Spiral bevel gears transmit motion between intersecting or offset shafts while combining rolling and sliding contact. Their load capacity, motion smoothness, and noise depend on a deliberately localized tooth contact pattern, which is created by coordinated cutter geometry and generating motion. In the completing process, also called the duplex helical, duplex spread-blade, or two-cut method, a single cutter head machines both the concave and convex flanks of the pinion. This raises productivity and reduces tool changes, but it also couples the two flanks: any change in cutter profile or machine motion affects both sides of the tooth. The control problem is therefore more constrained than in the conventional five-cut process, where separate operations allow each flank to be adjusted more independently.
The industrial origin of the completing process is usually traced to Gleason work in the 1930s. Calculation cards for small-module spiral bevel gears were released in 1965, and an improved version for larger modules followed in 1978. The underlying formulas remained proprietary and were expressed mainly as settings for cradle-type machines. Modern Free-form CNC gear cutting machines offer additional axes and greater freedom, but many existing formulations still convert cradle-machine settings into equivalent Free-form motions. Such conversions can require five-axis interpolation, including a workpiece swing axis, which is mechanically more demanding and harder to keep accurate. A direct four-axis formulation avoids that swing axis and uses the CNC machine’s freedom to impose the desired pinion surface and tooth depth.
Process principle and controlled quantities
In the completing process, the pinion is cut by one cutter head while the workpiece and cutter execute a linked generating motion. The concave pinion flank is usually the drive side, and the convex flank is the coast side. A practical four-axis arrangement uses three translational axes, X, Y, and Z, plus the workpiece rotation B. The installation root angle A is held constant during cutting, so no workpiece swing axis participates in the interpolation. In the example discussed below, B is the negative of the pinion rotation angle, B = −φp, and A is fixed at 17°.
The cutter head is not symmetric in its control function. Its outer blade is straight and primarily controls the pinion concave flank, which corresponds to the drive side. Its inner blade is a circular arc, which adds a free profile variable for controlling the convex flank in the tooth-height direction. The cutter-tip circle is constrained relative to the pinion root cone, so tooth depth is imposed geometrically rather than recovered through an additional workpiece swing motion. This is the main mechanical reason why the four-axis arrangement can preserve the designed tooth depth while avoiding the swing axis.
Active design of pinion microgeometry
The design starts from a specified wheel tooth surface and a specified gear-ratio function. Enveloping the wheel by the pinion under the theoretical ratio produces a fully conjugate pinion surface. Such a surface has high sensitivity to manufacturing and assembly errors and tends toward edge contact under load. The pinion is therefore modified so that the wheel and pinion become locally conjugate: contact remains instantaneous point contact, but the contact path, transmission error, and contact ellipse are controlled.
For the drive side, modification is imposed in two directions. The first is along the contact path and is defined by a transmission-error function TE(φp), where φp is the pinion rotation angle. For each discrete point on the wheel contact path, the conjugate pinion position and normal are obtained by coordinate transformation:
$$
\mathbf r_p=\mathbf M_{ps}\mathbf M_{su}\mathbf M_{uw}\mathbf r_w,
\qquad
\mathbf n_p=\mathbf m_{ps}\mathbf m_{su}\mathbf m_{uw}\mathbf n_w .
$$
Here, rw and nw are the wheel position and unit normal, Mps, Msu, and Muw are coordinate transformation matrices from the wheel to the pinion, and mps, msu, and muw are their 3 × 3 rotational submatrices. The roll relation used in this transformation contains the prescribed TE(φp), so the resulting pinion path points already include the first-direction modification.
The second modification direction is perpendicular to the contact path and is controlled through the curvature at the reference point. Let Kwplp be the induced normal curvature along the contact-path tangent and Gwplp the induced geodesic torsion. The induced curvature perpendicular to the path, Kwpvp, follows from the two principal induced curvatures Kwp1 and Kwp2:
$$
K^{wp}_{vp}=
\frac{K^{wp}_{lp}K^{wp}_{1}+(G^{wp}_{lp})^{2}-(K^{wp}_{1})^{2}}
{K^{wp}_{lp}-K^{wp}_{1}},
\qquad
K^{wp}_{1}=\frac{8\Delta}{L^{2}} .
$$
In the contact-ellipse convention used for the example, L is the major-axis length of the instantaneous contact ellipse and Δ is a small separation value; the design used L = 7.5 mm and Δ = 6.35 × 10−3 mm. The resulting pinion curvature targets, together with the path-point targets, define the drive-side ease-off. The coast side has fewer controlled quantities: the position and normal at its reference point are imposed, while its reference-point curvature is adjusted through the remaining cutter variables during the solution.
The numerical example targeted a straight drive-side contact path inclined at 45° to the pinion root cone, a contact-ellipse major axis of 7.5 mm at the reference point, and a peak-to-peak transmission error of 100 µrad. The transmission-error function was designed as a quadratic parabola on the drive side. Because the coast side is not controlled at every path point, its transmission-error curve in the example is not a standard parabola.
