Finding the spring constant of a rope

[erobz]

It cant both be 30% stretch independent of initial length AND a constant spring rate for a fixed rope diameter.

 

[B0B]

It cant both be 30% stretch independent of initial length AND a constant spring rate for a fixed rope diameter.

I think I made that clear long ago K depends on length, just like two spring in series make K 1/2. Spring rate is dependent on length as I said.

Maybe I'm not understanding your question or assertion

I think your saying, just like a spring, $k$ is dependent on length. Put two springs in series and and k prime is $k/2$

$k$ is dependent on length and radius of rope. Rope distributors sell ropes of a given radius but different lengths with the same WLL and MBS. I think MBS is invariant of length.

 

[erobz]

I think I made that clear long ago K depends on length, just like two spring in series have make K 1/2. Spring rate is dependent on length as I said.

What I'm saying is if that is the case, then we shouldn't expect a linear Force vs Deflection graph for a particular rope. The energy stored in the rope will not be $\frac{1}{2}kx^2$

 

[B0B]

What I'm saying is if that is the case, then we shouldn't expect a linear Force vs Deflection graph for a particular rope. The energy stored in the rope will not be $\frac{1}{2}kx^2$

I'm not following. Why won't the energy stored in a rope of length L using the K for a rope of length L not be $\frac{1}{2}kx^2$ ?

And I don't see how it differs for 2 springs in series, 3 springs in series, etc.
Sorry, I took a year of physics 45 years ago. I haven't done any LaTeX in 30 years so I can only copy/paste/edit your markup.

 

[erobz]

I'm not following. Why won't the energy stored in a rope of length L using the K for a rope of length L not be $\frac{1}{2}kx^2$ ?

Sorry, I took a year of physics 45 years ago.

Because its spring rate has dependency on the initial length of the rope.

For a linear spring we have that:

$$dF = k dx$$

Where $k$ is a constant.
$x$ is the displacement from the initial length.

If we integrate that we get the familiar linear spring force equation (neglecting the minus sign out front of course)

$$ \int_{0}^{F} dF = k \int_{0}^{x} dx \implies F = kx$$

And the work is given by:

$$W = \int_{0}^{x} F dx = \int_{0}^{x} kx dx = \frac{1}{2}kx^2$$

This rope has a $k$ value which is some non-constant function of its length. Its begins with a different differential equation:

$$dF = k(l)dl $$

The work of that type of a spring is given by:

$$W = \int_{l_o}^{l} \left( \int_{l_o}^{l} k(l) dl \right) ~dl $$

It's going to depend on whatever the function $k(l)$ is (which is not a constant if you get 30% stretch up to $F_{WLL}$ for ropes of different initial length)

 

[B0B]

Because its spring rate has dependency on the initial length of the rope.

For a linear spring we have that:

$$dF = k dx$$

Where $k$ is a constant.

If we integrate that we get that

$$ \int_{0}^{F} dF = k \int_{0}^{x} dx \implies F = kx$$

And the work is given by:

$$W = \int_{0}^{x} F dx = \int_{0}^{x} kx dx = \frac{1}{2}kx^2$$

This rope has a $k$ value which is some non-constant function of its length. Its begins with a different differential equation:

$$dF = k(l)dl $$

The work of that type of a spring is given by:

$$W = \int_{l_o}^{l} \left( \int_{l_o}^{l} k(l) dl \right) ~dl $$

It's going to depend on whatever the function $k(l)$ is (which is not a constant if you get 30% stretch up to $F_{WLL}$ for ropes of different initial length)

> (which is not a constant if you get 30% stretch up to $F_{WLL}$ for ropes of different initial length)

I know that part is wrong. $F_{WLL}$ is just $F_{MBS}/3$ and has nothing to do with how much the rope stretches.

The only thing we know is up to 30% stretch within the elastic limit. Perhaps $F_{MBS}$ is dependent on length and they publish the same values because of ignorance.

I still don't see why you assert that 2 springs in series are different than 2 ropes in series.

Do you disagree with - Spring constant of a rope ?

 

[Frabjous]

I can measure tensile stress (σ) but how would I measure tensile strain (ε)

Start off with engineering strain (Δl/l₀) and see where it takes you. Young‘s modulus for Nylon is around 0.2-0.6×10⁶ psi. I calculate Young's modulus of .16×10⁶, .14×10⁶, .14×10⁶psi for .875, 1, 1.25 inch cables so things are in the ballpark. The rope has open volume, so these numbers should be larger if this is taken into account.

