New model predicts the force required to tie simple knots

September 8, 2015 by Jennifer Chu, Massachusetts Institute of Technology

Got rope? Then try this experiment: Cross both ends, left over right, then bring the left end under and out, as if tying a pair of shoelaces. If you repeat this sequence, you get what's called a "granny" knot. If, instead, you cross both ends again, this time right over left, you've created a sturdier "reef" knot.

The configuration, or "topology," of a knot determines its stiffness. For example, a granny knot is much easier to undo, as its configuration of twists creates weaker forces within the knot, compared with a reef knot. For centuries, sailors have observed such distinctions, choosing certain knots over others to secure vessels—largely by intuition and tradition.

Now researchers at MIT and Pierre et Marie Curie University in Paris have analyzed the mechanical forces underpinning simple knots, and come up with a that describes how a knot's topology determines its mechanical forces.

The researchers carried out experiments to test how much force is required to tighten knots with an increasing number of twists. They then compared their observations with their theoretical predictions, and found that the theory accurately predicted the force needed to close a knot, given its topology and the diameter and stiffness of the underlying strand.

"This is the first time, to the best of our knowledge, that precision model experiments and theory have been tied together to untangle the influence of topology on the mechanics of knots," the researchers write in a paper appearing in the journal Physical Review Letters.

Pedro Reis, the Gilbert W. Winslow Career Development Associate Professor in Civil Engineering and Mechanical Engineering, says the new knot theory may provide guidelines for choosing certain knot configurations for a given load-bearing application, such as braided steel cables, or surgical stitching patterns.

"Surgeons, of course, have a great deal of experience, and they know this knot is better for this stitching procedure than this knot," Reis says. "But can we further inform the process? While maybe these knots are used, we might show that some other knots, done in a certain way, may be preferable."

A twisted theory

See how MIT researchers study the mechanics of knots. Credit: Melanie Gonick/MIT

Reis' colleague, French theoretician Basile Audoly, originally took on the problem of relating a knot's topology and mechanical forces. In previous work, Audoly, with his own colleague Sébastien Neukirch, had developed a theory based on observations of tightening a very simple, overhand knot, comprising only one twist. They then verified the theory with a slightly more complex knot with two twists. The theory, they concluded, should predict the forces required to tighten even more complex knots.

However, when Reis, together with his students Khalid Jawed and Peter Dieleman, performed similar experiments with knots of more than two twists, they found that the previous theory failed to predict the force needed to close the knots. Reis and Audoly teamed up to develop a more accurate theory for describing the topology and mechanics of a wider range of knots.

The researchers created knots from nitonol, a hyper-elastic wire that, even when bent at dramatic angles, will return to its original shape. Nitonol's elasticity and stiffness are well known.

To generate various topologies, the researchers tied knots with multiple overhand twists, creating increasingly longer braids. They then clamped one end of each braid to a table, used a mechanical arm to simultaneously pull the knot tight, and measured the force applied. From these experiments, they observed that a knot with 10 twists requires about 1,000 times more force to close than a knot with just one.

"When Pedro Reis showed me his experiments on knots with as much as 10 twists, and told me that they could resist such a high force, this first appeared to me to be far beyond what simple equations can capture," Audoly says. "Then, I thought it was a nice challenge."

From shoelaces to surgery

To come up with a theory to predict the forces observed, Reis and Audoly went through multiple iterations between the experiments and theory to identify the ingredients that mattered the most and simplify the model. Eventually, they divided the problem in two parts, first characterizing the knot's loop, then its braid. For the first part, the researchers quantified the aspect ratio, or shape of a loop, given the number of twists in a braid: The more twists in a braid, the more elliptical the loop.

The team then studied the forces within the braid. As a braid, or twist, is symmetric, the researchers simplified the problem by only considering one strand of the braid.

"Then we write an energy for the system that includes bending, tension, and friction for that one helical strand, and we are able to determine the shape," Audoly says. "Once we have the shape, we can match it to this loop, and ultimately we get the overall force displacement response of the system."

To test the theory, Reis plugged the experiments' measurements into the theory to generate predictions of force.

"When we put the data through the machinery of the theory, the predictions and the dataset all collapse onto this master curve," Reis says. "Once we have this master curve, you can give me a bending stiffness and diameter of a strand, and the number of turns in the knot, and I can tell you what force is required to close it. Also, we now understand how the knot locks itself up when more turns are added."

Reis envisions multiple applications for the group's theory, both significant and mundane.

"This theory helps us predict the mechanical response of knots of different topologies," Reis says. "We're describing the force it requires to close a loop, which is an indicator of the stiffness of the . This might help us to understand something as simple as how your headphones get tangled, and how to better tie your shoes, to how the configuration of knots can help in surgical procedures."

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11 comments

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Torbjorn_Larsson_OM
5 / 5 (5) Sep 08, 2015
To knot or not, a twisted tale.

