GM in GHS

GM, the distance from the center of gravity to the metacenter, is calculated in two different ways: 1) from the waterplane (BM - BG); and 2) from the slope of the righting-arm curve.
Note that the definition of GM implies a certain equilibrium; i.e., that the center of gravity and center of buoyancy lie on a line which is perpendicular to the waterplane. (By definition, the center of buoyancy and metacenter always lie on such a line.) If GM is considered to apply in a non-equilibrium condition, it is only by hypothetically moving the CG to a position in line with the CB.

GM from the GHS Command
The GHS command, when issued without a draft parameter, computes and displays GM at the present waterplane. It uses the waterplane method. If free surface is present, it is taken into consideration when calculating BM, not BG. If there is a formal FSM which differs from the true FSM, the formal FSM is used unless a /TRUEFSM parameter is present.
The GHS command does not produce or require equilibrium. If the present waterplane is not one of equilibrium, hypothetical TCG and LCG coordinates are assumed which put the CG in line with the CB. The VCG is not changed. Heel axis rotation is ignored: GMT is always in the ship's transverse direction and GML is always in its longitudinal direction.
The GHS command does not involve the magnitude of the ship's weight: only the vertical position of its CG. Therefore, the present waterplane need not have a realistic relationship with the weight or center of gravity in order to show GM.
In the presence of a heeling moment, equilibrium can exist in the sense that the heeling moment equals the righting moment. In order for there to be a righting moment, CG and CB must not be in line. Therefore, the sort of equilibrium demanded by GM does not exist. The GHS command ignores heeling moment. It creates its hypothetical equilibrium as described above regardless of whether equilibrium with the heeling moment exists.
At any given inclination, an equivalent of the heeling moment can be produced by moving the CG in a direction parallel to the waterplane until it comes into line with the CB. This usually involves shifting the VCG. Therefore, if the correct GM is to be obtained from the GHS command with respect to a heeling moment, the VCG must first be adjusted.

The GMTMMT Command
The GMTMMT command takes a GMT moment as its parameter. Holding the waterplane constant, it modifies the light ship weight and CG such that the given moment divided by the weight equals the GMT from the waterplane. The true free surface is always used.

GM from the STATUS Command
The STATUS WPL command also will compute and display GM using the same method as the GHS command, except that it does not show GM at all if equilibrium, in the strict sense of the righting arm being zero, is not present.

GM from the RA Command
The RA command computes and displays GM (in the transverse, or direction of heeling) when one of the GM LIMIT commands is in effect. It will also compute and display GM on the plot in some cases even when no GM LIMIT is in effect.
Two cases of GM are recognized by the RA command: GM at equilibrium and GM upright (zero heel). Whether the RA command is to deal with one or both of these GM cases is predetermined by LIMIT commands.
The RA process can compute GM from the waterplane or from the slope of the righting-arm curve. In some cases it blends a combination of both. Normally, it uses the waterplane only, unless the heel axis has been rotated.
When the RA process derives GM from the waterplane, it automatically projects the CG into line with the CB in a direction parallel to the waterplane. Therefore, it produces the correct GM when a heeling moment is involved insofar as the waterplane is concerned.
When the RA process derives GM from the righting arm curve, it samples small angles of heel on either side of the relevant angle in order to determine the slope of the curve.
The RA command will tend to use the righting-arm slope method if the parameter /GMRA is included. However, it will also look at the GM from the waterplane and use a blend of the two unless there is a large difference, in which case it uses the slope of the righting arm curve. This gives a smoother response in the presence of waterplane discontinuities while keeping as much agreement with the waterplane method as possible.
Waterplane discontinuities cause knuckles in the righting-arm curve at which its slope and the GM are discontinuous. By using the sampling method to determine the slope of the righting-arm curve, the GM discontinuity is smoothed, which helps when optimizing for a particular GM value (which may not in fact exist). Therefore, the /GMRA parameter can be used with the MAXVCG, SOLVE VCG and SOLVE WEIGHT commands to improve their performance when GM limits are in a controlling position.
When the heel axis has been rotated, the righting-arm slope is always used and the GM is actually in the direction of heel, not the transverse direction.
In the presence of a heeling moment, the GM which the RA process derives is always with respect to the absolute righting arm curve. Therefore, unless the heeling moment is constant at the angle at which GM is being considered, the residual righting arm curve will have a slope which differs from the reported GM.
While the RA process can involve two cases of GM, the RA plot can only show one GM line. If both cases are present, it shows GM upright. In the absense of GM limits, the RA plot shows GM only if the range of heel angles includes equilibrium.

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