larger errors may push the contact off the tooth, where the smooth response curves no longer apply. Within those limits, the main engineering conclusion is that convex and concave patterns can be made consistent by small assembly adjustments after cutting, provided both flanks are included in the target and the adjustment is solved as a two-flank problem rather than a single-flank one.