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Miter gears: designing a 1:1 bevel pair

A miter gear pair is a bevel gear pair with a ratio of 1:1. It turns a drive through an angle - usually 90° - and leaves the speed and the torque as they were. Because both gears have the same number of teeth, a few things are simpler than on any other bevel pair, and one thing is harder: nothing can be traded between the two members.

What makes a pair a miter pair

With equal tooth numbers the two pitch cones are equal too. On a 90° shaft angle each pitch angle is exactly 45°; on any other shaft angle each is half of it - a 1:1 pair on a 60° shaft angle has two 30° pitch cones. Pairs like that are sometimes called angular miter gears; the geometry is the same, and the designer takes any shaft angle.

The 45° cone has a second consequence: the separating force on each gear splits into equal axial and radial parts. On a pair with a ratio they differ - the pinion's radial force is the wheel's axial force and the other way round - but on a miter pair both bearings see the same.

One part, used twice

On a straight miter pair the pinion and the wheel are the same gear. In the designer every row of the dimension table is equal for the two, the backlash is taken half from each, and there is no profile shift to move addendum from one to the other - for a ratio of 1 the suggested shift is zero, since there is nothing to balance. So a straight miter pair needs one drawing, one part number and one STEP file, fitted twice.

A spiral miter pair is always one left-hand and one right-hand gear. The dimensions match, but the parts are mirror images and cannot be swapped.

A worked example

A straight 20/20 pair at 90°, calculated in the designer, rated to ISO 10300 at 40 Nm and 1 500 min^-1 (K_A 1.25, case hardened steel at 700 HV, quality MQ, accuracy grade B6, 20 000 h):

The face width is worth a second look. With the module fixed at the mean section, a wider face also moves the heel outwards, so the 30 % limit grows with it: at 15 mm this pair is already 0.02 mm over, and the designer says so. As on most hardened pairs pitting is the limit - S_H is close to its minimum of 1.0 while the root has plenty in hand - so if the torque had to go up, a larger module or a better material would help more than anything aimed at the root. What moves the safety factors has the numbers.

How few teeth

This is where a miter pair is at a disadvantage. On a pinion and wheel of different sizes, profile shift lengthens the pinion's addendum and shortens the wheel's, and a small pinion escapes undercut that way. On a 1:1 pair whatever one gear gains the other loses, so the shift cannot help both, and the tooth number alone has to be enough. Measured with ISO 23509 Clause 8's undercut check, at the largest recommended face width:

Below these the designer warns that no profile shift can prevent undercut on both gears. The check is deliberately conservative, and at 20° a 13/13 pair is still cut sound; at 12/12 the flank undercuts so far that it crosses itself, and the STEP build stops. The whole story is in minimum number of teeth and undercut.

Straight or spiral

The same trade-offs as on any bevel pair apply: a spiral miter pair runs more quietly and carries more load on its flanks for its size, at the price of two different parts and axial forces that are no longer equal on the two gears, their direction set by the hand and the direction of rotation. A straight pair is simpler to make, identical on both sides, and pushes each gear away from the cone apex. See straight vs spiral bevel gears for one pair worked both ways.

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