Small-module spiral bevel gears sit at the output of a growing class of compact actuators — robot joints, medical drives, camera gimbals and the terminal mechanisms of consumer smart equipment. Their tooth geometry is cut, not formed, and the cutting process has historically been inseparable from flood cooling. A 1–2 mm module gear pair has small chip loads and slender cutter blades, so the usual argument for coolant (heat removal from a heavy chip) does not really apply; coolant is used because the machine tools available for this size class were never stiff enough or fast enough to run otherwise. Removing the cutting fluid therefore becomes a machine-tool and tool-material problem as much as a cutting-mechanics problem.
Why small-module spiral bevel gears still run wet
Dry cutting — high-speed feed with no coolant and no lubricant — eliminates fluid purchase, filtration, disposal and mist, and it generally raises productivity because the workpiece no longer has to be washed between operations. The penalty is paid at the spindle and at the cutting edge: the process demands high static and dynamic stiffness, thermal stability of the machine frame, and tool materials that retain hardness at the temperatures generated in the shear zone.
The industrial record for medium and large modules is well established. Klingelnberg-Oerlikon introduced a dry-cutting bevel gear milling machine in 1998, and Gleason presented the Phoenix II CNC bevel gear cutting machine in 2003 with higher productivity and improved flank accuracy. Mitsubishi demonstrated dry hobbing at a cutting speed of 200 m/min and put the system into production. In China, work on dry gear cutting began in the 1990s; Changsha Zhongda Chuangyuan showed the YKA2260 fully CNC bevel gear dry-cutting machine in 2012 for equal-height and tapered-tooth designs, and Tianjin Jingcheng built the JCB-20 CNC spiral bevel gear dry-cutting machine.
That experience has not transferred to small modules. Because small-module gears are cheap per piece, the milling machines built for them were deliberately simple, and their spindles, thermal growth and feed drives could not support the cutting speeds dry machining requires. The situation changed with direct-drive technology: Tianjin Jingcheng and Zhongshan Mileite now offer small-module CNC gear milling machines with direct-drive motors on both the workpiece and cutter spindles, combining the stiffness and accuracy needed for dry cutting. What remains missing is the tooling side — coated cemented-carbide cutter heads developed specifically for small-module spiral bevel gears — and a validated process model that relates machine settings to the forces the cutter will see.
Oblique cutting as the physical model of gear milling
Spiral bevel gears are cut either by the forming method, where the wheel tooth profile replicates the cutter blade and the blank is held stationary while the cutter head rotates, or by the generating method, where cutter head and blank roll in a fixed ratio. Neither case is orthogonal cutting. Along the curved tooth the angle between the cutting edge and the cutting velocity vector changes continuously, so the process is oblique: a three-dimensional cutting geometry with a non-zero edge inclination angle λ, a chip that flows at an angle across the rake face, and a shear velocity that closes a velocity triangle with the cutting and chip velocities. Orthogonal (two-dimensional) cutting is far better documented in both theory and finite element practice, but it does not represent the tooth-milling operation, and this mismatch is the main reason cutting-force predictions for bevel gears are often treated as order-of-magnitude estimates.
Force is the quantity that links the process to everything else of engineering interest — tool life, flank quality, and chatter of a spindle running at high speed. Predicting it requires a shear-zone description. Merchant’s single-plane model (1944) assumed a zero-thickness shear plane; Oxley and Welsh (1963) introduced a parallel-sided shear zone with finite thickness, for which Shaw quoted a representative value of about 0.025 mm; Zorev proposed a fan-shaped zone. For high-speed dry cutting the non-equipartition model of Astakhov divides the zone into a wide region (0 ≤ y ≤ kh) and a narrow region (kh ≤ y ≤ h), which matches the strain distribution observed when the chip forms at high speed better than a uniform zone does.
