18
4
VESSEL
DYNAMICS:
LINEAR
CASE
X
A
number
of
coeﬃcients
can
be
discounted,
as
noted
in
the
last
chapter.
First,
in
a
homoge-
neous
sea,
with
no
current,
wave,
or
wind
effects,
{
X
x
, X
y
, X
δ
, Y
x
, Y
y
, Y
δ
, N
x
, N
y
, N
δ
}
are
all
zero.
We
assume
that
no
hydrodynamic
forces
depend
on
the
position
of
the
vessel.
1
Second,
consider
X
v
:
since
this
longitudinal
force
would
have
the
same
sign
regardless
of
the
sign
of
v
(because
of
side-to-side
hull
symmetry),
it
must
have
zero
slope
with
v
at
the
origin.
Thus
v
=
0.
The
same
argument
shows
that
{
X
r
, X
v
˙
, X
r
˙
, Y
u
, Y
u
˙
, N
u
, N
˙
=
0.
Finally,
since
u
}
ﬂuid
particle
acceleration
relates
linearly
with
pressure
or
force,
we
do
not
consider
nonlin-
ear
acceleration
terms,
or
higher
time
derivatives.
It
should
be
noted
that
some
nonlinear
terms
related
to
those
we
have
eliminated
above
are
not
zero.
For
instance,
Y
uu
=
0
because
of
hull
symmetry,
but
in
general
X
vv
=
0
only
if
the
vessel
is
bow-stern
symmetric.
We
have
so
far,
considering
only
the
linear
hydrodynamic
terms,
(
m
−
X
u
˙
)
˙
u
=
X
u
u
+
X
∗
(57)
(
m
−
Y
v
˙
)
˙
v
+
(
mx
G
−
Y
r
˙
)
˙
r
=
Y
v
v
+
(
Y
r
−
mU
)
r
+
Y
∗
(58)
(
mx
G
−
N
v
˙
)
˙
v
+
(
I
zz
−
N
r
˙
)
˙
r
=
N
v
v
−
(
N
r
−
mx
G
U
)
r
+
N
∗
.
(59)
The
right
side
here
carries
also
the
imposed
forces
from
a
thruster(s)
and
rudder(s)
{
X
∗
, Y
∗
, N
∗
}
.
Note
that
the
surge
equation
is
decoupled
from
the
sway
and
yaw,
but
that
sway
and
yaw
themselves
are
coupled,
and
therefore
are
of
immediate
interest.
With
the
state
vector
γs
=
{
v, r
}
and
external
force/moment
vector
γ
F
=
{
Y
∗
, N
∗
}
,
a
state-space
representation
of
the
sway/yaw
system
is
⎬
�
⎬
�
m
−
Y
v
˙
mx
G
−
Y
r
˙
dγs
Y
v
Y
r
−
mU
=
γ
s
+
γ
mx
G
−
N
v
˙
I
zz
−
N
r
˙
dt
N
v
N
r
−
mx
G
U
F
,
or
(60)
Mγ
s
˙
=
Pγs
+
γ
F
γs
˙
=
M
−
1
Pγs
+
M
−
1
γ
F
γs
˙
=
Aγs
+
B
γ
F.
(61)
The
matrix
M
is
a
mass
or
inertia
matrix,
which
is
always
invertible.
The
last
form
of
the
equation
is
a
standard
one
wherein
A
represents
the
internal
dynamics
of
the
system,
and
B
is
a
gain
matrix
for
the
control
and
disturbance
inputs.