Abstract

An analytical solution of the dynamic response of offshore wind turbines under wave load with nonlinear Stokes’s wave theory and wave–structure and soil–foundation interactions is developed. Natural frequencies and the corresponding modes are obtained. The effect of the wave–structure interaction, the added mass, the foundation stiffness, and the nacelle translational and rotational inertia on the motion of the structure is investigated. The nonlinear loading provided by the drag term of Morison’s equation is successfully handled. A parametric study to examine the effect of the structural parameters on the dynamic response is conducted, and the results of the proposed analytical solution are compared to numerical ones. The proposed method has the following advantages: (a) it is accurate and straightforward because of its analytical nature, (b) it does not ignore the drag term in the wave loading by keeping its nonlinearity nature, (c) the structure of the wind turbine is modeled as a continuous system, (d) it takes into account the effect of the rotational and translational inertia of the nacelle on the dynamic response, and (e) it provides an interpretation of the effect of the sea level variation in changing the natural frequencies.

References

1.
Wind Europe
,
2019
, “
The European Offshore Wind Industry—Key Trends and Statistics 2019
,” Wind Europe, https://windeurope.org/intelligence-platform/product/wind-energy-in-europe-in-2019-trends-and-statistics/
2.
Adeli
,
H.
, and
Kim
,
H.
,
2009
,
Wavelet-Based Vibration Control of Smart Buildings and Bridges
,
CRC Press, Taylor & Francis
,
Boca Raton, FL
.
3.
Kim
,
H.
, and
Adeli
,
H.
,
2004
, “
Hybrid Feedback-Least Mean Square Algorithm for Structural Control
,”
J. Struct. Eng.
,
130
(
1
), pp.
120
127
.
4.
Adeli
,
H.
, and
Saleh
,
A.
,
1997
, “
Optimal Control of Adaptive/Smart Bridge Structures
,”
J. Struct. Eng.
,
123
(
2
), pp.
218
226
.
5.
Saleh
,
A.
, and
Adeli
,
H.
,
1994
, “
Parallel Algorithms for Integrated Structural/Control Optimization
,”
J. Aerosp. Eng.
,
7
(
3
), pp.
297
314
.
6.
Saleh
,
A.
, and
Adeli
,
H.
,
1996
, “
Parallel Eigenvalue Algorithms for Large-Scale Control-Optimization Problems
,”
J. Aerosp. Eng.
,
9
(
3
), pp.
70
79
.
7.
Saleh
,
A.
, and
Adeli
,
H.
,
1997
, “
Robust Parallel Algorithms for Solution of Riccati Equation
,”
J. Aerosp. Eng.
,
10
(
3
), pp.
126
133
.
8.
El-Khoury
,
O.
, and
Adeli
,
H.
,
2013
, “
Recent Advances on Vibration Control of Structures Under Dynamic Loading
,”
Arch. Comput. Meth. Eng.
,
20
(
4
), pp.
353
360
.
9.
Gutierrez Soto
,
M.
, and
Adeli
,
H.
,
2017
, “
Recent Advances in Control Algorithms for Smart Structures and Machines
,”
Expert Syst.
,
34
(
2
), p.
e12205
.
10.
Ghaedi
,
K.
,
Ibrahim
,
Z.
,
Adeli
,
H.
, and
Javanmardi
,
A.
,
2017
, “
Invited Review: Recent Developments in Vibration Control of Building and Bridge Structures
,”
J. Vibroeng.
,
19
(
5
), pp.
3564
3580
.
11.
Jiang
,
X.
, and
Adeli
,
H.
,
2008
, “
Neuro-Genetic Algorithm for Non-Linear Active Control of Structures
,”
Int. J. Numer. Methods Eng.
,
75
(
7
), pp.
770
786
.
12.
Jiang
,
X.
, and
Adeli
,
H.
,
2008
, “
Dynamic Fuzzy Wavelet Neuroemulator for Non-Linear Control of Irregular Building Structures
,”
Int. J. Numer. Methods Eng.
,
74
(
7
), pp.
1045
1066
.
13.
Li
,
Z.
, and
Adeli
,
H.
