Friction Saver Friction -- some measurements

[moray]

...My hat is off to you.

But...it might have been easier for you to read a book on physics...

Don't give me too much credit, pdqdl; my little experiment certainly is not breaking any new ground! And as you may have guessed, I have read a book or two on physics. The exponential formula that describes friction of a rope around a post is very cool, and, as you say, flies in the face of how most people think about it. It was partly to test out the formula for myself that I did the experiment with the grooves. One of the great things about doing experiments, besides making you think hard about what you are doing, is that quite often they serve up a reminder that you don't know as much as you think you do. That was the case here.

Even though I vaguely expected the grooves would not reduce the friction, I assumed, without thinking much about it, that the remaining wood would make up the difference. That is, the lands between the grooves would now be supplying more friction than before the grooves were installed. But I was wrong!! That isn't what happens. More to come...

 

[moray]

Very interesting moray, personally wouldn't have guessed that contact area has almost no effect on friction...

Lumberjack, I need to be careful how I say things. Contact area has everything to do with friction. But when everything else is held constant, VARYING the contact area doesn't cause any change in friction. There. That's clearer.

 

[pdqdl]

Formula for friction of rope around a post ?

Now that's one they never told us about ! I know for a certainty that it is not in my old physics book. If you stumble across that one, please send it to me. If you do, I will actually study that a bit and learn some about that.

I'm getting pretty rusty on all that stuff, anyway.

Wouldn't it be cool if some of the makers of the friction devices would publish some useful tables that would let a treeman know how many wraps are needed for however much weight we were going to cut off ?

 

[lumberjack333]

So your saying that the friction on a set diameter (birch log) with the same rope and weight is equal between the grooved area and the untouched area, but if diameter increased and the contact area increased friction would then be greater? Thats what I was assuming before, should have worded my response better as well I supposed :yoyo: , none the less interesting that removing contact area while keeping everything else constant doesn't affect friction. Thanks for clarifying!

 

[moray]

So your saying that the friction on a set diameter (birch log) with the same rope and weight is equal between the grooved area and the untouched area...

Yes. This is what I measured.

...but if diameter increased and the contact area increased friction would then be greater?...

No. This is why I love this problem! This will not increase the friction.

 

[lumberjack333]

lol, well you've caught my interest now too moray, so increasing the contact area does nothing... just surface type/texture and rope design/material affect friction?

 

[moray]

And number of wraps. This is key.

 

[moray]

the equation

OK, here is the equation. Now a peculiar expression will come over a lot of people's faces when they see an equation--what I call the "equation look". It is precisely the same expression you'll see when they discover fresh wet dog sh*t stuck to their boot. Fortunately this equation, at least, can be explained pretty well in plain English.

In MathSpeak: The two T's are the tensions on the two legs of the line. The constant e is the base of the natural logarithms (2.71828...), A is the angle of wrap in radians, and C is the coefficient of friction between the rope and the post.

In English: This is just a growth equation, exactly like the equation that describes the growth of a savings deposit where the interest is compounded continuously. This is smooth type of growth such that any particular time interval always produces the same percentage of growth. If the savings grow 1% every 10 days, for example, then you can pick any arbitrary date, X, look at the value of the savings on that date, and know that 10 days later the value will have increased 1%. Obviously a bigger account will grow faster in absolute terms than a smaller one, but in percentage terms they both grow at the same rate.

In the case of the rope around the post, as you move around the post, let's say from the low- to the high-tension side, the tension increases by the same percentage each time you move one degree. The tension in the rope is smoothly growing, just as the savings did, by a constant percentage amount per degree of wrap. The percentage is governed by the coefficient of friction, and it is the accumulated friction around the post that causes the tension in the two legs to be different.

In summary: in the bank case we have an interest rate defined by the bank and growth occurring over intervals of time. In the rope case, we have a friction "rate" defined by nature, and growth in tension occurring over intervals of degrees.

 

[pdqdl]

Well ! That was a much simpler formula than I expected.

