Spiral bevel gears carry the final reduction in turboprop and accessory drive trains, where the pinion and gear are supported by housings that are deliberately thin to save weight. Under torque and thermal load those housings deflect, the shaft centre distance and the shaft angle change, and the theoretical mounting distance defined at assembly is no longer the mounting distance at which the teeth actually mesh. For a gear set whose contact pattern was tuned for nominal alignment, a few hundredths of a millimetre of axial displacement is enough to push the contact ellipse towards the heel or the toe, concentrate load on a corner of the flank, and raise both the flash-temperature risk and the transmission error. The engineering question is therefore not how to eliminate mounting error — it cannot be eliminated — but how to design the flank so that the contact pattern is insensitive to it.
Why conventional flank design is not robust by construction
Classical Gleason practice places the contact path roughly perpendicular to the root cone and controls the pattern through the second-order contact parameters: the angle between the contact path and the root cone, the first derivative of the transmission ratio (the slope of the predesigned parabolic transmission-error curve), and the ratio of the contact-ellipse major axis to the face width. High-contact-ratio variants deliberately tilt the path to lengthen the contact line and raise the number of teeth in mesh, which smooths the transmission error and reduces mesh excitation.
Both strategies, however, are evaluated at the nominal alignment. Whether they remain acceptable once the mounting distance drifts is a separate property, and it is this property — sensitivity, or robustness — that has to be optimized explicitly. The design procedure described here does exactly that: it treats the four mounting-error components as an input vector, computes how the contact pattern responds, and then searches the second-order contact parameters for a flank whose pattern barely responds at all.
Equivalent misalignment: four components in one frame
The error state of the mesh is reduced to four equivalent misalignments, each with a fixed sign convention:
- Pinion axial displacement Hp: negative when the pinion moves towards the toe (small end), positive towards the heel (large end).
- Gear axial displacement Hg: same sense as the pinion.
- Shaft offset V: referenced to the position at which the two cone apices coincide; viewed from the gear small end towards the large end, with the pinion on the right, upward displacement of the gear is positive.
- Shaft-angle deviation Σ: negative reduces the shaft angle, positive increases it.
Because Σ is an angle (rad) while the other three are lengths (mm), the components are not directly comparable. The deviation is converted to a length by multiplying it by the gear mid-cone distance, which allows the four columns of the sensitivity matrix to be read on a common footing.
Turning the contact pattern into numbers
The pattern is the primary acceptance criterion in production, but as a shape it is not directly differentiable. It is therefore reduced to three descriptors computed from the contact ellipses obtained by loaded tooth contact analysis (LTCA):
Area. The endpoints of the major axes of all contact ellipses are connected to form a polygon, which is decomposed into n triangles. With vertices (x1, y1), (x2, y2), (x3, y3) for triangle i,
$$S_i=\frac{1}{2}\left|\det\begin{bmatrix}x_1 & y_1 & 1\\ x_2 & y_2 & 1\\ x_3 & y_3 & 1\end{bmatrix}\right|,\qquad S=\sum_{i=1}^{n}S_i$$
Position. The polygon vertices are ordered counter-clockwise and the centroid is taken as the mean of the vertex coordinates:
$$\bar{x}=\frac{1}{n}\sum_{i=1}^{n}x_i,\qquad \bar{y}=\frac{1}{n}\sum_{i=1}^{n}y_i$$
Direction. The line joining the centre of the first ellipse entering mesh to the centre of the last ellipse leaving mesh defines the pattern direction d, which is the descriptor most directly linked to the second-order contact parameters.
Error-loaded tooth contact analysis and the sensitivity matrix
Introducing the misalignment vector E = (Hp, Hg, V, Σ) directly into the LTCA formulation gives an error-loaded tooth contact analysis (ELTCA), in which the four pattern descriptors become functions of E: Δx = fx(E), Δy = fy(E), ΔS = fS(E), Δd = fd(E), where each Δ denotes the deviation of a descriptor from its theoretical value.