 

[erobz]

> (which is not a constant if you get 30% stretch up to $F_{WLL}$ for ropes of different initial length)

I know that part is wrong. $F_{WLL}$ is just $F_{MBS}/3$ and has nothing to do with how much the rope stretches.

The only thing we know is up to 30% stretch within the elastic limit. Perhaps $F_{MBS}$ is dependent on length and they publish the same values because of ignorance.

I still don't see why you assert that 2 springs in series are different than 2 ropes in series.

Do you disagree with - Spring constant of a rope ?

You are telling me that two ropes of a same diameter one at 20 ft long and the other 30 ft long both have the same $F_{WLL}$. That is absolutely fine. What is not fine (for a linear spring ) is that the 20 ft one gets to $F_{WLL}$ deflecting only 6 ft, and the 30 ft long one gets there by deflecting 9 ft.

With a linear springs of constant $k$ the deflection for both up to a certain load is exactly the same independent of their initial unstretched length.

 

[Frabjous]

With a linear springs of constant k the deflection for both up to a certain load is exactly the same independent of their initial unstretched length.

Yes, but with n springs in series, 1/n of the displacement is associated with each spring. This is why I like stress-strain.

 

[erobz]

Yes, but with n springs in series, 1/n of the displacement is associated with each spring. This is why I like stress-strain.

I guess I need to get some rest!

Sorry @B0B please continue on without me. I apologize for any confusion I caused.

 

[B0B]

Yes, but with n springs in series, 1/n of the displacement is associated with each spring. This is why I like stress-strain.

How is that different than with N ropes in series where each rope has 1/n the displacement?

 

[B0B]

You are telling me that two ropes of a same diameter one at 20 ft long and the other 30 ft long both have the same $F_{WLL}$. That is absolutely fine. What is not fine (for a linear spring ) is that the 20 ft one gets to $F_{WLL}$ deflecting only 6 ft, and the 30 ft long one gets there by deflecting 9 ft.

With a linear springs of constant $k$ the deflection for both up to a certain load is exactly the same independent of their initial unstretched length.

I never said that and I've stated many times there's no relationship to $F_{WLL}$ and stretch. $F_{WLL}$ is just $1/3*F_{MBS}$

 

[Frabjous]

How is that different than with N ropes in series where each rope has 1/n the displacement?

It isn’t. I was trying to help erobz.

 

[B0B]

It seems very obvious now that $F_{MBS}$ is dependent on length. I'd venture to guess those numbers are for a 30' rope, which is 90% of the sales.

 

[Frabjous]

It seems very obvious now that $F_{MBS}$ is dependent on length. I'd venture to guess those numbers are for a 30' rope, which is 90% of the sales.

I can see why it depends on diameter (see post 37), why does it depend on length?

 

[erobz]

I never said that and I've stated many times there's no relationship to $F_{WLL}$ and stretch. $F_{WLL}$ is just $1/3*F_{MBS}$

The force was not the issue, it was the deflection that was the issue.I got screwed up and forgot that if you take a spring with constant $k$ and cut it in half, you have two springs each with constant $2k$.Just ignore the argument about the non-linearity of the spring constant. My apologies for that wild goose chase.

 

[B0B]

Shooting from the hip. If a spring has an elastic limit of 1 kg, 1,000 springs in series would have an elastic limit of 1,000 kg. I'm convinced that kinetic ropes act like springs and if for sure $F_{MBS}$ independent of length rules out a spring, I say it is dependent. WAUG (wild ass unscientific guess).

 

[B0B]

Can't I just plot deflection vs load for a fixed rope to see if it mimics a spring?

 

[Frabjous]

Shooting from the hip. If a spring has an elastic limit of 1 kg, 1,000 springs in series would have an elastic limit of 1,000 kg. I'm convinced that kinetic ropes act like springs and if for sure $F_{MBS}$ independent of length rules out a spring, I say it is dependent. WAUG (wild ass unscientific guess).

No, it is still 1 kg.

 

[Frabjous]

Can't I just plot deflection vs load for a fixed rope to see if it mimics a spring?

Yes.
I think you missed my post 37. The rope is performing like one would expect of nylon.