,
2016
, “
New Discrete-Time Robust H2/H∞ Algorithm for Vibration Control of Smart Structures Using Linear Matrix Inequalities
,”
Eng. Appl. Artif. Intell.
,
55
, pp.
47
57
.
14.
Gutierrez Soto
,
M.
, and
Adeli
,
H.
,
2017
, “
Many-Objective Control Optimization of High-Rise Building Structures Using Replicator Dynamics and Neural Dynamics Model
,”
Struct. Multidiscipl. Optim.
,
56
(
6
), pp.
1521
1537
.
15.
Soto
,
G.
, and
Adeli
,
H.
,
2017
, “
Multi-Agent Replicator Controller for Sustainable Vibration Control of Smart Structures Mariantonieta
,”
J. Vibroeng.
,
19
(
6
), pp.
4300
4322
.
16.
Gutierrez Soto
,
M.
, and
Adeli
,
H.
,
2018
, “
Vibration Control of Smart Base-Isolated Irregular Buildings Using Neural Dynamic Optimization Model and Replicator Dynamics
,”
Eng. Struct.
,
156
, pp.
322
336
.
17.
Li
,
Z.
, and
Adeli
,
H.
,
2018
, “
Control Methodologies for Vibration Control of Smart Civil and Mechanical Structures
,”
Expert Syst.
,
35
(
6
), p.
e12354
.
18.
Gutierrez Soto
,
M.
, and
Adeli
,
H.
,
2019
, “
Semi-Active Vibration Control of Smart Isolated Highway Bridge Structures Using Replicator Dynamics
,”
Eng. Struct.
,
186
, pp.
536
552
.
19.
Javadinasab Hormozabad
,
S.
, and
Soto
,
M. G.
,
2021
, “
Real-Time Damage Identification of Discrete Structures via Neural Networks Subjected to Dynamic Loading
,”
Proceedings Volume 11593, Health Monitoring of Structural and Biological Systems XV; 115932O
,
Virtual Online
.
20.
Azimi
,
M.
,
Eslamlou
,
A. D.
, and
Pekcan
,
G.
,
2020
, “
Data-Driven Structural Health Monitoring and Damage Detection Through Deep Learning: State-Ofthe- Art Review
,”
Sensors
,
20
(
10
), p.
2778
.
21.
Ngeljaratan
,
L.
,
Moustafa
,
M. A.
, and
Pekcan
,
G.
,
2021
, “
A Compressive Sensing Method for Processing and Improving Vision-Based Target-Tracking Signals for Structural Health Monitoring
,”
Comput.-Aided Civ. Infrastruct. Eng.
,
36
(
9
), pp.
1203
1223
.
22.
Long
,
J.
, and
Büyüköztürk
,
O.
,
2020
, “
Collaborative Duty Cycling Strategies in Energy Harvesting Sensor Networks
,”
Comput.-Aided Civ. Infrastruct. Eng.
,
35
(
6
), p.
534
548
.
23.
Sajedi
,
S.
, and
Liang
,
X.
,
2021
, “
Dual Bayesian Inference for Risk-Informed Vibration-Based Damage Diagnosis
,”
Comput.-Aided Civ. Infrastruct. Eng.
,
36
(
9
), pp.
1168
1184
.
24.
Sajedi
,
S. O.
, and
Liang
,
X.
,
2021
, “
Uncertainty-Assisted Deep Vision Structural Health Monitoring
,”
Comput.-Aided Civ. Infrastruct. Eng.
,
36
(
2
), pp.
126
142
.
25.
Huang
,
C. S.
,
Le
,
Q. T.
,
Su
,
W. C.
, and
Chen
,
C. H.
,
2020
, “
Wavelet-Based Approach of Time Series Model for Modal Identification of a Bridge With Incomplete Input
,”
Comput.-Aided Civ. Infrastruct. Eng.
,
35
(
9
), pp.
947
964
.
26.
Gil-Gala
,
F. J.
,
Mencía
,
C.
,
Sierra
,
M. R.
, and
Varela
,
R.
,
2021
, “
Learning Ensembles of Priority Rules for Online Scheduling by Hybrid Evolutionary Algorithms
,”
Integr. Comput. -Aided Eng.
,
28
(
1
), pp.
