Basics of Fluid Mechanics, 2014a

134 CHAPTER 4. FLUIDS STATICS
M
G c
G G ′
B B ′
Fig. -4.44. The effects of liquid movement on
the GM.
Ships and other floating bodies carry liquid
or have a load which changes the
mass location during tilting of the floating
body. For example, a ship that carries
wheat grains where the cargo is not properly
secured to the ship. The movement of
the load (grains, furniture, and/or liquid)
does not occur in the same speed as the
body itself or the displaced outside liquid.
Sometimes, the slow reaction of the load,
for stability analysis, is enough to be ignored.
Exact analysis requires taking into
account these shifting mass speeds. However,
here, the extreme case where the load reacts in the same speed as the tilting of
the ship/floating body is examined. For practical purposes, it is used as a limit for the
stability analysis. There are situations where the real case approaches to this extreme.
These situations involve liquid with a low viscosity (like water, alcohol) and ship with
low natural frequency (later on the frequency of the ships). Moreover, in this analysis,
the dynamics are ignored and only the statics is examined (see Figure 4.44).
A body is loaded with liquid “B” and is floating in a liquid “A” as shown in Figure
4.44. When the body is given a tilting position the body displaces the liquid on the
outside. At the same time, the liquid inside is changing its mass centroid. The moment
created by the inside displaced liquid is
M in = gρ lB βI xxB (4.165)
Note that I xxB isn’t the same as the moment of inertia of the outside liquid interface.
The change in the mass centroid of the liquid “A” then is
G 1 G ′ 1 = ✁g ✟ ρ lBβI ✟ xxB
✁gV B ✟ ρ lB
} {{ ✟ }
= I xxB
V B
(4.166)
Inside
liquid
weight
Equation (4.166) shows that GG ′
is only a function of the geometry. This quantity,
G 1 G ′ 1 , is similar for all liquid tanks on the floating body.
The total change of the vessel is then calculated similarly to center area calculations.
✁gm total GG ′ 0
= ✘ gm✘✘✘✿ body + gm ✁ f G 1 G ′ 1 (4.167)
For more than one tank, it can be written as
GG ′ =
g
n ∑
W total
i=1
G i G i ρ li V i =
g
n ∑
W total
i=1
I xxbi
V bi
(4.168)