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Bevel gear minimum number of teeth and undercut

A bevel pinion can only have so few teeth before the tool that generates it starts cutting away the flank it has just made. That is undercut. How few is "too few" depends on the ratio, the pressure angle and the profile shift - and for a 1:1 pair it is a hard limit. This guide explains why, with the limits measured in the designer.

What undercut is

A bevel tooth is generated: the cutter represents one tooth of an imaginary crown gear, and the blank rolls against it. On a pinion with few teeth the tip of that crown gear tooth reaches far down into the tooth space, and on its way through it removes part of the flank near the root. The flank that is left is shorter, so the contact is shorter, and the root is thinner, so it is weaker. In the extreme the cut flank crosses itself and no usable tooth remains.

A bevel pinion behaves like a larger spur gear

A bevel tooth, seen in its back cone, has the shape of a spur gear tooth with more teeth - the virtual (or equivalent) number of teeth, z_v = z / cos δ. The pitch angle δ of a pinion is small when the ratio is high and 45° on a 1:1 pair, so:

For a spur gear cut by a standard rack without profile shift, undercut begins below z = 2 / sin²α teeth: 31.9 at a 14.5° pressure angle, 17.1 at 20° and 11.2 at 25°. ISO 23509 Clause 8 - written for hypoid gears, of which a straight bevel pair is the simplest case - applies the same condition to the virtual spur gear, with a 10 % margin on the tool addendum. The table shows the two sides of it: at a high ratio a bevel pinion is barely larger than a spur gear of the same tooth count, while at 1:1 it has the most margin of any bevel pinion.

Profile shift: the usual cure

On a pinion and wheel of different sizes, the tooth depth is split unequally: the pinion gets a longer addendum and a shorter dedendum, the wheel the opposite. This profile shift moves the pinion's flank away from the region the tool undercuts, and the wheel, with many teeth, can afford to give it up. ISO 23509 Annex C suggests the shift from the ratio and the pinion tooth number, and Clause 8 bounds it from both ends - enough for the pinion, not so much that the wheel undercuts.

Why a 1:1 pair cannot be saved by shift

On a miter pair the two gears are the same size, so whatever one gains the other loses. If the tooth number is too small for both, no shift helps: the interval Clause 8 allows turns inside out. Measured at the largest face width ISO 23509 B.6 recommends (0.30 R_e):

A narrower face helps a little: at 25°, 9/9 passes with a face width of 0.20 R_e, while at 20° 13/13 does not pass even at 0.10 R_e.

Clause 8 is deliberately cautious: it assumes a sharp-cornered tool and adds its margin on top. The designer also cuts the actual profile with the cutter's edge radius and checks it. At 20° with the default ISO 53 type C tool, a 13/13 pair still comes out with a sound flank; at 12/12 the flank undercuts so far that it crosses itself, and the STEP build stops with "Severe undercut".

The practical floor

• Straight bevel pinions: 12 teeth. ISO 23509 Annex C tabulates its suggested profile shift from 12 pinion teeth; below that the standard suggests no value, and the designer asks you to enter one yourself - Clause 8 still bounds it.
• 12 or 13 teeth: consider 25°. ISO 23509 C.2.2 recommends a 25° pressure angle for pinions of 12 to 13 teeth, and the designer offers it. The price is a larger radial force on the bearings and a shorter contact ratio; and with the default ISO 53 type C tool the root fillets may no longer fit at the toe - the designer then gives the largest edge radius that does.
• Spiral bevel pinions go lower. ISO 23509 Annex C tabulates spiral bevel pinions down to 6 teeth, and the designer accepts a spiral pinion from 6 teeth.
• A large shift has its own limit. The more addendum the pinion takes, the thinner its tip; if the tooth would come to a point, the designer's build stops and says so.
• 14.5° is a legacy angle. It needs far more teeth, as both tables show.

What the designer tells you

Set the tooth numbers and the designer checks the pair against Clause 8 as you type. If no profile shift can protect both gears it says so, gives the interval each gear would need, and suggests the two ways out - a larger pressure angle or more teeth. If a shift you enter is outside the interval, it names the gear that undercuts. The suggested shift, the admissible range and the tooth section that results are shown on the page, so the effect of each change can be seen.

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