Bevel stages in mining machinery carry the highest specific load of any gear mesh in the drive train. In an armoured face conveyor reducer, the input bevel pair turns motor speed into high torque at a few tens of revolutions per minute, and the whole gearbox is sized around the contact and bending capacity of that pair. Improving the load capacity of the bevel stage therefore translates directly into a smaller, lighter or longer-lived reducer — a decisive advantage in underground equipment, where envelope, weight and service intervals are all constrained.
Why circular-arc gearing changes the contact problem
An involute spiral bevel gear brings two convex flanks together. The equivalent radius of curvature at the contact is small, so the Hertzian pressure for a given torque is high, and the load is carried on essentially a single line of contact that sweeps across the tooth. A double circular arc (DCA) spiral bevel gear replaces this by a conjugate pair in which the convex flank of one member runs against the concave flank of the other, in the tooth-height direction and along the tooth length. The resulting equivalent radius of curvature is much larger, the contact zone becomes an area rather than a line, and the contact stress at equal torque falls accordingly.
Two further features matter for heavy duty. The stepped (profile-shifted) variant thickens the root of the concave member, raising bending capacity; for soft-tooth-surface gearing, the bending strength of a double circular arc mesh has been put at roughly twice that of a single circular arc mesh. And a single cutter head generates both members of a mating pair, which keeps tooling and process cost low for a profile that is geometrically more demanding than an involute one.
The system-level payoff has been measured on a scraper conveyor reducer: fitting DCA spiral bevel gears reduced reducer volume by 9%, raised transmission efficiency from 92% to 94.5%, and extended service life from 900 h to 1900 h in bench testing.
| Quantity | Conventional bevel stage | Double circular arc stage | Change |
|---|---|---|---|
| Reducer volume | baseline | — | −9% |
| Transmission efficiency | 92% | 94.5% | +2.5 percentage points |
| Service life | 900 h | 1900 h | +111% |
Reference tooth profile: tangent-connected versus stepped
The reference profile of a DCA gear is assembled from four arc segments: a convex arc, a transition arc between convex and concave portions, a concave arc, and a root fillet. The relative placement of these segments is fixed by the coordinates of their centres, their radii, and the polar angles at which each arc begins. Two families are in production, and the difference between them governs field behaviour.
In the tangent-connected family, convex and concave arcs are joined by a common tangent, and the non-working transition is generated by the straight edge of the cutter. The gap between the non-working flanks of the mating pair is then very small. As the working flank wears, that gap closes, the non-working flanks eventually touch, and pitting is initiated on surfaces that were never intended to carry load. In the stepped family, the profile is tangentially shifted so that the concave member has a larger pitch-circle tooth thickness than the convex member; the transition is an arc, producing a visible step between the working and non-working zones. The clearance between non-working flanks is then generous enough that contact there is avoided over the wear life, and the thicker concave root adds bending capacity.
Generating the flank: face milling kinematics and the NC model
DCA spiral bevel gears are cut by the Gleason face-milling method. The generating gear and the machine cradle share a rotation centre, while the cutter head is mounted eccentrically on the cradle: it swings with the cradle and simultaneously rotates about its own axis, so that the locus of its cutting edges defines the generating gear’s tooth surface. Outer and inner edges produce the concave and convex flanks respectively. Cutter head and blank roll at a fixed ratio to cut one slot from one end of the tooth to the other; the blank then withdraws, indexes by one tooth space, and the cycle repeats.
The equivalent NC model has six axes: cutter spindle rotation, vertical slide, horizontal slide, saddle, cradle (rotary base), and workpiece rotation. These six motions fully determine the relative position of tool and blank. Virtual machining is then organised in four modules — machine model, blank model, cutter head model and NC program. Blank and cutter geometry are computed from the gear parameters, the cutter geometry fixes the machine setting parameters, and the NC program is written from the calculated tool-path points, cutting one slot and then indexing around the blank to produce the complete solid model.
| Parameter | Value |
|---|---|
| Wheel tooth count z2 | 35 |
| Pinion tooth count z1 | 12 |
| Shaft angle | 90° |
| Normal module at mid-face width mn | 6 mm |
| Face width | 46 mm |
| Mid-face spiral angle | 35° |
Finite element discretization
Meshing starts from a single tooth. The solid is exported in Parasolid format and imported into HyperMesh, where the geometry is cleaned — lost faces restored, duplicated adjacent surfaces merged — before the unit system and element type are set. A two-dimensional mesh is generated on the tooth surface and swept into three-dimensional hexahedra, and duplicate nodes within a set tolerance are eliminated with the equivalence function. The single-tooth mesh is exported in neu format to MSC.Patran, where a rotational copy about the gear axis produces the complete gear; the same procedure gives the mating wheel.
