Trans RINA, Vol 156, Part B2, Intl J Small Craft Tech, Jul-Dec 2014
the environment and/or damage which underlie the curve of limiting KG.
The lightship characteristics of RAN ships are regularly checked
by inclining experiments with the interval
determined by the expected time before any standard load condition will exceed the limiting KG, due to growth. Over the whole fleet, there is about one ship checked every three or four months.
2. THE CLASSIC INCLINING EQUATIONS
When a mass is moved across a ship so as to produce a shift of centre of gravity from G to G’ and a resulting heel angle , then tan is given by
tan ′
This can be re-arranged to derive GM0:
′ tan
(2)
If the mass w is moved a distance d across a vessel of displacement , then the shift of centre of gravity GG’ is given by:
′ ∆
∆ tan
(3)
This leads to the classic formula for evaluating GM0 from an inclining experiment:
(4)
Each value of / ∆ is plotted against the corresponding value of tan.
The slope of the linear line of best fit (trendline) through
the points gives /∆ tan which, from Equation 4, is equal to GM0.
Alternatively, values of w d may be plotted against the corresponding values of tan. The slope of the trendline through the points, when divided by then gives GM0.
(1)
The height of the transverse metacentre, KM0 can readily be found from the vessel’s hydrostatics corresponding to the draught and trim at the inclining. This, together with GM0, will provide the height of the vertical centre of gravity:
3. UP AGAINST THE WALL
The derivation of the formula for BM0 and its relationship to GM0 are not presented here as they are well documented in naval architectural texts, e.g. Reed [6], but examination of these shows that the accuracy of the classic KG calculation depends on several assumptions:
The ship is wall sided: that is, the sides of the ship are vertical in way of the waterline when the vessel is upright.
Throughout the length of the waterplane, the immersed and emerged wedges are identical.
The distance between the centroids of the immersed and emerged wedges is equal to 2/3 of the local waterline beam.
For small angles of heel, the metacentre is assumed to be fixed.
The last of these assumptions is generally incorrect.
Solids of revolution about a horizontal axis, of which a sphere and cylinder are two simple examples, are the only shapes for which the metacentre does not move, irrespective of draught, heel, or even submergence. In all other cases, including a perfectly wall sided solid, the metacentre is not fixed but will move with heel.
An elegant case which illustrates the behaviour of the metacentre is one of the most basic wall-sided shapes: a square-section prism floating at its half depth.
When upright, the metacentre is on the centreline at M0 (Figure 1a), a little (B/12) below the waterplane. As the shape rotates, the metacentre moves significantly and by 45o it has moved to a position M45 (Figure 1b), significantly (B/32) above the waterplane. With further rotation, M does a sharp U-turn and heads back again.
(5)
Figure 1a: Upright Metacentre
Figure 1b: 45o Heel
Figure 1c: Metacentrique
B-100
©2014: The Royal Institution of Naval Architects
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