Frequency-Domain Optimization of Multi-TMD Systems Using Hierarchical PSO for Offshore Wind Turbine Vibration Suppression

Abstract

With the rapid advancement of offshore wind power, structural vibration induced by multi-source excitations in complex marine environments is a critical concern. This study developed a multi-degree-of-freedom (MDOF) dynamic model of an offshore wind turbine using a lumped mass approach, which was then reduced to a first-order linear system to improve frequency-domain analysis and optimization efficiency. Given the non-stationary, broadband nature of wind and wave loads, a band-pass filtering technique was applied to extract dominant frequency components, enabling linear modeling of excitations within primary modal ranges. The displacement response spectrum, derived via system transfer functions, served as the objective function for optimizing tuned mass damper (TMD) parameters. Both single TMD and multiple TMD (MTMD) strategies were designed and compared. A hierarchical particle swarm optimization (H-PSO) algorithm was proposed for MTMD tuning, using the optimized single TMD as an initial guess to enhance convergence and stability in high-dimensional spaces. The results showed that the frequency-domain optimization framework achieved a balance between accuracy and computational efficiency, significantly reducing structural responses in dominant modes and demonstrating strong potential for practical engineering applications.

1. Introduction

With the continuous advancement of offshore wind power technology, the single-unit capacity and tower height of wind turbines have steadily increased [1], rendering their structures more susceptible to complex multi-source excitations from wind, waves, and currents in harsh marine environments [2]. Consequently, vibration issues in the tower and the entire turbine system have become increasingly pronounced. Frequent low-frequency vibrations not only exacerbate structural fatigue and damage [3] shortening service life, but also induce instability in power generation systems. This instability primarily arises from torque fluctuations in the drive train and variations in rotor speed caused by vibrations of the tower-nacelle structure, which can degrade power quality and increase the risk of protective shutdowns. Therefore, effective vibration suppression, operational stability enhancement, and service life extension have become critical research topics in the structural design and vibration control of offshore wind turbines [4].

Structural vibration control methods are typically categorized into passive, active, and semi-active control strategies [5,6]. Considering the severe operating conditions and maintenance challenges of offshore wind turbines, passive control—characterized by simple system architecture, high reliability, and cost-effectiveness—has become the predominant approach in engineering practice [7,8]. To achieve efficient vibration mitigation, it is essential first to establish a multi-degree-of-freedom (MDOF) dynamic model that balances accuracy with engineering applicability. Previous studies have developed various models: He et al. [9] constructed a flexible multibody dynamics model using SIMPACK to investigate TMD-based vibration control; Yang et al. [10] developed a 16-DOF aero-hydro-servo-structural coupled dynamic model to analyze offshore wind turbine vibrations comprehensively. Dai et al. [11] established multi-coordinate structural models incorporating wind, wave, gravitational, centrifugal, and inertial loads, and employed iterative methods to solve complex motion equations. Filho et al. [12] approximated the tower as an eight-DOF continuous beam model coupled with blade dynamics and soil-foundation interaction; Chen et al. [13] proposed an efficient aerodynamic model based on blade element momentum theory, validated via FAST simulations.

Tuned mass dampers (TMD), noted for their simple structure, clear working principle, and low cost, have been extensively applied in civil engineering structures such as tall buildings, bridges, and wind turbine towers [14,15,16,17,18,19]. For instance, Chen et al. [13] demonstrated that incorporating a TMD with a mass ratio of 1/100 in a 1.5 MW wind turbine nearly doubled the damping ratio of the tower’s first bending mode; Yang et al. [20] optimized TMD parameters via simplified models, significantly suppressing dynamic responses and stabilizing power output. He et al. [9] combined the Levenberg–Marquardt method with an improved artificial fish swarm algorithm (AFSA) to achieve global optimization of TMD parameters, verifying its effectiveness in vibration mitigation for floating wind turbines. Yang [21] introduced enhanced machine learning algorithms for both single- and multi-objective TMD optimization under seismic loading, outperforming traditional particle swarm optimization approaches. Deng et al. [22,23] proposed a composite structure with periodic additive acoustic black holes (ABHs), which achieves efficient low-frequency vibration suppression over a broad frequency band through local resonance modes and additional damping layers, while maintaining structural integrity.

Multiple tuned mass damper (MTMD) systems, capable of distributed multi-frequency vibration suppression [24,25,26], have attracted considerable attention. Iskandar et al. [27] compared series and parallel configurations of dual-mass TMD, showing superior performance over single-mass TMDs in short-period structures. The study by Shahraki et al. [28] combined the Park-Ang damage index with a hybrid particle swarm optimization algorithm to optimize the parameters of multiple tuned mass dampers (TMD) for seismic damage control in steel structures. Luo et al. [29] implemented three distributed TMD arranged in an equilateral triangle within a semisubmersible platform, employing a 9-DOF multibody dynamic model and H∞ optimization to significantly reduce pitch motion; Zhang et al. [30] developed a FAST v8-based framework coupling offshore wind turbines with multi-frequency tuned rotary inertia dampers (MRID-TMD), optimizing via intelligent algorithms to enhance robustness under wind-wave-earthquake loadings; Yang [10] optimized dual TMDs placed on platform and nacelle under mass and stroke constraints, achieving multi-frequency coordinated vibration reduction; Lara et al. [31] systematically evaluated multiple tuned inertia damper (TID) configurations for seismic response control in high-rise buildings using cultural algorithms, achieving up to 54% vibration reduction. Lu et al. [32]. further proposed a two-stage self-powered nonlinear low-frequency vibration isolation system, which significantly broadens the low-frequency isolation bandwidth and enhances isolation efficiency through a high-static-low-dynamic stiffness mechanism.

Despite these advances, current research exhibits notable limitations: (1) predominantly relies on simplified or lumped-parameter models, neglecting multi-DOF coupling effects and dynamic response variations under complex conditions, with insufficient validation against real turbine data, limiting engineering transferability; (2) TMD designs primarily focus on suppressing the dominant first-order bending mode of the tower, while control of the modal responses induced by typical wind–wave excitations remains relatively limited; (3) parameter optimization heavily depends on black-box metaheuristic algorithms like genetic and particle swarm optimization, which suffer from local optima entrapment, sensitivity to initial conditions, and lack hierarchical or structure-aware optimization frameworks.

Addressing these challenges, this study develops a multi-degree-of-freedom dynamic model for the integrated offshore wind turbine system based on the lumped mass method, incorporating tower, nacelle, and foundation coupling, validated with measured vibration data. The model is reduced to a first-order linear system via modal analysis to improve frequency-domain analysis and optimization efficiency. Considering the nonstationary and broadband nature of external excitations, a bandpass filter extracts dominant frequency components, enabling linearized modeling within the primary vibration mode range. A frequency-response-based objective function is formulated via system transfer functions to guide TMD parameter optimization. Both single and multiple TMD configurations are designed and compared, and a hierarchical optimization strategy is introduced to improve convergence robustness and efficiency in high-dimensional MTMD tuning.

Unlike existing hybrid PSO approaches—which typically enhance exploration by combining operators such as crossover, mutation, or local search—and adaptive PSO variants that dynamically adjust algorithmic coefficients without leveraging the physical structure of the system, the hierarchical PSO (H-PSO) proposed in this work adopts a physics-informed design philosophy. Specifically, the method decomposes the global MTMD optimization task into sequential, mode-targeted subproblems, uses the optimized single-TMD solution as a physically consistent warm start, and applies focused local refinements to account for modal coupling. This structure-aware framework reduces the effective search dimensionality, enhances convergence stability, and provides clearer modal-level interpretability of each damper’s tuning mechanism. The results demonstrate that the proposed approach achieves a favorable balance between modeling accuracy, computational efficiency, and practical applicability, showing strong potential for engineering implementation.

This study focuses on vibration control and optimization of offshore wind turbines, contributing to improved operational reliability and extended service life of wind energy systems [1]. These benefits align with the United Nations Sustainable Development Goals (SDGs), particularly SDG 7 “Affordable and Clean Energy” and SDG 13 “Climate Action” [33]. By suppressing tower and overall structural vibrations, the proposed framework enhances the stability of renewable power generation, thereby supporting the wider deployment of clean energy and climate-mitigation efforts.

2. Load Modeling and Mechanism Analysis of Wind Turbines

In the highly dynamic marine environment, offshore wind turbines are continuously exposed to multi-source external excitations such as wind and waves, posing significant challenges to their structural safety and operational stability. A schematic of the overall loading conditions is illustrated in Figure 1, with reference to CRRC Wind Power (Zhuzhou, China). Accurate modeling of the wind and wave loading mechanisms serves as the foundation for analyzing the dynamic response of the turbine, optimizing structural design, and developing effective vibration control strategies. Aerodynamic loads primarily arise from the nonlinear interaction between airflow and rotor blades, which requires modeling based on aerodynamic theory. In contrast, wave-induced loads exhibit strong randomness and broadband characteristics, and are typically modeled using linear or nonlinear wave theories combined with spectral analysis techniques. Therefore, developing a load modeling framework that captures the physical essence of wind–wave excitations while maintaining engineering applicability is essential for enhancing the fidelity of dynamic simulations and the effectiveness of control-oriented design.

2.1. Aerodynamic Loads

To accurately evaluate the aerodynamic loads exerted by the rotor on the tower and nacelle under varying wind conditions, the classical Blade Element Momentum (BEM) theory is adopted in this study for rotor modeling. This method combines the principles of momentum conservation with local aerodynamic analysis of blade sections and remains one of the most widely used and physically transparent approaches for steady-state aerodynamic modeling of wind turbines.

The rotor blades are discretized into multiple radial segments, with aerodynamic forces computed individually for each segment based on local flow conditions. These segmental forces are then integrated using global momentum theory to estimate key parameters such as thrust and torque. This method provides an effective basis for determining the aerodynamic loads applied to the structural system, and its effectiveness is further validated by quantitatively comparing the simulated tower-top and nacelle responses with reference field data under representative operating conditions.

The aerodynamic thrust generated by the rotor is calculated as Equation (1) [11]:

$$F_{fT}={\displaystyle \int d}{F}_{fT}=4\pi \rho V^2{\displaystyle {\int }_0^R{a}_1}(1-a_1)rdr$$

(1)

In this study, each blade is divided into 60 segments to calculate the aerodynamic loads, and the induction factor a₁ is solved using an iterative method.

