Timing-belt linear axis speed

What does this quantity describe?

Each revolution of the drive pulley, the positively engaged timing belt transports exactly z teeth spaced by pitch p, i.e. a distance z·p. At a rotational speed n this gives a feed speed v = z·p·n/60,000, with p in mm, n in rpm and v in m/s.

This works like a rack running over a pinion: each pinion tooth advances the rack by exactly one pitch, regardless of friction or contact pressure. That is why positioning is positive and slip-free, unlike a friction-wheel or flat-belt drive.

Formula and variables

v = z · p · n / 60,000

• v = z · p · n / 60,000
• z = v · 60,000/(p·n)
• n = v · 60,000/(z·p)
• p = v · 60,000/(z·n)

Choose the inputs correctly

Drive pulley tooth count z, belt pitch p (e.g. 2 mm for GT2, 3 mm for HTD 3M) and drive speed n set the feed speed. For a sizing question — what speed a target feed rate needs — n can be chosen as the target.

How to use the calculator

Choose v as the target to compute feed speed for a given pulley and speed. For motor sizing, choose n as the target and enter the desired maximum speed.

Worked example

20 teeth, a 2 mm GT2 pitch and 3,000 rpm give v = 20 · 2 mm · 3,000 / 60,000 = 2 m/s.

Understand the result and units

Given the motor's step count, z·p additionally gives the travel per motor revolution in mm — a key parameter for the steps-per-mm setting of a CNC or 3D printer controller.

Tooth count is dimensionless, belt pitch is given in mm, speed in rpm, and the resulting feed speed in m/s or mm/s for small axes.

Relation to steps per mm in motion control

In stepper-driven systems, the axis is often parameterized via the steps-per-millimeter setting. It follows from the motor's step count per revolution, multiplied by the driver's microstepping factor, divided by the distance traveled per revolution z·p. An error in this setting causes a proportionally wrong actual travel distance even though the controller reports a correct target position — a common cause of dimensional errors in 3D printers and CNC mills.

Typical applications

The relationship is used to design linear axes for 3D printers, laser cutters, pick-and-place systems and CNC gantries, to find the achievable travel speed from motor speed and belt geometry, or the motor speed needed for a target speed.

Assumptions, limits and common mistakes

The model assumes ideally slip-free, positive engagement between belt and pulley. Belt stretch under load, tooth skipping from overload or insufficient tension, and the stepper driver's microstep resolution are not captured separately and limit the actually achievable positioning accuracy.

Common mistake: A common mistake is confusing belt pitch with pulley diameter or using the driven side's tooth count — only the drive pulley matters for feed speed. It is also easy to overlook that different timing-belt profiles such as GT2, HTD or T5 use different, non-interchangeable pitches.