This capstan gear uses a sun pinion on the input shaft, three planet pinions carried by the winding drum, and an internal ring gear. Different pawl engagements select reduced drum speed or direct drive.
Capstan Gear Interactive Calculator
Choose sun and planet tooth counts, input speed, torque and drive mode. Slow mode holds the ring stationary. Direct mode locks the ring to the drum carrier. The drum inset turns at the calculated carrier speed.
Equation Used
- Three equally spaced planets; a common gear module is assumed. Tooth counts step by six so assembly phasing remains compatible.
- Ring teeth are derived from sun and planet teeth. The selected ranges also avoid adjacent-planet tip overlap for standard addendum proportions.
- Slow mode locks ring to frame; direct mode locks ring to drum carrier. Pawls are schematic ideal constraints, not a contact or load-rating model.
- Tooth outlines are illustrative. Pitch geometry and angular speed relationships are exact for the ideal simple planetary model.
- Playback is slowed by a factor of four. Zero input speed stops all gears. No efficiency, rope traction, bearings, stresses or rated load are calculated.
Hiscox figure 944 shows a planetary capstan gear with two selectable modes. Holding the ring reduces drum speed and increases ideal torque; locking it to the carrier gives direct drive. Rope friction is a separate effect described on the capstan-form page.
A planetary gear inside the capstan
Hiscox’s figure 944 shows a central pinion attached to the input shaft. Three intermediate pinions rotate on a carrier that is attached to the winding drum. An internal gear surrounds the planets.
For slow drum motion, external pawls hold the ring stationary. The sun turns the planets, and their orbit turns the carrier and drum at reduced speed. For quick motion, a pawl carried by the drum locks the ring to the carrier. The gearset then rotates together in direct drive.
The animation shows those two constraints separately. It does not shift pawls while a real load is moving. A drum inset carries the same output angle as the planet carrier.
Two operating ratios
The historical mechanism changes drum speed through gear constraints. Slow mode trades output speed for greater ideal output torque. Direct mode turns the drum at input-shaft speed.
This is separate from the friction between rope and drum. For that relation, use the capstan rope-friction calculator. Multiplying a gear torque ratio and a rope-friction ratio does not by itself give a rated lifting capacity.
Gear speeds and ideal torque
For common-module gears, the ring tooth count is Nr = Ns + 2 Np. The planetary speed relation is Ns(ns − nc) + Nr(nr − nc) = 0, where s is sun, r is ring and c is carrier.
Holding the ring gives nc = ns Ns/(Ns + Nr). The input-to-output speed reduction is therefore 1 + Nr/Ns. Locking the ring to the carrier gives nr = nc and hence nc = ns: direct drive.
The planet’s absolute spin is np = nc − (Ns/Np)(ns − nc). Its center rotates with the carrier. For ideal lossless operation, output torque magnitude is input torque multiplied by the reduction ratio.
Default four-to-one example
With 24 sun teeth and 24 planet teeth, the ring has 72 teeth. Slow mode has a reduction of 1 + 72/24 = 4. A 30 rpm input produces 7.5 rpm at the drum, and 10 N·m input torque corresponds to 40 N·m ideal output torque.
The planets spin at −15 rpm relative to the fixed frame while orbiting at 7.5 rpm. Selecting direct mode changes drum speed to 30 rpm and ideal torque to 10 N·m; the planets then turn at 30 rpm with the rest of the assembly.
Scope of this illustration
The selectable tooth counts maintain the basic geometry and three-planet assembly condition. The illustration shows the correct pitch-circle sizes and rotation relationships, with simplified tooth profiles and pawls.
It is a kinematic comparison, not a manufacturing drawing. Tooth strength, interference details, bearing loads, pawl stresses, lubrication, efficiency and drum dimensions require separate design information. Input torque is an entered operating value rather than a rating.
Questions about capstan gearing
Why is this different from the rope-friction calculator?
The gears set the drum’s speed and ideal torque. Friction between rope and drum sets a separate tension relation, explained on the linked capstan-form page.
What moves in slow mode?
The ring stays fixed. The sun rotates, the planets spin and orbit, and their carrier drives the drum at reduced speed.
Why do all gears turn together in direct mode?
Locking the ring to the carrier removes relative gear motion. The planetary speed equation then requires the sun, ring and carrier to have the same speed.
Does ideal drum torque give line-pull capacity?
No. Drum radius, losses, rope traction and hardware limits are not entered here.
Primary mechanism and gear references
Gardner D. Hiscox, Mechanical Movements, Powers, Devices and Appliances (1901), figure 944, page 240: capstan gear and its two pawl-selected modes.
KHK, Internal Gears technical information, page 209: common-module planetary tooth counts, assembly and planet spacing conditions.
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