Visualize the tension in a rope wrapped around a drum with the Euler-Eytelwein formula. Adjust friction, wrap angle and load tension to learn how wrapping amplifies force exponentially.
Parameters
Friction coefficient μ
—
Wrap angle β
°
Load tension T_load
N
Observation point (from hold side)
%
The "observation point" is a position along the contact arc, from the hold side (0%) to the load side (100%).
Results
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Required holding force T_hold
—
Force ratio T_load / T_hold
—
Tension at observation point T(φ)
—
Force reduction
Drum and Rope Tension
Color = tension (blue = small, red = large) / thin arrow = holding force, thick arrow = load tension, yellow dot = observation point
Tension Distribution Along the Arc T(φ)
X axis = angle φ from the hold side / Y axis = tension T (yellow dot = observation point, dashed line = load tension T_load)
Theory & Key Formulas
The ratio of the two end tensions of a rope wrapped around a cylinder grows exponentially with the wrap angle. This is the Euler-Eytelwein formula, also known as the capstan equation.
Ratio of the end tensions (force ratio). β is the wrap angle in radians, μ is the friction coefficient:
This ratio does not depend on the drum radius. The larger the wrap angle and the higher the friction coefficient, the larger the load a small holding force can support.
What is the Belt Friction Simulator
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When a ship is moored, the crew just seems to loop a thick rope around a post a few times. Does that really hold?
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That is exactly the power of belt friction. Roughly speaking, when you wrap a rope around a cylinder, the ratio between the holding force and the pulling force grows exponentially with the wrap angle. As a formula, $T_\text{load}/T_\text{hold} = e^{\mu\beta}$. In the simulator above, increase the "wrap angle β" and watch the required holding force shrink dramatically.
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Exponential? So even a little wrapping is very effective?
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Right. With a friction coefficient of 0.3, one full turn (360 degrees) gives a force ratio of about 6.6, and three turns about 290. That is why two or three turns around a bollard let one person hold a ship of many tonnes. Set β to 1080 degrees (three turns) in the simulator and look at the "force ratio" card — it becomes an enormous number.
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Wait, so does a thicker post make it work even better?
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That is the interesting part — the radius does not appear in the formula. A thicker post means a longer contact length, but the force the rope presses on the post with is spread thinner by the same factor, so the two cancel out exactly. Only the wrap angle and the friction coefficient matter. The simulator deliberately has no diameter slider, because it would not change the result.
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I see! When I move the "observation point" slider, the yellow dot moves along the curve in the graph.
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That shows the tension at a point partway along the contact arc. At an angle φ from the hold side, the tension is $T(\phi)=T_\text{hold}\,e^{\mu\phi}$. The graph is an upward exponential curve because, all along the wrapped section, friction is steadily "taking over" part of the tension. In real belt drive design, understanding this tension distribution is what makes the difference.
Frequently Asked Questions
It is set by the combination of materials of the rope (or belt) and the cylinder in contact. A dry hemp rope on a steel drum is roughly 0.2 to 0.3, and a rubber belt on a steel pulley is about 0.3 to 0.5. The value drops sharply when wet, so for safety-critical uses such as mooring, design with a conservatively low value. A V-belt's wedge effect makes the effective friction coefficient appear two to three times larger.
Physically there is no problem, and multiple turns are normal on mooring bollards and winches. The β in the formula is in radians: one turn is 2π, three turns is 6π, substituted directly. But because the force ratio grows exponentially, a few turns give an astronomically large ratio in theory. In practice the rope's own weight, stiffness and the layers biting into each other cause deviations from the ideal formula.
The load tension T_load is the larger one, and the holding force T_hold is the smaller side. Because friction acts opposite to the direction of impending slip, you can hold the loaded side with a smaller force. Conversely, if you pull the hold side even slightly harder than the load side, the rope starts to move toward the hold side. So this formula describes the limiting ratio right at the verge of slipping.
Yes — a band brake is exactly an application of the capstan equation. A band is wrapped around a rotating drum, one end fixed and the other pulled by a lever, and the difference of the end tensions (T_load minus T_hold) becomes the braking torque. Whether "self-energizing" (self-servo) action occurs depends on the relationship between the wrap direction and the rotation direction, so the wrap direction must be chosen carefully in design.
Real-World Applications
Mooring and winch operation: Securing a large ship to a bollard at a harbor is the most familiar application of belt friction. By wrapping the rope a few times, a worker can control a tension far exceeding their own body weight. An electric capstan winch works on the same principle — a rope is wrapped a few times around a rotating drum and a small holding force hoists a large load.
Belt and chain drive design: In a belt drive transmitting torque between pulleys, the maximum torque that can be transmitted without slipping is set by the difference between the tight-side and slack-side tensions, and the upper bound of their ratio is given by the capstan equation. Adding an idler pulley to gain wrap angle, or adopting a V-belt or timing belt to raise the friction coefficient, are all design decisions derived from this formula.
Band brakes and safety devices: Holding brakes on construction-equipment winches and cranes, and the classic automotive band brake, obtain braking torque by wrapping a band around a rotating drum. With a clever choice of wrap direction, the braking force itself tightens the band — a "self-energizing" action that yields a large braking torque from a small operating force.
Climbing and rescue rope work: A climber's belay and the braking device for a rappel pass the rope through a carabiner or descender to bend it, stopping a fall by belt friction. Because the braking force depends on how many places the rope is bent and at what angle it is wrapped, this is understood as a basic principle of rope work.
Common Misconceptions and Cautions
The most common misconception is to think a thicker drum or post "works better". As shown by the absence of the radius from the capstan equation, the tension ratio does not depend on the diameter at all. For a fixed wrap angle the contact length grows in proportion to the radius, but the normal force per unit length falls in inverse proportion, so the two cancel exactly. The simulator has no diameter slider precisely because it would be a "dead parameter". Remember that only the wrap angle and the friction coefficient matter.
The next most common error is to assume the force ratio grows "in proportion" to the wrap angle. In reality it is an exponential, so doubling the wrap angle does not double the ratio — it squares it. With a friction coefficient of 0.3, the ratio is about 2.6 at 180 degrees, 6.6 at 360 degrees, and 17 at 540 degrees; the rate of growth itself accelerates. Moving the wrap-angle slider in equal steps while watching the force-ratio card lets you feel how it jumps up faster toward the higher end. This "exponential amplification" is the essence of belt friction.
Finally, take care not to misread this formula as "a static relationship that always holds". The capstan equation $T_\text{load}/T_\text{hold}=e^{\mu\beta}$ describes the limiting state right at the verge of slipping. If the actual holding force is larger than this, the rope stays at rest (friction develops only as much as is needed); if it is smaller, the rope starts to slip. In other words, the formula gives the minimum holding force needed to prevent slipping, and in practice an ample safety factor is added to allow for scatter in the friction coefficient and the drop to kinetic friction.