Four-axis cutting model and parameter solution
The cutting model is written in a machine-fixed coordinate system. The cutter center has coordinates (Xoc, Yoc, Zoc) in this frame, the pinion rotates by φp, and the installation root angle A remains constant. The cutter head is defined by its outer straight blade, inner circular-arc blade, tip circle, and axial-section profile. The outer blade controls the concave drive flank; the inner arc provides an additional degree of freedom for the convex coast flank.
The drive-side targets are expressed as four vector constraints: equality of the pinion target position and cutter position, equality of the target unit normal and cutter unit normal, location of the cutter tip circle on the pinion root cone, and tangency of the cutter tip circle to the root cone. These give nine independent scalar equations:
$$
\mathbf r_{dp}=\mathbf r_{dc},\qquad
\mathbf n_{dp}=\mathbf n_{dc},\qquad
\mathbf r_{df}=\mathbf r_{db},\qquad
\mathbf n_{df}\cdot\mathbf t_{db}=0 .
$$
The coast-side targets are position and unit normal at the reference point, giving five independent scalar equations:
$$
\mathbf r_{cp}=\mathbf r_{cc},\qquad
\mathbf n_{cp}=\mathbf n_{cc}.
$$
The solution proceeds in two stages. For the drive side, the outer-blade pressure angle and the installation root angle are first assigned. The pinion rotation and cutter parameters are solved iteratively. At the reference point, the second-order curvature targets determine the cutter radius and the outer-blade tip radius. The cutter-center coordinates are then solved for each discrete contact-path point. These discrete positions are fitted by least squares as quintic polynomials in φp:
$$
X_{oc}=\sum_{k=0}^{5}c^{x}_{k}\varphi_p^{k},
\qquad
Y_{oc}=\sum_{k=0}^{5}c^{y}_{k}\varphi_p^{k},
\qquad
Z_{oc}=\sum_{k=0}^{5}c^{z}_{k}\varphi_p^{k}.
$$
For the coast side, the inner-arc radius and arc-tangent height are assigned, while A is kept at the drive-side value. The pinion rotation and cutter parameter at the coast reference point are solved, and the cutter-center coordinates are obtained from the already determined drive-side motion. The inner-blade pressure angle and inner tip radius then follow. The coast reference-point contact attributes—contact-path direction angle, instantaneous contact-ellipse major axis, and peak-to-peak transmission error—are checked. If they are unsatisfactory, the outer-blade pressure angle, inner-arc radius, arc-tangent height, or installation root angle can be changed and the complete drive-side and coast-side solution repeated.
Computed example and verification
The example pair has a 15-tooth left-hand pinion and a 45-tooth right-hand wheel, a 90° shaft angle, zero offset, and a 31° mean spiral angle. The wheel was produced by the formate method and the pinion by generating. The blank data are listed in Table 1.
| Member | z | Shaft angle Σ (°) | Offset a (mm) | Hand | Mean spiral angle βm (°) | Outer pitch diameter de (mm) | Pitch cone angle δ (°) | Face cone angle δa (°) | Root cone angle δf (°) |
|---|---|---|---|---|---|---|---|---|---|
| Pinion | 15 | 90 | 0 | Left | 31 | 93.333 | 18.435 | 22.086 | 16.915 |
| Wheel | 45 | 90 | 0 | Right | 31 | 280.000 | 71.565 | 73.085 | 67.914 |
The wheel cutter had a nominal radius of 95.25 mm, inner and outer pressure angles of 20°, and a tip width of 3.81 mm. Its machine settings were a horizontal cutter position of 80.76 mm, a vertical position of 75.74 mm, a root angle of 68.82°, and a horizontal offset of −2.27 mm. The pinion cutter used a straight outer blade with a tip radius of 100.64 mm and a pressure angle of 15°. Its inner blade was a circular arc with radius 37 mm, tangent height 6 mm, tip radius 98.54 mm, and pressure angle 20.82°. The resulting motion polynomials are given in Table 2; the A axis remains at 17°, close to the pinion root angle of 16.915°, and the B axis is slaved to −φp.
| Axis | c0 | c1 | c2 | c3 | c4 | c5 |
|---|---|---|---|---|---|---|
| X | −84.931 926 | 24.121 703 | 4.239 980 | −0.248 903 | −0.263 390 | −0.156 503 |
| Y | −0.188 649 | −0.094 379 | −0.010 120 | −0.321 459 | −0.825 279 | −0.734 557 |
| Z | −75.172 414 | −25.245 899 | 2.797 448 | 0.748 208 | 0.108 400 | 0.109 390 |
| A | 17 | 0 | 0 | 0 | 0 | 0 |
Using these cutter parameters and motion coefficients, the pinion surface is generated by the envelope relation
$$
\mathbf r_p=\mathbf M_{ps}\mathbf M_{sm}\mathbf M_{mc}\mathbf r_c,
\qquad
\mathbf n_p=\mathbf m_{ps}\mathbf m_{sm}\mathbf m_{mc}\mathbf n_c,
$$
where rc and nc are the cutter position and normal, and the transformation matrices include the machine motion and pinion rotation. Tooth contact analysis of the wheel and generated pinion gave a straight drive-side contact path at the intended 45° inclination and a parabolic drive-side transmission-error curve. The coast-side curve was not parabolic because fewer path constraints are imposed there.
