Three-Dimensional Prestressed Tuned Mass Damper for Passive Vibration Control of Coupled Multiple DOFs Offshore Wind Turbine

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Introduction
Te ofshore wind turbine (OWT) has become more popular in recent years due to advantages of being a cheap energy source, less visual impact, low noisy problem, and without land requirement [1].Taller and slender large-megawatts OWTs have been constructed ofshore [2] to capture more energy.However, the steel OWT structure has a long vibration period and light damping which make it more vulnerable to signifcant vibrations caused by wind or waves [3].Tese vibrations will further lead to adverse impacts on the efciency of energy generation and the fatigue life [4] of the tower.Consequently, it is necessary to mitigate the structural vibrations to enable continuous operation of the turbine in the harsh ofshore environments.
Tere are three types of vibration control methods for tall buildings and large span bridges, and they can be separated into three categories, such as active, semiactive, and passive vibration absorption technologies [5].Passive vibration absorbers have been applied widely for vibration control of OWT because of its low cost and high stability [6].Amongst these devices, tuned mass damper (TMD) has a wide range of application due to its simple confguration [7].Two methods are usually employed to analyze the TMDcontrolled OWT system.Te OWT is simplifed as a single degree-of-freedom (DOF) system for modelling the vibration behavior of the controlled OWT [8].Tis has the advantage of obtaining the analytical design parameters of the TMD from the harmonic balance method (HBM) or frequency response function (FRF) [9].Verma et al. [10] utilized an efcient framework to optimally design the TMD control system.Te optimal designed TMD was found effective for all cases of wind and wave loading studies.A multiple DOFs OWT system was also applied to numerically analyze the vibration energy dissipation capability of the TMD [11].Tis multiple DOFs OWT system can be employed to model the vital aerodynamic and hydrodynamic forces on an actual engineering structure [12].Sun and Jahangiri [13] utilized a three-dimensional pendulum TMD to mitigate the dynamic responses of the tower top.Results showed that the vibrations at the tower top under aerodynamic load induced by wind in both the fore-aft (F-A) and side-side (S-S) directions are reduced.
Te abovementioned literature review also shows that the efciency of the pendulum TMD decreases as the mass ratio decreases due to the limitation of available space inside the wind turbine tower [14].Te traditional pendulum TMD also has the disadvantage that the frequency of the damper can only be tuned by changing the pendulum length if the mass ratio is fxed [15].Several new TMDs have been proposed in recent years to remove this limitation.For instance, Chapain and Aly [16] presented a pendulum pounding the TMD with viscoelastic boundary for vibration suppression in wind turbines.Results indicated that this pendulum pounding TMD has the remarkable energy absorption competence over the conventional TMD since its robustness.Jahangiri et al. [12] proposed a threedimensional pounding pendulum TMD, which comprises of a pendulum mass damper along with a cylindrical pounding layer.Results showed that this device is more robust than the traditional pendulum TMD for vibration control of OWT.Subsequently, a novel prestressed tuned mass damper (PSTMD) was proposed by the authors with an additional cable beneath the mass for applying prestressing force in the cable of the traditional pendulum TMD [17].Te frequency of the PSTMD can then be tuned synchronously by changing the pendulum length and the tensile force of the prestressing cable.Numerical simulations indicated that vibration energy dissipation capability and the tuning efect of the PSTMD are greater than the corresponding conventional TMD when under harmonic load.
It should be noted that the PSTMD was considered as a single DOF system for obtaining its analytical design parameters on the basis of the virtual work principle [17], and only harmonic vibration in one direction is studied in the previous paper.However, the dynamic characteristics of an OWT at the F-A and S-S directions are clearly different, and the single DOF model of the wind turbine cannot prove the superiority of the PSTMD from the perspective of a theoretical analysis.Moreover, the blade, tower, and foundation should be taken as diferent DOFs because the aerodynamic force on the blade caused by wind and hydrodynamic force of foundation caused by wave action need to be considered separately to simulate better the actual environmental excitation.Tis paper developed a three-dimensional PS-TMD (3D-PSTMD) for vibration control of the OWT with the dynamic system modelled as coupling multiple DOFs of the blade, tower, and foundation and established the multiple source load model to simulate aerodynamic and hydrodynamic forces.More importantly, the superiorities of the PSTMD are proved from the perspective of a theoretical analysis in this paper when comparing with the traditional PTMD (see Section 3.5).
Te paper is organized as follows.Te dynamic model of OWT with the 3D-PSTMD is described in Section 2, and the equation of motion of the multiple DOFs OWT system with the 3D-PSTMD is derived in Section 3. Te aerodynamic and hydrodynamic forces are derived in Section 4 using the BEM theory and Morrison equation.Numerical simulation of a 5 MW OWT is studied with the simulated wind-wave loads and the 3D-PTMD and 3D-PSTMD as designed in Section 5. Te dynamic responses are computed by using the Wilson-θ method, and the energy dissipation competence of the 3D-PSTMD as compared to those from the traditional 3D-PTMD is assessed in Section 6. Te corresponding conclusions are presented in Section 7.

