2.019 Design of Ocean Systems
Lecture 11
Drift and Slowly-Varying Loads and Motions (II) March 14, 2011 Example of Slowly-Varying Drift Motion
Excitation:
Fx (t) = f0 cos(ωt) + 0.1f0 cos(0.1ωt)
d2 x
d t2
Equation of Motion:
M
= Fx (t)
So u o o
Solution
of motion:
o o
x((t)) =
=
½
0.1
f0
cos
ωt
+
cos((0.1ωt))
M ω2
0 12
0.1
f0
−
{cos ωt + 10 cos(0.1ωt)}
M ω2
−
¾
Responses of Floating Structures in Ocean
Natural Frequency:
ωn =
q
C
(M +Ma )
• For surge, sway, and yaw: hydrostatic restoring coefficients C11 , C22 , C66 = 0
→ ωn = 0 Large-amplitude responses can be excited by slowly-varying excitations.
• For structures with small water-plane area such as semi-submersibles, the
hydrostatic restoring for heave, pitch, and roll are small, the natural
frequencies are small. In this case, large-amplitude responses can also be
excited by slowly-varying excitations.
• In general, very little wave energy at low frequency is present in the ocean. Thus
quencyy wave excitation (based on linear wave theory)
y) is small. Thus,, from
low freq
linear theory, no large-amplitude slowly-varying responses can be caused by the
action of waves!!
• Source of slowlyy-varyying
g excitations:
–Nonlinear wave structure interaction
–Wind loads
Source: Faltinsen, O. M. Sea Loads on Ships and Offshore Structures.
Cambridge University Press, 1993. © Cambridge University Press.
All rights reserved. This content is excluded from our Creative
Commons license. For more information, see http://ocw.mit.edu/fairuse.
Slowly-Varying Wave Force/Moment
Incident wave:
ηI = A1 cos(ω1 t − k1 x) + A2 cos(ω2 t − k2 x)
z
y
x
Slowly-varying wave force/moment comes from:
(1) 2nd-ord
der h
hyd
drod
dynamiic pressure due tto tth
he fifirstt ord
der wave
(2) Interaction between the first-order motion and the first-order wave
(3) 2nd-order potential due to slowly-varying forcing on body surface and free-surface
2nd-order Slowly-Varying Hydrodynamic Pressure
C
Consider
id ttwo simple
i l plane
l
progressive
i waves iin d
deep water:
t
gA2 k2 z
1 k1 z
Φ(x, z, t) = − gA
e
sin(ω
t
−
k
x)
−
1
1
ω1
ω2 e
sin(ω2 t − k2 x)
η (x, t) = A1 cos(ω1 t − k1 x) + A2 cos(ω2 t − k2 x)
We look at the pressure field of the wavefield:
P (x,z,t)
ρ
1
= − ∂Φ
∂t − 2 ∇Φ · ∇Φ − gz
∇Φ
∇
Φ·∇
∇Φ
Φ = Φ2x + Φ2y + Φ2z
Φx =
Φ2x
=
=
gA1 k1 k1 z
ω1 e
cos(ω1 t − k1 x) +
gA2 k2 k2 z
ω2 e
cos(ω2 t − k2 x)
· · · + 2ω1 ω2 A1 A2 e(k1 +k2 )z cos(ω1 t − k1 x) cos(ω2 t − k2 x) + · · ·
· · · + ω1 ω2 A1 A2 e(k1 +k2 )z {cos[(ω1 − ω2 )t − (k1 − k2 )x) + cos[(ω1 + ω2 )t − (k1 + k2 )x)} + · · ·
Difference –frequency
nd
2 -order
(slowly-varying)
In wave amplitude
Sum-frequency
)z
cos[(ω
[( 1 − ω2 )t
) − (k1 − k2 )x]
) ]
2ndd-ord
der sllowlly-varyiing pressure component:
t ∼ A1 A2 e(k1 +k2 )z
Integration this slowly-varying pressure component over body surface to give
slowly-varying force/moment
Interaction Between Body Motion and First-Order Wave
F~ (t) =
R
P (t)~nds =
S(t)
2nd-ord
der sllowlly varyiing
pressure effect
R
Source: Faltinsen, O. M. Sea Loads on Ships and Offshore Structures.
Cambridge University Press, 1993. © Cambridge University Press.
