Quantifying friction for a rope coiled around a cylinder with external loading

Question

The Problem

Consider a massless rope that is coiled around a cylinder with radius $r$.

The shape of the rope's coiling is helical defined by variable curvature $\kappa(s)$ and torsion $\tau(s)$, where $s$ is the arc length parameter starting from the rope's initial contact with the cylinder at one end ($s=0$) up to the last point of contact at the other end ($s=l$).

The rope is initially pre-tensioned at both ends with a tension $T$. The rope is then rigidly fixed at the extremities. There are no discontinuities in the tangent of the rope at any point.

The friction coefficient between the rope and the cylinder is $\mu$. Friction can be assumed to be Columbic.

The Question

What is:

1. The maximum amount of longitudinal friction force $F_{fric}$ (i.e. along the cylinder's axis) that can be generated between the rope and the cylinder to counteract longitudinal motion of the cylinder due to to an external longitudinal force $F_{long}$ acting on it?
2. The maximum amount of rotational friction torque $\tau_{fric}$ (i.e. about the cylinder's axis) that can be generated between the rope and the cylinder to counteract rotational motion of the cylinder due to to an external rotational torque $\tau_{rot}$ acting on it?

Consider both cases occurring separately but also simultaneously.

Initial Attempt

An initial attempt was carried out building upon the work of Konyukhov and Schweizerhof to generalize their proposed model for the presented case. However, the model is limited to a constant curvature/torque profile and to a limiting geometrical case defined by a ratio between curvature and torsion.

Any insights or propositions would be appreciated.

Note: There is no need for a closed-form solution (although that would be highly beneficial). Numerical approaches are also welcome.

Answer 1

Firstly, if the rope ends are rigidly fixed, you would likely have zero friction and therefore, zero torque because the tension in the rope would be zero, despite being "preloaded".

So I'll just assume that the rope ends are not rigidly fixed and that there is some load applied to each side. In each half wrap, so $\theta=180$°, the friction force between the rope and the cylinder would be $2T\sin(\frac{\theta}{2})\mu$ where $2T$ can be replaced with $T_{Load} + T_{Hold}$ assuming you know what at least one of the applied loads is on one of the rope ends and that the applied loads are not the same. If they are the same, again, you have no friction, so let's assume they are not and you know what one of the loads is; and we'll consider this to be $T_{Load}$. The load on the other end is then, $T_{Hold}=\dfrac{T_{Load}}{e^{\mu \theta}}$. You can do this by wrap, or for the entire cylinder.

Finally, $\mu$ is material dependent, so you can't answer the question, "what is the maximum amount of... frictional anything", with the information provided.