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Nonlinear analysis of a floating offshore wind turbine with internal resonances

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Abstract

In this work, a coupled 6-DOF model of the NREL 5-MW floating offshore wind turbine is proposed. The model takes into account gravitational, aerodynamic, hydrostatic, hydrodynamic and mooring loads. Wind and sea waves influence is studied through a nonlinear dynamic analysis for two types of platform configurations: OC3-Hywind spar buoy and MIT/NREL TLP. Possible internal resonances were detected by calculating the natural frequencies of the system, with a one-to-one resonance condition for the OC3-Hywind spar buoy, and a two-to-one resonance condition for the MIT/NREL TLP, for the heave, roll and pitch motions. The application of the method of multiple scales to the heave and pitch dynamics confirmed the existence of these resonances. The obtained frequency response curves were validated by the use of a direct numerical method, based on the finite difference method combined with a pseudo-arclength continuation technique. Cyclic-fold and Hopf bifurcations are detected when the MIT/NREL TLP platform is used. For certain excitation frequencies, Poincaré sections generated smooth densely filled curves indicating the presence of higher-order quasiperiodic motion for the heave and pitch displacements. Large jumps of the motion amplitude for the frequency and force responses indicate the existence of potential damage to the mooring system.

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Data Availability

The data that support the findings of this study are available from the corresponding author upon reasonable request.

Abbreviations

\({{\textbf{q}}= \left[ u\;v\;w\;\phi \;\theta \;\psi \right] ^{T}}\) :

Generalized coordinate vector of the FOWT

\({{\textbf{q}_s}= \left[ u_s\;v_s\;w_s\;\phi _s\;\theta _s\;\psi _s\right] ^T}\) :

Static equilibrium vector of the FOWT

\({\textbf{Q}_{G}= \left[ \textbf{F}_G \; \textbf{M}_G \right] ^T}\) :

Gravitational load, force (N) and moment (N m)

\({\textbf{Q}_{A}= \left[ \textbf{F}_A \; \textbf{M}_A \right] ^T}\) :

Aerodynamic load, force (N) and moment (N m)

\({\textbf{Q}_{B}= \left[ \textbf{F}_B \; \textbf{M}_B \right] ^T}\) :

Buoyancy load, force (N) and moment (N m)

\({\textbf{Q}_{M}= \left[ \textbf{F}_M \; \textbf{M}_M \right] ^T}\) :

Mooring load, force (N) and moment (N m)

\({\textbf{Q}_{H}= \left[ \textbf{F}_H \; \textbf{M}_H \right] ^T}\) :

Hydrodynamic load, force (N) and moment (N m)

\({\textbf{Q}_{D}= \left[ \textbf{F}_D \; \textbf{M}_D \right] ^T}\) :

Damping load, force (N) and moment (N m)

\(\textbf{V}_{p,t,n,r}\) :

Center of mass velocity (m/s) of the platform, tower, nacelle and rotor, respectively

\({\textbf{U}_{a}=\left[ U_a \; 0 \; 0\right] ^T}\) :

Wind velocity (m/s)

\({\textbf{U}= \left[ U_x \; 0 \; U_z\right] ^T}\) :

Water particle velocity (m/s)

\({\varvec{\Gamma }}\) :

Angular velocity of the FOWT

\(\textbf{K}_{G,A,B,M}\) :

Stiffness matrix due to Gravitational, Aerodynamic, Buoyancy, Mooring loads

\(\textbf{K}_{nl}({\textbf{q}})\) :

Nonlinear stiffness vector of the FOWT

\(\textbf{D}_A\) :

Aerodynamic damping matrix

\(\textbf{M}\) :

Mass matrix of the FOWT

\(\textbf{A}\) :

Added mass matrix

\(\textbf{B}\) :

Added damping matrix

CG\(\left( x_g,y_g,z_g\right) \) :

Center of gravity of the FOWT

CB\(\left( x_B,y_B,z_B\right) \) :

Center of buoyancy of the FOWT

P\(_{1,2,3,4}\) :

Mooring attachment points

\(K_T\) :

Longitudinal stiffness of a single tether (N m\(^{-1}\))

G\(_{p,t,n,r,b}\) :

Center of mass of the platform, tower, nacelle, rotor and blades, respectively

T :

Kinetic energy of the FOWT

\(V_s\) :

Submerged volume at static equilibrium

\(V_q\) :

Variable submerged volume of the FOWT

\(A_{wp}\) :

Water plane area

\(J_{x,y}\) :

First moments of area of the water plane cross section

\(J_{xx,xy,yy}\) :

Second moments of area of the water plane cross section

g :

Gravitational constant (m s\(^{-2} \))

\( \rho _a \) :

Air density (kg m\(^{-3} \))

\( \rho _w \) :

Water density (kg m\(^{-3} \))

\( m_{p,t,n,r,T} \) :

Mass of platform, tower, nacelle, rotor and FOWT (kg)

\( \textbf{I}_{p,t,n,r} \) :

Inertia matrix of platform, tower, nacelle and rotor (kg m\(^2\))

\( \Omega \) :

Circular wave frequency (rad s\(^{-1}\))

\( \omega \) :

Circular natural frequency (rad s\(^{-1}\))

H :

Wave height (m)

k :

Wave number

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Authors and Affiliations

  1. Applied Mechanics and Systems Research Laboratory (LR03ES06), Ecole Polytechnique de Tunisie, University of Carthage, B.P. 743, 2078, Al Marsa, Tunis, Tunisia

    M. Ghozlane

  2. Department of Mechanical Engineering, College of Engineering at Al Kharj, Prince Sattam bin Abdulaziz University, 11942, Al-Kharj, Saudi Arabia

    F. Najar

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All authors contributed to the study conception and design. Material preparation, data collection and analysis were performed by Mehdia GHOZLANE and Fehmi NAJAR. The first draft of the manuscript was written by Mehdia GHOZLANE and Fehmi NAJAR. All authors read and approved the final manuscript.

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Correspondence to F. Najar.

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Ghozlane, M., Najar, F. Nonlinear analysis of a floating offshore wind turbine with internal resonances. Nonlinear Dyn 112, 1729–1757 (2024). https://doi.org/10.1007/s11071-023-09120-3

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