Stability of Semi-submerged Spheres

(Matias Volco) #21

A diametrical keel, like a heave plate, and its advantages or lackthereof, has been discussed indeed under the concept of “Lens Shape”, on the Marinea the Project thread, and indeed it makes more sense for surface dwelling. Underwater there is no “damping” only “over-correction” (permanent correction), that is ballast

If the structure is entirely underwater much like a Deep Vent Base, then indeed, check out Maximum Depth, Tubular Structures, Hydrostratic Load


I’m not posting any links or requesting any: I am simply discussing the engineering challenges to building a floating sphere which will remain stable.

As far as I’m concerned a floating sphere is a variant of the spar buoy design, both of which are very promising for seasteading in the high seas and thus relevant to this forum.


There is most certainly damping underwater: try dragging a large plate through water or rotating it (it will certainly dampen any movement).

I agree that a heave plate will not contribute to static stability underwater (which as you rightly point out is solely a question of ballast or, more precisely, center of gravity and center of buoyancy); however, it will dampen (resist) movement and thus contribute to dynamic stability.

(Matias Volco) #24

Indeed, it’s something in between a spar design and a submarine. The real estate advantages of a spherical shape are mathematically obvious

Of course you are correct, but if the purpose of the heave plate is dynamic positioning then why not just use a small engine and a slightly elongated blimp shape? if the purpose is permanent positioning, then why not an anchor on a perfectly Platonic sphere?


It’s not for dynamic positioning, but dynamic stability (ie: not moving).

Having trained in engineering, I have a strong tendency to favor passive failsafe solutions over active ones (which add another potential point of failure). The engine would be active and potentially fail while the anchor relies on the depth of the sea (and is not practical in high seas). Both are possibilities, though.

The sphere may be sufficient (hence my searching for an equation) without the damping; however, I’m unsure as to what dimensions are stable in what conditions.

(Matias Volco) #26

That would be the safer bet if the challenge is permanent dwelling at sea.
Consider ballast is even more passive than a break-away heave


I agree. I am completely in favor of a pure center of gravity/center of buoyancy solution if that is possible on a reasonable scale: again this comes down to the original question regarding the required dimensions for such a solution in given wave conditions.

Even though a sphere with an offset center of gravity will always return to its position of stability (center of gravity directly below the center of buoyancy), the time it takes to get there (period/frequency) and the maximal interim displacement (amplitude) are also critical.

Static stability is very easy to obtain and calculate: it is the dynamic stability that needs to be addressed. As you mentioned, a fully submerged structures skirts this issue; however, there are disadvantages to this (not the least being the pressures involved at depth and the ability to safely enter and exit the structure).


To delve into this further, my question revolves around structures withstanding waves on the order of magnitude of their size: when you have very large structures or very small waves you can use lumped analysis to focus on static stability and pretty much ignore the dynamic stability.

(Matias Volco) #29

But must the body be fully submerged to remain in constant equilibrium?

Could those disadvantages be completely offset per case? A small highly mobile sphere habitat can use natural beaches and outdoor environments. A large “destination” sphere can have a large enough occulus, or skylight, to recreate a tropical shaded garden.

How would a well ballasted, off center, sphere behave with only the arctic tip emerged when waves wash and topple it?


These are some of the questions I am seeking to answer as well.

The stability of a sphere in waves depends on its diameter and how much of it is submerged as well the waves in question (both wave amplitude/height, and wavelength).

Intuitively, when the waves are much smaller than the diameter of the sphere and its draft, they have little effect. Further, when the waves are much larger than the diameter of the sphere and its draft, they have a large effect (if the wave height were similar to the sphere diameter/draft, the sphere would alternate being fully submerged and completely out of water as the waves passed).

The question is at what size of waves relative to sphere diameter and draft, does its dynamic stability breakdown and how. Currently, I don’t have the answer to that.


