Why small-module spiral bevel gears are still cut wet
Spiral bevel gears transmit rotation between intersecting shafts with a contact ratio and load capacity that spur bevel pairs cannot match, and in the small-module range (roughly 1–3 mm) they sit inside the final drive trains of robotics, medical equipment, instruments and, increasingly, the actuation units of intelligent consumer hardware. As the number of such drives has grown, so has the demand for small-module pairs with tighter pitch and flank tolerances.
Milling these gears conventionally requires a generous flow of cutting fluid. The fluid carries away heat, flushes chips and extends tool life, but it also imposes costs that scale badly with volume: coolant purchase, filtration and disposal, mist extraction, workpiece cleaning, and the environmental burden of oil-contaminated swarf. Dry cutting removes the fluid entirely. The spindle runs at high speed with high feed, the chip leaves with most of the heat it generated, and the process becomes cleaner, cheaper per part and — because there is no fluid film or thermal shock — often more accurate on the flank.
The penalty is that every element of the process chain has to be upgraded. Dry cutting of gears demands machine tools with high static and dynamic stiffness, thermally stable spindles, fast chip evacuation and tool materials that retain hardness at the temperatures reached in the cutting zone. The technology matured first for medium and large modules. In 1998 Klingelnberg-Oerlikon built a dry-cutting bevel gear milling machine; in 2003 Gleason presented the Phoenix II CNC bevel gear cutting machine, which raised productivity and improved flank accuracy; Mitsubishi introduced a dry hobbing system operating at a cutting speed of 200 m/min and put it into production.
In China, research on dry gear cutting started in the 1990s and reached the machine-tool market in 2012, when Changsha Zhongda Chuangyuan launched the YKA2260, the country’s first fully CNC bevel gear dry cutting machine, able to cut both equal-height and tapered teeth; Tianjin Jingcheng Machine Tool Works developed the JCB-20 fully CNC spiral bevel gear dry cutting machine. By 2021 dry cutting of medium- and large-module spiral bevel gears was an established practice in Chinese production. Small-module milling, by contrast, remained almost entirely wet, for a simple economic reason: because the parts are small, the machines built for them were deliberately simple and inexpensive, and such machines cannot support high-speed dry cutting. That constraint has now relaxed. Direct-drive work and cutter spindles on small-module CNC gear milling machines from Tianjin Jingcheng and Zhongshan Maileite give the stiffness and accuracy the process needs. What is still missing is tooling — no coated carbide cutter system has been developed specifically for small-module dry milling — and a quantitative description of the cutting mechanics at that scale.
Oblique cutting as the correct kinematic description
Spiral bevel gears are cut either by the formate (forming) method or by the generating method. In formate cutting the wheel blank is held stationary while the cutter head rotates; the blade profile is reproduced directly in the tooth space, and the method is normally reserved for the large member of the pair. In generating cutting the cutter head and the blank rotate together in a fixed ratio, and the tooth flank is the envelope of successive blade positions. Neither case is a two-dimensional orthogonal cut: the cutting edge is inclined to the cutting velocity direction, and that inclination changes continuously along the path of the blade through the tooth space. The process is therefore a three-dimensional oblique cutting operation, and treating it as orthogonal cutting — the usual simplification, well supported by theory and finite element practice — misrepresents the force system that the tool and the machine actually see.
Descriptions of the shear zone have evolved through four stages. Merchant’s single-plane model (1944) treated the shear plane as having no thickness. Oxley and Welsh introduced a parallel-boundary shear zone with finite thickness, for which Shaw et al. gave a representative value of about 0.025 mm. Zorev proposed a fan-shaped shear zone. Astakhov et al. then split the parallel-boundary zone into two sub-layers of unequal thickness — a wider region from the tool tip, 0 ≤ y ≤ kh, and a narrower region, kh ≤ y ≤ h — and this non-equipartition model is the one that best matches high-speed dry cutting, where the strain and temperature gradients across the zone are steep.
