Multihull Seakeeping Dynamics
Approach
-
neglect hydrodynamic interaction between hulls
-
compute stability derivatives and forcing functions for each hull individually
-
transform stability derivatives to ship reference center
-
sum transformed stability
derivatives from each hull to form equivalent matrix for whole ship
-
add sail and hydrofoil contributions in like manner
-
solve six
-
degrees
-
of
-
freedom equations of motion
-
compute added resistance for inclusion in velocity prediction program
-
compute l
ocal RAO's for locations on ship to determine dynamic loads & ride quality
Variables
x
h
y
h
z
h
location of hull reference center in ship coordinate system
A
A
h
apparent mass matrix for ship and individual hull
B
B
h
damping matrix for ship and individual hull
C
C
h
stiffness matrix for ship and individual hull
F
F
h
wave influence matrix for ship and individual hull
M
M
h
mass and inertia matrix for ship and individual hull
T
6dof
six degree of freedom complex transfer function matrix
TV
TL
longitudinal and lateral 3D
OF transfer function matr
4
roll orientation of hull relative to ship coordinate system
5
pitch orientation of hull relative to ship coordinate system
6
yaw orientation of hull relative to ship coordinate system
h
wave phase difference between hull and
ship reference centers
wave phase angle
j
motion displacement of ship in ship coordinate system
h
j
motion displacement of hull in hull coordinate system
0
wave frequency
E
encounter frequency
Stability Derivative Transformations
SMP Equations of Mot
ion
j
a
j
cos
E
t
j
1
6
k
M
j
k
A
j
k
2
t
k
d
d
2
B
j
k
t
k
d
d
C
j
k
k
=
F
j
e
i
E
t
M
j
k
m
0
0
0
m
z
cg
0
0
m
0
m
z
cg
0
m
x
cg
0
0
m
0
m
x
cg
0
0
m
z
cg
0
I
xx
0
I
xz
m
z
cg
0
m
x
cg
0
I
yy
0
0
m
x
cg
0
I
xz
0
I
zz
A
j
k
A
1
1
0
A
3
1
0
A
5
1
0
0
A
2
2
0
A
4
2
0
A
6
2
A
1
3
0
A
3
3
0
A
5
3
0
0
A
2
4
0
A
4
4
0
A
6
4
A
1
5
0
A
3
5
0
A
5
5
0
0
A
2
6
0
A
4
6
0
A
6
6
B
j
k
B
1
1
0
B
3
1
0
B
5
1
0
0
B
2
2
0
B
4
2
0
B
6
2
B
1
3
0
B
3
3
0
B
5
3
0
0
B
2
4
0
B
4
4
0
B
6
4
B
1
5
0
B
3
5
0
B
5
5
0
0
B
2
6
0
B
4
6
0
B
6
6
