Spiral bevel gears are used in heavy-duty gas-turbine drivetrains because their curved teeth provide high contact ratio, smooth motion, and load capacity. In reversible service, the gear pair does not have a single working flank: the convex flank carries load in one direction, and the concave flank carries load in the other. Both patterns must therefore remain inside the tooth, near the intended position, and similar enough to preserve stable meshing in forward and reverse rotation.
Why Reversible Service Changes the Contact-Pattern Requirement
The contact pattern is the set of instantaneous contact points produced as the meshing point moves across the tooth. In a conjugate spiral bevel pair, contact is nominally point contact under light load, and the locus of those points forms the visible pattern used in gear inspection. A pattern near the tooth tip or near the toe or heel indicates a risk of edge contact, concentrated loading, noise, and shortened life. For a pair that operates in both directions, the design problem is not simply to center one pattern; the mounting position must keep the convex and concave patterns within acceptable limits at the same time.
Installation errors shift the relative position and orientation of the two members. The relevant errors are the offset displacement ΔE, the pinion axial displacement ΔP, the wheel axial displacement ΔG, and the shaft-angle error ΔΣ. Error tooth contact analysis, or ETCA, introduces these displacements into the coordinate transformations between the two tooth surfaces and solves for the contact point and the transmission error. The ETCA result is a geometric, unloaded prediction of the contact locus, not a loaded pressure distribution. Its value is that it exposes how individual mounting errors move the pattern before parts are assembled and adjusted.
Tooth-Surface Generation and ETCA Formulation
A Gleason-type spiral bevel pair is modeled from its blank geometry and cutting settings. The wheel is double-side generated, while the pinion is single-side generated with modified roll. The cutter head carries an external blade that cuts the concave flank and an internal blade that cuts the convex flank. In the cutter frame, the generating surface is described by the cutter radius, profile angle, and two surface parameters, \(u_{gi}\) and \(\theta_{gi}\):
$$
\mathbf r_{gi}(u_{gi},\theta_{gi})=
\begin{bmatrix}
(r_{ci}\pm u_{gi}\sin\alpha_{gi})\cos\theta_{gi}\\
(r_{ci}\pm u_{gi}\sin\alpha_{gi})\sin\theta_{gi}\\
-u_{gi}\cos\alpha_{gi}\\
1
\end{bmatrix}
$$
The cutter tip radius is \(R_{ci}=R_{ui}\pm0.5P_W\), where \(R_{ui}\) is the nominal cutter radius and \(P_W\) is the point width. The “±” sign distinguishes the concave and convex sides. The unit normal of the cutter surface is
$$
\mathbf n_{gi}=
\begin{bmatrix}
\cos\alpha_{gi}\cos\theta_{gi}\\
\cos\alpha_{gi}\sin\theta_{gi}\\
\pm\sin\alpha_{gi}
\end{bmatrix}
$$
Coordinate transformations take the cutter surface from frame \(S_{gi}\) through the machine and cradle frames into the blank frame \(S_i\). The generated tooth surface and the equation of meshing are
$$
\mathbf r_i(u_{gi},\theta_{gi},\phi_{ci})=[M_{ig}]\mathbf r_{gi}(u_{gi},\theta_{gi}),
\qquad
\mathbf n_{mi}\cdot\mathbf v_{mi}=0
$$
Here \(i=1\) denotes the left-hand pinion and \(i=2\) the right-hand wheel; \(\mathbf n_{mi}\) and \(\mathbf v_{mi}\) are the cutter-surface normal and the relative velocity at the cutting point in the machine frame. For the pinion, the modified-roll ratio varies with cradle rotation, which allows the tooth surface to be corrected without changing the basic cutter geometry.
Contact is solved in a common meshing frame \(S_h\). The two surfaces must have the same position and the same unit normal at the contact point:
$$
\mathbf r_h^{(1)}(\theta_{g1},\phi_{c1},\phi_1)=
\mathbf r_h^{(2)}(\theta_{g2},\phi_{c2},\phi_2),
\qquad
\mathbf n_h^{(1)}=\mathbf n_h^{(2)}
$$
The system gives five independent scalar equations for six unknowns. The pinion roll angle \(\phi_1\) is stepped through the mesh, and the remaining parameters are solved at each step. The transmission error follows as
$$
\Delta\phi_2=(\phi_2-\phi_{20})-\frac{z_1}{z_2}(\phi_1-\phi_{10})
$$
where \(z_1\) and \(z_2\) are the pinion and wheel tooth numbers, and \(\phi_{10}\) and \(\phi_{20}\) are the initial roll angles. Contact points are retained only inside the four tooth-surface boundaries derived from the pitch cone, root cone, face cone, and face width. The pattern is then the locus of those valid contact points.
