SailNet Community banner
  • SailNet is a forum community dedicated to Sailing enthusiasts. Come join the discussion about sailing, modifications, classifieds, troubleshooting, repairs, reviews, maintenance, and more!

Conservation of angular momentum

5K views 26 replies 13 participants last post by  BryceGTX 
#1 ·
I'm pretty sure I read it in Perry's Yacht Design book but I haven't been able to find the reference so I'm going by memory.

They were doing tank tests on hull stability when someone had the bright idea of putting a stick on the models to more accurately approximate real life.

They assumed that with the introduction of weight above the models would logically be more susceptible to capsize. What they found was the opposite.

In a purely static situation the boat without the mast would have the full lever arm of the keel to keep it upright.
In a real dynamic situation however the mass of the mast had a tendency to keep the boat from capsizing due to the conservation of angular momentum

Not totally intuitive but apparently true. People who have lost their mast have reported that their boat seemed even more prone to role subsequent times after the mast was cut away.

This has interesting consequences of course.
One I have never heard discussed is the possibly of hoisting something into the rigging to add weight.
It sounds counter productive as the goal is usually to reduce weight aloft. However maybe it is an appropriate technique in special circumstances.
For example would extra weight aloft stabilize a boat in certain kinds of rough anchorages.
 
See less See more
  • Like
Reactions: Bene505
#3 ·
I have a copy of "Desirable and Undesirable Characteristics of Offshore Yachts" that discusses this. You are right: people had previously thought of a capsize more like a static event with the large mass of ballast resisting, but found it is actually a dynamic event where distance and shape are important in addition to mass. Figure 3-1 in this book shows the contribution of the ballast, hull, and rig to resisting capsize in light and heavy yachts and the rig contributes about 60% and 70% respectively.

I took some related classes back in engineering school and am not sure about your weight aloft idea. Find a table of moment of inertia for different shapes and maybe plug in numbers to see what might be effective. For a "rod about it's end" like a mast, the formula is: I = (1/3)*M*L^2. The L squared tells you the length is the most important factor by far. Adding rigidly secured mass at the top, to move the effective M of that mast further out, thereby increasing L could add capsize resistance from an inertia perspective, but you sure wouldn't want it there when the boat is heeled far over and it's pulling it down further. So you'd want to calculate how much this idea would affect the overall roll resistance of the boat (i.e. Is it worth it? Is it possible to add enough weight to make a difference?) and how would it affect the boat's recovery from a capsize/knockdown (i.e. Is it counterproductive?) There are probably more issues.
 
#4 ·
Adding rigidly secured mass at the top, to move the effective M of that mast further out, thereby increasing L could add capsize resistance from an inertia perspective, but you sure wouldn't want it there when the boat is heeled far over and it's pulling it down further. So you'd want to calculate how much this idea would affect the overall roll resistance of the boat (i.e. Is it worth it? Is it possible to add enough weight to make a difference?) and how would it affect the boat's recovery from a capsize/knockdown (i.e. Is it counterproductive?) There are probably more issues.
That was exactly what was thinking.
I seriously doubt that hoisting an anchor to the top of the mast in a storm will help prevent a capsize.:)

I was thinking in perhaps a very specific environment. Maybe in a wave-less anchorage that has a ferry going by every hour. Maybe a little weight aloft would dampen the periodic slop from the ferry.
Not sure it would but that was the type of scenario I was thinking of.
 
#5 ·
I never did very well in physics, but I can attest to the comparative stability with/without a mast. We crossed Lake Michigan with our mast down and stowed along the centerline (long story, but we were on our way down to the Gulf); the 5' swells on the starboard quarter had us rolling to the point that I was a little worried we might lose the rig. Not fun...
 
#6 · (Edited)
I have a degree in Physics and taught most mechanical engineering classes including Dynamics. Adding weight aloft does not make a boat more stable, it will increase the angualr momentum which will reduce it's roll rate, it will make it take longer to turn turtle. Depending on the freqency of the forcing function (the wave spacing, angle of attack, wind, boat speed, etc.) it might make the boat react less to the forces and feel more stable. A sailboat without a mast is more stable but can react faster to the forces applied to it, it may toss you off the boat because you don't react fast enough or hold on well enough. It has less tendency to turn turtle but can do it faster and will right itself faster. Stabilty and angular momentum are two totally different things. Stability measures the desire to stay upright, (the distance the center of mass is hopefully below the center of boyancy), angular momentum (the amount the mass is spread out), measures the speed the boat can change angle of heel.
 