65
80
.
27.
Sørensen
,
R. A.
,
Nielsen
,
M.
, and
Karstoft
,
H.
,
2020
, “
Routing in Congested Baggage Handling Systems Using Deep Reinforcement Learning
,”
Integr. Comput.-Aided Eng.
,
27
(
2
), pp.
139
152
.
28.
Ni
,
F. T.
,
Zhang
,
J.
, and
Noori
,
M. N.
,
2020
, “
Deep Learning for Data Anomaly Detection and Data Compression of a Long-Span Suspension Bridge
,”
Comput.-Aided Civ. Infrastruct. Eng.
,
35
(
7
), pp.
685
700
.
29.
Ren
,
Q.
,
Li
,
M.
,
Li
,
H.
,
Song
,
L.
,
Si
,
W.
, and
Liu
,
H.
,
2021
, “
A Robust Prediction Model for Displacement of Concrete Dams Subjected to Irregular Water-Level Fluctuations
,”
Comput.-Aided Civ. Infrastruct. Eng.
,
36
(
5
), pp.
577
601
.
30.
Ghofrani
,
F.
,
Pathak
,
A.
,
Mohammadi
,
R.
,
Aref
,
A.
, and
He
,
Q.
,
2020
, “
Predicting Rail Defect Frequency: An Integrated Approach Using Fatigue Modeling and Data Analytics
,”
Comput.-Aided Civ. Infrastruct. Eng.
,
35
(
2
), pp.
101
115
.
31.
Tong
,
Z.
,
Yuan
,
D.
,
Gao
,
J.
, and
Wang
,
Z.
,
2020
, “
Pavement Defect Detection With Fully Convolutional Network and an Uncertainty Framework
,”
Comput.-Aided Civ. Infrastruct. Eng.
,
35
(
8
), pp.
832
849
.
32.
Kalenjuk
,
S.
,
Lienhart
,
W.
, and
Rebhan
,
M. J.
,
2021
, “
Processing of Mobile Laser Scanning Data for Large-Scale Deformation Monitoring of Anchored Retaining Structures Along Highways
,”
Comput.-Aided Civ. Infrastruct. Eng.
,
36
(
6
), pp.
678
694
.
33.
Zhu
,
M.
,
Zhu
,
H.
,
Guo
,
F.
,
Chen
,
X.
, and
Ju
,
J. W.
,
2021
, “
Tunnel Condition Assessment via Cloud Model-Based Random Forests and Self-Training Approach
,”
Comput.-Aided Civ. Infrastruct. Eng.
,
36
(
2
), pp.
164
179
.
34.
Amezquita-Sanchez
,
J. P.
,
Park
,
H. S.
, and
Adeli
,
H.
,
2017
, “
A Novel Methodology for Modal Parameters Identification of Large Smart Structures Using MUSIC, Empirical Wavelet Transform, and Hilbert Transform
,”
Eng. Struct.
,
147
, pp.
148
159
.
35.
Pavlou
,
D.
,
2022
, “
A Deterministic Algorithm for Nonlinear, Fatigue-Based Structural Health Monitoring
,”
Comput.-Aided Civ. Infrastruct. Eng.
,
37
(
7
), pp.
809
831
.
36.
Bjørheim
,
F.
,
Siriwardane
,
S. C.
, and
Pavlou
,
D.
,
2022
, “
A Review of Fatigue Damage Detection and Measurement Techniques
,”
Int. J. Fatigue
,
154
, p.
106556
.
37.
Bjørheim
,
F.
,
Pavlou
,
D. G.
, and
Siriwardane
,
S. C.
,
2022
, “
Nonlinear Fatigue Life Prediction Model Based on the Theory of the S-N Fatigue Damage Envelope
,”
Fatigue Fract. Eng. Mater. Struct.
,
45
(
5
), pp.
1480
1493
.
38.
Luan
,
M.
,
Qu
,
P.
,
Jeng
,
D. S.
,
Guo
,
Y.
, and
Yang
,
Q.
,
2008
, “
Dynamic Response of a Porous Seabed–Pipeline Interaction Under Wave Loading: Soil–Pipeline Contact Effects and Inertial Effects
,”
Comput. Geotech.
