Constant-depth spiral bevel gears have taken over a growing share of construction, mining, metallurgy and precision machinery drives, largely displacing earlier Oerlikon tooth forms that could not be ground. Klingelnberg HPG hard skiving and Gleason constant-depth grinding have pushed achievable accuracy steadily upward. One geometric constraint nevertheless survives in low-ratio pairs: where the pinion pitch cone angle is small and the adjacent bearing journal is large in diameter, the cutter head that generates the tooth slots collides with the journal shoulder. The interference leaves a gouge at the fillet, damages the seating face and removes material from the most heavily stressed section of the shaft.
The Interference Mechanism
A bevel cutter head is a disc of radius r carrying the blades. To cut a tooth space at the outer cone distance Re, the disc must be swung far enough out that its rim passes beyond the root cone. On a pinion with a small pitch cone angle the journal shoulder sits close to the tooth root and inside that swept envelope, so both the blades and the cutter body touch the shoulder. The consequences are threefold: the seating face loses squareness and surface quality, the shaft section at the shoulder is reduced exactly where bending stress peaks, and the damaged zone is usually blended by hand, which introduces further dimensional scatter.
The conventional production remedy is to insert a process land — a relief step between the tooth root and the bearing shoulder — or to reduce the journal diameter. Both move the journal out of the cutter path at the cost of a longer overhang and a weaker shaft. A longer overhang increases deflection under load, displaces the contact pattern and erodes the accuracy that modern hard-finishing operations are meant to deliver. Reducing the journal diameter lowers bearing capacity. Neither measure addresses the underlying geometry.
Rotating the Generating Cone About Mid-Face Width
Klingelnberg cutting machines, from the older AMK series to the current CNC C series, carry a tailstock tilting mechanism. If the part geometry is left untouched and the whole gear is tilted about the mid-point of the face width, the generating pitch cone shifts by a small angle μk while the shaft angle, the outer cone distance and the face width stay as designed. Applying opposite signs to the two members keeps the pair conjugate:
$$\delta_{E1}=\delta_{1}-\mu_{k},\qquad \delta_{E2}=\delta_{2}+\mu_{k}$$
Here δ1 and δ2 are the theoretical pitch cone angles of pinion and wheel, and δE1, δE2 are the generating pitch cone angles actually set on the machine. Because the rotation is taken about mid-face, the two members move coherently and mesh quality at mid-face is essentially preserved; the cutter head simply approaches the tooth space from a slightly rotated attitude and clears the shoulder. If the interference occurs at the opposite (left-hand) journal, the same procedure applies with the offset direction reversed. Unlike hypoid offset, this adjustment does not involve the hand of the spiral, because the displacement takes place on the imaginary crown gear of the pair rather than through a spiral angle change.
Rechecking Blank Dimensions and Bearing Loads
The offset leaves the outer cone distance and face width unchanged but shortens the cone distances measured to the mid-face and to the inner end:
$$R_{m}=R_{e}-0.5\,b\cos\mu_{k}$$
$$R_{i}=R_{e}-b\cos\mu_{k}$$
where Re is the outer cone distance, b the face width, Rm the mid-cone distance and Ri the inner cone distance; no other blank parameter is affected by μk.
Tilting the generating cone also rotates the line of action, so the axial and radial components carried by the bearings change. When the bearing life margin is thin, or when the offset is deliberately made large, both components must be recomputed with δE in place of the theoretical angle. For the convex flank acting as the meshing plane:
$$F_{x}=\frac{F_{t}}{\cos\beta_{m}}\left(\tan\alpha_{n}\sin\delta_{E}-\sin\beta_{m}\cos\delta_{E}\right)$$
$$F_{r}=\frac{F_{t}}{\cos\beta_{m}}\left(\tan\alpha_{n}\cos\delta_{E}+\sin\beta_{m}\sin\delta_{E}\right)$$
and for the concave flank as the meshing plane:
$$F_{x}=\frac{F_{t}}{\cos\beta_{m}}\left(\tan\alpha_{n}\sin\delta_{E}+\sin\beta_{m}\cos\delta_{E}\right)$$
$$F_{r}=\frac{F_{t}}{\cos\beta_{m}}\left(\tan\alpha_{n}\cos\delta_{E}-\sin\beta_{m}\sin\delta_{E}\right)$$
with Ft the tangential force at mid-face, αn the normal pressure angle and βm the mean spiral angle. These values, not the unoffset ones, feed the bearing life and shaft strength calculations.