 

[erobz]

Can't I just plot deflection vs load for a fixed rope to see if it mimics a spring?

Whatever you do, there is no reason to test this anywhere near WLL, don’t try to be a hero, and if you can’t measure the load remotely in a cleared area, don’t do it at all.

 

[Frabjous]

Whatever you do, there is no reason to test this anywhere near WLL, don’t try to be a hero, and if you can’t measure the load remotely in a cleared area, don’t do it at all.

Agree

 

[JT Smith]

I don't know about these particular kinds of ropes but climbing ropes are not quite like springs. They are spring-like to the first approximation but if you plot force vs. elongation it isn't linear, there is hysteresis, and there is memory. They are more complicated. I read a paper some years back that modeled them as a combination of two springs and a dashpot. Climbing rope manufacturers specify their products in terms of the number of a times they will survive a particular kind of extreme dynamic load. They aren't stupid or negligent in describing the properties of their ropes. There is a "static elongation" specification that is kind of like a spring constant for small loads. But it doesn't correlate very well to what you really want to know when you load the rope dynamically.

I don't know how much this applies to the ropes used to haul stuck vehicles or whatever. Ignore this if you think it doesn't.

 

[jrmichler]

I did a search using terms stress strain nylon rope, and found a document titled Review of Synthetic Fiber Ropes: https://apps.dtic.mil/sti/pdfs/ADA084622.pdf. It has a good discussion of what happens when a rope is subjected to repetitive loadings. That discussion is consistent with the observations of @B0B about ropes breaking after a number of uses. That document is highly suggested reading for anybody who finds this thread interesting.

The following stress strain curves are from that document:

Note that nylon does not have a linear stress strain curve. That means that the linear spring equations, such as $Energy = 0.5 Kx^2$ do not apply. It is necessary to integrate by measuring the area under the stress strain curve.

Since the OP wants to know the energy storage capability of one full length of rope, his best approach is to measure the deflection of a full length of rope. Measuring the strain of a short piece, and using that to calculate the deflection of the entire length, is not only a waste of effort, but also introduces errors. The above plot shows the need for a series of measurements. The resulting data plot would have the horizontal axis in feet and the vertical axis in pounds. It would apply to one specific rope. The results would be directly useful for the energy calculations desired by the OP.

 

[erobz]

I did a search using terms stress strain nylon rope, and found a document titled Review of Synthetic Fiber Ropes: https://apps.dtic.mil/sti/pdfs/ADA084622.pdf. It has a good discussion of what happens when a rope is subjected to repetitive loadings. That discussion is consistent with the observations of @B0B about ropes breaking after a number of uses. That document is highly suggested reading for anybody who finds this thread interesting.

The following stress strain curves are from that document:
View attachment 322629
Note that nylon does not have a linear stress strain curve. That means that the linear spring equations, such as $Energy = 0.5 Kx^2$ do not apply. It is necessary to integrate by measuring the area under the stress strain curve.

Since the OP wants to know the energy storage capability of one full length of rope, his best approach is to measure the deflection of a full length of rope. Measuring the strain of a short piece, and using that to calculate the deflection of the entire length, is not only a waste of effort, but also introduces errors. The above plot shows the need for a series of measurements. The resulting data plot would have the horizontal axis in feet and the vertical axis in pounds. It would apply to one specific rope. The results would be directly useful for the energy calculations desired by the OP.

The apparent stress/strain dependency on strain rate is going to be a further complication for the OP’s intentions…

 

[B0B]

I don't know about these particular kinds of ropes but climbing ropes are not quite like springs. They are spring-like to the first approximation but if you plot force vs. elongation it isn't linear, there is hysteresis, and there is memory. They are more complicated. I read a paper some years back that modeled them as a combination of two springs and a dashpot. Climbing rope manufacturers specify their products in terms of the number of a times they will survive a particular kind of extreme dynamic load. They aren't stupid or negligent in describing the properties of their ropes. There is a "static elongation" specification that is kind of like a spring constant for small loads. But it doesn't correlate very well to what you really want to know when you load the rope dynamically.

I don't know how much this applies to the ropes used to haul stuck vehicles or whatever. Ignore this if you think it doesn't.

Very useful, thanks for posting. From what I know, kinetic ropes are very similar, just bigger.

