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capstan equation, Eulers formula, power law friction, Problem with incoming rope force = 0



Dear colleages,


I am dealing with rope friction and the so-called Capstan equation.

Situation: A rope wraps around a cylinder with a wrap angle. It depends on the input force.

There are very comprehensive approaches by other colleagues, where the friction value depends on the normal force or pressure.

They are presented in the following publications.

Capstan equation including bending rigidity and non-linear frictional behaviour", Jae Ho Jung, Ning Pan, Taewook Kang, doi: 10.1016/j.mechmachtheory.2007.06.002  Equation 11

" Constraint ability of superposed woven fabrics wound on capstan " , Junpeng Liu Murilo Augusto Vaz Anderson Barata Custódio, doi: 10.1016/j.mechmachtheory.2016.05.014, equation 8.

Both equations have the problem described in the illustrations. If the incoming force becomes 0, then an outgoing force other than 0 will still be output. This is not possible if rope stiffness is neglected.

Symbols for Jung et al.: Incoming force T.1=0 kN. If the wrap angle theta and friction coefficient alpha are large enough, the outcoming force T.2 is already almost 0.8 kN.

Symbols for Liu et al.:  Incoming force T.0=0 kN. If the wrap angle theta and friction coefficient a.1 are large enough, T.1 is already almost 3 kN.

Does anyone know the problem?

Best regards

Paul Schumann


Jung power law friction.JPG

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Dear Paul,

I think the key to your dilemma is in the title of the first reference that you cite that speaks about "bending stiffness." The expression usually called "the capstan equation" is derived without considering bending stiffness. Without bending stiffness, the forward tension on the line is directly proportional to the back tension.

It is not easy to imagine the situation with zero back tension but including bending stiffness. If you imagine the rope somehow bent around the capstan with zero back tension, then there will be a contact force between the rope and the capstan. The friction developed by this contact force is the source of the forward tension, even in the absence of back tension.

I dare say this is all pretty iffy business, beyond the scope of the things where the capstan equation is usually applied.



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Dear DrD, Thank you for your reply. I can well imagine and understand the situation you describe. However, the posted version of Jung's equation is the version without bending stiffness. Liu's equation also does not take bending stiffness into account. The effect seems to be purely in the approach to the frictional forces. The power function approaches the y axis asymptotically for x towards 0. It thus becomes perfectly perpendicular, so that calculation programmes can only output the last value before the perfect perpendicular is reached. Does anyone know a workaround for this? Best regards Paul


Thema Green Plot is Ist Jungs/ lius equation. The Blue one ist the classical capstan equation. (Euler) you can see that this. Equation is completely useless in case of rubber friction.

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