Spiral bevel gears transmit motion and torque between intersecting shafts, usually at a 90° shaft angle. Unlike straight bevel gears, their teeth are curved and inclined along the face width. Contact therefore starts at one end of the tooth and moves progressively across the flank, raising the contact ratio and supporting high load capacity, low noise, and high-speed operation. These properties explain their use in machine-tool drives, aerospace transmissions, automotive differentials, and robotic joints. The spiral angle also generates axial thrust, so bearing selection and mounting stiffness are part of the design. The same curved flank makes the tooth surface a spatial geometry sensitive to module, tooth count, pressure angle, spiral angle, face width, and cone angle.
Parameter Set and Pitch Cone Geometry
A representative spiral bevel gear pair for CAD construction uses 25 pinion teeth and 54 wheel teeth. The pressure angle is 20°, the addendum coefficient is 1.0, the clearance coefficient is 0.25, the face width is 28 mm, and the spiral angle is 31°. The module is left as a driving parameter: it sets the absolute scale of the pitch cone and tooth height, while the tooth-count ratio sets the cone angles. The pinion and wheel must be of opposite hand to mesh.
| Parameter | Symbol | Pinion | Wheel |
|---|---|---|---|
| Number of teeth | z | 25 | 54 |
| Module | m | design input | design input |
| Pressure angle | α | 20° | 20° |
| Addendum coefficient | ha* | 1.0 | 1.0 |
| Clearance coefficient | c* | 0.25 | 0.25 |
| Face width | b | 28 mm | 28 mm |
| Spiral angle | β | 31° | 31° (opposite hand) |
| Angular pitch | 360°/z | 14.4° | 6.67° |
For a 90° shaft angle, the pitch cone angles follow from the tooth counts:
$$
\delta_1 = \arctan\left(\frac{z_1}{z_2}\right) = 24.8^\circ,
\qquad
\delta_2 = 90^\circ – \delta_1 = 65.2^\circ
$$
The gear ratio is
$$
i = \frac{z_2}{z_1} = \frac{54}{25} = 2.16
$$
The cone distance is common to both members:
$$
R = \frac{m z_1}{2\sin\delta_1} = \frac{m z_2}{2\sin\delta_2}
$$
At the large end, the standard diameters used to drive the Pro/E relations are
$$
d = mz, \quad
d_a = d + 2h_a^* m, \quad
d_f = d – 2(h_a^* + c^*)m, \quad
d_b = d\cos\alpha
$$
These diameters define the addendum, pitch, and base circles of the large-end section. The cone distance and tooth height scale with the module, so changing m rebuilds the entire model proportionally.
Involute Tooth Slot Construction
Pro/E modeling begins with user parameters for z, m, α, β, b, ha*, and c*. Relations compute the diameters and the pitch cone angle, and the large-end basic circles are sketched as construction geometry. Because the circles are relation-driven, a change in module or tooth count rebuilds the section instead of requiring a manual redraw.
At the large end, the involute is generated in the back-cone plane. The back-cone base radius is
$$
r_{bv} = \frac{d_b}{2\cos\delta}
$$
where δ is the pitch cone angle of the gear being modeled. The involute datum curve can be created with the Pro/E “From Equation” command:
$$
x = r_{bv}(\cos t + t\sin t), \qquad
y = r_{bv}(\sin t – t\cos t), \qquad
z = 0
$$
Here t is the roll angle in radians. If a Pro/E relation uses a degree-based roll parameter θ, the conversion t = θπ/180 must be applied to the linear terms. The first flank is then mirrored about the tooth center plane to form the opposite flank. The closed section defines the tooth slot. At the toe, the same construction is repeated with the smaller back-cone radius, and the toe and heel sections are connected by the tooth-trace guide curve.
Sweep, Mirror, and Pattern
The solid tooth slot is produced as a swept cut along a guide curve that represents the tooth trace. The pitch cone is drawn in the intended direction of rotation, and the trace is projected onto a reference plane to create the guide. The swept cut removes one slot; a mirror completes the section about the tooth center plane, and a circular pattern repeats the slot with count z and angular increment 360°/z. For the pinion, 25 teeth give a 14.4° increment; for the wheel, 54 teeth give a 6.67° increment. The pattern count is tied to the tooth-count parameter, so the model regenerates when z changes.
Parametric control extends beyond tooth count. Changing m scales the diameters and cone distance; changing β changes the guide curve and therefore the sweep; changing b moves the toe section and alters the face width. The model is robust only when the guide curve, section sketch, and pattern are tied to the same parameter set. If the guide curve is not driven by the spiral-angle parameter, a parameter change can invalidate the sweep even though the basic circles regenerate correctly.
Virtual Assembly and Motion Coupling
The two gear solids are inserted into an assembly and mounted on pin joints. Each pin joint aligns the gear axis with its shaft and fixes the axial position at the required mounting distance. The two axes are positioned to intersect at the pitch cone apex with the specified shaft angle; for the common 90° case, the axes form a right-angle pair. A fully constrained placement that removes all degrees of freedom locks the gears and prevents motion analysis. The pin joint preserves the rotational degree of freedom required by the gear pair.
For motion, the two joints are coupled by the tooth ratio:
$$
\frac{\omega_2}{\omega_1}
= -\frac{z_1}{z_2}
= -\frac{25}{54}
\approx -0.463
$$
The negative sign indicates opposite rotation; the sign convention depends on tooth hand and axis orientation. With this coupling, a servo motion or drag on the pinion produces the correct wheel rotation. Interference checks, motion envelopes, and bearing and shaft layout can be evaluated before machining. A change in a part parameter followed by regeneration updates the assembly, provided the mating constraints are referenced to stable datums.
Scope and Practical Limits
Parametric solid modeling is most valuable as a layout and assembly tool. It produces a closed solid for interference checks, motion studies, and preliminary finite element analysis. It does not, by itself, reproduce the generated spiral flank of a real cutting process. A sweep-cut model with a projected guide curve approximates the tooth trace; contact pattern, transmission error, and contact stress depend on the actual machine settings, cutter geometry, and flank modifications. For those outputs, the tooth surface should be generated from the manufacturing model or measured flank data, and the mesh and contact definition must be consistent with that surface.
Spiral bevel gears are also sensitive to mounting error and elastic deflection. If system-level behavior is the objective, the assembly model should include bearing stiffness, housing compliance, and the actual mounting distance. Within these limits, the parametric approach reduces repetitive modeling and makes it practical to compare tooth counts, face widths, and spiral angles before detailed analysis.