Diameter of rope or host doesn't matter to friction ; but rather the relationship between rope vs host diameters matter to efficiency/tensile support maintained i think.
Counterintuitively/ our eyes lie to us, collective arc frictions compound by degree of contact, not distance of contact. Thus, need to purposefully filter and adjust what eye hands to brain. A larger capstan/bollard easier on rope and dispersed heat byproduct, more substantial size/strength, possibly bigger heatsink if metal etc. host, NOT greater friction. To this model, nylon rope of any diameter (that rope can properly seat ) on aluminum pipe lends same brake force w/same amount of turns/degrees of contact in 2" or 2' diameter capstan(windlass) or bollard(fixed). Classically a windlass is horizontal vs. capstan vertical; but just a window dressing of same machine force to me.
. Thus, the Capstan/Bollard Effect, Stephen W. Attaway, Ph.D Att_Frict.pdf (link) shows effect of rope frictions compound arcs on Porty, Fig8., Rappel Rack, rocks etc. math in arcs on this; as an exponentially compounding math. The math used for calculating the compounding frictions in rope arcs is virtually same as calcs of exponential compounding growths of intere$t, population, disease spread, wave forms etc.
My own fledgling effort of Frictions CoF spreadcheat(link) now has a calculator can alter the CoF value in green dropdown shows how this would work with different mated materials in different amount of 180 arcs. 180 measurement as shown in ATT_FRICT.pdf preferred to me as also incorporates the single axis directional back and forth from a directional/linear input more recognizing directional factor maintained from input of Standing Part linear force converted to radial control of arcs. Thus not applicable to Round Binding against swell of force from all directions/axises in 2D dimension as radial/not linear force input to rope.
We take a focused, single direction 1D linear rope force and disperse to all axis/radial (2D) to control. Reverse of explosion radially 3D in all axises/directions, funneled/concentrated to one axis/ 1D, with 1 end blocked to force all force to now fold into now single direction on the single axis in gun barrel or engine cylinder type container. Axis contains equal and opposite pair, direction is output. Rope dilutes 1D linear to 2D radial, tbut gun/engine takes 3D radial and concentrates down to linear 1D (reverse utility).
A stone arc(h) bridge, is same miracle as rope arc to me. Only bridge uses all cosine and sine (all produced forces) expressing as compression to support, voiding the tension; because non-malleable masonry/stone/cement w/o modern binders has tension tolerance = ~10% of compression tolerance. Rope is reverse; cant use compression in rope for support so is opposite direction of all tension from cosine/sine thru rope arc vs compression in stone arc. Same principle of the same ruling geometry, just opposite direction of force flow of tension vs compression directions from the equal & opposite(less friction) end points force flow direction thru device/arc.
Tree Bones/wood has tension tolerance = ~70% compression tolerance but the Ancients wanted non decaying non-malleable structure to survive the ages. Steel a good choice as tension tolerance = ~100% compression tolerance, but they had not tamed nor really even found it back then.
Cycles to failure very relative, even in same day from elasticity point, that can fade and come back later from an initial dynamic hit. From more macho strength view cycles are more of long term to degradation that is permanent loss to static capacity. i think inside of it working rope less harder thru those cycles with less hits, lower leveraged force angles and more efficient knots that don't work rope as hard for same task can be a factor. If before Hitch or Bend had choice of Overhand or Fig.8 to pass thru, i'd think fig8 rope would not only be stronger but last longer to same tasks, just more efficient. Just like kink in chain or cable, would not last as long, as wears out faster, working itself harder.
Kenny, I don't think the bollard equation tells the whole story. When you get down to real small elements like in a rack, or the zigzag or unicender the radius at contact induces effects in/from the rope. In my data same material different bollard diameter shows this effect as different calculated effective mu's although mu should be the same due to identical materials and surfaces. I appreciate you reinforcing the exponential nature of the bollard equation to me. Real eye opener.
I think development of devices just deal with this empirically.
i agree on several levels - in this Direction !
To this counter-intuitive arc world(as flagged by Ancients);
Capstan Theory best start i think tho for our purposes.
And is truer to simpler continuous Bollard/Capstan arcs than more complicated/contradicting Rack even tho subtle changes from rule, they do pivotally define tho IMHO.