The assembled pair contains 29,380 hexahedral elements: 7,680 on the pinion and 21,700 on the wheel. Element quality was screened on aspect ratio, distortion, Jacobian and warp before the contact analysis was run.
Nonlinear contact setup
MSC.Marc solves the highly nonlinear meshing problem with either the full or the modified Newton–Raphson scheme, and describes rigid-body motion through prescribed displacement, prescribed velocity or applied load. Here the inner cone surfaces of both members were defined as rigid bodies: an angular velocity was imposed on the driving rigid body and a resisting torque on the driven one, which drives the pair through a dynamic meshing simulation.
- Deformable bodies: elements 1–7,680 form the pinion, elements 7,681–29,380 the wheel.
- Load: the input torque on the pinion shaft, computed from the motor power, is 100 N·m.
- Constraints: three translational degrees of freedom fixed at the large-end nodes of the wheel, y and z fixed at its small-end nodes, with the torque applied at the small-end node on the wheel axis.
- Material: elastic modulus 206,000 MPa, Poisson’s ratio 0.3, assigned as a 3D solid property to all elements of both members.
- Contact: a G-type connection between each deformable body and its rigid cone, where no relative sliding occurs, and a T-type connection between the two deformable tooth bodies at the actual mesh. The load steps use adaptive increments with the Newton–Raphson iteration.
Contact pattern and stress
The computed contact state is qualitatively different from an involute mesh. The DCA pair contacts at multiple points simultaneously: two contact lines form across the flanks, two instantaneous contact points can appear at a single moment, and as the pair rotates the number of teeth sharing load alternates between two and three pairs. Peak contact stress for the analyzed case is 518 MPa.
This multi-tooth, multi-point behaviour is the mechanical origin of the capacity advantage — load is spread over several flanks and over an area rather than a line — but it also means that load sharing is sensitive to manufacturing accuracy, mounting error and elastic deflection of the housing. A contact pattern that migrates toward the tooth ends under load will concentrate stress on a small number of the nominally engaged pairs.
Cross-check against the standard calculation
The semi-empirical relation used for double circular arc contact stress takes the form
$$\sigma_H=\left[\frac{T_1K_AK_vK_tK_{HZ}}{2^{\mu_\varepsilon}+K_{\delta\varepsilon}}\right]^{1/2}\frac{0.73\,Z_EZ_uZ_\beta Z_a}{z_1m_n^{2.19}}$$
where T1 is the nominal pinion torque; KA the application factor; Kv the dynamic factor; Kt the load-sharing factor along the contact path; KHZ the load-distribution factor within the contact path; με the integer part of the longitudinal contact ratio; Kδε the contact-path coefficient; ZE, Zu, Zβ and Za the elasticity, tooth-ratio, helix-angle and contact-arc-length coefficients; z1 the pinion tooth count; and mn the normal module. The constants 0.73 and the exponent 2.19 are empirical, specific to the double circular arc load-capacity formulation.
Evaluated with handbook coefficients for this pair, the relation gives 543 MPa, against 518 MPa from the contact simulation — a deviation of 4.6%.
| Method | Peak contact stress | Deviation relative to simulation |
|---|---|---|
| Nonlinear FE contact simulation (MSC.Marc) | 518 MPa | — |
| Semi-empirical standard calculation | 543 MPa | +4.6% |
The analytical result lies above the simulated one, which is consistent with a formula built on conservative assumptions about how load distributes among the engaged tooth pairs. The 4.6% gap is small enough to support the finite element model as a screening tool for profile and machine-setting variants, and large enough to remind the designer that the two numbers do not answer the same question: the simulation resolves where the load actually sits at a given instant, while the formula assumes a distribution.
Scope and limitations
Several boundaries should be kept in view when using these results. The stress field was computed for a single torque level, 100 N·m, and a limited set of rotational positions; a fatigue assessment needs a load spectrum rather than a rated point. The material model is linear elastic with no plasticity, residual stress or surface treatment, so the computed peak is a nominal contact stress rather than a predictor of subsurface initiation. Mesh density in the contact zone controls the peak value and should be checked by refinement. Most importantly, the reducer bench data in Table 1 validate a system-level benefit, not the computed stress field; no direct strain or photoelastic measurement of the simulated contact pressure is reported.
Implications for mining drive design
For low-speed, heavy-load duties such as scraper conveyor, crusher and mill drives, the DCA spiral bevel gear offers torque density that an involute bevel stage cannot match in the same envelope, and it does so with a manufacturing route — face milling on a Gleason-type machine with a single cutter head per pair — that is already familiar to gear shops. The design work that remains is concentrated in three areas: keeping the stepped profile’s non-working clearance adequate through the wear life so that pitting does not start on unloaded flanks; controlling the contact pattern under load, since multi-tooth contact distributes well only when the pattern stays inside the face width; and converting the computed contact stress into a qualified life through load-spectrum testing and component fatigue data. The validated contact model is the entry point for all three.