Spiral bevel gears dominate the final drive of aircraft, automotive and mining machinery because their curved, gradually engaging teeth give a large contact ratio, high load capacity, smooth and quiet running in a compact envelope. The tip of each tooth, however, is the part of the geometry that first pays for that performance: it is thin, it is the first feature to be bruised in handling, and it is the location where induction-hardening currents concentrate. Trimming a narrow band along the tip edge — removing the sharp crest and blending it into the flank and the top land — reduces the impact that occurs as a pair of teeth enters and leaves mesh, stabilises the mesh, and protects the flank from the raised burr that a deformed tip edge would otherwise press into the mating surface. On gears that are hardened after cutting, the same operation removes the acute corner that overheats during high-frequency induction heating, lowers the stress concentration at the tip, reduces the risk of quench cracking and gives a more uniform carburised case. The image below shows a typical spiral bevel gear of the kind these operations are applied to.
Why conventional tip trimming struggles with curved teeth
Hand fitting with an angle grinder is the default in low-volume shops. The operator follows the tip edge by eye, so the amount removed varies along the tooth and from tooth to tooth, the work is slow and physically demanding, and the wheel is applied close to the operator’s hands. Consistency between gears — the property that matters when trimmed parts are assembled into a matched pair — cannot be guaranteed.
Two patented machine-based alternatives address the productivity problem but inherit geometric ones. Chinese invention application CN101829815A (application 201010166040.5, published 15 September 2010, applicant Tianjin Jingcheng Machine Tool Co., Ltd.) describes a four-axis CNC chamfering machine with two linear and two rotary axes, in which a grinding wheel chamfers both tip edges of a tooth slot simultaneously. Its toolpath is not computed from the gear’s mathematical model: the operator jogs the axes and records the wheel-centre coordinates and wheel swing angles at a small number of positions, a curve is fitted through those points, and a quadratic polynomial supplies the swing angle between them. The result depends on how accurately the operator reaches each cutting position, and the sparse sampling leaves the path poorly conditioned for spiral bevel and hypoid teeth whose curvature changes along the face width. Because the concave and convex flanks of such gears have different pressure angles, the two chamfers produced in one pass are not guaranteed to be equal. The dedicated machine itself is inflexible and adds capital cost.
Chinese invention application CN101879635A (application 201010225340.6, published 10 November 2010, applicant Tianjin University) takes a different route for formate-cut gears, using a five-axis machine with three linked linear axes plus a wheel spindle and a gear indexing axis. One slot is indexed into position and held, and a right-angle conical wheel, whose axis is kept normal to the formate generating plane, grinds both tip edges of that slot as it traverses. The restriction is written into the kinematics: the method suits only the formate-cut wheel of a gear pair, not the pinion and not a generated wheel. Grinding also produces fine swarf that demands well-sealed guideways, which limits the method to machines built for it rather than general-purpose five-axis equipment.
| Approach | Machine configuration | Tool | Applicable workpieces | Principal constraint |
|---|---|---|---|---|
| Hand fitting | none | angle grinder | any | quality set by the operator; slow; hazardous |
| CN101829815A (2010) | 4-axis CNC chamfering machine: 2 linear, 2 rotary | grinding wheel, both tip edges of a slot per pass | spiral bevel and hypoid gears | path fitted to a few manually taught wheel positions; unequal chamfers from concave/convex pressure-angle difference; dedicated machine |
| CN101879635A (2010) | 5-axis, 3 linked linear axes plus wheel spindle and indexing axis | right-angle conical wheel, axis normal to formate plane | formate-cut wheel only | not applicable to pinions or generated wheels; fine swarf requires sealed guides |
| Generated-path trimming | 5-axis, 4 linked axes | tapered milling cutter | spiral bevel gears including generated wheels and pinions | CAM computation required; 0.3 mm trimming deviation reported |
Fixing the trimming band in a CAM coordinate frame
The starting point is the theoretical mathematical model of the spiral bevel gear, from which the flank surface and the boundary of the trimming band are defined directly rather than measured on a sample part. Two frames are used. The workpiece frame oxyz is the reference for modelling: its origin o lies at the centre of the large end face, the xy plane is coplanar with that face, and the z axis is collinear with the gear axis and points toward the small end. The machining frame ox’y’z’ shares the origin and the z axis and is the frame in which the G-code and the machine axis motions are expressed.