Expanding about a reference state E0 and retaining first-order terms gives a linear map between misalignment and pattern error, whose coefficients form the 4 × 4 sensitivity matrix. For each descriptor j ∈ {x, y, S, d}:
$$\mathbf{S}_j=\nabla f_j=\left[\frac{\partial f_j}{\partial H_p},\ \frac{\partial f_j}{\partial H_g},\ \frac{\partial f_j}{\partial V},\ \frac{\partial f_j}{\partial \Sigma}\right]^{\mathsf T}$$
$$\begin{bmatrix}\Delta x\\ \Delta y\\ \Delta S\\ \Delta d\end{bmatrix}=\begin{bmatrix}\mathbf{S}_x^{\mathsf T}\\ \mathbf{S}_y^{\mathsf T}\\ \mathbf{S}_S^{\mathsf T}\\ \mathbf{S}_d^{\mathsf T}\end{bmatrix}\begin{bmatrix}\Delta H_p\\ \Delta H_g\\ \Delta V\\ \Delta \Sigma\end{bmatrix}+\begin{bmatrix}f_x(\mathbf{E}_0)\\ f_y(\mathbf{E}_0)\\ f_S(\mathbf{E}_0)\\ f_d(\mathbf{E}_0)\end{bmatrix}$$
No closed-form expression is available for the derivatives, so they are evaluated numerically. One component at a time is perturbed within a ±10 % neighbourhood of its reference value while the others are held fixed, and a central difference is used for accuracy:
$$\frac{\partial f_j}{\partial e_i}\approx\frac{f_j\!\left(e_i+\tfrac{h}{2}\right)-f_j\!\left(e_i-\tfrac{h}{2}\right)}{h},\qquad e_i\in\{H_p,H_g,V,\Sigma\}$$
The sign of each entry carries meaning: a positive value means the descriptor grows relative to the theoretical pattern, a negative value means it shrinks. Large magnitudes identify the misalignment component to which the pattern is most exposed — the natural target for flank modification.
Formulating the optimization problem
The design variables are the three second-order contact parameters, x = [β, η, δ]T, where β is the angle between the contact path and the root cone, η is the first derivative of the transmission ratio, and δ is the ratio of the contact-ellipse major axis to the face width. The objective is the sum of the absolute values of all entries of the sensitivity matrix, a scalar proxy for overall robustness that avoids handling the matrix directly:
$$\min_{\beta,\eta,\delta}\ f=\sum_{i=1}^{m}\sum_{j=1}^{n}\left|a_{ij}\right|\qquad \text{s.t.}\quad \beta\in(30^\circ,\,80^\circ),\ \ \eta\in(-0.01,\,0.01),\ \ \delta\in(0.15,\,0.25)$$
The bounds keep the solution inside manufacturable and mechanically sensible territory: β well away from the degenerate cases, η small enough that the predesigned transmission error stays a mild parabola, and δ large enough that the contact ellipse does not collapse into a narrow band.
Solving an objective that has no formula
Every evaluation of f requires a full ELTCA plus eight perturbed ELTCA runs for the finite differences, which makes the objective implicit and comparatively expensive. The solution strategy therefore replaces it with a surrogate:
- Sample n design vectors Ei = [β, η, δ]T across the design space and compute the corresponding sensitivity Si by ELTCA and differencing, producing the training set.
- Train a back-propagation neural network with design variables as inputs and sensitivities as outputs; 75 % of the samples are used for training and 15 % for validation.
- Extract the trained network as an explicit analytical objective and minimize it with a genetic algorithm, which avoids dependence on the initial guess. With tanh activation, the network output takes the form
$$S=\tanh\left(\sum_{i=1}^{m}w^{(uv)}_{i1}\tanh\left(\sum_{j=1}^{n}w^{(wu)}_{ij}e_j+b_i^{(u)}\right)+b^{(v)}\right)$$
where w are interlayer weights, b are neuron biases, e are the error inputs, and u, v, w index the hidden, output and input layers respectively.
- Take the optimal second-order parameters back through the local-synthesis design step to obtain the cutting parameters of the modified flank, then verify by ELTCA. If the x-coordinate of the pattern centre misses the specified tolerance, the relevant parameters are adjusted and the loop repeats.
The final deliverable is thus a set of machine-tool settings, not merely an abstract optimum in parameter space.
Numerical example: an aviation accessory-drive pair
The procedure was applied to a 36/45-tooth pair with a 100° shaft angle, 1.5 mm large-end module, 20° pressure angle and 8 mm face width (Table 1), running at a pinion torque of 9.0265 N·m and a gear torque of 11.3 N·m, corresponding to a mesh force of 233.4139 N. The applied equivalent misalignment is listed in Table 2.
| Parameter | Value |
|---|---|
| Pinion / gear tooth number | 36 / 45 |
| Pressure angle | 20° |
| Large-end module | 1.500 mm |
| Mid-point 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 / minimum backlash | 0.28 mm / 0.2 mm |
| Gear hand (1 = right, 2 = left) | 2 |
| Quantity | Value |
|---|---|
| Pinion torque | 9.0265 N·m |
| Gear torque | 11.3 N·m |
| Mesh force | 233.4139 N |
| Pinion axial displacement Hp | −0.05 mm |
| Gear axial displacement Hg | 0.05 mm |
| Shaft offset V | 0.078 mm |
| Shaft-angle deviation Σ | −0.00045 rad |
Three designs were compared: a conventional layout with the contact path at 75° to the root cone; a high-contact-ratio layout at 45°; and the robust optimum. The optimized second-order parameters shifted to β = 60.5133°, η = 0.0047 and δ = 0.1601 (Table 3). Two features of that optimum deserve comment. The sign of η reverses relative to both baselines, so the predesigned transmission-error parabola is no longer the one used in conventional practice. And δ falls from 0.23 to 0.1601, i.e. the contact ellipse is deliberately shortened relative to the face width — a reduction in contact ratio accepted in exchange for pattern stability.