(Matias Volco) #31

Draft will be determined by ballast.
A high floating sphere will bob up and down with the waves, like a buoy, irrespective of its or the waves’ size.
A mostly submerged, ballasted sphere will stay put, like a mine imagen


Again, it’s always a question of relative size: a large enough sphere riding high will still likely have a large waterplane area and thus be relatively stable in small waves (like a barge).

Intuition is fine for general concepts, but the real test is mathematical analysis with the correct physics assumptions and equations/simulations.

(Matias Volco) #33

the most “real” test, is a physically occuring experiment, not a simulation. Empirical is not exactly intuitive :wink:

If it’s flat - which describes a “plate shape”, in which size and draft are indeed critical and unknown factors


Agreed! Unfortunately, I don’t currently have the budget to experiment with full-sized spheres in the open ocean! Further, analytical assessment allows rapid assessment of the sensitivity to variation of each variable.

And what is a plate but the bottom section of a large sphere? :wink:

(Matias Volco) #35

indeed! indeed indeed!

But you do have the resources for tank experiments, go back to @ellmer’s reply.

try the same with a ball

A spherical keel, is indeed a sphere. Cool slip of mind.

Along with the diagram you called it “circumferential keel” because it’s placed only along the equator.
Just as the ballast woud be more passive a solution if placed in the bottom section of the inside of the sphere instead of suspended; the equatorial heave plate is also mass inside the sphere, and the snorkel is the emerged top sectiion:

size and location (freeboard) determine size and type of surface access but those are economic considerations - adapted per case.

(Larry G) #36

Well, yes, of course. If it’s not fully submerged, then buoyancy changes with wave action. The degree to which this affects comfort and safety depend on the percentage of change. If the change is small enough you can call is “stable within a range”. Even completely submerged is not a constant equilibrium.

This can actually be pretty significant. Professional divers working shallow, for example scrubbing boat hulls, can experience more physiological problems due to the changing pressure gradient from waves than from submerging to deeper depths for longer periods. The rate of pressure change has a huge effect on bodies, but has some effect on machinery and structures as well. Diving shallow sucks because it is harder to maintain buoyancy when you move up or down a small bit in te water column, and it’s really hard on the body.

Picture a vessel floating in 100ft of still water, and submerge it to 67 feet from the bottom or 33 feet in depth (2 Atmospheres pressure). Wave conditions change from still water to 10 foot wave crests, you will note transient pressure changes equivalent to quickly moving from 23 ft (1.6945 atmospheres) to 43ft (2.3002 Atmospheres) or about 73.7% change.

The deeper it goes, the smaller the percentage of change, the higher the absolute pressure.

And masts provide leverage for righting moment. You’ve drawn a ballast that is not really a keel, because it’s not designed to provide a planing or cutting effect against the water for directional seakeeping. This is the damping effect you’re talking about, but damping is kind of a misnomer when applied to most hull shapes. Keels are designed to work against leeway using the difference between wind direction (or thrust from a motor) and the current; to lever the boat into a particular heading.

Precisely: what you’ve drawn might be better characterized as a ballast on an inverted mast rather than an extended keel. It’s not used for seakeeping. Sailboats rely upon the mast for more than hanging sails. They use the mast as inertial damping against buoyant and ballast-based righting moments. When a roundish-bottomed hull heels over due to wind or wave action, it does so at a certain speed which is not constant across the entire arc. As it reaches the end of the arc, more and more righting moment occurs because the ballast gets further out of zero (in relationship to gravity) and generally, because the hull shape is designed to float in a certain upright attitude so buoyancy is increased for that side of the vessel the further it heels, while buoyancy is simultaneously decreased for the upper side, opposite. As you reach the end of the arc, maximum accelerating force is applied to right the hull by moving back in the opposite direction. Sailors call this “stiffening up”. A de-masted sailing ship tends to “snap back” at this point with very little damping until nearly at the opposite end of the arc. A mast works as an inertial damper to slow the roll back. Otherwise the acceleration/deceleration at each end of the arc of the roll is VERY uncomfortable for passengers.