I presume that the units of tension are generic? Do the "T" variables represent static or kinetic loads, or does it matter (after all, the coefficient of friction would change when the load starts moving) ?

For the record: I always hated working with radians and natural logs, and I am not likely to work too long on bringing this into a useable table for a guy with a log, rope, and a chainsaw. However...I might figure out how much each wrap adds to the change in T1/T2. THAT would be really practical to know !

 

[moray]

...Wouldn't it be cool if some of the makers of the friction devices would publish some useful tables that would let a treeman know how many wraps are needed for however much weight we were going to cut off ?

I think you can come pretty close and still be on the safe side. If the ground guy is lowering a piece and the rope is wrapped around a limb, you could assume, as I discovered in my primitive experiments, that a half wrap has an efficiency of about .5. That means the groundie has a 2:1 advantage in controlling the load. Add another full wrap, and the advantage goes up to 8:1. Another full wrap and it's 32:1. Even a 1000 lb load would be easy to handle with 2 1/2 wraps. I'll bet 2 1/2 wraps is actually closer to 50:1 for most rope/wood combos.

Last fall I was the groundie on the lowering rope when we lowered a 2000 lb tree and laid it flat in a driveway. At one point the truck was pulling the butt, which was off the ground, while I controlled the top with the lowering rope, and the whole affair was suspended in midair. It was a 3/4 in nylon bull rope with 2 1/4 wraps around a maple tree right next to me, at waist height. At one point I added another 1/4 wrap and noticed a significant drop in tension.

In the case of metal devices like the Port-a-Wrap, I'll bet an efficiency of .5 for a half wrap is very close. I want to measure that one of these days, but it won't be as easy to set up as a chunk of wood.

 

[moray]

Wow, pdqdl, we seem to be on exactly the same page! Yeah, I'm not taking my slide rule into the woods either, but a rough rule of so much advantage per half wrap would seem pretty useful.

 

[moray]

Warning, math inside...

I have made a diagram that dissects my experiment with the grooves, ties everything together, and shows how close I came to walking away from this whole problem thinking I understood it when I really didn't. But first, here are a few preliminary ideas that seem necessary to understanding the problem. Some of these probably deserve discussion in their own right, but I will just toss them out there for now. Throughout all of this we are assuming that we are operating in a region where the "laws" of friction hold--the rope is not smoking, fibers are not snagging, bark is not being ripped from limbs, etc. Assume we are talking about sliding friction, and assume that the coefficient of friction between rope and wood is constant over the range of loads we apply. Static friction will be similar, but the numbers would be a bit larger.

1. Friction between two objects in contact is equal to the force holding them together multiplied by the mutual coefficient of friction between them. This is the standard law of friction. There is no mention of the area of contact!

2. A rope under tension is straight. To bend a rope under tension a force must be applied perpendicular to the rope.

3. Conversely, if a rope under tension is bent, it must have a perpendicular force acting on it.

5. For small amounts of bend (on the order of a degree or so) the force needed to bend a rope is proportional to the amount of bend multiplied by the tension. The force needed to bend a tensioned rope 1/2 degree is 1/2 the force needed to bend it a full degree. For a rope under 1000 lbs tension, the force needed to bend it 1 degree is twice what it would be if the tension was 500 lbs.

6. From (5) we can directly derive this: the perpendicular force per unit length of rope around a bend is reciprocally related to the radius of bend. This is a key relationship.
Imagine a horizontal log 360 inches in circumference deflecting a vertical tensioned rope 1 degree. Then one inch of the log circumference will be in contact with the rope. Let's say it takes 10 lbs. to cause this deflection.
Now halve the size of the log. Again push the log into the rope till there is a one degree deflection. From (5) we know this still takes 10 lbs. But, obviously, now there is only 1/2 inch of rope length in contact with the log. In the first case we have a total force of 10 lbs spread over one inch of circumference, but in the second case the 10 lbs is spread over 1/2 inch. We can loosely think of this as "pressure", and say the pressure of a tensioned rope against a surface is inversely proportional to the radius of curvature at the point of contact. A sharp bend means a lot more pressure than a gentle bend.