To accurately evaluate the aerodynamic load distribution on the wind turbine tower, this study develops a tower aerodynamic load model that considers fluid–structure interaction (FSI) between the wind field and the tower structure. As shown in Figure 1, the windward side of the tower generates significant aerodynamic forces under wind excitation, primarily including drag and lift components. These forces are strongly influenced by wind velocity, projected area, attack angle, and geometric properties of the tower. An empirical drag coefficient model based on the Reynolds number is typically employed, combined with local wind speed and surface area, to compute aerodynamic forces in both the along-wind and cross-wind directions. The aerodynamic force acting on each tower segment can be described by Equation (2) [11].

$$F_{fTdi}=\left({\displaystyle \frac{1}{2}}\rho C_{fTdi}{A}_{Ti}{(v_{Ti}-\dot{q}(x_i,y_i,z_i,t))}^2\right)l_i$$

(2)

Here, $\rho$ denotes the air density, C_fTdi is the drag coefficient, A_Ti represents the projected area of the i-th tower element, v_Ti is the incident wind velocity at the element location, and $\dot{q}(x_i,y_i,z_i,t)$ denotes the structural velocity of the tower element. This approach provides essential aerodynamic load input for the dynamic response analysis of the tower structure and plays a critical role in evaluating overall structural stability under coupled wind–wave excitations

2.2. Hydrodynamic Loads

Hydrodynamic loads in the ocean primarily consist of wave loads and current loads, with wave loads exhibiting stochastic characteristics. In this study, wave loads are modeled using linear wave theory, where random waves are represented as a linear superposition of regular waves with varying amplitudes, frequencies, and phases [11]. The wave kinematics can be described by Equation (3) [11].

$$\begin{array}{l}\eta (x,y,t)={\displaystyle \sum _{n=1}^{\infty }{a}_n\cos [\frac{{\omega }_n^2}{g}(x\cos {\theta }_n+y\sin {\theta }_n)-\omega t+{\epsilon }_n]}\\ \begin{array}{l}{u}_x(x,y,z,t)={\displaystyle {\sum }_{n=1}^{\infty }{a}_n{\omega }_{\mathrm{n}}\frac{ch(k_{n}z)}{sh(k_n{H}_w)}\cos (\frac{{\omega }_n^2}{g}(x\cos {\theta }_n+y\sin {\theta }_n)-{\omega }_{n}t+{\epsilon }_n)\cos {\theta }_n}\hfill \\ u_y(x,y,z,t)={\displaystyle {\sum }_{n=1}^{\infty }{a}_n{\omega }_n\frac{ch(k_{n}z)}{sh(k_n{H}_w)}\cos (\frac{{\omega }_n^2}{g}(x\cos {\theta }_n+y\sin {\theta }_n)-{\omega }_{n}t+{\epsilon }_n)\sin {\theta }_n}\hfill \end{array}\\ \begin{array}{l}{a}_x(x,y,z,t)={\displaystyle {\sum }_{n=1}^{\infty }{a}_n{\omega }_n^2\frac{ch(k_{n}z)}{sh(k_n{H}_w)}\cos (\frac{{\omega }_n^2}{g}(x\cos {\theta }_n+y\sin {\theta }_n)-{\omega }_{n}t+{\epsilon }_n)\cos {\theta }_n}\hfill \\ a_y(x,y,z,t)={\displaystyle {\sum }_{n=1}^{\infty }{a}_n{\omega }_n^2\frac{ch(k_{n}z)}{sh(k_n{H}_w)}\cos (\frac{{\omega }_n^2}{g}(x\cos {\theta }_n+y\sin {\theta }_n)-{\omega }_{n}t+{\epsilon }_n)\sin {\theta }_n}\hfill \end{array}\end{array}$$

(3)

Here, a_n, ω_n, ε_n, θ_n and k_n denote the amplitude, angular frequency, random phase, incident angle relative to the reference direction, and wave number of the n-th regular wave, respectively. H_w represents the elevation of the ocean surface relative to the seabed. The random phase ε_n is uniformly distributed within the range 0 to 2π. For fully developed deep-water waves, the wave number and angular frequency satisfy the dispersion relation as given in Equation (4) [34].

$$w_n^2=g{k}_{n}tanh(k_n{H}_{sea})$$

(4)

The distribution of ocean wave energy can be described using the Pierson–Moskowitz (P–M) wave spectrum, as given in Equation (5) [34].

$$S(\omega )={\displaystyle \frac{0.78}{{\omega }^5}}exp\left(-{\displaystyle \frac{3.11}{{\omega }^4{H}_{1/3}^{1/2}}}\right)$$

(5)

For vertical cylindrical foundations commonly used in offshore wind turbines, the wave load is typically calculated using the Morison equation. This formulation decomposes the total force into an inertial component, and a drag component. The offshore wind turbine considered in this study features a cylindrical foundation with a diameter of approximately 4 m. Under the representative wave conditions with a wave height of 1.1 m, the resulting H/D ratio of approximately 0.275 falls within the commonly accepted range for the “small wave” assumption in engineering practice. Based on this assumption, the wave force per unit length acting at a given height on the tower or foundation can be expressed as shown in Equation (6) [34].

$$F_{wavebi}(t)={\displaystyle \frac{1}{2}}\rho C_{Dbi}{D}_{bi}\left|v_{rbi}(t)\right|v_{rbi}(t)+\rho C_{Mbi}{\displaystyle \frac{\pi D_{bi}^2}{4}}{\dot{v}}_{rbi}(t)$$

(6)

where C_Dbi, C_Mbi, D_bi, and v_rbi(t) denote the drag coefficient, inertia coefficient, equivalent diameter, and fluid particle velocity at the location of the i-th foundation element, respectively. ρ_s represents the density of seawater. The total hydrodynamic load on the submerged structure can be obtained by performing coordinate transformations and integrating the hydrodynamic forces acting on the individual structural elements immersed in water.

3. Mechanism Modeling of Offshore Wind Turbine- Multiple Tuned Mass Dampers

3.1. Coordinate System Definition for the Coupled System

To accurately characterize the dynamic behavior of each structural component of an offshore wind turbine and its coupling interaction with multiple Tuned Mass Dampers (MTMD), it is essential to establish a consistent system of global and local coordinate frames. In this study, a unified dynamic formulation is developed for the coupled tower–nacelle–TMD system.

In the modeling process, a global spatial coordinate system X_OY_OZ_O is first established for the wind turbine structure, where point O is located at the intersection of the turbine’s foundation centerline with the seabed. The X_O-axis points in the horizontal upwind direction, the Y_O-axis is perpendicular to the wind and represents the crosswind direction, and the Z_O-axis is vertically upward, opposite to the direction of gravity and aligned with the tower centerline. This global coordinate system is used to describe the overall pitch and roll responses of the tower and serves as the reference frame for defining all local coordinate systems.

As a flexible structure, the wind turbine tower exhibits dynamic responses primarily in the form of lateral vibrations in the fore–aft (X) and side–side (Y) directions, along with torsional motion around the Z-axis. In this study, the tower is modeled as a flexible beam with a continuously distributed mass and is discretized using the lumped mass method. It is divided into a series of equivalent mass elements, each possessing both translational and rotational degrees of freedom. This work focuses on the lateral vibration characteristics of the tower in the X and Y directions.

To facilitate modeling and computation, a local coordinate system X_TiY_TiZ_Ti is established at the center of each mass element. Under static conditions, X_Ti and Y_Ti are aligned with the global X_O and Y_O axes, while Z_Ti always points vertically upward along the tower’s central axis. During dynamic motion, the local coordinate system rotates and translates with the corresponding mass element. The tower’s overall motion can be expressed by projecting each degree of freedom onto the global coordinate system. This study specifically focuses on the lateral displacement and tilt angle associated with the first bending mode. The transformation between the local coordinate systems of adjacent mass elements is described by the transformation matrix shown in Equation (7) [11].

$$\begin{array}{l}{A}_{Ei(i+1)}=\left[\begin{array}{ccc}\cos ({\theta }_{Ezi(i+1)})& -\sin ({\theta }_{Ezi(i+1)})& 0\\ \sin ({\theta }_{Ezi(i+1)})& \cos ({\theta }_{Ezi(i+1)})& 0\\ 0& 0& 1\end{array}\right]\\ \hspace{1em}\hspace{1em}\hspace{0.33em}\times \left[\begin{array}{ccc}1& 0& 0\\ 0& \cos ({\theta }_{Exi(i+1)})& \sin ({\theta }_{Exi(i+1)})\\ 0& -\sin ({\theta }_{Exi(i+1)})& \cos ({\theta }_{Exi(i+1)})\end{array}\right]\\ \hspace{1em}\hspace{1em}\hspace{0.33em}\times \left[\begin{array}{ccc}\cos ({\theta }_{Eyi(i+1)})& 0& -\sin ({\theta }_{Eyi(i+1)})\\ 0& 1& 0\\ \sin ({\theta }_{Eyi(i+1)})& 0& \cos ({\theta }_{Eyi(i+1)})\end{array}\right]\end{array}$$

(7)

Here, θ_Ezi(i + 1), θ_Exi(i + 1), θ_Eyi(i + 1) represent the relative rotation angles of the (i + 1) th structure element with respect to the i-th element about the X_Ti, Y_Ti, and Z_Ti axes, respectively. Based on these angles, the global coordinate position of the tower segment can be expressed as shown in Equation (8) [34].

$$\left[\begin{array}{c}{x}_{TiO}\\ y_{TiO}\\ z_{TiO}\end{array}\right]={\displaystyle \sum _{n=1}^N\left\{\left({\displaystyle \prod _{j=0}^{J=n}{A}_{Tj(j+1)}}\right)\left[\begin{array}{c}0\\ 0\\ l_{Tj}\end{array}\right]\right\}}+\left[\begin{array}{c}{x}_{bnO}\\ y_{bnO}\\ z_{bnO}\end{array}\right]$$

(8)

Here, x_TiO, y_TiO and z_TiO denote the global coordinates of the i-th tower segment, and l_Tj represents the length of the J-th tower section. x_bnO, y_bnO and z_bnO indicate the global coordinates of the top of the foundation, which also corresponds to the base of the tower.

In this study, the substructure of the offshore wind turbine is also modeled as a flexible body, using the same dynamic modeling approach as that of the tower. The foundation is discretized into a series of equivalent mass elements, each possessing translational and rotational degrees of freedom. The focus is placed on analyzing the lateral vibration behavior in the X and Y directions. A local coordinate system X_biY_biZ_bi is established at the center of each mass element. In the undeformed state, X_bi and Y_bi align with the global X_O and Y_O directions, respectively, while Z_bi always points upward along the tower axis. When the structure vibrates, the local coordinate system rotates and translates along with the corresponding foundation element. The coordinate transformation between adjacent foundation elements is described by Equation (7), and the global position of each foundation element is defined by Equation (9) [34].

$$\left[\begin{array}{c}{x}_{biO}\\ y_{biO}\\ z_{biO}\end{array}\right]={\displaystyle \sum _{n=1}^N\left\{\left({\displaystyle \prod _{j=0}^{J=n}{A}_{bj(j+1)}}\right)\left[\begin{array}{c}0\\ 0\\ l_{bj}\end{array}\right]\right\}}$$

(9)

Here, x_biO, y_biO, and z_biO denote the global coordinates of the i-th foundation element, and l_bj represents the length of the J-th foundation element.