Case Geometry and Pattern Descriptors
The example is a 28-tooth left-hand pinion meshing with a 33-tooth right-hand wheel at a 90° shaft angle. Its basic data are listed in Table 1. The cutter settings include profile angles of 19.834° and 20.063°, tip radii of 1.8 mm and 2.2 mm, and cutter radii of 113.7 mm and 114.3 mm. The pinion uses a second-order modified-roll coefficient \(C=0.035\) and a third-order coefficient \(D=0\).
Table 1. Basic parameters of the spiral bevel gear pair.
| Parameter | Pinion | Wheel |
|---|---|---|
| Number of teeth | 28 | 33 |
| Hand | Left | Right |
| Shaft angle | 90° | |
| Mean normal module | 6.1 mm | 6.1 mm |
| Face width | 39.6 mm | 39.6 mm |
| Outer cone distance | 131.998 mm | 131.998 mm |
| Mid-cone distance | 112.198 mm | 112.198 mm |
| Inner cone distance | 92.398 mm | 86.34 mm |
| 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 | 5.771 mm | 4.599 mm |
| Dedendum | 5.746 mm | 6.917 mm |
| Pitch diameter | 145.18 mm | 171.105 mm |
For optimization, the pattern is reduced to four descriptors: \(\lambda=[x,y,s,k]\). In the projected tooth-surface frame, \(x\) runs from the small end toward the large end, and \(y\) runs from the root toward the tip. The quantity \(s\) is the projected pattern area, and \(k\) is the angle between the pattern long axis and the pitch-cone direction. These four values make it possible to compare a calculated pattern with a target pattern without relying only on visual inspection.
Directional Effects of Installation Error
ETCA sweeps were made with ΔE and ΔP varied from −0.3 mm to +0.3 mm and with ΔΣ varied from −10′ to +10′. The resulting response curves were smooth and were fitted with trigonometric expressions, so the optimization could treat the installation errors as continuous variables. The qualitative trends for positive increments are summarized in Table 2.
Table 2. Pattern response to increasing installation error in forward and reverse rotation.
| Error variable | Forward: wheel convex working | Reverse: wheel concave working |
|---|---|---|
| ΔE, offset displacement | x moves from small end to large end; y moves from root to tip; s rises then falls; k shifts in the opposite sense to the ΔP effect | x moves from large end to small end; y moves from tip to root; s rises then falls; k decreases |
| ΔP, pinion axial displacement | x moves from large end to small end; y moves from tip to root; s rises then falls; k shifts in the opposite sense to the ΔE effect | x changes weakly; y moves from root to tip; s rises then falls; k increases |
| ΔΣ, shaft-angle error | x moves from large end to small end; y moves from root to tip; s rises then falls; k decreases | x moves from small end to large end; y moves from tip to root; s rises then falls; k increases |
Two results drive the correction strategy. First, the pattern area is non-monotonic: within the tested range it increases and then decreases as a single error grows. Second, the concave and convex flanks often respond in opposite directions. A positive ΔE, for example, moves the convex pattern toward the large end and the tooth tip, while it moves the concave pattern toward the small end and the tooth root. ΔE and ΔP also produce larger changes in area and long-axis inclination than ΔΣ. Consequently, an adjustment chosen only from the convex pattern can worsen the concave pattern, and the two flanks must be optimized together.