Bollard/capstan present a ' complimentary/radial list' of arcs in continuous direction arc is Native to.
Rack a more 'competing/linear list' of weaving/countering(direction) arcs not committed to Native radial but rather converted to linear progression less Naturally. RULE: Any conversion will have a co$t/loss.
So as we go to rack, fig.8 etc. we have added influences layered on top of the capstan effect, as more influence than over rule tho i think.
The closeness of the arcs as does the direction reverse intensifies the effects, of degree and direction of contact on linear span of arcs. Whether the mechanical closeness affecting angle is invoked by physical position and/or diameter to yield change of same angle of the reverse pull. Noticed mostly as we 'overcrowd' larger rope into what usually is more predictable in the smaller, more nominal sizes used for most. Still, as a base function, knowing the capstan effect brings close/closer than w/o i think. And then perhaps how to work it better, totally understanding to command more completely and confidently thru it's layered functions.
i always visualize a rope breaking in Standing Part, at most loaded/before friction reductions point, where it just deforms, then just before, as like internals drawn to stiffness of glass fiber and breaks then from back pressure of impending deformity in highest loading area. To this model capstan/bollard flex this once and maintain, while a rack or fig.8 bend these same areas backwards seems more 'disruptive'/inefficient to internal flow of force than continuous flow around bollard/capstan.
Carried on from Dr.Attaway's att_frict.pdf paper(link) i use counts in 180 arcs and i maintain same in calculator, table etc.(link) i believe this keeps linear concept by just 'reflecting' back and forth greatest compound force against host on the source force directional axis most properly(until that runs into a 90 flipping then force flow to what was the cross-axis). i set the compounding /apes of primary arc to pull same direction as input, then map from there on.
Compounding frictions math is key here. Arc uses BOTH sine and cosine in arc working together, that a focused linear uses these quantities separately for separate utility functions. Like, arc uses cosine and sine for support and for frictions , both for both/ALL tension forces produced, linear only uses cosine/inline for support and sine/deflection for frictions. MUCH more controlling friction in arc and w/o support loss(unless host too small/harsher rope deforming)...
Thus i visualize all knots and rope work falling between outer extreme of 2 reciprocals:
Capstan effect of more friction than see in a knot that can be forsaken to get more of the other/reciprocal extreme of pulley effect of compounding(increase of Capstan Effect decreases Pulley Effect vice/versa.). The compounding is VERY directional as compounds in pulleys only inline axis as power axis, and gives best nip under or cross over positions on turns as calculated from input directional axis. Unless redefined by 90degree shearing across the Achille's Heel of the power axis on the cross-axis, making that then power axis for rope calcs, and original input axis now a cross axis.
Direction is so important, we do NOT see same capstan and pulley effects in Round Binding against swell. (my)Reason is the source force starts as a dispersed radial force to then same radial controlling arcs(thus matching/mirroring, NOT degrading forces, as no conversion). Hitches, Bends, riggings etc. are linear force inputs thru Standing Part to controlling arcs. With focused linear input tension degrades around the host and yet compounds against host at apex. PROOF: In Round Binding the tension is the SAME thru the arcs until nipping ,NOT degrading around host, NOT compounding into host either. The same volume amount of source force is stronger in 1 axis, than dispersed to all axises, especially when only expressed in 1 direction on that axis, vs dispersed to both directions on all axises.
Direction is an overlooked quantity here to answer rack vs. bollard difference etc. i think. And then further to so much more, as its own quantity important as the tension itself; especially if focused linear vs. dispersed radial . Again i (personally)see all knots/rigs as arc controlled; linear just a connector/like electric wire trace to a component(arc). Straight/non arc rope part mostly might have some friction (and deformity) cost$ against efficient porting source force, reducing output(like resistance in electric wire).
In the end, rope doesn't have unique rules just to it's own island/world as a foreigner here,
but rather follows and exemplifies rules/lessons on how many things work with forces imposed/ported thru.
Thus can L-earn these things in rope to carry understanding to rest of mechanics etc., or in reverse L-earn science of mechanics and apply to ropes.
In broader view, in many ways; force is force, just might be electrical conducted thru wire, fluid thru pipe, physical force thru rope, temperature thru metal etc.; but there are rules..
(shortened version.. )