Page 49 - Introduction to Naval Architecture

36 FLOTATION AND STABILITY

Inclinations can be taken as fixed in position. The weight W = wig
acting downwards and the buoyancy force, of equal magnitude, acting
upwards are not in the same line but form a couple W X GZ, where
GZ is the perpendicular on to BjM drawn from G. As shown this
couple will restore the body to its original position and in this
condition the body is said to be in stable equilibrium. GZ = GM sin$>
and is called the righting lever or lever and GM is called the metacentric
height For a given position of G, as M can be taken as fixed for small
inclinations, GM will be constant for any particular waterline. More
importantly, since G can vary with die loading of die ship even for
a given displacement, BM will be constant for a given waterline, In
Figure 4.6 M is above G, giving positive stability, and GM is regarded
as positive in this case.
If, when inclined, the new position of the centre of buoyancy, B T ,
is directly under G, the three points M, G and Z are coincident and
there is no moment acting on the ship. When the disturbing force is
removed the ship will remain in the inclined position. The ship is said
to be in neutral equilibrium and both GM and GZ are zero.
A third possibility is that, after inclination, the new centre of
buoyancy will lie nearer to the centreline than G. There is then a
moment W X GZ which will take the ship further from the vertical.
In this case the ship is said to be unstable and it may heel to a
considerable angle or even capsize. For unstable equilibrium M is
below G and both GM and GZ are considered negative.
The above considerations apply to what is called the initial stability of
the ship, that is when the ship is upright or very nearly so. The criterion
of initial stability is the metacentric height. The three conditions can be
summarized as:

M above G GM and GZ positive stable
M at G GM and GZ zero neutral
M below G GM and GZ negative unstable

Transverse metacentre
The position of the metacentre is found by considering small
inclinations of a ship about its centreline, Figure 4.7. For small angles,
say 2 or 3 degrees, the upright and inclined waterlines will intersect at
O on the centreline. The volumes of the emerged and immersed
wedges must be equal for constant displacement.
For small angles the emerged and immersed wedges at any section,
W 0OW, and LoOLq, are approximately triangular. If y is the half-