Dynamic Model of OWT Coupled by 3D-PSTMD
2.1.OWT with 3D-PSTMD.Te OWT coupled by a 3D-PSTMD structure under the wind-wave excitations is shown in Figure 1.Te displayed OWT consists of three blades, nacelle, tower, and foundation.Te 3D-PSTMD composed of three assemblies, i.e., a mass block for generating the opposite control force, the prestressed cables for tuning own frequency via tensile force and the suspension height, and three viscosity dampers for absorbing the corresponding oscillation energy from the OWT vibration.Since the OWT tower is a typical high and thin-wall structure and its structural model damping ratio is generally less than 1%, this structure is usually viewed as the classical lower damping system.Hence, the dynamic vibration absorber (DVA) is always used to promote the structural model damping and reduce the dynamic responses.In this respect, the 3D-PSTMD may be considered as an improved DVA and oscillation dissipater on the basis of the conventional PTMD.As illustrated in Figure 1, the mass block is vertically assembled in position using the tensile force cables, and bottom cable is connected to the fange near the tower top.Within the horizontal plane, this mass block is linked to the tower tube wall via arranged viscosity dampers, as shown in Figure 1.

Mathematical Model.
To simulate the corresponding aerodynamic and hydrodynamic forces, the blade rotation and soil interaction efects need to be considered.A fully coupled three-dimensional OWT is modelled as a 12 DOFs system, including 6 DOFs for the three turbine blades in the edgewise and fapwise directions, 2 DOFs for the tower at F-A and S-S directions, and 4 DOFs for the translation and rotation of foundation at F-A and S-S directions.Te 3D-PSTMD is modelled with 2 DOFs in the fore-aft and sideside directions.Te 14 DOFs for all components are denoted by symbols q 1 ∼q 14 , respectively, and they are listed in Table 1.

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Structural Control and Health Monitoring Four coordinate systems (CSs), including a global CS (Oxyz) and three local CSs (O b x b y b z b for the blade, O t x t y t z t for the tower, and O P x P y P z P for the 3D-PSTMD), as shown in Figures 2 and 3 are designed to express the absolute displacement of all assemblies.Te DOFs q 1 ∼q 3 denote the generalized displacements of three blade tips relative to the root of the blade in the edgewise direction.DOFs q 4 ∼q 6 are similar displacements but in the fapwise direction.DOFs q 7 and q 8 denote the generalized displacements of the tower at F-A and S-S directions, respectively.Te corresponding DOFs q 9 and q 10 represent the translational and rotational displacements of monopile foundation in the F-A direction, and DOFs q 11 and q 12 are the translational and rotational displacements of monopile foundation in the S-S direction, respectively.
Te Q wind (t) and Q wave (t) in Figure 2 denote the integral aerodynamic and hydrodynamic forces, respectively.Point O is the centroid of foundation, and x, y, and z are the F-A, S-S, and vertical coordinate axes of the OWT, respectively, with point O as origin.Point O b is the centroid of the turbine hub of the OWT, and x b , y b , and z b are the fapwise, edgewise, and vertical coordinates, respectively, with point O b as origin.Point O t denotes the centroid of the section at the toe of the tower, and x t , y t , and z t are the F-A, S-S, and vertical coordinate axes, respectively, with point O t as origin.Point O P is the centroid of the 3D-PSTMD, and x P , y P , and z P are the F-A, S-S and vertical coordinates for the 3D-PS-TMD, respectively.φ t (z t ) represents the shape function corresponding to the fundamental mode of the tower along z t .
In Figure 3, dr is the infnitesimal unit of blade length, and r represents the distance from dr to the root of the blade.φ be (r) and φ bf (r) denote the shape function corresponding to the fundamental mode of the blade along r in the in-plane and out-plane, respectively.u b (r, t) is the generalized displacement of the infnitesimal unit length of the blade

Component Flapwise or fore-aft direction
Edgewise or side-side direction Blade 1 q 4 q 1 2 q 5 q 2 3 q 6 q 3 Tower q 7 q 8 Foundation Translation q 9 q 11 Rotation q 10 q 12 3D-PSTMD q 13 q 14 Note: Te fore-aft direction of the tower or foundation is the same as the fapwise direction of the blade, and the side-side direction of the tower or foundation is the same as the edgewise direction of the blade.
relative to the root of the blade, and θ b represents the rotational angle of the blade.h L represents the height suspending the 3D-PSTMD device, and h F represents the uppermost segment height of the tower.