The nacelle, mounted at the top of the tower, is modeled as a rigid body with concentrated mass and rotational inertia. Its dynamic response includes translational displacements along the X_O and Y_O directions, yaw rotation about the Z_O axis, and vibration coupled with the motion of the tower top. In this study, the nacelle is represented as a lumped mass located at the tower top, and a local coordinate system X_cY_cZ_c is established at its center of mass, aligned with the global coordinate axes, as illustrated in Figure 2. The dynamic behavior of the nacelle is directly coupled with the motion of the tower’s top node, and its inertial effects and dynamic boundary conditions are incorporated into the system model.

Tuned Mass Dampers (TMDs) are commonly installed at the tops of high-rise buildings to suppress structural vibrations in horizontal directions. From a practical engineering perspective, the implementation of an MTMD system in offshore wind turbines is feasible. The tower top generally provides sufficient space to accommodate compact damping devices, which can also be installed inside the nacelle or within the tower itself to achieve effective coupling. To ensure high-efficiency coupling with the tower’s primary vibration modes, this study similarly selects the tower top as the installation location for the MTMD system [35]. Each TMD is modeled as a single-degree-of-freedom mass-spring-damper system, with its relative displacement defined in the local coordinate system of the corresponding node. To maintain modeling consistency, the vibration control direction of each TMD is aligned with the predominant vibration direction at its installation location—typically along the fore–aft (X_O) axis. A local coordinate system, X_t1Y_t1Z_t1, is established at the center of mass of each TMD and is initially aligned with the global coordinate axes. Under dynamic excitation, this local coordinate system translates and rotates together with the TMD. All TMD displacements are subsequently transformed into the global coordinate system to ensure consistent coupling with the structural dynamic response of the wind turbine system.

3.2. Multibody Dynamics Modeling of Offshore Wind Turbine–Tuned Mass Dampers Coupled System

Offshore wind turbines represent complex multi-body flexible systems. To validate the accuracy of the proposed dynamic model against the actual operational behavior of wind turbines, this section aims to establish a comprehensive system dynamics framework that incorporates both the turbine structure and its coupling with tuned mass dampers (TMD).

3.2.1. Discretized Tower Dynamics

The wind turbine tower is modeled as a flexible beam structure and discretized along its height into a multi-body system consisting of n = 20 rigid elements. These elements are interconnected through equivalent elastic hinges, as illustrated in Figure 3a. Each rigid segment possesses an individual lumped mass mi and rotational inertia I_i, with translational and rotational degrees of freedom defined in their respective local coordinate systems. Equivalent stiffness and damping are applied at the hinge connections to characterize the tower’s bending flexibility in both the fore-aft (X-axis) and side-to-side (Y-axis) directions.

As wind flows over the wind turbine, the external aerodynamic loads acting on each tower segment, along with the structural loads transmitted from other segments, are described by Equation (10). In addition, eccentric bending moments and overturning moments arising from the relative positioning and mass distribution of different segments are calculated using Equation (11) [36].

$$\left\{\begin{array}{l}{F}_{Txi}=F_{Tfxi}+F_{Tgxi}+F_{TOxi}\hfill \\ F_{Tyi}=F_{Tfyi}+F_{Tgyi}+F_{TOyi}\hfill \\ F_{Tzi}=F_{Tfzi}+F_{Tgzi}+F_{TOzi}\hfill \end{array}\right.$$

(10)

$$\left\{\begin{array}{c}{M}_{Txi}=F_{Tfxi}{l}_{Ti}+F_{Tgxi}{l}_{Ti}+F_{TOxi}{l}_{T(i+1)}+q_{Ei}{l}_{Exi}+M_{TOxi}\\ M_{Tyi}=F_{Tfyi}{l}_{Ti}+F_{Tgyi}{l}_{Ti}+F_{TOyi}{l}_{T(i+1)}+q_{Ei}{l}_{Eyi}+M_{TOyi}\end{array}\right.$$

(11)

Here, F_Tfxi, F_Tgxi and F_Toxi denote the aerodynamic force, gravitational load and the elastic and damping forces from adjacent segments acting on the i-th tower segment, respectively. l_Ti is the length of the segment, and q_Eil_Exi represents the eccentric moment exerted by other structural components or adjacent tower segments, where l_Exi is the corresponding eccentric distance along the x-direction. M_TOxi denotes the moment applied to the i-th segment by its neighboring segments i + 1 and i − 1. Each tower segment is modeled as a rigid body. The momentum equations in the X_Ti and Y_Ti directions for the i-th segment are given in Equation (12).

$$\left\{\begin{array}{l}{I}_{Txi}{\ddot{\theta }}_{Txi}+C_{Txi}{\dot{\theta }}_{Txi}+K_{Txi}{\theta }_{Txi}-M_{Txi}=0\hfill \\ I_{Tyi}{\ddot{\theta }}_{Tyi(i-1)}+C_{Tyi}{\dot{\theta }}_{Tyi}+K_{Tyi}{\theta }_{Tyi}-M_{Tyi}=0\hfill \end{array}\right.$$

(12)

The damping coefficient C_Txi is determined using the Rayleigh damping formulation (Equations (13) and (14)), where K_TXi denotes the stiffness of the i-th tower segment, and θ_Tyi represents the change in rotational angle around the y-axis relative to the previous time step.

$$C=aM+bK$$

(13)

$$\left(\begin{array}{c}a\\ b\end{array}\right)={\displaystyle \frac{2{\omega }_i{\omega }_j}{{\omega }_j^2-{\omega }_i^2}}\left[\begin{array}{cc}{\omega }_j& -{\omega }_i\\ -{\displaystyle \frac{1}{{\omega }_j}}& {\displaystyle \frac{1}{{\omega }_i}}\end{array}\right]\left({\displaystyle \frac{{\xi }_i}{{\xi }_j}}\right)$$

(14)

Here, ω_i and ω_j denote the i-th and j-th natural frequencies of the structure, while ζ_i and ζ_j represent the corresponding damping ratios.

3.2.2. Nacelle

The nacelle of a wind turbine can be simplified as a rigid body fixed atop the tower. The external forces and moments acting on the nacelle are described by Equation (15).

$$\left\{\begin{array}{c}{F}_{cx}=F_{gxc}+F_{fxc}+F_{Txc}+F_{Oxc}\\ F_{cx}=F_{gxc}+F_{fxc}+F_{Txc}+F_{Oxc}\\ M_{cx}=F_{gxc}{l}_c+F_{fxc}{l}_c+F_{Oxc}{l}_c+F_{Txc}{l}_c+q_c{l}_{cx}+M_{px}\\ M_{cy}=F_{fyc}{l}_c+F_{gyc}{l}_{Tc}+F_{Oyc}{l}_c+F_{Txc}{l}_c+q_c{l}_{cx}+M_{py}\end{array}\right.$$

(15)

The terms F_fxc, F_gxc, F_Txc, and F_Oxc denote the aerodynamic load, gravitational force, rotor thrust, and the elastic and damping forces transmitted from the top tower segment acting on the nacelle, respectively. Here, l_c represents the distance between the nacelle’s center of mass and the topmost tower element. q_cl_cx represents the eccentric moment exerted on the tower by other loads acting on the nacelle, and l_cx is the corresponding eccentric distance relative to the tower axis. The bending moments M_px and M_py arise from the eccentricity of the nacelle–hub–rotor assembly’s mass distribution, inducing additional torsional effects on the tower structure due to gravity.

Due to the non-uniform mass distribution of the nacelle, hub, and blades, the center of mass of the nacelle is offset from the tower’s rotational axis. This eccentricity introduces additional bending moments during operation, which significantly affect the dynamic behavior of the system. The eccentric bending moment induced by the gravitational forces of the nacelle-hub-rotor assembly is expressed in Equation (16) [11].

$$\left\{\begin{array}{l}{G}_{rot}=(m_{nac}+m_{bld}+m_{hub})\times g_z\\ {ℝ}_{top}={\displaystyle \frac{{ℝ}_{\mathrm{nac}}{m}_{nac}+{ℝ}_{\mathrm{bld}}{m}_{bld}+{ℝ}_{\mathrm{hub}}{m}_{hub}}{m_{nac}+m_{bld}+m_{hub}}}\\ M_{px}={\left[G_{rot}\times ({ℝ}_{top}-{ℝ}_{cen})\right]}_x\\ M_{py}={\left[G_{rot}\times ({ℝ}_{top}-{ℝ}_{cen})\right]}_y\end{array}\right.$$

(16)

Let m_nac, m_bld, and m_hub represent the masses of the nacelle, blades, and hub, respectively. The coordinates of their centers of mass are denoted by ${ℝ}_{\mathrm{nac}}$, ${ℝ}_{\mathrm{bld}}$, and ${ℝ}_{\mathrm{hub}}$. The coordinate ${ℝ}_{cen}$ refers to the intersection point between the center axis of the tower and the top tower segment.

3.2.3. Fixed Foundation of Offshore Wind Turbine

The fixed foundation of the wind turbine is modeled as a flexible structure, discretized into a multi-body system composed of n = 5 rigid elements along its height. Each rigid element is assigned translational and rotational degrees of freedom within its local coordinate system, as illustrated in Figure 3b. When subjected to ocean waves and currents, each foundation element experiences external hydrodynamic loads as well as loads transmitted from adjacent elements, as defined in Equation (17). Additionally, eccentric and overturning moments generated by inter-element interactions are described in Equation (18).

$$\left\{\begin{array}{l}{F}_{bxi}=d{F}_{wvxi}{l}_{bi}+F_{bgxi}+F_{bOxi}\hfill \\ F_{byi}=d{F}_{wvyi}{l}_{bi}+F_{bgyi}+F_{bOyi}\hfill \\ F_{bzi}=d{F}_{wvzi}{l}_{bi}+F_{bgzi}+F_{bOzi}\hfill \end{array}\right.$$

(17)

$$\left\{\begin{array}{c}{M}_{bxi}=d{F}_{wvxi}{l}_{bi}^2+F_{bgxi}{l}_{bi}+F_{bOxi}{l}_{b(i+1)}+q_{Ei}{l}_{Exi}+M_{bOxi}\\ M_{byi}=d{F}_{wvyi}{l}_{bi}^2+F_{bgyi}{l}_i+F_{bOyi}{l}_{b(i+1)}+q_{Ei}{l}_{Eyi}+M_{bOyi}\end{array}\right.$$

(18)

Here, dF_wvxi, F_bgxi, and F_bOxi represent the hydrodynamic load, gravitational load, and the elastic and damping forces from adjacent foundation elements acting on the i-th foundation unit, respectively; l_bi denotes the length of the i-th unit, q_Eil_Exi represents the eccentric moment exerted by other structural components, where l_Exi is the corresponding eccentric distance along the x-direction. while M_bOxi is the resultant moment applied to the i-th unit by its adjacent i + 1 and i − 1 elements. Each foundation segment is modeled as a rigid body. The momentum equations in the X_bi and Y_bi directions for the i-th foundation unit share a similar form with those of the tower segments, as described in Equations (12)–(14). The global coordinate position of each foundation unit is determined using Equation (9).

