Righting Arms

Roger Long

The horizontal separation between the Center of Gravity and the Center of Buoyancy is known as the Righting Arm.

This is one of the most basic factors in stability and the one that, unlike the mythical Metacenter that we will discuss later, actually represents the forces involved. It's called an "Arm" because it is like a lever. The amount of stability a ship has at any particular angle to resist upsetting forces such as wind is the weight of the ship times the length of this arm. Say you are trying to pry something loose with a lever and you put your full weight on it to no effect. You then get a longer lever and put the same weight on the end and now you get results. Same weight, longer arm, more force. The ship's weight is constant so the length of the Righting Arm determines things like how much sail she can carry at a particular heel angle.

Righting Arms are usually put on a graph like this one which is known as a "Righting Arm Curve":

The length of the righting arm is usually called GZ and is shown here in feet although it would be meters in any of the civilized parts of the globe. The heel angle is self explanatory. This would be a typical righting arm for a powerboat or a wide shoal sailing type such as a catboat or coastal schooner. There are quite a few interesting things to point out on this graph so you might want to print it for easier reference as you scroll down.

Note first that the curve is nearly straight for the initial 20 - 30 degrees. We saw before that the the CB shifts because a wedge of hull is lifted out on one side and a wedge of additional hull is immersed on the other when the vessel is heeled. This change in volume changes the overall center of mass of the hole made by the boat in the water and the CB moves sideways. If you heel the boat twice as far, say from 5 to 10 degrees, the "In" and "Out" wedges will be twice as large. The righting arm curve will only be a perfectly straight line for ships whose sides are perfectly vertical all the way around but it will be very close to straight for most craft.

The curve starts at zero and it isn't always appreciated just how little zero is. I once stayed aboard a large schooner for a few days in preparation for conducting a stability test. A few of the crew and I were sitting on the rail with our feet on the dock watching people go by. I had my back up against a boat davit. As we talked, I tensed my legs and pushed back against the davit. I could feel the over 100 foot long ship move slightly. As she rolled back, I relaxed and then pushed again, getting into the rhythm without my companions noticing what I was doing. Just as you could get a very large and heavy pendulum swinging by pushing rhythmically, I soon had a good roll going as I sat there pretending to be just sitting and talking. The crew soon noticed and started looking around for the source of the motion. It was dead, flat, calm on a beautiful summer evening and there was no apparent explanation for the masts swinging against the sky and the booms creaking in their gear. They jumped up and started looking around. I finally let them in on the secret and said, "Well, I am here to check the stability, you know." Everyone was most helpful and interested as we conducted the stability test the next day just before departing for Bermuda.

After about 20 degrees, the curve no longer increases so quickly with heel. The reason is that parts of the deck begin to go under water. Buoyancy is no longer picked up as quickly on the low side. You can roughly estimate the angle the low point of the deck will first touch the water for many vessels by looking at a righting arm curve and finding the end of the initial straight portion of the curve.

"Range of Stability" is a term you will hear often in reference to vessel stability. It is simply the extent of the curve with a positive value. The curve above shows a vessel with a range of 82.5 degrees. Typical cruising sailboats have a range of stability well past 90 degrees.

To visualize how righting forces act, perform this thought experiment:

Imagine a large model boat with the righting arm curve shown above. Attach a very flexible shaft to the stem near the waterline with a crank on the other end. This will let you rotate the boat in the water without pushing up or down on it. It is initially very easy to move the crank. You can also start the model rolling back and forth rhythmically with almost imperceptible force on the crank. The boat will have a natural rolling period, like a pendulum, and you will only be able to make it roll at this speed.

If you exert a steady pressure on the crank, the force on the handle will increase at a constant rate up until about 25 degrees. It will then continue to increase but at a slower rate until the peak of the curve is reached at just over 40 degrees. It will then get easier and easier to turn the crank. The handle will still be pushing back on your hand but the force will be decreasing as you increase the heel angle. By the time you get the heel over 80 degrees, there will be almost no force at all. When you get the model rotated to the point that the curve crosses the zero line, no force at all will be required to hold it in that position. It will be like a pencil balanced on its point however, turn it a hair past the point at 82.5 degrees (for this vessel) and it will roll over in the water. A hair less, an it will roll back upright. Let go of it, and it will come all the way back to upright.

Bringing the model back to upright from 82 degrees with the crank will be sort of the mirror of heeling it over. You will be restraining the crank. Pressure will initially be very slight and steadily increase to about 42 degrees. It will then steadily drop off until the model is upright.

Now imagine that this is a very large model and the crank is a very large torque wrench. GZ, the righting arm, is 1 foot at 22 degrees and at 61 degrees over on the other side of the curve. If the boat weighs 10,000 pounds, the torque wrench will read 10,000 foot pounds when the vessel is held at either of these heel angles.

Next: The Mythical Metacenter

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