Generalized Absolute Displacement and Velocity.
If the blade rotation velocity is assumed as ω b , the azimuthal angle θ b of the j th blade can be expressed as shown in Figure 3 as According to the local coordinate transformation relationship in Figure 3, the absolute displacements of the tower top (nacelle) at F-A and S-S directions in the Oxyz system can be obtained as x nf � q 7 + q 9 + H tan q 10  ≈ q 7 + q 9 + Hq 10 , x ns � q 8 + q 11 + H tan q 12  ≈ q 8 + q 11 + Hq 12 , where x nf and x ns are the generalized absolute displacement of nacelle at the F-A and S-S directions, respectively.Symbol H denotes the height of nacelle above the sea bed.
Te generalized absolute velocities of nacelle in the Oxyz coordinate system at the F-A and S-S directions can be written as where v nf and v ns are the absolute generalized velocities of nacelle in F-A and S-S directions, respectively.As shown in Figure 3, the absolute generalized displacements of the infnitesimal unit dr of the blade in the Oxyz coordinate system can be obtained from equation (2) and the local coordinate transformation of the blade as x bej � x ns + r sin θ bj + q j φ bej cos θ bj , where x bej , x bfj , and x bzj represent the absolute displacements of the j th blade along the edgewise (in-plane), fapwise (outplane), and vertical directions, respectively.φ bej and φ bfj represent the shape function corresponding to the fundamental mode of the j th blade in the in-plane and outplane, respectively.Taking the frst derivative of equation ( 4), the absolute generalized velocity components of the infnitesimal unit dr of the j th blade in the Oxyz coordinate system are given as v bej � v ns + ω b r cos θ bj + _ q j φ bej cos θ bj − ω b q j φ bej sin θ bj , where v bej , v bfj , and v bzj are the absolute generalized velocity components along the edgewise (in-plane), fapwise (outplane), and vertical directions, respectively.Since the position of the mass block of the 3D-PSTMD is connected in the vertical direction as shown in Figure 3, the vertical motion is minimal and can be ignored.Consequently, the absolute generalized displacements of the 3D-PSTMD are formulated as where x Pf and x Ps are the absolute generalized displacements of the 3D-PSTMD along the F-A, S-S, and vertical directions, respectively.Consequently, the absolute generalized velocity components of the 3D-PSTMD along the F-A, S-S, and vertical directions can be, respectively, described as where v Pf and v Ps are the absolute generalized velocity components.

Equation of Motion and Dynamic Analysis
3.1.Lagrange Equation.Both the dashpot and servo-control (pitch controller or generator torque controller) technologies belong to the structural vibration control techniques of OWT [5,15].To not afect the tuning performance and adapting competence of dashpot for original environment loads, the servo-control system is ignored to establish the motion equation in this paper [11][12][13].Te Lagrange equation based on the Hamilton principle [18] is employed Structural Control and Health Monitoring to derive the structural dynamic equation of the OWT with and without the 3D-PSTMD device.Te dynamic equation may be formulated as follows: where T represents the entire kinetic energy of the uncontrolled OWT system or controlled OWT system by the 3D-PSTMD.Symbol V denotes the entire potential energy for the uncontrolled OWTsystem or controlled OWTsystem by the 3D-PSTMD, and t is the time instant.q i (t) represents the generalized displacement vector of each DOF, and _ q i (t) represents the generalized velocity vector.Q i (t) represents the generalized and nonconservative forces corresponding to i th DOF vector.Symbol _ ( ) denotes the frst derivative with regard to time.

Kinetic Energy.
Te resultant velocity of nacelle, v n , in the Oxyz coordinate system can be obtained according to equation (3) as Te resultant velocity v bj in the Oxyz coordinate system is obtained according to equation (5) as Owing to the structural motions of OWT are teeny and neglected at the vertical direction, the resultant velocity v t of an microunit dz of the tower in the Oxyz coordinate system can be expressed as At the same time, the resultant velocity v P of the 3D-PSTMD in the Oxyz coordinate system can be expressed according to equation (7) as Terefore, the entire kinetic energy of OWT coupled by the 3D-PSTMD system is formulated as where R represents the blade length.m b (r) denotes the distributed mass along with radial blade length, and M t (z) is the distributed mass along the vertical height of the tower.M P denotes the physical mass of the 3D-PSTMD.M n represents the physical mass integrated by the nacelle and hub, and M f represents the physical mass of monopile foundation.I f represents the moment of inertia for monopile foundation.

Potential Energy.
Considering the strain energy induced by the blade fexure, the centrifugal stifening, and the gravity efect, the potential energy of the three turbine blades is computed.It can be expressed [13] as 6

Structural Control and Health Monitoring
where V b is the potential energy of the three blades.k be and k bf represent the lateral stifness corresponding to the blade in the in-plane and out-plane, respectively.k gee and k gef represent the centrifugal stifness in the in-plane and outplane, respectively.k gre and k grf represent the blade stifness induced by the gravity efect in the in-plane and out-plane, respectively.Tey are formulated as where I be and I bf denote the inertial moments corresponding to the blade in the in-plane and out-plane, respectively, ω b denotes the angular rotational velocity of blades, and g denotes the acceleration of gravity.Symbols ( ) ′ and ( ) ″ represent the frst and second derivatives corresponding to radial length r of the blade.
Taking the static position of the 3D-PSTMD as reference, the corresponding potential energy V P of the 3D-PSTMD device caused by cable tension is described as where φ tP and φ tF denote the values of the tower shape function corresponding to the fundamental mode at the installation position of the 3D-PSTMD device and uppermost fange, respectively.h L is the pendulum length of the 3D-PSTMD.f P represents the tension of the prestressed cable.Hence, the entire potential energy V of the OWT system controlled by the 3D-PSTMD device is formulated as where k t denotes the lateral stifness corresponding to the tower in the F-A and S-S directions.k ft and k ft represent the translational and rotational stifness of monopile foundation at F-A and S-S directions, respectively.