3.2.4. Dynamic Modeling of Multi-Tuned Mass Damper

To effectively suppress structural vibrations of the wind turbine tower across different operational frequency bands, multiple tuned mass dampers (TMD) are introduced into the tower structure. A single-degree-of-freedom (SDOF) TMD system can be modeled as a mass-spring-damper subsystem, as illustrated in Figure 4a. Its dynamic behavior is governed by the differential Equation (19) [9].

$$\left\{\begin{array}{l}-c_{t1}({\dot{\theta }}_{t1}-{\dot{\theta }}_{Tx20})-k_{t1}({\theta }_{t1}-{\theta }_{Tx20})=F_{t1}(t)\\ m_{t1}{\ddot{\theta }}_{t1}+c_{t1}({\dot{\theta }}_{t1}-{\dot{\theta }}_{Tx20})+k_{t1}({\theta }_{t1}-{\theta }_{Tx20})=F_{e1}(t)\end{array}\right.$$

(19)

The second and third order TMD systems can be simplified and modeled as mass-spring-damper systems with two and three degrees of freedom, respectively, as shown in Figure 4b,c. Since no initial tension is applied in the multi-TMD (MTMD) system considered in this study, the external excitation forces Fe₁(t), Fe₂(t), and Fe₃(t) are all zero. Their dynamic behaviors are described by differential Equations (20) and (21) [9], respectively.

$$\left\{\begin{array}{l}-c_{t1}({\dot{\theta }}_{t1}-{\dot{\theta }}_{Tx20})-k_{t1}({\theta }_{t1}-{\theta }_{Tx20})=F_{t1}(t)\\ -c_{t2}({\theta }_{t2}-{\dot{\theta }}_{Tx20})-k_{t2}({\theta }_{t2}-{\theta }_{Tx20})=F_{t2}(t)\\ m_{t1}{\ddot{\theta }}_{t1}+c_{t1}({\dot{\theta }}_{t1}-{\dot{\theta }}_{Tx20})+k_{t1}({\theta }_{t1}-{\theta }_{Tx20})=F_{e1}(t)\\ m_{t2}{\ddot{\theta }}_{t2}+c_{t2}({\dot{\theta }}_{t2}-{\dot{\theta }}_{Tx20})+k_{t2}({\theta }_{t2}-{\theta }_{Tx20})=F_{e2}(t)\end{array}\right.$$

(20)

$$\left\{\begin{array}{l}-c_{t1}({\dot{\theta }}_{t1}-{\dot{\theta }}_{Tx20})-k_{t1}({\theta }_{t1}-{\theta }_{Tx20})=F_{t1}(t)\\ -c_{t2}({\dot{\theta }}_{t2}-{\dot{\theta }}_{Tx20})-k_{t2}({\theta }_{t2}-{\theta }_{Tx20})=F_{t2}(t)\\ -c_{t3}({\dot{\theta }}_{t3}-{\dot{\theta }}_{Tx20})-k_{t3}({\theta }_{t3}-{\theta }_{Tx20})=F_{t3}(t)\\ m_{t1}{\ddot{\theta }}_{t1}+c_{t1}({\dot{\theta }}_{t1}-{\dot{\theta }}_{Tx20})+k_{t1}({\theta }_{t1}-{\theta }_{Tx20})=F_{e1}(t)\\ m_{t2}{\ddot{\theta }}_{t2}+c_{t2}({\dot{\theta }}_{t2}-{\dot{\theta }}_{Tx20})+k_{t2}({\theta }_{t2}-{\theta }_{Tx20})=F_{e2}(t)\\ m_{t3}{\ddot{\theta }}_{t3}+c_{t3}({\dot{\theta }}_{t3}-{\dot{\theta }}_{Tx20})+k_{t3}({\theta }_{t3}-{\theta }_{Tx20})=F_{e3}(t)\end{array}\right.$$

(21)

3.3. Modeling of the OWT–TMDs Coupled System

To enable validation against measured vibration data from actual wind turbines, a comprehensive dynamic model of the wind turbine system is constructed. The system comprises key components such as the rotor, hub, and nacelle. A schematic of the overall dynamic model is shown in Figure 5, with the detailed mathematical formulation provided in Equation (22) [36].

$$\left\{\begin{array}{c}{I}_{bx1}\stackrel{..}{{\theta }_{bx1}}+c_{bx1}\stackrel{.}{{\theta }_{bx1}}+k_{bx1}{\theta }_{bx1}-c_{bx2}(\stackrel{.}{{\theta }_{bx2}}-\stackrel{.}{{\theta }_{bx1}})-k_{bx2}({\theta }_{bx2}-{\theta }_{bx1})=M_{bx1}\\ ⋮\\ I_{Tx1}\stackrel{..}{{\theta }_{Tx1}}+c_{Tx1}(\stackrel{.}{{\theta }_{Tx1}}-\stackrel{.}{{\theta }_{bx5}})+k_{Tx1}({\theta }_{Tx1}-{\theta }_{bx5})-c_{Tx2}(\stackrel{.}{{\theta }_{Tx2}}-\stackrel{.}{{\theta }_{Tx1}})-k_{Tx2}({\theta }_{Tx2}-{\theta }_{Tx1})=M_{Tx1}\\ I_{Tx2}\stackrel{..}{{\theta }_{x2}}+c_{Tx2}(\stackrel{.}{{\theta }_{Tx2}}-\stackrel{.}{{\theta }_{Tx1}})+k_{Tx2}\left({\theta }_{Tx2}-{\theta }_{Tx1}\right)-c_{Tx3}(\stackrel{.}{{\theta }_{Tx3}}-\stackrel{.}{{\theta }_{Tx2}})-k_{Tx3}({\theta }_{Tx3}-{\theta }_{Tx2})=M_{Tx2}\\ ⋮\\ I_{Tx19}\stackrel{..}{{\theta }_{Tx19}}+c_{Tx19}(\stackrel{.}{{\theta }_{Tx19}}-\stackrel{.}{{\theta }_{Tx18}})+k_{Tx19}\left({\theta }_{Tx19}-{\theta }_{Tx18}\right)-c_{Tx20}(\stackrel{.}{{\theta }_{Tx20}}-\stackrel{.}{{\theta }_{Tx19}})-k_{Tx20}({\theta }_{Tx20}-{\theta }_{Tx19})=M_{Tx19}\\ I_{Tx20}\stackrel{..}{{\theta }_{Tx20}}+c_{Tx20}(\stackrel{.}{{\theta }_{Tx20}}-\stackrel{.}{{\theta }_{Tx19}})+k_{Tx20}\left({\theta }_{Tx20}-{\theta }_{Tx19}\right)=M_{Tx20}\\ I_{cx}\stackrel{..}{{\theta }_{cx}}+c_{cx}\stackrel{.}{{\theta }_{cx}}+k_{cx}{\theta }_{cx}-c_{cx}(\stackrel{.}{{\theta }_{cx}}-\stackrel{.}{{\theta }_{Tx20}})-k_{cx}({\theta }_{cx}-{\theta }_{Tx20})=M_{cx}\end{array}\right.$$

(22)

Moreover, the offshore wind turbine—multiple tuned mass damper (OWT-MTMD) coupled system encompasses various deployment configurations of MTMD. Given the dynamic models of these deployment schemes exhibit close similarity in their dynamic expressions, only the dynamic coupling model for the 3-TMD deployment scheme is presented here, as shown in Equation (23).

$$\left\{\begin{array}{c}{I}_{bx1}\stackrel{..}{{\theta }_{bx1}}+c_{bx1}\stackrel{.}{{\theta }_{bx1}}+k_{bx1}{\theta }_{bx1}-c_{bx2}(\stackrel{.}{{\theta }_{bx2}}-\stackrel{.}{{\theta }_{bx1}})-k_{bx2}({\theta }_{bx2}-{\theta }_{bx1})=M_{bx1}\\ ⋮\\ I_{Tx19}\stackrel{..}{{\theta }_{Tx19}}+c_{Tx19}(\stackrel{.}{{\theta }_{Tx19}}-\stackrel{.}{{\theta }_{Tx18}})+k_{Tx19}\left({\theta }_{Tx19}-{\theta }_{Tx18}\right)-c_{Tx20}(\stackrel{.}{{\theta }_{Tx20}}-\stackrel{.}{{\theta }_{Tx19}})-k_{Tx20}({\theta }_{Tx20}-{\theta }_{Tx19})=M_{Tx19}\\ I_{cx}\stackrel{..}{{\theta }_{cx}}+c_{cx}\stackrel{.}{{\theta }_{cx}}+k_{cx}{\theta }_{cx}-c_{cx}(\stackrel{.}{{\theta }_{cx}}-\stackrel{.}{{\theta }_{Tx20}})-k_{cx}({\theta }_{cx}-{\theta }_{Tx20})=M_{cx}\\ I_{Tx20}\stackrel{..}{{\theta }_{Tx20}}+c_{Tx20}(\stackrel{.}{{\theta }_{Tx20}}-\stackrel{.}{{\theta }_{Tx19}})+k_{Tx20}\left({\theta }_{Tx20}-{\theta }_{Tx19}\right)+F_{t1}(t)+F_{t2}(t)+F_{t3}(t)=M_{Tx20}\\ -c_{t1}({\dot{\theta }}_{t1}-{\dot{\theta }}_{Tx20})-k_{t1}({\theta }_{t1}-{\theta }_{Tx20})=F_{t1}(t)\\ -c_{t2}({\dot{\theta }}_{t2}-{\dot{\theta }}_{Tx20})-k_{t2}({\theta }_{t2}-{\theta }_{Tx20})=F_{t2}(t)\\ -c_{t3}({\dot{\theta }}_{t3}-{\dot{\theta }}_{Tx20})-k_{t3}(x_{t3}-{\theta }_{Tx20})=F_{t3}(t)\end{array}\right.$$

(23)

4. Linearized Modeling of the OWT—TMD Coupled System

To achieve effective vibration control of offshore wind turbines, it is essential to develop a coupled dynamic model of the tower and tuned mass damper (TMD) that captures the turbine’s vibration characteristics under wind and wave loading and evaluates TMD control performance. Considering the coupling effects among the tower, rotor, and foundation, as well as the stochastic and broadband nature of wind-wave excitations, direct optimization using high-fidelity finite element models incurs prohibitively high computational costs. Therefore, this study linearizes the system while preserving its key dynamic characteristics and replaces the original time-domain model with a frequency-domain representation to significantly improve the computational efficiency of TMD parameter optimization.

