The stability of ships is of vital importance to the maritime world, and for a variety of reasons is likely to become even more important in the future. For example, ships in general are becoming ever larger, and a growing number of cruise ships have more decks above the waterline, potentially raising their centre of gravity and therefore decreasing their stability. There is also the growing economic pressure to "sail closer to the wind", and therefore potentially jeopardise stability and hence safety.
A floating ship has six degrees of freedom. In order to completely define the ship's motion it is necessary to consider movements in all these modes. In addition, allowance for specific hull structure and a host of non-linear effects must also be made. Many models, of varying sophistication, have been produced to predict the stability of a ship under a variety of conditions and give very good results when compared to the observed behaviour.
While sophisticated models produce many useful results, in general the problem may only be solved numerically (Silva et al, 2005). As such, the physical insight that these models provide, and therefore their pedagogical use, is limited. On the other hand, one can use a simple model that provides both physical insight and also be useful as a teaching tool.
In this paper we describe the basic concepts of the forces influencing ship stability and introduce the differential equation, Mathieu' s equation, describing the motion of a ship from the point of view of stability. We then give some typical numerical solutions of the equation, obtained using MATLAB, and investigate the conditions under which the stability may be aided or compromised.
2 BASIC THEORY
2.1 The forces
If we neglect forces such as wind and wave effects, which are clearly of great importance in all but the best weather conditions, there are two basic forces in action on a ship. These are: (i) the weight force, W = mg, acting vertically downwards, where m is the mass of the ship and g is the acceleration due to gravity; and (ii) the buoyancy force, [F.sub.B], acting vertically upwards (Paffett, 1990) (see figure 1). The weight force acts through the centre of gravity of the ship, the centre of gravity being coincident with the centre of mass in a uniform gravitational field. The buoyancy force (sometimes called the up-thrust) which, by Archimedes' principle, is equal to the weight of fluid displaced, acts through the centre of gravity of the displaced fluid.
The ship's centre of gravity is essentially fired (unless the mass distribution in the ship is altered), but the centre of buoyancy may move considerably if, for example, the ship heels (rolls) over. In general, the centre of gravity of the ship and the centre of gravity of the displaced fluid are not located at the same point. Furthermore, except when the ship is in the upright equilibrium position, they usually do not even lie in the same vertical line. If they are not in the same vertical line, then a moment or couple occurs, which may tend either to right the ship returning it to its equilibrium position or cause it to heel over further, potentially with catastrophic results.
[FIGURE 1 OMITTED]
Figure 1 shows a schematic diagram of the basic geometry of a heeling ship and introduces a number of relevant terms.
As a ship heels over, the original centre of buoyancy when the ship is in the upright equilibrium position, 131, shifts laterally. The point at which a vertical line through the new centre of buoyancy, [B.sub.2], intersects a line through the original centre of buoyancy and the ship's centre of gravity, G, is known as the metacentre, M (Paffett, 1990). The metacentre, like the centre of buoyancy, may move as the ship heels. The distance GM between the centre of gravity and the metacentre is known as the metacentric height, denoted by 1 in figure 1. …