Stability at large angles of heel for wall-sided inclinations
Figure shows a ship heeled to a large angle. When heeled to large angles it can no longer be assumed that the centre of buoyancy moves in an arc. The transfer of wedge of buoyancy from high side to low side is such that there is an increasing vertical movement of B; the vertical component of transfer of buoyancy increases at a faster rate than the horizontal component and B adopts a position at B2 rather than some position at B1 which it would have had if moving in an arc. The water plane area at the larger angle of heel is larger; consequently BM is larger as a result of the greater value of moment of inertia of the water plane area (I). This causes the metacentre to move at larger angles of heel such that it is termed the ‘prometacentre’ or moving metacentre (M2).
The righting lever arising from this higher position of the centre of buoyancy (B2) is:
GZ = GX + XZ
which is greater than the lever GX that would have existed if the upthrust due to buoyancy had been applied at B1 and passed through M.
The formula for this new GZ that applies for wall-sided inclinations only is:
where GM and BM are the values for the ship in the upright condition.
A more accurate definition of a small angle of heel is one where XZ is a small (or negligible) value when compared to the value of GX (where GX = GM x Sinθ, this being the value as calculated for a small angle of heel).
When using: GZ = GM x Sinθ it should be noted that for a ship that has a large initial GM, the error in using this formula would remain small up to a larger angle of heel than for a ship having a small initial GM value.
