FRC Timing Belt Calculator: How to Use It

The FRC timing belt calculator works out the belt length and pulley tooth counts you need for a custom timing belt drive on a FIRST Robotics Competition robot. You enter the distance between pulley centers and the tooth counts on the two pulleys, and the calculator returns the required belt length in teeth, along with the nearest standard belt size and the actual center distance for that belt. This page explains the math behind the calculator, how to use it, and the common mistakes teams make when sizing belts.

Formula

The standard timing belt length formula for two pulleys with a given center distance is:

L = 2C + (π/2)(D1 + D2) + (D2 – D1)² / (4C)

where:

• L = belt pitch length (in inches or mm)
• C = center distance between the two pulley shafts (same unit as L)
• D1 = pitch diameter of the driver pulley
• D2 = pitch diameter of the driven pulley

In FRC, most teams use 5mm pitch GT2 or HTD belts, or 0.2″ pitch (5.08mm) belts. The pitch diameter of a pulley is simply the number of teeth times the belt pitch divided by π. For a 5mm pitch belt, a 20-tooth pulley has a pitch diameter of 20 × 5 / π = 31.83mm.

Because the formula uses pitch diameters, you can also express it directly in terms of tooth counts and belt pitch (p):

L (in teeth) = 2C/p + T1 + T2 + (T2 – T1)² / (4π²C/p)

This is the form the calculator uses. The result is in belt teeth, which is how timing belts are sold (e.g., a 60-tooth belt).

Worked Example

Suppose you have a 3.5″ center distance, an 18-tooth driver, and a 36-tooth driven pulley, using a 0.2″ pitch belt (the defaults in the calculator).

1. Compute C/p = 3.5 / 0.2 = 17.5
2. 2C/p = 2 × 17.5 = 35
3. T1 + T2 = 18 + 36 = 54
4. (T2 – T1)² = (36 – 18)² = 324
5. 4π²C/p = 4 × 9.8696 × 17.5 = 690.87
6. Last term = 324 / 690.87 = 0.469
7. Total belt length = 35 + 54 + 0.469 = 89.47 teeth

So you need a belt with about 89.5 teeth. The nearest standard belt is 90 teeth (or 80 teeth if you can adjust the center distance). With a 90-tooth belt, the actual center distance would be slightly less than 3.5″ – about 3.47″ – which is fine if you have an adjustable motor mount.

How to Use the FRC Timing Belt Calculator

To get a useful answer, you need three pieces of information:

• Center distance – the distance between the center of the motor shaft and the center of the driven shaft. Measure this in inches or mm, but be consistent.
• Driver pulley teeth – the number of teeth on the pulley attached to the motor (or the input).
• Driven pulley teeth – the number of teeth on the pulley that drives the mechanism (wheel, arm, etc.).

Enter those values, and the calculator gives you the belt length in teeth and in inches (assuming 0.2″ pitch). If you are using a 5mm pitch belt, multiply the teeth result by 5 to get the length in mm.

Most FRC teams use a 5mm GT2 belt with a steel core. The calculator works for any pitch as long as you enter the center distance in the same unit as the pitch. For 5mm belts, convert the center distance to mm first.

Common Mistakes and Misconceptions

One frequent error is using the outside diameter of the pulley instead of the pitch diameter. The pitch diameter is the diameter at which the belt’s teeth engage the pulley, and it is always larger than the outside diameter. For a 5mm GT2 pulley, the pitch diameter is roughly the number of teeth × 5 / π, but the outside diameter is about 0.2mm less per tooth. Using the outside diameter gives a belt that is too short by a few mm.

Another mistake is forgetting that the belt must have an even number of teeth if you are using a belt with a steel tension cord – most GT2 belts are endless and can have any integer tooth count, but some belts are joined with a clip and require an even number. Check your belt type.

Also, the formula assumes the belt is taut and the two pulleys are in the same plane. If your pulleys are not aligned, the belt will rub and wear quickly, but the length calculation remains the same.

