How it works
A worm gear set pairs a worm (a screw with Z₁ starts) with
a worm wheel of Z₂ teeth. The ratio is the wheel tooth
count divided by the number of starts:
i = Z₂ / Z₁
so a single start gives a ratio equal to the wheel tooth count — a large reduction from
one mesh. The wheel pitch diameter is d₂ = m · Z₂ (axial module
m) and, with a worm pitch diameter d₁, the center distance is
a = (d₁ + d₂) / 2.
The lead is the axial advance of the worm per revolution,
L = Z₁ · π · m, and the lead angle is the helix angle of
the thread at the worm pitch diameter, λ = atan(L / (π · d₁)). A small lead
angle tends to be self-locking — below about 5–6°, the wheel cannot
back-drive the worm — but it is also less efficient, because the lower the lead angle the
more the teeth slide rather than roll.
Worked example
A single-start worm (Z₁ = 1) driving a Z₂ = 40 tooth wheel, with
axial module m = 2 mm and a worm pitch diameter d₁ = 20 mm, is a
40:1 reduction. The wheel pitch diameter is m · Z₂ = 80 mm,
the center distance is (20 + 80)/2 = 50 mm, and the lead is
1 · π · 2 ≈ 6.28 mm. The lead angle is
atan(6.28 / (π · 20)) = atan(0.1) ≈ 5.7° — below the friction angle, so this
set is likely self-locking. Those are the numbers the calculator shows.
Frequently asked questions
- How do you calculate a worm gear ratio?
- Divide the worm-wheel tooth count by the number of worm starts: ratio = Z₂ / Z₁. A single-start worm driving a 40-tooth wheel is a 40:1 reduction; a 2-start worm on the same wheel is 20:1.
- Why do worm drives give such a high reduction in one stage?
- Because the worm typically has just one start (one thread), each turn of the worm advances the wheel by only one tooth. With Z₂ wheel teeth that is a Z₂:1 reduction from a single mesh — a ratio that would take two or three stages of ordinary spur gears.
- What is the lead and lead angle of a worm?
- The lead L is the axial distance the worm advances per revolution, L = Z₁ · π · m, where m is the axial module. The lead angle λ is the angle of the thread helix at the worm pitch diameter d₁: λ = atan(L / (π · d₁)). A single-start worm has a small lead and a small lead angle.
- When is a worm gear self-locking?
- A worm set tends to be self-locking — the wheel cannot back-drive the worm — when the lead angle is below the friction angle, roughly λ ≲ 5–6°. Single-start worms usually fall in this range. It depends on the friction coefficient, lubrication and vibration, so treat it as a strong guide, not a guarantee.
- What is the efficiency of a worm gear?
- Worm-drive efficiency drops as the lead angle drops: a low lead angle (which gives the high reduction and self-locking) means more sliding and more friction loss. Multi-start worms with a larger lead angle are more efficient but reduce less and are less likely to self-lock.
- Does this work in metric and imperial?
- Yes — toggle SI/Imperial in the header to switch the module, diameters and center distance between mm and inches. The ratio and lead angle are unit-independent.
Method & assumptions
- Single-enveloping worm set described by the axial module of the worm; the wheel module matches.
- Self-locking is judged from the lead angle alone (λ ≲ 5–6°). Real back-driving depends on the friction coefficient, lubrication and vibration — this is a guide, not a guarantee.
- Efficiency falls as the lead angle drops; a low lead angle (high reduction, self-locking) means more sliding and more friction loss.
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