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How to design a spur or helical gear pair, step by step

A spur or helical gear pair is settled by a handful of choices - the ratio, the tooth numbers, the module, the helix angle, the face width and the material - and each one limits the others. This guide takes them in the order that works and ends with a worked example you can open in the spur and helical gear designer.

1. Ratio and tooth numbers

The ratio is u = z₂/z₁, and the pinion's tooth number is the choice with the most consequences. Few teeth give a small pinion, but its root is thin near the base circle and below a certain number the generating tool cuts into the flank - undercut. ISO/TR 4467 draws the line: with no profile shift a spur pinion stays inside its conventional limits down to 20 teeth; a 17-tooth pinion needs at least x = +0.11, a 14-tooth one +0.23 and a 12-tooth one +0.30 (3.4.2). The sum of the two tooth numbers should be at least 20, and is recommended at 24 or more (3.3.2).

Many designers also avoid a common factor between z₁ and z₂, so every pinion tooth meets every wheel tooth in turn and wear evens out. A 20/40 pair, as below, does not do this; 20/41 would.

2. Spur or helical

Straight teeth put no axial force on the bearings and are the simplest to make and inspect. A helix angle makes more teeth share the load and the engagement smoother, at the cost of an axial force of F_t tan β. In the same centre distance the gain is larger in the root than on the flank - the numbers for one pair designed both ways are in spur vs helical gears.

3. Module, size and centre distance

The normal module m_n is the size of the tooth - the one the hob is made for; ISO 54 lists the preferred values. With the tooth numbers and the helix angle it fixes the pair to ISO 21771:

• reference diameter d = z m_n / cos β;
• reference centre distance a = (z₁ + z₂) m_n / (2 cos β).

So a 20/40 spur pair of module 3 has diameters of 60 and 120 mm and a centre distance of 90 mm. If the housing fixes the centre distance at some other value, a profile shift makes up the difference (step 5). The module that is enough follows from the load: choose one from the space and let the rating in step 7 confirm or change it.

4. Face width

A wider face carries more, but not in proportion: the wider the face, the less evenly the load spreads across it, because the pinion shaft bends and twists and the helix is never perfect. ISO 6336-1 counts this in the face load factor K_Hβ. Measured on the example pair, with the pinion between its bearings on a 45 mm shaft:

Doubling the face from 30 to 60 mm raised S_H by only 17 %. A crowned or helix-corrected flank lowers K_Hβ - with crowning the 30 mm pair goes from S_H 1.09 to 1.20 - and a stiffer shaft or a pinion closer to mid-span helps too. For a helical pair the face width also sets the overlap ratio ε_β = b sin β / (π m_n); at ε_β of 1 or more there is always a full helix in contact.

5. Basic rack and profile shift

The tooth is generated by an ISO 53 basic rack: 20° pressure angle, addendum 1.0 m, dedendum 1.25 m. Types A, B and C differ only in the root fillet - 0.38, 0.30 and 0.25 m; type D deepens the root to 1.40 m with a 0.39 m fillet. Type A, the largest fillet and so the strongest root, is the designer's default.

A profile shift x moves the rack away from the gear (positive) or into it (negative). It cures undercut on small pinions, balances the root strength of pinion and wheel, and fits the pair to a centre distance that is not the reference one. How, and within which limits, is in profile shift in spur and helical gears. With x₁ = x₂ = 0 the pair runs on its reference circles, which is the designer's default.

6. Backlash and accuracy

Backlash does not come from the nominal geometry; it comes from thinning the teeth and from the centre distance tolerance. The designer takes both from DIN: the tooth thickness fit to DIN 3967 (a deviation series a–h and a tolerance series 21–30, default 26cd) and the housing's centre distance field to DIN 3964 (default js7). For the example the fit takes 70 to 130 µm off each tooth, and the pair ends up with a normal backlash of 0.120 to 0.256 mm.

Accuracy is an ISO 1328-1:2013 flank tolerance class from 1 to 11. At class 6 the example's pinion may have a single pitch deviation of ±9 µm and a cumulative pitch deviation of 26 µm. The class also enters the rating, through the dynamic factor and the load distribution. The finished thickness is checked by the span over k teeth W_k or the dimension over balls M_dK, both listed with their limits.

7. Load capacity to ISO 6336

ISO 6336 rates a cylindrical pair for pitting of the flank (part 2) and breakage of the tooth root (part 3), with the load factors of part 1 and the material strengths of part 5. Enter the pinion torque, speed and life, the application factor K_A (ISO 6336-1 Table 4 by the driving and driven machine), the material and the pinion shaft - its diameter, the bearing span and the pinion's offset from mid-span, which K_Hβ needs and the standard gives no default for. The result is a safety factor for each failure, S_H and S_F.

ISO 6336 sets no minimum safety factors (ISO 6336-1, 4.1.11): they are for the designer or the customer to agree. A breakage stops a machine at once and pitting grows gradually, so the root is usually asked for more. What moves the two factors is measured in spur gear strength: ISO 6336 safety factors explained.

8. From numbers to a 3D model

Once the numbers hold, look at the pair: the transverse section with its line of action and path of contact, the axial section with the dimensions you pick, and a 3D preview. The solids are built at the mean tooth thickness of the DIN 3967 fit, so the model has the backlash the finished gears will have; the STEP export writes the pinion, the wheel and the pair in mesh.

Worked example

A spur pair of ratio 2, driven by an electric motor at 1 450 min^-1 with 100 Nm on the pinion, driving a machine with light shocks for 20 000 h.

The root has a great deal of margin and the flank very little, which is normal for a case hardened pair. If S_H 1.09 is not enough, the cheapest steps are a crowned flank (1.20), class 5 instead of 6 (1.16) or module 3.5 (1.25) - not a wider face.

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