Small-module spiral bevel gears, generally with modules below 2.5 mm, are used in power tools, office equipment, sewing machines, instruments, and small reducers. Production runs are large and unit prices are low, so the manufacturing route has to be fast and inexpensive. The curved tooth flank is nevertheless sensitive to small setting errors: a contact pattern that drifts toward the tooth ends raises noise, concentrates stress, and shortens life.
These requirements conflict. The established route for small modules is duplex spread-blade cutting. One cutter head cuts both flanks of a tooth slot in a single generating pass, and the same cutter number is used for the pinion and the gear. The process is efficient, but its closed-form setting calculation only matches the spiral angle and pressure angle at the calculation point. Away from that point the flanks deviate, and the contact pattern can move off center. Correction has traditionally depended on experienced operators. Tooth contact analysis (TCA) offers a numerical alternative: calculate the initial settings, predict the contact pattern and transmission error, and adjust the machine settings before the first cut. The case below shows how TCA can correct the settings before the first cut and how the corrected surface can be carried into a solid model and a milling trial.
The duplex spread-blade route and its process window
The method dates to the 1940s, when Gleason introduced the G4 mechanical machine for small-module spiral bevel gears. The G4 and its copies—the HL-100B built in Japan, the YIS-10 from the Shanghai Machine Tool Research Institute, and the FG4B from Suzhou Jinyi Machine Tool—share a compact mechanical arrangement: a generating roll, a cutter head with inner and outer blades, and hand-set adjustments. Gleason later discontinued the G4 and introduced the duplex helical method on the No. 106 small-module machine in the 1960s. Duplex helical adds a helical feed synchronized with the generating roll, and it is closely related to the completing process now used for medium and large modules.
Whatever the machine generation, the duplex spread-blade principle is the same. The cutter head carries inner and outer blades. The blades are set to the full tooth depth, and the generating roll carries the cutter through the slot so that both flanks are formed at once. Because one cutter number serves both members, the blade pressure angle is a compromise between pinion and gear. Tooth thickness is controlled by the cutter offset and the machine settings. The double-contraction design used to match the two members tends to produce diagonal contact, and it places tight constraints on the cutter number. A small change in the root-angle sum shifts the contact across the flank.
The process window is narrow. A 1958 Soviet monograph on bevel-gear cutting machines described the method for modules below 2.5 mm and tooth counts above 16. In this range the tooth slot is narrow. Cutting the two flanks separately by a single-flank method would require a second setup and precise cutter alignment in a slot that is difficult to reach; duplex spread-blade cutting avoids that second operation.
Completing is not a drop-in replacement for small modules. It requires cutter tilt and swivel plus a helical feed, typically on a five-axis computer numerical control (CNC) machine, and the machine cost is high. Carbide insert blades with profile relief are difficult to scale down to the small integral cutter heads used for small modules. Small-module gears are also low-priced and often transmit in one direction only, so the economics of completing are unfavorable. The duplex spread-blade route remains the practical choice for high-volume small-module production.
| Feature | Duplex spread-blade | Completing / duplex helical |
|---|---|---|
| Cutter | One duplex head; both flanks of a slot cut simultaneously | One duplex head; both flanks cut in one pass |
| Generating motion | Generating roll; mechanical machine settings | Generating roll plus cutter tilt and helical feed; five-axis CNC |
| Typical machines | G4, HL-100B, YIS-10, FG4B | No. 106 for duplex helical; five-axis CNC for completing |
| Tooling | Small integral head; limited blade profiling | Insert-type carbide blades with profile relief |
| Efficiency | High; one pass per slot | High; one pass per slot |
| Main constraint | Contact correction by settings; sensitive to cutter number | Machine and tooling cost; small-size tooling difficult |
| Typical application | Small module, high volume, cost-sensitive | Medium and large module; small module uneconomical |
Geometry of the spiral-angle mismatch
Consider the pitch cone at the mean cone distance L. Let So be the circular tooth thickness at that distance, ru the nominal cutter radius, β the mean spiral angle, and α the pressure angle. The tooth trace converges toward the apex at the angle φo = So/L. The normal to the tooth trace converges at φu = So sinβ/ru. For the two flanks to mesh with the intended spiral angle, these convergence angles should be equal. In general they are not, and the difference is
$$
\Delta\beta=\varphi_o-\varphi_u=\frac{S_o}{L}\left(1-\frac{L\sin\beta}{r_u}\right).
$$
This expression gives the sign of the deviation: when L sinβ < ru, Δβ is positive; when L sinβ > ru, it is negative. A positive deviation means that the actual spiral angle at the ends is larger than the nominal value on one flank and smaller on the other. The result is a contact pattern that is not centered.
