You need helical blank diameters, lead, pair centre distance, contact ratios, or a first-pass Lewis check with axial thrust before you freeze a housing or order a hob. This pad covers six input modes — blank geometry, pair / centre distance, Strength (Lewis), lead & axial pitch, contact ratios, and force triangle — each with a distinct animated 2D Results sketch.
Defaults open on Blank geometry: mn = 2 mm, 20 teeth, β = 15°, αn = 20°, mating 40 teeth, face 25 mm. CALCULATE returns pitch diameter ≈ 41.41 mm, tip ≈ 45.41 mm, root ≈ 36.41 mm, lead ≈ 485.5 mm. Switch to Pair for a ≈ 62.12 mm and εγ ≈ 2.59, or Strength for Lewis FoS with Ft/Fa/Fr. Math stays in your browser.
It sits under Mechanical Calculators next to the spur gear calculator, gear ratio calculator, and gear train calculator.
Formula
- Normal system: mt = mn/cosβ, d = mt·Z = mn·Z/cosβ, da = d+2mn, df = d−2.5mn.
- Transverse pressure angle: tan αt = tan αn / cosβ. Base diameter db = d·cos αt.
- Lead L = π·mn·Z/sinβ. Axial pitch px = π·mn/sinβ. Virtual teeth zv = Z/cos³β.
- Pair (zero shift): a = mn(Z1+Z2)/(2 cosβ). εγ = εα + εβ with εβ = b·sinβ/(π·mn).
- Forces: Fa = Ft·tanβ, Fr = Ft·tanαn/cosβ. Lewis: σ = Ft/(Cv·b·mn·Y), Cv = 6/(6+v), Y from zv.
Reproduce the default Blank / Pair path on CALCULATE:
| Quantity | Value |
|---|---|
| mn / Z / β / αn | 2 mm · 20T · 15° · 20° |
| Mating teeth / face | 40 · 25 mm |
| mt / pitch d | ≈ 2.071 / 41.41 mm |
| Tip / root | ≈ 45.41 / 36.41 mm |
| Centre distance (Pair mode) | ≈ 62.12 mm |
| Lead / εγ | ≈ 485.5 mm · ≈ 2.59 |
How it works
Pick Blank, Pair, Strength (Lewis), Lead, Contact ratios, or Force triangle. ISO 21771-style normal system (mn, αn, β, Z), centre distance, εγ, lead/px, undercut warn on zv, Lewis with helical Cv, and Ft/Fa/Fr. Each mode draws a distinct animated 2D Results sketch.
Pick Input mode: Blank geometry (default), Pair / centre distance, Strength (Lewis), Lead & axial pitch, Contact ratios, or Force triangle. The animated 2D sketch in Results switches with the mode (Play/Pause). Enter teeth, normal module, helix (0–45°; β = 0 is the spur limit), and mode-specific fields. Mating teeth required for Pair and Contact. Face width feeds εβ and Lewis. CALCULATE fills Results. Editing clears Results. RESET restores mn2 · 20T · β15 · mate 40.
Normal module and helix angle set the pitch diameter
Shop references start from tooth size in the normal plane. The hob sees mn; the shaft sees d = mn·Z/cosβ because the transverse module grows with helix angle.
Defaults use mn = 2 mm, Z = 20, β = 15° so mt ≈ 2.071 mm and d ≈ 41.41 mm — larger than the spur twin at the same mn and Z. Set β = 0 to recover spur diameters on this pad.
Addendum and dedendum stay on mn (ha = mn, hf = 1.25mn) in the normal system with zero profile shift.
Lead L = π·mn·Z/sinβ is the axial advance for one full turn — useful for milling setup.
Centre distance grows with cosβ — and hands must oppose
Standard external spacing is a = mn(Z1+Z2)/(2 cosβ). Parallel shafts need the same mn and β but opposite helix hands.
Default mate 40T on mn2 β15 gives a ≈ 62.12 mm and i = 2 — about 3.5% longer than the spur a = 60 mm at the same teeth and module. Total contact ratio εγ = εα + εβ uses face width for the axial part.
Default helical pair vs spur twin:
| Case | Inputs | Key result |
|---|---|---|
| Helical default | 20T+40T · mn2 · β15° | a ≈ 62.12 mm · εγ ≈ 2.59 |
| β = 0 (spur limit) | Same teeth · mn2 | a = 60 mm |
| Single blank | Mating teeth = 0 | Diameters + lead only |
Pitch cylinders set shaft spacing; tip diameters do not.
Keep αn, mn, and |β| identical on both members for a standard parallel mesh.