For the geometry, the cutting velocity V, chip velocity Vc and shear velocity Vs form a velocity triangle, and the plane containing V and Vc — the equivalent plane — is used to reduce the three-dimensional problem to a tractable two-dimensional one. Working in this plane with inclination angle λ, normal rake angle α, shear-flow angle ηs, chip-flow angle ηc and normal shear angle φ gives the maximum shear strain rate in the zone as
$$\dot{\gamma}_m=\frac{2V\cos\lambda\cos\alpha}{h\cos\eta_s\cos(\phi-\alpha)}$$
and the strain distribution across the two sub-layers as
$$\gamma=\begin{cases}\dfrac{\dot{\gamma}_m\cos\eta_e\,y}{2khV\cos\lambda\sin\phi}, & y\in[0,\,kh]\\[8pt] \dfrac{\dot{\gamma}_m\cos\eta_e\left(-y^{2}+2ky-h^{2}\right)}{2(1-k)hV\cos\lambda\sin\phi}, & y\in[kh,\,h]\end{cases}$$
where ηe is the equivalent-plane angle, h the shear-zone thickness and k the non-equipartition coefficient.
Shear-zone stress and the cutting force components
Taking the shear stress τ as uniform over the primary shear plane, the shear force is obtained directly from the shear-plane area:
$$F_s=\tau A_{sh}=\frac{\tau\,b\,t_h}{\cos\lambda\sin\phi}$$
with b the width of cut and th the undeformed chip thickness. The normal force acting perpendicular to the primary shear plane includes the friction contribution through the mean friction angle β, and the chip-flow angle:
$$F_{no}=\cos\eta_s\,\frac{\tan(\phi-\alpha)+\tan\beta\cos\eta_c}{1-\tan\beta\cos\eta_c\tan(\phi-\alpha)}\,F_s$$
The resultant of Fs and Fno is then projected onto three orthogonal directions to give the main cutting force, the feed force and the thrust (back) force. The projection coefficients are direction cosines built from φ, ηs and λ, so the model predicts the three components independently of any empirical calibration. One further condition governs the high-speed regime: below about 90 m/min the inertia of the chip can be neglected, but above that threshold the momentum required to accelerate the chip out of the shear zone adds to the force on the shear plane and must be carried in the model. Since dry cutting runs at several hundred metres per minute, this term is always active.
Gear pair, cutter head and machine settings
The demonstration case is a 1:1 small-module spiral bevel pair with 30 teeth on each member, module 1.75 mm, shaft angle 90° and pressure angle 20°. Geometry and machining data are listed below; they define the blank, the cutter head and the machine setting used both in the finite element model and in the cutting trial.
| Parameter | Wheel (large member) | Pinion (small member) |
|---|---|---|
| Number of teeth | 30 | 30 |
| Hand of spiral | Right | Left |
| Module (mm) | 1.75 | 1.75 |
| Shaft angle (°) | 90 | 90 |
| Pressure angle (°) | 20 | 20 |
| Mid-point spiral angle (°) | 35 | 35 |
| Addendum coefficient | 0.85 | 0.85 |
| Clearance coefficient | 0.188 | 0.188 |
| Height modification coefficient | 0 | 0 |
| Tangential modification coefficient | 0 | 0 |
| Face cone angle (°) | 47.183 | 47.183 |
| Pitch cone angle (°) | 45 | 45 |
| Root cone angle (°) | 42.826 | 42.816 |
| Tip diameter (mm) | 54.207 | 54.207 |
| Crown-to-back distance (mm) | 25.24 | 25.24 |
| Outer cone distance (mm) | 37.123 | 37.123 |
| Machining parameter | Wheel | Pinion |
|---|---|---|
| Cutter head diameter (mm) | 50.8 | 50.8 |
| Cutter blade spacing (mm) | 0.81 | 0.81 |
| Blade designation (°) | 7.5 | — |
| Machine setting angle (°) | 42.9 | — |
| Radial cutter position (mm) | 26.34 | 26.34 |
| Angular cutter position (°) | 52.18 | −52.18 |
| Cutting roll ratio | 1.4142 | 1.4142 |
| Vertical wheel offset (mm) | 0 | 0 |
| Horizontal wheel correction (mm) | 0 | 0 |
| Bed correction (mm) | −0.6 | −0.6 |
Because the blade path repeats identically for every tooth space, the milling process can be represented by a single oblique cut of one blade through the blank, indexed by one tooth pitch for the next slot. That reduction is what makes a three-dimensional finite element analysis of the process affordable.
