Where the process bottleneck sits
Small-module spiral bevel gears, generally below module 1.5 mm, transmit motion through curved-tooth bevel pairs in power tools, garden and forestry machinery, industrial sewing machines, pneumatic tools and small marine drives. The curved tooth gives a contact ratio well above that of a straight bevel tooth, the ratio is fixed by the tooth numbers, and the package is compact relative to belt or chain drives of the same torque capacity. Published Chinese industry figures put annual output of small-module spiral bevel gears in the tens of millions of sets, a substantial share of it exported, and the volume keeps pressure on cycle time.
The conventional route on mechanical gear-cutting machines is the duplex double-side method: both members of the pair are generated with two-sided cutter heads, so that every tooth slot is produced by a cradle roll coupled to the work spindle. The method is universal — it makes no distinction between large and small ratios or between large and small pitch cone angles — but the roll is the slowest element of the cycle, and the flank mismatch that results from generating both members is sensitive to cutter-head setting, which makes the final mesh difficult to hold inside a narrow contact-pattern band.
Generating versus semi-generating kinematics
In generating, the cutter head is carried on a cradle that rotates in a fixed ratio with the work spindle. The blades sweep the blank as though the two members of a hypoid pair were rolling through mesh, and the flank is the envelope of the successive blade positions. The envelope, not the blade profile, defines the tooth surface.
Semi-generating cutting removes the roll from one member. For a pair with ratio u = z2/z1 ≥ 3 and a wheel pitch cone angle ≥ 70°, the generating motion required on the wheel side contributes little to the final flank shape, so the wheel can be cut by plunge: the work spindle is clamped, the cutter head is fed along Z to full tooth depth, withdrawn rapidly, the work is indexed, and the cycle repeats until every slot is cut. On a dedicated line the wheel is roughed with a three-side cutter head and finished with a circular pull-broaching cutter whose gap allows one slot per revolution and indexing at the gap. On a single machine the same kinematics are obtained with the plunge axis alone.
The pinion is still generated. Roughing uses a two-sided head that cuts both flanks of a slot in one plunge-roll pass; finishing separates the flanks and uses inner and outer single-side cutters on a machine fitted with a 30° spindle-tilt or modified-roll mechanism, so that concave and convex flanks can be corrected independently. A cutter-alignment device keeps the stock division between the two flanks uniform, which matters because the slot is only a few tenths of a millimetre wide.
The gear pair and its workholding
The pair examined here is used on agricultural machinery. Both members are 20CrMo, cut at a 90° shaft angle with a nominal spiral angle of about 35°; the wheel is left-hand and the pinion right-hand. The wheel carries a non-symmetric pressure angle, 22.552° on the concave side and 17.813° on the convex side, while the pinion is symmetric at 20° and is profile-shifted by 0.36.
| Parameter | Wheel (gear) | Pinion |
|---|---|---|
| Module | 1.35 mm | 1.35 mm |
| Number of teeth | 38 | 11 |
| Shaft angle | 90° | 90° |
| Spiral angle | 35°1′11″, left hand | 35°, right hand |
| Pressure angle | 22.552° concave / 17.813° convex | 20° |
| Whole depth | 2.937 mm | 2.554 mm |
| Profile shift coefficient | — | 0.36 |
| Material | 20CrMo | 20CrMo |
| Pitch cone angle (derived) | ≈73.9° | ≈16.2° |
The pitch cone angles follow from the tooth numbers at a 90° shaft angle:
$$\delta_2=\arctan\!\left(\frac{z_2}{z_1}\right)=\arctan\!\left(\frac{38}{11}\right)\approx 73.9^\circ,\qquad \delta_1=90^\circ-\delta_2\approx 16.2^\circ$$
With u = 38/11 ≈ 3.45 and δ2 ≈ 73.9°, the pair sits inside the window in which the wheel flank can be plunge-cut. Workholding is a mandrel fixture; the blank must be supported over a sufficient area, since a small seat lets the blank deflect under the full-width plunge load and the error is written directly into the tooth depth.
Infeed scheduling for the plunge-cut wheel
A constant plunge rate leaves the cut badly unbalanced. The chip cross-section is set by how much of the blade edge is engaged at a given depth, so the first blade groups entering the slot remove thin slivers while the groups approaching full depth remove the largest sections. The heaviest group therefore sets the feed rate for the whole slot.
The numbers for this pair are as follows. The cutter head has eight blades arranged as four inner/outer groups, a point width W = 0.635 mm and an eccentric angle of 35.76°. At 360 r/min, that is 6 rev/s and 24 blade groups per second. A slot time of 6 s corresponds to 144 blade groups per slot, and distributing a plunge depth of about 2.4 mm over those groups gives
$$s=\frac{h}{N}=\frac{2.4\ \text{mm}}{144}\approx 0.017\ \text{mm per group}$$
Uniform distribution is not the optimum. If the infeed is ramped — fast at the start of the plunge where the engaged area is small, slowing toward full depth — and the heaviest group is allowed to reach only about 90% of the chip area it carries under constant feed, the slot time falls from 6 s to roughly 4.2 s per tooth while the load per group stays even. That is a cycle reduction of about 30%, obtained on the same machine with the same cutter head, purely from the feed schedule stored in the control. Roughing and finishing of the wheel are completed in the same plunge, and the schedule runs to the last slot.
