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FrequencyFrequency--Domain Calculations of Moored Domain Calculations of Moored Vessel Motion Including Low Frequency EffectVessel Motion Including Low Frequency Effect
OMAE 2008OMAE 2008Estoril, PortugalEstoril, PortugalJune 15June 15--20, 200820, 2008
Cedric Le Cunff Cedric Le Cunff PrincipiaPrincipiaSam Ryu Sam Ryu SOFECSOFECJ.J.--M. HeurtierM. Heurtier PrincipiaPrincipiaArun DuggalArun Duggal SOFECSOFEC
2/28
Outline
1. Background
2. Summary of TD and FD Analyses
3. Case Studies
4. Comments on Modeling
5. Summary
3/28
Where We Are
1. Background
2. Summary of TD and FD Analyses
3. Case Studies
4. Comments on Modeling
5. Summary
4/28
Background“Derivation of CALM Buoy Coupled Buoy RAOs in FrequencyDomain and Experimental Validation” (Le Cunff et al., 2007)
deepwater CALM buoy
5/28
CALM Buoy System with Tanker Connected
FPSOTanker Buoy
Flowlines (OOLs)
Buoy Mooring
6/28
Validation with Model Test Results
7/28
Comparisons between TD and FD (Pitch Motion)
8/28
Where We Are
1. Background
2. Summary of TD and FD Analyses
3. Case Studies
4. Comments on Modeling
5. Summary
9/28
Time-Domain Analysis• Hydrodynamic Loads on the Bodies
– hydrodynamic calculations via BEM method
– added mass, radiation damping, first order wave force
• Loads on the Lines– FEM-based computer program (DeepLines)
– lines characteristics + Morison’s formulation
• Coupled System– extra node with six DOF for the floating body
– solve the coupled equation:
{ }0
(1) (2)
( ) ( )τ τ∞
+ ∞ + − +
= + + +
∫r r r&& &
r r r ra
d w w line
M M X R t Xd KX
F F F F
10/28
Frequency-Domain Analysis
• Equation of motion:
• Assuming that the position at time t is given by:
with
– Static equilibrium :
– Imposed displacement :
FxMxBKx =++ &&&
( ) ( ) ( )∑ ∑=
−
= ⎪⎭
⎪⎬⎫
⎪⎩
⎪⎨⎧
⎪⎭
⎪⎬⎫
⎪⎩
⎪⎨⎧
⎥⎥⎦
⎤
⎢⎢⎣
⎡++=
n
i
tjif
Nimp
impimpiimpstat e ixxaXtX
1 1
Re)(ω
ωω
0=impstatxK
statstatstat FxK =
11/28
(cont’d)
• Loads on body from hydrodynamic calculations
• Dependency of matrices on frequency
• Linearization of quadratic viscous damping (both body and lines)
– Linearization Coefficients
• Regular waves : (A: norm of velocity)
• Irregular waves: (σ: standard deviation)
( ) ( )( ) ( )[ ] ( ){ } { })()(2 ωωωωωωω FXMMBBjK aa =+−+−
relrelrelrel vvvv )(Ω≈
Aπ38=Ω
σπ 22=Ω
12/28
• wave elevation:
• low-frequency (LF) force, Molin (2002)
• LF force spectrum
Low-Frequency Wave Loading
[ ])expRe(1∑=
+−=n
llll jtjA θωη
( ) ( )[ ]))(exp,Re(1 1
)2()2( ∑∑= =
−+−−=n
l
n
mmlmlmlmlwave jtjfAAF θθωωωω
( ) ( ) ( ) ( ) ωωωωω dfSSSF
2
0
)2()2( ,8∫∞
−ΩΩ+=Ω
QTF:
Direct calculation
2) Newman’s approximation
13/28
Low-Frequency Wave Loading (QTF)
• Newman’s approximation
• Second order wave potential (x-direction for example)
( ) ( ) ( )mdldml fff ωωωω ×=2)2( ,
( ) ( )[ ] 2
2)2(
)()()(
)()(1,
mlmlml
mlmlmlxmlx hkkthkkg
kkVCmf
ωω
ωωωωρωω
−−−−
−−+=
( )( )[ ]( )[ ] βcos
hkkchhzkkch
jl
jl
−
+−×
14/28
• Wind speed driven from wind spectrum:
• Wind force:
• Evaluation of wind coefficients based on mean direction (wind w.r.t. body)
Wind Loading
[ ]⎟⎟⎠
⎞⎜⎜⎝
⎛+−= ∑
=
n
llllwind jtjAV
1
expRe θω
2, 2
1vesselwindxairwindx VVSCF −= ρ
15/28
Where We Are
1. Background
2. Summary of TD and FD Analyses
3. Case Studies
4. Comments on Modeling
5. Summary
16/28
FD/TD Calculation Results Comparison
0°
45°
90°
135°
180°
225°
270°
315°
Wave Direction (e.g. 180 deg)
earth-fixed+ X (North)
earth-fixed+ Y (West)
vessel-fixed+y (sway)
vessel-fixed+x (surge)
+Roll: Starboard Down+Pitch: Bow Down+Yaw: Bow to Port+Z: Upwards
Spread Moored FPSOComparison FD/TD:
FD - spectrum directly computed
TD - FFT of time series
17/28
Time-Domain Calculation
• 3-hour simulation
• LF and WF components
18/28
Case 1 (waves only; no wind)Results – FPSO Surge
19/28
Results – FPSO Sway (cont’d)
20/28
Results – FPSO Yaw (cont’d)
21/28
Case 2 (waves + wind)Results – FPSO LF Motions
22/28
Where We Are
1. Background
2. Summary of TD and FD Analyses
3. Case Studies
4. Comments on Modeling
5. Summary
23/28
1. Second-Order Wave Potential
• Effect negligible
• Induces a very small force spectrum at low frequency range
24/28
2. Reduction of LF Contribution by WF Components
Comparisons– first order waves (WF) only
– low frequency (LF) only
– LF + WF
25/28
3. Low Stiffness Mooring
• Wave + wind aligned: OK
• Otherwise TD required
26/28
Where We Are
1. Background
2. Summary of TD and FD Analyses
3. Case Studies
4. Comments on Modeling
5. Summary
27/28
Summary
1. Freq-domain analysis methodology is presented as a tool of motion estimate even including LF components.
2. Classical linearization methods applied to quadratic drag term
3. Case studies carried out with a spread moored FPSO
4. Comparison b/w FD and TD calculations shows that the results are in good agreement.
5. Second-order wave potential is not significant for the relatively low frequency range.
6. Unstable time-varying yaw motion can only be analyzed by using a TD analysis.
7. Comments on modeling presented.
28/28
Future Work: Validation against Model Test Results
29/28
Thank you very much!