A Weston differential pulley has two different-size upper chain wheels fixed together, one lower moving wheel and a single endless chain. Turning the upper pair takes up more chain on one side than it pays out on the other. The lower block rises by half that difference because two chain spans support it.
Differential Pulley Interactive Calculator
Follow the continuous chain around two upper wheels fixed together and one moving lower wheel. The difference between the upper circumferences raises the load. Compare wheel diameters and calculate hook travel for a chosen chain pull.
Equation Used
- Both upper wheels share one shaft and rotate through the same angle.
- The lower block is supported by two vertical chain spans.
- Effective pitch diameters are used in the smooth-chain model.
- The free chain bight is drawn as a rounded loop in projection, not a calculated hanging-chain shape.
- Ideal force ratio is not a load rating or proof that the hoist will hold a load.
Equal diameters produce zero hook travel, not a finite lifting ratio. If d exceeds D, the displayed input direction lowers the hook. Both entered diameters are used in the calculation.
Trace the endless chain
Starting on the hand-pull side, the chain passes over the driven upper wheel, down around the lower moving wheel, back over the companion upper wheel, and into the free hand-chain loop. The upper wheels rotate as one part. The lower wheel rotates while its centre moves vertically.
The drawing separates the loaded and free-loop paths by colour. Their crossings in this projected view are not joints. As the loaded loop shortens, the free loop gains the same chain length. The moving lower hook is connected to the lower wheel’s axle, not to a free chain end.
With D greater than d, pulling down on the driven wheel’s free span raises the hook. Reversing the demonstration reverses the motion. The drawing uses a short excursion so all parts remain visible. The result tiles give hook travel for the full entered chain pull.
The historical differential chain block
Henry R. Towne’s 1883 treatise describes the Weston block as an upper paired wheel, a lower pulley and an endless chain, with grooves that engage the chain to prevent slip. This page models that direct differential arrangement.
Other hand-chain hoists use reduction gears and brakes. Their internal mechanism should not automatically be identified as this two-diameter Weston arrangement merely because a hanging hand chain is visible.
Difference of circumferences
For one upper-shaft revolution, the driven wheel takes up πD of chain while the companion pays out πd. The loaded loop therefore loses π(D − d). Its two vertical spans shorten together, so hook rise is half that amount.
A chain pull P on the driven wheel turns the shaft through P/(πD) revolutions when the units agree. Multiplying by the rise per revolution gives hook rise P(D − d)/(2D). The calculator converts the entered pull in metres and diameters in millimetres consistently.
For a lossless device, equal input and output work gives the ideal force-ratio magnitude 2D/|D − d|. It is a kinematic ratio, not a permissible load. When the diameters match, motion of the chain produces no hook rise and no finite travel ratio can be reported.
100 mm and 90 mm upper wheels
One revolution takes up 314.16 mm on the 100 mm wheel and pays out 282.74 mm on the 90 mm wheel. Their 31.42 mm difference raises the lower block 15.71 mm. The ideal travel ratio is 20:1.
Pulling 20 m of chain therefore raises the hook 1 m in this ideal model. Setting the two diameters equal makes the rise zero. Setting the companion wheel larger reverses the signed rise for the same chosen input direction.
Travel advantage is not a lifting capacity
Bringing the two diameters closer increases chain travel per unit of hook movement. Real chain engagement also requires compatible pitch and pocket geometry; arbitrary slider values are a continuous geometric model rather than a manufacturable wheel specification.
Friction changes the force needed and can affect back-driving. This calculation does not predict self-locking, braking, efficiency, chain strength or support loads. Use the rated capacity and operating instructions of an actual hoist for lifting work.
Differential pulley questions
Why divide the circumference difference by two?
The lower wheel is supported by two chain spans; both shorten as it rises.
Why is there only one chain?
The loaded loop and hand loop are different portions of the same endless chain.
What happens when the diameters are equal?
Take-up equals payout, so the lower block stays at the same height.
Why can the rise be negative?
If d exceeds D, the stated pulling direction pays out more loaded chain than it takes up, lowering the block.
Does the animation show the full selected pull?
No. It shows a short, proportionate forward-and-return excursion. The calculator reports the result for the full entered pull.
Reference
Henry R. Towne, A Treatise on Cranes, 1883, printed pages 169–170, figures 66–67. The manufacturer’s account describes the paired upper wheels, endless-chain route and half-difference-of-circumferences displacement.
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