Belt-drive output speed from pulley diameters

Technical article

Understand Belt-drive output speed from pulley diameters

This calculator finds the missing quantity among output speed n₂, input speed n₁, and the two pulley diameters d₁ and d₂ from the other three. It answers, in one flexible calculator, both 'What speed does the driven pulley run at?' and 'What pulley diameter do I need for a desired output speed?'.

What does this quantity describe?

For a slip-free belt drive, output speed is inversely proportional to the driven pulley's diameter: n₂ = n₁·d₁/d₂. Larger pulleys correspondingly turn slower than smaller ones.

A combustion engine's accessory drive uses exactly this relationship: a small pulley on the crankshaft drives a larger or smaller pulley on the alternator via a serpentine belt. If the alternator pulley is smaller than the crankshaft pulley, the alternator spins considerably faster than the engine itself — often at ratios of 2:1 to 3:1, so enough charging current is generated even at idle.

Formula and variables

n₂ = n₁ · d₁ / d₂

n₂ = n₁ · d₁ / d₂
n₁ = n₂ · d₂ / d₁
d₂ = n₁ · d₁ / n₂

Symbol / input	Meaning
Output speed n₂	Speed of the driven pulley.
Input speed n₁	Speed of the driving pulley.
Driving pulley d₁	Pitch diameter of the driving pulley.
Driven pulley d₂	Pitch diameter of the driven pulley.

Choose the inputs correctly

Enter three of the four quantities to find the fourth. Use pitch diameters for d₁ and d₂.

How to use the calculator

Select the target quantity and enter the three known values with units. For very precise applications, account for a small slip margin below this ideal calculated value.

Worked example

Given n₁ = 1,500 rpm, d₁ = 100 mm and d₂ = 250 mm, substitution gives n₂ = 1,500 · 100/250 = 600 rpm.

Understand the result and units

An output speed of 600 rpm shows the actual speed of the larger, driven pulley at this input speed and these diameters. To change output speed without touching input speed, one of the two pulley diameters must instead be adjusted.

Speeds are given in revolutions per minute (rpm), diameters in millimetres.

Typical applications

The formula is used to design accessory drives, fan and pump drives, machine tool spindles, and to size a replacement pulley to adjust an output speed.

Assumptions, limits and common mistakes

The formula assumes slip-free force transmission. Real belt drives additionally show slight creep and, under overload, true slip, so actual output speed can sit slightly below the calculated value. For V-belts, a practical maximum ratio of about 1:15 also applies.

Common mistake: Do not substitute a pulley's plain outside diameter for its pitch diameter. Also, when replacing just one pulley, remember that this automatically changes the required belt length and wrap angle too — both should be rechecked with the corresponding calculators.

Frequently asked questions

How do I change output speed without replacing the drive motor?

By swapping one or both pulleys for a different diameter — a larger driven pulley lowers output speed, a smaller one raises it.

Why does a car's alternator spin faster than the engine?

Because its pulley is smaller than the crankshaft pulley. Per n₂ = n₁·d₁/d₂, a smaller driven pulley gives a higher output speed.

Does swapping a pulley also affect belt length and wrap angle?

Yes. A changed diameter at the same center distance alters both the required belt length and the wrap angle at the small pulley; both should be recalculated separately.