Shear-zone kinematics and force prediction
Working in the equivalent plane defined by the cutting velocity V and the chip velocity Vc, with h the shear-zone thickness, k the non-equipartition coefficient, α the normal rake angle, φ the normal shear angle, ηs the shear flow angle, ηe the equivalent plane angle and ηc the chip flow angle, the maximum shear strain rate is
$$\dot{\gamma}_m=\frac{2V\cos\lambda\cos\alpha}{h\cos\eta_s\cos(\phi-\alpha)}$$
and the shear strain distribution through the zone is piecewise:
$$\gamma=\begin{cases}\dfrac{\dot{\gamma}_m\cos\eta_e}{2khV\cos\lambda\sin\phi}\,y^{2}, & y\in[0,\,kh]\\[3mm]\dfrac{\dot{\gamma}_m\cos\eta_e\left(-y^{2}+2ky-h^{2}\right)}{2(1-k)hV\cos\lambda\sin\phi}, & y\in[kh,\,h]\end{cases}$$
Taking the shear stress τ as uniform on the primary shear plane, with b the width of cut and th the undeformed chip thickness, the shear force follows directly from the shear-plane area:
$$F_s=\tau A_{sh}=\frac{\tau\,b\,t_h}{\cos\lambda\sin\phi}$$
Above roughly 90 m/min the chip’s inertia can no longer be neglected, and it contributes to the normal force on the primary shear plane. With β the mean friction 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 projected onto the cutting, feed and thrust directions through a transformation that depends on λ, φ and ηs:
$$\begin{bmatrix}F_c\\F_r\\F_a\end{bmatrix}=\mathbf{M}(\lambda,\phi,\eta_s)\begin{bmatrix}F_s\\F_{no}\end{bmatrix}$$
The practical content of this projection is that the edge inclination angle redistributes force between components rather than simply scaling them: the thrust (back) force Fa is the component most sensitive to λ, while the main cutting force Fc and feed force Fr respond mainly to chip thickness and cutting speed. The derivation of φ, ηs, the strain rate, k, β, ηe and Fno follows the oblique-cutting thermal-mechanical formulation of Li et al. (2010).
Gear pair and machine settings used in the trial
The demonstration case is a 1:1 small-module spiral bevel gear pair — 30 teeth on each member, module 1.75 mm, shaft angle 90°, pressure angle 20°, mid-point spiral angle 35°. Both members have zero height and tangential modification, which keeps the blank simple while still exercising the cutter-head geometry. Tables 1 and 2 list the geometric and machining parameters; they define the blank, the cutter head and the machine settings that the simulation and the cutting trial share.
| Parameter | Wheel | Pinion |
|---|---|---|
| Number of teeth | 30 | 30 |
| Hand | 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 |
| Parameter | Wheel | Pinion |
|---|---|---|
| Cutter-head diameter (mm) | 50.8 | 50.8 |
| Blade spacing (mm) | 0.81 | 0.81 |
| Blade number (°) | 7.5 | 7.5 |
| Machine installation 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 |
Finite element representation
The blade–blank interaction reduces to a single oblique cut: the cutter head repeats an identical engagement for each tooth slot, and after one slot is finished the blank is indexed by one tooth pitch, so a representative cutting edge entering a representative workpiece segment captures the mechanics without modelling the whole blank. The blank is quenched and tempered 45 steel (0.45% C), the tool is YT15 cemented carbide, and because the tool is far harder than the workpiece it is treated as a rigid body — its elastic deflection is negligible compared with the plastic flow of the workpiece. The workpiece mesh is refined in the cutting zone and coarsened away from it, with the bottom and left boundaries constrained to mimic the fixture. Table 3 gives the physical properties assigned to both materials.