""This is the first time, to the best of our knowledge, that _precision model experiments and theory have been tied together to untangle the influence_ of topology on the mechanics of knots,""

Double (triple?) pun intended, I'm sure!
wiggers
5 / 5 (2) Sep 08, 2015
In boat handling the type of knot used is determined by how it performs under load and how easy it is to undo after being under load. The force to close the knot is neither here nor there. In fact I would say a high closing force is a disadvantage as it makes the knot harder to tie. The four knots most used on boats are round-turn-two-half-hitches, reef/sheet bend, clove hitch and bowline. In each of these knots there is a main loop that grips the object being tied to and a locking mechanism to prevent it coming undone accidentally. As the 'lock' is never under the full load it can be released easily.
antigoracle
not rated yet Sep 08, 2015
The four knots most used on boats are round-turn-two-half-hitches, reef/sheet bend, clove hitch and bowline.

Wow. Who woulda thunk, sailors are "knotters".
[Yeah....I know....showing myself out..]
Earthman
1 / 5 (2) Sep 09, 2015
"If, instead, you cross both ends again, this time right over left, you've created a sturdier "reef" knot."

No, you've created a "square knot".

Uncle Ira
3 / 5 (4) Sep 10, 2015
"If, instead, you cross both ends again, this time right over left, you've created a sturdier "reef" knot."

No, you've created a "square knot".



Skippy, a reef is a square knot. The same knot has had both names for hundreds of years.
Nik_2213
5 / 5 (1) Sep 10, 2015
Uh, a lot of this work was done empirically waaay back when 'monofilament' came into use for fishing & angling. The number of twists etc was crucial for making 'monofil' grab and hold.
Captain Stumpy
5 / 5 (1) Sep 11, 2015
Double (triple?) pun intended, I'm sure!
@Torbjorn_Larsson_OM
Yeah, rather liked that myself!

I wonder if they've applied this to the "family of 8's" used in Rescue?
One reason they're [8's] used is purely out of safety and load bearing… a knot that creates a bight removes integrity from a rope. The "8's" use bends and allow for 80% rated load capacity on a rope, so they are required for rescues

Skippy, a reef is a square knot
@Ira
Yeah, but a lot of people still don't know that… it isn't well taught anymore, not even in the scouts! (at least, not around here, anyway)

on a tug etc you learn a lot about knots because they're heavily used, like being the brakes headed into a lock, etc- especially on the rivers.
same in the FD. (or fishing, right Nik_2213?)
Captain Stumpy
5 / 5 (1) Sep 11, 2015
Uh, a lot of this work was done empirically waaay back ...
@Nik
forgot to post this earlier... sorry

but per the article above... even though we know a lot about ropes and knots, we haven't actually studied ! we knew about certain things like the fact that a bight creates pressures that will reduce the integrity of the rope, reducing the load capacity... but we didn't have anything describing the mechanical forces
Now researchers at MIT and Pierre et Marie Curie University in Paris have analyzed the mechanical forces underpinning simple knots, and come up with a theory that describes how a knot's topology determines its mechanical forces.
now, i can't say for absolute certain, but even with the extreme amount of study, science and knowledge that we used to determine the family of 8's were the best to use to insure rope integrity, i've never seen anything that described the mechanical underpinnings for the 8's knots.
Uncle Ira
3.7 / 5 (3) Sep 11, 2015
Skippy, a reef is a square knot
@Ira
Yeah, but a lot of people still don't know that… it isn't well taught anymore, not even in the scouts! (at least, not around here, anyway)


Maritime related Skippys are usually the only ones who know that today. It got it's name back in the old-time days. It was the standard knot for when you were "taking in a reef" on square rigged sail. The sails had rows of "reef points" sown into them across the sails, so when you wanted to shorten sail (make him smaller if the wind was up), one of the "reef points" (a pair of short ropes) on opposite sides of the sail) would dangle down on each side of the bunched up (reefed) sail and you used a reef knot to tie him up.
Captain Stumpy
3 / 5 (2) Sep 11, 2015
Maritime related Skippys are usually the only ones who know that today. It got it's name back in the old-time days. It was the standard knot for when you were "taking in a reef" on square rigged sail. The sails had rows of "reef points" sown into them across the sails, so when you wanted to shorten sail (make him smaller if the wind was up), one of the "reef points" (a pair of short ropes) on opposite sides of the sail) would dangle down on each side of the bunched up (reefed) sail and you used a reef knot to tie him up
@Ira
you know what? i love sailing, and subscribe to Cruising World... even have a lot of sailing texts around...

but i have never heard it (reefing, reef points, reef knot) actually described better or more clearly than your post.
simple, clear, concise and to the point.
THANKS

(I am saving that one for my grandkids to read)
antigoracle
1 / 5 (1) Sep 11, 2015
Luke: Dammit!! My shoelaces always come undone.
Obi-Wan: Use the force Luke... use the force.

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