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Profile shift in spur and helical gears

Profile shift - addendum modification - is the one free parameter of an involute gear: the same tool and the same tooth number give a whole family of teeth, thicker or thinner at the root, longer or shorter in the addendum. This guide explains what the shift does, the limits ISO/TR 4467 sets on it, how it splits a sum between pinion and wheel, and what it does to the centre distance, the contact ratio and the strength - measured on one pair.

What a profile shift is

The gear is generated by a basic rack - in practice a hob or a shaper cutter - rolling on the gear's reference circle. If the rack's datum line is moved away from the gear by x m_n, the involute flanks are cut further out on the same base circle: the tooth gets thicker at the root and thinner at the tip, the tip and root diameters grow by 2 x m_n, and the reference and base diameters do not change. A negative x moves the rack in, with the opposite effect.

The pressure angle of the tool does not change - it is still 20° for an ISO 53 rack. What changes, when the shifts of the two gears do not cancel, is the working pressure angle α_wt at which the pair runs, together with its centre distance.

Undercut and pointed teeth: the limits

A shift that is too small lets the tool's tip cut into the flank near the base circle - undercut, which weakens the root and shortens the usable involute. One that is too large narrows the tip to a point. ISO/TR 4467 replaces these two geometric limits with simple lines in the virtual number of teeth z_v = z / cos³β (3.4): conventional limits, which are not to be exceeded in any circumstances, and the narrower recommended limits, within which the standard says no check of the operating conditions is needed.

So 20 teeth is the smallest spur pinion that may be cut without a shift; below that the conventional lower limit is positive and the designer warns until x reaches it.

Shifting with the centre distance unchanged

With x₂ = −x₁ the pair keeps its reference centre distance and its 20° working pressure angle; only the teeth change shape. On a 20/40 pair of module 3 at 100 Nm:

Without a shift the smaller pinion has the weaker root. Shifting it positive and the wheel negative moves strength from the wheel to the pinion; the lower of the two is highest where they cross, near x₁ = +0.2 - which is where ISO/TR 4467's split puts it (below). Going further only moves the weak point to the wheel, thins the pinion's tip and lowers the contact ratio; at −0.5 the wheel is also outside its conventional limit of −0.333.

Shifting to reach a centre distance

When the housing fixes the centre distance, the sum of the shifts follows from it (ISO 21771): a larger centre distance needs a positive sum, and the pair runs at a larger working pressure angle. The same pair at three centre distances, the sum split by ISO/TR 4467:

A positive sum helps the flank - the larger working pressure angle gives larger radii of curvature at the pitch point - and costs contact ratio. ISO/TR 4467 says the same in two lines (3.3.4): every increase of the sum raises the breaking load capacity, every decrease raises the contact ratio. For this pair of 60 teeth in total, the sum's conventional limits are −0.100 to +1.333 and the recommended ones 0 to +1.000 (3.3.2, 3.3.3).

Splitting the sum: ISO/TR 4467

ISO/TR 4467 3.2.2 shares a chosen sum between the two gears as

x₁ = λ (u − 1)/(u + 1) + Σx / (u + 1),   x₂ = Σx − x₁

with u the ratio of the virtual tooth numbers (capped at 5) and λ between 0.5 and 0.75 for a speed-reducing pair, 0 and 0.5 for a speed-increasing one (3.2.3). The first term gives the pinion more of the shift the larger the ratio, which is what balances the roots. The designer takes the middle of the band, λ 0.625, unless you enter one; it offers three ways to give the shift - x₁ and x₂ directly (the default, 0 and 0), the sum, or the centre distance.

The tip, and when to shorten it

A positive shift thins the tip. ISO/TR 4467 3.4.4 asks for a normal tip thickness s_an of at least 0.2 m_n - 0.6 mm at module 3 - and 3.5 gives the addendum reduction that restores it. The designer warns when a tip falls below the limit and proposes the reduction. At x₁ +0.5 the example's pinion still has 1.419 mm.

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