Virtual Work Done by Damping Force.
Te damping force belongs to nonconservative force, and the corresponding virtual work done by these forces needs to be considered.When letting the static balance point of the 3D-PSTMD device as the reference position, the virtual work done by the damping force from the viscous damper in the 3D-PSTMD is obtained as Structural Control and Health Monitoring where W P is the work done and c Pf and c Ps represent the integral damping coefcients from the three viscosity dampers in the 3D-PSTMD.
Te total works done by the damping force in the OWT system coupled by the 3D-PSTMD device are gained as where c be and c bf are the damping coefcients of the tower in the edgewise and fapwise directions, respectively.I t is the inertial moment corresponding to the tower in the F-A and S-S directions.α denotes the material damping constant.c ft and c fr are, respectively, the translational and rotational damping coefcients for monopile foundation in the F-A and S-S directions.Te equation of motion can be written in the matrix form by substituting equations ( 13), (17), and ( 19) into equation (8) as where Q wind (t) and Q wave (t) represent the aerodynamic and hydrodynamic forces, respectively, which will be discussed later.€ q, _ q, and q represent, respectively, the column vectors corresponding to dynamic responses of the uncontrolled OWT system or controlled system by the 3D-PSTMD.
M represents the mass matrix of the uncontrolled OWT structure or OWT coupled by the 3D-PSTMD device system, and C and K represent the corresponding damping and stifness matrices, respectively.Teir detailed expressions are shown in Appendixes A and B, respectively.An inspection shows that only the mass, stifness, and damping matrices of the tower at F-A and S-S directions (DOFs q 7 and q 8 ) have changed after the inclusion of the 3D-PSTMD inside the tower.Tis is because that the local motion of the 3D-PSTMD is directly related to the motion of the nacelle and tower, as shown in equations ( 7) and ( 16).

Advantages of 3D-PSTMD.
According to equations (A7), (A10), (B3), and (B6), the tower stifness and damping coefcients of the OWT structure with and without the 3D-PSTMD device are written as where subscript U and C denote, respectively, the uncontrolled OWT structure and OWT coupled by the 3D-PSTMD system.Symbols ( ) ′ and ( ) ″ represent the frst and second derivatives with regard to tower height z.
A comparison of equation (21) shows that the 3D-PSTMD can provide an additional local stifness to the OWT tower, and this means that the 3D-PSTMD can reduce the static displacement of the tower.It is worth to note that this is diferent from the traditional pendulum TMD because the 3D-PSTMD has an extra prestressed tensile force from the prestressed cables that acts directly within the topmost segment of the tower only.Its efect on reducing the static displacement is very small, as the prestressed tensile force is very small relative to elastic restoration force from the lateral stifness of the tower.
Moreover, like the traditional pendulum TMD, the 3D-PSTMD can also provide additional damping to the OWT tower, as shown in equation (22).Te mode damping coefcient of OWT coupled by the 3D-PSTMD system will be larger than that of the traditional pendulum TMD.Tis will greatly mitigate the excessive vibration of the OWT tower.More importantly, the 3D-PSTMD frequency is synchronously tuned by changing the cable tension and the suspension height, as shown in equation (B8).shows that the suspension height of the 3D-PSTMD is shorter than that of the traditional pendulum TMD.Tis illustrates that the 3D-PSTMD is closer to the tower top relative to the traditional pendulum TMD, and this leads to a large shape function value φ t for the 3D-PSTMD.Te abovementioned features of the OWT structure coupled by the 3D-PSTMD device are responsible for the better vibration mitigation efects compared to an OWT with the traditional pendulum TMD.

