Aviation accessory drives use spiral bevel gears to transfer power between shafts that intersect at large angles, often in a compact, weight-critical housing. The supporting structure is deliberately light, so bearing and shaft deflection under load changes the relative position of the pinion and gear. That change shifts the tooth contact pattern toward the toe or heel, alters the loaded contact area, and can compromise load sharing and transmission smoothness. A tooth surface that performs well only at the nominal mounting distance is therefore not sufficient for an aircraft accessory gearbox. The design target is a contact pattern that remains close to its nominal position, size, and orientation across the expected mounting-error envelope.
The mounting-error problem
In a spiral bevel gear pair, the contact pattern is the practical indicator of meshing quality. Its position shows where the load is concentrated, its area indicates how many teeth share the load, and its direction reveals whether the contact path follows the intended orientation relative to the root cone. Under torque, tooth compliance spreads the contact; under misalignment, the initial contact point moves and the pattern can migrate toward an edge. The resulting local stress concentration is not captured by a nominal geometric design alone.
Loaded tooth contact analysis (LTCA) solves the contact problem with tooth compliance and load distribution. Error-loaded tooth contact analysis (ELTCA) extends this by introducing the gear pair’s installation error as a rigid-body misalignment before the loaded contact solution. The actual initial contact point is then determined under the combined action of load and error, rather than assumed from the nominal geometry. This makes ELTCA a suitable basis for relating the contact pattern to mounting error.
Equivalent misalignment and the error vector
Four equivalent misalignment components are used: pinion axial offset \(H_p\), gear axial offset \(H_g\), shaft-center distance offset \(V\), and shaft-angle offset \(\Sigma\). The first three are lengths in millimetres; \(\Sigma\) is an angle in radians. Because an angular deviation cannot be compared directly with a linear offset, \(\Sigma\) is converted into an equivalent linear quantity by multiplying it by the gear midpoint cone distance. The error vector is then
\[
E=[H_p,\;H_g,\;V,\;\Sigma]^T .
\]
The sign convention defines the direction of each offset along the face width, the center distance, and the shaft angle. Reversing a sign reverses the corresponding first-order response of the contact pattern. In the full-error check, all four components are applied simultaneously at their stated magnitudes, which represents a combined worst-case misalignment rather than a statistical distribution.
Quantifying the contact pattern
To make sensitivity a quantitative property, the contact pattern is reduced to three descriptors: area, center position, and direction. The area is approximated by a polygon formed by connecting the endpoints of the major axes of the contact ellipses. If the polygon has \(n\) triangular elements, the area of the \(i\)-th triangle with vertices \((x_1,y_1)\), \((x_2,y_2)\), and \((x_3,y_3)\) is
\[
A_i=\frac{1}{2}\left|\det
\begin{bmatrix}
x_1 & y_1 & 1\\
x_2 & y_2 & 1\\
x_3 & y_3 & 1
\end{bmatrix}\right|,
\]
and the total pattern area is
\[
A=\sum_{i=1}^{n}A_i .
\]
The pattern center is approximated by the mean coordinates of the ordered polygon vertices,
\[
\bar{x}=\frac{1}{n}\sum_{i=1}^{n}x_i,\qquad
\bar{y}=\frac{1}{n}\sum_{i=1}^{n}y_i .
\]
The pattern direction \(d\) is defined by the line joining the centers of the first and last contact ellipses in the meshing sequence. This gives a simple directional descriptor that can be related to the second-order contact parameters. The four observable pattern quantities are the center coordinates \(x\) and \(y\), the area \(A\), and the direction \(d\). Their deviations from the nominal pattern form the error vector
\[
\Delta \mathbf{p}=[\Delta x,\;\Delta y,\;\Delta A,\;\Delta d]^T
=\mathbf{f}(E).
