A pulley combined with a differential gear is a power transmission mechanism that takes a single rotating input through belt or rope pulleys and feeds it into a differential gear set, which then splits torque between two output shafts while allowing them to rotate at different speeds. You see this layout in chain hoists like the classic Yale Weston differential block and in some band saw resaw drives. The arrangement gives you huge mechanical advantage on lifting jobs or balanced torque on twin-output drives — a 200 lb pull on the hand chain can hold 4,000 lbs of load.
Pulleys Combined with a Differential Gear Interactive Calculator
Vary the two Weston pulley diameters and hand pull to see mechanical advantage, diameter difference, ideal lifted load, and lift per turn.
Equation Used
The worked example sizes the ideal mechanical advantage of a Weston differential pulley from the two upper pulley diameters. A small diameter difference produces a large force multiplier: MA = 2D1/(D1-D2). Ideal lifted load is the hand pull multiplied by MA, and the hook rise per full revolution is half the circumference difference.
- Ideal Weston differential pulley calculation with no friction losses.
- D1 is the larger upper pulley diameter and D2 is the smaller upper pulley diameter.
- Pull is entered as kgf-equivalent mass, matching the worked example notation.
How the Pulleys Combined with a Differential Gear Actually Works
The mechanism stacks two ideas on top of each other. The pulley side gives you speed reduction or speed increase through a ratio of pulley diameters — wrap a rope or chain around a small sheave coupled to a large sheave, and the difference in circumferences becomes a force multiplier. The differential side takes that single rotating input and feeds it through bevel gears or planetary gears into two output shafts that share torque but can spin independently. In a Weston differential pulley block, the two pulleys sit on the same axle but have slightly different diameters — say 100 mm and 95 mm — and a single endless chain wraps both. Pull the chain on one side and the load hook rises by exactly half the difference in circumferences per turn. That tiny differential is where the mechanical advantage comes from.
Why build it this way? Because you get torque multiplication and load-holding in one compact unit. The differential gear splits the input torque into two equal output torques regardless of relative output speed, which means if one output stalls the other still receives the full input torque. That's exactly the behaviour you want in a hoist that must hold a load when the operator stops pulling, and it's the behaviour you want on a vehicle or machine drive where two wheels or two cutter heads must share power but turn at different rates around a curve or under uneven cut load.
Tolerances matter more than people expect. On a Weston block, the two pulley diameters must differ by the design value within about ±0.2 mm — go tighter and the mechanical advantage shrinks, go looser and the chain skips off the smaller sheave under load. Bevel gear backlash on the differential side should sit around 0.05 to 0.10 mm; tighter and the gears bind under thermal expansion, looser and you'll hear a sharp clack every time the load reverses. Common failure modes are chain elongation past 2% of pitch (which causes the chain to ride high and skip teeth), bevel gear tooth pitting from running without oil, and spider gear pin wear that lets the differential lock up intermittently.
Key Components
- Input Pulley (drive sheave): The larger of the two coupled sheaves on a Weston block, or the driven pulley on a belt-fed differential drive. Diameter sets the input lever arm — typically 100 to 300 mm on hand-chain hoists. Surface must be machined to the chain pitch within ±0.1 mm or the chain rides unevenly.
- Output Pulley (load sheave): The slightly smaller coupled sheave that determines the differential ratio. On a 1-tonne Yale block the load sheave runs about 5 mm smaller in diameter than the drive sheave, giving roughly 40:1 mechanical advantage. Hardness should be 45-50 HRC to resist chain peening.
- Differential Bevel Gears (side gears): Two bevel gears keyed to the output shafts, meshing with the spider gears. They split input torque equally between the two outputs. Module typically 2 to 5 mm on industrial units; backlash 0.05-0.10 mm.
- Spider Gears (planet gears): Two or four small bevel gears mounted on a cross pin inside the differential carrier. They allow the two output shafts to rotate at different speeds while still receiving equal torque. Run on needle bearings or bronze bushings rated for 5,000 hours.
- Differential Carrier: The housing that holds the spider gears and rotates as a unit driven by the input pulley. Must run true within 0.05 mm TIR or the spider gears will see uneven loading and pit prematurely.
- Load Chain or V-Belt: Transmits force from the operator or motor to the input pulley. On hoists, calibrated load chain to EN 818-7 grade T(8); on belt drives, typically a B or C-section V-belt sized for 1.5× the rated input torque.
Who Uses the Pulleys Combined with a Differential Gear
You find this combination wherever a single input has to drive two outputs that may turn at different rates, or where you need extreme force multiplication in a self-locking package. The differential side handles the speed-difference problem; the pulley side handles the force-multiplication problem. Together they show up in hoisting, vehicle drivetrains with belt-driven differentials, and some specialty machine tools.