Aerodynamic and Hydrodynamic Loads
In this study, the exponential wind profle is adopted to calculate the mean velocity as where z r is the reference height, and v r is the mean velocity at the reference height.α win is the exponent of the wind profle.Te turbulent wind velocity is calculated by using the Davenport spectral model [19], which can be described as where n denotes the frequency of turbulent wind in Hz. x represents an intermediate coefcient, which is described as where v 10 represents the average wind speed at the reference altitude 10 m.
To account for the spatial dependency S c of wind velocity at diferent points, the cross spectra between two points p 1 and p 2 are defned as where r is the distance between p 1 and p 2 , and Coh (r, n) is the coherence function.
According to the Davenport spectral mode, the spatial coherence function [20] is expressed as where z 1 and z 2 are the height at the reference points p 1 and p 2 , respectively.C z is the attenuation coefcient of the spectral mode, and it is usually taken as 10.
Te turbulent wind velocity can be obtained from equations (25), (27), and (28), as where H (ω win ) is the matrix norm, and ω win denotes the frequency of fuctuating wind in rad/s.θ win represents the random phase angle, which is averagely distributed from 0 to 2π.
4.1.2.Generalized Aerodynamic Force.According to the BEM method [21], an integral blade section may be dispersed into N elements as shown in Figure 5(a), where R is the radial length of the rotor and ω b is the angular rotation speed of the blade.Te BEM method assumes that no radial dependency exists along the blade span, and thus, the blade elements can be analyzed independently with the momentum theory.
In Figure 5(a), dr is an increment along the span length and c (r) is the chord length at the mid-depth section of the blade.An arbitrary blade element having local velocities and aerodynamic load is shown in Figure 5(b).Te relative wind velocity v rel corresponding to the blade element dr in Figure 5(b) is described as where a and a′ represent the induction factors corresponding to the axial and tangential velocities, which are obtained via iterations [22].Te fow angle ϕ can also be calculated on the basis of the BEM theory as According to Figure 5, the wind attack angle between relative wind velocity and chord line in Figure 5(b) can be expressed as where θ represents the integrated angle between the pitch and twist angles, which is obtained from the airfoil data of the blade.Te lift and drag coefcients C L and C D can be obtained with the wind attack angle in equation (32) from airfoil data of the blade.Meanwhile, the lift force p L perpendicular to and the drag force p D parallel to the relative velocity can be computed as where ρ represents the density of the blade, and c is the corresponding chord length.Te normal and the tangential coefcients C N and C T are defned as Structural Control and Health Monitoring Hence, the normal and tangential forces p N and p T can be calculated as Combining Figures 3 and 5(b) with equation (35) on the basis of the virtual work principle, the virtual work done δW wind by the normal and tangential forces p N and p T is obtained as where p Tj (r, t) and p Nj (r, t) represent the tangential and normal distribution wind load on the j th blade.Te generalized aerodynamic force Q wind (t) then is obtained on the basis of the principles of the virtual work [23], and it is expressed as where k represents the DOFs corresponding to diferent components of the OWT structure.
After combining equation (36) with equation (37), the aerodynamic force caused by wind load can be obtained as where the symbol j represents the j th blade.

Hydrodynamic Force.
In this study, the Morrison equation [24] is adopted to calculate the hydrodynamic force on circular cylindrical monopile of the OWT induced by wave action, and it can be expressed as where ρ is the sea water density, and its value is usually taken as 1025 kg/m 3 .d e is the monopile diameter of the OWT tower.C M and C D are the mass and drag coefcients, and their values are taken as 1.0 and 1.2, respectively, in the present study [12,13].€ u and _ u are, respectively, the horizontal acceleration and velocity of monopole induced by wave action.
To simulate the random wave time history, the Pierson-Moskowitz (P-M) spectrum [25] is adopted as where α wa and β represent two constants of the P-M wave spectrum, and the corresponding parameter values are, respectively, taken as 0.0081 and 0.74 in this paper.ω wa denotes the frequency of the random wave in rad/s, U 19.5 represents the average wind speed corresponding to the reference altitude 19.5 m.
Based on the P-M spectrum representation method, the wave elevation η (t), the wave velocity _ u, and the acceleration € u can be expressed as where d w is the water depth, and ε is a random phase angle uniformly distributed from 0 to 2π [12,13].
After combining equation (43) with equation (37), the generalized hydrodynamic forces caused by wave action can be obtained as where N z represents the number of discrete segments for monopile foundation, and the symbol △z represents the segment length in the tower wetted portion.It is also noted that the wind speed acting on each discrete blade element will change with the blade angle of rotation.Tus, considering the blade rotation speed in equation ( 30), the aerodynamic load on blades and tower corresponding to the simulated wind speed can then be obtained from equations ( 30)-(38), as shown in Figure 7. Te load amplitude induced by wind in the F-A direction and S-S direction is notably diferent.Tis may be because the distinguishing normal and tangential coefcients in equation ( 34) are used to generate the corresponding aerodynamic force.

Numerical Simulation
Te Morrison equation is employed to simulate the hydrodynamic force based on the P-M spectrum in Section 4.2, and the random phase angle of the wave is uniformly distributed from 0 to 2π [12,13].Ten, the distribution of wave velocity and wave acceleration in time and space can be obtained according to equations (40) and (41) using the MATLAB software.
Te wave height and hydrodynamic load on the foundation and tower corresponding to the simulated wave velocity and acceleration can then be obtained from equation (44), as shown in Figure 9.
It is noted in Figures 8 and 9 that the acceleration, velocity, height, and hydrodynamic load of the wave action will change with the physical distance between the point considered and the monopile foundation.Tis is because of the notable infuence of physical distance to the phase angle of stochastic wave loading when the harmonicsuperposition theory is used in the computation.Subsequently, the wave number and hydrodynamic load of tower-changing trends are plotted in Figure 10.In contrast, the wave number depends more on the circular frequency of the random wave than the water depth; this point can be also seen in equation (42).In addition, unlike the aerodynamic load amplitude, the hydrodynamic load amplitude of the tower at F-A and S-S directions induced by wave has only a small diference since the properties of the circular monopile foundation are the same in both directions.

