Underground mining drives often turn slowly while carrying high torque and shock loads. Right-angle transfer in scraper conveyors, crushers, and other compact reducers relies on spiral bevel gears. A double circular arc spiral bevel gear replaces the usual involute or single-arc flank with convex and concave circular arcs in both the tooth profile and the face-width direction. The two flanks are conformal: a convex arc on one member runs against a concave arc on the other. This conformity lowers the equivalent curvature and spreads contact pressure, while a staged profile thickens the root. One cutter head can generate both members of a mating pair, so the tooth form remains compatible with established face-milling practice. These features are the basis for the interest in double circular arc spiral bevel gears in confined, high-torque drives.
Tooth geometry and face-milling generation
The generating process follows the face-milling method used for spiral bevel gears. A cutter head is mounted eccentrically on a cradle that rotates about the generating-gear center. The head turns with the cradle and also spins about its own axis. Its inner and outer cutting edges sweep the generating surfaces that form the concave and convex flanks. The workpiece and cutter roll at a set ratio while a tooth slot is cut from one end to the other. The workpiece then withdraws, indexes by one pitch, and the cycle repeats until the gear is complete.
The basic tooth profile is built from a convex arc, a transition arc, a concave arc, and a root arc. The relative positions of these arcs are defined by the coordinates of their centers, their radii, and the polar angles of their starting points. The arc parameters can be adjusted for strength and for the intended application, which is one reason the profile is not a fixed geometric standard.
Two profile variants are used in production. In the tangent-connected form, the non-working transition region is generated by a straight cutter edge, leaving little clearance between the non-working flanks. Slight wear can close that clearance, bring the non-working flanks into contact, and initiate pitting. In the staged form, the convex and concave arcs are tangentially offset so that the concave pitch thickness is larger than the convex pitch thickness. The transition becomes a step, leaving a larger gap between non-working flanks. The thicker concave root also raises bending strength; analyses for soft tooth surfaces have indicated roughly twice the bending strength of a single circular arc profile.
Virtual generation of the gear pair
Contact analysis requires flank geometry that follows the actual cutting motion, not a simplified CAD tooth. A virtual machining model therefore represents four elements: the CNC milling machine, the gear blank, the cutter head, and the NC program. The machine model includes cutter-spindle rotation, vertical and horizontal travel, carriage travel, cradle rotation, and workpiece rotation. These six motions determine the relative position of the cutter and the blank.
From the part data, the blank, cutter, and machine settings are calculated. Cutter-path points are converted into an NC program, one slot is cut, and the blank is indexed around its axis. The resulting pair used for contact analysis is listed in Table 1.
Table 1. Basic parameters of the analyzed gear pair.
| Parameter | Value |
|---|---|
| Pinion teeth | 12 |
| Wheel teeth | 35 |
| Theoretical shaft angle | 90° |
| Normal module at the mid-face | 6 mm |
| Face width | 46 mm |
| Spiral angle at the mid-face | 35° |
Finite element discretization
The single-tooth solid is imported into HyperMesh, where lost faces and duplicated adjacent surfaces are cleaned before meshing. A two-dimensional mesh is generated first and then swept into three-dimensional hexahedral elements. Equivalence is used to merge duplicate nodes within the specified tolerance. The single-tooth mesh is rotated about the gear centerline in MSC Patran to produce the complete gear, and the same procedure gives the mating wheel.
The full gear pair contains 29,380 hexahedral elements: 7,680 for the pinion and 21,700 for the wheel. Element quality is checked through aspect ratio, distortion, Jacobian, and warpage. These checks matter because a contact solution is sensitive to poor element shape near the loaded flank.
Nonlinear contact simulation
The meshing problem is solved in MSC Marc as a transient dynamic contact analysis with a full Newton–Raphson iteration and adaptive load increments. The deformable tooth rings are tied to rigid inner cones. The pinion is driven by an angular velocity, and the wheel carries a resisting torque of 100 N·m derived from the motor input power. The wheel is constrained at its large end in three translational directions and at its small end in the y and z directions; the resisting torque is applied at a small-end node on the wheel axis. The material is linear elastic with an elastic modulus of 206,000 MPa and Poisson’s ratio of 0.3.
Contact is defined between the two deformable gear bodies. The analysis resolves the movement of the contact zone as the teeth roll through mesh, rather than assuming a fixed elliptical contact area.
Contact pattern and stress
At a selected meshing instant, the predicted contact zone is not a single ellipse. Two contact lines form, and two instantaneous contact points can coexist. As the pair rotates, two or three tooth pairs may share the load at different moments. This multi-point, multi-tooth behavior follows from the double circular arc geometry and is central to the load capacity of the design.
The maximum contact stress at the selected instant is 518 MPa. A conventional contact-stress calculation for the same nominal conditions uses the pinion torque, application and dynamic factors, contact-trace load distribution, elastic coefficient, tooth-ratio coefficient, spiral-angle coefficient, and arc-length coefficient, with 12 pinion teeth and a 6 mm normal module. That calculation returns 543 MPa. The finite element value is 25 MPa lower, a deviation of 4.6%.
Table 2. Contact-stress estimates for the analyzed pair.
| Method | Maximum contact stress | Difference from conventional calculation |
|---|---|---|
| Nonlinear finite element contact analysis | 518 MPa | −4.6% |
| Conventional contact-stress calculation | 543 MPa | Reference |
The deviation is calculated as (518 − 543)/543. A 4.6% difference between a nonlinear contact solution and a semi-empirical contact formula is a useful consistency check. It supports the modeling route, but it does not by itself validate absolute fatigue life.
Engineering reading and limits
Bench testing of a scraper conveyor reducer with a double circular arc spiral bevel gear pair reduced reducer volume by 9%, raised efficiency from 92% to 94.5%, and extended reducer life from 900 h to 1,900 h. These are system-level outcomes from one reducer configuration. They reflect the combined effect of tooth form, manufacturing quality, alignment, lubrication, and operating profile, and should not be read as universal gains for every application.
For low-speed, heavy-load mining drives, contact stress controls pitting risk and root stress controls bending fatigue. The simulation represents nominal geometry, nominal torque, a homogeneous elastic material, and ideal alignment. It does not by itself establish how pitch error, shaft deflection, surface finish, lubrication, or thermal distortion will redistribute load among the active tooth pairs. Those effects matter because the benefit of multi-tooth contact depends on maintained load sharing; a small misalignment or wear-induced change in the non-working clearance can concentrate load on fewer contacts.
Within those limits, the workflow—virtual generation of the actual flanks, all-hex finite element meshing, and nonlinear contact analysis—gives a practical way to estimate contact stress and identify multi-tooth engagement before bench testing. For design selection, its main value is comparative: it allows profile variants and alignment assumptions to be screened against the same load case, with the conventional contact calculation serving as an independent check on the predicted stress level.