ABSTRACT
The stability of a ship depends on the relative position of (i) the center of gravity of the ship or center of mass, where the weight is applied; (ii) the center of gravity of the displaced fluid or center of flotation, where the flotation force, equal and opposite to the weight, is applied. If the center of gravity lies below (above) the center of flotation (Section 8.3), the analogy with a suspended (inverted) pendulum (Section 8.2) shows that (Section 8.1) the equilibrium position, with weight and flotation force opposite along the same straight line, is stable (unstable). The implication is that, if the ship or pendulum is displaced from the position of stable (unstable) equilibrium, the weight and flotation force remain equal and opposite, but act on distinct parallel lines, giving rise to a torque that causes a return toward (further deviation from) the position of equilibrium. Thus the motion following an initial disturbance, will be an oscillation (Section 8.8) [(monotonic divergence Section 8.9)], with a frequency (rate-of-growth in time) that is calculated most simply in the case of motion of small amplitude (Section 8.6), for which the differential equation of the motion can be linearized. For motions with large amplitude, the equation of motion is nonlinear, but the conservation of energy holds in both cases (Section 8.5), since all forces present are associated with gravity that is conservative. The analogy of the ship and pendulum allows the choice of the length of the latter so that it reproduces the motion of the former (Section 8.4); the length of the equivalent pendulum (Section 8.7) is independent (dependent) of angular displacement for small (large) motions. The monotonous divergence (oscillation) is typical of unstable (stable) systems, of which the simple pendulum and transverse motion of a ship are just two examples; various kinds of mechanical, electrical, and other vibrations and waves are also perturbations of an equilibrium state.
