The frequency method: T = 4 · m · L² · f²

A tensioned belt span behaves like a guitar string: the tighter it is, the higher it vibrates. For a flexible strand fixed at both ends, string theory gives the fundamental frequency f = (1/2L)·√(T/m), where T is strand tension, m is mass per unit length, and L is the free span. Solving for tension inverts one line of algebra into the working formula:

T = 4 · m · L² · f²

with T in newtons when m is in kg/m and L is in meters. This is exactly the relation commercial sonic tension meters implement — the Gates manual linked above documents it as T = 4 × S² × M × W × f² × 10⁻⁹ with S in millimeters and mass scaled per belt width, which is the same equation in different units. The practical procedure: pluck the span sharply at its middle, measure the ring-down with a phone’s spectrum app or a dedicated meter, and use the dominant low frequency — not the harmonics above it. Free spans on small industrial drives typically ring between roughly 30 and 120 Hz; if your app shows a forest of peaks, the lowest strong peak is usually the fundamental.

A worked example with the first preset: a 400 mm span (0.4 m) of belt measuring 0.20 kg/m rings at 70 Hz. Then T = 4 × 0.20 × 0.16 × 4900 = 627 N, or about 141 lbf per strand. The same arithmetic in imperial inputs — a 20 in (0.508 m) span at 0.13 lb/ft (0.193 kg/m) ringing at 60 Hz — gives T = 4 × 0.193 × 0.258 × 3600 = 718 N ≈ 161 lbf. Note how sensitive the result is to the square terms: a 10% error in span length shifts tension 21%, and the same error in frequency shifts it 21% in the other direction. Measure both carefully; the formula punishes sloppiness quadratically.

Getting the span mass right

The mass input is the easiest to get and the most often faked. The honest method takes thirty seconds: weigh the belt on any scale accurate to a few grams and divide by the belt’s pitch length in meters. A 2240 mm SPB belt weighing 0.52 kg is 0.232 kg/m. Datasheet “weight per belt” figures work too, but weigh the actual part if you have it — worn belts have lost rubber and run lighter than the catalog value, which biases computed tension low and invites over-tensioning on the next pass.

The deflection method: 1.6 mm per 100 mm of span

The older shop method needs no electronics. Press the span at its midpoint with a spring scale until the belt deflects by 1.6 mm for every 100 mm of free span — one sixty-fourth of an inch per inch — and read the force at that deflection. That force is the value compared against the manufacturer’s deflection-force table. The “deflection calculator” mode above computes the target depth for your span so you are not doing fractions on the shop floor: a 400 mm span wants 6.4 mm of press, a 20 in span wants 5/16 in.

What the method deliberately does not do here is hand you a force table. Correct deflection forces depend on belt section, number of ribs, span length, sheave diameter and manufacturer construction, and the published numbers differ meaningfully between brands. Hard-coding one vendor’s table would quietly mislead users of every other brand, so this tool treats the recommended force as an input: look up your belt section in the manual that shipped with the drive (or the manufacturer’s published tension table), then compare your scale reading at the computed depth. The two methods cross-check each other nicely — a drive whose scale-force and frequency-derived tension agree within roughly 15% is set correctly.

Where each method wins

The frequency method wins on long, accessible spans, on drives where a scale cannot push squarely at mid-span, and for tracking tension drift over time — the same span, same mass, re-plucked weekly gives a trend line for nothing but the cost of a phone app. The deflection method wins on short stiff spans (below about 200 mm the ring-down is too fast to read reliably), on very heavy multi-belt bands where one strand’s vibration couples into its neighbors, and in noisy shops where a microphone picks up everything except the belt. Neither method reads tension inside a guard or on a running drive — both are static measurements made with the drive locked out.

What tension costs the bearings: shaft-load estimation

Tension does not disappear into the belt — it loads every shaft the belt wraps. With strand tension T on both the tight and slack side (approximately true for a static, non-running drive) and wrap angle θ, the radial load on that pulley’s shaft is P = 2 · T · sin(θ/2). The “Shaft load” mode above evaluates it: 600 N of strand tension across a 120° wrap — the classic minimum for a well-designed open drive — puts 1039 N (234 lbf) of radial force on the shaft. Shrink the wrap to 90° and the same tension loads the shaft with 849 N; grow it to 180° and it is 1200 N.

Two design consequences follow. First, idlers and small wrap angles are bearing-friendly but slip-prone: the pulley with the smallest wrap is the first to slip when load rises, which is why the multi-pulley solver flags wraps below about 120°. Second, tension is a tax every bearing pays continuously — a drive run at twice its required tension doubles the belt-induced share of bearing load around the clock, and bearing life scales roughly with the inverse cube of load, so “just a bit tight” is a measurable shortening of the bearing’s calendar. When you have computed tension here and wrap angles from the geometry tools, the numbers feed directly into any bearing-life calculation your machine’s manual prescribes.

For spring-loaded tensioners the static estimate changes: the spring preloads one strand, the drive’s working load unbalances the two strands, and the sum — not the average — stays close to constant. Treat the frequency-method result as the average strand tension, and let the tensioner specification handle the dynamic split. If the belt geometry on a tensioned serpentine drive is still being decided, solve the layout first in the multi-pulley calculator — its wrap-angle table is the direct input to the shaft-load formula here.

Reading the drive: over-tension and under-tension signatures

The calculator tells you the number; the drive tells you whether the number is right. Under-tension leaves a clear signature: squeal on startup, shiny glazed belt faces, black rubber dust collecting under the drive, and — measured — a frequency well below the specification, falling further within days as the belt elongates into its new seated position. New belts should be re-checked after the first run-in window, typically within the first day of service, because seat-in and elongation drop static tension 2–5% on a fresh install.

Over-tension hides better, which is why it does more damage. There is no squeal — the opposite, drives run quiet and grip hard — so the failure surfaces elsewhere: bearing housings hot to the touch after an hour of running, belt cords whitened and stretched at the pitch line, cracks along the belt’s underside from flex fatigue at excess load, and in the frequency numbers a reading far above the manufacturer’s span. If a drive needs re-tensioning weekly, the belt is either stretched past its service life or the span geometry changed — measure the belt and pulleys with the guides on this site (belt and pulley) before turning the tensioner again.

Every tool referenced here — length, speed, tension and the measurement guides behind them — is indexed on the calculators hub alongside the measurement guides. Bookmark it; the shortest path back to any part of the drive-design workflow runs through it.

Frequently asked questions