Q:

Why does SOLVE sometimes give different answers for the angle of equilibrium? With a certain vessel in a particular condition it even gives different heel angles when I issue the same SOLVE command repeatedly!

A:

SOLVE will give you a different heel/trim answer depending on where the heel/trim was when the SOLVE command was issued. The reason that the program is designed this way is to allow you to access more than one solution. Most vessels have at least two heel/trim equilibrium positions: one upright and the other inverted. Some vessels in some conditions have more.

Another phenomenon is where the righting arm holds essentially to zero for a range of angles. In such cases there is no "stable" equilibrium angle in the sense that the angle is fixed. The vessel will lie at rest anywhere in that range but not resist any moment to change its angle either.

SOLVE is supposed to find the heel/trim equilibrium that is closest to the initial heel/trim when the command was issued.

If the righting arm is below a certain tolerance (which depends on the size of the vessel, among other factors) over a range of angles, SOLVE can settle on any angle in that range.

The case you mention has the waterplane just barely below the top of the lower hulls with the VCG raised to a point where, at zero heel, the GM is positive, but a slight inclination makes it negative as the waterplane greatly diminishes. From there the righting arm falls only slightly below the axis, then comes back up to the axis at around one degree. Then it rises only slightly, reaching a very small maximum at 5 degrees, then descends and comes back up to the axis at about 18 degrees. But its career there is short too as it rises only to a small maximum before going negative at about 21 degrees.

So we have a total of four equilibrium angles: 0, 1, 18 and 180 degrees. But the ones at 0, 1 and 18 are very tenuous with very slight energy. In fact the righting arms are so small that much of that range qualifies as equilibrium.

This case is technically stable but there are many angles of equilibrium. In real life it is unstable because the energy is so small. As the VCG is lowered, the energy increases and the equilibrium angle at zero becomes more defined while the ones at 1 and 18 degrees disappear.

This ambiguity gives the MAXVCG process fits, and it doesn't find a VCG with zero margin. Perhaps this is just as well because it flags the condition as being something you will need to investigate. In order to have a valid maximum VCG in that range you will have to add something to the criterion which looks at energy.

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