Choosing the Right Belt and Pulleys

When selecting components for an FRC drive, consider these factors:

• Belt pitch – 5mm GT2 is the most common for FRC because it is strong, lightweight, and available in many lengths. HTD 5mm is also compatible with GT2 pulleys in many cases, but check the tooth profile.
• Belt width – 9mm is typical for light loads, 15mm for high torque. The calculator does not account for width, but wider belts handle more load without stretching.
• Pulley material – aluminum pulleys are standard; plastic ones wear faster. Use metal pulleys for high-speed or high-torque applications.
• Number of teeth in mesh – the more teeth engaged, the better the power transmission. For a small pulley (12-18 teeth), ensure the belt wraps at least 120° around it.

If the calculated belt length is not a standard size, you have two options: adjust the center distance (if your mount allows it) or choose a different pulley tooth count. The calculator’s output can guide you – try changing the center distance by 0.1″ and see how the belt length changes.

When to Use a Timing Belt vs. Other Drive Options

Timing belts are excellent for FRC because they do not slip, they are quiet, and they require no lubrication. They are the go-to for transferring power from a motor to a gearbox or wheel. However, for very long distances (more than 12″), a chain or a shaft might be more efficient. For very high torque at low speed, a gearbox is usually better.

If you need a speed reduction, a timing belt can do that with different pulley sizes. For example, an 18-tooth driver and a 36-tooth driven gives a 2:1 reduction. The calculator does not compute the ratio, but you can easily divide the tooth counts.

Frequently Asked Questions

What is the difference between GT2 and HTD belts?
GT2 (Gates Tooth 2) and HTD (High Torque Drive) belts have similar tooth profiles, but GT2 teeth are slightly different to reduce backlash. In practice, GT2 pulleys and HTD pulleys are often interchangeable for 5mm pitch, but it is safer to use matching brands. The calculator works for both.

Can I use this calculator for a 5mm belt?
Yes. Convert the center distance to mm (1 inch = 25.4mm) and use the tooth count result. For example, a 3.5″ center distance is 88.9mm. The belt length in mm is the tooth count multiplied by 5.

What if my belt length is not a whole number?
You cannot have a fractional tooth. Round up to the nearest whole tooth and then adjust the center distance slightly. The calculator’s center distance output (which is the same as your input) is a starting point; for the actual belt, the center distance will be a bit less than the nominal value. Use a belt tensioner or slotted motor mount.

How do I measure the center distance on the robot?
Use a caliper or ruler to measure from the center of the motor shaft to the center of the driven shaft. If the shafts are not accessible, measure from the outside of one pulley to the outside of the other and subtract half the diameter of each pulley.

Is a longer belt better than a shorter one?
Not necessarily. A longer belt adds weight and may flex more, but it allows a larger center distance. A shorter belt is lighter and stiffer, but if it is too tight, it will put extra load on the motor bearings. Follow the manufacturer’s recommended tension.

Why does the calculator give a belt length in teeth?
Because timing belts are sold by tooth count. For example, a 90-tooth 5mm belt is 450mm long. Knowing the tooth count lets you order the correct part number directly.

Related

• Jellyfin vs Streaming Services: Which Is Better?
• OneNote Handwriting: Does It Make It Prettier?
• History of Google: From Garage to Global

in c | Center distance (in) | in | 3.5 | 0.1 | 60
in t1 | Driver pulley teeth | teeth | 18 | 6 | 120
in t2 | Driven pulley teeth | teeth | 36 | 6 | 120
out teeth | Belt length (teeth) | teeth | 2 * c / 0.2 + t1 + t2 + (t2 - t1)^2 / (4 * pi^2 * c / 0.2) | 1
out belt_in | Belt length (in) | in | (2 * c / 0.2 + t1 + t2 + (t2 - t1)^2 / (4 * pi^2 * c / 0.2)) * 0.2 | 2
out center_in | Actual center distance (in) | in | c | 2