- Material Handling: Yale Weston-pattern differential chain hoist used in fabrication shops for lifting 500 kg to 5 tonne loads with hand-chain effort under 35 kg pull
- Agricultural Machinery: Belt-driven rear-axle differential on the Gravely Model L two-wheel walking tractor, where a flat belt feeds the input pinion of a small bevel differential
- Stage and Theatre Rigging: Differential pulley winches used at venues like the Royal Opera House for batten lifts where load self-holds without a brake
- Marine Deck Equipment: Manual anchor windlasses on classic sailing yachts such as the Hinckley Bermuda 40, using a differential pulley to multiply hand-crank force
- Automotive Restoration: Belt-driven differential on the Morgan three-wheeler from 1909 to 1936, where a chain-and-belt system drove a bevel differential at the rear wheel
- Mining and Construction: CM Lodestar-style chain hoists with differential reduction used for underground equipment lifts in potash and salt mines
The Formula Behind the Pulleys Combined with a Differential Gear
The headline number for a differential pulley combined with a gear differential is the mechanical advantage — how much the input force gets multiplied at the load. At the low end of the practical range, with pulley diameters nearly equal (D1 within 1 mm of D2), the mechanical advantage climbs above 100:1 but the lift speed becomes painfully slow and chain wrap errors dominate. At the nominal sweet spot, with D1 roughly 5% larger than D2, you get 30-50:1 mechanical advantage and a usable 0.5 to 1 m of lift per minute of chain pulling. At the high end, where D1 is 15-20% larger than D2, MA drops to about 6:1 and you start needing a brake because the system loses its self-locking behaviour.
Variables
| Symbol | Meaning | Unit (SI) | Unit (Imperial) |
|---|---|---|---|
| MA | Mechanical advantage of the combined system (output force divided by input force) | dimensionless | dimensionless |
| D1 | Diameter of the larger (drive) pulley | mm | in |
| D2 | Diameter of the smaller (load) pulley | mm | in |
| η | Combined efficiency of pulley friction and differential gear losses | dimensionless (0 to 1) | dimensionless (0 to 1) |
Worked Example: Pulleys Combined with a Differential Gear in a 2-tonne Weston differential chain hoist for a custom motorcycle workshop
You are specifying a Weston-pattern differential chain hoist for a custom motorcycle workshop in Portland Oregon, where the heaviest lift is a fully dressed Indian Chief at 1,800 kg being pulled off a build bench. You want the operator to hold the bike steady with a 25 kg hand-chain pull, and you need to verify the pulley diameter pair plus the differential efficiency will deliver that.
Given
- D1 = 120 mm
- D2 = 114 mm
- η = 0.85 dimensionless
- Load = 1800 kg
Solution
Step 1 — at the nominal design point, compute the ideal mechanical advantage from the pulley diameter pair:
Step 2 — apply the combined efficiency to get the real mechanical advantage at the hand chain:
Step 3 — compute required hand-chain pull for the 1,800 kg load (load weight = 1800 × 9.81 = 17,658 N):
That's higher than your 25 kg target. At the low end of the practical diameter range, tighten D1−D2 to 3 mm and MAideal jumps to 80, giving Fhand ≈ 26 kg — right on target, but you'll be pulling roughly 25 m of chain to lift the bike 300 mm, and chain wrap errors on the smaller sheave become noticeable. At the high end, opening the diameter pair to D1−D2 = 12 mm drops MA to 17 and Fhand climbs to 105 kg — well past what an operator can sustain, and the system loses self-locking so you need a load brake.
Result
At the as-given 6 mm diameter difference and 85% efficiency, the operator needs about 53 kg of hand-chain pull to hold the 1,800 kg motorcycle — too much for one person on a long lift. Tightening the diameter pair to a 3 mm difference brings the pull down to a manageable 26 kg but doubles the chain travel; opening it to 12 mm drops mechanical advantage to 17 and forces you to add a separate load brake. The sweet spot for a 1.5-2 tonne shop hoist sits at 3-5 mm diameter difference. If your measured hand pull comes in 30% higher than predicted, check three things in this order: (1) chain elongation past 2% pitch, which makes the chain ride on tooth tips and effectively shrinks D2; (2) dry pulley pockets where loss of grease drops η from 0.85 to about 0.65; and (3) carrier runout above 0.10 mm TIR, which causes the spider gears to bind on every revolution and adds a lumpy resistance the operator feels as a dead spot every half turn.
