Underwater Robots - Gianluca Antonelli.pdf

2.5 Gravity and Buoyancy
2.5 Gravity and Buoyancy 31
“Ses deux mains s’accrochaient à mon cou; elles ne se seraient pas accrochées
plus furieusement dans un naufrage. Et je ne comprenais pas si elle voulait
que je la sauve, ou bien que je me noie avec elle”.
Raymond Radiguet, “Le diable au corps” 1923.
When arigid body is submerged in afluid under the effect ofthe gravity
two more forces have to be considered: the gravitational force and the
buoyancy. The latter is the only hydrostatic effect, i.e., it is not function of
arelative movement between body and fluid.
Let usdefine as
g I ⎡
= ⎣ 0
⎤
0 ⎦ m/s
9 . 81
2
the acceleration of gravity, ∇ the volume of the body and m its mass.
The submerged weight ofthe body is defined as W = m � � g I � � while its
buoyancy B = ρ ∇ � � I g � � .
The gravity force, acting in the center of mass r B C is represented in body-
fixed frame by:
f G ( R B I )=R B I
⎡
⎣ 0
⎤
0 ⎦ ,
W
while the buoyancy force, acting in the center of buoyancy r B B is represented
in body-fixed frame by:
f B ( R B I )=− R B ⎡
⎣
I
0
⎤
0 ⎦ .
B
The (6×1) vector of force/moment due to gravity and buoyancy in bodyfixed
frame, included in the left hand-side of the equations of motion, is
represented by:
g RB( R B �
f G ( R
I )=−
B I )+f B ( R B I )
r B G × f G ( R B I )+r B B × f G ( R B �
.
I )
In the following, the symbol r B G =[x G y G z G ] T (with r B G = r B used for the center of gravity.
C )will be
The expression of g RB in terms of Euler angles is represented by:
⎡
( W − B ) s θ
⎢
− ( W − B ) c θ s φ
⎢
− ( W − B ) c θ c φ
g RB( η 2 )= ⎢ − ( y G W − y B B ) c θ c φ +(z G W − z B B ) c θ s φ
⎣
( z G W − z B B ) s θ +(x G W − x B B ) c θ c φ
− ( x G W − x B B ) c θ s φ − ( y G W − y B B ) s θ
⎤
⎥ , (2.45)
⎥
⎦