Dynamic Model Validation.
To analyze the rationality of the developed dynamic model in Sections 3 and 4, a 5 MW OWTsimulation model is established by using OpenFAST to calculate the fundamental frequencies and dynamic responses of the blade and tower and compared them with the calculation results of the developed model from MATLAB.Te geometric and physical parameters of 5 MW OWT can be found in the studies in [28]; then, substituting these   Structural Control and Health Monitoring parameters into equations (A1) to (A12) of Appendix A, the generalized mass and stifness of the blade and tower for uncontrolled OWT and their fundamental frequencies are calculated as listed in Table 3.
As is shown in Table 3, the maximum frequency tolerance between OpenFAST and the developed model is only 2.244%, so this developed DOFs coupling model can refect the structural characteristic and dynamic behavior in the dynamic analysis.Subsequently, the aerodynamic and hydrodynamic loads simulated by OpenFAST are replaced as the load vectors Q wind (t) and Q wave (t) to calculate the displacement responses at the tower top, respectively, and compared these responses of uncontrolled OWT with the calculation results of OpenFAST, as shown in Figure 11.When the servo-dynamic system is ignored, it can be found that the bidirectional displacement responses of the tower top from MATLAB calculation agree well with the Open-FAST simulation at the vast majority of time domain, so this developed DOFs coupling model can be efectively used to examine the vibration suppression competence of the 3D-PSTMD for OWT.

Dynamic Parameters of the OWT with 3D-PSTMD.
Te fundamental mode shapes of the blade in the in-plane and out-plane and the tower in the F-A or S-S directions are formulated [13] as where shape functions of three blades corresponding to the fundamental vibration mode are the same, and r � r/61.5 and h � h/87.6 represent the normalized radial length of the blade and vertical height of the tower, respectively.
Te soil and hydrodynamic efect is also modelled by linear springs and dashpots, as illustrated in Figure 3. Te foundation translation stifness coefcient k ft is 1.03 × 10 10 N/m, and the foundation rotational stifness Structural Control and Health Monitoring coefcients at F-A and S-S directions are assumed the same with k fr equals 1.14 × 10 12 N/m at F-A and S-S directions [29].Te foundation translation and rotational damping ratio are assumed the same with c ft equals 0.6% to denote the clay soil condition.

Devices' Parameters.
A traditional 3D pendulum TMD (3D-PTMD) is also studied for a comparison.Te optimal frequency ratio and damping ratio of the 3D-PTMD can be obtained [13,30] as where L represents the physical pendulum length of the 3D-PTMD device, and f t denotes the fundamental frequency of the OWT tower.α T is the mass ratio between the  Te cable tension of the 3D-PSTMD device can be described [17] as where M P and M O denote the generalized mass of the 3D-PSTMD and OWT structure, respectively.k t represents the lateral stifness of the tower structure for the uncontrolled OWT, and its expression is listed in equation (A12) in Appendix A.
A single DOF (degree-of-freedom) wind turbine tower with a concentrated mass (representing the blades) is established, and the parameters' design method of the unidirectional PSTMD is derived in the studies in [17].Tis paper developed a three-dimensional PS-TMD (3D-PSTMD) for vibration control of the OWT with the dynamic system modelled as coupling multiple DOFs of the blade, tower, and foundation.Te tower member models from these two methods are same, and the PSTMD designs only depend on the dynamic parameters of the tower.Tis point can be also found in equations (B1) and (B7) of Appendixes A and B; only the DOFs q 7 and q 8 (representing the tower) are coupled with the PSTMD.Hence, to achieve the best tuning condition of the designed 3D-PSTMD, the optimal frequency ratio and damping ratio of the 3D-PSTMD can be taken as follows [17]: where μ P and ζ P represent the optimal frequency ratio and damping ratio of the 3D-PSTMD device, respectively.Symbols β P and c P represent the tuning coefcients of the 3D-PSTMD device, which is described as  Structural Control and Health Monitoring Since the mass ratio of the TMD device is generally 1%-3% of the primary structure [31], the mass ratios of 3D-PTMD and PS-TMD are set as 1% in this study.According to the OWT parameters mentioned above and equation (46), the pendulum length L of the 3D-PTMD is calculated as 7.2 m.Te physical suspension height h L of the 3D-PSTMD device is taken as 5 m for a better performance of the 3D-PSTMD.Te design parameters of the 3D-PTMD and 3D-PSTMD with the same mass and frequency ratio can then be computed and are shown in Table 4.
In terms of the abovementioned design parameters, the frequency-response curves including uncontrolled OWT tower, OWT tower with 3D-PTMD, and 3D-PSTMD are plotted in Figure 12.Since these two dashpots are designed by using the optimal parameter design method from theoretical derivation [13,17], their dynamic magnifcation factors (DMFs) between left and right fxed points are same.Meanwhile, the DMF of OWT tower with the 3D-PSTMD is smaller than the traditional 3D-PTMD; this is because the PSTMD well suppresses the dynamic displacement responses, and this point can be also found in Section 6.2.

Vibration Control Effect
6.1.Evaluation Indices R 1 and R 2 .Two metrics are used to study quantitatively the vibration control performance, the maximal value index R 1 , and the root mean square (RMS) index R 2 , which may be described as follows: where the symbol max represents the maximal value of the dynamic response, and the rms represents the corresponding RMS values.Subscripts C and U represent the dynamic responses with and without the 3D-PTMD or 3D-PSTMD, respectively.

