Spiral bevel gears transmit rotation between intersecting shafts with a curved tooth trace that can carry high torque smoothly. Their final meshing behavior depends on the coordinated motion of the cutter and workpiece during generation: small changes in machine motion alter the contact path, the transmission error, and the load distribution across the flank. The completing process, also called the duplex helical or two-cut method, is attractive because one cutter head finishes the pinion concave and convex flanks in one setup. It raises productivity and can improve tooth strength, but the same cutter and the same machine motion influence both flanks at once. Controlling a completing-process pinion is therefore more constrained than controlling a pinion cut by the conventional five-cut method.
Gleason introduced the completing process in the 1930s. Calculation cards for small-module spiral bevel gears appeared in 1965, and an improved setting method for larger modules followed in 1978. The underlying formulas remained proprietary, and the machine settings were tied to cradle-type mechanical machines. Later work converted cradle settings to free-form CNC machines, but the conversion still reproduced the logic of a cradle machine and generally required five-axis linkage, including a moving workpiece swing axis. That swing axis is difficult to manufacture and its motion accuracy is difficult to maintain. A more direct route is to solve the completing-process motions for a four-axis free-form machine, using three translational axes and the pinion spindle while keeping the root-angle axis fixed.
Active design of the pinion flank
The starting point is a fully conjugate pinion obtained by enveloping a given wheel flank with a prescribed transmission-ratio function. A fully conjugate flank is sensitive to machining and assembly errors and can promote edge contact. The pinion is therefore modified so that wheel and pinion remain in local conjugate contact, with point contact at each instant. The drive side is designed in two directions. Along the contact path, the position and normal vector of every discrete point are prescribed through a transmission-error function \(TE(\varphi_p)\), where \(\varphi_p\) is the pinion rotation angle. Perpendicular to the contact path, the curvature at the reference point is prescribed to set the instantaneous contact ellipse. The coast side prescribes the position and normal vector at its reference point; its reference curvature is adjusted during the solution of the cutting parameters.
The wheel-to-pinion transformation gives the target pinion position and normal vectors:
$$\mathbf r_p=\mathbf M_{ps}\mathbf M_{su}\mathbf M_{uw}\mathbf r_w,\qquad
\mathbf n_p=\mathbf m_{ps}\mathbf m_{su}\mathbf m_{uw}\mathbf n_w,$$
where the \(\mathbf M\) matrices transform coordinates from the wheel to the pinion and the \(\mathbf m\) matrices are their \(3\times3\) rotational submatrices. Because the transmission-error function enters the construction of every contact-path point, the drive-side modification along the path is already embedded in the target position and normal vectors.
At the reference point, the induced normal curvature and geodesic torsion along the contact-path tangent are fixed by the transmission-error function. The induced curvature normal to the path is then chosen from the contact-ellipse target. If \(K^{wp}_{lp}\) and \(G^{wp}_{lp}\) are the induced normal curvature and geodesic torsion along the path tangent, and \(K^{wp}_{1}\) is the larger principal induced curvature, the cross-direction induced curvature is
$$K^{wp}_{vp}=
\frac{K^{wp}_{lp}K^{wp}_{1}+(G^{wp}_{lp})^{2}-(K^{wp}_{1})^{2}}
{K^{wp}_{lp}-K^{wp}_{1}}.$$
This construction allows the contact path to be a straight line or a curve, and the transmission error to be a second-order, higher-order, or piecewise function. In the numerical example, the wheel drive-side contact path is designed as a straight line in the projection plane at \(45^\circ\) to the root cone, the transmission error is a parabolic function, and the instantaneous contact-ellipse major axis at the reference point is \(7.5\ \mathrm{mm}\).
Four-axis free-form cutting model
The cutting model is built directly in the coordinate system of a free-form CNC machine. The machine uses three translational axes \(X,Y,Z\) and a pinion spindle \(B\). The installation root angle \(A\) is fixed during cutting, so no workpiece swing axis participates in the linkage. The cutter center has coordinates \((X_{oc},Y_{oc},Z_{oc})\) in the machine frame. The pinion rotation is \(\varphi_p\), and the spindle rotation is \(B=-\varphi_p\).
The pinion cutter uses a straight outer blade for the concave flank and a circular-arc inner blade for the convex flank. The straight outer blade controls the drive-side concave surface. The inner arc introduces an additional free variable for controlling the convex flank in the tooth-height direction, which is needed for the coast-side contact target. Tooth depth is controlled by a cutter top circle tangent to the pinion root cone, so the depth is set geometrically rather than by a large motion of a workpiece swing axis.