This is the crux of many arguments other folks have had with the proponents of this idea. Note- I do not personally think it’s a great idea, but I am not discounting it entirely. But if I were championing this idea, I would develop some basic formulae for scaling the dimensions. If one had been researching and developing a particular concept for years, it would be rather odd to not have these figures at one’s fingertips, have published them, and be able to cite them at a moment’s notice.

Entraining water should help with increasing inertial dampening, absolutely. The more surface area of the heave plate, and possibly shaping edges with caps, the more water resistance it has. The heave plate, the hull, and all internal fittings also serve as ballast to get the sphere down to hit the marks on waterline. At a minimum placement should be at or below the center line. The trade-off is the lower it is the better for ballast effect, the higher it is, the bigger it can be. You could have multiple heave plate rings at additional cost and complexity. At some chord of the sphere, depending upon gross overall size, the interior space becomes less usable for activity and better suited to using ti for ballast. The engineering of a ring around the sphere adds considerable stresses and complexity, I imagine. One might do better by building a ball-cupped-in-a-cylinder concept. The cylinder provides a skirt around a lower chord of the sphere for adding ballast weight, and simultaneously entraining more water for “free” inertial weight as well as significant water resistance due to surface area. During certain stages of construction, movement, and theoretically for placement in shallow areas, the cylindrical skirt also forms a gravity base. The additional weight of the skirt requires some significant engineering because it is a tensile connection rather than in compression (as is the “ballast keel” pictured above).

For really simple ballast, concrete can simply be poured on the inside, but concrete is not particularly excellent ballast because it is not dense enough- sailboat builders often include lead or scrap iron “pigs” in the concrete pour to increase the average density of their ballast.

A sphere also has a limited range of sizes where the emergent disc is useful. As you get further from the ‘north pole’ it gets steeper and steeper. The smaller the sphere, the faster that happens. If it only has a snorkel tube, you start to run into problems making it an attractive place to live and work.

I see the term “pressure vessel” in this thread. That is not a correct term. Pressure vessels contain pressure. Keeping pressure out is not the same thing. If you were pressurizing the interior to match outside pressure, then you have LOTS of problems not only with maintaining the interior pressure, but with transitioning from interior pressure to the outside. If you don’t pressurize the interior to match the outside pressure, then you have a force gradient from outside water pressure that is much greater down lower (adds another atmosphere of pressure roughly every 33 feet). The spherical nature of the structure is good for distributing that force as “evenly as possible”, but that is not actually the same as “evenly distributed”.

Designing Access From the Water & Other Human Factors

If I am correct, this does not, however, affect the buoyancy of a structure (presuming the structure is rigid enough so as not to lose volume). The change in pressure may be important from an integrity point of view but from a stability point of view it seems the primary issue is the changing fraction of the structure submerged and thus changing buoyancy as well as the actual energy imparted by wave impacts. Is this correct?

To clarify, I was referring the to circumferential ring at the equator as a “keel” (rather than the suspended ballast on the bottom of the inverted mass). I am not sure the exact term but it seems somewhere between a keel and heave plate. What term would be used for this structure (its purpose is to create additional drag with rotation about any non-vertical axis)?

Yes, this is a topic I was considering: the question becomes what is the correct balance between righting moment and rotational inertia. I drew the suspended ballast/inverted mass with the idea that enough righting moment could be acheived so as to minimize the angular motion (although perhaps at a cost of increased frequency). Are there any design guidelines on tolerable angular accelerations or something to that effect?

I am just starting out with this idea (I know others have promoted variations of it in the past): perhaps I can round up a few of my engineering friends and work on addressing this deficit if no one has yet done so!

Yes, my intention with this design was to load the bottom fraction of the sphere with ballast so as to acheive as great a separation between the center of buoyancy (assuming deep submersion this would be approximately at the equator) and the center of gravity.

Agreed. I am a fan of concrete but not for ballast: dense ballast is critical to maximize usable space in a mostly submerged structure!

Can you please clarify what you meant here?

