Timing belt length calculator
Belt length from the centre distance, or centre distance from the belt you have, for two pulleys on GT2, HTD, T5, AT5, XL, MXL and the rest of the 34 profiles the pulley generator makes. Exact pitch-line maths, whole-tooth belt sizes, and a check on wrap and teeth in mesh.
The generator opens with these pulleys, the belt and the centre distance set. Add bores, hubs and flanges there, and download STL, STEP, DXF or SVG.
The exact open-belt length
Each pulley's pitch radius is R = teeth × pitch ÷ 2π. With the smaller radius R1, the larger R2 and the centre distance C, the angle of the straight spans is sin α = (R2 − R1) ÷ C, and the belt's pitch length is
L = 2·C·cos α + R1·(π − 2α) + R2·(π + 2α)
That is the same formula the Timing Pulley Generator uses for its drives, so a drive opened from here shows the same belt and centre distance. Going the other way, from a belt to a centre distance, the calculator solves the formula numerically.
A belt is a whole number of teeth, so the length you get at an arbitrary centre distance rarely matches a real belt. The calculator offers the whole-tooth belt just shorter and just longer, each with the exact centre distance it needs. The generator picks the longer one when it snaps a drive.
The wrap angle on the smaller pulley is 180° − 2α, and the teeth in mesh are that share of its teeth, rounded down. Gates recommends at least 6 teeth in mesh.
Timing belt questions
How is timing belt length calculated?
For an open belt on two pulleys, the pitch length is L = 2·C·cos(α) + R1·(π − 2α) + R2·(π + 2α), where C is the centre distance, R1 and R2 are the pitch radii (R = teeth × pitch ÷ 2π, R1 the smaller) and sin(α) = (R2 − R1) ÷ C. This calculator uses that exact formula, the same one the Timing Pulley Generator uses, not the common approximation.
Why does the belt have to be a whole number of teeth?
A timing belt is a closed loop of teeth, so its pitch length is always a whole number of pitches: 200 teeth of GT2 is exactly 400 mm. Pick the centre distance you want, then choose the whole-tooth belt just shorter or just longer; the calculator gives the exact centre distance for each.
How many teeth should be in mesh?
Gates recommends at least 6 teeth in mesh on the smaller pulley; below that the belt can jump teeth and its rated torque drops. Teeth in mesh is the wrap angle on the small pulley times its tooth count, divided by 360, rounded down.
What is the pitch diameter of a timing pulley?
Pitch diameter = teeth × pitch ÷ π. It is the diameter of the belt's pitch line, which is larger than the pulley's outside diameter by twice the pitch line differential (for example 0.254 mm each side for GT2). A 20-tooth GT2 pulley has a 12.73 mm pitch diameter and a 12.22 mm outside diameter.
Does this work for GT2 belts in 3D printers?
Yes. Choose GT2 (2 mm), enter the motor and driven pulley tooth counts, and set the centre distance. GT2 closed loops are sold by length in 2 mm steps, so any whole-tooth length below is a real belt size; check your supplier stocks it.
How do I tension the belt?
Design for the exact centre distance of a stock belt, then allow a few millimetres of adjustment (slotted motor holes or an idler) to take up tolerance and stretch. Don't shorten the centre distance to tension a belt: it reduces wrap.
Now make the pulleys
Bores, set screws, hubs, spokes and flanges, checked against the standards, as STL, STEP, DXF or SVG.