| Design | β (°) | η | δ |
|---|---|---|---|
| Conventional (75° path) | 75 | −0.01 | 0.23 |
| High contact ratio (45° path) | 45 | −0.01 | 0.23 |
| Robust optimum | 60.5133 | 0.0047 | 0.1601 |
Reading the sensitivity matrices
Each cell below lists the coefficient for the conventional design / the high-contact-ratio design / the robust optimum, with rows corresponding to the four descriptors and columns to the four misalignment components (Σ expressed as an equivalent length).
| Descriptor | Hp | Hg | V | Σ (equivalent length) |
|---|---|---|---|---|
| Δx | −1.487 / −2.124 / −0.923 | 0.335 / −0.011 / 0.158 | −5.441 / −2.471 / −0.481 | −0.476 / −0.299 / −0.203 |
| Δy | 4.510 / 4.694 / 2.322 | 0.789 / 0.859 / 0.566 | 0.376 / 2.582 / 0.762 | 0.384 / 0.117 / 0.134 |
| ΔS | 7.840 / 12.034 / 4.755 | 0.215 / −1.789 / 0.389 | 8.414 / −0.369 / 0.117 | −0.301 / −0.608 / −0.242 |
| Δd | −31.026 / −57.032 / −12.612 | 5.829 / 2.569 / 9.475 | 19.343 / −27.941 / 8.136 | 9.462 / 5.706 / 3.907 |
Three observations follow. First, the dominant term in every design is ∂Δd/∂Hp, the rotation of the contact path caused by axial displacement of the pinion; the optimized design cuts it from −31.026 and −57.032 to −12.612. Second, tilting the path to 45° to gain contact ratio makes matters worse rather than better: the largest coefficient nearly doubles, and the pattern becomes strongly sensitive to shaft offset as well (∂Δd/∂V = −27.941 versus 19.343 for the conventional layout). High contact ratio and robustness are therefore independent attributes, and one does not buy the other. Third, the area descriptor is markedly stabilized, with ∂ΔS/∂Hp falling from 7.840 and 12.034 to 4.755, while its sensitivity to shaft offset almost vanishes (8.414 and −0.369 down to 0.117).
Full-error behaviour of the contact pattern
With all four misalignments applied simultaneously rather than one at a time, both baseline flanks showed a pronounced migration of the contact pattern towards the heel on pinion and gear alike — the qualitative signature of a design tuned only for nominal alignment. The optimized flank held its position, with no comparable shift.
Quantifying the descriptor errors confirms the ranking implied by the matrix: for a given error state the descriptors move in the same direction in all three designs, the largest variation occurring in the axial coordinate of the pattern centre, followed by the pattern area, with the radial coordinate and the path direction changing least.
| Design | Max. variation (pattern-centre axial coordinate) | Second-largest variation (pattern area) |
|---|---|---|
| Conventional (75° path) | 1.1431 mm | 0.4893 |
| High contact ratio (45° path) | 1.1054 mm | 0.3275 |
| Robust optimum | 0.5391 mm | 0.1034 |
Relative to the conventional design, the maximum positional drift drops by about 52 % and the area variation by 78.87 %; the latter figure is the headline sensitivity reduction, and it should be read as referring to the second-largest descriptor variation rather than to the norm of the whole matrix. Even the 45° high-contact-ratio design, which already reduces the area variation to 0.3275, is outperformed by a factor of roughly three on that measure after optimization.
Scope and limitations
The sensitivity matrix is a first-order object, evaluated by central differences over a ±10 % perturbation band around the reference state. Its validity is therefore local: the misalignments considered here are of the order of 0.05–0.08 mm in translation and 4.5 × 10−4 rad in shaft angle, and the linearization should not be extrapolated to gross misalignment or to loss of contact. The surrogate route — sampling, BP network, genetic algorithm — also inherits the coverage of its training set, so the optimum is only as trustworthy as the sampling density in the region of interest; the ELTCA re-check and iterative adjustment of the pattern-centre coordinate are what close that gap.
The result is established for a single load case and a single gear pair geometry, without experimental confirmation of the predicted pattern stability on a test rig. It also carries an explicit trade-off that any application must weigh: robustness is purchased by shortening the contact ellipse relative to the face width and by reshaping the transmission-error curve, both of which act on load capacity and mesh excitation. The practical value of the method lies less in the specific angles it returns than in making that trade-off visible and quantifiable at the design stage, so that a flank can be chosen for the misalignment range the airframe will actually impose rather than for the alignment that exists only on the assembly drawing.