To me, i have learned much of capstans/bollards and pulleys that have carried into knot internal secrets and then back to the mechanical devices and back to knots in cyclic growth. The
att_frict paper is a very shining example; i always thought so, especially after years of reading it, and then finally really 'got it'. They all are studies from different angles of the same tensioned rope arcs. The linears in knotting are just connectors to arc, input and termination. Linears controlling frictions are by sine only, and compound by distance. Arcs use all of tensions, the cosine and the sine for controlling frictions and compound by degrees. Then the arcs also have the compound point of pressure onto, not around host. BUT only if from linear force input, that then maintains the focused directional axis thru the system but not if diffused directional radial force of round binding against expansion that has no focused direction, therefore no compounding point w/o other strand. For all tensions equal not degrading to Nipping point. i could not have found these things in knotting without these other supporting views, so offer them all as the more complete tangible fabric to grab, instead of collection of individual points.
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Knots are machines to me, just in stop/locked position. Poldo and Truckers can take from static to dynamic machine leveraged control, but then lock as static machine again. The linear focused force thru a SPart is diffused to greater area that is more controllable in arcs. Trading in leverages focused force to diffused, as also avails to more frictions in an arc to leverage control with the system. If use lever as a machine to lift dynamically /motion, and then hold the position, it is as knot in a static position, but still a machine taking a linear gravity force and diffusing out to wider arc to control easier with final nip of effort to control remaining force as a ballast against chaining backwards thru the system against load.
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Stone doesn't take tension well, but that is what they wanted to make bridges out of to last longest. Had to find the arc(h) in architecture to do that. Arc(h) in architecture to me means not to try to support with Zer0 cosine of cross axis. Necessity being such a mother produced the stone arch allowed to use the borne forces pretty much all in compression that stone can take very well. Rope tension on arc is reverse, all tension used to target as can't use compression for support. Same science. Bollard/capstan and pulley are same arc science in knots, just showing different faceted aspects of the same jewel. Thus can get a view here and there in one and trace in another. But also, can 'parity check' in same way. If think have principle in one, must persist fairly in rest that apply : capstan,bollard, pulley, knot. If can't cross-verify forwards and backwards against own self and between these members then not seeing correctly. Things i present have run and persisted thru all these gauntlets back and forth over the decades of study. 2 spread pulleys in system only use 180 arc of each given pulley at any time, just as a host mount for knot does, only closer together at higher friction and at end of travel. other wise the same as you take out the straight linear connecting parts.
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COSINE !To me cosine is the focused linear support against load, both directions on that loaded axis of Equal & Opposite pair of load vs. support.
>>sine is anything not on that simple, single dimension. On tv you don't stand in front or behind a rocket launcher. That is the loaded axis of target force direction and it's E&O as a loaded axis. Guns and ropes can have recoil on that E&O axis etc. Sine is anything not on that single dimension axis, more compound/complex.
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Cosine as a number x 100 gives the efficiency of the support column as a percentage form.
>> if supporting column of rope against imposed load is purely aligned cosine= 1.00, 100% efficient
>>500# tension will hold 500# load, cuz 500/1=500, 100% efficient use of borne tensions
>>if however line deformed or angled to 30 degrees cosine .866 is 86.6% efficient
>>577.35# tension needed to hold against 500# 500/.866 for 86.6% efficiency
>>577.35 tension x 86.6% efficiency = 500#
>>577.35 tension x 50% (sine of 30 = .5) gives 288.68 side force also carried
77.35 more on support axis needed as also handle 288.68 across
>>to carry same 500# that with pure inline support column at most efficiency only was same 500# and only 1 force axis
>>our friend rope only works in tension
>> helpful here as any side force expressed will try to groom system more inline proper.
>> whereas a rigid support may use compression
>> and compression side force pushes the Equal & Opposite more out of line towards less efficiency
>> they are running towards each other and have no place else to go!