Cutter Clearance Check According to KN3029
Interference is verified against the offset generating angle rather than the nominal one, following the KN3029 design standard. Two distances are compared with the clearance envelope of the blade tip, r + ha0tanαn, where r is the cutter head radius and ha0 the blade addendum. Non-interference requires both
$$\sqrt{\left(X_{m}-X_{A}\right)^{2}+\left(Y_{m}-Y_{A}\right)^{2}}<r+h_{a0}\tan\alpha_{n}$$ $$\sqrt{\left(x_{m}-x_{i}\right)^{2}+\left(y_{m}-y_{i}\right)^{2}}<r+h_{a0}\tan\alpha_{n}$$=""
to hold simultaneously, where subscript A refers to the blade position on the outer cone and I to that on the inner cone. The reference point M on the cutter head is located from the cutter head dimension Md and the angular setting ψe − λ:
$$X_{m}=M_{d}\sin\left(\psi_{e}-\lambda\right),\qquad Y_{m}=M_{d}\cos\left(\psi_{e}-\lambda\right)$$
The coordinates of A and I follow from the cone distance (Re or Ri), the generating angle δE and the blade height h. Because δE appears throughout, offsetting the cone changes the calculated clearances directly, and a pair that fails the check at the nominal angle may pass at δE.
Setting the Offset on the Machine
Once the offset angle is chosen, the workpiece spindle is displaced vertically by the amount that moves the generating cone apex into its new position:
$$h_{v}=R_{m}\tan\mu_{k}$$
The resulting tailstock setting dimension is built up from the machine constant Kb read from the data card, the fixture offset Ab, the distance LD from the chuck locating face to the gear, and projection terms of the offset cone of the form Rmcosζ′mcosδE and hvsinδE. On AMK machines the spindle is raised or lowered by a hydraulic cylinder controlled through a manual valve, and the actual angle is confirmed from the arbor height, the Kb value on the data card and the chuck datum distance. On the C series the rotary slide is driven through a control valve commanded by the KIMOS software, so the offset is entered as a parameter and the mechanical setting follows automatically.
| Quantity | Plain design | With offset μk | Remark |
|---|---|---|---|
| Pinion generating pitch cone angle | δ1 | δ1 − μk | reduced to clear the journal |
| Wheel generating pitch cone angle | δ2 | δ2 + μk | opposite sign preserves conjugacy |
| Outer cone distance / face width | Re, b | Re, b | unchanged |
| Mid-cone distance | Re − 0.5b | Re − 0.5bcosμk | shortened |
| Inner cone distance | Re − b | Re − bcosμk | shortened |
| Axial and radial bearing loads | from δ | from δE | recalculate if margin is small |
| Cutter head versus journal | interference | clearance | primary objective |
| Contact pattern | nominal | slight shift | corrected through cutter position Ex |
| Machine generation | Offset actuator | Setting route |
|---|---|---|
| AMK series | tailstock tilt driven by hydraulic cylinder | manual valve; angle confirmed from arbor height, Kb on the data card and chuck datum distance |
| C series (CNC) | rotary slide driven through control valve | offset entered as a KIMOS parameter, actuator commanded by software |
Contact Pattern and Practical Limits
The offset changes the axial force direction and magnitude, but the angles used in this method are small — around 2° is usually enough to give a pronounced improvement in cutter-to-journal clearance — and because pinion and wheel are offset together, the meshing quality of the pair is largely unaffected. At such magnitudes the contact pattern barely moves from its unoffset position.
Two operating limits matter. First, correction of the contact pattern during cutting is straightforward while the offset is small: the conventional adjustment of cutter position Ex restores the pattern. Second, when the offset must be large and the contact pattern sits at an extreme position — running off the toe or concentrated at the heel edge — adjusting Ex alone is no longer sufficient and blade wedge shims must be added to or removed from the cutter head. Beyond that point the strength of the pair and the bearing life have to be recalculated rather than assumed.
One point of workshop practice deserves recording. Several domestic design handbooks recommend solving this interference by increasing the plane tilt angle of the cutter head. Klingelnberg machines of either generation possess a cutter eccentricity mechanism only; the cutter tilt mechanism found on Oerlikon equipment is not available on them. The correction must therefore be made through the tailstock offset described here, not through a cutter tilt that the machine cannot execute.
Conclusions
- On Klingelnberg spiral bevel gears, the tailstock offset function of the cutting machine can eliminate, or at least substantially reduce, interference between the cutter head and the bearing journal, improving the surface quality and positional strength of the shoulder that carries the bearing.
- Because pinion and wheel are offset in opposite directions about mid-face width, a small offset leaves the contact pattern essentially intact; a large offset requires a fresh check of gear strength and bearing life using the offset generating angle.
- Small offsets are corrected by conventional adjustment of cutter position Ex; at large offsets with extreme contact patterns, blade wedge shims must additionally be added or removed on the cutter head.