Pulleys
Pulleys
A pulley is a wheel that carries a flexible rope, cord, cable, chain, or belt on its rim. Pulleys are used singly or in combination to transmit energy and motion.
Pulleys for Mechanical Advantage
One or more independently rotating pulleys can be used to gain mechanical advantage, especially for lifting heavy objects.
The shafts about which the pulleys turn may affix them to frames or blocks, and a combination of pulleys, blocks, and rope or other flexible material is referred to as a block and tackle.
Archimedes is reported to have used compound pulleys to pull a ship onto dry land.
Mechanical Advantage
To calculate the mechanical advantage of a pulley, you simply need to determine the number of supporting sections of rope that are part of the system. Looking at the magnificent gif of the cargo pulley system, you can see 4 supporting sections of rope. The effort section being pulled by the user is not counted, (neither are pulleys used for guidance that are not directly supporting the load as some exam questions may try to trick you with).
On an old sailing vessel, there is a pulley system for loading cargo. There are 4 pulleys in the system, there are 5 sections of rope, with 4 of them supporting the load. The cargo weighs 200kg.
$$ \text{MA} = \text{4 Supporting Ropes} = 4 $$
We can now use this multiplier, plus our known forces to determine what the input force is going to be.
We know the mass of the load is 200kg - so our force (N) is going to be $F = Ma$. We generally treat gravity as 10, so
$$ F_{\text{Load}} = 200 \times 10 $$ $$ F_{\text{Load}} = 2000\text{N} $$
Because our mechanical advantage is greater than 1, the output force is increased. That means our input force must be lower than the output force, so lets divide the output force by our mechanical advantage to determine what the minimum input force is to move the load.
$$ F_{\text{Effort}} = \frac{F_{\text{Load}}}{\text{MA}} $$ $$ F_{\text{Effort}} = \frac{2000}{4} $$ $$ F_{\text{Effort}} = 500 {\text{N}} $$
The pulley's advantage comes from distributing the force equally across the supporting ropes. Whatever force is applied, is applied to each supporting rope. The distance moved by the effort is divided equally among the supporting ropes.
Pulleys for Transfer of Motion
Two or more pulleys connected by a belt can transmit rotary motion and energy from one shaft to another. The pulley driven by the power source (usually a motor) is called the driver, and the pulley receiving the motion is called the driven pulley.
Unlike a block and tackle, a belt drive does not lift a load directly. Instead it changes the rotational speed and the torque delivered to the output shaft, and can transfer motion across a distance between two shafts that are not touching.
Early factories ran entire floors of machinery from a single steam engine using long overhead "line shafts" and dozens of belt drives dropping down to each machine.
Next time you're in Ballarat, drop into Sovereign Hill and see the Proctor's Wheelwright Factory in action.
How a Belt Drive Transmits Motion
A belt does not lock into the pulley — it grips it. Motion is transferred by friction between the belt and the rim of the pulley. If the friction is insufficient, the belt slips, the driver keeps spinning and the driven pulley slows or stops.
This is both a weakness and a feature:
- Weakness: slip means a belt drive is never perfectly precise, and some energy is always lost as heat.
- Feature: slip acts as built-in overload protection. If the output jams, the belt slips rather than snapping the shaft or burning out the motor.
Where slip cannot be tolerated (e.g. a 3D printer axis), a toothed belt (timing belt) is used, which meshes with matching teeth on the pulley and transmits motion by interlocking rather than friction.
Flat belt – simple, cheap, high slip.
V-belt – wedge shape jams into a grooved pulley, increasing friction and grip.
Round belt – small drives, can turn corners.
Toothed / timing belt – no slip, maintains exact timing between shafts.
Pulley Ratio
The pulley ratio describes how much a belt drive changes rotational speed. Because the belt is a fixed length looping around both rims, a large pulley must turn slowly and a small pulley must turn quickly to keep up with the same belt speed.