I did a search using terms stress strain nylon rope, and found a document titled Review of Synthetic Fiber Ropes: https://apps.dtic.mil/sti/pdfs/ADA084622.pdf. It has a good discussion of what happens when a rope is subjected to repetitive loadings. That discussion is consistent with the observations of @B0B about ropes breaking after a number of uses. That document is highly suggested reading for anybody who finds this thread interesting.

The following stress strain curves are from that document:
View attachment 322629
Note that nylon does not have a linear stress strain curve. That means that the linear spring equations, such as $Energy = 0.5 Kx^2$ do not apply. It is necessary to integrate by measuring the area under the stress strain curve.

That's super useful. However to a discount engineer or a retired drywall contractor where close enough counts, at high V, linear is a good approximation.

Since the OP wants to know the energy storage capability of one full length of rope, his best approach is to measure the deflection of a full length of rope. Measuring the strain of a short piece, and using that to calculate the deflection of the entire length, is not only a waste of effort, but also introduces errors. The above plot shows the need for a series of measurements. The resulting data plot would have the horizontal axis in feet and the vertical axis in pounds. It would apply to one specific rope. The results would be directly useful for the energy calculations desired by the OP.

The only way I could measure is with a full rope, but I'm guessing your suggesting that my measurements with a 20' rope cannot be applied to a 30' rope using 2/3*K for the spring constant. I'm also unwilling to apply more than 12K lbs to a rope rated at 43K lbs breaking strength as my current meter requires me to be near the winch line.

EDIT: Update. I've got a portable backup camera I can point at the current meter and then get a safe distance from the ropes under tension. Using a snatch ring to double, I'd probably be OK pulling up to 16K lbs.

Thanks everyone for helping out.

 

[erobz]

Very useful, thanks for posting. From what I know, kinetic ropes are very similar, just bigger.
That's super useful. However to a discount engineer or a retired drywall contractor where close enough counts, at high V, linear is a good approximation.The only way I could measure is with a full rope, but I'm guessing your suggesting that my measurements with a 20' rope cannot be applied to a 30' rope using 2/3*K for the spring constant. I'm also unwilling to apply more than 12K lbs to a rope rated at 43K lbs breaking strength as my current meter requires me to be near the winch line.

Thanks everyone for helping out.

The 20 ft rope would have a larger effective $k$ value in comparison to the 30 ft rope.

 

[B0B]

The 20 ft rope would have a larger effective $k$ value in comparison to the 30 ft rope.

If it was a spring, the 30' rope would have 2/3*K of the 20'. But it's not a spring.

When I said at high V it's close to linear, I'd be measuring at a very slow speed which is the least linear.

 

[erobz]

When I said at high V it's close to linear, I'd be measuring at a very slow speed which is the least linear.

That doesn’t help you though, it’s a hinderance.

You are trying not to exceed the WLL. The faster you attempt to yank to get the vehicle out, means you have less available deflection before WLL is reached because of this strain rate dependency. The faster you go, the stiffer the spring. That’s opposite of ideal for this situation. It seems like it's going to act to limit the max initial velocity in a non-obvious way.

 

[B0B]

I did a search using terms stress strain nylon rope, and found a document titled Review of Synthetic Fiber Ropes: https://apps.dtic.mil/sti/pdfs/ADA084622.pdf. It has a good discussion of what happens when a rope is subjected to repetitive loadings. That discussion is consistent with the observations of @B0B about ropes breaking after a number of uses. That document is highly suggested reading for anybody who finds this thread interesting.

The following stress strain curves are from that document:
View attachment 322629
Note that nylon does not have a linear stress strain curve. That means that the linear spring equations, such as $Energy = 0.5 Kx^2$ do not apply. It is necessary to integrate by measuring the area under the stress strain curve.

Since the OP wants to know the energy storage capability of one full length of rope, his best approach is to measure the deflection of a full length of rope. Measuring the strain of a short piece, and using that to calculate the deflection of the entire length, is not only a waste of effort, but also introduces errors. The above plot shows the need for a series of measurements. The resulting data plot would have the horizontal axis in feet and the vertical axis in pounds. It would apply to one specific rope. The results would be directly useful for the energy calculations desired by the OP.

From the cited PDF

I read that as shock loads, aka, high speed loads from a vehicle going 15 MPH yanking on a vehicle in mud is more elastic with better recovery than slow speed loads (aka, my testing approach with a winch). Am I reading that right?