Why small-module spiral bevel gears are a process problem
Spiral bevel gears carry the final drive in agricultural implements, industrial sewing machines, electric and pneumatic power tools, and small marine drives. Their appeal is mechanical: a curved-tooth bevel pair transmits a large torque in a small envelope, holds a fixed ratio over life, runs with a high contact ratio and low noise, and wastes little energy in the mesh. In the small-module class — modules of roughly 1 mm to 1.5 mm, produced in the tens of millions of sets per year in China alone — the economics of the process matter as much as the geometry, because these are consumer-grade components made in very large batches.
On mechanical small-module gear cutting machines, both members of the pair are conventionally cut by full generating in the double-duplex arrangement (双重双面法): a duplex cutter head cuts both flanks of a slot at once while the cradle and work spindle roll through a geared ratio. Generating both members is geometrically general — it works for any ratio and any pitch angle — but it is slow, because every slot on every part is enveloped by a long roll, and the flank shape that can be produced is tied to the machine’s generating train. The alternative examined here is semi-generating cutting, in which the gear member is cut without generating and only the pinion is generated, with the pinion flanks corrected to match the resulting straight-sided gear.
Generated versus semi-generated tooth slots
In the generating process the cutter head is mounted on a cradle. Work spindle and cradle rotate at a fixed ratio about their own axes, and the rotating blades sweep out the slot as the roll proceeds; the cutting action resembles nothing so much as a hypoid pair running in mesh, with the cutter head playing the part of the mating gear. Because the envelope depends on the roll, changes in ratio or cone angle are absorbed by the machine settings rather than by the tooling.
Semi-generating cutting removes the envelope from the gear member. The gear blank is held stationary in the generating sense, the cutter head is plunged along the Z axis to full depth, withdrawn quickly, the work is indexed, and the next slot is cut. The flank produced is a copy of the straight-sided blade rather than a generated surface, which is why the process is often called the Formate or straight-profile method. In high-volume production the two steps are separated: a dedicated roughing machine with a three-side blade cutter head plunges and indexes the gear slots, and a circular broach (Revacycle-type) cutter head finishes them — the broach carries a notch, one revolution completes one slot, and the work indexes while the cutter passes the notch.
The pinion cannot be cut this way, because its flanks must conjugate with the straight-sided gear. It is generated on a machine fitted with a 30° spindle-tilt or modified-roll mechanism, with single-side cutter heads used separately for the concave and convex flanks, and a cutter-setting (对刀) device that keeps the stock allowance even on both sides of the slot.
The method is not universal. It is applied where the transmission ratio is at least 3 and the gear pitch cone angle is at least 70°, the regime in which the sliding conditions across the face width stay tolerable when the gear flank is a straight profile. The pair studied below sits inside that window: with 38 and 11 teeth at a 90° shaft angle, tan δp = 11/38 gives a pinion pitch angle of about 16.1° and a gear pitch angle of about 73.9°, while the ratio of 3.45 clears the ratio threshold.
Workpiece pair and fixturing
The trial pair is a small-module spiral bevel set for an agricultural machine, cut from 20CrMo. The gear is the larger, asymmetric-pressure-angle member; the pinion is a small, positively modified member. Both are mounted on dedicated fixtures, and the blank support area of the fixture is deliberately generous: on a plunge cut the axial thrust is carried largely through the blank and fixture, and a small support face allows the blank to deflect, which shows up directly as depth variation around the gear.
| Parameter | Gear (wheel) | Pinion |
|---|---|---|
| Module | 1.35 mm | 1.35 mm |
| Number of teeth | 38 | 11 |
| Pressure angle, concave flank | 22.552° | 20° |
| Pressure angle, convex flank | 17.813° | 20° |
| Spiral angle | 35°1′11″, left hand | 35°, right hand |
| Whole depth | 2.937 mm | 2.554 mm |
| Modification coefficient | — | 0.36 |
| Pitch cone angle (from zp/zg) | ≈73.9° | ≈16.1° |
| Transmission ratio | 38/11 = 3.45 | |
Cutting tool profile
The blade profile is computed in the same way as the hob profile for a special-shaped circular-arc instrument gear: the conjugate rack profile is established from the required tooth form, and the cutter curve is derived from it and then discretized for grinding and inspection. For this pair the resulting curve is given in three segments — an 18-point flank curve, a 7-point tip curve and a 16-point back curve — in a plane whose x axis runs across the tooth and whose y axis runs along the tooth height. The flank coordinates are listed below; the tip and back segments serve to close the blade and to set the grinding wheel path.
