Spiral bevel gears transmit torque between intersecting shafts through teeth that are curved in both the lengthwise and the profile direction. Their flanks are not machined to a fixed profile: the surface is the envelope of a rotating cutter head that stands for a generating gear, while the workpiece rolls with the cradle at a prescribed ratio. Because the flank is defined by that relative motion rather than by the cutter alone, the same nominal gear — identical module, pressure angle, spiral angle and face width — can be produced with different flank topologies simply by changing how the cradle, the sliding base and the workpiece move with respect to one another. This is the property that makes systematic tooth-surface correction possible, and it is also the property that makes machine settings difficult to interpret without a quantitative model.
The unused degrees of freedom on CNC bevel gear machines
Production CNC machines for spiral bevel gears implement universal motions — modified roll, helical motion and vertical motion — each driven as a function of cradle rotation. In the calculation that precedes cutting or grinding, however, only the basic settings and the first-order helical motion coefficient are normally evaluated, while the second- to sixth-order coefficients are left at zero. The machine can therefore execute considerably more than the setup sheet asks of it.
The consequence becomes visible at the rolling tester. When the contact pattern is too long, too short or skewed diagonally, the operator alters machine settings and re-tests, relying on experience rather than on a quantitative map between a coefficient and the resulting change of the flank. Without such a map the additional motion axes are hard to use deliberately: a correction that a small, well-localized change of a second- or third-order coefficient could produce is instead pursued through settings that also displace the contact pattern. The question is therefore specific: when one helical motion coefficient at a time is varied, how does the normal deviation field over the flank respond — in direction, in magnitude, and in the type of geometric error it represents?
Kinematics of the generating motion
In the machine coordinate system \(\sigma=\{O;\mathbf{i},\mathbf{j},\mathbf{k}\}\), \(O\) is the machine center, the \(\mathbf{i}\)–\(\mathbf{j}\) plane is the machine plane and \(\mathbf{k}\) points into the cradle. The cutter head axis passes through the intersection \(O_c\) of the tool tip plane with the cutter axis; its projection on the machine plane is \(\mathbf{b}\) and its unit direction is \(\mathbf{c}\). The remaining quantities are the radial setting \(S\), the cradle angle \(q\), the vertical offset \(E_m\), the workpiece axis \(\mathbf{p}\), the sliding base \(X_B\), the horizontal correction \(X_p\) and the blank mounting angle \(\delta_m\), with \(O_1\) denoting the design crossing point. For a left-hand gear the arrangement of these quantities is the one shown in the standard vector formulation of bevel gear generation.
A point \(M\) on the conical cutting surface is located by the vector \(\mathbf{r}\) from \(O\) and by \(\mathbf{r}_c\) from \(O_1\); its unit normal is \(\mathbf{n}\) and the unit tangent along the cone generatrix is \(\mathbf{t}\). The cutting surface is thus a two-parameter family in the cradle angle \(q\) and the cutter phase angle \(\theta\), and the generated flank is the surface conjugate to it.
The helical motion acts on the sliding base. Writing \(\Delta q\) for the cradle rotation increment measured from the reference position, the sliding base becomes a truncated series in \(\Delta q\):
$$X_B = X_{B0} – H_1\Delta q + \frac{H_2\Delta q^{2}}{2!} – \cdots + \frac{H_6\Delta q^{6}}{6!}$$
and the corresponding velocity of the workpiece along the generating-gear axis is its derivative:
$$v_H = -H_1 + H_2\Delta q – \frac{H_3\Delta q^{2}}{2!} + \cdots + \frac{H_6\Delta q^{5}}{5!}$$
where \(X_{B0}\) is the sliding base at the setup position and \(H_1,\dots,H_6\) are the first- to sixth-order helical motion coefficients. The first-order term is the linear axial translation already contained in conventional setup calculations; the higher-order terms add progressively steeper, cradle-dependent axial displacements, so the instantaneous rate of axial motion is no longer constant over the roll.
