Spiral bevel gears transmit torque between intersecting shafts with a curved, oblique tooth form. Compared with straight bevel gears, the spiral tooth engages progressively along the face width, so the contact ratio is higher and the motion is smoother. These properties explain their use in high-speed and heavy-load drives, but they also make the gear difficult to represent geometrically. The flank is not a simple cone or an involute surface. It is the envelope of a cutter surface generated by the timed rotation of the cutter head, cradle, and workpiece. A CAD model built from approximate cones may look correct but will not reproduce the local curvature, tooth thickness, or contact pattern needed for finite-element contact analysis.
Parameter set and blank geometry
The modeling chain separates the blank from the tooth flank. The blank is bounded by the pitch, root, and face cones and by the toe and heel planes. Its dimensions follow from the module, tooth numbers, pressure angle, shaft angle, spiral angle, face width, tooth depth, and hand. For the 20/22-tooth example in Table 1, the module is 2.5 mm, the shaft angle is 90°, the nominal spiral angle is 35°, and the face width is 33 mm. The pinion is left-hand and the wheel is right-hand, a combination that must be preserved when the two members are assembled.
| Parameter | Pinion | Wheel |
|---|---|---|
| Module, m (mm) | 2.5 | 2.5 |
| Number of teeth, z | 20 | 22 |
| Pressure angle, α (°) | 20 | 20 |
| Shaft angle, Σ (°) | 90 | — |
| Spiral angle, β (°) | 35 | 35 |
| Face width, b (mm) | 33 | 33 |
| Whole tooth depth, h (mm) | 4.75 | 4.75 |
| Hand | Left | Right |
| Addendum circular thickness, sa (mm) | 4.461 | 4.305 |
The table gives only the primary inputs and selected derived dimensions. Root and face cone angles, addendum and dedendum, and the axial positioning of the cutter are calculated from these values by the blank routine. Scripting this calculation avoids repeated manual table work and reduces transcription errors when the design variables change.
Generating the tooth flank from cutter motion
In spiral bevel gear cutting, the inner and outer cutting edges rotate with the cutter head and sweep two conical surfaces. The cutter head rotates about its own axis, the cradle rotates it about the machine axis, and the workpiece rotates about its own axis in a timed relationship. The tooth flank is the envelope of the cutter surfaces under this relative motion. In the machine coordinate system Sm, with origin at the machine center Om and the XmOmYm plane normal to the cradle axis, a cradle-fixed system Sq rotates about Zm. A point on a cutting edge is
$$ \mathbf{r}_c(u)=\begin{bmatrix}x_c(u)\\ y_c(u)\\ z_c(u)\\ 1\end{bmatrix} $$
and its position in gear coordinates is
$$ \mathbf{r}_g(u,\phi)=\mathbf{M}_{gc}(\phi)\,\mathbf{r}_c(u). $$
Here u is a parameter along the cutting edge, φ is the cradle rotation, and Mgc is the homogeneous transformation from cutter coordinates to gear coordinates, including the timed workpiece rotation. The envelope condition is the meshing equation
$$ \mathbf{n}_g(u,\phi)\cdot \mathbf{v}_g^{(cg)}(u,\phi)=0, $$
where ng is the surface normal in gear coordinates and vg(cg) is the relative velocity between the cutter and the gear. For each cradle position φ, the meshing equation selects the points on the cutting edge that belong to the envelope; repeating over φ gives a family of contact lines whose union is the tooth surface. The inner and outer edges generate the two flanks, one concave and one convex, with the same transformation structure but different cutter geometry.
The solution is not a single closed-form surface. It is a family of contact lines. Sampling the edge parameter u and the cradle angle φ produces a point cloud for each flank. The points are exported as a DAT file and imported into UG NX, where they are fitted as surface patches. The concave and convex patches are stitched around a tooth space, the space is cut from the blank, and the tooth is patterned by the tooth count. Because the point generator is driven by the basic parameters, a change in module, tooth number, spiral angle, or face width produces a new point cloud without manual re-drawing.