The target conditions are written as vector constraints:
$$\mathbf r_{dp}=\mathbf r_{dc},\quad
\mathbf n_{dp}=\mathbf n_{dc},\quad
\mathbf r_{df}=\mathbf r_{db},\quad
\mathbf n_{df}\cdot\mathbf t_{db}=0,\quad
\mathbf r_{cp}=\mathbf r_{cc},\quad
\mathbf n_{cp}=\mathbf n_{cc}.$$
The first four equations apply to the drive side: the position and unit normal of the pinion target must match the cutter-generated values, the root-cone position must match the cutter top circle, and the cutter top-circle tangent must be perpendicular to the root-cone normal. These give nine independent scalar equations. The last two equations apply to the coast-side reference point and give five independent scalar equations: three from position and two from the unit normal.
Solving the motions and cutter profile
The drive side is solved first. The outer-blade pressure angle and the installation root angle are preset. The pinion rotation and cutter angular parameter are obtained iteratively. At the reference point, the prescribed second-order parameters determine the cutter radius and the outer-blade tip radius. The cutter-center coordinates are then solved for each point on the contact path. These discrete center positions are least-squares fitted as quintic polynomials in the pinion rotation:
$$X_{oc}=\sum_{i=0}^{5}c_i^x\varphi_p^i,\qquad
Y_{oc}=\sum_{i=0}^{5}c_i^y\varphi_p^i,\qquad
Z_{oc}=\sum_{i=0}^{5}c_i^z\varphi_p^i.$$
This gives the machine motion law for the drive side: three translational axis polynomials, a spindle rotation tied to the pinion rotation, and a fixed root-angle axis. The remaining outer-blade geometry is also determined.
The coast side is solved after the drive side. The inner-arc radius and the arc tangent height are preset, and the installation root angle is retained. The pinion rotation and cutter parameter at the coast reference point are solved together with the corresponding cutter-center position, using the drive-side motion law as the base. The inner-blade pressure angle and tip radius are then obtained from the cutter profile equations. At this stage the complete cutter geometry and the complete four-axis motion law are defined.
The coast-side contact condition is checked through the contact-path direction angle at the reference point, the instantaneous contact-ellipse major axis, and the peak-to-peak transmission error. If these are not satisfactory, four free inputs are adjusted: the outer-blade pressure angle, the inner-arc radius, the inner-arc tangent height, and the installation root angle. The drive-side and coast-side solutions are then repeated. This loop determines a completing-process solution that satisfies the drive-side contact path and the coast-side reference-point targets while producing the required tooth depth.
Numerical example
The example is a zero-offset spiral bevel gear pair with a \(90^\circ\) shaft angle. The wheel is cut by the formate method and the pinion by generating. The blank data are listed in Table 1.
| Member | \(z\) | \(\Sigma\) (°) | Offset \(a\) (mm) | Hand | \(\beta_m\) (°) | \(d_e\) (mm) | \(\delta\) (°) | \(\delta_a\) (°) | \(\delta_f\) (°) |
|---|---|---|---|---|---|---|---|---|---|
| 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 has a nominal radius of \(95.25\ \mathrm{mm}\), inner and outer pressure angles of \(20^\circ\), and a tip width of \(3.81\ \mathrm{mm}\). The wheel is positioned with a horizontal cutter position of \(80.76\ \mathrm{mm}\), a vertical cutter position of \(75.74\ \mathrm{mm}\), an installation root angle of \(68.82^\circ\), and a horizontal wheel-position correction of \(-2.27\ \mathrm{mm}\). The pinion cutter has an inner circular arc with radius \(37\ \mathrm{mm}\) and tangent height \(6\ \mathrm{mm}\). Its inner tip radius is \(98.54\ \mathrm{mm}\), its inner-blade pressure angle is \(20.82^\circ\), its outer tip radius is \(100.64\ \mathrm{mm}\), and its outer-blade pressure angle is \(15^\circ\). The installation root angle on the pinion machine is fixed at \(17^\circ\).
The fitted motion places most of the generating action in the \(X\) and \(Z\) axes. The \(Y\) axis stays close to zero because the installation root angle is nearly equal to the pinion root cone angle and the cutter top face is almost tangent to the root cone; only a small \(Y\) range is needed to satisfy tooth depth. The \(A\) axis remains constant and does not participate in the linkage.
