The Geometry of Ships

44 THE PRINCIPLES OF NAVAL ARCHITECTURE SERIES
Fig. 35 Illustration of various types of equilibrium. (a) Unconditionally
stable. (b) Unconditionally unstable. (c) Neutral. (d) Conditionally stable,
globally unstable.
The same ball resting at a maximum of a convex surface
is a typical unstable equilibrium. Following a small
displacement in any direction, the ball tends to accelerate
away from its initial position. In an unstable system,
a small disturbance produces a large result.
On the boundary between stable and unstable behavior,
there is neutral stability, represented by a ball on a
level plane. Here, there is no tendency either to return to
an initial equilibrium, or to accelerate away from it.
Stability can depend on the nature of the disturbance.
Picture the ball resting at the saddle point on a saddleshaped
surface. In this situation, the system is stable
with respect to disturbances in one direction and simultaneously
unstable with respect to disturbances in other
directions. A ship can be stable with respect to a change
of pitch and unstable with respect to a change in roll, or
(less likely) vice versa. In order to be globally stable, the
system must be stable with respect to all possible “directions”
of disturbance, or degrees of freedom.
A 3-D rigid body has in general six degrees of freedom:
linear displacement along three axes and rotations
with respect to three axes. Let us first examine hydrostatic
stability with respect to linear displacements.
When a floating body is displaced horizontally, there is
no restoring force arising from hydrostatics. This results
in neutral stability for these two degrees of freedom.
Likewise, rotation about a vertical axis results in no
change in volume or restoring moment, so is a neutrally
stable degree of freedom.
The vertical direction is more interesting. In the case
of a fully submerged neutrally buoyant rigid body, the
equilibrium is neutral; a small displacement in z does
not change the vertical (buoyant) force, since the volume
is constant. (Note, however, that a submerged
compressible body will always be unstable with respect
to vertical displacement. If the disturbance is a slight
downward displacement, the increased pressure induces
a decrease in volume; this reduces the buoyant
force, so the body tends to sink. Conversely, if the disturbance
is a slight upward displacement, the body expands,
displacing more fluid, so it tends to rise toward
the surface.)
A rigid body floating in equilibrium with positive waterplane
area A wp is always stable with respect to vertical
displacement. If the disturbance is a small positive
(upward) displacement in z, say dz, the displaced volume
decreases (by A wp dz), decreasing buoyancy relative
to the fixed weight, so the imbalance of forces will
tend to return the body to its equilibrium flotation. gA wp
is the coefficient of stiffness with respect to the vertical
degree of freedom: the hydrostatic restoring force per
unit of displacement distance, exactly like a “spring constant”
in mechanics.
The two remaining degrees of freedom are rotations
about horizontal axes; for example, for a ship, trim (rotation
about a transverse axis) and heel (rotation about
a longitudinal axis). For a fully submerged rigid body,
the stability of these degrees of freedom depends entirely
on the vertical position of the center of gravity
(CG) with respect to the center of buoyancy (CB).
Archimedes’ principle states that equilibrium requires
that the center of gravity and the center of buoyancy lie
on the same vertical line. If the two centers are coincident,
the submerged body can assume any attitude, with
neutral stability. If they are distinct, there will be exactly
one attitude of stable equilibrium, with the CG below the
CB, and exactly one attitude of unstable equilibrium
with the CG above the CB.
For floating bodies, the rotations about horizontal
axes are generally very important, and hydrostatically interesting,
degrees of freedom. The question is, will the
vessel return to an upright attitude following a small
displacement in heel or trim? And, how strong is her tendency
to do so? In Moore (2009), it is shown that the centroid
of waterplane area, also known as the center of
flotation (CF), is a pivot point about which small rotations
can take place with zero change of displacement;
and the stability of these degrees of freedom depends on
the moments of inertia of the waterplane area about
axes through the CF:
dL/d g[I xx (z B z G )] (106)
dM/d g[I yy (z B z G )] (107)