Optimization of Installation Settings
The correction problem treats \(\lambda=[x,y,s,k]\) as the output and ΔE, ΔP, and ΔΣ as the adjustable inputs; the wheel axial displacement ΔG is held outside the adjustment set. A target pattern \(\lambda_0=[x_0,y_0,s_0,k_0]\) is specified. The long-axis inclination is the objective, while the midpoint coordinates are constrained within the face width and tooth height and the area is bounded:
$$
\min_{\Delta E,\Delta P,\Delta\Sigma}|k-k_0|
$$
The constraints keep the pattern inside the tooth and prevent the optimizer from buying a favorable inclination with an unacceptable midpoint shift or an oversized contact zone. With zero added installation adjustment, the convex pattern was \([19.0425,\ 0.6631,\ 388.884,\ -8.8605]\) and the concave pattern was \([19.1065,\ 0.8458,\ 342.706,\ 11.0183]\). Both lay too close to the tip edge, their inclinations differed substantially, and the transmission-error amplitude at the mesh transition was 21.4858.
The optimized adjustment was ΔE = 0.0113 mm, ΔP = 0.0095 mm, and ΔΣ = 0. These values are small relative to the ±0.3 mm sweep range. The resulting pattern descriptors are given in Table 3. The convex pattern became \([17.064,\ 0.0606,\ 332.8346,\ -3.9825]\), and the concave pattern became \([16.627,\ -0.07955,\ 360.4073,\ 2.8149]\). The patterns moved to the mid-height region, their long-axis inclinations moved closer to zero, and their midpoints became more consistent. The area term was constrained rather than maximized: the convex area decreased from 388.884 to 332.835, while the concave area increased from 342.706 to 360.407.
Table 3. Pattern descriptors before and after installation-error optimization.
| Direction | Working flank | Setting | x | y | s | k |
|---|---|---|---|---|---|---|
| Forward | Convex | Baseline | 19.0425 | 0.6631 | 388.884 | −8.8605 |
| Forward | Convex | Optimized | 17.064 | 0.0606 | 332.8346 | −3.9825 |
| Reverse | Concave | Baseline | 19.1065 | 0.8458 | 342.706 | 11.0183 |
| Reverse | Concave | Optimized | 16.627 | −0.07955 | 360.4073 | 2.8149 |
The four descriptors use the model’s projected tooth-surface scale; no unit conversion is applied to x, y, s, and k. Baseline means zero added installation adjustment. The optimized setting is ΔE = 0.0113 mm, ΔP = 0.0095 mm, ΔΣ = 0′.
The transmission-error amplitude at the mesh transition fell from 21.4858 to 16.1163, a reduction of about 25%. The corrected pattern was also more nearly elliptical, with transition traces at its ends rather than a concentration at the tooth-tip edge. These changes are ETCA results; the subsequent rolling check addressed pattern location and shape, not the transmission-error curve.
Rolling-Test Verification
The corrected settings were checked on a 500 mm universal bevel gear rolling tester. The gear pair was ground, the pinion flanks were coated with red lead powder, and the pair was mounted with the optimized installation values. Patterns were read from the wheel at four angular positions, 90°, 180°, 270°, and 360°, for forward rotation on the convex flank and reverse rotation on the concave flank.
The experimental patterns agreed with the simulated patterns in position and general shape. Both lay near the small end in the face-width direction and near the middle in the tooth-height direction, with a date-kernel-like outline that met the design requirement for this pair. The comparison supports the ETCA trends and the optimized adjustment, but its scope should be stated clearly: it is a rolling contact-pattern check of one 28/33 Gleason pair over limited error ranges. It is not a loaded endurance, thermal, or lubrication test, and the numerical trends should not be transferred to another tooth geometry without recalculating the response surfaces.
Implications for Installation Adjustment
The correction method turns installation adjustment into a quantified pattern-target problem. Instead of moving the pinion until one flank looks acceptable, the ETCA model evaluates both working flanks, extracts midpoint position, area, and long-axis inclination, and solves for small mounting changes that keep those descriptors within the tooth and close to the target. For the case examined, ΔE = 0.0113 mm and ΔP = 0.0095 mm were sufficient to move both patterns away from the tip edge, reduce the inclination mismatch, and lower the transmission-error amplitude by about 25%.
The broader lesson is that concave and convex contact patterns in reversible spiral bevel gears cannot be corrected independently. A single mounting error generally shifts the two flanks in opposite directions, and the area response is non-monotonic. The practical adjustment therefore needs to treat the two flanks as simultaneous constraints, use pattern descriptors rather than visual judgment alone, and verify the result with a rolling check. For production, the fitted error-response surfaces provide a starting point for setting the offset, pinion axial position, and shaft angle; the final values remain specific to the gear geometry, cutter settings, and acceptable pattern window.