Mathematic approach to anchoring scope

[MicHughV]

I was trying to think about how to explain water forces on a boat to my wife....so I told her to think of it like trying to stir a tub of jello with a wooden spoon or how a bowl of jello could support a heavy mug....water is not jello off course....but as she is an avid cook, the jello story made total sense to her....and could comprehend that water is also a "substance"...

[barnakiel]

Yes. Explaining to others things that we seem to understand natively ... from thousands of repetitions (aka experience). Not easy.



Studying cryptography (basics) I came across this example:


Take your oats and add one spoon of honey. Easy, eh?


Now try to take that honey out of the oats.


Easy?


Etc.


We could benefit from some such analogies for better understanding of anchoring basics. Eh?


barnakiel

[barnakiel]

@ OP


Have you allowed for the fact that the rode is most of the time not just 'in catenary' but also curved in the horizontal plane?


I mean, we are getting some amount of cushion effect from things other than catenary - namely, in this example, from having to pull the horizontal curve out of the rode. Water is quite dense. I would get there is quite some amount of this 'oh, drag me, drag me, please' cushioning effect to consider, adding to the cushioning ('spring') we get from catenary.


???


barnakiel

[MicHughV]

in my neck of the woods, there is a weekly barge service to Puerto Rico. The barge is big, close to 700' long and the size of a ship. The barge gets pulled there by an ocean going tug. The barge has a massive chain bridle hanging from the bow. The tug attaches a wire cable to this chain bridle and off they go.

I've often seen them leave and return. The chain bridle will be hanging near vertically from the front of the barge, often times dragging in the water, yet the tug is pulling this barge merrily along. You can see the barge towing cable clearly stretched tout, but the barge chain, while not quite vertical, is certainly "hanging there".
The towing cable off course, is nearly parallel to the water, being that the tug is low. The barge is quite high, I would guess 50-60', maybe higher, but in all the many times I've seen this barge I have never seen the chain bridal pulled tout. It is always hanging down. I've have seen this barge offshore as well as near shore.

it's incredible to me, that this tug can pull this massive barge to Puerto Rico and back and that barge bridal chain is always hanging down. That tug, I believe, has somewhere around 7,000 hp....it seems to me, that the tug is trying to pull the chain bridle horizontally, but it can't, and the barge is just trying to catch up to it.

I don't know all the dynamics behind it all, but it is fascinating...for me...to see.

[MathiasW]

Originally Posted by barnakiel

@ OP

barnakiel

Who is OP?

[MicHughV]

I've been trying to mull the tug/barge thing thru' my befuddled mind....so here goes....

the chain hanging from the barge is substantive...ie, bloody heavy.....
the full weight of the chain is borne by the barge when it is standing still, but as soon as the tug pulls on it, a portion of the weight must be transferred to the tug towing cable. The tug is not pulling the chain up, but sideways, away from the barge.
were the barge to be tied a dock. it's possible that the tug could pull the chain tout....ie, it has to become the third leg of the triangle supporting the chain in the air. well, maybe not bar tout, but certainly away from the barge, however, once the barge starts to move, the chain will start to droop back down.
it then becomes a balancing act between the forward motion of the barge, the weight of the chain and the towing stress put on the chain by the tug.
it's like anchoring, but in reverse.....

that's as far as my thinking goes at this time...to go further, would require knowing how heavy the barge is, how quick it's moving, it's displacement, etc....but yet, if the weight and length of the chain is known, and the distance the tug can pull it away from the barge..before the barge starts to move, might give some insight to the "mathematical approach to anchoring"

I'm just musing here.....trying to find a reasonable explanation that I could explain to my wife...and she would understand it...

[barnakiel]

Originally Posted by MathiasW

Who is OP?

Oooppss.


'The German engineer'. But is that not YOU?


I mean the person who did all the mathematical wizardy with chain.


barnakiel

[MathiasW]

Originally Posted by barnakiel

Oooppss.


'The German engineer'. But is that not YOU?


I mean the person who did all the mathematical wizardy with chain.


barnakiel

Yes, that would be me. But I did not start this thread and only hijacked it...

Not sure what OP stands for, but I thought it refers to the original poster?

Will look to give a proper answer tomorrow. Kind of late here now.

[MathiasW]

Originally Posted by barnakiel

@ OP


Have you allowed for the fact that the rode is most of the time not just 'in catenary' but also curved in the horizontal plane?


I mean, we are getting some amount of cushion effect from things other than catenary - namely, in this example, from having to pull the horizontal curve out of the rode. Water is quite dense. I would get there is quite some amount of this 'oh, drag me, drag me, please' cushioning effect to consider, adding to the cushioning ('spring') we get from catenary.