#7 · (Edited)
One other thing related to this topic, Dynamic Stabilty, it enters in here too. It is the measure of the change in the location of the center of boyancy versus angle of heel, Multihulls have very high dynamic stability, the center of boyancy moves out very quickly and makes the boat stable even though the center of mass is above the center of boyancy. A standard hull has some dynamic stabilty, more beam, more stable. This is what can make a boat have two stable points, rightside up and pefectly upside down, or a cat be more stable upside down.

Also to get terminology correct, we should be using Moment of inertia instead of angular momentum, Momemt of inertia is the measure of the distribution of mass away from the center of mass (the mast being raised vs on deck, has it's center farther away from the total center of mass, therefore increasing the moment of inertia.)

Quick dynamics lesson. Momentum is mass times velocity, it's a linear thing, (acting in a straight line.) Conservation of momentum, means when something moving crashes into something not moving, they end up moving at an amount equal to the initial mass moving divided by the total mass moving after the crash. Conservation of angular momnetum is the same thing in a rotational sense, Moment of inertia before times angular velocity before equals amount after.
 
#8 · (Edited)
What JH said.

Depending on the frequency of the wave train, a mast-intact boat may have a better reaction to that frequency that a mastless boat. Having anchored beam-to a small swell, I can attest that there isn't a lot of roll dampening in modern sailboats. Otherwise we wouldn't care or notice the defference much.

Note that the wave train frequency that sets up roll oscillations would be limited to some very specific frequencies. At the dock, jump off the side of your boat and time how long the side of the boat takes to go down and back up again (start the timer when the side of the boat is at its highest). For more accuracy, time over several cycles and divide the time by the number of cycles. The answer you get is the same as the wave train frequency that will set-up the worst roll oscillations of your mast-intact boat. (Actually, harmonics of that frequency would likely have a similar effect, e.g. a wavetrain frequency that was a third of what you measured.)

With your mast gone, the boat will have a higher roll oscillation frequency, and would react to a different wave train frequency, perhaps the one currently facing your just-dismasted boat.

If anyone who is taking their stick down wants to do this test before-and-after, it would be a great YouTube video.


Regards,
Brad

P.S. Of course, if you have forward movement, you could change your heading to alter the apparent wave train frequency. If hove-to with a sea anchor, changing to a smaller/larger sea anchor -- to add/subtract downwind drift speed -- could theoretically affect the apparent frequency, conceivably enough to matter. Perhaps a better and easier tact would be to let the sea achor act as a roll dampener by running some of it's (characteristically dampening) force to an off-centerline cleat, via a snatch block on the sea anchor line.

P.P.S. Theoretically, when not moving, altering the angle of the boat to the waves doesn't fully alleviate the runaway roll problem, since the underlying wave train has the same frequency, if not the same strength-per-individual-cycle. This is why people buy rocker-stoppers and other roll dampeners.
 
#9 ·
Note that the wave train frequency that sets up roll oscillations would be limited to some very specific frequencies.
So, say you are motoring in a seaway where the wave train (or a harmonic) is the same frequency as your boats roll rate. Would it reduce roll if you hoist some weight up the mast until the roll rate of the boat is different than the waves?
 
#17 · (Edited)
I think Bryce just answered that one extremely well. The answer (tongue in cheek) is to send a friend aloft. A 100 kg person at the top of Bryce's mast has this effect:

With mast:
Sqrt(100000/38400) rad/s = 1.61 rad/s = 0.257 hz
so the natural period of my boat is about 4 seconds.

Without the mast, the natural frequency is:
Sqrt(100000/13400) = 2.73 rad/s = 0.434 hz or 2.3 second period
With 100kg buddy at the top, this adds an additional 100 * 18^2 = 32,400 kgm^2
Sqrt(100000/(38400+32400)) rad/s = 1.19 rad/s = 0.189 hz
so the new natural period is about 5.29 seconds.

