Belts and Pulleys
Flat, V and timing belts: speed ratio, belt length, wrap and tension, power rating, sizing and the pulley and tooth counts for timing drives.
Belt types
A belt drive carries power between shafts that may be meters apart, quietly and without lubrication, and damps shocks and vibration. The belt also acts as a mechanical fuse and is cheap to replace. The price is a lower power density than gears and, for friction belts, a little slip.
| Type | How it transmits | Typical use | Notes |
|---|---|---|---|
| Flat belt | Friction on the pulley rim | Conveyors, older machinery, high-speed light drives | Simple; needs high tension; polyamide or fabric-reinforced types are efficient |
| V-belt (classical, narrow) | Friction in a groove: wedging multiplies the grip | Fans, pumps, compressors, general industry | Compact, tolerates a short center distance; multiple belts side by side for more power |
| Poly-V (ribbed) belt | Friction on many small ribs | Car accessories, appliances | Flexible, runs on small pulleys at high speed |
| Timing (synchronous) belt | Teeth mesh with the pulley: no slip | Cams, 3D printers, CNC, robots, positioning | Constant speed ratio; low tension; needs correct pitch (HTD, GT2, T5) |
| Round belt | Friction on a grooved sheave | Light drives, small machines | Cheap, allows twists and angles |
Speed ratio, belt length and wrap
i = D2 ÷ D1 = n1 ÷ n2 v = π D n ÷ 60 L = 2C + π (D1 + D2) ÷ 2 + (D2 − D1)² ÷ (4C) θ = π − 2 asin((D2 − D1) ÷ (2C))
D1 and D2 are the pitch diameters of the small and large pulleys, C the center distance, L the belt length and θ the wrap angle on the smaller pulley (in radians; it is always less than 180° for an open belt). The speed ratio of a friction belt is slightly below the diameter ratio because of slip of 1 to 2 percent; a timing belt has the exact ratio of the tooth counts. To find the center distance from a standard belt length use C = [B + √(B² − 32 (D2 − D1)²)] ÷ 16 with B = 4L − 2π (D1 + D2). Belts are sold in standard lengths, so choose the closest and recompute C (allowing an adjustment of 2 to 3 percent for installation and tensioning).
Tension, friction and power
A friction belt carries power because the tight-side tension T1 is higher than the slack-side tension T2. The difference is the effective pull Fe = T1 − T2, and the power is P = (T1 − T2) × v. The ratio is limited by friction: before the belt slips on the pulley, T1 ÷ T2 = eμθ (the capstan equation), with μ the friction coefficient and θ the wrap angle in radians.
- V-belts: the wedge multiplies the friction: μeff = μ ÷ sin(β ÷ 2), with β the groove angle (typically 34° to 40°). With β = 38° the factor is 3.1, so a V-belt grips about three times better than a flat belt at the same tension.
- Centrifugal tension: a fast belt is pulled outward by its own mass: Tc = m v² (m in kg/m, v in m/s). It reduces the grip and adds to the stress, which is why belts have a speed limit (about 25 to 30 m/s for V-belts).
- Initial tension is set so that the slack side stays tight enough that the belt does not slip at the maximum load: T2 must not fall below the value from the equation.
- Bending stress from wrapping the small pulley is usually the largest stress in the belt, and why a minimum pulley diameter is specified for each belt section.
Worked example: a V-belt drive
- Duty: a 3.0 kW motor at 1 450 rpm drives a pump through a V-belt on a 125 mm driver and a 250 mm driven pulley at a center distance C = 500 mm.
- Speed: ratio 2.0 so the pump turns at 1 450 ÷ 2.0 = 725 rpm (about 1 to 2 percent less from slip). Belt speed v = π × 0.125 × 1 450 ÷ 60 = 9.49 m/s, well below the limit.
- Belt length: L = 2 × 500 + π × 375 ÷ 2 + (125)² ÷ (4 × 500) = 1 597 mm: the nearest standard belt length is used and the center distance adjusted to suit (check: for L = 1 597 mm the formula returns C = 500.0 mm).
- Wrap angle: θ = π − 2 asin(125 ÷ (2 × 500)) = 2.891 rad = 165.6°. Above 120° is acceptable; below that the capacity drops.