The two frames are tied together by a single reference point, the centre P of a virtual probe sphere. Its z coordinate and its perpendicular distance from the z axis are chosen first; a practical choice places the probe centre at one third of the face width from the small end and one third of the tooth height below the tip. The x and y coordinates of P then follow from the condition that the probe sphere is tangent to the concave flank. With P located, the machining frame is completed by taking the direction from P perpendicular to the z axis as the negative y’ direction. Because P is a single, reproducible contact point on a mathematically defined surface, it serves both as the CAM reference and as the physical touch-off point on the machine.
From flank points to cutter locations
The trimming band is bounded by two curves: Γ1, where the trimming surface meets the flank, and Γ2, where it meets the top land. Γ1 is discretised into a point set whose coordinates are extracted in ox’y’z’. For each point Pn on Γ1 the method constructs a virtual cutting cone whose apex coincides with Pn and whose surface is tangent to Γ2 at a point Qn. The vector from that apex toward the centre of the large end is the tool axis vector Tn for that position; it is obtained numerically by successive bisection rather than in closed form, which keeps the solution stable where the tangent condition has no simple algebraic expression.
The cutter location point CLn is the apex Pn displaced by two offsets: a radius compensation R and a tool length compensation H. The radius compensation vector runs from the large-end tangent point Mn perpendicular to the virtual axis Ln, and its length is the cutting radius — the small-end diameter of the cone band that actually removes material. The length compensation acts along the reverse of Tn and is fixed by the cutter geometry and the cutting radius. For a flat-end tapered cutter of tip diameter d and cone angle α, the axial offset to the cutting circle is
$$H=\frac{R-d/2}{\tan\alpha}$$
and for a ball-nose tapered cutter of ball diameter D the spherical tip adds a second term:
$$H=\frac{R-\frac{D}{2}\cos\alpha}{\tan\alpha}+\frac{D}{2}\left(1-\sin\alpha\right)$$
The cutter is therefore fully specified by its cone angle α, its tip diameter d or ball diameter D, and the cutting radius R. Joining successive cutter location points by linear interpolation produces the toolpath, which is post-processed into an NC program in the usual way.
Alignment and the cutting cycle on the machine
The distinguishing feature of the setup is that only the rotary axis is aligned on the machine. The offsets of the three linear axes are established once from the fixture, which carries the clamping position, so the part does not have to be indicated in with a dial gauge on every cycle. The probe centre is driven to the computed position of P in the machine work frame, and the gear is then rotated until the probe touches the concave flank; the angle swept by the rotary axis is the rotary offset. Because the contact condition is defined on the theoretical flank, the alignment inherits the accuracy of the model rather than the skill of the setter.
Cutting proceeds continuously through a repeating sequence: approach, cut, retract, index. Tip edges on the same flank side are fed in the same direction and tip edges on opposite flanks in opposite directions. In the preferred arrangement the gear is clamped with its small end upward, the right-hand edges of the teeth are cut from the small end toward the large end, and the left-hand edges from the large end toward the small end. After each edge the cutter retracts, the gear indexes to the next tooth, and the next edge of the same side is cut; once all right-hand edges are finished the same routine is repeated for the left-hand edges. Because every pass follows a computed cutter location path in a single machining frame, the trim on each tooth is nominally identical, which is precisely the property that hand fitting cannot deliver.
Verified performance and operating envelope
Trial machining on a general-purpose five-axis, four-linkage CNC machine tool — three linear axes and one rotary axis interpolated together — held the trimming deviation within 0.3 mm and produced a clean, uniform edge along the tooth tips. Within that envelope the method offers three practical gains: the toolpath and NC program are generated automatically from the gear model, so manual part programming and point teaching disappear; alignment is confined to the rotary axis with the linear offsets guaranteed by the fixture; and no dedicated chamfering machine is needed. A patent application covering the method was filed under number 201910140609.1 by First Tractor Company Limited.
The 0.3 mm figure also indicates what the process is and is not for. Tip trimming is an edge-conditioning operation: its purpose is a repeatable edge break that improves meshing impact, hardening behaviour and handling robustness, not a correction of flank form. Judged against that purpose, the tolerance is adequate, and the value of the approach lies less in the absolute number than in the repeatability it brings to a feature that was previously finished by hand or by a machine dedicated to a single gear type. Milling rather than grinding is a further part of that practicality: a tapered carbide cutter produces chips rather than the fine swarf that forces sealing precautions on a general-purpose machine, which is what allows the method to run on standard five-axis equipment already present in a transmission plant.