4.1. Input Excitation Linearization Based on Bandpass Filtering

Wind and wave loads exhibit pronounced broadband stochastic characteristics. Direct modeling introduces numerous high-frequency and non-critical frequency components, complicating subsequent structural response analyses. To address this, a bandpass filter is applied to limit the frequency bandwidth of the input excitation, emphasizing the structure’s primary response frequency range. The excitation signal F(ω) after processing by the bandpass filter H_bp(ω) can be expressed in the frequency domain as Equation (24) [29]:

$$F_{filt}\left(\omega \right)={\displaystyle {\sum }_{i=1}^N{H}_{bp}\left(\omega ,i\right)F(\omega )}$$

(24)

Typically, a second-order bandpass filter H_bp(ω) is used for the frequency-domain transfer function, which is expressed as shown in Equation (25) [27]:

$$H_{bp}\left(\omega ,i\right)={\displaystyle \frac{{(\omega /{\omega }_{ni})}^2}{{[1-{(\omega /{\omega }_{ni})}^2]}^2+{(2{\zeta }_i\omega /{\omega }_{ni})}^2}}$$

(25)

The parameter ω_ni represents the target structural response frequency, while ζ_i denotes the damping ratio of the filter, generally ranging from 0.05 to 0.2. This filtering process effectively attenuates non-target frequency components in the excitation, enabling focused analysis within the desired frequency band.

4.2. Linear Modeling of the Tower Structure

In frequency-domain vibration control modeling [37], although the tower structure exhibits multiple vibration modes, its dynamic response is typically dominated by the lower-order modes—particularly the first bending mode, which plays a critical role in controlling tower-top vibrations. Therefore, a model order reduction approach is adopted in this study, retaining only the primary first-order mode of the tower structure for simplification, as shown in Equation (26) [36].

$$x_{\mathrm{Ts}}(t)\approx {\varphi }_1{q}_1(t)$$

(26)

Here, ϕ₁ denotes the normalized first mode shape, q₁(t) is the corresponding modal coordinate, and x_Ts(t) represents the approximate displacement at the tower top. The first-order modal equation of the tower structure can be expressed as a first-order linear differential equation with constant coefficients, as shown in Equation (27) [36].

$${\ddot{q}}_1(t)+2{\zeta }_1{\omega }_1{\dot{q}}_1(t)+{\omega }_1^2{q}_1(t)={\displaystyle \frac{1}{M_1}}{F}_{filt}\left(t\right)$$

(27)

Here, ω₁ represents the first modal frequency of the tower, ζ₁ denotes the damping ratio of the first mode, M₁ is the generalized mass, and F_filt(t) is the equivalent force acting on this mode.

4.3. Linearized Modeling of the Coupled Offshore Wind Turbine System

In this study, the Tuned Mass Damper (TMD) is installed at the 20-th finite element segment of the tower structure. This location corresponds closely to the peak amplitude region of the tower’s first mode shape, effectively enhancing the TMD’s ability to absorb and suppress vibrational energy at the main structure’s resonance frequency. Each TMD is modeled as a classical mass-spring-damper system, generating coupling forces through its relative displacement with respect to the main tower structure, thereby facilitating the transfer and dissipation of vibrational energy.

Based on these physical assumptions, the dynamic behavior of the entire Offshore Wind Turbine-Multi-Tuned Mass Damper (OWT-MTMD) system can be described by a set of linear differential equations in matrix form. This model incorporates the dynamic response of the tower as well as the coupled interactions with each TMD unit. The detailed mathematical formulation is given in Equation (28), which accounts for the system’s mass, damping, and stiffness matrices, along with the external excitations and coupling forces between the TMD and the main structure [9].

$$\left\{\begin{array}{c}{m}_{eq}\ddot{\theta }(t)+c_{eq}\dot{\theta }(t)+k_{eq}\theta (t)-{\displaystyle {\sum }_{i=1}^n{c}_{ti}(\stackrel{.}{{\theta }_{ti}}-\stackrel{.}{{\theta }_{Tx20}})-k_{ti}({\theta }_{ti}-{\theta }_{Tx20})}=F_{filt}\left(t\right)\\ I_{ti}\stackrel{..}{{\theta }_{ti}}+c_{ti}(\stackrel{.}{{\theta }_{ti}}-\stackrel{.}{{\theta }_{Tx20}})+k_{ti}\left({\theta }_{ti}-{\theta }_{Tx20}\right)=0\end{array}\right.$$

(28)

The linear transfer function describing the dynamic response of the OWT-MTMD system is given by Equation (29) [27].

$$H(s)={\displaystyle \frac{{\displaystyle \prod _{i=1}^n(m_{ti}{s}^2+c_{ti}s+k_{ti})}}{(M_1{s}^2+2{\zeta }_1{\omega }_1{M}_{1}s+M_1{\omega }_1^2){\displaystyle \prod _{i=1}^n(m_{ti}{s}^2+c_{ti}s+k_{ti})}-{\displaystyle \sum _{i=1}^n{\xi }_{H}L{\left(c_{ti}s+k_{ti}\right)}^2{\displaystyle \prod ^{n\ne i}(m_{ti}{s}^2+c_{ti}s+k_{ti})}}}}$$

(29)

4.4. Input Excitation Parameter Correction

Although the applied band-pass filter effectively concentrates on the target frequency range of the structure, the spectral width of the actual excitation may still be attenuated during the modeling process. To ensure consistency between the filtered excitation energy and the original excitation energy, an equivalent energy correction factor α is introduced. This factor adjusts the filtered excitation energy, enabling the linearized system’s frequency-domain response to more accurately represent the spectral characteristics of the actual excitation. The correction factor α is calculated based on the integral energy ratio, as defined in Equation (30).

$$\alpha =\sqrt{{\displaystyle \frac{{\displaystyle {\int }_{{\omega }_1-\mathrm{\Delta }}^{{\omega }_1+\mathrm{\Delta }}|}{F}_{\mathit{raw}}(\omega ){|}^{2}d\omega }{{\displaystyle {\int }_{{\omega }_1-\mathrm{\Delta }}^{{\omega }_1+\mathrm{\Delta }}|}{F}_{\mathit{filt}}(\omega ){|}^{2}d\omega }}}$$

(30)

Here, F_raw(ω) represents the frequency-domain spectrum of the actual nacelle-top displacement response of the offshore wind turbine. The corrected linearized input excitation is then defined by Equation (31).

$$F_{\mathit{fc}}(\omega )=\alpha \cdot F_{\mathit{filt}}(\omega )$$

(31)

This ensures that the energy level of the input excitation and the resulting frequency-domain response remain as consistent as possible with the actual conditions during the linearization process, thereby improving the accuracy and reliability of the model.

5. MTMD Parameter Design Based on Hierarchical Particle Swarm Optimization

To effectively mitigate the structural vibration of offshore wind turbine towers under combined wind–wave excitations, a Multiple Tuned Mass Damper (MTMD) system is introduced in this study. The key parameters of the MTMD—including mass, damping ratio, and tuning frequency—are optimized to enhance vibration suppression across multiple structural modes. Given the high-dimensional design space, strong modal coupling, and nonlinear interactions within the MTMD system, traditional optimization techniques such as Genetic Algorithms (GA) and Particle Swarm Optimization (PSO) often operate as black-box methods, which are prone to local optima and sensitive to initial conditions, lacking physical interpretability and hierarchical control. To address these challenges, a hierarchical particle swarm optimization (H-PSO) framework is adopted, in which the global MTMD design task is decomposed into sequential, mode-specific optimization stages. The optimal single-TMD solution is used as a physically consistent initialization for the multi-TMD search, and each subsequent stage targets residual modal peaks left by previously tuned dampers. This structure-aware strategy reduces the effective search dimensionality, enhances convergence stability, and provides clearer modal-level interpretability compared with conventional hybrid or adaptive PSO variants. The resulting approach enables more robust and computationally efficient MTMD parameter design for offshore wind turbine vibration control.

5.1. Objective Function for Multiple Tuned Mass Dampers Parameter Optimization

The system includes different TMD deployment schemes, with 1 to 3 TMD configured. The parameters to be optimized for each TMD include mass m_tj, stiffness k_tj, and damping ratio ζ_tj. Thus, the design parameter vector for each TMD is defined as in Equation (32).

$$F_{\mathit{fc}}(\omega )=\alpha \cdot F_{\mathit{filt}}(\omega )$$

(32)

Here, μ_j represents the mass ratio between the j-th TMD and the tower, ζ_j is the equivalent damping ratio of the j-th TMD, and ω_tj denotes the tuning frequency of the j-th TMD.

The optimization objective is to minimize the response energy peak or the amplitude of the tower top acceleration at a specific frequency under typical operating conditions, as defined in Equation (33) [29]:

$$\left\{\begin{array}{l}J={\left|Y{(\omega )}_{\mathrm{\Delta }{\omega }_n}\right|}_{\max }\\ Y(\omega )=F_{fc}(\omega )H(j\omega )\end{array}\right.$$

(33)

Δω_n denotes the frequency bandwidth centered at a specific frequency, while H(jω) represents the system’s frequency response function as defined in Equation (29). J corresponds to the peak value of the frequency-domain response within this frequency band.

5.2. Hierarchical Particle Swarm Optimization

To address the high-dimensional search space and the complex coupling among TMD parameters, this study adopts a hierarchical optimization strategy, decomposing the overall process into two sequential layers:

Modal Decomposition Layer: For the primary excitation modal response of the structure, the tuning frequencies of the TMDs are initially distributed around the excitation frequency (±10%). The objective is to minimize the peak of the frequency-domain response within the range of the main frequency ±20%. Standard Particle Swarm Optimization (PSO) is then employed to optimize the first TMD parameter set P_j⁽¹⁾. Once optimal values are obtained, the performance contribution of this TMD is recorded, and its parameters are fixed for subsequent stages.

Higher-Order Tuning Layer: The remaining TMDs are then allocated to suppress secondary response peaks introduced by the first TMD. The optimization proceeds using updated objective functions J₂, J₃, each targeting different modal peaks. During each stage, previously optimized TMD parameters remain unchanged, and the algorithm iteratively approaches the global optimum. The overall optimization workflow is illustrated in Figure 6.

