Mathematic approach to anchoring scope

[MathiasW]

Now, this is an interesting technique, trying to avoid killing corals whilst anchoring: Floating kellets:

https://www.spiritofargo.com/2019/08...-what-is-that/

I am going to look into building a model for that...

[carstenb]

Originally Posted by MathiasW

Now, this is an interesting technique, trying to avoid killing corals whilst anchoring: Floating kellets:

https://www.spiritofargo.com/2019/08...-what-is-that/

I am going to look into building a model for that...

yeah - welcome to Tuamotus anchoring. This is the technique used there. The way to do this is to keep the distance between the fenders/buoys short enough that the weight of the chain does not drag the buoy below the surface.

For the caternary freaks - here you end up with a series of short caternarys, not sure exactly how to judge how much chain to put out. WE tried to err on the side side and put out more than we usually would for the depth. I suspect that anchoring like this is akin to anchoring in shallow water - in terms of the caternary.

It takes a fair bit longer to raise our anchor since you have to disengage all the floats as they come close to your roller.

[Dsanduril]

This is one where I'd love to see the math, because my opinion, based on experience, is that pulling the float(s) underwater can provide a significant amount of "elasticity" in the whole system. Of course sizing of the float(s), placement, weight of chain make for a complicated model.... But on empirical evidence, having a float out in the chain does provide additional energy absorption. Now for the math.

[MathiasW]

Originally Posted by Dsanduril

This is one where I'd love to see the math, because my opinion, based on experience, is that pulling the float(s) underwater can provide a significant amount of "elasticity" in the whole system. Of course sizing of the float(s), placement, weight of chain make for a complicated model.... But on empirical evidence, having a float out in the chain does provide additional energy absorption. Now for the math.

I have no experience with this kind of anchoring at all, but yes, I would think so too.

It is going to be a complicated model, so please bear with me...

[MicHughV]

yes, i've seen that multiple float anchoring system before. it's not very common here, but in the Florida Keys where there is a concern about anchoring in coral beds, it has been in use, somebody likely got the idea from elsewhere.

nowadays, coral areas that are popular with divers, etc, all use pre-installed moorings...instead of anchoring.

for that matter...mooring fields now proliferate around here, and there is barely room for anchoring anymore....one must take a mooring, instead of anchoring...for a fee off course...

mooring fields started popping up in the Bahamas as well, another way of milking a few dollars out of a cruiser..but recent hurricanes have more or less put the kibosh on that idea...

[MathiasW]

Originally Posted by Dsanduril

This is one where I'd love to see the math, because my opinion, based on experience, is that pulling the float(s) underwater can provide a significant amount of "elasticity" in the whole system. Of course sizing of the float(s), placement, weight of chain make for a complicated model.... But on empirical evidence, having a float out in the chain does provide additional energy absorption. Now for the math.

So, as a start, this is the model I am currently floating the idea of in this forum, before trying to work it out in more detail. All buoys need to be balanced to support just the chain to their left, except for the last buoy before the bow, which needs to support a bit more. Some constraints need to be adhered to so that this system is stable and not some of the buoys are drowning. This way the buoys all just float on the surface and keep the chain tidy, away from the seabed. If the buoys are chosen to have a larger buoyancy, it will impact the pulling angle at the anchor negatively.

More buoys can be inserted between buoy 1 and 3.

The last buoy, closest to the bow, should be considerably larger, and there should be quite a bit more chain between it and the bow. This chain can hang down in a breeze and get used in a storm. It must never touch the seabed, of course.

The pull at the anchor will always be horizontal, as long as this last buoy floats. When it starts to dip under water, the angle at the anchor will start to increase.

What do you think?

Calculating the elasticity of this model will be a nightmare, I am afraid...

[Dsanduril]

The diagram is about what I have experienced in coral sites. Usually more buoys, but small, so that is variable depending on the boat.

The values you are proposing are the size of the buoys? How much flotation they should have? So b1 has enough buoyancy to float the chain length L1? And the last buoy is then supporting L3 + half the chain to the boat? This seems right for selecting the flotation/buoyant value for each buoy.

In essence then, the chain has almost zero weight (or a very little weight near the anchor = L0-L1). This becomes quite interesting.

One other thing from real-world conditions, when yachts are yawing, having the buoys can create a significant amount of drag through the water, reducing the magnitude of yawing. Another piece solely from empirical observation, but it does seem to make a (noticeable) difference.

[waterman46]

Originally Posted by MathiasW

Note that at any point along the chain the down-ward pointing force Fg is equal to the weight of the chain to its left. After a little calculus one finds that the load in the chain at that point is equal to Fw plus the weight of a virtual chain hanging vertically down at this point, ending at the depth Y2 where the catenary becomes horizontal. This is kind of neat and easy to remember.

If I understand Fg and Fw, then the "load", which is actually tension, at any point in the chain is the vector sum of Fw and Fg. Just not a simple addition. So tension varies not only with depth due to Fg but also the angle of the catenary at that point.

[dfelsent]

Ok I need to apologize. I read this thread as someone trying to control what they did not understand. Now I read this thread as me not understanding in depth what I thought I knew. Thanks to all, especially Mathias for his analytical work.

[BjarneK]

Regarding floats on the anchor chain, my online calculator also does that. On the front page, go to the bottom and click "Floats on chain" for an example. You can experiment with different float and chain if you want.

[MathiasW]

Originally Posted by Dsanduril

The diagram is about what I have experienced in coral sites. Usually more buoys, but small, so that is variable depending on the boat.