Constructing the finite element model
The workpiece is quenched and tempered 45 steel and the tool is YT15 cemented carbide; the contrast in hardness is large enough that the tool is treated as a rigid body, its elastic deflection being negligible against the plastic flow of the workpiece. The blank is constrained along its bottom and left edges to reproduce the fixturing, both bodies are meshed with quadrilateral elements, and the mesh is refined in the cutting zone and coarsened away from it. Chip formation is allowed to develop into a C-shaped curl, which favours breaking and evacuation and reduces the hazard of long chips to the operator.
| Property | Tool: YT15 carbide | Workpiece: 45 steel |
|---|---|---|
| Density (kg/m³) | 11 500 | 7 890 |
| Elastic modulus (GPa) | 530 | 209 |
| Poisson ratio | 0.3 | 0.269 |
| Specific heat (J/(kg·K)) | 480 | 480 |
| Thermal conductivity (W/(m·K)) | 50 | 50 |
| Inelastic heat fraction | 0.9 | 0.9 |
Contact between chip and rake face is modelled with a sticking–sliding law, in which the shear stress saturates at the material’s flow shear stress once the frictional stress reaches it:
$$\tau=\begin{cases}\mu\sigma_n, & \mu\sigma_n\le\tau_{max}\\ \tau_{max}, & \mu\sigma_n\ge\tau_{max}\end{cases}$$
with μ the friction coefficient, σn the contact normal stress and τmax the maximum shear stress. The constitutive response of the workpiece follows the Johnson–Cook formulation, which couples strain hardening, strain-rate sensitivity and thermal softening — the three effects that dominate when neither coolant nor lubricant is present:
$$\sigma=\left[A+B\bar{\varepsilon}^{\,n}\right]\left[1+C\ln\frac{\dot{\bar{\varepsilon}}}{\dot{\bar{\varepsilon}}_0}\right]\left[1-\left(\frac{T-T_r}{T_m-T_r}\right)^{m}\right]$$
where σ is the equivalent stress, ε̄ the equivalent plastic strain, ε̄̇ the plastic strain rate, ε̄̇0 the reference plastic strain rate (1.0 s⁻¹), T the current temperature, Tm the melting point of the workpiece and Tr room temperature; A is the yield stress, B the hardening modulus, C the strain-rate sensitivity, n the strain-hardening exponent and m the thermal-softening exponent.
| Johnson–Cook parameter | A (MPa) | B (MPa) | n | C | m |
|---|---|---|---|---|---|
| 45 steel | 507 | 320 | 0.28 | 0.28 | 1.06 |
Chip separation uses a physical rather than a geometric criterion, so that the stress and force history are governed by material behaviour instead of by a pre-set distance. Damage accumulates according to
$$\omega=\sum_{j=1}^{n}\frac{\Delta\bar{\varepsilon}_{pl}}{\bar{\varepsilon}_{pl}^{\,f}}$$
where Δε̄pl is the equivalent plastic strain increment and ε̄plf the critical equivalent plastic strain; the material fails and the chip separates once ω exceeds unity. The critical strain depends on stress triaxiality, strain rate and temperature:
$$\bar{\varepsilon}_{pl}^{\,f}=\left[d_1+d_2\exp\left(d_3\frac{p}{q}\right)\right]\left[1+d_4\ln\frac{\dot{\bar{\varepsilon}}}{\dot{\bar{\varepsilon}}_0}\right]\left[1+d_5\left(\frac{T-T_r}{T_m-T_r}\right)\right]^{m}$$
with p/q the ratio of compressive to deviatoric stress and d1–d5 the failure parameters.
| Failure parameter | d1 | d2 | d3 | d4 | d5 |
|---|---|---|---|---|---|
| 45 steel | 0.10 | 0.76 | 1.57 | 0.005 | −0.84 |
Simulated influence of feed, inclination angle and cutting speed
With the model assembled, three process variables were swept one at a time under dry conditions. Increasing the feed from 0.2 mm to 0.4, 0.6 and 0.8 mm at a cutting speed of 300 m/min and an inclination angle of 5° produced a clear and substantial rise in the main cutting force: the uncut chip section grows almost in proportion to feed, and the shear-plane area with it.