\]
Sensitivity matrix and optimization model
Expanding the pattern-error function about a reference misalignment \(E_0\) and retaining the first-order term gives
\[
\Delta \mathbf{p}\approx \mathbf{f}(E_0)+\mathbf{J}(E_0)(E-E_0),
\]
where \(\mathbf{J}\) is the Jacobian of the pattern error with respect to the four misalignment components:
\[
J_{ij}=\frac{\partial f_i}{\partial E_j},
\qquad i\in\{x,y,A,d\},\quad
j\in\{H_p,H_g,V,\Sigma\}.
\]
The absolute value of \(J_{ij}\) measures how strongly pattern descriptor \(i\) responds to misalignment component \(j\). Because analytical derivatives are not available, the entries are evaluated numerically by a central difference. Each component is perturbed by a small amount \(h_j\), normally within about \(\pm 10\%\) of its reference value, while the other components are held fixed:
\[
J_{ij}\approx
\frac{f_i(E_0+h_j\mathbf{e}_j)-f_i(E_0-h_j\mathbf{e}_j)}{2h_j}.
\]
The optimization uses the second-order contact parameters of local synthesis as design variables: \(\beta\), the angle between the contact path and the root cone; \(\eta\), the first derivative of the transmission ratio; and \(\delta\), the ratio of the contact-ellipse major axis to the face width. These parameters control the contact path orientation, the slope of the transmission error, and the contact-ellipse size. The low-sensitivity design problem is written as
\[
\min_{\beta,\eta,\delta}\;F=\sum_{i=1}^{4}\sum_{j=1}^{4}|J_{ij}|
\]
subject to
\[
30^\circ<\beta<80^\circ,\qquad
-0.01<\eta<0.01,\qquad
0.15<\delta<0.25.
\]
The objective is the sum of the absolute sensitivity coefficients. It is a screening metric: it reduces the overall first-order response without assigning separate engineering weights to center position, area, and direction.
The solution procedure uses a neural-network surrogate because each objective evaluation requires an ELTCA calculation plus four finite-difference perturbations. A set of design-variable samples is generated, the corresponding sensitivity matrices are computed, and a back-propagation network is trained to map \((\beta,\eta,\delta)\) to the sensitivity response. In the reported implementation, 75% of the samples were used for training and 15% for validation. A genetic algorithm then searches the network response within the design bounds. The optimized second-order parameters are converted back into cutting settings through local synthesis, and the resulting tooth surface is checked again by ELTCA. If the \(x\)-coordinate error of the pattern center does not meet the requirement, the parameters are adjusted and the loop is repeated.
Aviation accessory-drive case
The method was applied to an aviation accessory-drive spiral bevel gear pair with the basic data in Table 1 and the load and misalignment data in Table 2. The gear pair has a 100° shaft angle, an 8 mm face width, and a 1.5 mm large-end module. The applied torques are modest, as expected for an accessory drive, but the mounting-error magnitudes are large enough to move the contact pattern substantially.
| Parameter | Value |
|---|---|
| Pinion teeth | 36 |
| Gear teeth | 45 |
| Pressure angle | 20° |
| Large-end module | 1.500 mm |
| Mean spiral angle | 20° |
| Shaft angle | 100° |
| Face width | 8 mm |
| Addendum coefficient | 0.85 |
| Clearance coefficient | 0.18800 |
| Radial modification coefficient | 0.1737 |
| Tangential modification coefficient | 0 |
| Maximum backlash | 0.28 mm |
| Minimum backlash | 0.2 mm |
| Gear hand | Left |
| Quantity | Value |
|---|---|
| Pinion torque | 9.0265 N·m |
| Gear torque | 11.3 N·m |
| Mesh force | 233.4139 N |
| Pinion axial offset \(H_p\) | −0.05 mm |
| Gear axial offset \(H_g\) | 0.05 mm |
| Shaft-center offset \(V\) | 0.078 mm |
| Shaft-angle offset \(\Sigma\) | −0.00045 rad |
Optimized surface and predicted response
Three designs were compared. The first represents a conventional Gleason-type design with \(\beta=75^\circ\), \(\eta=-0.01\), and \(\delta=0.23\). The second is a high-contact-ratio design with \(\beta=45^\circ\) and the same \(\eta\) and \(\delta\). The third is the optimized low-sensitivity surface, for which the design variables moved to \(\beta=60.5133^\circ\), \(\eta=0.0047\), and \(\delta=0.1601\). The optimization therefore selected a contact path between the two baselines, a slightly positive transmission-ratio slope, and a smaller contact ellipse.