The values you are proposing are the size of the buoys? How much flotation they should have? So b1 has enough buoyancy to float the chain length L1? And the last buoy is then supporting L3 + half the chain to the boat? This seems right for selecting the flotation/buoyant value for each buoy.

In essence then, the chain has almost zero weight (or a very little weight near the anchor = L0-L1). This becomes quite interesting.

One other thing from real-world conditions, when yachts are yawing, having the buoys can create a significant amount of drag through the water, reducing the magnitude of yawing. Another piece solely from empirical observation, but it does seem to make a (noticeable) difference.

Yes, absolutely correct understanding!

Each Fb1, Fb2, Fb3 is the buoyancy of the respective buoy and it supports the 'local' chain near it. The motivation for this choice was that a single catenary, at any given point along the chain, has a down-ward pointing force that is equal to the total weight of the chain that goes from here to the seabed. Only till that point, so not including any chain that might be still lying on the seabed.

I wanted the chain to just float, of hover in the water, as anything more would quickly result in a vertical pull at the anchor.

In reality one will use more buoys than I had sketched out, but that diagram was already difficult enough to adjust it all... The curves are not hand-drawn or some bezier curves in a paint program, but rather real cosh(x)!

It is straight-forward to add more buoys to this minimal system later.

[MathiasW]

Originally Posted by waterman46

If I understand Fg and Fw, then the "load", which is actually tension, at any point in the chain is the vector sum of Fw and Fg. Just not a simple addition. So tension varies not only with depth due to Fg but also the angle of the catenary at that point.

Absolutely correct! But if you do this vector analysis, with Pythagoras and all, you will find that the load along the chain, or tension as you call it, is not only the Pythagoras of the vertical force at that point along the chain and the wind force, but it just so happens that it can also be written as a simple sum of the wind force and the weight of an imaginary chain hanging down vertically from the bow and ending at the same water depth as the anchor. This is the result of the vector analysis and not a fool trying to add two vectors that he shouldn't!

This is why I find this mathematics so beautiful! It leads to surprisingly simple results. And this result in particular I can remember and can do even in my head.

[MathiasW]

Originally Posted by dfelsent

Ok I need to apologize. I read this thread as someone trying to control what they did not understand. Now I read this thread as me not understanding in depth what I thought I knew. Thanks to all, especially Mathias for his analytical work.

Oh, thank you!

I have to say that I find this thread most enlightening for me. SO much very useful input from so many sailors so much more experienced in anchoring than I am. I have learned a lot and it has helped me to sharpen my thinking. I often find that it is when I am trying to explain to somebody that I realise the gaps in my analysis where I had smudged over subconsciously. When you read through this entire thread you will see that I had to correct myself here and there. With some delay I keep up on my web page and rewrite my pitch to take advantage of the learnings in this thread. If in doubt, what I have written there supersedes anything I might have said before...

So, once again, thanks to all!

Overall, I think the originally seemingly different worlds and opinions in the early phase of this thread were more often than not just looking at two different sides of the same coin. The work and analysis developed here shows that most of the time folks do have a point, they just had not looked at it in a slightly different context or model. But when that is done, a lot of it does fit together.

Clearly, we will never get a really accurate model of the anchoring, but knowing as many as possible of the relevant factors and how they play together in a model is something that can help us to make better informed choices for selecting a safe anchoring technique for the site at hand. And this is what I am after. In the three months we have been locked down at anchor here, one vessel was lost because of anchorage problems, and another nearly so.

My personal anchoring experience really only started when we started sailing around the world a little less than a year ago. Before that we have been anchoring in the Baltic Sea with our little tri there, but only during our too short holidays, or in the Adria. And that was only for some 6 years. Before that it was beach catamaran for me only. So, I am a newbie and I am very grateful to hear and learn from folks with many decades of anchoring experience!

[MathiasW]

Originally Posted by BjarneK

Regarding floats on the anchor chain, my online calculator also does that. On the front page, go to the bottom and click "Floats on chain" for an example. You can experiment with different float and chain if you want.

Oh, how marvellous! I had not seen that! You beat me to it, again!

I will more focus on the model and trying to obtain analytical result, leaving the precise numerical results to Bjarne's calculator...

Note to all: There is a link to Bjarne's calculator at the end of my web page.

[MathiasW]

Originally Posted by waterman46

If I understand Fg and Fw, then the "load", which is actually tension, at any point in the chain is the vector sum of Fw and Fg. Just not a simple addition. So tension varies not only with depth due to Fg but also the angle of the catenary at that point.

Concretely:

Catenary equation L = sqrt(Y(Y+2a)), where a = F/(m g), F being the anchor load.

Total weight of chain at bow: m g L

So, we have the weight m g L pointing down, and the anchor load F = a m g pointing horizontally.

Vector addition of perpendicular vectors:

Chain Load = sqrt(m^2 g^2 L^2 + a^2 m^2 g^2) = m g sqrt(L^2 + a^2)

Now, when solving the catenary equation for Y, one finds

Y = sqrt(L^2 + a^2) - a

Hence we have

Chain Load = m g sqrt(L^2 + a^2) = m g (Y + a) = m g Y + F

So, it is simply the SUM of the anchor load F and the weight of a virtual chain of length Y, which I like to visualise as a chain hanging down vertically to the depth of the anchor, m g Y. This is, of course, only a simple picture that helps me to remember this result. It is not a real chain hanging down there. I am sure there is a deeper meaning to this and a great Physicist would have a very intuitive explanation for it, but I have not found it yet.

qed.