3D printer belt tension calculator: GT2 frequency to newtons and back

The formula

A free belt span vibrates like a guitar string. Its lowest natural frequency depends on tension, span length and the belt’s mass per meter:

f = (1 / 2L) × √(T / μ)

Solved for tension, and for the frequency to tune to:

T (N)  = 4 × μ × L² × f²
f (Hz) = √(T / (4 × μ × L²))

• T is the tension in newtons.
• μ is the belt mass in kg per meter (8.4 g/m is 0.0084 kg/m).
• L is the free span in meters, from where the belt leaves one pulley or idler to where it meets the next.
• f is the pluck frequency in hertz.

Gates writes the same formula for its sonic tension meter with the span in millimeters and the mass per millimeter of belt width: T = 4 × M × W × S² × f² × 10⁻⁹. Prusa’s firmware uses it too, citing the taut-string formula.

Frequency rises with the square root of tension. A 10% higher pitch means 21% more tension. Double the pitch and the tension quadruples.

How do you calculate belt tension from frequency?

Multiply four by the belt’s mass per meter in kg/m, by the squared free span in meters and by the squared frequency: T = 4 × μ × L² × f². A 6 mm GT2 belt (8.4 g/m) plucked at 110 Hz over a 150 mm span carries 9.15 N, about 0.93 kgf.

Worked example: a 6 mm belt at 110 Hz

These are the calculator’s defaults, which match Voron’s A/B belt target.

1. Mass per meter: 1.4 g/m per mm × 6 mm = 8.4 g/m = 0.0084 kg/m.
2. Span: 150 mm = 0.15 m, so L² = 0.0225 m².
3. Frequency squared: 110² = 12,100.
4. Tension: 4 × 0.0084 × 0.0225 × 12,100 = 9.15 N.
5. In other units: 9.15 / 9.807 = 0.93 kgf, and 9.15 / 4.448 = 2.06 lbf.

The same 110 Hz on a 9 mm belt (12.6 g/m) is 13.7 N, because the heavier belt needs more tension to ring at the same pitch. Pick the right belt width before you trust the result.

How to measure the frequency

1. Park the axis so the span you measure has a fixed, known length. Voron moves the X extrusion until 150 mm of A/B belt is free. Prusa moves the gantry to the front and the Nextruder to the right on the CORE One.
2. Pluck the middle of the span with a finger and hold the phone’s microphone close to it.
3. Read the lowest strong peak on a spectrum app. Voron’s docs list Sound Spectrum Analysis for iOS, Spectroid for Android and Gates Carbon Drive for both. Prusa’s belt tuner runs in a phone browser.
4. Measure the span from contact point to contact point with a ruler or calipers.

Klipper’s TEST_RESONANCES does not read a pluck frequency. On a CoreXY printer (two belts that drive X and Y together), run TEST_RESONANCES AXIS=1,1 OUTPUT=raw_data and TEST_RESONANCES AXIS=1,-1 OUTPUT=raw_data, then plot both files with graph_accelerometer.py. Similar curves mean the two belts carry similar tension.

What frequency should a GT2 belt be tuned to?

Tune to your printer maker’s figure, because the right frequency depends on the span. Voron sets A/B belts to about 110 Hz over 150 mm. Prusa’s belt tuner targets 98 Hz upper and 92 Hz lower on the CORE One, and 85 Hz X and 95 Hz Y on the MK4/S.

Pick a row under “Published target” in the calculator to load its frequency and span.

Printer and belt	Frequency	Span	Tension
Voron A/B belts	110 Hz	150 mm	9.15 N (6 mm), 13.7 N (9 mm)
Voron V2 Z belts	140 Hz	150 mm	14.8 N (6 mm), 22.2 N (9 mm)
Prusa CORE One, upper	98 Hz	262 mm	20.0 N
Prusa CORE One, lower	92 Hz	272 mm	19.0 N
Prusa CORE One L, upper	92 Hz	311.5 mm	24.9 N
Prusa CORE One L, lower	88.5 Hz	324.5 mm	25.0 N
Prusa MK4/S, X / Y	85 / 95 Hz	not published	needs your span
Prusa XL, XY belts	80 Hz	not published	needs your span
Prusa MK3/S/+, X / Y	100 / 120 Hz	not published	needs your span
Prusa MINI, X / Y	110 / 110 Hz	not published	needs your span

Where the targets come from:

• Voron: the docs give 110 Hz over 150 mm for A/B belts and call it roughly 2 lb, “on the lower end of the range”. For V2 Z belts, 140 Hz over 150 mm is the starting point.
• Prusa CORE One: frequencies are from Prusa’s belt tuner. The spans come from Prusa’s firmware, which aims for 19 N on both belts and puts the upper span at 262 mm and the lower at 272 mm. The help center accepts 90 to 98 Hz with no more than 8 Hz between the two belts.
• Prusa CORE One L: the firmware aims for 25 N and converts it to 92 Hz (upper) and 88.5 Hz (lower). The help center gives a range of 85 to 95 Hz.
• Prusa tensions in the table use the firmware’s own belt mass, 7.569 g/m.
• Prusa MK4/S, XL, MK3 and MINI: the belt tuner gives a frequency per axis but no span, so the calculator can’t turn these into newtons unless you measure your span.
• Bambu Lab: the X1, P1 and H2D series publish no frequency. Their tensioners are spring-loaded: loosen the screws, move the toolhead as the wiki shows, and tighten them again. Then rerun the printer’s resonance calibration.

Same tension, different span

Frequency scales with 1 / L at a fixed tension. If you can only pluck 200 mm of belt where your target was set at 150 mm, tune to 110 × 150 / 200 = 82.5 Hz for the same 9.15 N. Use the tension-to-frequency direction in the calculator with your own span.

Printer targets against the belt’s rating

Printer targets sit near the bottom of what Gates publishes for its 2 mm pitch belts. Gates sells its 2MGT belts as PowerGrip GT3, which runs in GT2 sprockets. These figures come from its design manual, converted from pounds.

Gates 2MGT GT3 figure	6 mm	9 mm
Minimum static tension per span (Table 7)	9.3 N	14.2 N
General static tension per span, power drives (Table 8)	44.5 N	75.6 N
Rated working tension, 20-groove pulley (Table 6)	173 N	289 N
Minimum breaking strength, fiberglass cord (Table 6)	556 N	845 N

Gates’ minimum is the least tension that makes a lightly loaded belt wrap and mesh with its sprockets. Voron’s 110 Hz target gives 9.15 N on a 6 mm belt, 0.98 times that minimum and 5.3% of the working rating. Prusa’s 19 to 25 N targets are 11 to 14% of the 6 mm rating. The belt’s strength doesn’t set the target, so use the printer maker’s figure.

Belt mass per meter

The calculator’s presets come from two published values:

• Gates 2 mm pitch belts: 1.4 g/m per mm of width, from the mass constant table in the Gates 550C sonic tension meter manual. That gives 8.4 g/m for 6 mm and 12.6 g/m for 9 mm. For another width, multiply 1.4 by the width in millimeters.
• Prusa’s firmware: 7.569 g/m for one meter of belt. The firmware uses this one value for the CORE One, CORE One L and iX.

Belts from other makers can differ. Tension is proportional to mass, so a belt 10% lighter than the preset reads 10% less tension at the same pitch. To check yours, weigh a measured length of spare belt on a scale that reads to 0.01 g and divide the grams by the meters. Then pick “Other” and type the result.

Gates also notes that a belt is stiffer than a string, so a frequency-based reading can come out above the real tension. When the exact figure matters, Gates recommends a calibration test.