7. Finally we can get at the friction. Remember from (1) that friction is the product of force times coefficient of friction. Remember also that the coefficient of friction for a particular combination of rope and wood is more or less a constant. Then the 10 lbs of force in (6) produces the same amount of friction between rope and wood in both the big log and the half-size log. The quantity that is the same in both cases (and the source of the 10 lbs) is the one-degree bend of the rope. The size of the log is not a factor. Conclusion: over any small-angle section of tensioned rope around a post, the friction generated by that section is the product of tension X angle X coefficient of friction. We're done.

This is all the background for my diagram, which I'll try to post tomorrow. All the hard ideas are here, so tomorrow will be a breeze...

 

[pdqdl]

Damn the complexities !

There may be another stumble in the path to our plan to calculate the number of wraps needed for any load: almost always (in the real world of tree work) the rope will be feeding off the axis of rotation at a variable angle from perpendicular, as in wraps taken around a vertical tree trunk to control an overhead load.

Do you suppose that influences the holding force of x(radians) rotation ? I haven't thought about it enough yet, and it was a LONG time ago that I studied any physics.

 

[rahtreelimbs]

All interesting stuff!!!

I go by what my hands and body tell me. A 2 ring friction saver is nice because it lowers friction and is consistant fro tree to tree. A rope guide setup with a pulley is a whole differant animal. With the pulley setup you have zero friction to hold you while hip thrusting and also you hitch has to be tailored for it. So far I wil stick with the 2 ring setup.........although the jury is still out on the pulley deal!!!

 

[moray]

There may be another stumble in the path to our plan to calculate the number of wraps needed for any load: almost always (in the real world of tree work) the rope will be feeding off the axis of rotation at a variable angle from perpendicular, as in wraps taken around a vertical tree trunk to control an overhead load.

Do you suppose that influences the holding force of x(radians) rotation ? I haven't thought about it enough yet, and it was a LONG time ago that I studied any physics.

I think this is an excellent question. And I am almost positive the answer will be: it makes no difference. A nice tight circular wrap around a trunk should behave the same as the spiral wrap you describe as long as they both involve the same number of degrees (or loathesome radians).

 

[moray]

The spiral wrapped rope will have more surface contact with the tree per radian so I would expect there to be some difference. As the rope pulls tight and the circular wraps go spiral due to the overhead loading angle there might be some beneficial shock load dissipation.

True enough, there will be more contact per radian. But the curvature of the rope is less in the spiral case, since it takes a longer piece of rope to make one circuit of the tree, so there is less pressure at each point of contact. A wash.

The second idea is an interesting one. How does the rope decide whether to create a long spiral or a compact one? Does increasing the load cause the spiral to elongate? One of you experienced riggers no doubt has the answer to this.

 

[moray]

diagram one

This diagram shows a section view of the rope on the surface of the log crossing a groove. The curved dashed line at the top represents the contours of the rope on the ungrooved original surface. The meaning of the various labels is as follows:

C is the center of the log. Moving clockwise around the log, starting at far left, we have:
X is any arbitrary point on the ungrooved log surface.
A represents the last point on the original surface before the groove begins. Beyond A the wood smoothly drops away towards the bottom of the groove.
B represents the last point of wood-rope contact before the rope begins crossing empty groove space.
Line RC divides the groove, and the rope span, in half.
P is the mirror of B. It is the first point where the rope regains contact with the wood after crossing the groove.

When a tensioned rope is draped over the log, grooved or not, the wood obviously is supporting the full load. In my experiment, the load was 20 lbs on one leg and my pull (about 40 lbs) on the other. It seemed pretty obvious that this 60-lb load had to be supported by the lands between the grooves once the grooves had been cut. This would mean that an arbitrary point like X, situated in one of the lands, would now be under more pressure than before. Not only would this extra pressure, distributed over all the remaining surface, be enough to support the 60 lbs, but the extra pressure would mean extra friction, thus supplying the friction previously supplied by the missing groove wood. This seemed pretty obvious. But it is completely wrong!!