Tooth contact analysis (TCA) gives a drive-side contact pattern with a near-parabolic transmission error, as designed. The coast-side transmission error is not a standard parabola because the coast side has fewer controlled parameters: only its reference-point position and normal are prescribed, while the contact path is not controlled point by point. The ease-off modification of the pinion concave and convex flanks is correspondingly different.
A virtual cutting check was performed in Vericut. A four-axis free-form machine model and the computed cutter were built, and the axis motion polynomials were converted into cutting G-code. The theoretical pinion flank was represented by a \(9\times5\) grid. After simulation, the theoretical grid points fell on the simulated cut surface. This supports the geometric consistency of the computed cutter geometry and machine motion, although it is a numerical and virtual check rather than a physical cutting test.
Sensitivity to machine motion errors
The sensitivity study perturbs the constant and linear coefficients of the translational-axis polynomials. A constant-term error shifts the axis position uniformly; a linear-term error changes the rate of motion with pinion rotation. For each perturbation, an error flank is generated and compared with the theoretical flank. The raw error includes pitch error and flank-form error. Pitch error is removed by rotating the error surface about the gear axis until its midpoint coincides with the theoretical midpoint. The remaining normal deviation is
$$\delta_r(i,j)=\left[\mathbf r_r(i,j)-\mathbf r_t(i,j)\right]\cdot\mathbf n_t(i,j),
\qquad i=1,\ldots,J_c,\quad j=1,\ldots,J_R,$$
where \(\mathbf r_r\) and \(\mathbf r_t\) are the error and theoretical position vectors, \(\mathbf n_t\) is the theoretical unit normal, \(J_c\) is the number of columns, and \(J_R\) is the number of rows. A positive \(\delta_r\) means material addition relative to the theoretical flank; a negative value means material removal.
The perturbations used in the example are \(0.1\) in the coefficient value: \(0.1\ \mathrm{mm}\) for a constant translational term and \(0.1\ \mathrm{mm/rad}\) for a linear translational term when \(\varphi_p\) is in radians. The resulting dominant flank-form deviations are summarized in Table 2.
| Axis term | Pinion concave / drive side | Pinion convex / coast side |
|---|---|---|
| \(X\), constant term | Spiral angle error | Spiral angle error |
| \(X\), linear term | Diagonal error | Diagonal error |
| \(Y\), constant term | Spiral angle error | Pressure angle error |
| \(Y\), linear term | Diagonal error | Diagonal error |
| \(Z\), constant term | Diagonal error | Spiral angle error |
| \(Z\), linear term | Diagonal error | Diagonal error |
Three general rules emerge. First, a constant offset and a linear rate error produce different topographic signatures: the constant term tends to shift the contact pattern in the spiral or pressure-angle direction, while the linear term tends to produce a diagonal pattern across the flank. Second, for the same coefficient increment, the linear term has a larger effect than the constant term because its influence accumulates over the generating roll. Third, reversing the sign of a perturbation produces nearly the same magnitude of normal deviation with the opposite trend. Among the three translational axes, \(Y\) has the smallest influence on the pinion flank for the same coefficient increment.
These signatures are geometric sensitivities. They show how individual axis errors map onto spiral-angle, pressure-angle, and diagonal flank errors, but they are not a complete error budget for a real machine. Thermal distortion, cutter wear, fixture error, and elastic deflection under load are separate effects. Within the geometric scope, the sensitivity map offers a compensation route: if the measured flank deviation has a known spiral-angle, pressure-angle, or diagonal component, the corresponding axis motion can be adjusted in the opposite direction to reduce the normal deviation. The correction is implemented by modifying the motion polynomial, not by changing the theoretical tooth-surface design.
Scope and practical implications
The direct four-axis completing process uses the flexibility of a free-form machine without converting the problem back into a cradle-machine setting. It solves the cutter profile and the machine motion together, produces a prescribed drive-side contact path and coast-side reference condition, and controls tooth depth through the cutter top circle rather than through a moving workpiece swing axis. This avoids the manufacturing and motion-accuracy burden of a swing axis and gives a clear geometric path from a desired meshing behavior to machine motion.
The main limitation is on the coast side. Because the coast side has fewer prescribed parameters than the drive side, its contact path is not controlled point by point, and its transmission error is not necessarily the intended parabolic form. Improving coast-side contact remains a design problem. The numerical example establishes geometric consistency through TCA and virtual cutting; production validation would require cutting tests, flank measurement, and compensation of the actual machine and cutter errors. Within that scope, the axis-error sensitivity table provides a practical starting point for flank-topography correction in spiral bevel gear completing processes.