Speed and diameter are therefore inversely proportional:
$$ \text{pulley ratio} = \frac{\text{diameter of driven}}{\text{diameter of driver}} = \frac{\text{speed of driver (rpm)}}{\text{speed of driven (rpm)}} $$
Or, written as the exam formula sheet does:
$$ \frac{\text{Pulley A rpm}}{\text{Pulley B rpm}} = \frac{\text{diameter of Pulley B}}{\text{diameter of Pulley A}} $$
A motor spins at 1200 rpm and drives a 50 mm pulley. It is belted to a 200 mm pulley on the output shaft. What is the output speed?
Steps
- Work out the pulley ratio
- Calculate the RPM based of the pulley ratio
$$ \text{pulley ratio} = \frac{200}{50} = 4 \quad (4:1) $$
$$ \text{speed}_{\text{driven}} = \frac{\text{speed}_{\text{driver}}}{\text{ratio}} = \frac{1200}{4} $$
$$ \text{speed}_{\text{driven}} = 300\text{ rpm} $$
The big pulley turns four times slower than the little one.
A belt drive is a trade, not a free gain. A 4:1 speed reduction is also a 4:1 torque increase (ignoring losses). Whatever you divide the speed by, you multiply the torque by.
Small pulley driving a big pulley = slower but stronger.
Big pulley driving a small pulley = faster but weaker.
Mechanical Advantage in a Belt Drive
Because the output torque is multiplied by the same factor the speed is divided by, the pulley ratio is the mechanical advantage of the drive:
$$ \text{MA} = \text{pulley ratio} = \frac{\text{diameter of driven}}{\text{diameter of driver}} $$
A 4:1 drive has a mechanical advantage of 4 — it produces four times the input torque at a quarter of the input speed.
Direction of Rotation
- Open belt (the normal arrangement): both pulleys turn in the same direction.
- Crossed belt (the belt is twisted in a figure-8): the driven pulley turns in the opposite direction.
- A pulley pressed against the outside of the belt turns opposite to the pulleys it runs between.
Idler and Tensioner Pulleys
An idler (or tensioner) is a third pulley that presses on the belt. It exists to take up slack, keep the belt tight enough to grip, increase the wrap angle around a small pulley, or route the belt around an obstacle.
An idler does not change the ratio between the driver and the driven pulley — the drive still behaves as if the idler were not there.
It does, however, have a speed of its own. Every point on the belt is moving at the same belt speed, so the idler's rpm is set by its diameter. A small idler spins quickly; a large one spins slowly.
Compound Belt Drives
Where a single stage cannot produce a large enough change in speed, two or more stages are combined. Two pulleys of different sizes are locked to the same intermediate shaft, so the driven pulley of stage 1 and the driver pulley of stage 2 must rotate together at the same rpm.
The ratios multiply:
$$ \text{ratio}_{\text{final}} = \text{ratio}_{1} \times \text{ratio}_{2} $$
A motor runs at 1500 rpm turning a 40 mm pulley, belted to a 120 mm pulley on a shaft. Locked to that same shaft is a 30 mm pulley, belted to a 150 mm pulley on the output.
$$ \text{ratio}_{1} = \frac{120}{40} = 3 \qquad \text{ratio}_{2} = \frac{150}{30} = 5 $$
$$ \text{ratio}_{\text{final}} = 3 \times 5 = 15 \quad (15:1) $$
$$ \text{speed}_{\text{output}} = \frac{1500}{15} = 100\text{ rpm} $$
The output shaft turns at 100 rpm with fifteen times the motor's torque.
Advantages and Disadvantages
| Advantages | Disadvantages |
|---|---|
| Quiet and smooth running | Belt slip means the ratio is not exact |
| Cheap, simple, easy to replace | Lower efficiency than gears or chain |
| Absorbs shock and vibration | Belts stretch and wear over time |
| Can span a long distance between shafts | Needs tensioning to work reliably |
| Slip protects the motor during an overload | Cannot transmit very high torque |