C
j
k
0
0
0
0
0
0
0
0
0
0
0
0
0
0
C
3
3
0
C
5
3
0
0
0
0
C
4
4
0
0
0
0
C
3
5
0
C
5
5
0
0
0
0
0
0
0
Moment Transfer from hull reference center to ship reference center
Mom
Mom
h
r
h
F
h
Mom
4
Mom
5
Mom
6
Mom
h
4
Mom
h
5
Mom
h
6
x
h
y
h
z
h
F
h
1
F
h
2
F
h
3
Mom
4
Mom
5
Mom
6
Mom
h
4
F
h
3
y
h
F
h
2
z
h
Mom
h
5
F
h
1
z
h
F
h
3
x
h
Mom
h
6
F
h
2
x
h
F
h
1
y
h
F
1
F
2
F
3
Mom
4
Mom
5
Mom
6
1
0
0
0
z
h
y
h
0
1
0
z
h
0
x
h
0
0
1
y
h
x
h
0
0
0
0
1
0
0
0
0
0
0
1
0
0
0
0
0
0
1
F
h
1
F
h
2
F
h
3
Mom
h
4
Mom
h
5
Mom
h
6
Motion of hull
h
1
h
2
h
3
1
2
3
x
h
y
h
z
h
4
5
6
x
h
y
h
z
h
h
4
h
5
h
6
4
5
6
4
5
6
h
1
h
2
h
3
h
4
h
5
h
6
1
0
0
0
0
0
0
1
0
0
0
0
0
0
1
0
0
0
0
z
h
y
h
1
0
0
z
h
0
x
h
0
1
0
y
h
x
h
0
0
0
1
1
2
3
4
5
6
x
h
y
h
z
h
4
5
6
C
h
j
k
h
1
h
2
h
3
h
4
h
5
h
6
0
0
0
0
0
0
0
0
0
0
0
0
0
0
C
h
3
3
0
C
h
5
3
0
0
0
0
C
h
4
4
0
0
0
0
C
h
3
5
0
C
h
5
5
0
0
0
0
0
0
0
1
0
0
0
0
0
0
1
0
0
0
0
0
0
1
0
0
0
0
z
h
y
h
1
0
0
z
h
0
x
h
0
1
0
y
h
x
h
0
0
0
1
1
2
3
4
5
6
x
h
y
h
z
h
4
5
6
C
h
j
k
h
1
h
2
h
3
h
4
h
5
h
6
0
0
0
0
0
0
0
0
0
0
0
0
0
0
C
h
3
3
0
C
h
5
3
0
0
0
C
h
3
3
y
h
C
h
4
4
C
h
5
3
y
h
0
0
0
C
h
3
3
x
h
C
h
3
5
0
C
h
5
3
x
h
C
h
5
5
0
0
0
0
0
0
0
1
2
3
4
5
6
0
0
C
h
3
3
z
h
C
h
3
5
5
C
h
4
4
4
C
h
5
3
z
h
C
h
5
5
5
0
transferring to ship moment reference center
1
0
0
0
z
h
y
h
0
1
0
z
h
0
x
h
0
0
1
y
h
x
h
0
0
0
0
1
0
0
0
0
0
0
1
0
0
0
0
0
0
1
C
a
j
k
h
1
h
2
h
3
h
4
h
5
h
6
1
0
0
0
z
h
y
h
0
1
0
z
h
0
x
h
0
0
1
y
h
x
h
0
0
0
0
1
0
0
0
0
0
0
1
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
C
h
3
3
0
C
h
5
3
0
0
0
0
C
h
4
4
0
0
0
0
C
h
3
5
0
C
h
5
5
0
0
0
0
0
0
0
1
0
0
0
0
0
0
1
0
0
0
0
0
0
1
0
0
0
0
z
h
y
h
1
0
0
z
h
0
x
h
0
1
0
y
h
x
h
0
0
0
1
1
2
3
4
5
6
x
h
y
h
z
h
4
5
6
C
0
0
0
0
0
0
0
0
0
0
0
0
0
0
C
h
3
3
C
h
3
3
y
h
C
h
3
3
x
h
C
h
5
3
0
0
0
C
h
3
3
y
h
C
h
3
3
y
h
2
C
h
4
4
C
h
3
3
x
h
C
h
5
3
y
h
0
0
0
C
h
3
3
x
h
C
h
3
5
C
h
3
3
x
h
C
h
3
5
y
h
C
h
3
3
x
h
2
C
h
5
3
C
h
3
5
x
h
C
h
5
5
0
0
0
0
0
0
0
trim contribution
0
0
C
h
3
3
z
h
C
h
3
5
5
C
h
3
3
z
h
C
h
3
5
5
y
h
C
h
4
4
4
C
h
3
3
x
h
C
h
5
3
z
h
x
h
C
h
3
5
C
h
