5.13 Large angle stability
Stability has been considered so far in relation to small angles of inclination, up to about 5 deg,where certain assumptions are made and the metacentric height, GM, is used as the measure of stability.
For larger angles of inclination the upright and inclined waterplanes cannot be assumed to intersect on the waterline, and the metacentre, M, does not remain in a fixed position on the centerline. The righting lever, GZ, which is the perpendicular distance between vertical lines through the centre of gravity and the inclined centre of buoyancy, is instead used as a measure of stability.
Fig 5.8: Large angle stability
Consider a ship which is inclined to some large angleθfrom the vertical, see Fig
5.8.If WL is the initial waterline and W1L1 the waterline when inclined, the volume of displacement in each case will be the same. If the sides of the ship were vertical along its length then the two waterlines would intersect at the centerline. However, the ship’s side is often at an angle to the waterline, particularly in the region of the bow, and thus the waterlines will intersect at some point, P. The volume WPW1, which has emerged will be equal to the volume L1PL, which has been immersed. If this volume is considered to be v and the centroids of each are denoted by g and g1 located a distance, d, apart on the waterline W1L1, the horizontal movement of the centre of buoyancy can be found.
Thus BC = v×d / ▽
where ▽ is the volume of displacement of the ship.
Also GZ = BC – BG sinθ= v×d / ▽ – BG sinθ.
This equation is called Atwood’s formula, after the work done by George Atwood in the eighteenth century. If v and d are evaluated for a range of angles of inclination, a curve of righting lever, GZ, to a base of angle of inclination, θ, can be drawn, as shown in Fig 5.9. This is called a curve of statical stability. The righting lever can be seen to rise to a maximum value and then fall to zero. A ship inclined beyond the point of zero GZ will be unstable. The angle up to the zero GZ point is the range of stability of the ship, at that particular loaded condition. Ship operators need to know the maximum value of GZ and range of stability for a range of loaded conditions.
Fig 5.9:Curve of statical stability
Angle of inclination , θ =
Righting lever , GZ =
Maximum GZ =
Range of stability =
θ=橫傾至1/2乾舷時之傾側角度或14°，取兩者之小者。
Stability = 船舶穩度
Bow = 艏
G = 船舶重心
W = 船重
centre of buoyancy = 浮力中心
displacement = 船舶排水量
W1L1 = 船橫傾dø角後之水線
GZ = 扶正力臂
load = 負荷
metacentric = 穩定
metacentre = 定傾中心
▽ = 排水體積
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