| Point | x (mm) | y (mm) | Point | x (mm) | y (mm) |
|---|---|---|---|---|---|
| 1 | −0.2447 | 0.0956 | 10 | −0.3513 | −0.3542 |
| 2 | −0.4286 | 0.0624 | 11 | −0.3268 | −0.4007 |
| 3 | −0.4303 | 0.0386 | 12 | −0.2985 | −0.4467 |
| 4 | −0.4306 | 0.0338 | 13 | −0.2657 | −0.4925 |
| 5 | −0.4337 | 0.0086 | 14 | −0.2286 | −0.5376 |
| 6 | −0.4151 | −0.1656 | 15 | −0.1859 | −0.5817 |
| 7 | −0.4040 | −0.2127 | 16 | −0.1368 | −0.6255 |
| 8 | −0.3898 | −0.2598 | 17 | −0.0798 | −0.6679 |
| 9 | −0.3723 | −0.3072 | 18 | −0.0122 | −0.7089 |
Machine stiffness for a plunge cut
Plunge cutting loads the machine differently from generating. The cutter is fed axially into a stationary work rotation, so the thrust on the work spindle and on the Z-axis drive is higher and more sustained than in a rolling cut. On a small mechanical machine the spindle housing overhangs its support, and that overhang is the weak link. Finite-element checks of the housing point to two measures: a support rib under the overhang, and a double-wall section where the layout allows it. The double wall is the better answer structurally and also looks cleaner, but on most gear cutting machines the specification leaves no room to place the rib below the spindle, so the reinforcement has to be arranged around the existing outline.
Bearing selection is the second lever. Enlarging the spindle bearings is not necessarily possible within the headstock envelope, so the axial capacity is raised instead by combining tapered roller bearings or deep-groove angular-contact ball bearings on the work spindle, and by using the same combination on the Z-axis ball screw. The third measure is backlash: the worm gear pair of the work spindle has to be adjusted so that the plunge and the rapid withdrawal do not introduce lost motion that would appear as a depth step in the slot. With these three changes — ribbed or double-walled housing, axially stiffer bearing sets, and controlled worm backlash — a mechanical machine can hold the tolerances the process needs.
Load-levelled plunge schedule
A constant plunge rate is a poor schedule for a copying cut. The area swept by each successive blade group grows as the cutter approaches full depth, so the last group removes the most material and the first groups remove very little; the tool is either underloaded for most of the cycle or overloaded at the end. The trial used a Gleason No.102 small-module machine, the only mechanical small-module machine on which the 1:1 roll ratio setting is available to suppress generating during the gear cut, with the following data: cutter head 1.5N15, point width W = 0.635 mm, eccentric angle 35.76°, 8 blades arranged as four inner/outer groups, spindle speed 360 r/min.
At 6 r/s and four groups per revolution the group rate is 24 groups per second. With a nominal 6 s per slot, one slot is cut by 144 blade groups:
$$N=\frac{h}{s},\qquad t=\frac{N}{n\,z_g}$$
where h is the plunge depth, s the feed per group, n the cutter speed in revolutions per second and zg the number of blade groups per revolution. Taking the depth used in the timing calculation as about 2.4 mm and N = 144 gives an increment of roughly 0.017 mm per group, and t = 144/(6 × 4) = 6 s, consistent with the setting. (The drawing whole depth of the gear, 2.937 mm, is larger than the depth used for this calculation; the schedule is set from the cutting depth, not from the finished slot depth.)
Sizing every pass against 90% of the area removed by the last group levels the load across the whole plunge. Under that constraint the slot can be completed in about 4.2 s instead of 6 s, a 30% reduction in cutting time and a factor of about 1.4 in throughput, while the cut stays uniform enough for roughing and finishing to be combined in a single plunge. The same levelling reduces peak tooth load on the blades, which is where the tool-life benefit comes from.
Pinion generation: two-direction cutting and flank correction
A straight-sided gear flank changes the conjugate pinion. Because the gear root is cut by a copying plunge rather than generated, the root no longer lies on the root cone, and the slot geometry varies along the face width: the slot narrows from mid-face toward both ends and its depth changes with it. The gear flank is a straight profile standing in for the involute, and the pinion has to be corrected on two counts — the change from involute to straight profile in the normal section, and the change of slot depth and width along the face width.