,
35
(
2
), pp.
173
186
.
39.
Li
,
X.-J.
,
Gao
,
F.-P.
,
Yang
,
B.
, and
Zang
,
J.
,
2011
, “
Wave-Induced Pore Pressure Responses and Soil Liquefaction Around Pile Foundation
,”
Int. J. Offshore Polar Eng.
,
21
(
3
), pp.
233
239
.
40.
Chang
,
K. T.
, and
Jeng
,
D. S.
,
2014
, “
Numerical Study for Wave-Induced Seabed Response Around Offshore Wind Turbine Foundation in Donghai Offshore Wind Farm, Shanghai, China
,”
Ocean Eng.
,
85
, pp.
32
43
.
41.
Chen
,
L. F.
,
Zang
,
J.
,
Hillis
,
A. J.
,
Morgan
,
G. C. J.
, and
Plummer
,
A. R.
,
2014
, “
Numerical Investigation of Wave-Structure Interaction Using OpenFOAM
,”
Ocean Eng.
,
88
, pp.
91
109
.
42.
Sui
,
T.
,
Zhang
,
C.
,
Guo
,
Y.
,
Zheng
,
J.
,
Jeng
,
D.
,
Zhang
,
J.
, and
Zhang
,
W.
,
2015
, “
Three-Dimensional Numerical Model for Wave-Induced Seabed Response Around Mono-Pile
,”
hips Offshore Struct.
,
11
(
6
), pp.
667
678
.
43.
Zhang
,
C.
,
Zhang
,
Q.
,
Wu
,
Z.
,
Zhang
,
J.
,
Sui
,
T.
, and
Wen
,
Y.
,
2015
, “
Numerical Study on Effects of the Embedded Monopile Foundation on Local Wave-Induced Porous Seabed Response
,”
Math. Probl. Eng.
,
2015
, pp.
1
13
.
44.
Lin
,
Z.
,
Pokrajac
,
D.
,
Guo
,
Y.
,
Jeng
,
D.-s.
,
Tang
,
T.
,
Rey
,
N.
,
Zheng
,
J.
, and
Zhang
,
J.
,
2017
, “
Investigation of Nonlinear Wave-Induced Seabed Response Around Mono-Pile Foundation
,”
Coastal Eng.
,
121
, pp.
197
211
.
45.
Bazeos
,
N.
,
Hatzigeorgiou
,
G. D.
,
Hondros
,
I. D.
,
Karamaneas
,
H.
,
Karabalis
,
D. L.
, and
Beskos
,
D. E.
,
2002
, “
Static, Seismic and Stability Analyses of a Prototype Wind Turbine Steel Tower
,”
Eng. Struct.
,
24
(
8
), pp.
1015
1025
.
46.
Murtagh
,
P. J.
,
Basu
,
B.
, and
Broderick
,
B. M.
,
2005
, “
Along-Wind Response of a Wind Turbine Tower With Blade Coupling Subjected to Rotationally Sampled Wind Loading
,”
Eng. Struct.
,
27
(
8
), pp.
1209
1219
.
47.
Lavassas
,
I.
,
Nikolaidis
,
G.
,
Zervas
,
P.
,
Efthimiou
,
E.
,
Doudoumis
,
I. N.
, and
Baniotopoulos
,
C. C.
,
2003
, “
Analysis and Design of the Prototype of a Steel 1-MW Wind Turbine Tower
,”
Eng. Struct.
,
25
(
8
), pp.
1097
1106
.
48.
Alkhoury
,
P.
,
Soubra
,
A.-H.
,
Rey
,
V.
, and
Aït-Ahmed
,
M.
,
2022
, “
Dynamic Analysis of a Monopile-Supported Offshore Wind Turbine Considering the Soil-Foundation-Structure Interaction
,”
Soil Dyn. Earthquake Eng.
,
158
, p.
107281
.
49.
Graff
,
K. F.
,
2012
,
Wave Motion in Elastic Solids
,
Dover Publications
,
Mineola, New York.
50.
Meirovitch
,
L.
,
1967
,
Analytical Methods in Vibrations
,
Macmillan
,
New York
.
51.