Traditional practice corrects the deviation by tilting the pitch cone through the root angle γ, defined as the angle between the pitch cone and the root cone rather than the absolute root-cone angle measured from the axis. The tilt changes the tooth thickness at a section x from the mean by 2x tanγ tanα. Since the convergence angle is the thickness change divided by x and projected by cosβ, the correction is
$$
\Delta\beta\approx 2\tan\gamma\tan\alpha\cos\beta.
$$
Setting the geometric deviation equal to the tilt correction gives
$$
\tan\gamma\approx\gamma=\frac{S_o}{2L\tan\alpha\cos\beta}\left(1-\frac{L}{r_u}\sin\beta\right).
$$
Summing for pinion and gear gives
$$
\gamma_1+\gamma_2=\frac{S_1+S_2}{2L\tan\alpha\cos\beta}\left(1-\frac{L}{r_u}\sin\beta\right).
$$
The cutter number is then set from the sum of root angles:
$$
N_o=\frac{(\gamma_1+\gamma_2)\sin\beta}{20}.
$$
Here γ1+γ2 is in minutes and No is the cutter-number designation, quoted in degrees in the setting table. The relation shows why the cutter number is not a free variable: it is tied to the root-angle sum, which in turn is tied to the tooth thickness and the spiral angle. A change in mounting angle that corrects the contact position also changes the root angle and therefore requires a new cutter number.
These closed-form relations are fast and have been used since the 1950s. Their weakness is that they do not predict the contact pattern. They ensure the calculation point, but the shape of the contact ellipse, the position of the contact path, and the transmission error remain unknown until the gear is cut and rolled. TCA fills that gap. It solves the meshing equations for the two generated surfaces under the assembly constraints and returns the contact path, contact width, and transmission-error curve.
TCA correction of a 30/30 pair
Table 1 lists the geometry of the test pair: a 30/30, module 1.75 mm, face width 11 mm, shaft angle 90°, pressure angle 20°, mean spiral angle 35°, addendum coefficient 0.85, clearance 0.2, no height or tangential modification, pitch cone angle 45°, face cone angle 47.034°, root cone angle 42.966°, tip diameter 54.495 mm, crown distance 25.252 mm, and outer cone distance 37.123 mm. The pinion is right-hand and the gear left-hand.
| Parameter | Pinion | Gear |
|---|---|---|
| Number of teeth | 30 | 30 |
| Module | 1.75 mm | 1.75 mm |
| Face width | 11 mm | 11 mm |
| Shaft angle | 90° | 90° |
| Pressure angle | 20° | 20° |
| Mean spiral angle | 35° | 35° |
| Addendum coefficient | 0.85 | 0.85 |
| Clearance coefficient | 0.2 | 0.2 |
| Height modification | 0 | 0 |
| Tangential modification | 0 | 0 |
| Pitch cone angle | 45° | 45° |
| Face cone angle | 47.034° | 47.034° |
| Root cone angle | 42.966° | 42.966° |
| Tip diameter | 54.495 mm | 54.495 mm |
| Crown distance | 25.252 mm | 25.252 mm |
| Outer cone distance | 37.123 mm | 37.123 mm |
| Hand | Right | Left |
Table 2 lists the initial cutter and generating settings. The cutter diameter is 50.8 mm, so ru = 25.4 mm. The cutter offset is 0.75 mm, the cutter number is 7.3°, the machine mounting angle is 42.821°, the radial cutter position is 26.903 mm, the angular cutter position is ±50.659°, the cutting roll ratio is 1.4142, and the vertical, horizontal, and bed corrections are zero. The sign of the angular cutter position is opposite for pinion and gear.
| Parameter | Initial | Corrected |
|---|---|---|
| Cutter diameter | 50.8 mm | 50.8 mm |
| Cutter offset | 0.75 mm | 0.75 mm |
| Cutter number | 7.3° | 6° |
| Machine mounting angle | 42.821° | 43.2° |
| Radial cutter position | 26.903 mm | 26.903 mm |
| Angular cutter position | +50.659° (gear), −50.659° (pinion) | +50.659° (gear), −50.659° (pinion) |
| Cutting roll ratio | 1.4142 | 1.4142 |
| Vertical wheel position | 0 | 0 |
| Horizontal correction | 0 | 0 |
| Bed position | 0 | 0 |
TCA of the initial settings predicted a contact pattern slightly toward the small end and an asymmetric transmission-error curve. The calculation point was acceptable, but the contact did not meet the intended centered pattern. The correction was made in two linked steps. First, the machine mounting angle was increased by 0.379° to move the contact toward the large end. Because the mounting angle is tied to the root angle, this reduced each root angle by 0.379°. Second, the cutter number was recalculated from the new root-angle sum. The sum decreased by 0.758°, and with sin 35° ≈ 0.574 the cutter number fell by about 1.3, from 7.3 to 6. The other settings were left unchanged.