Enforcing the meshing condition between the cutter surface and the workpiece yields the combination of \(q\), \(\theta\) and workpiece rotation \(\phi\) at which a point of the cutter surface contacts the flank. The contact point and its normal in the workpiece frame then follow from a rigid-body rotation about the workpiece axis:
$$\mathbf{r}_w = (\mathbf{p}\cdot\mathbf{r}_c)\mathbf{p} + \cos\phi\,(\mathbf{p}\times\mathbf{r}_c)\times\mathbf{p} + \sin\phi\,(\mathbf{p}\times\mathbf{r}_c)$$
$$\mathbf{n}_w = (\mathbf{p}\cdot\mathbf{n})\mathbf{p} + \cos\phi\,(\mathbf{p}\times\mathbf{n})\times\mathbf{p} + \sin\phi\,(\mathbf{p}\times\mathbf{n})$$
With \(H_2=\dots=H_6=0\) the model reduces to the conventional generating motion; switching individual coefficients on perturbs the surface without altering the nominal gear data.
A 45-point grid and the normal deviation
Deviation is evaluated on a grid laid out in the rotation-projection plane of the flank, with 9 columns along the face width and 5 rows along the tooth height, giving 45 points. The layout follows the practice of gear measuring centers, where the probe approaches each point along the theoretical normal in an S-shaped path from the heel and the toe. Edge relief follows AGMA practice: 10% of the face width is trimmed at toe and heel, and 5% of the full tooth depth at tip and root, with a minimum of 0.6 mm, so that corner radii and chamfers do not influence the measurement.
For a point \(M\) on the flank, let \(x\) be its distance along the gear axis from \(O_1\) and \(y\) its distance from the axis \(\mathbf{p}\). Then
$$y = \left|\mathbf{r}_w\times\mathbf{p}\right|, \qquad x = -\mathbf{r}_w\cdot\mathbf{p}$$
Both are functions of \(q\) and \(\theta\). Given a grid point \((x,y)\), the corresponding pair \((q,\theta)\) is recovered by two-variable iteration, after which the position vector \(\mathbf{r}_w\) and normal \(\mathbf{n}_w\) of that point follow from the equations above and are transformed into the measurement frame. Recomputing the same procedure with a perturbed coefficient gives \(\mathbf{r}’_w\) and \(\mathbf{n}’_w\), and the deviation at each grid point is taken along the theoretical normal:
$$\delta = (\mathbf{r}’_w-\mathbf{r}_w)\cdot\mathbf{n}_w$$
A positive value means that the actual flank lies outside the theoretical one (material remaining), a negative value that it lies inside. Mapping \(\delta\) over the grid produces the deviation topology of the flank for that perturbation.
Five descriptors of the deviation field
A deviation map contains far more numbers than an operator can act on, so it is condensed into a second-order surface. With the origin at the grid center, \(X\) along the face width, \(Y\) along the tooth height and \(Z\) the deviation,
$$Z = a_1X + a_2Y + a_3XY + a_4X^{2} + a_5Y^{2}$$
The five coefficients are interpreted as follows. The first-order terms describe the tilt of the surface: \(a_1\) is the spiral-angle error sensitivity and displaces the contact pattern along the face width, while \(a_2\) is the pressure-angle error sensitivity and displaces it along the tooth height. The second-order terms describe the curvature: \(a_3\) is the torsion sensitivity and governs how strongly the pattern runs diagonally, \(a_4\) is the lengthwise curvature sensitivity and controls the length of the contact pattern in the face-width direction, and \(a_5\) is the profile curvature sensitivity and controls its width in the tooth-height direction.
For \(m\) grid points the system is written as
$$\{Z\} = [S]\{a\}$$
with \([S]\) built from the coordinates of the grid points. Since the system is overdetermined, the coefficients are obtained in the least-squares sense:
$$\{a\} = \left([S]^{T}[S]\right)^{-1}[S]^{T}\{Z\}$$
Each perturbation of a helical motion coefficient produces its own deviation field and therefore its own set of five coefficients, which makes the effect of that coefficient comparable across the different geometric error types.