The geometry was also checked by virtual cutting. A four-axis Free-form machine, cutter model, and G-code derived from the motion polynomials were built in Vericut. The theoretical pinion surface was represented by a 9-column by 5-row grid. After simulation, all 45 theoretical grid points fell on the simulated cut surface, indicating that the solved cutter and motion parameters reproduce the intended pinion geometry within the numerical model. This is a geometric verification, not a physical cutting test; it does not by itself include cutting forces, thermal distortion, fixture compliance, or measured machine geometric errors.
The axis motions show the expected structure. A stays constant at 17° and therefore has zero first derivative with respect to φp. Y remains close to zero because the installation root angle is nearly equal to the pinion root angle and the cutter tip plane is nearly tangent to the root cone; the tooth-height direction therefore needs only a small translation range. The generating action is carried mainly by X, Z, and B. As the cutter moves from the large end of the tooth slot toward the small end, B increases, X decreases, and Z increases. The first derivative of X is positive and that of Z is negative over the working range.
Motion-error sensitivity
Sensitivity was evaluated by perturbing the zeroth- and first-order coefficients of the machine-axis polynomials and recomputing the pinion surface. The resulting error surface was first rotated about the gear axis until its center point coincided with the theoretical center, which removes pitch error and isolates tooth-surface topography. The normal deviation at grid point (i, j) is
$$
\delta_r(i,j)=\left[\mathbf r_r(i,j)-\mathbf r_t(i,j)\right]\cdot\mathbf n_t(i,j),
$$
where rr is the error-surface position, rt is the theoretical position, and nt is the theoretical unit normal. A positive value means material addition relative to the theoretical flank; a negative value means removal.
Table 3 summarizes the peak absolute deviations obtained with a +0.1 perturbation of one coefficient at a time. For the linear axes, the zeroth-order coefficient is in millimetres and the first-order coefficient is in millimetres per radian. The opposite sign produces a nearly mirror-image error map with almost equal magnitude and reversed trend.
| Axis | Zeroth-order peak |δ| (µm) | First-order peak |δ| (µm) | Observed topography |
|---|---|---|---|
| X | 18.4 | 55.1 | Zeroth order: spiral-angle error on both flanks; first order: diagonal error on both flanks. |
| Y | 6.5 | 26.8 | Zeroth order: spiral error on the concave flank and pressure-angle error on the convex flank; first order: diagonal error on both flanks. |
| Z | 16.5 | 42.1 | Zeroth order: diagonal error on the concave flank and spiral error on the convex flank; first order: diagonal error on both flanks. |
Three regularities stand out. First, the three translational axes can all produce spiral-angle, diagonal, and pressure-angle deviations, but their signatures differ. X is associated mainly with spiral error at zeroth order and diagonal error at first order. Y produces spiral and pressure-angle effects at zeroth order and diagonal effects at first order. Z produces diagonal and spiral effects at zeroth order and predominantly diagonal effects at first order. Second, for the same numerical perturbation, a first-order coefficient change has a larger effect than a zeroth-order change. A constant offset shifts the cutter center, whereas a slope change alters the generating relation along the whole path. Third, among the three linear axes, Y has the smallest influence on the pinion tooth surface. This is consistent with the small Y motion range and its role in the tooth-height direction.
These signatures provide a basis for topography compensation. If a measured or predicted error map has a known combination of spiral, diagonal, and pressure-angle components, the machine motion can be adjusted in the opposite sense to reduce the deviation. The sensitivity maps indicate which axes and which polynomial terms are most effective for a given error component, and they show that X and Z require tighter motion control than Y when the same coefficient tolerance is specified. This is a compensation principle rather than a completed closed-loop correction; practical implementation still requires measured machine behavior and a validated error model.
Engineering implications and limits
The direct four-axis completing process uses the Free-form machine’s freedom at the motion-generation level rather than reproducing cradle-machine settings. Because the workpiece swing axis is eliminated, the mechanical error sources associated with that axis are avoided, and tooth depth is imposed through the cutter-tip circle and root-cone constraint. The pinion drive side can be actively designed along the contact path and across it, with the transmission-error function controlling the first direction and the reference-point curvature controlling the second. The coast side is controlled at its reference point and through additional cutter variables, but it has fewer constraints than the drive side; its contact pattern and transmission-error curve are therefore less completely prescribed.
The error analysis shows that constant and slope errors in the translational axes are not equivalent. First-order coefficient errors are more damaging for the same numerical perturbation, and opposite perturbations produce nearly mirrored topography. Y is the least sensitive linear axis in this example, while X and Z are stronger levers for spiral and diagonal correction. These results are specific to the stated blank geometry, cutter design, and motion polynomials. They establish a numerical relationship between machine-axis motion and pinion-topography error, but they do not replace physical cutting trials, thermal and load-deflection testing, or measurement of actual machine-axis error maps.