Timing belt drive calculation
Design an open two-pulley timing belt drive: profile (trapezoidal T/AT or HTD), tooth counts and speed, together with either the desired centre distance or a given belt tooth count, yield the ratio, the belt length with the exact centre distance, the wrap angle and the teeth in mesh on the small pulley, plus optionally the working force and a pre-tension guideline.
Calculation
Design rating is an approximation: tooth load capacity and required belt width remain manufacturer-specific (catalogue) and are not included here.
Geometry and kinematics
- Pitch diameter d1
- 56.02 mm
- Pitch diameter d2
- 112.05 mm
- Ratio i
- 2
- Speed n2
- 725 1/min
- Belt speed v
- 4.253 m/s
Belt length and centre distance
- Belt tooth count z_R (selected)
- 109
- Belt length L
- 872 mm
- Centre distance a (back-calculated)
- 302.7 mm
- Wrap angle beta1 (small pulley)
- 169.4 °
- Teeth in mesh z_e (small pulley)
- 10
Forces
No power or torque entered - working force and pre-tension are not calculated.
Sketch: pulley pitch diameters with belt spans and centre distance (to scale)
Your inputs stay in your browser - all calculations run locally, nothing is sent to a server.
Formulas and fundamentals
Pitch Diameter and Ratio
The timing belt wraps the pulley as a chordal polygon with pitch p, so the pitch diameter follows exactly as for a sprocket:
The ratio is i = n1/n2 = z2/z1, giving the large pulley speed n2 = n1/i. For the small pulley there are guideline minimum tooth counts per profile (T5/AT5/3M: 10, T10/AT10/5M: 12, 8M: 16, 14M: 20); below these the tooth edge pressure rises and service life drops.
Belt Length from the Desired Centre Distance
In centre-distance mode, the approximation formula gives the calculated belt length from the desired centre distance a0:
Because timing belts are manufactured as endless belts with a fixed tooth count, L/p is rounded up to the next whole belt tooth count z_R (unlike a roller chain, an even number is not required). The real, slightly larger centre distance is then back-calculated from the selected tooth count.
Centre Distance from the Belt Tooth Count
In belt-tooth-count mode - for example when a catalogue belt with a fixed tooth count is given - the belt length follows directly as L = z_R·p. The matching centre distance follows from the length formula solved for a (the calculator uses the same inversion in centre-distance mode to back-calculate the real value):
Wrap Angle and Teeth in Mesh
The wrap angle on the small pulley follows from the real centre distance:
From this the number of teeth in mesh on the small pulley follows (rounded down to the next whole number):
Fewer than 6 teeth in mesh is considered critical: with too few load-carrying teeth, the load per tooth rises, and in the extreme case the belt can skip teeth. From 6 teeth in mesh the drive is uncritical; at 4 to 5 the margin for shock loads and manufacturing tolerances is tight.
Belt Speed and Forces
The belt speed on the small pulley is:
Up to about 40 m/s is common; beyond that, centrifugal effects and the demands on belt material and pulley balancing increase noticeably. From an optionally entered power P or torque M1, the working force follows:
The pre-tension force per span is given as a guideline between 0.5 and 1.0 times the working force (F_v ≈ 0.5 … 1.0·F_t); the exact pre-tension depends on the belt material and operating case and should be taken from manufacturer data (tension meter, pre-tension force chart).
Worked example
Reference example: A drive with an HTD 8M profile (p = 8 mm), z1 = 22, z2 = 44 (ratio i = 2), a catalogue belt with z_R = 112 teeth and n1 = 1450 rpm. The pitch diameters are d1 = 22·8/pi = 56.02 mm and d2 = 112.05 mm, the belt length L = 112·8 = 896 mm. The centre distance formula gives a = 314.75 mm, the wrap angle on the small pulley beta1 = pi − 2·arcsin(56.02/629.51) = 2.963 rad (169.8°), and hence z_e = 22·2.963/(2·pi) = 10.38, rounded down to z_e = 10 teeth in mesh - well above the critical limit of 6 (green).
At n1 = 1450 rpm, v = pi·56.02·1450/60000 = 4.25 m/s, far below the 40 m/s guideline. At a power of P = 3 kW, the working force is Ft = 1000·3/4.25 = 705 N; the pre-tension per span is guided between 353 N and 705 N. All three evaluation criteria (teeth in mesh, speed, minimum tooth count z1 = 22 >= 16) are in the green range.
For comparison of the two modes: choosing a desired centre distance of a0 = 300 mm in centre-distance mode instead, the approximation formula gives a calculated belt length of 866.6 mm, i.e. z_R = 108.3, rounded up to z_R = 109. The real centre distance after back-calculation is then a = 302.7 mm - slightly larger than the desired value, because it was rounded up to a whole belt tooth count.
Frequently asked questions
What is the difference between an HTD profile and a trapezoidal profile (T/AT)?
Trapezoidal profiles (T, AT per DIN 7721 / ISO 5296) have trapezoid-shaped teeth and are the classic, low-cost design for light to moderate loads. HTD profiles (High Torque Drive per ISO 13050) have a rounded, parabola-like tooth root that distributes the load more evenly across the tooth, transmitting higher working forces at a smaller size - the standard choice for high-torque drives. AT profiles are a trapezoidal evolution with a taller tooth for higher loads than the standard T profile.
How many teeth must be in mesh at minimum?
Guideline: from 6 teeth in mesh on the small pulley the drive is uncritical (green); at 4 to 5 teeth the margin for shock loads and manufacturing tolerances is tight (amber); below that, tooth skipping or increased wear becomes a risk (red). Teeth in mesh depend on the wrap angle and hence on the ratio of centre distance to pulley diameter difference - a centre distance that is too small combined with very different pulley sizes reduces it.
Timing belt or chain - which is the better choice?
A timing belt runs quietly, needs no lubrication, and positions backlash-free (positive engagement like a chain, but without joint wear) - hence the standard choice in servo axes and precision drive applications. A roller chain transmits higher forces, tolerates heat, oil and contamination better, and is more resistant to shock loading, but needs regular lubrication and elongates through joint wear. In a clean environment with moderate loads, the timing belt is usually the lower-maintenance solution.
How is the pre-tension of a timing belt drive set?
The calculator only outputs a rough guideline range (0.5 to 1.0 times the working force per span). In practice, pre-tension is usually set via the natural frequency of the free span using a belt tension meter and compared against the belt manufacturer's pre-tension force/frequency chart - the exact value is always belt-type- and application-specific.
Does the centre distance have to be adjustable?
Yes, as a rule: to fit the endless belt, the centre distance must be reducible, and setting the pre-tension requires an adjustment travel. Common solutions are slotted bearing mounts, a tensioner rail, or a spring-loaded idler pulley in the slack span. Without an adjustment provision, neither assembly nor correcting the pre-tension after the belt has settled is possible.
What does the belt tooth count z_R mean and how do I choose it?
z_R is the total number of teeth on the closed (endless) belt; the belt length is L = z_R·p. Timing belts are manufactured with fixed, catalogued tooth counts (no custom lengths as with a chain), so z_R must always be a whole number. In centre-distance mode the calculator automatically rounds up to the next whole number; in belt-tooth-count mode you enter an available catalogue value directly.
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