5
5
5
0
Velocities
t
h
1
h
2
h
3
d
d
t
1
2
3
d
d
t
4
5
6
d
d
x
h
y
h
z
h
t
h
4
h
5
h
6
d
d
t
4
5
6
d
d
t
h
1
h
2
h
3
h
4
h
5
h
6
d
d
1
0
0
0
0
0
0
1
0
0
0
0
0
0
1
0
0
0
0
z
h
y
h
1
0
0
z
h
0
x
h
0
1
0
y
h
x
h
0
0
0
1
t
1
2
3
4
5
6
d
d
B
h
j
k
t
h
1
h
2
h
3
h
4
h
5
h
6
d
d
B
h
1
1
0
B
h
3
1
0
B
h
5
1
0
0
B
h
2
2
0
B
h
4
2
0
B
h
6
2
B
h
1
3
0
B
h
3
3
0
B
h
5
3
0
0
B
h
2
4
0
B
h
4
4
0
B
h
6
4
B
h
1
5
0
B
h
3
5
0
B
h
5
5
0
0
B
h
2
6
0
B
h
4
6
0
B
h
6
6
1
0
0
0
0
0
0
1
0
0
0
0
0
0
1
0
0
0
0
z
h
y
h
1
0
0
z
h
0
x
h
0
1
0
y
h
x
h
0
0
0
1
t
1
2
3
4
5
6
d
d
B
h
j
k
t
h
1
h
2
h
3
h
4
h
5
h
6
d
d
B
h
1
1
0
B
h
3
1
0
B
h
5
1
0
0
B
h
2
2
0
B
h
4
2
0
B
h
6
2
B
h
1
3
0
B
h
3
3
0
B
h
5
3
0
B
h
1
3
y
h
B
h
2
2
z
h
B
h
2
4
B
h
3
3
y
h
B
h
4
2
z
h
B
h
4
4
B
h
5
3
y
h
B
h
6
2
z
h
B
h
6
4
B
h
1
1
z
h
B
h
1
3
x
h
B
h
1
5
0
B
h
3
1
z
h
B
h
3
3
x
h
B
h
3
5
0
B
h
5
1
z
h
B
h
5
3
x
h
B
h
5
5
0
B
h
1
1
y
h
B
h
2
2
x
h
B
h
2
6
B
h
3
1
y
h
B
h
4
2
x
h
B
h
4
6
B
h
5
1
y
h
B
h
6
2
x
h
B
h
6
6
t
1
2
3
4
5
6
d
d
transferring to center hull moment reference cen
ter
B
B
h
1
1
0
B
h
3
1
y
h
B
h
3
1
z
h
B
h
1
1
x
h
B
h
3
1
B
h
5
1
y
h
B
h
1
1
0
B
h
2
2
0
z
h
B
h
2
2
B
h
4
2
0
x
h
B
h
2
2
B
h
6
2
B
h
1
3
0
B
h
3
3
y
h
B
h
3
3
z
h
B
h
1
3
x
h
B
h
3
3
B
h
5
3
y
h
B
h
1
3
y
h
Bha
1
3
z
a
B
h
2
2
B
h
2
4
y
h
B
h
3
3
z
h
2
B
h
2
2
y
h
2
B
h
3
3
z
h
B
h
4
2
B
h
2
4
B
h
4
4
z
h
B
h
1
3
x
h
B
h
3
3
B
h
5
3
y
h
x
h
z
h
B
h
2
2
z
h
B
h
6
2
y
h
2
B
h
1
3
x
h
B
h
2
4
B
h
6
4
z
h
B
h
1
1
B
h
1
3
x
h
B
h
1
5
0
B
h
3
1
z
h
B
h
3
5
x
h
B
h
3
3
B
h
3
1
z
h
B
h
3
3
x
h
B
h
3
5
y
h
z
h
2
B
h
1
1
x
h
2
B
h
3
3
B
h
3
1
B
h
1
3
x
h
z
h
B
h
5
1
B
h
1
5
z
h
B
h
5
3
B
h
3
5
x
h
B
h
5
5
z
h
B
h
1
1
B
h
1
3
x
h
B
h
1
5
y
h
y
h
B
h
1
1
x
h
B
h
2
2
B
h
2
6
y
h
B
h
3
1
y
h
2
B
h
3
1
B
h
4
6
x
h
z
h
Bha
2
2
z
h
B
h
2
6
x
h
B
h
4
2
z
h
B
h
1
1
x
h
B
h
3
1
B
h
5
1
y
h
x
h
2
B
h
2
2
y
h
2
B
h
1
1
B
h
6
2
B
h
2
6
x
h
B
h
6
6
Acceleration
2
t
h
1
h
2
h
3
d
d
2
2
t
1
2
3
d
d
2
2
t
4
5
6
d
d
2
x
h
y
h
z
h
t
4
5
6
d
d
t
4
5
6
d
d
x
h
y
h
z
h
2
t
h
1
h
2
h
3
h
4
h
5
h
6
d
d
2
1
0
0
0
0
0
0
1
0
0
0
0
0
0
1
0
0