Virtual rolling and static contact analysis
Two checks are used on the assembled pair. The first is a virtual rolling run. The pinion is given a prescribed rotation and the wheel is driven through the mesh. In this step the bodies are effectively rigid, so the resulting footprint is a kinematic contact pattern. It shows where the flanks touch geometrically, not the stress under load. For the 20/22 pair, the footprint formed a diagonal band running from the toe (small end), near the root, toward the heel (large end), near the addendum.
The second check is a static contact analysis in ANSYS Workbench. The CAD pair is imported, a contact pair is defined between the mating flanks, a torque of 150 N·m is applied at the wheel, and the pinion is rotated through 25°. The contact stress field again formed a diagonal band in the same region, with the highest intensity near the middle of the face width. The location and orientation of the static contact zone were consistent with the unloaded rolling footprint.
| Assessment | Loading and kinematics | Contact indication |
|---|---|---|
| Virtual rolling | Prescribed pinion rotation; rigid bodies; no applied torque | Unloaded geometric footprint; diagonal band from toe-root toward heel-addendum |
| Static FEA | 150 N·m torque on wheel; pinion rotated 25°; elastic contact | Contact stress band in the same region; concentration near mid-face |
The static result also provides a stress distribution that the kinematic run cannot. Regions of elevated contact stress can be identified and used to adjust the design or machine settings before cutting. The comparison is most meaningful as a check on contact location and shape; the stress magnitudes belong only to the stated torque, mesh, and boundary conditions. The 150 N·m torque is a single load case, and the resulting stress values are not a general allowable load for this gear size.
What the comparison establishes
Agreement between the rolling footprint and the static contact band supports the geometric model, the assembly orientation, and the contact setup. If the flank patches, hand, or coordinate systems were inconsistent, the two contact zones would be displaced or differently oriented. The comparison therefore provides an internal consistency check on the parametric chain.
It is not independent proof that a manufactured gear will show the same pattern. Both analyses use the same nominal CAD surface. The rolling step is unloaded and rigid, while the static step includes elastic deformation at one torque and one angular position. The result is also sensitive to point-cloud density, surface-fitting tolerance, finite-element mesh, contact stiffness, friction, and boundary conditions. The model represents nominal geometry; it does not include cutter wear, machine-tool error, heat-treatment distortion, or lapping.
For those reasons, the appropriate use of the model is comparative. It can rank design variants, indicate the direction in which a contact pattern will move, and provide a starting point for finite-element stress studies. Final acceptance of a spiral bevel gear set still requires a physical tooth-contact check and, where transmission error or dynamics matter, measurement on a gear tester.
Engineering value of the parametric chain
The main engineering value is not the particular 20/22 model but the traceability from design inputs to analysis geometry. Once the blank calculation and flank-point generation are scripted, the same chain can be run for different tooth numbers, modules, spiral angles, and face widths. This makes it practical to examine how a design change affects the contact pattern before committing to a cutter setup. It also gives the finite-element analyst a surface that carries the generated curvature rather than a simplified cone.
Spiral bevel gears remain a manufacturing-sensitive product. The generated flank depends on the cutter head, blade geometry, and machine settings, and small setting errors can shift the contact pattern. A parametric CAD model is therefore most useful when it is tied to the actual generator settings and checked against a rolled test. Used in that way, it shortens the loop from parameter selection to contact analysis and reduces the number of physical trials needed to reach an acceptable pattern.
In the 20/22 case, the chain produced a closed sequence: blank parameters, generated flank points, solid tooth space, virtual rolling footprint, and static contact stress. The two contact assessments agreed in location, which supports the model as a nominal analysis geometry. The limits are equally clear: the result is a numerical consistency check, not a substitute for experimental validation or for the machine-setting corrections used in production.