The velocity and position of each particle are updated in every generation according to the standard PSO update rule, as defined by Equation (34) [38].

$$\begin{array}{l}{v}_i^{(t+1)}=w{v}_i^{(t)}+c_1{r}_1\left(p_i^{\mathit{best}}-x_i^{(t)}\right)+c_2{r}_2\left(g^{\mathit{best}}-x_i^{(t)}\right)\\ x_i^{(t+1)}=x_i^{(t)}+v_i^{(t+1)}\end{array}$$

(34)

Here, x_i^(t) denotes the current position of the i-th particle in the t-th iteration (a candidate set of TMD parameters), and v_i(t) represents its velocity vector. p_i^best is the historical best position found by the i-th particle (personal best), while g denotes the global best position identified by the entire swarm. The parameter ω is the inertia weight controlling the balance between exploration and exploitation. c₁ is the cognitive coefficient (set to 1.5 in this study), representing the particle’s tendency to return to its own best experience, and c₂ is the social coefficient (also set to 1.5), reflecting the particle’s tendency to be influenced by the global best solution.

In the hierarchical particle swarm optimization (H-PSO) algorithm, each optimization stage features a distinct objective function and parameter dimensionality, while maintaining a consistent iterative structure. The output of each stage is used as either the initial condition or boundary constraint for the subsequent stage, enabling a stepwise refinement of the solution space. To ensure the practical feasibility and structural rationality of the designed TMD system, the following physical constraints of Equation (35) are incorporated into the optimization process.

$$\left\{\begin{array}{c}{\mu }_j=2\%\\ 0.001\le {\zeta }_j\le 0.3\\ 0.9{\omega }_i\le {\omega }_{tj}\le 1.1{\omega }_i\\ Y(0.8{\omega }_n)\le Y(\omega )\le Y(1.2{\omega }_n)\end{array}\right.$$

(35)

Here, μ denotes the total mass ratio, ζ represents the damping ratio, ω indicates the initial frequency range of the TMD, and Y(ω) defines the frequency range over which the optimization function searches for the maximum value. In this study, the modal frequencies are close to the first primary modal frequency, and Y(ω) is used to prevent interference with the optimization process.

6. System Model Validation and Multiple Tuned Mass Dampers Optimization

In this section, the engineering applicability of the model is evaluated from three perspectives: first, by validating the multi-degree-of-freedom dynamic model under real-world operating conditions to demonstrate its practical relevance; second, by performing sensitivity analyses on structural stiffness, damping coefficients, and the number of modeling elements in the lumped-mass representation to examine the numerical robustness of the model; and third, by assessing the computation time under different scenarios to discuss the feasibility of its computational efficiency. Key Structural Parameters of the Operational Offshore Wind Turbine are Presented in

Table 1. Key Structural Parameters of the OWT.

Property	Value	Property	Value
Rated power [MW]	3.2	Tower height [m]	91.37
Number of blades	3	tower base above sea level [m]	10
Rotor diameter [m]	153	Tower mass [t]	304.5
Cut-in wind speed [m·s−1]	2.5	Nacelle mass [t]	41
Rated wind speed	11.0	Cut-out wind speed	20

.

6.1. Dynamics Model Validation

6.1.1. Free Decay

Based on field monitoring data obtained during the startup and shutdown phases of an operating offshore wind turbine, the displacement response during the harmonic vibration stage was extracted for model validation. The natural frequencies of the actual turbine were identified using the Pencil Matrix Method, yielding the first and second fore-aft (along the X_O axis, rotating around the Y_O axis) natural frequencies as 0.2217 Hz and 0.4345 Hz, respectively.

To evaluate the accuracy of the proposed dynamic model, the free-decay time-history responses computed from the model were compared with the measured data in terms of frequency, damping, and amplitude attenuation trends, as shown in Figure 7. Under no external excitation, the dynamic model of the tower—initialized with a unit displacement perturbation—was used to simulate the free-decay response. The simulation results revealed that the model-predicted first and second fore-aft modal frequencies were 0.2213 Hz and 0.446 Hz, respectively, both of which showed relative errors of less than 3% compared to the measured values (see

Table 2. Verification of free-decay response in the tower-top x direction.

	1st Mode	2nd Mode
Actual Frequency	0.2217	0.4345
Simulated Frequency	0.2213	0.446
Relative Error	0.18%	2.65%

).

In addition to the comparison of modal frequencies, quantitative error metrics were employed to further evaluate the agreement between the simulated and measured free-decay responses. The root-mean-square error (RMSE) between the two time-history signals is 0.0014; the normalized RMSE relative to the standard deviation of the measured signal (0.014) is 10%, and the normalized RMSE relative to the peak amplitude (0.0216) is 6.5%. These error levels indicate reasonably good agreement [39], suggesting that the proposed model can adequately reflect the intrinsic vibration characteristics of the structure under unforced conditions.

6.1.2. Model Validation Under Typical Conditions

To assess the dynamic simulation accuracy of the proposed model under realistic wind-wave conditions, time-history comparisons were conducted using structural response data from specific operating scenarios. Detailed simulation parameters are summarized in

Table 3. Actual operating conditions parameters.

Parameter	Value
Water depth/m	17.86
Significant wave height/m	1.1
Average rotor speed/rpm	4.9
Average wind speed (m/s)	9.6
Wave period/s	6.2
Simulation duration/s	150

. The excitation signal was generated by superimposing mean wind speed and turbulence components using a wind field model based on the standard Kaimal spectrum. Time-domain wind speed sequences were applied at the tower top and rotor load points to serve as the system excitation.

Simulation outputs, including lateral displacement at the tower top and horizontal acceleration at the nacelle, were generated through Bladed simulations using SCADA data, including wind speed, rotor speed, blade pitch angle, and turbine power, from an offshore wind farm dataset provided in collaboration with our institution. The acceleration data have been validated in both time and frequency domains against actual measured acceleration data, and the corresponding displacement data can thus be regarded as baseline data. Figure 8 presents a comparison between the simulated responses and 150 s of these baseline data from the structural monitoring system.

$$RMSE=\sqrt{{\displaystyle \frac{1}{n}}{\displaystyle \sum _{i=1}^n{\left(y_i^{\mathit{sim}}-y_i^{\mathit{obs}}\right)}^2}}$$

(36)

The Root Mean Square Error (RMSE) metric was employed to quantify discrepancies between the two time series, with the calculation detailed in Equation (36). The time-domain response analysis demonstrates excellent agreement between simulation and measurement in waveform trends, frequency content, and major peak amplitudes, with an average amplitude deviation within 6%, thereby validating the model’s accuracy and reliability. Detailed results are presented in

Table 4. Statistical results of tower top displacement response.

Location	Measured Average Vibration Amplitude	Simulated Average Vibration Amplitude	RSME	Relative Deviation
Tower top	0.2615	0.2693	0.0152	5.64%

.

The combined validation results demonstrate that the developed tower dynamic model achieves a controllable level of structural simplification while maintaining high accuracy. It effectively captures the dynamic behavior of the wind turbine during free vibration and under typical wind field conditions. This model provides a reliable foundation for subsequent parameter optimization and vibration control performance evaluation of the system with the integration of tuned mass damper (TMD) structures.

6.1.3. Parameter Sensitivity and Model Robustness

This section aims to evaluate the effects of different modeling parameters on the final model results, as well as the impact of the number of tower modeling elements on computational efficiency.

Figure 9 shows the responses of all degrees of freedom of the tower and the foundation under perturbations of stiffness parameters and damping ratios. It can be seen from Figure 9 that the tower-top displacement is significantly more sensitive to stiffness perturbations than to damping ratio variations. The unperturbed model results are used as a reference, and the relative changes in tower displacement responses are quantified using the mean relative error (MRE) metric. The specific results are presented in Figure 10.

As shown in Figure 10, the relative error of the model’s displacement response under parameter perturbations exhibits an approximately proportional variation. The impact of stiffness perturbations on the results is significantly greater than that of damping ratio perturbations, with a difference of about two orders of magnitude. The relative change induced by stiffness perturbations is roughly consistent with the variability of the model results, whereas the relative change caused by damping ratio perturbations is much larger than the corresponding change in the model response. Therefore, it can be concluded that the proposed dynamic model is more sensitive to stiffness parameters, while relatively insensitive to damping ratios.

Figure 11 illustrates the relative error in the dynamic response and the corresponding computation time of the tower and full turbine models under identical operating conditions and a fixed simulation duration of 250 s, for different element lengths and numbers. Specifically, the discretization schemes for Cases 1–5 are as follows: Case 1—1 element for the foundation and 4 elements for the tower; Case 2—2 and 8 elements for the foundation and tower, respectively; Case 3—3 and 12 elements; Case 4—4 and 16 elements; and Case 5—5 and 20 elements. In this study, the results of Case 5 are used as the reference to quantify the relative error of the dynamic responses for the other cases.

From Figure 11, it can be seen that as the number of elements increases, the model results gradually converge. Starting from Case 3, the changes in error begin to level off, indicating that further mesh refinement provides very limited improvement in the response. At the same time, overly fine discretization significantly increases the computational cost.

6.2. Tower Damping Frequency Under Typical Conditions

During the entire service life of offshore wind turbines, the structure is subjected to various complex operating conditions, among which the resonance amplification effects caused by wind and wave loads are critical factors leading to tower fatigue damage and ultimate loads. The proper determination of the tuned mass damper (TMD) tuning frequency is fundamental for achieving effective vibration mitigation [40]. Based on the previously established dynamic model of the wind turbine, this section selects typical operating parameters from Table 3 to analyze the frequency characteristics of the tower’s 1000 s vibration time history under the given condition. The primary response frequency range of the structure is identified, providing a basis for the design of the TMD tuning frequency. Considering the significant impact of low-frequency loads on the stability of wind turbines, the analysis focuses on the 0–1 Hz frequency band.

As shown in the spectral results of Figure 12, two prominent peaks appear at 0.1621 Hz and 0.2242 Hz. The 0.2242 Hz peak corresponds to the first natural frequency of the tower and represents the dominant dynamic response frequency, while the 0.1621 Hz peak is associated with the wave load frequency component. This study selects the response peak near 0.1621 Hz as the target tuning frequency of the TMD, aiming to effectively suppress low-frequency modal responses while ensuring the overall system stability.

6.3. Control Target Curve Construction

To enable efficient frequency-domain response analysis and parameter optimization of the OWT–TMD coupled system, the excitation signal is linearized to construct an input that aligns with the system’s transfer function. This approach aims to produce a frequency-domain response that closely approximates the actual structural response spectrum. Compared to the original wind field excitation—which exhibits broadband and non-stationary characteristics—the linearized input conforms more closely to the assumptions of linear system analysis, enhancing both the interpretability and physical consistency of the model output.

The processing procedure is as follows: first, Fast Fourier Transform (FFT) is applied to the measured structural response signal to extract the frequency characteristics of the tower under the target operating condition. Then, based on a simplified first-order modal model of the tower, the system transfer function H(ω) is computed (see Figure 13). This transfer function is used as a deconvolution reference to derive a matching excitation spectral envelope F_fc(ω) (see Figure 14), where the applied correction factor α is 1.171. Finally, the derived spectral envelope is applied to the original wind load signal using band-pass filtering, resulting in a linearized excitation. The corresponding linear system frequency response is then obtained by computing F_fc(ω) × H(ω).