All rights reserved. This content is excluded from our Creative Commons license. For more information, see http://ocw.mit.edu/fairuse. P
S0
(2)
(t)~nds +
R
(1)
P
(t)~nds
∆S(t)
∆S(t) ∼ A1 cos(ω
( 1 t + A2 cos(ω
( 2 t)
P (1) (t) ∼ A1 cos(ω1 t + A2 cos(ω2 t)
Interaction gives 2nd-order
slowly-varying force/moment
2nd-Order Slowly-Varying Potential Due to Forcing on Body Surface and Free-Surface I id t wave:
Incident
z
ηI = A1 cos(ω1 t − k1 x) + A2 cos(ω2 t − k2 x)
y
x
~n
Body has a first-order motion resulting from the action of the incident wave, for example, the heave motion:
ζ3 (t) = a1 cos(ω1 t) + a2 cos(ω2 t)
General body boundary condition imposed on instantaneous body position SB(t):
∂Φ
∂n
=
dζ3 (t)
( )
dt
· nz = −nz [a1 ω1 cos(ω1 t) − ω2 a2 sin(ω2 t)]
Applying Taylor series expansion of the body boundary condition about the mean body position S̄B
¯
¯
∂Φ
¯
∂n SB ((t))
=
¯
∂Φ
¯
∂n S̄B
+ ζ3 (t) ·
∂2Φ ¯
∂z∂n ¯
S̄B
S
+
· · · This terms gives slow-varying terms:
Similar forcing terms are also obtained on the free-surface boundary condition.
• These lead to a 2nd-order potential:
Φ(2) ∼ cos[(ω1 − ω2 )t]and sin[(ω1 − ω2 )t]
From Bernoulli equation, this potential gives a slowly-varying pressure
(2)
P (2) = −ρ ∂Φ∂t
∼ cos[(ω
[( 1 − ω2 )t]
)]
∼ sin[(ω1 − ω2 )t]
Determination of Slowly-Varying Wave Force/Moment
z
ηI = A1 cos(ω1 t − k1 x) + A2 cos(ω2 t − k2 x)
y
x
~n
Fj (
(t),
) J = 1, · · · , 6
js
Fjsv (t) = A1 A2 {Qjc
cos(ω
−
ω
)t
+
Q
1
2
12
12 sin(ω1 − ω2 )t}
Qjc
12 (ω1 , ω2 )
and
j = 1, · · · , 6
Qjs
12 (ω1 , ω2 ) are the slowly-varying force/moment transfer functions.
How to find the slowly-varying force/moment transfer functions??
• By experiments ⎯ accurate measurement of 2nd-order slowly-varying
force/moment is challenge in laboratory
• By numerical computation ⎯ using WAMIT or other nonlinear computational tools
(state-of-the-are research in this area is still going on….)
Determination of Slowly Varying Force/Moment in Irregular Seas
• Incident wave travels in x direction in deep water:
η I (x, t) =
N
X
`=1
A` cos(ω` t − k` x + ²` )
A` (ω` ) =
p
2S(ω` )∆ω
and
∆ω = (ωmax − ωmin )/N
S(ω) is the spectrum of the irregular waves
• Slowly-varying force//moment on a floating body is given by:
n
o
1 XX
jc
js
=
A` Ak Q`k cos[(ωk − ω` )t + (²k − ²` )] + Q`k sin[(ωk − ω` )t + (²k − ²` )]
2
N
Fjsv
N
`=1 k=1
j = 1, 2, . . . , 6
j
jc
j
Q`k
= Qjc
k`
j
js
j
and Q`k
= −Qjs
k`
• Applying Newman’s Approximation:
jc
jc
jc
Qjc
=
Q
=
Q
+
Q
`k
k`
``
kk
and
js
Qjs
=
Q
`k
k` = 0
jc
where Qjc
and
Q
``
kk are the transfer function for drift force/moment, i.e.
F¯j (ω` ) = A2` Qjc
``
and F¯j (ωk ) = A2k Qjc
kk
• Spectrum of the slowly
slowly-varying
varying force/moment:
S F j (µ) = 8
Z
0
∞
2
S (ω )S (ω + µ)[Qjc
S(
(ω
+
µ/2)]
dω
``
Slowly-Varying Motion
Moored system in waves:
Mass-S
Spring-Dashpot
i D h t system:
t
sv
F (t)
x(t)
F sv (t)
( t)
c: mooring line spring equivalent
b: total damping in the system
Equation of motion:
2
(M + Ma ) ddt2 x(t) + b ddt x(t) + cx(t) = F sv (t)
X/f (ω)
In the freq
quency
ydomain:
−ω2 (M + Ma )X(ω) + iωbX(ω) + cX(ω) = f (ω)
X(ω) =
ωn =
f (ω)
[c−ω 2 (M +M
q
c
c
M +Ma
a )]+iωb
ωn ∼ 0.1 rad/s
1
c
Much lower than wave freq.
ωn
Spectrum of slowly-varying motion:
Sx (ω) =
Variance:
σx
=
Z
SF sv (ω)
( )
[c−ω 2 (m+Ma )]2 +b2 ω 2
∞
Sx (ω)dω
Z0 ∞
SF sv (ω))
=
dω
2
2
2
2
[c − ω (m + Ma )] + b ω
0
Z ∞
dω
≈ SF sv (ωn )
[c − ω 2 (m + Ma )]2 + b2 ω 2
0
π
SF sv (ωn )
=
2cb
Source of c: mooring lines
Source of b: (i) related to hull from ⎯ friction,
friction flow separation
separation, current/wind
current/wind,
wave drift damping
(ii) mooring lines
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2.019 Design of Ocean Systems
Spring 2011
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