???


barnakiel

Hmm, not sure I understand you completely. What is the horizontal curve? Yes, a bridle will be at an angle to the anchor chain, but what curve are you referring to? A curve caused by a bridle leg being dragged through the water?

So far, I have not included any dissipative effects, like friction in water, be it the anchor chain, a rope, a bridle, or the vessel for that matter. My thoughts on this had always been that they are benign and help me. They will reduce adverse effects, and by ignoring them I can focus on the worst case and be more likely on the safe side.

The way I 'bolted' the bridle / snubber onto my equations was to take the force at the bow and require this to be the same force pulling at the bridle / snubber. I always assume the bridle / snubber to be in the same direction as the anchor chain at the bow. If this is not the case, one would need to vector decompose the forces there.

Now, coming back to the connection bridle / chain. The continuity of force between chain and bridle is the basic requirement. This force will then elongate the bridle according to F = c * s, and thereby store energy as F^2/(2 * c). I then add this energy to the potential energy of the chain and check the total energy balance, whether all swell energy has been absorbed or not. By adjusting the total length of the chain I can then increase or reduce this force until I have matched the energy conservation law to sufficient numerical accuracy.

[barnakiel]

Originally Posted by MathiasW

Hmm, not sure I understand you completely. What is the horizontal curve?


(...)


So far, I have not included any dissipative effects, like friction in water, be it the anchor chain, a rope, a bridle, or the vessel for that matter. My thoughts on this had always been that they are benign and help me. They will reduce adverse effects, and by ignoring them I can focus on the worst case and be more likely on the safe side.


(...)

Mathias,


Imagine you are filming a nice anchorage with clear water FROM A DRONE - on a very windy day.


What will you see? Well, as boats sail from side to side, most of the time their rodes will be see from the drone as curves - due to exactly what you mentioned - drag (friction) of the chain and rope being forced to move left, then again right, in endless cycles.


The more a boat tends to sail (most cruising boats sail a lot due to accumulated cruising garbage) the more curve there will be. Also the diameter of the chain and rope will matter - more diameter creates more drag and more drag gives more curve.


This horizontal plane curve will add dumping / cushioning effect to the rode. I am almost certain the amount will be no less than that of a well designed and deployed nylon snubber.


Another way to look at it is to see the catenary (induced by rode weight) as a 3D experience - for most of the time the rode will have to be pulled not just upwards 'against' its (weight induced) catenary but at the same time it will need to be pulled sidewise, acroos the water - against its 'horizontal catenary vector' resulting from friction of the rode dragged, sidewise, by the boat.


What do you think?



barnakiel

[MicHughV]

yes, I see Barna's point of view....I think

I use a rope/chain (75' chain) combo on my boat, and have dove on the anchor numerous times.
The anchor rope coming of the bow roller will be straight ...all the way to the chain connection, but the anchor chain is often laying in an arc on the seabed... caused by wind shifts, etc..that arc would need to be straightened out in addition to any catenary curve...so you have to add frictional forces to the equation.
Strangely, the arc does not seem to affect anchor holding power as the arc will be laying on the seabed, and only 10' or so of chain is off the seabed.
Kinda like a train on a curved track....the loco will pull all the carriages around the curve, not in a straight line....ever seen that....the loco starts pulling...and one by one the carriages will start moving.....but each each carriage is pulling the next following carriage along a slightly different arc angle.....it would seem chain links act in the same way.....ie, each individual link pulls on the next link on a slightly different angle as the the link before it.. it is not a continuous line.
Another analogy might be to be in a long line of cars at a traffic light.....you can see the lead cars moving and turning, but if you are 20 cars back, you are still standing still.

There is clearly more to this than one might think....

[MathiasW]

Originally Posted by barnakiel

Mathias,


Imagine you are filming a nice anchorage with clear water FROM A DRONE - on a very windy day.


What will you see? Well, as boats sail from side to side, most of the time their rodes will be see from the drone as curves - due to exactly what you mentioned - drag (friction) of the chain and rope being forced to move left, then again right, in endless cycles.


The more a boat tends to sail (most cruising boats sail a lot due to accumulated cruising garbage) the more curve there will be. Also the diameter of the chain and rope will matter - more diameter creates more drag and more drag gives more curve.


This horizontal plane curve will add dumping / cushioning effect to the rode. I am almost certain the amount will be no less than that of a well designed and deployed nylon snubber.


Another way to look at it is to see the catenary (induced by rode weight) as a 3D experience - for most of the time the rode will have to be pulled not just upwards 'against' its (weight induced) catenary but at the same time it will need to be pulled sidewise, acroos the water - against its 'horizontal catenary vector' resulting from friction of the rode dragged, sidewise, by the boat.


