What the calculation gives you
Once a gear pair's ratio and size are decided, every dimension of the teeth follows from two numbers: the module (or its inverse, the diametral pitch) and the number of teeth. The calculation gives the pitch, base, tip and root diameters, the tooth depth and the centre distance, which is what a drawing, a hob or a CAD model needs. For a bevel pair, which turns a shaft through an angle, it also gives the cone angles and the dimensions along the cone, which are what make bevel gears harder to draw than spur gears.
Spur gear basics
d = m z
d pitch (reference) diameter · m module in mm, the pitch diameter per tooth · z number of teeth. In inch practice the diametral pitch P = z / d is used, with m = 25.4 / P
Centre distance a = m (z1 + z2) / 2 · Ratio i = z2 / z1 · Base diameter db = d cos α
α pressure angle, 20° standard, 14.5° in older designs
Addendum ha = m · Dedendum hf = 1.25 m · Tip diameter da = d + 2 m · Root diameter df = d − 2.5 m
standard full-depth proportions; circular pitch p = π m, tooth thickness at the pitch circle p / 2
Two gears mesh only if they share the module and the pressure angle. The ratio is the tooth count ratio and nothing else; the module sets the size, and with it the strength, for a given ratio. Below about 17 teeth at 20° a standard tooth is undercut by the hob and weakened; profile shift (addendum modification) or more teeth avoids it.
Straight bevel gears
A bevel pair with shaft angle Σ (90° in most cases) is defined on the pitch cones, which meet at the apex. The module and tooth counts give the reference diameters at the large end as for spur gears; the cone angles follow from the ratio:
tan δ1 = sin Σ / (z2 / z1 + cos Σ), δ2 = Σ − δ1
δ1, δ2 pitch cone angles of pinion and gear; for Σ = 90°, tan δ1 = z1 / z2
Cone distance R = d2 / (2 sin δ2) · Face width b ≤ R / 3
the teeth taper toward the apex; a face width over a third of the cone distance leaves the small end too thin to cut and to carry load
The tooth proportions are split unequally between the two members (the Gleason system gives the gear a shorter addendum and the pinion a longer one, to balance their strength), and each addendum and dedendum becomes an angle seen from the apex, θa = atan(ha / R) and θf = atan(hf / R). The tip angle δa = δ + θa and root angle δf = δ − θf of each member, the outside diameter da = d + 2 ha cos δ, the distance from apex to crown and the inner tip diameter at the small end are the figures a drawing and a lathe need. For strength and for the base circle, each bevel gear is treated as the spur gear of its back cone, with the virtual tooth count zv = z / cos δ (Tredgold's approximation).
Worked example
The worksheet below lays out a straight bevel pair at a 90° shaft angle, module 0.47 mm, 20° pressure angle, 14 and 24 teeth. From those four inputs it computes the reference diameters, the two cone angles, the cone distance and the face width, the unequal addenda and dedenda of the Gleason system, the addendum and dedendum angles, the tip and root angles, the tip diameters, the apex-to-crown distances, the axial face widths, the inner tip diameters, the back-cone radii, the virtual tooth counts and the base circle radii. Change the tooth counts and every dimension of both members updates; change the module to scale the whole pair.
Common mistakes
- Module and diametral pitch confused. A module 2 gear and an 8 DP gear are not far apart in size, but they will not mesh. One system per machine.
- Tooth counts too low. Below 17 teeth (20°) or 32 teeth (14.5°) the standard profile is undercut; the worksheet's 14-tooth pinion relies on the bevel geometry's larger virtual count, 14 / cos δ1.
- Bevel face width too large. More than R / 3 does not add capacity, because the small end does not carry its share, and it makes the gear hard to cut.
- Pitch diameter measured at the wrong end. Bevel dimensions are given at the large end (the heel); the small end (the toe) is smaller by b / R.
- Mounting distance ignored. A bevel pair meshes correctly only when each member is at its designed distance from the apex; a shim error changes the backlash and the contact pattern.
- Geometry taken for strength. These formulas size the teeth; the load they carry needs the Lewis or the AGMA/ISO bending and pitting checks, with the material and the duty.
Questions
- Straight, spiral or hypoid bevel?
- Straight teeth are the easiest to make and to calculate and are fine at low speed. Spiral bevels run quieter and carry more load at speed; hypoids add an offset between the axes, as in an automotive axle, and are a different calculation.
- How is the backlash set?
- By the mounting distance, for a bevel pair, or the centre distance, for spur gears, rather than by the tooth geometry. Typical backlash is 0.03 to 0.05 times the module.
- Where does the pressure angle matter?
- In the base circle, the tooth thickness at the root and the undercut limit. 20° gives a stronger tooth and allows fewer teeth; 14.5° runs quieter and was the older standard.