The correction cannot be made in one bidirectional pass. The two flanks of the pinion are generated in opposite roll directions, so the correction for the concave flank and the correction for the convex flank run in opposite senses of the generating motion, and a single pass cannot satisfy both. Nor is a second setup a way out: small-module machines of this class carry no cutter-setting device, and the slots are too small for the operator to split the allowance evenly by eye between two mountings.
The trial resolved this with a two-direction generating sequence on a duplex cutter head. One flank (concave or convex) is roughed and finished while the opposite flank is only roughed; the work is then not withdrawn, the machine parameters are changed, and the opposite flank is cut in the reverse direction; the work is again not withdrawn, the indexing is performed, and the cycle repeats. Because the part stays on its arbor throughout, the allowance problem of a second mounting disappears. The correction values for the two flanks and for the depth at the toe and heel are written into the part program, and the CNC axes apply them through the servo drives, so the correction per tooth is identical around the ring.
The same arrangement allows a spiral angle correction that the double-duplex method does not provide. The gear’s nominal spiral angle βc is specified at mid-slot, mid-face; the two-direction cut lets the pinion be corrected about that reference, with the correction offsets for the concave and convex flanks taken from the cutter radius Rμ and the point width, and with the reference distance evaluated at the mid-face position:
$$L_c = L – \frac{b}{2}$$
where L is the cone distance to the heel, b the face width and Lc the cone distance to mid-face where βc is defined. Correcting the pinion spiral angle against the gear gives a contact pattern that can be placed deliberately rather than inherited from the cutter settings.
Verification on the shop floor
The cutting trials were run against a drawing that fixes both the setup and the acceptance conditions: fixture mandrel radial and face runout ≤ 0.005 mm, blade runout ≤ 0.012 mm, fixture replacement every 200 pieces or whenever accuracy drifts, and, before running, a check against a master gear on a rolling meshing tester for mounting distance and contact position, with the contact required to sit in the middle of the tooth. Three samples were taken. The convex flank contact sat at mid-face and the concave contact slightly toward the toe, matching the standard pattern used as the reference.
| Characteristic | Drawing requirement | Measured | Result |
|---|---|---|---|
| Circumferential backlash | 0.10–0.12 mm | 0.05–0.9 mm | Accepted |
| Tooth-to-tooth runout | ≤ 0.015 mm | 0.01 mm | Pass |
| Total gear runout | ≤ 0.05 mm | 0.034 mm | Pass |
| Contact pattern | Per drawing | Convex mid-face, concave slightly to toe | Pass |
The runout figures sit comfortably inside the drawing limits, and the contact pattern is where the correction programming put it. Backlash is the exception: the measured band across the three samples is far wider than the 0.10–0.12 mm drawing band, which makes backlash the characteristic this process controls least well. That is consistent with the mechanism — backlash in a copying cut is set by plunge depth, and depth is the quantity most sensitive to blank support stiffness, fixture runout and worm backlash in the work spindle train, all of which the process measures only indirectly through the master gear check.
What the method does and does not settle
Semi-generating cutting is a real option for small-module spiral bevel gears inside its window — ratio of 3 or more, gear pitch cone angle of 70° or more — and the trial pair falls inside it. The gain reported here is a 30% cut in cutting time per slot, from 6 s to about 4.2 s, obtained purely by levelling the plunge load rather than by changing the tool material or the machine. Because the spindle and feed drives are unchanged, cutting energy per gear falls roughly in proportion to machine time, which is where the energy claim for the process comes from.
Two limits are worth stating plainly. The first is machine availability: among mechanical small-module machines, only the Gleason No.102 provides the 1:1 roll setting needed for the copying plunge, while rigid domestic CNC machines can also perform the Z-axis plunge and, with the correction values programmed, the pinion sequence. The second is that the correction work is not optional. Once the gear flank is straight-sided, the pinion flank, the depth taper along the face width and the spiral angle all have to be corrected in the part program, and the quality of the result depends on that programming as much as on the machine. Within those limits, the obvious next step is the tool: blade coatings or indexable carbide inserts would allow a higher cutting speed, and the gear and pinion times quoted here should be read as a floor set by high-speed steel blades rather than as a limit of the process.