Figure 15 shows a comparison of the frequency spectra before and after signal linearization. The output spectrum of the processed input exhibits two primary modal frequencies at 0.162 Hz and 0.225 Hz, with peak amplitudes and energy distributions generally consistent with the trends of the reference data. Specifically, the amplitude at the primary excitation modal frequency is 101.42 for the simplified model, compared with 118.17 for the reference, corresponding to a relative error of approximately 14.2%; at the first tower modal frequency, the amplitude is 91.94 for the simplified model versus 94.88 for the reference, with a relative error of approximately 3.1%. The overall mean relative error within the 0.1 Hz–0.35 Hz frequency range is 22.26%. In general, the simplified model produces a smoother fitted curve, lacking higher-order modal features and nonlinear wind–wave coupling effects, which leads to deviations in specific frequency bands and may cause discrepancies in absolute accuracy between the optimized vibration reduction in the frequency domain and the actual reduction performance.

To further quantify the frequency-domain fitting performance, the coefficient of determination R² between the simulated and measured response spectra was calculated using Equation (37). The results show R² = 0.831, with a Pearson correlation coefficient of r = 0.9173 indicating that the simplified model captures the dominant trends within the low-order modal frequency range well. Based on this, the simplified model can serve as a reference objective function for TMD parameter optimization, representing the vibration suppression trends of different schemes.

$$R^2=1-{\displaystyle \frac{{\displaystyle {\sum }_{i=1}^n{(y_i-{\widehat{y}}_i)}^2}}{{\displaystyle {\sum }_{i=1}^n{(y_i-{\overline{y}}_i)}^2}}}$$

(37)

Here, y_i denotes the measured data values, ${\widehat{y}}_i$ represents the corresponding values calculated by the model, and ${\overline{y}}_i$ is the mean of the model-predicted values.

Therefore, the linearized model response curve can be adopted as the target curve for the optimization of TMD control parameters.

6.4. Multiple Tuned Mass Dampers Parameter Optimization

6.4.1. Single Multiple Tuned Mass Dampers Scheme

Building upon the coupled OWT–TMD dynamic model and the constructed vibration control target curve, further simulations were conducted to optimize the parameters of a multiple tuned mass damper (MTMD) system, aiming to enhance vibration suppression performance within the target frequency band under typical operating conditions. Compared with a single TMD system, the MTMD configuration offers broader frequency coverage and better adaptability to structural modal shifts, effectively addressing issues such as frequency drift and mode coupling.

To identify the optimal parameter configuration of the MTMD system, this study employs the Hierarchical Particle Swarm Optimization (H-PSO) algorithm. Under the constraint of a fixed total mass ratio, the key structural parameters—tuning frequency (f_t1) and damping coefficient (c_t1)—of each TMD are simultaneously optimized to minimize the peak spectral amplitude of tower-top displacement within the vibration control frequency range. The system under optimization consists of 3-TMD placed at the same height along the tower. Initial parameter bounds are defined based on constraints derived in Section 6.3. The total TMD-to-structure mass ratio is set at 2%, with damping ratios ranging from 0.001 to 0.25, and tuning frequencies restricted to the target control range of 0.15–0.176 Hz. The objective function is defined as the maximum spectral peak within this range (see Equation (33)).

The optimization begins with the single TMD case. The PSO parameters are set as follows: population size = 150, cognitive and social learning factors = 1.5, inertia weight = 0.7, initial reference frequency = 0.1621 Hz, and initial damping ratio = 0.01254. The optimization trajectory is shown in Figure 16, where the X-axis represents the dimensionless tuning frequency. The initial point is (1.04, 0.01254), and the final optimized point is (1.0147, 0.0145). The evolution of the objective function during iterations is shown in Figure 17, where convergence is observed after approximately 25 iterations, yielding an optimal value of 0.25447. The corresponding optimized frequency response is shown in Figure 18, demonstrating effective suppression of the spectral peak near 0.1621 Hz. Specifically, the uncoupled system exhibits a peak of 111.43 in the 0.1–0.2 Hz band, while the single TMD system reduces it to 93.56, achieving a peak reduction of approximately 16.04%.

To evaluate the robustness and generalizability of the PSO algorithm for single TMD parameter optimization, 100 independent optimization runs were performed. The optimization trajectories are illustrated in Figure 19. Although the initial conditions differ in each run, the final solutions converge to similar regions. The distribution of the optimized frequency ratios and damping ratios from the 100 runs is summarized in Figure 20. The average frequency ratio across all runs is 0.9736 with a standard deviation of 3.38 × 10⁻⁵, indicating that the PSO algorithm demonstrates satisfactory robustness, convergence, and consistency in the single TMD optimization scenario.

6.4.2. Dual Multiple Tuned Mass Dampers Scheme

Parameter optimization for the 2-TMD scheme is performed using both the PSO and H-PSO algorithms. For the PSO algorithm, the population size was set to 300, the cognitive and social learning factors were set to 1.5, inertia weight to 0.7, and the initial reference frequencies were 0.1621 Hz and 0.126 Hz. The total mass ratio constraint for the 2 TMD system was approximately 2%, with the first TMD accounting for 80% and the second TMD for 20% of the total TMD mass [27]. The H-PSO algorithm used the same parameter settings for learning factors, inertia weight, and mass distribution. The initial reference frequency for the first TMD was set to 0.1621 Hz. The hierarchical PSO (H-PSO) only requires the initial frequency of the single TMD system (0.1621 Hz) to begin optimization.

The parameter optimization processes for the 2-TMD scheme are shown in Figure 21: (a) represents the PSO-based optimization, and (b) shows the H-PSO-based optimization. The objective function value variations during iterations are presented in Figure 22, where the values stabilize after the 36th iteration for PSO and the 38th for H-PSO. The optimized frequency response results at convergence are shown in Figure 23. The spectral peak near 0.1621 Hz is effectively suppressed. Without TMD, the coupled system exhibits a peak value of 111.43 within the 0.1–0.2 Hz band; after PSO, the peak reduces to 108.64 (approx. 2.5% reduction). H-PSO achieves a peak of 90.57, representing an 18.72% reduction. This suggests that PSO may have ultimately converged to a local optimum due to the initial values and the settings of the optimization parameters. In the high-dimensional MTMD design problem, purely random initialization lacks structural information, causing particles to explore large flat regions in the search space and increasing the likelihood of premature convergence to suboptimal local minima. In contrast, the H-PSO framework uses the optimized single-TMD solution as a physics-informed initial guess, which significantly improves the probability of reaching the global optimum.

To compare the stability of H-PSO and PSO, the PSO algorithm was run with five sets of parameters (see

Table 5. Initial Frequency Parameters for 2-TMD Scheme.

	Initial Frequency of TMD1	Initial Frequency of TMD2
P1	0.1621 Hz	0.106 Hz
P2	0.1621 Hz	0.146 Hz
P3	0.1621 Hz	0.166 Hz
P4	0.1621 Hz	0.186 Hz
P5	0.1621 Hz	0.226 Hz

), each repeated 100 times [41]. The statistical results of these runs are shown in Figure 24. Under identical initial parameters, the optimized frequencies for TMD2 and damping ratios for both TMD exhibited considerable fluctuations. Across different initial parameters, the average frequency optimization results for both TMD showed significant variability. Detailed parameter results are listed in

Table 6. 2-TMD Parameter Optimization Results Using PSO Algorithm.

Parameter	P1	P2	P3	P4	P5	Standard Deviation	Mean Value	Variation
TMD1 fmean	0.1217	0.1485	0.1721	0.1686	0.1643	0.0207	0.155	13.4%
TMD2 fmean	0.1902	0.1560	0.1351	0.1373	0.1395	0.0231	0.152	15.2%
TMD1 Cmean	0.0232	0.0237	0.0237	0.0233	0.0229	3.44 × 10−4	0.0234	1.5%
TMD2 Cmean	0.0135	0.01372	0.0142	0.0139	0.0135	2.96 × 10−4	0.0138	2.1%
Y(ω)	108.64	107.65	90.99	90.78	107.40	9.33	101.1	9.2%

. The mean and standard deviation were used to quantify the variability of TMD parameter optimization across different initial values. The frequency parameters showed fluctuations exceeding 10%, and the optimized peak values of the objective function in the 0.14–0.2 Hz band fluctuated by more than 9%, as detailed in Table 6.

Conversely, the H-PSO algorithm was executed 100 times, requiring only the initial reference frequency of the single TMD setup to optimize the 2 TMD parameters following the schematic flow. The results after 100 runs are presented in Figure 25. The mean objective function value was 90.70 with a standard deviation of 0.1849 (0.2% fluctuation). Overall, the H-PSO algorithm produces more stable results for the objective function optimization.

These findings indicate that the PSO algorithm is more sensitive to initial values of high-dimensional parameters, whereas the H-PSO algorithm demonstrates superior overall stability and generalizability in high-dimensional parameter optimization scenarios.

6.4.3. Triple Multiple Tuned Mass Dampers Scheme

The 3-TMD configuration was further optimized using both the PSO and H-PSO algorithms. For the PSO approach. For the PSO approach, the population size was set to 600, the inertia weight w to 0.5, and the cognitive and social learning factors c₁ and c₂ to 2.the initial reference frequencies were set to 0.1621 Hz, 0.126 Hz, and 0.1752 Hz. The total mass ratio of the TMD system was constrained to 2%, with the first TMD accounting for 80% of the total mass, and the second and third each accounting for 10%. The H-PSO algorithm adopted the same settings for cognitive and social learning factors, inertia weight, and TMD mass distribution. Unlike PSO, the H-PSO algorithm only requires the initial reference frequency for the single-TMD configuration (0.1621 Hz).

The parameter optimization processes for both algorithms are shown in Figure 26, and the corresponding frequency response spectra confirm that the dominant response peaks remain near 0.1621 Hz. The iterative evolution of the objective function values is illustrated in Figure 27, where convergence is observed at the 74th iteration for PSO and the 43rd iteration for H-PSO. The optimized spectral results are presented in Figure 28. Without any TMDs, the peak amplitude of the structural response in the 0.1–0.2 Hz frequency range is 111.43. After PSO, the peak reduces to 91.41 (a 17.97% decrease), while H-PSO achieves a further reduction to 89.64 (a 19.55% decrease), indicating superior suppression of low-frequency resonance.

It can be seen that increasing from two to three TMDs results in only a relatively limited improvement in peak vibration suppression within the target frequency band. This is mainly due to the following factors: the primary TMD, which accounts for 80% of the total mass, already absorbs the majority of the controllable energy of the first bending mode; allocating the remaining 20% of the mass to two secondary TMDs reduces the individual mass ratios, limiting their effectiveness in absorbing energy over a broader frequency range. Nevertheless, the addition of a third TMD helps smooth the response peaks within the target frequency band and enhances the system’s robustness against variations in wave excitation or slight detuning.