I used the word snorkel for convenience: I was envisioning a larger “tower” similar to the SeaOrbiter (with some useful space or, at the very least, a climbable tower possibiltiy with an above water bridge.

Fair enough. I was simplifying the terms and applying my knowledge of pressure vessels to explain the principle of greater wall stress with greater pressures (and hence the need for thicker walls with deeper spheres), which I believe is correct (although as you point out perhaps not the ideal terminology).

No, the intention is certainly to maintain surface atmospheric pressure. I understand that there will be a gradient and the pressure will be outside in (rather than inside out): which actually lends itself quite nicely to a concrete (or partially concerete) design (since it will be under constant compression and no tension).

Thanks for all your input, it is much appreciated! I by no means claim to be a naval engineer (just trying my best to learn and apply my general engineering training in this situation).

Designing Access From the Water & Other Human Factors
(Larry G) #38

Likewise, I’m just an amateur trying to learn something through extensive reading and application of exsperience gained in other technical pursuits.

We’re moving into non-standard areas, but personally I think heave plate is the most appropriate term.

You’re correct… the point I was getting at is a bit peripheral perhaps. Buoyant force is equal to weight of water displaced. As water is relatively incompressible, it only matters a little bit if you’re displacing water at 1 atmosphere or 2. The weight is very close to the same. Buoyancy control based upon dynamic displacement (changes in volume for the same weight, like lungs moving, buoyancy compensators “BCD” etc) has large feedback in the first 33 feet because the change in pressure is 100% across that gradient. It goes from 1 atmosphere (surface) to 2 atmospheres (33ft). From 33-66 feet, it adds one more atmosphere, going from 2 atmospheres to 3 atmospheres. Likewise from 66-99 feet, one more atmosphere for 4 total. So the biggest change in buoyancy is in the top, where waves exist and make rapid changes in buoyancy and pressure unavoidable. I take the same volume of air into my lungs at 66 ft as I did at 33, but it’s a lot denser. My volume is the same but my displacement is slightly less. Now- my buoyancy compensator (BCD) bladder gets compressed as I descend- it has the same number of molecules, packed closer together, so it displaces less water the more pressure there is, and my buoyancy changes- I sink faster unless I add balancing pressure to re-inflate it to a new relative density. As I ascend, unless I let some molecules out of my BCD, they will over-expand and either blow out the bladder, or cause me to ascend too quickly, or both. The outside pressure actually changes my density dynamically, thus changing my buoyancy, and all of it is during my submerged time, no air exposure required. A cement structure should be rigid enough to not experience such massive changes in density due to outside pressure, but on a large enough structure, some small percentage might be significant. Concrete DOES stretch and compress to mathematically and structurally significant percentages, this is obvious from pre-and post-tensioning techniques even if we rarely observe it with the naked eye.

Thank you, this is a question more people should be asking:

That is the kind of design requirement statement I wold hope to see more people emulating.

The part of the sphere that emerges from the water has a curvature. Once that curvature goes a bit vertical, it’s no longer useful for occupation by people. Just to take a WAG at where it starts to be a trade-off, most human activities don’t take place on a surface more than 30 degree or so- that’s where (for example) vineyards start to become impractical for machinery to navigate and for people to stand and do manual labor. It takes a pretty large sphere to appear flat- even the Earth’s curvature is observable from a moderately tall structure like a tall sailing mast.

I should have said, “A sphere also has a minimum size where the emergent disc is useful.”



I had assumed the usable space would be inside the sphere. This is an interesting idea though: it could always include a flat deck on top built from a porous material (perhaps perforated steel) to prevent significant loading in the event it is swamped.

(Larry G) #40

Certainly, but interfacing with the world around you is a critical piece of the design that has been discussed in many of the design proposals. One problem with a spar platform with a low wave zone interaction is it places the platform far above the water, and requires special mechanisms and techniques to access the water for transportation, fishing, aquaculture, and recreation.

Designing Access From the Water & Other Human Factors