28
FLOATING
STRUCTURE
IN
WAVES
92
28
Floating
Structure
in
Waves
We
consider
the
pitch
and
heave
dynamics
of
a
large
ﬂoating
structure
in
a
random
sea.
You
can
consider
this
a
twodimensional
problem.
The
structure
has
two
main,
identical
struts
that
pierce
the
water:
each
has
area
A
w
of
hundred
square
meters,
and
their
centers
are
separated
by
a
distance
L
of
ﬁfty
meters.
The
mass
center
of
the
structure
is
at
the
midpoint.
The
mass
m
is
1000
tons,
and
the
mass
moment
of
inertia
about
the
centroid
is
J
=4
.
0
×
10
5
ton
·
m
2
.
Each
hull
has
an
apparent
linear
damping
in
the
vertical
direction
of
b
=60
kN
·
s/m
.
The
horizontal
motion
of
the
structure
is
nearly
zero.
The
vertical
excitation
force
exerted
at
each
of
the
struts
may
be
approximated
as
the
stiﬀness
(provided
by
the
strut’s
waterplane
area)
times
η
−
ζ
,where
η
is
the
wave
elevation
at
the
location
of
the
strut’s
center,
and
ζ
is
the
vertical
displacement
of
the
strut.
Make
linearizations
where
needed.
Note
we
do
not
take
into
account
any
added
mass
forces
in
this
problem.
Also,
we
assume
that
the
mass
center
is
low
on
the
water,
so
that
the
pitching
moment
is
generated
exactly
by
the
net
loss
of
ﬂotation
on
one
side
and/or
the
increase
of
ﬂotation
on
the
other.
For
the
description,
we
use
the
Bretschneider
spectrum;
it
is
given
by
A
−
B/ω
4
S
(
ω
)=
e
,
where
ω
5
ω
m
=
modal
(or
peak)
frequency,
rad/s
B
=1
.
25
ω
4
;
A
BE
S
;
=
H
2
/
16
.
m
E
S
1
/
3
In
SeaState
5,
we
take
the
modal
period
as
9.7
seconds,
and
the
signiﬁcant
height
H
1
/
3
as
3.3m.
assume
that
the
waves
are
all
traveling
in
the
same
direction,
from
negative
x
toward
positive
x
.
1.
Write
a
pair
of
diﬀerential
equations,
that
express
the
heave
motion
of
the
center
of
mass
(say
z
(
t
)),
and
the
pitch
motion
(say
φ
(
t
)),
in
terms
of
the
elevations
at
the
struts.
Hint:
use
the
fact
that
ζ
(
t,
−
L/
2)
=
z
(
t
)
−
φ
(
t
)
L/
2
,
and
so
on.
Solution:
have,
using
the
hint,
mz
¨
=
ρgA
w
[
η
(
−
L/
2)
−
(
z
−
Lφ/
2)
+
η
(
L/
2)
−
(
z
+
Lφ/
2)]
−
b
[(
˙
z
−
L
˙
z
+
L
˙
φ/
2)
+
( ˙
φ/
2)]
=
ρgA
w
[
η
(
−
L/
2)
+
η
(
L/
2)
−
2
z
]
−
2
bz
˙
Jφ
¨
=
ρgA
w
L
[
−
η
(
−
L/
2)
+
(
z
−
Lφ/
2)
+
η
(
L/
2)
−
(
z
+
Lφ/
2)]
+
2
L
b
[(
˙
z
−
L
˙
z
+
L
˙
φ/
2)
−
(˙
φ/
2)]
2
L
L
2
=
ρgA
w
[
η
(
L/
2)
−
η
(
−
L/
2)
−
Lφ
]
−
bφ.
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 Spring '05
 GeorgeKocur
 Frequency, sea state, ρg Aw

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