Small-module spiral bevel gears are compact, high-volume components used in office equipment, power tools, instruments, sewing machines, and small reducers. Their tooth slots are narrow, their unit value is low, and their production is usually organized around single-pass cutting. In a small-module pinion, the two flanks cannot be milled separately and then matched to a gear cut by the duplex spread-blade process. The practical route is to cut both members by duplex spread-blade, a combination often called double-duplex spread-blade. One cutter head cuts both flanks of a tooth slot simultaneously. Productivity is high, but the contact pattern is difficult to control because both members share one cutter number and the generating geometry has limited freedom.
Process Window and Historical Development
Duplex spread-blade cutting is commonly applied to spiral bevel gears with module below 2.5 mm and tooth count above 16. The method ensures that the two members have approximately the same spiral angle and pressure angle at the calculation point, but it cannot guarantee the same conditions over the whole flank. Because the gear and pinion use the same cutter number, the associated taper design has little freedom to avoid diagonal contact. The cutter number is therefore a strict constraint rather than a free design variable.
Gleason introduced the duplex spread-blade process for small-module spiral bevel gears in the 1940s on the G4 gear milling machine. The machine was later discontinued by Gleason, but the concept was copied and adapted by other machine builders. A Japanese version was designated HL-100B; the Shanghai Machine Tool Research Institute produced the YIS-10, and Suzhou Jinyi Machine Tool produced the FG4B. These machines are compact and mechanically ingenious, but they place high demands on the operator. The discontinuation of the G4 also meant the withdrawal of the simple duplex spread-blade design and process from Gleason’s product line, although the process remained widely used in Chinese production.
In the 1960s, Gleason developed a more mature small-module process called Duplex Helical, implemented on the No. 106 small-module gear milling machine. Duplex Helical is closely related to the Completing process, and Completing can be regarded as a later development of the same concept. Completing, also called the two-cut method in Chinese industry, cuts both gear and pinion in one duplex operation without separate roughing and finishing. It requires complex machine motion: cutter tilt, cutter rotation, and helical feed, usually realized by five-axis simultaneous CNC motion. This gives the cutter and workpiece arbitrary relative positioning, but the machine cost, tooling requirements, and process complexity are high.
For small-module spiral bevel gears, Completing is not yet an economical replacement for duplex spread-blade. First, the machine motion is complex and the five-axis equipment is expensive. Second, it is difficult to apply carbide insert trimming to a small integral cutter head of small module. Third, small-module spiral bevel gears are low-priced and are often used for unidirectional drives, so the economic case for a high-cost process is weak. The duplex spread-blade route therefore remains a rational choice for mass-produced small gears, provided that contact quality can be pre-controlled rather than left to shop-floor experience.
Geometric Condition at the Pitch Cone
At the calculation point, the gear and pinion must share the same spiral angle and pressure angle. In duplex spread-blade cutting, both flanks are generated by one cutter head, so the geometry of the pitch cone and the cutter radius must be matched. For a simplified pitch-cone model, assume the root angle \(\gamma=0\) and the shift coefficient \(\zeta=0\). The tooth-trace convergence angle at the pitch apex is
$$
\varphi_o=\angle P_iOP_e=\frac{P_iP_e}{L}=\frac{S_o}{L}
$$
where \(S_o\) is the tooth thickness, \(L\) is the midpoint cone distance, and \(P_i\), \(P_e\) are the inner and outer endpoints of the tooth trace. The corresponding convergence angle at the cutter center is
$$
\varphi_u=\angle P_iO_uP_e=\frac{P_iP_e\sin\beta}{r_u}=\frac{S_o\sin\beta}{r_u}
$$
where \(r_u\) is the nominal cutter radius and \(\beta\) is the midpoint spiral angle. Correct meshing would require \(\varphi_u=\varphi_o\). In practice the two angles differ, producing a spiral-angle deviation
$$
\Delta\beta=\varphi_o-\varphi_u=\frac{S_o}{L}-\frac{S_o\sin\beta}{r_u}
=\frac{S_o}{L}\left(1-\frac{L\sin\beta}{r_u}\right).
$$
The sign of \(\Delta\beta\) depends on the ratio \(L\sin\beta/r_u\). When \(L\sin\betar_u\), it is negative. A standard way to correct the deviation is to tilt the pitch cone by the root angle. The tilt changes the distance between the tangent lines to the convex and concave tooth traces. At a section a distance \(x\) from the midpoint, the distance changes by \(2x\tan\gamma\tan\alpha\), where \(\alpha\) is the pressure angle. The resulting correction is
$$
\Delta\beta\approx 2\tan\gamma\tan\alpha\cos\beta.
$$
Combining the two expressions for \(\Delta\beta\) gives the root-angle condition
$$
\tan\gamma\approx\gamma=
\frac{S_o}{2L\tan\alpha\cos\beta}
\left(1-\frac{L\sin\beta}{r_u}\right).
$$
For a gear pair, the sum of the two root angles is therefore
$$
\gamma_1+\gamma_2=
\frac{S_1+S_2}{2L\tan\alpha\cos\beta}
\left(1-\frac{L\sin\beta}{r_u}\right).
$$
This sum determines the cutter number required for the pair. Because the gear and pinion share the same cutter number, the corrected root angles and the cutter number must be recalculated together. The classical calculation route, developed in the 1950s, is still used in shop-floor adjustment, but its output is not visually intuitive. Contact control therefore depends heavily on experienced adjusters unless a numerical contact analysis is introduced.