What do you think?



barnakiel

Barnakiel, ok, I think I get your point. In order to assess how big this effect could be, I would look at the drag force for a rope pulled through water. Quite generally, this can be written as F_D = 1/2 rho v^2 C_D A, where rho is the density of water, v the velocity of the dragging, C_D a coefficient accounting for the shape of the object to be dragged, and A its cross section.

Now, water density is roughly 1 kg/m^3, and a reasonable dragging speed could be 1 m/sec. The drag coefficient D_C needs to be estimated, but by setting it equal to 1, one is not likely to underestimate the force by much. If I have a bridle of 2 x 10 m of rope in the water, with a diameter of 25 mm, then A is 0.5 m^2. This gives me F_D = 0.25 N, which is not a lot. Dragging this rope by, say, 20 m from one side to the other at anchor, requires a work of 20 m x 0.25 N = 5 J, and this is not accounting for this dragging actually happening in a semi circle.

At the same time, a vessel of 10 t with a speed of 1 m/sec in water has a kinetic energy of 1/2 M v^2 = 5000 J, so three orders of magnitude larger. This difference in magnitudes is not going to change if the dragging speed is increased, as this would also increase the speed of the vessel, necessarily.

So, it would appear to me that the energy dissipation via this mechanism is very small and certainly within the errors of all other assumptions made.

[MathiasW]

Originally Posted by MicHughV

yes, I see Barna's point of view....I think

I use a rope/chain (75' chain) combo on my boat, and have dove on the anchor numerous times.
The anchor rope coming of the bow roller will be straight ...all the way to the chain connection, but the anchor chain is often laying in an arc on the seabed... caused by wind shifts, etc..that arc would need to be straightened out in addition to any catenary curve...so you have to add frictional forces to the equation.
Strangely, the arc does not seem to affect anchor holding power as the arc will be laying on the seabed, and only 10' or so of chain is off the seabed.
Kinda like a train on a curved track....the loco will pull all the carriages around the curve, not in a straight line....ever seen that....the loco starts pulling...and one by one the carriages will start moving.....but each each carriage is pulling the next following carriage along a slightly different arc angle.....it would seem chain links act in the same way.....ie, each individual link pulls on the next link on a slightly different angle as the the link before it.. it is not a continuous line.
Another analogy might be to be in a long line of cars at a traffic light.....you can see the lead cars moving and turning, but if you are 20 cars back, you are still standing still.

There is clearly more to this than one might think....

Sure, but the chain links can only pull at an angle if there is some force perpendicular to the chain. It is this very feature of forces only working along the local direction of the chain, which results in the catenary shape in the first place. So, what you describe can only happen when the seabed is providing such force perpendicular to the chain's direction. This boils down to friction force of the chain lying on the seabed. If there are no obstacles / kinks / hooks on the seabed where the chain get caught, this is simply given by a rather smallish fraction of the chain's weight. Exceeding that would get the chain moving and straightening out.

This can only be observed in calmer situations where the wind load is so small that the friction of the chain on the seabed is enough to absorb it. The load on the anchor will be tending to zero in such a case.

[MathiasW]

Originally Posted by MathiasW

Barnakiel, ok, I think I get your point. In order to assess how big this effect could be, I would look at the drag force for a rope pulled through water. Quite generally, this can be written as F_D = 1/2 rho v^2 C_D A, where rho is the density of water, v the velocity of the dragging, C_D a coefficient accounting for the shape of the object to be dragged, and A its cross section.

Now, water density is roughly 1 kg/m^3, and a reasonable dragging speed could be 1 m/sec. The drag coefficient D_C needs to be estimated, but by setting it equal to 1, one is not likely to underestimate the force by much. If I have a bridle of 2 x 10 m of rope in the water, with a diameter of 25 mm, then A is 0.5 m^2. This gives me F_D = 0.25 N, which is not a lot. Dragging this rope by, say, 20 m from one side to the other at anchor, requires a work of 20 m x 0.25 N = 5 J, and this is not accounting for this dragging actually happening in a semi circle.

At the same time, a vessel of 10 t with a speed of 1 m/sec in water has a kinetic energy of 1/2 M v^2 = 5000 J, so three orders of magnitude larger. This difference in magnitudes is not going to change if the dragging speed is increased, as this would also increase the speed of the vessel, necessarily.

So, it would appear to me that the energy dissipation via this mechanism is very small and certainly within the errors of all other assumptions made.

Oh, I just see my mistake, water density is 1000 kg / m^3, this changes something... Sorry, was a bit too quick with this.

Then this is indeed something to be reckoned with!!!

[barnakiel]

OK. I see. The energy of the boat is this order much bigger. We will feel plenty of drag when handling the rode (e.g. rowing an anchor off in a dinghy,) but when the rode is pulled by the boat, this cushion effect will be negligible.



Makes plenty of sense thinking of all the fingers neatly chopped off while trying to handle anchor rodes and berthing lines.


b.