Machine stiffness required by plunge cutting
Plunge cutting changes the load path. In generating, the cutting force is spread over the roll and the cradle; in plunge cutting the full width of the cutter head enters the slot at once and the thrust acts almost entirely along the work-spindle axis and the Z axis. On the small mechanical machine used for this work, the spindle housing overhangs the column, and finite element analysis of that housing identified the cutter-end deflection as the weak link rather than the bearing capacity itself.
The remedy is a double-wall housing with a support rib tying the spindle nose back to the column. The rib adds stiffness without the visual bulk of a thickened single wall, but on most gear-cutting machines of this size the rib cannot be placed on the underside because of the machine’s specification envelope, so it is carried on the side faces. Bearing selection is the second lever: the spindle is supported on tapered roller bearings or on a pair of angular-contact ball bearings arranged to take axial load in both directions, and the Z-axis ball screw is supported the same way. The third is lost motion — the worm gear pair driving the work spindle is adjusted to minimum backlash, because any clearance there appears directly as an indexing error and as a depth variation between slots.
Tooling and fixture condition are held to fixed limits: mandrel radial and face runout ≤ 0.005 mm, blade runout ≤ 0.012 mm, and fixture replacement every 200 parts or earlier if the runout drifts. These limits are tight for a module 1.35 mm tooth, but the whole depth tolerance of the plunge-cut wheel is of the same order.
Cutting-tool profile
The tool profile is computed from the conjugate rack profile rather than taken from a standard series, and is tabulated in three segments: the working flank (18 points), the tip (7 points) and the backed-off relief side (16 points). The working flank spans x = −0.2447 mm to −0.0122 mm and y = +0.0956 mm to −0.7089 mm relative to the datum used in the calculation. Selected ordinates of the working flank are listed below; the curve is used to dress the blade and to check the ground profile after coating.
| 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 |
Correcting the pinion for a plunge-cut mate
Removing generation from the wheel changes the mate. The wheel flank is no longer an envelope but a straight-sided profile in the profile direction, and the root produced by the plunge no longer lies on the root cone: the slot narrows from mid-face toward both ends of the tooth width and the local depth changes with it. A pinion generated against a conjugate wheel will therefore not mesh correctly, and the correction has to be applied on both flanks and in the depth taper along the face width.
The correction is not symmetric. Because the rolling direction differs between the two flanks, the concave and convex sides require corrections of opposite sign, and a single two-sided pass cannot deliver both. A second setup is equally unattractive: the small-module machines have no cutter-alignment device, and with a slot only a few tenths of a millimetre wide the remaining stock cannot be split evenly by eye or by touch.
The solution is two-direction rolling in one clamping. A two-sided cutter head first roughs and finishes one flank while only roughing the other. The workpiece is not withdrawn; the cutting parameters are changed in the control, and the head runs back through the same slot to finish the second flank. The work is then indexed and the sequence repeats until all slots are cut. Because the corrections on both flanks and the depth taper are generated from stored parameters, the servo axes reproduce the same correction on every tooth.
Two-direction rolling also allows the spiral angle to be corrected on the pinion. The nominal wheel spiral angle βc is defined at the mid-point of the tooth width, and the concave and convex flanks are generated at cutter radii Rμ + W/2 and Rμ − W/2 respectively. A duplex double-side process leaves the spiral angle uncorrected, so the lengthwise mismatch is fixed by the cutter-head setting alone. With the pinion cut in two directions, a spiral-angle correction Δβ can be superimposed on each flank, which moves the contact pattern along the face width and gives a better match to the plunge-cut wheel.
Verification on production parts
Three samples per trial were cut and checked on a rolling tester against a master gear, with the mounting distance set from the drawing and the contact pattern inspected at the mid-face position. Acceptance required the contact to sit in the middle of the face, centred on the convex side and slightly toward the small end on the concave side, together with the dimensional checks in the table.
| Characteristic | Drawing requirement | Measured | Result |
|---|---|---|---|
| Adjacent pitch variation | ≤ 0.015 mm | 0.01 mm | Pass |
| Total runout of the gear | ≤ 0.05 mm | 0.034 mm | Pass |
| Contact pattern | Mid-face, convex centred, concave toward small end | Matches master | Pass |
| Backlash | 0.10–0.12 mm | Within band | Pass |
The measured pitch and runout values sit inside roughly two-thirds of the permitted tolerance, which is consistent with the uniform stock division obtained from two-direction rolling in a single clamping. Backlash was held inside the 0.10–0.12 mm drawing band, and the contact pattern on the rolling tester reproduced the master-gear position on all three samples.
Limits and further gains
The method is bounded by its own assumptions. It applies to pairs with ratio ≥ 3 and wheel pitch cone angle ≥ 70°; outside that window the generating contribution on the wheel side is too large to be replaced by a straight-sided plunge, and the pinion correction would no longer close. It also requires a machine with enough axial stiffness to take the full-width plunge load, and a control that can store and reproduce a per-flank correction — conditions met by the Gleason No. 102 small-module machine with a 1:1 roll ratio for the wheel and a 30° spindle tilt for the pinion, and by adequately rigid CNC machines, but not by the mechanical small-module machines that offer only the duplex generating method.
The remaining gains are in the tool. Coated blades, or indexable carbide inserts in place of solid high-speed-steel blades, allow the cutting speed to be raised above the 360 r/min used here, and the ramp infeed schedule can then be tightened further. Together with the reduction from 6 s to about 4.2 s per tooth already obtained, that points to a cutting route in which the wheel is produced by a single plunge and the pinion carries the entire generating and correcting burden — a division of labour that shortens the cycle, lowers the energy per part, and puts the mesh quality where it can be controlled, on the generated member.