A vertical windlass or capstan carries rope around an upright drum. Traditional bars turn the head, and a pawl engages the circular rack at the base. This calculator examines the rope-friction limit, separately from the leverage at the handle bars.
Capstan or Vertical Windlass Interactive Calculator
Adjust tail tension, rope turns and friction coefficient. See the rope path and compare the ideal high-side tension at impending slip. Animation highlights points along a stationary rope; it does not simulate hauling speed.
Equation Used
- Flexible rope on a circular surface, with constant friction coefficient and negligible bending stiffness.
- The entered turns give contact angle theta = 2 pi N. The small helix pitch separates turns visually and is neglected by the planar belt-friction equation.
- Both tensions are forces in pounds-force (lbf). No load mass is calculated.
- No rated capacity, actual load, slip rate, heat, rope strength or drum torque is calculated.
Each complete rope turn contributes 2 pi radians of contact. The exponential friction relation gives the limiting tension ratio at impending slip; multiplying it by the tail tension gives an ideal high-side force limit in lbf.
Drum, handle bars and base ratchet
Hiscox figure 52 identifies this mechanism as a capstan or vertical windlass. It has an upright rope barrel, bars through the head and a pawl at the circular rack on its base.
The bars supply drive torque. Rope friction is a separate relationship between the tensions at the free ends. The calculator evaluates rope friction. Handle force and drum torque also require the handle and drum radii, which are not entered here.
The drawing separates the entered turns and shows tangent free ends. A highlight follows the stationary rope contact to explain limiting tension. Its movement is explanatory, not a prediction of hauling speed.
What this model explains
The capstan equation explains how friction distributed around a curved contact permits different tensions at the two ends of a flexible rope or belt. A capstan drum is one application of that principle.
The calculator explores that tension relationship. It does not select a marine capstan, rigging system or lifting machine, and it does not establish the capacity of their hardware.
Use the physical contact angle
Each full turn contributes 2 pi radians. With N turns, theta = 2 pi N, and the ideal limiting tension ratio is exp(mu theta). The ratio grows exponentially with friction coefficient and contact angle.
Multiplying tail force by this ratio gives the high-side force at impending slip under the model. Before slip, static friction can support a lower tension ratio. The actual load need not equal the displayed limit.
The tail-percentage result is 100 divided by the limiting ratio. It expresses the low-side force as a percentage of this ideal high-side limit, not equipment efficiency or a capacity margin.
Four turns at the default friction value
Four full turns give 8 pi, approximately 25.133 radians. At mu=0.30, the limiting ratio is exp(0.30 × 8 pi), approximately 1881.50. A 50 lbf tail force therefore gives an ideal high-side limit of about 94,075 lbf.
This large mathematical result is not a rating for any rope, drum or ratchet. Their strengths are absent from the equation. Check the load limits of the complete rope and hardware assembly independently of this friction estimate.
Friction limit and equipment capacity are different
The exponential expression can return very large values. Actual equipment is also limited by rope strength, drum structure, bearings, brakes, supports and operating conditions. None of those limits is entered here.
The drawing uses a small, constant pitch to keep turns visible. Its proportions are illustrative. Rope migration, rope-on-rope contact, bending stiffness and changing friction are outside the simple circular-contact model.
Questions about the vertical windlass
How is rope wrap converted to contact angle?
Multiply the number of complete rope turns by 2 pi to obtain radians. Four turns give 8 pi radians, approximately 25.133.
Is the calculated force a lifting capacity?
No. It is an ideal impending-slip force limit. Rope and hardware strength are not included.
Does this calculate the handle-bar effort?
No. The handle and drum radii are not entered. The current controls describe rope friction.
Why are the drum and rope stationary?
The model describes a limiting static condition. The moving highlight traces contact position and tension; it does not represent rope speed.
Drawing and equation references
Gardner D. Hiscox, Mechanical Movements, Powers, Devices and Appliances (1901), figure 52, page 29: vertical capstan form, handle bars and base ratchet.
Engineering Statics, section 9.5, Flexible Belt Friction: tangent free ends, contact angle and limiting tension relation.
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