Effect of moving ballast

The whole question of ship stability is further complicated by the problem of moving ballast. The most common form of moving ballast is liquid that is free to move, either without constraint, for example bilge water, or liquid that must be carried but is free to move in a tank which is not completely full[1].  Figure 2-25a shows a vessel that is shaped like a cuboid with an inner tank which is partly filled with liquid of density r_b and is floating level in a liquid of density r. If the vessel is made to tilt through a small angle q the liquid moves to the new position shown in figure 2-25b. As a consequence of the tilt the centre of mass of the vessel moves from G to G' and the centre of buoyancy moves from B to B¢, as shown in figure 2-26.

 

The righting couple is now :-

                                                                     

where Q is the point of intersection of the line of action of  and the centre line. The vessel is less stable than it would have been if the liquid had been prevented from moving. Using the method we used to find the horizontal shift of B we can see that the horizontal shift of G is given by :-

                                                               

 where  is the second moment of area of the free surface of the moving liquid ballast about the axis of the vessel. Then the righting couple is equal to:-

                                     or

 

                       righting couple = m.g.((I/V) - (r_b.I_b/r.V).tanq)                                            

 

The importance of this equation is that it tells us that it is not the quantity of moving liquid contained in the ship that matters but how much can move from side to side. Liquids are carried in ships and, as liquid can move fore and aft just as easily as it can move transversely, the liquid is carried in many tanks each having a small cross section and for preference these are filled. This problem is also a headache for designers and operators of vehicle ferries because the vehicles are carried on an unobstructed floor in the ship. Should water get on to this floor, as it has in several well-known incidents[2], the ship becomes unstable with only a little water and may capsize.

 

The metacentric height as a measurable quantity is used very extensively to quantify the stability of vessels of all sorts. It can obviously be used to classify vessels as stable, unstable or having neutral equilibrium according to whether MG  is positive, negative or zero. Some vessels are clearly stable both when upright and capsized but lifeboats are designed to have a negative metacentric height when inverted so that they are self-righting.

 

The metacentric height can also be used to give a quite good estimate of the natural frequency of rolling of a ship. If the motion is limited to small amplitudes and if it is regarded as taking place about a longitudinal axis through the centre of mass and the ship has a second moment of mass about this axis of   we can say that :-

                               righting couple =  times the angular acceleration or:-

                                                           

 and, as q  is small and can be put equal to sinq this is the expression for S.H.M.for which the period of roll T will be given by :-

                                                   T = 2pÖ(I_G/(m.g.MG))                                                

It will be obvious that the whole process of rolling involves the movement of the water as well as the ship and this expression could be regarded as much too simple to be of any value. Nevertheless, in practice, it gives a very useful guide to the observed period of rolling.