Reference gear pair and machine settings
The evaluation was carried out on a 17-tooth pinion and its mating 17-tooth wheel, both with a 90° shaft angle, a 40° mean spiral angle, module 3 mm, pressure angle 17.5° and face width 12 mm; the pinion is left-hand and the wheel right-hand. The pinion is produced by the duplex helical method, and the machine and cutter settings used as the reference state are listed below.
| Parameter | Wheel | Pinion |
|---|---|---|
| Hand of spiral | Right | Left |
| Number of teeth | 17 | 17 |
| Shaft angle (°) | 90 | 90 |
| Mean spiral angle (°) | 40 | 40 |
| Module (mm) | 3 | 3 |
| Pressure angle (°) | 17.5 | 17.5 |
| Face width (mm) | 12 | 12 |
| Parameter | Wheel | Pinion |
|---|---|---|
| Radial setting (mm) | 28.7080 | 28.6176 |
| Horizontal wheel position (mm) | 0 | −0.2618 |
| Vertical offset (mm) | 0 | −0.7194 |
| Sliding base (mm) | 0 | −1.7016 |
| Cutter tilt (°) | 2.58 | 7.42 |
| Cutter swivel (°) | 219.31 | 84.06 |
| Blank mounting angle (°) | 37.48 | 35.99 |
| Roll ratio | 1.402053 | 1.411378 |
| Center roll position (°) | 81.13 | 86.87 |
| Nominal cutter diameter (mm) | 76.2 | 77.55 |
| Inner–outer blade offset (mm) | 1.25 | 1.16 |
| Inner blade edge angle (°) | 25 | 25 |
| Outer blade edge angle (°) | 10 | 10 |
| First-order helical motion coefficient | 0 | 4.9717 |
To isolate the contribution of each order, one coefficient at a time was changed from the reference state, in increments of \(+0.2\), \(+2\), \(+20\), \(+200\), \(+2000\) and \(+20000\) for \(H_1\) through \(H_6\). The increments grow by a factor of ten per order because a given numerical change of a high-order coefficient in the series above produces a much smaller displacement than the same change of a low-order one; scaling them keeps the induced deviations visible on the same micrometre scale. A dedicated tooth-surface analysis program was used to compute the deviation at all 45 grid points for each case and to fit the five coefficients by least squares.
How the flank responds to each order
The deviation maps use a thick solid line for the theoretical flank, a thin solid line where the theoretical flank lies below the actual one, and a thin dashed line where it lies above it; the four corner points of the concave and convex flanks carry the local deviation in micrometres. Four features stand out.
- The helical motion coefficients act essentially as a diagonal modification of the flank. The effect is markedly stronger on the convex flank than on the concave one, and stronger at the toe and heel than in the middle of the face width.
- On the concave flank the behavior is regular. Changing the first-, third- or fifth-order coefficient produces the same sense of distortion: the actual flank at the toe tip lies below the theoretical surface, while at the heel root it lies above it. Changing the second-, fourth- or sixth-order coefficient reverses both: the toe tip lies above and the heel root below the theoretical surface.
- On the convex flank the heel tip lies above the theoretical surface for every order. The first-, third- and fifth-order coefficients put the toe root below the theoretical surface, whereas the second-, fourth- and sixth-order coefficients put it above.
- The dominant geometric error differs between flanks. On the concave flank every order affects the pressure-angle term most and the profile-curvature term least. On the convex flank the first-order coefficient affects the spiral-angle term most and the profile-curvature term least, while the second- to sixth-order coefficients affect the pressure-angle term most and the lengthwise-curvature term least.
Sensitivity coefficients on the two flanks
Applying the least-squares fit to each single-coefficient perturbation gives the coefficient sets below. Because the imposed increments differ by a factor of ten between successive orders, the values are best read within a row — as a ranking of which geometric error component responds most strongly to that order — rather than as a direct comparison of one order against another.