But if you are motoring and your friend won't go aloft, you could change your angle to the waves, thereby changing their apparent frequency and preventing oscillation. It's David's anchoring scenerio that makes you want to hoist something aloft.

Good thread.

Regards,
Brad
 
#12 ·
pendulum effect

Conservation of angular momentum can be visualised in a playground by watching how kids slow themselves spinning by shifting their centre of mass outward on those spinning poles.

On a sailing vessel the principles are the same but you're not rotating continuously you're oscillating like a pendulum. Polar moment of inertia is the term used in vehicle dynamics but dynamic oscillation of sailing vessels is less well understood.

While you could in theory dynamically balance your vessel by changing its mass moment of inertia you are subject to somewhat random frequency and amplitude inputs in a real ocean making it a tail chasing exeecise. You already have sails with near infinite adjustment. When you've dropped all canvas you are left with the vessels design and configuration.
 
#14 · (Edited)
No one has addressed the specific condition.
Your at a protected anchorage. There is no danger of large continuous waves.
On occasion a single 1 foot roller comes in caused by the wake of of boat.

In this specific case would some extra weight in the rigging have the effect of damping the sloshing?

Any natural motion would have the probability of causing the boat to oscillate so weight aloft would only help for the first couple of seconds then would cause even worse oscillation.

But with a single wave caused by a boat wake, what do you think?
 
#15 ·
I haven't done the math, but I can tell you that riding on a mooring with no mast is much worse than with a mast--the frequency of the roll is much higher, and so is the acceleration in response to a boat wake--enough to slide the coffee cups off the table. Seasickness is a strong function of frequency of oscillation, and it is one of the common complications of dismasting at sea.

I also think that the most common stability scenario is the dynamic response to being hit by a breaking wave from the side, where the taller and heavier masts give a distinct advantage. At one point I contemplated going to a carbon fiber mast, but the disadvantages at anchor and in storms made me decide to stick with the heavier aluminum.
 
#16 · (Edited)
In a real dynamic situation however the mass of the mast had a tendency to keep the boat from capsizing due to the conservation of angular momentum
No doubt you could calculate this using momentum, energy or inertia calculations. I find inertia calculations to be more intuitive.

If I look at my boat, I see the predominant inertias (around waterline) as:

Keel at 8200 lbs (3727 kg) , at about 1.5 m gives 8400 kgm^2 inertia
Mast 60 feet, 550 lb, 250 kg, at 10 m gives 25000 kgm^2 inertia
Fiberglass hull 11,000 lb or 5000 kg at 1 meter gives 5000 kgm^2

First I notice that the mast and rigging determine the predominant inertia, so yes the mast is the most important inertia that will resist acceleration. So if we have a fixed torque applied to this inertia, it will cause an angular acceleration much higher if the mast is missing.

The sum of inertias is 38400 kgm^2

Now lets put a 100 lb mass or 45 kg at the top of the 18 m mast. This gives an inertia of 45 * 18^2 = 14580 kgm^2

This added inertia is about 38%. So yes, this added inertia will also reduce the acceleration.

Just for kicks:

The derivative of my stability diagram is about 100,000 Nm/rad at zero. This is the effective spring rate of the water against the boat.

So the natual frequency of my boat is:

Sqrt(100000/38400) rad/s = 1.61 rad/s = 0.257 hz
so the natural period of my boat is about 4 seconds.

Without the mast, the natural frequency is:
Sqrt(100000/13400) = 2.73 rad/s = 0.434 hz or 2.3 second period

So if we just consider inertias, the angular acceleration will higher when the mast is missing. On the other hand, the angular displacement is the same with or without the mast because the frequency has changed.

However, the problem comes that we have neglected damping. When you reduce the inertia, the acceleration is higher, the velocity is higher, so the damping is higher. Since damping is proportional to angular velocity, the damping is going to be higher by about the ratios of natural frequencies: 0.434/0.257 = 1.7 times higher without the mast.

When we consider the higher damping and the higher frequency without the mast we find that the heeling angle goes down without the mast because the damping is higher.

Here is a simulink model that illustrates the effects.
 

