Stability

Floating or submerged bodies such as boats, ships etc. are sometime acted upon by certain external forces. Some of the common external forces are wind and wave action, pressure due to river current, pressure due to maneuvering a floating object in a curved path, etc. These external forces cause a small displacement to the body which may overturn it. If a floating or submerged body, under action of small displacement due to any external force, is overturn and then capsized, the body is said to be in unstable. Otherwise, after imposing such a displacement the body restores its original position and this body is said to be in stable equilibrium. Therefore, in the design of the floating/submerged bodies the stability analysis is one of major criteria.

Stability of a Submerged body

Consider a body fully submerged in a fluid in the case shown in figure (Fig. L-11.1) of which the center of gravity (CG) of the body is below the centre of buoyancy. When a small angular displacement is applied a moment will generate and restore the body to its original position; the body is stable.

However if the CG is above the centre of buoyancy an overturning moment rotates the body away from its original position and thus the body is unstable (see Fig L-11.2). Note that as the body is fully submerged, the shape of the displaced fluid remains the same when the body is tilted. Therefore the centre of buoyancy in a submerged body remains unchanged.

Stability of a floating body

A body floating in equilibrium ( ) is displaced through an angular displacement . The weight of the fluid W continues to act through G. But the shape of immersed volume of liquid changes and the centre of buoyancy relative to body moves from B to B 1 . Since the buoyant force  and the weight W are not in the same straight line, a turning movement proportional to '  ' is produced.

The moment is a restoring moment and makes the body stable. In figure (Fig. L-11.2) an overturning moment is produced. The point ' M ' at which the line of action of the new buoyant force intersects the original vertical through the CG of the body, is called the metacentre. The restoring moment.

Provided  is small;  (in radians).

The distance GM is called the metacentric height. We can observe in figure that

Stable equilibrium : when M lies above G , a restoring moment is produced. Metacentric height GM is positive.

Unstable equilibrium : When M lies below G an overturning moment is produced and the metacentric height GM is negative.

Natural equilibrium : If M coincides with G neither restoring nor overturning moment is produced and GM is zero.