0
0
z
h
y
h
1
0
0
z
h
0
x
h
0
1
0
y
h
x
h
0
0
0
1
2
t
1
2
3
4
5
6
d
d
2
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
t
4
y
h
d
d
t
5
x
h
d
d
t
6
x
h
d
d
t
4
z
h
d
d
0
0
0
t
4
y
h
d
d
t
5
x
h
d
d
0
t
5
z
h
d
d
t
6
y
h
d
d
0
0
0
t
6
x
h
d
d
t
4
z
h
d
d
t
5
z
h
d
d
t
6
y
h
d
d
0
0
0
0
t
1
2
3
4
5
6
d
d
A
h
j
k
2
t
h
1
h
2
h
3
h
4
h
5
h
6
d
d
2
A
h
1
1
0
A
h
3
1
0
A
h
5
1
0
0
A
h
2
2
0
A
h
4
2
0
A
h
6
2
A
h
1
3
0
A
h
3
3
0
A
h
5
3
0
0
A
h
2
4
0
A
h
4
4
0
A
h
6
4
A
h
1
5
0
A
h
3
5
0
A
h
5
5
0
0
A
h
2
6
0
A
h
4
6
0
A
h
6
6
1
0
0
0
0
0
0
1
0
0
0
0
0
0
1
0
0
0
0
z
h
y
h
1
0
0
z
h
0
x
h
0
1
0
y
h
x
h
0
0
0
1
2
t
1
2
3
4
5
6
d
d
2
A
h
1
1
0
A
h
3
1
0
A
h
5
1
0
0
A
h
2
2
0
A
h
4
2
0
A
h
6
2
A
h
1
3
0
A
h
3
3
0
A
h
5
3
0
0
A
h
2
4
0
A
h
4
4
0
A
h
6
4
A
h
1
5
0
A
h
3
5
0
A
h
5
5
0
0
A
h
2
6
0
A
h
4
6
0
A
h
6
6
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
t
4
y
h
d
d
t
5
x
h
d
d
t
6
x
h
d
d
t
4
z
h
d
d
0
0
0
t
4
y
h
d
d
t
5
x
h
d
d
0
t
5
z
h
d
d
t
6
y
h
d
d
0
0
0
t
6
x
h
d
d
t
4
z
h
d
d
t
5
z
h
d
d
t
6
y
h
d
d
0
0
0
0
t
1
2
3
4
5
6
d
d
transferring to center hull moment reference center
A
A
h
1
1
0
A
h
3
1
A
h
3
1
y
h
A
h
1
1
z
h
x
h
A
h
3
1
A
h
5
1
A
h
1
1
y
h
0
A
h
2
2
0
A
h
2
2
z
h
A
h
4
2
0
A
h
2
2
x
h
A
h
6
2
A
h
1
3
0
A
h
3
3
A
h
3
3
y
h
z
h
A
h
1
3
A
h
3
3
x
h
A
h
5
3
A
h
1
3
y
h
A
h
1
3
y
h
A
h
2
2
z
h
A
h
2
4
A
h
3
3
y
h
z
h
2
A
h
2
2
y
h
2
A
h
3
3
z
h
A
h
4
2
A
h
2
4
A
h
4
4
y
h
A
h
1
3
z
h
A
h
3
3
x
h
A
h
5
3
y
h
2
A
h
1
3
z
h
A
h
2
2
x
h
A
h
6
2
z
h
x
h
A
h
2
4
A
h
6
4
A
h
1
3
x
h
A
h
1
1
z
h
A
h
1
5
0
A
h
3
1
z
h
A
h
3
3
x
h
A
h
3
5
y
h
A
h
3
1
z
h
A
h
3
3
x
h
A
h
3
5
z
h
2
A
h
1
1
x
h
2
A
h
3
3
A
h
3
1
A
h
1
3
x
h
z
h
A
h
5
1
A
h
1
5
z
h
A
h
5
3
A
h
3
5
x
h
A
h
5
5
y
h
A
h
1
1
z
h
A
h
1
3
x
h
A
h
1
5
A
h
1
1
y
h
A
h
2
2
x
h
A
h
2
6
A
h
3
1
y
h
z
h
A
h
2
2
x
h
z
h
A
h
2
6
y
h
2
A
h
3
1
A
h
4
2
x
h
A
h
4
6
z
h
A
h
1
1
x
h
A
h
3
1
A
h
5
1
y
a
x
h
2
A
h
2
2
y
h
2
A
h
1
1
A
h
6
2
A
h
2
6
x
h
A
h
6
6
Mass and inertia transfer to ship reference center
M
1
0
0
0
0
0
0
1
0
0
0
0
0
0
1
0
0
0
0
z
h
y
h
1
0
0
z
h
0
x
h
0
1
0
y
h