Virtual teeth decide undercut risk
Helical gears behave like a virtual spur with zv = Z/cos³β. Undercut screening uses that count against Z_min ≈ 2/sin²αn. A low real tooth count can still clear the warn if β is large enough to lift zv.
Profile shift and ISO span / ball measurement live on deeper rating pads. This pad warns only — it does not size xn or export tooth form.
Lewis helical screen includes axial thrust
Cutting and force calculators separate geometry from load. Strength mode uses σ = Ft/(Cv·b·mn·Y) with Cv = 6/(6+v), Y from round(zv), plus Fa = Ft·tanβ and Fr = Ft·tanαn/cosβ.
Example path: mn3, 30T, β15°, b = 35 mm, 5 kW at 1200 rpm, allowable 140 MPa → Ft ≈ 854 N, Fa ≈ 229 N, σ ≈ 49.3 MPa, FoS ≈ 2.84 (Pass at ≥ 1.5). Full ISO 6336 / AGMA rating is out of scope.
Axial thrust Fa loads bearings along the shaft — size them for Fa, not only Ft.
Use FoS as a preliminary go/no-go before a catalog rating or ISO 6336 worksheet.
What this pad does not generate
CAD tooth-profile scripts, span/ball dimensions, and crossed-axis helical layouts are out of scope. Use this pad for normal-system blank sizing, pair a/εγ, lead, undercut warn on zv, and Lewis screening with thrust — then hand off to CAD or a signed rating.
Worked example
Default Blank geometry: mn = 2 mm, Z = 20, β = 15°, αn = 20°, mating Z = 40, b = 25 mm. Reproduce on CALCULATE.
- Leave Input mode on Blank geometry, 20° normal pressure angle.
- Confirm teeth 20, normal module 2, helix 15°, mating 40, face 25.
- CALCULATE. d ≈ 41.41 mm, da ≈ 45.41 mm, df ≈ 36.41 mm, L ≈ 485.5 mm. Results shows the 2D blank sketch.
- Switch to Pair / centre distance → a ≈ 62.12 mm, i = 2. Contact ratios → εγ ≈ 2.59.
- Optional: Strength mn3 · 30T · β15 · b35 · 5 kW · 1200 rpm · 140 MPa → FoS ≈ 2.84 Pass.
Result: Default blank: d ≈ 41.41 mm. Pair: a ≈ 62.12 mm, i 2:1. Strength example: FoS ≈ 2.84 on the mn3/30T path.
When to use
- Sizing a helical blank OD before turning or hobbing
- Setting parallel-shaft centre distance for a helical pair
- Computing lead and axial pitch for milling setup
- Screening εα / εβ / εγ before housing tryout
- Checking undercut risk via virtual teeth
- First-pass Lewis FoS or Ft/Fa/Fr before catalog selection
Limitations
- External parallel-shaft helical only. No crossed-axis, herringbone, bevel, worm, or internal gears.
- Zero profile shift; no tip relief or backlash design.
- No span, measurement-over-balls, or tooth-form export.
- Lewis + helical Barth Cv is a preliminary bending screen, not ISO 6336 / AGMA rating.
- Lewis Y uses the common 20° table on virtual teeth for all αn.
- At β = 0 the pad reports spur diameters and centre distance; lead and axial pitch are undefined.
- Hand (RH/LH) on the sketch is a mesh rule reminder — not a signed CAD twist export.
- Not a substitute for a signed gearbox selection on safety-critical drives.
FAQ
- How do I calculate helical gear pitch diameter?
- d = mn·Z/cosβ (or mt·Z with mt = mn/cosβ). Defaults: 2×20/cos15° ≈ 41.41 mm.
- What is the helical centre distance formula?
- For two external gears with no profile shift, a = mn(Z1+Z2)/(2 cosβ). Use Pair / centre distance mode with mating teeth set.
- What do the six input modes change?
- Each mode picks which fields matter and which 2D Results sketch you see: blank diameters, pair a/εγ, Lewis FoS, lead/px unwrap, contact-ratio stack, or Ft/Fa/Fr triangle. Geometry formulas stay the same underneath.
- Why is helical centre distance larger than spur?
- Because pitch diameters inflate with 1/cosβ. Same mn and teeth at β = 15° sit farther apart than at β = 0°.
- What is virtual number of teeth?
- zv = Z/cos³β. Lewis Y and undercut screening use this equivalent spur count.
- Do parallel helical gears use the same hand?
- No. Same mn and β, opposite hands (RH with LH). Same-hand pairs are for crossed shafts — not this pad.
- Is this ISO 6336 strength?
- No. Strength mode is Lewis with helical Cv and Ft/Fa/Fr. ISO 6336 / AGMA 2001 dynamic and pitting factors are not computed.
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