Imagine we have a researcher ant who will inspect the rope for us with 2 instruments. The ant can measure the rope tension at any point and the rope curvature at any point. Assume there is a 40-lb load on the left leg, 20 lbs on the right, and the rope is slowly moving downward on the left. We send the ant crawling up the left leg. Her instructions are to record frequent readings as she hikes the rope.

In the ungrooved case, the ant's bendometer shows a perfectly steady reading all the way across the top of the log. The tensionometer shows a maximum at the beginning, but declines in a steady fashion from the initial 40 lbs to 20 lbs when she reaches the free-hanging right leg.

Now we move the rope to the grooved section, and send the ant up again. Contrary to expectations, when she reaches point X, the bend reading and the tension reading are exactly the same as in the ungrooved case! Things start to change at A, though. The bend reading jumps by a factor of about 6! Moreover, the tension starts plummeting. After a few more steps, the ant reaches point B. Her bendometer suddenly drops to zero, and the tension becomes constant. Both stay perfectly steady all the way across the groove until she reaches P. There the tensionometer resumes dropping very rapidly and the bendometer jumps again to a high value. A few more steps and she reaches the undisturbed section of log, whereupon the tension resumes dropping at a steady, much slower rate, and the bendometer reading becomes perfectly steady.

When the ant analyses the data later, she realizes that all the missing friction and missing support previously supplied by the missing groove wood are now supplied by the edges of the groove! The little section between A and B now does all the work previously done by the surface from A to R.

But it gets better. See next post...

 

[moray]

diagram two

As the ant discovered in traversing the rope across the groove, nothing is happening there. No friction, no bend, no change in tension, no rope-wood contact. This section plays no role at all in the experiment! So let's remove it...

Here the section has been removed. Note points P and B, previously at the extreme edges of the groove, now coincide. Every atom of wood in the previous diagram that was in contact with wood is still there. Every tiny section of curved rope is still present. Only the groove is gone. All the tension and friction and bent sections of rope are still there. But where there was a groove, now there is a ridge. For a rope wrapped around a post, a groove and a ridge are the same thing! How cool is that?

 

[pdqdl]

... How does the rope decide whether to create a long spiral or a compact one? Does increasing the load cause the spiral to elongate? One of you experienced riggers no doubt has the answer to this.

If you are holding a load overhead by wrapping the holding line around a nearby tree, it has a huge tendency to form a spiral wrap. It is not uncommon sometimes to do a spiral wrap on the trunk of the tree being removed. For large limb lowering, a good understanding of the holding power of spiral wraps might be helpful.

Having done spiral wraps a few times myself, I am not sure. Sometimes it worked better than I wanted, and other occasions, the load had a tendency to pull harder than expected. Since every load is different, I suspect that the angle of the rope was not as significant as the judgement of the holder of the rope. Certainly, it is physically harder to hold the rope against a spiral direction than it is perpendicular to the tree, unless it happens to be an overhead pull.

 

[pdqdl]

As the ant discovered ... How cool is that?

Pretty cool indeed. My analysis would have been different, however. Since the surface area of the rope contact with the log had been reduced by the groove, the pressure applied per unit of area would have been correspondingly greater. With no theoretical difference in the coefficient of friction, and the radians of rotation being the same, I would expect the same amount of resistance.

Your analysis is much more fun, however, and probably makes more sense to the typical non-mathematical thinker. I gotta give you some points for that.

I do have a sneaking feeling that somewhere we are missing a practical application for our theoretical analysis. Somewhere there MUST be a consideration for the radius of the object being wrapped. I suspect that the coefficient of friction must vary acording to the radius that the rope is bending over, because we all know that a rope wrapped around a square bar has much more resistance than a round bar. (it's a lot more destructive to the rope, too!) The only difference between the two objects is the miniscule radius for each of the 4 corners and the deformation of the rope as it passes over those tiny arcs.