Spur gear geometry with profile shift

Formulas and fundamentals

Transverse and normal section

For helical gears the normal section (tool, manufacturing) and the transverse section (meshing conditions) must be distinguished. The normal module gives the transverse module and the transverse pressure angle:

For spur gears β = 0, hence mt = mn and αt = αn. Pitch and base circle follow as d = z·mt and db = d·cos αt, the reference centre distance of the pair as ad = mn·(z1 + z2)/(2·cos β).

Operating pressure angle via the involute function

The involute function inv α = tan α − α (α in radians) links the sum of the profile shift coefficients to the operating pressure angle at backlash-free mesh:

The angle αwt has to be recovered from inv αwt. The equation has no closed-form inverse; classically αwt was read from an involute function table. The calculator solves it with Newton's method from f(α) = tan α − α − inv αwt and f'(α) = tan²α, starting at α0 = (3·inv αwt)^(1/3) from the series expansion inv α ≈ α³/3. It stops as soon as the Newton step falls below 1e-12 rad, which in the usual range takes fewer than ten steps.

Centre distance and sum of profile shifts

With the operating pressure angle the centre distance of the modified gear pair follows:

Conversely, a centre distance dictated by the design can be reached through profile shift. Then αwt is determined first from cos αwt = cos αt·ad/a, and from it the required sum:

Splitting the sum between the two gears is a design decision: the aim is equal tooth root capacity on both gears without causing undercut on the pinion. The calculator follows the DIN 3992 recommendation in its approximate form (u = z_large/z_small, using the virtual tooth numbers zn for helical gears):

The second coefficient follows necessarily from x2 = Σx − x1; what matters is that the sum is kept. Both coefficients can afterwards be adjusted by hand in the calculator's "coefficients x1/x2" mode.

Addendum modification and tip clearance

On an external gear pair the centre distance grows more slowly than the sum of the profile shifts. Without a countermeasure the tip clearance shrinks. The addendum modification compensates for this:

On an external pair it is always negative and therefore reduces both tip circles. With the DIN 867 basic rack profile (addendum factor 1, tip clearance factor 0.25) the diameters read:

The root circle is unaffected by the addendum modification because it only shortens the tip circles. The resulting tip clearance is c = a − 0.5·(da1 + df2); with the addendum modification applied it comes out at exactly c = 0.25·mn again. For a small sum of profile shifts it can often be omitted, because the deeper tool infeed used to create the backlash partly compensates for it - the calculator offers a toggle and shows the resulting tip clearance in either case.

Undercut and limiting profile shift

Below a certain tooth count the tool cuts away part of the usable involute during generation. The theoretical limiting tooth number follows from zg = 2/sin²αn (about 17 for αn = 20°); 14 teeth count as the practical limit. For a profile-shifted gear this limit moves:

zn is the virtual tooth number of the equivalent spur gear in the normal section. If a gear's profile shift coefficient falls below x_limit, undercut has to be expected; the calculator's rating additionally flags the band between the practical and the theoretical limit as marginal.

Contact ratios

The transverse contact ratio is the ratio of the length of path of contact to the base pitch and states how many tooth pairs are in mesh on time average:

It should be at least 1.1, and more than 1.25 is recommended. Helical gears add the overlap ratio from the helix offset across the face width, and together the two give the total contact ratio:

The normal tooth thickness on the pitch circle, as it appears in the drawing data, follows from sn = mn·(π/2 + 2·x·tan αn).

Calculation basis

All equations follow Roloff/Matek "Maschinenelemente", chapters 21.1 (spur gear geometry, straight teeth) and 21.2 (helical gears). Only the external gear pair at backlash-free mesh is calculated; internal gearing (annulus), tooth thickness allowances and backlash as well as the load capacity per ISO 6336 are not part of this calculation.

Worked example

Reference example, straight teeth: a gear pair z1 = 18, z2 = 50 with module m = 3 mm has the reference centre distance ad = 3·68/2 = 102 mm. The design, however, calls for a = 106.5 mm. From cos αwt = (102/106.5)·cos 20° = 0.899987 the operating pressure angle αwt = 25.8436° follows, hence inv αwt = 0.0333 and, with inv 20° = 0.014904, the required sum of profile shifts Σx = (0.0333 − 0.014904)/(2·tan 20°)·68 = +1.72. The DIN 3992 split gives x1 ≈ +0.70 for the pinion and therefore x2 = 1.72 − 0.70 = +1.02 for the gear.

This yields d1 = 54 mm and d2 = 150 mm, with root circles df1 = 50.7 mm and df2 = 148.62 mm. Computing the tip circles without addendum modification as da1 = 64.2 mm and da2 = 162.12 mm leaves a tip clearance of only c = 106.5 − 0.5·(64.2 + 148.62) = 0.09 mm - far too little. With the addendum modification k = 106.5 − 102 − 3·1.72 = −0.66 mm the tip circles are shortened to da1 = 62.88 mm and da2 = 160.8 mm, and the tip clearance is back at the usual c = 0.75 mm = 0.25·m.

Reference example, helical teeth: for z1 = 26, z2 = 86, mn = 4 mm, β = 15° and b = 50 mm as unmodified gears, αt = arctan(tan 20°/cos 15°) = 20.6469°, d1 = 107.67 mm, d2 = 356.14 mm and ad = 231.90 mm. The overlap ratio is εβ = 50·sin 15°/(π·4) = 1.03, the transverse contact ratio εα = 1.64 and hence the total contact ratio εγ = 2.67. Here the helix offset alone already covers one full tooth, which is why helical gears run considerably more quietly than comparable spur gears.

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