Belt and chain drives: types, speed ratio and design

When two shafts are too far apart for gears, a belt or a chain carries the power between them. Both are cheaper than gears over a long distance, both tolerate some misalignment, and both can change speed. Belts are quiet, need no lubrication and slip safely under overload; chains are compact, don’t slip and carry high torque at low speed.

Belt drives

Types of belt

How a friction belt transmits power

The driver pulls the belt in on one side (the tight side, tension T₁) and lets it out on the other (the slack side, T₂). The power transmitted is the difference in tension times the belt speed:

$$P = (T_1 - T_2)\,v, \qquad v = \frac{\pi D N}{60}$$

The largest possible ratio of the two tensions before the belt slips is set by friction and by how far the belt wraps around the pulley (the capstan equation):

$$\frac{T_1}{T_2} = e^{\mu\theta} \ \ \text{(flat belt)}, \qquad \frac{T_1}{T_2} = e^{\mu\theta/\sin(\beta/2)} \ \ \text{(V-belt)}$$

where $\mu$ is the coefficient of friction, $\theta$ the wrap angle in radians and $\beta$ the groove angle. The wedge of a V-belt multiplies the effective friction about three times, which is why it grips at much lower tension than a flat belt.

Geometry of an open belt drive

$$\frac{N_1}{N_2} = \frac{D_2}{D_1}, \qquad L = 2C + \frac{\pi (D_1 + D_2)}{2} + \frac{(D_2 - D_1)^2}{4C}, \qquad \theta = 180^\circ - 2\sin^{-1}\frac{D_2 - D_1}{2C}$$

Friction belts slip slightly (about 1–2 %), so the real driven speed is a little lower. Timing belts and chains have no slip.

In practice the belt is tensioned well above this minimum so it doesn’t slip, and catalogues from belt makers give the power per belt for each section, speed and pulley size, with correction factors for wrap angle and belt length.

Chain drives

A roller chain is a series of pins, bushes and rollers that mesh with toothed sprockets. It gives an exact speed ratio like a gear, over a centre distance like a belt.

$$\frac{N_1}{N_2} = \frac{z_2}{z_1}, \qquad d = \frac{p}{\sin(180^\circ / z)}, \qquad v = \frac{z\,p\,N}{60\,000}$$

where $z$ is the number of sprocket teeth, $p$ the chain pitch in mm, $d$ the sprocket pitch diameter in mm and $v$ the chain speed in m/s.

Design points for chains:

• Use at least 17 teeth on the small sprocket. With fewer teeth the chain rises and falls as it wraps (the polygonal or chordal effect), which causes vibration and wear.
• Use an even number of links so no offset link is needed, and odd tooth numbers on sprockets to spread wear.
• Keep the centre distance around 30–50 chain pitches.
• Lubricate and adjust. Chains wear and stretch; a tensioner or adjustable centres take up the slack.

Belt vs chain vs gear