Dynamic Time History Analysis.
Te dynamic responses are computed by using the Wilson-θ method in this paper.It is a numerical method with unconditional convergence, and the intermediate parameter θ is usually taken as 1.4 to ensure the unconditional convergence.With the OWT parameters obtained in Section 5.1, the corresponding matrices of equation ( 20) are calculated from Appendices A and B. Similar matrices of OWT coupled by the 3D-PTMD system are obtained from the studies in [23].Considering the simulated aerodynamic and hydrodynamic loads in Section 5.2, the acceleration, velocity, and displacement responses at the top of the tower of the uncontrolled OWT, OWT controlled by 3D-PTMD and 3D-PSTMD are calculated, and they are plotted in Figures 13 and 14.
All the responses at the tower top of the OWT with the 3D-PTMD or 3D-PSTMD are obviously mitigated compared with the uncontrolled OWT structure when under the aerodynamic or hydrodynamic loads.

Performance Comparison.
Te efectiveness of vibration control performances of the 3D-PTMD and 3D-PSTMD is computed by using equation (50) when under the aerodynamic or hydrodynamic loads, as shown in Tables 5 and 6.
When under the aerodynamic force caused by wind loading, the 3D-PSTMD device outperforms the 3D-PTMD in reducing the peak acceleration response (F-A and S-S directions) with an improvement of approximately 9.7% and 11.0%, respectively.Te index on the standard deviation of the acceleration response (F-A and S-S directions) can, respectively, be improved by approximately 7.0% and 6.8% by the 3D-PSTMD as compared to the 3D-PTMD.Te proposed 3D-PSTMD can also improve the efectiveness of vibration mitigation on the peak-velocity response by 12.9% and 7.3% in fore-aft and side-side direction, respectively, Structural Control and Health Monitoring relative to the 3D-PTMD.Te index on the standard deviation of the velocity response is also increased by 4.1% and 5.8% relative to the 3D-PTMD in the fore-aft and side-side direction, respectively.Te improvement in the efectiveness of vibration control of the peak-displacement response is 5.9% and 4.9% in the F-A and S-S directions, respectively, compared to the 3D-PTMD device.Te index on the standard deviation of the displacement response is also increased by 4.4% and 6.5% in the fore-aft and side-side directions, respectively, compared to the 3D-PTMD.
When under the hydrodynamic force caused via the wave action, the 3D-PSTMD device outperforms the 3D-PTMD in reducing the peak acceleration response (F-A and S-S directions) with an improvement of approximately 6.1% and 13.2%, respectively.Te index on the standard deviation of the acceleration response (F-A and S-S directions) can also be improved by around 7.9% and 8.5%, respectively, when compared to the 3D-PTMD.Te proposed 3D-PSTMD can also promote the efectiveness of vibration control on the peak velocity response by 8.4% and 11.2% relative to the 3D-PTMD in F-A and S-S directions, respectively.Te index on the standard deviation of the velocity response can also be increased by 8.5% and 8.8% in the F-A and S-S directions, respectively, relative to the 3D-PTMD device.Furthermore, the 3D-PSTMD can also enhance the vibration mitigation efectiveness of the peak displacement response by 8.6% and 12.3% in F-A and S-S directions, respectively, compared to the 3D-PTMD device.Te index on the standard deviation of the displacement responses can also be enhanced by 8.4% and 8.9% in F-A and S-S directions, respectively, compared to the 3D-PTMD device.

Conclusions
Te one-dimensional PSTMD developed by the authors for vibration mitigation of the ofshore wind turbine tower (OWT) is extended to a 3D-PSTMD for vibration mitigation of a more realistic OWTsystem modelled as a multiple DOFs system with full coupling of the turbine blades, tower, and foundation.Tis enables separate consideration of the aerodynamic and hydrodynamic forces in the study of vibration mitigation at F-A and S-S directions.Te dynamic model of multiple DOFs OWT system with the 3D-PSTMD is derived with the Lagrangian equation to explain the advantages of the 3D-PSTMD compared to the traditional tuned mass damper.Te aerodynamic and hydrodynamic forces are generated via the BEM theory and Morrison equation, and the dynamic responses are computed by using the Wilson-θ method.Te vibration mitigation capability of the 3D-PSTMD is evaluated in comparison to those from traditional three-dimensional pendulum TMD (3D-PTMD).Te following major conclusions can be obtained as follows: (1) Unlike the traditional 3D-PTMD, the 3D-PSTMD can provide additional lateral stifness to the OWT which can slightly diminish the structural static displacement of OWT.(2) Since the own frequency of the 3D-PSTMD device is synchronously tuned by changing cable tension and suspension height, the 3D-PSTMD system can be tuned with a larger modal damping coefcient than that from the traditional 3D-PTMD.
(3) Under the aerodynamic force caused by wind loading, the OWT with 3D-PSTMD performs better than the OWT with 3D-PTMD in the mitigation of the peak dynamic responses with over 5.9% and 4.9% improvements in the F-A and S-S directions, respectively.Te standard deviation of dynamic responses (F-A and S-S directions) can also be improved by more than 4.1% and 5.8%, respectively, by the 3D-PSTMD when compared to those from the 3D-PTMD.(4) Under the hydrodynamic force caused by the sea wave loading, the 3D-PSTMD outperforms the 3D-PTMD in reducing the peak dynamic responses (F-A and S-S directions) with an improvement of over 6.1% and 11.2%, respectively, compared to those from 3D-PTMD.Te standard deviation of dynamic responses can also be improved by 7.9% and 8.5% in where m ij is the infuence coefcient of the mass matrix, which can be, respectively, described as Structural Control and Health Monitoring Te damping matrix of the uncontrolled OWT system can be expressed as where c ij is the infuence coefcient of the damping matrix, which are, respectively, described as Te stifness matrix of the uncontrolled OWT system can be expressed as where k ij is the infuence coefcient of the stifness matrix, which is, respectively, described as where k t denotes the structural stifness of the tower, and it is, respectively, obtained based on the structural generalized displacement theory [17] as