| Coefficient changed (increment) | \(a_1\) spiral angle | \(a_2\) pressure angle | \(a_3\) torsion | \(a_4\) lengthwise curvature | \(a_5\) profile curvature |
|---|---|---|---|---|---|
| \(H_1\) (+0.2) | 0.0001489 | −0.0008759 | −0.0002049 | 0.0001305 | 0.0001092 |
| \(H_2\) (+2) | −0.0105070 | 0.0113974 | −0.0005773 | 0.0005052 | 0.0000306 |
| \(H_3\) (+20) | 0.0094212 | −0.0103516 | 0.0001602 | −0.0002695 | 0.0001505 |
| \(H_4\) (+200) | −0.0078969 | 0.0084628 | −0.0002366 | 0.0002847 | −0.0000734 |
| \(H_5\) (+2000) | 0.0048817 | −0.0052221 | 0.0000960 | −0.0001506 | 0.0000699 |
| \(H_6\) (+20000) | −0.0023761 | 0.0025240 | −0.0000547 | 0.0000779 | −0.0000299 |
| Coefficient changed (increment) | \(a_1\) spiral angle | \(a_2\) pressure angle | \(a_3\) torsion | \(a_4\) lengthwise curvature | \(a_5\) profile curvature |
|---|---|---|---|---|---|
| \(H_1\) (+0.2) | −0.0155950 | 0.0103966 | 0.0002281 | 0.0001788 | −0.0001620 |
| \(H_2\) (+2) | 0.0511271 | −0.0513838 | −0.0028206 | 0.0001809 | 0.0026612 |
| \(H_3\) (+20) | 0.0135064 | −0.0154027 | −0.0022695 | 0.0007755 | 0.0015837 |
| \(H_4\) (+200) | 0.0221120 | −0.0223736 | −0.0019513 | 0.0004423 | 0.0015221 |
| \(H_5\) (+2000) | 0.0080402 | −0.0083500 | −0.0010432 | 0.0003231 | 0.0007344 |
| \(H_6\) (+20000) | 0.0047823 | −0.0048448 | −0.0005228 | 0.0001459 | 0.0003797 |
The tables quantify what the deviation maps show qualitatively. On the concave flank the leading coefficients of the odd orders (\(H_1\), \(H_3\), \(H_5\)) carry one sign and those of the even orders (\(H_2\), \(H_4\), \(H_6\)) the opposite, which is consistent with the alternating signs of the terms in the sliding-base series and explains why the diagonal orientation of the pattern flips from one order to the next. The same flank is comparatively insensitive to the first-order coefficient: its spiral-angle term is two orders of magnitude below the corresponding value on the convex flank, so a linear axial motion during the roll is felt mainly on one side of the tooth.
The convex flank responds several times more strongly to the same coefficient changes. There the first-order coefficient is dominated by the spiral-angle term, while from the second order upward the pressure-angle term clearly dominates and the lengthwise-curvature term is the smallest of the five. In every row the torsion term sits between the leading term and the two curvature terms, which is the signature of a modification that runs diagonally across the flank rather than along one of its principal directions.
Grinding and measurement check
The computed behavior was checked by grinding pinions on an H350GH CNC spiral bevel gear grinder with the same coefficient perturbations, and measuring the flanks on a gear measuring center with the 45-point grid described above. The measured deviation topologies reproduced the predicted direction of the diagonal distortion and the sign pattern at the corner points, with the predicted and measured values in general agreement; residual differences were attributed to machining error and measurement uncertainty. Only the case of a second-order coefficient change of \(+2\) is reported in detail, the remaining cases having been omitted for space.
The check is a geometric one. It establishes that the deviation field computed from the surface model corresponds to the field left on a ground flank for at least this perturbation, but it does not address loaded contact, transmission error under torque, or the running behavior of the pair.
What the coefficients can and cannot do
Two practical consequences follow. First, the order of the coefficient selects the diagonal sense of the modification: odd and even orders tilt the deviation field in opposite directions, so a combination of two orders can be used to trade one diagonal bias against the other and to place the correction where it is needed, while a single order produces a one-sided diagonal bias. Second, because the response is concentrated at the toe and heel and is weak in the middle of the face width, the higher-order coefficients are better suited to localized end correction than to a uniform shift of the contact pattern, which remains the domain of the first-order settings.
The scope of the numbers has to be kept in view. The coefficients above belong to one pinion geometry — 17 × 17 teeth, 40° mean spiral angle, 12 mm face width, duplex helical generation — and to the specific reference settings listed, so they cannot be transferred to another gear set without recomputation; the sensitivity coefficients depend on the reference state. The analysis concerns unloaded flank geometry only, and the experimental confirmation covers a single perturbation case. Within those limits, the results give the operator a quantitative basis for deciding which order to change when a contact pattern has to be rotated diagonally, and how much of that change will appear at the ends of the tooth rather than at its center.