Stability of Semi-submerged Spheres


I understand that there has been much discussion regarding spherical designs.

I have a simple question: is there a mathematical relationship between a semi-submerged sphere’s dimensions and mass distribution and its stability?

Obviously proportionally less freeboard, larger diameter, and higher metacentric height all increase stability; however, I am wondering if there is a formula which can be applied to determine the wave sizes this design could handle based on the above properties?

Thanks in advance!

(.) #2

I guess, a semi-submerged sphere would roll in any direction.
May be a keel would help. I have no idea about a formula.
I think, it is the right direction.


Every object will roll in every direction: the question is the extent to which it will do so in response to different conditions.

A keel would provide additional damping but theoretically, given enough righting moment relative to the absorbed wave energy, it should also be possible without a keel.

(Bob LLewellyn) #4

My brother used to install stabilizers on boats. They made them quit large for ships, I wonder how they would be to dampen out roll on a sphere? Maybe use two of them to work on a vector system?

The structure would weight the bottom for extra strength and to add mass.
The problem that I am not clear about is angular momentum. If it starts to spin, the outside may tend to throw people into the sea. Bad for the travel brochure.

(.) #5

I accept your ides for your project. Good luck.
Let us know, how that works for you.

(.) #6

Yeah, being thrown into the sea. People frown on that.

(Wilfried Ellmer) #7

@MasonWilliamL You will find the discussion about this topic under the term “ocean sphere” - all depends on the ballast. Submerged to 80% with a deep ballast center it is extremly stable. The idea “because it is round is will roll” is a misperception. It depends on the forcearms and the intermittent air liquid contacts.You get the best insight by doing experiments with a small ballasted sphere in your bathtube, keep an eye on the relation wave size - sphere size when doing the tank experiments. I have been there and done this in real world size, being on board in storms personally, so you can take my word for granted. A storm that makes similar sized boats send out a distress call - you will not notice it - in such a sphere. (been there done that).

Kindest Regards

The key post to understand this properly is here - a mine does not roll…


Stabilizers are always a possibility; however, intuitively, they may destabilize a structure (with respect to maintaining an even keel) in large waves since they will have a tendency to level the structure relative to the water’s surface rather than perpendicularly to gravity’s pull (as a righting moment would). Catamarans are a perfect example of this: they are extremely stable up until waves of a certain size, after which their geometric stability is actually a disadvantage since it tends to keep them horizontal to the water’s surface.

In terms of angular momentum, this is a property of all objects and depends only upon the axis of rotation and the width of the structure about that axis. Luckily, larger structures are less apt to rotate quickly about their axis (given that they would be most subject to rapid accelerations at their extremes secondary to rotation about that axis).


@ellmer Yes, I found those forms and agree with the stability empirically and intuitively; however, I am looking for a quantitative relationship between the above-mentioned parameters and stability.

I have no problems with the mathematical hydrostatic stability of the sphere, but I am having trouble finding the correct hydrodynamic equations to apply in this situation (and thus quantify the dimensions required for stability in given sea conditions).

Thanks for your input!

(Matias Volco) #10

Unlike an ocean surface solution, whose initial size is one of those “Draupner imposed” design criteria, an Ocean Sphere at snorkel depth can be as small as desired.

A very small, studio or family apartment-sized, ocean sphere will be able to move very easily, while a very large semi submerged ocean sphere will permit a skylight large enough to simulate an inland valley.

Wil @ellmer would it be fair to say the quantitavie relationship Mason is looking for involves this three factors and the concept of snorkel depth?


The problem can be easily corrected by modifying the underwater configuration of the hull (sphere). Bilge keels is the first thing that comes to mind.



With a snorkel the factors I mentioned may become irrelevant: so long as the snorkel is sufficiently small diameter (relative to the sphere) and high (relative to the waves), it can cause the sphere to act as a submarine and thus be completely stable in any seas (up to the height proportional to snorkel height/sphere depth).

The more challenging question is the quantitive relationship which determines the maximum wave size a given semi-submerged sphere (at the surface) would remain stable under.

(Matias Volco) #13

If the emerged area is small enough in relation to a well ballasted body, then the exposed beach will act as a snorkel. This is the most relevant information I can link if you’re looking for @MasonWilliamL for the Stability or Seaworthiness of a Semi Submerged Ocean Sphere or Blimp but of course searching some of the existing threads for Ocean Sphere or Underwater Habitat will yield relevant results too


Yes, there is always the potential for keels. I have rendered a simple model including a circumfrential keel as well as a snorkel and suspended mass/ballast (the small sphere opposite the snorkel). The interesting question is which combination of sphere diameter, keel diameter, snorkel depth, and suspended ballast depth/mass allow for stability in a given set of conditions for minimal cost?

The simple answer is the @ellmer sphere; however, the question of efficiency dictates that it may be more efficient to build smaller spheres with additional stabalizing features (since larger spheres require additional structural integrity to withstand the greater pressures at the depths they will reach).

(Matias Volco) #15

What would be the advantage of the diametrical keel? A larger emerged percentage?

Could you achieve the same desired results with ballast inside the antarctic pole of the Ellmer Sphere?


Thanks, will have a read through that (although on first glance it does again appear to be empirical and intuitive rather than theoretical and mathematically based: it will work but the question remains what sizes are required/efficient)!

(Matias Volco) #17

do they? in any case the larger the D of any sphere, the more structuraly sound with a thinner shell (in relation to the volume).


Keels and ballast approach the problem in two different ways: keels provide damping while ballast provides righting moment. The way to think of this is that damping slows and movement and righting moment actively reverses it (similar to a hydraulic damper vs spring).

As far as I understand (which I’ll admit is limited), damping is more valuable in dynamic situations (waves) and righting moment is more valuable in static ones (calm seas), although it will also help with righting in waves.

The spherical keel was my idea since it would act to oppose rotation in all directions symmetrically and add minimal horizontal translational drag (assuming you want to have a moveable sphere, this is important).


If y’all are drifting into the Concrete Submarine stuff, wouldn’t it make more sense to take that conversation there, instead of Spamming TSI with Wil’s links?


Yes, it’s not a question of the spherical integrity so much as the pressures at any given depth (one ATM per 10m deep in water). Further, a pressure vessel of a larger diameter must withstand more wall stress for the same contained pressure. These factors both combine to make larger/deeper spheres geometrically more challenging to construct.