Pavlou
,
D. G.
,
2021
, “
Soil–Structure–Wave Interaction of Gravity-Based Offshore Wind Turbines: An Analytical Model
,”
ASME J. Offshore Mech. Arct. Eng.
,
143
(
3
), p.
032101
.
52.
Wang
,
J.
,
Qin
,
D.
, and
Lim
,
T. C.
,
2010
, “
Dynamic Analysis of Horizontal Axis Wind Turbine by Thin-Walled Beam Theory
,”
J. Sound Vib.
,
329
(
17
), pp.
3565
3586
.
53.
Adeli
,
H.
, and
Karim
,
A.
,
1997
, “
Neural Network Model for Optimization of Cold-Formed Steel Beams
,”
J. Struct. Eng.
,
123
(
11
), pp.
1535
1543
.
54.
Karim
,
A.
, and
Adeli
,
H.
,
1999
, “
Global Optimum Design of Cold-Formed Steel Hat-Shape Beams
,”
Thin-Walled Struct.
,
35
(
4
), pp.
275
288
.
55.
Tashakori
,
A.
, and
Adeli
,
H.
,
2002
, “
Optimum Design of Cold-Formed Steel Space Structures Using Neural Dynamics Model
,”
J. Constr. Steel Res.
,
58
(
12
), pp.
1545
1566
.
56.
Arany
,
L.
,
Bhattacharya
,
S.
,
Adhikari
,
S.
,
Hogan
,
S. J.
, and
Macdonald
,
J. H. G.
,
2015
, “
An Analytical Model to Predict the Natural Frequency of Offshore Wind Turbines on Three-Spring Flexible Foundations Using Two Different Beam Models
,”
Soil Dyn. Earthquake Eng.
,
74
, pp.
40
45
.
57.
Arany
,
L.
,
Bhattacharya
,
S.
,
Macdonald
,
J. H. G.
, and
John Hogan
,
S.
,
2016
, “
Closed Form Solution of Eigen Frequency of Monopile Supported Offshore Wind Turbines in Deeper Waters Incorporating Stiffness of Substructure and SSI
,”
83
, pp.
18
32
.
58.
Amar Bouzid
,
D.
,
Bhattacharya
,
S.
, and
Otsmane
,
L.
,
2018
, “
Assessment of Natural Frequency of Installed Offshore Wind Turbines Using Nonlinear Finite Element Model Considering Soil-Monopile Interaction
,”
J. Rock Mech. Geotech. Eng.
,
10
(
2
), pp.
333
346
.
59.
Adeli
,
H.
,
Gere
,
J. M.
, and
Weaver
,
W.
,
1978
, “
Algorithms for Nonlinear Structural Dynamics
,”
ASCE J. Struct. Div.
,
104
(
2
), pp.
263
280
.
60.
Alkhoury
,
P.
,
Soubra
,
A. H.
,
Rey
,
V.
, and
Aït-Ahmed
,
M.
,
2021
, “
A Full Three-Dimensional Model for the Estimation of the Natural Frequencies of an Offshore Wind Turbine in Sand
,”
Wind Energy
,
24
(
7
), pp.
699
719
.
61.
Damgaard
,
M.
,
Ibsen
,
L. B.
,
Andersen
,
L. V.
, and
Andersen
,
J. K. F.
,
2013
, “
Cross-Wind Modal Properties of Offshore Wind Turbines Identified by Full Scale Testing
,”
J. Wind Eng. Ind. Aerodyn.
,
116
, pp.
94
108
.
62.
Prendergast
,
L. J.
,
Gavin
,
K.
, and
Doherty
,
P.
,
2015
, “
An Investigation Into the Effect of Scour on the Natural Frequency of an Offshore Wind Turbine
,”
Ocean Eng.
,
101
, pp.
1
11
.
63.
Prendergast
,
L. J.
,
Reale
,
C.
, and
Gavin
,
K.
,
2018
, “
Probabilistic Examination of the Change in Eigenfrequencies of an Offshore Wind Turbine Under Progressive Scour Incorporating Soil Spatial Variability
,”
Mar. Struct.
,
57
, pp.
87
104
.
64.
Dong
,
X.
,
Lian
,
J.
,
Wang
,
H.
