Capstan Static Friction Bound

Static admissibility bounds

For an ideal flexible massless rope wrapped on a fixed rough cylinder with static Coulomb coefficient mu and contact angle theta = 2 π × turns, the applied tension (T_applied) and reference tension (T_reference) must satisfy the Capstan equation inequality for static equilibrium:

Both tensions are assumed positive. Equality represents the impending-slip bound, which is not reached in every static state. Both directions matter: the applied tension can be greater than or less than the reference tension. The log admissibility margin is calculated as mu × theta - |ln(T_applied / T_reference)|. If the margin is non-negative, the state is statically admissible.

Calculating bounds and log ratios

Consider a rope wrapping halfway around a cylinder with a friction coefficient of 0.3. A reference tension of 2 N is applied to one end, and an applied tension of 5 N is pulled on the other end.

1. The contact angle theta is 0.5 turns × 2π = 3.141592653589793 rad.
2. The maximum admissible tension is 2 × exp(0.3 × 3.141592653589793) ≈ 5.13266479041627 N.
3. The minimum admissible tension is 2 × exp(-0.3 × 3.141592653589793) ≈ 0.7793222747506936 N.
4. The required mu is |ln(5 / 2)| / 3.141592653589793 ≈ 0.2916643985741246.
5. The applied tension of 5 N lies within the bounds, so the state is static admissible.

Exploring limits with real controls

Investigate how the wrap angle and friction coefficient expand the admissible region.

1. Open the interactive module. Set Friction coefficient to 0.3 and Wrap turns to 0.5.
2. Set Reference tension to 2 and Applied tension to 5. Notice the margin is positive, so it is static admissible.
3. Increase the Wrap turns to 1. The maximum admissible tension jumps up to over 13 N, significantly increasing the log admissibility margin.
4. Swap the tensions by setting Reference tension to 5 and Applied tension to 1. The required mu becomes roughly 0.51, which exceeds the available 0.3, resulting in a Below lower bound violation.
5. Set Wrap turns to 0. The required friction coefficient becomes Unavailable if tensions are unequal. Only equal tensions are permitted.

Limitations of the ideal model

This model uses an ideal flexible massless rope with no stretch. It assumes a uniform static Coulomb coefficient mu and contact angle theta without accounting for dynamic acceleration or the actual failure of real equipment. The cylinder is perfectly fixed and rough, without a changing-contact model. Weight is ignored.

Primary references

These primary references detail the capstan tension equation and the associated mathematical bounds utilized in this physics model.

• MIT OpenCourseWare 2.007 Design and Manufacturing I, Lecture 18: Provides the foundational capstan equation derivation, exponential bounding, and considerations for impending slip.