- Pull: Fe = P ÷ v = 3 000 ÷ 9.49 = 316 N.
- Tensions: μ = 0.30 with a 38° groove gives μeff = 0.92 and T1 ÷ T2 = e0.92 × 2.891 = 14.4. Then T2 = Fe ÷ (14.4 − 1) = 24 N and T1 = 340 N. Adding the centrifugal tension Tc = 0.12 × 9.49² = 10.8 N gives a peak of 351 N in the belt.
- Design power and belt count: with a service factor of 1.3 (a pump with a long daily duty) the drive is designed for 3 900 W. A single A- or SPZ-section belt of this size usually carries 2 to 4 kW; the manufacturer’s tables give the exact number of belts.
- For comparison: a flat belt on the same pulleys would give T1 ÷ T2 = e0.3 × π = 2.57 at 180° (less at 166°) and would need a far higher initial tension: the V-belt’s advantage.
Timing belts
A timing belt has teeth that mesh with the pulley: it cannot slip, so the ratio is exactly the ratio of tooth counts, and it needs only a light tension (just enough to keep the teeth engaged). It is used wherever position matters: engine cams, printers, CNC machines and robots. The belt and pulley must have the same tooth profile and pitch (an HTD 5M belt does not fit a GT2 pulley).
| Profile | Pitch (mm) | Typical use |
|---|---|---|
| GT2 | 2 | 3D printers, small robots, light CNC |
| GT3 | 3 | Larger printers, positioning |
| T5 (metric trapezoidal) | 5 | Older designs, light industrial |
| HTD 3M, 5M, 8M | 3, 5, 8 | Industrial servo drives (rounded teeth, high torque) |
| XL, L, H (inch) | 5.08, 9.525, 12.7 | Inch-system drives |
- Teeth in mesh: at least 6 teeth should be engaged on the smaller pulley to carry the rated load; for a 20-tooth pulley that is a wrap of 108°.
- Belt width and power: power capacity scales with the belt width and the speed; use the manufacturer’s table. For a drive of 150 W at 3 000 rpm on a 20-tooth, 2 mm pitch pulley (pitch diameter 12.7 mm), the torque is 0.48 N·m and the tooth force is 75 N: GT2 6 mm belts are rated for about 100 to 150 N continuous, so a wider belt or a bigger pitch is needed.
- Idlers and tensioners help wrap and tension, but add bending cycles: use smooth idlers on the back of the belt, at least as large as the smallest pulley.
- Stiffness and resonance: the stretch of a long belt limits the accuracy and the servo bandwidth (see Mechanical Transmission); use steel or aramid cord belts and keep the span short.
Worked example: a GT2 axis for makers
- A stepper (200 steps per revolution, 16 microsteps) drives a GT2 belt over a 20-tooth pulley (pitch 2 mm): one revolution moves the belt 20 × 2 = 40 mm.
- Steps per mm = 200 × 16 ÷ 40 = 80 steps/mm, so one microstep moves 12.5 µm. Enter this in the firmware (Marlin, Klipper, GRBL) as the steps per millimeter.
- For a 16-tooth pulley the travel is 32 mm per turn and the value is 100 steps/mm; for a 20-tooth it is 80 steps/mm. The pulley with fewer teeth gives finer resolution but a smaller wrap and more flexing.
- See Stepper Motors and Drivers for the pulse rate at speed (at 200 mm/s the axis needs 16 000 steps per second).
Setting the tension
Too slack and the belt slips (friction belts) or skips teeth (timing belts) and wears fast; too tight and the bearings, shafts and belt are overloaded and the life drops sharply. Tension is set with a gauge (a tension meter or deflection-force tester) or by sound: a belt span vibrates like a string at a frequency f = (1 ÷ 2L) √(T ÷ m), where L is the free span, T the tension and m the mass per unit length. A frequency meter app can read it.
Worked example: tuning a timing belt by frequency
A GT2 belt of mass 5.2 g/m (6 mm wide, typical) over a free span of 400 mm at a tension of 30 N vibrates at f = (1 ÷ (2 × 0.40)) × √(30 ÷ 0.0052) = 95 Hz; at half the tension (15 N) the frequency is 67 Hz. Pluck the span and adjust until the tone reaches the target.
