Forces of rope arcs

[TheTreeSpyder]

Rope arcs happen inside of knots whar ya can't see'em.  But, they also happen in rigging.  Because a pulley has 2 legs of loaded line on it, it has a potential loading of 2x Load.  But, because the sling/rope holding the pulley, only resists inline not perpendicular tensions, we only need to calcualte the inline forces of the rope angle from the sling X 2 legs.  This means that we take 2x Load X cosine of half the spread.  And we are done.

This also explains a pulley with weight on a horizontal clothesline, or leveraging tighter by swigging or sweating purchases.  For, this is the same math, in reverse, we are now just initiating force by the bend, not the end, into the system.

Thus; if we place 175 degree spread in angle box, we will see very little load on pulley; so pulley has leverage over the line at that angle.  Thus, if we anchor leg and pull bend, we can incite those kind's of forces.  The calculator crunches by radians, so calcs by hand in degrees all the way through might be slightly different.  For code testers/breakers i allowed negative/floating load weights, but not angles< Zer0 nor >= 180.

calcPulleyLoadings.swf

[Bob Thrun]

Thus; if we place 175 degree spread in angle box, we will see very little load on pulley; so pulley has leverage over the line at that angle.
calcPulleyLoadings.swf

If you consider stretch in the rope, you will find that an angle of 175 degrees is impossible, even for high modulus ropes.

[TheTreeSpyder]

Now, i'd think that the top pulley could be adjusted up and down to force any angle from carried line?? 

And on such short slings elasticity would matter less?

[Dan_Lehman]

If you consider stretch in the rope, you will find that an angle of 175 degrees is impossible, even for high modulus ropes.

 
I recall Ellen MacArthur exclaiming at the near frightening tautness of her hi-mod
shrouds--like steel.  Seems like a measurable force would needed to deflect them.

--dl*
====

[TheTreeSpyder]

i think this would be more like BobT speaks of.  Hear the pulley floats free to make the bend.  Above, the pulley is supported, to adjust the bend.

So, here without the direct overhead pulley support, (and higher leveraging here); lines are more likely to deform to relieve forces.  but, still that has to do with loading range, length as well as materials and construction.  Also, these same forces would take place inside of knots and hitchings, show the forces of sweating lines for more purchase etc.

calcLineLeveraging.swf

[TheTreeSpyder]

here are some applications

Intense sweating
Leveraging tiedowns for more power

[squarerigger]

Hi TS!

Speaking as one who does not have access to the .swf file opening technique, I can say that what you have to share, as important as it may be, is nonsensical to me.  Without the picture I am guessing that this whole thing makes no sense _ (I cannot make sense of it anyway) - would you be able to explain what you mean in regular English what you mean for shlubs like me please?  Your use of phrases like "Hear [sic] the pulley floats free to make the bend" - what pulley, what bend and where does it float free and what do you mean by float?  Then you go on to say "...here without the direct pulley overhead support (and higher leveraging here) lines are more likely to deform to relieve forces" What forces, what overhead pulley and what is the higher leveraging?  I cannot be the only one lost with your comments, as erudite as you may think them to be (and I feel sure they must be great comments!).  PLEASE explain (or at least tell us where to read your files) in regular English.  Thank you.

SR

[TheTreeSpyder]

The first calculator shows loads tied to the end of a line, routed on supporting pulleys with rope spread at different angles.  You input the angle of the rope spread, and the load (that becomes the line tension) into the white boxes; and the calculator crunches the numbers. 
calcPulleyLoadings.jpg

The second calculator, is the inverse though you still set the angle of deflection, but here you input force at the pulley/bend and it gives you the line tension instead(where line tension was an input b4).  The 'drawing' in the swf is more dynamic, in that it adjusts to show the angle of the rope spread you input.  The floating pulley reference, is to the pulley riding the line, unanchored.

calcLineLeveraging.jpg

The inverse in the math; is that the former uses cosine of half the spread; whereby the latter uses 1/cosine of half the spread.  We also work with forced placed on 2 legs to the pulley perspective, then also that pulley's force shared by 2 supports perspective, respectively in the calcs.  We only use the cosine, and not add sine forces, because the lines only resist the inline/adjacent force:distance ratios.

i think these rigging lessons, give us magnified views of forces inside of knots; akin to looking at them thru a microscope. 

[squarerigger]

Thank you TS - I greatly appreciate your putting the information in as a jpg file...

SR

[TheTreeSpyder]

NP, i thought only drLehman had a problem with the  freeFlash Player.  i do publish in ver.8(ver. 9 is latest out of Beta), not sure what ver. you have.

calcLineLeveraging.xls

Here is same formulae calculator, but done in free Open Office. Org's starCalc spreadsheet, then saved in msOffice 2003 Excel Spreadsheet.
Hopefully this will give wide range of compatibility for anyone interested in manipulating the numbers  for angle and load;
and seeing the effects on line tension as i tried to lend in Flash.