Pulleys Combined with a Differential Gear vs Alternatives
Pulleys with a differential gear compete with simpler hoist and torque-split mechanisms. The pulley-plus-differential combination wins on self-locking behaviour and torque sharing, but loses on speed and on cost when you need only one of those features. Here's how it stacks up against a plain block-and-tackle and a worm-gear hoist on the dimensions builders actually compare.
| Property | Pulley + Differential Gear | Block and Tackle (rope) | Worm Gear Hoist |
|---|---|---|---|
| Mechanical advantage range | 20:1 to 60:1 typical | 2:1 to 8:1 typical | 30:1 to 200:1 typical |
| Self-locking under load | Yes, inherent (no brake needed) | No, requires cleat or operator | Yes, inherent below ~5° lead angle |
| Lift speed per metre of input pull | 0.025 to 0.05 m/m (slow) | 0.125 to 0.5 m/m (fast) | 0.005 to 0.033 m/m (very slow) |
| Capital cost (1 tonne capacity) | $200-400 USD | $50-100 USD | $300-600 USD |
| Efficiency | 80-88% | 92-96% | 40-65% |
| Service life under intermittent shop use | 20+ years, chain replacement at 10-15 yr | 5-10 years on rope | 15-25 years |
| Best application fit | Mid-load shop hoists, theatre rigging, marine windlasses | Light occasional lifts, sailing rigs | Heavy precision lifts, overhead cranes |
Frequently Asked Questions About Pulleys Combined with a Differential Gear
True self-locking on a differential pulley depends on chain friction in the pulley pockets being greater than the residual torque from the load trying to back-drive the system. If your hoist creeps, the most likely cause is glazed or oily pulley pockets — somebody hit the unit with degreaser or hung it in a paint booth and the chain now slides instead of grips. The fix is to clean both sheaves with a wire brush and let them re-oxidise lightly.
The second cause is a chain that has worn past its service limit. Once chain pitch elongates past 2%, the links no longer seat in the pocket properly and the friction coefficient drops by half. Measure 11 links end-to-end and compare to the original pitch × 11 — if you're 2% longer, replace the chain.
Work backwards from the operator pull. Decide what's sustainable — 20-25 kg for repeated lifts, up to 35 kg for occasional. Then MArequired = load weight / operator pull, and D1 − D2 = (2 × D1 × η) / MArequired. Pick D1 from the chain pitch first (load chain manufacturers publish minimum sheave diameters — typically 20× chain link diameter) then solve for D2.
Rule of thumb: for shop hoists in the 0.5 to 3 tonne range, the diameter difference lands between 3 and 8 mm. Below 3 mm you fight chain wrap errors; above 8 mm you lose self-locking and need to add a brake.
A Weston unit has no gears at all on the lifting side — the mechanical advantage comes entirely from the diameter difference between two coupled sheaves on the same axle. It's mechanically dead simple, fully self-locking, and tolerates dirt and weather. The downside is low efficiency around 85% and slow lift speed.
A planetary chain block uses a reduction gearbox between the hand wheel and the load sheave, achieving the same mechanical advantage in less chain travel but at higher cost and with mandatory load brake. Pick Weston for outdoor, occasional, or harsh-environment use; pick planetary for production lifting where speed matters.
That's classic spider-gear tooth contact under torque. When the differential is loaded, the side gears push the spider gears against their cross pin and the tooth flanks load up; if the bevel gear pattern is set too deep or the backlash is below 0.05 mm, you get a pitch-line whine that disappears the moment torque drops to zero on coast.
Pull the carrier, check side-gear backlash with a dial indicator on a tooth, and shim the side gears outboard until you see 0.05-0.10 mm backlash. Also check the spider pin for galling — a worn pin lets the spider gears tilt under load and creates the same noise.
No, and this trips up a lot of designers. A differential gear shares torque equally between two outputs but the sum of their speeds is fixed by the input — if one runs faster, the other runs slower by the same amount. It's a speed-difference allocator, not an independent two-speed drive.
If you need two outputs at fixed but different speeds, use a layshaft with separate pulley ratios for each output. Reserve the differential for cases where the outputs naturally want to vary in relative speed (turning a corner, uneven cutter load) but should share torque equally.
If pull effort climbs gradually over months, the chain has elongated. If it jumps overnight, suspect bearing failure on the pulley axle or the differential carrier. A seized needle bearing in the carrier converts what should be smooth rotation into a stick-slip drag that adds 30-50% to operator effort.
Quick diagnostic: lift the load 100 mm and listen. A healthy unit makes a smooth tick-tick from chain entering the pocket. A bearing-failed unit makes a rougher grinding sound and the hand chain feels notchy through each revolution. Strip and inspect the axle bearings — they're usually 6000-series radial bearings and replacement is under $20.
Measure it directly. Hang a known load (a 50 kg kettlebell works), measure the hand-chain pull with a fish scale, compute MAreal = load / pull, then divide by MAideal from the diameter formula. The ratio is your efficiency.
For a clean, properly chained Weston-pattern unit, expect 0.80 to 0.88. If you measure below 0.70, you have either a fouled pulley pocket, misaligned sheaves on the axle, or a chain that doesn't match the pocket pitch. Anything below 0.60 means the unit isn't safe to use because the self-locking margin has collapsed and the load may back-drive when you let go.
References & Further Reading
- Wikipedia contributors. Differential pulley. Wikipedia
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