Attachments

#20 · (Edited)
Even though the heeling angle is reduced without the mast, you may be more sensitive to the accelerations. In that case, without the mast, the accelerations are about three times higher than with the mast.

And since these accelerations are such low frequency, the higher accelerations may actually be more uncomfortable. And since the heeling angle is only slightly reduced, but a higher frequency, we may also find this more uncomfortable.
 

Attachments

#21 · (Edited)
Last, but not least. If we hoist the 100 pound weight to the top of the mast and apply the same 5 degree forcing step, we see that the acceleration is reduced even further to below 0.2 rads/s^2. And we see that the frequency reduces a bit more.

Now given all these graphs.. would I hoist a 100 pound weight to the top of my mast during a storm? Not likely. These graphs consider the dynamic effects of the added mass. WE need to also consider the static effects of the 100 pounds at the top of a 60 foot mast that produces a heeling moment of 6000 foot pounds. This is a quite significant heeling moment to add to a boat that may have a righting moment of 70000 foot pounds.

Even worse is that this heeling moment is a mass heeling moment rather than a hull moment. So as they say on tv.. "Don't try this at home.."
 

Attachments

#24 · (Edited)
Now given all these graphs.. would I hoist a 100 pound weight to the top of my mast during a storm? Not likely.
Me neither not during a storm and I'm not strong enough to deal with a hundred pounds and I don't like the idea of anything all the way at the top of the mast.

However if you hoisted say 50 lbs to the middle of the mast and only in quiet water to reduce rocking due to boat wakes. What does your graph say would happen then?

It is obviously no danger assuming you trust the halyard and the weather.
So the only question is would it reduce rocking and if so would it be enough so the exercise would be worth it.
 
#22 ·
Maybe you can calculate a co-efficient of comfort. A slow roll ,even if extreme is easier to tolerate than than the quick action of a power vessel or cat in heavy action. The older west coast trollers were narrow gutted with mast and trolling poles. Rolled like a pig in the wrong period swell but they were great sea boats and comfortable.even without a steady sail .
 
#23 · (Edited)
Great posts Bryce. Really brilliant.

During this thread, I've been wondering of you could tie an anchor to a pair of 60 foot lines, suspended from each midship cleats, to act as sort of an underwater mast when anchored. The weight of the anchor keeping the lines taught, combined with the angle of the lines would provide increased rotational inertia, especially for low angles of motion. There would be some additional dampening added by the anchor in the water (which I'm visualizing would keep the anchor from enscribing perfect arcs during rolling).

And the lower center of gravity of this "mast" wouldn't make for increased roll amplitude due to top-heaviness.

What do you think?

----- EDIT -------

I think I answered my own question. Without a rigid mast underwater, the configuration would act very similar to simply placing the anchor on top of a midships cleat. If you could re-purpose your spinnaker pole for this purpose, it could be very interesting. (Clip one end to your anchor roller, run taut lines to forward cleats on both sides, and clip the other end to your anchor.)

But if you are going to that trouble, hoist the anchor aloft with those 60 foot lines tied to the bottom of it and then tied to the midships cleats. The lines would keep the anchor from banging around up there, and you'd have your full 60 foot distance for maximum effect.

Regards,
Brad
 
#25 ·
Great posts Bryce. Really brilliant.

During this thread, I've been wondering of you could tie an anchor to a pair of 60 foot lines, suspended from each midship cleats, to act as sort of an underwater mast when anchored. The weight of the anchor keeping the lines taught, combined with the angle of the lines would provide increased rotational inertia, especially for low angles of motion. There would be some additional dampening added by the anchor in the water (which I'm visualizing would keep the anchor from enscribing perfect arcs during rolling).

And the lower center of gravity of this "mast" wouldn't make for increased roll amplitude due to top-heaviness.

Regards,
Brad
There is a common trick I think they call it the flopper stopper or some such. You can make your own and they have commercial units.
It is basicaly a bucket with a bottom with a one way flap. The bucket sinks easily as the flap opens inward but when you try to lift the bucket the flap seals and it is hard to lift.
The bucked is suspected by a line from a boom which is set off the beam of the boat.

Supposed to work pretty well.
 
This is an older thread, you may not receive a response, and could be reviving an old thread. Please consider creating a new thread.
Top