Varying the inclination angle over 0°, 5°, 10°, 15° and 20° at 300 m/min and 0.2 mm feed gave a markedly different response. The main cutting force and the feed force changed only slightly, while the thrust force — the component normal to the machined surface — grew steadily with λ. This asymmetry is exactly what the direction-cosine structure of the oblique cutting force model predicts, and it matters practically: the thrust component loads the blank and the fixture rather than the spindle, and on a small-module tooth it is the component most likely to deflect the work and bias the flank.
Sweeping the cutting speed at fixed feed (0.2 mm) and inclination (5°) showed all three force components falling as speed rose. The trend is consistent with the thermal softening term of the Johnson–Cook model outweighing the strain-rate hardening term once the cutting zone is hot enough, and it is the mechanical basis for the productivity argument in favour of dry cutting: within this window, higher speed does not cost force.
Dry cutting trial and gear inspection
The trial pair was cut by the double-duplex method, in which the convex and concave flanks of both wheel and pinion are generated in a single plunge of a duplex cutter head — the most productive of the bevel gear cutting cycles. Blanks and fixture were prepared to the geometry above, the machining parameters were loaded into the CNC system, the blank was mounted and aligned on the work spindle, and the teeth were cut with no coolant and no lubricant.
Meshing quality was checked on a dedicated small-module rolling tester. A film of red lead roughly 0.00635 mm thick was spread evenly on the working flanks — the convex flank of the wheel and the concave flank of the pinion — and the pair was rolled at its specified mounting distance. The contact mark appeared as an ellipse centred on the mid-face of the tooth, which is the intended contact pattern for a pair of this size; had the pattern been displaced towards the toe, the heel or the root, the machine settings would have been corrected and the pair re-cut.
Tooth accuracy was then measured on a Gleason 650 gear measuring machine and evaluated against DIN 3965/86.
| Flank | Total cumulative pitch deviation (μm) | Grade | Single pitch deviation (μm) | Grade | Adjacent pitch deviation (μm) | Grade |
|---|---|---|---|---|---|---|
| Convex | 46.0 | 8 | 9.0 | 6 | 7.1 | 5 |
| Concave | 32.9 | 7 | 6.0 | 5 | 7.8 | 5 |
The single and adjacent pitch deviations fall in grades 5–6 on both flanks, while the cumulative deviation is coarser, at grade 7 on the concave flank and grade 8 on the convex flank. The spread between the short-range and long-range measures is the signature of a thermally loaded process: local tooth-to-tooth geometry, which is set within a single indexing cycle, comes out tight, whereas the accumulated error over the full circumference reflects the gradual growth and drift of the machine and workpiece during the cut. Coolant would normally suppress that drift; in dry cutting it has to be handled by machine thermal design and by the cutting sequence. Even so, the accuracy achieved is sufficient for the small-module drive trains the process targets.
What the results establish, and what they do not
Three conclusions follow from the combination of mechanics, simulation and trial. First, oblique cutting is the appropriate framework for spiral bevel gear milling: because the inclination angle varies continuously through the cut, an orthogonal model cannot represent the thrust component that grows with λ, and the simulation reproduces the direction-cosine trend of the analytical force model rather than merely a monotonic force rise. Second, the shear-zone description, the sticking–sliding friction law, the Johnson–Cook constitutive model and a physical chip separation criterion together form a workable prediction chain for a process in which no coolant carries heat away. Third, and most directly, a 30-tooth, module 1.75 mm pair can be milled dry with a YT15 carbide cutter to DIN grades 5–6 on single and adjacent pitch, with an acceptable mid-face contact ellipse.
The evidence should not be stretched further than that. The comparison between simulated and published forces was qualitative — the trends agreed, the magnitudes were not reported as a calibrated error band — so the model is validated as a trend predictor, not as a quantitative design tool. Tool wear, tool life and cost per part, which decide whether dry cutting actually displaces wet cutting in a small-module shop, were not measured here; neither was the thermal drift of the machine over a full production run, although the cumulative pitch results point to it. And the cutter itself remains the bottleneck: a coated carbide system designed specifically for small-module dry milling, with geometry matched to the oblique cutting conditions described above, is still the missing piece between a demonstrated feasibility and a production process.