,
Yu
,
T.
, and
Zhao
,
Y.
,
2018
, “
Structural Vibration Monitoring and Operational Modal Analysis of Offshore Wind Turbine Structure
,”
Ocean Eng.
,
150
, pp.
280
297
.
65.
Norén-Cosgriff
,
K.
, and
Kaynia
,
A. M.
,
2021
, “
Estimation of Natural Frequencies and Damping Using Dynamic Field Data From an Offshore Wind Turbine
,”
Mar. Struct.
,
76
, p.
102915
.
66.
Natarajan
,
A.
,
2014
, “
Influence of Second-Order Random Wave Kinematics on the Design Loads of Offshore Wind Turbine Support Structures
,”
Renewable Energy
,
68
, pp.
829
841
.
67.
Wang
,
Y.
,
2020
, “
Bottom Effects on the Tower Base Shear Forces and Bending Moments of a Shallow Water Offshore Wind Turbine
,”
Mar. Struct.
,
70
, p.
102705
.
68.
Wang
,
S.
,
Larsen
,
T. J.
, and
Bredmose
,
H.
,
2021
, “
Ultimate Load Analysis of a 10 MW Offshore Monopile Wind Turbine Incorporating Fully Nonlinear Irregular Wave Kinematics
,”
Mar. Struct.
,
76
, p.
102922
.
69.
Hirdaris
,
S. E.
,
Bai
,
W.
,
Dessi
,
D.
,
Ergin
,
A.
,
Gu
,
X.
,
Hermundstad
,
O. A.
,
Huijsmans
,
R.
,
Iijima
,
K.
,
Nielsen
,
U.D.
,
Parunov
,
J.
,
Fonseca
,
N.
,
Papanikolaou
,
A.
,
Argyriadis
,
K.
, and
Incecik
,
A.
,
2014
, “
Loads for Use in the Design of Ships and Offshore Structures
,”
Ocean Eng.
,
78
, pp.
131
174
.
70.
Bachynski
,
E.
,
Thys
,
M.
, and
Delhaye
,
V.
,
2019
, “
Dynamic Response of a Monopile Wind Turbine in Waves: Experimental Uncertainty Analysis for Validation of Numerical Tools
,”
Appl. Ocean Res.
,
89
, pp.
96
114
.
71.
Darvishi-Alamouti
,
S.
,
Bahaari
,
M. R.
, and
Moradi
,
M.
,
2017
, “
Natural Frequency of Offshore Wind Turbines on Rigid and Flexible Monopiles in Cohesionless Soils With Linear Stiffness Distribution
,”
Appl. Ocean Res.
,
68
, pp.
91
102
.
72.
Veritas
,
D. N.
,
2010
, “
Environmental Conditions and Environmental Loads,” DNV-RP-C205
.
73.
Morison
,
J. R.
,
Johnson
,
J. W.
, and
Schaaf
,
S. A.
,
1950
, “
The Force Exerted by Surface Waves on Piles
,”
J. Pet. Technol.
,
2
(
5
), pp.
149
154
.
74.
Bak
,
C.
,
Zahle
,
F.
,
Bitsche
,
R.
,
Kim
,
T.
,
Yde
,
A.
,
Henriksen
,
L. C.
,
Natarajan
,
A.
, and
Hansen
,
M.
,
2013
, "Description of theDTU 10 MW Reference Wind Turbine.", DTU Wind Energy Report-I-0092:1-138. https://dtu-10mw-rwt.vindenergi.dtu.dk.
75.
Zuo
,
H.
,
Bi
,
K.
, and
Hao
,
H.
,
2018
, “
Dynamic Analyses of Operating Offshore Wind Turbines Including Soil-Structure Interaction
,”
Eng. Struct.
,
157
, pp.
42
62
.
76.
Søren
,
P. H. S.
, and
Ibsen
,
L. B.
,
2013
, “
Assessment of Foundation Design for Offshore Monopiles Unprotected Against Scour
,”
Ocean Eng.
,
63
, pp.
17
25
.
77.
Wolfram Mathematica
,
2021
,
2021
, “
The World’s Definitive System for Modern Technical Computing
,” https://www.wolfram.com/mathematica
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