| Property | YT15 carbide (tool) | 45 steel (workpiece) |
|---|---|---|
| Density (kg/m³) | 11 500 | 7 890 |
| Elastic modulus (GPa) | 530 | 209 |
| Poisson ratio | 0.30 | 0.269 |
| Specific heat (kJ/(kg·°C)) | 480 | 480 |
| Thermal conductivity (W/(m·K)) | 50 | 50 |
| Thermal expansion coefficient (K⁻¹) | 6.5×10⁻⁶ | 1.309×10⁻⁵ |
| Inelastic heat fraction | 0.9 | 0.9 |
Friction on the rake face is split into a sticking region and a sliding region, with the shear stress capped at the material’s shear strength:
$$\tau=\begin{cases}\mu\sigma_n, & \mu\sigma_n\le\tau_{\max}\\ \tau_{\max}, & \mu\sigma_n\ge\tau_{\max}\end{cases}$$
The workpiece constitutive response uses the Johnson-Cook form, which represents the combined influence of strain, strain rate and temperature that dominates at the speeds and temperatures of dry cutting:
$$\sigma=\left[A+B\bar{\varepsilon}^{\,N}\right]\left[1+C\ln\frac{\dot{\bar{\varepsilon}}}{\dot{\bar{\varepsilon}}_0}\right]\left[1-\left(\frac{T-T_r}{T_m-T_r}\right)^{m}\right]$$
with A the yield stress, B the hardening modulus, N the strain-hardening exponent, C the strain-rate sensitivity, m the thermal softening exponent, Tm the melting temperature and Tr room temperature; the initial plastic strain rate is 1.0 s⁻¹. For 45 steel the parameters are A = 507 MPa, B = 320 MPa, N = 0.28, C = 0.28 and m = 1.06, as listed in Table 4.
| A (MPa) | B (MPa) | N | C | m |
|---|---|---|---|---|
| 507 | 320 | 0.28 | 0.28 | 1.06 |
Chip separation uses the physical criterion rather than the geometric one, because the geometric criterion ignores the material’s constitutive behaviour while the physical criterion compares stress and force against a critical value and so reproduces chip formation more faithfully. Damage accumulates as
$$\omega=\sum_{j=1}^{n}\frac{\Delta\bar{\varepsilon}_{pl}}{\bar{\varepsilon}_{pl}^{\,f}}$$
and the material fails when ω exceeds unity, with the critical equivalent plastic strain given by
$$\bar{\varepsilon}_{pl}^{\,f}=\left[d_1+d_2\exp\left(d_3\frac{p}{q}\right)\right]\left[1+d_4\ln\frac{\dot{\bar{\varepsilon}}}{\dot{\bar{\varepsilon}}_0}\right]\left[1+d_5\left(\frac{T-T_r}{T_m-T_r}\right)\right]$$
where p/q is the ratio of compressive to deviatoric stress and d₁–d₅ are the shear failure parameters of Table 5. The resulting chip curls into a C shape, which aids chip breaking and evacuation — a practical advantage when no coolant is available to flush the cutting zone.
| d₁ | d₂ | d₃ | d₄ | d₅ |
|---|---|---|---|---|
| 0.10 | 0.76 | 1.57 | 0.005 | −0.84 |
Parametric trends from the cutting simulation
With the baseline cutting speed fixed at 300 m/min and the edge inclination angle at 5°, increasing the feed from 0.2 mm to 0.4, 0.6 and 0.8 mm raises the main cutting force substantially and monotonically. This is the expected consequence of a larger undeformed chip thickness acting over a shear-plane area that grows with th, and it sets the practical ceiling on feed for a small-module cutter head whose blades are thin: force, blade deflection and flank error all rise together.
Varying the edge inclination angle over 0°, 5°, 10° and 20° at a feed of 0.2 mm and a cutting speed of 300 m/min produces a different pattern. The main cutting force and the feed force change little, whereas the thrust force grows steadily with λ. This is the component-level behaviour implied by the projection of the shear-plane forces onto the cutting, feed and thrust directions, and it is the reason a small inclination angle is retained rather than eliminated: too large a value loads the slender blade in the direction it is least able to resist.
