9.3 Timing Belts and Pulleys

A toothed belt transmits torque without slip, tolerates misalignment that would destroy a gear train, needs no lubrication, and can be printed around. For the RC car project it carries power from the 775 motor to the drive shaft at a ratio we choose, and it does so through a compliant member that absorbs the torque step the BTS7960 delivers when the throttle is opened. That compliance is why we prefer a belt here over a spur gear pair, and it is also why belt tension turns out to matter more than any other single number in the drivetrain.

We treat two profiles. The DKS-Pro chassis the project is based on drives its gearbox with a 70XL belt, 6 mm wide.¹ You are instructed to use GT2 instead, because it is what the workshop stocks and what every printed motion system in the building already runs. The two are not interchangeable, and the reasons are worth understanding before you commit a pulley to a print.

Profiles: trapezoidal and curvilinear

The imperial series MXL, XL, L, H, XH and XXH carry a trapezoidal tooth, the oldest synchronous belt form and the one described by ISO 5296. The tooth is easy to cut and easy to draw, but load concentrates at the tooth root, and the belt tooth has to deflect as it enters and leaves the groove, which shows up as a whine and as backlash under reversing load.

The curvilinear profiles that replaced it, HTD and then the GT family, put a rounded tooth into a rounded groove. Contact spreads over the whole flank instead of concentrating at the root, so the same belt width carries more torque, and the tooth seats with less radial motion, which is why positioning machines use them. GT2 is the Gates PowerGrip GT2 profile; the parts sold as 2GT on the open market are dimensionally the same 2 mm pitch curvilinear system.

	XL	GT2 (2GT)
Pitch \(p\)	5.08 mm (0.200 in)	2.00 mm
Tooth form	trapezoidal	curvilinear
Belt tooth height	TODO	0.75 mm
Belt thickness	TODO	1.38 mm
Pitch line differential \(U\)	TODO, see below	0.254 mm
Common widths	6.35, 9.53, 12.7 mm	6, 9, 15 mm

Geometry

Pitch diameter

The belt’s tensile cord sits a little above the root of its teeth, and the circle that cord follows when the belt is wrapped on a pulley is the pitch circle. That circle, not the metal, is what the kinematics see. Since \(N\) teeth at pitch \(p\) have to fit around it,

\[ d_p = \frac{N p}{\pi} \tag{9.3.1}\]

A 20 tooth GT2 pulley therefore has \(d_p = 20 \cdot 2/\pi = 12.732\) mm, and travel per revolution is exactly \(N p = 40\) mm.

Outside diameter and the pitch line differential

The pulley you actually machine or print is smaller than its pitch circle, because the cord rides above the tooth tips by a distance \(U\), the pitch line differential,

\[ d_o = d_p - 2U \tag{9.3.2}\]

For 2 mm GT2 the PowerDrive catalogue tabulates \(d_o = d_p - 0.020\) in for every tooth count from 12 to 120, giving

\[ U_{\text{GT2}} = 0.010\ \text{in} = 0.254\ \text{mm} \]

so a 20 tooth GT2 pulley turns 12.732 mm at the pitch line into an outside diameter of 12.224 mm. Getting this wrong is the most common error in a home made pulley: draw the tooth tips on the pitch circle and the belt runs 0.5 mm too large in diameter, the pitch no longer matches, and the belt climbs.

Warning

Numbers of the form “GT2 20T outside diameter 13.73 mm” circulate widely on marketplace listings and in machine generated buying guides. They are wrong: they add \(2U\) instead of subtracting it. Check any belt dimension you find online against a manufacturer catalogue before you cut metal or start a print.

Belt length and centre distance

For two pulleys of pitch diameters \(d_1 \le d_2\) at centre distance \(C\), the belt leaves each pulley along a common tangent inclined at

\[ \alpha = \arcsin\frac{d_2 - d_1}{2C} \]

so the large pulley is wrapped through \(\pi + 2\alpha\) and the small one through \(\pi - 2\alpha\), and the pitch length is

\[ L = 2C\cos\alpha + \frac{d_2}{2}\left(\pi + 2\alpha\right) + \frac{d_1}{2}\left(\pi - 2\alpha\right) \tag{9.3.3}\]

The approximation usually quoted,

\[ L \approx 2C + \frac{\pi}{2}\left(d_1 + d_2\right) + \frac{\left(d_2 - d_1\right)^2}{4C} \]

comes from expanding 9.3.3 for small \(\alpha\) and is good to well under a millimetre for the ratios we use.

Belts come in discrete lengths, and \(L\) must be a whole number of belt teeth, \(L = N_b p\). In practice we do not solve 9.3.3 for \(C\); we pick a stock belt, compute the nominal \(C\) it implies, and then give the design enough adjustment to take up the difference. A 70XL belt is 70 tenths of an inch of pitch length, that is 7.000 in = 177.8 mm, and therefore \(177.8/5.08 = 35\) teeth.

