Calculations Required


Compiled By : Binoy Perumpalath

Heights of Centers

The relative heights of the centers of gravity and buoyancy and the metacenter govern the magnitude and sense of the moment arms developed as the ship inclines. They are, therefore, the primary indicators of a ship’s initial stability. Nominally, the symbols KG, KB, and KM indicate the heights of the centers of gravity and buoyancy and the metacenter above the bottom of the keel, while the symbols VCG and VCB indicate the vertical positions of the centers of gravity and buoyancy, measured from the baseline. In practice, KG/KB and VCG/VCB are used almost interchangeably; in steel ships with flat plate keels, the difference in height above baseline and keel for any point is generally less than two inches and is not significant.

Height of the Center of Gravity

The height or vertical position of the center of gravity above the keel (KG or VCG) is defined by weight distribution. KG can be varied considerably without change of displacement by shifting weight up or down in the ship. Conversely, it is possible to add or remove weight without altering KG. In most ships, the center of gravity lies between six-tenths of the depth above the keel and the main deck:

0.6D < KG < D
Where,  
 D = Hull depth , Keel to Main deck.

If the height of the ship's center of gravity is known for any condition of loading (lightship, for example), and the location of added or removed weights is known, the new height of the center of gravity can be calculated:

Where,

Height of the Center of Buoyancy

The height of the center of buoyancy above the keel (KB) is solely a function of the shape of the underwater volume. As the centroid of the underwater hull, the center of buoyancy is lower in flat-bottomed, full-bodied ships, such as tankers and ore carriers, than in finer lined ships like destroyers or frigates. Disregarding changes in the shape of the immersed hull due to trim and heel, KB of any ship is a function of displacement, and therefore of draft. The height of the center of buoyancy can be calculated by summing incremental waterplane areas (aWP) multiplied by their heights above the keel (z) and dividing the result by the displacement volume:

This expression can be evaluated by numerical integration methods if accurate drawings or offsets are available. In practice, KB can be approximated with sufficient accuracy for salvage work as 0.52T for full-bodied ships and 0.58T for fine-lined ships. At very light drafts, KB is closer to the given waterline because the lower waterlines are usually much finer than the waterlines in the normal draft range. As a vessel’s underwater hull form approaches a rectangular prism (CB = 1.0), KB approaches 0.5T. The following empirical relationships give estimates for KB that are very close to calculated values for merchant vessels of ordinary form at normal drafts

Where,

Metacentric Height

The transverse metacentric height (GMT), commonly called the metacentric height, of a ship is the vertical separation of the center of gravity and the transverse metacenter (see Figure 1-4) and is a primary indicator of initial stability. A ship with a positive metacentric height (G below M) will tend to right itself by developing righting arms as soon as an inclining force is applied. A ship with a negative metacentric height (G above M) will list to either port or starboard with equal facility until the centers of buoyancy and gravity are on the same vertical line, and thereafter develop positive righting arms. This condition, known as lolling, is a serious symptom of impaired initial stability. Metacentric height is calculated by subtracting the height of the center of gravity from the height of the metacenter above the keel:

Transverse Metacentric Radius. The transverse metacentric radius (BMT) is the vertical distance between the center of buoyancy and the metacenter. This distance is termed a radius because for small heel angles, the locus of successive centers of buoyancy approximates a circular arc, with the transverse metacenter as its center. Metacentric radius is equal to the moment of inertia of the waterplane about its longitudinal centerline (transverse moment of inertia, IT) divided by the underwater volume of the hull:

For a rectangular water plane,

And,

Where,

If the waterplane shape can be accurately defined, the moment of inertia can be determined by numerical integration. If not, the transverse moment of inertia of most ships' waterplanes can be approximated by:
where CIT is the transverse inertia coefficient and is approximated by CWP2/11.7 or 0.125CWP-0.045. These provide reasonable approximations for ships with CWP<0.9. For ships with CWP>0.9, LB3/12 is a closer approximation of the transverse moment of inertia of the waterplane.

References

1.    http://web.nps.navy.mil/~me/tsse/NavArchWeb/1/toc.htm
2.    http://www.bodrum-bodrum.com/vorteks/arsenal/stability.htm
3.    http://persweb.direct.ca/tbolt/stabilit.htm
4.    http://homepages.tesco.net/~robin.coles/stabilit.htm
5.    http://www.guillemot-kayaks.com/Design/StabilityArticle.html
6.    http://web.nps.navy.mil/~me/tsse/NavArchWeb/1/module6/basics.htm
7.    http://www.math.ubc.ca/~ttsai/m317/buoyancy.pdf

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