To assess the robustness and generalizability of the two algorithms, the PSO method was tested with five different parameter sets (see

Table 7. Five parameter sets for the 3-TMD scheme.

	Initial Frequency of TMD1	Initial Frequency of TMD2	Initial Frequency of TMD3
P1	0.1621 Hz	0.106 Hz	0.1732 Hz
P2	0.1621 Hz	0.146 Hz	0.1732 Hz
P3	0.1621 Hz	0.166 Hz	0.1732 Hz
P4	0.1621 Hz	0.186 Hz	0.1732 Hz
P5	0.1621 Hz	0.226 Hz	0.1732 Hz

), each repeated 100 times. The results are summarized in Figure 29. Significant variability was observed in the optimized frequency and damping ratios of TMD2 across runs, particularly in comparison to the 2-TMD scheme. Detailed data in

Table 8. 3-TMD Parameter Optimization Results Using PSO Algorithm.

Parameter	P1	P2	P3	P4	P5	Standard Deviation	Mean Value	Variation
TMD1 fmean	0.1233	0.1235	0.1557	0.1483	0.1267	0.0153	0.1355	11.3%
TMD2 fmean	0.2332	0.1458	0.1530	0.1579	0.1650	0.0355	0.171	20.8%
TMD3 fmean	0.1735	0.1128	0.1340	0.1251	0.1671	0.0266	0.1425	18.7%
TMD1 Cmean	0.0231	0.0227	0.0227	0.0230	0.0233	2.6 × 10−4	0.023	1.1%
TMD2 Cmean	0.0143	0.0393	0.0100	0.0397	0.0703	0.0242	0.0347	69.7%
TMD3 Cmean	0.0101	0.0390	0.0680	0.0101	0.0101	0.0259	0.0275	94.2%
Y(ω)	106.57	107.86	87.23	88.38	107.1	10.6280	99.43	10.7%

show that parameter fluctuations in the 3-TMD scheme are considerably more pronounced, especially for TMD2 and TMD3, whose damping ratios varied by nearly 100%. These results suggest that when the total mass is fixed, allocating smaller masses to additional TMDs can lead to greater variability in the optimized parameters.

In contrast, the H-PSO algorithm requires only the initial frequency reference from the single-TMD configuration. After 100 repeated runs following the proposed optimization flow (Figure 30), the statistical results demonstrate improved stability. The average final objective function value was 89.68, with a standard deviation of 0.1881 (variation: 0.21%).

These results indicate that while the performance of H-PSO for TMD2 and TMD3 is comparable to that of PSO, it provides significantly greater stability for TMD1 and for the overall objective function. Therefore, H-PSO demonstrates better reliability and robustness in high-dimensional parameter optimization for MTMD systems.

The simulation results indicate that, in high-dimensional parameter optimization scenarios, the PSO algorithm is more sensitive to the selection of initial parameter values, which can significantly affect the optimization outcomes. In contrast, the H-PSO algorithm is capable of autonomously determining initial values for high-dimensional parameters during the iteration process, thereby reducing the instability caused by manual initialization. Under the constraint of a total mass ratio of 2% for the MTMD system, the three sets of optimized MTMD parameters obtained through H-PSO are summarized in

Table 9. Optimized parameters of three M-TMD configurations.

Configurations	Mass	Frequency	Damping Ratio
S1	6198 kg	0.1581 Hz	0.0247
S2	4958.4 kg	0.1592 Hz	0.0122
	1239.6 kg	0.1554 Hz	0.023
S3	4958.4 kg	0.1575 Hz	0.0209
	619.8 kg	0.1638 Hz	0.01
	619.8 kg	0.1517 Hz	0.0167

.

It can be seen that the optimized TMD frequencies are closely clustered around the target frequency band (0.151–0.164 Hz), while the optimal damping ratios remain relatively low. These observations indicate that, although the multi-TMD system considering structural and aerodynamic damping falls outside the strict applicability range of Den Hartog’s formulas [42], the solutions obtained via H-PSO remain physically meaningful and stable. This further validates the robustness of the H-PSO framework for multi-dimensional MTMD parameter optimization.

7. Multiple Tuned Mass Dampers Scheme Vibration Control Analysis for the Tower

Building upon the optimized configurations of MTMD systems, this study further evaluates the vibration mitigation performance of different TMD array schemes on the tower structure. Three representative configurations are investigated: a single TMD (S1), a dual TMD system (S2), and a triple TMD system (S3). Under identical excitation conditions, comparative simulations are conducted to quantitatively assess the control efficacy of each setup.

The performance evaluation focuses on three key metrics:

(1)

the reduction ratio of the response amplitude at the dominant modal frequency, representing the effectiveness of resonance suppression.

(2)

the attenuation of total response energy within the primary frequency range, reflecting the energy dissipation capability of the system.

(3)

The spectral value at the target frequency reflects the system’s effectiveness in suppressing vibrations at that specific frequency.

The simulation results are presented in Figure 31 and Figure 32. Figure 31 shows the time-domain response of the tower-top displacement in the x-direction, while Figure 32 illustrates the corresponding frequency spectra. The results indicate that the implementation of TMDs leads to a substantial decrease in displacement amplitude, confirming the effectiveness of the control strategy. In the frequency domain, significant attenuation is observed around the primary modal frequency of 0.1621 Hz, with the most notable reduction occurring precisely at this frequency. Adjacent frequency ranges also exhibit moderate suppression, consistent with the optimization trends of the control objective function discussed earlier.

To further examine whether the observed reductions are statistically significant, a paired t-test was conducted using RMS values extracted from 20 s non-overlapping segments of the time-domain responses. Compared with the baseline case (no TMDs), all three TMD configurations exhibit highly significant vibration reduction (p < 0.01). Pairwise comparisons among S1, S2, and S3 yield p-values below 0.05, indicating statistically distinguishable—though relatively moderate—differences in performance among the TMD designs. These findings confirm that (1) the application of TMDs leads to a statistically significant reduction in tower-top vibration, and (2) the performance differences across different TMD configurations are statistically identifiable.

A summary of the key numerical results is provided in

Table 10. Statistical results of dynamic displacement response at tower top.

Scheme	Peak Reduction	Frequency Band Energy Reduction	Target Frequency Reduction
S1	12.64%	3.03%	17.00%
S2	14.28%	3.6%	18.80%
S3	17.95%	4.32%	24.62%

. Compared to the baseline (no TMDs), the S1 system achieves a 12.64% reduction in peak response within the 0.1–0.2 Hz band, a 3.03% reduction in overall energy across 0.1–1 Hz, and a 17.00% reduction in amplitude at the target frequency of 0.1621 Hz. The S2 configuration further improves performance, with peak and energy reductions reaching 14.28% and 3.60%, respectively, and a 18.80% drop at the target frequency. The S3 setup demonstrates the best performance, with a 17.95% peak reduction, 4.32% total energy reduction, and a 24.62% decrease at 0.1621 Hz, indicating superior resonance suppression and broadband energy dissipation capabilities.

It is worth noting, however, that while the incremental improvements in peak reduction from S2 to S3 are evident, the marginal gains in total energy dissipation are relatively limited. This suggests that the advantages of higher-order MTMD configurations lie primarily in their ability to finely tune to specific modal frequencies, rather than offering proportional benefits across the entire frequency spectrum. Due to the concentration of structural response energy near the dominant mode, the additional TMDs mainly enhance local resonance absorption but offer diminishing returns for broadband control.

In conclusion, the S3 configuration offers the most robust vibration mitigation performance, achieving effective control in both resonance suppression and energy dissipation. The S2 system provides a balanced trade-off between control efficacy and system complexity, making it suitable for applications with stringent performance and cost requirements. The S1 setup, with its simple structure and ease of implementation, remains a practical choice where moderate control effectiveness is sufficient. Therefore, the degree of TMD configuration can be flexibly adapted to meet specific engineering demands and vibration control objectives, ensuring an optimal balance between control performance and implementation feasibility.

8. Conclusions

This study addresses the low-frequency vibration issues of offshore wind turbine towers under wind and wave excitations. To overcome limitations in existing control methods—such as inadequate adaptability to structural dynamic characteristics and low parameter optimization efficiency—a novel vibration control approach is proposed. This method integrates multi-body dynamics modeling, input excitation linearization, and Multiple Tuned Mass Damper (MTMD) parameter optimization. The effectiveness of the proposed approach is validated through numerical simulations.

(1) A multi-degree-of-freedom dynamic model of the coupled wind turbine tower and TMD system is developed, considering the mass distribution and modal coupling of key components such as the tower, nacelle, hub, and TMDs. The model provides a practical and computationally efficient framework for evaluating structural responses and design-ing vibration control strategies. Comparison of the simulation results with measured vi-bration data from an operational offshore wind turbine demonstrates the model’s stability and reasonable accuracy. Nevertheless, the model has certain limitations: it primarily captures low-frequency excitation modes relevant to operational loads, while higher-order modes, geometric and material nonlinearities of the tower, and some three-dimensional effects are not fully resolved. These aspects will be addressed with higher-fidelity modeling approaches in future work.

(2) To enhance the accuracy and efficiency of frequency-domain analysis, the input excitation signal is linearized by reconstructing its spectrum using a band-pass filtering strategy. This linearized input better matches the system’s transfer function response and closely fits the measured structural response spectrum. Simulation results demonstrate that the linearized excitation not only concentrates the modal energy in the dominant frequencies but also achieves high fitting precision in both frequency and time domains, providing a solid foundation for subsequent frequency-domain vibration control optimization.

(3) Focusing on the first-order modal response of the tower, this study introduces a Hierarchical Particle Swarm Optimization (H-PSO) algorithm to jointly optimize the damping ratios and tuning frequencies of the Multiple Tuned Mass Damper (MTMD) system. An optimization framework is established with the goal of minimizing the response energy within the target frequency band. Under typical operating conditions, the three-TMD system achieves approximately an 18% reduction in peak response and a 4.32% decrease in energy within the target frequency band compared to the undamped system, demonstrating the method’s effectiveness in modal cooperative control and practical engineering applications. Furthermore, comparative experiments with the conventional Particle Swarm Optimization (PSO) algorithm using multiple sets of initial parameters show that H-PSO exhibits superior stability and generalizability.

(4) Finally, Comparative analyses of multiple TMD configurations reveal that increasing the number of TMDs significantly enhances suppression of the main structural modal frequency, although the marginal gains in overall energy attenuation diminish. Therefore, TMD configuration should balance engineering cost and control objectives.

In summary, the developed coupled tower-TMD modeling and control method effectively controls the dominant modal responses while maintaining model simplicity. This approach provides reliable technical support for vibration mitigation and structural health management of offshore wind turbines.