Raising the cutting speed at fixed feed and inclination angle lowers all three force components. The mechanism is thermal: the shear-zone temperature rises, the workpiece flow stress falls according to the thermal-softening term of the constitutive model, and the shear-plane stress τ decreases. For a dry process this is the favourable direction of the trade-off, since higher speed is also where productivity is gained — provided the tool material and the machine’s thermal stability can absorb the temperature rise.
Predicted trends were checked against previously published oblique-cutting results and found to be consistent, which supports using this model to select process windows rather than to certify absolute force values.
Dry cutting trial and tooth-flank inspection
The trial used the duplex double-side method: both the concave and the convex flanks of the wheel and of the pinion are cut by a duplex cutter head in a single pass through each slot, which is the most productive arrangement for a small-module pair. The blank and fixture were prepared to the geometry of Table 1, the settings of Table 2 were entered into the CNC system, the blank was mounted on the work-holding fixture, the cutter was aligned, and cutting proceeded with no coolant or lubricant.
Meshing quality was first assessed by a rolling test, since dimensional inspection alone does not reveal whether the contact pattern is correctly placed. A uniform film of red lead powder — about 0.00635 mm thick — was applied to the working flanks (wheel convex, pinion concave), and the pair was rolled in a dedicated small-module rolling tester at the specified mounting distance. The resulting contact patch sits in the middle of the tooth flank with an elliptical footprint, the configuration regarded as desirable for load distribution and insensitivity to deflection under torque.
Gear accuracy was then measured on a Gleason 650 measuring machine, with the tooth-flank deviation and pitch deviation results evaluated to DIN 3965/86 (Table 6).
| Flank | Cumulative pitch deviation (μm) | Grade | Single pitch deviation (μm) | Grade | Adjacent pitch deviation (μm) | Grade |
|---|---|---|---|---|---|---|
| Convex | 46 | 8 | 9 | 6 | 7.1 | 5 |
| Concave | 32.9 | 7 | 6 | 5 | 7.8 | 5 |
The pattern in Table 6 is typical of a dry-cut part: the short-range errors (single and adjacent pitch deviation) reach grades 5–6 on both flanks, while the long-range cumulative pitch deviation is the weakest measure at grade 8 on the convex flank and grade 7 on the concave flank. Cumulative error is the quantity most exposed to thermal growth of the machine and blank during an uninterrupted dry run, so it is exactly where a coolant-free process would be expected to lose accuracy first. For applications that specify grade 7 or better overall, this points to thermal compensation of the workpiece spindle rather than to changes in the cutting parameters as the development priority.
Assessment and open issues
The trial supports the central claim: a small-module spiral bevel gear pair can be milled at 300 m/min without cutting fluid using a YT15 carbide cutter on a machine with direct-drive spindles, and the result meshes correctly and reaches grade 5–8 accuracy depending on the measure. Two factors make this feasible where earlier attempts were not — machine-tool stiffness and spindle dynamics of the direct-drive generation, and a force model that lets the process window be chosen from the edge inclination angle, feed and speed rather than by trial and error.
The demonstration is a single gear pair in one material combination, and several questions remain outside its scope. Tool life under dry conditions was not quantified, although it is the dominant economic variable once coolant costs are removed; crater and flank wear rates for uncoated YT15 at 300 m/min in 45 steel will set the real limit on speed. Coated carbide grades developed specifically for small-module cutter heads are not yet available domestically, and blade geometry for modules below 1.75 mm has not been examined. The cumulative pitch deviation also indicates that thermal behaviour over a full production run, not just a single tooth slot, needs measurement before the process can be released for volume manufacturing. Finally, the force model predicts trends reliably but was not calibrated against measured cutting forces in this configuration, which is the natural next step toward a closed process-design loop.
Where those gaps are filled, the economics are straightforward: no cutting fluid purchase, filtration or disposal, no workpiece washing between operations, higher material removal rate, and flank quality that already meets the requirements of the compact transmissions now being designed into intelligent equipment.