After correction, the contact pattern moved to the central part of the flank and the transmission-error curve became nearly symmetric. Table 3 summarizes the change. A machine-setting change of less than four-tenths of a degree was enough to shift the contact from the small end to the center, and the cutter number had to be adjusted with it. The correction is a pre-correction: it is applied to the theoretical settings before the first cut, not by trial and error on the machine. The example is a 1:1 pair; other tooth combinations follow the same calculation but need their own TCA correction.
| Item | Initial | Corrected |
|---|---|---|
| Contact position | Slightly toward the small end | Central region |
| Transmission-error curve | Asymmetric | Nearly symmetric |
| Machine mounting angle | 42.821° | 43.200° |
| Root-angle change | — | −0.379° |
| Cutter number | 7.3° | 6° |
Generated surface, solid model, and contact simulation
The same generating parameters define the tooth surface. The cutter blade is a cone; as the cutter head rotates and the workpiece rolls, the blade family envelopes the tooth surface. A coordinate transformation from the cutter frame to the workpiece frame gives the surface points. Sampling the blade edge and the generating motion produces a point cloud. Imported into a computer-aided design (CAD) environment such as UG, the point cloud is fitted into a surface patch, and the patches are trimmed to the tooth boundaries. The result is a solid model of the gear pair.
Because the model is generated from the same settings used in TCA, it is consistent with the analyzed surface by construction. It can serve as the reference for tooth-flank deviation measurement, finite-element strength analysis, meshing simulation, and mold manufacture. The point cloud can also be written in the format required by a gear measuring center, so the theoretical points become the nominal grid for deviation maps.
For a visual check, the pinion and gear are assembled at the correct mounting distance with the pitch cone apexes coincident, and the two flanks are given a numerical interference of 0.00635 mm. The overlap region is displayed as the contact patch. This is not a load simulation; it is a geometric way to visualize the unloaded contact area. The patch obtained in this way agreed with the TCA prediction, which confirms that the solid model and the TCA model describe the same surface. A rigid-body motion simulation can then run the pair at a constant pinion speed and show how the contact shifts from one tooth pair to the next. Loaded contact, load sharing, and tooth-root stress still require a separate loaded tooth contact analysis (LTCA) or finite-element model.
Milling trial and measurement
A milling trial was run with the geometry of Table 1 and the corrected pinion settings of Table 2. The blank and fixture were prepared to the table dimensions, and the pinion was cut by the duplex spread-blade method. The pair was checked on a bevel-gear rolling tester. The contact pattern after cutting was located in the central region and did not run off the ends, matching the TCA prediction.
Tooth-flank deviation and pitch were measured on a gear measuring center. According to DIN 3695/86, the pitch accuracy of both flanks was grade 6–7. This is consistent with a small-module gear cut with corrected mechanical settings. The trial supports the feasibility of the TCA-based pre-correction, but it covers one 30/30 pair; it does not establish a general process capability. Wear, noise, and fatigue were not part of the trial.
What the approach changes—and what it does not
The value of the approach is that it separates two issues that are often mixed in production: the efficiency of the duplex spread-blade cut and the accuracy of its settings. The cut itself remains a one-pass, two-flank operation. The settings are no longer chosen only by closed-form geometry and operator experience; they are screened by TCA before the first cut. This reduces the number of trial cuts, shortens lead time, and makes the result less dependent on the skill of the adjuster.
The approach also has clear limits. TCA is an unloaded, rigid-body analysis. It predicts the contact pattern and transmission error of the theoretical surface, not the effect of machine errors, cutter wear, heat treatment, or load deflection. The 0.00635 mm interference model is a visualization aid, not a substitute for loaded contact analysis. The trial is a single pair, and long-term consistency still depends on machine condition and tool maintenance.
From a process-economics view, completing remains attractive for medium and large modules, where five-axis machines and insert tooling are justified. For small modules, machine cost, the difficulty of profiling small insert blades, and the low unit price of the gear keep the duplex spread-blade route competitive. The practical path is therefore to improve the setting calculation and the pre-correction, not to replace the machine. Numerical simulation and helical-feed correction on CNC machines are natural extensions: the same geometric model can optimize the settings, and the machine can execute the corrected motion.
For small-module spiral bevel gears, duplex spread-blade cutting is not likely to disappear soon. Its efficiency is intrinsic; its weakness is the lack of a direct link between the setting numbers and the contact pattern. TCA supplies that link, and the generated solid model carries it into measurement, analysis, and tooling. The 30/30 case shows that a sub-degree change in mounting angle, coupled with a recalculated cutter number, can move the contact from the small end to the center while keeping the one-pass process. The practical gain is not a new cutting principle but a more reliable way to set the existing one.