Worm Gear Calculator

Gear Ratio · Torque · Efficiency · Lead Angle · Output Speed

Worm Gear Calculator

Calculation type:

Gear ratio i

Output torque M₂

Efficiency η

Lead angle γ

Output speed n₂

Worm wheel teeth z₂:

Worm starts z₁:

Formulas & Legend

1. Gear ratio:

\[i = \frac{z_2}{z_1}\]

2. Output torque:

\[M_2 = M_1 \cdot i \cdot \eta\]

3. Efficiency:

\[\eta = \frac{\tan(\gamma)}{\tan(\gamma + \varphi)}\]

γ = lead angle, φ = friction angle = arctan(μ)

4. Lead angle:

\[\gamma = \arctan\!\left(\frac{z_1 \cdot m}{d_1}\right)\]

m = module, d₁ = worm pitch circle diameter

5. Output speed:

\[n_2 = \frac{n_1}{i}\]

Legend:

z₁	Number of worm starts
z₂	Number of worm wheel teeth
i	Gear ratio
M₁/M₂	Input/output torque (Nm)
η	Efficiency (0–1)
γ	Lead angle (°)
φ	Friction angle = arctan(μ)
m	Module (mm)
d₁	Worm pitch diameter (mm)
n₁/n₂	Input/output speed (rpm)

Detailed Description

What is a worm gear?

A worm gear (worm drive) consists of two main elements: the helical worm (input) and the worm wheel (output). The axes are typically perpendicular (90°). Worm gears achieve high gear ratios in a single stage within a very compact envelope.

Working principle

The worm resembles a screw with one or more starts (z₁ = 1, 2, 4, …). For each full revolution of the worm, the wheel advances by exactly z₁ teeth. This gives the gear ratio:

\[i = \frac{z_2}{z_1}\]

A 2-start worm with z₂ = 40 yields i = 20: the wheel rotates 20 times slower than the worm.

Efficiency and self-locking

Due to sliding contact between worm and wheel, friction losses are higher than in rolling-contact gears. Efficiency depends on the lead angle γ and the friction angle φ = arctan(μ):

\[\eta = \frac{\tan\gamma}{\tan(\gamma+\varphi)}\]

Self-locking: When γ < φ the gear cannot be back-driven — a useful safety feature for lifting equipment and actuators.

Typical values

Parameter	Typical range	Note
Gear ratio i	5 … 100 (up to 500)	Single-stage achievable
Efficiency η	0.50 … 0.92	Depends on γ and lubrication
Worm starts z₁	1, 2, 4, 6	More starts → higher η, lower i
Friction coeff. μ (steel/bronze)	0.04 … 0.12	With good lubrication
Lead angle γ	2° … 30°	<6° usually self-locking

Practical example

Given: z₁ = 2, z₂ = 40, M₁ = 5 Nm, η = 0.80

Gear ratio: i = 40 / 2 = 20

Output torque: M₂ = 5 · 20 · 0.80 = 80 Nm

Output speed (at n₁ = 1400 rpm): n₂ = 1400 / 20 = 70 rpm

Advantages & disadvantages

✔ Advantages

Very high single-stage ratios
Quiet, smooth operation
Self-locking possible
Compact form factor
Right-angle power transmission

✘ Disadvantages

Lower efficiency (sliding friction)
Heat generation under high load
Worm wheel typically bronze (costly)
Good lubrication required
Limited power density

Applications

Elevators & hoists – self-locking as a safety feature
Conveyors – low-speed material handling
Steering gears – automotive rack-and-pinion steering
Actuators – solar trackers, antenna pointing
Machine tools – feed axes, indexing tables

FAQ

Why is worm gear efficiency lower?

Because worm and wheel slide rather than roll. Sliding friction produces heat and losses that decrease as the lead angle increases.

How do I calculate the lead angle?

γ = arctan(z₁ · m / d₁). A larger z₁ or module m gives a steeper worm and higher efficiency.

When does self-locking occur?

When the lead angle γ is less than the friction angle φ = arctan(μ). For μ = 0.1, φ ≈ 5.7° — any γ below that yields self-locking.

Key Takeaways

✓ i = z₂ / z₁ (high ratios in one stage)
✓ η = tan(γ) / tan(γ+φ) (depends on lead angle and friction)
✓ M₂ = M₁ · i · η (output torque)
✓ Self-locking when γ < φ
✓ More starts (z₁↑) → better efficiency, lower ratio

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