B.
Te OWT coupled by the 3D-PSTMD system is a coupled 14 DOFs dynamic system, and its mass matrix in terms of equation ( 20 where the infuence coefcients of the mass matrix for OWT coupled by the 3D-PSTMD system are formulated as equations (B2) and (B3), and the other infuence coefcients of the mass matrix are consistent with equation (A1).
where the infuence coefcients of the damping matrix for OWT coupled by the 3D-PSTMD can be formulated as equations (B5) and (B6), and the other infuence coefcients of the damping matrix are consistent with equation (A6).
Figure 4 also 8 Structural Control and Health Monitoring

5. 2 .
Load Design.According to the Davenport spectral model in Section 4.1.1 and the OWT parameters in Section 5.1, the aerodynamic force is simulated by the BEM method.Te mean wind speed v(t) is taken as 12 m/s at the height of the turbine hub, and the phase angle of random wind averagely distributed from 0 to 2π.Te wind velocity of the blade and wind design spectrum are plotted in Figures6(a) and 6(b) from equations (23)-(29).

Figure 6 (
Figure 6(b)  shows that the wind design spectrum agrees very well with the Davenport spectrum indicating that the simulated wind record is capable of representing the wind velocity in the wind feld.It is also noted that the wind speed acting on each discrete blade element will change with the blade angle of rotation.Tus, considering the blade rotation speed in equation (30), the aerodynamic load on blades and tower corresponding to the simulated wind speed can then be obtained from equations (30)-(38), as shown in Figure7.Te load amplitude induced by wind in the F-A direction and S-S direction is notably diferent.Tis may be because the distinguishing normal and tangential coefcients in equation (34) are used to generate the corresponding aerodynamic force.Te Morrison equation is employed to simulate the hydrodynamic force based on the P-M spectrum in Section 4.2, and the random phase angle of the wave is uniformly distributed from 0 to 2π[12,13].Ten, the distribution of wave velocity and wave acceleration in time and space can be obtained according to equations (40) and (41) using the MATLAB software.Te wave height and hydrodynamic load on the foundation and tower corresponding to the simulated wave velocity and acceleration can then be obtained from equation (44), as shown in Figure9.

0 Figure
Figure (a) Wave number and (b) hydrodynamic load.

Figure 13 :
Figure 13: Dynamic responses of OWT tower under the aerodynamic load: (a) fore-aft direction and (b) side-side direction.

Figure 14 :
Figure 14: Dynamic responses of OWT tower under the hydrodynamic load: (a) fore-aft direction and (b) side-side direction.
is the wave number per meter distance, and parameters ω wa and κ represent the dispersion equation when the water depth z is given as According to equation (39) and the virtual work principle, the virtual work δW wave done by the hydrodynamic force corresponding to the virtual displacement δq is written as δW wave �  dQ wave φ t δq 7 + δq 8  + δq 9 + δq 11  + z δq 10 + δq 12   .
[26,27]monopile OWT from the National Renewable Energy Laboratory (NREL) is adopted in the present study.Te Young modulus of the material for the OWT structure is 2.06 × 10 5 MPa, and its Poisson ratio and density are 0.3 and 7850 kg/m 3 , respectively.Te mean diameter and thickness of the tower linearly changes from the tower base to the tower top, and the blades' length and weight are 61.5 m and 1.774 × 10 4 kg, respectively.Te main physical and geometrical parameters of this 5 MW OWT are shown in Table2, according to the study in[26,27].

Table 2 :
Main parameters of the NREL 5 MW OWT.

Table 3 :
Fundamental frequencies of blade and tower.
Structural Control and Health Monitoringwhere the infuence coefcients of the stifness matrix for OWT coupled by the 3D-PSTMD system are formulated as equations (B8) and (B9), and the other infuence coefcients of the damping matrix are consistent with equation (A9).k713� k 137 � k 814 � k 148 � − f P φ tP − φ tF  h F − h L .(B.9)