Capstan equation, Euler's formula, power law friction

[capstan1]

TL;DR Summary

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

Dear colleagues,

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

Bastian

 

[Dullard]

Those equations (obviously) don't work for negative tension (you can't push a rope). It wouldn't surprise me if the authors state/imply that 'zero' (or effectively zero) tension was also not in the domain of this function.

 

[capstan1]

Those equations (obviously) don't work for negative tension (you can't push a rope). It wouldn't surprise me if the authors state/imply that 'zero' (or effectively zero) tension was also not in the domain of this function.

Hello, thank you for your quick reply. We may have misunderstood each other. I didn't mean that I want to push the rope. By the way this situation would work with the classical Euler equation.
I was wondering that with the equations at hand, near zero or at zero does not come out zero as with the classical Euler equation. I did not want to reverse the sign of the incoming rope force. Maybe you have an idea for this. Best regards and have a nice evening
Paul

 

[capstan1]

Hello, thank you for your quick reply. We may have misunderstood each other. I didn't mean that I want to push the rope. By the way this situation would work with the classical Euler equation.
I was wondering that with the equations at hand, near zero or at zero does not come out zero as with the classical Euler equation. I did not want to reverse the sign of the incoming rope force. Maybe you have an idea for this. Best regards and have a nice evening
Paul

I am referring to the graph. The green plot shows what I mean.

 

[Baluncore]

Does anyone know the problem?

The different functions you give by Jung and by Liu appear to be numerical approximations that are not designed to pass through zero. They are special case adaptions for restricted applications.

The Capstan Equation involves the exponential function which is a transcendental function, it transcends simple algebra, yet the approximations employ normal algebra, they are not transcendental.

For those reasons, you should abandon the approximations and revert to the original exponential Capstan Equation.
https://en.wikipedia.org/wiki/Capstan_equation