| Case | \(\beta\) (°) | \(\eta\) | \(\delta\) | Largest \(|J_{ij}|\) | Largest \(|\Delta x|\) (mm) | Second-largest \(|\Delta A|\) |
|---|---|---|---|---|---|---|
| 75° baseline | 75 | −0.01 | 0.23 | 31.026 | 1.1431 | 0.4893 |
| 45° high-contact-ratio baseline | 45 | −0.01 | 0.23 | 57.032 | 1.1054 | 0.3275 |
| Optimized | 60.5133 | 0.0047 | 0.1601 | 12.612 | 0.5391 | 0.1034 |
The largest absolute sensitivity coefficient in each matrix occurs in the direction response to pinion axial offset \(H_p\). Its magnitude is 31.026 for the 75° design, 57.032 for the 45° high-contact-ratio design, and 12.612 after optimization. The 45° design therefore has a larger first-order directional response than the conventional 75° design, even though it is intended to provide higher contact ratio. Increasing contact ratio by changing the contact path is not, by itself, a guarantee of robustness.
Under the full misalignment vector, both baseline patterns migrate noticeably toward the heel. The optimized pattern remains much closer to its nominal position. The largest \(x\)-coordinate error of the pattern center falls from 1.1431 mm for the 75° baseline to 0.5391 mm after optimization, a reduction of about 52%. The area-error response, which is the second-largest pattern error, falls from 0.4893 to 0.1034, a reduction of 78.87% relative to the 75° baseline. The 45° baseline has a smaller area error than the 75° baseline, but its center error remains near 1.1 mm and its largest sensitivity coefficient is higher than that of the 75° design.
The sensitivity-matrix comparison also shows which response dominates. In all three cases the direction term is the largest single coefficient, but its absolute value is reduced by about 59% from the 75° design and by about 78% from the 45° design. This matters because a direction change alters the orientation of the loaded contact path, while a center shift controls proximity to the tooth edges. The optimized surface reduces both, but it does not make the response uniform across all four pattern descriptors.
Interpretation and limits
The case shows that low sensitivity is a curvature-and-contact-path problem rather than a simple contact-ratio target. The optimized design reduces \(\delta\) from 0.23 to 0.1601, producing a smaller contact ellipse, and moves \(\beta\) to 60.5133° with a slightly positive \(\eta\). That combination lowers the first-order pattern response under the specified misalignment vector. A smaller contact ellipse, however, can raise contact pressure for the same load. Because the optimization objective contains only pattern sensitivity, the low-sensitivity surface still has to be checked against contact-stress, tooth-root-stress, and scoring requirements before it can be released.
The quantitative scope is also specific. The evidence comes from a single 36/45-tooth accessory-drive pair with an 8 mm face width, a 100° shaft angle, and torques of roughly 9–11 N·m. The misalignment input is one maximum vector, not a measured statistical distribution, and the sensitivity matrix is evaluated locally with finite-difference perturbations of about ±10%. The 78.87% area-error reduction and the 52% center-error reduction are therefore model-based predictions for this load and error set, not a fleet-wide guarantee.
Within those limits, the workflow provides a direct design route for spiral bevel gears in stiffness-limited aviation drives: introduce the equivalent mounting error into loaded contact analysis, quantify the contact pattern by area, center, and direction, form the Jacobian with respect to the four misalignment components, and optimize the second-order contact parameters against the absolute sensitivity sum. The result is a tooth surface whose contact pattern is designed to stay near its nominal location when the gear pair is subjected to the mounting errors that lightweight aircraft structures inevitably produce.