A driven capstan uses friction around a rotating drum to support different tensions at the two ends of a flexible band. Explore the principle and calculate the ideal tension ratio and corresponding drum torque for the entered wrap and friction.
Ideal Capstan Tension and Drum Torque
Enter low-side tension, coefficient of friction, total wrap angle and drum radius. Results are ideal Euler–Eytelwein limits, not a real amplifier rating or cascade design.
Equation Used
- The entered coefficient is constant around the full wrap.
- The band is flexible and normal pressure follows the capstan derivation.
- Torque is only the ideal tension difference times entered drum radius; speed, power, wear, heat and stability are excluded.
The calculation covers one ideal capstan stage. The entered friction coefficient and wrap angle determine the limiting tension ratio; drum radius converts the tension difference into torque.
Driven capstan as a mechanical amplifier
MIT's model describes a flexible band sliding around a driven capstan. Friction permits different tensions at the two ends; the motor driving the drum supplies the power associated with their difference. Bush's differential analyzer used opposed rotating drums and two stages so a very small integrator torque could control a much larger output torque in either direction.
The animation is a one-stage principle view, not a reconstruction of the opposed two-stage machine.
One ideal relation, not a device rating
Use the screen to inspect exponential sensitivity to entered friction and wrap. Do not use it to choose materials, pretension, stages, motor power or safe load.
Euler–Eytelwein relation
The MIT mechanical-amplifier model states the same relation under continuous slip and adds a drum-rim-speed condition. This page reports only the static ratio and ideal tension-difference torque.
Arithmetic example
For T1=5 N, mu=0.25, theta=180 degrees and r=50 mm, the ideal tension ratio is 2.19, the high-side limit is 11.0 N, the difference is 6.0 N and corresponding drum torque is 0.298 N m.
Limits of the single-stage calculation
The tension relation alone does not determine a cascade design or the attainable performance of a driven amplifier. Drum speed, slip, material behavior, heat, wear and stability require additional device-specific analysis.
What the ideal relation does not diagnose
The capstan equation cannot diagnose pretension, glazing, overlap, seal leakage, drift or any other device fault. Use the actual manufacturer documentation, service data and an instrumented inspection for a real mechanism.
Validation boundary
A real driven-capstan amplifier needs a device-specific, instrumented test plan covering its materials, drive, speed, heat, wear, stability and loads. This ideal calculator prescribes no operating or repair test.
Capstan-amplifier questions
Is high-side tension guaranteed?
No. It is an ideal limiting value for the entered assumptions.
Does the calculator size multiple stages?
No.
Where does amplified power come from?
In a driven capstan amplifier it comes from the drum drive, subject to speed, slip and loss conditions not solved here.
Primary and institutional references
- MIT OpenCourseWare, Capstan—a mechanical amplifier: force relation, drum torque, continuous-slip and speed conditions.
- Bush and Hazen, The Differential Analyzer (1931): original two-stage opposed-drum torque-amplifier description.
- MIT, Vannevar Bush's Differential Analyzer: museum and reconstruction context for the contra-rotating capstan amplifier.
- Harvard Natural Sciences Lecture Demonstrations, Friction: Euler–Eytelwein relation and radians convention.
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