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Cosine is not a set direction nor axis, it is the benchmark axis to me. Many books show it as horizontal which very much confused my self study of vertical/gravity fed usage of climbing, rigging and felling. Cosine of support is how efficiently the support aligns to the load. So can benchmark force or support and measure to the other. Many times in talking to school trained folk on cosine, they could only see cosine as horizontal then calc from there and may times struggling to do in the vertical force context. My imagery now is much more fluid than that.
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i follow the force into the dormant (rope)structure of responding force against the imposed loading.
The linear line of force imposed is my benchmark cosine axis and direction.
Anything to sides, the other 2 dimensions are sine.
Mnemonic of cosine is to my cos(cause), and Sine the distraction sin from, to the sides
>>to control the load takes the cosine, the sine is just born as extra costly baggage not to target against the porting device.
Less cosine efficiency needs more tension to give same support sum to ballast against same load.
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So the cosine in Hitches as terminations and Bends as couplings type node deformities in an otherwise straight rope run is that line itself. The SPart is the input to the controlling arcs. It ends in my model where the Primary Arc begins as 180 around host or 90 around other rope part as a Half Hitch pre-fix does in Kellig. The input endpoint of this arc must point in direction of force feeding it.
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The controlling frictions degrade the tension as you say in a LINEAR decreasing fashion of it running out of gas to the final Nip(s). Noting, in Binding is radial force induced, thus no conversion loss, and not degrading tensions to the final Nip(s).
The radial is diffused evenly around w/o direction in Binding, the focused linear induced maintains the directional axis (horiz or vert) until 90 conversion to cross axis or force termination in Hitch or Bend usage, even of the same knot, just by the type and direction of force imposed.
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Knots can express compression on a host to use, but only tensioned parts of rope give force internally except for firmer frictions. Cannot give lift, pull etc. If on smaller arc get compression on the interior side as tension pulling on outside, there must be a section between this sliding gradient that is neither stretched nor compressed as rope area segregates to these 3 zones. Only the tension zone is carrying load pull, that is now leveraged to that smaller section, of greater tension to compensate.
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To me, a pulley system tries to maintain conservation of energy towards end output, while a bollard trades it away for friction before, but in total both conserve the same amount, just that bollards are mostly in friction/heat in a physical to heat force exchange.
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Trucker's Hitch efficiency loss is from rope on rope and other hosts. It will always have 3 legs of pull for standard POTENTIALS of:
>> 3x as a compression jig or 2x as a lift if input is downwards where can use body weight/up to as input
>>unless grab and pull up on something to essentially make self heavier
>>can reverse trucker hitch vertically, to picking up 3x, or compressing 2x
>>this way can use strength, like more powerful leg force as input, but more fatiguing than hanging/sitting or rope as input
'My' method is for the 2nd pull inside the Truckers Hitch as also hang on it
>>this takes us from 2xEffort on one end and 3xEffort on the other
>>to 4xEffort on both, then if can hang on end as input add 3xBodyweight to upper and 2xBodyweight to lower as additives
>>if cannot hang on end as input, hang on the interior input to then give 2x upper and 1x lower bodyweight additives to the 4xEffort.
In normal operation, the Truckers could choose between leg or gravity input. In this modified version, it can use both and to higher multiplier for the effort.
The 3xEffort potential of the Truckers is converted to (3x2)-2 for 4xEffort as a potential for both ends
>> 5xEffort potential of a 5xcompress is converted to (5x2)-2 for 8xEffort as a potential for both ends
>>plus bodyweight affects for each.
i would say this conservation of forces, to include those assumed to be outside of the system now folded in
>>is in itself a more efficient Trucker's!
ALSO, with the dual input system of effort and weight, can hold tension in Truckers with effort part and impact with bodyweight thru system(or reverse that strategy) to tweak linearly before leveraging/swig across as potential input.
>>get as tight a lever as possible linearly first, to make less rubbery a lever as come across rope.
So, even tho i can't offer an improvement to this 'sledgehammer' of force; please let me offer to use both hands in it's style of application for the more you seek.
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i totally agree on imaging camera but not in budget, and have not seen all could with what have already yet also; but see more than i did before journey started for sure.
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Thank you for your fine works and also questions here too!