What is the correct friction coefficient and theta (capstan equation) for a strap/rope in a tie down mechanism? I am curious about the mechanism of the tie-down strap.
I understand that these devices typically will use a ratchet-prawl mechanism to incrementally wrap a (typically polymer based) strap around itself.
I am interested in the tension/force that the mechanism (i.e, the pawl) must provide to prevent the strap from slipping.
Specifically, I first considered the Capstan equation:
$$T_{2}=T_{1}e^{\mu \theta}$$
Where $T_{2}$ is the $load$ force needed to be applied so the ratchet mechanism can hold against a force $T_{1}$.
However, when I review the equation I cannot figure out what would be the correct $\mu$ to apply. It appears $\mu$ is the coefficient between the rope/strap and the cylinder or in this case the windlass.
However, in the tie down mechanism the strap/rope is wrapped around itself. The initial piece of rope/strap goes around the windlass and the rest wraps on itself.
So my question is: which is the appropriate $\mu$ to use? Does $\theta$ and $\mu$ only apply to the rope/strap that contacts the windlass? How do you accommodate the rope on rope / strap on strap friction?
Could the answer be a superposition of multiple Capsan equations each with $\theta = 2\pi$ such that the first term is the strap / cylinder and the rest are strap on strap until you reach the total number of windings in the tie down mechanism?
 A: I think you are on the right track, but it may get complicated to follow the the different tensions as the number of layers increases. I am pretty sure that for metal ratchet-strap tie-downs, the important friction is strap-on-strap. The strap-on-metal friction is negligible in comparison. (I spent some time thinking about this when trying to figure out - in the days before handy online videos - why I often had such a hard time undoing tie-downs.)
A closely related mechanism is discussed in "Novel frictional-locking-mechanism for a flat belt: Theory, mechanism, and validation". I believe that the tie-down is equivalent to the mechanism in this paper, if we assume zero strap-metal friction and that $\theta<\pi$ as in this diagram:

Equation 17 of the paper then implies us that the tie-down will not slip as long as $\mu\theta >\ln{2}$, where $\mu$ is the strap-strap friction and $\theta$ is the angular region where the upper strap is holding the lower strap onto the metal. I believe this follows from having the total load tension split between the upper and lower straps, so effectively we have a hold tension $T_{hold}=T_{load}/2$ and the strap should slip if
$$T_{load}<T_{hold}e^{\mu\theta}=\frac{1}{2}T_{load}e^{\mu\theta}$$
This result should not be trusted for real tie-downs, however, since according to this article on Tie-Down Ratchet Strap Safety:

The … Web Sling & Tie Down Association, …
found that less than two wraps of webbing around the mandrel may result in webbing slippage under load. If more than four wraps are used, it can place unnecessary strain on the ratchet device and actually reduce the working load limit (WLL) of the unit.

Our simple model ignores issues such as vibration reduced friction, non-linear friction for soft materials such as straps, stresses at sharp bends in the mandrel, …. Never trust simple models over results established by real-life measurements and testing.
