Empirical Formula to Calculate the Center of Gravity

Paint, lagging, welds, and fasteners account for 7-10% of lightship weight. Don't let anyone tell you that they don't need to file/include it in a weight report.

"But it is just a couple of feet of electrical cable...."

 

Without going into complete detail, I think I know what you are implying by using buoyancy and weights. I'm going to leave out a few steps.

I blithely assume you have a way to seal the openings in the boat off, so no water comes in when the ship is submerged.

1. You need to first find the total weight. Possibly that can be done out of the water. If not, you could adding weights to the deck until it is exactly fully submerged - except that might sink the ship. (I assume the boat is strong enough to take that weight!

So you go just short of that, in several steps, and extrapolate the curve until it is fully submerged.

You do this in a large tank, and you check how much water is displaced by full immersion.

The weight of that water = the weight of the ship + weights.
The volume of the ship = the weight of that water / its density.

(You want to make sure none of the weights are in the water - which may mean you need to build a lightweight scaffolding to hold them. Or you use somewhat more complicated formula including the volume of the weights.)

2. Now you change the amount of weight added until you have only displaced 1/2 the volume corresponding to the weight of ship. At that point the COG lies in the plane of the surface of the water. I.E., you have the vertical axis COG.

3. You change the front/rear tilt of the vessel, by redistributing the weights somewhat, fore to aft (front to rear). The COG still lies in the the new plane of the surface of the water.

4. You again redistribute them again, this time tilting it to the left (port) or right (starboard). You get a new plane containing the COG. (Alternately, with most ships, you could just assume it is symmetric left to right and take the 3rd plane to be the one that symmetrically bisects left and right.)

5. The intersection of those 3 planes is the COG.

Did I do that right?

This procedure might be OK for a small sturdy boat, or for a small scale model of the ship, with the same density distribution.

But for a large ship, sealing off all the deck openings, hoping the ship is strong enough to hold the weights, building a tank big enough for the whole ship and putting it inside, and having a way to measure the amount of water that flows in and out of the tank (if the tank is rectangular, I guess you could just mark off the volume on the sides), makes this a pretty involved thing to do.

I assume people don't really do that for large ships - that if you have the plans, it is easier to calculate COG from them numerically. Right?

BTW, how common is it to construct a small scale model of a ship with the same density distribution, before building the ship full size? That's a lot of work too, but for ships costing millions of dollars or more, maybe it is plausible? Though I'm not sure how useful it would be. Since surface water waves and airflow are non-linear, and don't scale, you can't really use it determine exactly how well the ship would handle in waves and wind, or how much drag there would be when the ship moves through the water.

I suppose you could add chemicals to get the same scaled viscosity, and hope it was close enough. Then when you have the parameters right, you could build a somewhat larger scale model, in several steps. Finally, you build the real ship, and hope it works the same.

Or is computational fluid dynamics now good enough you don't need physical scale models like that?

 

For that you must place such a large amount of weights that you should first check that the deck supports them.

How can you get, with added weights, to displace 1/2 the volume corresponding to the weight of ship?

 

I mentioned the problem with the first objection - which is why it might only work for a small scale model.

But I see your problem with the second - that many ships already displace at least half their weight in water, even if they have no cargo or crew. For such ships, that completely destroys my ideas. I keep thinking in terms of little kayaks, where the boat is usually lighter than the person who uses it. (Though I suppose you could argue that the most useful COG is the one that applies when it is loaded, including that person. Just as you could argue that the most useful COG for a ship is the one that applies when when it is loaded with cargo, crew, and fuel.)

So how would YOU find the COG by direct measurement, using buoyancy?

 

When the ship is out of the water you could get the hull shape from photogrammetry. When the ship is in the water you could get the waterline, which defines the liquid displaced which will give you the weight and the lateral and longitudinal position of the center of gravity. You can get an estimate of the vertical position of the of the center of gravity by applying a moment and measuring the movement of the floating vessel. You could also calculate the vertical position from the natural frequency of the ship roll.

When a craft is designed they do a summation of all the weights and positions like Gonzo said. When the boat is finished and put in the water they could use the process above to validate. I have wondered if (why not) all ships have a small accelerometer that monitors the natural frequency of the roll. The chips to do this cost less than an hour of crew time, and vessels are still lost due to capsize.

 

Oh. It makes sense that people have worked out practical ways to do these things.

I suspect some yacht captains wouldn't know what to do with accelerometer or gyroscope measurements.

 

The logic would be pretty straight forward so it could report an action plan along with what it actually reads.
Examples
-after departing a port the natural frequency is low, outside the design specification. The load it took on may be top-heavy and need to be redistributed unless smooth sailing can be assured.
-the natural frequency has been steadily increasing during a trip, check for leaks or load shifting.

Some captains can feel the ship going crank and order checks to investigate what they feel with not data to back it up.