x
h
0
0
0
1
T
m
0
0
0
m
z
cg
0
0
m
0
m
z
cg
0
m
x
cg
0
0
m
0
m
x
cg
0
0
m
z
cg
0
I
xx
0
I
xz
m
z
cg
0
m
x
cg
0
I
yy
0
0
m
x
cg
0
I
xz
0
I
zz
1
0
0
0
0
0
0
1
0
0
0
0
0
0
1
0
0
0
0
z
h
y
h
1
0
0
z
h
0
x
h
0
1
0
y
h
x
h
0
0
0
1
M
m
0
0
0
m
z
h
z
cg
y
h
m
0
m
0
m
z
h
z
cg
0
m
x
h
x
cg
0
0
m
y
a
m
m
x
a
x
cg
0
0
m
z
h
z
cg
y
h
m
m
z
h
2
2
z
h
z
cg
y
h
2
I
xx
m
x
h
x
cg
y
a
m
z
h
x
h
x
h
z
cg
z
h
x
cg
I
xz
m
z
h
z
cg
0
m
x
h
x
cg
y
h
m
x
h
x
cg
m
z
h
2
2
z
h
z
cg
x
h
2
2
x
h
x
cg
I
yy
y
h
m
z
h
z
cg
y
h
m
m
x
h
x
cg
0
m
z
h
x
h
x
h
z
cg
z
h
x
cg
I
xz
m
z
h
z
cg
y
h
m
y
h
2
x
h
2
2
x
h
x
cg
I
zz
Froude
-
Krylov Excitation
The wave phase is taken as the phase at the ship reference center. Each hull is at a differen
t place in the wave, so wave influence terms must be
phase shifted to produce the same forces and moments as when the wave phase was taken at the hull reference center.
E
0
0
2
g
U
0
cos
(
)
k
0
2
g
F
a
j
e
i
E
t
h
x
y
t
(
)
cos
k
x
cos
(
)
y
sin
(
)
(
)
E
t
h
k
x
h
cos
(
)
y
h
sin
(
)
h
0
2
g
x
h
cos
(
)
y
h
sin
(
)
F
j
F
h
j
e
i
E
t
0
2
g
x
h
cos
(
)
y
h
sin
(
)
F
1
F
2
F
3
F
4
F
5
F
6
1
0
0
0
z
h
y
h
0
1
0
z
h
0
x
h
0
0
1
y
h
x
h
0
0
0
0
1
0
0
0
0
0
0
1
0
0
0
0
0
0
1
F
h
1
F
h
2
F
h
3
F
h
4
F
h
5
F
h
6
e
i
0
2
g
x
h
cos
(
)
y
h
sin
(
)
F
1
F
2
F
3
F
4
F
5
F
6
F
h
1
cos
0
2
g
x
h
cos
(
)
y
h
sin
(
)
i
sin
0
2
g
x
h
cos
(
)
y
h
sin
(
)
F
h
2
cos
0
2
g
x
h
cos
(
)
y
h
sin
(
)
i
sin
0
2
g
x
h
cos
(
)
y
h
sin
(
)
F
a
3
cos
0
2
g
x
h
cos
(
)
y
h
sin
(
)
i
sin
0
2
g
x
h
cos
(
)
y
h
sin
(
)
z
h
F
a
2
y
h
F
a
3
F
h
4
cos
0
2
g
x
h
cos
(
)
y
h
sin
(
)
i
sin
0
2
g
x
h
cos
(
)
y
h
sin
(
)
z
h
F
h
1
x
h
F
h
3
F
h
5
cos
0
2
g
x
h
cos
(
)
y
h
sin
(
)
i
sin
0
2
g
x
h
cos
(
)
y
h
sin
(
)
y
h
F
h
1
x
h
F
h
2
F
h
6
cos
0
2
g
x
h
cos
(
)
y
h
sin
(
)
i
sin
0
2
g
x
h
cos
(
)
y
h
sin
(
)
Solving Equations of Motion
SMP motion solved as two 3DOF solutions
TV
h
1
h
3
h
5
F
h
1
F
h
3
F
h
5
TL
h
2
h
4
h
6
F
h
2
F
h
4
F
h
6
Multihull requires 6DOF solution. Individual matrices need not be formed, but complete matrix can be formed from SMP 3DOF ma
trices.