- Tension a new belt, run it for a few hours and re-tension (the belt beds in).
- Align the pulleys: both the axes (parallel) and the faces (in one plane). Misalignment is the main cause of V-belt wear and of timing belt tracking off the flange.
- Keep belts clean and dry: oil and grease swell the rubber and cause slip.
Selecting a belt drive
- Design power = nominal power × service factor (1.0 smooth, 1.3 normal, 1.5 to 2 for heavy shock or long hours).
- Choose the belt section or pitch from the design power and speed (the manufacturer’s selection chart).
- Choose the pulley diameters: at least the minimum for the section (for a classical V-belt A section about 75 mm, B 125 mm; for a timing belt 15 or more teeth), and a ratio below about 7:1.
- Compute the length and the center distance (see above), with adjustment for tension.
- Number of belts or the width from the power rating per belt, corrected for the arc of contact and the length.
- Check the shaft loads: the bearing load is about T1 + T2 pulling the shafts together (see Shafts, Keys and Couplings and Bearings and Bearing Selection).
- Guard the drive and provide for replacement without stripping the machine.
| Need | Choose |
|---|---|
| General power transmission, motor to fan or pump | Classical or narrow V-belts, or a poly-V belt |
| Exact speed ratio or positioning | Timing belt (HTD, GT2) |
| High speed, light loads, long spans | Flat belt |
| Heavy shock or dirty places | Roller chain (see Chains and Sprockets) or gears |
| Large torque in a small space | Gears (see Gear Types and Selection) |
Troubleshooting belt drives
| Symptom | Likely cause | Check |
|---|---|---|
| Squeal at start-up or under load | Slipping (low tension, oil, worn belt) | Tension, clean the belt and pulleys, check the pulley grooves |
| Belt wears quickly on one edge | Misaligned pulleys | Straight edge across the pulley faces; parallel shafts |
| Belt cracks and hardens | Heat, small pulleys, ozone, age | Ambient temperature, pulley diameters, replace as a set |
| Belt flaps or whips | Resonance of a long slack span, low tension | Add an idler, adjust the tension |
| Timing belt skips teeth | Low tension, overload, a worn pulley, a small pulley | Tension, torque, wear; larger pulley or wider belt |
| Rapid bearing wear | Belt over-tensioned | Reduce to the minimum that does not slip |
| Noise from a timing belt | Misalignment, a too-tight belt, wrong profile | Alignment, tension; matching belt and pulley profile |
Related
- Gears and Power Transmission: the overview of this guide.
- Chains and Sprockets and Gear Trains, Ratios and Efficiency: the alternatives, gears and chains.
- Shafts, Keys and Couplings and Bearings and Bearing Selection: the shaft and bearing loads from belt tension.
- Mechanical Transmission: belts on servo axes, stiffness and resonance.
- Gear Ratio, Torque and Speed Generator (speed and ratio apply to pulleys too).
FAQ
V-belt or timing belt?
A V-belt is cheaper, tolerant of shock and slightly slipping, suitable for power transmission. A timing belt keeps an exact ratio and position and needs less tension, so is the choice for positioning and synchronized shafts.
How tight should a belt be?
As loose as possible without slipping (friction belts) or skipping (timing belts): use the manufacturer’s tension values and a gauge or frequency meter. Over-tightening is the commonest mistake.
Why does the belt length formula have a square term?
The belt leaves each pulley tangent to both, so the straight spans are slightly longer than the center distance: (D2 − D1)² ÷ (4C) is the correction for the difference in pulley sizes.
How many belts do I need?
Divide the design power (power × service factor) by the power rating of one belt (from the catalog, corrected for the wrap angle and the belt length), and round up.
References
- Shigley, J. E. and Mischke, C. R., Mechanical Engineering Design, McGraw-Hill: flexible mechanical elements.
- Gates, Optibelt, Contitech and Gates Mectrol design manuals (V-belt, timing belt, GT2, HTD selection).
- ISO 4184 (classical V-belts), ISO 5294 (synchronous belt drives), ISO 9608 (poly-V belts).
Last reviewed: October 2026. Ratings, masses and friction coefficients are typical; use the belt manufacturer’s tables for real selection.