Teeth in mesh

Torque capacity is set by how many teeth share the load, not by the belt width alone. The number in mesh on the small pulley is

\[ z_e = N_1 \frac{\pi - 2\alpha}{2\pi} \]

and suppliers rate their belts assuming at least six. Below that the rating must be derated, and the failure mode is ratcheting: the belt climbs a tooth under peak torque, which on an RC car means the wheels step out of phase with the motor.

Making the pulley in CAD

Three routes, in decreasing order of how much we recommend them.

Import the supplier model. If the pulley is bought, the manufacturer’s STEP file is correct by definition and costs nothing. Use it for the interface and do not redraw it.

Generate it. GT2 pulley generators produce a STEP from tooth count, width, bore and flange settings.² They encode the groove profile so you do not have to, which matters because the GT2 groove is a specific curvilinear form, not an arc you can guess.

Draw it. Only when you need something the generators will not give you, such as a pulley integrated into a printed wheel hub. Construct the pitch circle from 9.3.1, the tip circle from 9.3.2, lay one groove on the centreline, and pattern it \(N\) times, exactly as we did for the involute tooth in the gear chapter. The groove profile itself should be traced from the manufacturer’s dimensioned drawing rather than approximated.

Printing the pulley

Here the two profiles stop being interchangeable, and this is the part worth thinking about before you follow the instruction to use GT2.

Our standard printer profile is a 0.4 mm nozzle at 0.2 mm layer height, with roughly \(\pm 0.1\) mm of surface deviation per face.³ Set that against the two tooth forms. A GT2 groove is 2 mm from neighbour to neighbour and about 0.75 mm deep, so it is around two extrusion widths wide and fewer than four layers deep, and \(\pm 0.1\) mm is a 13% error on tooth depth. An XL groove at 5.08 mm pitch is roughly two and a half times larger in every direction, and the same absolute deviation is a much smaller fraction of it.

The conclusion is not that a printed GT2 pulley cannot work. It is that a printed GT2 pulley is operating close to the resolution limit of the process, whereas a printed XL pulley has margin. So:

Buy the GT2 pulleys that sit on the motor shaft, where torque per tooth is highest and the pulley is smallest, and print only what you cannot buy.

Print with the pulley axis vertical. The teeth are then built from stacked perimeters, and the tangential force that loads them acts in the plane of the layers rather than across them, where the part is roughly three times stronger.

Chamfer the bottom of the tooth band. Elephant foot fattens the first layers, and a fat first tooth binds the belt.

Give the bore the same treatment as any printed shaft interface: a hexagonal or square hole rather than a circular press fit, or crush ribs undersized by about 0.2 mm in a bore oversized by about 0.4 mm, and a heat set insert for any grub screw that will be undone more than once.

Fitting the pulley to a printed wheel

Tensioning

A toothed belt does not need tension to transmit torque the way a flat belt does. It needs just enough to keep the belt seated in the grooves so it cannot climb, and no more, because everything above that goes straight into the shaft bearings.

Measuring tension by frequency

The free span between two pulleys is a string under tension, so its fundamental frequency is

\[ f = \frac{1}{2L_s}\sqrt{\frac{T}{\mu}} \tag{9.3.4}\]

with \(L_s\) the free span length, \(T\) the tension and \(\mu\) the belt mass per unit length. Inverting 9.3.4,

\[ T = 4\mu L_s^2 f^2 \tag{9.3.5}\]

which turns a tension specification into something you can check with a phone. Pluck the span, hold the microphone near it, and read the frequency off a guitar tuner app.

9.3.5 is also the reason to be sceptical of the frequency targets that circulate in the printer community. Values such as 40 to 60 Hz for a 200 to 250 mm span on a Cartesian machine, or 110 to 140 Hz for the longer spans of a CoreXY,⁴ are tension specifications in disguise, and they are only transferable to another machine if \(L_s\) and \(\mu\) match. Quoting a frequency without the span is meaningless. Measure \(\mu\) by weighing a known length of your belt, decide the tension you want, and compute your own target.

Rules of thumb and mechanisms

A belt at roughly the right tension deflects 2 to 3 mm at midspan under firm finger pressure. This is worth knowing because it takes two seconds, but it is a check, not a specification.

Tension is taken up either by moving a pulley or by pressing on the span. Sliding the motor on slotted holes is simplest and is what the DKS-Pro chassis does. An eccentric idler gives fine adjustment in little space. A spring loaded idler holds tension as the belt beds in and as the printed parts creep, which on a plastic chassis is a real effect over the first few hours of running, and it is the reason a car that was tight on Monday ratchets on Wednesday. Whatever the mechanism, press on the slack side, and use a toothed idler if it runs on the toothed face.