T
6dof
1
2
3
4
5
6
F
1
F
2
F
3
F
4
F
5
F
6
T
6dof
1
0
0
0
z
h
y
h
0
1
0
z
h
0
x
h
0
0
1
y
h
x
h
0
0
0
0
1
0
0
0
0
0
0
1
0
0
0
0
0
0
1
TV
11
0
TV
21
0
TV
31
0
0
TL
11
0
TL
21
0
TL
31
TV
12
0
TV
22
0
TV
32
0
0
TL
12
0
TL
22
0
TL
32
TV
13
0
TV
33
0
TV
33
0
0
TL
13
0
TL
23
0
TL
33
1
0
0
0
z
h
y
h
0
1
0
z
h
0
x
h
0
0
1
y
h
x
h
0
0
0
0
1
0
0
0
0
0
0
1
0
0
0
0
0
0
1
T
T
6dof
TV
11
0
TV
21
y
h
TV
21
z
h
TV
11
x
h
TV
21
TV
31
y
h
TV
11
0
TL
11
0
z
h
TL
11
TL
21
0
x
h
TL
11
TL
31
TV
12
0
TV
22
y
h
TV
22
z
h
TV
12
x
h
TV
22
TV
32
y
h
TV
12
y
h
TV
12
z
h
TL
11
TL
12
y
h
TV
22
z
h
2
TL
11
TL
21
TL
12
z
h
TL
22
y
h
2
TV
22
y
h
z
h
TV
12
x
h
TV
22
TV
32
x
h
z
h
TL
11
z
h
TL
31
y
h
2
TV
12
x
h
TL
12
TL
32
z
h
TV
11
TV
12
x
h
TV
13
0
TV
21
z
h
x
h
TV
22
TV
33
y
h
TV
21
z
h
x
h
TV
22
TV
33
z
h
2
TV
11
z
h
x
h
TV
21
z
h
TV
31
x
h
z
h
TV
12
x
h
2
TV
22
x
h
TV
32
z
h
TV
13
x
h
TV
33
TV
33
y
h
z
h
TV
11
TV
12
x
h
TV
13
y
h
TV
11
x
h
TL
11
TL
13
y
h
TV
21
y
h
2
TV
21
x
h
z
h
TL
11
x
h
TL
21
z
h
TL
13
TL
23
y
h
z
h
TV
11
x
h
TV
21
TV
31
y
h
2
TV
11
x
h
2
TL
11
x
h
TL
31
TL
13
TL
33
Added Resistance from Kochin Functions
H
h
H
h
7
H
V
T
h
1
h
3
h
5
H
L
T
h
2
h
4
h
6
H
H
h
7
H
V
1
H
L
1
H
V
2
H
L
2
H
V
3
H
L
3
T
h
1
h
2
h
3
h
4
h
5
h
6
H
H
h
7
cos
0
2
g
x
h
cos
(
)
y
h
sin
(
)
i
sin
0
2
g
x
h
cos
(
)
y
h
sin
(
)
H
V
1
H
L
1
H
V
2
H
L
2
H
V
3
H
L
3
T
1
0
0
0
0
0
0
1
0
0
0
0
0
0
1
0
0
0
0
z
h
y
h
1
0
0
z
h
0
x
h
0
1
0
y
h
x
h
0
0
0
1
1
2
3
4
5
6
H
H
h
7
cos
0
2
g
x
h
cos
(
)
y
h
sin
(
)
i
sin
0
2
g
x
h
cos
(
)
y
h
sin
(
)
H
V
1
H
L
1
H
V
2
H
L
1
z
h
H
V
2
y
h
H
L
2
H
V
1
z
h
H
V
2
x
h
H
V
3
H
V
1
y
h
H
L
1
x
h
H
L
3
T
1
2
3
4
5
6
F
x
1
2
E
2
cos
(
)
Im
H
(
)
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