TCA Correction of a 30/30 Pair
Tooth contact analysis, or TCA, builds a meshing coordinate system for the gear pair, solves the contact points under the generated tooth surfaces, and outputs the contact imprint position and width, the contact path, and the shape and amplitude of the transmission error curve. TCA is an unloaded numerical simulation, but it is effective for checking whether a set of machine settings will produce a favorable contact pattern before the first cut. It can also be used iteratively to optimize the generating parameters.
A representative case is a 30/30 spiral bevel gear pair with module 1.75 mm, face width 11 mm, shaft angle 90°, pressure angle 20°, midpoint spiral angle 35°, addendum coefficient 0.85, clearance coefficient 0.2, and zero profile and tangential shifts. The pinion is right-hand and the gear is left-hand. The main blank geometry is listed below.
| 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° |
| Midpoint spiral angle | 35° | 35° |
| Addendum coefficient | 0.85 | 0.85 |
| Clearance coefficient | 0.2 | 0.2 |
| Profile shift coefficient | 0 | 0 |
| Tangential shift coefficient | 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-hand | Left-hand |
The initial cutter and generating parameters used a cutter diameter of 50.8 mm, cutter spacing of 0.75 mm, cutter number 7.3, machine setting angle 42.821°, radial cutter position 26.903 mm, angular cutter position of magnitude 50.659°, and cutting ratio 1.4142. The vertical wheel position, horizontal wheel correction, and bed position were zero. TCA of the initial settings showed a contact pattern slightly toward the small end and an asymmetric transmission error curve. The contact was close to acceptable, but it was not centered and the transmission error was not balanced.
| Parameter | Initial | TCA-corrected |
|---|---|---|
| Cutter diameter | 50.8 mm | 50.8 mm |
| Cutter spacing | 0.75 mm | 0.75 mm |
| Cutter number | 7.3 | 6 |
| Machine setting angle | 42.821° | 43.2° |
| Radial cutter position | 26.903 mm | 26.903 mm |
| Angular cutter position | ±50.659° | ±50.659° |
| Cutting ratio | 1.4142 | 1.4142 |
| Vertical wheel position | 0 | 0 |
| Horizontal wheel correction | 0 | 0 |
| Bed position | 0 | 0 |
The correction was made by increasing the machine setting angle by 0.379° and reducing the root angle by the same amount. The cutter number was reduced from 7.3 to 6. After recalculation, TCA showed a favorably positioned contact pattern and a nearly symmetric transmission error curve. The correction used the same single-pass duplex spread-blade strategy, so the productivity advantage of the original process was retained.
Exact 3D Modeling and Contact Simulation
Once the TCA-corrected generating parameters are known, the tooth surface can be modeled exactly. The cutter blade edge defines a cutting cone as the cutter head rotates. By establishing the workpiece and cutter coordinate systems and applying the corresponding transformation matrices, a set of points on the cutting cone can be mapped to the generated tooth surface. The resulting point cloud is imported into a CAD environment such as UG and fitted into a surface patch. The model follows the actual generating motion rather than an approximate geometric construction, so it is theoretically identical to the real tooth surface when machining errors are absent.
Such a model is useful beyond visualization. It can provide reference points for tooth-surface deviation measurement on a gear measuring center, supply a geometry for finite element strength analysis, support meshing and motion simulation, and provide the tooth surface for mold forming. The same exact surface can be produced by mold forming or by conventional milling because both are based on the same generating geometry.
For contact simulation, the gear and pinion models are assembled at the correct mounting distance with the pitch-cone apexes coincident. An interference of 0.00635 mm is imposed between the two tooth surfaces to obtain an approximate contact imprint. This is not a loaded contact analysis; actual loaded contact is more complex. The interference simulation is nevertheless useful for checking the size and location of the contact zone, the backlash, the presence of interference, and the risk of edge contact. For the 30/30 pair, the simulated imprint was consistent with the TCA result, which confirms that TCA is a practical tool for pre-control of the contact pattern.
In a motion simulation environment, the pinion can be driven at constant speed about its own axis while the gear follows the meshing motion. This allows the alternating contact between the two flanks to be observed and provides a further check of the assembly and the generated tooth surfaces.
Milling Trial and Measurement
Milling trials used blanks and a fixture prepared according to the 30/30 geometry. The pinion was cut with the TCA-corrected duplex spread-blade parameters. The resulting gear pair was checked on a bevel gear rolling tester to inspect the contact pattern. Tooth-surface deviation and pitch error were also measured. According to DIN 3695/86, the pitch accuracy of both flanks reached grade 6–7. The trial therefore supports the feasibility of the proposed cutting theory: a process that retains the efficiency of duplex spread-blade cutting can produce a contact pattern that is consistent with the TCA prediction.
Implications and Limits
The main value of TCA-assisted duplex spread-blade cutting is that it separates contact quality from operator experience. The initial geometric calculation gives a starting point, but it is approximate and does not directly show the contact pattern. TCA checks the initial settings, identifies the direction of correction, and adjusts the machine setting angle, root angle, and cutter number before cutting. This shortens the trial cycle, reduces trial cost, and makes the process more repeatable for high-volume small-module spiral bevel gears.
The approach is not a universal replacement for advanced Completing. It is a cost-effective compromise for small-module gears that are produced in large batches, often for unidirectional drives. The 30/30 case is a specific geometry; other tooth ratios, modules, and shaft angles require recalculation of the root angles, cutter number, and machine settings. TCA also addresses unloaded meshing only. Loaded contact, strength, and durability still require loaded tooth contact analysis, finite element analysis, and physical bench testing. Within these limits, TCA-assisted duplex spread-blade cutting offers a practical route to better contact patterns and more stable production of small-module spiral bevel gears.